I. Dynamics of living systems

• Understanding the dynamics at the molecular level. • Understanding the dynamics at the cellular level• Filling the gap between these two levels

Dynamics Dynamics Function Function

Life’s complexity pyramid

Oltvai & Barabasi, Science 2002, 298, 763-764

“The complexity pyramid might not be specific only to cells”

Different levels of structural organization:

Residues Proteins

10

102 1

103 2-10

104 10-100

-

microtubulesmicrotubules

Incre

asin

g s

pecifi

cit

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istr

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Dom

inan

ce o

f m

ole

cu

lar

mach

inery

Challenge: to understand the long-time dynamics of large systems

Model: Coarse-grained

Method: Analysis of principal modes of motion (Frame transformation: Cartesian collective coordinates)

What is the optimal (realistic, but computationally efficient) model for a given scale (length and time) of representation?

Which level of details is needed for representing global (collective) motions?

How much specificity we need for modeling large scale systems and/or motions?

What should be the minimal ingredients of a simplified (reductionist) model?

Protein dynamics

Folding/unfolding dynamics

Passage over one or more energy barriersTransitions between infinitely many conformations

Fluctuations near the folded state

Local conformational changesFluctuations near a global minimum

Gaussian Network Model

Bahar, I., Atilgan, A.R., & Erman, B. (1997) Folding & Des. 2, 173.Flory, P.J. (1976) Proc. Roy. Soc. London A. 351, 351.

FOR MORE INFO…

Can we predict fluctuations dynamics from native state topology only?

“A single parameter potential is sufficient to reproduce the slow dynamics in good detail”

Detailed specific potentials

Approximate uniform potential

Rouse chain

R1

R2

R3

R4

Rn

=

1-1

-1 2-1

-1 2

-1

.. ...-1

2-1

-1 1

Connectivity matrix

Vtot = (/2) [ (R12)2 + (R23)2 + ........ (RN-1,N)2 ]

= (/2) [ (R1 - R2)2 + (R2 - R3)2 + ........

Kirchhoff matrix of contacts

==

1 if rik < rcut

0 if rik > rcutik==

ii = = - k ik

Vtot = (/2) RT

R

Comparison with X-ray Temperature Factors

Bahar, I., Atilgan, A.R., & Erman, B. (1997) Folding & Design 2, 173-181

FOR MORE INFO...

30

25

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75

100

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(b) 1omftheory

experiments

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80

0 20 40 60 80 100 120

(a) 2ccya

Debye-Waller factors:

Bk = 8 2 <Rk Rk> /3

Comparison with H/D Exchange – NMR data

Bahar, I., Wallquist, A., Covell, D.G., and Jernigan, R.L. (1998) Biochemistry 37, 1067.

Si = k ln W(Ri) = - (Ri)2/ (2T [-

1]ii)

Covariance matrix

(directly found from MD or MC trajectories)

R1 . R1> R1 . R2> R1 . R3>

R2. R1> R2 . R2>

RN . RN>

C =

Ri = instantaneous fluctuation in the position vector Ri of atom i= Ri - <Ri>

<R1 . R1> = ms fluctuation of site 1 averaged over all snapshots.

Eigenvalue decomposition of C

U is the matrix of eigenvectors, is the diagonal matrix of eigenvalues. The ith

column (eigenvector) of U is given by a linear combination of Cartesian

coordinates and represents the axis of the ith collective coordinate (principal

axis) in the conformational space.

C = U U-1

The ith eigenvalue represents the mean-square fluctuation along the ith principal axis. The motion along the ith principal axis is the ith mode.

Time-consuming

aspect of PCA

Decomposition into normal modes

Bahar, I., Atilgan, AR, Demirel MC, Erman B. (1998) Physical Review Lett. 80, 2733. Demirel MC, Atilgan AR, Jernigan RL, Erman B. & Bahar, I. (1999) Protein Science 7, 2522.

• Slowest (global) modes function• Fastest (local) modes stability

FOR MORE INFO...

Compare experimental B-factors with theoretical B-factors

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40

50

0 50 100 150 200 250 300

theoretical B-factorexperimental B-factor

residue number

http://www.ccbb.pitt.edu/CCBBResearchDynHemRel.htm

Comparison of the slowest modes of T and R2

0 50 100 150 200 250 300

T

R2

residue number

chain chain

b2

2

1

37-4484-94

132-141

145-146

92-100

35-40

T R transitions in Hb

Experimental T

Experimental R2

Computed (R2)Reference...

Xu & Bahar, submitted.

0 30 60 90 120 150

-1 deoxygenated Hb

-1 CO-Hb

Ord

er

par

ame

ter

residue index

Order parameters for CO-bound and unliganded Hb

0 50 100 150

CO-Hb ()deoxygenated ()

Ord

er

par

am

ete

r

residue index

Haliloglu & Bahar, Proteins 1999, 37, 654-667

Si = 3/2 <cos2i> - 1/2

For details on theory see...

(A)

(B)

(I) (II)

(I) (II)

Fluctuations of the nevirapine-bound (A) and unliganded (B) forms of RT

• Fluctuating conformations of the nevirapine-bound (A) and unliganded (B) forms of RT. The p66 subdomains are colored cyan (fingers), yellow (palm), red (thumb), green (connection) and pink (RNase H).

• See the difference in the mechanism of global fluctuations for the liganded and unliganded RTs. This difference is significant given that the sizes or distributions of fluctuations are unaffected by ligand binding

http://www.ccbb.pitt.edu/CCBBResearchDomMot.htm

Two hinge bending sites

• (A) Two hinge-bending centers on RT forming minima in Figure 1: (I) near the NNRTI binding site, involving residues 107-110 (cyan), 161-165 (green), 180-188 (red) and 219-231 (blue), and (II) near the p66 connection and RNase H interface, comprising residues 363-366 (cyan), 394-408 (green), 410-423 (loop, magenta), 424-429 (interdomain linker, red), and 504-512 (yellow).

•(B) A closer view of region II, showing explicitly the side chains near the hinge site. Close tertiary contacts are indicated by the yellow dots.

Topology-based models• Near-native fluctuations

(springs acting on effective centroids, usually C atoms)

• Ben-Avraham (1993)• Tirion (1996)• Bahar et al. (1997)• Hinsen (1998)• Sanejouand, Tama (2000)• Wriggers, Brooks (2001)• Ma (2002)

• Folding/unfolding processes (folding loss of configurational

entropy)

• Micheletti et al, PRL (1999)• Cecconi et al. Proteins (2001)• Go & Scheraga Macromolecules (1976)• Galzitskaya & Finkelstein, PNAS (1999)• Munoz et al. PNAS (1999)• Alm & Baker, PNAS (1999)• Klimov & Thirumalai, PNAS (2000)• Clementi et al (Onuchic), JMB (2000)

“Native topology determines force-induced unfolding pathways”

Protein folding kinetics examined by a Go-like model

Koga, N. & Takada, S. J Mol. Biol. 2001, 313, 171-180

Topological and Energetic Factors: What determines the transition state ensemble, and folding intermediates?

Clementi, C. Nyemeyer, H. & Onuchic, J. N. J Mol. Biol 2000, 298, 937.

Simulations with Go-like potential

“ Topology plays a central role in determining folding mechanisms”

Can we use such simplified approaches for estimating amyloidogenic intermediates?

Applied to CI2, SH3 (2-state folders) and barnase, RNase H and CheY (have intermediates)