Bayesian reconstruction of free energy profiles from umbrella samples

Transcription

1 Bayesian reconstruction of free energy profiles from umbrella samples Thomas Stecher with Gábor Csányi and Noam Bernstein 21st November 212

2 Motivation Motivation and Goals Many conventional methods implicitly assume a smooth free energy Thermodynamic Integration Umbrella Integration Improve by making smoothness assumption explicit? Aim to develop an efficient technique to analyse data Try and use as much information as possible Alternative to WHAM, UI Bayesian framework: consistent way to analyse noisy data

3 Background Theory Umbrella Sampling Umbrella sampling in multiple windows Free energy in collective variable u = s(x): F (u) = 1 ln P (u) β P (u) = 1 e βv (x) δ(s(x) u)dx Z Thermal barriers make direct sampling impossible Torrie and Valleau (1974): add umbrella potential(s) w(u) to sample specific region: w(u) F (u) Repeat F (u) = F b (u) w(u) + C Complication: C will vary from window to window

4 Background Theory Gaussian Process Regression Bayes Theorem Tackles inverse probability problems: given some data, what are the model parameters? One-line derivation: P (f y)p (y) = P (y f)p (f) = P (y, f) P (f y) = P (y f)p (f) P (y) posterior = likelihood prior evidence P (y) = P (y f)p (f)df is a normalisation constant

5 Background Theory Gaussian Process Regression Gaussian Process Prior Prior distribution over functions A collection of random variables, any finite number of which have a joint Gaussian distribution Think: infinite-dimensional multivariate Gaussian Defined by a mean function, and a covariance function, m(x) = f(x) k(x 1, x 2 ) = (f(x 1 ) m(x 1 ))(f(x 2 ) m(x 2 ))

6 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σ 2 f exp( 1 2 (x 1 x 2 ) 2 /l 2 )

7 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σf 2 exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f =

8 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σf 2 exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f =

9 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σf 2 exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f =

10 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σf 2 exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f =

11 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σ 2 f exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f = 1. l = 2., σ f =

12 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σ 2 f exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f = 1. l = 2., σ f =

13 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σ 2 f exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f = 1. l = 2., σ f =

14 Background Theory Gaussian Process Regression Samples from a Gaussian Process Prior m(x) = k(x 1, x 2 ) = σ 2 f exp( 1 2 (x 1 x 2 ) 2 /l 2 ) l = 1., σ f = 1. l = 2., σ f =

15 Background Theory Gaussian Process Regression Likelihood and posterior distribution Observations with Gaussian noise: y f(x) N(f(x), Σ y ) Results in Gaussian process posterior: f(x ) = k T (x )(K + Σ y ) 1 y cov(f(x 1), f(x 2)) = k(x 1, x 2) k T (x 1)(K + Σ y ) 1 k(x 2) where (k(x )) i = k(x, x i ) K ij = k(x i, x j )

16 Background Theory Gaussian Process Regression Samples from the posterior

17 Background Theory Gaussian Process Regression Samples from the posterior

18 Background Theory Gaussian Process Regression Samples from the posterior

19 Background Theory Gaussian Process Regression Samples from the posterior

20 Background Theory Gaussian Process Regression Samples from the posterior

21 Background Theory Gaussian Process Regression Samples from the posterior

22 New methodology Histograms Learning from umbrella sampled histograms Bin probabilities biased free energies unbiased free energies F (u) = F b (u) w(u) + C Use unbiased free energies in GPR. Two complications: 1 True likelihood (noise) not Gaussian: approximate: Multinomial bin probabilities Gaussian propagate to first order to free energies 2 Need to account for unknown constants C

23 New methodology Histograms Accounting for the unknown free energy offsets Two equivalent ideas: 1 Model unknown constants explicitly with a flat (uninformative) prior: f(x ) = k T (x )Ky 1 (y H T β) cov(f(x 1), f(x 2)) = k(x 1, x 2) k T (x 1)Ky 1 k(x 2) where + k T (x 1)K 1 y β = [HK 1 y H T [HKy 1 H T ] 1 HKy 1 k(x 2) H T ] 1 HKy 1 y 2 Use free energy differences as input Linearity of difference operation crucial

24 New methodology UI Derivative information (UI) Mean forces from either constraint (TI) or restraint simulations (UI: Kästner and Thiel, 25) Harmonic umbrella: F (u) u κ i (ū i u i ) ūi Derivative (linear) of Gaussian process is another Gaussian process: k f,f (x 1, x 2 ) = f(x 1 ) f(x 2 ) x 2 = f(x 1 )f(x 2 ) = k(x 1, x 2 ) x 2 x 2 k f,f (x 1, x 2 ) = 2 x 1 x 2 k(x 1, x 2 )

25 New methodology UI Derivative information (UI) Results in posterior: k f,f (x 1, x 2 ) = k f,f (x 1, x 2 ) = x 2 k(x 1, x 2 ) 2 x 1 x 2 k(x 1, x 2 ) f(x ) = k T f,f (x )(K f,f + Σ y ) 1 y cov(f(x 1), f(x 2)) = k(x 1, x 2) k T f,f (x 1)(K f,f + Σ y ) 1 k f,f (x 2) where (k f,f (x )) i = k f,f (x, x i ) (K f,f ) ij = k f,f (x i, x j )

26 New methodology Hybrid Using derivative information in conjunction with histograms Define a joint Gaussian process of function values and its derivatives ( ) K(xy, x y ) K K f f = ff (x y, x y ) Kff T (x y, x y ) K f f (x y, x y ) Assume no correlation between function and derivative noise: Σ y y = Σ y Σ y

27 New methodology Hybrid Using derivative information in conjunction with histograms Define a joint Gaussian process of function values and its derivatives ( ) K(xy, x y ) K K f f = ff (x y, x y ) Kff T (x y, x y ) K f f (x y, x y ) Assume no correlation between function and derivative noise: Σ y y = Σ y Σ y BRUSH: Bayesian Reconstruction of free energy from Umbrella Samples using Histograms BRUSH(h) BRUSH(d) BRUSH(h+d)

28 Performance 1D 1D performance Slice of alanine dipeptide as test system Reference reconstruction: 5 x 5 ns MD umbrella strength: 1 kcal/mol Compare methods by comparing average RMS errors (over 1 reconstructions) for an optimised set of parameters: Umbrella strength: 5, 15, 25, 5, 1 kcal/mol Number of windows: 24, 12, 8 Number of bins: 2, 4, 8 (WHAM); 2-1 (GP)

29 Performance 1D 1D performance RMS error/kbt.1 BRUSH(h+d) BRUSH(h) BRUSH(d) UI, spline UI, K&T WHAM sampling time/ps

30 Performance 1D Representative reconstructions (6 ps) 3 Reference WHAM BRUSH(h+d) F/kBT -3 π π Ψ

31 Performance 1D Representative reconstructions (12 ps) 3 Reference WHAM BRUSH(h+d) F/kBT -3 π π Ψ

32 Performance 1D Representative reconstructions (12 ps) 3 Reference WHAM BRUSH(h+d) F/kBT -3 π π Ψ

33 Performance 1D Representative reconstructions (12 ps) 3 Reference WHAM BRUSH(h+d) F/kBT -3 π π Ψ

34 Performance 1D 1D Conclusions Ouperform conventional methods such as WHAM and UI Can use different types of data systematically Uncertainty estimates intrinsic to the method and accurate Smooth reconstructions

35 Performance 2D Two and more collective variables π Want to reconstruct from derivative information Conventional approach: least-squares fit of of radial basis (RB) functions Known problems with high condition numbers Ψ π π Φ π

36 Performance 2D Varying the number of windows (derivative readings) 5 ps/window RMS error/(kcal/mol) BRUSH(d) RB, opt. l RB, l = π/3 RB, l = π/3, SVD RMS noise in data number of windows

37 Performance 2D Varying the number of windows (derivative readings) π 18 = windows, 5ps 72 = windows, 5ps GP Ψ π 234 = 48 2 windows, 5ps π π Ψ RB Ψ π π Φ π π π Φ π π Φ π

38 Performance 2D Varying the simulation length RMS error/(kcal/mol) trajectory length/ps BRUSH(d) RB, l = π/3

39 Performance 2D Varying the simulation length π.25ps/window.5ps/window GP Ψ π 234 = 48 2 windows 5ps/window π π Ψ RB Ψ π π Φ π π π Φ π π Φ π

40 Conclusions and Outlook Conclusions and Outlook More efficient use of data in Bayesian framework Concurrent use of different information possible Histograms Derivatives Outperforms WHAM and UI Outperforms radial basis functions in several regimes

41 Conclusions and Outlook Conclusions and Outlook More efficient use of data in Bayesian framework Concurrent use of different information possible Histograms Derivatives Outperforms WHAM and UI Outperforms radial basis functions in several regimes Go beyond Gaussian noise approximation? Learn from MD trajectory directly? (Density estimation)