Abstract

Membrane channel proteins control the diffusion of ions across biological membranes. They are closely related to the processes of various organizational mechanisms, such as: cardiac impulse, muscle contraction and hormone secretion. Introducing a membrane region into implicit solvation models extends the ability of the Poisson–Boltzmann (PB) equation to handle membrane proteins. The use of lateral periodic boundary conditions can properly simulate the discrete distribution of membrane proteins on the membrane plane and avoid boundary effects, which are caused by the finite box size in the traditional PB calculations. In this work, we: (1) develop a first finite element solver (FEPB) to solve the PB equation with a two-dimensional periodicity for membrane channel proteins, with different numerical treatments of the singular charges distributions in the channel protein; (2) add the membrane as a dielectric slab in the PB model, and use an improved mesh construction method to automatically identify the membrane channel/pore region even with a tilt angle relative to the z-axis; and (3) add a non-polar solvation energy term to complete the estimation of the total solvation energy of a membrane protein. A mesh resolution of about 0.25 Å (cubic grid space)/0.36 Å (tetrahedron edge length) is found to be most accurate in linear finite element calculation of the PB solvation energy. Computational studies are performed on a few exemplary molecules. The results indicate that all factors, the membrane thickness, the length of periodic box, membrane dielectric constant, pore region dielectric constant, and ionic strength, have individually considerable influence on the solvation energy of a channel protein. This demonstrates the necessity to treat all of those effects in the PB model for membrane protein simulations.

Highlights

  • The Poisson–Boltzmann (PB) equation is one of the most popular implicit models to describe the solvent effect through the Boltzmann distribution [1,2,3,4,5,6,7,8,9,10,11,12]

  • There are three main numerical techniques based on the discretization of the domain of interest into small regions: the finite difference method (FDM), the boundary element method (BEM), and the finite element method (FEM)

  • Application of Lateral Periodic Boundary Condition In Section 2.2.1, We use a DNA molecule to examine the effects of lateral periodic boundary conditions and compare different numerical methods for solution of the PB equation; In Section 2.2.2, We study the effects of the ion strength and membrane thickness on the electrostatic solvation energy of the ion channel gramicidin A (gA), and further discuss the importance of element recognition for the channel region

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Summary

Introduction

The Poisson–Boltzmann (PB) equation is one of the most popular implicit models to describe the solvent effect through the Boltzmann distribution [1,2,3,4,5,6,7,8,9,10,11,12]. A primitive way to deal with the issue is to manually define the pore region as a combination of multiple spheres or cylinders This method is neither efficient nor practical because any change on the radius of each membrane channel protein needs to be made by hand. There are multiple proteins scattered on the membrane, a lateral periodic boundary condition can used as a good approximation to simulating the real membrane environment The use of this type of boundary conditions can avoid boundary effects caused by the finite box size in the traditional PB calculations by using a fixed boundary potential value. The final state of a molecule is a result of the balance between the non-polar solvation free energy and the electrostatic solvation free energy In this way, we can estimate the possible tilt angle of the membrane channel protein by calculating the solvation energy in membrane environment

Validation for the Treatments of Fixed Singular Charge
Application of Lateral Periodic Boundary Condition
Validation of Lateral Periodic Boundary Conditions
Application to the Channel Protein
Non-Polar Contribution to Solvation Energy
Finite Element Method of Poisson–Boltzmann Equation
Mesh Construction for Membrane Protein System
Treatments of Fixed Singular Charges
Conclusions
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