Lesson 2

Modelli Coarse Grained per proteine e acidi nucleici

Valentina TozziniNEST-CNR-INFM Scuola Normale Superiore

Didactic material available athttp://homepage.sns.it/tozzini/public_files/

Lesson 2

In the previous lesson…

un-biased CG models and in particular one-bead models can be build that describecomplex processes such as structuraltransitions, with a great gain incomputational cost and the possibility ofefficiently sample the phase space

Processes that involve inter-strand hbondsare more difficult to accurately reproduce,because of the h-bond directionality

NH

Cα

CO

HN

Cα

R

R

α

θ

Cα

RNH

Cα

CO

HN

R

ωψФ

How to include this in the one bead mode il work in progress …

In the meanwhile, the problem of correctly reproduce the structure ofhydrogen bonds network can be solved locally maintaining a weak biastowards a reference structure

Lesson 2 One bead Models - Parameterization

P(Q1) = exp(-βF(Q1))/Z = ∫ dQ2 … dQN exp(-βU(Q)) /Z Parameterization of a CG model based on the Boltzmann inversion

Probabilitydistribution of theinternal variable Q1

Free energy surface with respectto Q1 ~ Potential of mean force

“True” potentialenergy

If U(Q)= U(Q1) … U(QN) then U(Qi)= -kt ln (P(Qi)/P0) i.e. if the internal variables are completely uncorrelated, the “true” CG potentialcoincides with the potential mean forceOtherwhise, the potential of mean force is an approximation for the truepotentialThe potential of mean force of an internal variable can be evaluated based on theprobability distribution of that variable P(Qi)

P(Qi) can be evaluated based on experimental dataU(Qi)= -kt ln (P(Qi) /P0) = Boltzmann inverse of the probability distribution= a first approximation for the CG potential

Probabilitydistribution of thenon interacting orreference system

Lesson 2

Non bonded partof the distribution

Non localLocal

Locally biased one bead model

The g(r) (and its inverse, the potential of mean force for non bonded interaction) showsa net separation between a local peak, due to secondary structure dependent localinteractions, mainly h-bonds inter or intra chain, and a “non local” peak, mainly due tosecondary structure independent hydrophobic like interactions

Unb= Ulocal + Unon local

Eloc=E(rij)[1-exp(-α (r-rij))]2 Enonloc=E0[1-exp(-α (r-r0))]2

The function E(rij) is monotonically decreasing (exponential) to account for the largerstrength of the local bonds with respect to the non localAll the parameters are fitted on the Boltzman inverse of g(r) and then reoptimized

Unb~-kt ln g(r)

g(r)= P(r)/P0= P(r)/4πr2

=pair distribution function

Lesson 2

Non bonded partof the distribution

Non localLocal

Locally biased one bead model

The bias is local:it maintains the correct local hbond network (accurate structures) anddoes not prevent the large fluctuations far from the reference structure

Half-way between Go-like models and the unbiased model

Electrostatics:included with screened effective charges on charged amino-acids

Lesson 2 Locally biased one bead model

An example: the HIV-1 protease action mechanismThe HIV-1 protease cleaves the gag andpol viral polyproteins in functional proteinsnecessary for the virus maturationSubstrate capture mechanism: unknownBUT it is known that the active site openingoccurs on the micro-millisecond timescale

HIVpr

Lesson 2 Locally biased one bead model

Step 1 From expt: The active site (flaps) opening occurs on the microsec timescale

In the simulation:The opening time scaledepends on which kind ofdynamics is used (MD, LD orBD) → the stochastic effects(solvent and other) must beincluded

Lesson 2 Locally biased one bead model

Langevin dynamics =damped dyn + external stochastic force

Brownian dynamics = Overdamped limit of LD

Langevin and Brownian dynamicsSimulate implicitly the effect ofthe solventSlow down the dynamics

V Tozzini, J Trylska, C-E Chang, J A McCammon Flapopening dynamics in HIV-1 protease explored with a coarse-grained model J Stuct Biol, 157 606-615 (2007)

Lesson 2

Allosteric inhibition effectThe principal mode analysis gives a set of principal modes and their correlations

Mode # 7 describesthe opening of the siteenclosed betweenresidues 17 and 39

There is a correlationbetween mode #7 andmode #1 (open-closemode)

If the 17-39 site isblocked, the flapcannot open.

⇒ Site to design newinhibitor drugs

Locally biased one bead model

V Tozzini, J A McCammonA coarse grained model for the dynamics of flap opening in HIV-1 Protease Chem Phys Lett 413 123-128 (2005)

Lesson 2 Locally biased one bead model

The model is flexible enough to reproduce the flap opening,but accurate enough to reproduce the experimentally knownstructures (close and semi-open)

Lesson 2 Locally biased one bead model

Transferability to different temperatures

The sigmoid transition curve from close to open state of the active sitecan be evaluatedThe transition is seen to occur at about 330K

This can be indirectly verifiedcomparing with the affinity dataof different drugs, assumingthat the most relevantcomponent of the affinity is theopening rate of the active site

Lesson 2 Locally biased one bead model

Effect of crowders molecules

QuickTime™ and a3ivx D4 4.5.1 decompressor

are needed to see this picture.

DDL Minh, C-E Chang, J Trylska, VTozzini, and JA McCammon TheInfluence of Macromolecular Crowdingon HIV-1 Protease Molecular DynamicsJ Am Chem Soc, 128 6006-6007(2006)

Lesson 2

The interaction with the substrate favors opening, but when the substrate hasentered, the close conformation is stableAfter cleavage the complex is unstable and the substrate is released

Locally biased one bead modelStep 2 and 3, 4Substrate capture, cleavage and release

J Trylska, V Tozzini, C-E Chang, J A McCammon HIV-1 protease substrate binding and product release pathways exploredwith coarse-grained molecular dynamics Biophys J, 92 4179-4187 (2007)

Lesson 2 Locally biased one bead model

Lesson 2 Locally biased one bead model

Dynamical correlation matrix

Lesson 2

In summary

GC models can be used to simulate long time scale processesor large scale systems or both

The one-bead models are particularly efficient, but theirparameterization is quite difficult

The locally biased one-bead models are a good compromisebetween accuracy and predictive power

However the goal is to build completely unbiased models

… A very open field of investigation …

Locally biased one bead model

Lesson 2 One Bead Models for Nucleic Acids - hints

Developing a model for DNA-RNA forstructural transitions in nucleic acids

The attractive - repulsive “doublefeature” interaction between theconjugated bases

Similar for non bonded interactions

Denatured state is entropically favored

Lesson 2 CG models model

Problems related with the CG models

The potential of mean force IS NOT the true interacting potential

For instance, consider a pure LJ liquid: the “true”inter-bead potential is a LJ (single well) potential,while the Boltzmann inverse of the g(r) is amodulated function, with several wells correspondingto the first, second, third… shells of first neighbours(the structure of the liquid)

The difference between the two is due to the multi-body correlations that are neglected in theBoltzmann inversion procedure

How to solve the problem?1. Iteratively re-fitting the effective potential until the “true” g(r) is

reproduced2. Use the force matching method, i.e. fit the GC forces on the forces

calculated on all-atom trajectories

Lesson 2 Force Matching Methods

Least squares fitting the CG forces over the all-atom forces

The method ensures the mechanical consistency

Is it equivalent to the thermodynamic consistency?

Emerging methods for multiscale simulations of biomolecularsystemsJW Chu GS Ayton GA Voth J Mol Phys 105 167-175 (2007))

Lesson 2 Force Matching Methods

Difference between the force matching and the PMF

FM

PMFJ Zhou et al Coarse-Grained Peptide Modeling Using a Systematic Multiscale ApproachBiophys J 92 4289-4303 (2007)

Lesson 2 Multi-scale approaches

With FM method, mixed all atoms-CG simulations or multi-scaleapproaches to large systems canbe consistently matched

Hybrid CG all-atom models

Lesson 2 Multi-scale approaches

A different way tomulti-scaleapproaches:

CG for the slowprocesses

All atom for the fastprocesses

C-E Chang, J Trylska, V Tozzini, J A McCammon Binding pathways of ligands to HIV-1 Protease: Coarse-grainedand atomistic simulations Chem Biol Drug Des, 69 5-13 (2007)

Lesson 2 CG models model

Matching the“particle”representationwith continuumfield theories

Elastic membrane (quasi) particle representation

Matching the “particle” representationwith continuum field theories

The matching can be done throughthe Free Energy function-functional

This approach can reproduce thestructure of liposomes, vesicles andtubules

GS Ayton GA VothMultiscale simulations of transmembrane proteinsJ Struct Biol 157 570-578 (2007)

Lesson 2 Fictitious dynamics acceleration

Problems connected with the CG models: the artificial acceleration

The unphysical acceleration of the dynamics

Native structure

Free energy landscape

Go Model

Frustration

Landscape smoothening

Less locally metastable states ⇒ less barriers to overcome ⇒ fasterdynamics

This effect is particularly enhanced in Go models (where the simulationtime is meaningless), less enhanced in un-biased or partially biasedmodels

Lesson 2 Fictitious dynamics accelerationMissing degrees of freedom can be reintroduced by coupling to anoscillator bath

In general, a relationshipcan be found between thememory function and thefrequency of the oscillators

However, in the simplestapproximation the memoryfunction is a delta (i.e.completely uncorrelatedoscillators) and one getsback the LangevinDynamics

In conclusion to use the LD is essential with CGs, not only to treat implicitlythe solvent, but also to treat implicitly the effect of the hidden degrees offreedomThe LD parameters can in principle be evaluated, but a simpler way is to fitthe dynamics using known kinetic constants

Lesson 2 A peculiar example of CG models…

Protein docking methods

Representation of the molecule:molecular surface (not always)

Type of interactionssurface complementarityElectrostatichydrophobicity

FF: scoring functions determinedstatistically

Phase space sampling

Stochastic methods

Lesson 2

CG methods can be used to simulate large systems and/or slowprocesses

CG can be done at various levels (number of beads). The smaller thenumber of beads, the larger the computer time gain, the more difficultthe parameterization

The parameterization can be based on different approaches, but inany case (thermodynamic and/or mechanical) consistency isdesirable

The fictitious acceleration of the dynamics due to the smoothening ofthe FES must be corrected, possibly using LD

In many case, multi-scale approaches must be used

The techniques are not standard and there are a many openquestions⇒ there is still space for development and advancements

Summary

Lesson 2

Modelli Coarse Grained per proteine e acidi nucleici

Valentina TozziniNEST-CNR-INFM Scuola Normale Superiore

Didactic material available athttp://homepage.sns.it/tozzini/public_files/