Developing Modern Mathematical Methods and Computational Tools

for Biological Physics

Bo Li Math and CTBP, UCSD

CTBP EAC meeting, Rice Univ., Jan. 10 & 11, 2013

2 fast algorithms

multiscale models surfaces/interface

simulations soln’s of diff. eqns

0 2000 4000 6000 8000 100002

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stochastic models

Projects §  Variational implicit solvation (with McCammon) §  Electrostatics: Dielectric boundary forces, Ionic

size effects, etc. § A two-scale model: Brownian particles and

diffusion equation (with McCammon) § Diffusion of RNA molecules (with Levine) § Cell shapes and dynamics (with Levine/Rappel) §  Fast algorithms: multigrid, GPU computing, etc.

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Free-energy functional

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

€

c j∞,

€

q j , wρG[Γ]= Pvol(Ωm )+γ0 (1− 2τH )dSΓ

∫

€

+ρw ULJ ,ii∑

Ωw

∫ (| r − r i |)dV + Gelec[Γ]

1. Variational Implicit Solvation

(McCammon group, 2006)

The level-set method

5 !! !" # " ! $ % &#"#!

"#$

"#%

"&#

"&"

"&!

"&$

'()*+,

-./010.(

'()*+,23/240+5(627./010.(

0(010542.8219.2/:5442)(3)4.7)0(010542.82.()245*+)2)(3)4.7)

PMF

wall-particle distance

A receptor-ligand system p53/MDM2

uncharged charged

A host-guest system

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Dielectric boundary force

€

Fn = −δΓGelec[Γ]

€

r i

€

Ωm

Γ

€

Qi

€

Ωw

€

c j∞,

€

q j , wρ€

εm =1

€

εw = 80

€

∇ ⋅εε0∇ψ − χwB'(ψ) = −ρ f

€

Gelec[Γ] = −εε02|∇ψ |2 +ρ fψ − χwB(ψ)

)

* + ,

- . ∫ dV

δΓGelec[Γ]=ε02

1εm

−1εw

#

$%

&

'( |ε∂nψ |

2 +ε02εw −εm( ) (I − n⊗ n)∇ψ 2

+B(ψ)

Electrostatic force points to solutes!

2. Electrostatics

Surface energy vs. electrostatic energy Stability of water-protein interface

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8R

K: r=u(z)

LO

x

y

z1_

1+ 1 2 3 4 5k

!1.5

!1.0

!0.5

0.5

"2"1"

0 1 2 3 4 5 6

1.342

1.344

1.346

1.348

1.35

1.352

1.354

1.356

1.358

1.36

t = 0

t = 2

t = 4

t = 6

z

r

water

protein

Water molecules inside a protein are unhappy!

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Ionic size effects: A mean-field model

€

F[c] =12ρψ + β−1 ci ln(ai

3ci)i= 0

M

∑ − µicii=1

M

∑'

( )

*

+ , ∫ dV

€

ρ = ρ f + qicii=1

M

∑

€

∇ ⋅εε0∇ψ = −ρ

€

a03c0 =1− ai

3cii=1

M

∑

€

δiF[c] = 0 Generalized Boltzmann distributions

With a uniform size

€

ci =ci∞e−βqiψ

1+ a3 c j∞ e−βq jψ −1( )j=1

M∑

With nonuniform sizes

€

aia0

"

# $

%

& '

3

ln a03c0( ) − ln ai3ci( ) = β qiψ −µi( )

No explicit formulas. Constrained optimization!

Mean-field Theory and Monte Carlo Simulations

0 5 10 150

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25

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Distance to the charged surface (A)

Rad

ialde

nsity

! i(r

)(M

)

z1=+1, R

1=3.0, N

1=100

z2=+2, R

2=2.5, N

2=100

z3=+3, R

3=3.5, N

3=100

5 10 15 200

5

10

15

20

25

Distance to the charged surface (A)

Radia

ldensity

c i(r)(M

)

z1=+1, R

1=3.0, N

1=100

z2=+2, R

2=2.5, N

2=100

z3=+3, R

3=3.5, N

3=100

"   counterion stratification "   key parameters:

€

Zi /ai3

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Cluster 1 §  Housed in San Diego Supercomputer center §  1500+ processors, ~ 600 nodes §  HHMI donation of 200 2-year old nodes in 2012 §  Serial and small parallel jobs

Cluster 2 §  Housed in the Physics Server Room §  48 cores, 24 GPUs

Software: AMBER, APBS, AutoDock, CHARMM, GAMESS, Gaussian, Gromacs, MMTSB, NAMD, Python, Rasmol, VMD

CTBP Computational Resources at UCSD

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§  377th in the Top 500 supercomputer list of June 2012

§  84 teraflops of performance §  24,576 cores §  Massively parallel

IBM Blue Gene/P at Rice University

6 racks 192 cards 6,144 cards 24,576 cores

MBB (Math & Biochem-Biophys) Group §  Initiated in 2007. §  Current: 4 graduate students, 2 postdocs, and 2

visitors, 2 faculty members. §  Former members: postdocs, graduate students,

2 undergrads, and 2 high school students. §  Graduate students: San Diego Fellowship

through CTBP, UCOP Computational Science Fellowship, HHMI Fellowship UCSD nominee.

§  Weekly MBB seminars. §  Brainstorming workshops (with CTBP groups).

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Thank you !

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