Identification of slow molecular order parameters for Markov model construction

Abstract

A goal in the kinetic characterization of a macromolecular system is the description of its slow relaxation processes, involving (i) identification of the structural changes involved in these processes, and (ii) estimation of the rates or timescales at which these slow processes occur. Most of the approaches to this task, including Markov models, Master-equation models, and kinetic network models, start by discretizing the high-dimensional state space and then characterize relaxation processes in terms of the eigenvectors and eigenvalues of a discrete transition matrix. The practical success of such an approach depends very much on the ability to finely discretize the slow order parameters. How can this task be achieved in a high-dimensional configuration space without relying on subjective guesses of the slow order parameters? In this paper, we use the variational principle of conformation dynamics to derive an optimal way of identifying the "slow subspace" of a large set of prior order parameters -either generic internal coordinates (distances and dihedral angles), or a user-defined set of parameters. It is shown that a method to identify this slow subspace exists in statistics: the time-lagged independent component analysis (TICA). Furthermore, optimal indicators-order parameters indicating the progress of the slow transitions and thus may serve as reaction coordinates-are readily identified. We demonstrate that the slow subspace is well suited to construct accurate kinetic models of two sets of molecular dynamics simulations, the 6-residue fluorescent peptide MR121-GSGSW and the 30-residue natively disordered peptide KID. The identified optimal indicators reveal the structural changes associated with the slow processes of the molecular system under analysis.

Estimation and uncertainty of reversible Markov models

“…Thus, this prior enforces P to have the same sparsity structure as the count matrix, like the choice b i j = −1 for nonreversible sampling. Note that the reversible MLE has the same sparsity structure as can be seen from update rule (24). We will choose proposal densities q(x ′ k l |X) that are also scale-invariant.…”

“…We prepared data as follows: C α atom positions were oriented to the mean structure and saved every 10 ns, resulting in about 100 000 configurations with 174 dimensions. Time-lagged independent component analysis (TICA) 24,32 was applied to reduce this 174-dimensional space to the two dominant ICs as a spectral gap was found after the second nontrivial eigenvalue. k-means clustering with k = 100 was used to discretize this space.…”

“…The appropriate choice of discretization is a topic of ongoing research. 24,31,32 The error incurred by the discretization and by the subsequent approximation of the jump-process as a Markov process can be systematically controlled and evaluated. 7,[33][34][35] a) B. Trendelkamp-Schroer and H. Wu contributed equally to this work.…”

“…[19][20][21] A variety of complex molecular processes have been successfully described using MSMs. Examples include the folding of proteins into their native folded structure, 20,22,23 the dynamics of natively unstructured proteins, 24,25 and the binding of a ligand to a target protein. 21,[26][27][28][29][30] There are two key steps in the construction of a MSM.…”

Markov modeling of peptide folding in the presence of protein crowders

“…In this article, we use Markov modeling, along with time-lagged independent component analysis (TICA), [19][20][21][22] to analyze data from MC simulations of a test peptide in the presence of interacting protein crowders, for two different types of crowder proteins. We show that the major free-energy minima and slow dynamical modes of these high-dimensional systems can be identified in a systematic manner using TICA and Markov state models (MSMs).…”