Bioinformatics: Practical Application of Simulation and Data

Mining

Protein Folding II

Prof. Corey O’HernDepartment of Mechanical Engineering

Department of PhysicsYale University

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 minimumtransition

•Transitions are temperature dependent

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

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

B=hydrophobicN=neutralL=hydrophilic

Nsequences= 3 ~ 1022

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

Nm Number of sequences forNm=46

different mapping?

and dynamics

Molecular Dynamics: Equations of Motion

rFi =m i

ri =m id 2rridt2

rri t( ) for i=1,…Natoms

rFi =−

∂V∂rri

Coupled 2nd order Diff. Eq.

How are they coupled?

(iv) Bond length potential

Pair Forces: Lennard-Jones Interactions

ij rrij

rrj

rri rrj +

rrij=rri

rrij =

rri −rrj

Parallelogramrule

rFij =−

dVdrij

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

force on i due to j

rFi =

rFij

j∑

‘Long-range interactions’

BB

V(r)

r/

NB, NL, NNLL, LB

r*=21/6

hard-core

attractions-dV/dr < 0

Bond Angle Potential

Vb θijk( )=k02

θijk −θ0( )2

θ0=105

i jk

cosθijk =rji grjk

θijk

rFjb =−

dV b

dθijk

dθijk

drrj=−k0 θijk −θ0( )

dθijk

drrj

θijk=[0,]

Dihedral Angle Potential

Vd(ijkl)

Vd(ijkl)Vd jijkl( )=A 1+ cojijkl( )+ B 1+ co3jijkl( )

cosjijkl=

rrij×rrkj( )g

rrjk ×rrlk( )

rrij×rrkj

rrjk ×rrlk

rFjd =−

dV d

djijkl

djijkl

drrj= Ainjijkl+ 3Bin3jijkl( )

djijkl

drrj

ijkl

Successive N’s

Bond Stretch Potential

Vbs =kb2

rij−( )2

rFijbs =−kb rij−( )rij

i j

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

rFitot =

rFi

lr +rFi

b +rFi

d +rFi

b =m iri

Equations of Motion

xi t + Δt( )=xi t( )+ vi t( )Δt+12i t( ) Δt( )2

vi t+ Δt( )=vi t( )+i t( )+ i t+ Δt( )

2Δt

velocityverletalgorithm

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

Collapsed Structure

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

Native State

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

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start end

native states

Total Potential Energy

slow quench

unfolded

native state

Rg2 =

12N2 rij

2

i, j∑

Radius of Gyration

Tf

(1)Construct the backbone in 2D

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

(3) 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 LandscapeRg

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

5 contacts4 contacts 3 contacts

E/CE/C

Rate DependenceEC

CT

5 contacts

4 contacts

3 contacts2 contacts

Misfolding

Reliable Folding at Low Rate

log10 rη / T( )

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Slow rate

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Fast rate

Next…

•Thermostats…Yuck!•More results on coarse-grained models•Results for atomistic models•Homework•Next Lecture: Protein Folding III (2/15/10)

So far…

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