Monte Carlo Particle Simulation of Ionic Channels

1. Monte Carlo Particle Simulation of Ionic Channels Trudy van der Straaten Umberto Ravaioli Beckman Institute University of Illinois at Urbana-Champaign DARPA Simbiosys Review 1

2. Outline • Introduction to Transport Monte Carlo simulation • Extensions for the treatment of ionic transport • inclusion of ions as charged spheres • Validation of the transport model for microfluidics: • pair correlation function results • Contact injection issues and model development • Future work DARPA Simbiosys Review 2

3. Introduction to Transport Monte Carlo simulation A Transport Monte Carlo model simulates transport of charged particles as a sequence of trajectories interrupted by random scattering events. The free flights have random duration, obtained by selecting a uniform random number “r”. For uniform scattering rate , the flight has duration given by DARPA Simbiosys Review 3

4. Neuman Boundary Conditions Dirichlet Boundary Conditions Introduction to Transport Monte Carlo simulation The charged particle trajectories are evaluated from a detailed electric field distribution in space, obtained by solving the Poisson equation. A charge assignment procedure associates the carrier charge to the mesh, to generate the right-hand side of the discretized equation. DARPA Simbiosys Review 4

5. Introduction to Transport Monte Carlo simulation Short-range charge-charge interaction may be introduced by evaluating the Coulomb force in a range surrounding the particles. The total force is obtained by adding the short-range Coulomb force (particle-particle) with the Poisson equation force (particle-mesh), plus a correction to eliminate double counting in the overlap region between the two domains. DARPA Simbiosys Review 5

6. Gramicidin - Successful Na+ trajectory - Point particle model Extension for the treatment of ionic channels Water is assumed to be a continuum background with a dielectric permittivity distribution. Interaction between charged ions and water is modeled by the scattering rate. DARPA Simbiosys Review 6

7. Lennard-Jones 6-12 potential Lennard-Jones potential point-particle Coulomb potential Extension for the treatment of ionic channels Ion size is introduced by dressing the particles with a Lennard-Jones model potential. This creates an additional force that maintains the ions separated from each other and from the boundary. DARPA Simbiosys Review 7

8. Validation of the transport model for microfluidics The Transport Monte Carlo model needs to be validated by comparing with other models that treat water interaction in greater detail. We compare against benchmarks obtained from Molecular Dynamics simulations (Rush group) and Metropolis Monte Carlo simulations (Utah group), which calculate the ion-ion pair correlation function in space. Reproducing the ion-ion pair correlation function is a crucial prerequisite to obtaining the correct thermodynamic properties of the system. DARPA Simbiosys Review 8

9. Validation of the transport model for microfluidics The pair correlation function is the radial distribution function that measures how atoms organize themselves around one another. It is a measure of the probability of finding two atoms separated by a distance . It can be measured from x-ray and neutron diffraction experiments and is readily computed from simulations of trajectories We compared results for simulation of a bulk monovalent electrolyte solution. DARPA Simbiosys Review 9

10. Validation of the transport model for microfluidics The benchmarks were provided for a simplifed (shifted-truncated) Lennard-Jones potential DARPA Simbiosys Review 10

11. Validation of the transport model for microfluidics DARPA Simbiosys Review 11

12. Magnified view Validation of the transport model for microfluidics DARPA Simbiosys Review 12

13. Validation of the transport model for microfluidics Simulation conditions 1 Molar monovalent electrolyte solution q+ = +1; q-= -1 Lennard-Jones distance parameter + = -= 3Å Simulation cell LLL DARPA Simbiosys Review 13

14. Validation of the transport model for microfluidics DARPA Simbiosys Review 14

15. Validation of the transport model for microfluidics DARPA Simbiosys Review 15

16. Validation of the transport model for microfluidics DARPA Simbiosys Review 16

17. Validation of the transport model for microfluidics DARPA Simbiosys Review 17

18. Validation of the transport model for microfluidics DARPA Simbiosys Review 18

19. Validation of the transport model for microfluidics DARPA Simbiosys Review 19

20. Validation of the transport model for microfluidics DARPA Simbiosys Review 20

21. Validation of the transport model for microfluidics DARPA Simbiosys Review 21

22. Validation of the transport model for microfluidics DARPA Simbiosys Review 22

23. Validation of the transport model for microfluidics DARPA Simbiosys Review 23

24. Validation of the transport model for microfluidics DARPA Simbiosys Review 24

25. Validation of the transport model for microfluidics DARPA Simbiosys Review 25

26. Results obtained using untruncated Lennard-Jones Validation of the transport model for microfluidics DARPA Simbiosys Review 26

27. Contact injection issues and model development Work in progress is addressing the formulation of contact boundary conditions in the full ion channel simulation by Transport Monte Carlo. For realistic simulation, one has to reach a trade-off between computational cost and model complexity. The actual ion population in the simulation domain is small and the real computational bottleneck in 3-D simulation becomes the solution of Poisson equation. The most natural way to implement contacts, avoiding excessive fluctuations or spurious effects, is to provide buffer layers where the bath concentration is maintained constant. DARPA Simbiosys Review 27

28. Contact injection issues and model development Gramicidin x ( Å ) z ( Å ) DARPA Simbiosys Review 28

29. Future Work Completion of comparisons for pair-correlation function over a wide range of benchmarks. Inclusion of parallelized Poisson solver (several developed and tested on SGI Origin and PC clusters). Completion and testing of buffer layer contacts. Design of a reduced Monte Carlo algorithm based on Local Iterative procedure. Optimization of the overall Transport Monte Carlo algorithm, based on different grids for Poisson solution and ion dynamics. DARPA Simbiosys Review 29