Fokker-Planck Equations for Transport through Ion … · Fokker-Planck Equations for Transport through Ion Channels. Ion Channels and Nanopores 2 ... Cf. Poster of Bärbel Schlake

Martin BurgerInstitute for Computational and Applied

Mathematics

European Institute for Molecular

Imaging

Center for Nonlinear

ScienceCeNoS

Fokker-Planck Equations for Transport through Ion Channels

Ion Channels and Nanopores 2

29.6.2009Martin Burger

Joint Work / Discussions withBärbel Schlake, MünsterKattrin Arning, LinzMary Wolfram, Cambridge / Münster / LinzBob Eisenberg, ChicagoHeinz Engl, LinzZuzanna Siwy, Irvine



DisclaimerFirst part is mainly reviewing previous work, in particularresults by Eisenberg, Nonner, Gillespie, et al

Second part is speculative and (hopefully) provocative



Ion Channels and LifeMost of human life occurs in cells

Transport through cell membraneis essential for biological function

The transport or blocking of ionsis controlled by channels

Ion channels = proteins with ahole in their middle

~5 µm



Electrostatic InteractionFlow of ions creates / modifies electricpotentialElectrical field determinesflow direction of ions

Proteins in the channel walls create ahuge charge in the channel

Additional effects due to size exclusionin narrow channels ~30 Å

K+



Polymer NanoporesAnalogous modelling issues in polymer nanopores,made by track etching(see Zuzanna Siwy, Christina Trautmann, Veronika Bayer)



Channel FunctionExperimental setup:

Bath of ions and water on both sidesof channel

Bath concentrations controlledVoltage applied across channel



SelectivityObserved current-voltage curves

Curves for a range ofdifferent bathconcentrations

OmpF KCl1M 1M

||

OmpF CaCl21M 1M

||



ModellingMicroscopic model based on equations of motions

Forces include interaction between ions, and withprotein

( ); 2kp

k x q pp pkk k

f kTm mx x wγγ− = − +%

&& & &Positive cat ion, e.g., p = Na+

( );

Newton'sLaw Friction & Noise

2kn

k x q nn nkk k

f kTm mx x wγγ− = − +%

&& & &14243 1442443

Negative an ion,

e.g., n = Cl¯



ModellingForce fk includes

- Excess „chemical“ force- Electrical force: Electrical potential to be computedfrom Poisson equation with sources from all charges(ions, protein)

Forces to be estimated or possibly computed (ab-initio), as well as protonization states



Modelling and SimulationMolecular-dynamics / Monte Carlo Simulation, particle simulationen with reduced degrees of freedomPossible for (parts of small channels), e.g. Na / K+ :Roux et al, Aqvist et al, Boda et al, Kleinekathöfer …

Important Input (?): Protein structure, charge patterns, pKa-computationKnapp et al, Morra et al

Doyle et al, 96



LimitationsGrenzen der Partikelsimulation bei großen Systemen

- Bath ions cannot be included, no robust prediction of voltage-current measurements over a range of concentrations

- Larger channels (L-type Ca, Ryanodine R, Polymer nanopores) cannot be simulated



Reduced Models: BiosensorsAdditional motivation:

Future perspective of design

Even if microscopic simulations can be made practical, they will still be far from being used in rational designtechniques (with many runs)

Macroscopic models can at least suggest ideas (fast)- cf. Synthetic Ca channels by H.Miedema



Reduced ModelsAre there reduced macroscopic models that allow to compute important features in a large channel-bathsystem ?

What are the basic ingredients needed ?

How far can we go with macroscopic models ? (Selectivity ! Single File ? Gating ?)

How do we get closure relations ?



Classical Macroscopic Model for Open State

Standard Mean-Field (Vlasov) limit leads to Poisson-Nernst-Planck(Poisson-drift-diffusion) system for potential and ionconcentrations

Similar issues as in Semiconductor Simulation



ModellingAdditional issues due to finite radius (steric effects)

Excess chemical potential includes- Chemical interaction between the ions- Chemical interaction between ions and proteins



ModellingComputation of the macroscopic excess chemicalpotential is a hard problem

Various models and schemes at different resolution

We currently use density functional theory (DFT) of statistical physics. Consequence are many nonlinearintegrals to be computed with fine resolution and self-consistency iterations: lead to enormous computationaleffort



ModellingDue to narrow size of channels in two dimensions and predominant flow in one direction, use of effectivespatially one-dimensional models becomes attractive

Appropriate averaging still point of discussion



ModellingStructure is not frozen at the working temperature.

Hence, the concentration of the protein charges(modelled as half-charged oxygens for L-type Ca) needsto be modelled as an additional unknownBinding forces of the protein on its charges are encodedin a confining potential

Structure can be represented via confining potentials in a unified way (almost infinite to include rigid structures)



Numerical Simulation: PNP-DFTMixed finite element method, symmetric discretization in entropy variablesmb-Carrillo-Wolfram 09Wolfram (PhD, 08)

L-type Ca channel with four freecharges in the proteinStructure averaged to 1Dmb-Eisenberg-Engl, SIAP 07

Voltage 50mV



Numerical Simulation: PNP-DFT3D channel simulation with rotational symmetry,Wolfram (PhD, 2008)



Numerical Simulation: PNPAnalogous approaches for polymer nanoporesWolfram (PhD, 2008)



ModellingContinuum simulation with structural information encodedvia geometry and confining potential applies to the openstate in several channels, quantitative prediction of I-V curves

Parameters like diffusion coefficients fit at one specific setupof concentrations, then used for a wide range of concentrations without further adjustment

Ca channels: Chen-Eisenberg et al 95-98, Boda, Gillespie, Eisenberg, Nonner- et al 2001-2009

Ryanodine Receptors: Gillespie et al 2005-2008



Beyond large open channelsTwo common prejudices:

- Continuum models cannot treat single-file

- Continuum models cannot treat gating

True ? Maybe there was just something wrong with our math …



Back to Model DerivationBack to the derivation of continuum models:

Starting from Newton / Langevin equations for N particles we can write the Fokker-Planck equation in 3N +1 (or 6N +1) dimensions for the joint probability density

f (x1; x2; x3; : : : ; xN ; t)



Fokker Planck EquationTake Langevin for simplicity:

Change of position xj determined by

- Size exclusion (potentials µij ) - electrostatics (via V)

- mobility ηj (from Newton‘s equation)- random diffusion



Fokker Planck EquationJoint probability density satisfies

G being the Greens function of the Laplace Operator, u applied potential (voltage)



Fokker Planck EquationDerivation of Nernst-Planck equations by simplest closurerelations

where k(i) denotes the type of ion i Rigorous mean-field limit for smooth bounded interactions, butinvolved potentials have singularities (Poisson, Lennard-Jones) and may be non-smooth (hard-core)

f (x1; : : : ; xN ; t) ¼Y

i

½k ( i ) (x i ; t)



Fokker Planck EquationStatistical properties of the simple closure:

Means stochastic independence to leading order, onlymean-field interaction

Not true in narrow channels, in particular for single filetransport several pairs of (succeeding) ions are highlycorrelated

f (x1; : : : ; xN ; t) ¼Y

i

½k ( i ) (x i ; t)



Fokker Planck EquationNeed to invoke higher-order closures, based on m-particlecorrelation functions (stay with single type of ion to keepnotation reasonable)

In single file transport m=2 or m=3 should be appropriate(direct interaction with ions in front and maybe behind)

½(m ) (x1; : : : ; xm ; t) :=Z

: : :Z

f (x1; : : : ; xN ; t) dxm + 1 : : : dxN



Fokker Planck EquationHigher-order closure, e.g. Kirkwood closure relation

Improvement:-Possible correction factors-or similar factorization of quadruplet correlation in tripletcorrelation

½(3) (x1; x2; x3) =½(2) (x1; x2)½(2) (x1; x3)½(2) (x2; x3)

½(x1)½(x2)½(x3)



Fokker Planck EquationBut 2-particle correlation function does not satisfy

Messy mathematics, but more appropriate

½(2) (x1; x2; t) = ½(x1; t)½(x2; t)



Pair correlationAnsatz for single-file transport: pair-correlation or triplet-correlation dependent on single density and distance between particles

Leads to additional nonlinearity in the equation for thesinglet density



Essence: Modified PNPTransient version of standard PNP

Modified versions in case of correlation effects

Equilibria unchanged, but different dynamical behaviour

@t ½i = r ¢J i

@t ½i = r ¢(Â(½1; : : : ; ½m )J i )



Modified PNPAdditional mobility must dependmontonically decreasing on the occupied-volume density

and vanish at some critical volume

Hence flux has a bimodal structure (vanishing at zero and some positive maximum value)Cf. Poster of Bärbel Schlake for a detailed example

Â(½1; : : : ; ½m )

v =X

Vi ½i



Mathematical analogiesFor bimodal fluxes, metastable solutions are possible(consisting of empty bubbles and some ion clusters)

This could provide a qualitative reproduction of recentlyproposed bubble gating mechanisms Eisenberg et al 2006-09



Quantitative IssuesDifficult if not impossible to find closed form of χAd-hoc closures can provide qualitative, but not quantitative predictions

Future Approach:Exploit multiple scales, compute effective mobility by localmicroscopic simulationOnly small systems to be computed locally (randomlysampled)Microscopic information can be incorporated



Voltage GatingCan we also get conclusions on voltage gating usingFokker Planck equations ? Probably yes – cf. Sigg, Bezanilla, 2003

Investigate gating current measured when changingapplied potentials

Gatting current attributed to position change of charges in the voltage sensor, Bezanilla et al 1974



Gating CurrentShaker K + channels

From:Stefani, Toro, Perozo, Bezanilla, 1994



Gating CurrentsStandard models of voltage gating are nowadays Markovtransition models

Based on jumps of some „charges“ by random processMany parameters to fit reality, but limited physics

Markov transition models can be derived from Fokker-Planck equations with steep potentialsPhysical models via Fokker-Planck equations for chargesSigg, Bezanilla, 2003



Gating CurrentsCan the main features measured in ensemble gatingcurrents be explained from Fokker-Planck equation ?

- inital jump (dependence on voltage change) - appearance (or non-appearance) of rising phase (smalltime)- exponential decay (large time)



Gating CurrentsStart with the same model as before, include all possiblecharges (in solution and in protein)



Gating CurrentsVoltage changes modeled by (applied voltage U )



Gating CurrentsSetup: - At t < 0, voltage is U0, system is in equilibrium- at time t =0, U is raised from U0 to U1



Ramo-Shockley Theorem Computation of the garing current from the Ramo-ShockleyTheorem (Nonner, Peyser, Eisenberg, Gillespie, 2004)

Gating current is scalar product of scaled field in absenceof charges and the charge flux



Ensemble currentEnsemble gating current related to the expected value (lawof large numbers)



Ensemble currentFlux as appearing in Fokker-Planck equation

Note: at time t=0 only the red part is changing immediately, f needs some time to evolve from the equilibriumdistribution f0 at U = U0



Ensemble currentFlux as appearing in Fokker-Planck equation

Note: at time t=0 only the red part is changing immediately, f needs some time to evolve from the equilibriumdistribution f0 at U = U0



Gating currentInitial jump

Note: much smaller dielectric coefficient inside protein(steeper gradient of u), hence major contribution by mobile charges in the protein !



Initial JumpSimplification for (locally) constant mobilities, dielectriccoefficient and field

Independent of forces to leading order !



Rising Phase Rising phase can be investigated with a similar analysis:Appearance of rising phase related to the sign of the time derivative of formula for the ensemble gating current- Compute time derivatives of the ensemble gatingcurrents at time t =0

- Insert Fokker-Planck equation and equilibrium relation

Tedious computations, but can predict appearance in somecases. Now dependent on potentials / forces !



DecayExponential decay can be shown in the above setup(gating current is linear functional of the solution of a Fokker-Planck equation)

Quantitative: decay time scale characterized by the leadingeigenvalue of the Fokker-Planck operator

To be computed and compared …



ConclusionFokker-Planck equations and continuum models can beused to describe several effects in ion channels, but notwith standard derivations and standard mathematics

Various open and challenging mathematical questions, which can have practical impact

Route towards design



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Fokker-Planck Equations for Transport through Ion … · Fokker-Planck Equations for Transport through Ion Channels. Ion Channels and Nanopores 2 ... Cf. Poster of Bärbel Schlake