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Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates.

ABSTRACT: The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform particle sizes. The numerical tractability of the model is demonstrated as well. The main contributions of this study are as follows. 1), We show that an (arbitrarily) size-modified PB model is indeed implied by the SMPNP equations under certain boundary/interface conditions, and can be reproduced through numerical solutions of the SMPNP. 2), The size effects in the SMPNP effectively reduce the densities of highly concentrated counterions around the biomolecule. 3), The SMPNP is applied to the diffusion-reaction process for the first time, to our knowledge. In the case of low substrate density near the enzyme reactive site, it is observed that the rate coefficients predicted by SMPNP model are considerably larger than those by the PNP model, suggesting both ions and substrates are subject to finite size effects. 4), An accurate finite element method and a convergent Gummel iteration are developed for the numerical solution of the completely coupled nonlinear system of SMPNP equations.

SUBMITTER: Lu B

PROVIDER: S-EPMC3093557 | biostudies-literature | 2011 May

REPOSITORIES: biostudies-literature

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Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates.

Lu Benzhuo B Zhou Y C YC

Biophysical journal 20110501 10

The effects of finite particle size on electrostatics, density profiles, and diffusion have been a long existing topic in the study of ionic solution. The previous size-modified Poisson-Boltzmann and Poisson-Nernst-Planck models are revisited in this article. In contrast to many previous works that can only treat particle species with a single uniform size or two sizes, we generalize the Borukhov model to obtain a size-modified Poisson-Nernst-Planck (SMPNP) model that is able to treat nonuniform ...[more]

PMID: 21575582

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Project description:The Poisson-Nernst-Planck (PNP) model is based on a mean-field approximation of ion interactions and continuum descriptions of concentration and electrostatic potential. It provides qualitative explanation and increasingly quantitative predictions of experimental measurements for the ion transport problems in many areas such as semiconductor devices, nanofluidic systems, and biological systems, despite many limitations. While the PNP model gives a good prediction of the ion transport phenomenon for chemical, physical, and biological systems, the number of equations to be solved and the number of diffusion coefficient profiles to be determined for the calculation directly depend on the number of ion species in the system, since each ion species corresponds to one Nernst-Planck equation and one position-dependent diffusion coefficient profile. In a complex system with multiple ion species, the PNP can be computationally expensive and parameter demanding, as experimental measurements of diffusion coefficient profiles are generally quite limited for most confined regions such as ion channels, nanostructures and nanopores. We propose an alternative model to reduce number of Nernst-Planck equations to be solved in complex chemical and biological systems with multiple ion species by substituting Nernst-Planck equations with Boltzmann distributions of ion concentrations. As such, we solve the coupled Poisson-Boltzmann and Nernst-Planck (PBNP) equations, instead of the PNP equations. The proposed PBNP equations are derived from a total energy functional by using the variational principle. We design a number of computational techniques, including the Dirichlet to Neumann mapping, the matched interface and boundary, and relaxation based iterative procedure, to ensure efficient solution of the proposed PBNP equations. Two protein molecules, cytochrome c551 and Gramicidin A, are employed to validate the proposed model under a wide range of bulk ion concentrations and external voltages. Extensive numerical experiments show that there is an excellent consistency between the results predicted from the present PBNP model and those obtained from the PNP model in terms of the electrostatic potentials, ion concentration profiles, and current-voltage (I-V) curves. The present PBNP model is further validated by a comparison with experimental measurements of I-V curves under various ion bulk concentrations. Numerical experiments indicate that the proposed PBNP model is more efficient than the original PNP model in terms of simulation time.

Project description:The Poisson Nernst-Planck (PNP) theory is a simplified continuum model for a wide variety of chemical, physical and biological applications. Its ability of providing quantitative explanation and increasingly qualitative predictions of experimental measurements has earned itself much recognition in the research community. Numerous computational algorithms have been constructed for the solution of the PNP equations. However, in the realistic ion-channel context, no second order convergent PNP algorithm has ever been reported in the literature, due to many numerical obstacles, including discontinuous coefficients, singular charges, geometric singularities, and nonlinear couplings. The present work introduces a number of numerical algorithms to overcome the abovementioned numerical challenges and constructs the first second-order convergent PNP solver in the ion-channel context. First, a Dirichlet to Neumann mapping (DNM) algorithm is designed to alleviate the charge singularity due to the protein structure. Additionally, the matched interface and boundary (MIB) method is reformulated for solving the PNP equations. The MIB method systematically enforces the interface jump conditions and achieves the second order accuracy in the presence of complex geometry and geometric singularities of molecular surfaces. Moreover, two iterative schemes are utilized to deal with the coupled nonlinear equations. Furthermore, extensive and rigorous numerical validations are carried out over a number of geometries, including a sphere, two proteins and an ion channel, to examine the numerical accuracy and convergence order of the present numerical algorithms. Finally, application is considered to a real transmembrane protein, the Gramicidin A channel protein. The performance of the proposed numerical techniques is tested against a number of factors, including mesh sizes, diffusion coefficient profiles, iterative schemes, ion concentrations, and applied voltages. Numerical predictions are compared with experimental measurements.

Project description:Protegrin peptides are potent antimicrobial agents believed to act against a variety of pathogens by forming nonselective transmembrane pores in the bacterial cell membrane. We have employed 3D Poisson-Nernst-Planck (PNP) calculations to determine the steady-state ion conduction characteristics of such pores at applied voltages in the range of -100 to +100 mV in 0.1 M KCl bath solutions. We have tested a variety of pore structures extracted from molecular dynamics (MD) simulations based on an experimentally proposed octomeric pore structure. The computed single-channel conductance values were in the range of 290-680 pS. Better agreement with the experimental range of 40-360 pS was obtained using structures from the last 40 ns of the MD simulation, where conductance values range from 280 to 430 pS. We observed no significant variation of the conductance with applied voltage in any of the structures that we tested, suggesting that the voltage dependence observed experimentally is a result of voltage-dependent channel formation rather than an inherent feature of the open pore structure. We have found the pore to be highly selective for anions, with anionic to cationic current ratios (I(Cl-)/I(K+)) on the order of 10(3). This is consistent with the highly cationic nature of the pore but surprisingly in disagreement with the experimental finding of only slight anionic selectivity. We have additionally tested the sensitivity of our PNP model to several parameters and found the ion diffusion coefficients to have a significant influence on conductance characteristics. The best agreement with experimental data was obtained using a diffusion coefficient for each ion set to 10% of the bulk literature value everywhere inside the channel, a scaling used by several other studies employing PNP calculations. Overall, this work presents a useful link between previous work focused on the structure of protegrin pores and experimental efforts aimed at investigating their conductance characteristics.

Project description:Introduction: In addition to gap junction conduction, recent reports implicate possible ephaptic coupling contributions to action potential (AP) propagation between successive adjacent cardiomyocytes. Here, AP generation in an active cell, withdraws Na+ from, creating a negative potential within, ephaptic spaces between the participating membranes, activating the initially quiescent neighbouring cardiomyocyte. However, sustainable ephaptic transmission requires subsequent complete recovery of the ephaptic charge difference. We explore physical contributions of passive electrodiffusive ion exchange with the remaining extracellular space to this recovery for the first time. Materials and Methods: Computational, finite element, analysis examined limiting, temporal and spatial, ephaptic [Na+], [Cl-], and the consequent Gaussian charge differences and membrane potential recovery patterns following a ΔV∼130 mV AP upstroke at physiological (37°C) temperatures. This incorporated Nernst-Planck formalisms into equations for the time-dependent spatial concentration gradient profiles. Results: Mammalian atrial, ventricular and purkinje cardiomyocyte ephaptic junctions were modelled by closely apposed circularly symmetric membranes, specific capacitance 1 μF cm-2, experimentally reported radii a = 8,000, 12,000 and 40,000 nm respectively and ephaptic axial distance w = 20 nm. This enclosed an ephaptic space containing principal ions initially at normal extracellular [Na+] = 153.1 mM and [Cl-] = 145.8 mM, respective diffusion coefficients D Na = 1.3 × 109 and D Cl = 2 × 109 nm2s-1. Stable, concordant computational solutions were confirmed exploring ≤1,600 nm mesh sizes and Δt≤0.08 ms stepsize intervals. The corresponding membrane voltage profile changes across the initially quiescent membrane were obtainable from computed, graphically represented a and w-dependent ionic concentration differences adapting Gauss's flux theorem. Further simulations explored biological variations in ephaptic dimensions, membrane anatomy, and diffusion restrictions within the ephaptic space. Atrial, ventricular and Purkinje cardiomyocytes gave 40, 180 and 2000 ms 99.9% recovery times, with 720 or 360 ms high limits from doubling ventricular radius or halving diffusion coefficient. Varying a, and D Na and D Cl markedly affected recovery time-courses with logarithmic and double-logarithmic relationships, Varying w exerted minimal effects. Conclusion: We thereby characterise the properties of, and through comparing atrial, ventricular and purkinje recovery times with interspecies in vivo background cardiac cycle duration data, (blue whale ∼2000, human∼90, Etruscan shrew, ∼40 ms) can determine physical limits to, electrodiffusive contributions to ephaptic recovery.

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Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates.

Publications

Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates.

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