Controlling fluctuations

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1 Controlling fluctuations Michele Parrinello Department of Chemistry and Applied Biosciences ETH Zurich and ICS, Università della Svizzera Italiana, Lugano, Switzerland

2 Today s menu Introduction Fluctuations and rare events Enhancing fluctuation via metadynamics Enhancing fluctuations with VES Entropy as a collective variable Crystallisation Calculating rates Coarse graining with VES

3 An analogical model for a complex energy landscape

4 Ancient wisdom Isaiah 40:4 Every valley shall be raised up, every mountain and hill made low; the rough ground shall become level, the rugged places a plain. Switzerland Tuscany

5 Fluctuations and rare events A B F (s) F P (s) s s k / e F F (s) = 1 logp(s)

6 Umbrella sampling Given a set of collective variables s=s(r) and their associated free energy F (s) = 1 log Z dr (s s(r))e U(R) U(R) U(R)+V (s(r)) ho(r)i = ho(r)e V (s(r)) i V he V (s(r)) i V Torrie and Valleau, J. Comp. Phys. (1977)

7 Which bias? Best choice? V (s) = F (s) Choose V (s) = (1 1 )F (s) then P V (s) / (P (s)) 1 P (s) p(s) / [P (s)] 1/

8 Constructing the bias Molecular Dynamics Metadynamics A B t4 Free Energy ΔG*>>kT CV Free Energy Laio and Parrinello PNAS (2002) Barducci, Bussi and Parrinello PRL (2008) t2 t1 A = bias potential CV B t3 P (s)! P (s) 1 R dsp (s) 1

9 The miracle of metadynamics One can build a stochastic iterative described rigorously by the Ordinary Differential Equation: where dv (s, t) dt = Z ds 0 G(s s 0 )e V (s 0 ) 1 PV (s 0,t) P V (s, t) = e (F (s)+v (s,t)) R dse (F (s)+v (s,t)) and P V (s, t!1) / P (s) 1 Laio and Parrinello PNAS (2002) Barducci, Bussi and Parrinello PRL (2008) Dama, Parrinello and Voth PRL 2014

10 Some benefits A position dependent free energy estimator Easy reweighing Tiwary and Parrinello JPC B 2014 Y. Kevrikidis, W. van Gunsteren

11 Constructing a thermodynamics functional H = E + pv E! F = F V F F V = F = 1 Z log 1 Z log dse dse Reversible work (F (s)+v (s)) F (s) pv! Z dsp(s)v (s) p(s) = F V (s) = e (F (s)+v (s)) R dse (F (s)+v (s))

12 Valsson and Parrinello, PRL (2015) A variational approach This leads to the convex functional p(s) = arbitrary normalized target distribution The functional is convex and at the minimum p(s) = e (F (s)+v (s)) R dse (F (s)+v (s)) or F (s) = V (s) 1 log p(s)

13 Optimizing the functions The force that drives the system to the minimum is:!ω!!" =!V!!"! +!V!!"! One possible choice is to expand in a convenient basis set Find the minimum V (s; ) = X i i f i (s) The free energy surfaces tend to be smooth and the expansion converges rapidly

A stochastic optimization Naive (t + t) = (t) t @ (t) (t) Bach and Moulines 2013 Expand the force to linear terms

14 A stochastic optimization Naive (t + t) = (t) (t) (t) Bach and Moulines 2013 Expand the force to linear terms around the average value (t) = 1 t Z t 0 dt 0 (t 0 ) (t + t) = (t) t ( (t) ( (t)@ (t) ( (t) (t))

15 The role of p(s) At the minimum the biased system will sample the distribution p(s) P V (s) = e (F (s)+v (s)) R e (F (s)+v (s)) ds = p(s) The freedom of selecting p(s) gives a lot of flexibility It can tailor sampling to our needs It is a natural way of introducing restrains on CVs Its choice is essential for a successful sampling

16 Two examples of p(s) Metadynamics!! =!"#$%!! =!! Well Tempered Metadynamics!! =!!!!!!!"!!!!!!!! = 1 1!!!

Using p(s) to accelerate convergency Alanine tetrapeptide (Ala3) in vacuum 1, 2, 3 as CVs 1 1 2 2 3 3 12 FES convergence 8

17 Using p(s) to accelerate convergency Alanine tetrapeptide (Ala3) in vacuum 1, 2, 3 as CVs FES convergence 8 4 UNIFORM p(s) WELL TEMPERED p(s) 0 0 ns 10 ns 20 ns 30 ns 40 ns 50 ns One can easily converge biases with ~10000 parameters!

18 What are collective variables? The notion of collective variables means different things to different people: Slow variables. Reaction paths. Order parameters. Coarse grained variables etc.. To us, in this talk, they will be a set of variables s = s(r) whose fluctuations we want to enhance

19 The CVs that I like Simple Transparent Lead to understanding and not only to a calculation They have a clear physical meaning They can be measured

20 Chemical potential Phase transitions Chemical equilibrium Electrochemistry µ T,V µ ' F (N +1,V,T) F (N,V,T)

21 Chemical potential The chemical potential is the difference in free energy between a N+1 and an N particle systems. is the energy of adding a particle at R* In Widom s method particles are added at random positions. {Easy for the reds difficult for the blues} W. Kob and H. C. Andersen Phys. Rev. Lett. (1994)

22 A collective variable In a homogeneous fluid is position independent thus: This suggests the collective variable In terms of s the chemical potential becomes:

23 The role of fluctuations Z µ ex = 1 log e s p(s)ds e s p(s) p(s) e s p(s) p(s)

24 Calculating µ ex With the standard method the convergence is slow or impossible!

Use the expansion in terms of multi particle correlation function Z 1 S 2 = 2 k B [g(r)

25 Crystallisation In a first order phase transition there is an interplay between entropy and enthalpy Enthalpy is easy H = E + PV But entropy? Use the expansion in terms of multi particle correlation function Z 1 S 2 = 2 k B [g(r) log g(r) 0 S ex = 1X n=2 The first term depends on the pair correlation function and gives 95% of a liquid entropy S n g(r) + 1]r 2 dr

26 Sodium vs. Aluminum Na! BCC Al! FCC Liquid FCC BCC HCP

27 There is much more to it The variational approach is more than an efficient sampling method! By choosing appropriately the variational function new types of simulations can be performed. We shall show some examples: Kinetics from biased simulation Bespoke bias for nucleation Bespoke bias for free energy differences Very high dimensional biases

28 Rates for rare events!!!!!!"!!! Without bias!!!!!!"!!! With bias H. Grubmüller PRE 1995 A.Voter JCP 1997 P. Tiwary and M.P. PRL 2013 Acceleration factor of rates!!!!!! =!!!! =!!!!

29 Building a bias for kinetics Write the variational bias as V (s) =v(s) S( v(s) F c ) 1 S(x) switching function 0 0 optimize V(s) using the target distribution p(s) = S(F (s) F c ) R ds S(F (s) F c ) F (s) v(s) (iteratively updated) J. McCarthy, O. Valsson, and M.P. PRL 2015

* Zhang and Brüschweiler, JACS 2002 Entropy for proteins Using NMR relaxation methods the square of the generalised order parameter S 2 of the NH group can be measured.

30 * Zhang and Brüschweiler, JACS 2002 Entropy for proteins Using NMR relaxation methods the square of the generalised order parameter S 2 of the NH group can be measured. A similar order parameter can be associated to the methyl axis. S 2 measures the degree of spacial restriction of the motion. (Lipari and Szabo, JACS 1982). S 2 has been empirically related to conformational entropy. S 2 = X n S 2 n S 2 n tanh ( X k (e ro n 1,k + e r NH n 1,k ) 0.1

31 Unfolding times for chignoline D. E. Shaw el al., Science 2011

32 Discovering cryptic binding sites Human Interlukin-2

33 Ginzburg-Landau theory: a primer F = g(t ) Close to a second order described by an order parameter ψ the free energy is written a a functional of ψ Z (~r )r 2 (~r ) d~r + a(t ) Z (~r ) 2 d~r + b(t ) Z (~r ) 4 d~r g(t ) const a(t )=a c (T T c ) b(t ) const F(T, ) F(T, ) F(T, ) T>T c T=T c T<T c

34 Ginzburg-Landau theory as a coarse grained model In Ginzburg-Landau theory is is implicit a coarse graining procedure in which only the long wavelength fluctuations are considered. The rational is that close to a second order phase transition the properties of the systems are dominated by long wavelengths fluctuations. The short range fluctuations are integrated out and their effect is lumped into the parameters g(t), a(t), and b(t). (~r )= X c( ~ k)e i~ k.~r ~ k apple 2 = coarse graining length

35 From atoms to fields We will use Variational Enhanced Sampling to compute g(t), a(t), and b(t) At equilibrium F (s) = V (s) 1 log p(s) We take Simplifying the notation we write F = g(t ) Z s = (~r ) (~r )r 2 (~r ) d~r + a(t ) Z (~r ) 2 d~r + b(t ) Z (~r ) 4 d~r F = g(t )I 1 ( (~r )) + a(t )I 2 ( (~r )) + b(t )I 4 ( (~r ))

36 The variational expression F = g(t )I 1 ( (~r )) + a(t )I 2 ( (~r )) + b(t )I 4 ( (~r )) F (s) = V (s) 1 log p(s) s = (~r ) We take : V ( (~r )) = g v I 1 ( (~r )) + a v I 2 ( (~r )) + b v I 2 ( (~r )) 1 log p( (~r )) = g0 I 1 ( (~r )) + a 0 I 2 ( (~r )) At the minimum F( (~r )) (g 0 g v )I 1 ( (~r )) + (a 0 a v )I 2 ( (~r )) b v I 2 ( (~r ))

37 The role of p(s) F(T, ) F(T, ) F(T, ) T>T c Helps getting a better estimate of the quartic term coefficient b T=T c T<T c Helps reducing pbc and symmetry breaking effects.

38 The Lennard Jones case As predicted by Landau a depends linearly and rather markedly on T. T c 6= T c a(t )=a c (T T c ) but due to finite size effects T c 6= T c However all values of a can be collapsed into a single line if T is properly scaled.

39 How good is the Ginzburg-Landau Hamiltonian? Z = Z e F[ ] Binder Cumulant N=128 N=512 N=2048 N=8192 U(T )=1 I 4 3I 2 2 Use Binder cumulant to eliminate finite size effects Tc is correctly predicted! Temperature T/T c

40 Temperature Ginzburg-Landau theory in action Z = Z e F[ ] gas supercritical fluid liquid solid 0.6 non homogeneous Density T>T c T = T c T<T c

41 Acknowledgements OMAR VALSSON

42 Acknowledgements Pratyush Tiwari

43 Acknowledgements Jay MaCarty

44 Acknowledgements Ferruccio Palazzesi

45 Acknowledgements Claudio Perego

46 Acknowledgements Pablo Piaggi

47 Acknowledgements Michele Invernizzi

48 The end Thank you for your attention

Making the best of a bad situation: a multiscale approach to free energy calculation

Making the best of a bad situation: arxiv:1901.04455v2 [physics.comp-ph] 15 Jan 2019 a multiscale approach to free energy calculation Michele Invernizzi,, and Michele Parrinello, Department of Physics,