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Free energy calculations usingmolecular dynamics
simulations
Anna Johansson 2007-03-13
Outline• Introduction to concepts• Why is free energy important?• Calculating free energy using MD
– Thermodynamical Integration (TI)– Free energy perturbation (FEP)– PMF– Umbrella sampling
• Example• Summary
Thermodynamical concepts
• Internal energy: U
• Enthalpy: H = U + PV
• Entropy: dS = ∆Q/TS = kB ln W
Free energy
• Gibbs free energy:G(N,P,T) = U - TS + PV
• Helmholtz free energy: F(N,V,T) = U - TS!
G = µiN
i
N
"
Every system seeks to achievea minimum of free energy
Favorable
Unfavorable
!
"G < 0
"G = 0
"G > 0
Statistical mechanics• A system with N interacting particles can be
described using a HamiltonianH(p1,p2…pN,r1,r2…rN)
• Ensembles are defined of which quantitiesthat are kept fixed– Canonical ensemble (N,V,T)– NPT-ensemble (N,P,T)
Solvation free energy
Binding free energy
Conformational free energy
Calculation of Free energy?• Experimentally
– Probability of finding a system at a given state
– Reversible work required to transform the systemfrom one state to another
• Computationally– Both can be used, but the second approach is
most efficient
!
"G = #RT ln(SA/S
B)
Thermodynamic cycles
!
"Ghyd = "G1#"G
3#"G
2= "G
1#"G
2
Statistical mechanics description offree energy in the canonical ensemble
!
A = "kBT lnQNVT
!
QNVT =1
h3NN!
exp["1
kBTH(x, px )]# dxdpx#
!
A = kBT ln exp1
kBH(x, px )
"
# $
%
& '
!
A = "kBT lnQNVT
!
QNVT =1
h3NN!
exp["1
kBTH(x, px )]# dxdpx#
!
A = kBT ln exp1
kBH(x, px )
"
# $
%
& '
Statistical mechanics description offree energy in the canonical ensemble
!
A = "kBT lnQNVT
!
QNVT =1
h3NN!
exp["1
kBTH(x, px )]# dxdpx#
!
A = kBT ln exp1
kBTH(x, px )
"
# $
%
& '
Statistical mechanics description offree energy in the canonical ensemble
Problems• Accurate calculations of absolute free energy is not
possible due to insufficient sampling during finitelength simulations.
• But free energy differences can be calculated usingstatistical simulations. Most used methods include:
Thermodynamical integrationFree energy perturbation
Umbrella samplingPotential of mean force
Thermodynamical integration
• Make the Hamiltonian a function of acoupling parameter
!
"
!
H(x, px;"a ) = H(x, px;" = 0)
!
H(x, px;"b ) = H(x, px;" =1)
Derivation of TI
!
"Aa#b = A($b ) % A($a ) =dA($)
d$d$
$a
$b
&
dA($)
d$=
'H(x, px;$)
d$exp%
1
kBTH(x, px;$)dxdpx&
exp%1
kBTH(x, px;$)dxdpx&
"Aa#b ='H(x, px;$)
'$$a
$b
&$
d$
Slow growth vs. intermediate values
• Either the integration can be obtained fromone simulation with a varying , “slowgrowth”
• Or, the value of is accuratelydetermined for a number of intermediatevalues of , the total free energy isdetermined with numerical integrationmethods based on these values
!
"
!
dA /d"
!
"
Single vs. double topology
Error estimation• Convergence criterion is that the is
smooth enough.• Slow growth
– Often results in insufficient sampling, thehysteresis can for some applications be used as ameasure of fluctuations
• Intermediate values– Estimated from the fluctuations in for each
value of
!
dA /d"
!
dH /d"
!
A(")
Free energy perturbation
!
"Aa#b = A($b ) % A($a ) = %kBT lnQNVT ($b )
QNVT ($a )
!
"Aa#b = $kBT ln exp $1
kBTH(x, px;%b ) $H(x, px,%a )[ ]
& ' (
) * +
%a
Free energy perturbation
!
"Aa#b = A($b ) % A($a ) = %kBT lnQNVT ($b )
QNVT ($a )
!
"Aa#b = $kBT ln exp $1
kBTH(x, px;%b ) $H(x, px,%a )[ ]
& ' (
) * +
%a
Number of intermediate states
• The perturbation formula only holds for smallchanges between the states
• Reaction pathway often broken up into intermediatestates, such that the configuration sampled in state Aalso have a high probability in state B which is thecriterion for the ensemble average to converge
!
"Aa#b = $kBTk=1
N$1
% ln exp $1
kBTH(x, px;&b ) $H(x, px,&a )[ ]
' ( )
* + ,
&k
Error estimation
• Convergence may be probed by thetime-evolution of the ensemble average
• Statistical error may be estimated by afirst order expansion of the free energy
Potential of mean force
• According to the concept of PMF, if aforce depending on some reactioncoordinate can be extracted, then
!
"
"#$A
a%b= & F# #
Umbrella sampling
!
A(") = #kBT lnP(") + A0
• Confine the system to a small region by applying abiasing potential to ensure a uniform distributionof
• The reaction pathway often broken down inwindows where the free energy is determined
!
P(")!
P(") = # " $"(x)[ ]exp $1
kBTH(x, px )
%
& '
(
) * dxdpx+
Error estimation
• Convergence is probed by two criteria:– Convergence of individual windows. The
statistical error can be measured throughblock-averaging over sub-runs
– Appropriate overlap of free energy profilesbetween adjacent windows
Statistical precision vs. accuracy
• The approaches to estimate errors forthe different methods based on a singlesimulation only reflect the statisticalprecision of the method
• Statistical accuracy can be derived froman ensemble of simulations startingfrom different regions in phase space
Membrane proteins• α-helical membrane
proteins account for 25%of all proteins andpossibly as much as 50%of drug targets.
• Polar residues in trans-membrane segments areboth existing andimportant.
• Little is known about theinteractions betweenindividual residues andthe surroundingmembrane environment
A lipid bilayer is aheterogeneous solvent, andpositional differences areimportant when studyinginteractions between aminoacids and lipid membranes
Free energy ofsolvating aminoacids analogs ina membrane
Potential of mean force
Potential of mean force
!
PMF(z) = Fconstr
(z)dz"
Summary• Free energy is a very useful measurement of
the preferred direction of different kind ofreaction
• In most cases the free energy differencebetween states is most easily calculated andalso most interesting
• A number of different MD-based methodsexist to calculate free energy and there is aconstant development of these and new ones
References.• Understanding Molecular Simulation, Frenkel D. & Smith
• Free energy calculations in Biological systems. How useful are they in practice? Christophe Chipot. http://www.cirm.univ-rs.fr/videos/2006/exposes/02_LeBris/Chipot.pdf
• Molecular dynamics lecture notes 2003, Olle Edholm, Course inComputational Physics at KTH, http://courses.theophys.kth.se/SI2530/
• "Calculating free energy using average force", Eric Darve and AndrewPohorille, http://ctr.stanford.edu/ResBriefs01/darve2.pdf
• Free Energy calculations: a breakthrough for modeling organic chemistryin solution. W.L. Jorgensen. ACC Chem Res, 22(1989) 184-189
• Avoiding singularities and numerical instabilities in free energycalculations based on molecular simulations. Thomas C. Beutler, Alan E.Mark, Rene C. van Shaik, Paul R. Gerber, Wilfred F van Gunsteren. ChemPhys Letters 222(1994) 529-539
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