Reac%on coordinates and LSDMap

Transcription

1 Reac%on coordinates and LSDMap

2 The challenges in molecular biophysics: a broad range of interconnected length and %me scales Organism Cell E.Coli ~10 20 atoms ~10 10 atoms ~1-10 µm Thermodynamics Macroscale Thermodynamics Mesoscale System ~ atoms ~ nm Mesoscale Mul%scale Biomolecule (Macromolecule) ~ atoms ~ 1-10 nm Mesoscale Mul%scale Molecule ~10 1 atoms ~ 1-10 Ǻ Quantum Chemistry Atom ~1 atom ~ 1 Ǻ Quantum Physics

3 What we study Dynamics Function 10 9 Cells Number of atoms Systems Macromolecules Molecules Atoms Length scale (m)

4 Way do we care about dynamics? func&on requires dynamics! from

5 Main goals and challenges Trajectories in equilibrium distribution Observables to compare with the experiment Mechanism from the dynamics New predictions dd 1. Force Field a set of parameters and equations describing the interactions between atoms ? 2. Sampling Can the simulations cover the rare events that we are interested in? 3. Data analysis How do we understand the mechanism in the data?

6 Protein Representa%ons Cartesian coordinate representa%on x, y, z coordinate for each single atom Internal coordinate representa%on: b - bond length α - angle between two consecu%ve bonds θ - angle between three consecu%ve bonds Idealized geometry model dihedral angles - only DOFs 2 backbone dihedrals 4 sidechain dihedrals

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8 All- Atom Molecular Mechanics Force- Field A biomolecule is considered a collec%on of masses (atoms) connected by springs (bonds). The associated effec%ve energy (including electronic effects) is parameterized in a classical force- field. Generally, a force- field is in the form:

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11 What we are interested in.

12 What is relevant and what is noise? in between bulk water one water molecule quantum chemistry gives molecular orbitals Large water clusters Wet/Dry interfaces Interac%on with solutes what are the relevant variables? what is the intrinsic dimensionality? thermodynamics describes the system What is the minimal set of variables needed to describe realis%cally the dynamics of a macromolecule? C.Clemen%, Curr. Opin. Struct. Biol. 2008, vol.18(1), 10-15

13 The problem of reac%on coordinates?

14 The problem of reac%on coordinates Reac%on Coordinates iden%fy the minimum free energy path along which the reac%on take place, and allow to locate reactants, products, and transi%on states Example: H + D 2 O Figures from B.R.Strazisar et al., Science 290, 958(2000)

15 How to choose good reac%on coordinates Physical- relevant variables Q, Rg, z automated ways: gene%c neural network, maximum likelihood a priori knowledge of the system Iso- commitor based on some other reac%on coordinate Dimensionality reduc%ons? linear dimensionality reduc%on (PCA, MDS) usually around equilibrium nonlinear dimensionality reduc%on Isomap, Diffusion Map, Sketch- map require correla%on to the physical- relevant variables

16 Reaction coordinates Gauge the progress of a reaction Cluster the (meta)stable states Preserve the barrier height Free Energy reaction coordinate Physical (intuitive) collective variables C5 Iso-commitor (Pfold) αp αr αl Dimensionality reduction?

17 The problem of reac%on coordinates p- fold ( commitment probability, isocommidor ): probability for a path star%ng in a par%cular point on the landscape to visit the folded state before the unfolded state 50% 50% Folded Unfolded

18 Empirical reac%on coordinates: how good are they? Folding trajectory of CI2 (Go- like model) at T f C.Clemen%, P.A. Jennings, J.N. Onuchic J.Mol.Biol. 311, (2001)

19 Is p- fold the ul%mate reac%on coordinate? S.S. Cho, Y. Levy, P.G. Wolynes, P versus Q: Structural reac:on coordinates capture protein folding on smooth landscapes PNAS 103:3, (2006)

20 Mathema%cally this is a problem of non- linear dimensionality reduc%on Example: a set of points on a torus in 3d defines a 2d embedded surface

21 Reduc%on coordinates for macromolecular mo%ons The problem consists in finding the best low dimensional descrip%on of a collec%on of macromolecular conforma%ons? Main mo%on 2 Main mo%on 1 We need to introduce a measure of similarity between configura%ons

22 Similar problems arise in different research fields (computer science, engineering, applied math, sta%s%cs, biology, ) Examples: classifica%on of documents, image recogni%on Courtesy of M.Maggioni

23 Linear dimensionality reduc%on: Principle component analysis (PCA) Pearson, Philos. Mag, 2, 559, 1901

24 Idea: Use geodesics to define the surface A geodesic is the shortest path between two points If we know the geodesics between any couple of points then we know everything about the geometry of the system Isomap algorithm - Tenenbaum, de Silva, & Langford, (2000) Science 290,

25 Basic ideas of ISOMAP Define a low- dimensional hyper- surface preserving as best as possible geodesic distances between all pairs of data points in the sample Geodesic distance and Euclidean distance

26 Use geodesics to define the manifold Network of nearest neighbors can serve to approximate geodesics between any pairs of points P. Das, M. Moll, H. Stama:, L.E. Kavraki, & C.Clemen: Proc. Natl. Acad. Sci. USA 103, (2006)

27 Applica%on of nonlinear dimensionality reduc%on to SH3 folding dynamics ScIMAP Na%ve structure of SH3 protein P. Das, M. Moll, H. Stama:, L.E. Kavraki, & C.Clemen: Proc. Natl. Acad. Sci. USA 103, (2006)

28 Applica%on of nonlinear dimensionality reduc%on to SH3 folding dynamics Free energy as a func%on of the first collec%ve coordinate P. Das, M. Moll, H. Stama:, L.E. Kavraki, & C.Clemen: Proc. Natl. Acad. Sci. USA 103, (2006)

29 Applica%on of nonlinear dimensionality reduc%on to SH3 folding dynamics U unfolded state N na%ve state TS transi%on state P. Das, M. Moll, H. Stama:, L.E. Kavraki, & C.Clemen: Proc. Natl. Acad. Sci. USA 103, (2006)

30 The limits of Isomap Example: Polymer reversal inside a narrow pore Huang and Makarov, J. Chem. Phys (2008) Empirical reac%on coordinate: distance between the first bead and the last bead projected on the z direc%on, z = z N - z 1 Is z a good reac%on coordinate to describe the polymer reversal dynamics?

31 How can we es%mate the reversal rate? 1. Direct method ( experiment ) Measuring the wai%ng %me between reversal events: p(t) dt α e - kt dt 2. Transi%on state theory If we have an accurate free energy profile, we can es%mate the rate: Huang and Makarov, J. Chem. Phys (2008)

32 Transi%on state theory Trajectories from simula%ons Weighted Histogram Analysis Method (WHAM) to get free energy profile as a func%on of the reac%on coordinate Transi%on State Theory (TST) to get the reversal rate Huang and Makarov, J. Chem. Phys (2008)

33 Recrossing of the transi%on events TST overes%mates the rate constant k Transmission factor Langevin equa%on Kramers theory

34 Comparison TST with κ Kramer both overes%mate the rate constant k if reac%on coordinate is not well chosen Huang and Makarov, J. Chem. Phys (2008) Is z a good reac%on coordinate to describe the transloca%on of polymer inside the pore? No.

35 Rate constant as obtained by using the 1 st ISOMAP coordinate S%ll huge gap between ISOMAP and sta%s%cal results The geodesic distance is not the best way to describe the dynamics of polymer reversal inside the pore.

36 Sketch- map RMSD distribu%on: Short range matches gaussian noise and long range matches uniformly- distributed points for alanine- 12. Define a set of coordinates best preserve the distances in the medium range. Minimize the sum of differences between the sigmoid func%ons (F and f) of distances in high and low dimensional space. Cerio^, Tribello & Parrinello, Proc. Nat. Acad. Sci. (USA), 108, 13023, 2011

37 Similarity measure Mul%dimensional scaling: Euclidean distance (black) Isomap: Geodesic distance (purple) Diffusion map: Diffusion distance

38 Diffusion Map If the data {x} are obtained from the sampling of a diffusion process with a poten%al energy func%on E(x), the associated probability distribu%on p(x,t) is expected to sa%sfy the Fokker- Planck equa%on: A natural distance measure can be defined on the data It measures how easily x 0 and x 1 transform into each other RR Coifman, S Lafon, A Lee, M Maggioni, B Nadler, FJ Warner, and SW Zucker, Proc. of Natl. Acad. Sci. USA, 102, , 2005

39 Diffusion Map The Fokker- Planck equa%on has a discrete eigenvalue spectrum 0 = λ 0 < λ 1 < λ 2 < λ 3. Boltzmann distribu%on (equilibrium) > 0 eigenfunc%ons If there is a separa%on of %mescales: λ k << λ k+1 Diffusion distance = 0 GOOD REACTION COORDINATES

40 Diffusion Map A discrete approxima%on of these eigenvalues and eigenvectors can be obtained by considering the kernel: eigenvalues and eigenfunc%ons of M are the discrete approxima%on of R. R. Coifman, S. Lafon, A.B. Lee, M. Maggioni, B. Nadler, F. Warner, S.W. Zucker Geometric diffusions as a tool for harmonic analysis and structure defini:on of data: Diffusion maps Proc.Natl.Acad.Sci.USA 102(21) , 2005

41 Idea of local scale The spread of Gaussian distribu%on is different for different points, which tells the level of the flatness of the manifold. Within the local scale, the manifold should be approximately flat. Only noises in green circle. Curvature in blue circle. Jung, Licle and Maggioni, Proc. AAAI, (2009) Rohrdanz, Zheng, Maggioni and Clemen:. J Chem. Phys., 134(12), (2011)

42 Determina%on of the local scale Find the smallest local scale above the noise in which PCA captures the dynamics reasonably well. PCA on increasing scales Local Principle Component Analysis Local intrinsic dimensionality the k- th neighbor Noise Jung, Licle and Maggioni, Proc. AAAI, (2009) Rohrdanz, Zheng, Maggioni and Clemen:. J Chem. Phys., 134(12), (2011)

43 Locally scaled diffusion map (LSDMap) RMSD To extract Point- specific local scale from a discrete data set (i.e. molecular dynamics data) Rohrdanz, Zheng, Maggioni and Clemen:. J Chem. Phys., 134(12), (2011)

44 Diffusion eigenspectrum Sta%onary solu%on Pore radius large gap %mescale separa%on Zheng, Rohrdanz, Maggioni and Clemen:, J Chem. Phys., (2011)

45 Free energy landscape

46 Local PCA spectra Minimum Barrier

47 Local scales Different regions of the configura%on space have different local scales.

48 Local heterogenei%es Barrier: Large spectra gap, small intrinsic dimension, large local scale Minimum: Small spectra gap, large intrinsic dimension, small local scale

49 We test the goodness of the first diffusion coordinate as reac%on coordinate by es%ma%ng the rates From Kramers theory of escape rates we have: Free Energy Escape rate D(x) = diffusion coefficient, it s NOT a constant Reac%on Coordinate, x

50 Reversal rate Zheng, Rohrdanz, Maggioni and Clemen:, J Chem. Phys., (2011)

51 Z vs. 1 st DC Zheng, Rohrdanz, and Clementi, J Chem. Phys., 134, (2011) Z 1 st DC = 0 surface is different from z=0 surface. 1 st DC

52 Correla%on to the contact probabili%es z 1 st DC

53 LSDMap code h4p://sourceforge.net/projects/lsdmap in Fortran90 and MPI

54 Clemen& s group Dr. Mary Rohrdanz Dr. Jordane Preto Lorenzo Boninsegna Wenwei Zheng Fernando Yrazu Alex Kluber Amarda Shehu (now: GMU) Payel Das (now: IBM) Silvina Matysiak (now: U. Maryland) Brad Lambeth (now: Shell) Collaborators: Prof. Mauro Maggioni Miles Crosskey (Duke Math) $$ NSF CHE CHE $$ Welch Founda%on C- 1570