Statistical Mechanics Statistical Mechanics of Proteinsof Proteins

!! Equilibrium and nonEquilibrium and non--equilibrium equilibrium properties of proteinsproperties of proteins!! Free diffusion of proteinsFree diffusion of proteins

! Coherent motion in proteins: temperature echoes

! Simulated cooling of proteins

Ioan KosztinDepartment of Physics & AstronomyUniversity of Missouri - Columbia

Molecular ModelingMolecular Modeling

1. Model building

2. Molecular Dynamics Simulation

3. Analysis of the

� model � results of the simulation

Collection of MD DataCollection of MD Data

� DCD trajectory file! coordinates for each atom

! velocities for each atom

� Output file! global energies! temperature, pressure, �

Analysis of MD DataAnalysis of MD Data

1. Structural properties2. Equilibrium properties3. Non-equilibrium properties

Can be studied via both equilibrium and non-equilibrium MD simulations

Equilibrium (Thermodynamic) Equilibrium (Thermodynamic) PropertiesProperties

MD simulation

microscopic information

macroscopic properties

Statistical Statistical MechanicsMechanics

Γ[r(t),p(t)]

Phase space trajectory

Ensemble average overprobability density

ρ(Γ)

Statistical Ensemble Statistical Ensemble

Collection of large number of replicas (on a macroscopic level) of the system

Each replica is characterized by the same macroscopic parameters (e.g., NVT, NPT)

The microscopic state of each replica (at a given time) is determined by Γ in phase space

Time Time vsvs Ensemble AverageEnsemble AverageFor t"∞, Γ(t) generates an ensemble with

tddt

/lim( τ=Γ)Γρ∞→

Ergodic Hypothesis: Time and Ensemble averages are

equivalent, i.e., ρ⟩Γ⟨=⟩⟨ )(),( AprA t

Time average:

Ensemble average:

∫=⟩⟨T

t ttAdtT

A0

)](),([1 pr

∫ ΓΓρΓ=⟩⟨ )()( AdA

Thermodynamic Properties Thermodynamic Properties from MD Simulationsfrom MD Simulations

∑=

≈⟩⟨N

iitAA

N 1

)(1

MD simulationtime series

Thermodynamicaverage

Finite simulation time means incomplete sampling!

Thermodynamic (equilibrium) averages can be calculated via time averaging of MD simulation time series

Common Statistical EnsemblesCommon Statistical Ensembles1. Microcanonical (N,V,E):

2. Canonical (N,V,T):

3. Isothermal-isobaric (N,p,T)

])([)( EHNVE −Γδ∝Γρ

}/)](exp{[)( TkHF BNVT Γ−=Γρ

}/)](exp{[)( TkHG BNPT Γ−=Γρ

Different simulation protocols [Γ(t)# Γ(t+δt )] sample different statistical ensembles

$Newton�s eq. of motion

$Langevin dynamics

$Nose-Hoover method

Examples of Thermodynamic Examples of Thermodynamic ObservablesObservables

� Energies (kinetic, potential, internal,�)� Temperature [equipartition theorem]� Pressure [virial theorem]

� Specific heat capacity Cv and CP� Thermal expansion coefficient αP� Isothermal compressibility βT� Thermal pressure coefficient γV

Thermodynamic derivatives are related to mean square fluctuations of thermodynamic quantities

Mean EnergiesMean Energies

Total (internal) energy:

Kinetic energy:

Potential energy:

You can conveniently use namdplot to graph the time evolution of different energy terms (as well as T, P, V) during simulation

Note:

∑=

=N

iitEE

N 1

)(1

∑∑= =

=N

i

M

j j

ij

mt

KN 1 1

2

2)(1 p

TOTAL

KINETIC

BONDANGLEDIHEDIMPRPELECTVDW

KEU −=

TemperatureTemperature

From the equipartition theorem

⟩⟨= KTBNk3

2

Note: in the NVTP ensemble N#N-Nc, with Nc=3

TkpH/p Bkk =⟩∂∂⟨

Instantaneous kinetic temperature

BNk

K

3

2=T namdplot TEMP vs TS …

PressurePressureFrom the virial theorem TkrH/r Bkk =⟩∂∂⟨

⟩⟨+= WTNkPV BThe virial is defined as

∑∑>=

−=⋅=iji

ij

M

jjj rwW

,1

)(3

1

3

1 fr

Instantaneous pressure function (not unique!)

VWTkB /+ρ=P

drrdrrw /)()( v=pairwise

interactionwith

Thermodynamic Fluctuations (TF)Thermodynamic Fluctuations (TF)

[ ]∑=

⟩⟨−≈⟩δ⟨N

ii AtAA

N 1

)(1

According to Statistical Mechanics, the probability distribution of thermodynamic fluctuations is

δ⋅δ−δ⋅δ∝ρTk

STVPB

fluct 2exp

2222 )( ⟩⟨−⟩⟨=⟩⟩⟨−⟨=⟩δ⟨ AAAAA

Mean Square Fluctuations (MSF)

TF in NVT EnsembleTF in NVT EnsembleIn MD simulations distinction must be made between properly defined mechanical quantities (e.g., energy E, kinetic temperature T, instantaneous pressure P ) and thermodynamic quantities, e.g., T, P, �

VB CTkE 222 =⟩δ⟨=⟩δ⟨ HFor example:

TB VTkP β=⟩δ⟨≠⟩δ⟨ /22P!!!!

""""But:

22 )(2

3 TkK BN=⟩δ⟨

)2/3(22BVB NkCTkU −=⟩δ⟨

)(2BVB kTkU ρ−γ=⟩δδ⟨ P

Other useful formulas:

VV TEC )/( ∂∂=VV TP )/( ∂∂=γ

How to Calculate How to Calculate CCV V ??

VV TEC )/( ∂∂=1. From definition

22 / TkEC BV ⟩δ⟨=

2. From the MSF of the total energy E

222 ⟩⟨−⟩⟨=⟩δ⟨ EEEwith

Perform multiple simulations to determine ⟩⟨≡ EEthen calculate the derivative of E(T) with respect to T

as a function of T,

Analysis of MD DataAnalysis of MD Data

1. Structural properties2. Equilibrium properties3. Non-equilibrium properties

Can be studied via both equilibriumand/or non-equilibrium MD simulations

NonNon--equilibrium Propertiesequilibrium Properties

1. Transport properties2. Spectral properties

Can be obtained from equilibrium MD simulations by employing linear response theory

Linear Response TheoryLinear Response Theory

Equilibriumstate

Non-equilibriumstate

)(tVext

Thermal fluctuationsautocorrelation function C(t)

External weak perturbationResponse function R(t)

Relaxation(dissipation)

Related through the Fluctuation Dissipation Theorem

(FDT)

Time Correlation FunctionsTime Correlation Functions⟩−⟨=⟩⟨=− )0()'()'()()'( BttAtBtAttCAB

since ρeq is t independent !

BA ≠BA =

cross-auto- correlation function

Correlation time: ∫∞

=τ0

)0(/)( AAAAc CtCdt

Estimates how long the �memory� of the system lasts

In many cases (but not always): )/exp()0()( ctCtC τ−=

Response FunctionResponse Functionor generalized susceptibility

External perturbation: )()( tfAtV extext ⋅−=

Response of the system: )'()'(')(0

tfttRdttA ext

t

−=⟩⟨ ∫Response function: ⟩⟨∂β−=⟩⟨= AtAAtAtR tPB )(}),({)(

with TkB/1=βGeneralized susceptibility: ∫

∞ω=ω≡ωχ

0

)()()( tRedtR ti

Rate of energy dissipation/absorption:

tieftfftAQdt

df ω−ω =ωχω=⟩⟨≡ 0

20 Re)(,||)(")(

2

1

FluctuationFluctuation--Dissipation TheoremDissipation Theorem

Relates R(t) and C(t), namely:

)()2/()(" ωβω=ωχ C

Note: quantum corrections are important when ω≤ !TkB

In the static limit (t → ∞): )0()0( 2 RTkAC B=⟩⟨=

)()2/tanh()(" 1 ωωβ=ωχ − C!!

Diffusion CoefficientDiffusion Coefficient

Generic transport coefficient: ∫∞

⟩∂⟨∂=γ0

)0()( AtAdt tt

Einstein relation: ⟩−⟨=γ 2)]0()([2 AtAt

Example: self-diffusion coefficient

∫∞

⟩⟨=0

)0()(3

1 vv tdtD

[ ] 26 ( ) (0)Dt t= −r r

Free Diffusion (Brownian Motion) of ProteinsFree Diffusion (Brownian Motion) of Proteins

! in living organisms proteins exist and function in a viscous environment, subject to stochastic (random) thermal forces

! the motion of a globular protein in a viscous aqueous solution is diffusive

2 ~ 3.2R nm

! e.g., ubiquitin can be modeled as a spherical particle of radius R~1.6nm and mass M=6.4kDa=1.1x10-23 kg

Free Diffusion of Ubiquitin in WaterFree Diffusion of Ubiquitin in Water! ubiquitin in water is subject to two forces:

� friction (viscous drag) force:

� stochastic thermal (Langevin) force:

f γ= −F v

( )L t=F ξξξξ ( ) 0t =ξξξξoften modeled as a �Gaussian white noise�

( ) (0) 2 ( )Bt k T tγ δ=ξ ξξ ξξ ξξ ξ

6 Rγ πη=friction (damping) coeff

viscosity

(Stokes law)

231.38 10 ( ),Bk J K Boltzmann constant T temperature−= × / =

The The DiracDirac delta functiondelta function

00 0

for tt

for tδ

∞ =( ) = ≠

In practice, it can be approximated as:

( )1( ) exp , 02

t t t asτδ δ τ ττ

≈ ( ) = − / →

Useful formulas:

0( ) ( ') ( ') '

tf t f t t t dtδ= −∫ ( ) ( ) /a t t aδ δ=

-0.1 0.1

10

20

30

40

50

0.00

3.0

0.05

1

τττ

===

0

⇒ δ(t) describes τ=0 correlation time (�white noise�) stochastic processes

Equation of Motion and SolutionEquation of Motion and Solution

( )LfdvMa F M vdt

F tξγ= + ⇒ = − +Newton�s 2nd law:

Formal solution (using the variation of const. method):

/0 0

/ '/( 1 ( ') ) 't tt tv e t e dtM

t v e ττ τ ξ− −= + ∫( ) ( )/ (/

0 0' ) /

0( ') 1 '( )

ttt t tx e t e dv eM

t tx τ τ ττ τ ξ− − −−= + 1− + ∫Mτγ

= = velocity relaxation (persistence) time

The motion is stochastic and requires statistical descriptionformulated in terms of averages & probability distributions

Statistical AveragesStatistical Averages( ) 0t =ξξξξ ( ) (0) 2 ( )Bt k T tγ δ=ξ ξξ ξξ ξξ ξ

/0( ) 0tv t v e as tτ−= → → ∞

Exponential relaxation of x and v with characteristic time τ

( )/0 0 0 0( ) 1 tx t x v e x v as tττ τ−= + − → + → ∞

( )22 /( ) ( ) 1 tD Dv t v t e as tτ

τ τ−= + − → → ∞

( )22

2

/

2

( ) (

)

) 3

(

2

2

tx t x t

x t x

Dt D

x D t

e

s

O

t a

ττ −= + − +

= =⇒ ∆ − → ∞

Diffusion coefficient:(Einstein relation) /BD k T γ=

Typical Numerical EstimatesTypical Numerical Estimatesexample: ubiquitin - small globular protein

2 10 21

3

3 / 29.60.560.16

25.4

1.6 10

6 0.

?

?

?

?9

?

?

B

T

T B

T

k Tv

v k T M m sps

d vpN s m

D m s

R mPa s

τγ

τ γτ

γ

η γ π

−=

= /= /=

⋅ /

= × /

= / ⋅

SimulationProperty Theory

ΜA

23

3 3

8.6 1.42 10 , 1.6 ,10 , 310

M kDa kg R nmM V kg m T Kρ

−≈ ≈ × ≈= / ≈ / =

mass : size :

density : temperature :

Thermal and Friction ForcesThermal and Friction Forces

! Friction force:

! Thermal force:

94.5 10 4.5f TF v N nNγ −= ≈ × =

2 2 4.53

BT T f

k TF v F nNγ γτ

= = ≈∼

For comparison, the corresponding gravitational force:14~ 10 ~g f TF Mg nN F F−= #

Diffusion can be Studied by MD Simulations!Diffusion can be Studied by MD Simulations!

solvate

ubiquitin in water

1UBQPDB entry:

total # of atoms: 7051 = 1231 (protein) + 5820 (water) simulation conditions: NpT ensemble (T=310K, p=1atm), periodic BC, full electrostatics, time-step 2fs (SHAKE) simulation output: Cartesian coordinates and velocities of all atoms saved at each time-step (10,000 frames = 40 ps) in separate DCD files

How ToHow To: : vel.dcdvel.dcd ——> > vel.datvel.dat! namd2 produces velocity trajectory (DCD) file if in the

configuration file containsvelDCDfile vel.dcd ;# your file namevelDCDfreq 1 ;# write vel after 1 time-step

! load vel.dcd into VMD [e.g., mol load psf ubq.psf dcd vel.dcd]note: run VMD in text mode, using the: -dispdev text option

! select only the protein (ubiquitin) with the VMD commandset ubq [atomselect top “protein”]

! source and run the following tcl procedure: source v_com_traj.tclv_com_traj COM_vel.dat

! the file �COM_vel.dat� contains 4 columns: time [fs], vx, vy and vz [m/s]

70 12.6188434361 -18.6121653643 -34.7150913537

note: an ASCII data file with the trajectory of the COM coordinates can be obtained in a similar fashion

the the v_COM_trajv_COM_traj TclTcl procedureprocedureproc v_com_traj {filename {dt 2} {selection "protein"} {first_frame 0} {frame_step 1} {mol top} args} {

set outfile [open $filename "w"]

set convFact 2035.4

set sel [atomselect $mol $selection frame 0]

set num_frames [molinfo $mol get numframes]

for {set frame $first_frame} {$frame < $num_frames} {incr frame $frame_step} {

$sel frame $frame

set vcom [vecscale $convFact [measure center $sel weight mass]]

puts $outfile "$frame\t $vcom"

}

close $outfile

}

Goal: calculate D and Goal: calculate D and ττττττττ

! theory:

by fitting the theoretically calculated center of mass (COM) velocity autocorrelation function to the one obtained from the simulation

2 /0

20

( ) ( ) (0) tvv

B

C t v t v v ek T DvM

τ

τ

−= =

= =

! simulation: consider only the x-componentreplace ensemble average by time average

( )xv v→

1

1( )N i

vv i n i nn

C t C v vN i

−

+=

≈ =− ∑

( ), , #i n nt t i t v v t N≡ = ∆ = = of frames in vel.DCD

(equipartition theorem)

Velocity Autocorrelation FunctionVelocity Autocorrelation Function

0.1 psτ ≈

2 11 2 13.3 10B

x

D k Tv m s

γτ − −

= / == ≈ ×

0 200 400 600 800 1000

050

100150200250300

100 200 300 400

50100150200250300

100 200 300 400

50100150200250300

0 2 4 6 8 10-40-20

02040

time [ps]

V x(t

) [m

/s]

( )vvC t

[ ]time fs

[ ]time fs

/( ) tvv

DC t e τ

τ−=

Fit

Probability distribution of Probability distribution of

( ) ( )( )

1/ 22 2 2

2

( ) 2 exp

exp

p v v v v

D v D

π

τ π τ

−= − /2

= /2 − /2

, ,x y zv

, ,x y zv v≡with

Maxwell distribution of Maxwell distribution of vvCOMCOM

COM velocity [m/s]

( ) ( )3/ 2 2 2

3/ 2 22

( )

ex

(

p 4

2 ex

( )

2

( )

p

)x y z x y

B B

zP v

D v D

dv p v p v p v dv dv dv

dv

M Mvvk T

v

dk

vT

τ π τ π

π

/2 − /2

−

=

=

=

What have we learned ?What have we learned ?

2 10 2 10 21

3

3 / 29.6 31.60.56 0.10.16 0.03

25.4 141.6

1.6 10 0.3 10

6 0.9 4.7

B

T

T B

T

k Tv

v k T M m s m sps ps

d vpN s m pN s m

D m s m s

R mPa s mPa s

τγ

τ γτ

γ

η γ π

− −=

= / /= /=

⋅ / ⋅ /

= × / × /

= / ⋅ ⋅

Property Theory Simulation

ΜA A

soluble, globular proteins in aqueous solution at physiological temperature execute free diffusion (Brownian motion with typical parameter values:

How about the motion of parts of How about the motion of parts of the protein ?the protein ?

! parts of a protein (e.g., side groups, a group of amino acids, secondary structure elements, protein domains, �), besides the viscous, thermal forces are also subject to a resultantforce from the rest of the protein

! for an effective degree of freedom x (reaction coordinate) the equation of motion is

( ) ( )mx x f x tγ ξ= − + +$$ $In the harmonic approximation ( )f x kx≈ −and we have a 1D Brownian oscillator