continuum modeling of electrodiffusion around single molecule benzhuo lu institute of computatuonal...
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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