Free energy calculations using molecular dynamics simulations. Anna Johansson

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1 Free energy calculations using molecular dynamics simulations Anna Johansson

2 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

3 Thermodynamical concepts Internal energy: U Enthalpy: H = U + PV Entropy: ds = Q/T S = k B ln W

4 Free energy Gibbs free energy: G(N,P,T) = U - TS + PV G = " N µ i N i Helmholtz free energy: F(N,V,T) = U - TS

5 Every system seeks to achieve a minimum of free energy "G < 0 Favorable "G = 0 "G > 0 Unfavorable

6 Statistical mechanics A system with N interacting particles can be described using a Hamiltonian H(p 1,p 2 p N,r 1,r 2 r N ) Ensembles are defined of which quantities that are kept fixed Canonical ensemble (N,V,T) NPT-ensemble (N,P,T)

7 Solvation free energy

8 Binding free energy

9 Conformational free energy

10 Calculation of Free energy? Experimentally Probability of finding a system at a given state "G = #RT ln(s A /S B ) Reversible work required to transform the system from one state to another Computationally Both can be used, but the second approach is most efficient

11 Thermodynamic cycles "G hyd = "G 1 # "G 3 # "G 2 = "G 1 # "G 2

12 Statistical mechanics description of free energy in the canonical ensemble A = "k B T lnq NVT Q NVT = 1 h 3N N! # # exp[" 1 k B T H(x, p x)] dxdp x " A = k B T ln exp 1 % $ H(x, p x )' # k B &

13 Statistical mechanics description of free energy in the canonical ensemble A = "k B T lnq NVT Q NVT = 1 h 3N N! # # exp[" 1 k B T H(x, p x)] dxdp x " A = k B T ln exp 1 % $ H(x, p x )' # k B &

14 Statistical mechanics description of free energy in the canonical ensemble A = "k B T lnq NVT Q NVT = 1 h 3N N! # # exp[" 1 k B T H(x, p x)] dxdp x " A = k B T ln exp 1 k B T H(x, p % $ x) ' # &

15 Problems Accurate calculations of absolute free energy is not possible due to insufficient sampling during finite length simulations. But free energy differences can be calculated using statistical simulations. Most used methods include: Thermodynamical integration Free energy perturbation Umbrella sampling Potential of mean force

16 Thermodynamical integration Make the Hamiltonian a function of a coupling parameter " H(x, p x ;" a ) = H(x, p x ;" = 0) H(x, p x ;" b ) = H(x, p x ;" =1)

17 Derivation of TI "A a #b = A($ b ) % A($ a ) = $ b & $ a da($) d$ d$ da($) d$ = & 'H(x, p x ;$) d$ & exp% 1 k B T H(x, p x;$)dxdp x exp% 1 k B T H(x, p x ;$)dxdp x "A a #b = $ b & $ a 'H(x, p x ;$) '$ $ d$

18 Slow growth vs. intermediate values Either the integration can be obtained from one simulation with a varying ", slow growth da /d" Or, the value of is accurately determined for a number of intermediate values of ", the total free energy is determined with numerical integration methods based on these values

19 Single vs. double topology

20 Error estimation Convergence criterion is that the A(") is smooth enough. Slow growth Often results in insufficient sampling, the hysteresis can for some applications be used as a measure of fluctuations Intermediate values Estimated from the fluctuations in for each value of da /d" dh /d"

21 Free energy perturbation "A a #b = A($ b ) % A($ a ) = %k B T ln Q NVT ($ b ) Q NVT ($ a ) & "A a #b = $k B T ln exp ' $ 1 k B T H(x, p x;% b ) $ H(x, p x,% a ) ( [ ] ) * + %a

22 Free energy perturbation "A a #b = A($ b ) % A($ a ) = %k B T ln Q NVT ($ b ) Q NVT ($ a ) & "A a #b = $k B T ln exp ' $ 1 k B T H(x, p x;% b ) $ H(x, p x,% a ) ( [ ] ) * + %a

23 Number of intermediate states The perturbation formula only holds for small changes between the states Reaction pathway often broken up into intermediate states, such that the configuration sampled in state A also have a high probability in state B which is the criterion for the ensemble average to converge N$1 ' "A a #b = $k B T% ln exp ( $ 1 k B T H(x, p x;& b ) $ H(x, p x,& a ) ) k=1 [ ] * +, &k

24 Error estimation Convergence may be probed by the time-evolution of the ensemble average Statistical error may be estimated by a first order expansion of the free energy

25 Potential of mean force According to the concept of PMF, if a force depending on some reaction coordinate can be extracted, then " "# $A a %b = & F # #

26 Umbrella sampling A(") = #k B T ln P(") + A 0 P(") = + % #[" $"(x) ]exp $ 1 k B T H(x, p ) ( ' x * dxdp x & ) Confine the system to a small region by applying a biasing potential to ensure a uniform distribution of P(") The reaction pathway often broken down in windows where the free energy is determined

27 Error estimation Convergence is probed by two criteria: Convergence of individual windows. The statistical error can be measured through block-averaging over sub-runs Appropriate overlap of free energy profiles between adjacent windows

28 Statistical precision vs. accuracy The approaches to estimate errors for the different methods based on a single simulation only reflect the statistical precision of the method Statistical accuracy can be derived from an ensemble of simulations starting from different regions in phase space

Polar residues in transmembrane segments are both existing and important.

29 α-helical membrane proteins account for 25% of all proteins and possibly as much as 50% of drug targets. Polar residues in transmembrane segments are both existing and important. Little is known about the interactions between individual residues and the surrounding membrane environment Membrane proteins

30 Free energy of solvating amino acids analogs in a membrane A lipid bilayer is a heterogeneous solvent, and positional differences are important when studying interactions between amino acids and lipid membranes

31 Potential of mean force

33 Potential of mean force PMF(z) = " F constr (z)dz

37 Summary Free energy is a very useful measurement of the preferred direction of different kind of reaction In most cases the free energy difference between states is most easily calculated and also most interesting A number of different MD-based methods exist to calculate free energy and there is a constant development of these and new ones

38 References. Understanding Molecular Simulation, Frenkel D. & Smith Free energy calculations in Biological systems. How useful are they in practice? Christophe Chipot. Molecular dynamics lecture notes 2003, Olle Edholm, Course in Computational Physics at KTH, "Calculating free energy using average force", Eric Darve and Andrew Pohorille, Free Energy calculations: a breakthrough for modeling organic chemistry in solution. W.L. Jorgensen. ACC Chem Res, 22(1989) Avoiding singularities and numerical instabilities in free energy calculations based on molecular simulations. Thomas C. Beutler, Alan E. Mark, Rene C. van Shaik, Paul R. Gerber, Wilfred F van Gunsteren. Chem Phys Letters 222(1994)

Free energy simulations

Free energy simulations Marcus Elstner and Tomáš Kubař January 14, 2013 Motivation a physical quantity that is of most interest in chemistry? free energies Helmholtz F or Gibbs G holy grail of computational