Multiscale Multiphysics and Multidomain Models I: Basic Theory

Abstract

This work extends our earlier two-domain formulation of a differential geometry based multiscale paradigm into a multidomain theory, which endows us the ability to simultaneously accommodate multiphysical descriptions of aqueous chemical, physical and biological systems, such as fuel cells, solar cells, nanofluidics, ion channels, viruses, RNA polymerases, molecular motors and large macromolecular complexes. The essential idea is to make use of the differential geometry theory of surfaces as a natural means to geometrically separate the macroscopic domain of solvent from the microscopic domain of solute, and dynamically couple continuum and discrete descriptions. Our main strategy is to construct energy functionals to put on an equal footing of multiphysics, including polar (i.e., electrostatic) solvation, nonpolar solvation, chemical potential, quantum mechanics, fluid mechanics, molecular mechanics, coarse grained dynamics and elastic dynamics. The variational principle is applied to the energy functionals to derive desirable governing equations, such as multidomain Laplace-Beltrami (LB) equations for macromolecular morphologies, multidomain Poisson-Boltzmann (PB) equation or Poisson equation for electrostatic potential, generalized Nernst-Planck (NP) equations for the dynamics of charged solvent species, generalized Navier-Stokes (NS) equation for fluid dynamics, generalized Newton's equations for molecular dynamics (MD) or coarse-grained dynamics and equation of motion for elastic dynamics. Unlike the classical PB equation, our PB equation is an integral-differential equation due to solvent-solute interactions. To illustrate the proposed formalism, we have explicitly constructed three models, a multidomain solvation model, a multidomain charge transport model and a multidomain chemo-electro-fluid-MD-elastic model. Each solute domain is equipped with distinct surface tension, pressure, dielectric function, and charge density distribution. In addition to long-range Coulombic interactions, various non-electrostatic solvent-solute interactions are considered in the present modeling. We demonstrate the consistency between the non-equilibrium charge transport model and the equilibrium solvation model by showing the systematical reduction of the former to the latter at equilibrium. This paper also offers a brief review of the field.

I Introduction

An important trend in contemporary life sciences is that with the availability of modern biotechnologies, traditional disciplines, such as physiology, plant biology, neuroscience etc, are undergoing a fundamental transition from macroscopic phenomenological ones into molecular based biosciences. In parallel with this development, a major feature of life sciences in the 21st Century is their transformation from phenomenological and descriptive disciplines to quantitative and predictive ones. Ample opportunities have emerged for mathematically driven advances in biological research. Experimental exploration of self-organizing molecular biological systems, such as HIV viruses, molecular motors, ribosomes, RNA polymerase and proteins in Alzheimer's disease, are examples of dominating driving forces in scientific discovery and innovation in the past few decades. However, the emergence of excessive complexity in self-organizing biological systems poses fundamental challenges to their quantitative description, because of their excessively high dimensionality and the complexity of processes involved. Mathematical approaches that are able to efficiently reduce the number of degrees of freedom, and model complex biological systems, are becoming increasingly popular in molecular biosciences. Multiscale modeling, intrinsic manifold extraction, dimensionality reduction and machine learning techniques are introduced to reduce the complexity of biomolecular systems while maintaining an essential and adequate description of the biomolecular observables of interest.

Recently, multiscale and multiphysics modeling and computation have become some of the most powerful approaches in chemistry, physics, biology, nanoscience and engineering.^{1,29,56,95,146,148,162,164,166,180} Most of these approaches are aimed at the understanding of complex systems, such as complex fluids, turbulent flows, micro-fluidics,^29,180 soft material, solids, interface problems, structure and fluid interactions, wave propagation in random media, stochastic processes, deoxyribonucleic acid (DNA) nanowires, molecular junctions, solar cells, fuel cells, battery cells, molecular switches, nanotubes, field effect transistors, nanofibers, thin films, ion channels, ATPases, neuron synapses, and self-similar problems. A main purpose of developing multiscale models is to maintain efficient descriptions of key physical measurements in multiphysical problems while avoid detailed descriptions of some physical components that do not significantly contribute to the behavior of physical observations of interest, so that the resulting computations are feasible with the current computer capability. In the past two decades, a large variety of multiscale models and algorithms has been proposed. Among them, many multiscale models, such as Boltzmann theory kinetic theory,^{141–143,158} describe multiphysics with multiple governing equations, such as microscopic laws for atoms and molecules at microscopic settings, and transport equations for the conservation of mass, momentum, and energy at macroscopic settings. Multiscale approaches that bridge macro-micro scales and couple macro-micro domains are commonly used.¹⁴⁸ An interesting class of multiscale models has their origin from earlier wavelet multiresolution analysis. Yet the other class of multiscale approaches is heterogeneous multiscale models.⁵⁶ Multiscale coarse-grained methods¹⁰² and quantum mechanical/molecular mechanical (QM/MM) approaches^46,65,77,78 are developed for bimolecular systems. Multigrid methods which extract and utilize information at different time or spatial scales governed by one or a few equations can also be regarded as multiscale approaches. An elegant example is the homogenization method.

A new class of multiscale models, differential geometry based multiscale approaches, has been introduced by the present author¹⁶⁴ for large chemical, physical and biological systems and nano devices, such as fuel cells, solar cells, nanofluidics, ion channels, molecular motors, subcellular organelles, and virus complexes. A common feature in these systems is that they have aqueous environment. We seek multiscale models which provide microscopic descriptions of chemical and biological subjects of interest, while maintain macroscopic descriptions of the aqueous environment, so as to significantly reduce the number of degrees of freedom of the original complex system. To this end, we make use of the differential geometry theory of surfaces and the geometric measure theory as a natural means to separate macroscopic and microscopic domains.^{8–10,163,165} A variational strategy developed in our earlier work for the minimal molecular surfaces^9,10 is generalized to cast our multiscale modeling of multiphysics in a self-consistent manner. Our differential geometry based multiscale paradigm provides variational formulations to a number of physical phenomena, including polar and nonpolar solvations, molecular dynamics, fluid dynamics, electrokinetics, electrohydrodynamics, electrophoresis, and elastic dynamics. By using the Euler-Lagrange variation, coupled Laplace-Beltrami equation and Poisson-Boltzmann equation are obtained for solvation. For non-equilibrium systems, additional generalized Poisson-Nernst-Planck equations and/or Navier-Stokes equations are derived for the charged species. Multiscale Newton's equations are obtained to allow the molecular mechanics (MM) description of biomolecular systems. Finally, for excessively large chemical and biological systems, the linear elastic dynamics is employed to replace expensive MD simulations and further reduce the dimensionality.

Differential geometry based multiscale models have been intensively validated in the past three years.^{21,23,31–34,166} The first series of efforts was given to the multiscale solvation analysis.^{31–34,79,149,179} We implement differential geometry based solvation models in both the Eulerian formulation³¹ and the Lagrangian formulation³² for real-world problems. We analyze the equivalence of these formulations for both small and large molecules. An immediate consequence of this development is a significant reduction in the number of free parameters that users must “fit” or adjust in applications to real-world systems. The surfaces generated by our methods are free of geometric singularities, which commonly occur in conventional molecular surfaces and cause computational instabilities.^43,132 Very good consistency between our theoretical predictions and experimental solvation energies has been found for tens of compounds.^31,32 The robustness of the approach was confirmed.^149,176 Our differential geometry based nonpolar model gives rise to some of the best predictions of nonpolar solvation energies.³⁴ To further improve the accuracy of our multiscale models, we have introduced the quantum density functional theory (DFT) description of solute molecules,³³ which significantly improves the predictive power of our earlier solvation models. In a series of parallel efforts, we have developed differential geometry based multi-scale quantum models for proton transport.^21,23 Proton transport underpins the molecular mechanism in biological energy transduction, sensory systems and reproduction of influenza A viruses.²⁷ Due to significant quantum effects, proton permeation across membrane proteins needs to be treated by quantum mechanical means.^116,125 In our multiscale and multiphysics model, we have constructed a new density functional theory based on the Boltzmann statistics, rather than the Fermi-Dirac statistics, to describe proton dynamics quantum mechanically, while implicitly treat numerous solvent molecules as a dielectric continuum. The membrane proteins are described in the atomistic detail to enable the gating effect. An interesting aspect is that the densities of all the other ions in the solvent are treated with the Boltzmann distributions following an approach introduced in our earlier work,¹⁷⁸ which was independently confirmed by using Monte Carlo simulations.⁹⁴ We have explored non-electrostatic van der Waals interactions among all the ions, and between ions and proteins, including size (steric) effects.²³ Our method provides excellent predictions of experimental current-voltage curves. Most recently, we have developed differential geometry based multiscale models for heterogeneous chemical and biological systems that are far from equilibrium.¹⁶⁶ In this new theory, the consistency between the equilibrium model and the non-equilibrium is established at equilibrium as demonstrated both theoretically and numerically. Our main focus in such a development lies in the understanding of ion channel gating mechanism in membrane proteins. With our variational multiscale framework, we consider both nonpolar and polar (electrostatic) solvation effects, chemical potential and the associated free energy, continuous modeling of solvent species and discrete representation of membrane proteins. Once again, we found very good agreements between our model predictions and experimental measurements.¹⁶⁶

The objective of the present work is to extend our earlier differential geometry based multiscale and multiphysics models into multiscale, multiphysics and multidomain models. Indeed, in our previous formulations, only one solute domain is considered, although it can be described either by using a set of discrete point charges, molecular dynamics, quantum mechanics or with elastic dynamics. In this work, we consider arbitrarily many solute domains so as to allow simultaneously multiphysical descriptions for different macromolecular domains and complex nano-bio devices. For example, in solvation analysis, our multidomain models assign different part of macromolecular complexes with different dielectric functions.^28,127 This approach is potentially useful to the theoretical analysis of metalloproteins, which are crucial to many cellular different functions in cells, such as signal transduction, oxygen carrier (hemoglobin), and electron transfer (cytochrome). Approximately half of all proteins are metalloproteins. Another example is that in ion channel analysis, we can model the ion channel protein by molecular mechanics while describe the bending and vibration of membrane bilayers with elastic dynamics. Figure 1 illustrates a membrane protein complex, in which the transmembrane protein can be described by molecular mechanics or quantum mechanics, while the solvent can be treated by fluid mechanics. The lipid bilayer can be represented by elastic theory. In fact, hydrophilic head groups in the lipid bilayer can have dielectric functions different from those of hydrophobic tails. Other distinct chemistry or biology in the complex can be described by necessary means as well. Finally, for multidomain or multi-subunit proteins, different domains or subunits can be treated with different approaches, depending on physical observables and practical needs.

Figure 1.

Illustration of multiscale, multiphysics and multidomain models with a protein-membrane complex. Multiphysical descriptions at multiscales are employed in multidomains, which are labeled with different colors.

The rest of this paper is organized as follows. Section II is devoted to the theory and formulation of our new models. We first describe the notation and scope of the present work. The interaction potentials for different physical laws, including global Coulombic interactions, solvent-solute interactions and interactions between different solvent species, are considered. Based on this preparation, we introduce three new multiscale, multiphysics and multidomain models. Our first model is for solvation analysis. We introduce a multidomain representation of macromolecular complexes, in which each domain has its own surface tension, pressure, dielectric function, and charge density distribution. The energy variation leads to a set of Laplace-Beltrami equations, each for one solute domain. A generalized Poisson-Boltzmann equation is also derived from the present formulation. We expect this model to provide an efficient description for ion microstructures near the interface. Our second model is for charge transport in chemical, physical and biological systems. This model is a natural extension of the multiscale, multiphysics and multidomain solvation model. The chemical energy functional is employed to allow the description of non-equilibrium charge transport. The standard gradient flow procedure is employed to construct the generalized Nernst-Planck equation. A set of coupled Laplace-Beltrami, Poisson-Nernst-Planck equations are obtained from the variational analysis. This model is relevant to ion channel, nanofluidic and fuel cell systems. Finally, we construct a chemo-electro-fluid-MD-elastic model. This model allows simultaneously three different treatments, discrete point charges, molecular mechanics and elastic dynamics, of charge transfer macromolecular complexes, fuel cells and solar cells. In the solvent domain, multiple solvent species and their fluid flows are considered. This paper ends with concluding remarks.

II Variational multiscale multiphysics and multidomain models

In this section, we discuss a family of variational multiscale multiphysics and multidomain models. Our formulation extends the theory of the differential geometry based multiscale models¹⁶⁴ with the multidoain consideration. The novelty of our new models is the use of multiple domains to accommodate multiphysics descriptions of large biomolecular complexes and nano-bio systems.

We first discuss the scope of the present multiscale multiphysics and multidomain modeling. Three different models, a solvent model, a charge transport model, and a chemo-electro-fluid-MD-elastic model, are developed to illustrate our ideas.

II.A Scope of the present formulation

We denote $\Omega \subset {\mathbb{R}}^3$ as the total domain. Assume that there are a total of N macromolecular domains, denoted as, Ω_I, I = 1, 2, ..., N. We characterize these domains by a set of hypersurface functions {S_I}, I = 1, 2, ..., N, such as r ∈ Ω_I, if S_I(r) > 0. Obviously, these domains overlap each other, ${\cap }_I{\Omega }_I>0$. Additionally, we denote $S_S=1-S=1-{\sum }_{I=1}^N{S}_I$ the solvent characteristic function. The solvent domain is labeled as Ω_S. Obviously, the sum of all characteristic functions is the partition of unity ${\sum }_{I=1}^N{S}_I+S_S=1$.

Electrostatic interactions are fundamental in nature and ubiquitous in all biomolecules, including proteins, nucleic acids, lipid bilayers, sugars, etc. Electrostatic interactions are inherently of long range, which leads to computational difficulties. Since 65-90 percent of cellular mass is water under physiological condition, biomolecules live in a heterogeneous environment, where they interact with a wide range of aqueous ions, counterions, and other molecules. As a result, electrostatic interactions often manifest themselves in a vast variety of different forms, due to polarization, hyperpolarization, vibrational and rotational averages, screening effects, etc, to mention just a few. The importance of electrostatics in biomolecular systems and nano devices cannot be overemphasized because they underpin the molecular mechanism for almost all important biological processes, including signal transduction, DNA recognition, transcription, post-translational modification, translation, protein folding and protein ligand binding. In general, electrostatics is often the fundamental mechanism for macromolecular structure, function, dynamics and transport. In the chemical and biophysical literature, it is a convention that only those interactions that directly obey the Coulomb's law are referred to the electrostatic (or polar) interactions. All other interactions, including dispersion interactions, steric effects,^{11,17,71,80,103,155} ion-water dipolar interactions, hyperpolarizations, ion-water cluster formation or dissociation, ion spin effects, ion-protein interaction, hydrogen bonds and van der Waals interactions, are referred as non-electrostatic (or nonpolar) interactions, although they are ultimately electrostatic in origin.^23,164,166 This convention was adopted in our earlier work and is employed in the present work as well.

As in our earlier work, electrostatic modeling is the main focus of the present work. Due to their long range characteristic, electrostatic interactions are modeled as a global quantity that penetrates across domains. Therefore, electrostatic interactions between all domains are considered in the present work. Additionally, although non-electrostatic interactions are relatively short range, their influences near the interfaces can be significant. In particular, the mobile ions in the solvent domain are extremely sensitive to non-electrostatic interactions near the solvent-solute interfaces. Therefore, non-electrostatic interactions between the solvent domain and all macromolecular domains are of primary concern in the present work. Finally, it is well-known that solvent-solvent non-electrostatic interactions, including ionion non-electrostatic interactions, play a major role in determining ion microstructures near the solvent-solute interfaces. Therefore, we consider ion-ion non-electrostatic interactions as well. We denote U^S all the non-electrostatic (or nonpolar) interactions involving the solvent (S)

$$U^S=\sum _{\alpha }{\rho }_{\alpha }{U}_{\alpha }^S$$ (1)

where ρ_α is the density of solvent species α and $U_{\alpha }^S$ is given by

$$U_{\alpha }^S=\sum _I\sum _j{U}_{\alpha j}^{SI}\left(\mathbf{r}\right)+\sum _{\beta }{U}_{\alpha \beta }^{SS}\left(\mathbf{r}\right),$$ (2)

where $U_{\alpha j}^{SI}\left(\mathbf{r}\right)$ is the pairwise nonpolar interaction potential between the αth solvent species and jth component in the Ith solute domain. Similarly $U_{\alpha \beta }^{SS}\left(\mathbf{r}\right)$ are the pairwise nonpolar interaction potentials between the αth solvent species and βth solvent species in the solvent domain. These interactions, particularly the ones between solvent species, are important for the understanding of a number of phenomena, such as the solvent polarization, size effects, ζ potentials and solvent microstructures near the solvent-solute interfaces.

For generality, we do not specify the form of $U_{\alpha }^S$ in the present work. We assume that various interactions such as dipole,⁶⁴ multipole,^89,134ζ potential and steric effects,¹⁵ are modeled in the present theory by appropriate selections of $U_{\alpha }^S$. As an example, Lennard-Jones potentials, which have already been employed for $U_{\alpha j}^{SI}\left(\mathbf{r}\right)$³¹ and $U_{\alpha \beta }^{SS}\left(\mathbf{r}\right)$^23,166 in our earlier work, can be utilized. As pointed out in our earlier work,^23,31,32,166 these Lennard-Jones potentials involve one or two continuum variables and are significantly different from the conventional Lennard-Jones potential, which traditionally represents short-range interactions between two explicitly labeled particles. Therefore, the resulting formulations involve integral-differential equations. Note that integral equation approaches,^{13,20,50,123,128,157,174} including classical density functional theory,^{68,93,96,115,120,133,161,170,172,173} are quite popular in solvation analysis. Our formulations therefore connect both integral equation and differential equation approaches.

In the rest of this section, we first construct a new multiscale multiphysics and multidomain solvation model in Section II.B. Based on this new model, we further develop corresponding differential geometry based models for charge transport in Section II.C. Finally, we illustrate the flexibility and robustness of the proposed theory by constructing a multiscale multiphysics and multidomain model involving electro-statics, multiple charge species, fluid dynamics, molecular dynamics and elastic dynamics in Section II.D. In the solvent domain, flow convection and viscosity are considered. In the molecular mechanical description, all the bonding and nonbonding interactions in the implicit MD level⁶⁷ are accounted. In the elastic domain, the stress-strain relation is considered.

II.B Multiscale-multiphysics-multidomain model for solvation

Due to the ubiquitous nature of electrostatic interactions and the aqueous environment common to chemical, and biomolecular systems, analysis of molecular solvation is of significant importance in chemistry, biophysics, and medicine. Solvation is a physical process which involves a variety of solvent-solute interactions, such as the electrostatic, dipolar, induced dipolar, hydrogen bonding and van der Waals interactions between the solvent and the solute. Both explicit^30,121 and implicit models are used to describe the solvation process. Implicit solvent models that treat the solvent as a dielectric continuum, and describe the solute molecule as a static atomistic charge distribution^{3,49,76,85,131,138,160,169} have become popular recently, due to their simplicity and efficiency. Generalized Born,^{7,25,51,66,70,97,114,118,150,152,182} polarizable continuum,^{6,18,38,45,82,113,147,151} Poisson-Boltzmann (PB) models^{49,62,101,138} and nonlocal dielectric methods¹⁷¹ are commonly used. Among them, the PB models are the most popular and can be formally derived from Maxwell's theories.^12,75,117 A relatively comprehensive descriptions of the solvation process, solvation models and various applications of solvation methods can be found in our two review-style papers.^31,32

II.B.1 Total energy functional for solvation

Main direct experimental measurements of solvation include solvation free energy and solvent microstructures near the solvent-solute interface. These measurements validate solvation models. Typically, a solvation model offers either a description of the solvation free energy or the solvent microstructure. In the present work, we try to develop models for both physical observations.

Nonpolar free energy functional

The solvation free energy can be divided into polar (i.e., electro-static) and nonpolar contributions.^31,55,156 We propose the following multidomain nonpolar free energy functional

$$G_{\text{nonpolar}}=\sum _I({\gamma }_I{\mathrm{Area}}_I+p_I{\mathrm{Vol}}_I)+{\int }_{{\Omega }_S}{U}^{S}d\mathbf{r},\mathbf{r}\in {\mathbb{R}}^3,$$ (3)

where “Area_I” and “Vol_I” are respectively the solute surface area and volume of the Ith solute domain, γ_I(r) is the surface tension, and p_I is the pressure associated with the Ith solute domain. Here the integration is over the solvent domain Ω_S. In Eq. (3), the first two terms come from the scaled particle theory (SPT), which describes the surface free energy and the mechanical work of creating a cavity of the solute size in the solvent,^124,144 and the third term describes the solvent-solute interactions.^{31,55,156,164}

To bring the first two terms of Eq. (3) into an Eulerian representation, we utilize the mean surface area for domain Ω_I¹⁶⁴ and the coarea formula⁵⁹

$${\mathrm{Area}}_I={\int }_0^1{\int }_{S_I^{-1}\left(c\right)\cap {\Omega }_I}d\sigma dc={\int }_{\Omega }\mid \nabla S_I\left(\mathbf{r}\right)\mid d\mathbf{r},\mathbf{r}\in {\mathbb{R}}^3,$$ (4)

where 0 ≤ S_I ≤ 1 is a characteristic function or hypersurface function of the solute domain _I. Therefore, the volume of a macromolecular domain in Eq. (3) can be given by

$${\mathrm{Vol}}_I={\int }_{{\Omega }_I}d\mathbf{r}={\int }_{\Omega }{S}_I\left(\mathbf{r}\right)d\mathbf{r},$$ (5)

where Ω_I is the Ith macromolecular domain. Note that for adjacent domains, ${\Omega }_I\cap {\Omega }_J$ is not empty because each hypersurface function S_I is a smooth function, which leads to the overlapping between Ω_I and Ω_J. We rewrite the last term in Eq. (3) as

$${\int }_{{\Omega }_S}{U}^{S}d\mathbf{r}={\int }_{\Omega }\left(1-\sum _I{S}_I\left(\mathbf{r}\right)\right)U^{S}d\mathbf{r}={\int }_{\Omega }{S}_S\left(\mathbf{r}\right)U^{S}d\mathbf{r}.$$ (6)

Polar free energy functional

In our solvation models, the polar solvation free energy is modeled with the Poisson-Boltzmann theory. Sharp and Honig¹³⁷ introduced a variation formulation of the Poisson-Boltzmann equation. The derivation of solvation forces has been given by Gilson et al⁶⁹ and others.^67,164 In our multidomain formalism, we express the polar solvation free energy as

$$G_{\text{polar}}=\int \left\{\sum _I{S}_I\left[-\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I\right]+S_S\left[-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2-k_{B}T\sum _{\alpha }{\rho }_{\alpha 0}\left(e^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}}-1\right)\right]\right\}d\mathbf{r},$$ (7)

where Φ is the electrostatic potential, ε_S and ε_I are the dielectric functions of the solvent and the Ith solute, respectively, and ϱ_I represents the charge density of the Ith solute. The form of ϱ_I depends on the level of the physical description. For example, in the static atomistic description, one has ${\varrho }_I={\sum }_j{Q}_j^I\delta (\mathbf{r}-{\mathbf{r}}_j^I)$, with $Q_j^I$ denoting the partial charge of the jth atom in the Ith solute. Whereas, charge density ϱ_I takes a continuous form when the domain I is described in a continuous representation. It can also be computed with the density functional theory as demonstrated in our recent work.³³ Here k_B is the Boltzmann constant, T is the temperature, ρ_α0 denotes the reference bulk density of the αth solvent species, and q_α is the charge valence of the αth solvent species, which is zero for an uncharged solvent component, such as water molecules.

The Boltzmann distribution in Eq. (7) is of the same form as that in our earlier work and can be derived from the equilibrium condition of the generalized electrochemical potential.¹⁶⁶ Similar to integral equation theories,^68,129 the potentials $U_{\alpha }^S$ involve the integration of the continuum variable. By modifying $U_{\alpha }^S$ term in the Boltzmann distribution, one can easily take into the consideration of dipole,⁶⁴ multi-pole,^89,134 steric effects,¹⁵ and van der Waals interactions in a generalized Poisson-Boltzmann equation. In the present solvation model, the focus is on a simple description of the solvation free energy and solvent microstructures near the solvent-solute interface. The non-electrostatic interactions between different macromolecular domains have little impart to the equilibrium solvation properties, and thus, are neglected, for simplicity.

The direct combination of polar and nonpolar solvation free energy functionals does not lead to the desirable total free energy functional for solvation. Instead, a modification of the nonpolar energy functional is necessary because the solvent-solute interactions have been accounted in the Boltzmann distribution

$$G_{\text{total}}^{\mathrm{LB}-\mathrm{PB}}[\left\{S_I\right\},\Phi ]=\int \left\{\sum _I\left[{\gamma }_I\mid \nabla S_I\mid +p_I{S}_I+S_I\left(-\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I\right)\right]+S_S\left[-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2-k_{B}T\sum _{\alpha }{\rho }_{a0}\left(e^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}}-1\right)\right]\right\}d\mathbf{r}.$$ (8)

Here, the first row is the solvation free energy of the solute molecules and the second row is the polar solvation free energy of the solvent. Equation (8) provides a starting point for our variational analysis.

II.B.2 Governing equations for solvation

Established in a series of work,^{21,31,32,164,166} the variation of the solvation free energy functional (8) is quite standard. The derivation of two governing equations is discussed below.

Multidomain Laplace-Beltrami equations

In our formalism, the surfaces of biomolecular complexes are described by hypersurface functions {S_I}, which are governed by generalized Laplace-Beltrami equations. The total solvation free energy{in Eq. (8) is a functional of hypersurface functions {S_I} and electrostatic potential Φ. By using the Euler-Lagrange equation, we have

$$\frac{\delta G_{\text{total}}^{\mathrm{LB}-\mathrm{PB}}}{\delta S_I}\Rightarrow -\nabla \cdot \left({\gamma }_I\frac{\nabla S_I}{\mid \nabla S_I\mid }\right)+p_I-\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I+\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+k_{B}T\sum {\rho }_{\alpha 0}\left(e^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}}-1\right)=0.$$ (9)

It is convenient to introduce an artificial time^10,31,32,164 to arrive at the following generalized Laplace-Beltrami equations

$$\frac{\partial S_I}{\partial t}=\mid \nabla S_I\mid \left[\nabla \cdot \left({\gamma }_I\frac{\nabla S_I}{\mid \nabla S_I\mid }\right)+V_I^{\mathrm{LB}-\mathrm{PB}}\right],I=1,\cdots ,N,$$ (10)

where the potential driven terms are given by

$$V_I^{\mathrm{LB}-\mathrm{PB}}=-p_I+\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2-\Phi {\varrho }_I-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2-k_{B}T\sum _{\alpha }{\rho }_{\alpha 0}\left(e^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha }0}{k_{B}T}}-1\right).$$ (11)

Generalized Laplace-Beltrami equations (10) determine not only the solvent-solute interfaces but also the solute-solute interfaces. It is possible for many domains to overlap at one particular point of the space. Technically, there is some similarity between the present multidomain Laplace-Beltrami flows and the phase field theory based multi-component multiphase fluid flows,⁹² despite of their major conceptual differences.

Multidomain Poisson-Boltzmann equation

Variation with respect to electrostatic potential Φ leads to

$$\frac{\delta G_{\mathrm{total}}^{\mathrm{LB}-\mathrm{PB}}}{\delta \Phi }\Rightarrow \nabla \cdot \left(\left[S_S{∊}_S+\sum _I{S}_I{∊}_I\right]\nabla \Phi \right)+\sum _I{S}_I{\varrho }_I+S_S\sum _{\alpha }{q}_{\alpha }{\rho }_{\alpha 0}{e}^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}}=0.$$ (12)

From Eq. (12), one has the multidomain Poisson-Boltzmann equation

$$-\nabla \cdot (∊\left(S\right)\nabla \Phi )=\sum _I{S}_I{\varrho }_I+S_S\sum _{\alpha }{q}_{\alpha }{\rho }_{\alpha 0}{e}^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}},$$ (13)

where $∊\left(S\right)=S_S{∊}_S+{\sum }_I{S}_I{∊}_I$ is the generalized permittivity function for a multidomain setting. It reduces to the form discussed in our earlier work^31,164 when there is only one solute domain. As described in our earlier work, ε(S) is a smooth dielectric function. The Boltzmann factor $e^{-\frac{U_{\alpha }^S}{k_{B}T}}$ in Eq. (13) gives rise to a non-electrostatic correction to the charge density near the interface. Therefore, it can be used to describe solvent microstructures near the solvent-solute interface.

In our formulation, Eqs. (10) and (13) govern the surface evolution and the electrostatic potential, respectively. We denote these coupled equations the Laplace-Beltrami and Poisson-Boltzmann (LB-PB) equations. When there is only one solute domain, these equations reduce to their corresponding forms obtained in our earlier work.¹⁶⁶

The solvation model describes the system at equilibrium. However as the charge density is approximated by the Boltzmann distribution, which involves appropriate integrals in the potential $U_{\alpha }^S$,^23,166 the generalized Poisson-Boltzmann equation, Eq. (13), is in fact an integral-differential equation. The solution of Eq. (13) needs to carried out iteratively for realistic problems.

II.C Multiscale-multiphysics-multidomain model for charge transport

Solvation models are for systems at equilibrium, in which ion densities can be efficiently approximated by the Boltzmann distribution. However, for non-equilibrium systems, such as charge transport in fuel cells, solar cells, nano-fluidics, ion channels, gap junctions and neuron synapses, an independent description of ion densities is required. A wide variety of transport theories have been developed, ranging from Boltzmann kinetics,^{16,37,73,141,158} Monte Carlo approach,⁸⁴ Fokker-Planck and Master equations,^60,83 non-equilibrium Green's function,^{22,47,48,87,100,136,145} coupled Navier-Stokes and Poisson-Boltzmann (PB) equations,¹²⁶ to Poisson-Nernst-Planck (PNP) equations.^{4,26,57,61,98,110,140} Among these approaches, the PNP model is relatively simple, and able to offer very good predictions of current-voltage curves for many channel proteins.^19,98,177 However, the PNP theory neglects the finite size effect due to its continuum representation of ion densities.^{17,44,80,86,90,106,110,139} Advantages and limitations the aforementioned transport models have been discussed in the literature.2,5,35,36,40–42,42,54,57,58,99,104,105,111,130,135,154 The reader to referred to Ref¹⁶⁶ for a recent review.

Differential geometry based charge transport models have been introduced in our earlier work.¹⁶⁴ Extension to proton transport and validation with experimental data have been carried out recently.^21,23,166 A major feature of our differential geometry based charge transport models is that they combine the transport modeling with the geometric flow based surface modeling so as to generate self-consistent solvent-solute interfaces.¹⁶⁴ Another important feature of our transport models is that they unify the transport modeling with the solvation modeling. As a result, our nonequilibrium transport models reduce to corresponding equilibrium solvation models at equilibrium.¹⁶⁶ In the next subsection, we provide a multidomain generalization of our earlier transport formalism.

II.C.1 Total energy functional for a system with charged species

Charge transport involves material exchange and thus chemical potential(s). In nonequilibrium thermodynamics, chemical potential related free energy can be expressed as¹⁶⁶

$$G_{\text{chem}}=\int \sum _{\alpha }\left\{({\mu }_{\alpha }^0-{\mu }_{\alpha 0}){\rho }_{\alpha }+k_{B}T{\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-k_{B}T({\rho }_{\alpha }-{\rho }_{\alpha 0}\right\}d\mathbf{r},$$ (14)

where ${\mu }_{\alpha }^0$ is a reference chemical potential of the αth species at which the associated ion density is ρ_0α given $\Phi =U_{\alpha }^S={\mu }_{\alpha 0}=0$. The second term is associated with entropy of mixing, and the last term is for relative osmotics.¹¹²

To construct a total energy functional for charge transport, we need to recognize that the densities of ion species do not obey the Boltzmann distribution. We therefore utilize the nonpolar free energy functional (3), and modify the source term of the solvent polar energy functional (7). Together with chemical potential related free energy (14), the total free energy functional for the charge transport is given by

$$G_{\text{total}}^{\mathrm{LB}-\mathrm{PNP}}[\left\{S_I\right\},\Phi ,\left\{{\rho }_{\alpha }\right\}]=\int \left\{\sum _I({\gamma }_I\mid \nabla S_I\mid +p_I{S}_I)+S_S{U}^S+\sum _I{S}_I\left[-\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I\right]+S_S\left[-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }\right]+S_S\sum _{\alpha }\left[({\mu }_{\alpha }^0-{\mu }_{\alpha 0}){\rho }_{\alpha }+k_{B}T{\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-k_{B}T({\rho }_{\alpha }-{\rho }_{\alpha 0})+{\lambda }_{\alpha }{\rho }_{\alpha }\right]\right\}d\mathbf{r},$$ (15)

where the first row is the nonpolar solvation free energy functional for our multidomain description, the second row is the polar (electrostatic) solvation free energy functional, and the last row is the chemical potential related energy functional for charge transport. We denote λ_α a Lagrange multiplier for ensuring appropriate physical properties at equilibrium.^61,166

II.C.2 Governing equations

As functionals of hypersurface functions {S_I}, electrostatic potential and ion densities {ρ_α}, the total free energy (15) can be minimized by using the variational principle, which gives rise to desirable governing equations for the system, namely, partial differential equations (PDEs) for {S_I}, Φ and {ρ_α. The solution of these PDEs in turn minimizes the total free energy in Eq. (15).

Multidomain Laplace-Beltrami equations

As discussed in the last subsection, hypersurface functions {S_I} are governed by generalized Laplace-Beltrami equations. These equations are different in their source terms. We apply the Euler-Lagrange equation to {S_I} to have

$$\frac{\delta G_{\text{total}}^{\mathrm{LB}-\mathrm{PNP}}}{\delta S_I}\Rightarrow \nabla \cdot \left({\gamma }_I\frac{\nabla S_I}{\mid \nabla S_I\mid }\right)+p_I-U^S-\sum _I\left(\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I\right)+\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2-\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }-\sum _{\alpha }\left[-{\mu }_{\alpha 0}{\rho }_{\alpha }+k_{B}T{\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-k_{B}T({\rho }_{\alpha }-{\rho }_{\alpha 0})\right]=0.$$ (16)

By using the same procedure as that used in our earlier work,^8,164 we arrive at a set of N Laplace-Beltrami equations:

$$\frac{\partial S_I}{\partial t}=\mid \nabla S_I\mid \left[\nabla \cdot \left({\gamma }_I\frac{\nabla S_I}{\mid \nabla S_I\mid }\right)+V^{\mathrm{LB}-\mathrm{PNP}}\right],I=1,\dots ,N,$$ (17)

where

$$V^{\mathrm{LB}-\mathrm{PNP}}=-p_I+U^S+\sum _I\left(\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2-\Phi {\varrho }_I\right)-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }+\sum _{\alpha }\left[k_{B}T\left({\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-{\rho }_{\alpha }+{\rho }_{\alpha 0}\right)-{\mu }_{\alpha 0}{\rho }_{\alpha }\right].$$ (18)

Multidomain Poisson equation

We first carry out the variation of the total free energy functional with respect to the electrostatic potential Φ

$$\frac{\delta G_{\text{total}}^{\mathrm{LB}-\mathrm{PNP}}}{\delta \Phi }\Rightarrow \nabla \cdot \left([S_S{∊}_S+\sum _I{S}_I{∊}_I]\nabla \Phi \right)+\sum _I{S}_I{\varrho }_I+S_S\sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }=0.$$ (19)

This gives rise to the desirable multidomain Poisson equation

$$-\nabla \cdot (∊\left(S\right)\nabla \Phi )=\sum _I{S}_I{\varrho }_I+S_S\sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha },$$ (20)

where $∊\left(S\right)=S_S{∊}_S+{\sum }_I{S}_I{∊}_I$ is an interface dependent dielectric profile, which is a continuous function. Obviously, Eq. (20) involves hypersurface functions S_I and densities of ions ρ_α. The latter is determined by a set of PDEs, instead of a Boltzmann factor, in the present formulation.

Multidomain Nernst-Planck equations

The derivation of Nernst-Planck equations is similar to that described in our earlier work.¹⁶⁶ Let us carry out the variation with respect to ion densities {ρ_α}

$$\frac{\delta G_{\text{total}}^{\mathrm{LB}-\mathrm{PNP}}}{\delta {\rho }_{\alpha }}\Rightarrow {\mu }_{\alpha }^{\mathrm{gen}}={\mu }_{\alpha }^0-{\mu }_{\alpha 0}+k_{B}T\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}+q_{\alpha }\Phi +U_{\alpha }^S+{\lambda }_{\alpha }$$ (21)

where ${\mu }_{\alpha }^{\mathrm{gen}}$ is the relative generalized potential of species α and vanishes at equilibrium, which gives rise to

$${\lambda }_{\alpha }=-{\mu }_{\alpha }^0\text{and}{\rho }_{\alpha }={\rho }_{\alpha 0}{e}^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}}.$$ (22)

It is seen that the generalized Boltzmann factor used in the last subsection is justified by equilibrium condition (22).

For charge transport, the system is of non-equilibrium in general due to inhomogeneities in ion densities and electrostatic potential. In many situations, these inhomogeneities originate from boundary conditions, such as concentration gradient between intercellular and extracellular ions, and electrostatic potential gradient due to applied voltages in nanofluidic devices and patch clamps. We construct a set of ion flux equations by using Nernst-Einstein equation ${\mathbf{J}}_{\alpha }=-D_{\alpha }{\rho }_{\alpha }\nabla \frac{{\mu }_{\alpha }^{\mathrm{gen}}}{k_{B}T}$ with D_α being the diffusion coefficient of species α. In fact, D_α needs to be a position dependent function in many applications, such as ion channels.¹⁷⁷ By taking into consideration of Eq. (22), the relative generalized potential ${\mu }_{\alpha }^{\mathrm{gen}}$ is given by ${\mu }_{\alpha }^{\mathrm{gen}}=k_{B}T\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}+q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}$. We further make use of Fick's law of diffusion $\frac{\partial {\rho }_{\alpha }}{\partial t}=-\nabla \cdot {\mathbf{J}}_{\alpha }$ to arrive at desirable Nernst-Planck equations

$$\frac{\partial {\rho }_{\alpha }}{\partial t}=\nabla \cdot \left[D_{\alpha }\left(\nabla {\rho }_{\alpha }+\frac{{\rho }_{\alpha }}{k_{B}T}\nabla (q_{\alpha }\Phi +U_{\alpha }^S)\right)\right],$$ (23)

where q_αΦ + U_α can be regarded as a mean field potential, which gives rise a generalized convection (force) term in the ion density dynamics. This approach, namely, the construction of a flow flux from the energy minimization, is often called a gradient flow method in the literature.

At steady state, Eq. (23) becomes

$$\nabla \cdot \left[D_{\alpha }\left(\nabla {\rho }_{\alpha }+\frac{{\rho }_{\alpha }}{k_{B}T}\nabla (q_{\alpha }\Phi +U_{\alpha }^S)\right)\right]=0.$$ (24)

The solution of Eq. (23) depends on Φ, i.e., the solution of the multidomain Poisson equation (20). We call Eqs. (20) and (23) the generalized Poisson-Nernst-Planck (PNP) equations. Note that the solution of Eq. (20) depends on hypersurface functions S_I.

The multidomain Laplace-Beltrami equations (17) and PNP equations (20) and (23) constitute a coupled system, and are called generalized LB-PNP equations. Their solution minimizes the total free energy functional (15) for charge transport. Unlike the traditional PNP equations, the present LB-PNP equations self-consistently couple biomolecular surfaces and dielectric profiles with electrostatic potentials and ion densities. Additionally, the general interaction potential U^S include possible solvent-solute and ion-ion interactions, which endows the present LB-PNP formalism with the capability of predicting ion microstructures near solvent-solute interfaces. Finally, the multidomain setting in the present formulation further allows the flexibility of modeling multiple material compositions.

II.C.3 Consistency with the multidomain solvation model

In our earlier work, we have shown both theoretically and numerically that the non-equilibrium LB-PNP model reduces to the equilibrium solvation model at equilibrium.¹⁶⁶ This consistency between a non-equilibrium theory and an equilibrium one is essential in theoretical modeling and our understanding of non-equilibrium dynamics. In the present multidomain modeling, we demonstrate further that this important consistency can also be established.

To this end, we apply the constraints in Eq. (22) to the total free energy functional in Eq. (15)

$$G_{\text{total}}^{\mathrm{LB}-\mathrm{PNP}}=\int \left\{\sum _I({\gamma }_I\mid \nabla S_I\mid +p_I{S}_I)+S_S{U}^S+\sum _I{S}_I\left[-\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I\right]+S_S\left[-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }\right]+S_S\sum _{\alpha }\left[({\mu }_{\alpha }^0-{\mu }_{\alpha 0}){\rho }_{\alpha }+k_{B}T{\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-k_{B}T({\rho }_{\alpha }-{\rho }_{\alpha 0})+{\lambda }_{\alpha }{\rho }_{\alpha }\right]\right\}d\mathbf{r}\stackrel{\mathrm{Eq}.\left(22\right)}{\to }\int \left\{\sum _I\left[{\gamma }_I\mid \nabla S_I\mid +p_I{S}_I+S_I\left(-\frac{{∊}_I}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_I\right)\right]+S_S\left[-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2-k_{B}T\sum _{\alpha }{\rho }_{\alpha 0}\left(e^{-\frac{q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}}{k_{B}T}}-1\right)\right]\right\}d\mathbf{r}=G_{\text{total}}^{\mathrm{LB}-\mathrm{PB}}.$$ (25)

Additionally, for the surface driven functions of the generalized LB equation, it is easy to show that under the constraints of Eq. (22), one has V^LB–PNP → V^LB–PB. Furthermore, under the constraints of Eq. (22), ${\mathbf{J}}_{\alpha }=-D_{\alpha }{\rho }_{\alpha }\nabla \frac{{\mu }_{\alpha }^{\mathrm{gen}}}{k_{B}T}=0$ and PNP equations (20) vanish. Therefore, we fully recover the equilibrium LB-PB model from the non equilibrium LB-PNP theory at equilibrium.

II.D Chemo-Electro-Fluid-MD-Elastic model

Many chemical, physical and biological systems are excessively large and involve a huge number of degrees of freedom. As such, their atomic description is intractable with current computer capability. Subcellular organelles, molecular motors, virus particles and fuel cells are examples of excessively large aqueous systems. Theoretical modeling, analysis and simulation of these systems pose a fabulous challenge to the research community. Multiscale, multiphysics and multidomain methods proposed in this work provide potential tactics and strategies for this class of excessively large problems.

In this subsection, we demonstrate the use of the present multidomain theory by considering a system with three distinguished domains, a solvent domain and two molecular domains. The solvent domain is described in terms of the fluid mechanics. One of the molecular domain is treated with the molecular dynamics (or coarse grained dynamics) and the other molecular domain is furnished by using the elastic dynamics. Two molecular domains are directly coupled to each other via electrostatic interactions. Additionally, both molecular domains are strongly coupled to the solvent domain via electrostatic interactions as well as general solvent-solute non-electrostatic interactions described by U^S.

II.D.1 The action functionals

Fluid dynamics

Fluid flows play an important role in nanofluidic devices and fuel cell systems. In nanofluidics, fluids are controlled and manipulated at submicrometer and nanometer scales to study the behavior of molecular and biological systems. At such scales, the characteristic length scale of the fluid coincides with the length scale of the biomolecule and the scale of the Debye length. As a result, fluids show interesting behaviors which are not observed in larger scales. Mirco/nano fluidic apparatuses have been developed for basic measurements, ranging from molecular diffusion coefficients,⁸⁸ pH values,^109,167 chemical binding affinities,⁸⁸ to enzyme reaction kinetics.^53,72 As a new technology, nanofluidics has been devised for polymerase chain reaction (PCR) amplifications,¹⁴ macromolecule accumulator,^39,168 electrokinetics,¹¹ biomaterial separation⁹¹ membrane protein crystallization,¹⁰⁷ single nucleotide polymorphism genotyping,¹⁵⁹ and gene expression analysis via DNA computing.¹⁷⁵ The influence domain of electrostatic potentials in nanofluidic systems is characterized by the Debye length ${\lambda }_D=\sqrt{{∊}_S{k}_{B}T∕{\sum }_{\alpha }{\rho }_{\alpha 0}{q}_{\alpha }^2}$, which varies dramatically from the channels of transmembrane proteins, pores of proton exchange membranes, to clefts of neuron synapses. To model electro-osmosis and electrophoresis, it is necessary to combine fluid mechanics with electrostatic analysis.

We consider multicomponent homogeneous incompressible flows. The Lagrangian of an incompressible viscous flow was discussed in our earlier work.¹⁶⁴ It consists of kinetic energy, potential energy and viscous energy lost due to friction¹⁶⁴

$$L_{\text{Fluid}}=\int (1-S)\left[\rho \frac{{\mathbf{v}}^2}{2}-\left(\Psi +p_S+U^S-\frac{{\mu }_f}{8}{\int }^t{\left(\frac{\partial {\mathbf{v}}_i}{\partial {\mathbf{x}}_j}+\frac{\partial {\mathbf{v}}_j}{\partial {\mathbf{x}}_i}\right)}^{2}d{t}^{\prime }\right)\right]d\mathbf{x},$$ (26)

where $\rho ={\sum }_{\alpha }{\rho }_{\alpha }$ is the total solvent density for a multicomponent homogeneous flow, p_S is the hydrodynamic pressure, v is the flow stream velocity, v_i are velocity components and μ_f is the viscosity of the fluid. Here, Ψ is the potential energy mostly due to the gravitation. The last term in Eq. (26) is the stress energy density

$$E_{\text{Stress}}=\frac{{\mu }_f}{8}{\int }^t{\left(\frac{\partial {\mathbf{v}}_i}{\partial {\mathbf{x}}_j}+\frac{\partial {\mathbf{v}}_j}{\partial {\mathbf{x}}_i}\right)}^{2}d{t}^{\prime }=\frac{1}{2{\mu }_f}{\int }^t{\mathbb{T}}^{2}d{t}^{\prime },$$ (27)

where $\mathbb{T}$ is the stress tensor. The Einstein summation convention is used in the above expression. Obviously, the stress tensor is symmetric

$${\mathbb{T}}_{ij}={\mathbb{T}}_{ji}.$$ (28)

The reader is referred to Ref.¹⁶⁴ for more detailed discussion of the fluid energy functional. In the present work, we slightly modify Eq. (26) to avoid redundancy in energy density functional.

Molecular dynamics

Unlike the macroscopic fluid dynamics, the molecular mechanics seeks microscopic atomistic or coarse grained descriptions of a solute component, which is typically crucial to the physics of interest. For example, the structure and dynamics of an ion channel protein is the key to the understanding of the channel gating mechanism. Molecular dynamics can be employed to obtain structure information due to mutations. The molecular mechanics in the present formulation is akin to the implicit molecular dynamics proposed in the earlier work by Gilson et al⁶⁹ and others,^81,108 including ours.⁶⁷

The energy functional of a molecular mechanics was introduced in our earlier work.¹⁶⁴ The Lagrangian of molecular mechanics includes the kinetic energy of each individual atom or particle and the potential energy due to various microscopic interactions¹⁶⁴

$$L_{\mathrm{MD}}=\int \int S\sum _j\left[{\rho }_j\frac{{\stackrel{.}{\mathbf{z}}}_j^2}{2}-U^M\left(\mathbf{z}\right)\right]d\mathbf{x}d\mathbf{z}$$ (29)

where ρ_j = m_jδ(z_j – x_j) is the mass density of the jth atom or particle in a coarse grained description, m_j and x_j are the mass and the macroscopic position of the jth atom or particle, respectively. Here ${\rho }_j\frac{{\stackrel{.}{\mathbf{z}}}_j^2}{2}$ and U^M(z) are respectively the kinetic and the potential energy densities of the jth atom or particle with ${\stackrel{.}{\mathbf{z}}}_j=\frac{d{\mathbf{z}}_j}{dt}$. Additionally, we denote $\mathbf{z}=({\mathbf{z}}_1,{\mathbf{z}}_2,\cdots ,{\mathbf{z}}_{N_{\alpha }})\in {\mathbb{R}}^{3N_{\alpha }}$ as the microscopic variable of N_a atoms or particles and dz = dz₁dz₂ · · · dz_Na. In principle, the potential interactions U^M include all bonding and nonbonding components used in implicit MD calculations.^67,108

Elastic dynamics

The microscopic domain described above is complemented with an elastic domain to dramatically reduce the number of degrees of freedom. Both fluid dynamics and electrostatic interactions will be coupled to molecular dynamics and elastic dynamics. Some pioneer work on fluid-structure coupling was due to Peskin.¹²² The coupling of electrostatics and elasticity was considered by Zhou et al¹⁸¹ for biomolecules. Alternatively, phase field⁵² and Helfrich curvature¹¹⁹ approaches of the elastic bending energy for vesicle membranes were discussed in the literature.

Let us consider a point r in ${\mathbb{R}}^3$ that is deformed to r̄ due to a displacement w, i.e.,

$$\mathbf{w}=\stackrel{‒}{\mathbf{r}}-\mathbf{r}.$$ (30)

The deformation can be characterized by a strain tensor¹⁶⁴

$${\sigma }_{ij}=\frac{1}{2}\left[\frac{\partial {\mathbf{w}}_i}{\partial {\mathbf{r}}_j}+\frac{\partial {\mathbf{w}}_j}{\partial {\mathbf{r}}_i}\right],$$ (31)

where the nonlinear term in w is omitted for relatively small deformations.¹⁶⁴ This linear elasticity analysis has been widely used. The Einstein summation notation is used in the above expression.

For an isotropic system, the elastic potential energy density takes the form

$$E_{\text{Elastic}}=\frac{1}{2}[{\lambda }_E{\sigma }_{ii}^2+{\mu }_E{\left({\sigma }_{ij}\right)}^2],$$ (32)

where λ_E is the elastic modulus or stress/strain ratio, and μ_E is the shear modulus. Both the elastic modulus λ_E and the shear modulus μ_E are connected to the atomic or molecular interaction strengths.

Additionally, the kinetic energy density can be expressed as $\frac{{\rho }_E}{2}{\stackrel{.}{\mathbf{w}}}^2$, where ρ_E is the mass density of the elastic macromolecule and ẉ is the velocity of the displacement. Therefore, the Lagrangian of the elastic system is given as the difference of the kinetic energy and the potential energy

$$L_{\text{Elastic}}=\int S\left[\frac{{\rho }_E}{2}{\stackrel{.}{\mathbf{w}}}^2-\frac{1}{2}({\lambda }_E{\sigma }_{ii}^2+{\mu }_E{\left({\sigma }_{ij}\right)}^2)\right]d\mathbf{r}.$$ (33)

Total action functional

Total action functional of the present multidomain system involves energy densities from differential physics, namely, polar and nonpolar solvation, chemical mixing, fluid dynamics, molecular dynamics and elastic dynamics. However, we need to eliminate any redundancy in energy densities. We consider the following total action functional

$$G_{\text{total}}^{\mathrm{MD}-\mathrm{FD}-\mathrm{ED}}[S,\Phi ,\left\{{\rho }_{\alpha }\right\}]=\int \int \int \left\{{\gamma }_M\mid \nabla S_M\mid +{\gamma }_E\mid \nabla S_I\mid +p_M{S}_M+p_E{S}_E+S_S{U}^S+S_M\left[-\frac{{∊}_M}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_M\right]+S_E\left[-\frac{{∊}_E}{2}{\mid \nabla \Phi \mid }^2+\Phi {\varrho }_E\right]+S_S\left[-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }\right]+S_S\sum _{\alpha }\left[({\mu }_{\alpha }^0-{\mu }_{\alpha 0}){\rho }_{\alpha }+k_{B}T{\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-k_{B}T({\rho }_{\alpha }-{\rho }_{\alpha 0})+{\lambda }_{\alpha }{\rho }_{\alpha }\right]-S_S\left[\rho \frac{{\mathbf{v}}^2}{2}-p_S+\frac{{\mu }_f}{8}{\int }^t{\left(\frac{\partial {\mathbf{v}}_i}{\partial {\mathbf{r}}_j}+\frac{\partial {\mathbf{v}}_j}{\partial {\mathbf{r}}_i}\right)}^{2}d{t}^{\prime }\right]-S_{\mathrm{M}}\sum _j\left[{\rho }_j\frac{{\stackrel{.}{\mathbf{z}}}_j^2}{2}-U^M\left(\mathbf{z}\right)\right]-S_E\left[\frac{{\rho }_E}{2}{\stackrel{.}{\mathbf{w}}}^2-\frac{1}{2}({\lambda }_E{\sigma }_{ii}^2+{\mu }_E{\left({\sigma }_{ij}^{}\right)}^2)\right]\right\}d\mathbf{r}d\mathbf{z}dt,$$ (34)

where expressions from the first to the last row in Eq. (34) are respectively the multidomain nonpolar free energy, electrostatic energy, chemical potential related energy, fluid dynamics energy, molecular dynamics energy and elastic dynamics energy.

II.D.2 Governing equations

We derive governing equations by a total variation in the present case.¹⁶⁴ A number of coupled PDEs are obtained as described below.

Generalized Laplace-Beltrami equation

By using the same procedure as that used in the earlier sections, we end up with two generalized Laplace-Beltrami equations, one for the molecular mechanics domain and the other for the elastic domain

$$\frac{\partial S_I}{\partial t}=\frac{\mid \nabla S_I\mid }{}\left[\nabla \cdot \left({\gamma }_I\frac{\nabla S_I}{\mid \nabla S_I\mid }\right)+V_I\right],I=M,E,$$ (35)

where driven terms V_M and V_E are respectively given by

$$V_M=-p_M+U^S+\frac{{∊}_M}{2}{\mid \nabla \Phi \mid }^2-\Phi {\varrho }_M-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }+\sum _{\alpha }\left[k_{B}T\left({\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-{\rho }_{\alpha }+{\rho }_{\alpha 0}\right)-{\mu }_{\alpha 0}{\rho }_{\alpha }\right]-\left[\rho \frac{{\mathbf{v}}^2}{2}-p_S+\frac{{\mu }_f}{8}{\int }^t{\left(\frac{\partial {\mathbf{v}}_i}{\partial {\mathbf{r}}_j}+\frac{\partial {\mathbf{v}}_j}{\partial {\mathbf{r}}_i}\right)}^{2}d{t}^{\prime }\right]+\sum _j\left[{\rho }_j\frac{{\stackrel{.}{\mathbf{z}}}_j^2}{2}-U^M\left(\mathbf{z}\right)\right].$$ (36)

and

$$V_E=-p_E+U^S+\frac{{∊}_E}{2}{\mid \nabla \Phi \mid }^2-\Phi {\varrho }_E-\frac{{∊}_S}{2}{\mid \nabla \Phi \mid }^2+\Phi \sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }+\sum _{\alpha }\left[k_{B}T\left({\rho }_{\alpha }\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}-{\rho }_{\alpha }+{\rho }_{\alpha 0}\right)-{\mu }_{\alpha 0}{\rho }_{\alpha }\right]\left[\rho \frac{{\mathbf{v}}^2}{2}-p_S+\frac{{\mu }_f}{8}{\int }^t{\left(\frac{\partial {\mathbf{v}}_i}{\partial {\mathbf{r}}_j}+\frac{\partial {\mathbf{v}}_j}{\partial {\mathbf{r}}_i}\right)}^{2}d{t}^{\prime }\right]+\left[\frac{{\rho }_E}{2}{\stackrel{.}{\mathbf{w}}}^2-\frac{1}{2}({\lambda }_E{\sigma }_{ii}^2+{\mu }_E{\left({\sigma }_{ij}\right)}^2)\right].$$ (37)

The above two expressions differ in their electrostatic energies and their last terms, which are associated with specific dynamic descriptions. Solution to these equations determines S_M and S_E, as well as S_S, because of S_S = 1 – S_M – S_E.

Generalized Poisson equation

The variation of the total action functional (34) with respect to Φ leads to the generalized Poisson equation

$$-\nabla \cdot (∊\left(S\right)\nabla \Phi )=S_M{\varrho }_M+S_E{\varrho }_E+S_S\sum _{\alpha }{\rho }_{\alpha }{q}_{\alpha }$$ (38)

where ε(S) = S_Sε_S + S_Mε_M + S_Eε_E is the generalized permittivity function. Obviously, Eq. (38) is a special case of Eq. (20).

Generalized Nernst-Planck equation

With a non-vanishing flow velocity, the derivation of the generalized Nernst-Planck is slightly different from that in Section II.C.2, but is very similar to that discussed in our earlier work.¹⁶⁶ The variation of the total action functional (34) with respect to ion densities ρ_α gives rise to a generalized relative potential ${\mu }_{\alpha }^{\mathrm{gen}}$

$${\mu }_{\alpha }^{\mathrm{gen}}=k_{B}T\ln \frac{{\rho }_{\alpha }}{{\rho }_{\alpha 0}}+q_{\alpha }\Phi +U_{\alpha }^S-{\mu }_{\alpha 0}-\frac{{\mathbf{v}}^2}{2}.$$ (39)

We use Eq. (39) to construct a generalized flux

$${\mathbf{J}}_{\alpha }=-D_{\alpha }{\rho }_{\alpha }\nabla \frac{{\mu }_{\alpha }^{\mathrm{gen}}}{k_{B}T}.$$ (40)

The generalized Fick's law, which takes care of chemical reactions and incompressible fluid flows, gives^164,166

$$\frac{\partial {\rho }_{\alpha }}{\partial t}+\mathbf{v}\cdot \nabla {\rho }_{\alpha }=-\nabla \cdot {\mathbf{J}}_{\alpha }+\sum _j{\stackrel{‒}{\nu }}_{\alpha j}{J}^j$$ (41)

where ${\stackrel{‒}{\nu }}_{\alpha j}{J}^j$ is the density production of α species per unit volume in the jth chemical reaction.¹⁶⁴ Explicitly, the generalized Nernst-Planck equation reads

$$\frac{\partial {\rho }_{\alpha }}{\partial t}+\mathbf{v}\cdot \nabla {\rho }_{\alpha }=\nabla \cdot D_{\alpha }\left[\nabla {\rho }_{\alpha }+\frac{{\rho }_{\alpha }}{k_{B}T}\nabla \left(q_{\alpha }\Phi +U_{\alpha }^S-\frac{{\mathbf{v}}^2}{2}\right)\right]+\sum _j{\stackrel{‒}{\nu }}_{\alpha j}{J}^j.$$ (42)

Equation (42) provides a generalized mass conservation where the rate of change of the αth ion species is balanced by the convective transport of the incompressible multicomponent fluid flow, density gradient, electrostatic gradient, potential forces due to solvent-solute interactions, solvent-solvent interactions, the flux of the fluid kinetic energy, and finally, chemical reactions. Equation (42) reduces to Eq. (23) when the velocity and the chemical reaction flux vanish.

Generalized Navier-Stokes equation

The conservation equation for flow stream velocity of incompressible flows can also be derived from variational principle.^164,166 The total variation of total action functional (34) leads to the generalized Navier-Stokes equation

$$\rho \left(\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}\cdot \nabla \mathbf{v}\right)=-\nabla p_S+\frac{1}{S_S}\nabla \cdot S_S\mathbb{T}+{\mathbf{F}}_{\mathrm{E}},$$ (43)

where flow stress tensor $\mathbb{T}$ can be expressed as

$$\mathbb{T}=\frac{{\mu }_f}{2}\left(\frac{\partial {\mathbf{v}}_i}{\partial {\mathbf{r}}_j}+\frac{\partial {\mathbf{v}}_j}{\partial {\mathbf{r}}_i}\right)=\frac{{\mu }_f}{2}[\nabla \mathbf{v}+{(\nabla \mathbf{v})}^T],$$ (44)

where symbol T denotes the transpose. In Eq. (43), F_E is the total force given by

$${\mathbf{F}}_{\mathrm{E}}=\frac{1}{1-S_M-S_E}\left(-S_M\nabla p_M-S_E\nabla p_E-S_S\sum _{\alpha }{\rho }_{\alpha }\nabla U_{\alpha }^S+{\varrho }_M\nabla \left(S_M\Phi \right)+{\varrho }_E\nabla \left(S_E\Phi \right)\right)$$ (45)

Equation (45) reduces to the standard Navier-Stokes at the inner solvent domain (i.e., S_M = S_E = 0), except for an extra force term $-{\sum }_{\alpha }{\rho }_{\alpha }\nabla U_{\alpha }^S$, which is due to the consideration of solvent-solvent interactions. In our earlier work,¹⁶⁴ we discussed the reduction of the two-domain version of Eq. (45) to the Stokes equation, which is relevant for biomolecular systems. The connection of the two-domain version of Eq. (45) with the Navier-Stokes equation for classical electroviscous flows was also discussed.¹⁶⁶

Generalized Newton equation

As discussed in our earlier work,¹⁶⁴ part of the total variation is associated with δz_j, which gives rise to the desirable Newton's equation for the molecular dynamics (MD)

$${\rho }_j{\ddot{\mathbf{z}}}_j={\mathbf{f}}^j,j=1,2,\cdots ,N^a,$$ (46)

where {f^j} are a set of forces associated with solvent-solute interaction near the interfaces and molecular interactions. We have that ${\mathbf{f}}^j={\mathbf{f}}_{\mathrm{SSI}}^j+{\mathbf{f}}_{\mathrm{RF}}^j+{\mathbf{f}}_{\mathrm{PI}}^j$ with the components given as

$${\mathbf{f}}_{\mathrm{SSI}}^j=-\frac{S_S}{S_M}{\nabla }_j{U}^S$$ (47)

$${\mathbf{f}}_{\mathrm{RF}}^j=\frac{1}{S_M}({\varrho }_M{\nabla }_j\left(S_M\Phi \right)+{\varrho }_E{\nabla }_j\left(S_E\Phi \right))$$ (48)

$${\mathbf{f}}_{\mathrm{PI}}^j=-{\nabla }_j{U}^M\left(\mathbf{z}\right),$$ (49)

where ${\mathbf{f}}_{\mathrm{SSI}}^j$, ${\mathbf{f}}_{\mathrm{RF}}^j$ and ${\mathbf{f}}_{\mathrm{PI}}^j$ are respectively, solvent-solute interaction force, reaction field force and potential interaction force due to atomic or particle interactions.

Elastic dynamics

The variation of the total action functional also leads to the governing equation for the elastic dynamics of the macromolecule

$${\rho }_E\ddot{\mathbf{w}}=\frac{1}{S_E}[({\lambda }_E+{\mu }_E)\nabla S_E\nabla \cdot \mathbf{w}+{\mu }_E\nabla \cdot S_E\nabla \mathbf{w}]+{\mathbf{f}}^E.$$ (50)

where ${\mathbf{f}}^E={\mathbf{f}}_{\mathrm{SSI}}^E+{\mathbf{f}}_{\mathrm{RF}}^E$ are the forces acting on the elastic macromolecule. The fluid-structure interaction (FSI) force ${\mathbf{f}}_{\mathrm{FSI}}^E$ and reaction field (RF) force ${\mathbf{f}}_{\mathrm{RF}}^E$ are given by

$${\mathbf{f}}_{\mathrm{FSI}}^E=-\frac{S_S}{S_E}\sum _{\alpha }{\rho }_{\alpha }{\nabla }_{\mathbf{w}}{U}_{\alpha }^S$$ (51)

$${\mathbf{f}}_{\mathrm{RF}}^E=-\frac{1}{S_E}\Phi (S_M{\nabla }_{\mathbf{w}}{\varrho }_M+S_E{\nabla }_{\mathbf{w}}{\varrho }_E).$$ (52)

To understand Eq. (50), we define the stress tensor of the elastic material as¹⁶⁴

$${\mathbb{T}}_{ij}^E={\lambda }_E{\sigma }_{ii}{\delta }_{ij}+2{\mu }_E{\sigma }_{ij}.$$ (53)

We therefore finally rewrite Eq. (50) as

$${\rho }_E\ddot{\mathbf{w}}=\frac{1}{S_E}\nabla \cdot S_E{\mathbb{T}}^E+{\mathbf{f}}^E.$$ (54)

Equation (54) is a generalized version of the classical elastic dynamics. Its steady state is given by

$$\frac{1}{S_E}\nabla \cdot S_E{\mathbb{T}}^E+{\mathbf{f}}^E=0.$$ (55)

Here, Eq. (55) describes the possible bending of the biomolecule due to the solvent-solute interaction potential force ${\mathbf{f}}_{\mathrm{FSI}}^E$ and the RF force ${\mathbf{f}}_{\mathrm{RF}}^E$, which originates from non-uniform charge distributions. The bending and curvature of vesicle membranes due to protein interactions are popular research topics.^52,119 However, most work in the literature is essentially qualitative. The Helfrisch curvature and phase field models provide interesting phenomenological descriptions to relatively simple geometries.^52,74,119 The proposed multiscale multiphysical models have a potential to provide new insights to the bending and curvature of macromolecular complexes.

In the present multiscale, multiphysics and multidomain theory, the generalized Laplace-Beltrami equation (35), Poisson equation (38), Nernst-Planck equation (42), Navier-Stokes equation (43), Newton's equations (46) and elastic equation (54) are directly or indirectly coupled to each other to form a system of governing equations for aqueous macromolecular complexes. Solution to these equations minimizes the total action functional and determines physical variables {S_I}, Φ, {ρ_α}, v, {z_j}, and w.

III Concluding remarks

Last decade has witnessed the continuous miniaturization of mechanical, chemical, thermal, optical, and electronic devices in the engineering sciences, meanwhile, an increased ability to manipulate large biomolecular complexes and subcellular organelles in biological sciences. These developments have led us to the exciting era of nanoscience and nanotechnology. However, nanoscale chemical, physical and biological systems pose fundamental challenges in theoretical modeling and numerical computation due to their excessively large number of degrees of freedom. Multiscale approaches are efficient strategies for dimensionality reduction of the aforementioned problems. The goal of multiscale analysis lies in developing new methodologies which sufficiently describe all the key physical observations, while dramatically reduce the total number of degrees of freedom so that the resulting systems are tackleable with the contemporary computer capability.

One of multiscale paradigms that is particularly suitable for the modeling and computation of aqueous chemical, physical and biological systems was introduced by the present author.¹⁶⁴ A major feature of this multiscale formalism is the use of differential geometry theory of surfaces as a versatile tool to geometrically divide the total computational domain into a macroscopic solvent domain and a microscopic solute domain, and dynamically coupling the continuum mechanics in the solvent domain and the discrete mechanics in the solute domain. An essential strategy of our approach is the use of energy functional to put multiphysics on an equal footing. Subsequently, key physical observables are served as the main variables of the energy functional and their variations give rise to coupled governing partial differential equations (PDEs). The solution of these PDEs results in the minimization of the energy functional. The incorporation of quantum descriptions further enhances the power of our multiscale formulation.^21,23,33 Our multiscale paradigm has been extensively validated with experimental measurements, such as solvent free energies,^{31–34,149,176} binding affinities,³² current-voltage curves,^21,23,166 etc. A limitation of our earlier theory is that only two domains, namely, a solvent domain and a solute domain, are employed. However, for large macromolecular complexes and sophisticated nano-bio devices, it is desirable to simultaneously invoke a number of differential physical descriptions for appropriate parts of the macromolecular complexes and/or nano-bio devices. The present work formulates such a multiscale, multiphysics and multidomain theory.

In the present formulation, the solvent domain is either represented with a dielectric continuum or equipped with fluid dynamics. A total of N different domains is assumed for macromolecular complexes. Depending on the need, these solute domains can be furnished with different physical descriptions, such as static atomistic point charges, molecular dynamics (MD), coarse grained dynamics, quantum mechanics, elasticity, etc. In all cases, the electrostatic interactions among all domains, which are delocalized and of long range, are carefully considered. Additionally, all the solvent-solute non-electrostatic interactions, including potential dipolar, multipole, dispersion and van der Waals interactions, are accounted. Moreover, ion-ion non-electrostatic interactions in the solvent domains are included. Appropriate force fields or interactions are assumed for MD, coarse-graining and quantum descriptions within their domains. To demonstrate these ideas, we have explicitly studied three multiscale, multiphysics and multidomain models. The first model is for solvation analysis. In this case, we consider a simple equilibrium system with dielectric continuum representation of the solvent and static charge density presentations of the solute complex, which is divided into multiple domains with different surface tensions, pressures, dielectric functions, and charge density distributions. The interactions among solvent components and between the solvent and the solute are accounted. Generalized Boltzmann distributions are used for ion densities. The morphology of each solute domain is governed by one generalized Laplace-Beltrami (LB) equation (i.e., geometric flow equation). Additionally, the multidomain Poisson-Boltzmann (PB) equation is obtained for the electrostatic potential. The second model is for charge transport in chemical, physical and biological systems. We formulate our theory for a non-equilibrium system whose generalized electrochemical potential does not vanish due to spatial inhomogeneities in densities and/or electrostatic potentials. For the dynamics of ion densities, the gradient flow approach is employed to construct a set of multidomain Nernst-Planck equations, which are coupled to a multidomain Poisson equation for the electrostatic potential. These equations are further coupled with a total of N LB equations, one for each macromolecular domain. We have illustrated that this LB-PNP model recovers the LB-PB theory at equilibrium. Finally, we develop a fluid-electro-MD-elastic model. In this model, additional Navier-Stokes equations, Newton's equations, and elasticity equation are systematically derived for electrostatic fluid dynamics, molecular dynamics, and elastic dynamics, respectively. Force balances within or between domains are obtained via the total variation.

Although quantum mechanical treatment is not explicitly described in the present work, it is straightforward to add the quantum description in one or few domains. The related quantum formulation in the framework of our differential geometry based multiscale models has been developed in our earlier work.³³ For simplicity, we have omitted quantum description in the present work.

It is worthwhile to point out that our earlier quantum dynamics in continuum formalism^21,23,24 is in fact a quantum density functional theory (qDFT). This method treats protons in the solvent quantum mechanically. The variation of its energy functionals results in a non-conventional Kohn-Sham equation for proton transport. Additionally, the LB-PNP model developed in our earlier work¹⁶⁶ and the present multidomain LB-PNP model are essentially non-convention density functional theory (DFT). Classical DFT of complex fluids^68,170 has found its success in microstructure prediction. Unlike the classical DFT, which depends on hard-sphere approximations for correlations, our DFT utilizes realistic potentials. It will be interesting to compare the performance of our DFT methods with that of the classical DFT for real world problems. This aspect will be investigated in our future work.

The numerical validation of the present multiscale multiphysics and multidomain models is under our consideration. An interesting numerical issue is the verification that the proposed integral-differential LBPB model is capable of predicting solvent microstructures near the solvent-solute interfaces. Currently, more expensive integral equation theories, including hyper-netted chain equation, Carnahan-Starling equation, Percus-Yevick equation and density functional theory of liquids, are employed to deliver solvent microstructures at equilibrium.^{13,63,68,129,153}

The multidomain methodology proposed in the present work, in conjugation with our multiscale and multiphysics paradigm, is potentially useful in many chemical, physical and biological systems, including deoxyribonucleic acid (DNA) nanowires, molecular junctions, fuel cells, solar cells, battery cells, molecular switches, nanotubes, field effect transistors, nanofibers, thin films, ion channels, ATPases, neuron synapses, etc.