Chicago, July 23, 2002DARPA Simbiosys Review 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

Chicago, July 23, 2002DARPA Simbiosys Review 2 Outline Discussion on continuum simulation Hierarchy of diffusivity models Progress and milestones Recent results on porin channels Work in progress and future work

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 6 Discussion on Continuum Drift-Diffusion Simulation Drift-diffusion model

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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: CaCl 2 in ompF porin

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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.

Chicago, July 23, 2002DARPA Simbiosys Review 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

Chicago, July 23, 2002DARPA Simbiosys Review 15 Progress and milestones Implementation of physical diffusivity model inferred from molecular dynamics simulations Molecular dynamics calculations using GROMACS [ S.-W. Chiu, E. Jakobsson ]

Chicago, July 23, 2002DARPA Simbiosys Review 16 Progress and milestones Large scale model of ompF Porin completed Equilibrium K + density ompF mutation (G119D) >0.75M 0M

Chicago, July 23, 2002DARPA Simbiosys Review 17 Recent Simulation Results for ompF Porin PROTEIN STRUCTURE  IV CURVE Generate customized PROPHET mesh Define protein surface flag grid points (UHBD) in protein, lipid bilayer Download protein structure (protein databank) atomic coordinates, radii amino acid sequence Assign charge to each atomic coordinate Interpolate charge to grid  fixed PROPHET script –Specify solver parameters –Input physical domain –Assemble PDE system –Input BCs & physical parameters –Solve (Newton’s method) –Write output Postprocessor reconstruct j ± and I current to electrodes. D,  fluctuation analysis of MD ion trajectories (GROMOS)

Chicago, July 23, 2002DARPA Simbiosys Review 18 Recent Simulation Results for ompF Porin

Chicago, July 23, 2002DARPA Simbiosys Review 19 Recent Simulation Results for ompF Porin

Chicago, July 23, 2002DARPA Simbiosys Review 20 Recent Simulation Results for ompF Porin electrodes lipid (  =2) protein (  =20) 96Å electrolyte (  =80) 96Å lipid (  =2) protein (  =20) aqueous pores (  =80) side view cross-section 3-D computational domain generated for PROPHET

Chicago, July 23, 2002DARPA Simbiosys Review 21 Recent Simulation Results for ompF Porin D(K+) = D(Cl-) D pore = D bulk /3 D pore = D bulk /4 z [ Å ] K+K+ Cl -

Chicago, July 23, 2002DARPA Simbiosys Review 22 Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath D pore = D bulk /3 D pore = D bulk /4 I (pA) V bias (mV) experimental

Chicago, July 23, 2002DARPA Simbiosys Review 23 Recent Simulation Results for ompF Porin

Chicago, July 23, 2002DARPA Simbiosys Review 24 Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath

Chicago, July 23, 2002DARPA Simbiosys Review 25 Recent Simulation Results for ompF Porin

Chicago, July 23, 2002DARPA Simbiosys Review 26 Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath

Chicago, July 23, 2002DARPA Simbiosys Review 27 Recent Simulation Results for ompF Porin K+K+ Cl -

Chicago, July 23, 2002DARPA Simbiosys Review 28 Recent Simulation Results for ompF Porin 100 mM KCl - symmetric bath 3-D Drift-Diffusion

Chicago, July 23, 2002DARPA Simbiosys Review 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)