Abstract
Biomolecular dynamics play an important role in numerous biological processes. Markov State Models (MSMs) provide a powerful approach to study these dynamic processes by predicting long time scale dynamics based on many short molecular dynamics (MD) simulations. In an MSM, protein dynamics are modeled as a kinetic process consisting of a series of Markovian transitions between different conformational states at discrete time intervals (called "lag time"). To achieve this, a master equation must be constructed with a sufficiently long lag time to allow interstate transitions to become truly Markovian. This imposes a major challenge for MSM studies of proteins since the lag time is bound by the length of relatively short MD simulations available to estimate the frequency of transitions. Here, we show how one can employ the generalized master equation formalism to obtain an exact description of protein conformational dynamics both at short and long time scales without the time resolution restrictions imposed by the MSM lag time. Using a simple kinetic model, alanine dipeptide, and WW domain, we demonstrate that it is possible to construct these quasi-Markov State Models (qMSMs) using MD simulations that are 5-10 times shorter than those required by MSMs. These qMSMs only contain a handful of metastable states and, thus, can greatly facilitate the interpretation of mechanisms associated with protein dynamics. A qMSM opens the door to the study of conformational changes of complex biomolecules where a Markovian model with a few states is often difficult to construct due to the limited length of available MD simulations.
Highlights
Biological molecules often need to dynamically change their shapes or conformations in order to perform their functions
The Markov State Models (MSMs) for this three-state model is fully characterized by its Markovian transition count matrix, which only becomes well defined for times equal to or longer than the lag time, τT, which sets the minimal time step that allows for the dynamics of the reduced model to be described as a Markovian process
We have shown that quasi-Markov State Models (qMSMs) can be used to capture short, intermediate, and long-time dynamics of a variety of systems, including alanine dipeptide and the WW Domain Fip[35], with a memory kernel lifetime, τK, that is 5–10 times shorter than the lag time, τT, needed to obtain accurate MSM dynamics
Summary
Biological molecules often need to dynamically change their shapes or conformations in order to perform their functions. To model the millisecond folding of the NTL9 peptide, an MSM containing 2000 states (with a lag time of 12 ns) is required.[25] our work on a 37-residue intrinsically disordered peptide has shown that to allow for Markovian evolution, an MSM with as many as 10 000 states is needed.[26] While MSMs containing thousands of states are useful to make quantitative predictions to be tested against experiments, the large number of states often prevents intuitive interpretations and obscures biological insights.[27] In contrast, MSMs built using only a few states facilitate understanding and interpretation, but require very long individual simulations to reach the lag times that enable a Markovian description of the dynamics.[27]. We demonstrate that to construct qMSMs, one needs MD simulations that are 5–10 times shorter than those required by the analogous MSMs to describe the dynamics of these model systems, suggesting that the GME framework can provide an efficient approach to interrogate the long-time dynamics associated with protein folding
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