Continuum Drift-Diffusion Simulation of Ionic Channels

1. Continuum Drift-Diffusion Simulation of Ionic Channels Trudy van der Straaten Umberto Ravaioli Narayan Aluru Beckman Institute University of Illinois at Urbana-Champaign Robert S. Eisenberg Rush Medical College DARPA Simbiosys Review 1

2. Outline • Discussion on continuum simulation • Hierarchy of diffusivity models • Progress and milestones • Recent results on porin channels • Work in progress and future work DARPA Simbiosys Review 2

3. Discussion on Continuum Drift-Diffusion Simulation • 3-D drift-diffusion is an essential analysis tool to represent globally the input-output characteristics of the system. No other available approach provides the same level of detail and fast turn-around time. • Contacts are naturally incorporated by merely imposing a density boundary condition. • For future multi-scale applications, the drift-diffusion model is the necessary link between molecular scale and system level simulation. • Of the available state-of-the-art TCAD simulators, PROPHET is currently the only package flexible enough to handle the complex multi-material geometries characteristic of ion channel systems. DARPA Simbiosys Review 3

4. Discussion on Continuum Drift-Diffusion Simulation • Drift-diffusion is applicable to a charged fluid system even under strong bias, because ion transport is highly damped by collisions with water. • The use of a continuum conduction model raises some concerns when applied to flow through openings of restricted dimensionality. This is due to the fact that the assumption of point particles is implicit in the traditional drift-diffusion formalism. • However, corrections that modify the local electro-chemical potential may be formulated to introduce the effect of finite size of the ions in the flow model. DARPA Simbiosys Review 4

5. Discussion on Continuum Drift-Diffusion Simulation • PROPHET is a robust simulation platform based on state-of-the-art numerical engines. The correctness of the solvers and of the simulation models have been exhaustively tested and validated over at least two decades by industrial and university groups. • In addition, the simulation environment is based on a sophisticated scripting language approach that allows users to specify any PDE-based model, unlike any other commercial tools. • The non-linear solution procedure uses a robust Newton iteration, and discretization of the flux equations uses the correct exponential interpolation realized by the Scharfetter-Gummel approximation. DARPA Simbiosys Review 5

6. Discussion on Continuum Drift-Diffusion Simulation • Drift-diffusion model DARPA Simbiosys Review 6

7. Discussion on Continuum Drift-Diffusion Simulation • Scharfetter-Gummel approximation Under the assumption of linear potential (constant field) between two adjacent mesh points, the mobile charge distribution must follow an exponential law. DARPA Simbiosys Review 7

8. Discussion on Continuum Drift-Diffusion Simulation • A survey of existing ion channel literature reveals that continuum simulations conducted by other groups ignore the need for the proper use of exponential interpolation but assume a linear behavior between nodes. DARPA Simbiosys Review 8

9. Discussion on Continuum Drift-Diffusion Simulation • It is straightforward to include several multi-valent species in the PROPHET framework without modifying the source-code. Example: CaCl2 in ompF porin DARPA Simbiosys Review 9

10. Hierarchy of diffusivity models • For the purpose of fitting current measurements, a uniform diffusion coefficient is the simplest approach. This approach is suitable for treating the channel as a “black box” at system level simulation. • A physical approach requires the specification of space-dependent diffusivity for all ion species. In this way the experimentally known diffusivity in the bath (contacts) can be applied. • There is the need for coupling with higher order physical models (e.g., Molecular Dynamics). • Fitting of experimental curves using a more physical model of diffusivity is inherently more difficult due to the restricted tunability of the parameter space. DARPA Simbiosys Review 10

11. Progress and milestones • Completion of PROPHET framework for generalized diffusivity in ionic channel model • Major improvement in accessing output data structure for flux quantities (e.g., current) has increased productivity and flexibility of the tools • Implementation of improved model of charged states for ionizable amino acid residues, accounting for realistic representation of protein environment. • Implementation of physical diffusivity model inferred from molecular dynamics simulations • Large scale model of ompF Porin completed DARPA Simbiosys Review 11

12. Progress and milestones • Completion of PROPHET framework for generalized diffusivity in ionic channel model We have now the capability to specify an arbitrary diffusivity as a function of space, for different ionic species. DARPA Simbiosys Review 12

13. Progress and milestones • Major improvement in accessing output data structure for flux quantities (e.g., current) has increased productivity and flexibility of the tools Because of the complexity of the multi-material structure in the model, post-processing of results involves a large amount of data manipulation in 3-D. PROPHET has been augmented with new batch commands to access flux information anywhere on the mesh. DARPA Simbiosys Review 13

14. Progress and milestones • Implementation of improved model of charged states for ionizable amino acid residues, accounting for realistic representation of protein environment. The probability of ionization depends on local potential, pH, and salt concentration. Data from S. Varma and E. Jakobsson DARPA Simbiosys Review 14

15. Progress and milestones • Implementation of physical diffusivity model inferred from molecular dynamics simulations Molecular dynamics calculations using GROMACS [ S.-W. Chiu, E. Jakobsson ] DARPA Simbiosys Review 15

16. Equilibrium K+ density >0.75M 0M ompF mutation (G119D) Progress and milestones • Large scale model of ompF Porin completed DARPA Simbiosys Review 16

17. PROTEIN STRUCTURE  IV CURVE Assign charge to each atomic coordinate Download protein structure (protein databank) atomic coordinates, radii amino acid sequence • PROPHET script • Specify solver • parameters • Input physical domain • Assemble PDE system • Input BCs & physical • parameters • Solve (Newton’s method) • Write output Interpolate charge to gridfixed Define protein surface flag grid points (UHBD) in protein, lipid bilayer D,  fluctuation analysis of MD ion trajectories (GROMOS) Generate customized PROPHET mesh Postprocessor reconstruct j± and I current to electrodes. Recent Simulation Results for ompF Porin DARPA Simbiosys Review 17

18. Recent Simulation Results for ompF Porin DARPA Simbiosys Review 18

19. Recent Simulation Results for ompF Porin DARPA Simbiosys Review 19

20. 96Å 96Å lipid (=2) lipid (=2) 96Å protein (=20) protein (=20) electrolyte (=80) aqueous pores (=80) electrodes cross-section side view Recent Simulation Results for ompF Porin 3-D computational domain generated for PROPHET DARPA Simbiosys Review 20

21. D(K+) = D(Cl-) Cl- K+ Dpore= Dbulk/3 Dpore= Dbulk/4 z [ Å ] Recent Simulation Results for ompF Porin DARPA Simbiosys Review 21

22. experimental Dpore= Dbulk/3 I (pA) Dpore= Dbulk/4 Vbias (mV) Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath DARPA Simbiosys Review 22

23. Recent Simulation Results for ompF Porin DARPA Simbiosys Review 23

24. Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath DARPA Simbiosys Review 24

25. Recent Simulation Results for ompF Porin DARPA Simbiosys Review 25

26. Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath DARPA Simbiosys Review 26

27. Recent Simulation Results for ompF Porin K+ Cl- DARPA Simbiosys Review 27

28. Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath 3-D Drift-Diffusion DARPA Simbiosys Review 28

29. Future plans for continuum simulation • Excess Chemical Potential Correction A correction in the electrochemical potential, to account for the finite volume and charge distribution associated to the conducting ions, is required to prevent ion densities from reaching unphysically high values. (Collaboration with W. Nonner’s group) DARPA Simbiosys Review 29