Bioinformatics: Practical Application of Simulation and Data

Mining

Protein Folding II

Prof. Corey O’HernDepartment of Mechanical Engineering & Materials

ScienceDepartment of Physics

Yale University

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What did we learn about proteins?•Many degrees of freedom; exponentially growing # of energy minima/structures•Folding is process of exploring energy landscape to find global energy minimum•Need to identify pathways in energy landscape; # of pathways grows exponentially with # of structures•Coarse-graining/clumping required

energy minimum

transition

•Transitions are temperature dependent 2

J. D. Honeycutt and D. Thirumalai, “The nature of foldedstates of globular proteins,” Biopolymers 32 (1992) 695.

T. Veitshans, D. Klimov, and D. Thirumalai, “Protein folding kinetics: timescales, pathways and energy landscapes

in terms of sequence-dependent properties,” Folding & Design 2 (1996)1.

Coarse-grained (continuum, implicit solvent, C) models for proteins

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3-letter C model: B9N3(LB)4N3B9N3(LB)5L

B=hydrophobic

N=neutral

L=hydrophilic

Nsequences= 3 ~ 1022

Np ~ exp(aNm)~1019 Number of structuresper sequence

Number of sequences forNm=46

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different mapping?

and dynamics

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Molecular Dynamics: Equations of Motion

for i=1,…Natoms

Coupled 2nd order Diff. Eq.

How are they coupled?

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(iv) Bond length potential

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Pair Forces: Lennard-Jones Interactions

ij

Parallelogramrule

-dV/drij > 0; repulsive-dV/drij < 0; attractive

force on i due to j

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‘Long-range interactions’

BB

V(r)

r/

NB, NL, NN

LL, LB

r*=21/6

hard-core

attractions-dV/dr < 0

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Bond Angle Potential

0=105

i jkijk

ijk=[0,]

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Dihedral Angle Potential

Vd(ijkl)

Vd(ijkl)

ijkl

Successive N’s

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Bond Stretch Potential

i j

for i, j=i+1, i-1

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Equations of Motion

velocityverletalgorithm

Constant Energy vs. Constant Temperature (velocity rescaling, Langevin/Nosé-Hoover thermostats)

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Collapsed Structure

T0=5h; fast quench; (Rg/)2= 5.48

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Native State

T0=h; slow quench; (Rg/)2= 7.78

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start end

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native states

Total Potential Energy

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slow quench

unfolded

native state

Radius of Gyration

Tf

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Construct the backbone in 2D

Assign sequence of hydrophobic (B) and neutral (N) residues, B residues experience an effective attraction. No bond bending potential.

Evolve system under Langevin dynamics at temperature T

Collapse/folding induced by decreasing temperatureat rate r.

BN

2-letter C model: (BN3)3B

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Energy Landscape

end-to-end distance end-to-end distance

5 contacts4 contacts 3 contacts

E/CE/C

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Rate Dependence

5 contacts

4 contacts

3 contacts2 contacts

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Misfolding

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Reliable Folding at Low Rate

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Slow rate

Fast rate

Next…

•Thermostats…Yuck!•More results on coarse-grained models•Results for atomistic models•Homework

So far…

•Uh-oh, proteins do not fold reliably…•Quench rates and potentials

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