introduction to: modelling molecular interactions and dynamics bioinformatics ii m. meuwly
TRANSCRIPT
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