Coarse Graining in Conformational Space. Feliks Nüske and Cecilia Clementi Rice University Sep 14, 2017

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1 Coarse Graining in Conformational Space and Cecilia Clementi Rice University Sep 14, 2017

2 Coarse-Graining coarse-graining in conformation space coarse-graining in structural space

3 Biophysical systems are often characterized by different timescales, separated by large gaps Example: helix-coil transition coil helix typical timescales: helix-coil transition t slow µs diffusion within a state t fast ns

4 Reaction Coordinates should capture the (rare) barrier-crossing events. Courtesy of Michele Ceriotti

5 I. Optimal Reaction Coordinates

6 Basic Assumptions Dynamical Model is a Markov process X t,t 0 The process is ergodic if spatial and temporal averages coincide: E µ (f(x 0,X,...,X L )) = = lim K!1 K 1 L K L 1 X k=0 f(x k ),...,X (k+l) ) The process is reversible if there is no preferred direction: P µ (X 0 2 A, X t 2 B) =P µ (X 0 2 B,X t 2 A).

7 Basic Assumptions Dynamical Model is a Markov process X t,t 0 A distribution is stationary if it is invariant under the dynamics: P µ (X t 2 A) =µ(a) For an ergodic process, there is a unique stationary distribution.

8 Transfer Operators Transfer Operators describe the evolution of probability distributions: Z Z A [T t f](x)µ(dx) = S p t (x, A)f(x)µ(dx). Here, A is a set, µ is a measure, f is a density w.r.t. µ, and p t (x, A) is the stochastic transition function. Lasota, Mackey: Chaos, Fractals and Noise, Springer (1993)

9 Properties If µ is stationary, transfer op. are defined on all spaces L p µ. Semigroup property: T s+t = T t T s,s,t 0. For reversible processes, the transfer operators are self-adjoint on. L 2 µ Lasota, Mackey: Chaos, Fractals and Noise, Springer (1993)

10 Spectral Decomposition Using the infinitesimal Generator: Lf =lim t!0 T t f f t and its spectral components:, L j = apple j j we have the spectral mapping theorem: e (L) (T ) e (L) [ {0}, Pazy: Semigroups of linear operators and applications to partial differential equations, Springer (1983) and the spectral decomposition: T f = 1X j=1 e apple j hf, j i µ j

11 Slow Processes In many biophysical applications, we have: such that, for 0=apple 0 < apple 1 <...<apple M apple M+1 1 apple M+1 T f MX j=1 e apple j hf, j i µ j The presence of separated timescales is directly linked to the structure of the spectrum.

12 Slow Processes Using the spectral decomposition, we find for the auto-correlation function: E t (f(x t )f(x t+ )) = E µ (f(x 0 )f(x )) = ht f,fi µ 1X = e applej hf, j i 2 µ j=1

13 Variational Theorem Variational Theorem for self-adjoint operators: MX e apple j =sup j=1 j=1 MX ht f j,f j i µ, =sup MX j=1 E t (f j (X t )f j (X t+ )), hf j,f j 0i µ = j,j 0. Dominant eigenfunctions are slowest to de-correlate. They are optimal descriptors of slow dynamics. Noé and Nüske, SIAM Multiscale Model. Simul. (2013)

14 Galerkin Projection Proposition 3.2. Let D be a space of N linearly independent ansatz functions f i, i = 1,..., N. The set of M Nmutuallyorthonormalfunctionsf am, m = 1,..., M which maximize the Rayleigh trace restricted to D, is given bythefirstmeigenvectors of the generalized eigenvalue problem where the matrices C τ, C 0 are given by C τ a m = ˆλ m C 0 a m, (3.1.7) C τ (i, j) = T τ f i, f j µ (3.1.8) C 0 (i, j) = f i, f j µ. (3.1.9) Noé and Nüske, SIAM Multiscale Model. Simul. (2013)

15 Galerkin Projection By ergodicity of the process: Theorem 3.4. Let τ = L t beanintegermultipleofthediscretetimestep. Ifthe Markov process is initially distributed according to the unique invariant measure µ, then for P t -a.s. all trajectories (X k t ) k=0,wehave: C τ (i, j) = lim K 1 K L C 0 (i, j) = lim K 1 K L K L 1 k=0 K L 1 k=0 f i (X k t ) f j (X (k+l) t ), (3.2.1) f i (X k t ) f j (X k t ). (3.2.2) Noé and Nüske, SIAM Multiscale Model. Simul. (2013)

16 MSMs Markov State Models (MSM): Apply previous formalism with a basis of step functions: Schütte et. al. J. Comput. Phys. (1999) Prinz et. al. J. Chem. Phys. (2011)! c ij = 1 T i(x) = X T k=1 i(x k ) j (X k+ ) Normalize to stochastic matrix: T( ) = Schütte and Sarich, Metastability and Markov State Models in Molecular Dynamics: Modeling, Analysis, Algorithmic Approaches, Courant Lecture Notes (2013) Bowman et. al. (Eds) An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, Springer (2014) ( 1, x2 S i 0, else cij c i ij

17 MSMs Identification of metastable states: and the reaction field method for computing nonbonded forces. The simulation setup is described in detail in the SI Appendix. The simulated structures were aligned onto the native structure and then clustered finely into 1,734 kinetically connected and wellpopulated clusters. A transition matrix T(τ) was constructed by counting transitions between these clusters at a lag time of τ = 2 ns (see Theory). It was verified that T(τ) is a good model for the long-time kinetics (details on the Markov model construction and validation are given in the SI Appendix). All properties computed from the Markov model are associated with statistical uncertainty resulting from the fact that only a finite amount of simulation data has been used to construct the model. These uncertainties are computed by using a Bayesian inference method Transition Path Theory 6 h-bonds in the native state, has 3 h-bonds in the native sta the transition matrix T(τ) betw tor probabilities and the foldin in Theory. In order to obtain a view unbiased by defining reaction must be considered individua decomposed into individual p of them the times when hairp were computed. Formation average number of hydrogen b state, but variations of this th in high-dimensional spaces where binning of all relevant coordithe space spanned by the US coordinate and the second IC was discretized into 100 Voronoi cells with the k-means algorithm (Fig. nates is not an option. We first analyze pure US simulations with 150 umbrella windows 5A). This number of states is far smaller than the number of bins used to sample the position of benzamidine between the bound pose that would be required with a binned estimator such as WHAM or discrete TRAM. A count matrix ckij was estimated for every umand a prebinding site (Fig. 5A, structures i iv; details in Appendix). brella at a lag time of 11 ns, and the largest strongly To detect rare events in the unbiased coordinates, time-lagged P connected component S of the summed count matrix cij = k ckij was deindependent component analysis (TICA) (29, 30) was used with termined. The initial set was strongly disconnected, and we therethe Cartesian coordinates of residues around the binding site. fore adaptively started new umbrella simulations in nine rounds, to The first independent component (IC) is strongly correlated with improve the connectivity (Appendix). In the complete dataset, some the US coordinate. From the remaining ICs, two had timescales clusters are still disconnected (red clusters in Fig. 5A). implied by the TICA eigenvalues larger than the trajectory In particular, these disconnected states include structures in length, indicating undesirable metastable transitions orthogonal which the binding site is occluded by a tryptophan side chain, while to the umbrella coordinate. The second IC corresponds to benzamidine is still inside, and structures in which the binding site closing of the binding pocket by the Trp 215 side chain (Fig. 5D, attempts to close during the exit pathway. TRAM is applied on the structures i and iii). The third IC corresponds to an isomerization connected subset of states (white clusters in Fig. 5A). The TRAM of the disulfide bond between Cys 191 and Cys 220. An analysis results show that the Trp-occluded conformation is a local miniusing MBAR or WHAM is thus unfeasible or inefficient, as the mum inofthe free-energy landscape This is confirmed by ustrative Markov state model analysis of a pentapeptide. (a) Projection a trajectory onto the(fig. slow5c). collective coordinates (independent global equilibrium assumption is strongly violated. refs. 13 and 61 where the Trp-occluded conformation is shown to Markov model, ICs) shows rare transitions between different metastable states. (b) Implied relaxation time scales as a function of the One strategy to deal with this very common problem is to re- be a metastable conformation of the protein. In contrast, this local hows that strain a lag coordinates time τ = 1orthogonal ns is suitable. Shown coordinate, as shadedtoregions 95% confidence intervals. (c) several A Chapman Kolmogorov test shows to the umbrella avoid are minimum is not found by MBAR, and disconnected minundesirable degrees of freedom switchingon (45). Although this scales. 1 ns model accurately predicts thefrom behavior longer time (d)spuriously The free energy(boxes landscape is computed from the MSM as a ima are estimated in Fig. 5B). approach is useful for computing energy differences between end To analyze full high-dimensional binding mechanism and the two slowest ICs. The metastable states can be visually distinguished as freethe energy minima. (e) Probability distributions are given for states, it may change or restrain the transition mechanism and estimate unbiased kinetics, we must go beyond US simulations. gest livingartificially metastable states. The resulting assignment between energy minima and state numbers is shown in panel d. (f) The rate model increase free-energy barriers along the pathway. With We therefore used TRAM to combine the US data with up to from a hidden model based of the Rates areunbiased given MD in nanoseconds; 10 structures have been sampled for TICA Markov and TRAM, we now have coarse-graining the possibility to allow thesemsm. or μs of data (details in Appendix). The unbiased started the unbound state, such that many dynamics to happen,shown and to treat these events able state thogonal from the distributions in panel c. Allexplicitly. subfigurestrajectories except for the in molecular structure images havebinding been generated with Deuflhard and Weber, Linear Algebra and its Applications (2005) Multi-Ensemble MSMs: Molecular structures were rendered with VMD.134 D connected set disconnected (iii) or generalized ensemble simulations (e.g., replica-exchange MD), and combination of such data with direct MD via MSMs. Will contain estimators such as WHAM104 and TRAM.97,98,100 mtms (planned). Markov transition models.93 Continuous state kinetic models with a Markovian kernel, i.e., a means to compute the transition density p(x,y) or rate density k(x,y) between20000 two continuous60points of state space. IC2 (i) time scales 4as a function of some model parameter, (iii) representation of the kinetic model as a network. 3 (planned). Discrete state models with multiple ther(i) c states. This2 package is currently developed separately thub.com/markovmodel/thermotools) and will be 1 in PyEMMA in the near future. It will enable the 0 data from biased simulations (e.g., umbrella sampling), 10 B US coordinate / Å C 18 C IC (iv) (ii) free energy / kt US coordinate / Å US coordinate / Å 0 (ii) (iv) / cgi / doi / / pnas Noé et al. PNAS (2009) E TRAM MSM % of unbiased data used Fig. 5. Thermodynamics and kinetics of all-atom protein ligand binding model for trypsin benzamidine. (A C) US simulations. (D and E) MEMM using both unbiased and US simulations. (A) Trajectories projected on the space of the umbrella sampling (US) coordinate and the second independent component (IC). The US coordinate describes a transition from benzamidine bound to Asp-189 to benzamidine located outside the binding pocket on the surface of trypsin. The second IC corresponds to concerted opening of loop (Trp-215-Gln-221) and flipping of Trp-215. The Voronoi centers of the Markov states are shows as disks. Markov states that are irreversibly connected to the data set are shown as red disks and are excluded from the MEMM. (B) Potential of mean force (PMF) in the same coordinate space computed with unbounddoi: /acs.jctc.5b00743 J. Chem. Theory Comput. XXXX, XXX, XXX XXX 1000 probab. of finding koff A5 Wu et al. PNAS (2016)

18 Diffusion Maps Courtesy of Ralf Banisch

19 Diffusion Maps Pick a symmetric, positive kernel k. Define a Markov chain by p(x,y) = k(x,y) d(x). d(x) = X k(x,y)dµ(y) Introduce the diffusion distance: { } D t (x, y) 2 p t (x, ) p t (y, ) 2 L 2 (X,dµ/π) = By spectral expansion: ( ( D t (x, y) = ( that as { l 1 λ 2t l X ( pt (x, u) p t (y, u) ) 2 dµ(u) π(u). ( ψl (x) ψ l (y) ) ) In the following space, Euclidean distance equals diffusion dist. Ψ t (x) is constant, we have omitted the λ t 1 ψ 1(x) λ t 2 ψ 2(x).. λ t s(δ,t) ψ s(δ,t)(x). Coffman et. al. Appl. Comput. Harmon. Anal. (2006)

20 Diffusion Maps (1) Fix α R and a rotation-invariant kernel k ε (x, y) = h ( x y 2 ε (2) Let q ε (x) = k ε (x, y)q(y) dy ). X and form the new kernel k (α) ε (x, y) = k ε(x, y) q α ε (x)qα ε (y). (3) Apply the weighted graph Laplacian normalization to this kernel by setting d ε (α) (x) = k ε (α) (x, y)q(y) dy X and by defining the anisotropic transition kernel p ε, α (x, y) = k(α) ε (x, y) d (α) ε (x). Coffman et. al. Appl. Comput. Harmon. Anal. (2006)

21 Diffusion Maps Theorem 2. Let L ε,α = I P ε,α ε be the infinitesimal generator of the Markov chain. Then for a fixed K>0,wehaveonE K lim L ε,αf = (f q1 α ) ε 0 q 1 α (q1 α ) q 1 α f. For =0.5, this operator becomes the generator of a diffusion process Coffman et. al. Appl. Comput. Harmon. Anal. (2006)

22 II. Effective Dynamics on Reaction Coordinates

23 How can we define dynamics on the space of reaction coordinates? Assume the process is given by an SDE dx t = b(x t )+ 2β 1 σdb t. Then the generator is a differential operator: Lf(x) = n i=1 b i (x) f x i (x)+ 1 β n i=1 σ 2 ii 2 f x 2 i (x). Reaction coordinates are represented by smooth mapping: : R n 7! R m

24 Projection Formalism Projection operator: Pf(x) = 1 ν(z) ˆ Σ z f(x)µ(x)j 1/2 (x)dσ z (x). Projection is an orthogonal projection onto the space: H 0 = { } f L 2 µ : f = f(z) =f(ξ(x)) Zhang et. al., Faraday Discuss. (2016)

25 Projection Formalism Projected generator is again the generator of a diffusion: L ξ = PLP, m L ξ f(z) = P(Lξ l )(z) f l=1 + 1 β m l,r=1 P z l ( n i=1 σ 2 ii ξ l x i ξ r x i ) 2 f z l z r. Effective drift and diffusion can be estimated from data: e bl (z) = lim E l (x(s)) z l x(0) µ z, 1 apple l apple m, s!0+ s ea lk (z) = lim E ( l (x(s)) z l )( k (x(s)) z k ) x(0) µ z 2 s!0+ s Legoll and Lelièvre, Nonlinearity (2010) Zhang et. al., Faraday Discuss. (2016)

26 Approximation Quality Do these effective dynamics preserve the dominant timescales of the original process? If we use the optimal reaction coordinates, yes! More precisely, if the eigenfunctions only depend on z: j = j (z) the dominant timescales are exactly preserved: PLP j (z) = apple j j (z) Zhang et. al., Faraday Discuss. (2016)

27 Approximation Quality What if we only have an approximation? Proposition 1 If, for each dominant eigenfunction ψ i,i=2,...,m,thereisafunctiong i H 0 that approximates ψ i in H 1 -norm, i.e. g i ψ i 2 H 1 = g i ψ i 2 L 2 + N j=1 g i x j ψ i x j 2 L 2 ϵ2, (31) then the maximal relative error between the dominant eigenvalues of L ξ and L is bounded by max i=2,...,m ω i κ i ω i (M 1)κ 1/2 2 σ β 1 ϵ. (32) Nüske and Clementi (in preparation)

28 Numerical Example - Two-d landscape - Dynamics in x-direction is slow. - Study several rotations of the system. - Always use x as reaction coordinate. Nüske and Clementi (in preparation)

29 Numerical Example Nüske and Clementi (in preparation) Figure 36. Implied timescales of the one-dimensional effective dynamics obtained by projecting the dynamics Eq. (C9) onto the original coordinate x 0,fordifferent values of θ shown on the left. The columns correspond

30 Numerical Example II Use the polar angle as reaction coordinate. Nüske and Clementi (in preparation)

31 Numerical Example II Figure 26. Implied timescale tests for the effective dynamicssimulationsusingthedriftanddiffusion estimated at different values of the offset s (shown on top of each plot). The dashed lines correspond to the sixleading Nüske and Clementi (in preparation)

32 ACKNOWLEDGMENTS Prof. Cecilia Clementi Dr. Giovanni Pinamonti Lorenzo Boninsegna Alexander Kluber Justin Chen Eugen Hruska Our collaborators: Dr. Ralf Banisch (Freie Universität Berlin) Dr. Péter Koltai (Freie Universität Berlin) Prof. Frank Noé (Freie Universität Berlin)