Stability of a Cylindrical Solute-Solvent Interface: Effect of Geometry, Electrostatics, and Hydrodynamics

Abstract

The solute-solvent interface that separates biological molecules from their surrounding aqueous solvent characterizes the conformation and dynamics of such molecules. In this work, we construct a solvent fluid dielectric boundary model for the solvation of charged molecules and apply it to study the stability of a model cylindrical solute-solvent interface. The motion of the solute-solvent interface is defined to be the same as that of solvent fluid at the interface. The solvent fluid is assumed to be incompressible and is described by the Stokes equation. The solute is modeled simply by the ideal-gas law. All the viscous force, hydrostatic pressure, solute-solvent van der Waals interaction, surface tension, and electrostatic force are balanced at the solute-solvent interface. We model the electrostatics by Poisson’s equation in which the solute-solvent interface is treated as a dielectric boundary that separates the low-dielectric solute from the high-dielectric solvent. For a cylindrical geometry, we find multiple cylindrically shaped equilibrium interfaces that describe polymodal (e.g., dry and wet) states of hydration of an underlying molecular system. These steady-state solutions exhibit bifurcation behavior with respect to the charge density. For their linearized systems, we use the projection method to solve the fluid equation and find the dispersion relation. Our asymptotic analysis shows that, for large wavenumbers, the decay rate is proportional to wavenumber with the proportionality half of the ratio of surface tension to solvent viscosity, indicating that the solvent viscosity does affect the stability of a solute-solvent interface. Consequences of our analysis in the context of biomolecular interactions are discussed.

A multi-scale method for dynamics simulation in continuum solvent models. I: Finite-difference algorithm for Navier–Stokes equation

“…al presented an interesting method to calculate the translational friction and intrinsic viscosity, and found a good agreement with experiment [17]. Interestingly, viscosity also plays an important role in the stability of solute-solvent interface, which is recently demonstrated by a model with cylindrical solute-solvent interface [18]. …”

Stochastic level-set variational implicit-solvent approach to solute-solvent interfacial fluctuations

“…79-81. Within our dielectric-boundary based VISM, which is efficient in the description of electrostatic effects, we can possibly include the solvent fluctuation through a fluid mechanics approach using the Landau-Lifshitz random stress tensor, 82,83 which, though, can be more computationally expensive. Another possible approach is to design those spatial "basis" functions ψ i (x) in the noise (3.4) to describe the solvent fluctuation, similar to a gas-lattice model, 76,77 somewhat ad hoc but can be straightforward and efficient.…”

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

“…Our governing equations and boundary conditions are as follows [13, 26]: Interface motion: $$\dot{\mathbf{x}}=\mathrm{boldu}false(\mathrm{boldx}false(tfalse)false)\text{for}\mathrm{boldx}=\mathrm{boldx}false(tfalse)\in \mathrm{normal\Gamma },$$where a dot donates the time derivative.The Stokes equation for incompressible flow: $$\{\begin{array}{c}\mu \mathrm{\Delta }\mathbf{u}-\nabla p+\mathbf{G}=\mathbf{0}\text{in}{\mathrm{normal\Omega }}_{+},\\ \nabla ·\mathbf{u}=0\text{in}{\mathrm{normal\Omega }}_{+}.\end{array}$$The ideal-gas law: $$p_{-}\text{Vol}false({\mathrm{\Omega }}_{-}false)=C_m.$$Traction interface conditions for the fluid velocity and pressure: $$\{\begin{array}{c}\mu \mathbf{n}·D(\mathbf{u})\mathbf{n}-p+p_{-}=-\mathbf{f}·\mathbf{n}\text{for}\mathbf{x}\in \mathrm{\Gamma },\\ \mu \mathit{\tau }·D(\mathbf{u})\mathbf{n}=-\mathbf{f}·\mathit{\tau }\text{for}\mathbf{x}\in \mathrm{\Gamma }.\end{array}$$Boundary conditions on ∂ Ω for the velocity and pressure: $$\mathrm{boldu}={\mathbf{u}}_0\text{and}p=p_{\infty }\text{on}\partial \mathrm{normal\Omega }.$$…”

“…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.…”

“…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.…”