Fokker-Planck Equations for Transport through Ion Channels

Transcription

1 Fokker-Planck Equations for Transport through Ion Channels CeNoS European Institute for Molecular Imaging Institute for Computational and Applied Center for Nonlinear

2 Ion Channels and Nanopores 2 Joint Work / Discussions with Bärbel Schlake, Münster Kattrin Arning, Linz Mary Wolfram, Cambridge / Münster / Linz Bob Eisenberg, Chicago Heinz Engl, Linz Zuzanna Siwy, Irvine

3 Ion Channels and Nanopores 3 Disclaimer First part is mainly reviewing previous work, in particular results by Eisenberg, Nonner, Gillespie, et al Second part is speculative and (hopefully) provocative

4 Ion Channels and Nanopores 4 Ion Channels and Life Most of human life occurs in cells Transport through cell membrane is essential for biological function The transport or blocking of ions is controlled by channels Ion channels = proteins with a hole in their middle ~5 µm

5 Ion Channels and Nanopores 5 Electrostatic Interaction Flow of ions creates / modifies electric potential Electrical field determines flow direction of ions Proteins in the channel walls create a huge charge in the channel Additional effects due to size exclusion in narrow channels K + ~30 Å

6 Ion Channels and Nanopores 6 Polymer Nanopores Analogous modelling issues in polymer nanopores, made by track etching (see Zuzanna Siwy, Christina Trautmann, Veronika Bayer)

7 Ion Channels and Nanopores 7 Channel Function Experimental setup: Bath of ions and water on both sides of channel Bath concentrations controlled Voltage applied across channel

8 Ion Channels and Nanopores 8 Selectivity Observed current-voltage curves Curves for a range of different bath concentrations OmpF KCl 1M 1M OmpF CaCl 2 1M 1M

9 Ion Channels and Nanopores 9 Modelling Microscopic model based on equations of motions Positive cation, p k f p k x; q p 2γ kt p ( % k ) m n n f ( x% ; q k k k m ) e.g., p = Na + Negative anion, e.g., n = Cl && x = γ x& + w& Newton's Law Forces include interaction between ions, and with protein k n k m 2γ kt m && x = γ x& + w& Friction & Noise k n k

10 Ion Channels and Nanopores 10 Modelling Force f k includes - Excess chemical force - Electrical force: Electrical potential to be computed from Poisson equation with sources from all charges (ions, protein) Forces to be estimated or possibly computed (ab-initio), as well as protonization states

11 Ion Channels and Nanopores 11 Modelling and Simulation Molecular-dynamics / Monte Carlo Simulation, particle simulationen with reduced degrees of freedom Possible 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-computation Knapp et al, Morra et al Doyle et al, 96

12 Ion Channels and Nanopores 12 Limitations Grenzen 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

13 Ion Channels and Nanopores 13 Reduced Models: Biosensors Additional motivation: Future perspective of design Even if microscopic simulations can be made practical, they will still be far from being used in rational design techniques (with many runs) Macroscopic models can at least suggest ideas (fast) - cf. Synthetic Ca channels by H.Miedema

14 Ion Channels and Nanopores 14 Reduced Models Are there reduced macroscopic models that allow to compute important features in a large channel-bath system? What are the basic ingredients needed? How far can we go with macroscopic models? (Selectivity! Single File? Gating?) How do we get closure relations?

15 Ion Channels and Nanopores 15 Classical Macroscopic Model for Open State Standard Mean-Field (Vlasov) limit leads to Poisson- Nernst-Planck (Poisson-drift-diffusion) system for potential and ion concentrations Similar issues as in Semiconductor Simulation

16 Ion Channels and Nanopores 16 Modelling Additional issues due to finite radius (steric effects) Excess chemical potential includes - Chemical interaction between the ions - Chemical interaction between ions and proteins

17 Ion Channels and Nanopores 17 Modelling Computation of the macroscopic excess chemical potential is a hard problem Various models and schemes at different resolution We currently use density functional theory (DFT) of statistical physics. Consequence are many nonlinear integrals to be computed with fine resolution and selfconsistency iterations: lead to enormous computational effort

18 Ion Channels and Nanopores 18 Modelling Due to narrow size of channels in two dimensions and predominant flow in one direction, use of effective spatially one-dimensional models becomes attractive Appropriate averaging still point of discussion

19 Ion Channels and Nanopores 19 Modelling Structure is not frozen at the working temperature. Hence, the concentration of the protein charges (modelled as half-charged oxygens for L-type Ca) needs to be modelled as an additional unknown Binding forces of the protein on its charges are encoded in a confining potential Structure can be represented via confining potentials in a unified way (almost infinite to include rigid structures)

20 Ion Channels and Nanopores 20 Numerical Simulation: PNP-DFT Mixed finite element method, symmetric discretization in entropy variables mb-carrillo-wolfram 09 Wolfram (PhD, 08) L-type Ca channel with four free charges in the protein Structure averaged to 1D mb-eisenberg-engl, SIAP 07 Voltage 50mV

21 Ion Channels and Nanopores 21 Numerical Simulation: PNP-DFT 3D channel simulation with rotational symmetry,wolfram (PhD, 2008)

22 Ion Channels and Nanopores 22 Numerical Simulation: PNP Analogous approaches for polymer nanopores Wolfram (PhD, 2008)

23 Ion Channels and Nanopores 23 Modelling Continuum simulation with structural information encoded via geometry and confining potential applies to the open state in several channels, quantitative prediction of I-V curves Parameters like diffusion coefficients fit at one specific setup of 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 Ryanodine Receptors: Gillespie et al

24 Ion Channels and Nanopores 24 Beyond large open channels Two common prejudices: - Continuum models cannot treat single-file - Continuum models cannot treat gating True? Maybe there was just something wrong with our math

25 Ion Channels and Nanopores 25 Back to Model Derivation Back 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 (x 1 ; x 2 ; x 3 ; : : : ; x N ; t)

26 Ion Channels and Nanopores 26 Fokker Planck Equation Take Langevin for simplicity: Change of position x j determined by - Size exclusion (potentials µ ij ) - electrostatics (via V) - mobility η j (from Newton s equation) - random diffusion

27 Ion Channels and Nanopores 27 Fokker Planck Equation Joint probability density satisfies G being the Greens function of the Laplace Operator, u applied potential (voltage)

28 Ion Channels and Nanopores 28 Fokker Planck Equation Derivation of Nernst-Planck equations by simplest closure relations f (x 1 ; : : : ; x N ; t) ¼ Y i ½ k (i ) (x i ; t) where k(i) denotes the type of ion i Rigorous mean-field limit for smooth bounded interactions, but involved potentials have singularities (Poisson, Lennard- Jones) and may be non-smooth (hard-core)

29 Ion Channels and Nanopores 29 Fokker Planck Equation Statistical properties of the simple closure: f (x 1 ; : : : ; x N ; t) ¼ Y i ½ k (i ) (x i ; t) Means stochastic independence to leading order, only mean-field interaction Not true in narrow channels, in particular for single file transport several pairs of (succeeding) ions are highly correlated

30 Ion Channels and Nanopores 30 Fokker Planck Equation Need to invoke higher-order closures, based on m-particle correlation functions (stay with single type of ion to keep notation reasonable) ½ (m ) (x 1 ; : : : ; x m ; t) := Z Z : : : f (x 1 ; : : : ; x N ; t) dx m + 1 : : : dx N In single file transport m=2 or m=3 should be appropriate (direct interaction with ions in front and maybe behind)

31 Ion Channels and Nanopores 31 Fokker Planck Equation Higher-order closure, e.g. Kirkwood closure relation ½ (3) (x 1 ; x 2 ; x 3 ) = ½(2) (x 1 ; x 2 )½ (2) (x 1 ; x 3 )½ (2) (x 2 ; x 3 ) ½(x 1 )½(x 2 )½(x 3 ) Improvement: -Possible correction factors -or similar factorization of quadruplet correlation in triplet correlation

32 Ion Channels and Nanopores 32 Fokker Planck Equation But 2-particle correlation function does not satisfy ½ (2) (x 1 ; x 2 ; t) = ½(x 1 ; t)½(x 2 ; t) Messy mathematics, but more appropriate

33 Ion Channels and Nanopores 33 Pair correlation Ansatz for single-file transport: pair-correlation or tripletcorrelation dependent on single density and distance between particles Leads to additional nonlinearity in the equation for the singlet density

34 Ion Channels and Nanopores 34 Essence: Modified PNP Transient version of standard t ½ i = r J i Modified versions in case of correlation t ½ i = r (Â(½ 1 ; : : : ; ½ m )J i ) Equilibria unchanged, but different dynamical behaviour

35 Ion Channels and Nanopores 35 Modified PNP Â(½ 1 ; : : : ; ½ m ) Additional mobility must depend montonically decreasing on the occupied-volume density v = X V i ½ i 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

36 Ion Channels and Nanopores 36 Mathematical analogies For bimodal fluxes, metastable solutions are possible (consisting of empty bubbles and some ion clusters) This could provide a qualitative reproduction of recently proposed bubble gating mechanisms Eisenberg et al

37 Ion Channels and Nanopores 37 Quantitative Issues Difficult if not impossible to find closed form of χ Ad-hoc closures can provide qualitative, but not quantitative predictions Future Approach: Exploit multiple scales, compute effective mobility by local microscopic simulation Only small systems to be computed locally (randomly sampled) Microscopic information can be incorporated

38 Ion Channels and Nanopores 38 Voltage Gating Can we also get conclusions on voltage gating using Fokker Planck equations? Probably yes cf. Sigg, Bezanilla, 2003 Investigate gating current measured when changing applied potentials Gatting current attributed to position change of charges in the voltage sensor, Bezanilla et al 1974

39 Ion Channels and Nanopores 39 Gating Current Shaker K + channels From: Stefani, Toro, Perozo, Bezanilla, 1994

40 Ion Channels and Nanopores 40 Gating Currents Standard models of voltage gating are nowadays Markov transition models Based on jumps of some charges by random process Many parameters to fit reality, but limited physics Markov transition models can be derived from Fokker- Planck equations with steep potentials Physical models via Fokker-Planck equations for charges Sigg, Bezanilla, 2003

41 Ion Channels and Nanopores 41 Gating Currents Can the main features measured in ensemble gating currents be explained from Fokker-Planck equation? - inital jump (dependence on voltage change) - appearance (or non-appearance) of rising phase (small time) - exponential decay (large time)

42 Ion Channels and Nanopores 42 Gating Currents Start with the same model as before, include all possible charges (in solution and in protein)

43 Ion Channels and Nanopores 43 Gating Currents Voltage changes modeled by (applied voltage U )

44 Ion Channels and Nanopores 44 Gating Currents Setup: - At t < 0, voltage is U 0, system is in equilibrium - at time t =0, U is raised from U 0 to U 1

45 Ion Channels and Nanopores 45 Ramo-Shockley Theorem Computation of the garing current from the Ramo-Shockley Theorem (Nonner, Peyser, Eisenberg, Gillespie, 2004) Gating current is scalar product of scaled field in absence of charges and the charge flux

46 Ion Channels and Nanopores 46 Ensemble current Ensemble gating current related to the expected value (law of large numbers)

47 Ion Channels and Nanopores 47 Ensemble current Flux 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 equilibrium distribution f 0 at U = U 0

48 Ion Channels and Nanopores 48 Ensemble current Flux 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 equilibrium distribution f 0 at U = U 0

49 Ion Channels and Nanopores 49 Gating current Initial jump Note: much smaller dielectric coefficient inside protein (steeper gradient of u), hence major contribution by mobile charges in the protein!

50 Ion Channels and Nanopores 50 Initial Jump Simplification for (locally) constant mobilities, dielectric coefficient and field Independent of forces to leading order!

51 Ion Channels and Nanopores 51 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 gating currents at time t =0 - Insert Fokker-Planck equation and equilibrium relation Tedious computations, but can predict appearance in some cases. Now dependent on potentials / forces!

52 Ion Channels and Nanopores 52 Decay Exponential 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 leading eigenvalue of the Fokker-Planck operator To be computed and compared

53 Ion Channels and Nanopores 53 Conclusion Fokker-Planck equations and continuum models can be used to describe several effects in ion channels, but not with standard derivations and standard mathematics Various open and challenging mathematical questions, which can have practical impact Route towards design

54 Ion Channels and Nanopores 54 Downloads / Contact imaging.uni-muenster.de/ martin.burger@uni-muenster.de