Continuum Modeling of Electrodiffusion around single molecule

Benzhuo LuInstitute of Computatuonal Mathematics and

Scientific/Engineering Computing

KITPC, Beijing, July 29, 2009

Outline

• Introduction

• Continuum model of Electro-Diffusion-reaction Processes

– Numerical strategies

– Applications

– Model extensions

• Summary

• Future work

Outline

• Introduction

• Continuum model of Electro-Diffusion-reaction Processes

– numerical strategies

– Applications

– Model extensions

• Summary

• Future work

Substrate consumption

electrostatic steering

Rate and binding affinity decrease with [NaCl] has been attributed to screening effects.

An example: neurotransmission in synapse

Radic Z, et al. 1997. J Biol Chem 272: 23265.

Computational approach

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

Gra

nu

lari

ty

Continuum model

http://mccammon.ucsd.edu/gallery

Solvent Continuum dielectric media -> save ~90% degrees of freedom

Protein-protein/ligand association

Solution system: free energy and dynamics

Solute charges

ρfixed f, εin

Diffusive/reactive particles

+/- salt

Density pi, εout

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Interactions:

Electrostatics

van der Waals

Outline

• Introduction

• Continuum model of Electro-Diffusion-reaction Processes

– numerical strategies

– Applications

– Model extensions

• Summary

• Future work

rate coefficient: or

boundary conditions

PNPE

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

Poisson-Nernst-Planck equations

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;

,,0)(-D)(-D= ii i

iii

i qio

iqqii epppeepqpJ

Mesh generation

MSMSSurface triangular mesh

ADVENTURE_TetMeshSurface mesh smoothing

TetGenVolume tetrahedral mesh

Sanner et al., 1996; Yagawa et al., 1995; Si and Gaertner, 2005.

MSMS Smoothed

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. 2007

Solution procedure

• Time-dependent:

• Steady-state

tn NP in Ωs

PE in Ω

tn+1

NP in Ωs

PEin Ω

Under-relaxation iteration

Charge density distributions

Electric potential

Realistic spatiotemporal resolution

Domain mesh info transfer

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

Backward Euler method

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

AFMPB: Adaptive Fast Multipole PB Solver

B. Lu, XL Cheng, JF Huang, JA McCammon, J Comput. Theor Chem, 2009

Ribosome(30S)

21 peptides and a 1540 nucleotides RNA subunit

Atoms: 88431

Size: 211 × 177 × 200 A

Decompose = s + h+ r

The singular part

The harmonic part

The regular part

Solution of the Poisson equation-- Regularization scheme II

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

• To discretize the PDE, construct a FE base function space {vi} defined at each vertex, and assume the solution is expanded in the space

,)r(va=)ru(i

ii

• Solution in the sense of the week form:

Finite element method for PDE solution

sΓsΩ

udsvDdxfvvubvuDvF(u), ))((=

0= vF(u), find u, such that

Newton iteration

Linearize (Bilinear form): vF(u),

Damped-inexact –Newton algorithm

Holst, M, 2001

sΓsΩ

l

wdsvDdxwvubvwD=

vlw),+F(udl

d=vDF(u)w,

))('(

| 0

Time-dependent solution

Backward Euler method

The weak form

sΩ

qqi dxvt

ueuv

teqvuDvF(u),

ii

)'(=

Time-dependent expansion (the method of line):

Matrix notation

an+1=an+Δan

NPE:

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

PNP versus Nonlinear Poisson-Boltsmann

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

Outline

• Introduction

• Continuum model of Electro-Diffusion-reaction Processes

– numerical strategies

– Applications

– Model extensions

• Summary

• Future work

Potential and Counter-ion density

50 mM 1:1 salt

Compensate charge 21.3e

25 mM 2:1 salt

Compensate charge 21.1e

Surface potential

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

An application to synapse

Substrate consumption

electrostatic steering

ACh consumption by AChE

AChE with a reactive patch

System:

ions, a swam of ACh (+), one AChE molecule

ACh AChE acetate + choline

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. J. Chem. Phys., 2007Zhou YC, Lu BZ, Huber G, Holst M, McCammon JA. J. Phys. Chem. B, 2008

A time-dependent diffusion-reaction process

Lu BZ, Zhou YC, Huber G, Bond S, Holst M, McCammon JA. JCP, 2007

Rate coefficients for AChE monomer-- Modification on Debye-Huckel limiting law (?)

Zhou YC, et al. JPCB, 2008

• PNP• LPB + NP (uncoupled)

Lu et al. submitted

Debye-Huckel limiting law: H

onIH

ononon k+)k(k=k SEzz1.180 10

Radic Z, et al. 1997.

Enzymatic activity versus structural dynamics

Varying rate coefficients of

different snapshots from MD

simulation trajectory

Occlusion count for each subunit.A (black), B (red), C (green), and D (blue).

Gorfe A, Chang C, Ivanov I, McCammon JA, Biophysical J. 2008

AChE

tetramer

Gorfe A, B. Lu, McCammon JA, Biophysical J. 2009

Outline

• Introduction

• Continuum model of Electro-Diffusion-reaction Processes

– numerical strategies

– Applications

– Model extensions

• Summary

• Future work

Extensions of the standard PNP model

• vdw interaction A main nonpolar interaction

Molecular surface-free model

• Size effects

Incorporating van der Waals interaction--- Molecular surface-free method

Modified diffusion equations

where

Lu BZ, McCammon JA, Chem. Phys. Lett. 2008

Water density around two atoms

One species is water!

Potentials and ionic densities around a sphere

Lu BZ, McCammon JA, Chem. Phys. Lett. 2008

Size effects on potential and diffusion process

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unreasonable packing X

Size modified PNP model

• A modified PNP model

• Add a solvent entropy term

• 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, 1997Chu V. et al., Biophysical J., 2007

Lu BZ, Zhou YC, et al. in preparation

Model extension-- Particle size modified PNP

Ionic density around a sphere Reaction coefficient

Lu BZ, Zhou YC, et al. in preparation

Outline

• Introduction

• Continuum model of Electro-Diffusion-reaction Processes

– numerical strategies

– Applications

– Model extensions

• Summary

• Future work

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.

Future work

• Aim for faster, converged and stable algorithms!

Future work: where PNP fails

B Corry, S Kuyucak, and SH Chung, Biophysical J, 2000

• PB: equilibrium situation, involve bulk conditions with system sizes much larger than the Debye length; shielding is largely overestimated in PB theory

• PNP: Nonequilibrium

G Moy, B Corry, S Kuyucak, and SH Chung, Biophysical J. 2000

Radial distribution of ion pair with two types repulsion potentials: PB solid line, dotted line BD.

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).

Ion channel

Like charge attraction

+ +

Larsen AE and Grier DG, 1997

Like-charge attraction

Vojko Vlachy, 1999, Annu. Rev. Phys. Chem. 50:145-65

Macroion and ion pair distribution functions

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)

Thanks

International Workshop on Continuum Modeling of Biomolecules

September 14-16, 2009, Beijing, China

http://lsec.cc.ac.cn/~wcmb