statistical mechanics of proteins equilibrium and non-equilibrium properties of proteins free...
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Statistical Mechanics Statistical Mechanics of Proteinsof Proteins
Equilibrium and non-equilibrium Equilibrium and non-equilibrium properties of proteinsproperties of proteins Free diffusion of proteinsFree diffusion of proteins
Coherent motion in proteins: temperature echoes
Simulated cooling of proteins
Ioan Kosztin Department of Physics & Astronomy
University 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 properties
2. Equilibrium properties
3. Non-equilibrium properties
Can be studied via both equilibrium and non-equilibrium MD simulations
Equilibrium Equilibrium (Thermodynamic) (Thermodynamic)
PropertiesPropertiesMD simulation
microscopic information
macroscopic properties
Statistical Statistical MechanicsMechanics
[r(t),p(t)]
Phase space trajectory
Ensemble average overprobability density
()
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 in phase space
Time vs Ensemble Time vs Ensemble AverageAverage
For t, (t) generates an ensemble with
tddt
/lim(
Ergodic Hypothesis: Time and Ensemble averages are
equivalent, i.e., )(),( AprA t
Time average:
Ensemble average:
T
t ttAdtT
A0
)](),([1
pr
)()( AdA
Thermodynamic Thermodynamic Properties from MD Properties from MD
SimulationsSimulations
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 Ensembles
1. Microcanonical (N,V,E):
2. Canonical (N,V,T):
3. Isothermal-isobaric (N,p,T)
])([)( EHNVE
}/)](exp{[)( TkHF BNVT
}/)](exp{[)( TkHG BNPT
Different simulation protocols [(t) (tt )] sample different statistical ensembles
Newton’s eq. of motion
Langevin dynamics
Nose-Hoover method
Examples of Examples of Thermodynamic Thermodynamic
ObservablesObservables• Energies (kinetic, potential, internal,…)• Temperature [equipartition theorem]• Pressure [virial theorem]
• Specific heat capacity Cv and CP • Thermal expansion coefficient P
• Isothermal compressibility T
• Thermal pressure coefficient V
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
m
tK
N 1 1
2
2
)(1 p
TOTAL
KINETIC
BONDANGLEDIHEDIMPRPELECTVDW
KEU
TemperatureTemperature
From the equipartition theorem
KTBNk3
2
Note: in the NVTP ensemble NN-Nc, with Nc=3
TkpH/p Bkk
Instantaneous kinetic temperature
BNk
K
3
2T namdplot TEMP vs TS …
PressurePressureFrom the virial theorem TkrH/r Bkk
WTNkPV B
The virial is defined as
iji
ij
M
jjj rwW
,1
)(3
1
3
1fr
Instantaneous pressure function (not unique!)
VWTkB /P
drrdrrw /)()( vpairwise
interactionwith
Thermodynamic Fluctuations Thermodynamic Fluctuations (TF)(TF)
N
ii AtAA
N 1
)(1
According to Statistical Mechanics, the probability distribution of thermodynamic fluctuations is
Tk
STVP
Bfluct 2
exp
2222 )( 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 HFor example:
TB VTkP /22P
But:
22 )(2
3TkK B
N
)2/3(22BVB NkCTkU
)(2BVB kTkU P
Other useful formulas:
VV TEC )/( VV TP )/(
How to Calculate How to Calculate CCV V ??
VV TEC )/( 1. From definition
22 / TkEC BV
2. From the MSF of the total energy E
222 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 properties
2. Equilibrium properties
3. Non-equilibrium properties
Can be studied via both equilibrium and/or non-equilibrium MD simulations
Non-equilibrium PropertiesNon-equilibrium 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 Time Correlation FunctionsFunctions
)0()'()'()()'( BttAtBtAttCAB
since eq is t independent !
BA BA
cross-auto-
correlation function
Correlation time:
0
)0(/)( AAAAc CtCdt
Estimates how long the “memory” of the system lasts
In many cases (but not always): )/exp()0()( ctCtC
Response FunctionResponse Functionor generalized susceptibility
External perturbation: )()( tfAtV extext
Response of the system: )'()'(')(0
tfttRdttA ext
t
Response function: AtAAtAtR tPB )(}),({)(
with TkB/1Generalized susceptibility:
0
)()()( tRedtR ti
Rate of energy dissipation/absorption:
tieftfftAQdt
df 0
20 Re)(,||)(")(
2
1
Fluctuation-Dissipation Fluctuation-Dissipation TheoremTheorem
Relates R(t) and C(t), namely:
)()2/()(" C
Note: quantum corrections are important when TkB
In the static limit (t ): )0()0( 2 RTkAC B
)()2/tanh()(" 1 C
Diffusion CoefficientDiffusion Coefficient
Generic transport coefficient:
0
)0()( AtAdt tt
Einstein relation: 2)]0()([2 AtAt
Example: self-diffusion coefficient
0
)0()(3
1vv tdtD
26 ( ) (0)Dt t r r
Free Diffusion (Brownian Motion) of Free Diffusion (Brownian Motion) of ProteinsProteins
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 Free Diffusion of Ubiquitin in WaterWater
ubiquitin in water is subject to two forces:• friction (viscous drag) force:
• stochastic thermal (Langevin) force:
f F v
( )L tF ( ) 0t often modeled as a “Gaussian white noise”
( ) (0) 2 ( )Bt k T t
6 R friction (damping) coeff
viscosity
(Stokes law)
231.38 10 ( ),Bk J K Boltzmann constant T temperature
The Dirac delta functionThe Dirac delta function
0
0 0
for tt
for t
In practice, it can be approximated as:
1( ) exp , 0
2t t t as
Useful formulas:
0( ) ( ') ( ') '
tf t f t t t dt ( ) ( ) /a t t a
0.1 0.1
10
20
30
40
50
0.00
3.0
0.05
1
0
(t) describes =0 correlation time (“white noise”) stochastic processes
Equation of Motion and Equation of Motion and SolutionSolution
( )Lf
dvMa F M v
dtF t
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 / (/
0 0' ) /
0( ') 1 '( )
ttt t tx e t e dv eM
t tx M
= velocity relaxation (persistence) time
The motion is stochastic and requires statistical description formulated in terms of averages & probability distributions
Statistical AveragesStatistical Averages
( ) 0t ( ) (0) 2 ( )Bt k T t
/0( ) 0tv t v e as t
Exponential relaxation of x and v with characteristic time
/0 0 0 0( ) 1 tx t x v e x v as t
22 /( ) ( ) 1 tD Dv t v t e as t
22
2
/
2
( ) (
)
) 3
(
2
2
tx t x t
x t x
Dt D
x D t
e
s
O
t a
Diffusion coefficient: (Einstein relation) /BD k T
Typical Numerical EstimatesTypical Numerical Estimatesexample: ubiquitin - small globular protein
2 10 21
3
3 / 29.6
0.56
0.16
25.4
1.6 10
6 0.
?
?
?
?9
?
?
B
T
T B
T
k Tv
v k T M m s
ps
d v
pN s m
D m s
R mPa s
SimulationProperty Theory
Μ
A
23
3 3
8.6 1.42 10 , 1.6 ,
10 , 310
M kDa kg R nm
M V kg m T K
mass : size:
density: temperature:
Thermal and Friction ForcesThermal and Friction Forces
Friction force:
Thermal force:
94.5 10 4.5f TF v N nN
2 24.5
3B
T T f
k TF v F nN
For comparison, the corresponding gravitational force:
14~ 10 ~g f TF Mg nN F F
Diffusion can be Studied by MD Diffusion can be Studied by MD Simulations!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.dcd —> vel.dat: vel.dcd —> vel.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 v_COM_traj Tcl procedurethe v_COM_traj Tcl procedureproc 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
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 D
vM
simulation: consider only the x-component
replace 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 Velocity Autocorrelation FunctionFunction
0.1 ps
2 11 2 13.3 10B
x
D k Tv m s
0 200 400 600 800 1000
050
100150200250300
100 200 300 400
50100150200250300
100 200 300 400
50100150200250300
0 2 4 6 8 1040
20
0
20
40
time [ps]
Vx(t
) [
m/s
]
( )vvC t
[ ]time fs
[ ]time fs
/( ) tvv
DC t e
Fit
Probability distribution of Probability distribution of
1/ 22 2 2
2
( ) 2 exp
exp
p v v v v
D v D
, ,x y zv
, ,x y zv vwith
Maxwell distribution of Maxwell distribution of vvCOMCOM
COM velocity [m/s]
3/ 2 2 2
3/ 2 22
( )
ex
(
p 4
2ex
( )
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 Mvv
k T
v
dk
vT
What have we learned ?What have we learned ?
2 10 2 10 21
3
3 / 29.6 31.6
0.56 0.1
0.16 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 s
ps ps
d v
pN s m pN s m
D m s m s
R mPa s mPa s
Property Theory Simulation
Μ
A 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 How about the motion of parts of the protein ?of 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 resultant force from the rest of the protein
for an effective degree of freedom x (reaction coordinate) the equation of motion is
( ) ( )mx x f x t In the harmonic approximation ( )f x kx
and we have a 1D Brownian oscillator