Introduction to: Modelling Molecular Interactions and Dynamics Bioinformatics II M. Meuwly

Introduction to:

Modelling MolecularInteractions and Dynamics

Bioinformatics II

M. MeuwlyDepartment of Chemistry

University of Basel

1 Introduction

Experimental

techniquesTheoretical

methods

(dynamics and structure)Light/X−ray/neutron scattering

(dynamics and structure)X−ray, NMR

(dynamics and structure)Imaging/Cryo−EM

Development ofmathematical models v(r)

Exploration of modelphenomenology andproperties

structure, dynamics and function

Understanding biomolecular

Development of newtheories and modelsto rationalize andpredict experimental observations

Calorimetry, pKas,thermodynamics,physical measurements

methods toDevelopment of

explore models

1

2 Force field

• A force field is an empirical approximation for expressing structure-

energy relationships in biopolymers

• It is a compromise between speed and accuracy

• Common form (CHARMM):

E(r1, r2, . . . , rN) =∑

bonds

1

2kb

i (di − d0i )

2

+∑

angles

1

2kθ

i (θi − θ0i )

2

+∑

torsions

kφi [1 + cos(niφi − δi)]

+1

2

∑

nonbond

ǫminij

(

dminij

dij

)12

− 2

(

dminij

dij

)6

+qiqj

ǫdij

2

2.1 Energy terms

Energy

Bond

Angle

Energy

Energy

Dihedral angle

Energy

Distance

Energy

Distance

Bonds

θAngles

φDihedrals

Van der Waals

Electrostaticsδ δ−+

From quantum chemistry, thermodynamics

From specctroscopy, IR, NMR vbond = 12k

bi (di − d0

i )2

vangle = 12k

θi (θi − θ0

i )2

vdihedral = kφi [1 + cos(3φi)] +

kφ′

i [1 − cos(φi − π)]

vvdW = ǫminij

[(dmin

ij

dij

)12

− 2(

dminij

dij

)6]

vCoulomb =qiqj

ǫdij

3

2.2 Parametrization

4

6 Molecular dynamics

6.1 Basics

• Atomic positions (coordinate file) −→

• Covalent structure (topology file) −→

• Potential energy function (parameter file) −→

• Additional atoms (solvent, counterions) −→

• Special features (PBC, constant T and/or P) −→

• Atomic velocities −→

• Effective temperature

(through kinetic energy)

• Forces on each atom

13

6.2 Equations

• Fi =miai Fi = −gradiE E = Ebonding + Enon−bonding

1. solve for ai at t −dEdri

= Fi = miai(t)

2. update vi at t + ∆t/2 vi(t + ∆t/2) = vi(t − ∆t/2) + ai(t)∆t

3. update ri at t + ∆t ri(t + ∆t) = ri(t) + vi(t + ∆t/2)∆t

4. go to 1.

• Timestep controls accuracy of numerical solution.

• Fundamental timestep is determined by high frequency vibrations (co-

valent bonds −→ ∆t = 10−15 sec).

• Highest frequency motions, i.e., hydrogen atom vibrations, can be re-

moved with holonomic constraints.

14

6.3 Thermodynamic variables T and P

• Statistical ensembles connect microscopic to macroscopic

Microcanonical (NVE, entropy)

Canonical (NVT, Helmhotz free-energy)

· T =∑

m⟨v2⟩/(3kb)

Isothermal-isobaric (NPT, Gibbs free-energy)

· P = kinetic + virial contributions

• Thermostats, barostats allow to choose the appropriate ensemble.

· Andersen, Nose, Hoover.

15

4 Sampling techniques

4.1 Energy minimization

Reaction coordinate

Potential Energy

• Minimization follows gradient

• Reaches the nearest local minimum

• Steepest descent, conjugate gradient

8

4.2 Metropolis Monte Carlo (Boltzmann statistics)

• Metropolis Monte Carlo yields an ensemble (Boltzmann statistics).

• Ergodicity: every accessible point in configuration space should be reached in a finite

number of Monte Carlo steps from any other point.

• Kinetics are usually not meaningful.

9

4.3 Simulated annealing (good for sampling but no ensemble)

High Temperature

Cooling

Reaction coordinate

Potential Energy

10

4.4 Parallel tempering (equilibrium Monte

Carlo scheme)

• M non-interacting copies of the system at different Tm

• A state is defined by

X =

T1, T2,..., TM

︷ ︸︸ ︷(

x1m(1), . . . , x

Mm(M)

)

, xim ≡ (qi, pi)m

• In order to converge toward equilibrium the detailed

balance should be satisfied. Therefore:

w(X → X ′) =

1, ∆ ≤ 0,

exp(−∆), ∆ > 0.

where ∆ ≡ [βn − βm](E(xi

m) − E(xjn)).

3 2

0 IV

V

32

360330 390300

0

0

0

1

1

1 2

2

2

3

3

3

01

1

Cycle

I

II

III

300

320

340

360

380

400

420

440

460

480

0 200

400 600

800 1000

T(K)fram

e

High T replicas jump from basin to basin (inter-basin)

Low T replicas explore a single valley (intra-basin)

Rao and Caflisch, J. Chem. Phys. 119, 4035, 2003

11

8 Free-energy barriers and timescales

Reaction coordinate

Free Energy

barrier crossing

G∆

• To cross a free-energy barrier τ = τ0 exp(∆G‡/kBT ) with τ0 ∼ 10−12

s:

1 kcal/mol : ∼ ps, 5 kcal/mol : ∼ ns, 10 kcal/mol : µs or longer

• Sampling should exceed timescales of interest by ∼ 10-fold.

• System size and complexity increase required timescales (equilibration

of ions, complex landscapes, multiple minima)17

9 Approximations in molecular dynamics

• Approximations inherent to the force field (E) −→

Systematic error:

Calculations of free energy differences is still very difficult.

• Time scale and sampling problem −→

Statistical error:

Conformational transitions that require > 0.1 − 1µs cannot be

simulated (yet) by conventional molecular dynamics techniques.

• Other simulation approaches:

– MD with implicit solvent (approximate)

– Brownian dynamics

– Monte Carlo (move definitions are difficult for macromolecules)

18

Differences between Force Fields

Differences between Force Fields

Differences between Force Fields

Force Field Ab initio

System Size Several 10´000 atoms 20 heavy atoms (correl.)1000 atoms (HF, [DFT])

Application StructuresConformational SearchNon-covalent interactions

StructuresEnergeticsReactions

Limitations Bond-breakingFixed atomic charges[Quantitative Information]

Very time consumingDynamics often impossible

Practical Considerations for Calculating Energies

Free energy: classical definition

+

Enthalpic Entropic

! Hydrogen bonds! Polar interactions! Van der Waals interactions! ...

! Loss of degrees of freedom! Gain of vibrational modes! Loss of solvent/protein structure! ...

Theoretical Predictions: ! Approximate: empirical formula for all contributions

! Exact: using statistical physics definition of G

G = -KBT ln(Z)

�

!G = !H "T!S

The free energy is the energy left for once you paid the tax to entropy:

Free energy: statistical mechanics definition

G = !kBT ln(Z ) Z = e

!"Ei

i#where

is the partition function

Free energy differences between 2 states (bound/unbound, É)

are, therefore, ratios of partition functions

!G = GA"G

B= "k

BT ln

ZA

ZB

#

$%&

'(

Free energy simulations aim at computing this ratio using various

techniques.

Relation with chemical equilibrium

A + B "# AÕBÕ

A + B"#AÕBÕ

Kb : binding constant, Kd : dissociation constant, Ki : inhibition constant

KD (mol/l)

"Gbinding (kcal/mol) -2 -4 -6 -8 -10 -12 -14 -16

10 -1210

-910 -610

-3

Weak asso. Strong asso.

KD= K

i=

A[ ] B[ ]A'B'[ ]

KA

= Kb=

A'B'[ ]A[ ] B[ ]

!Gbinding = "RTlnKA = RTlnKD = !H " T!S

Connection micro/macroscopic: thermodynamics and kinetics

Free Energy Association Constant

e - RT!G = KA

Microscopic Structure Biological function

Relative bindingfree energies: !!G

" KAÕ / KA

Absolute binding free energies: !G

" KA

Binding free energyprofiles: !G(#)" KA, Kon, Koff

The free energy is the main function behind all process

A) Chemical equilibrium

B) Conformational changes

C) Ligand binding

D) É

+!Gbinding = RTlnKA

!Gconf

= kBTlnPConf 1

PConf 2

!Gbinding = kBTlnPUnbound

PBound

KA=

AB[ ]A[ ] B[ ]

A B ABKD= 1 / K

A

R = kBN

A

Free energy: computational approaches

!G = GA"G

B= "k

BT ln

ZA

ZB

#

$%&

'(

Free energy simulations techniques aim at computing ratios of

partition functions using various techniques.

Z = e!"Ei

i#

Sampling of important

microstates of the system

(MD, MC, GA, É)

Computation of energy

of each microstate

(force fields, QM, CP)

Connection micro/macroscopic: intuitive view

E1, P1 ~ e-$E1

E2, P2 ~ e-$E2

E3, P3 ~ e-$E3

E4, P4 ~ e-$E4

E5, P5 ~ e-$E5

Where

is the partition function

Expectation value

�

O =1

ZOie!"Ei

i

#

�

Z = e!"Ei

i#

Central Role of the Partition Function

G = -kBT ln(Z)

. . .

Expectation Value

Internal Energy Pressure Gibbs free energy

Z = e!"Ei

i#

O =1

ZOie!"Ei

i#

E =!

!"ln(Z ) =U p = kBT

! ln(Z )!V

"#$

%&'N ,T

Binding free energy decomposition: MM-PBSA, MM-GBSA

Lig + Prot Lig:Prot

Lig:Prot!Gbind

Lig + Prot

Gaz

Sol

Averaged over an MD simulation trajectoryof the complex (and isolated parts)

B. Tidor and M. Karplus, J. Mol. Biol., 1994, 238, 405

Molecular mechanics Ð Poisson-Boltzmann Surface Area (MM- PBSA)

Molecular mechanics Ð Generalized Born Surface Area (MM- GBSA)

J. Srinivasan, P.A. Kollmann et al., J. Am. Chem. Soc., 1998, 120, 9401

H. Gohlke, C. Kiel and D.A. Case, J. Mol. Biol., 2003, 330, 891

Depending on the way !Gsolv,elec is calculated:

�

!Gbind = !Egaz + !Gdesolv "T !S

�

Egaz = Eelec + Evdw + !Eint ra

�

!Gdesolv = !Gsolv

comp" !Gsolv

lig+ !Gsolv

prot( )

�

!T"S = !T(Scomp

! (Sprot

+ Slig))

�

!Gsolv

lig

�

!Gsolv

prot

�

!Gsolv

comp

�

S = Strans

+ Srot

+ Svib

�

!Gsolv = !Gsolv,elec + !Gsolv,np

�

!Gdesolv = !Gsolv,elec

comp" !Gsolv,elec

lig+ !Gsolv,elec

prot( ) + # SASAcomp

" SASAlig

+ SASAprot( )( )

�

!Egaz

Summary

Force Field

Energy minimization

Molecular Dynamics

Monte Carlo

Macroscopic Properties

Normal Mode

Structural Optimization

F. Schotte et al., Science 300, 1944 (2003)

Example from Research: CO motion in myoglobinExperiments

Simulation techniques:

Classical Molecular Dynamics

Conventional Force Field for Protein (CHARMM)

Use realistic model for CO (CO is neutral, has small dipole but large quadrupole moment)

Use explicit solvation with water

Extended simulation times (several ns)

Analysis via Fourier Transformation of dipole autocorrelation function

Ligand Dynamics in Mb

M. Lim et al., J. Chem. Phys., 102 4355 (1995)

IR spectrum for dissociated CO in native Mb

Ligand Dynamics in Mb

Experiment Simulations

Simulations

D. R. Nutt and M. Meuwly, PNAS , 101, 5998 (2004)

Ligand Dynamics in MbProtocol: 50 ps of MD simulations

CO treated at B3LYP/6-31G**Stochastic boundary conditions

M. Lim et al., J. Chem. Phys., 102 4355 (1995)

IR spectrum for dissociated CO in native Mb

Ligand Dynamics in Mb

Experiment Simulations

Possible Reaction Pathways in MbCO

Advanced Topic: Ligand Rebinding in Mb

A, B, Xe4 Ostermann et al., Nature, 404, 205 (2000)

A, B, Xe4, Xe1, BFrauenfelder et al., PNAS, 98, 2370 (2001)Nienhaus et al., Biochem., 42, 9647 (2003)

A, B, Xe4, Xe3, Xe1Bossa et al., Biophys. J., 87, 1537 (2004)

A, B, Xe1Srajer et al., Biochem, 40, 13802 (2001)

A, B, C competing with A, B, SScott+Gibson, Biochem., 36, 11909 (1997)

Simulation techniques:

Classical Molecular Dynamics

Conventional Force Fields for initial and final state

Detect Crossing via energy criterion

Carry out crossing using a „mixing algorithm“

Use explicit solvation with water

Extended simulation times (several ns)

Advanced Topic: Ligand Rebinding in Mb

Explicit rebinding dynamics in MbNO

B

A

R(Fe-X)

Ener

gy

Method:

Propagate on surface BLocating of crossing(EA = EB)BackpropagationSurface crossing to surface A

BA

A

Ligand Rebinding in Mb

Explicit rebinding dynamics in MbNO

Ligand Rebinding in Mb

Rebinding is nonexponential in timeTime constants: τ1 = 3.8 ps (5.5 to 28 ps)

τ2 = 18.0 ps (50 to 280 ps)

• Extended recrossing region• Rebinding into secondary

minimum

Rebinding dynamics in MbCO on free energy curves

• Precalculate the free energy curve for bound and unbound CO motion using umbrella sampling.

• Solve the Smoluchowski equation for G(q).

Ligand Rebinding in Mb

Data derived from experiment

J. S. Olson and G. N. PhillipsJBC, 271 17593 (1996)

R(Fe-CM)

G(k

cal/m

ol)

A

B Xe4

Rebinding dynamics in MbCO on free energy curves

Ligand Rebinding in Mb

unbound

bound

R(Fe-X)

Ener

gy

Rebinding dynamics in MbCO on free energy curves

From Xe4

Ligand Rebinding in Mb

Δ(kcal/mol)

τns

4.0 100

5.0 2807.5 1770

For native MbCO: inner barrier 4.3 kcal/mol vs 4.5 kcal/mol from experimentFor L29F mutant: rebinding time 10 ps vs rapid escape to Xe4 pocket