RLDOCK method for predicting RNA-small molecule binding modes

Abstract

RNA molecules play critical roles in cellular functions at the level of gene expression and regulation. The intricate 3D structures and the functional roles of RNAs make RNA molecules ideal targets for therapeutic drugs. The rational design of RNA-targeted drug requires accurate modeling of RNA-ligand interactions. Recently a new computational tool, RLDOCK, was developed to predict ligand binding sites and binding poses. Using an iterative multiscale sampling and search algorithm and a energy-based evaluation of ligand poses, the method enables efficient and accurate predictions for RNA-ligand interactions. Here we present a detailed illustration of the computational procedure for the practical implementation of the RLDOCK method. Using Flavin mononucleotide (FMN) docking to F. nucleatum FMN riboswitch as an example, we illustrate the computational protocol for RLDOCK-based prediction of RNA- ligand interactions. The RLDOCK software is freely accessible at http://https://github.com/Vfold-RNA/RLDOCK.

1. Introduction

RNA molecules play essential roles in cellular functions at the level of protein synthesis,¹ gene regulation,² nucleotide modification,³ and functional response to environmental changes.⁴ In particular, non-coding RNAs directly participate in tumorigenesis and neurological, cardiovascular and many other human diseases.⁵ For example, RNAs are implicated in a number of diseases such as Huntington’s disease and AIDS.

RNA molecules fold up to form complicated tertiary structures that consist of different motifs at various levels of complexity, such as stem-loop, hairpins, bulges, and pseudoknots. Highly structured regions of RNA with an array of different structural motifs can serve as an ideal receptor and druggable target for small molecules (ligands).^8-10 The drugability of RNA is particularly appreciated if the protein target lacks suitable ligand-binding pockets.

The drugability of RNA structures has inspired tremendous efforts to develop RNA-based therapeutic strategies.^{6, 7} So far, many ligands have been discovered to target RNA through various mechanisms. For example, amino- glycoside antibiotics target bacterial ribosomal RNA with high affinity and specificity to inhibit protein synthesis,^11-13 ribocils selectively bind to Flavin mononucleotide (FMN) riboswitch to terminate gene expression and subsequently inhibit further bacterial infection,¹⁴ anthraquinone derivatives target HIV transactivation response element to inhibit viral replication.¹⁵ Moreover, designed ligand-RNA aptamer complexes can potentially enhance therapeutic applications. For example, experiments indicated that a complex of a modified RNA aptamer and tetramethylrosamine (a fluorescent malachite green analogue) can regulate the cell cycle of S.cerevisiae.¹⁶ An accurate computational tool for predicting ligand-RNA interactions can greatly facilitate in vitro selection of RNA aptamers that bind to a specific ligand and ligands that bind to a specific RNA aptamer to optimize intended aptamer structures.¹⁷

Over the past decades, various experimental methods, such as X-ray crystallography,¹⁸ nuclear magnetic resonance (NMR) spectroscopy,¹⁹ and cryo-electron²⁰ microscopy, have been used to determine biologically important RNA-ligand complex structures, As of December 2020, there have been more than 400 experimentally determined RNA-ligand complexes structures deposited in the Protein Data Bank²¹ (PDB) database, of which more than 75% were discovered over the past decade. These structures have provided much needed data for understanding RNA-ligand interactions and developing structure-based discovery of drugs.

In parallel with the experimental advances in structure determination of RNA-ligand complexes, substantial efforts have been devoted to the computational modeling of RNA-ligand binding. The computational efforts can be mainly classified into two categories: data-driven and physics-based models. Machine learning as a data-driven approach has been extensively applied to the study of RNA-ligand docking. Additionally, other data-driven models, such as DrugScore^{RNA22, 23} and LigandRNA,²⁴ predict ligand binding using statistical potentials derived from the structural data of the known RNA-ligand complexes. Physics-based approaches such as DOCK6²⁵ and MORDOR,²⁶ however, employ physical energy functions for ligand-RNA interactions and predict the ligand-RNA complex by minimizing the energy. The different approaches have shown significant success for different RNA-ligand systems.²⁷ However, in general, the accuracy for an RNA-ligand model is lower than that of protein-ligand docking models, and the performance cannot meet the requirement for drug design and other applications such as virtual screening and selection of ligands and RNA aptamers for the intended aptamer-ligand complex structures.¹⁷ One of the key problems is that, compared with protein-ligand complexes, we have far less known RNA-ligand complex structures.²⁸ The insufficient number and diversity of known RNA-ligand complex structures can directly impact the reliability of both the data-driven and the physical approaches, which rely on the structural data to optimize model parameters. In recent years, as more and more RNA-ligand complexes structures are determined,²⁹ we can realistically expect continuous improvements in the accuracy of computational models for RNA-ligand binding.

Given the limited availability of known structures, physics-based approaches become an attractive alternative. For a physics model, complete sampling and accurate scoring for the binding modes (also referred to as binding poses) are two key bottlenecks.^{25, 30} To tackle these bottleneck problems, we recently developed the RLDOCK model³⁰ (http://https://github.com/Vfold-RNA/RLDOCK). The RLDOCK model has two key components. First, the model employs a novel multi-tier screening algorithm that enables an iterative exhaustive search for the binding sites and ligand conformations. Second, the scoring of the different binding modes is based on a comprehensive physics-based energy function. In this Methods paper, we focus on the detailed illustration of the computational procedure for the practical implementation of RLDOCK. As an application of the RLDOCK, we show the computational prediction for the binding mode of FMN (ligand) docking to F.nucleatum FMN riboswitch (RNA; PDB²¹ identifier: 2yie³¹).

In Fig. 1 we show the pipeline of the RLDOCK method. The main algorithm of RLDOCK has two components: the sampling of the possible ligand binding modes and the scoring of the different binding modes. In RLDOCK, sampling is guided by scoring (energy) function. Therefore, in what follows, we first describe the scoring function then introduce the sampling method.

Figure 1:

The workflow of the RLDOCK model.

2. Method

2.1. Preparation of the system

The RLDOCK model uses the 3D structure of the RNA and the chemical structure of the ligand as the input information. Depending on the flexibility of the ligand molecule, the model generates an ensemble of 3D conformers for the ligand.

1. An initial structure of a ligand can be generated from its 2D chemical information using cheminformatics tools such as Open Babel³² and OMEGA TK.³³ Staring from the initial ligand structure, we generate diverse 3D conformers for the ligand and group structurally similar ligand conformers into clusters. Here the structural similarity is measured by the root-mean-square deviation (RMSD) of the heavy atoms between the structures. On a workstation powered with an AMD Ryzen Threadripper 1950X 16-Core Processor and 64 GB RAM, we generate an ensemble of 30 diverse conformers for a ligand.

2. Using UCSF Chimera,³⁴ we prepare the input RNA and ligand structures as mol2 files. The files contain not only the atomic coordinates but also the partial charges carried by the atoms. Unlike proteins, RNAs are highly charged, therefore, electrostatic interaction is critical for ligand-RNA binding and the charge assignments are important for the model.

2.2. Scoring function

The scoring function in RLDOCK is based on the free energy change upon ligand binding to the RNA. The total free energy of the system comprises the following components: (a) the mutual van der Waals (VDW) U_lj and (b) Coulomb U_e interaction energies between ligand and RNA atoms, (c) the polar hydration energy, which is decomposed into self-polarization energy U_self of the RNA and ligand atoms and the mutual polarization energy U_pol between the different charged atoms, (d) the nonpolar hydration energy U_sa, (e) the hydrogen-bond energy U_h, and (f) the intramolecular VDW interaction energy between ligand atoms U_internal. The hydration energies are calculated based on the generalized Born approximation with the solvent-accessible surface area (GB/SA model).^35-40 For a given RNA-ligand binding mode i, the total energy score is given by the following formula; See Appendix A for a detailed illustration of each energy term.

$$\begin{gathered} S_i=c_{lj}\times \Delta U_{lj}+c_e\times \Delta U_e+c_h\times \Delta U_h+c_{sa}\times \Delta U_{sa}+c_{pol}\times \Delta U_{pol}+ \\ {c}_{self}^R\times \Delta U_{self}^R+c_{self}^L\times \Delta U_{self}^L+c_{internal}^L\times \Delta U_{internal}^L, \end{gathered}$$ ##(1)

where the weight coefficients⁴¹ c are introduced to account for the correlation between the different components, and the superscripts R and L denote the RNA and the ligand, respectively.

The evaluation of the scoring function (energy) for each sampled ligand binding mode is computationally demanding. Therefore, an effective method to speed up energy calculation is a critical ingredient in the RLDOCK model. RLDOCK uses two methods to achieve a fast energy calculation.

2.2.1. Grid-based Lennard-Jones (LJ) energy map

One of the approaches used in RLDOCK is to pre-tabulate the energy values for pairwise interactions, specifically, the VDW interaction energy, which, as shown in Appendix A, is in the form of a Lennard-Jones (LJ) potential. The basic strategy is to discretize the 3D space and pre-compute the LJ interaction energy between all the RNA atoms and a ligand atom placed at a grid site. In practice, we discretized the space with a grid spacing of 0.2Å and place common atom types (C, N, O, P, S, etc) at each grid. The LJ energy values for the different atom types on each grid site gives the grid energy map. For a given binding mode, the mutual VDW energy can be quickly evaluated by summing up the grid energies over the grids occupied by the ligand atoms.

2.2.2. A simplified scoring function

The solvent-accessible surface area (SASA) and the Born radii of the atoms are sensitive to the structure of the RNA-ligand complex and hence need to be computed/updated for each ligand-RNA binding mode generated in the sampling process. Therefore, the time-consuming SASA and Born radii calculations become the most time-demanding steps in the whole computational process. In RLDOCK, the problem is resolved by applying an initial crude screening process where the following simplified and fast calculations for the SASA and Born radii can be used.

1. The calculation of SASA, which is determined by the molecular shape, is intrinsically a many-body problem. As an approximation, we simply add up the SASA changes of each pair of ligand-RNA atoms and ignore the existence of other atoms in the calculation of each ligand-RNA atom pair.

2. We neglect the ligand docking-induced changes in the Born radii and the self-polarization energy $\Delta U_{self}^R$ for RNA.

3. We use the VDW radii to approximate the Born radii of the bound ligand atoms.

The above approximations can lead to an increase in the computational efficiency of thousands of folds.

2.2.3. Method for parameter optimization

The scoring function contains 8 weight coefficients. We determine the coefficients by minimizing the difference between the predicted and the experimentally determined binding modes for a training set. Our training set contains 30 RNA-ligand complexes deposited in the PDB; See Appendix B. The 30 cases are selected to cover a diverse range of different RNA and ligand types. In the training set, the RNA sizes range from 516 to 2337 heavy atoms and the ligand sizes vary from 10 to 52 heavy atoms, with an average of 1318 and 25 atoms for RNA and ligand, respectively.

For each of the 30 ligand-RNA complexes, a ligand binding mode ensemble is generated. The coordinated descent method⁴² is applied to optimize the weight coefficients. Repeated application of the coordinate descent method results in multiple sets of putative weight coefficients, and the set that corresponds to the minimum RMSD between the predicted and the experimentally determined ligand pose are selected: c_lj = 3.30, c_e = 1.32, c_h = 1.32, c_sa = 1.26(0.30), c_pol = 1.38(0.36), $c_{self}^R=4.98(0.00)$, $c_{self}^L=2.78(0.58)$, $c_{internal}^R=0.66$. The values in the parentheses refer to the parameter used for the simplified scoring function above.

2.3. Sampling and scoring ligand-RNA binding modes

We use four variables, R, L, A, and O, to describe a ligand pose. Here the ligand atom A (referred to as the anchor atom) is fixed at position R (referred to as the anchor site), and the ligand pose is generated by the 3D rotation O of the ligand conformer L about A. As shown below, RLDOCK uses a multi-tier sieving process to search for the ligand binding pose efficiently.

2.3.1. Global sampling of the anchor sites

1. We configure the RNA structure in a box whose six boundaries are 3Å away from the outermost atoms of the RNA, and discretize the box space with a simple cubic lattice of grid size 0.5Å.

2. We search for all the possible anchor sites that involve no steric clashes with a ligand or an RNA atom and reside inside a pocket of the RNA structure.

   1. To probe the steric clash, we place a virtual sphere of radius 2 Å the grid sites and let the sphere traverse the RNA surface to detect the atomic overlap. The clash-free grid sites are kept as the viable anchor sites R.
   2. To identify the RNA pockets, on each anchor site selected above, we move the sphere 6Å along the six directions of the egocentric coordinates: left and right; front and back; up and down. If the test probe meets any RNA atom in at least five directions, the anchor site is considered to be inside a pocket and would be forwarded to the next step; see Fig.2A.

Figure 2:

(A) Visualization of the binding sites for F.nucleatum FMN riboswitch (PDB²¹ identifier: 2yie³¹). Dots in yellow denote the possible binding sites based on consideration of steric clash. Dots in red denotes the top-300 selected candidate binding sites based on LJ energy. (B) is an enlarged view of a region surrounded by a red rectangle of (A). The crystal structure of the ligand is displayed with green as a reference.

2.3.2. Sampling binding modes based on the ligand-RNA van der Waals interactions

In this step, we use the RNA-ligand VDW interaction energy (LJ potential) to sample and select plausible poses. We note that this step is primarily a shape-based selection as the LJ potential is a soft potential for the volume exclusion.

1. For each anchor site R selected in the previous step, we enumerate all the possible L, A, and O, and find the minimum LJ energy LJ₁(R). We keep 300 anchor sites R from the top-300 lowest LJ₁(R) energies.

2. For each R site selected above, for each given ligand conformer L, we sample all the possible A and O, and find the minimum LJ energy LJ₂(R, L). We keep the 3 ligand conformers L from the top-3 lowest LJ₂(R, L) energies.

3. For each (R, L) pair selected above, for each ligand atom A as the anchor atom, we sample all the possible rotations O about A and find the minimum LJ energy LJ₃(R, L, A). We keep the anchor atoms A from the top-3 lowest LJ₃(R, L, A).

Here we rotate the ligand around 500 uniformly orientated axes with a 10° increment in the rotation angle. The rotations result in a total of 36×500 = 18000 ligand orientations. In summary, the LJ potential-guided sampling leads to a total of 300 × 3 × 3 × 18000 ~ 5 × 10⁷ binding modes.

2.3.3. Scoring binding modes based on the full ligand-RNA interaction energy.

As shown below, we use a two-step approach to efficiently score the ~ 5 × 10⁷ binding modes.

1. Initial coarse-grained scoring of the ligand orientations. For each of the 300 anchor sites R selected above, using the aforementioned simplified energy function, we quickly select the top-10 poses. This step leads to a pool of 300 × 10 = 3000 potential binding modes; See Fig. 3A.

2. Further refinement using the rigorous energy function. We re-score the 3000 binding modes using the original rigorous energy function; See Fig. 3B.

Figure 3:

The intermediate results obtained from the scoring module for F.nucleatum FMN riboswitch (PDB²¹ identifier: 2yie³¹). (A) The top-3 poses scored by the simplified scoring function. (B) The top-3 poses scored by the complete, rigorous scoring function. (C) The percentage of the near-native poses among the top-100 poses predicted by the simplified and the complete scoring functions, respectively. Here “near-native” means the RMSD is less than 2.0 Å with respect of the experimentally determined pose. (D) The correlation between the final score given by RLDOCK and the RMSD with respect of crystal structure for the predicted poses after clustering.

2.3.4. Clustering of the binding modes.

Starting from the top-ranked binding mode, we cluster the ligand poses according to the structural similarity. We use 2 Å as the RMSD cutoff a cluster. The top-scored pose in each cluster is chosen to represent the cluster. This step leads to a list of ranked binding modes (ligand poses). The top ligand pose is output as a mol2 file for visualization.

3. Application of the RLDOCK model

As an illustration, we apply RLDOCK to predict the FMN (ligand) pose in the FMN-F. nucleatum FMN riboswitch (RNA) complex. The ligand FMN contains 31 heavy atoms. We prepare an input ensemble of 30 conformers for the ligand. The global sampling procedure predicts 2205 anchor sites. Anchoring each of the 31 heavy atoms to the 2205 sites for each of the 30 ligand conformers results in a total of 2205 × 30 × 31 × 18000 ~ 3.7 × 10¹⁰ binding modes. Subsequent LJ energy-based sampling and ranking of above binding modes leads to a list of top-300 anchor sites. As shown in Fig. 2, the selected binding sites are indeed in the pocket region.

For each of the 300 selected anchor sites, the simplified scoring function selects the top-10 binding poses from the ~ 5 × 10⁷ candidates; See Fig. 3A. Subsequent re-scoring using the rigorous energy function gives the re-ranked 3000 binding modes. Finally, the clustering procedure leads to the final 527 ranked binding modes; See Fig. 3B. The predicted top-ranked FMN ligand pose is within 2.0Å (RMSD) from the crystal structure; See Fig. 3C and D.

The executable file for RLDOCK is available at https://github.com/Vfold-RNA/RLDOCK. Here are some tips for a successful implementation of RLDOCK.

• For a larger RNA (atoms ~ 3 × 10⁴) RLDOCK requires a computing power with larger RAM (≥ 128G) and more CPU threads (≥ 32).

• For a large flexible ligand (rotatable bonds > 12), we suggest generating multiple ensembles of ligand conformers instead of a single large ensemble.

4. Conclusion

Using a novel multi-scale method for global sampling and energy-guided search for ligand binding pose, the RLDOCK method can successfully predict RNA-ligand near-native binding modes; See Tables 1 and 2. As shown in Table 1, RLDOCK can successfully predict the binding mode within the top-10 poses with a success rate of 70% for all the three data sets tested. For the training set and Test set 2, the top-ranked binding mode can give successful hits for more than 50% of the cases. For Test set 1, which contains 200 RNA-ligand cases, the success rate of the top-ranked pose is less than 40%, suggesting the need for further refinement of the method.

Table 1:

The success rate of RLDOCK for different data sets^a

Data set	Number of cases	Top 1	Top 3	Top 10
Training set30	30	50.0%	70.0%	86.7%
Test set 130	200	39.0%	56.5%	72.5%
Test set 224b	38	55.3%	60.5%	71.0%

^aIn this table, a prediction is successful if the best RMSD of the top binding modes is less than 2.0Å with respect to the native binding mode.

^bThe four ribosomal RNA cases are excluded from the 42 RNA-ligand complexes.

Table 2:

The success rate of Test set 2²⁴ for the different docking models^a

Docking model	Top 1	Top 3
RLDOCK30	55.3%	60.5%
DOCK625b	36.8%	44.7%
rDock43c	28.9%	47.4%
rDock_solv43c	39.5%	55.3%
AutoDock Vina44d	31.6%	44.7%

^aA prediction is successful if the best RMSD of the top binding modes is less than 2.0Å with respect to the native binding mode.

^bThe data is obtained from Ref. 24.

^cFor rDock and rDock solv, the cavity is defined using the reference ligand method, the radius of outer sphere is set as 5, and a final 50 runs-per-ligand rDock job is performed.

^dThe centroid of the native ligand binding pose is set as the center of the docking box. The size of the docking box is set as 20Å ×20Å ×20Å

As shown in Table 2, compared with other docking models, RLDOCK has a better performance on a validation set of 38 RNA-ligand complexes. The promising performance of RLDOCK indicates that it may serve as a new valuable tool for predicting RNA-ligand interactions in the discovery of lead compounds as RNA-targeted drugs and the selection of ligand-bound RNA aptamers.

However, the applicability of the RLDOCK method is challenging for large RNAs and ligands. For systems with a large RNA such as the ribosomal RNA (~ 5 × 10⁴ atoms) or a large flexible ligand (rotatable bonds > 12), The time-consuming sampling of the binding modes causes prohibitive low computational efficiency of the method. Further improvement in the computational efficiency is possible. For example, the rDock model applies the genetic algorithm to generate initial ligand conformers and refine the conformer ensemble “on-the-fly” using Monte Carlo simulation.⁴³ Another major challenge for the application of the RLDOCK model stems from RNA conformational flexibility. Unlike a protein, an RNA molecule often folds into multiple conformers with comparable stabilities. RNA conformational multiplicity and the resultant flexibility of RNA binding pockets can affect ligand binding affinity. The conformational heterogeneity can also negatively influence the crystal packing of RNA structures and challenge the structure determination for ligand-RNA complexes. The current version of RLDOCK assumes a rigid RNA structure and cannot treat RNA conformational changes upon ligand binding. The RLDOCK method here, combined with an RNA folding model, however, may provide a promising new method to treat ligand-induced RNA conformational changes.

Highlights.

• RLDOCK is a novel computational tool that predicts RNA-ligand interactions using a multiscale sampling method.

• A global search algorithm enables the complete sampling of ligand binding sites.

• An energy-guided coarse-grained sampling method facilitates a fast search for ligand conformations and orientations.

• A physics-based energy function successfully scores and ranks different binding modes.

• A two-step scoring approach leads to a substantial speed-up in the computational prediction of the ligand-binding mode.

• RLDOCK is a valuable new tool for RNA-targeted drug discovery.

Appendix

Appendix A. Energy terms in the scoring function

(a). VDW interaction energy U_lj

The Lennard-Jones (LJ) potential ΔU_lj is applied to represent VDW interaction:

$$\Delta U_{lj}=\sum _r\sum _l\left[{\left(\frac{{\sigma }_{rl}}{r_{rl}}\right)}^{12}-{\left(\frac{{\sigma }_{rl}}{r_{rl}}\right)}^6\right].$$ #(2.)

Here the subscripts r and l denote the atom of RNA and ligand, respectively. r_rl represents the distance between the two atoms and σ_rl = 0.8(R_r + R_l) is the equilibrium distance, where R_r (R_l) is the radii of RNA (ligand) atom. A cut-off distance r_cut = 2.5(R_r + R_l) is applied in the LJ potential calculation.

(b). Coulomb interaction U_e

The electrostatic interaction ΔU_e is the Coulomb interaction between RNA and ligand atoms:

$$\Delta U_e=\sum _r\sum _l\frac{Z_r{Z}_l{e}^2}{{\epsilon }_c{r}_{rl}}.$$ #(3.)

Here Z_r and Z_l denote the electric charges of the atoms r in RNA and l in ligand, respectively, r_rl is the distance between atoms, e is the electronic charge, ε_c (=20 in our calculation) is the dielectric constant of the RNA-ligand complex.

(c). Polar hydration energy

Polar hydration interaction is decomposed into the mutual polarization energy U_pol and self-polarization energy U_self of the RNA and ligand atoms.

Mutual-polarization energy

The mutual polarization energy change ΔU_pol is obtained as below:

$$\Delta U_{pol}=U_{pol}^{\text{complex}}-\left(U_{pol}^{\text{RNA}}+U_{pol}^{\text{ligand}}\right),$$ #(4.)

where $U_{pol}^{\text{complex}}$, $U_{pol}^{\text{RNA}}$, and $U_{pol}^{\text{ligand}}$ are the mutual polarization of the complex, the RNA alone, and the ligand alone, respectively. We estimate the three mutual polarization energies from the GB model:^35-40

$$U_{pol}=\frac{1}{2}\left(\frac{1}{{\epsilon }_w}-\frac{1}{{\epsilon }_c}\right)\sum _{ij}\frac{Z_i{Z}_j{e}^2}{\sqrt{r_{ij}^2+B_i{B}_j\text{exp}\left(-\frac{r_{ij}^2}{4B_i{B}_j}\right)}},$$ #(5.)

where ε_w (=78) denotes the dielectric constant of water. The dielectric constant ε_c is assumed to be the same for the bound and the unbound RNA and ligand. The subscripts i and j (i ≠ j) represent respective molecule (complex, RNA alone, or ligand alone). r_ij denotes the distance between these two atoms. B_i and B_j are the Born radii of atoms i and j.

For an atom i in the RNA-ligand complex, RNA alone, or ligand alone, its Born radius is calculated as follows:

$$\frac{1}{B_i}=\frac{1}{a_i}{}_{{}_{-}}\frac{1}{2}\sum _j{A}_j$$ #(6.)

$$\text{and}\frac{1}{A_i}=\left(\frac{1}{L_{ij}}-\frac{1}{U_{ij}}\right)+\left(\frac{S_j^2{a}_j^2}{4r_{ij}}-\frac{r_{ij}}{4}\right)\left(\frac{1}{L_{ij}^2}-\frac{1}{U_{ij}^2}\right)+\frac{1}{2r_{ij}}\text{ln}\frac{L_{ij}}{U_{ij}},$$ #(7.)

$$\text{where}{L}_{ij}=\{\begin{array}{cc}1\hfill & if{a}_i\ge r_{ij}+S_j{a}_j\hfill \\ \max (a_i,r_{ij}-S_j{a}_j)\hfill & if{a}_i<r_{ij}+S_j{a}_j\hfill \end{array}\phantom{\}}$$ #(8)

$$\text{and}{U}_{ij}=\{\begin{array}{cc}1\hfill & if{a}_i\ge r_{ij}+S_j{a}_j\hfill \\ r_{ij}+S_j{a}_j\hfill & if{a}_i<r_{ij}+S_j{a}_j\hfill \end{array}\phantom{\}}$$ #(9)

Here a_i and a_j denote the VDW radii of atoms i and j, respectively. r_ij is the distance between atoms i and j. S_j is the structural scaling factor and is equal to 1 if there is no overlap between the atoms. In general, S_j < 1 in the RNA-ligand complex, RNA alone, or ligand alone.

Self-polarization energy

The self-polarization energies $\Delta U_{self}^R$ of the RNA and $\Delta U_{self}^L$ of the ligand are calculated as the following:

$$\begin{gathered} \Delta U_{self}^R=\left(\frac{1}{{\epsilon }_w}-\frac{1}{{\epsilon }_c}\right){\sum }_r\left(\frac{1}{B_r^a}-\frac{1}{B_r^b}\right)Z_r^2{e}^2 \\ \Delta U_{self}^L=\left(\frac{1}{{\epsilon }_w}-\frac{1}{{\epsilon }_c}\right){\sum }_l\left(\frac{1}{B_l^a}-\frac{1}{B_l^b}\right)Z_l^2{e}^2, \end{gathered}$$ #(10.)

Here $B_{r(orl)}^b$ and $B_{r(orl)}^a$ denote the Born radii of atom r(or l) in the RNA (or ligand) before and after the ligand-RNA docking, respectively.

(d). Nonpolar hydration energy U_sa

The nonpolar hydration energy ΔU_sa is evaluated according to the change in the solvent-accessible surface area (SASA):^45-47

$$\Delta U_{sa}=\sigma \times \Delta SA,$$ #(11)

where ΔSA is the total SASA change before and after the ligand docking.

$$\Delta S{A}_{complete}=S{A}_{complex}-(S{A}_{RNA}+S{A}_{ligand}),$$ #(12)

Here SA_complex denotes the SASA of the RNA-ligand complex for the given pose (R, L, A, O). SA_RNA and SA_ligand are the SASA of the RNA alone and ligand alone, respectively.

In the simplified scoring function, the sum of the SASA change for each ligand-RNA atom pair gives the approximate total SASA change:

$$\Delta S{A}_{simply}=\sum _r\sum _l\Delta S{A}_{rl},$$ #(13.)

where ΔSA_rl denotes the SASA changes of the RNA atom r and the ligand atom l upon binding.

We choose σ = 0.0054 kcal/(mol · Ų) for the empirical atomic solvation parameter σ.⁴⁸

(e). Hydrogen-bond interaction energy U_h

The hydrogen-bond interaction energy ΔU_h between the RNA and ligand is calculated as:

$$\Delta U_h=\sum _r\sum _l{u}_h(r_{rl}),$$ #(14)

where u_h(r_rl) is the hydrogen-bond energy of an RNA-ligand atom pair. We evaluate the hydrogen-bond energy via an empirical formula:⁴⁹

$$u_h(r_{rl})=\{\begin{array}{cc}-1\hfill & r_{rl}<r_{min}\hfill \\ -1+\frac{r_{rl}-r_{min}}{r_{max}-r_{min}}\hfill & r_{min}<r_{rl}<r_{max}\hfill \\ 0\hfill & r_{rl}\ge r_{max}\hfill \end{array}\phantom{\}}.$$ #(15.)

Here r_min = 0.8(R_r + R_l and r_max = 1.3(R_r + R_l).

(f). Ligand intramolecular VDW interaction energy U_internal

We also use LJ potential to evaluate ligand intramolecular VDW interaction:

$$\Delta U_{internal}^L={\sum }_i^L{\sum }_{j(j\ne i)}^L\left[{\left(\frac{{\sigma }_{ij}}{r_{ij}}\right)}^{12}-{\left(\frac{{\sigma }_{ij}}{r_{ij}}\right)}^6\right].$$ #(16.)

Here i and j denote a non-bonded heavy atom pair in the ligand, r_ij is the distance between the two atoms, and σ_ij = 0.8(R_i + R_j) is the equilibrium distance, where R_i and R_j are the radii of atom i and j, respectively. A cut-off distance r_cut = 2.5(R_i + R_j) is applied here for the LJ potential.

The VDW radii and the structural scaling factors for various atom types can be obtained from http://www.rbvi.ucsf.edu/chimera/current/docs/UsersGuide/midas/vdwtables.html and Ref. 39, respectively.

Appendix B. List of data sets

List of PDB IDs for RNA-ligand complexes in the data sets

Training set
1AKX	1ET4	1F27	1LVJ	1PBR	1J8G
1KOC	1QD3	1Y26	2ET4	2FD0	2BE0
2BEE	2F4T	2KTZ	2O3X	2XO1	3DIX
3FO4	3GES	3SUH	3SUX	3SKL	4LVW
4LW0	4FEJ	4FEO	4NYB	4KQY	5C45
Test set 1
1AJU	1AM0	1ARJ	1BYJ	1DDY	1EHT
1EI2	1EVV	1F1T	1FMN	1FUF	1FYP
1I7J	1I9V	1J7T	1KOD	1LC4	1MWL
1NBK	1NEM	1NTA	1NTB	1O15	1O9M
1Q8N	1RAW	1TN1	1TN2	1TOB	1UTS
1UUD	1UUI	1XPF	1YKV	1YLS	1YRJ
1ZZ5	292D	2A04	2AU4	2B57	2EES
2EET	2EEU	2EEW	2ESI	2ESJ	2ET3
2ET5	2ET8	2F4S	2F4U	2FCX	2FCY
2FCZ	2G5K	2G5Q	2G9C	2GCV	2GDI
2GIS	2GQ5	2HOP	2JUK	2KD4	2KGP
2KU0	2KX8	2KXM	2L1V	2L8H	2MIY
2MXS	2N0J	2OE5	2OE8	2PWT	2QWY
2TOB	2W89	2XNW	2XNZ	2XO0	2YDH
2YIE	3B4B	3B4C	3C3Z	3C44	3C5D
3C7R	3D0U	3D2X	3DIG	3DIL	3DIM
3DIO	3DIY	3DIZ	3DJ0	3DJ2	3DS7
3DVV	3DVZ	3DW4	3DW6	3E5C	3E5E
3E5F	3F2Q	3F2T	3F4G	3F4H	3FO6
3FU2	3G4M	3GAO	3GCA	3GER	3GOG
3GOT	3GX2	3GX3	3GX5	3GX6	3GX7
3IQN	3IQR	3IRW	3K1V	3LA5	3NPN
3NPQ	3OWI	3OWZ	3Q3Z	3Q50	3RKF
3S4P	3SD3	3SKI	3SKR	3SKT	3SKW
3SKZ	3SLM	3SLQ	3TD1	3TZR	3WRU
4AOB	4B5R	4ERL	4F8U	4F8V	4FE5
4FEL	4FEN	4FEP	4FRG	4GPW	4GPX
4GPY	4JF2	4K32	4L81	4LVV	4LVX
4LVY	4LVZ	4LX5	4LX6	4NYA	4NYC
4NYD	4NYG	4OQU	4P20	4PDQ	4QK8
4QK9	4QKA	4QLM	4QLN	4RZD	4TS0
4TS2	4TZX	4TZY	4WCQ	4WCR	4XNR
4YAZ	4YB0	4YB1	4ZC7	5C7U	5C7W
5NDH	5NEF
Test set 2
1AJU	1AM0	1BYJ	1EHT	1EI2	1F1T
1F27	1FMN	1FYP	1J7T	1KOC	1KOD
1MWL	1NBK	1NEM	1PBR	1Q8N	1TOB
1UTS	1UUD	1UUI	1XPF	1Y26	2BE0
2BEE	2ET8	2F4U	2FCZ	2FD0	2GDI
2O3X	2OE5	2PWT	2TOB	3D2X	3GX2
3SUX	4P20