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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/
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