1
|
Ding J, Sun H, Zhou S. Hysteresis and linear stability analysis on multiple steady-state solutions to the Poisson-Nernst-Planck equations with steric interactions. Phys Rev E 2020; 102:053301. [PMID: 33327140 DOI: 10.1103/physreve.102.053301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2020] [Accepted: 09/14/2020] [Indexed: 06/12/2023]
Abstract
In this work, we numerically study linear stability of multiple steady-state solutions to a type of steric Poisson-Nernst-Planck equations with Dirichlet boundary conditions, which are applicable to ion channels. With numerically found multiple steady-state solutions, we obtain S-shaped current-voltage and current-concentration curves, showing hysteretic response of ion conductance to voltages and boundary concentrations with memory effects. Boundary value problems are proposed to locate bifurcation points and predict the local bifurcation diagram near bifurcation points on the S-shaped curves. Numerical approaches for linear stability analysis are developed to understand the stability of the steady-state solutions that are only numerically available. Finite difference schemes are proposed to solve a derived eigenvalue problem involving differential operators. The linear stability analysis reveals that the S-shaped curves have two linearly stable branches of different conductance levels and one linearly unstable intermediate branch, exhibiting classical bistable hysteresis. As predicted in the linear stability analysis transition dynamics, from a steady-state solution on the unstable branch to one on the stable branches, are led by perturbations associated to the mode of the dominant eigenvalue. Further numerical tests demonstrate that the finite difference schemes proposed in the linear stability analysis are second-order accurate. Numerical approaches developed in this work can be applied to study linear stability of a class of time-dependent problems around their steady-state solutions that are computed numerically.
Collapse
Affiliation(s)
- Jie Ding
- Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, 1 Shizi Street, Suzhou 215006, Jiangsu, China
| | - Hui Sun
- Department of Mathematics and Statistics, California State University, Long Beach, California 90840, USA
| | - Shenggao Zhou
- Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, 1 Shizi Street, Suzhou 215006, Jiangsu, China
| |
Collapse
|
2
|
Zhou S, Sun H, Cheng LT, Dzubiella J, Li B, McCammon JA. Stochastic level-set variational implicit-solvent approach to solute-solvent interfacial fluctuations. J Chem Phys 2016; 145:054114. [PMID: 27497546 PMCID: PMC4975753 DOI: 10.1063/1.4959971] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2016] [Accepted: 07/05/2016] [Indexed: 12/29/2022] Open
Abstract
Recent years have seen the initial success of a variational implicit-solvent model (VISM), implemented with a robust level-set method, in capturing efficiently different hydration states and providing quantitatively good estimation of solvation free energies of biomolecules. The level-set minimization of the VISM solvation free-energy functional of all possible solute-solvent interfaces or dielectric boundaries predicts an equilibrium biomolecular conformation that is often close to an initial guess. In this work, we develop a theory in the form of Langevin geometrical flow to incorporate solute-solvent interfacial fluctuations into the VISM. Such fluctuations are crucial to biomolecular conformational changes and binding process. We also develop a stochastic level-set method to numerically implement such a theory. We describe the interfacial fluctuation through the "normal velocity" that is the solute-solvent interfacial force, derive the corresponding stochastic level-set equation in the sense of Stratonovich so that the surface representation is independent of the choice of implicit function, and develop numerical techniques for solving such an equation and processing the numerical data. We apply our computational method to study the dewetting transition in the system of two hydrophobic plates and a hydrophobic cavity of a synthetic host molecule cucurbit[7]uril. Numerical simulations demonstrate that our approach can describe an underlying system jumping out of a local minimum of the free-energy functional and can capture dewetting transitions of hydrophobic systems. In the case of two hydrophobic plates, we find that the wavelength of interfacial fluctuations has a strong influence to the dewetting transition. In addition, we find that the estimated energy barrier of the dewetting transition scales quadratically with the inter-plate distance, agreeing well with existing studies of molecular dynamics simulations. Our work is a first step toward the inclusion of fluctuations into the VISM and understanding the impact of interfacial fluctuations on biomolecular solvation with an implicit-solvent approach.
Collapse
Affiliation(s)
- Shenggao Zhou
- Department of Mathematics and Mathematical Center for Interdiscipline Research, Soochow University, 1 Shizi Street, Jiangsu, Suzhou 215006, China
| | - Hui Sun
- Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112, USA
| | - Li-Tien Cheng
- Department of Mathematics, University of California, San Diego, La Jolla, California 92093-0112, USA
| | - Joachim Dzubiella
- Soft Matter and Functional Materials, Helmholtz-Zentrum Berlin, 14109 Berlin, Germany and Institut für Physik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany
| | - Bo Li
- Department of Mathematics and Quantitative Biology Graduate Program, University of California, San Diego, La Jolla, California 92093-0112, USA
| | - J Andrew McCammon
- Department of Chemistry and Biochemistry, Department of Pharmacology, Howard Hughes Medical Institute, University of California, San Diego, La Jolla, California 92093-0365, USA
| |
Collapse
|
3
|
Sun H, Zhou S, Moore DK, Cheng LT, Li B. Numerical Treatment of Stokes Solvent Flow and Solute-Solvent Interfacial Dynamics for Nonpolar Molecules. JOURNAL OF SCIENTIFIC COMPUTING 2016; 67:705-723. [PMID: 27365866 PMCID: PMC4922513 DOI: 10.1007/s10915-015-0099-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2015] [Revised: 09/02/2015] [Accepted: 09/05/2015] [Indexed: 06/06/2023]
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.
Collapse
Affiliation(s)
- Hui Sun
- Department of Mathematics, University of California, San Diego, CA 92093
| | - Shenggao Zhou
- School of Mathematical Sciences and Mathematical Center for Interdiscipline Research, Soochow University, 1 Shizi Street, Suzhou, Jiangsu 215006, China
| | - David K. Moore
- Department of Physics, University of California, San Diego, CA 92093
| | - Li-Tien Cheng
- Department of Mathematics, University of California, San Diego, CA 92093
| | - Bo Li
- Department of Mathematics and Quantitative Biology Graduate Program, University of California, San Diego, CA 92093
| |
Collapse
|
4
|
Xiao L, Cai Q, Li Z, Zhao H, Luo R. A Multi-Scale Method for Dynamics Simulation in Continuum Solvent Models I: Finite-Difference Algorithm for Navier-Stokes Equation. Chem Phys Lett 2014; 616-617:67-74. [PMID: 25404761 DOI: 10.1016/j.cplett.2014.10.033] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Abstract
A multi-scale framework is proposed for more realistic molecular dynamics simulations in continuum solvent models by coupling a molecular mechanics treatment of solute with a fluid mechanics treatment of solvent. This article reports our initial efforts to formulate the physical concepts necessary for coupling the two mechanics and develop a 3D numerical algorithm to simulate the solvent fluid via the Navier-Stokes equation. The numerical algorithm was validated with multiple test cases. The validation shows that the algorithm is effective and stable, with observed accuracy consistent with our design.
Collapse
Affiliation(s)
- Li Xiao
- Department of Biomedical Engineering, University of California, Irvine, CA 92697 ; Department of Molecular Biology and Biochemistry, University of California, Irvine, CA 92697
| | - Qin Cai
- Department of Biomedical Engineering, University of California, Irvine, CA 92697 ; Department of Molecular Biology and Biochemistry, University of California, Irvine, CA 92697
| | - Zhilin Li
- Department of Mathematics, North Carolina State University, Raleigh, NC 27695
| | - Hongkai Zhao
- Department of Mathematics, University of California, Irvine, CA 92697
| | - Ray Luo
- Department of Biomedical Engineering, University of California, Irvine, CA 92697 ; Department of Molecular Biology and Biochemistry, University of California, Irvine, CA 92697 ; Department of Chemical Engineering and Materials Science, University of California, Irvine, CA 92697
| |
Collapse
|