This package contains basis sets, estimators and solvers for the variational approach for conformation dynamics, a theory that has been proposed in [1] and was further developed in [2] and [3]. The variational approach is analogous to the Ritz method [4] that is employed in computational quantum chemistry. It differs in the way how the involved matrices are computed and in the meaning of the involved operators, eigenfunctions and eigenvalues - see [3] for a comparison.
Roughly, the idea of the VAC is as follows: Given a (classical) molecular dynamics trajectory with configurations {x_1, ..., x_T}, and a set of basis functions defined on the space of configurations {chi_1(x), ..., chi_n(x)}, we compute the two correlation matrices:
c_ij (0) = < chi_i(x_t) chi_j(x_t) >_t c_ij (tau) = < chi_i(x_t) chi_j(x_t+tau) >_t
where < . >_t is average over time t. Of course this can be generalized to many trajectories. Then we solve the generalized eigenvalue problem
C(tau) r = C(0) r l(tau)
where the eigenvalues l(tau) approximate the dominant eigenvalues of the Markov propagator or Markov backward propagator of the underlying dynamics. The corresponding eigenfunction of the backward propagator is approximated by
psi(x) = sum_i r_i chi_i(x)
This package aims at providing code to help addressing a number of key problems:
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Basis sets for molecular dynamics (MD), and in particular protein dynamics. See [5] for a first approach in this direction.
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Estimators for the corration matrices C(0), C(tau). The trivial time-average that is usually employed has a number of problems especially for many short simulation trajectories that are initiated far from the equilibrium distribution (the usual case!).
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Solvers for accurately solving the eigenvalue problem above, even for huge basis sets.
At this time only a few of the above functionalities are implemented and we will go step by step. This package will undergo heavy development and there is currently no date for an official release, so don't be surprised if the API (the look + feel of functions and classes) change. At the moment this package is purely intended for development purposes, so use it at your own risk.
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The time-lagged independent component analysis (TICA) method originally developed in [6] and proposed as an optimal data transformation method for building Markov state models of MD in [3,7] is a VAC with mean-free basis functions. Therefore you can easily implement TICA with this package.
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By transforming the internal coordinates such as torsion angles or interatomic distances into suitable basis functions, you can approximate experimentally-measurable relaxation timescales and determine the corresponding structural rearrangements for peptides and proteins [2,5]
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... more will follow.
[1] Noe, F. and Nueske, F. (2013): A variational approach to modeling slow processes in stochastic dynamical systems. SIAM Multiscale Model. Simul. 11, 635-655.
[2] Nueske, F., Keller, B., Perez-Hernandez, G., Mey, A.S.J.S. and Noe, F. (2014) Variational Approach to Molecular Kinetics. J. Chem. Theory Comput. 10, 1739-1752.
[3] Perez-Hernandez, G., Paul, F., Giorgino, T., De Fabritiis, G. and Noe, F. (2013) Identification of slow molecular order parameters for Markov model construction. J. Chem. Phys. 139, 015102.
[4] Ritz, W. (1909): Ueber eine neue Methode zur Loesung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math., 135, 1–61.
[5] Vitalini, F., Noé, F. and Keller, B. (2015): A basis set for peptides for the variational approach to conformational kinetics. (In review).
[6] Molgedey, L. and Schuster H. G. (1994): Phys. Rev. Lett. 72, 3634.
[7] Schwantes, C. R. and Pande, V. S. : J. Chem. Theory Comput. 9, (2013)