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