Statistical Mechanics Statistical Mechanics of Proteinsof Proteins

! Equilibrium and non-equilibrium properties of proteins! Free diffusion of proteins

! Coherent motion in proteins: temperature echoes

!! Simulated cooling of proteinsSimulated cooling of proteins

Ioan KosztinDepartment of Physics & AstronomyUniversity of Missouri - Columbia

Simulated Cooling of UbiquitinSimulated Cooling of Ubiquitin� Proteins function in a narrow (physiological)

temperature range. What happens to them when the temperature of their surrounding changes significantly (temperature gradient) ?

� Can the heating/cooling process of a protein be simulated by molecular dynamics ? If yes, then how?

0T

1T � What can we learn from the simulated cooling/heating of a protein ?

Nonequilibrium (Transport) PropertiesNonequilibrium (Transport) Properties

! macromolecular properties of proteins, which are related to their biological functions, often can be probed by studying the response of the system to an external perturbation, such as thermal gradient

! �small� perturbations are described by linear response theory (LRT), which relates transport (nonequilibrium) to thermodynamic (equilibrium) properties

! on a �mesoscopic� scale a globular protein can be regarded as a continuous medium ⇒ within LRT, the local temperature distribution T(r,t) in the protein is governed by the heat diffusion (conduction) equation

2( ) ( )T t D T tt

∂ , = ∇ ,∂r r

Atomic Atomic vsvs MesoscopicMesoscopic

� each atom is treated individually

� length scale ~ 0.1 Ǻ

� time scale ~ 1 fs

� one partitions the protein in small volume elements and average over the contained atoms

� length scale ≥ 10 Ǻ = 1nm

� time scale ≥ 1 ps

Heat Conduction EquationHeat Conduction Equation2( ) ( )T t D T t

t∂ , = ∇ ,

∂r r

D K cρ= /thermal diffusion coefficient

thermal conductivityspecific heat

mass density

� approximate the protein with a homogeneous sphere of radius R~20 Ǻ

� calculate T(r,t) assuming initial and boundary conditions:

0( ,0)( , ) bath

T r T for r RT R t T

= <=

R0r

Thermal diffusion coefficient D=?Thermal diffusion coefficient D=?

D is a phenomenological transport coefficient which needs to be calculated either from a microscopic (atomistic) theory, or derived from (computercomputer) experimentexperiment

�Back of the envelope� estimate:

2 202

( ) ( )T t T TD T t D D Rt R

ττ 0

∂ , ∆ ∆= ∇ , ⇒ ⇒ /∂

∼ ∼r r

9 110

2 7 2 3 20

~ 10 10 , ~ 10 10

~ 10 10 /

R m ps sD R m s cm s

ττ

− −

− −

=

⇒ / / =

A =∼

???

From MD simulation !From MD simulation !

Solution of the Heat EquationSolution of the Heat Equation

( )2 22 2

12

6 1( ) (0) exp

( ) ( ) ,n

bath

T t T n tn

T t T t T R D

π τπ

τ

∞

0=

0

∆ = ∆ × − /

∆ ≡ − = /

∑where

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1.0

( , )(0, )

T t rT r

∆∆

rR

time

protein

coola

nt

averaged over the entire protein!averaged over the entire protein!

How to simulate cooling ?How to simulate cooling ?� In laboratory, the protein is immersed in a coolant and the

temperature decreases from the surface to the center

� Cooling methods in MD simulations:

1.1. Stochastic boundary methodStochastic boundary method

2. Velocity rescaling (rapid cooling, biased velocity autocorrelation)

3. Random reassignment of atomic velocities according to Maxwell�s distribution for desired temperature (velocity autocorrelation completely lost)

2

1( )

dN

i ii new

i isimd B old

m vTT t v v

N k T= ′= ⇒ =∑

NAMD User Guide: Temperature ControlNAMD User Guide: Temperature Control

6.3 Temperature Control and Equilibration . . . .50 6.3.1 Langevin dynamics parameters . . . . . 6.3.2 Temperature coupling parameters . . . 6.3.3 Temperature rescaling parameters . . . 6.3.4 Temperature reassignment parameters .

Stochastic Boundary MethodStochastic Boundary Method

2R δ

coolant layer of atomscoolant layer of atomsmotion of atoms is subject to stochastic Langevindynamics

FF H f L

FF

H

f

L

m = + + +

→→→→

r F F F FFFFF

force fieldharmonic restrainfriction Langevin force

""

FFm =r F""

Heat transfer through mechanical coupling between atoms in the two regions

atoms in the inner region follow Newtonian dynamics

22--66--heat_diffheat_diff: Simulated Cooling of UBQ : Simulated Cooling of UBQ

Start from a pre-equilibrated system of UBQ in a water sphere of radius 26Ǻ mol load mol load psfpsf ubq_ws.psfubq_ws.psf namdbinnamdbin ubq_ws_eq.restart.coorubq_ws_eq.restart.coor

Create the a coolant layer of atoms of width 4Ǻ

Select all atoms in the system:set set selALLselALL [[atomselectatomselect top all]top all]

Find the center of the system:set center [measure center $set center [measure center $selALLselALL weight mass]weight mass]

Find X, Y and Z coondinates of the system's center:foreachforeach {{xmassxmass ymassymass zmasszmass} $center { break }} $center { break }

22--66--heat_diffheat_diff: Simulated Cooling of UBQ: Simulated Cooling of UBQ

Select atoms in the outer layer:set set shellSelshellSel [[atomselectatomselect top "not ( top "not ( sqr(xsqr(x--$xmass$xmass) + ) + sqr(ysqr(y--$ymass$ymass) + ) + sqr(zsqr(z--$zmass$zmass) <= sqr(22) ) "]) <= sqr(22) ) "]

Set beta parameters of the atoms in this selection to 1.00:$$shellSelshellSel set beta 1.00set beta 1.00

Select the entire system again:set set selALLselALL [[atomselectatomselect top all]top all]

Create the pdb file that marks the atoms in the outer layer by 1.00 inthe beta column:$$selALLselALL writepdbwritepdb ubq_shell.pdbubq_shell.pdb

NAMD configuration file: NAMD configuration file: ubq_cooling.confubq_cooling.conf# Spherical boundary conditions# Spherical boundary conditions# Note: Do not set other bondary conditions and PME if spherical # boundaries are usedif {1} {sphericalBC onsphericalBCcenter 30.30817, 28.80499, 15.35399sphericalBCr1 26.0sphericalBCk1 10sphericalBCexp1 2}

# this is to constrain atoms# this is to constrain atomsif {1} {constraints Onconsref ubq_shell.pdbconsexp 2conskfile ubq_shell.pdbconskcol B}

NAMD configuration file: NAMD configuration file: ubq_cooling.confubq_cooling.conf# this is to cool a water layer this is to cool a water layer if {1} {tCouple ontCoupleTemp 200tCoupleFile ubq_shell.pdbtCoupleCol B}

RUN THE SIMULATION FOR 10 ps

(5000 steps; timestep = 2 ps)The (kinetic) temperature T(t) is extracted from the simulation log (output) file; it can be plotted directly with

namdplot TEMP ubq_cooling.logIs this procedure of getting Is this procedure of getting T(tT(t) correct ?) correct ?

Determine D by Fitting the DataDetermine D by Fitting the Data

0 0.2 0.4 0.6 0.8 1

220

240

260

280

300

Tem

pera

ture

[K]

time [x10ps]

( )22 2

1

2

00

( ) 6 1 exp(0)

10 , ,

n

T tn x t t

T n

R xt ps x fitting parameter Dt

π

π

∞

0=

∆= − /

∆

= = =

∑

3 21.41 0.97 10x D cm s−≈ ⇒ ≈ × /

Thermal Conductivity of UBQThermal Conductivity of UBQ

K D cρ=

( )2 2 2 2 2/ /V B BC E k T E E k Tδ= ⟨ ⟩ = ⟨ ⟩ − ⟨ ⟩

3 20.97 10D cm s−≈ × /3 31 10 kg mρ ≈ × /