Boundary conditions for- single-ion diffusion

McGill, Paul; Schumaker, Mark F.

doi:10.1016/s0006-3495(96)79374-8

Cited by 42 publications

(58 citation statements)

References 26 publications

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“…Framework models have been constructed to describe the permeation of single Na + ions (16) and single protons (2) through gramicidin. These models describe the dynamics of permeation in simplified configuration spaces that are designed to incorporate potentials of mean force and diffusion coefficients calculated by the molecular dynamics simulations.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…The exponential distribution also underlies rate theory models (for example, ref. 18) and is an appropriate approximation when the concentration of excess protons in the surrounding baths is not too large (16,19). The boundary regions of the defect segment are modeled as the lumped states b I and b II .…”

Section: Lumped State Approximation Modelmentioning

confidence: 99%

“…These considerations motivate the development of methods to construct and solve numerical framework models which can be more elaborate than the one dimensional models constructed so far (2,6,10,16). This article introduces a method which involves finding the steady states of a random walk constructed using a trapezoid rule closely related to the rule for numerical integration.…”

Section: Introductionmentioning

confidence: 99%

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Numerical framework models of single proton conduction through Gramicidin

Schumaker¹

2003

Front Biosci

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Section: Summary and Discussionmentioning

confidence: 99%

Section: Lumped State Approximation Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical framework models of single proton conduction through Gramicidin

Schumaker¹

2003

Front Biosci

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“…The stochastic Brownian motion of the multiion system was implemented as a continuous-time Markov chain with discrete states corresponding to the ion positions, and the state-to-state random walk was constructed by generating exponentially distributed random survival times. Such a Markov random walk satisfies the condition of detailed balance under equilibrium conditions in the absence of net flux, and the stochastic evolution of the system obeys a multidimensional Smulochowsky (Nernst-Planck) diffusion equation as ␦Z becomes increasingly small (30)(31)(32). The forward and backward transition rates are given by (e.g., for ion 1)…”

Section: Theory and Methodsmentioning

confidence: 99%

A microscopic view of ion conduction through the K+ channel

Bernèche

Roux²

2003

Proc. Natl. Acad. Sci. U.S.A.

227

260

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Recent results from x-ray crystallography and molecular dynamics free-energy simulations have revealed the existence of a number of specific cation-binding sites disposed along the narrow pore of the K ؉ channel from Streptomyces lividans (KcsA), suggesting that K ؉ ions might literally ''hop'' in single file from one binding site to the next as permeation proceeds. In support of this view, it was found that the ion configurations correspond to energy wells of similar depth and that ion translocation is opposed only by small energy barriers. Although such features of the multiion potential energy surface are certainly essential for achieving a high throughput rate, diffusional and dissipative dynamical factors must also be taken into consideration to understand how rapid conduction of K ؉ is possible. To elucidate the mechanism of ion conduction, we established a framework theory enabling the direct simulation of nonequilibrium fluxes by extending the results of molecular dynamics over macroscopically long times. In good accord with experimental measurements, the simulated maximum conductance of the channel at saturating concentration is on the order of 550 and 360 pS for outward and inward ions flux, respectively, with a unidirectional flux-ratio exponent of 3. Analysis of the ionconduction process reveals a lack of equivalence between the cation-binding sites in the selectivity filter. molecular dynamics ͉ Brownian dynamics ͉ potential of mean force ͉ membrane potential ͉ Poisson-Boltzmann equation P otassium channels are transmembrane proteins that have the ability to conduct K ϩ ions at nearly the diffusion limit while remaining very selective (1). The availability of high-resolution crystallographic structures (2-4), together with the development of sophisticated computer models (5-10), is providing us with a unique opportunity to refine our understanding of these systems at an unprecedented level. Although the complexity of these channels does present a formidable challenge to biomolecular modeling, it is particularly encouraging to note that many of the recent results from molecular dynamics (MD) simulations based on realistic all-atom models have been consistent with the information emerging from higher-resolution structural data (for a recent review, see ref. 11). In particular, the free-energy profile, or potential of mean force (PMF), associated with the position of three K ϩ along the axis of the pore was calculated with umbrella sampling simulations of the K ϩ channel from Streptomyces lividans (KcsA) embedded in a phospholipid membrane with explicit solvent (5). This MD calculation reproduced the four cation-binding sites (S 1 -S 4 ) located in the narrow selectivity filter that were already known from x-ray diffraction data at 3.2-Å resolution (3) and also anticipated the existence of two additional sites (S ext and S 0 ) located on the extracellular side of the channel, which were observed independently in diffraction data at 2.0-Å resolution (2). This result increased the confidence in the abili...

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“…However, ion permeation is traditionally discussed in terms of the free-energy profile of ions along the channel axis (Hille, 2001). In particular, such quantity is an essential input to simple kinetic rate models (Läuger, 1973) and in the 1D Nernst-Planck (1D-NP) electrodiffusion equation (Levitt, 1986 ;McGill & Schumaker, 1996). From a dynamical point of view, the reduction in dimensionality from r to z is based on the assumption that all motions perpendicular to z reach equilibrium rapidly and that z is the only relevant slow variable in the system, i.e.…”

Section: Reduction To a One-dimensional (1d) Free-energy Profilementioning

confidence: 99%

Theoretical and computational models of biological ion channels

Roux

Allen

Bernèche

et al. 2004

Quart. Rev. Biophys.

383

419

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Abstract. The goal of this review is to establish a broad and rigorous theoretical framework to describe ion permeation through biological channels. This framework is developed in the context of atomic models on the basis of the statistical mechanical projection-operator formalism of Mori and Zwanzig. The review is divided into two main parts. The first part introduces the fundamental concepts needed to construct a hierarchy of dynamical models at different level of approximation. In particular, the potential of mean force (PMF) as a configuration-dependent free energy is introduced, and its significance concerning equilibrium and non-equilibrium phenomena is discussed. In addition, fundamental aspects of membrane electrostatics, with a particular emphasis on the influence of the transmembrane potential, as well as important computational techniques for extracting essential information from all-atom molecular dynamics (MD) simulations are described and discussed. The first part of the review provides a theoretical formalism to 'translate ' the information from the atomic structure into the familiar language of phenomenological models of ion permeation. The second part is aimed at reviewing and contrasting results obtained in recent computational studies of three very different channels : the gramicidin A (gA) channel, which is a narrow one-ion pore (at moderate concentration), the KcsA channel from Streptomyces lividans, which is a narrow multi-ion pore, and the outer membrane matrix porin F (OmpF) from Escherichia coli, which is a trimer of three b-barrel subunits each forming wide aqueous multi-ion pores. Comparison with experiments demonstrates that current computational models are approaching semi-quantitative accuracy and are able to provide significant insight into the microscopic mechanisms of ion conduction and selectivity. We conclude that all-atom MD with explicit water molecules can represent important structural features of complex biological channels accurately, including such features as the location of ion-binding sites along the permeation pathway. We finally discuss the broader issue of the validity of ion permeation models and an outlook to the future.

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Boundary conditions for- single-ion diffusion

Cited by 42 publications

References 26 publications

Numerical framework models of single proton conduction through Gramicidin

Numerical framework models of single proton conduction through Gramicidin

A microscopic view of ion conduction through the K+ channel

Theoretical and computational models of biological ion channels

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