# Numerical Treatment of Stokes Solvent Flow and Solute–Solvent Interfacial Dynamics for Nonpolar Molecules

## Abstract

We design and implement numerical methods for the incompressible Stokes solvent flow and solute–solvent interface motion for nonpolar molecules in aqueous solvent. The balance of viscous force, surface tension, and van der Waals type dispersive force leads to a traction boundary condition on the solute–solvent interface. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of a computational domain through a consistency condition. We use a finite difference ghost fluid scheme to discretize the Stokes equation with such boundary conditions. The method is tested to have a second-order accuracy. We combine this ghost fluid method with the level-set method to simulate the motion of the solute–solvent interface that is governed by the solvent fluid velocity. Numerical examples show that our method can predict accurately the blow up time for a test example of curvature flow and reproduce the polymodal (e.g., dry and wet) states of hydration of some simple model molecular systems.

### Keywords

Nonpolar molecules Solute–solvent interface The Stokes equation Ghost fluid method Level-set method Interface motion Change of volume Traction boundary conditions## 1 Introduction

Aqueous solvent plays a significant role in dynamical processes of biological molecules, such as conformational changes, molecular recognition, and molecular assembly, that control cellular functions of underlying biological systems [2, 12, 21, 22]. Implicit-solvent models are efficient descriptions of such dynamical processes. In such descriptions, the solvent is treated implicitly as a continuum and the effect of individual solvent molecules is coarse grained [3, 15, 23]. One of the successful dielectric boundary based implicit-solvent approaches is the variational implicit-solvent model (VISM) [7, 8]. In VISM, one minimizes a macroscopic solvation free-energy functional of all possible dielectric boundaries, coupling both nonpolar and polar contributions, and the solute–solvent van der Waals (vdW) interaction. Implemented by a robust level-set numerical method [4, 5], VISM can predict polymodal states of hydration, such as wet and dry states, subtle electrostatic effects, and solvation free energies of an underlying bimolecular system [4, 9, 18, 25, 28, 29].

While dielectric boundary based implicit-solvent models, including VISM, have been successful in many cases, they treat the solvent as a structureless dielectric medium, neglecting other solvent effects, such as the solvent hydrodynamic effect. Recent experimental and theoretical studies have indicated that the solvent shear motion can induce protein conformational changes and the solvent viscosity can affect the kinetics of such changes [1, 10, 11, 16, 17, 19, 20, 21, 24].

In several recent works [13, 14, 26, 27], the authors have initiated the development of a fluid mechanics approach to treat the solvent fluid in molecular systems. The key features of such a new approach include: (1) the aqueous solvent (i.e., water or salted water) is treated as an incompressible fluid and its motion is by the Stokes or Navier–Stokes equation; (2) the solute pressure is simply described by the ideal-gas law; (3) the electrostatic interactions are modeled by the Poisson or Poisson–Boltzmann equation; and (4) all viscous force, electrostatic force, and vdW force are balanced on the solute–solvent interface that moves with solvent velocity. White [26] proposed to add the Landau–Lifshitz random stress tensor in the Stokes equation to model the solvent fluctuations. They also propose to describe the electrostatic interaction through the dielectric boundary force, without introducing ionic charge densities in the solvent. They further applied their model, termed dynamical implicit-solvent model (DISM), to a charged spherical molecule to derive a generalized Rayleigh–Plesset equation, a stochastic ordinary differential equation for the fluctuating radius. With the same deterministic model, Li et al. [13] study the linear stability of a cylindrical solute–solvent interface, and conclude that the viscosity can modify the critical wavelength of such stabilities. Luo et al. [14, 27] make a connection of solvent fluid mechanics model with statistical mechanics theory. They also develop numerical methods to solve the solvent fluid equations. In particular, they design boundary conditions on the boundary of a computational domain to allow solutes to change their volumes. They also implement an augmented immersed interface method for the Navier–Stokes flow with moving interface.

- (1)
We discretize the Stokes equation using a ghost fluid finite difference scheme. We show that our scheme is second-order accurate;

- (2)
We refine the method proposed in [14, 27] to design artificial boundary conditions on the boundary of computational domain, allowing the change of solute volume;

- (3)
We couple the level-set method with our Stokes solver to track the motion of solute–solvent interface. Our method captures both dry and wet states for some simple model systems.

The rest of the paper is organized as follows: In Sect. 2, we present the solvent fluid model. In particular, we describe the boundary conditions for the Stokes equation. In Sect. 3, we describe our numerical schemes. In Sect. 4, four test examples are provided to show the accuracy of our numerical schemes. In Sect. 5, we compute the interface around two nonpolar spherical solute atoms and the interface around two nonpolar plates. In Sect. 6, we draw conclusions and discuss our future work.

## 2 A Solvent Fluid Model

*t*. At a given time

*t*, we denote by \(\mathbf{u}=(u,v): \Omega _+\rightarrow \mathbb {R}^2\) and \(p:\Omega _+\rightarrow \mathbb {R}\) the velocity and pressure of the solvent fluid, respectively. We also denote by \(p_-:\Omega _-\rightarrow {R}\) the pressure inside the solute region \(\Omega _-\).

- Interface motion:where a dot donates the time derivative.$$\begin{aligned} \dot{\mathbf{x}}=\mathbf{u}(\mathbf{x}(t))\quad \quad \text {for }\quad \mathbf{x}=\mathbf{x}(t)\in \Gamma , \end{aligned}$$(2.1)
- The Stokes equation for incompressible flow:$$\begin{aligned} {\left\{ \begin{array}{ll} \quad \mu \Delta \mathbf{u}-\nabla p+ \mathbf{G}=\mathbf{0}\quad &{}\text { in }\quad \Omega _+,\\ \quad \nabla \cdot \mathbf{u}=0\quad \quad \quad \quad \quad &{}\text { in }\quad \Omega _+. \end{array}\right. } \end{aligned}$$(2.2)
- The ideal-gas law:$$\begin{aligned} p_-\text {Vol}(\Omega _-)=C_m. \end{aligned}$$(2.3)
- Traction interface conditions for the fluid velocity and pressure:$$\begin{aligned} {\left\{ \begin{array}{ll} \mu \mathbf{n}\cdot D(\mathbf{u})\mathbf{n}-p+p_-=-\mathbf{f}\cdot \mathbf{n}\quad \quad &{}\text { for }\mathbf{x}\in \Gamma ,\\ \mu {\varvec{\tau }}\cdot D(\mathbf{u})\mathbf{n}=-\mathbf{f}\cdot {\varvec{\tau }}\quad \quad \quad \quad &{}\text { for }\mathbf{x}\in \Gamma .\\ \end{array}\right. } \end{aligned}$$(2.4)
- Boundary conditions on \(\partial \Omega \) for the velocity and pressure:$$\begin{aligned} \mathbf{u}=\mathbf{u}_0\quad \text { and }\quad p=p_{\infty }\quad \quad&\text {on }\quad \partial \Omega . \end{aligned}$$(2.5)

*i*. As usual, we denote \(D(\mathbf{u})=\nabla \mathbf{u}+\nabla \mathbf{u}^T\) to be the rate-of-strain tensor. The force \(\mathbf{f}\) in the traction boundary condition is given by

*H*is the mean curvature. The surface vdW force \(\mathbf{f}_\mathrm{vdW}\) is defined by [7, 8, 13, 29]

*i*. We have implicitly assumed that there are

*I*solute atoms inside the solute region \(\Omega _-\). The boundary velocity \(\mathbf{u}_0\) will be specified later. The boundary pressure \(p_{\infty }\) is a given function.

- A parabolic profile:$$\begin{aligned} {\left\{ \begin{array}{ll} \quad u(0,y)=C(t)(l_y-y)y,\quad \quad &{}v(0,y)=0,\\ \quad u(l_x,y)=-C(t)(l_y-y)y,\quad &{}v(l_x,y)=0,\\ \quad u(x,0)=0,\quad \quad &{}v(x,0)=C(t)(l_x-x)x,\\ \quad u(x,l_y)=0,\quad \quad &{}v(x,l_y)=-C(t)(l_x-x)x. \end{array}\right. } \end{aligned}$$(2.7)
- A circular profile:$$\begin{aligned} {\left\{ \begin{array}{ll} \quad u(0,y)=C(t)\frac{2l_x}{l_x^2+(2y-l_y)^2},\quad \quad &{}v(0,y)=C(t)\frac{2l_y-4y}{l_x^2+(2y-l_y)^2},\\ \quad u(l_x,y)=-C(t)\frac{2l_x}{l_x^2+(2y-l_y)^2},\quad &{}v(l_x,y)=C(t)\frac{2l_y-4y}{l_x^2+(2y-l_y)^2},\\ \quad u(x,0)=C(t)\frac{4x-2l_x}{(2x-l_x)^2+l_y^2},\quad \quad &{}v(x,0)=C(t)\frac{2l_y}{(2x-l_x)^2+l_y^2},\\ \quad u(x,l_y)=C(t)\frac{4x-2l_x}{(2x-l_x)^2+l_y^2},\quad \quad &{}v(x,l_y)=-C(t)\frac{2l_y}{(2x-l_x)^2+l_y^2}. \end{array}\right. } \end{aligned}$$(2.8)

*C*(

*t*) is to be determined by the consistency condition (2.6). The idea is that, under the assumption of the absence of external flow, the error of the boundary velocity prescribed by the profiles given in (2.7) and (2.8) would have relatively minor effect on the interface motion. This assumption is justified in Sect. 4 for a shrinking circle, whose analytical solution is well known.

*t*, i.e., \(\Gamma =\{\mathbf{x}\in \Omega :\phi (\mathbf{x},t)=0\}\). We also assume that \(\Omega _-=\{\mathbf{x}\in \Omega :\phi (\mathbf{x},t)<0\}\) and \(\Omega _+=\{\mathbf{x}\in \Omega :\phi (\mathbf{x},t)>0\}\). The level-set equation is

In summary, we couple the Stokes Eq. (2.2) in \(\Omega _+\) with the traction interface conditions (2.4) and the ideal-gas law (2.3), the consistency condition (2.6), and the Dirichlet boundary condition (2.5) for *p*, and (2.7) or (2.8) for \(\mathbf{u}\). The fluid velocity dictates the motion of \(\Gamma \), which is tracked by the level-set Eq. (2.9).

## 3 Numerical Methods

In this section, we introduce our numerical methods. We divide our computational domain \(\Omega =(0, l_x)\times (0, l_y)\) into \(n_x\times n_y\) grid cells, with \(n_x\) and \(n_y\) two positive integers. We denote \(h_x=l_x/n_x\) and \(h_y=l_y/n_y\), and define \(\mathbf{x}_{i,j}=\left( (i+1/2)h_x, (j+1/2)h_y\right) \), \(\mathbf{x}_{i\pm 1/2,j}=\left( (i+1/2\pm 1/2)h_x, (j+1/2)h_y\right) \), and \(\mathbf{x}_{i,j\pm 1/2}=\left( (i+1/2)h_x, (j+1/2\pm 1/2)h_y\right) \).

### 3.1 Discretization of the Stokes Equation

*u*,

*v*, and

*p*. We approximate

*p*at the center \(\mathbf{x}_{i,j}\) of each cell,

*u*at the midpoints of vertical cell edges \(\mathbf{x}_{i-1/2,j}\), and

*v*at the midpoints of horizontal cell edges \(\mathbf{x}_{i,j-1/2}\); cf. Fig. 2. For convenience, we define the regular points, boundary points, and ghost points as follows:

- (1)
The regular velocity points are those points on the edges of the cells and are located inside the fluid region \(\Omega _+\).

- (2)
The regular pressure points are those points on the center of the cells, of which at least one edge contains a regular velocity point.

- (3)
The boundary velocity points are those points on \(\partial \Omega \), or on an edge that intersects \(\partial \Omega \).

- (4)
The boundary pressure points are the center points \(\mathbf{x}_{i,j}\) of those boundary cells, each of which has at least one edge entirely on \(\partial \Omega \).

- (5)
A ghost velocity point is a point located either inside \(\Omega _-\) or on the interface \(\Gamma \), and is a neighbor to a regular velocity point. Two velocity points are neighbors of each other if the edges they are on share a same vertex or if they are on the edges of the same cell.

*C*(

*t*), which is determined by discretizing (2.6). We approximate the left-hand side of (2.6) by numerical integrating along the lines \(x=0\), \(l_x\) and \(y=0\), \(l_y\):

*p*at \(\mathbf{y}_{i+1/2,j}\) and \(\mathbf{y}_{i,j+1/2}\), respectively. Thus, we obtain a third-order approximation to the traction interface conditions at the projection points. This third-order approximation scheme on the interface covers various cases of the local geometry of interface. In some other cases, where the geometry of \(\Gamma \) does not allow enough points for a third order scheme, we switch to a second order or even first order scheme, which causes the solution to be reverted to a lower order approximation. We include a detailed discussion and formulation of the discretization scheme for ghost velocity points in the “Appendix”.

*u*,

*v*at the fluid points,

*p*at the pressure points, and

*C*(

*t*). The discretization matrix \(\mathcal {A}\) is asymmetric and sparse. It takes the following form

*u*,

*v*, and

*p*points and introduces many nonzero off-diagonal entries. This linear problem is similar in structure to a saddle point problem. It may suggests a solution method involving the Schur complement reduction. However, the highly coupled interface conditions together with the geometry dependent discretization make the conditional number of \(\mathcal {A}\) very high. Furthermore, submatrices such as \(A_u\), \(A_v\) are far away from diagonal dominant. At a few ghost points, the diagonal entries have their magnitude smaller than off-diagonals. As a consequence, it is not efficient to apply any Krylov subspace solver to even a subproblem. Leaving the development of numerical algorithms to future work, as a first step, we use UMFPACK, an implementation of a direct multifrontal sparse LU factorization method, to solve this system [6]. The time cost for UMFPACK to solve a sparse linear system of size \(\sim 30,000\)-by-30, 000 is

*O*(0.1) seconds.

### 3.2 Solving the Level-Set Equation

*t*, then \(\Gamma \) no longer moves, and a steady state interface is reached.

To approximate the normal velocity \(u_n(\mathbf {x})\), we use the level-set function, to interpolate the points on the interface, calculate the normal fluid velocity on these points, and use a fast marching algorithm to extend the values in the normal direction to all spatial grid points. Note that we here use the same grid as that for the discretization of the pressure *p*. For spatial derivatives, we use a fifth-order WENO method to approximate \(|\nabla \phi ^{(k)}(\mathbf {x})|\). We use the homogeneous Neumann boundary conditions at the outer boundary of the computational box.

*t*is different from that in the original level-set equation.

## 4 Numerical Tests

### 4.1 Flow Outside a Circular Region

*u*,

*v*, and

*p*to this test example are plotted in Fig. 3 on an \(N\times N\) spatial grid with \(N=400\). We analyze the error between the numerical solutions and the analytical solutions (4.2), by generating in Fig. 4 six log-log plots of the \(L^2\)-norm and \(L^{\infty }\)-norm of the error for

*u*,

*v*, and

*p*versus

*N*, the number of grid points in both

*x*and

*y*directions. The log-log plots show that the error for

*u*,

*v*, and

*p*all decay in an order of \(O(N^{-2})\) in both \(L^2\)-norm and \(L^{\infty }\)-norm, with spikes intermittently. These spikes arise due to the insufficient grid resolution for the curved interfacial geometry, resulting unpredicted sudden increase of the conditional number of discretization matrix. In average, the conditional number increases in the order of \(O(N^3)\). We believe that such spikes can be reduced by discretizing the interface condition using a least-squares approach.

### 4.2 Flow Outside a Clover Shaped Region

*u*,

*v*, and

*p*are plotted in Fig. 5, with the spatial discretization parameter \(N=400\) in both

*x*and

*y*directions. An analysis of the error between the numerical and analytical solutions shows similar behavior than the previous example in terms of \(L^2\)-norm and \(L^{\infty }\)-norm of the error on the three fluid components

*u*,

*v*, and

*p*, as shown in Fig. 6. We get an average second-order convergence in

*u*,

*v*, and

*p*.

### 4.3 Flow Around a Disk with Designed Numerical Boundary Conditions

*p*on \(\Omega =(0,1)\times (0,1)\) with \(p_{\infty }=0\), and the boundary velocity profile given by (2.7) or (2.8). In Fig. 7, we plot the solution with profile (2.7) and \(n_x=n_y=N=101\). The tendency of shrinking of the volume of \(\Omega _-\) can be observed from the inward velocity field.

*N*, the flux \(\zeta C(t)\) approaches a constant. In Fig. 8, we plot the flux \(\zeta C(t)\) as a function of

*N*. We see that for both (2.7) and (2.8), \(\zeta C(t)\) converges to a certain value. For (2.8), \(\zeta C(t)\) approaches the analytical value \(0.04\pi \), because the circular velocity profile is exact in this case. For (2.7), \(\zeta C(t)\) approaches to a value which is slightly less than, but reasonably close to \(0.04\pi \). This shows that even though the assumptions of boundary velocity profiles are artificial, they can be expected to work relatively accurately. To test the convergence of \(\zeta C(t)\) with respect to increasing

*N*, we take the value of \(\zeta C(t)\) at \(N=600\) as a reference value for both (2.7) and (2.8). We then compute the absolute values of the differences between the fluxes at the other values of

*N*and the reference values. This process gives us the convergence plot in Fig. 8, from which we observe a convergence rate of roughly first order for (2.7) and second order for (2.8).

### 4.4 Moving Interface Driven by Solvent Fluid Flow with Curvature Force

*p*and \(\mathbf{u}\) solve the Stokes Eq. (2.2) with \(\mu =1\) and \(\mathbf{G}=\mathbf{0}\) in the region \(\mathbb {R}^2\setminus \overline{\Omega _-(t)}\), where

*p*and \(\mathbf{u}\) satisfy the interface condition (2.4) on \(\Gamma (t)=\partial \Omega _-(t)\) with \(p_-=0\) and \(\mathbf{f}=-\mathbf{n}/r(t)\) (i.e., the curvature flow). The time needed for the circle \(\Gamma (t)\) to shrink to the center (0.5, 0.5) is \(t=0.4\).

*H*is the curvature, and the boundary conditions \(p_{\infty }=0\) and boundary profile (2.7) and (2.8). Our numerical solution for \(\Gamma (t)\) as plotted in Fig. 9 exhibits circular shrinking with constant speed. The center is slightly shifted due to small numerical error. The numerical critical time \(t_c\) for both boundary profiles (2.7) and (2.8) agree well with the analytic value.

## 5 Applications

In this section, we present two examples of application of our method to two model nonpolar molecular systems. One is a two-particle system and the other is a two plate system.

### 5.1 A Two-Particle System

*c*: (1) \(l_x=1\), \(c=0.05\); (2) \(l_x=1\), \(c=0.2\); (3) \(l_x=2\), \(c=0.25\). The final steady state for \(\Gamma \) is plotted in Fig. 10. We see that the interface breaks into two parts as the distance between the two particles is large enough; cf. case (3).

### 5.2 A Two-Plate System

*c*an adjustable constant. With certain parameter choices, one observe dry/wet polymodal states depending on the initial solute region \(\Omega _-\). We choose \(c=0.15\), \(\gamma =5\), \(\varepsilon = 25\), \(\sigma =0.1\), \(C_m=0.001\) in (5.2). Moreover, for a loose initial condition of \(\Gamma \), we define

## 6 Conclusions

In this paper, we model the aqueous solvent by the incompressible Stokes flow and treat the solute with the ideal-gas law. The solute–solvent interface moves with the solvent fluid flow. All the viscous, pressure, surface tension, and the solute–solvent vdW forces are balanced on the interface, leading to a traction boundary condition. To allow the change of solute volume, we design special numerical boundary conditions on the boundary of our computation domain. We design a second-order ghost fluid method for solving the Stokes equation. We also couple the level-set equation for the moving solute–solvent interface. Our methods accurately predict the blowup time of a shrinking bubble under surface tension. Moreover, our methods capture the dry and wet polymodal interfaces for the two-plate system.

Some existing issues of this model include: (1) The boundary conditions assume a velocity profile, which may not be realistic; (2) The numerical error is sensitive to the location of the interface; (3) We solve the linear system using a direct LU factorization, which can be problematic in extension to three dimensions.

As a first step in the future, we need to propose more realistic boundary conditions. We also plan to decrease the sensitivity of error dependence on the interface by considering a least square problem. Moreover, an extension of our two dimensional fluid solver to three dimensions is necessary. After we develop such a fluid solver, we can add noise to the solvent and observe the fluctuations of the interface. Through a combination of such a three dimensional fluid solver with fluctuations and a robust Poisson–Boltzmann solver, one can better observe and describe the dynamics of a solvent–solute system.

## Notes

### Acknowledgments

This work was supported by the US National Science Foundation (NSF) through grant DMS-1319731 and the US National Institutes of Health (NIH) through grant R01GM096188. Work in McCammon’s group is supported in part by NSF, NIH, HHMI, and NBCR. The authors thank Dr. Robert Krasny, Dr. Ray Luo, and Mr. Li Xiao for helpful discussions.

### References

- 1.Alexander-Katz, A., Schneider, M.F., Schneider, S.W., Wixforth, A., Netz, R.R.: Shear-flow-induced unfolding of polymeric globules. Phys. Rev. Lett.
**97**, 138101 (2006)CrossRefGoogle Scholar - 2.Baron, R., McCammon, J.A.: Molecular recognition and ligand association. Annu. Rev. Phys. Chem.
**64**, 151–175 (2013)CrossRefGoogle Scholar - 3.Chen, J., Brooks III, C.L., Khandogin, J.: Recent advances in implicit solvent based methods for biomolecular simulations. Curr. Opin. Struct. Biol.
**18**, 140–148 (2008)CrossRefGoogle Scholar - 4.Cheng, L., Dzubiella, J., McCammon, J.A., Li, B.: Application of the level-set method to the implicit solvation of nonpolar molecules. J. Chem. Phys.
**127**, 084503 (2007)CrossRefGoogle Scholar - 5.Cheng, L., Li, B., Wang, Z.: Level-set minimization of potential controlled hadwiger valuations for molecular solvation. J. Comput. Phys.
**229**, 8497–8510 (2010)MathSciNetCrossRefMATHGoogle Scholar - 6.Davis, T.A.: Algorithm 832: UMFPACK v4.3-an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw.
**30**, 196–199 (2004)MathSciNetCrossRefMATHGoogle Scholar - 7.Dzubiella, J., Swanson, J., McCammon, J.: Coupling hydrophobicity, dispersion, and electrostatics in continuum solvent models. Phys. Rev. Lett.
**96**, 087802 (2006)CrossRefGoogle Scholar - 8.Dzubiella, J., Swanson, J., McCammon, J.: Coupling nonpolar and polar solvation free energies in implicit solvent models. J. Chem. Phys.
**124**, 084905 (2006)CrossRefGoogle Scholar - 9.Guo, Z., Li, B., Dzubiella, J., Cheng, L.-T., McCammon, J.A., Che, J.: Heterogeneous hydration of p53/MDM2 complex. J. Chem. Theory Comput.
**10**, 1302–1313 (2014)CrossRefGoogle Scholar - 10.Hagen, S.J.: Solvent viscosity and friction in protein folding dynamics. Curr. Protein Pept. Sci.
**11**, 385–395 (2010)CrossRefGoogle Scholar - 11.Klimov, D.K., Thirumalai, D.: Viscosity dependence of folding rates of protein. Phys. Rev. Lett.
**79**, 317–320 (1997)CrossRefGoogle Scholar - 12.Levy, Y., Onuchic, J.N.: Water mediation in protein folding and molecular recognition. Annu. Rev. Biophys. Biomol. Struct.
**35**, 389–415 (2006)CrossRefGoogle Scholar - 13.Li, B., Sun, H., Zhou, S.: Stability of a cylindrical solute–solvent interface: effect of geometry, electrostatics, and hydrodynamics. SIAM J. Appl. Math.
**75**, 907–928 (2015)MathSciNetCrossRefMATHGoogle Scholar - 14.Li, Z., Cai, Q., Zhao, H., Luo, R.: A semi-implicit augmented IIM for Navier–Stokes equations with open and traction boundary conditions. J. Comput. Phys.
**297**, 182–193 (2015)MathSciNetCrossRefGoogle Scholar - 15.Roux, B., Simonson, T.: Implicit solvent models. Biophys. Chem.
**78**, 1–20 (1999)CrossRefGoogle Scholar - 16.Schneider, S.W., Nuschele, S., Wixforth, A., Gorzelanny, C., Alexander-Katz, A., Netz, R.R., Schneider, M.F.: Shear-induced unfolding triggers adhesion of von Willebrand factor fibers. Proc. Natl. Acad. Sci. USA
**104**, 7899–7903 (2007)CrossRefGoogle Scholar - 17.Sekhar, A., Latham, M.P., Vallurupalli, P., Kay, L.E.: Viscosity-dependent kinetics of protein conformational exchange: microviscosity effects and the need for a small viscogen. J. Phys. Chem. B
**118**, 4546–4551 (2014)CrossRefGoogle Scholar - 18.Setny, P., Wang, Z., Cheng, L.T., Li, B., McCammon, J.A., Dzubiella, J.: Dewetting-controlled binding of ligands to hydrophobic pockets. Phys. Rev. Lett.
**103**, 187801 (2009)CrossRefGoogle Scholar - 19.Siedlecki, C.A., Lestini, B.J., Kottke-Marchant, K.K., Eppell, S.J., Wilson, D.L., Marchant, R.E.: Shear-dependent changes in the three-dimensional structure of human von Willebrand factor. Blood
**88**, 2939–2950 (1996)Google Scholar - 20.Singh, I., Themistou, E., Porcar, L., Neelamegham, S.: Fluid shear induces conformation change in human blood protein von Willebrand factor in solution. Biophys. J.
**96**, 2313–2320 (2009)CrossRefGoogle Scholar - 21.Szymczak, P., Cieplak, M.: Hydrodynamic effects in proteins. J. Phys.: Condens. Matter
**23**, 033102 (2011)Google Scholar - 22.Tanford, C.: The Hydrophobic Effect: Formation of Micelles and Biological Membranes. Wiley, New York (1973)Google Scholar
- 23.Tomasi, J., Persico, M.: Molecular interactions in solution: an overview of methods based on continuous distributions of the solvent. Chem. Rev.
**94**, 2027–2094 (1994)CrossRefGoogle Scholar - 24.Vergauwe, R.M.A., Uji-i, H., De Ceunynck, K., Vermant, J., Vanhoorelbeke, K., Hofkens, J.: Shear-stress-induced conformational changes of von Willebrand factor in water-glycerol mixture observed with single molecule microscopy. J. Phys. Chem. B
**118**, 5660–5669 (2014)CrossRefGoogle Scholar - 25.Wang, Z., Che, J., Cheng, L., Dzubiella, J., Li, B., McCammon, J.A.: Level-set variational implicit-solvent modeling of biomolecules with the coulomb-field approximation. J. Chem. Theory Comput.
**8**, 386–397 (2012)CrossRefGoogle Scholar - 26.White, M.: Mathematical Theory and Numerical Methods for Biomolecular Modeling. PhD thesis, University of California, San Diego (2015)Google Scholar
- 27.Xiao, L., Cai, Q., Li, Z., Zhao, H., Luo, R.: A multi-scale method for dynamics simulation in continuum solvents I: finite-difference algorithm for Navier-Stokes equation. Chem. Phys. Lett.
**616**, 67–74 (2014)CrossRefGoogle Scholar - 28.Zhou, S., Cheng, L., Sun, H., Che, J., Dzubiella, J., Li, B., McCammon, J.A.: LS-VISM: a software package for analysis of biomolecular solvation. J. Comput. Chem.
**36**, 1047–1059 (2015)CrossRefGoogle Scholar - 29.Zhou, S., Cheng, L.-T., Dzubiella, J., Li, B., McCammon, J.A.: Variational implicit solvation with Poisson–Boltzmann theory. J. Chem. Theory Comput.
**10**, 1454–1467 (2014)CrossRefGoogle Scholar