Multiscale modeling of crystalline dynamics and the corresponding diffuse X-ray scatter from

biological molecules

Demian Riccardi

Qiang Cui and George N. Phillips, Jr.

Advanced Light Source User Meeting October 9, 2013

Lawrence Berkeley National Laboratory

anl.gov “The difference is comparable to skipping through an open field or being crammed into a crowded elevator” -Lee Makowski

Molecular Context

Crystal as metaphor Phillips, Fillers, and Cohen, BiophysJ (1980)

30-70% water Similar to cytoplasm Record Jr., M. T. et. al. TiBS (1998)

Structure/Dynamics/Function

A. B.

Molecular Context

Zhang, Wozniak, and Matthews JMB (1995) T4 Lysozyme

Elas%c'Network'Model'Typical representation of potential energy

Just use simple springs between interacting atoms!

Harmonic approximation: in terms of Hessian Matrix

MM Tirion, PRL (1996)

V = Vbonds + Vangles + Vdihedrals + Vnonbond

Vsprng(ra, rb) =C

2(|ra � rb|�| r0a � r0b|)2 if |r0a � r0b| ⇥ Rcut

V =�

a,b

Vsprng(ra, rb)

V (r) =12rT �r

ENM variations

Cutoff: 8Å Cutoff: 16Å

Isolated Lysozyme

k(R) =⇢

R�c if R Rcut

0 if R > Rcut

- The network - coarse-graining - more or less “chemistry”

- The force constant - relative interaction strength - can be more detailed

Surprising efficacy: low frequency modes dominate No energy minimization

Most common:

Normal Mode Analysis

Eigen values and vectors

But try and do this on the entire crystal...

Variance-covariance matrix!

LT ⇥L = �

⇥uuT ⇤ = kBT � L��1LT

Go, Noguti, and Nishikawa, PNAS (1983) Brooks and Karplus, PNAS (1983) Levitt, Sander, and Stern, JMB (1985)

Ichiye and Karplus, Proteins (1991)

Variance-covariance matrix Temperature Factors

Atom-Atom correlations

“Liquid like” correlations CDOS

Diffuse X-ray scattering

Some useful information from ENM NMA

huiuTi i =

0

@h�xi�xii h�xi�yii h�xi�ziih�yi�xii h�yi�yii h�yi�ziih�zi�xii h�zi�yii h�zi�zii

1

A

�ij =q

u2i u

2j

h�ri�rjiqh�r2

i ih�r2j i

G(!f ) =fX

i

g(!i)C(r) = C0(1� �0)e�

r� + �0

Caspar, Clarage, Salunke, and Clarage Nature (1988) !

Meinhold and Smith Proteins (2007) !

Example: Tropomyosin

PDBID: 2TMA Phillips, JMB (1986) !

Isotropic Temperature Factor

Isolated〈

u2 k 〉[Å

2 ]

0

0.5

1.0

Residue0 100 200 300 400 500

Tropomyosin

symmetry reduced P1 Unit Cell

•  pdb'structure'•  assymetric'unit'•  unit'cell'•  crystal'

Super Cell

symmetry reduced

Tropomyosin

Crystalline Context: BVK boundary conditions

�Thermal vibrations in Crystallography�,Willis and Pryor (1975)

Sample wavevectors to construct variance-covariance matrix

Dimensions: 3nN to 3n

Construct the dynamic matrix from Hessian

⇥uuT ⇤ = kBT � ⇥L(q)��1L(q)†⇤q

L(q)T D(q)L(q) = �(q)

D(kk0,q) =X

l0

�(kk0

0l0 )eiq·(R(k0l0)�R(k0))

Isotropic Temperature Factor

IsolatedPBC〈

u2 k 〉[Å

2 ]

0

0.5

1.0

Residue0 100 200 300 400 500

PBC: q = 0

Covariances and correlations

Riccardi, Cui, Phillips, Biophys J (2009)

Riccardi, Cui, Phillips, Biophys J (2010)

Variances and DOS

0.00

0.25

0.50

0.75

1.00

Cu

mu

lativ

e D

en

sity

of S

tate

s

Frequency (THz)

ANM

ANM

HCA

0 20 40 60 80 100 120 140

10

16

83 high resolution structures

Atom

-Ato

m C

orre

latio

n

-0.2

0

0.2

0.4

0.6

0.8

1.0

Distance (Å)0 10 20 30 40 50

0814253700ANM2

50TLSλ=10 Åλ= 2 Å

Footline Author PNAS Issue Date Volume Issue Number 11

0 4000 8000 e.

a b c

d e f

Footline Author PNAS Issue Date Volume Issue Number 13

Staph Nuclease

REACH Liquid-like TLS

PFANM2 100 a vs d 8 a vs d -Some popular ENMs that are unphysical. -Relative interaction strength! !Simplest: 1/R6

-Crystal context for crystalline observables

Wall, Ealick, Gruner, PNAS (1997)

Crystallographers Tricks

Subtract the background: IBragg

= ITotal

� IDiffuse

Multiscale modeling of Diffuse X-ray scattering

From the entire crystal:

ID = IT � IB

ID =X

kl

X

k0l0

fk(Q)e�12QT huklu

TkliQfk0(Q)e�

12QT huk0l0u

Tk0l0 iQeiQ·(rkl�rk0l0 )

⇥(eQT hukluTk0l0 iQ � 1)

Equations simplified in the following:

ID =X

kl

X

k0l0

Blk,l0k0(Q)(eQT huklu

Tk0l0 iQ � 1)

Independent'unit'cells'

Atoms not correlated if in different unit cells

Correlations within unit cell

ID(Q) = NX

k

X

k0

Bk,k0(Q)(eQT huku

Tk0 iQ � 1)

hukuk0i = huk,luTk0,l0i ⇥ �l,l0

Independent'Blocks'of'Atoms'

Correlations within blocks

ID,blocks

(Q) = NblocksX

m

X

k,k

02m

Bk,k

0(Q)(eQT huku

Tk0 iQ � 1)

ID,atoms

(Q) = NX

k

f2k

(Q)(1� e�QT hukuTk iQ)

E.g. One atom per block

hukuk0i = huk,muTk0,m0i ⇥ �m,m0

Atoms not correlated if in different blocks

(⇥�k,k0)

Example:'Block'NMA'of'Tropomyosin'

PBC: q = 0 Project Hessian into space of rigid block TransRots Tama, Gadea, Marques, Sanejouand, Proteins (2000) Li and Cui Biophys J (2002)

10 atoms/block

To all atom To all atom

50 atoms/block

To all atom To all atom

100 atoms/block

To all atom To all atom

Correla%ons'between'blocks'

ID,interblock

(Q) = NblocksX

m

X

k2m

blocksX

m

0 6=m

X

k

02m

0

Bk,k

0(Q)(eQT huku

Tk0 iQ � 1)

Contribution to the diffuse scattering from interblock correlation

ID,interresidue(Q) = ID(Q)� ID,intraresidues(Q)

ID,inter2ndary(Q) = ID(Q)� ID,intra2ndary(Q)

ID,interatomic

(Q) = ID

(Q)� ID,atoms

(Q)

Remix it

ID,blocks

(Q) = NblocksX

m

X

k,k

02m

Bk,k

0(Q)(eQT huku

Tk0 iQ � 1)

0 0.2 0.4 0.6

=10TLS1Atomself

ENM63:10

ENM250

-2

0

2

4

0 0.2 0.4 0.6

ANM250

REACH=10

TLS1Atomself

q (Å-1)

a. b. c.

-50

0

50

100

0 0.2 0.4 0.6

BNM63:10

BNM250

TLS1=10

Atom

-Ato

m C

orre

latio

n

-0.2

0

0.2

0.4

0.6

0.8

1.0

Distance (Å)0 10 20 30 40 50

0814253700ANM2

50TLSλ=10 Åλ= 2 Å

Footline Author PNAS Issue Date Volume Issue Number 11

Riccardi, Cui, Phillips, Biophys J (2010)

Wal

l, Ea

lick,

Gru

ner,

PNA

S (1

997)

M

einh

old

and

Smith

Pro

tein

s (20

07)

!C-alpha All atom BNM

1 Urea per block 3.5 Å cutoff

1/R6

Wavevectors!

Swaminathan, Craven, and McMullan, Acta D (1984)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

24000

26000

28000

30000

ID hk0 One block Tipped a little to visualize

0

30000

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

Iatom,self Iurea,self Irows

ID

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

Iplanes Ixblox

ID

Minus!

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

Minus!

Minus! Minus!

ID

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

0 5000 10000 15000 20000 25000 30000

Minus!

Minus! Minus!

ID,block

ID,blocks

(Q) = NblocksX

m

X

k,k

02m

Bk,k

0(Q)(eQT huku

Tk0 iQ � 1)

Multiscale Modeling of ID

TMTOWDI Vary the model VCOV Vary the atom groups of VCOV Dissect the pattern

Conclusions

Acknowledgements

NLM Grant: 5T15LM007359

•  George N. Phillips, Jr.

•  Qiang Cui

•  Jeremy C. Smith

•  Juan Rodriguez-Carvajal (CrysFML)