Foldamer dynamics expressed via Markov state models. II. State space decomposition

Abstract

The structural landscape of poly-phenylacetylene (pPA), otherwise known as m-phenylene ethynylene oligomers, has been shown to consist of a very diverse set of conformations, including helices, turns, and knots. Defining a state space decomposition to classify these conformations into easily identifiable states is an important step in understanding the dynamics in relation to Markov state models. We define the state decomposition of pPA oligomers in terms of the sequence of discretized dihedral angles between adjacent phenyl rings along the oligomer backbone. Furthermore, we derive in mathematical detail an approach to further reduce the number of states by grouping symmetrically equivalent states into a single parent state. A more challenging problem requires a formal definition for knotted states in the structural landscape. Assuming that the oligomer chain can only cross the ideal helix path once, we propose a technique to define a knotted state derived from a helical state determined by the position along the helical nucleus where the chain crosses the ideal helix path. Several examples of helical states and knotted states from the pPA 12-mer illustrate the principles outlined in this article.

Estimation and uncertainty of reversible Markov models

“…As a result, they are an excellent tool to integrate the data of multiple simulation trajectories that have been run independently and from different initial states into a single informative model. [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.…”

Using Markov state models to study self-assembly

“…[46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93] In cases where a set of collective coordinates describing all of the slow degrees of freedom is known (or guessed) a priori, biased sampling can be used to determine the free energy landscape as a function of these coordinates. States can then be defined based on local free energy minima (e.g., Refs.…”

Maximum margin clustering for state decomposition of metastable systems

“…The first relaxation allows elements of M and D to take values in [0, 1] instead of {0, 1} and the second relaxation is to replace M = DD with a convex inequality M DD . By using these two relaxations, (16) can be relaxed to a convex optimization problem:…”

“…(The first implied timescale ITS i (τ ) ≡ ∞) It can be proved that the value of ITS i (τ ) is a constant independent of τ and equal to the dominant relaxation timescales of the original system if the transitions between metastable states are exactly Markovian [12,16]. Thus, we can check if a Markov chain on metastable states can accurately approximate the system dynamics through comparing implied timescales with different τ .…”

“…For some low-dimensional systems, metastable states can be found manually by exploring their energy landscapes (see e.g., [13][14][15][16]). But for high-dimensional and complex systems, the intuitive partitioning of the whole phase space is generally infeasible, and the metastable state decomposition can only be performed through statistical analysis of the simulation or experimental data.…”