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
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Molecular ModelingMolecular Modeling
1. Model building
2. Molecular Dynamics Simulation
3. Analysis of the
• model
• results of the simulation
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Collection of MD DataCollection of MD Data
• DCD trajectory file
coordinates for each atom
velocities for each atom
• Output file global energies temperature, pressure, …
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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
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Equilibrium Equilibrium (Thermodynamic) (Thermodynamic)
PropertiesPropertiesMD simulation
microscopic information
macroscopic properties
Statistical Statistical MechanicsMechanics
[r(t),p(t)]
Phase space trajectory
Ensemble average overprobability density
()
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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
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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
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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
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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
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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
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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
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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 …
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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
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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)
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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 )/(
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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,
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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
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Non-equilibrium PropertiesNon-equilibrium Properties
1. Transport properties2. Spectral properties
Can be obtained from equilibrium MD simulations by employing linear response theory
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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)
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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:
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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
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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
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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
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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
}
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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)
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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
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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
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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
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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:
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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