Upper-Order TICA and Fractional Non-Markovian Process to Model Anomalous Dynamic Regimes

The coupling of time-lagged independent component analysis (TICA) with the Markov state model (MSM) technique has become a well-established route to study dynamics in complex molecular systems. Identification of the slow modes relevant to the molecular functions, quantification of the characteristic times involved in the slow dynamics, and prediction of dynamic properties are the basic frame of application of such methods. Among the current research developments in the field, great activity is devoted to the formulation of methods to improve approximation of the leading eigenfunctions of the transfer operator of a dynamical system from trajectory data and to include memory effects into MSM analysis. Along these lines of research, various developments are proposed here, in the framework of TICA-MSM approaches: a criterion to select dynamically informative intramolecular distances and a method to use them to build optimal nonlinear basis sets for TICA (upper-order TICA) are presented to overcome the limitations of linear approximations to the transfer operator eigenfunctions. Then, a fractional, non-Markovian process is introduced to deal with anomalous dynamic regimes characterized by nonexponential relaxations. The fractional process is described in terms of a time derivative of noninteger order α > 0 in the master equation of the temporal evolution of the states' probabilities, which replaces the exponential decay in time, typical of Markovian processes, with Mittag-Leffler functions in the temporal variable. This kind of temporal dependency is more appropriate to capture the characteristics of anomalous dynamics, often observed in proteins and peptides by both experiments and simulations. The theory is cast in a form that the researchers are familiar with when applying MSM analysis, allowing direct manipulations over the transition probability matrix. Moreover, the technique allows us to check whether the dynamics encoded in the molecular dynamics (MD) trajectory occur in an anomalous regime or not, and, in case, permits to quantify and treat the anomaly by identifying the appropriate fractional order α of the non-Markovian process. Purely Markovian dynamic regimes are special cases of the proposed theory and can be recovered by letting α = 1. The benchmark MD trajectories of chignolin (a), villin (b), and Trp-cage (c) proteins, provided by D.E. Shaw Research (DESRES), are revisited in light of the proposed developments, and cases (a) and (c) show that the ability to describe the system dynamics in terms of fractional non-Markovian processes is necessary to obtain a more accurate qualitative and quantitative picture of molecular dynamics occurring in anomalous regimes.