Molecular Modeling of Proteins - Bioinformatics · Molecular Modeling of Proteins Lecture Plan: -...

O. Michielin, SIB/LICR

Molecular Modeling of Proteins


Molecular Modeling of Proteins

Lecture Plan:

- Central role of partition function- Review molecular interactions- Modeling of molecular interactions: CHARMM force field- Recent techniques: implicit solvent models

long range electrostatic treatment- Molecular dynamics simulations

- elements of statistical mechanics- microcanonical sampling - canonical sampling- isothermal-isobaric sampling- Langevin dynamics

- Other sampling techniques- Monte-Carlo- Simulated Annealing (SA) - Genetic Algorithms (GA)


Molecular Modeling: Introduction

What is Molecular Modeling?

What is it good for?

Molecular Modeling is concerned with the description of the atomicand molecular interactions that govern microscopic and macroscopicbehaviors of physical systems

The essence of molecular modeling resides in the connection between the microscopic world and the macroscopic world provided by the theory of statistical mechanics

Macroscopicobservable

(Solvation energy,affinity between two

proteins, H-H distance,conformation, ... )

Average of observableover selected microscopic

states


Connection micro/macroscopic: intuitive view

⟨O ⟩= 1Z∑i

Oi e−E i

E1, P1 ~ e-bE1

E2, P2 ~ e-bE2

E3, P3 ~ e-bE3

E4, P4 ~ e-bE4

E5, P5 ~ e-bE5

Z=∑ie−E iWhere

is the partition function

Expectation value


Central Role of the Partition function

The determination of the macroscopic behavior of a system from athermodynamical point of vue is tantamount to computing a quantitycalled the partition function, Z, from which all the properties can bederived.

Z=∑ie−E i

⟨O ⟩= 1Z∑i

Oi e−E i

⟨E ⟩= ∂∂

ln Z =U p=kT ∂ ln Z ∂V

N ,T

A = -kT ln(Z)

. . .

Expectation Value

Internal Energy Pressure Helmoltz free energy


Computation of the Partition function

The partition function is a very complex function to compute, and, in most cases, only numerical approximations are possible

Numerical approximations require:

1) the computation of the energy of the system for microstate i- performed using semi-empirical force fields

CHARMM / Amber / Gromos / ...

2) a method to sample all (or a representative portion) of the microstates accessible to the system in a given macroscopicstate, i.e:

- microcanonical sampling for fixed N,V,E systems- canonical sampling for fixed N,V,T systems- isothermic-isobaric sampling for fixed N,P,T systems- ...

1)

2)

Z=∑ie−E i


Introduction & historical note

Theoretical milestones:

Molecular dynamics milestones:

Newton (1643-1727): Classical equations of motion: F(t)=m a(t)Schrödinger (1887-1961): Quantum mechanical equations of motion:

-ih t (t)=H(t) (t)Boltzmann(1844-1906): Foundations of statistical mechanics

Metropolis (1953): First Monte Carlo (MC) simulation of a liquid (hard spheres)

Wood (1957): First MC simulation with Lennard-Jones potentialAlder (1957): First Molecular Dynamics (MD) simulation of

a liquid (hard spheres)Rahman (1964): First MD simulation with Lennard-Jones potential

Karplus (1977) & First MD simulation of proteinsMcCammon (1977)Karplus (1983): The CHARMM general purpose FF & MD programKollman(1984): The AMBER general purpose FF & MD programCar-Parrinello(1985): First full QM simulationsKollmann(1986): First QM-MM simulations

LiquidsProteins


Molecular Interactions (I)

Bonded Interactions:- Intramolecular energy terms associated with the deformation of the electronic structure of the molecule. 3 terms are introduced in all FF:

1) Bond stretch

2) Angle bend

3) Dihedral torsion

and a fourth one is often used to maintain planarity

4) Improper torsion


Molecular Interactions (II)

Non-Bonded Interactions:

- Inter and intramolecular energy terms arising from electrostatics

1) Electrostatic interactions

Coulomb law:

where e is the dielectric constant, 1 for vacuum, 4-20 for protein core,and 80 for water

2) van der Waals interactions

Attractive part: due to induced-dipole/induced-dipoleRepulsive part: due to Pauli exclusion principle

Usually represented by the Lennard-Jones potential

V Ele=∑i j

qi q j

41

r i , j

V vdW=∑i j

4ij [ ij /r ij 12− ij /r ij

6]

eij=(ei ej)1/2, sij = 1/2(si + sj) are obtained from the single atom param. e and s


Derived Interactions

Some interactions are often referred to as particular interactions, but theyresult from the two interactions previously described, i.e. the electrostaticand the van der Waals interactions.

1) Hydrogen bonds (Hb)

- Interaction of the type D-H ··· A- The origin of this interaction is a dipole-dipole attraction- Typical ranges for distance and angle:

2.4 < d < 4.5Å and 180º < f < 90º

2) Hydrophobic effect

- Collective effect resulting from the energeticallyunfavorable surface of contact between the water and an apolar medium (loss of water-water Hb)- The apolar medium reorganizes to minimize thewater exposed surface

D H

Af-d

+d

d

Water

Oil

-d


Semi-Empirical Force Fields

1) Goals of semi-empirical force fields- Definition of empirical potential energy functions V(r) to model the molecular interactions described previously- These functions need to be differentiable in order to compute the forces acting on each atom: F=-V(r)

2) Ways to compute semi-empirical potential energy functions- First, theoretical analytical functional forms of the interactions are derived- The system is divided into a number of atom types that differ by their atomic number and chemical environment, e.g. the carbons in C=O or C-C are not of the same type- Parameters are determined so as to reproduce the interactions between the various atom types by fitting procedures

- experimental enthalpies (CHARMM)- experimental free energies (GROMOS, AMBER)

Parametrization available for proteins, lipids, sugars, ADN, ...


The CHARMM Force Field

∑Impropers

K−02

∑Dihedrals

K [1−cosn−]

∑i j


6]

∑i j

qi q j

41

r i , j

V r =∑Bonds

K bb−b02∑

AnglesK −0

2


The CHARMM 22 Parameter Set (Sample)BONDS!V(bond) = Kb(b - b0)**2!Kb: kcal/mole/A**2!b0: A!atom type Kb b0C C 600.000 1.3350 ! ALLOW ARO HEM ! Heme vinyl substituent (KK, from propene (JCS))CA CA 305.000 1.3750 ! ALLOW ARO ! benzene, JES 8/25/89CE1 CE1 440.000 1.3400 !

! for butene; from propene, yin/adm jr., 12/95CE1 CE2 500.000 1.3420 !

! for propene, yin/adm jr., 12/95

ANGLES!V(angle) = Ktheta(Theta - Theta0)**2!V(Urey-Bradley) = Kub(S - S0)**2!Ktheta: kcal/mole/rad**2!Theta0: degrees!Kub: kcal/mole/A**2 (Urey-Bradley)!S0: A!atom types Ktheta Theta0 Kub S0CA CA CA 40.000 120.00 35.00 2.41620 ! ALLOW ARO ! JES 8/25/89CE1 CE1 CT3 48.00 123.50 !

! for 2-butene, yin/adm jr., 12/95CE1 CT2 CT3 32.00 112.20 !

! for 1-butene; from propene, yin/adm jr., 12/95CE2 CE1 CT2 48.00 126.00 !

! for 1-butene; from propene, yin/adm jr., 12/95CE2 CE1 CT3 47.00 125.20 !

! for propene, yin/adm jr., 12/95

DIHEDRALS!V(dihedral) = Kchi(1 - cos(n(chi) - delta))!Kchi: kcal/mole!n: multiplicity!delta: degrees!atom types Kchi n deltaC CT1 NH1 C 0.2000 1 180.00 ! ALLOW PEP ! ala dipeptide update for new C VDW Rmin, adm jr., 3/3/93cC CT2 NH1 C 0.2000 1 180.00 ! ALLOW PEP ! ala dipeptide update for new C VDW Rmin, adm jr., 3/3/93cC N CP1 C 0.8000 3 0.00 ! ALLOW PRO PEP ! 6-31g* AcProNCA CA CA CA 3.1000 2 180.00 ! ALLOW ARO ! JES 8/25/89CA CPT CPT CA 3.1000 2 180.00 ! ALLOW ARO ! JWK 05/14/91 fit to indole

IMPROPER!V(improper) = Kpsi(psi - psi0)**2!Kpsi: kcal/mole/rad**2!psi0: degrees!note that the second column of numbers (0) is ignored!atom types Kpsi psi0CPB CPA NPH CPA 20.8000 0 0.0000 ! ALLOW HEM ! Heme (6-liganded): porphyrin macrocycle (KK 05/13/91)CPB X X C 90.0000 0 0.0000 ! ALLOW HEM ! Heme (6-liganded): substituents (KK 05/13/91)CT2 X X CPB 90.0000 0 0.0000 ! ALLOW HEM ! Heme (6-liganded): substituents (KK 05/13/91)CT3 X X CPB 90.0000 0 0.0000 ! ALLOW HEM ! Heme (6-liganded): substituents (KK 05/13/91)HA C C HA 20.0000 0 0.0000 ! ALLOW PEP POL ARO ! Heme vinyl substituent (KK, from propene (JCS))HA CPA CPA CPM 29.4000 0 0.0000 ! ALLOW HEM


The CHARMM 22 Topology File (Tyr)RESI TYR 0.00GROUP ATOM N NH1 -0.47 ! | HD1 HE1 ATOM HN H 0.31 ! HN-N | | ATOM CA CT1 0.07 ! | HB1 CD1--CE1ATOM HA HB 0.09 ! | | // \\GROUP ! HA-CA--CB--CG CZ--OHATOM CB CT2 -0.18 ! | | \ __ / \ATOM HB1 HA 0.09 ! | HB2 CD2--CE2 HHATOM HB2 HA 0.09 ! O=C | | GROUP ! | HD2 HE2 ATOM CG CA 0.00GROUP ATOM CD1 CA -0.115ATOM HD1 HP 0.115GROUP ATOM CD2 CA -0.115ATOM HD2 HP 0.115GROUP ATOM CE1 CA -0.115ATOM HE1 HP 0.115GROUP ATOM CE2 CA -0.115ATOM HE2 HP 0.115GROUP ATOM CZ CA 0.11ATOM OH OH1 -0.54ATOM HH H 0.43GROUP ATOM C C 0.51ATOM O O -0.51BOND CB CA CG CB CD2 CG CE1 CD1 BOND CZ CE2 OH CZ BOND N HN N CA C CA C +N BOND CA HA CB HB1 CB HB2 CD1 HD1 CD2 HD2 BOND CE1 HE1 CE2 HE2 OH HHDOUBLE O C CD1 CG CE1 CZ CE2 CD2 IMPR N -C CA HN C CA +N O

DONOR HN N DONOR HH OH ACCEPTOR OH ACCEPTOR O C IC -C CA *N HN 1.3476 123.8100 180.0000 114.5400 0.9986IC -C N CA C 1.3476 123.8100 180.0000 106.5200 1.5232IC N CA C +N 1.4501 106.5200 180.0000 117.3300 1.3484IC +N CA *C O 1.3484 117.3300 180.0000 120.6700 1.2287IC CA C +N +CA 1.5232 117.3300 180.0000 124.3100 1.4513IC N C *CA CB 1.4501 106.5200 122.2700 112.3400 1.5606IC N C *CA HA 1.4501 106.5200 -116.0400 107.1500 1.0833IC N CA CB CG 1.4501 111.4300 180.0000 112.9400 1.5113IC CG CA *CB HB1 1.5113 112.9400 118.8900 109.1200 1.1119IC CG CA *CB HB2 1.5113 112.9400 -123.3600 110.7000 1.1115IC CA CB CG CD1 1.5606 112.9400 90.0000 120.4900 1.4064IC CD1 CB *CG CD2 1.4064 120.4900 -176.4600 120.4600 1.4068IC CB CG CD1 CE1 1.5113 120.4900 -175.4900 120.4000 1.4026IC CE1 CG *CD1 HD1 1.4026 120.4000 178.9400 119.8000 1.0814IC CB CG CD2 CE2 1.5113 120.4600 175.3200 120.5600 1.4022IC CE2 CG *CD2 HD2 1.4022 120.5600 -177.5700 119.9800 1.0813IC CG CD1 CE1 CZ 1.4064 120.4000 -0.1900 120.0900 1.3978IC CZ CD1 *CE1 HE1 1.3978 120.0900 179.6400 120.5800 1.0799IC CZ CD2 *CE2 HE2 1.3979 119.9200 -178.6900 119.7600 1.0798IC CE1 CE2 *CZ OH 1.3978 120.0500 -178.9800 120.2500 1.4063IC CE1 CZ OH HH 1.3978 119.6800 175.4500 107.4700 0.9594


Treatment of long range interactions

∑i j , r ijvdW


6] ∑

i j , r ijEle

qi q j

41

r i , j

Introduction of dynamical cutoffs

d

Shift/Switch

V r =∑Bonds

k r r−r02∑

Anglesk −0

2 ∑Impropers

k−02 ∑

Dihedralsk[1−cos n−]


Effect of cutoff values on energy calculations

8 Å cutoff No cutoff

ElecVdWTotal

ElecVdWTotal


Major limitations of current force fields & perspectives

1) No electronic polarization:- fixed partial charges allowfor conformational polarizationbut not electronic polarization

Problems Solutions

• Extended Lagrangian• charges treated as dynamical

parameters• QM-MM • Full QM simulations

• Car-Parrinello

2) Quadratic form of potentials:- problematic far from equilibriumvalues- no bond creation or deletion

• QM-MM • Full QM simulations

• Car-Parrinello


Introduction to molecular surfaces

Solvent accessibleSurface

Connolly Surface

Van der WaalsSurface

Rotating probe: radius 1.4 Å

Definitions:

- Van der Waals: ensemble of van der Waals sphere centered at each atom- Connolly: ensemble of contact points between probe and vdW spheres- Solvent: ensemble of probe sphere centers


Examples of molecular surfaces

Van der Waals

Solvent accessible

Connolly(Contact)


e=1

Mapping properties on molecular surfaces

1) Electrostatic potential: surface elements are colored accordingto the value of f(x) with a linear scaling

f(x) = -0.3 V Red f(x) = 0.0 V Gray f(x) = +0.3 V Blue

where f(x) is the solution ofthe Poisson-Boltzmann eq.

2) Other Properties:i) surface curvatureii) residue chargeiii) residue conservationiv) ...

∇r ∇r =Macror ∑i

qi ni0 exp−qir

e=80


Implications of surface properties for ligand binding

Acetate (Negative charge) Toluene (Hydrophobic) Meguan (Positive charge)


Treatment of the solvent contribution

1) Explicit water molecule model: TIP3P, ...

2) Implicit solvent model:

- Based on Poisson-Boltzmann Equation:

- or an approximation...-ACE potential (Schaeffer & al.)-SASA potential (Caflish & al.)-EEF1 potential (Lazaridis & al.)

∇r ∇r =Macror ∑i

qi ni0 exp −qir

e=80e=1

OH H

-d

+d +d

For a discussion of theoretical aspects of implicit solvent models, see Roux & Simonson (Biophys. Chem. 1999, 78:1-20)


Minimal Input for Molecular Modeling

1) Topological properties:

2) Structural properties:

3) Energetical properties:

description of the covalent connectivity of the molecules to be modeled

the starting conformation of the molecule, provided by an X-ray structure, NMR data or a theoretical model

a force field describing the force acting on each atom of the molecules

4) Thermodynamical properties:

a thermodynamical ensemble that corresponds to the experimental conditions of the system, e.g. N,V,T , N,P,T, ...

T0

NV


Setup of a MD Simulation


Examples of MD Simulations

2) Vacuum Simulation of Melan-A Peptide

3) Explicit Solvent Simulation of Melan-A in p-MHC

1) Simulation of b2 microglobuline


Energy Landscape

Landscape for / plane of dialanine

Landscape for a protein

E

3N Spatial coordinates


Review of Statistical Mechanics

Thermodynamical Ensembles

Definition: a thermodynamical ensemble is a collection of microscopicstates that all realize an identical macroscopic state

A microscopic state of the system is given by a point (r, p) of the phasespace of the system, where r= (r

1, ..., r

N ) and p= (p

1, ..., p

N ) are the positions

and the momenta of the N atoms of the system.

Examples of thermodynamical ensembles:

- Microcanonical: fixed N, V, E often used in MD- Canonical: fixed N, V, T often used in MD- Constant P-T: fixed N, P, T often used in MD- Grand Canonic: fixed m, P, T



Boltzmann Distribution

Boltzmann showed that the canonical probability of the microstate i is given by

P i=e−E i

∑j

e−E j= 1

Ze−E i

where Z is the partition function, b = 1/KBT with K

B the Boltzmann constant.

The partition function is a very complex function to compute, and in most cases only approximations can be obtained. All the properties of the system, micro or macroscopic, are contained in this function.

Illustration:If a system can have two unique states, state (1) and state (2),then the ratio of systems in state (1) and (2) is

P1

P2= e−E1

e−E2=e−E1−E2=e−E at 300K, a DE of 1.3 kcal/mol

results in a P1/P2 of 1 log10

.

(1)

(2)

Cave: if state (1) and (2) are composed of several microscopic states, DE DG



Expectation value

the macroscopic value of an observable, or expectation value, is givenby the weighted average over all microscopic states of the thermodyna-mical Ensemble. In the case of a continuous microscopic ensemble,

⟨O ⟩=∫Ens

O e−E p , r d p d r

∫Ens

e−E p , r d p d r=∫Ens

e−K pd p

∫Ens

e−K pd p

∫Ens

O e−V r d r

∫Ens

e−V r d r=∫Ens

O e−V r d r

∫Ens

e−V r d r= 1

Z ∫EnsO e−V r d r

if O and V are not dependent upon pand E(p,r)=K(p)+V(r)

In the case of a discret set of microscopic states

⟨O ⟩= 1Z∑i Oi e

−V i


Ergodic Hypothesis

The ergodic hypothesis is that the ensemble averages used to compute expectationvalues can be replaced by time averages over the simulation.

The microstates sampled by molecular dynamics are usually a small subset of the entire thermodynamical ensemble.

The validity of this hypothesis depends on the molecular modeling techniqueand on the quality of the sampling produced. The sampling should reach allimportant minima and explore them with the correct probability,

- NVE simulations Microcanonic ensemble P = cst.- NVT simulations Canonic ensemble P(E) = e-bE

- NPT simulations Isothermic-isobaric ensemble P(E) = e-b(E+PV)

Note that the statistical weights are not present in the time average because ofthis fact.

⟨O ⟩Ensemble =1Z∫O p , qe−E p , qd p d q = 1

∫t=0

O t dt = ⟨O ⟩Time

Ergodicity


E

3N Spatial coordinates

Ergodic Hypothesis: Intuitive view

NVE simulation in alocal minimum

⟨O ⟩Ensemble =1Z∫O ,e−E ,d d = 1

∫t=0

O t dt = ⟨O ⟩Time?

MD Trajectory


Lagrangian and Hamiltonian Formalism

Generalized coordinates

Lagrangian Hamiltonian

qi , qi qi , pi=∂ L∂ qi

Equations of Motion

ddt ∂ L∂ qi= ∂ L∂qi

qi=∂H∂ pi

Definition

L q t , q t , t =K−V H q t , p t , t =∑ipi qi−Lq t , q t , t

pi=−∂H∂qi

Important properties

i=1 , ,3 N

∂ L∂qi=0 ⇒ ∂ L

∂ qiis a constant of motion

if KK(t) and V=V(q,t) than H=K+V is the total energy

if HH(t), the total energy is a constant of motion

i=1 , ,3 N


1) Microcanonical ensemble (constant N,V,E)sampling is obtained by finite difference method integrations of the dynamics equations:- Verlet, Leap-Frog, Velocity Verlet, Gear

3) Isothermic-isobaric ensemble (constant N,P,T)sampling is obtained analogously to canonicalsampling by 2 additional degrees of freedomand scaling of time ( T) and space ( P)

2) Canonical ensemble (constant N,V,T)sampling is obtained by two methods

1) Nose: additional degree of freedom s acting as an external system on the physical system. s introduces a scalingof time: dt = s dt.

2) Nose-Hoover: rewriting of the extendedsystem equations to remove time scaling(which is hard to implement in MD)

Integration of extended system by Verlet

MD Techniques (I): Sampling of the various ensemblesMD Techniques (I): Sampling of the various ensembles

NVE

T(Infinite

Reservoir)NV

T(Infinite

Reservoir)

N P(Infinite

Reservoir)


MD Techniques (II): Microcanonical sampling

Several numerical methods have been developed to integrate these equations.One of the most stable integrator is that of Verlet: for a small time incrementdt, one can use a Taylor expansion of the function r(t):

mi ai=−∂∂ r ir i=1, ..., N

r i t t =r i t v i t t1/2 a i t t2r i t− t =r i t −v i t t1/2 a i t t2

Adding those equations, one gets r(t+dt) as a function of r(t) and r(t-dt). r i t t =2 r i t −r i t− t a i t t2

- In practice, this scheme is applied iteratively, starting from the initial conditions. - Velocities are postcomputed as v(t) = [r(t+dt)-r(t-dt)] / 2dt. - Positions are correct up to dt4 and velocities to dt2.- This scheme conserves energy with very good accuracy.

For an Hamiltonian of the form

in cartesian coordinates, the Hamilton equations of motion reduce to the Newton equations

H p , r =∑i=1

3 N pi2

2 mir1 , , r3 N

r i=pi

mi


MD Techniques (III): Canonical sampling

The system is here coupled to an external infinite constant temperature bath.The coupling to the physical system is done by an additional degree of freedoms and a scaling of the time such that dt' = dt / s. Therefore q

i'=q

i and p

i'=p

i/s,

where (qi',p

i',t') are the real variables and (q

i,p

i,t) the virtual variables.

The Hamiltonian for the extended system is

H p , q , ps , s=∑i

N pi2

2 mi s2qps

2

2Q3 N1kT ln s

and the equations of motion are

qi=∂H∂ pi=

pi

mi s2 pi=

−∂H∂qi

=−∂∂qi

(Equations for p and q)

s=∂H∂ ps=

ps

Qps=−∂H∂ s=1

s∑i=1

N

[pi

2

mi s2−3 N−1kT ] (Equations for ps and s)

See Nose 1984, J. Chem. Phys. 81(1): 511-19


MD Techniques (IV): Derivation of canonical ensemble for HNose

The microcanonical partition function associated to the Nose extended Hamiltonianis of the form

Z=∫dps∫ds∫ d p∫ d q [H 0 p / s , q ps2/2Q3 N1kT ln s−E ]

where i s the unextended physical Hamiltonian.

The volume element dpdq = s3N dp'dq', therefore

H 0 p , q=∑i=1

N pi2

2 miq

Z=∫dps∫d p '∫d q '∫ds s3 N [H 0 p ' , q ' ps2/2Q3 N1kT ln s−E ]

and using d[f(s)] = d(s-s0) / f'(s

0) with s

0 the root of f(s), on gets

Z=C∫d p '∫d q ' exp [−H 0 p ' , q ' /kT ]

which is the partition function of the physical system in the canonical ensembleexcept for a constant factor


H=∑i

N pi2

2 mi s2 V 2/3V

1/3 qps

2

2Q3 N1kT ln s

pV2

2WPex V

MD Techniques (V): Isothermic-Isobaric sampling

The system is here coupled to an external infinite thermostat and to a barostat. Thecoupling to the physical system is done by two additional degrees of freedom s and V. Analogously, a scaling of time and of the spatial coordinates is performed. The virtual variables (q

i,p

i,s,V,t) are related to the real variables (q

i',p

i',s,V,t') by the relations

dt' = dt / s, qi'= V1/3q

i and p

i'= p

i / V1/3s.

The Hamiltonian for the extended system is

T(Infinite

Reservoir)

P(Infinite

Reservoir)

s

V


MD Techniques (VI): NTP ensemble equations of motion

qi=∂H∂ pi=

pi

mi V2/3 s2

s=∂H∂ ps=

ps

Q

pi=−∂H∂qi

=−∂∂qi=−∂∂qi '

V 1/3

ps=−∂H∂ s=1

s∑i=1

N

[pi

2

mi V2 /3 s2−3 N−1kT ]

H=∑i

N pi2

2 mi s2 V 2 /3V

1/3 qps

2

2Q3 N1kT ln s

pV2

2WPex V

Using the extended Hamiltonian with thermostat and barostat coupling

the following equations of motion are derived

(Equations for p and q)

(Equations for ps and s)

dVdt= ∂H∂ pV

=pV

W(Equations for p

V and V)pV=

−∂H∂V

= 13V−Pex

[∑i=1

N

pi

2

mi V2/3 s2−

∂∂qi '

qi ' ]


MD Techniques (VII): Derivation of NTP ensemble for HNose

The microcanonical partition function associated to the Nose extended NTPHamiltonian is of the form

Z=∫dpV∫dV∫dps∫ds∫d p∫d q

[H 0 p / sV 1/3 ,V 1/3 qps

2

2Q

pv2

2WPex V3 N1kT ln s−E ]

where again is the unextended physical Hamiltonian.

Similarly to the case of the canonical ensemble, on can show that

Z=C∫d p '∫ d q ' exp [−H 0 p ' , q ' Pex V /kT ]

which is the partition function of the physical system in the NTP ensembleexcept for a constant factor

H 0 p , q=∑i=1

N pi2

2 miq


Other sampling methods: I

Langevin Dynamics (LD)In Langevin Dynamics, two additional forces are added to the standard force field:

- a friction force: -gi p

i

whose direction is opposed to the velocity of atom i

- a stochastic (random) force: z(t) such that <z (t)> = 0.

This leads to the following equation for the motion of atom i:

This equation can for example simulate the friction and stochastic effect of the solvent in implicit solvent simulations. The temperature is adjusted via g and z, using the dissipation-fluctuation theorem. The stochastic term can improve barrier crossing and hence sampling.

Cave: LD does not produce a canonical ensemble

qi=pi

m, pi=F i q ,−i pit


Other sampling methods: II

Monte Carlo Simulations and the Metropolis criterion

In this sampling method, instead of computing the forces on each atom tosolve its time evolution, random movements are assigned to the system and the potential energy of the resulting conformer is evaluated.

To insure Boltzmann sampling, additional criteria need to be applied on thenew conformer. Let C be the initial conformer and C' the randomly modified:

- if V(C') < V(C), the new conformer is kept and C' becomes C for next step

- if V(C') > V(C), a random number, r, in the [0,1] interval is generated and

if e-b(V'-V) > r, the new conformer is kept and C' becomes C for next step

Using this algorithm, one insures Boltzmann statistics,

P C ' P C

=e−V '−V


Other sampling methods: III

Simulated annealing (SA) and simulated quenching (SQ)

?

In these techniques, the temperature of the system is raised and cooled several times during a standard MD, LD or MC simulation.

Two different types of cooling can be achived:

- a slow (logarithmic) protocol: annealing

- a fast (exponential) protocol: quenching

This allows to pass over free energy barriers and to find other minima. The sampling obtained by SA generates of an ensemble that tends to a Boltzmann distribution as the cooling time tends to infinity.The sampling obtained by SQ does not follow a Boltzmann distribution.


Conclusion

Molecular dynamics and other sampling techniques allow:

Sampling of the conformational space of the molecule according to variousthermodynamical systems:

- following an isoenergetic surface (microcanonic)

- following an isothermic and/or isobaric surface (canonic, constant TP)

- probability of state is relatedto its free energy

- macroscopic quantities can beobtained as averages over the trajectory, the ergodic hypothesis

⟨O ⟩Ensemble=⟨O ⟩Time=1 ∫t=0

O t dt


Some Applications of Molecular Dynamics Techniques

1) Equilibrium MD simulations

• Expectation values (macroscopic observables)• using ergodic hypothesis

• Fluctuations• correlation RMSD/B-Factors

• Free energies• absolute values

3) Non-Equilibrium MD simulations• Study of large conformational changes

• Domain dynamics• Docking

• Protein Folding / Stucture predictions

2) Quasi-Equilibrium MD simulations• Free energy differences between closely related molecules

• conformational free energy change• alchemical mutations


References

1) Online- http://cmm.info.nih.gov/intro_simulation/course_for_html.html- http://www.ch.embnet.org/MD_tutorial- http://www.biophysics.org/btol/img/Beard.D.pdf

2) Textbooks

Statistical Mechanics- D. A. McQuarrie: Statistical Mechanics- R. K. Pathria: Statistical Mechanics- D. Chandler: Introduction to Modern Statistical Mechanics- R. P. Feynman: Statistical Mechanics. A Set of Lectures

Molecular Dynamics- Allen & Tildesley: Computer Simulations of Liquids- Tidor & Karplus: Proteins

Electrodynamics- J. D. Jackson: Classical Electrodynamics

Protein Structure- T. E. Creighton: Proteins. Structures and molecular properties

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