Chaos in Protein Dynamics

Michael Braxenthaler, Ron Unger, DitzaAuerbach, James A Given and John Moult

Presented by: Trilce Estrada

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Folding Pathways

Protein folding mustfollow a pathwayfrom any unfoldedconformation to thenative conformation

To what extent theevents along thefolding pathway areinvariant?

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Folding Pathways (2)

Difficulty to characterize experimentally: Transient folding events Partially folding intermediates

Early folding events seem to occur locally insmall regions of the polypeptide chain

Study of small regions is more tractable thanthe study of complete proteins

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MD algorithm

MD simulations consist on iteratively solvesystems of nonlinear differential equations.

Many such systems are known to exhibitchaotic behavior

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Chaotic systems

A system is defined as chaotic if small perturbationsin its initial configuration are amplified exponentiallywith time

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Three objectives for this paper

Show that MD simulations of protein foldingexhibit chaotic behavior

Calculate the divergence rate of the systemat different stages of the folding process

Study the effect of this chaotic behavior onthe occurrence of certain folding events

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Measurement of chaos in MDsimulations The standard measure

for determining whetheror not a system ischaotic is the Lyapunovexponent

There are as manyLyapunov exponentsas degrees of freedomin the system

A system is chaotic if ithas at least 1 positiveLyapunov exponent

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Proving that MD simulations ofprotein folding are chaotic Calculate the maximum Lyapunov exponent

in the MD folding simulations by introducingsmall perturbations at many points along thetrajectory

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MD simulations (1)

13-residue barnasefragment. Explicit solventtreatment

1000 ps MD simulationstarting from the foldedstate

600 ps MD simulationstarting from arepresentative unfoldedconformation

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MD simulations (2)

MD simulationswere executedusing the programDISCOVER

Native-like system 1598 water molecules in

a box of 37x37X37 A3

Denatured system 2042 water molecules in

a box of 40x40x40 A3

Both systems wereequilibrated

Restart files were savedevery 10ps

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Chaotic behavior of MD simulations

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Effect of Magnitude of Perturbation

Exponential growth with similar slopes. The plateau is reached at a constant RMSD

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Behavior at Different Stages ofFoldingCan major folding events be avoided if thesimulation is slightly perturbed just before theyoccur?

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Distance from the transition: 1.5 ps

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Distance from the transition: 0.8ps

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Distance from the transition: 10.8ps

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Effects of Chaotic Behavior onFolding Events

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Numerical Stability (1) Experiment 1

Cray Y-MP Double precision DISCOVER CVFF force field A cutoff radius for non bonded interactions of 8.5 Å in a

water box All atom representation

Experiment 2 SGI Indigo2 workstation Single precision GROMOS software and force field Instead of an all-atom representation the GROMOS force

field uses a united atom representation for non polar CHxgroups.

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Numerical Stability (2)

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Is MD really numerically Stable?

It has been a common frustrating experiencethat when a computer or a compiler hasbeen slightly changed, trajectories of MDcannot be reproduced. The chaotic behaviorof our system offers an explanation for theseexperiences.

Is a simulation of 2ps enough to conclude thestability of MD simulations?

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Discussion (1)

1.- Show that MD simulations of protein foldingexhibit chaotic behavior

Individual MD trajectories of folding are toosensitive to small perturbations to havesignificant predictive quality.

Single trajectories may provide insight intothe type of events possible in a system, butaveraging over a number of independentruns is essential to obtain data on eventlikelihood.

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Discussion (2)

2.- Calculate the divergence rate of the systemat different stages of the folding process

Observed a rate of exponential divergenceof 1A RMSD

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Discussion (3)

3.- Study the effect of this chaotic behavior onthe occurrence of certain folding events

Given two identical polypeptide chains startto fold at the same time from the same initialconformation, except for one atom differingin position by 10-9 Å

The results of the paper suggest that alongthe way the folding pathways will rapidlydiverge and become significantly distinct.

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Transient chaos

How can reconcile the chaotic nature of thefolding trajectory and the fact that alltrajectories end at essentially the same foldedfunctional conformation?

This combination of properties can be foundin dissipative systems and is known astransient chaos.

Thank you!!