Methodology-Centered Review of Molecular Modeling, Simulation, and Prediction of SARS-CoV-2

Abstract

Despite tremendous efforts in the past two years, our understanding of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), virus–host interactions, immune response, virulence, transmission, and evolution is still very limited. This limitation calls for further in-depth investigation. Computational studies have become an indispensable component in combating coronavirus disease 2019 (COVID-19) due to their low cost, their efficiency, and the fact that they are free from safety and ethical constraints. Additionally, the mechanism that governs the global evolution and transmission of SARS-CoV-2 cannot be revealed from individual experiments and was discovered by integrating genotyping of massive viral sequences, biophysical modeling of protein–protein interactions, deep mutational data, deep learning, and advanced mathematics. There exists a tsunami of literature on the molecular modeling, simulations, and predictions of SARS-CoV-2 and related developments of drugs, vaccines, antibodies, and diagnostics. To provide readers with a quick update about this literature, we present a comprehensive and systematic methodology-centered review. Aspects such as molecular biophysics, bioinformatics, cheminformatics, machine learning, and mathematics are discussed. This review will be beneficial to researchers who are looking for ways to contribute to SARS-CoV-2 studies and those who are interested in the status of the field.

1. Introduction

Since its first case was identified in Wuhan, China, in December 2019, coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), has expeditiously spread to as many as 226 countries and territories worldwide and led to over 433 million confirmed cases and over 5.9 million fatalities as of February 2022. This pandemic has also brought a massive economic recession globally. Countries all around the world have implemented a variety of policies to tackle the COVID-19 pandemic (https://stip.oecd.org/covid/).

Many SARS-CoV-2 vaccines and monoclonal antibodies (mAbs) have already obtained use authorization worldwide (https://www.nytimes.com/interactive/2020/science/coronavirus-vaccine-tracker.html). Additionally, the U.S. Food and Drug Administration (FDA) has given emergency use authorization to the oral SARS-CoV-2 Mpro inhibitor PAXLOVID (PF-07321332) developed by Pfizer^1,2 (https://www.pfizer.com/news/press-release/press-release-detail/pfizer-receives-us-fda-emergency-use-authorization-novel). However, COVID-19 has a high infection rate, high prevalence, long incubation period,³ asymptomatic transmission,⁴⁻⁶ and potential seasonal patterns.⁷ SARS-CoV-2 keeps evolving into new infectious and antibody resistant variants.⁸⁻¹⁰ Therefore, it is imperative to understand the viral molecular mechanism,¹¹ to track its genetic evolution,¹² and to continuously improve the efficacy of its antiviral drugs and antibody therapies.

Belonging to the β-coronavirus genus and coronaviridae family, SARS-CoV-2 is an unsegmented positive-sense single-stranded RNA (+ssRNA) virus with a compact 29,903 nucleotide-long genome, and the diameter of each SARS-CoV-2 virion is about 50–200 nm.¹⁴ In the first 20 years of the 21st century, β-coronaviruses have triggered three major outbreaks of deadly pneumonia: SARS-CoV (2002), Middle East respiratory syndrome coronavirus (MERS-CoV) (2012), and SARS-CoV-2 (2019).¹⁵ Like SARS-CoV and MERS-CoV, SARS-CoV-2 also causes respiratory infections, but at a much higher infection rate.^16,17 The complete genome of SARS-CoV-2 comprises 15 open reading frames (ORFs), which encodes 29 structural and nonstructural proteins (nsps), illustrated in Figure 1. The 16 nonstructural proteins nsp1–nsp16 get expressed by protein-coding genes ORF1a and ORF1b, while four canonical 3′ structural proteins, spike (S), envelope (E), membrane (M), and nucleocapsid (N) proteins, as well as accessory factors, are encoded by another four major ORFs, namely ORF2, ORF4, ORF5, and ORF9 (see Figure 1).¹⁸⁻²¹

Figure 1.

Genomics organization and proteins of SARS-CoV-2. Adapted with permission from ref (13). Copyright 2021 John Wiley and Sons.

The viral structure of SARS-CoV-2 can be found at the upper right corner of Figure 2. This structure is formed by the four structural proteins: the N protein holds the RNA genome, the S protein helps the virus enter into the host cell, and the M and E proteins define the shape of the viral envelope.²² The studies on SARS-CoV-2 as well as previous SARS-CoV and other coronaviruses have mostly identified the functions of these structural proteins, nonstructural proteins, as well as accessory proteins, which are summarized in Table 1. Their 3D structures are also largely known from experiments or predictions, which can be found in Figure 1.

Figure 2.

Six stages of the SARS-CoV-2 life cycle. Stage I: Virus entry. I(a): Virus can enter the host cell via plasma membrane fusion. I(b): Virus can enter the host cell via endosomes. Stage II: Translation of viral replication. Stage III: Replication. Here, nsp12 (RdRp) and nsp13 (helicase) cooperate to perform the replication of the viral genome. Stage IV: Translation of viral structure proteins. Stage V: Virion assembly. Stage VI: Release of a virus.

Table 1. Descriptions of SARS-CoV-2 Proteins and 3D Structures.

proteins	3D structures
nsp1 (180 residues) inhibits host innate immune functions by interaction with the human 40S subunit in ribosomal complexes. Its carboxyl terminus (C-terminus) binds to and obstructs the ribosomal mRNA entry tunnel, thereby inhibiting the antiviral response triggered by innate immunity or interferons. The nsp1–40S ribosome complex further induces an endonucleolytic cleavage near the 5′ untranslated region (5′ UTR) of host mRNAs, targeting them for degradation. By suppressing host gene expression, nsp1 facilitates efficient viral gene expression in infected cells and evasion from the host immune response.27	nsp1: 7K7P (X-ray 1.77 Å),28 7K3N (X-ray 1.65 Å),29 7EQ4 (X-ray 1.25 Å),30 7OPL (cryo-EM 4.12 Å);31 nsp1–40S: 6ZOK (cryo-EM 2.80 Å), 6ZOL (cryo-EM 2.80 Å), 6ZOJ (cryo-EM 2.80 Å),27 etc.
nsp2 (638 residues) may play a role in the modulation of the host cell survival-signaling pathway by interacting with the host factors, prohibitin 1 and prohibitin 2, which are involved in maintaining the functional integrity of the mitochondria and protecting cells from various stresses. It appears that nsp2 could change the intracellular milieu and perturb host intracellular signaling.32	7MSX (cryo-EM 3.15 Å), 7MSW (cryo-EM 3.76 Å),32 etc.
nsp3 (1945 residues) includes the papain-like protease (PLpro) and some multipass membrane proteins. PLpro is responsible for cleaving and releasing nsp1, nsp2, and nsp3. PLpro also possesses a deubiquitinating/delSGylating activity and processes both “Lys-48”- and “Lys-63”-linked polyubiquitin chains from cellular substrates. It cleaves preferentially interferon stimulated gene 15 (ISG15) from substrates in vitro, which can play a role in host ADP-ribosylation by binding ADP-ribose. In addition, nsp3 participates together with nsp4 in the assembly of virally induced cytoplasmic double-membrane vesicles necessary for viral replication and antagonizes innate immune induction of type I interferon by blocking the phosphorylation, dimerization, and subsequent nuclear translocation of host interferon regulatory factor 3 (IRF3). It also prevents host NF-κB signaling.33,34	Residues 819–929: 7KAG (X-ray 3.21 Å). Residues 1024–1192 (ADP-ribose phosphatase domain): 6WEY (X-ray, 0.95 Å),35 6WCF (X-ray 1.06 Å).36 Residues 1570–1877 (PLpro): 6W9C (X-ray 2.70 Å).37 Etc.
nsp4 (500 residues) is a multipass membrane protein. Together with nsp3, it participates in the assembly of virally induced cytoplasmic double-membrane vesicles, which is necessary for viral replication.38	AlphaFold prediction.39
nsp5 (306 residues) is the main protease (Mpro/3CLpro) of SARS-CoV-2. It takes charge of cleaving and releasing nsp4–nsp16. Additionally, it recognizes substrates containing the core sequence [ILMVF]-Q-|-[SGACN] and is also able to bind an ADP-ribose-1″-phosphate. Moreover, it plays a role in nsp maturation.22	6LU7 (X-ray 2.16 Å),40 etc.
nsp6 (290 residues) is a multipass membrane protein. Working with nsp3 and nsp4, it induces double-membrane vesicles (autophagosomes) in infected cells from their endoplasmic reticulum. It also limits the expansion of these autophagosomes that are no longer able to deliver viral components to lysosomes.41,42	AlphaFold prediction.39
nsp7 (83 residues) plays a role in viral RNA synthesis. It forms a hexadecamer with nsp8 that may participate in viral replication by acting as a primase. Alternatively, it may synthesize substantially longer products than oligonucleotide primers.43	6M5I (X-ray 2.50 Å), 7BV1 (cryo-EM 2.80 Å), 7BV2 (cryo-EM 2.50 Å)44
nsp8 (198 residues) plays a role in viral RNA synthesis. It forms a hexadecamer with nsp7 that may participate in viral replication by acting as a primase. Alternatively, it may synthesize substantially longer products than oligonucleotide primers.43	6M5I (X-ray 2.50 Å), 7BV1 (cryo-EM 2.80 Å), 7BV2 (cryo-EM 2.50 Å).44
nsp9 (113 residues) functions in viral replication as a dimeric ssRNA-binding protein.45	6W4B (X-ray 2.95 Å),46 6WXD (X-ray 2.00 Å),47 etc.
nsp10 (139 residues) plays a pivotal role in viral transcription. It forms a dodecamer and interacts with both nsp14 and nsp16 to stimulate their respective 3′-5′ exoribonuclease and 2′-O-methyltransferase activities in viral mRNAs that cap methylation.45	6ZCT (X-ray 2.55 Å),48 6ZPE (X-ray 1.58 Å),48 etc.
nsp11 (13 residues) is a pp1a cleavage product at the nsp10/11 boundary. For pp1ab, it is a frameshift product that becomes the N-terminal of nsp12. Its function, if any, is currently unknown.45	None
nsp12 is the RNA-dependent RNA polymerase (RdRp) performing both replication and transcription of the viral genome. Specifically, it catalyzes the synthesis of the RNA strand complementary to a given RNA template. The RdRp of SARS-CoV-2 can be inhibited by the nucleoside analogue Remdesivir.45	6M71 (cryo-EM 2.90 Å),49 6YYT (cryo-EM 2.90 Å),50 etc.
nsp13 (Helicase) (932 residues) is a multifunctional superfamily 1 helicase capable of using both double-stranded DNA (dsDNA) and double-stranded RNA (dsRNA) as substrates with 5′-3′ polarity. In addition to working with nsp12 in viral genome replication, it is also involved in viral mRNA capping. It associates with nucleoprotein in membranous complexes.51	5RLH (X-ray 2.38 Å),49 6ZSL (X-ray 1.94 Å),52 etc.
nsp14 (601 residues) possesses two different activities: (1) an exoribonuclease activity on both single-strand RNA (ssRNA) and dsRNA in a 3′ to 5′ direction; (2) a N7-guanine methyltransferase (viral mRNA capping) activity. It acts as a proofreading exoribonuclease for RNA replication, thereby lowering the sensitivity of the virus to RNA mutagens.53 It always interacts with nsp10.45	Exoribonuclease domain: 7DIY (X-ray 2.69 Å),54 7QGI (X-ray 1.65 Å), 7QIF (X-ray 2.53 Å), etc.
nsp15 (346 residues) is the nidoviral RNA uridylate-specific endoribonuclease (NendoU) that favors the cleavage of RNA at the 3′-ends of uridylates. The loss of nsp15 affects both viral replication and pathogenesis. It is also required for the evasion of host cell dsRNA sensors.55	6WLC (X-ray 1.82 Å),56 etc.
nsp16 (298 residues) is activated by and interacts with nsp10. Its 2′-O-methyltransferase activity mediates mRNA cap 2′-O-ribose methylation to the 5′-cap structure of viral mRNAs. Since N7-methyl guanosine cap is a prerequisite for binding of nsp16, it plays an important role in viral mRNAs cap methylation which is essential to evade the immune system. It may also work against host cell antiviral sensors.45	6W4H (X-ray 1.80 Å),57 etc.
ORF2 (Spike (S) protein) (1273 residues) binds to the host ACE2 receptor and internalizes the virus into the endosomes of host cells with the usage of human TMPRSS2 for priming in human lung cells. S protein consists of three chains and its stalk domain contains three hinges, giving the head unexpected orientational freedom. One single chain can be cleaved into two subunits, S1 and S2. S1 interacts with the host receptor, initiating the infection. S2 mediates fusion of the virion and cellular membranes by acting as a class I viral fusion protein. The current model of S protein has at least three conformational states: prefusion native state, prehairpin intermediate state, and postfusion hairpin state. During viral and target cell membrane fusion, the coil regions (heptad repeats) assume a trimer-of-hairpins structure, positioning the fusion peptide in close proximity to the C-terminal region of the ectodomain. The formation of this structure appears to drive apposition and subsequent fusion of viral and target cell membranes.25	Spike protein: 6VSB (cryo-EM 3.46 Å),17 etc. RBD-ACE2 complex: 6VW1 (X-ray 2.68 Å),58 etc.
ORF3a (275 residues) is a multipass membrane protein that forms homotetrameric potassium sensitive ion channels (viroporin). It upregulates expression of fibrinogen subunits fibrinogen alpha (FGA), fibrinogen beta (FGB), and fibrinogen gamma (FGG) in host lung epithelial cells. It also induces apoptosis in cell culture. Additionally, ORF3a downregulates the type 1 interferon receptor by inducing serine phosphorylation within the interferon (IFN) alpha-receptor subunit 1 (IFNAR1) degradation motif and increases IFNAR1 ubiquitination. More importantly, it activates both NF-κB and NLRP3 inflammasome and contributes to the generation of the cytokine storm. It may also modulate viral release.59	6XDC (cryo-EM 2.9 Å),60 7KJR (cryo-EM 2.08 Å).60
ORF3b (22 residues) along with nucleocapsid protein and ORF6 and ORF3b (22 residues) appears to block induction of type I interferons (IFN-I). This 22-residue variant is also present in SARS-CoV-2-related viral genomes in bats and pangolins.61	None
ORF4 (Envelope (E) protein) (75 residues) is a single-pass type III membrane protein playing a central role in virus morphogenesis and assembly; it acts as a viroporin and self-assembles in host membranes forming pentameric protein–lipid pores that allow ion transport. It is also involved in the induction of apoptosis.62	Residues 8–38: 7K3G (solid-state NMR)63
ORF5 (Membrane protein) (222 residues) is the most abundant structural component of the virion and very conserved. It mediates morphogenesis, assembly, and budding of viral particles through the recruitment of other structural proteins to the ER-Golgi-intermediate compartment (ERGIC). It also interacts with N for RNA packaging into the virion.64	AlphaFold prediction39
ORF6 (61 residues) appears to be a virulence factor. It disrupts cell nuclear import complex formation by tethering karyopherin alpha 2 and karyopherin beta 1 to the membrane. Retention of import factors at the ER/Golgi membrane leads to a loss of transport into the nucleus, thereby preventing STAT1 (signal transducer and activator of transcription 1) nuclear translocation in response to interferon signaling and, thus, blocking the expression of ISGs that display multiple antiviral activities.65	7VPH (X-ray 2.8 Å)66
ORF7a (121 residues) is a type I membrane protein that plays a role as an antagonist of bone marrow stromal antigen 2 (BST-2), disrupting its antiviral effect. As BST-2 tethers virions to the host’s plasma membrane, ORF7a binding inhibits BST-2 glycosylation and interferes with this restriction activity. ORF7a may suppress small interfering RNA (siRNA) and also may bind to host integrin, alpha L (ITGAL), thereby playing a role in attachment or modulation of leukocytes.67	Residues 16–82: 6w37 (X-ray 2.90 Å)
ORF7b (43 residues) is a type III integral transmembrane protein in the Golgi apparatus. In SARS-CoV-2, it appears to be a viral attenuation factor. It may be involved in the human infectivity of SARS-CoV-2.68	None
ORF8 (43 residues) might be a luminal ER membrane-associated protein. It may trigger the activating transcription factor 6 (ATF6) activation and affect the unfolded protein response (UPR). Like ORF7b, it may be involved in the human infectivity of SARS-CoV-2.68−70	7JTL (X-ray 2.04 Å),71 etc.
ORF9a (N protein) (419 residues) packages the positive strand viral genome RNA into a helical ribonucleocapsid (RNP) and plays a fundamental role during virion assembly through its interactions with the viral genome and membrane protein. It also plays an important role in enhancing the efficiency of subgenomic viral RNA transcription as well as viral replication. It may modulate transforming growth factor-beta signaling by binding host smad3.21	Residues 41–174: 6M3M (X-ray 2.70 Å),72 etc.. Residues 247–364: 6ZCO (X-ray 1.36 Å).73 Etc.
ORF9b (97 residues) plays a role in the inhibition of the host’s innate immune response by targeting the mitochondrial antiviral-signaling protein (MAVS). Mechanistically, it usurps the itchy E3 ubiquitin protein ligase (ITCH) to trigger the degradation of MAVS, TNF receptor associated factor 3 (TRAF3), and TRAF6. In addition, it can cause mitochondrial elongation by triggering ubiquitination and proteasomal degradation of dynamin-1-like (DNM1L) protein.74	6Z4U (X-ray 1.95 Å)
ORF9c (70 residues) is located in the N-coding region and interacts with various host proteins including Sigma receptors, implying involvement in lipid remodeling and the ER stress response. It might also target NF-κB signaling.75	None
ORF10 (38 residues) interacts with factors in the Cullin 2 (CUL2) RING E3 ligase complex and thus may modulate ubiquitination.75	7BYF (X-ray 2.5 Å)76

With these SARS-CoV-2 proteins, the intracellular viral life cycle of SARS-CoV-2 can be realized.²³ This life cycle has six stages as shown in Figure 2. The first stage is the entry of the virus. SARS-CoV-2 enters the host cell via either endosomes or plasma membrane fusion. In both ways, the S protein of SARS-CoV-2 first attaches to the host cell-surface protein, angiotensin converting enzyme 2 (ACE2). Then, the cell’s protease, TMPRSS2, cuts and opens the S protein of the virus, exposing a fusion peptide in the S2 subunit of S protein.²⁴ After fusion, an endosome forms around the virion, separating it from the rest of the host cell. The virion escapes when the pH of the endosome drops or when cathepsin, a host cysteine protease, cleaves it. The virion then releases its RNA into the cell.²⁵ After the RNA release, polyproteins pp1a and pp1ab are translated. Notably, facilitated by viral papain-like protease (PLpro), nsp1, nsp2, nsp3, and the amino terminus of nsp4 from the pp1a and pp1ab are released. Moreover, nsp5– nsp16 are also cleaved proteolytically by the main protease.²⁶ The next stage of the life cycle is the replication process, where nsp12 (RdRp) and nsp13 (helicase) cooperate to perform the replication of the viral genome. Stages IV and V are the translation of viral structural proteins and the virion assembly process. In these stages, structural proteins S, E, and M are translated by ribosomes and then present on the surface of the endoplasmic reticulum (ER) and are transported from the ER through the Golgi apparatus for the preparation of the virion assembly. Meanwhile, multiple copies of the N protein package the genomics RNA in the cytoplasm, which interacts with another three structural proteins to direct the assembly of virions. Finally, virions will be secreted from the infected cell through exocytosis.

Since the initial outbreak of COVID-19, the raging pandemic caused by SARS-CoV-2 has lasted over two years. We do have many promising vaccines, but they might have side effects and their full side effects, particularly, long-term side effects, remain unknown. To make things worse, nearly 29208 unique mutations have been recorded for SARS-CoV-2 as shown by Mutation Tracker (https://users.math.msu.edu/users/weig/SARS-CoV-2_Mutation_Tracker.html). All of these reveal the sad reality that our current understanding of life science, virology, epidemiology, and medicine is severely limited. Ultimately, the core of the challenges is the lack of molecular mechanistic understanding of many aspects, namely coronavirus RNA proofreading, virus–host cell interactions, antibody–antigen interactions, protein–protein interactions, protein–drug interactions, viral regulation of host cell functions, including autophagocytosis and apoptosis, and irregular host immune response behavior such as cytokine storm and antibody-dependent enhancement. Molecular-level experiments on SARS-CoV-2 are both expensive and time-consuming and require heavy safety measures. Moreover, disparities among reported experimental binding affinities can be more than 100-fold for the receptor-binding domain (RBD) of S protein binding to ACE2 or antibodies (see Table 1 of ref (77)). All these complicated realities make the understanding of the viral evolution and transmission mechanism one of the most challenging tasks.

On the other hand, computational tools provide alternative approaches in understanding viral evolution and transmission with higher efficiency and lower costs. The increasing computer power, the accumulation of molecular data, the availability of artificial intelligence (AI) algorithms, and the development of new mathematical tools have paved the road for mechanistic understanding from molecular modeling, simulations, and predictions. RBD residues 452 and 501 were predicted to “have very high chances to mutate into significantly more infectious COVID-19 strains” in summer 2020⁷⁸ and were later confirmed in the prevailing SARS-CoV-2 variants Alpha, Beta, Gamma, Delta, Theta, Epsilon, Kappa, Lambda, Mu, and Omicron. These predictions,⁷⁸ achieved via the integration of deep learning, biophysics, genotyping, and advanced mathematics, are some of the most remarkable events. Additionally, 3,696 possible RBD mutations were classified into three categories with different appearance likelihoods, namely, 1149 most likely, 1912 likely, and 625 unlikely.⁷⁸ The predicted “most likely” partition successfully contained all the newly observed RBD mutations, until the recent appearance of S371L from Omicron BA.1. Most remarkably, the mechanism governing SARS-CoV-2 evolution and transmission, i.e., natural selection via mutation-strengthened infectivity, was discovered in July 2020⁷⁸ when there were only 89 RBD mutations with the highest observed frequency of merely 50 globally.⁷⁸ In April 2021, this mechanism was confirmed beyond any doubt. By using 506,768 sequences isolated from patients, the authors demonstrated that the predicted binding free energy (BFE) changes of the 100 most observed RBD mutations out of 651 existing RBD mutations are all above the BFE change of −0.28 kcal/mol, indicating evolution favors variants having higher infectivity.⁷⁹ Moreover, using network-based modeling for drug repurposing, Baricitinib was found to be a potential treatment for COVID-19.⁸⁰ These extraordinary results prove that computational approaches spearhead the discovery of new drugs and the mechanisms of SARS-CoV-2 evolution and transmission.

Considering intensive research activities in molecular modeling, simulations, and predictions of SARS-CoV-2, it has become essentially impossible for experts and researchers to go through the literature. It is important to present a methodology-centered review to enable readers to grasp the current status of SARS-CoV-2 modeling, simulations, and predictions. In this review, the purpose is to provide a general introduction of molecular-level methodologies for SARS-CoV-2 modeling, simulations, and predictions from the aspects of biophysics, mathematical approaches, and machine learning, including deep learning, bioinformatics, and cheminformatics. A wide variety of molecular-level methodologies is described, followed by their applications to SARS-CoV-2. Comments and discussions are presented. Future perspectives are provided in the Concluding Remarks.

2. Methods and Approaches

2.1. Biophysics

The molecular modeling of viruses and their interactions with host cells involves a variety of aspects of biology, biophysics, biochemistry, virology, immunology, computer science, statistics, and mathematics. This section starts with thermodynamics and electrostatics, followed by discussions on molecular dynamics, normal-mode analysis, Monte Carlo methods, molecular docking, and binding free energy analysis, and ends with density-functional theory and quantum mechanics/molecular mechanics methods.

2.1.1. Thermodynamics

Thermodynamics is a foundation of biological science. The laws of thermodynamics are basic principles of biology that govern biological, chemical, and physical processes in all living organisms as well as viruses. The relations among internal energy, Helmholtz free energy, Gibbs free energy, enthalpy, entropy, temperature, volume, and pressure underpin biophysics. The Gibbs–Helmholtz equation describes the thermodynamics calculating changes in the Gibbs energy of a system as a function of temperature. It is a separable differential equation that is given as

1

where ΔG is the change in Gibbs free energy, ΔH is the enthalpy change, T is the absolute temperature, and P is the constant pressure.

In the study of the N protein of SARS-CoV, it was shown that the N protein shows its maximum conformational stability near pH 9.0. The oligomer dissociation and protein unfolding occur simultaneously.⁸¹ In the denaturation of the N protein by chemicals, the Gibbs free energy change (ΔG) of unfolding at temperature (T) is calculated by the solution of the Gibbs–Helmholtz equation

2

where T_m is the transition temperature, ΔH_m is the enthalpy of unfolding at T_m, and ΔC_p is the heat capacity change.

2.1.2. Electrostatic Modeling

In biomolecular studies, electrostatic interactions are important due to their ubiquitous existence in solvation, molecular recognition, molecular interactions, protein–ligand binding, antibody–antigen binding, intramolecular interactions, etc. Electrostatics can be computed using explicit solvent or implicit solvent models as shown in Figure 3. However, including explicit solvent models in free energy calculation is computationally expensive due to their detailed description of solvent molecules. Using an atomic description of the solute molecule, implicit solvent models describe the solvent as a dielectric continuum.⁸²⁻⁸⁶ A wide variety of two-scale implicit solvent models has been developed for electrostatic analysis, including Poisson–Boltzmann (PB),^83,87 generalized Born (GB),⁸⁸⁻⁹¹ polarized continuum,^92,93 and differential geometry-based models.^94,95 GB models give an efficient approximation of PB models but provide only heuristic estimates for electrostatic energies, while PB methods offer more accurate methods for electrostatic analysis.^{86,89,95−99}

Figure 3.

(a) Illustration of the PB model, in which the molecular surface separates the computational domain into the solute region Ω₁ and solvent region Ω₂. (b) Electrostatic potential of the SARS-CoV-2 Mpro based on the PB model.

2.1.2.1. Generalized Born Model

The GB model is devised to offer a relatively simple and computationally efficient approach to calculate electrostatic solvation free energy.⁸⁸⁻⁹¹ Under an appropriate parametrization for a given system, a GB solver can achieve accuracy as a PB solver.¹⁰⁰ The GB approximation of electrostatic solvation free energy can be given as

3

where R_i is the effective Born radius of the ith atom, d_ij is the distance between atoms i and j, ϵ₁ is the dielectric constant of the solute, ϵ₂ is the dielectric constant of the solvent, q_i is the partial charge of atom i, β = ϵ₁/ϵ₂, α = 0.571412, and A is the electrostatic size of the molecule. Additionally, the function f_ij is given as

4

The effective Born radius R_i is calculated via a boundary integral:

5

where ∂Ω₁ is the molecular surface, such as the solvent-excluded surface, dS is the infinitesimal surface element vector, r_i represents the position of atom i, and r shows the position of the infinitesimal surface.

2.1.2.2. Poisson–Boltzmann Model

As illustrated in Figure 3, the PB model describes a two-scale treatment of electrostatics. The interior domain Ω₁ contains the solute biomolecule with fixed charges, and the exterior domain Ω₂ contains the solvent and dissolved ions separated by the interface Γ. While various surface models are available, the most commonly used ones are the molecular surface or solvent excluded surface. A biomolecule in domain Ω₁ consists of a set of atomic charges q_k located at atomic centers r_k for k = 1, ..., N_c, with N_c as the total number of charges. On the other hand, domain Ω₂ contains the mobile ions, whose charge source density is approximated by the Boltzmann distribution. The linearized PB model is always applied:

6

where ϕ(r) is the electrostatic potential, ϵ(r) is a dielectric constant given by ϵ(r) = ϵ₁ for r ∈ Ω₁ and ϵ(r) = ϵ₂ for r ∈ Ω₂, and κ is the inverse Debye length representing the ionic effective length. Interface conditions are involved on ∂Ω₁, which are given as

7

where ϕ₁ and ϕ₂ are the limit values when approaching the interface from inside or outside the solute domain and n is the outward unit normal vector on ∂Ω₁. For the far-field boundary condition of the PB model, lim_|r|→∞ϕ(r) = 0 is implied. Therefore, the electrostatic solvation free energy can be calculated by

8

where ϕ₀(r_k) is the solution of the PB equation as if there were no solvent–solute interface.

Due to their success in describing biomolecular systems, the PB and GB models have attracted wide attention in both the mathematical and biophysical communities.^101−103 Meanwhile, much effort has been given to the development of accurate, efficient, reliable, and robust PB solvers. A large number of methods have been proposed in the literature, including the finite difference method (FDM),¹⁰⁴ finite element method (FEM),¹⁰⁵ and and boundary element method (BEM).¹⁰⁶ The emblematic solvers in this category include Poisson–Boltzmann surface area (PBSA),^107,108 Delphi,^109,110 adaptive Poisson–Boltzmann solver (ABPS),^97,111 matched interface and boundary-based Poisson–Boltzmann (MIBPB),^98,102,112 chemistry at Harvard macromolecular mechanics (CHARMM) PBEQ-Solver,¹⁰⁴ and treecode-accelerated boundary integral (TABI) PB solver.^113,114

The PB and GB models have been applied to SARS-CoV-2 studies including protein–ligand binding and protein–protein binding energetics. The surface electrostatic potential values of S protein and Mpro were calculated for SARS-CoV and SARS-CoV-2 with almost the same values for both viruses,¹¹⁵ as well as the SARS-unique domain.¹¹⁶ When focusing on the RBD of S protein binding to ACE2, slightly higher binding energy was revealed for SARS-CoV-2 compared to SARS-CoV because of enhanced electrostatic interactions with the negative electrostatic potential of ACE2 and positive electrostatic potential of RBD.¹¹⁷ For the fusion cleavage site on S protein, mutations near the cleavage site caused changes in the electrostatic distribution of the S protein surface.¹¹⁸ Antigens targeting SARS-CoV-2 from T cells were studied using the electrostatic surface potentials.¹¹⁹ By studying the surface potential of S protein, it was shown that the pH and salt concentration changed dramatically in terms of scale and sign for electrostatic interactions.¹²⁰ Recently, Dung et al. proposed a theoretical model to elucidate intermolecular electrostatic interactions between a virus and a substrate. Their model treats the virus as a homogeneous particle having surface charge and the polymer fiber of the respirator as a charged plane. The electric potentials surrounding the virus and fiber are influenced by the surface charge distribution of the virus. The PB equation was used to calculate the electric potentials. Then, Derjaguin’s approximation and a linear superposition of the potential function were extended to determine the electrostatic force.¹²¹

2.1.3. Molecular Dynamics (MD) Simulation

Macromolecular structures are highly dynamic rather than static. X-ray crystallography and nuclear magnetic resonance (NMR) reveal that even the same molecule can adopt multiple conformations.^122,123 On the other hand, conformational change plays a significant role in biomolecular functions. While NMR is limited to small biomolecules and X-ray crystallography can only provide static structures, MD simulation is an effective way to investigate biomolecular conformational changes.^124,125 The Poisson–Boltzmann-based MD is also studied.¹²⁶ Furthermore, thanks to high-performance computing platforms such as graphical processing units (GPUs), current MD simulations can reveal conformational changes of biomacromolecules such as proteins, DNA, and RNA in the time scale of milliseconds (ms).¹²⁷

MD simulation is becoming an invaluable computational method commonly used for understanding the biomolecular structure and dynamics of atoms in macromolecules (proteins and RNA). Describing internal forces in the structure with simple mathematical functions, the motions are determined by using Newton’s second law.¹²⁸ In Figure 4a, a general MD algorithm is demonstrated, where potential energy functions (force field), energy minimization, environment settings, ensembles, and solvation are included. In the prediction of atom positions and velocities, equations are given by a standard Taylor expansion. Classical interatomic potentials or quantum mechanisms are applied to calculate forces, which is followed by the correction of positions and velocities with some functions f, g of a, and Δt by energy minimization. Among all their effects on MD simulations, a collection of suitable force field functions is of fundamental importance to all other dynamics methodologies, which will be introduced first.

Figure 4.

(a) Workflow of molecular dynamics simulations. (b) Workflow of the metropolis Monte Carlo method.

Molecular Mechanics Force Field. A molecular mechanics force field is a set of functions equipped with an associated set of parameters, describing the interactions between atoms. The energy function for nonbonded interactions is associated with simple pairwise additive functions, van der Waals and electrostatic forces, of nuclear coordinates only, while for bonded groups, that is the forms of chemical bonds, bond angles, and bond dihedrals.¹²⁹ For example, the functional form of a typical force field such as AMBER¹³⁰ is given as

9

where k_r and k_θ are the force constants for the bond lengths and bond angles, respectively. Here, r and θ are a bond length and a bond angle, r₀ and θ₀ are the equilibrium bond length and bond angle, ω is the dihedral angle, k_n is the corresponding force constant, n is the multiplicity, and phase angle γ_n takes values of either 0° or 180°. The nonbonded part of the potential is represented by the Lennard-Jones repulsive A_ij and attractive B_ij terms for Coulomb interactions between partial atomic charges (q_i and q_j). Here, R_ij is the distance between atoms i and j. Finally, ϵ is the dielectric constant that considers the medium effect that is not explicitly represented and usually equals 1.0 in a typical solvated environment where the solvent is represented explicitly. The nonbonded terms are calculated for atom pairs that are either separated by more than three bonds or not bonded. CHARMM is another popular force field.¹³¹ Polarizable models, such as the AMOEBA force field,¹³² have been developed to achieve higher accuracy. To tackle systems with an excessive number of atoms, coarse-grained models are also developed. In these models, a group of atoms is represented by a “pseudo-atom”, so the number of atoms is largely reduced.¹³³ Popular coarse-grained models are the Go̅ model,¹³⁴ MARTINI force field,¹³⁵ united-residue (UNRES) force field,¹³⁶ etc.

Energy Minimization. Energy minimization methods are applied to efficiently optimize molecular structures. In a complex system of N atoms, the potential energy function, such as eq 9U(r^N), has its global minimum. It is extremely computationally expensive to provably locate the global minimum. With an unrefined molecular structure equipped with a force field, energy minimization methods can be iteratively applied to molecular systems. The steepest descent method is one of the most popular iterative descent methods, which uses derivatives of various orders and points the path toward the nearest energy minimum. Thus, the force is calculated by

10

where r is the vector of the atomic coordinates.

Solvation. In an aqueous environment, a protein is solvated in a pure or ion-containing water environment, and explicit solvent models are computationally expensive. To avoid using water explicitly in modeling, numerous implicit solvent models have been developed.⁸²⁻⁹³ Two implicit solvent models, the Poisson–Boltzmann method and the generalized Born model, have been described in previous sections. In addition, there are explicit solvent models such as the transferable intermolecular potential with 5 points model,¹³⁷ the extended simple point charge model,¹³⁸ and the flexible simple point charge model water models.

Since MD simulations can provide many samplings, one can calculate the free energy change between different states from these samplings. Typical binding free energy calculation methods based on MD simulations are the molecular mechanics energies combined with Poisson–Boltzmann or generalized Born and surface area continuum solvation (MM/PBSA and MM/GBSA),^108,139,140 free energy perturbation (FEP),¹⁴¹ thermodynamic integration,¹⁴² metadynamics,¹⁴³ and steered MD simulations.¹⁴⁴ Recently, a method that is more efficient than normal-mode analysis, called WSAS (work and social adjustment scale),¹⁴⁵ was developed to estimate the entropic effect in the free energy calculation.

2.1.3.1. MD Simulations Revealing Conformational Changes

The most important application of MD simulations is to investigate the dynamical properties of SARS-CoV-2 proteins and the interactions between proteins and inhibitors. Moreover, Mpro and S protein are the two main targets of SARS-CoV-2 proteins to investigate, while some studies focused on PLpro. For larger systems, coarse-grained MD simulations are implemented.

Mpro. To enhance sampling of conformational space, a microsecond-scale Gaussian accelerated MD simulation to SARS-CoV-2 Mpro was performed,¹⁴⁶ where the simulations identified cryptic pockets within Mpro, including some regions far from the active site. The 2 μs MD trajectories of the apo form of the SARS-CoV-2 Mpro indicated that the long loops, which connect domains II and III and provide access to the binding site and the catalytic dyad, carried out large conformational changes.¹⁴⁷ Additionally, MD simulations were applied to compare the dynamical properties of the SARS-CoV-2 Mpro and SARS-CoV Mpro, which suggests that the SARS-CoV Mpro has a larger binding cavity and more flexible loops¹⁴⁸ and reveals the key interactions and pharmacophore models between the Mpro and its inhibitors.¹⁴⁹ Recently, Sanjeev et al. used MD simulations to study the impact of a crowded environment on drug–Mpro complexes, suggesting that crowding enhances the difference in the dynamics of apo- vs drug-bound complexes.¹⁵⁰ Lamichhane et al. not only ran MM/PBSA calculation but also used the analysis by dihedral angle distribution and radial distribution functions to confirm the strong interactions between inhibitor N3 and Mpro.¹⁵¹

S Protein. MD simulations were performed to study the binding of S protein and ACE2,^152−156 which were the most important studies of S protein. Notably, a 100 ns MD simulation of the complexes of human ACE2 and S protein from SARS-CoV-2 and SARS-CoV showed that the SARS-CoV-2 complex was more stable.¹⁵⁶ Grishin et al.’s MD simulation suggests disulfide bonds play a critical role in S protein–ACE2 binding and the flexibility of the surface loops increases when the four disulfide bonds of the domain are reduced.¹⁵⁷ As for temperature, MD simulations at different temperatures suggested S protein had a stronger binding at a low temperature.¹⁵⁸ Abdalla et al. investigated the effects of mutations on S protein stability and solubility through MD simulations in a 100 ns period.¹⁵⁹ Inhibitor and antibody binding to ACE2 was studied by MD simulations.^160−162 One of the studies suggested that the SARS-CoV-2 S protein can interact with a nicotinic acetylcholine receptors (nAChRs) inhibitor.¹⁶¹ Moreover, ACE2-Fc fusion proteins with the SARS-CoV-2 S protein RBD were simulated by a glycosylated molecular model.¹⁶³ In a steered MD model, a semiopen intermediate state was observed of the transition between closed and open states of S protein.¹⁶⁴ Further study was about the motion of glycans in S protein by a 1 μs MD simulation, uncovering the detail of the S protein glycan shield.¹⁶⁵ A recent interesting study by Lupala et al. performed an MD simulation of the SARS-CoV-2 S protein with ACE2 from different species. Their findings suggest that the ACE2 proteins of bovine, cat, and panda form strong binding interactions with RBD, while in the cases of rat, least horseshoe bat, horse, pig, mouse, and civet, the ACE2 proteins interact weakly with RBD.¹⁶⁶

PLpro and Other Proteins. MD simulations were also used to investigate the conformational changes of other SARS-CoV-2 proteins. For PLpro, researchers performed pH replica-exchange CpHMD (constant pH molecular dynamics) simulations to estimate the pK_a values of Asp/Glu/His/Cys/Lys side chains and assessed possible proton-coupled dynamics in SARS-CoV, SARS-CoV-2, and MERS-CoV PLpros.¹⁶⁷ They also suggested a possible conformational-selection mechanism by which inhibitors bind to the PLpro. Supervised MD simulations were employed to investigate the unbinding pathways of GRL0617 and its derivates from PLpro.¹⁶⁸ Sun et al. applied MD simulations and topological and electrostatic analyses to study the effects of palmitoylation on an E protein pentamer. Their results indicated that the structure of the palmitoylated E protein pentamer was more stable while the loss of palmitoylation caused the pore radius reduced and even collapsed, which might help the drug design for the treatment of COVID-19.¹⁶⁹

2.1.3.2. Coarse-Grained MD Simulations

Modeling the whole SARS-CoV-2 in a fine grid is extremely time-consuming, if not impossible. To study the behavior of SARS-CoV-2, a coarse-grained model based on the data from a combination of cryo-electron microscopy and X-ray crystallography was employed in a complete virion model.¹⁷⁰ More importantly, the binding between S protein and ACE2 or antibodies can be studied by coarse-grained MD simulations.¹⁷¹ Bai et al. also used their own coarse-grained models to predict mutation-induced binding affinity changes between Mpro and human ACE2.¹⁷² With replica-exchange umbrella sampling MD simulations, a comparison of the binding of SARS-CoV-2 S protein and SARS-CoV S protein to human ACE2 revealed that the SARS-CoV-2 binding to human ACE2 is stronger than that of SARS-CoV.¹⁷³ The infectivity induced by mutations on S protein is another problem after studying the binding of ACE2 and S protein, where coarse-grained MD simulations were employed to reveal the dynamics impact of mutations T307I and D614G¹⁷⁴ or SARS-CoV-2 variants B.1.1.7 and P1.¹⁷⁵ The impacts of mutations to antibodies CR3022 and CB6 were also predicted.¹⁷⁶ Lastly, glycan shield effects on drug binding are also studied via multiscale coarse-grained MD simulations.¹⁷⁷

2.1.4. Normal-Mode Analysis

Compared to molecular dynamics simulations, normal-mode analysis (NMA) has its advantages in dealing with the flexible motions accessible to a protein system at a steady-state position. Based on the equation of motion, normal-mode analysis studies molecular structure conformation by a restoring force acting on a vibrational system perturbed slightly at its equilibrium. Achieving its efficiency for large proteins and protein complexes, NMA is widely applied in large molecules and homology modeling studies, as well as evolutionary and stability analysis of proteins. Additionally, for computational efficiency, the elastic network model (ENM), Gaussian network model (GNM), and anisotropic network model (ANM) are developed and applied to similar problems.

The derivation of normal-mode analysis starts from the equations of motion. It is formulated from Lagrange’s second kind equation, with the Lagrangian , where E_k and U are the kinetic and potential energies of the molecular system, respectively.^178−180 The potential energy U is calculated by eq 9. The system is defined to be in a potential minimum of equilibrium where the generalized forces acting on it are eliminated. In this Lagrangian mechanical system , M is a configuration space and Lagrangian , where r ∈ M, v is the velocity vector at position r, and t is the time. They thus define the generalized coordinates r̂ and apply the Taylor expansion to the potential energy. The equation can be formulated as

11

where r_i is the ith component of the instantaneous configuration and η_i = r_i – r̂_i. Here, the Einstein summation convention is used. Note that eq 11 studies the mechanism system at equilibrium, and therefore, the first term can be set to zero in terms of the minimum value of the potential with the second term zero at any minimum. We can rewrite the potential energy to be

12

where U_ij is the Hessian matrix. As for the kinetic energy E_k, it is given as

13

where M is a diagonal matrix of the mass of each particle. By applying Lagrange’s equation, the equations of motion can be given as

14

with one oscillatory solution η_i = a_ij cos(ω_jt + δ_j), where a_ij is the amplitude of oscillations, ω_j is the frequency, and δ_j is a factor. Here, it is easy to obtain the eigenvalue problem UA = λA, where A is an eigenvector of the aforementioned Hessian matrix and λ is an eigenvalue which is the square of frequency ω_k. The eigenvector is also referred to as a normal-mode vector for a particle’s movement in terms of direction and distance with a certain frequency.

2.1.4.1. Elastic Network Model

Though NMA has computational efficiency compared to MD simulation, its computation is still expensive, especially for proteins containing tens of thousands of atoms. Additionally, one assumption of NMA requires energy minimization to ensure the starting conformation is at equilibrium. This minimization process might distort the structure, leading to a different structure from the experimental one. Therefore, a variety of coarse-grained approximate algorithms has been developed to overcome these limitations.^181,182 Among them, the elastic network model (ENM) is widely applied.

The ENM simplifies the force fields used in standard NMA by a harmonic potential¹⁸³

15

where d_ij is the distance between the ith and jth atoms, d_ij⁰ stands for the distance at the initial structure, and k mimics the spring constant in Hooke’s law. R_C is a cutoff and usually is set between 7.0–8.0 Å, according to the distances between nonbonded atoms.¹⁸⁴ More studies consider the C_α atoms only, for their sufficiency in backbone motion investigation. Many generalizations implemented the ENM’s idea to reformulate potential functions.

2.1.4.2. Gaussian Network Model and Its Generalization

One of the generalizations is the Gaussian network model (GNM),¹⁸⁵ which is considered the most efficient one, using the discrete Laplacian matrix instead of the Hessian matrix. The expected residue fluctuations constructed by the GNM are in great agreement with the Debye–Waller factor (a.k.a. B factor). More precisely, the B factor of the ith α carbon atom (C_α) in an N-particle coarse-grained representation of a biomolecule can be obtained by the generalized GNM (gGNM) method^186,187

16

where a_gGNM is a fitting parameter and (Γ^–1)_ii is the ith diagonal element of the matrix inverse Γ^–1. Here, Γ is the generalized Kirchhoff matrix

17

where r_i are positions of C_α, the kernel functions Φ can be generalized exponential functions or generalized Lorentz functions, and η_j are the characteristic distances.^186,187 If the kernel functions are set to be {0, 1} with distance ∥r_i – r_j∥ outside or inside a fixed cutoff distance, then the Kirchhoff matrix becomes the Laplacian matrix and the GNM is recovered.

2.1.4.3. Anisotropic Network Model and Its Generalization

Another popular method is the anisotropic network model (ANM),^183,188 which gives extra information about the directionality of the fluctuations. The B factor of ith C_α in an N-particle coarse-grained biomolecule can be displayed by the generalized ANM method^186,187

18

where a_gANM is a fitting parameter and (H^–1)_ii is the ith diagonal element of the matrix inversion of Hessian matrices.

2.1.4.4. Applications to SARS-CoV-2

The SARS-CoV-2 Mpro is used as one of the most popular target proteins for drug repurposing in applications. According to the stability analysis by NMA, the inhibitor repurposing of SARS for COVID-19 may be challenging.¹⁴⁸ With further investigations of Mpro by the ENM, possible noncompetitive inhibiting binding sites were suggested¹⁸⁹ (see Figure 5). Moreover, the SARS-COV-2 S protein is another target for inhibitor repurposing. In the comparison of the S proteins of SARS-CoV-2, SARS-CoV, and MERS-CoV, the ANM was employed to study the dynamic modes of the S proteins, revealing that the receptor binding motif had high vertically upward motion.¹⁹⁰ In ref (191) ENM-based analysis tools were applied for the allosteric modulation region of the S protein with ACE2, and it was indicated that hepcidin induces an inhibitory effect on the binding affinity of the S protein and ACE2. The stability and flexibility of mutations can be examined by normal-mode analysis, especially the mutations on RBD or high-frequency mutation D614G of the S protein^175,192,193 and on other SARS-CoV-2 proteins.^193,194

Figure 5.

Illustration of the ENM on SARS-CoV-2 Mpro.¹⁸⁹ Reproduced with permission from ref (189). Copyright 2021 Dubanevics and McLeish under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/. (a) Mpro secondary structure. (b) Elastic model of Mpro. Cα atoms are in blue, and node-connecting springs are in black. (c) The first real vibrational mode eigenvectors are in yellow.

2.1.5. Monte Carlo Methods

Monte Carlo (MC) methods rely on repeated random sampling to obtain optimized numerical results. In principle, Monte Carlo methods can be used to solve any problems having a probabilistic distribution.¹⁹⁵ When the probability distribution of the variable is parametrized, researchers often use a Markov chain Monte Carlo (MCMC) sampler,¹⁹⁶ whose central idea is to design a judicious Markov chain model with a prescribed stationary probability distribution. By the ergodic theorem, the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. Recently, a machine-learning-based implicit solvent Monte Carlo method was developed to predict the molecular structure.¹⁹⁷

Importantly, metropolis Monte Carlo methods¹⁹⁸ are popular in molecular modeling. As shown in Figure 4b, the essential idea is that, if the energy of a trial conformation is lower than or equal to the current energy, it will always be accepted. If the energy of a trial conformation is higher than the current energy, then it will be accepted with a probability determined by the Boltzmann (energy) distribution,

19

where j is the current conformation, i is the new conformation, is the probability to accept the new conformation, ΔU_ij is the energy difference between i and j, k_B is the Boltzmann constant, and T is temperature. Therefore, the evolution of molecular conformations can be simulated. There are several aspects of MC applications regarding SARS-CoV-2. An MC simulation of ionizing radiation damage to the SARS-CoV-2 found that γ-rays produced significant S protein damage but much less membrane damage.¹⁹⁹ Thus, the γ-rays were proposed as a new effective tool to develop inactivated vaccines. A metropolis MC sampling process was applied to simulate a pharmacokinetic model of the human immunodeficiency virus (HIV) drug darunavir against SARS-CoV-2.²⁰⁰ MC modeling was also implemented in the analysis of SARS-CoV-2 PLpro²⁰¹ and N protein.²⁰² Studies focusing on Mpro and S protein are introduced as follows.

Mpro. Liang et al. used protein energy landscape exploration (PELE) Monte Carlo simulations for a blind binding site search and the best binding poses for these binding sites.²⁰³ Their simulations found that compounds such as cyanidin-3-O-glucoside and hypericin have the strongest interactions with the active sites. Their PELE also identified additional binding sites for hypericin with comparable interaction energies.²⁰³ A coarse-grained Monte Carlo simulation was integrated with other computational methods to reveal the relationship between the rigidity and enzymatic function for Mpro,²⁰⁴ while the Mpro inhibitory activity of aromatic disulfide compounds was studied by the weight search of MC simulations for the QSAR (quantitative structure–activity relationship) model.²⁰⁵

S Protein. Two major directions of the S protein are mutation studies,^206,207 namely G614D and N501Y, and binding problems about ACE2,^117,208 antibodies,¹⁷¹ and peptide-based inhibitors.^171,208,209 To estimate the density of states of the S protein system, the Wang and Landau Monte Carlo method was applied to study the human ACE2 complexes with SARS-CoV-2 and SARS-CoV S protein RBDs,²⁰⁸ while the difference between SARS-CoV-2 and SARS-CoV was studied.²¹⁰ In the study of the heterogeneity of glycosylation on the S protein trimer make up, the MC approach was applied to calculate the glycan mass distribution because of the enormous number of possible glycoforms.²¹¹

2.1.6. Molecular Docking

As shown in Figure 6a, molecular docking, which can predict the binding conformation of a ligand on its binding site, is one of the most popular methods in the structure-based drug design.^213,214 A typical docking program includes two key components: a scoring function to calculate the binding energies of different conformations and a search algorithm to sample the conformational degrees of freedom and locate the global energy minimum from all the sampled conformations.²¹⁵ Traditional scoring functions are derived from physical models such as molecular mechanism force fields. In recent years, more and more machine-learning-based scoring functions have been developed, which outperform traditional scoring functions in many cases.^216−219 In addition to regular docking, ensemble docking²²⁰ considers the dynamics of the receptor and docks a ligand to various receptor conformations (often yielded from molecular dynamics simulation). Molecular docking is well-established in early stage drug discovery. As a result, during the pandemic, to seek drug leads, docking studies have been performed targeting a variety of SARS-CoV-2 proteins.

Figure 6.

(a) Procedure of molecular docking simulation. (b) Procedure of quantum mechanics/molecular mechanics (QM/MM) calculation.²¹² Reproduced with permission from ref (212). Copyright 2017 Royal Society of Chemistry.

Mpro. One important source for searching for SARS-CoV-2 treatments is existing drugs. In a quite extensive drug repurposing work, 7173 purchasable drugs, including 4574 unique compounds and stereoisomers, were docked and their binding affinities to Mpro were predicted.²²¹ As a result, diosmin, hesperidin, and MK-3207, with a docking score of −10.1 kcal/mol, were suggested as the most potent inhibitors. A collection of 8625 drugs or compounds from FDA, drugbank, and Zinc data sets were docked to Mpro,²²² and seven drugs such as metyrapone could maintain key interactions within the active site of the enzyme suggested by the crystallographic complex structures, revealing their repurposing potential. Additionally, Sencanski et al.²²³ and Gurung et al.²²⁴ both screened about 1400 FDA-approved drugs with docking, predicting that dihydroergotamine has a promising affinity. Docking was used to evaluate the potency of around 100 approved protease inhibitors, and it suggested that faldaprevir has the strongest binding affinity.²²⁵ In a screening of roughly 7100 molecules, several natural molecules such as δ-viniferin, myricitrin, taiwanhomoflavone A, lactucopicrin 15-oxalate, nympholide A, afzelin, biorobin, hesperidin, and phyllaemblicin B were identified.²²⁶ Many other studies^{222,227−236} have also docked and repurposed existing drugs against SARS-CoV-2 Mpro.

Natural products are popular inhibition candidates. In a study of 1000 active phytochemicals from Indian medicinal plants by molecular docking, rhein and aswagandhanolide were predicted to have binding affinities over −8.0 kcal/mol.²³⁷ In a study of 100 natural and nature-inspired products from an in-house library to Mpro, leopolic acid A is predicted to have the highest affinity of −12.22 kcal/mol.²³⁸ From mushrooms and other herbal or natural compounds, colossolactone VIII²³⁹ and eugenin²⁴⁰ were identified as having a high affinity to Mpro. It is predicted that from Amphilophium paniculatum leaves, luteolin 7-O-b-glucopyranoside (cynaroside) has the highest affinity of −9.54 kcal/mol.²⁴¹ All of them also predicted ADMET (absorption, distribution, metabolism, excretion, and toxicity) properties of their compounds. Moreover, a variety of natural products were also investigated by docking-based virtual screening, including syzygium aromaticum, cassia acutifoliaaloe vera, rhus spp., moroccan medicinal plants, fungal metabolite, millet, tannins, neem leaves, nigella sativa, etc.^242−256

Some researchers focus on peptides and small compounds from other sources. Tsuji et al.²⁵⁷ screened compounds from the ChEMBL (chemical database of bioactive molecules with drug-like properties) database²⁵⁸ to Mpro via docking, suggesting that the compound CHEMBL1559003 with a binding affinity of −10.6 kcal/mol is the most potent. In a set of 13535 Mpro compounds, the consensus of the docking scores from the five different pieces of docking software evaluates the potency.²⁵⁹ Udrea et al.²⁶⁰ predicted the binding affinities of 15 phenothiazines, where the compound sulphoridazine (SPZ) was reported as the most effective. Ghaleb et al.²⁶¹ also studied some pyridine N-oxide compounds, indicating the most potent one with a predicted pIC₅₀ (the negative log of the IC₅₀ when converted to molar) value of 5.294. More similar works can be found in the literature.^262−270

S Protein. Many researchers focus on the binding interactions of S protein and the host ACE2 or antibodies. A docking calculation was applied to construct the complexes of SARS-CoV-2 S protein and SARS-CoV S protein binding to ACE2,²⁷¹ which suggested that there were more residue interactions between SARS-CoV-2 S protein and ACE2 than that of SARS-CoV S protein. It also leads to a higher binding affinity and is consistent with a recent experiment.²⁷² In a study of ACE2s from cats, tigers, hamsters, dogs, and ferrets via homology modeling and docking, it was shown that the cat and tiger ACE2s could potentially interact with S protein RBD and share the same virus-binding interface with ACE2, whereas the dog, ferret, and hamster ACE2 were not predicted to establish stable interactions with S protein RBD.²⁷³ A docking simulation suggested ACE2 polymorphisms from different human races could change the ACE2 affinity to SARS-CoV-2 S protein.²⁷⁴ For other receptors, S protein cannot strongly bind to the human dipeptidyl-peptidase 4 (DPP4) receptor,²⁷⁵ while S protein RBD was able to bind to many amyloidogenic proteins, initiating aggregation of these proteins and leading to neurodegeneration in the brain.²⁷⁶ Interestingly, a recent docking study by Kazybay et al. suggested that the Omicron variant EGFR (epidermal growth factor receptor) was one of the potential binary partners of the S RBD that binds almost with equal affinity as the RBD–hACE2 complex.²⁷⁷ Hanai et al. used docking studies to suggest that Omicron’s binding to ACE2 was stronger than Delta’s and Alpha’s.²⁷⁸

More investigations were carried out to repurpose existing drugs to S protein or find inhibitors for S protein. In a study of screening FDA-approved drugs, iron oxide nanoparticles were suggested for COVID-19 treatment.²⁷⁹ Existing drugs, such as amentoflavone, ledipasvir, tenofovir, levodopa, lopinavir, and ubrogepant, against S protein were investigated.^280−283 For natural products inhibiting S protein, studies focusing on indigenous food additives, herbal constituents, antioxidants, traditional medicinal plants, tea, and others^284−289 suggested potent inhibitors such as phycocyanobilin, phycoerythrobilin, phycourobilin, folic acid, hinokiflavone, and phytochemicals. Lastly, some other compounds are repurposed to inhibit the S protein. Mohebbi et al.²⁹⁰ screened more than 1 billion compounds from the databases ZINC Pharmer and Pharmit in silicon. The docking of dermaseptin-based antiviral peptides to the S protein was studied.²⁹¹ Some drugs were identified with high binding potential against the ACE2–S protein interaction pocket, such as Atazanavir, Grazoprevir, Saquinavir, Simeprevir, Telaprevir, and Tipranavir.²⁹²

RdRp. RdRp is another target for docking inhibitors from existing drugs or traditional medicines. In a data set of 7922 approved or experimental drugs, Nacartocin has the highest binding affinity.²⁹³ Beg et al.²⁹⁴ screened 70 anti-HIV (human immunodeficiency virus) or anti-HCV (hepatitis C virus) drugs, reporting that the drug paritaprevir has the highest binding affinity. Aftab et al.²⁹⁵ studied 10 antiviral drugs and revealed that Remdesivir’s docking score was the highest, but Padhi et al.¹⁹⁴ showed that the docking affinity of remdesivir is relatively low. Meanwhile, many researchers screened potential drugs in traditional medicinal compounds. Theaflavin was reported to have the highest binding affinity among a data set of 83 traditional Chinese medicinal compounds plus their similar structures from the ZINC15 database.²⁹⁶ Pandeya et al.²⁹⁷ also investigated some biologically active alkaloids of argemone mexicana. Other RdRp drug repurposing works can be found in the literature.^298−301

Other Targets. Some docking studies selected the SARS-CoV-2 PLpro as their targets. Choudhury et al.³⁰² docked 27 existing drugs to PLpro and predicted stallimycin to be the best inhibitor. Similarly, Li et al.³⁰³ repurposed 21 drugs to inhibit SARS-CoV-2 PLpro and reported neobavaisoflavone as the most potent candidate. Mohideen et al.³⁰⁴ revealed that the binding affinity of the natural product thymoquinone to the E protein is −9.01 kcal/mol. Borgio et al.³⁰⁵ screened 23 FDA-approved drugs to target the helicase of SARS-CoV-2 and reported vapreotide having a binding affinity of −11.58 kcal/mol as the most potent candidate. Mahmud et al.³⁰⁶ showed that drugs such as valrubicin, aprepitant, and saquinair have excellent docking scores to SARS-CoV-2 nsp15. Khan et al.³⁰⁷ used docking to study the interaction between N protein and nsp3.

Multiple Targets. Many researchers studied the whole SARS-CoV-2 or multiple SARS-CoV-2 proteins for more potent inhibitors. In an analysis of therapeutic targets for SARS-COV-2 involving homology modeling and molecular docking, a data set of 78 commonly used antiviral drugs for SARS-CoV-2 proteins was selected.²² By using molecular docking on 2631 US FDA-approved small molecules, five drugs (avapritinib, bictegravir, ziprasidone, capmatinib, and pexidartinib) were suggested as candidates against SARS-CoV-2 proteins. In a study of docking 11 antiviral drugs to Mpro, S protein, PLpro, nsp10, nsp16, and nsp9, ritonavir, lopinavir, and remdesivir were selected as drug candidates against SARS-CoV-2.³⁰⁸ Other approved structural analogs, such as telbivudine, tenofovir, amprenavir, and fosamprenavir, were identified as potent drugs for SARS-CoV-2 by molecular docking.³⁰⁹ On a data set consisting of 2285 FDA-approved drugs and 1478 Taiwan National Health Insurance-approved drugs (https://covirus.cc/drugs/), a virtual screening targeting S protein, Mpro, PLpro, RdRp, N protein, hACE2, and human cellular TMPRSS2 were conducted.³¹⁰ Chandel et al.³¹¹ repurposed about 2000 FDA-approved compounds targeting S protein and nsp9, reporting that Tegobuvir was the most potent candidate to S protein and Conivaptan was the most potent candidate to nsp9. Many works considered two different protein targets. Elmezayen et al.³¹² virtually screened 4500 approved or experimental drugs against Mpro and human TMPRSS2, finding out that ZINC000103558522 has the highest binding affinity to Mpro and ZINC000012481889 has the highest binding affinity to the TMPRSS2. Via the DockThor-VS platform, Guedes et al.³¹³ predicted the binding affinities of over 40 approved drugs to SARS-CoV-2 Mpro, S protein, PLpro, RdRp, N protein, and nsp15. The binding affinities of the compounds protoporphyrin IX, verteporfin, and chlorin e6 to Mpro, S protein, ORF3a, ORF9b, and ORF7a were studied.³¹⁴ In addition, some components such as essential oil components,³¹⁵ organosulfur compounds,³¹⁶ and methisazones³¹⁷ were investigated. It was exhibited that rutin has some inhibitory effect on SARS-CoV-2 proteins.³¹⁸ Phytochemicals from the traditional medicines were also investigated by docking, such as those from traditional Himalayan medicinal plants,³¹⁹ Indian traditional medicinal plants,^320−333 Chinese traditional medicines,³³⁴ and Brazilian herbal medicines.³³⁵

Targeting Mpro, S protein, and RdRp, Parvez et al.²⁹⁸ studied some plant metabolites. Maurya et al.³³⁶ investigated yashtimadhu (glycyrrhiza glabra) active phytochemicals, and Alexpandi et al.³³⁷ simulated quinoline-based inhibitors. Flavonoids might inhibit Mpro, S protein and RdRp as well.^338,339 Against the S protein and nsp15S, Sinha et al.³⁴⁰ screened 23 saikosaponins and reported that saikosaponin V was potent to both targets. Montelukast was predicted to be potent to both Mpro and RdRp³⁴¹ as well as Camptotecin.²⁸⁷ Srikanth et al.³⁴² and Agrawal et al.³⁴³ virtually explored the potential of andrographolide as well as other antivirals, antibiotics, antiparasitics, flavonoids, and vitamins in inhibiting S protein and RdRp. Another molecular docking of compounds such as coumarins, porphyrins, propolis, or existing drugs can be found in refs (344−349).

2.1.7. Binding Free Energy Calculations

In the study of SARS-CoV-2, protein–protein and protein–ligand interaction processes are essential. Some of these processes are investigated through molecular biophysics, such as the binding free energies for protein–ligand and protein–protein complexes. To estimate the binding free energies, classical methods such as FEP and thermodynamic integration (TI) methods are computationally expensive, while many other methods are developed considering efficiency, such as the molecular mechanics/Poisson–Boltzmann surface area (MM/PBSA) method,^350,351 the molecular mechanics/generalized Born surface area (MM/GBSA) method,^351,352 the linear interaction energy (LIE) method,³⁵³ the chemical Monte Carlo/molecular dynamics (CMC/MD) method,^354,355 the pictorial representation of free energy components (PRO-FEC) method,³⁵⁶ etc. Among these methods, MM/PBSA and MM/GBSA are widely applied for their accuracy and efficiency.

2.1.7.1. MM/PBSA and MM/GBSA

In the MM/PBSA and MM/GBSA approaches,¹⁰⁸ the binding free energy ΔΔG_bind for binding between a ligand and a protein receptor in the form of a protein–ligand complex can be calculated by

20

where ΔG_complex is the total free energy of the complex and ΔG_protein and ΔG_ligand are the total free energies of the protein and ligand in solvent, respectively (see Figure 7). The free energy for each individual body can be calculated by

21

where G_* are referring to the total free energies of the complex, protein, and ligand, ⟨V_MM⟩ is the average molecular mechanical potential energy in a vacuum of eq 9, and TS is the entropic contribution to the free energy with the temperature T and the entropy S in a vacuum. ΔG_sol is the free energy of solvation

22

The polar solvation energy ΔG_sol^polar is calculated by solving the PB equation or the GB equation, and the nonpolar solvation energy ΔG_sol is evaluated by cavity formation in the solvent and van der Waals interactions between solvent and solute. More details about the PB and GB models can be found in section 2.1.2.

Figure 7.

Illustration of the thermodynamic cycle of MM/PB(GB)SA calculations. ΔG_complex is the total free energy of the complex, and ΔG_protein and ΔG_ligand are the total free energies of the protein and ligand in solvent, respectively. ΔG_bind,sol and ΔG_bind,vac are the total free energies in solvent and in vacuum, respectively.

2.1.7.2. MD-Based Methods

Besides MM/PBSA or MM/GBSA, other binding free energy calculation methods such as FEP, metadynamics, and steered MD simulations were also applied to evaluate the binding affinities of inhibitors to SARS-CoV-2 Mpro or S protein.

FEP. MD simulations and FEP calculations were considered to uncover the mechanism of the stronger binding of SARS-CoV-2 S protein to ACE2 by Wang et al.³⁵⁷ They compared the hydrogen-bonding and hydrophobic interaction networks of SARS-CoV-2 S protein and SARS-CoV S protein to ACE2 and calculated the free energy contribution of each residue mutation from SARS-CoV to SARS-CoV-2. FEP calculations indicated that the N501Y mutation on the SARS-CoV-2 S protein enhanced the binding to host ACE2.³⁵⁸ FEP calculations indicated the E484Q/L452R mutations significantly reduce the binding affinity between the RBD of the Kappa variant and the antibody LY-CoV555.³⁵⁹ Ngo et al.³⁶⁰ first docked about 4600 drugs or compounds to Mpro and then used steered MD simulations to rescore the top 35 compounds. They reevaluated the top three compounds using FEP free energy calculations. Zhang et al.³⁶¹ docked remdesivir and ATP to RdRp and used FEP to calculate the binding free energy, indicating that the binding of remdesivir was about 100 times stronger than that of ATP and it can inhibit the ATP polymerization process. FEP calculations were performed by Hassan et al.,³⁶² Allam et al.,³⁶³ and Alhadrami et al.³⁶⁴

Thermodynamic integration (TI). In a recent work,³⁶⁵ the authors used MD simulations to explore the structural coordination and dynamics associated with the SARS-CoV-2 nsp13 apo enzyme, as well as its complexes with natural ligands. The binding free energy and the corresponding mechanism of action by TI calculations were presented for three small molecules that are revealed as efficient inhibitors of the previous SARS-CoV nsp13 enzyme.

Metadynamics. Researchers docked 16 artificial-intelligence generated compounds by Bung³⁶⁶ to Mpro and then ran metadynamics to calculate their binding affinity and predicted some potential inhibitors.³⁶⁷ Metadymics simulations were used to predict the affinities of three neem tree extracts to S protein.³⁶⁸

Steered MD Simulations. Steered MD simulations were used to dock drugs and marine compounds to Mpro, to infer the top inhibitors by Tam et al.^369,370 MM/PBSA and steered MD simulations suggest that the adsorption of the ACE2 on specific silane monolayers could increase its affinity toward the S protein RBD, which could help develop biosensing tools efficient toward any variants of the SARS-CoV-2 S protein.³⁷¹

Semiempirical Free Energy Force Field Methods. A semiempirical free energy force field³⁷² was also adopted to calculate the affinities of 16 drugs to S protein.³⁷³

In most applications, MM/PBSA or MM/GBSA calculations are used to select the trajectories or rank the drugs. The applications of MM/PBSA or MM/GBSA on SARS-CoV-2 proteins are discussed below.

Mpro. Docking, MD simulations, and MM/PBSA binding free energy calculations were applied to investigate around 10000 drugs or experimental drugs from the DrugBank.³⁷⁴ These compounds were first screened by docking, and then MD-based MM/PBSA binding free energy calculations were performed on the top 36 compounds. The reported binding data were the consensus of docking and MM/PBSA prediction, and leuprolide was the top one. Very similar work also screened thousands of compounds from the DrugBank by docking and MM/GBSA MD simulations.³⁷⁵ Gahlawat³⁷⁶ implemented an MM/GBSA procedure on 2454 FDA-approved or experimental drugs, 138 natural products, and 144 other inhibitors to predict the MM/GBSA binding free energy of the top ones screened by docking. MM/PB(GB)SA was also implemented for trajectory selection.^377,378 In addition, Sharma et al.³⁷⁹ also implemented MM/PBSA to screen 2100 drugs as well as 400 other compounds and reported that cobicistat had the highest binding affinity of −11.42 kcal/mol. Cobicistat, flavin adenine dinucleotide, and simeprevir were suggested through drug ranking according to binding energy by MM/PB(GB)SA calculations. Other assessments based on docking and MM/PB(GB)SA calculations focused on specific drugs or compounds such as ravidasvir, lopinavir, ritonavir, saquinavir, teicoplanin, GC-376, calpain XII, calpain II, anti-HIV drugs, doxorubicin, chloroquine, quinoline, hydroxychloroquine, noscapine, echinocandins, coumarins, and their derivatives.^380−417

MM/PBSA and MM/GBSA methods were also applied to predict the potency of natural products to Mpro. Ibrahim et al.⁴¹⁸ virtually screened the MolPort database containing 113,756 natural or natural-like products (https://www.molport.com) by docking. The top 5,000 compounds were selected and subjected to MD simulations combined with MM/GBSA binding affinity calculations, and the compound MolPort-004-849-765 was predicted to have the highest binding free energy. Kapusta et al.⁴¹⁹ also performed docking on 13,496 natural or natural-like products from MolPort, and the top 15 were chosen for rescoring by MM/GBSA calculations. The authors reported MolPort-039–338–330 as the most potent one. Prajapati et al.⁴²⁰ investigated 1830 secondary metabolites of fungal via docking, and additional MM/GBSA calculations were performed on the top from compounds. Mahmud et al.⁴²¹ screened 1480 natural plant products from the literature initially by docking scores, and then the best 10% were rescored by MM/GBSA. Other natural product sources screened by docking and MM/PB(GB)SA against Mpro were flavonoid-based phytochemical constituents of calendula officinalis, phytochemicals in Indian ginseng, food compounds, marine natural polyketides, malaria-box compounds, cressa cretica compounds, strychnos nux-vomica products, ayurvedic compounds, moringa oleifera compounds, withania sp. products, stilbenolignans from plants, acridinedione analogs, alkaloids from justicia adhatoda, tea plant products, neem compounds, turmeric compounds, echinacea angustifolia products, Withania somnifera (ashwagandha) products, cyperus rotundus Linn products, salvia plebeia products, lichen compounds, curcuma longa products, and polyphenols from broussonetia papyrifera.^422−458

There are more compounds screened by docking and MM/PBSA or MM/GBSA against Mpro. Andrianov et al.⁴⁵⁹ first virtually screened over 213.5 million chemical structures from http://pharmit.csb.pitt.edu/ to select the ones satisfying the pharmacophore model from the known X77 potent main protease inhibitor. Then they docked them to Mpro and ran MM/GBSA simulations of the docking complexes to calculate binding free energy. Through this procedure, the authors reported some potent inhibitors such as Pub-chem-22029441. Jimenez et al.⁴⁶⁰ docked 4858 flavonoids to Mpro, and MM/PBSA calculations were performed on the top six compounds. The pharmacophore procedure was also performed where the top three by docking scores were subjected to MM/GBSA calculations, reporting macimorelin acetate as the best one.⁴⁶¹ In a docking of the 15754 compounds in their in-house data set to Mpro, compounds were rescored by MM/GBSA calculations, reporting the most potent one, dimethyl lithospermate.⁴⁶² Khan et al.⁴⁶³ used docking to screen approximately 8000 compounds in their in-house database and applied MM/GBSA MD simulations to calculate the binding affinities of the top five inhibitors, with remdesivir being the best. Fakhar et al.⁴⁶⁴ screened 3435 anthocyanin substructure compounds by docking and MM/GBSA calculations, reporting the best compound to be 44256921. Some other compounds, such as α-ketoamide covalent inhibitors, macrolactin compounds, echinocandins, essential oil compounds, glucocorticoids, angucycline compounds, hydroxychloroquine derivatives, aminoglycosides, imidazole derivatives, Se-containing heterocyclic compounds, circadian clock modulating compounds, oxazine substituted 9-anilinoacridines, nitric oxide donor furoxan, nitric oxide donor heterocyclic vasodilators, withanone caffeic acid phenethyl ester, tetracycline, β-glutamyl-S-allylcysteine peptides, and others, were also screened by docking and MM/PB(GB)SA simulations.^465−484

S Protein. Both SARS-CoV and SARS-CoV-2 infect humans through S protein binding to the human ACE2, and many investigations focused on the interaction between the S protein and the ACE2. In the comparison of the binding affinities of S protein from SARS-CoV and SARS-CoV-2 to the human ACE2 by MM/PB(GB)SA,^485,486 the calculations indicated that SARS-CoV-2 S protein bound to ACE2 much more tightly than SARS-CoV S protein. The mechanism of tighter binding of the SARS-CoV-2 S-protein was studied by using MD simulations and MM/GBSA or MM/PBSA calculations by Xue et al.,⁴⁸⁷ Jafary et al.,⁴⁸⁸ Spinello et al.,⁴⁸⁹ and Bhattacharyya et al.⁴⁹⁰ Interestingly, MM/PBSA calculations at different temperatures suggested that the SARS-CoV-2 RBD was more resistant to temperature changes than the SARS-CoV RBD.⁴⁹¹ Researchers ran MM/PBSA calculations and found that some mutations on S protein could facilitate stronger interactions with human ACE2.^492−496

MD simulations and MM/PBSA calculations revealed that the formation of disulfide bonds, prevalent during oxidative stress, created a conformation more ready to bind to the receptor, which offered future clues for alternate therapeutic possibilities.⁴⁹⁷ Similar work was also performed by Ghasemitarei et al.⁴⁹⁸ One interesting study performed MM/PBSA calculations to reveal the binding affinities of SARS-CoV-2 S protein to the ACE2s from different species.^499,500 This study showed that chimpanzees’ binding affinity was even higher than humans, cats, pangolin, dogs, and monkeys, and chimpanzees had a similar affinity to humans, which suggested some mammals were also vulnerable to SARS-CoV-2. Additionally, a recent MM/GBSA study suggested SARS-CoV-2 Omicron RBD shows weaker binding affinity than the Delta variant to human ACE2.⁵⁰¹

Drug repurposing against S protein was also implemented by MM/GBSA or MM/PBSA. De Oliveira et al.⁵⁰² docked 9091 approved or experimental drugs to S protein and selected the top three to perform MM/PBSA calculation, revealing that suramin sodium had the highest binding affinity. Following a similar scheme, a study repurposed 8770 approved or experimental drugs by docking and MM/GBSA, identifying 31h-phthalocyanine as the most potent candidate.⁵⁰³ Padhi et al.⁵⁰⁴ performed docking and MM/PBSA calculation studies on the inhibition of umifenovir (Arbidol) to the RBD/ACE2 complex. Moreover, MM/GBSA or MM/PBSA approaches were also applied to calculate the efficacy of other compounds to S protein. For example, a study performed docking to 330 galectin inhibitors against S protein and ran MM/GBSA calculations to some active ones, revealing that ligand No.213 had the highest binding free energy.⁵⁰⁵ Rane,⁵⁰⁶ Singh et al.,⁵⁰⁷ and Li et al.⁵⁰⁸ calculated the potency of diaryl pyrimidine derivatives, some bioactive molecules, and the MERS-CoV receptor DPP4, respectively. Lastly, some MM/GBSA or MM/PBSA investigations were about the use of natural products against S protein. For example, docking and MM/PBSA calculations were performed for 11 phytochemicals, suggesting that quercetin had the highest affinity.⁵⁰⁹ Other studies that applied MM/PB(GB)SA calculations on natural compounds to block S protein were from Indian medicinal plants,⁵¹⁰ the NPACT (naturally occurring plant-based anticancer compound-activity-target) database,⁵¹¹ luteolin,⁵¹² and curry.⁵¹³

RdRp. In the study of 7496 approved or experimental drugs against both SARS-CoV-2 and SARS-CoV RdRp, lonafarnib, tegobuvir, olysio, filibuvir, and cepharanthine were screened with high potency by docking and MM/GBSA calculations.⁵¹⁴ Doharey et al.⁵¹⁵ predicted the affinities of amodiaquine, hydroxychloroquine, chloroquine, as well as ATP to RdRp by docking and MM/GBSA calculations, suggesting that these three drugs have higher affinities than ATP. Pirzada et al.⁵¹⁶ and Arba et al.⁵¹⁷ studied the binding mechanism of remdesivir, ledipasvir, and paritaprevir to S protein through docking, MD, and MM/PBSA simulations. Furthermore, as to natural products and other compounds, Khan et al.⁵¹⁸ screened 6842 South African natural products against RdRp using docking and selected the top four for further investigation by MD simulations and MM/GBSA calculations. Their most potent one was Genkwanin 8-C-beta-glucopyranoside ranked by MM/GBSA calculations. In another study of 100 natural polyphenols by docking, the leading eight compounds were used in MD simulations and MM/GBSA calculations, showing that the compound TF3 was the best.⁵¹⁹ Nakinadine B and ormycalamide A were subjected to MM/GBSA studies in docking of 51 marine sponge metabolites to RdRp.⁵²⁰ Sonousi et al.⁵²¹ and Jena et al.⁵²² evaluated the efficacy of adenosine derivatives and synthetic nucleotides to RdRp via docking and MM/GBSA calculations. Molecular docking, MD simulations, and MM/GBSA approaches have also been used to examine the role of several short ionic peptides in inhibiting RdRp.⁵²³ Other similar studies include refs (524−531).

PLpro. In a repurposing of 1697 approved drugs against PLpro by docking, the top 10 were studied by MD simulations and MM/GBSA, with the drug phenformin being their best one.⁵³² Mitra et al.⁵³³ screened tens of natural compounds from Vitex negundo L. by docking and ADMET predictions and performed MM/GBSA calculations on the top four compounds. By docking, MM/GBSA calculations, and interaction analysis, the prediction of the potency of six fungal metabolites to PLpro found GRL0617 is the only potent one.⁵³⁴ Via a similar procedure, a binding free energy analysis suggests that human ub-like interferon-stimulated gene product 15 binds more strongly with SARS-CoV-2 PLpro compared to SARS-CoV or MERS-CoV.⁵³⁵ Pitsillou et al.⁵³⁶ studied dietary compounds and naphthalene-based inhibitors. Bosken et al.⁵³⁷ assessed the potential effectiveness of one naphthalene-based inhibitor 3k and one thiopurine inhibitor 6MP through docking, MD simulations, and MM/PBSA calculations.

Other Targets. Many MM/GBSA or MM/PBSA investigations focused on the SARS-CoV-2 N protein. For instance, Khan et al.⁵³⁸ studied the mechanism of RNA recognition by the N-terminal RNA-binding domain of the SARS-CoV-2 N protein as well as mutation-induced binding affinity changes by docking, MD simulations, and MM/GBSA calculations. In a collection of 8987 compounds from the Asinex and PubChem databases, one study was targeting against the N protein and assessed the potency of the top 10 by MM/GBSA calculations.⁵³⁹ Meanwhile, SARS-CoV-2 helicase (nsp13) was another attractive target.⁵⁴⁰ Vivek-Ananth et al.⁵⁴¹ estimated the docking scores of 10510 drug-like phytochemicals from PubChem to helicase, and the top five compounds were further evaluated by MM/PBSA calculations. In another work, 131 compounds were docked to helicase. More importantly, via MM/GBSA and MM/PBSA calculations, the best one from docking, nilotinib, was used as a probe to detect its affinities to different binding sites of helicase.⁵⁴² Research about targets such as nsp16 or nsp10 can be found.^543,544 Chandra et al.⁵⁴⁵ studied 2895 approved or experimental drugs against NendoU (nsp15) and selected the top three compounds from docking results and ran MM/PBSA calculations for these three, identifying glisoxepide, with a MM/PBSA binding free energy, as the most potent one. The docking of 123 antiviral drugs to NendoU found simeprevir had the highest binding energy, where the MM/PBSA calculations also confirmed this finding.⁵⁴⁶ Encinar et al.⁵⁴⁷ used docking to screen 8696 approved or experimental drugs against the nsp16(methyltransferase)/nsp10 protein complex and, through MM/PBSA calculations, discovered that the presence of nsp10 strengthens the ligand binding to nsp16. In the repurposing against nsp16 involving 4200 drugs or compounds, the best one predicted from MM-PBSA was Carba-nicotinamide-adenine-dinucleotide.⁵⁴⁸ The potency of hundreds of bioactive compounds to methyltransferase was also studied via docking and MM/PBSA calculations.^549,550 Moreover, El Hassab et al.⁵⁵¹ designed a new methyltransferase inhibitor AP-20 based on fragments and calculated the binding affinity using MM/PBSA. Since the MMLFA-1/SARS-CoV-2 Orf7a complex contributes to SARS-CoV-2 infectivity and pathogenicity, Ongaro et al.⁵⁵² used MM/GBSA calculations to study the interactions inside the MMLFA-1/SARS-CoV-2 Orf7a complex. Other SARS-CoV-2 targets under MM/PB(GB)SA studies also include nsp1⁵⁵³ and nsp14.⁵⁵⁴

Multiple Targets. Some researchers adopted MM/PB(GB)SA calculations to screen drugs or compounds against multiple targets of SARS-CoV-2. Nunes et al.⁵⁵⁵ docked 24 approved drugs to SARS-CoV-2 Mpro, PLpro, and the ADP ribose phosphatases of nsp3, nsp9, nsp12, nsp15, and nsp16. The MM/GBSA calculations were further performed on the top three drugs from docking tests. 131 quinoline-based drugs were targeted to Mpro, S protein RBD, PLpro, RdRp, and N protein, and the best drug for each target was further evaluated by MM/PBSA calculations.⁵⁵⁶ Famotidine was shown to have a high binding affinity to PLpro in similar studies.⁵⁵⁷ Targeting SARS-CoV-2 Mpro, S protein, RdRp, PLpro, nsp14, Mpro, N protein, human ACE2, and TMPRSS2, Eweas et al.⁵⁵⁸ in silico screened chloroquine, hydroxychloroquine, ivermectin, remdesivir, favipiravir, lopinavir, and camostat via docking. Their docking simulations identified that ivermectin and remdesivir were potent to all nine targets, and the MM/PBSA calculations confirmed it. Targeting Mpro and PLpro, Jade et al.⁵⁵⁹ screened 4182 drugs and 321 other compounds through ADMET and docking predictions. Many works aimed to repurpose compounds from other sources to inhibit Mpro and S protein. Panda et al.⁵⁶⁰ screened 640 compounds through docking and MD simulations and identified that PC786 had high docking scores to both S protein and Mpro. Moreover, their MD simulations and MM/PBSA calculations revealed that the binding of PC786 can change the conformation of the S protein and weaken the S protein’s binding interactions to ACE2. The MM/PBSA binding free energy was calculated on five PLpro–compound and 6 Mpro–compound complexes. Similarly, Naidoo et al.⁵⁶¹ investigated the potency of cyanobacterial metabolites against Mpro and S protein. Thurakkal et al.⁵⁶² predicted the binding affinities of tens of organosulfur compounds to Mpro, S protein, PLpro, RdRp, and helicase. The top six compounds were further investigated by MM/PBSA. The repurposing potential of 34 bioactive terpenes and their derivatives to Mpro and PLpro was also predicted by docking and MM/PBSA calculations.⁵⁶³ Other MM/PB(GB)SA-based drug repurposing works considering two or more targets include studies in the literature.^564−580

Natural compounds are another source of drug repurposing. Targeting Mpro, PLpro, and RdRp, a collection of 14492 marine-derived natural bioactive compounds by the criteria of Lipinski’s RO5, predicted ADMET properties, and docking scores, the best 14 compounds were subjected to MM/PBSA calculations.⁵⁸¹ In a similar studying, Al-Sanea et al.⁵⁸² performed docking simulations on about 30 strawberry and ginger silver nanoparticles, and the top four were selected to be studied by MM/GBSA calculations. In ref (583) 56 licorice major components and metabolites were docked to Mpro, S protein RBD, PLpro, RdRp, nsp15, and human ACE2. MM/GBSA calculations were performed on the top six compounds. In the investigation of four compounds targeting four different proteins in SARS-CoV-2, i.e., Mpro, S protein–ACE2 complex, RdRp, and PLpro, through docking, MD simulations, and MM/GBSA, it was found that AGP-3 had potency for all four targets.⁵⁸⁴ In a collection of 100,000 natural compounds against SARS-CoV-2 proteins, compounds were investigated by MD simulations and MM/PBSA, reporting that Baicalin was potent against RdRp, nsp4, and NendoU.⁵⁸⁵ Kar et al.⁵⁸⁶ studied Mpro, S protein, and RdRp. Their ligands were natural products from Clerodendrum spp. After docking and rescoring the top ones by MM/GBSA, these authors found taraxerol to be effective to all three targets. Using docking and MM/GBSA or MM/PBSA calculations, Alajmi et al.⁵⁸⁷ and Sasidharan et al.⁵⁸⁸ evaluated the potency of around 40 compounds, including some existing drugs and the protein azurin secreted by the bacterium Pseudomonas aeruginosa as well as its derived peptides, against Mpro, PLpro, and S protein. Prasanth et al.⁵⁸⁹ studied 48 isolated compounds from cinnamon by docking and MD-simulation-based MM/PBSA calculations, suggesting that the compounds tenufolin and pavetannin C1 were potent to both Mpro and S protein. Other similar works about natural products are reported in the literature.^590−604

2.1.8. Density-Functional Theory (DFT) and Quantum Mechanism (QM) Methods

Density-functional theory (DFT) is utilized whenever the electronic structure is important, which is the typical case for chemical reactions. DFT is a computational quantum mechanics modeling method widely used in computational physics, computational chemistry, and computational material science to investigate the electronic structure of atoms, molecules, and condensed phases.^605−607 Using this theory, the properties of a many-electron system are represented by functionals (functions of another function) of the spatially dependent electron density.⁶⁰⁵ Because of the development of DFT, Walter Kohn won the Nobel Prize in Chemistry in 1998.⁶⁰⁸ DFT is constructed on the total electronic charge density ρ(r) given as

23

of an N-electron problem, where r_i are positions and Ψ(r, r₂, ..., r_N) is the wave function satisfying the many-electron time-independent Schrödinger equation. Here, Ψ* is the complex conjugate of Ψ. DFT, as a successor of the Schrödinger equation and the Thomas–Fermi model, studies a representative of the N-electron problem as a set of N one-electron problems, whose foundations are the Hohenberg–Kohn and Kohn–Sham theorems. For the Hohenberg–Kohn theorem, if the density function ρ(r) of a quantum system is known at the ground state ρ₀, then the wave function is determined as Ψ₀ = Ψ[ρ₀]. The Kohn–Sham equation for the orbitals φ_i(r) can be written as

24

Here ℏ is the reduced Planck constant. The Kohn–Shan potential is given as

25

where V_ext(r) is the external potential, the second term is the Hartree term of the electron–electron Coulomb repulsion, and V_xc[ρ(r)] is the exchange-correlation potential which is

26

where E_xc is the exchange-correlation energy.

DFT calculations were employed to simulate the attack of cysteine to covalent inhibitors by Nogara et al.,⁶⁰⁹ Madabeni et al.,⁶¹⁰ Wang et al.,⁶¹¹ and Shekh et al.⁶¹² For example, DFT simulations by Nogara et al. and Madabeni et al. suggested the mechanism of cysteine attacking, and the energy barrier of its attacking to ebselen was around 30 kcal/mol. Many studies applied DFT to predict electronic properties and chemical reactivity. For instance, via DFT calculations, Khan et al.⁶¹³ suggested the electrophilicity ranking of their six isolated compounds is apigenin > luteolin > diosmetin > quercetin > spinacetin > eriodictoyl. Yadav et al.⁶¹⁴ suggested the higher reactivity of favipiravir toward Mpro since the presence of electron donor and receptor groups in favipiravir displayed the capability of forming a complex with the external target molecule. Other similar studies include refs (443 and 615−632). DFT calculations by Aitouna et al.^633,634 indicated the epoxidation reaction of parthenolide and himachalene derivatives presented high chemoselectivity. This can explain how parthenolide and himachalene derivatives form. More studies used DFT to preoptimize the structures of compounds.^{609,635−640} Additionally, the PM7 semiempirical quantum-chemical method was applied to binding calculations of Mpro,^641,642 RdRp,⁶⁴³ PLpro, and S protein.⁶⁴²

2.1.9. Quantum Mechanics/Molecular Mechanics (QM/MM)

As illustrated in Figure 6b, the QM/MM approach is a hybrid molecular simulation method that combines the accuracy of QM and the speed of MM. For a biological system, the computational region where the chemical process takes place is treated at an appropriate level of QM,⁶⁴⁴ while the remainder is described by a MM force field.⁶⁴⁵ This approach can be used to study chemical processes in solution and proteins. The Nobel Prize in Chemistry in 2013 was awarded to Arieh Warshel and Michael Levitt for the introduction of QM/MM. There are two different ways to calculate the energy of the combined system. The subtractive scheme calculates the energy of the entire system by using a molecular mechanics force field added by the energy of the QM system and subtracted by the MM energy of the QM system

27

where V^MM(QM) is the energy of the quantum mechanics region using molecular mechanics. More widely, the additive scheme is applied and given as

28

where V(QM) is the QM energy of the QM region, V(MM) is the MM energy in the molecular mechanics region, and V(QM/MM) is the energy of interactions between the two systems given as

29

where each term is calculated as a similar formula as eq 9 and V_bond(QM/MM), V_elec(QM/MM), and V_VDW(QM/MM) are calculated in both systems.

The covalent binding of PF-07321332 to Mpro was elaborately investigated via QM/MM calculations, suggesting the reaction energy barrier is −16.3 kcal/mol.⁶⁴⁶ Ramos-Guzmán et al.⁶⁴⁷ and Arafet et al.⁶⁴⁸ also performed QM/MM-based simulations to reveal the mechanism of the Michael reaction to Mpro, and Ramos et al. suggested some strategies to improve inhibitor design. Ramos et al.⁶⁴⁹ considered aldehyde derivatives. The covalent binding of inhibitor PX-12 and peptides to Mpro was simulated by the QM/MM method.^650−655 Regarding the S protein, the interactions of human ACE2 and hydroxychloroquine to S protein RBD were analyzed through QM/MM calculations.^656,657

2.2. Mathematical Approaches

Various mathematical tools, including different geometries,^94,658−660 algebraic topology,^219,661 and graph theory,⁶⁶² have been applied to the modeling and prediction of biomolecules.⁶⁶³ In this review, network analysis, the flexibility–rigidity index, and topological data analysis are discussed. These approaches become very powerful when paired with deep learning.

2.2.1. Graph Network Analysis

A network is a graph in which vertices represent objects and edges represent relationships between objects. Networks can be used to represent biological systems as sets of biological objects and interactions between biological objects. For example, protein–protein interactions (PPIs) give rise to both protein-scale networks and atom-scale networks. In the protein-scale PPI networks, proteins are regarded as vertices and protein–protein interactions as edges. In the atom-scale PPI networks, such as a RBD-ACE2 complex, the atoms in the RBD and ACE2 can be regarded as vertices and the interactions between atoms in the RBD and atoms in the ACE2 form edges. Other biological systems, such as atomic interactions, drug–target interactions, disease–protein associations, and drug–disease relations, can also be represented as networks. Thanks to the recent development of biological technologies, such as high-throughput affinity purification combined with mass spectrometry and the yeast two-hybrid assay, interactome data are increasing rapidly, and a large number of interactome networks can be constructed. Understanding biology from the perspective of networks is important for many purposes. For instance, the knowledge of a PPI network can shed light on the putative roles of uncharacterized proteins. Graph theory is a well-established mathematical field that is readily applicable to the study of biological networks. In this section, we first recapitulate some basic notions of graph theory; then we introduce several network measures, which are proposed to characterize the local or global properties of a network, such as “irregularity”, “centrality”, and “communicability”.

An undirected simple graph (V, E) consists of a set V of vertices and a set of edges E connecting pairs of vertices, without self-loops or multiple edges between vertices. We denote the number of vertices and the number of edges as n_v and n_e, respectively. The edge connecting vertices i and j is denoted as e_ij. A simple graph with V = {v₁, ..., v_n} can be represented by its adjacency matrix A, where A_ij is 1 if there is an edge connecting v_i and v_j and is 0 otherwise. The degree of a vertex i, sometimes denoted as k_i, is the number of edges that are incident to the vertex i. It is clear that the average degree can be calculated by the formula 2n_e/n_v. A regular graph is a graph where each vertex has the same degree. A walk of length k is a series of vertices i₁, i₂, ..., i_k+1 such that for all 1 ≤ l ≤ k there is an edge connecting i_l and i_l+1. A path is a walk in which all vertices are distinct. It is well-known that the (i, j) entry of the kth power of the adjacency matrix, (A^k)_ij, is equal to the number of walks of length k starting at vertex i and ending at vertex j. A graph is connected if there is a path between every pair of two vertices, and a tree is an undirected graph in which every pair of two vertices is connected by exactly one path.

Degree Heterogeneity. The degree heterogeneity^664,665 measures how heterogeneous the degrees of vertices are. Considering a star graph S_k, its only internal node has degree k and each of its external node has degree 1. The degree heterogeneity of a graph such as S_k reflects its “irregularity”. It is defined as

30

where is the set of edges, k_i is the number of neighbors of vertex i, and e_ij is the edge connecting vertices i and j. The degree heterogeneity of a regular graph is 0, since each vertex has the same degree. The degree heterogeneity of the star S_k is .

Edge Density. The edge density is defined as

31

where n_e is the number of edges and n_v is the number of nodes. For a complete network in which each pair of network vertices is connected, the edge density is equal to one. A noncomplete network has an edge density smaller than one. If n_e ≈ (n_v)^k with 1 < k < 2, we say that this graph is dense. If n_e ≈ (n_v)^k with k ≤ 1, we say that this graph is sparse.

Average Path Length. The average path length can be seen as a measure of the efficiency of information transport and is typically used to characterize the “small-worldness” of a network.⁶⁶⁶ A network with shorter average path length facilitates quicker transfer of information. Let d(i, j) denote the shortest path length between vertices i and j, then the average path length ⟨L⟩⁶⁶⁴ is defined as

32

Betweenness Centrality. If a vertex v falls on the shortest paths between two vertices i and j, by control of the vertex v one can control the transmission of information between vertices i and j. The notion of the betweenness centrality illustrates this potential.⁶⁶⁷ The betweenness centrality of a vertex v is defined as a sum over all (unordered) pairs of vertices i and j such that i ≠ v ≠ j

33

where σ_ij is the number of shortest paths between vertices i and j and σ_ij(v) is the number of those paths that passes the vertex v (v is not an end point). The probability that the vertex v falls on a randomly chosen shortest path connecting vertices i and j is . The average betweeness centrality is defined as the average of betweenness centralities over all vertices.

Eigenvector Centrality. The eigenvector centrality⁶⁶⁴ takes account of not only the shortest paths but also any path connecting two vertices. Let N_l(v) be the number of walks of length l that start at v and end elsewhere. If the given network is not bipartite, one can define the eigenvector centrality of a vertex

34

which can be regarded as the ratio of the number of infinite length walks starting at v over the number of all infinite length walks. One can define the average eigenvector centrality as the average over all vertices.

Subgraph Centrality. Let A be the adjacency matrix and G = exp(A); then the subgraph centrality^664,668 of the ith vertex is defined as

35

The subgraph centrality is closely related to closed walks. To see this, rewrite G_ii as ∑_l = 0^∞(A^l)_ii/k!. As (A^l)_ii is the number of closed walks of length l starting and ending at the same ith vertex, the subgraph centrality is indeed a weighted sum of closed walks of all lengths starting and ending at the same node, in which shorter closed walks are given more weight. To get a global characterization of a network, one may also consider the average subgraph centrality.

Communicability. There are many different ways to measure the communicability of two vertices. Estrada and Hatano⁶⁶⁹ proposed to define the communicability between vertices i and j as G_ij, which is indeed a weighted sum of walks of all lengths starting at vertex i and ending at vertex j. This definition of communicability is justified because communication between two vertices can take place through nonshortest paths. Estrada and Hatano⁶⁷⁰ also defined the communicability angle between the ith and jth vertices

36

Taking the average over all pairs of vertices, one can define the average communicability and the average communicability angle. The average communicability angle evaluates the efficiency of a network transmitting information between its pairs of vertices with all possible paths.

Closeness Centrality. The closeness centrality⁶⁷¹ measures vertices’ connecting efficiency through the network. In a connected graph, the closeness centrality of the ith vertex is defined as

37

The sum ∑_j≠id(i, j), also referred to as the farness of a vertex i, is the sum of the shortest path distance to the ith vertex over all n_v – 1 reachable vertices. The normalized form of the closeness centrality is given by . If a vertex has a larger closeness centrality, it has a greater “centrality” in the sense of being more independent of other vertices.⁶⁶⁷

Topological Coefficient. The topological coefficient⁶⁷²C_i^t measures the extent to which the ith vertex shares neighbors with other vertices, which is defined as

38

where J(i, j) is the number of joint neighbors of the ith and jth vertices (plus one if there is an edge between i and j), o_i is the degree of the ith vertex, and ⟨J(i, j)⟩ is the average over all vertices that share a neighbor with the ith vertex.

2.2.1.1. Network-Based Biomolecular Structure Analysis

Using networks to analyze the structural similarities is important for drug repurposing and understanding functional mechanisms. Estrada applied the aforementioned network measures to analyze the interaction networks between SARS-CoV-2 Mpro and various inhibitors.⁶⁶⁴ Chen et al.⁷⁸ applied a similar strategy to predict binding affinity changes induced by mutations. A variety of studies using the network measures on protein residue/atom networks followed the same path.^{6,10,77,673−675} Moreover, Chen et al. employed the network analysis of antibody–antigen complexes on Cα atoms⁷⁷ as illustrated in Figure 8. Lata and Akif⁶⁷⁶ implemented network analysis on 3CLpro from SARS-CoV and SARS-CoV-2. Amamuddy et al.⁶⁷⁷ used five independent criteria of network centrality to study the allosteric effects of potential allosteric modulators for the SARS-CoV-2 3CL protein. Focusing on the correlations between the RBD and residues distant to it in the S protein, Ray et al.⁶⁷⁸ built a protein graph connectivity network and calculated the betweenness centrality. A modification of the average shortest path length was used in ref (679). Saha et al.⁶⁸⁰ employed various network measures to identify the spreader nodes in the SARS-CoV-human protein–protein interaction network, hoping to find possible lineage with the disease propagation pattern of the COVID-19 pandemic.

Figure 8.

Cα network analysis of three antibody–antigen complexes. Here, circle markers represent antigen (S protein RBD), and cube markers represent antibody or ACE2. The PDB IDs of the three antibody–antigen complexes are 3D0G, 6M0J, and 6W41. The rows represent (a) betweenness centrality, (b) eigencentrality, and (c) subgraph centrality.⁷⁷ (d) Illustration of the S protein and ACE2 interaction. The RBD is displayed in green, the ACE2 is given in pink, and mutation D614G is highlighted in red. (e) Difference of FRI of the S protein between the network with wild type and the network with mutant type. (f) Difference of the subgraph centrality between the network with wild type and the network with mutant type.¹⁰

2.2.2. Flexibility–Rigidity Index (FRI)

The FRI is a geometric graph-based method that utilizes weighted graphs to model molecular interactions.^187,681 The multiscale FRI,⁶⁸² the colored (i.e., element-specific) FRI,⁶⁸³ and their algebraic graph counterpart¹⁸⁶ have also been proposed. The atomic rigidity index at position r_i is defined as a summation of all the weighted edges around it:

39

where r_j is the position of the ith atom, w_ij is a weight, N_c′ is the number of atoms in the neighborhood of r_i, and η is a characteristic scale. Element-specific rigidity⁶⁸³ and molecular rigidity⁶⁸¹ can be obtained by an appropriate collection of atomic rigidity indices. The FRI has been applied to protein and nucleic acid flexibility and fluctuation analysis⁶⁸¹ and protein–ligand binding affinity prediction.⁶⁸⁴ Protein–protein interactions, such as the elasticity between antibody and antigen, especially long-range impacts, were studied by calculating the FRI of the network consisting of Cα atoms. The FRI is an important feature for machine learning models to predict the binding affinity changes on mutations⁶⁸⁵ and the protein folding energy changes on mutations.⁶⁸⁶

Some studies applied the machine learning models based on the FRI to study the SARS-CoV-2 proteins combined with network analysis. Wang et al.¹⁰ calculated the FRI and investigated the folding stability changes of the S protein (see Figure 8a, e, and f) and other proteins caused by mutations. The FRI-based binding affinity change between the S protein and human ACE2 due to mutations was also calculated by Chen et al. and Wang et al.^{10,77−79,687}

2.2.3. Topological Data Analysis (TDA)

Recent years have witnessed rapid development in TDA and its applications to a wide variety of scientific and engineering problems.^689,690 The main workhorse of TDA is persistent homology,^691,692 a new branch of algebraic topology. This approach has been applied to characterize biomolecular systems.^661,693,694 More powerful methods that provide simultaneous topological persistence and spectral analysis have been proposed.^{660,695−697} In TDA, molecular atoms can be treated as a point cloud and a filtration of simplicial complexes can be constructed. We first recall some basic notions of algebraic topology. A simplicial complex is a finite collection of simplices σ, where a k-simplex σ^k = [v₀, ..., v_k] is a convex hull of k + 1 points {v₀, ..., v_k} in (n ≥ k). A simplicial complex K is valid if any face τ of a simplex σ in K is also in K, and the nonempty intersection of any two simplices is a face for both. Given a simplicial complex K, a k-chain is a finite formal sum of k-simplices ∑_iα_iσ_i^k with coefficients in a ring (usually a field such as ). The set of all k-chains forms an abelian group C_k(K). The boundary operator ∂_k:C_k(K) → C_k–1(K) is a group homomorphism defined by ∂_kσ^k = ∑_i = 0(−1)ⁱ[v₀,...,v̂_i,...,v_k], where [v₀,...,v̂_i,...,v_k] is a (k – 1)-simplex excluding v_i. A k-cycle is a k-chain whose image is 0 under the boundary operator ∂_k. An important property of boundary operators is that ∂_k–1∂_k = 0, so we have the following chain complex

40

and the kth homology group H_k is defined as H_k = Z_k/B_k where Z_k = ker ∂_k = {c ∈ C_k|∂_kc = 0} and B_k = im ∂_k+1 = {∂_k+1c|c ∈ C_k+1}. The kth Betti number is defined by the rank of the kth homology group H_k which counts the k-dimensional holes. In particular, β₀ = rank(H₀) reflects the number of connected components, β₁ = rank(H₁) reflects the number of loops, and β₂ = rank(H₂) reveals the number of voids or cavities. Together, the set of Betti numbers {β₀, β₁, β₂, ...} indicates the topology of a simplicial complex.

Persistent homology is devised to track the multiscale topological information along a filtration.⁶⁹⁸ A filtration of simplicial complex K is a nested sequence of subcomplexes {K^t}_t=t0,...,tm of K such that

41

Moreover, the inclusion map X^t_i ⊆ X^t_j induces a homomorphism f_k^t_i,t_j between homology groups H_k(K^t_i) → H_k(K^t_j) for each dimension k. The p-persistent kth homology group of K^t is defined by

42

where Z_k^t = ker ∂_k and B_k^t + p = im ∂_k+1. Intuitively, this homology group records the k-dimensional homology classes of X^t that are persistent at least until X^t+p. The birth and death of homology classes can be encoded by a barcode, a set of intervals. Given a molecule, a filtration can be constructed, and hence, a barcode can be calculated. Feature vectors can be constructed from barcodes for machine learning models.⁶⁹⁹ One can use the persistent barcode to distinguish the randomness from the structure of the growing graphs. For instance, as illustrated in Figure 9, a filtration was introduced to a graph G to create multiple simplex complexes. We can distinguish the topological differences among complexes along filtration by analyzing the persistent barcode and Betti curves.

Figure 9.

Illustration of persistent homology filtration. Reused with permission from ref (688). Copyright 2020 Anand et al. under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/. (a) Simplicial complex at radius 1.2 has 0-simplexes (black dots), 1-simplexes (red edges), 2-simplexes (yellow triangles), and 3-simplexes (purple tetrahedral). The barcode shows β₀ and β₁. (b) Persistent homology filtration at radius 0.6, 0.8, and 1.2. (c) 0-, 1-, 2-, and 3-simplex. (d) Topological invariants of three examples.

Since the first integration of persistent homology and machine learning,⁷⁰⁰ topology-based approaches have found much success in biomolecular modeling and prediction.^{219,663,699,701} Combined with large datasets and machine learning algorithms, TDA is a powerful tool for predicting biomolecular properties such as protein–ligand binding affinity^219,699 and drug discovery.⁷⁰² According to the biomolecular properties, complexes are constructed as an atomic-specific strategy or bipartition graph. For instance, when studying the protein folding energy of the ACE2 and SARS-CoV-2 S protein, one can use element-specific and/or site-specific persistent homology to simplify the structural complexity of the protein structure and encode vital biological information into topological invariants.^687,699 Wang et al.⁶ applied topological features to protein folding studies on the energy changes on mutations of the SARS-CoV-2 nsp6 protein. Moreover, in the complex formed in a bipartite graph, the features of the protein–protein interaction can be studied where the atoms of the antibody and antigen consist of two disjointed and independent sets. Chen et al.⁷⁷ used this idea to predict the binding free energy changes on mutations of the protein–protein interactions between the S protein and antibodies. Nguyen et al.⁷⁰³ studied the potency and molecular mechanism of the main protease inhibition from 137 crystal structures by integrating mathematics, deep learning methods, and applied persistent homology. Topological data analysis is not only applied to studying protein–protein interactions. Chen et al.^10,78,79,687 further studied the mutations that strengthened SARS-CoV-2 infectivity where persistent homology plays a key role in analyzing the interactions between the S protein and human ACE2. Moreover, Pérez-Moraga et al. applied TDA to identify the drug repurposing targeting SARS-CoV-2 proteins (3CLpro, nsp15, and nsp12).⁷⁰⁴

2.3. Machine Learning

Machine learning (ML), including deep learning (DL), is a transformative technique in artificial intelligence (AI). ML and DL can be categorized into four major tasks, namely regression, classification, clustering, and dimensionality reduction. The first two involve supervised learning using labeled data, and the last two rely on unsupervised learning using unlabeled data. All of these methods are widely used in computational biology, computational chemistry, and computational biophysics.

2.3.1. Dimensionality Reduction

As raw data often exist in high-dimensional space, dimensionality reduction techniques can be applied to transform raw data into a low-dimensional representation, making it easy for visualization and analysis. Various dimensionality reduction algorithms can be divided into two categories, namely matrix factorization and neighbor graphs. The matrix factorization maintains the pairwise distance among the data samples. Techniques such as principal component analysis (PCA),⁷⁰⁵ multidimensional scaling (MDS),⁷⁰⁶ linear autoencoder,⁷⁰⁷ Sammon mapping,⁷⁰⁸ latent Dirichlet allocation,⁷⁰⁹ non-negative matrix factorization,⁷¹⁰ etc. fall into this category. Neighbor graphs seek to preserve the global distance among the data sample, which includes Laplacian eigenmaps,^711,712 Hessian eigenmaps,⁷¹³ local tangent space alignment,⁷¹⁴ Isomap,⁷¹⁵ t-distributed stochastic neighbor embedding (t-SNE),^716,717 uniform manifold approximation and projection (UMAP),⁷¹⁸ etc. In the following, popular methods, such as PCA, t-SNE, and UMAP, are briefly introduced.

PCA. PCA⁷⁰⁵ aims to find an orthogonal linear transformation such that the projection of transformed data on the first coordinate has the largest variance, the projection on the second coordinate has the second largest variance, and so on. We can take the first several coordinates in the new coordinate system to get a low-dimensional projection of the original data.

t-SNE. t-SNE^716,717 is a nonlinear dimensionality reduction technique. It first calculates the pairwise distribution over pairs of high-dimensional data such that a pair of near data points is assigned with a higher probability, while a pair of dissimilar points is assigned with a lower probability. Second, a similar probability distribution over the low-dimensional data is calculated as well. Last, the minimization of the Kullback–Leibler (KL) divergence is carried out between the two distributions to map the high-dimensional data to a low-dimensional space appropriately.

UMAP. Another nonlinear dimensionality reduction technique is UMAP,⁷¹⁸ which is computationally faster than t-SNE. UMAP first constructs a weighted graph from the original data, where edge weights represent distances and then projects the weighted graph to low-dimensional space. UMAP is motivated by category theory and Riemannian geometry.

Many dimensionality reduction techniques have been applied in SARS-CoV-2 research. Some research employed PCA to analyze the dynamics of proteins. For instance, Islam et al. examined PCA as a part of the techniques to seek the best candidates that can be used as potent inhibitors against the main protease of SARS-CoV-2. They first selected candidates with strong binding affinities and interactions between the main protease and the phytochemicals with AutoDock Vina and GOLD. Next, MD simulations are applied to validate five top-ranked inhibitors. Among them, three inhibitors, baicalin, cyanidin 3-glucoside, and a-ketoamide-11r, are selected by applying PCA, which has structural similarity with the apo-form of the main protease.⁷¹⁹ In addition, PCA was also used to access the chemical space of a given SARS-CoV-2 data set. For example, PCA was set along the SARS-CoV-2 molecular fingerprint descriptors to show that the SARS-CoV-2 chemical space is well distributed with inactive and active molecules.⁷²⁰ Moreover, PCA was also applied to study the motions of the protein during the binding of the ligand by Prasad et al.⁵⁶⁵ In the initial process of the SARS-CoV-2 entering the host cell, TMPRSS2 and Cathepsins B/L activate the S protein and enable SARS-CoV-2 to invade the host cell through two independent pathways. Therefore, seeking a simultaneous target to both entry pathways would be a good idea to block the virus from entering host cells. Prasad et al. applied PCA techniques to study the significant motions of the drug candidates during the binding of TMPRSS2 and Cathepsins B/L. The results showed that cyclosporin A (CsA), one of the drug candidates, is quite stable with TMPRSS2 in the complex when the dynamics of this structural conformation is increased. A similar pattern can also be observed in the cathepsins L (CTSL)-CsA complex. Furthermore, due to the capacity of PCA to reduce the dimensionality to maximize the data set variance, PCA is used as a metric for analyzing conformational the diversity from Gaussian accelerated MD (GaMD) and conventional MD simulations of the SARS-CoV-2 main protease by Sztain et al.¹⁴⁶

Dimensionality reduction techniques are often coupled with K-means clustering. Gussow et al.⁷²¹ tried to identify genomic determinants of coronavirus that are related to high case fatality rates (CFRs). They performed multiple sequence alignment for 944 human coronavirus genomes and recoded aligned sequences as sequences consisting of 0 and 1. Then they applied PCA and t-SNE on recoded aligned sequences and performed K-means clustering, identifying 11 regions of nucleotide alignments as predictive of the high CFRs of coronaviruses.

UMAP is often used for visualizing gene expression data (e.g., refs (722 and 723)). Zhang et al.⁷²³ tried to identify the susceptible cell types and potential infection routes of SARS-CoV-2, since the coexpression of ACE2 and TMPRSS2 is critical for viral entry. Therefore, they analyzed five data sets with single-cell transcriptomes of human tissues to study the coexpression pattern of ACE2 and TMPRSS2 in five different cell types, consisting of esophageal cells, gastric mucosa cells, ileal cells, colon cells, and lung cells. First, UMAP is applied to get the landscape of five different types of cells. Next, the expression of ACE2 (blue) and TMPRSS2 (red) was marked on the UMAP plots. Meanwhile, the UMAP plots were merged as well to show the coexpression of ACE2 and TMPRSS2. Such single-cell analysis indicated that ACE2 and TMPRSS2 may have coexpressed not only in lung alveolar type 2 cells but also in upper epithelial and gland cells from esophageal and absorptive enterocytes from the ileum and colon.

Moreover, a rich reference data set that describes the transcriptional landscape at the single-cell level of the lung and subsegmental bronchial in a total of 16 individuals was established by Lukassen et al.⁷²⁴ The gene expression patterns of ACE2, TMPRSS2, and FURIN in the lung can also be observed by UMAP (see Figure 10a). Furthermore, a work by Ravindra et al. applied UMAP to analyze the gene expression levels of the ACE2, CTSL, TMPRSS2, and TMPRSS4 protease of human bronchial epithelial cell (HBEC) samples as illustrated in Figure 10b and c.

Figure 10.

(a) Expression level of ACE2 in the lung plotted on top of the UMAP coordinates. Expression level of TMPRSS2 in the lung plotted on top of the UMAP coordinates. Expression level of FURIN in the lung plotted on top of the UMAP coordinates. Reproduced with permission from ref (724). Copyright 2020 Lukassen et al. (b and c) UMAP visualization of HBEC samples, colored by expression (normalized and square-root transformed counts) of the ACE2 receptor, CTSL, TMPRSS2, and TMPRSS4 proteases. Reproduced with permission from ref (725). Copyright 2021 Ravindra et al. Under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/.

2.3.2. Linear Regression

Linear regression is one of the basic algorithms in machine learning and can be used to solve the regression problem. Assume the training set is . Here, n is the number of samples, and m represents the number of features. Then, the prediction corresponding to x_i is defined as

43

where represents the weights, is the bias, and w^T represents the transpose of w. The loss function of the linear regression model is

44

The aim of the linear regression is to minimize the loss function, which is demonstrated as eq 45.

45

Additionally, a regularization term can also be taken into account in the case of overfitting:

46

where λ represents a penalty constant.

Applications are involved in applying linear regression to calculate the correlation coefficient and predict different types of dependent variable values, where most applications study the experimental data. As for theoretical analysis, studies focus on implementations of QSAR, which will be discussed later. To study the SARS-CoV-2 fatality rates, the CFRs of SARS-CoV-2 variants were compared by linear regression of worldwide data between wild-type and mutant-type virus on the S protein, resulting in G614 being shown to be a more pathogenic strain.²⁰⁶ In the study of gene evolution of SARS-CoV-2, linear regression was applied to the line between the number of viral variants and the gene length, suggesting that the mutation frequency was proportional to the length of the gene.⁷²⁶ A regression model was designed and related the the experimental binding affinity for antibodies by applying structural features, where several mutations at the S binding motif were identified.⁷²⁷ Moreover, a linear regression model was applied by Israel et al. to quantify the association between the logarithm of antibody levels and the elapsed time, in both fully vaccinated and convalescent individuals.⁷²⁸ This study suggests that individuals who received the Pfizer-BioNTech mRNA vaccine had higher initial antibody levels but a much faster exponential decrease compared to patients who had been infected by SARS-CoV-2.

2.3.3. Logistic Regression

Logistic regression is an algorithm designed for solving classification problems. Assume the training set is . Here, n is the number of samples, m represents the number of features, and represents different categories. Then, the prediction of the logistic regression corresponding to the point x_i is

47

where represents the weights, is the bias, and w^T represents the transpose of w. When y_i ∈ [0, 1], the loss function can be defined by

48

Ayouba et al.⁷²⁹ used logistic regression to represent the dynamics of the immunoglobulin G (IgG) response to the S protein, N protein, or both antigens at the same time since onset of symptoms. In addition, researchers stated that the publicly shared CD8⁺ (Cytotoxic T cells with CD8 surface protein) might be used as a potential biomarker of SARS-CoV-2 infection at high specificity and sensitivity by applying the logistic regression.⁷³⁰ In addition, only subtle differences were observed from the initial MD simulations of the two RBD-ACE2 complexes by Pavlova et al. Later, logistic regression was used to successfully identify the individual residues with the most distinctive ACE2 interactions, many of which have been highlighted in previous experimental studies.⁷³¹

2.3.4. k-Nearest Neighbors

The k-nearest neighbors algorithm (k-NN) is a nonparametric technique proposed by Thomas Cover and P. Hart in 1967.⁷³²k-NN can be used for solving both regression and classification problems,⁷³³ and it is sensitive to the local structure of the data. The flow chart of the k-NN algorithm can be found in Figure 11. The features of the training set are {x_i}_i=1ⁿ with , k shows the number of the nearest neighbors, and is a feature representation of the training set. Different distance metrics can be employed in the k-NN algorithm, such as Euclidean distance, Manhattan distance, Minkowski distance, Chebyshev distance, natural log distance, generalized exponential distance, generalized Lorentzian distance, Canberra distance, quadratic distance, and Mahalanobis distance.

Figure 11.

Flow chart of the k-NN algorithm. The features of the training set are {x_i}_i=1ⁿ with , k shows the number of nearest neighbors, and is a feature representation of the training set.

The classifier can be built by using the k-NN algorithm. Setting classifiers as k-NN models, an automated system can distinguish the SARS-CoV-2 genome from the SARS-CoV genome and MERS genome by using the genomic sequences from the National Center for Biotechnology Information (NCBI) GenBank for accelerating the diagnosis process and improving the accuracy of disease detection.⁷³⁴ Moreover, AllerTOP v.2.0 classified allergens and nonallergens based on the k-NN method with an accuracy of 88.7%.⁷³⁵ Furthermore, the k-NN algorithm has been employed to classify the human protein sequences of COVID-19 according to country.⁷³⁶ Moreover, k-NN was also applied on the transcriptomics data. The recently reported transcriptomics data of upper airway tissue with acute respiratory illnesses is integrated with some machine learning algorithms such as the k-NN algorithm to identify effective qualitative biomarkers and quantitative rules for the distinction of SARS-CoV-2 infection from other infectious diseases.⁷³⁷ Furthermore, with the implementation of the k-NN algorithm, as well as the GRU (gated recurrent unit) neural networks and LSTM (long short-term memory) autoencoder models by Liang et al.,⁷³⁸ the analysis of the nanosecond backbone root-mean-square deviation (RMSD) of the S protein assisted in the prediction of the long-term properties of SARS-CoV-2 S proteins.

2.3.5. K-Means Clustering

K-means clustering is an unsupervised learning algorithm, aiming to partition a set of observations into K subsets or clusters. It typically begins with selecting K observations as centroids of clusters and assigning observations to clusters according to their distances with centroids. Then, the algorithm recursively updates centroids by minimizing the within-cluster sum of squares.

In the study of SARS-CoV-2, a common method for discovering evolution patterns and transmission pathways is to cluster mutations. Wang et al. retrieved 31,421 genome samples from GISAID and rearranged them according to the reference SARS-CoV-2 genome.⁹ Then they computed the Jaccard distance matrix (Jaccard distance measures the dissimilarity between two genomes) and clustered genomes using the Jaccard distance matrix. In this work, the authors also analyzed how mutations would impact the efficacy of certain COVID-19 diagnostic kits. As directly applying K-means clustering is often time-consuming, it is often coupled with dimensionality reduction techniques. In refs (12 and 739) the authors followed a similar method as in ref (9) and used PCA or UMAP to reduce the dimensionality of the Jaccard distance matrix before applying K-means clustering. K-means clustering can also be found in works such as ref (721). Additionally, in ref (740) by Assis et al., the authors first constructed a coronavirus antigen microarray (COVAM). Such a model included 11 SARS-CoV-2, 5 SARS1, 5 MERS, and 12 seasonal coronavirus recombinant proteins, which could cluster COVID-19 convalescent plasma (CCP) based on their antibody reactivity patterns against 11 SARS-CoV-2 antigens. Then, K-means analysis, gap statistics, and hierarchical clustering were applied, revealing three main clusters with distinct reactivity intensities and patterns as illustrated in Figure 12.

Figure 12.

Group 1 cluster analysis and PCA demonstrate two subgroups. Reproduced with permission from ref (740). Copyright 2021 Assis et al. under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/. (a) Reactivity to the SARS-CoV-2 antigens. Samples were clustered using hierarchical clustering analysis. (b) Bar plot of the mean reactivity and the standard error of each cluster to each individual SARS-CoV-2 antigen. (c) Distribution of the samples that were clustered into three groups by PCA.

Moreover, K-means clustering was also used to select conformations that represent the overall conformational heterogeneity of molecular dynamics simulation data. For instance, a team led by Albert Y. Lau developed a machine learning algorithm called TACTICS (trajectory-based analysis of conformations to identify cryptic sites), aiming to address the difficulties in seeking druggable sites.⁷⁴¹ First, by applying the K-means clustering algorithm in TACTICS on multiple molecular dynamics simulation data, a small number of conformations was found. Next, such conformational data was integrated into a random forest model in TACTICS to identify possible druggable sites in each conformation based on its protein motion and geometry. Last, the scores of potential binding pockets were given based on the fragment docking analysis. This approach provided a way to predict the locations of binding sites that cannot be viewed in the experimentally determined structures.

2.3.6. Support Vector Machine

The support vector machine (SVM) was developed by Vapnik and his colleagues and can be used for both classification and regression analysis.^742,743 For the classification problem, assume the training set is . The prediction of the SVM at point x_i will be ŷ = w^Tx_i + b. Here, is the weights and b is the bias. If the training set is linearly separable, the aim is to minimize ∥w∥ subject to y_i(w^Tx_i – b) ≥ 1. If the training set is not linearly separable, then the hinge loss function max(0, 1 – y_i(w^Tx_i – b)) will be involved. The aim of the SVM is to minimize

49

where λ is the regularization term (a.k.a. penalty). For the regression problem, the aim is to minimize ∥w∥ subject to |y_i – ⟨w^T, x_i⟩ – b| ≤ ϵ.

The SVM mentioned above is a linear classifier. To design a nonlinear classifier, the kernel trick is employed to maximize margin hyperplanes. The feature of the kernel SVM will be Φ(x, z), where the commonly used kernels are the linear kernel Φ(x, z) = x^Tz, the polynomial kernel defined by Φ(x,z) = (αx^Tz + r)^d, the radial basis function kernel (RBF) Φ(x,z) = e^{–(∥x–z∥/σ)μ}, and the sigmoid kernel denoted as Φ(x,z) = 1/(1 + e^–γxTz), where r, α, σ, γ, and μ are constants.

There is a large number of SVM applications on SARS-CoV-2, and some of these focus on the biomolecular level and are summarized as follows. A collection of 100,000 FDA-registered chemicals and approved drugs, as well as about 14 million other purchasable chemicals against multiple SARS-CoV-2 targets, was screened for drug repurposing,⁷⁴⁴ where SVM was applied for identifying the top features. Additionally, Dutta et al.⁷⁴⁵ predicted a novel peptide analog of the S protein using SVM models implemented by the AVPred antiviral peptide prediction server.

The SVM was applied to identify inhibitors for Mpro as well. Mekni et al.⁷⁴⁶ applied the SVM on a data set with two million commercially available compounds to classify two hundred novel chemotypes that are potentially active against the viral protease. Sun et al.⁷⁴⁷ implemented a hybrid support vector machine classification model to find the viral entry inhibitors using a collection of publicly available SARS-CoV-2 pseudotyped particle entry assay repurposing screen data. An SVM-based Web server called CellPDD was applied to determine the cell penetrating peptides (CPPs) in the proteome of SARS-CoV-2, such as S protein, M glycoprotein, N phosphoprotein, E protein, ORF1ab polyprotein, ORF3a protein, ORF6 protein, ORF7a protein, ORF8 protein, and ORF10 protein. The results showed that CPPs were not found in E protein, one CPP was identified in ORF6, and one CPP was primarily found in the proteome of ORF1ab. Such work may be valuable to some studies in the nuclear localization sequence (NLS) for vaccine development or drug discovery.⁷⁴⁸

2.3.7. Decision Trees, Random Forest, and Gradient Boosting Decision Trees

The decision tree (DT) method is a basic ML method, which is used to perform both classification and regression models by representing the attribute of the data using a flow-chart-like structure. As the fundamental architecture of tree structure methods, decision trees are further developed to a series of ensemble methods, such as the random forest method, extremely randomized tree method, AdaBoost methods, and the gradient boosting decision tree method. Among them, random forest and gradient boosting decision trees are widely applied. Random forest (RF)⁷⁴⁹ is an ensemble learning method, which is designed to reduce the overfitting in the original decision trees. Both classification and regression problems are suitable for random forest models. Gradient boosting decision tree (GBDT) is a machine learning technique for regression and classification problems, which produces a prediction model in the form of an ensemble of decision trees.⁷⁵⁰ This ensemble of decision trees is built in a stagewise fashion like other boosting methods. That is, algorithms optimize a cost function over function space by choosing a function that points in the negative gradient direction iteratively. These methods can be applied using packages such as scikit-learn in Python⁷⁵¹ or R packages.⁷⁴⁹

In applications, DT, RF, and GBDT were used commonly in the diagnosis of COVID-19 or the analysis of virus spreading. In this review, we focus on molecule-based studies. Decision trees and the ensemble methods can handle the small-size data set well and, therefore, were implemented widely at the early stage of the SARS-CoV-2 pandemic, when databases were not well-established. Investigations applying neural networks on large databases are booming and will be introduced in the next section. In ref (752) the authors used GBDT to predict which proteins would likely make up an effective vaccine for COVID-19. A GBDT model repurposed 8565 approved or experimental drugs targeting Mpro, suggesting that 20 FDA-approved drugs could be effective.⁷⁵³ Wang et al.⁶ used topology-based features and GBDT models to analyze the nsp6 protein stability upon mutation. Bocci et al.⁷⁵⁴ constructed a machine learning platform to estimate anti-SARS-CoV-2 activities, where a total of 22 feature types are created according to chemical and biophysical information such as topological fingerprints, molecular weights, etc. Another RF classification was conducted for geographic-specific SARS-CoV-2 mutations using genetic sequences.⁷⁵⁵ In addition, an RF classifier was applied to the analysis of multiple isotype-specific responses to identify infected individuals.⁷⁵⁶ Rola et al.⁷⁵⁷ studied different docking protocols and applied structures from 2D and 3D at the molecular mechanics level as features for the random forest for docking studies of the SARS-CoV-2 S protein binding to ACE2. RF models were also applied to identify SARS-CoV-2 drug inhibitors and antibodies for SARS-CoV-2 S protein and N protein.^758,759

2.3.8. Artificial Neural Network (ANN)

ANN or deep neural network (DNN) is a ML model inspired by biological neural networks that constitute animal brains.⁷⁶⁰ ANN can be viewed as a weighted directed graph in which artificial neurons can be considered as nodes and weights can be considered as the links between input and output nodes. ANN is designed for both regression and classification problems. We assume the training set is . Here, n is the number of samples, m represents the number of features, and shows the number of classes. If l = 1, then the training set is designed for the regression problem. If l > 1, then we say the ANN model is designed for a classification problem. There are two main procedures in the ANN algorithm, the feed-forward the back-propagation procedures. Assume is a feature representation in the training set; then the feed-forward starts from the input layer to the first hidden layer that is defined as

50

where represents the weights from the input layer to the first hidden layer, represents the bias from the input layer to the first hidden layer. Here h₁ is the number of the neurons in the first hidden layer and function f represents the activation functions such as the ReLu or Sigmoid function. Next, from the first hidden layer to the second hidden layer, a similar function is defined as

51

where and . Here, h₂ is the number of neurons in the second hidden layer. A similar procedure goes until it gets to the output layer. The predictor from the last hidden layer (the jth hidden layer) to the output layer is

52

where , , and h_j is the number of neurons in the last hidden layer in the ANN. The cross-entropy loss describes the cost function, which is defined as

53

The ANN algorithm obtains the prediction via the feed-forward procedure and then minimizes the cross-entropy loss through the back-propagation procedure. The back-propagation procedure applies the loss function evaluated at the output layer and propagates it back through the network to update the weights and bias. In the calculation of the gradient for back-propagation, the stochastic gradient descent (SGD) with momentum method is one of the most popular approaches which evaluates a small part of the training data and contributes to the next iteration with different weights. The process of the SGD with momentum can be expressed as

54

where η is the learning rate and β ∈ [0, 1] is a scalar coefficient for the momentum. Fully connected layers inducing a large number of degrees of freedom cause an overfitting issue in the training process. The dropout technique can prevent the network overfitting,⁷⁶¹ which randomly sets partial hidden units zero values to their connected neurons in the next layer. ANN has numerous applications in molecular biology.⁷⁶² For example, deep learning methods were used to predict the mutation-induced changes of protein stability,⁷⁶³ protein–ligand binding affinity,²¹⁹ and protein–protein binding affinity.⁶⁸⁵

Two directions of ANN/DNN application on SARS-CoV-2 are studying the infectivity of SARS-CoV-2 and the efficacy of SARS-CoV-2 antibodies and repurposing existing drugs and compounds or even generating new ones to treat SARS-CoV-2. The former focuses on the binding energy of protein–protein interactions or protein–ligand interactions. Chen et al. applied a GBDT and neural network-integrated method to calculate the BFE changes between SARS-CoV-2 S protein- and ACE2-induced by mutations.⁷⁸ Assuming that SARS-CoV-2 infectivity is proportional to the BFE of S protein and ACE2, one can quantitatively predict mutation-induced impacts on the infectivity of the SARS-CoV-2 virus. Similar work can be found to study the mutation impacts on the efficacy of SARS-CoV-2 antibodies.⁷⁶⁴ Moreover, considering the spreading of SARS-CoV-2 variants, Wang et al.^79,765 applied the DNN model results about BFE changes induced by mutations of bindings of SARS-CoV-2 S protein and antibodies and discovered the escape mutations and emerging vaccine-breakthrough variants. A deep docking (DD) model provided a fast prediction of docking scores from Glide or any other docking program, hence, enabling structure-based virtual screening of billions of purchasable molecules in a short time.⁷⁶⁶ The DD model relies on a deep neural network trained with docking scores of small random samples of molecules extracted from a large database to predict the scores of remaining molecules. A deep neural network method was developed and validated by using a docking algorithm on existing drugs for drug repurposing.⁷⁶⁷ A pretrained deep learning-based drug–target interaction model called molecule transformer–drug target interaction (MT-DTI) identified commercially available drugs that can act on viral proteins of SARS-CoV-2.⁷⁶⁸ Another deep neural network model was designed to find the protein–ligand interactions for drug repurposing.⁷⁶⁹ With a deep neural network model, 33 potential compounds were identified as ideal inhibitors against Mpro,³⁶⁶ as well as a similar work for SARS-CoV-2 inhibitors.⁷⁷⁰ Moreover, Izumi et al.⁷⁷¹ used the FASTA file for a deep neural network to predict sequence-based super secondary structure codes. Furthermore, to identify essential physicochemical and structural characters for SARS-CoV-2 Mpro inhibition, a nonlinear QSAR model assisted by ANN and SVM was designed with 69 structurally diverse chemicals with potential SARS-CoV-2 Mpro inhibitory property as descriptors.⁷⁷² Furthermore, ComboNet was designed to predict (1) the interaction between a drug and multiple biological targets, (2) the intrinsic antiviral activity of a drug, and (3) the synergy of drugs.⁷⁷³ ComnoNet is composed of two subnetworks: a drug–target interaction (DTI) and a target–disease association network, which enabled an effective in silico search for synergistic combinations against SARS-CoV-2 (see Figure 13).

Figure 13.

Structure of ComboNet. ComboNet consists of two networks: a DTI and a target–disease association network. Reused with permission from the authors.⁷⁷³ Copyright 2021 Jin at el. under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/. (a) Workflow of ComnoNet for single-drug synergy. First, a single drug is fed into the DTI network to get its molecular representation z_A. Then, such a molecular representation will be the input of the target–disease association network, and its output will be the predicted antiviral effect of a single drug. (b) Workflow of ComnoNet for drug combination synergy. First, drugs will be fed into the DTI network to get their molecular representations z_A and z_B. Then, the combination of such molecular representations z_AB, as well as z_A and z_B, will be fed into the target–disease association network to get the predicted antiviral effect of a combination of drugs.

2.3.9. Convolutional Neural Network (CNN)

CNN⁷⁷⁴ is a specialized type of neural network model originally designed to analyze visual imagery, but it can also be applied to many areas. Since the first successful CNN was developed in the late 1990s, CNN has achieved much success in image and video recognition, natural language processing, etc. CNN has had success in biophysics such as protein structure prediction and protein–ligand binding.^219,699 The core of CNN is the convolutional layer where its name comes from (see Figure 14). In the context of CNN, convolution is a linear operation that involves the multiplication of a set of weights with the input. This multiplication is always called a filter or a kernel. Using a filter smaller than the input is intentional as it allows the same filter to be multiplied by the input array multiple times at different points on the input. Specifically, the filter is applied systematically to each overlapping part or filter-sized patch of the input data, left to right and top to bottom, which allows the filter an opportunity to discover that feature anywhere in the input.

Figure 14.

Structure of 2D CNN. The feature extraction process includes multiple convolutional layers and pooling layers. The convolutional layer extracts the local features of the initial input, and the average pooling layer increases the translational invariances of the network and reduces the parameters that need to be trained. The output of the last pooling layer is a 2D array. Next, the flattened layer reshapes a 2D array to a 1D array to feed the feature into a fully connected layer. Last, the integrated information will be fed into a regressor for the final prediction.

In the antibody and vaccine research, algebraic-topology-based features were used to build a CNN-GBT hybrid model for predicting mutation-induced binding affinity change, investigating the impact of S protein mutations on the ACE2^10,78 and 27 antibodies (see Figure 15),⁷⁷ as well as suggesting some highly risky ones to vaccine design.²²¹ In the inhibitor research, Nguyen et al.⁷⁰³ used algebraic-topology-based features and CNN models to predict the potency of ligands from the 137 crystal structures of Mpro. Critical assessment of protein structure prediction (CASP) also proved a domain for the application of powerful CNN methods in protein structure prediction. For example, CNN-based AlphaFold by Google DeepMind obtained the highest accuracy in CASP13.^783,784 During this epidemic, DeepMind applied the AlphaFold to predict the 3D structures of SARS-CoV-2 Mpro, PLpro, nsp2, nsp4, and nsp6.³⁹ Meanwhile, the CNN-based C-I-TASSER algorithm developed by the Zhang lab was implemented to predict as many as 24 SARS-CoV-2 proteins.⁷⁸⁵ Yang et al.⁷⁸⁶ implemented the CNN model on secondary structure information with other biophysics properties to construct a multiepitope vaccine. Lastly, the graph convolutional neural network is considered as one of the graph neural network (GNN) variants,⁷⁸⁷ which applied convolutional ideas on a graph as the network structure and is a recursive direct-contacting aggregation algorithm.⁷⁸⁸ In ref (789) Haneczok and Delijewski applied the graph convolutional neural network to study the potential SARS-CoV-2 Mpro inhibitors.

Figure 15.

3D alignment of the available unique 3D structures of SARS-CoV-2 S protein RBD in binding complexes with 19 antibodies as well as ACE2. (a) ACE2 (6XDG⁷⁷⁵), CT-P59 (7CM4⁷⁷⁶), and CB6 (7C01⁷⁷⁷). (b) C135 (7K8Z⁷⁷⁸), C110 (7K8 V⁷⁷⁸), REGN10933 (6XDG⁷⁷⁹), and REGN10987 (6XDG⁷⁷⁹). (c) C119 (7K8W⁷⁷⁸), C144 (7K90⁷⁷⁸), and C121 (7K8Y⁷⁷⁸). (d) LY-CoV481 (7KMI⁷⁸⁰), LY-CoV555 (7KMG⁷⁸⁰), and LY-CoV488 (7KMH⁷⁸⁰). (e) C002 (7K8T⁷⁷⁸), C104 (7K8U⁷⁷⁸), C105 (6XCM⁷⁷⁸), and C102 (7K8M⁷⁷⁸). (f) S309 (6WPS⁷⁸¹), AZD1061 (7L7E⁷⁸²), and ACD8895 (7L7E⁷⁸²).

2.3.10. Natural Language Processing (NLP) Methods

The recurrent neural network (RNN) (see Figure 16a) is a class of artificial neural networks where connections between nodes form a directed graph along a temporal sequence,⁷⁹⁰ which allows it to exhibit temporal dynamic behavior in the data. Derived from the feed-forward neural network, RNN can use its internal state (memory) to process variable-length sequences of inputs. RNN was originally designed for language processing tasks, but it can also be applied to other circumstances, such as molecular sequence data.

Figure 16.

(a) Workflow of a RNN cell. Here, t represents an object at time-step t. x_t, y_t, and a_t denote the input x, output y, and activation at time-step t, respectively. ŷ_t represents the prediction at time-step t. (b) Workflow of a LSTM cell. t represents an object at time-step t. x_t, y_t, a_t, and c_t denote the input x, output y, activation, and cell state at time-step t, respectively. ŷ_t represents the prediction at time-step t. f_t, u_t, o_t, c_t, and c̃_t denote the forget gate state, update gate state, output gate state, cell state, and previous cell state at time-step t. σ is the activation function such as tanh function. (c) Workflow of a GRU cell. Here, t represents an object at time-step t. x_t, y_t, and a_t denote the input x, output y, and activation state at time-step t, respectively. ŷ_t represents the prediction at time-step t. r_t, o_t, and u_t denote the reset gate state, output gate state, and update gate state at time-step t. (d) Illustration of the generative network complex.⁷⁹³ SMILES strings are encoded into latent vector space through a gated recurrent neural network (GRU)-based encoder.

The long short-term memory (LSTM) shown in Figure 16b and the gated recurrent unit (GRU) in Figure 16c are two popular variants of RNN. LSTM⁷⁹¹ is designed to avoid the vanishing gradient problem and is normally augmented by recurrent gates called “forget gates”, and so errors can flow backward through unlimited numbers of virtual layers unfolded in space. GRU is a gating mechanism in recurrent neural networks introduced in 2014.⁷⁹² Its performance was found to be similar to that of LSTM. However, as it lacks an output gate, its parameters are fewer than LSTM, so it is easier and faster to train.

Their application to SARS-CoV-2 includes the following: Hofmarcher et al.⁷⁷⁰ utilized “ChemAI” to screen and rank around one billion molecules from the ZINC database for favorable effects against SARS-CoV-2. In more detail, the network is of the type SMILES LSTM.⁷⁹⁴ Bung et al.³⁶⁶ employed RNN-based generative and predictive models for de novo design of new small molecules capable of inhibiting SARS-CoV-2 Mpro.

COVID-DeepPredictor is another work using long short-term memory as a recurrent neural network to identify unknown sequences of virus pathogens. Hie et al.⁷⁹⁵ designed a bidirectional LSTM (BiLSTM) to predict structural escape patterns of influenza hemagglutinin, HIV-1 envelope glycoprotein (HIV Env), and SARS-CoV-2 S proteins.

2.3.11. Autoencoder and Transformer

Autoencoder is an advanced DL model built from RNN, GRU, or LSTM to learn efficient codings of unlabeled data. In molecular sciences, Autoencoders are designed to generate effective low-dimensional molecular representations. As shown in Figure 16d, a generative network complex⁷⁹⁶ is an autoencoder-based technique designed to automatically generate new drug candidates with desirable properties. It consists of an encoder, a latent space, a latent-space molecular generator, a decoder, and a 2D fingerprint-binding predictor (2DFP-BP). Gao et al.⁷⁹³ used this AI technology to generate some potential Mpro inhibitors judged by the consensus of a latent-space prediction and the 2DFP-BP.

Transformer is one of the frequently used models in the field of natural language processing, which was introduced in 2017 for sequential data analysis.⁷⁹⁷ A transformer model is equipped with an encoder–decoder structure. Typically, a stack of multiple identical layers consists of a transformer encoder. Each layer is constructed by a multihead self-attention pooling and a position-wise feed-forward network. Similarly, a transformer decoder is also a stack of multiple identical layers. Such a layer is called the encoder–decoder attention. Specifically, the encoder processes the input literately and generates encodings that contain information from the input, and the decoder then takes all the encodings as input and generates decoded sequences as output.

Transformer has been widely applied to seek potential drug candidates. For instance, Beck et al., designed a pretrained model called transformer–drug target interaction (MT-DTI) to find commercially available drugs that target SARS-CoV-2 viral proteins.⁷⁶⁸ Such a model used the BindingDB database⁷⁹⁸ as a training set and K_i, K_d, and IC₅₀ as evaluation metrics to predict the interaction between viral proteins and antiviral drugs. Besides, transformer is also integrated into a model called AlphaFold/AlphaFold2⁷⁹⁹ to predict highly accurate protein structures such as Mpro, OFR8 protein, and ORF3a protein of SARS-CoV-2.^800−802

2.4. Topics in Bioinformatics and Cheminformatics

Bioinformatics and cheminformatics are of paramount importance in modeling and analysis of SARS-CoV-2. Bioinformatic and cheminformatic approaches are integrated with experiments, biochemistry, biophysics, ML, statistics, and mathematics. In this section, we illustrate several topics from the methodology-centered perspective. Sequence alignment, homology modeling, and network-based bioinformatics are discussed, followed by cheminformatics methods, namely QSAR and pharmacophore models.

2.4.1. Sequence Alignment

Sequence alignment is a method in which one can arrange DNA, RNA, or amino acid sequences to identify their similar regions.⁸⁰³ Such similar regions may arise from functional, structural, geometrical, or evolutionary similarities. Though sequence alignment offers the best accuracy, it is not practical to be used for a large sample size. There are two main categories of sequence alignment, namely pairwise sequence alignment and multiple sequence alignment. The former only compares two sequences at a time, while the latter compares many sequences. There are many popular tools for sequence alignment such as BLAST (basic local alignment search tool), for pairwise alignment, and MAFFT, Clustal Omega, ClustalW, and MUSCLE, for multiple sequence alignment. The following section describes BLAST first, followed by several multiple sequence alignment tools.

2.4.1.1. Pairwise Sequence Alignment

One of the popular pairwise sequence alignment tools is BLAST. BLAST is a local similarity search tool that is commonly used to find similar DNA, RNA, and amino acid sequences to the sequence in question. BLAST was created in 1990 based on the k-tuple method and has since been implemented in the GenBank and had numerous updates to increase efficiency and accuracy. The k-tuple method⁸⁰⁴ is a fast heuristic method for pairwise alignment and is commonly used as an initial step for a large sample size. The similarity score S_ij between sequences i and j is defined as the number of k-tuple matches in the best pairwise alignment minus a fixed gap penalty term. For DNA and RNA, k usually ranges from 2 to 4, and for amino acids, k is 1 or 2. S_ij is calculated as the number of identities divided by the number of residues compared between i and j. The distance is defined as

55

Note that this method does not guarantee optimal alignment, but it is a fast heuristic method and can be used for the initialization of BLAST and multiple sequence alignment.

BLAST begins by first creating a list of k-letter words. It then searches for possible matching k-letter words in the databank and scores them, and any words that score above a threshold are kept. The high-scoring words are kept in a search tree. This process is then extended to high-scoring pairs (HSPs), which also looks for similar words, rather than only looking at exact matching words. After searching for HSPs, the significance of the HSPs score is considered by utilizing Gumbel extreme value distribution (EVD). Further details can be found in the literature.^805,806 The GenBank tutorial can be found in ref (807). As a basic tool for sequence alignment, it is utilized to detect, identify, or search sequences in a database. For example, similar coronavirus strands in other organisms, such pangolins^808,809 and bats,⁸¹⁰ were found. This tool is also used to detect SARS-CoV-2 virus in the environment^811,812 such as in wastewaters.^813,814

2.4.1.2. Multiple Sequence Alignment (MSA)

Unlike pairwise sequence alignment, MSA arranges three or more DNA, RNA, or protein sequences by identical regions. Through multiple sequence alignment, one can further analyze sequence homology to find evolutionary origins. In many cases, one uses a reference sequence, which is the first sequenced data, to observe mutation in the SARS-CoV-2 genome.⁸¹⁵ There are several popular tools, Clustal,⁸¹⁶ MUSCLE,⁸¹⁷ MAFFT,^818,819 etc.

Clustal. Clustal is a series of multiple sequence alignment tools for sequence analysis. With the first version Clustal released in 1988,⁸¹⁶ its package has been developed for several generations based on different methods. ClustalW is the third generation and is updated to ClustalW2 currently, which aligns sequences with the best similarity score first and progressively aligns more distant scores.^820,821 This is achieved by first obtaining a rough pairwise sequence alignment using the k-tuple method,⁸⁰⁴ followed by a neighbor-joining method,⁸²² which uses midpoint rooting to create a guided tree. ClustalW2 is used as the basis for global alignment.

As for Clustal Omega, unlike ClustalW, it uses a guided tree approach, rather than a progressive alignment method. Clustal Omega begins with first producing a pairwise alignment using the k-tuple method. This, however, does not guarantee finding optimal alignment, but it is time-efficient. Then, the sequences are clustered using the mBed method,⁸²³ which calculates pairwise distance using the embedding method. Afterward, K-means clustering is used to further cluster the sequence. Then, a guided tree is formed utilizing the UPGMA method.⁸²⁴ Lastly, MSA is produced using the HHAlign package from HH-Suite.⁸²⁴ Clustal Omega’s advantage comes from the large-scale MSA. The accuracy and time complexity are average for a low number of samples. For a large number of samples with a long sequence, Clustal Omega produces high accuracy and is time-efficient. ClustalW is the updated version of the original Clustal MSA tool.

Multiple Alignment Using Fast Fourier Transform (MAFFT). MAFFT is a MSA package based on fast Fourier transform (FFT). Given two sequences v₁ and v₂, the correlation c_v(s) of volume between the two sequences with positional lag of s sites can be defined as

where v̂₁ and v̂₂ are the FFT of the two sequences. If homologous regions exist, through Fourier analysis, there will be a peak in similar regions. For amino acid sequences, MAFFT also calculates correlation between polarity:

where ρ(s) is the polarity of each amino acid, N is the length of v₁, and M is the length of v₂. Then, a scoring function can be calculated through the sum of the two correlations

To reduce the computational complexity, only peaks above some threshold are considered. Note that the peak does not tell the location of the homologous region directly and only shows the lag. Therefore, neighboring regions at the peak must be analyzed carefully. Further details of MAFFT can be found in the literature.^818,819

Multiple sequence comparison by Log-Expectation (MUSCLE). MUSCLE is an MSA tool that utilizes progressive alignment. There are three stages in MUSCLE: draft progressive, improved progressive, and refinement.⁸¹⁷ During the draft progressive stage, a distance matrix is constructed by computing the pairwise distance of each sequence through using k-mer counting or by constructing global alignment of pairs and determining the fractional identity. Then, a tree is constructed using UPGMA (unweighted pair group method with arithmetic mean)⁸²⁴ or neighbor-joining⁸²² in which the root of the tree is determined. Lastly, a progressive alignment is built by tracing the branches of the tree, yielding the first MSA of the sequence. In the improved progressive stage, a new progressive alignment is constructed from iteratively refining the previous tree. First, a new similarity matrix is constructed from their mutual alignment in the current multiple alignments. Then, a new tree is constructed, similar to the draft progressive stage. Each tree is compared to identify any changes in the nodes or branching pathway. These steps are repeated until conversion or a maximum iteration is reached. In the progressive alignment stage, a new alignment is computed for only the set of changed nodes. More details can be found in refs (817 and 825).

Sequence alignment methods are widely used in SARS-CoV-2 analysis. Many applications focus on identifying mutations and comparing virus sequences from species and organisms. Yin used sequence alignment to understand the evolution and transmission of SARS-CoV-2.⁸ Wang et al. employed Clustal Omega to decode asymptomatic COVID-19 infection and transmission⁶ and to study mutational impacts on COVID-19 diagnosis, vaccines, and medicine.¹² Sequence alignment is an indispensable approach for SARS-CoV-2 modeling and analysis.^826−829

2.4.2. Homology Modeling

Homology modeling constructs an atomic-resolution model of the target protein from its amino acid sequence based on experimental 3D structures of related homologous proteins (i.e., templates).⁸³⁰ Homology modeling relies on identifying one or more known protein structures likely to resemble the structure of the query sequence and producing an alignment that maps residues in the query sequence to residues in the template sequence.⁸³¹

Because the 3D experimental structures of SARS-CoV-2 proteins were largely unknown at the early stage of the epidemic, homology modeling was widely applied to predict 3D structures of SARS-CoV-2 proteins, such as Mpro,^{221,602,832−838} S protein or variants,^{271,839−861} RdRp,^{294−296,299,361,514,517,529,862−864} PLpro,⁸⁶⁵ E protein,^{61,304,866−869} N protein,⁸⁶⁹ and others.^{22,305,554,602,838,870−886} Some host proteins that interact with SARS-CoV-2 were also predicted, such as ACE2,^273,887,888 TMPRSS2,^889−894 B cell epitopes,⁸⁹⁵ and CD147.⁸⁹⁶ Some 3D structures of vaccine proteins^897−899 were also built by homology modeling. Due to the worldwide attention on this virus, experimental structures of various SARS-CoV-2 proteins and their variants can be found in the Protein Data Bank as discussed in Table 1.

2.4.3. Network-Based Bioinformatics

Drug repurposing methods require comparing the unique features, such as chemical components or proteomic, metabolomic, or transcriptomic data, of a drug candidate with existing drugs, diseases, or clinical phenotypes. Compared to de novo drug discovery being time-consuming and costly, drug repurposing methods are considered as a more effective drug discovery strategy, which could shorten the time and reduce the cost. One idea of drug repurposing is that one drug currently working for one disease may also work for other diseases if these diseases share some similar protein targets.^901,902 Thus, integrated disease–human–drug interactions could form a network with nodes such as drugs, diseases, and proteins, with weighted edges referring to interactions between them, e.g., the number of drugs with a certain target. Novel drug usage can be discovered based on shared treatment profiles from disease connections, and the weight between two disease connections determines the possibility of repurposing drugs.⁹⁰¹ Common pathways between different viruses or diseases are already identified on a large scale.⁹⁰³ Meanwhile, another way to define drug repurposing is based on the structural similarities of two drugs: two drugs may work on the same therapeutic target if the two drugs have similar structures. Causal network models of SARS-CoV-2 expression and aging have been applied to drug repurposing⁹⁰⁰ (see Figure 17). Synthetic analysis of drug repurposing is carried out by network analysis, which provides the relationships within biological data sets such as protein–protein interaction networks and genomic and/or phenotypic data sets. These biological networks integrate many different data types from different resources such as experiments or in silico methods. Computational drug repurposing can be concluded as four parts: network constructions, computational analyses, validations, and applications, where the network constructions and computational analyses are illustrated as follows.

Figure 17.

Illustration of the drug repurposing. Reproduced with permission from ref (900). Copyright 2021 Belyaeva et al. under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/. (a) Hypothesis of the relation between SARS-CoV-2 and aging of individuals. Ciliated cells are in blue, stromal/fibroblast cells are in orange, and SARS-CoV-2 viral cells are in red. (b) RNA-seq/GTEx, protein–protein interactions network, drug–target data, and CMap are integrated as a data set. (c and d) Pipeline of the drug discovery/repurposing platform. First is mining relevant drugs by using an autoencoder with blue and orange points in the latent space representing data from the drug screen and the SARS-CoV-2 infection studies. Second is identifying the disease interactome within the protein–protein interaction network by implementing Steiner tree analysis. Last is investigating the drug mechanism from the first step (green diamond).⁹⁰⁰

2.4.3.1. Network Constructions

Biological networks have a variety of types or formats such as protein–protein interaction networks, knowledge graphs, or transcriptomic databases. Each of them is a large repository of medical information constructed via dry and wet laboratories.

Protein–Protein Interaction (PPI) Networks. To build the protein–protein interactome, the detection methods are summarized into three types: in vitro, in vivo, and in silico methods. The in vitro methods are performed outside a living organism. There are tremendous methods for PPI networks, such as affinity purification–mass spectrometry (AP-MS) methods, coimmunoprecipitation, affinity chromatography, protein arrays, protein fragment complementation, NMR spectroscopy, viral protein pull-down assay, and X-ray crystallography. For example, the AP-MS experiment is one of the most popular methods to build PPI networks starting to select interesting proteins, called baits, for the coassociating “prey” proteins to build PPI networks.^904,905 Then, a 2D bait–prey matrix is generated for analyzing and drug repurposing. Published sample data sets are available such as the host and HIV proteins⁹⁰⁶ and yeast protein–protein interaction network.⁹⁰⁷ Gordon et al.⁷⁵ implemented AP-MS to identify 332 high-confidence protein–protein interactions between SARS-CoV-2 and human proteins based on SARS-CoV-2 proteins in human cells. The in vivo methods are done in a living organism, such as yeast two-hybrid (Y2H) and synthetic lethality, where the Y2H method studies a bait against a random library of potential prey proteins and synthetic lethality studies functional interactions. In the work of drug repurposing by Zhou et al.,⁹⁰⁸ methods such as Y2H and AP-MS are implemented to build the virus–host and protein–protein interactome. Popular methods are sequence- or structure-based analysis, gene fusion, phylogenetic tree, gene expression analysis, and in silico 2 hybrid.

Knowledge Graphs. A knowledge graph is a large repository including the syntheses of small organic molecules, retrosynthetic steps, pathophysiology, and other biological information from the scientific literature collected via traditional computer-aided search methods or machine learning methods. Several knowledge graphs regarding COVID-19 have been built, including the CovidGraph (https://covidgraph.org/) and the Blender Lab COVID-KG (http://blender.cs.illinois.edu/covid19/).⁹⁰⁹ Many knowledge graphs were created by extracting entities and their associations from scientific publications. For example, Domingo-Fernández et al.⁹¹⁰ retrieved scientific literature related to COVID-19 and manually encode information in the format of biological expression language (BEL). Knowledge graphs can also be constructed via other methods. For example, Monte Carlo tree search was applied and combined with neural networks to guide chemical synthesis.⁹¹¹ Later, this knowledge graph was applied to search approved drugs for COVID-19, resulting in Baricitinib as a potential treatment.⁸⁰

Genome and Phenome. Genomics gives large volumes of biological data such as disease samples, animal models, cell lines, tissue samples, etc., while the phenome is the collection of phenotypic information.⁹¹² Genomic and genetic profiles have been studied for drug repurposing such as the connectivity map (CMap),⁹¹³ drug versus disease (DvD),⁹¹⁴ the database for annotation, visualization, and integrated discovery (DAVID),⁹¹⁵ etc. Meanwhile, transcriptome (noncoding RNAs) data sets are recently developed for drug repurposing.⁹¹⁶ Data sets of microRNAs are used as predictive biomarkers and therapeutic targets in prostate cancer.⁹¹⁷ At the early stage of the COVID-19 pandemic, a high-resolution map of the SARS-CoV-2 transcriptome and epitranscriptome was constructed, which gave a better understanding of the life cycle and pathogenicity of SARS-CoV-2.²³ Based on transcriptome data, differentially expressed genes (DEGs) were screened out between SARS-CoV-2-infected cells and the control group and analyzed the changes of the relevant molecular pathway, and it turned out that 136 of 145 genes were upregulated and 9 of 145 genes were downregulated.⁹¹⁸ Other transcriptome data sets are applied to study the SARS-CoV-2 infection.^{737,919−923}

2.4.3.2. Computational Analyses

Many computational methods regarding networks have been applied. In this section we briefly describe the most popular ones.

Mass Spectrometry Interaction Statistics (MiST). MiST⁹⁰⁶ was invented to identify protein–protein interactions that have biological significance from AP-MS data. Roughly speaking, a MiST score is a weighted sum of three features (abundance, reproducibility, and specificity). There are multiple ways to determine the weights, and in ref (906) the weights are determined by performing PCA. MiST was applied to identify high-confidence PPI between SARS-CoV-2 proteins and human proteins by Gordon et al.⁷⁵ They identified 332 high-confidence PPIs and revealed how SARS-CoV-2 interacts with human immune pathways and host translation machinery. They also performed a cheminformatics search for drugs that modulate human proteins and disrupt the SARS-CoV-2 interactome. Similar applications of MiST can also be found in the literature⁹²⁴ (see Figure 18).

Figure 18.

Illustration of the discovery of PPI inhibitors for SARS-CoV-2. Reproduced with permission from ref (924). Copyright 2021 Elsevier.

Centrality. Different notions of centrality have already been described in section 2.2.1. To analyze the network of the differentially expressed genes (DEGs), degree centrality, closeness centrality, and betweenness centrality are computed for each node in ref (925). Fagone et al.⁹²⁵ studied the high infectivity of SARS-CoV-2 and observed gender differences from the clinical data. They identified a gene signature that characterizes SARS-CoV-2 infection and compared it with the gene signature induced by SARS-CoV infection. To predict drugs, they carried out an antisignature perturbation analysis based on the DEGs identified for SARS-CoV-2. Network analysis via closeness centrality and betweenness centrality is applied to data analysis.⁹²⁵ Maroli et al. implemented three measures, closeness centrality, betweenness centrality, and eigenvector centrality for the network analysis and examined potential drugs.⁶⁰³ Sheik Amamuddy et al. targeted Mpro and used the betweenness centrality to analyze Mpro,²⁰⁴ while Ghorbani et al. targeted S protein.¹⁶⁵ Similar notions of centrality also appear in the literature.^926,927

Proximity. Supposing C is a set of host genes associated with a HCoV and T is a set of drug targets, one version of network proximity of C with T can be calculated by

where d(c, t) is the shortest distance between c and t in the human protein interactome.⁹⁰⁸ Zhou et al.⁹⁰⁸ retrieved HCoV-associated host protein from scientific publications and calculated network proximity between drug targets and HCoV-associated proteins to search for drug candidates. They also sought possible drug combinations, guided by the principle that each drug in a drug combination should target separate neighborhoods in the human interactome network. Notions of network proximity are also used in papers such as ref (928).

Random Walk with Restart (RWR). Given a network, simulating a particle walking from a node to a randomly chosen nearby node can reveal the topology of the network. If one repeatedly starts a random walk from the same node or a set of nodes called seed(s), one can obtain nodes that are proximal to the seed(s). In the study of PPI networks, RWR is a method for identifying the most likely interactions. RWR is used by Messina et al.⁹²⁹ to better understand the SARS-CoV-2 pathogenesis. In their paper, since the SARS-CoV-2 genome is similar to the SARS-CoV genome, they assumed that the SARS-CoV-2 interactome and the SARS-CoV interactome share several interactions. They extracted virus–host interactomes of SARS-CoV, MERS-CoV, and HCoV-229E from public databases and scientific publications, assembling a large PPI network. S-glycoproteins of SARS-CoV, MERS-CoV, and HCoV-229E were taken as seeds for RWR, discovering 200 closest proteins to S-glycoproteins. With a similar assumption, an unbalanced birandom walk with Laplacian regularized least-squares was implemented on a virus–drug association network⁹³⁰ called VDA-RWLRLS. Compared to other state-of-the-art prediction models, their methods showed better VDA prediction performance.

Tremendous network-based applications were performed on SARS-CoV-2. Many of them provide an essential PPI network dataset between SARS-CoV-2 and humans,⁷⁵ an architecture of SARS-CoV-2 transcriptome,²³ or effective candidates.^75,80,908 Gordon et al.⁷⁵ screened drugs targeting the human proteins in the SARS-CoV-2 human interactome based on the PPI network data set and considering the features of drugs such as drug status, drug selectivity, drug availability, and the statistical calculations of the protein interactions. They identified 29 drugs already approved by the United States Department of Agriculture (USDA), 12 investigational new drugs, and 28 preclinical compounds. Zhou et al.⁹⁰⁸ studied the antiviral drug repurposing methodology targeting SARS-CoV-2. A systematic pharmacology-based network medicine platform was implemented to identify the interplay between the virus–host interactome and drug targets where they investigated the network proximity of the SARS-CoV-2 host and drug target interaction. Based on that, they reported three potential drug combinations. In the study by Sadegh et al.,⁹³¹ CoVex was developed for SARS-CoV-2 host interactome exploration and drug (target) identification, which also explored the virus–host interactome and potential drug target. The network was constructed based on PPIs, drug–protein–protein interactions, etc. for repurposing drug candidates. Additionally, Srinivasan et al.⁹³² developed a network of the comprehensive structural gene and interactome of SARS-CoV-2. Messina et al.⁹²⁹ investigated the host–pathogen interaction model through the PPI network. Das et al. explored the host protein for SARS-CoV-2 by observing central protein associations in the PPI network.⁹²⁶ Analyses were done via PPI networks between viral and host protein for host biological responses.^933,934 Kumar et al. studied the SARS-CoV-2 pathogenesis through the network.⁹²⁷ Bellucci et al. studied the meta-interactome of SARS-CoV-2 via the network analysis.⁹³⁵ Drug repurposing of SARS-CoV-2 is one of the hottest topics in the applications of the network of SARS-CoV-2. In addition to the aforementioned works, other similar works were done for identifying drug repurposing of SARS-CoV-2.^{310,928,936−940}

2.4.4. Quantitative Structure–Activity Relationship (QSAR) Models

QSAR models refer to regression or classification models to predict the physicochemical, biological, and environmental properties of compounds from the knowledge of their chemical structures.⁹⁴¹ In QSAR modeling, the predictors consist of physicochemical properties and theoretical molecular descriptors of chemicals. The QSAR response variable can be the biological activity of the chemicals. There are two steps to build a QSAR model. First, the relationship between chemical structures and biological activity is summarized from a data set of chemicals. Second, QSAR models are implemented to predict the activities of new chemicals.⁹⁴² After features are extracted from a data set, the learning method of QSAR can be the ML methods mentioned in section 2.3, such as linear regression, logistic regression, support vector machine, and random forest.

To identify potential Mpro inhibitors, Ghaleb et al.²⁶¹ and Acharya et al.⁹⁴³ applied 3D QSAR models where Ghaleb et al.’s 3D model was based on comparative molecular similarity indices analysis (CoMSIA) and Acharya et al.’s 3D model was based on pharmacophores. More works used 2D QSAR models, such as Alves et al.⁹⁴⁴ using the random forest algorithm to build the model. Kumar et al.²⁶³ and Masand et al.^945,946 used genetic algorithms. SVM models were applied to a classification-based QSAR model for structural and physicochemical interpretation analysis to identify potential Mpro inhibitors.⁹⁴⁷ Ghosh et al.⁹⁴⁷ built a Monte Carlo-based classification model involving classification QSAR-based data mining. Under the framework of QSAR,⁹⁴² to identify potential SARS-CoV-2 main protease inhibitors, other works adopted multiple linear regression with QSAR models,^{205,719,948−954} where Borquaye et al.⁹⁵⁵ used multiple linear regressions. Moreover, QSAR models that predict inhibitors to other SARS-CoV-2 proteins were also constructed. Targeting S protein, Khaldan et al.⁹⁵⁶ built a 3D-QSAR model. Against PLpro, QSAR models based on Monte Carlo classification and multiple linear regression were constructed by Amin et al.^201,957 Blocking both Mpro and RdRp, Ahmed et al.⁵⁷³ built a QSAR model following partial-least-squares regression. The QSAR model of Ahmed et al.⁵⁷³ was based on a partial-least-squares regression.

2.4.5. Pharmacophore Models

A pharmacophore is an abstract description of molecular features that are necessary for the molecular recognition of a ligand by a biological macromolecule. A pharmacophore model represents the binding patterns of bioactive molecules with the target binding site, by a distinct 3D arrangement of abstract interaction features accounting for different types of noncovalent interactions.⁹⁵⁸ Thus, a pharmacophore model explains the process of structurally diverse ligands binding to a common receptor site and the identification of ligands binding to the same receptor.

In the work of searching COVID-19 therapeutics, most pharmacophore models focused on Mpro.^{390,415,461,879,959−962} Pharmacophore models were used to screen inhibitor compounds to SARS-CoV-2 from FDA-approved drugs,^461,959 DrugBank,⁸⁷⁹ or HIV inhibitors.^960,961 A fragment-based pharmacophore model was built from 22 noncovalent fragments cocrystallized with Mpro.⁹⁶³ More pharmacophore models were created for Mpro,^964−967 PLpro,⁹⁶⁸ and nsp16.⁵⁵¹

2.5. Miscellaneous

2.5.1. Molecular Modeling of Peptides, Proteins, or Graphene Binding to SARS-CoV-2 Targets

According to the large RBD of S protein, small-molecule drugs may not efficiently block the entire RBD. The entire RBD of S protein needs to be blocked by peptides.⁹⁶⁹ In a study of 1070 peptide-based drugs docking to S protein, one high binding affinity was Sar9Met (O2) 11-Substance P.⁹⁷⁰ Basit et al.⁹⁷¹ also designed a truncated version of the ACE2 receptor covering the binding residues. They performed protein–protein docking and MD simulations to analyze its binding affinity to RBD and complex stability. One study predicted the affinities of the peptide analogues Seq12, Seq12m, and Seq13m to S protein through molecular docking, MD simulation, and MM-PB/GBSA calculations.⁷⁴⁵ Other docking and MD studies of peptides to S protein include refs (876 and 972−981). In addition, potency of peptides to other targets was also anticipated. The 37 peptides from the antimicrobial peptide database were docked to N protein, and the peptides with the highest docking scores were further studied by MD simulations.⁹⁸² Porto et al.⁹⁸³ exhaustively performed docking of over 70000 peptides to Mpro. Yathisha et al.’s MM/GBSA studies suggested three angiotensin-I converting enzyme (ACE-I) inhibitory peptides are potent to Mpro.⁹⁸⁴ Zhao et al.’s docking studies identified GSRY among lactoferrin-derived peptides is potent to Mpro.⁹⁸⁵ Behzadipour et al. evaluated the SARS-CoV-2 Mpro inhibitory activity of 326 di- and tripeptides from the proteolysis of bovine milk proteins by docking.⁹⁸⁶ Some works^974,987,988 aimed to repurpose peptides from seed proteins to targets Mpro, S protein, and PLpro. Some molecular modeling studies of the protein–protein binding were also contributed to treating SARS-CoV-2. Shaheer et al.⁹⁸⁹ employed protein–protein docking and MD simulations to design degraders of SARS-CoV-2. They first docked Mpro to E3 ligase and predicted the possible complementarity between them. Then, they generated the ternary complexes of Mpro, E3 ligase, and possible linkers and ran MD simulations on these complexes. Recently, Wang et al. studied graphene interacting with SARS-CoV-2 Mpro via MD simulations. They showed that Mpro can be absorbed onto the surfaces of investigated materials and defective graphene and graphene oxide interact with Mpro more intensely.⁹⁹⁰ Similar works were also performed in the literature.^991,992 Through MM/GBSA calculations, Mehranfar et al. also suggested phosphorene can be a good candidate for designing new nanomaterials for selective detection of SARS-CoV-2.⁹⁹³

2.5.2. Molecular Modeling Studies on Vaccines

Inspired by the composition of mRNA-based COVID-19 vaccines as a lipid mixture, Paloncýová et al.⁹⁹⁴ first studied the behavior of the lipids and their mixtures in preassembled bilayers and then modeled the self-assembly of IL-containing (interleukin-containing) lipid nanoparticles (LNPs) for mRNA delivery by MD simulations with and without the presence of an RNA fragment. Additionally, they investigated the effect of the IL’s charge (i.e., the pH) on the stability of the lipid phase. In other MD simulation studies, the effect of glycan microheterogeneity was evaluated, which could impact the epitope exposure of the S protein.⁹⁹⁵ This study indicated that glycans shield approximately 40% of the underlying protein surface of the S protein from epitope exposure. Another two studies^898,996 generated tens of epitode vaccine candidates originating from S protein by the online servers NetCTL, IEDB, and FNepitope. The tertiary structures of these epitodes were predicted and docked to the toll-like receptor 3, where MD simulations were run for these complexes and the immune reactions also were simulated. Sikora et al.¹⁶² performed microsecond-scale MD simulations of a 4.1 million atom system containing a patch of viral membrane with four full-length, fully glycosylated and palmitoylated S proteins. By mapping steric accessibility, structural rigidity, sequence conservation, and generic antibody binding signatures, they recovered known epitopes on S protein and predicted promising epitope candidates for vaccine design. They also discovered that the flexible glycan coat shielded a surface area larger than expected from static structures. Through docking and MD simulations, De Moura et al.⁹⁹⁷ identified epitopes from the S protein that were able to elicit an immune response mediated by the most frequent MHC-I alleles in the Brazilian population. Both epitodes from S protein and epitodes from other targets were investigated. Rahman et al.⁹⁹⁸ also investigated some epitodes from the S protein, Mpro, and E proteins. Ezaj et al.,⁹⁹⁹ Chauhan et al.,¹⁰⁰⁰ Kalita et al.,¹⁰⁰¹ Waqas et al.,¹⁰⁰² Ranga et al.,¹⁰⁰³ and Sarkar et al.¹⁰⁰⁴ designed and simulated epitodes from other proteins of SARS-CoV-2. Other similar reports can be found in the literature.^{899,1005−1008}

2.5.3. Molecular Modeling of Blocking Host Targets with Controversy

Some researchers investigated potential damages of inhibition aimed at host receptor proteins. Parts of molecular modeling studies aimed to block host proteins relating to SARS-CoV-2 attachment, such as ACE2,^{315,324,876,1009−1029} TMPRSS2,^{479,892,1019,1030−1032} furin,^1015,1031 and glucose regulated protein 78 (GRP78).¹⁰³³ However, according to some reports, it is controversial to design SARS-CoV-2 inhibitors targeting host ACE2s or related proteins. For instance, ACE2 is an important enzyme attached to cell membranes in the lungs, arteries, heart, kidney, and intestines. It is critical for lowering blood pressure in a human body.¹⁰³⁴ It is unclear whether drugs that can inhibit ACE2 or related targets are more beneficial than harmful. Further investigation is needed.¹⁰³⁵

2.5.4. Combination of Docking and MD Simulation

Many works combine docking and MD simulation. For example, molecule docking predicts binding poses, and MD simulation further optimize and stabilize the confirmations of complexes. Other researchers follow an ensemble docking procedure to dock compounds to multiple conformations of the protein extracted from MD simulations.

Ensemble Docking. An ensemble docking of the SARS-CoV-2 Mpro was performed by Sztain et al.¹⁴⁶ They docked almost 72000 compounds to over 80 conformations of the main protease generated from MD simulations and screened these compounds through the ensemble docking strategy. To obtain extensive conformational samplings of Mpro, a Gaussian accelerated MD simulation¹⁰³⁶ was run. Another ensemble docking work of Mpro was implemented by Koulgi.¹⁰³⁷ They carried out long-time MD simulations on the apo form of Mpro. Sixteen representative conformations were collected from these MD simulations by clustering analysis and Markov state modeling analysis.¹⁰³⁸ Targeting these 16 conformations, ensemble docking was performed on some FDA-approved drugs and other drug leads, suggesting some potent candidates such as Tobramycin. Additionally, Novak et al.³⁹¹ first screened 8756 approved or experimental drugs by regular docking; then the best 10 drugs from regular docking were further evaluated by an ensemble docking strategy. The 10 drugs were docked to 5 conformation representatives of Mpro obtained from MD simulations and cluster analysis. Other ensemble docking studies were implemented on the targets Mpro^1039−1043 and RdRp.¹⁰⁴⁴ Other investigations focused on docking and optimizing by MD simulations.

Mpro. Conivaptan and azelastine were suggested as potential repurposed drugs after applied docking to systematically predict the binding affinities of 1615 FDA-approved drugs to Mpro.¹⁰⁴⁵ Then, MD simulations were performed on these two drugs for revealing their interactions with Mpro. In the docking studies of 18 derivatives of hydroxychloroquine (HCQ), remdesivir, and tetrahydrocannabinol, two derivatives give excellent docking scores and higher stability than the parent molecules. Their strong inhibition toward Mpro was validated by MD simulations.¹⁰⁴⁶ With the same strategy, Cardoso et al.¹⁰⁴⁷ in silico repurposed 10 different HIV protease inhibitors to Mpro, and the binding free energy surfaces of the best 3 drugs to Mpro were depicted by long-time MD simulations. Other existing drugs, such as lopinavir, oseltamivir, ritonavir, atazanavir, darunavir, plitidepsin, testosterone, progesterone, hydroxychloroquine, tetracyclines, flaviolin, hydroxyethylamine analogs, buriti oil (mauritia flexuosa L.) like inhibitors, etc., were also investigated specifically by docking and MD simulations.^{397,614,1048−1073}

Another important inhibitor source is natural products. Jairajpuri et al.¹⁰⁷⁴ performed the most extensive virtual screening of natural products. They screened about 90000 compounds by ADMET, drug-like, and docking score predictions. The best one, ZINC02123811, was further studied by MD simulations. Following the workflow of ligand docking, MD optimization, and rescoring, the library including 14064 marine natural products was screened.⁹⁶¹ The best one, heptafuhalol A, was predicted to have a docking score as high as −18.0 kcal/mol. Ul Qamar et al.¹⁰⁷⁵ used docking and MD simulation to screen a medicinal plant library containing 32,297 potential antiviral phytochemicals/traditional Chinese medicinal compounds. Potent inhibitors such as 5,7,3′,4′-tetrahydroxy-2′-(3,3-dimethylallyl)isoflavone with a docking score of −16.35 kcal/mol were predicted. Virtual screening was also performed toward other natural products such as marine products, Indian medicinal herbs, and plant products.^{238,612,613,620,621,719,1048,1075−1090}

Other small molecules were also screened to inhibit the SARS-CoV-2 Mpro. Ton et al.⁷⁶⁶ identified potential Mpro inhibitors by docking 1.3 billion compounds and suggested that compound ZINC000541677852 had the highest binding affinity of −11.32 kcal/mol. Its interactions with Mpro were studied by MD simulations. The docking and MD simulations tested 4384 molecules from the Zinc data set,¹⁰⁹¹ and some of them are FDA-approved drugs. Among them, the best one was bisoctrizole.⁸³³ Additionally, some diaminobenzophenone derivatives, prototypical-ketoamide inhibitors, HIV protease inhibitors, leucoefdin, nutraceuticals, and others were also studied.^{619,625,636,1047,1092−1107} Notably, Mohammad et al.¹¹⁰⁸ optimized some complexes of Mpro with ligands in the protein data bank (PDB) and rescored them by AutoDock Vina, with the best PDB structure being 6M2N.

S Protein. By docking and MD simulations to identify potential inhibitors against SARS-CoV-2 S protein from 1582 FDA-approved drugs, lumacaftor was predicted to have the highest binding affinity.¹¹⁰⁹ Moreover, to evaluate the binding interactions between S protein and compounds, steered MD simulations¹¹¹⁰ were performed on the top compounds. Bharath et al.¹¹¹¹ and Choudhary et al.⁸⁵¹ screened 4015 and 1280 small compounds, respectively, and predicted that phytic acid and GR 127935 hydrochloride hydrate had the highest energies. Shoemark et al.¹¹¹² applied docking and MD simulations to bind some vitamins, retinoids, and steroids as to the free fatty acid pocket of S protein. Gupta et al. studied the interactions between some functional spike-derived peptides and S protein by docking and MD simulations.¹¹¹³ Heparin has been found to have antiviral activity against SARS-CoV-2. Paiardi et al. investigated the binding of heparin to the SARS-CoV-2 spike glycoprotein by docking and MD simulations. Their results suggest that heparin can inhibit SARS-CoV-2 infection by three mechanisms: by allosterically hindering binding to the host cell receptor, by directly competing with binding to host heparan sulfate proteoglycan coreceptors, and by preventing S cleavage by furin.¹¹¹⁴ Other works evaluating potential inhibitors against S protein include refs (630, 634, 975, and 1115−1123).

RdRp. RdRp is another important target of SARS-CoV-2. Nagar et al.¹¹²⁴ performed ADMET and docking-based screening on 267324 compounds, and the best two of them were subjected to MD-based further study. Another extensive screening based on ADMET, docking, and MD simulations was implemented by Ghazwani et al., which covers 13840 compounds.¹¹²⁵ Singh et al.,¹¹²⁶ Mondal et al.,³⁰¹ Begum et al.,¹¹²⁷ and Pokhrel et al.¹¹²⁸ evaluated the potency of thousands of existing drugs via docking and MD simulations. Tens of fungal secondary metabolites were docked to RdRp by Ebrahimi et al., and MD simulations were performed on the top five compounds.¹¹²⁹ Elghoneimy’s docking and MD simulations¹¹³⁰ aimed to repurpose some HCV NS5B palm subdomain binders to inhibit SARS-CoV-2 RdRp.

PLpro. Targeting PLpro, in a study aiming to repurpose all current FDA-approved drugs, the best drug, ergotamine, among their predictions was reevaluated by MD simulations.¹¹³¹ Baildya et al.¹¹³² predicted the potency of 19 compounds from Neem, and MD simulation was performed on the best one, desacetylgedunin. Other similar studies can be found in refs (537 and 1133).

Other Targets. Dihydroergotamine and irinotecan were predicted to be the best drugs of the study of 3000 FDA-approved and experimental drugs against 2′-O-methyltransferase in nsp16 of SARS-CoV-2.¹¹³⁴ De Lima Menezes et al.¹¹³⁵ screened 8694 approved and experimental drugs from DrugBank against nsp1 and predicted that tirilazad was the most potent one. Tazikeh-Lemeski et al.¹¹³⁶ docked 1516 FDA-approved drugs from DrugBank to nsp16 and investigated the drug–protein interactions by MD simulation, finding that raltegravir had the best predicted binding affinity. Following a similar scheme, a collection of 22122 Chinese traditional medicines from TCM Database@Taiwan against nsp14 was screened,⁸⁸⁴ with the best one being TCM57025. Liang et al.¹¹³⁷ predicted the affinities of 300 potential molecule inhibitors to the nsp10–nsp16 methyltransferase complex via docking and MD simulations. Sixto-López et al.¹¹³⁸ and Kumar et al.¹¹³⁹ predicted the potency of more than 2000 ligands to nsp15. Tatar et al.¹¹⁴⁰ aimed to repurpose 34 antiviral compounds to inhibit N protein.¹¹⁴⁰ Following the workflow of ADMET screening, docking, optimizing of the top predictions by MD simulations, and redocking, Rampogu et al.¹¹⁴¹ screened potential nsp16 inhibitors from 59619 natural compounds.

Multiple Targets. In many reports, the same drugs were tested against multiple targets. In the work of El-Demerdash et al.,¹¹⁴² the potentials of 15 marine polycyclic guanidine alkaloids to block Mpro, S protein, N protein, M protein, and nsp15 were evaluated by docking and MD simulations. Many works cover three or two targets. Dwarka et al.¹¹⁴³ docked 14 South African medicinal plants to Mpro, RdRp, and S protein RBD. They also investigated the dynamics and interactions inside the complexes using MD simulations. However, no potent inhibitors were found. Bhowmik et al.¹¹⁴⁴ virtually screened more than 200 antiviral natural compounds and 348 antiviral drugs targeting the E, M, and N proteins responsible for envelope formation and virion assembly. Molavi et al.¹¹⁴⁵ repurposed 1760 FDA-approved drugs to Mpro and RdRp, and Rajpoot et al.¹¹⁴⁶ repurposed 291 drugs to Mpro and PLpro. More similar works include refs (599, 609, 622, 624, 628, 632, 637, 838, 875, 889, and 1147−1173).

2.5.5. Accuracy Tests of Molecular Modeling Methods on SARS-CoV-2 Targets

To assess the predictive power of molecular modeling methods on SARS-CoV-2 targets, accuracy tests were performed on some SARS-CoV-2 inhibitors with known experimental binding affinities. Ngo et al.¹¹⁷⁴ systematically evaluated four popular binding-affinity calculation approaches, namely, FEP, steered MD, LIE, and MM/PBSA. They tested 20 Mpro inhibitors with available affinity values and found FEP was the most accurate with a correlation of 0.85, while the correlations of steered MD, LIE, and MM/PBSA are 0.74, 0.73, and 0.32, respectively. A test of the affinity prediction of 19 Mpro inhibitors revealed docking can achieve a correlation of 0.50 and steered MD can raise the correlation to 0.65.¹¹⁷⁵ Another test of the docking poses predictions from several leading docking programs, namely, Glide, DOCK, AutoDock, AutoDock Vina, FRED, and EnzyDock.¹¹⁷⁶ In the scope of 193 experimental Mpro–inhibitor complexes, the best performance is that 26% of noncovalently bound ligands (from glide) and 45% of covalently bound ligands (from EnzyDock) can reach RMSAs < 2 Å. According to their tests, the work suggested good poses may be provided but the affinity predicted for each pose may not be reliable.¹¹⁷⁷ Fan et al.¹¹⁷⁸ added some structural restraints deduced from experiments that could improve the accuracy of docking predictions.

2.5.6. Combined MD Simulation and Deep Learning

Gupta¹¹⁷⁹ first selected 92 potential Mpro inhibitors from FDA-approved drugs by docking and then further evaluated their potency by MD simulations and MM/PBSA calculations using the hybrid of the Accurate NeurAl networK engINe for molecular energies (ANAKIN-ME) deep learning force field¹¹⁸⁰ and a conventional molecular mechanics force field. Their results suggested that targeting was the most potent drug against the main protease. A similar work integrated a deep neural network model with docking and MD simulations.¹¹⁸¹ Moreover, AI-driven multiscale simulations provided analyses of the spike’s full glycan shield elucidation, the role of spike glycans in the viral infection, the interactions between the S and the human ACE2, etc.¹¹⁸²

2.5.7. Combined MD Simulation and Experiment

The MD simulations revealed three flexible hinges within the stalk, coined hip, knee, and ankle of the S protein, which were consistent with tomographic experiments.¹¹⁸³ In the implementation of virtual screening of 6218 drugs targeting Mpro and RdRp,¹¹⁴⁷ the best ones and some of their combinations were verified by cell-based assays. Combining experiments and calculation, Zaidman et al.¹¹⁸⁴ developed a fragment-based and automatic pipeline to design potent Mpro inhibitors. Dwivedi et al.’s docking and MD simulation confirmed their experimental results that holothurian sulfated glycans show potential effects against SARS-CoV-2 S protein RBD.¹¹⁸⁵ Similar works include refs (1093, 1153, 1154, and 1186−1213). Other studies applied experiments to identify potent compounds and used docking to reveal interactions.^1214−1218

2.5.8. Combined MD Simulation and Data Analysis

Some of the popular methods to analyze dynamics characteristics in MD simulation are principal component analysis (PCA) and normal-mode analysis, which can extract the principal modes of motion from MD simulations.^1219,1220 Toward SARS-CoV-2, some works applied PCA to reveal the internal motions of Mpro.^{407,421,481,719,1061,1074} Rane et al.⁵⁰⁶ and Dehury et al.¹²²¹ used such analysis to investigate the dynamics of the S protein. Bera et al.,⁵⁶⁹ Henderson et al.,¹⁶⁷ and Chandra et al.⁵⁴⁵ used PCA to elucidate the motions of the PLpro and NendoU, respectively. Bhattacharya et al.¹⁰⁰⁵ applied NMA to display the mobility of the human TLR4/5 protein and SARS-CoV-2 vaccine component complex. Shaikh et al.¹²²² used elastic network models to identify the residue cross-correlation matrix.

2.5.9. Combinations of Multiple Molecular Modeling Methods

In a comprehensive study in predicting potent Mpro inhibitors, it first screened their compounds by molecular docking, and then the top four compounds from the docking prediction were more accurately calculated by MM/PBSA simulation.¹²²³ In the third step, the binding of the top two compounds to Mpro was further studied by QM/MM simulations and the chemical properties of these two compounds, such as highest the occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), were predicted by DFT calculations. These studies also involve multiple molecular modeling methods.^1224−1226

2.5.10. Molecular Modeling of Mutational Impacts

Through docking and MD simulations, the drug candidate PF-0083523, under clinical trial now, was predicted, which was potent to four reported Mpro mutants.¹²²⁷ Muhammad et al.’s¹²²⁸ virtual screening suggested that some potent phytochemical inhibitors were effective to two Asian mutants of Mpro. Sheik Amamuddy et al.²⁰⁴ studied the impact of Mpro mutations on its 3D conformation as well. More studies focused on the mutations of S protein. Some works applied docking, MD simulations, MM/PBSA, or MM/GBSA calculations to investigate S protein mutation impacts on the interactions and affinities between S protein RBD and ACE2 protein.^1229−1234 Additionally, via MD simulations, it was found that mutations on the distal polybasic cleavage sites of S protein could weaken the binding between S protein and human ACE2, which means these distal polybasic cleavage sites were critical to the binding.¹²³⁵ Virtual alanine scanning mutagenesis by FEP MD simulations was also performed to uncover key S protein residues in its ACE2 binding.¹²³⁶ Dehury et al.¹²²¹ compared the interactions of mutated S proteins to the wild type of S protein binding to ACE2. Calcagnile et al.²⁷⁴ and Hadi-Alijanvand et al.¹²³⁷ assessed impacts of ACE2 polymorphisms to their affinities to S protein by docking and MM/GBSA calculations. Furthermore, the impact of S protein mutations on inhibitor and antibody efficacy was also simulated. For example, the docking and MD simulations indicated the Alpha variant did not compromise the efficacy of catechins.¹²³⁸ However, mutations H49Y, D614G, and T573I were suggested to considerably affect the binding of cepharanthine, nelfinavir, and hydroxychloroquine into their respective binding sites.¹²³⁹ Wu et al.’s MM/GBSA calculation suggested that most antibodies (about 85%) have weaker binding affinities to the E484 K mutated S protein than to the WT, indicating the high risk of immune evasion of the mutated virus from most current antibodies.¹²⁴⁰ Lastly, Hossain et al.¹²⁴¹ performed docking to investigate the impacts of the mutations on nsp1, nsp3, and PLpro. Wu et al.¹²⁴² systematically predicted mutation impacts to the conformations of multiple SARS-CoV-2 proteins and the interactions between them.

2.5.11. Protein Pocket Detection

The detection and characterization of protein pockets and cavities are critical issues in molecular biology studies. Pocket detection algorithms can be classified as grid-based and grid-free approaches.^1243,1244 Grid-based approaches embed proteins in 3D grids and then search for grid points that satisfy some conditions. Grid-free ones include methods based on the probe (sphere) or the concepts of Voronoi diagrams.

The plug-in Pockets 2.0 combined the pocket-detecting algorithms Fpocket¹²⁴³ and PLANTS¹²⁴⁵ to characterize the SARS-CoV-2 druggable binding pockets.¹²⁴⁶ Zimmerman et al.¹²⁴⁷ launched large-scale Folding@home simulations from their FAST-pockets adaptive sampling to aid in the discovery of cryptic pockets on various SARS-CoV-2 proteins. Another server, CavityPlus (www.pkumdl.cn/cavityplus), was implemented to search druggable cavities.²⁹⁰ Vithani et al.¹²⁴⁸ adopted FAST sampling algorithms to quickly explore cryptic pockets on nsp16 and the nsp10/nsp16 complex. Manfredonia et al.¹²⁴⁹ predicted the 3D structure of SARS-CoV-2 RNA by coarse-grained modeling and detected potential pockets. Sheik Amamuddy et al.²⁰⁴ predicted potential binding pockets of Mpro.

2.5.12. AlphaFold

Chains of amino acids spontaneously fold to form biologically active proteins with their native 3D structures. However, it is challenging to understand how amino acid sequences determine the 3D structures of the proteins, and such a topic is called the “protein folding problem”. Protein structures can be identified through multiple techniques such as cryo-electron microscopy, X-ray crystallography, and NMR. However, such techniques are expensive and time-consuming, and only 0.085% (170 K out of 200M) of the 3D structures of proteins have been determined over the past 60 years.⁷⁸⁴ Therefore, seeking a more efficient and accurate way to predict the 3D structure of proteins is crucial. Starting from 1994, a worldwide experiment for protein structure prediction called CASP has taken place every two years. Researchers worldwide have put efforts into this experiment to help achieve high correlations between the experimental structures and the 3D structures predicted by theoretical-based methods. However, such predicted structures had accuracy without experimental precision, which limited their utility for many biological applications.

Developed by Google’s DeepMind in 2018, an AI-based program called AlphaFold that was designed to predict protein structures achieved high accuracy competitive with experiments in CASP13 assessment.⁷⁹⁹ As a result, AlphaFold was ranked first among 98 teams. Specifically, AlphaFold made the best prediction for 25 out of 43 targets. It was the first time that AlphaFold drew the attention of scientists worldwide. In 2020, AlphaFold2, a new version of the AI-based model, entered into the CASP14 assessment.⁷⁹⁹ The predicted protein structures by AlphaFold2 were vastly more accurate than other competing methods. Overall, 88 out of 97 protein structures predicted by AlphaFold2 achieved the most precise structure for targets, which was a significant improvement compared to AlphaFold in 2018.

AlphaFold2 consists of two major modules: the evoformer module and the structure module. First, by applying the genetic database search and structure database search, the primary amino acid sequences are encoded to embed MSAs and pairwise features, which can be treated as the inputs of the evoformer module. The evoformer module consists of attention-based and nonattention-based transformers. Such a module produces a processed MSA array with shape N_seq × N_res and a processed residue pair feature with an array size of N_res × N_res, which are then fed into the structure module to get the explicit predicted 3D structures. Here, N_seq and N_res represent the number of sequences and the number of residues, respectively. Moreover, an iterative refinement called “recycling” is involved in AlphaFold2 to minimize the final loss and achieve higher accuracy as shown in Figure 19.

Figure 19.

AplhaFold2 architecture. Reproduced with permission from ref (799). Copyright 2021 Jumper et al. under Creative Commons Attribution 4.0 International License https://creativecommons.org/licenses/by/4.0/. Arrows show the information flow among the various components described in this paper. Array shapes are (s, r, c), where s shows the number of sequences (N_seq in the main text), r represents the number of residues (N_res in the main text), and c is the number of channels.

AlphaFold2 has been used to predict highly accurate SARS-CoV-2 protein structures since 2020. For instance, the AlphaFold-predicted structure of ORF8 was used as a search model, which provided a phase solution by molecular replacement (MR) by Flower et al.⁸⁰¹ Pandey et al. applied AlphaFold to predict the secondary structure of nsp6, which showed 91.7% and 8.3% of residues are located in the most favored and additionally allowed regions, respectively.¹²⁵⁰ Slavin et al. employed AlphaFold2 to generate a single consistent all-atom model of SARS-CoV-2 nsp2, which indicated three putative metal binding sites and further suggested that nsp2 may have a role in zinc regulation.¹²⁵¹ Furthermore, a team led by Ad Bax introduced three models to evaluate their concordance with residual dipolar couplings (RDCs) measured in 2021. They are (1) a standard AlphaFold2 model; (2) an implementation of AlphaFold2 that excluded all candidate template X-ray structures after Jan. 1, 2020; and (3) an implementation of the AlphaFold2 model that excluded structures homologous to coronaviral Mpro.⁸⁰⁰

3. Discussion

Since the outbreak of the COVID-19 epidemics in December 2019, enormous effort has been devoted to scientific research relating to SARS-CoV-2, leading to significant breakthroughs, such as the development of vaccines and experimental determination of protein structures. Vaccines, drugs, and antibody therapies are in emergency use authorization. Along with the rapid development of high-performance computers, biophysical methods, and AI algorithms in recent decades, plenty of theoretical and computational studies were carried out against SARS-CoV-2. Theoretical and computational studies are significant for combating urgent epidemics and pandemics since they are faster and cheaper.¹²⁵² This review strives to summarize the existing SARS-CoV-2 related theoretical and computational works and inspire future ones. SARS-CoV-2 protein structure predictions also play an important role, especially at the early stage of the epidemics when experimental structures were largely unavailable. At this point, besides traditional homology modeling, a more popular solution is the high-level deep-learning-based models such as AlphaFold³⁹ and C–I-TASSER,⁷⁸⁵ both making use of deep CNNs.

3.1. Drug Repurposing

Many research efforts covered by this review are about repurposing current drugs or inhibitors to target SARS-CoV-2 because the drug development has been one of the most urgent issues in combating COVID-19. A variety of drug repurposing approaches has been applied, from molecular docking and MD simulation to network analysis and machine/deep learning, as summarized below. (1) The most straightforward approach is molecular docking, which provides both binding poses and corresponding scores. (2) In many studies, docking poses were further optimized by MD simulations, and these optimized poses were rescored by docking programs. (3) More accurate binding free energies can be achieved by MD-simulation-based or even QM-based calculations, such as MM/PB(GB)SA, free energy perturbation, metadynamics, QM/MM, and DFT. (4) Other than traditional molecular docking and MD simulations, the development of AI, machine/deep learning technologies opens a new approach to discover SARS-CoV-2 drugs, as well as network analysis. With existing drugs as training sets, machine/deep learning can predict the potency of a large number of potential SARS-CoV-2 inhibitors in a short time.⁷⁵³ 3D models also provided binding poses.⁷⁰³ Moreover, AI has the potential to create new drugs to combat COVID-19.^796,1253 For example, Bung et al.³⁶⁶ employed RNN-based networks, and Gao et al.⁷⁹³ used GRU-based generative networks to design new potential main protease inhibitors. (5) Network-based approaches were also performed in SARS-CoV-2 drug repurposing, promising solutions to identify drugs associated with certain diseases. These networks can be based on proteomic, transcriptomic, or metabolomic data. The basic idea is that one drug currently curing one disease may also work for other diseases if sharing some similar protein targets.^75,80,913 Thus, integrated disease–human–drug interactions form a network connecting drugs, diseases, and targets. Novel drug usage can be discovered based on shared treatment profiles from disease connections. At the level of proteomic networks, Gordon et al.⁷⁵ and Zhou et al.⁹⁰⁸ systematically explored the host dependencies of the SARS-CoV-2 virus to identify host proteins that are already targeted by existing drugs. Therapies that target the host–virus interface could potentially present durable, broad-spectrum treatments. At the level of transcriptomic networks, Belyaeva et al.⁹⁰⁰ used the transcriptomic data from the CMap database to search for compounds that may cause genomic changes opposite to the changes caused by SARS-CoV-2, so as to identify novel and potentially effective drugs with antiviral properties. Ahmed¹²⁵⁴ adopted a transcriptomic network to reveal the mechanism of Vitamin D treating the cytokine storm caused by SARS-CoV-2. (6) Traditional QSAR approaches were implemented in many calculations for drug rediscovery.

3.2. Mutational Impacts on SARS-CoV-2 Infectivity

SARS-CoV-2 infectivity plays a paramount important role in SARS-CoV-2 transmission and COVID-19 prevention and control. The experimental methods to evaluate viral infectivity are expensive and time-consuming considering the rapid spread of COVID-19. For all SARS-CoV-2 variants around the world, it is even more challenging to experimentally measure infectivity. However, a computational platform based on theoretical analysis, which quantitatively predicts the BFE changes of the RBD-ACE2 complex induced by mutations on the S protein RBD, will deliver a consistent measurement of SARS-CoV-2 infectivity. A computational platform of RBD-ACE2 BFE changes induced by RBD mutations is given in ref (78). It integrates a series of well-established methods, including the genotyping of SARS-CoV-2 genetic sequences,¹² protein sequence alignment,⁷⁸ the biophysics of PPIs, the algebraic topology representation of proteins,^219,663 the deep learning modeling of RBD-ACE2 BFE changes induced by mutations,⁶⁹⁵ and the training with existing experimental mutational data sets. According to the genotyping, the authors theoretically revealed that the SARS-CoV-2 diagnostic target mutations had false-negative test results.⁹ Experimentally, this finding is confirmed.¹²⁵⁵ Similarly, the impacts of mutations on vaccines and antibody therapies can be decoded.¹² In addition, by applying the large data set of SARS-CoV-2 spreading around the world, the top eight mutations in the United States are from two groups in which one group of five concurrent mutations is prevailing and another group has three concurrent mutations that faded out gradually.¹⁰ Additionally, many of the MD-based studies are devoted to the mutation-induced conformational changes of S protein and its corresponding binding affinity changes to the human ACE2.^485−490

3.3. Mechanisms of SARS-CoV-2 Evolution

The understanding of the mechanism of SARS-CoV-2 evolution is a prerequisite for the prediction of emerging variants and essential for combating COVID-19. SARS-CoV-2 evolution is driven by competing mutations at molecular and organism scales. Molecular-scale mutations are mainly caused by a series of random errors in replications, transcriptions, and translations.¹²⁵⁶ Unlike other RNA viruses such as the flu virus and HIV, SARS-CoV-2 has a genetic proofreading mechanism¹²⁵⁷ and has more fidelity. Additionally, at the organism scale, host gene editing is found to be the dominating source for mutations.¹¹ Moreover, viral genetic recombination may happen at the organism scale as well. In Figure 20a, it is shown that SARS-CoV-2 had over 28,912 unique mutations by November 2021, where each SARS-CoV-2 nucleotide had one known mutation on average. Nonetheless, most mutations have little to do with virus evolution, which is regulated by a population-scale mechanism. In summer 2020, Chen et al. hypothesized that “natural selection favors those mutations that enhance the viral transmission”.⁷⁸ The authors further hypothesized that mutations that strengthen the binding of the RBD-ACE2 complex will enhance the SARS-CoV-2 infectivity and transmission.⁷⁸ To investigate this natural selection mechanism of evolution from a theoretical perspective, the authors applied the single-nucleotide polymorphism calling of over 15,000 SARS-CoV-2 genomes from GISAID and identified 725 nondegenerated mutations on the SARS-CoV-2 S protein at the early stage of the pandemic (i.e., May 27, 2020). Among them, 89 RBD mutations are important to the binding of SARS-CoV-2 S protein and ACE2. Despite the highest frequency of 89 mutations being only about 50, the theoretical model resulted in Figure 20c, showing the first evidence of natural selection. There was a dramatic increase in the frequencies of a few infectivity-strengthening RBD mutations as the pandemic unfolded in the following months.⁷⁹Figure 20d shows that all the 100 most observed RBD mutations have their predicted BFE changes above the average values of −0.28 kcal/mol.⁷⁹ The chance for this to occur accidentally is one in 2¹⁰⁰, which undoubtedly confirms the natural selection mechanism of SARS-CoV-2 evolution.

Figure 20.

(a) Illustration of SARS-CoV-2 mutations given by Mutation Tracker. The interactive version is available at the Web site: https://users.math.msu.edu/users/weig/SARS-CoV-2_Mutation_Tracker.html. (b) Illustration of the analysis of SARS-CoV-2 mutations given by interactive Mutation Analyzer (https://weilab.math.msu.edu/MutationAnalyzer/). (c) Reproduction of Figure 3 of ref (78). The time evolution of 89 SARS-CoV-2 S protein RBD mutations. The green lines represent the mutations that strengthen the infectivity of SARS-CoV-2, and the red lines represent the mutations that weaken the infectivity of SARS-CoV-2. Many mutations overlap their trajectories. Here, the collection date of each genome sequence deposited in GISAID was applied according to the information recorded in June 2020.⁷⁸ (d) Reproduction of Figure 2 of ref (79). Illustration of SARS-CoV-2 mutation-induced BFE changes for the complexes of S protein and ACE2. Here, the 100 most observed mutations out of 651 mutations on S protein RBD and their frequencies are illustrated as recorded in April 2021.⁷⁹ The highest frequency was 168,801, while the lowest frequency was 28. Therefore, the frequencies of the rest of the 551 mutations were lower than 28. (e) Reproduction of the right chart of Figure 11 of ref (764). Illustration of SARS-CoV-2 RBD mutation-induced binding free energy changes for the complexes of S protein and antibody LY-CoV555. Here, mutations L452R, V483F/A, E484K/Q, F486L, F490L/S, Q493K/R, and S494P could potentially disrupt the binding of antibodies and S protein RBD.

Currently, the prevailing SARS-CoV-2 variants are known as Alpha, Beta, Gamma, Delta, Epsilon, Theta, Kappa, Lambda, and Mu, involving RBD (co)mutations [N501Y], [K417N, E484K, N501Y], [K417T, E484K, N501Y], [L452R, T478K], [L452R], [E484K, N501Y], [L452R, E484Q], [L452Q, F490S], and [R346K, E484K, N501Y], respectively. More recently, the Omicron variant, starting in late November 2021, includes 15 RBD mutations [G339D, S371L, S373P, S375F, K417N, N440K, G446S, S477N, T478K, E484A, Q493R, G496S, Q498R, N501Y, Y505H]. Remarkably, two common RBD residues, 452 and 501, for these variants were predicted to “have high chances to mutate into significantly more infectious COVID-19 strains” in May 2020,⁷⁸ many months before any variants were identified.

Meanwhile, efforts from researchers worldwide provide more genome sequence information of SARS-CoV-2 and experimental data on ACE2/antibodies binding to S protein. Some deep mutational data on RBD and ACE2 were available in late 2020,^1258,1259 which can be used to improve the mathematical model in terms of accuracy.⁶⁸⁷ A computational workflow was presented to predict impacts induced by mutations on the binding between RBD and ACE2.¹²⁶⁰ Computational mutagenesis work suggested mutations L455D/W, F456 K/W, Q493K, N501T, and Y505W on RBD enhance the ACE2/RBD binding.¹²⁶¹ By docking, Omicron and Delta variants were studied, whose mutations Q493R, N501Y, S371L, S373P, S375F, Q498R, and T478K contribute significantly to the high binding affinity with human ACE2.¹²⁶²

3.4. Mutational Impacts on SARS-CoV-2 Antibodies and Vaccines

As the vaccination rate increasing and SARS-CoV-2 variants erupting, a more intriguing question is whether vaccination and antibody therapies protect us from the SARS-CoV-2 variants. More precisely, it is imperative to understand how these variants affect vaccines and antibody therapies. Therefore, a statistical analysis of a variety of antibodies can guide the efficacy analysis of vaccination in a population. As the collection of antibody–RBD complexes accumulates, a library of 130 antibody–RBD complex structures⁷⁶⁵ was used as a statistical ensemble to analyze the RBD mutation impacts on antibodies and vaccines. Additionally, the theoretical model can be applied to study these emerging variant impacts on mAbs, especially the mAbs obtained EUA from the FDA such as Regeneron and Eli Lilly antibodies.⁷⁶⁴Figure 20e illustrates the BFE changes induced by the 100 most observed RBD mutations of the complex binding of RBD and antibody LY-CoV555. One noted that mutations L452R, V483F/A, E484K/Q, F486L, F490L/S, Q493K/R, and S494P could weaken the binding ability of Eli Lilly mAbs. Coincidentally, LY-CoV555 was taken off from the EUA (Emergency Use Authorization) in March 2021 due to its low efficacy on the Beta [K417N, E484K, N501Y] and Gamma [K417T, E484K, N501Y] variants. Soon afterward, it was allowed to return to the market in September 2021 for its efficacy on the Delta variant [L452R, T478K]. The prediction in Figure 20e is a good explanation behind these events. More results for other mAbs as well as a series of confirmations by experimental data were given in the literature.⁷⁶⁴ A complete analysis of all RBD mutations is provided at the Web site Mutation Analyzer (https://weilab.math.msu.edu/MutationAnalyzer/) as shown in Figure 20b.

Some computational investigations were devoted to vaccine design. MD simulations were employed to simulate vaccine-related immune reactions, such as the binding of the MCH (major histocompatibility complex) II–epitope complexes.¹⁰⁰⁸ Liu et al.’s³⁵⁹ FEP calculations suggested the E484Q and L452R mutations significantly reduce the binding affinity between the RBD of the Kappa variant and the antibody LY-CoV555 as well. Accurate computational predictions of Omicron’s vaccine breakthrough and antibody resistance were made available before any experiments and were all confirmed by experimental data.¹²⁶³

4. Concluding Remarks

Since the first COVID-19 case was reported in December 2019, this pandemic has led to five waves of infections, over 400 million reported cases globally, and near 6 million deaths. Despite the exciting progress in the developments of vaccines and monoclonal antibodies, their potential side effects, such as allergic reactions to COVID-19 vaccines, are not very clear. Additionally, the latest Omicron variant is able to evade current vaccines and compromise essentially all monoclonal antibodies. Although the Omicron variant may be less deadly than the original virus, there is no guarantee that future variants will be less virulent. Our present understanding of SARS-CoV-2 and COVID-19 is still quite poor.

Molecular modeling, simulation, and prediction of SARS-CoV-2 have had tremendous contributions to the development of effective vaccines, drugs, and antibody therapies. Their role in combating COVID-19 is indispensable. For example, thanks to an approach that integrates genotyping, biophysics, artificial intelligence, advanced mathematics, and experiment data, it is now well-understood that the SARS-CoV-2 evolution and transmission are governed by natural selection.⁷⁸ This means the next SARS-CoV-2 variant will be increasingly more transmissible through high infectivity, robust vaccine breakthrough, and strong antibody resistance.^765,1264 This understanding cannot be achieved through individual experiments. Therefore, it is imperative to provide a literature review for the study of the molecular modeling, simulation, and prediction of SARS-CoV-2. Since the related literature is huge and varies in quality, we cannot collect all of the existing literature for the topic. However, we try to put forward a methodology-centered review in which we emphasize the methods used in various studies. To this end, we gather the existing theoretical and computational studies of SARS-CoV-2 concerning aspects such as molecular modeling, biophysics, bioinformatics, cheminformatics, machine learning including deep learning, and mathematical approaches, aiming to provide a comprehensive, systematic, and indispensable component for the understanding of the molecular mechanism of SARS-CoV-2 and its interactions with host cells. This review provides a methodology-centered description of the status of the molecular model, simulation, and prediction of SARS-CoV-2. We discuss the traditional molecular theories, models, and methods and emergent machine learning algorithms and mathematical approaches.

Although various vaccines have been approved and in use, vaccine-breakthrough mutations have become a serious problem. Even with the promising news of new vaccines, COVID-19 as a global health crisis may still last for years before it is fully stopped globally.

The research on SARS-CoV-2 will also last for many years. It will take researchers many more years to fully understand the molecular mechanism of coronaviruses, such as RNA proofreading, virus–host cell interactions, antibody–antigen interactions, protein–protein interactions, protein–drug interactions, viral regulation of host cell functions, and immune response. Even if we could control the transmission of SARS-CoV-2 in the future, newly emergent coronaviruses may still cause similar pandemic outbreaks. Therefore, the coronaviral studies will continue even after the current pandemic is fully under control.

Currently, epidemiologists, virologists, biologists, medical scientists, pharmacists, pharmacologists, chemists, biophysicists, mathematicians, computer scientists, and many others are called to investigate various aspects of COVID-19 and SARS-CoV-2. This trend of a joint effort on COVID-19 investigations will continue beyond the present pandemic.

The urgent need for the molecular mechanistic understanding of SARS-CoV-2 and COVID-19 will further stimulate the development of computational biophysical, artificial intelligence, and advanced mathematical methods. The theoretical, computational, and mathematical communities will benefit from this endeavor against the pandemic.

The year 2020 has witnessed the birth of human mRNA vaccines for the first time—a remarkable accomplishment in science and technology. Although there are more dark days ahead of us, humanity will prevail in a post-COVID-19 world. Science will emerge stronger against all pathogens and diseases in the future.

Biographies

Kaifu Gao obtained his Ph.D. degree in physical chemistry from the Chinese Academy of Sciences and completed his postdoctoral studies at the University of California, San Diego, under the guidance of Prof. Michael Gilson. His Ph.D. and postdoctoral studies focused on molecular dynamics (MD) simulations of protein conformational transitions and binding free energy calculations. Now, he is a Research Associate in Prof. Guo-Wei Wei’s group at Michigan State University. His current research concerns the application of deep learning to biological science and drug discovery, especially automated druglike compound generation and drug property prediction. His deep learning models have already been applied to the design and screening of SARS-CoV-2 Mpro inhibitors.

Rui Wang received her B.S. degree in mathematics from Xian Jiaotong University in 2017. Later, she joined Dr. Guo-Wei Wei’s group at Michigan State University. Currently, she is a fifth-year Ph.D. candidate in the Department of Mathematics, and she expects to finish her Ph.D. study in July 2022. Her methodological research focuses on developing mathematical tools for the descriptive and predictive modeling of biomolecules. She has also worked on genomics analysis and mathematical modeling of infectious diseases. She is highly interested in integrating AI, mathematics, genomics, biophysics, bioinformatics, and experimental data to tackle challenges in biological sciences, human diseases, and infectious pathogens.

Jiahui Chen received his Ph.D. degree in computational and applied mathematics from Southern Methodist University, where his research focused on the implementation of mathematical methods for biophysics, including the Poisson–Boltzmann equation, boundary element method, Treecode method, fast multipole method, and parallel computing. After graduation, he joined the Department of Mathematics at Michigan State University as a Research Associate in Prof. Guo-Wei Wei’s group. His current research focuses on topological and geometrical data analysis and machine learning algorithms with their modeling of and application to biomolecules. His model studies were applied to the prediction of mutation-induced binding free energy changes of the SARS-CoV-2 spike protein binding to ACE2 and antibodies.

Limei Cheng is an accomplished scientist with 15+ years of experience in developing novel algorithms, modeling, and simulations for healthcare applications such as quantitative systems pharmacology (QSP). She is currently the QSP-CV Lead of Clinical Pharmacology and Pharmacometrics at (CP&P) at Bristol Myers Squibb (BMS) in New Jersey. In her position, Dr. Cheng uses model-based QSP approaches to integrate clinical and nonclinical data in a quantitative and mechanistic way to generate actionable predictions. Dr. Cheng has numerous recent presentations and publications demonstrating the utility of Systems Pharmacology in Pharmaceutical R&D. Dr. Cheng earned a B.E. in Biomedical Electronic Engineering from Xi’an Jiao Tong University, an M.S. in Electrical Engineering from the University of California, Los Angeles, and a Ph.D. in Biomedical Engineering from the University of Southern California. Her key areas of interest include QSP modeling, virtual population simulation, novel algorithm development, and personalized precision medicine.

Jaclyn Frishcosy is an undergraduate student from the Professorial Assistantship Program at Michigan State University. She will be graduating in May 2022 with a B.S. degree in Data Science. She will then start work at Epic Careers in Verona, Wisconsin.

Yuta Huzumi received his B.S. in Applied Mathematics in 2018 from Case Western Reserve University, where he worked under the mentorship of Dr. Weihong Guo and Dr. Michael Hinczewski. He is currently a Ph.D. candidate at Michigan State University under Dr. Guo-Wei Wei, where he is studying machine learning and its application towards biological science. His current research concerns the dimensional reduction of high-dimensional biological data, such as mutation and gene expression datasets.

Yuchi Qiu obtained his Ph.D. degree in mathematics from the University of California, Irvine. Currently, he is a research associate in the Department of Mathematics at Michigan State University. Now, he uses topological data analysis and machine learning models to study protein mutations and evolutions.

Thomas Schluckbier was born in Schaumburg, Illinois, in 2001. He is currently pursuing B.S. degrees in Mathematics and Computer Science at Michigan State University, working under Professor Guo-Wei Wei.

Xiaoqi Wei was born in Heilongjiang, China, in 1993. He received his B.S. in Mathematics in 2015 from Jilin University and his M.S. in Mathematics in 2018 from ETH Zurich. He is currently a Ph.D. candidate in Prof. Guo-Wei Wei’s group at Michigan State University. He conducted research on using advanced mathematics to encode biological data.

Guo-Wei Wei earned his Ph.D. degree from the University of British Columbia in 1996. He was awarded a fellowship from the NSERC of Canada to pursue his postdoctoral work at the University of Houston. In 1998, he joined the faculty of the National University of Singapore and was promoted to Associate Professor in 2001. In 2002, he relocated to Michigan State University, where he is an MSU Foundation Professor of Mathematics, Electrical and Computer Engineering, and Biochemistry and Molecular Biology. His current research interests include mathematical biosciences, deep learning, drug discovery, and computational geometry, topology, and graphing. He has advised over 150 research students, postdoctoral associates, and visiting scientists. Dr. Wei has served extensively on a wide variety of national and international panels, committees, and journal editorships.

Author Contributions

^⊥ K.G. and R.W. contributed equally to this paper.

The authors declare no competing financial interest.