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Continuum Modeling of Electrodiffusion around single molecule

Continuum Modeling of Electrodiffusion around single molecule. Benzhuo Lu Institute of Computatuonal Mathematics and Scientific/Engineering Computing KITPC, Beijing, July 29, 2009. Outline. Introduction Continuum model of Electro-Diffusion-reaction Processes Numerical strategies

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Continuum Modeling of Electrodiffusion around single molecule

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  1. Continuum Modeling of Electrodiffusion around single molecule Benzhuo Lu Institute of Computatuonal Mathematics and Scientific/Engineering Computing KITPC, Beijing, July 29, 2009

  2. Outline • Introduction • Continuum model of Electro-Diffusion-reaction Processes • Numerical strategies • Applications • Model extensions • Summary • Future work

  3. Outline • Introduction • Continuum model of Electro-Diffusion-reaction Processes • numerical strategies • Applications • Model extensions • Summary • Future work

  4. An example: neurotransmission in synapse Substrate consumption electrostatic steering Rate and binding affinity decrease with [NaCl] has been attributed to screening effects. Radic Z, et al. 1997. J Biol Chem272: 23265.

  5. Computational approach Granularity all things are made of atoms, … everything that living things do can be understood in terms of the jigglings and wigglings of atoms.---- Feynman lecture in physics, 1963

  6. Continuum model Solvent  Continuum dielectric media -> save ~90% degrees of freedom Protein-protein/ligand association http://mccammon.ucsd.edu/gallery

  7. + + + + + + + - + + - - - + + + + - - - + - + + + + - - + + Solution system: free energy and dynamics Diffusive/reactive particles Interactions: Electrostatics van der Waals Solute charges fixed ρf, εin +/- salt Density pi, εout

  8. Outline • Introduction • Continuum model of Electro-Diffusion-reaction Processes • numerical strategies • Applications • Model extensions • Summary • Future work

  9. boundary conditions or Poisson-Nernst-Planck equations PNPE rate coefficient: Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

  10. Special cases • Flux PNPE --> nonlinear Poisson-Boltzmann equation • Pure diffusion equation • Partially coupled case (Smoluchowski equation): PBE + NP Tai, KS, et al., 2003; Song, YH, et al, 2004;

  11. ADVENTURE_TetMesh Surface mesh smoothing TetGen Volume tetrahedral mesh Mesh generation MSMS Surface triangular mesh MSMS Smoothed Sanner et al., 1996; Yagawa et al., 1995; Si and Gaertner, 2005. Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. 2007

  12. Charge density distributions NP in Ωs PE in Ω Under-relaxation iteration Electric potential PE in Ω NP in Ωs tn tn+1 Realistic spatiotemporal resolution Solution procedure • Steady-state Domain mesh info transfer • Time-dependent: Backward Euler method Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

  13. Solution of the Poisson equation-- Regularization scheme I Decompose  into singular and regular parts: = s + r The singular part is easy to solve using boundary element method Hybrid FD/BEM for nonlinear PBE, Boschitsch and Fenley, 2004 Hybrid FEM/BEM Lu BZ, Zhou YC, et al, 2007

  14. AFMPB: Adaptive Fast Multipole PB Solver Ribosome(30S) 21 peptides and a 1540 nucleotides RNA subunit Atoms: 88431 Size: 211 × 177 × 200 A B. Lu, XL Cheng, JF Huang, JA McCammon, J Comput. Theor Chem, 2009

  15. Solution of the Poisson equation-- Regularization scheme II Decompose = s + h+ r The singular part The harmonic part The regular part Chern IL, Liu JG, Wang WC, 2003; Chen, L, Holst M, Xu J, 2007; Geng WH, Yu SN, Wei GW, 2007; Lu BZ, Zhou YC, Holst M, and McCammon JA, 2008

  16. Finite element method for PDE solution • To discretize the PDE, construct a FE base function space {vi} defined at each vertex, and assume the solution is expanded in the space • Solution in the sense of the week form: find u, such that

  17. Newton iteration Linearize (Bilinear form): Damped-inexact –Newton algorithm Holst, M, 2001

  18. Time-dependent solution NPE: The weak form Time-dependent expansion (the method of line): Matrix notation Backward Euler method an+1=an+Δan Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

  19. PNP versus Nonlinear Poisson-Boltsmann Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

  20. Outline • Introduction • Continuum model of Electro-Diffusion-reaction Processes • numerical strategies • Applications • Model extensions • Summary • Future work

  21. Potential and Counter-ion density 50 mM 1:1 salt Compensate charge 21.3e Surface potential 25 mM 2:1 salt Compensate charge 21.1e Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

  22. An application to synapse Substrate consumption electrostatic steering

  23. ACh consumption by AChE System: ions, a swam of ACh (+), one AChE molecule ACh AChEacetate + choline AChE with a reactive patch Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. J. Chem. Phys., 2007 Zhou YC, Lu BZ, Huber G, Holst M, McCammon JA. J. Phys. Chem. B, 2008

  24. A time-dependent diffusion-reaction process Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

  25. Rate coefficients for AChE monomer-- Modification on Debye-Huckel limiting law (?) • LPB + NP (uncoupled) • PNP Debye-Huckel limiting law: Radic Z, et al. 1997. Lu et al. submitted Zhou YC, et al. JPCB, 2008

  26. Enzymatic activity versus structural dynamics Varying rate coefficients of different snapshots from MD simulation trajectory AChE tetramer Occlusion count for each subunit. A (black), B (red), C (green), and D (blue). Gorfe A, B. Lu, McCammon JA, Biophysical J. 2009 Gorfe A, Chang C, Ivanov I, McCammon JA, Biophysical J. 2008

  27. Outline • Introduction • Continuum model of Electro-Diffusion-reaction Processes • numerical strategies • Applications • Model extensions • Summary • Future work

  28. Extensions of the standard PNP model • vdw interaction A main nonpolar interaction  Molecular surface-free model • Size effects

  29. Incorporating van der Waals interaction--- Molecular surface-free method Modified diffusion equations where One species is water! Water density around two atoms Lu BZ, McCammon JA, Chem. Phys. Lett. 2008

  30. Potentials and ionic densities around a sphere Lu BZ, McCammon JA, Chem. Phys. Lett. 2008

  31. + + + + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - Size effects on potential and diffusion process unreasonable packing X

  32. Size modified PNP model • Add a solvent entropy term • A modified PNP model Lu BZ, Zhou YC, et al. in preparation • The size effect is easier to be formulated in PNP model for multiple species with different sizes, but difficult in modified PBE, Borukhov, D. Andelman, and H. Orland, PRL, 1997 Chu V. et al., Biophysical J., 2007

  33. Model extension-- Particle size modified PNP Ionic density around a sphere Reaction coefficient Lu BZ, Zhou YC, et al. in preparation

  34. Outline • Introduction • Continuum model of Electro-Diffusion-reaction Processes • numerical strategies • Applications • Model extensions • Summary • Future work

  35. Summary Systematically developed variable rigorous, efficient, and/or accurate numerical methods for PB electrostatics and force calculations  AFMPB solver Developed fully-continuum diffusion models and the corresponding numerical methods PNP solver. Possibly applied to enzyme kinetics, transportation, ion-channel … Predicted strong dependence of reaction rate coefficients on high substrate concentration in addition to the ionic strength, which was ignored in the Debye-Huckel limiting law.

  36. Future work • Aim for faster, converged and stable algorithms!

  37. Future work: where PNP fails Ion channel • PB: equilibrium situation, involve bulk conditions with system sizes much larger than the Debye length; shielding is largely overestimated in PB theory • PNP: Nonequilibrium Changes in the concentration profiles in PNP (solid lines) and BD (dotted lines) as the channel radius is increased progressively from r = 4 to 6, 8, and 12 Å. The concentration of sodium ions is shown in (A) and of chloride ions in (B). Radial distribution of ion pair with two types repulsion potentials: PB solid line, dotted line BD. G Moy, B Corry, S Kuyucak, and SH Chung, Biophysical J. 2000 B Corry, S Kuyucak, andSH Chung, Biophysical J, 2000

  38. Macroion and ion pair distribution functions Like-charge attraction + + Larsen AE and Grier DG, 1997 Vojko Vlachy, 1999, Annu. Rev. Phys. Chem. 50:145-65 Like charge attraction

  39. Acknowledgments Yongcheng Zhou (UCSD --> Colorado State University) Stephen Bond (Univ. of Illinois at Urbana-Champaign) Xiaolin Cheng (UCSD -> Oak Ridge National Lab) Jingfang Huang(Univ. of North Carolina) J. Andrew McCammon (Univ. Cal. San Diego) Michael Holst (UCSD)

  40. Thanks

  41. International Workshop on Continuum Modeling of Biomolecules September 14-16, 2009, Beijing, China http://lsec.cc.ac.cn/~wcmb

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