Ext

ract

ing

coar

se-g

rain

edel

astic

beh

avio

rof

capsid

prot

eins

from

mol

ecula

rdyn

amic

s

sim

ula

tion

s

Ste

phen

D.H

icks

Christ

opher

L.H

enle

y

Corn

ell

Univ

ers

ity

Gor

don

Con

f.on

Phys

ical

Virol

ogy

Feb

.20

09

Outlin

e

IG

oal:

Whol

e-ca

psid

elas

tici

tyfrom

all-at

omM

Dof

singl

edom

ains

I1.

Set

-up:

gener

aliz

edsp

rings

I2.

Met

hod

toex

trac

tst

atic

san

ddyn

amic

sfrom

sim

ula

tion

I3.

Applic

atio

nto

HIV

I4.

Mor

eon

the

dyn

amic

s

Mot

ivat

ions

forge

ttin

gel

astici

ty

ICap

sid

shap

e(r

ounded

orfa

cete

d?)

dep

ends

onben

d/s

tret

chra

tio

ofel

astic

const

ants

[Lid

mar

etal

,PRE

2003

].

IAto

mic

-for

ce-m

icro

scop

e(A

FM

)in

den

tation

mea

sure

men

ts:

pred

ict

valu

eof

spring

const

ant?

IG

uid

ance

infitt

ing

prot

ein

units

into

hyp

othet

ical

model

sof

the

full

stru

cture

IM

uta

nt

buddin

gm

orphol

ogie

sdue

toa

poi

nt

muta

tion

:bec

ause

itch

ange

san

elas

tic

oran

gle

par

amet

erre

lating

two

capsid

units?

[Vog

tla

b]

IK

eyin

gred

ient

inou

rol

dirre

vers

ible

asse

mbly

model

[Hic

ksan

dH

enle

yPRE

2006

]:ben

d/s

tret

chra

tio

det

erm

ined

rate

offa

talm

ista

kes;

ther

mal

fluct

uat

ions

insh

ape

shou

ldgo

vern

grow

thra

te.

Gen

eral

ized

Spr

ings

ITre

atea

chpr

otei

nas

one

ora

few

rigi

dunits

IStu

dy

each

pai

rwise

inte

ract

ion

bysim

ula

ting

just

2units⇒

smal

l,tr

acta

ble

MD

sim

ula

tion

s

IIn

tera

ctio

nm

aybe

eith

era

flex

ible

cova

lent

linke

r,or

non

-cov

alen

tdock

ing

ID

eter

min

ein

tera

ctio

ns

with

no

assu

mption

sex

cept

smal

ldev

iation

sfrom

afree

ener

gym

inim

um

)

IPre

dic

tm

easu

rem

ents

with

no

adju

stab

lepar

amet

ers!

Gen

eral

form

of“s

prin

g”in

tera

ctio

ns

PSfrag

repla

cem

ents

stre

tch

shea

rben

dtw

ist

IG

ener

aliz

esp

ring

todep

end

onal

l6

deg

rees

offree

dom

Fre

een

ergy

(exp

and

tose

cond

order

)

Isp

ring

const

ant

bec

omes

6×

6m

atrix

F=

1 2~ u

TK

uu~ u

+~ u

TK

uξ~ ξ

+1 2~ ξ

TK

ξξ~ ξ

tran

slat

ional

stiff

nes

ses

orie

nta

tional

stiff

nes

ses

tran

slat

ion-o

rien

tation

couplin

g!

PSfrag

repla

cem

ents

Act

ual

pos

itio

n

Equili

briu

mpos

itio

n

~ u~ ξ

What

we

could

do

with

this?

ICoa

rse-

grai

ned

sim

ula

tion

sof

spring

net

wor

ks−→

trac

table

I1000-fold

reduct

ion

indim

ension

Ider

ive

latt

ice

spac

ings,

continuum

elas

tic

par

amet

ers,

...

[Im

age:

whole

-cap

sid

sim

ula

tion

Fre

ddolin

oet

al...]

Apply

tore

trov

irus

(HIV

):in

tera

ctio

ns?

Pos

sible

inte

ract

ions

inge

ner

icca

psid

IQ

uas

iequiv

alen

ceal

low

s3

clas

ses

ofin

tera

ctio

ns

bet

wee

nca

psid

prot

eins

(show

nas

trap

ezoi

ds)

ITrim

er,H

exam

er/P

enta

mer

,an

dD

imer

units

and

inte

ract

ions.

Mat

ure

HIV

capsid

has

4in

tera

ctio

ns

CA

has

N-t

erm

inal

dom

ain

(NT

D,blu

e);

C-t

erm

inal

dom

ain

(CT

D,ye

llow

)

I(0

)N

TD

-CT

Dlin

ker

I(1

)N

TD

-CT

D(T

rim

er)

I(2

)N

TD

-NT

D(P

ent./h

ex.)

I(3

)CT

D-C

TD

(Dim

er)

Sim

ula

tion

det

ails

Sim

ula

tion

det

ails

IN

AM

Dso

ftwar

epac

kage

I1,

500,

000

step

s@

2fs

=3n

sI

2dom

ains

+4–

7A

wat

er

linke

r

CT

Ddim

er

NT

Dhet

erodim

er

Exa

mple

tim

eser

ies:

linke

r

Rot

atio

nal

direc

tion

s

PSfrag

repla

cem

ents

tim

e

sin(θx/2)

PSfrag

repla

cem

ents

tim

esin(θy/2)

PSfrag

repla

cem

ents

tim

e

sin(θz/2)

Tra

nslat

ional

direc

tion

s

PSfrag

repla

cem

ents

tim

e

rxPSfrag

repla

cem

ents

tim

e

ry

PSfrag

repla

cem

ents

tim

erz

Met

hod:

stat

ics

IW

em

easu

reth

eequilib

rium

fluctu

ations

and

use

the

equip

art

itio

nth

eore

mto

extr

act

entire

6×

6st

iffnes

sm

atrix

K:

〈xαxβ〉

=T

K−

1α

β

Pro

ble

m:

slow

rela

xation

s

XX

X

Art

ifac

tof

isol

atin

gon

epai

r

IRea

lco

nfigu

ration

iseq

uili

briu

mof

forc

esac

ting

onal

lsides

ICut

other

inte

ract

ions⇒

drift

sto

war

ds

unre

alistic

stat

e

IU

neq

uili

brat

edm

odes

hav

ela

rge〈x

2〉,

hen

ceap

pea

ras

spuriou

sso

ftdirec

tion

sin

Km

atrix

Equili

brat

edvs

.re

laxi

ng

direc

tion

s

Im

ostly

equili

brat

edaf

ter∼

1ns

PSfrag

repla

cem

ents

tim

e

sin(θx/2)

Ire

laxi

ng

during

entire

sim

ula

tion

PSfrag

repla

cem

ents

tim

e

sin(θz/2)

Sto

chas

tic

dyn

amic

s:tw

oke

ym

atrice

sK

and

Γ

d~ x dt

=Γ~f(x

)+

~ ξ(t)

,

I~ f(x

)=

forc

e=−

K~ x

I~ ξ(

t)=

noi

se,sa

tisf

ying〈ξ

i(t)

ξ j(t

′)〉

=D

ijδ(

t−

t′)

ID

=2k

BT

Γ=

noi

se(d

iffusion

)m

atrix

IB

y“fl

uct

uat

ion

dissipat

ion”

rela

tion

(Ein

stei

n),

the

sam

em

atrix

(Γ−

1)

tells

the

visc

ous

dam

pin

gfo

rce

IW

eca

nm

easu

reth

enoi

se/d

ampin

gm

atrix

gij(τ

)≡

⟨

δxi(

τ)δ

xj(τ

)⟩

∼D

ijτ,

for

short

τ

Idrift

velo

city

~ v d=

Γ~f,te

llsfo

rce

tobe

cance

led.

ICom

bin

ein

to“r

elax

atio

nm

atrix”

W=

Γ1/2K

Γ1/2;it

has

dim

ension

s(t

ime)

−1

and

its

eige

nva

lues

are

rela

xation

rate

sof

the

(ove

rdam

ped

)ei

genm

odes

.

IChec

kth

atre

laxa

tion

tim

esar

esu

ffici

ently

shor

ter

than

run

tim

e.

Res

ults

Tec

hnic

ality

Ico

ordin

ates

hav

ediff

eren

tunits

(3pos

itio

n+

3an

gle)

soca

n’t

dia

gonal

ize

6×

6m

atrix

I...

inst

ead,ge

teff

ective

3×

3m

atrice

sby

lett

ing

the

other

3co

ordin

ates

vary

free

ly.

Ave

rage

s:

Diff

usion

Stiffnes

s

rot

(1/s

)tr

ans

(A2/s

)ro

t(k

BT

)tr

ans

(kBT

/A2)

NT

D–N

TD

.028

1017

609.

4CT

D–C

TD

.057

1222

02.

5Lin

ker

(U)

.067

2814

01.

6Lin

ker

(F)

.067

2814

10.

85

Con

clusion

s

IG

ener

aliz

edsp

rings

are

ause

fulan

dtr

acta

ble

inte

rmed

iate

bet

wee

nfu

llat

omistic

MD

and

continuum

model

s

IA

pply

stoch

astic

dyn

amic

sfo

rmal

ism

(ove

rdam

ped

vers

ion)

tosim

ula

tion

dyn

amic

s⇒

ID

ynam

ics

give

s(1

)cr

iter

ion

for

equili

brat

ion

ofsim

ula

tion

(2)

tim

esc

ale

for

quan

tita

tive

com

puta

tion

sof

all-ca

psid

dyn

amic

susing

the

net

wor

km

odel

(no

adju

stab

lepar

amet

ers)

IT

han

ksto

DO

Efu

ndin

g,Cor

nel

lCCM

RCom

puting

Fac

ility

,an

duse

fuldiscu

ssio

ns

with

Dav

idRou

ndy

Mol

ecula

rdyn

amic

ssim

ula

tion

s:m

ore

HIV

mat

ure

capsid

(sou

rce

stru

cture

s)

IFull-

lengt

hCA

(CT

D–N

TD

linke

rst

iffnes

s)(G

anse

r-Por

nill

oset

al,20

07)

ICT

D–C

TD

dim

erbon

d(G

amble

etal

,19

97)

IN

TD

–NT

Dhex

amer

bon

d(T

ang

etal

,20

02;

Mor

tuza

etal

,20

04)

Sim

ula

tion

det

ails

IN

AM

Dpac

kage

ICH

ARM

Mfo

rce

fiel

d+

explic

itwat

er

I1,

500,

000

step

s@

2fs/

step

=3n

s

IN

TD

+CT

Ddom

ains

(mm

+nnn

residues

),7A

box

I(s

omet

imes

)ex

t.fo

rces

ondom

ains

Slo

wre

laxa

tion

s:m

ore

NT

D-C

TD

linke

rbef

ore/

afte

r:m

otio

nis

mai

nly

angu

lar

(untw

isting)

Res

ults:

mor

e

Stiffnes

sva

lues

IRat

ioof

angu

lar/

coor

din

ate

par

tis∼

20nm

–co

mpar

able

with

inte

r-unit

dista

nce

Noi

se/d

issipat

ion

mat

rix

IRat

ioof

coor

din

ate/

angu

lar

par

tsis∼

0.6

nm

:co

mpar

able

tora

diu

sof

each

prot

ein

unit.

Con

sist

ent

with

hyd

rodyn

amic

pred

iction

sfo

rpos

itio

nal

and

rota

tion

aldiff

usion

const

ants

ofa

spher

e