Statistical Mechanics Statistical Mechanics of Proteins of Proteins ! ! Equilibrium and non Equilibrium and non - - equilibrium equilibrium properties of proteins properties of proteins ! ! Free diffusion of proteins Free diffusion of proteins ! Coherent motion in proteins: temperature echoes ! Simulated cooling of proteins Ioan Kosztin Department of Physics & Astronomy University of Missouri - Columbia
39
Embed
Statistical Mechanics of Proteins...Statistical Mechanics of Proteins! Equilibrium and non-equilibrium properties of proteins! Free diffusion of proteins! Coherent motion in proteins:
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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, �
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
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 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