hyonseok hwang et al- incorporation of inhomogeneous ion diffusion coefficients into kinetic lattice...

8/3/2019 Hyonseok Hwang et al- Incorporation of Inhomogeneous Ion Diffusion Coefficients into Kinetic Lattice Grand Canoni

1/7

Incorporation of Inhomogeneous Ion Diffusion Coefficients into Kinetic Lattice Grand

Canonical Monte Carlo Simulations and Application to Ion Current Calculations in a

Simple Model Ion Channel

Hyonseok Hwang*

Department of Chemistry, Kangwon National UniVersity, Chuncheon, Kangwon 200-701, Republic of Korea

George C. Schatz and Mark A. Ratner

Department of Chemistry, Northwestern UniVersity, 2145 Sheridan Rd., EVanston, Illinois 60208-3113

ReceiVed: July 24, 2007; In Final Form: September 20, 2007

To deal with inhomogeneous diffusion coefficients of ions without altering the lattice spacing in the kineticlattice grand canonical Monte Carlo (KLGCMC) simulation, an algorithm that incorporates diffusion coefficientvariation into move probabilities is proposed and implemented into KLGCMC calculations. Using thisalgorithm, the KLGCMC simulation method is applied to the calculation of ion currents in a simple modelion channel system. Comparisons of ion currents and ion concentrations from these simulations with Poisson -Nernst-Planck (PNP) results show good agreement between the two methods for parameters where the lattermethod is expected to be accurate.

I. Introduction

Kinetic or dynamic lattice Monte Carlo (KLMC or DLMC)simulation algorithms1-4 and Brownian dynamics (BD) simula-tion methods5-8 are useful tools for studying ion transport incomplex systems such as an ion channel.9-12 The moleculardynamics (MD) simulation method,13-16 which can describe iontransport at an atomistic level, is computationally too demandingto reach the time scales needed for ion transport in ion channels.On the other hand, Poisson-Nernst-Planck (PNP) theory,17-20

based on a dielectric continuum model, can be used to calculateion currents at necessary time scales, but several important issuessuch as finite ion size and correlation effects are ignored in thattheory. As alternative computational methods, the KLMC andBD simulation methods overcome limitations of both MD andPNP methods by using a simple description of the proteins andmembranes and employing explicit ions. Consequently, KLMCsimulations and BD simulations can be performed for time scalesrelevant to ion transport in ion channels.

In a lattice-based approach, the ions move on a fixed latticefor each time step. Consequently, when ions have a spatiallyvarying diffusion coefficient such as occurs inside a channel,6,15

the implementation of an inhomogeneous diffusion coefficientinto a lattice-based method causes difficulty.

Cheng et al. proposed a method to incorporate the inhomo-

geneous diffusion coefficients of ions into DLMC simulations.3

In that method, the lattice spacing is varied depending on thediffusion coefficients, and the electrostatic interaction energiesat the ion positions are calculated with a multilinear interpolationalgorithm. This use of interpolation is a computational limitationfor this method.

Natori and Godby and later Righi et al. have shown a linearrelationship between the time and the mean square displacement

of an adatom on a surface when the diffusion coefficient hasan Arrhenius form.21,22 Oum and co-workers revealed that whenthe diffusion coefficient is expressed in terms of the sojourntime of a particle at a position it is equivalent to the Arrheniusform.23 Farnell and Gibson performed off-lattice Monte Carlo(MC) simulations of diffusion with a spatially varying diffu-sion coefficient, and showed that accurate results can beproduced by altering both the MC step size and the steppingprobability.24

In this paper, we introduce a probability-based algorithm todeal with inhomogeneous diffusion coefficients of ions without

altering the lattice spacing and implement this algorithm into akinetic lattice grand canonical Monte Carlo (KLGCMC) simula-tion method, a lattice-based simulation approach. By maintaininga constant lattice spacing, this algorithm does not require anyinterpolation scheme and is easy to implement into a lattice-based method. We also calculate ion currents in a simple mod-el ion channel using this probability-based KLGCMCalgorithm.

This probability-based algorithm is basically the same as otherkinetic Monte Carlo (KMC) schemes proposed by Gillespie25

or by Fichthorn and Weinberg26 in that all of them describe thestochastic processes of multiple events, but those KMC schemesare used to simulate the reaction process while the algorithmin this paper is used to describe the diffusion process of

ions.This paper is organized as follows. In the next section, the

probability-based algorithm to treat inhomogeneous diffusioncoefficients of ions is proposed. A simple model system isintroduced, and applications of the KLGCMC simulation methodfrom Hwang et al.4 with this algorithm to the model system arepresented in section III. Comparisons of KLGCMC simulationresults with PNP calculations are also made in the same section.Further studies of the KLGCMC simulation method with theprobability-based algorithm are discussed in the finalsection.

Part of the Giacinto Scoles Festschrift.* To whom correspondence should be addressed. E-mail: hhwang@

kangwon.ac.kr.

12506 J. Phys. Chem. A 2007, 111, 12506-12512

10.1021/jp075838o CCC: $37.00 2007 American Chemical SocietyPublished on Web 10/26/2007


2/7

II. Methods

A. Kinetic Lattice Grand Canonical Monte Carlo (KLGC-

MC) Simulations with the Local Control Method. Becausethe details of the KLGCMC simulation with the local controlmethod are discussed elsewhere,4,27-30 we only briefly describethe simulation method here. In this method, the creation anddeletion of ions are restricted to local control (LC) regions, andonly diffusion occurs in the diffusion region. For the ion channelin this work, the LC regions are located at the bottom and topof the simulation box shown in Figure 1.

In this simulation method, ion creation is performed byrandomly selecting a lattice site in one of the two LC regionsand by attempting to create a cation or anion on that latticepoint. A creation attempt is accepted or rejected with a

probability,P

NLf

NL+1):

where FV VL/V where VL and V are the volumes of a localcontrol region and the simulation box, respectively. In eq 1, NL(or NL + 1) is the ion number in that local control region before(or after) an attempt, UNL (or UNL + 1) is the total interactionenergy of ions before (or after) an attempt, qcre and rcre are thecharge and position of the created ion, ext(r) is an externalfield explained later, and is 1/kBTwhere kB is the Boltzmann

constant and Tis a temperature. The constant B in eq 1 is givenas

where is the chemical potential of the created ion and h/(2mkBT)1/2 where m is the mass of the created ion. In theapplications in this paper, we assume that cations and anionshave the same chemical potential and mass.

For ion deletion, an ion in one of the two LC regions israndomly selected and an attempt to delete that ion is acceptedor rejected by generating and comparing a random number witha probability, PNLfNL-1):

where qdel and rdel are the charge and position of the deletedion.

An ion move is performed by sequentially selecting an ionand moving it to one of the six nearest-neighbor sites in thecase of 3D. The move is accepted or rejected with a probability,Pafb given by

where UNa (or UNb) is the total interaction energy before (orafter) a move and a (or b) is the ion position before (or after)a move.

A time step, which is required to calculate ion currents, isgiven as4

where L is the lattice spacing in the KLGCMC simulationand Dref is a reference ion diffusion coefficient, which will beexplained in the next subsection. Then, ion currents arecalculated using the equation

where Nioncross is the number of ions (cations or anions) crossing

the cross section per time step.B. Implementation of Inhomogeneous Diffusion Coef-

ficients. In the KLGCMC simulation, a constant lattice spacingis employed in all regions of the simulation box. To treatinhomogeneous ion diffusion coefficients without altering thelattice spacing, we introduce an algorithm based on moveprobability.

Natori and Godby21 and later Righi et al.22 showed a linearrelationship between time and mean square displacement whenthe ratio of two diffusion coefficients is written as

where Di(ri) and Eiact(ri) are a space-dependent diffusion

coefficient and an activation barrier energy of ion i at ri. Thereference diffusion coefficient Drefis chosen as the largest value

among the bulk diffusion coefficients of the cation and anion.Oum et al. showed that when the ratio of two different diffusioncoefficients of ion i is written as

then eq 8 is equivalent to eq 7.23 In eq 8, i(ri) is a dimensionlesssojourn time of ion i at ri. We now introduce a diffusionprobability of ion i as

Figure 1. 2D cross section of a 3D model ion channel system. Thelocal control regions in the KLGCMC are located at the bottom (-60e z < -50 ) and the top (50 < z e 60 ). Lz

diff) 100 and Lchn )

40 . The bulk salt concentration at both LC regions is 0.1 M.

PNLfNL+1) min{1,

FV

NN + 1exp[B -(UNL+1 - UNL -

qcrecre(rcre))]} (1)

B + lnV

3

(2)

PNLfNL-1) min{1,

NL

FVexp[-B -(UNL-1 - UNL +

qdelext(rdel))]} (3)

Pafb ) min{1, exp[-(UNb - UNa)]} (4)

t)(L)2

6Dref(5)

Iion )Nion

cross

t)

6DNioncross

(L)2(6)

Di(ri)

Dref

) exp[- Eiact(ri)

kBT ] (7)

Di(ri)

Dref

)1

i(ri)(8)

Pidif(ri)

1i(ri)

)Di(ri)

Dref

(9)

Inhomogeneous Diffusion Coefficients of Ions J. Phys. Chem. A, Vol. 111, No. 49, 2007 12507


3/7

where it is assumed that Drefg Di(ri). Equation 9 implies thatthe ion i moves to one of the nearest-neighbor lattices with aprobability of Pi

dif(ri) ) 1/i(ri) when the ion diffusion coef-ficient is smaller than Drefbut moves unconditionally when thediffusion coefficient is Dref. Thus to make a move attempt forion i due to the diffusion coefficient Di(r), a random number isgenerated and compared with the diffusion probability Pi

dif(r)from eq 9.

C. Interaction Energy Calculation. To test the proposed

KLGCMC simulation method and the algorithm for implement-ing inhomogeneous ion diffusion coefficients, we consider aproblem with very simple interactions. First of all, no electro-static interaction between ions is assumed and ions only interactwith an applied external field. A hard-sphere potential betweenions is employed as a short range interaction. Second, we assumethe dielectric constants of the channel and the membrane to bethe same as the water dielectric constant 800 to avoid anyreaction field effect. Then, the total interaction energy UN inthe system can be written as1,31,32

Here the direct interaction between ions i and j, uij(ri,rj) is ahard-sphere potential given as

where i and j is the radii of the ions i and j, respectively. Theinteraction energy hi(r) in the second term represents the hard-wall interaction of ion i with the boundaries between water andthe channel or membrane region and is given by

We assume the external field ext(r) to be linear and expressedas

where Lzdif is the length of the diffusion region along the z axis

and V0 is a voltage applied to the bottom of the model ionchannel system (see Figure 1).

To describe inhomogeneous diffusion in the KLGCMCsimulation correctly, the diffusion probability Pi

dif(ri) in (9)must be taken into account in the interaction energyUN(r1,r2, ,rN). The last term in eq 10 incorporates this diffusionprobability into the KLGCMC simulation.3,6,24

III. Data and Results

A. Description of the Model System and Details of

KLGCMC Simulations. A 2D cross section of the 3D modelion channel system under investigation is shown in Figure 1.The model ion channel is depicted as a cylinder whose radiusand length are 5 and 40 , respectively. The LC regions arelocated at the bottom (-60 e z < -50 ) and the top (50 Daniblk,bot, Dref

) Dcatblk,bot

) 2.0 10-5 cm2 /s, and the diffusion probabilitiesfor the cation and anion are written as

Figure 9 shows the total, cation, and anion current density-voltage curves. The KLGCMC simulation results show goodagreement with the PNP calculations. Due to the asymmetricdiffusion coefficients of the ions in both baths and inside thechannel, the current densities for the cation and anion are notsymmetric with respect to V0 ) 0 V. However, note that thetotal current density-voltage curve is symmetric with respect

Figure 5. Continuously changing diffusion coefficients of ions as afunction ofz. The cation and anion are assumed to have the same bulkand inside-channel diffusion coefficients. The locations of the channelentrances at z ) (20 and the local control regions at z ) (50 are

also shown. The diffusion coefficients are in units of 10-5

cm2

/s.

Figure 6. Dependence of cation current density-voltage curves onthe inside-channel diffusion coefficients in the case of continuouslychanging diffusion coefficients. Comparison between the KLGCMCsimulation results and the PNP calculations is also shown. For the samereason as in Figure 3, only the cation current densities are presented.The bulk salt concentration is 0.1 M. The diffusion coefficients (seeinset) are in units of 10-5 cm2/s.

Dcat(or ani)(r) )

{Dcat(or ani)

blk,top

z gLchn/2,Dcat(or ani)

chn,upp+ (Dcat(or ani)

chn,upp- Dcat(or ani)

chn,low )g(z) |z| < Lchn/2,

Dcat(or ani)blk,bot z e - Lchn/2

Figure 7. Dependence of cation and anion concentration profiles onthe inside-channel ion diffusion coefficients at V0 ) -0.2 V. (a) Dchn

) 2.0 10-5 cm2/s, (b) Dchn ) 1.0 10-5 cm2/s, and (c) Dchn ) 0.5 10-5 cm2/s. Comparison between the KLGCMC simulation resultsand the PNP calculations is also presented. The locations of the channelentrances at z ) (20 and the local control regions at z ) (50 arealso shown. The bulk salt concentration at both LC regions is 0.1 M.

Figure 8. Asymmetric diffusion coefficients for the cation and anionas a function ofz. The location of the channel entrances at z ) (20 is shown. The diffusion coefficients are in units of 10-5 cm2/s.

Pcat(or ani)dif (r) )

Dcat(or ani)(r)

Dref

(15)

12510 J. Phys. Chem. A, Vol. 111, No. 49, 2007 Hwang et al.


6/7

to V0 ) 0 V. The higher diffusion coefficient of the cation thanthat of the anion in all the regions leads to the higher cationcurrent densities than the anion current densities. There is nonet current density for the cation or anion at V0 ) 0 V despitethe asymmetric diffusion coefficients. It appears that at V0 ) 0V, the decrease in cation diffusion coefficient along the channelaxis (+z direction) causes downward movement of cations,which produces a negative current for the cation. However, dueto the downward movement of cations, the cation concentrationincreases in the lower part of the channel. The increase in cationconcentration in the lower part leads to a positive cationconcentration gradient, which produces a positive current. Asthe result, the negative and positive currents cancel each other.

Cation and anion concentration profiles at V0 ) -0.2, 0, and0.2 V are shown in panels a-c of Figures 10. At V0 ) -0.2 V,cations move downward and anions move upward. Because thediffusion coefficients of ions decrease along the channel axis(+z direction), anions move slower and pile up inside thechannel more than cations. On the contrary, the concentrationof cations increases inside the channel at V0 ) 0.2 V. The cationconcentration inside the channel at V0 ) 0.2 V is smaller thanthe anion concentration at V0 ) -0.2 V because the cationdiffusion coefficient is larger than the anion diffusion coefficientinside channel and bath. In Figure 10c, there is no increase ofcation or anion concentration inside the channel.

IV. Conclusions

In this paper, we have presented an algorithm to implement

inhomogeneous ion diffusion coefficients into a KLGCMCsimulation method and calculated ion currents in a simple modelion channel system using the KLGCMC simulation methodequipped with this algorithm. Based on move probability, thisalgorithm does not require altering the lattice spacing andinterpolating electrostatic potentials, and as a result, is easy toimplement. In addition, this probability-based algorithm can beeasily incorporated into more realistic KLGCMC simulationmethods because it requires modifying few parts of thoseKLGCMC simulations. The examples indicate that this algo-rithm can be employed both for discontinuously or continuouslychanging ion diffusion coefficients and for asymmetric diffusioncoefficients of cations and anions in bulk as well as inside thechannel.

In the examples in this paper, a simple static field was usedand fixed charges inside the channel were ignored. Also ignoredis the reaction field effect which can play an important role inion transport through an ion channel. Also, small ions were usedto reduce finite size effects for the ions33,34 and thereby to make

it possible to test the KLGCMC method by comparison withPNP results for the same problem. The good agreement betweenresults which we found indicates that the move probabilityalgorithm is working correctly. In addition, the results provideinsights concerning the influence of spatial anisotropy in thediffusion constant on the ion currents.

Combining this algorithm with a more realistic KLGCMCsimulation method that we have previously developed,4 providesa method that can be used for realistic ion channel problemswhere the inhomogeneous diffusion of ions inside the channelare important. This algorithm can also be used to describe theinhomogeneous diffusion of particles in other lattice-basedsimulations such as dip-pen nanolithography (DPN) simula-tions.35,36

Acknowledgment. This work was supported by the Networkfor Computational Nanotechnology (NCN) through a grant fromthe National Science Foundation, and by NSF Grant No.0550497. H.H. is grateful to Prof. YounJoon Jung for a usefuldiscussion.

References and Notes

(1) Graf, P.; Nitzan, A.; Kurnikova, M. G.; Coalson, R. D. J. Phys.Chem. B 2000, 104, 12324.

(2) van der Straaten, T.; Kathawala, G.; Ravaioli, U. J. Comput.Electron. 2003, 2, 231.

(3) Cheng, M. H.; Cascio, M.; Coalson, R. D. Biophys. J. 2005, 89,1669.

Figure 9. Total, cation, and anion current density-voltage curves whencations and anions have asymmetric diffusion coefficients in both bathsas well as inside the channel. Comparison between the KLGCMCsimulation results and the PNP calculations is also shown. The bulksalt concentration is 0.1 M.

Figure 10. Cation and anion concentration profiles at V0 ) -0.2, 0.0,and 0.2 V. Comparison between the KLGCMC simulation results andthe PNP calculations is also presented. The locations of the channelentrances at z ) (20 and the local control regions at z ) (50 arealso shown. The bulk salt concentration at both LC regions is 0.1 M.

Inhomogeneous Diffusion Coefficients of Ions J. Phys. Chem. A, Vol. 111, No. 49, 2007 12511


7/7

(4) Hwang, H.; Schatz, G. C.; Ratner, M. A. J. Chem. Phys. 2007,127, 024706.

(5) Corry, B.; Kuyucak, S.; Chung, S. H. Biophys. J. 2000, 78, 2364.(6) Im, W.; Roux, B. J. Mol. Biol. 2002, 322, 851.(7) Burykin, A.; Schutz, C. N.; Villa, J.; Warshel, A. Proteins 2002,

47, 265.(8) Marreiro, D.; Tang, Y.; Aboud, S.; Jakobsson, E.; Saraniti, M. J.

Comput. Electron. 2007, 6, 377.(9) Hille, B. Ionic Channels of Excitable Membranes, 3rd ed.; Sinauer

Associates, Inc.: Sunderland, MA, 2001.(10) Jakobsson, E. Methods 1998, 14, 342.(11) Roux, B. Curr. Opin. Struct. Biol. 2002, 12, 182.

(12) Eisenberg, B. Biophys. Chem. 2003, 100, 507.(13) Smith, G. R.; Sansom, S. P. Biophys. J. 1998, 75, 2767.(14) Warshel, A. Acc. Chem. Res. 2002, 35, 385.(15) Allen, T. W.; Andersen, O. S.; Roux, B. Proc. Natl. Acad. Sci.

U.S.A. 2004, 101, 117.(16) Hwang, H.; Schatz, G. C.; Ratner, M. A. J. Phys. Chem. B 2006,

110, 26448.(17) Chen, D.; Lear, J. D.; Eisenberg, R. S. Biophys. J. 1997, 72, 97.(18) Kurnikova, M. G.; Coalson, R. D.; Graf, P.; Nitzan, A. Biophys. J.

1999, 76, 642.(19) van der Straaten, T. A.; Tang, J.; Ravaioli, U.; Eisenberg, R. S.;

Aluru, N. R. J. Comput. Electron. 2003, 2, 29.(20) Hwang, H.; Schatz, G. C.; Ratner, M. A. J. Phys. Chem. B 2006,

110, 6999.

(21) Natori, A.; Godby, R. W. Phys. ReV. B 1993, 47, 15816.(22) Righi, M. C.; Pignedoli, C. A.; Felice, R. D.; Bertoni, C. M. Phys.

ReV. B 2005, 71, 075303.(23) Oum, L.; Parrondo, J. M. R.; Martinez, H. L. Phys. ReV. E 2003,

67, 011106.(24) Farnell, L.; Gibson, W. G. J. Comput. Phys 2005, 208, 253.(25) Gillespie, D. J. Phys. Chem. 1977, 81, 2340.(26) Fichthorn, K.; Weinberg, W. H. J. Chem. Phys. 1991, 95, 1090.(27) Valleau, J. P.; Cohen, L. K. J. Chem. Phys. 1980, 72, 5935.(28) Torrie, G. M.; Valleau, J. P. J. Chem. Phys. 1980, 73, 5807.

(29) Torrie, G. M.; Valleau, J. P. J. Phys. Chem. 1982, 86, 3251.(30) Papadopoulou, A.; Becker, E. D.; Lupkowski, M.; van Swol, F. J.Chem. Phys. 1993, 98, 4897.

(31) Hoyles, M.; Kuyucak, S.; Chung, S.-H. Phys. ReV. E 1998, 58,3654.

(32) Im, W.; Roux, B. J. Chem. Phys. 2001, 115, 4850.(33) Gillespie, D.; Nonner, W.; Eisenberg, R. S. Phys. ReV. E 2003, 68,

031503.(34) Gillespie, D.; Valisko, M.; Boda, D. J. Phys. Condens. Matter2005,

17, 6609.(35) Jang, J.; Hong, S.; Schatz, G. C.; Ratner, M. A. J. Chem. Phys.

2001, 115, 2721.(36) Lee, N.-K.; Hong, S. J. Chem. Phys. 2006, 124, 114711.

12512 J. Phys. Chem. A, Vol. 111, No. 49, 2007 Hwang et al.

hyonseok hwang et al- incorporation of inhomogeneous ion diffusion coefficients into kinetic lattice...

Documents