A mathematical framework for Exact Milestoning

Transcription

1 A mathematical framework for Exact Milestoning David Aristoff (joint work with Juan M. Bello-Rivas and Ron Elber) Colorado State University July 2015 D. Aristoff (Colorado State University) July / 22

2 Introduction Milestoning is a technique for estimating mean first passage times (MFPTs). Exact Milestoning is a variant which yields exact times in a certain limit. Milestoning and Exact Milestoning are both practical algorithms. Both algorithms are appropriate for systems with rough energy landscapes. Important application: in silico drug design. Estimating characteristic time for a drug to dissociate from a protein target. D. Aristoff (Colorado State University) July / 22

3 Introduction Efficiency is based on the use of short trajectories simulated in parallel. These trajectories start on a milestone and end at a neighboring milestone. The milestones are usually codimension 1 hypersurfaces. Examples: 1D milestoning: milestones = level sets of a scalar reaction coordinate Network milestoning: milestones = faces of Voronoi cell boundaries D. Aristoff (Colorado State University) July / 22

4 Introduction The problem. Let (X t ) be a stochastic dynamics and R, P disjoint subsets of state space. We want to compute the mean first passage time of (X t ) from R to P. Source and sink. When (X t ) reaches P, it immediately restarts at R. This assumption does not affect the MFPT but it is useful for theory. Examples to keep in mind: State space is a torus, (X t ) is a diffusion; State space is discrete, (X t ) is a continuous time Markov process. Below we think of R as a point, though it can be a distribution too. D. Aristoff (Colorado State University) July / 22

5 Setup Let (X t ) take values in a standard Borel space. Our goal. To efficiently compute E R [τ P ], where τ P is the first time for (X t ) to hit P. τ P P R D. Aristoff (Colorado State University) July / 22

6 Setup We coarse-grain (X t ) using closed sets called milestones. Milestones. The milestones are closed sets with pairwise disjoint interiors. τ P P R D. Aristoff (Colorado State University) July / 22

7 Setup We coarse-grain (X t ) using closed sets called milestones. Milestones. P and R are two of the milestones. M is the union of the milestones. τ P P R D. Aristoff (Colorado State University) July / 22

8 Setup Assume (X t ) is strong Markov with càdlàg paths. Jump chain. Keep the first hit points, J n, on the milestones. (J n ) is a Markov chain on M. J 4 J 3 J 5 P J 2 J 1 R = J 0 = J 6 D. Aristoff (Colorado State University) July / 22

9 Setup Assume (X t ) is strong Markov with càdlàg paths. Sojourn times. Keep the times, τ n, between the first hit points. τ n depends only on J n 1 and J n. τ 5 τ 4 P τ 3 τ 2 τ 1 τ P = τ τ 5 R D. Aristoff (Colorado State University) July / 22

10 Theory Semi-Markov viewpoint Define Y t = J n for τ τ n t < τ τ n+1, and let (Y t ) start on M. Theorem. (Y t ) is a semi-markov process on M with the same FPT to P as (X t ). Proof: This is clear from construction... Definition. Let τ P = inf{t > 0 : Y t P} and σ P = min{n 0 : J n P}. Assumption. E ξ [τ P ] and E ξ [σ P ] are finite for all initial distributions ξ on M. This ensures that (Y t ) has finite MFPTs to P and is nonexplosive. D. Aristoff (Colorado State University) July / 22

11 Theory The invariant measure Definition. Let K(x, dy) be the transition kernel for (J n ) and K(x, dy) = { K(x, dy), x / P 0, x P. K and K have left/right actions on measures/bounded functions in the usual way. Theorem. (J n ) has an invariant probability measure µ, defined by µ = Z 1 δ R K n, n=0 where Z is a normalization constant. Moreover, µ(p) > 0. Proof: Show that the residence time E R [ σ P n=0 1 J n ] is invariant. D. Aristoff (Colorado State University) July / 22

12 Theory The exact milestoning equation Definition. Let M x be the milestone containing x and τ x M = inf{t > 0 : Y t M \ int(m x )}. P τ x M x R D. Aristoff (Colorado State University) July / 22

13 Theory The exact milestoning equation Definition. Let M x be the milestone containing x and τm x = inf{t > 0 : Y t M \ int(m x )}. Theorem. With µ the invariant distribution for (J n ), we have µ(p)e R [τ P ] = E µ [τ M ] := µ(dx)e x [τm]. x M Proof: Write τ P = (τ P τ x M ) + τ x M, condition on Y τ x M = y, integrate w.r.t. µ(dy). Note that E x [τm x ] can be obtained from short trajectories running in parallel. So if we can sample efficiently from µ, we can efficiently estimate E R [τ P ]. D. Aristoff (Colorado State University) July / 22

14 Theory Convergence to stationarity Theorem. Assume (J n ) is aperiodic in the following sense: Then for any initial distribution ξ, g.c.d.{n 1 : P R (σ P = n 1) > 0} = 1. lim ξk n µ TV = 0. n Proof: Coupling argument, using the fact that R is recurrent for (J n ). This suggests how to estimate µ: start with a guess ξ = µ 0, and iterate. D. Aristoff (Colorado State University) July / 22

15 Theory Convergence to stationarity Aside: 1D milestoning. The milestones are R = M 1, M 2,..., M m = P. If J n M 1, then J n+1 M 2. And if J n M j for j = 2,..., m 1, then J n+1 M j 1 or J n+1 M j+1, both w.p. > 0. M 1 = R M 2 M 3 M 4... M m = P Here, ξk n µ TV 0 for all ξ the number of milestones is odd. Why? Say there are m milestones. Then (J n ) can go from R to P in m 1 or m + 1 steps, and m, m + 2 are coprime when m is odd. Conversely, if m is even and ξ is supported on the odd indexed milestones, then after an even (resp. odd) number of steps the distribution of (J n ) is supported on the odd (resp. even) indexed milestones. D. Aristoff (Colorado State University) July / 22

16 Theory Convergence to stationarity Error Analysis. The error in milestoning has two sources: Error in the approximation µ of µ; Error due to time discretization X nδt of (X t ). Define τ x M = min{n > 0 : line segment from X (n 1)δt to X nδt intersects M \ int(m x )}. Theorem. The error in the Milestoning approximation of the MFPT satisfies E R [τ P ] µ(r) 1 E µ [ τ M ] c 1 µ(p) 1 µ(p) 1 + µ(p) 1 (c 2 µ µ TV + φ(δt)), where φ is a function depending only on the time step error, and c 1 = E µ [τ M ], c 2 = sup E x [τm]. x x M Proof: Triangle inequalities. D. Aristoff (Colorado State University) July / 22

17 Theory Numerical error Is an iteration scheme based on powers of a numerical approximant K consistent? Theorem. (Ferré et. al., 2013) Suppose (J n ) is geometrically ergodic (GE): there exists κ (0, 1) such that sup δ x K n µ TV = O(κ n ). x M Given ɛ > 0, if K is sufficiently close to K in operator norm, then sup δ x K n µ TV = O( κ n ), µ µ TV < ɛ, x M where µ is some probability measure on M and κ (κ, 1). Lemma. (J n ) is GE if there is a uniform lower bound to reach P in n steps. Lemma. (J n ) is GE if it is strong Feller and aperiodic, and state space is compact. D. Aristoff (Colorado State University) July / 22

18 Example Example. Let (X t ) be Brownian dynamics: dx t = V (X t ) dt + 2β 1 db t, and for N = 10 let V (x 1, x 2 ) = N 1 N 1 k 1= N k 2= N C k1,k 2 f k1,k 2 (x 1, x 2 ), (1) where C k1,k 2 is either 0 or sampled uniformly from ( 1 2π, ) 1 2π, each w.p. 1 2, and 1 cos(2πk 1 x 1 ) cos(2πk 2 x 2 ), w.p. 3, 1 f k1,k 2 (x 1, x 2 ) = cos(2πk 1 x 1 ) sin(2πk 2 x 2 ), w.p. 3, 1 sin(2πk 1 x 1 ) sin(2πk 2 x 2 ), w.p. 3. V defines a rough energy landscape on the torus R 2 /Z 2. D. Aristoff (Colorado State University) July / 22

19 Example x2 x 1 Figure: Source R (center bottom), sink P (center), and other milestones (line segments). D. Aristoff (Colorado State University) July / 22

20 Example x x1 Figure: Contour map of the canonical Gibbs density Z 1 e βv. D. Aristoff (Colorado State University) July / 22

21 Example x2 µ x 1 Figure: The stationary measure µ, superimposed on contour lines of V. D. Aristoff (Colorado State University) July / 22

22 Example MFPT Iteration Figure: The MFPT vs. the number of iterations, starting at µ 0(dx) = Z 1 e βv (x) dx. D. Aristoff (Colorado State University) July / 22