>> DIEP Seminar: Peter Bolhuis (U. Amsterdam)

Understanding emergent rare event behavior in high dimensional systems | 3rd of February 2022

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Microscopic dynamics governs the macroscopic behavior of complex molecular systems, from materials to living cells. To predict this emerging behavior, physics-based atomistic models combined with well-established equations of motion provide a high dimensional dynamical system, which, given an initial condition, can be time-integrated to yield the long-time behavior for the system, and hence its macroscopic properties. In principle, one could use accurate quantum mechanics for this time evolution, but in all but a few cases this is utterly unfeasible and intractable. Fortunately, coarse-graining in time and/or space, neglecting or integrating out degrees of freedom leads to simpler descriptions, which make the problem tractable.  This leads to molecular dynamics (MD) based on effective (empirical) force fields, which can be designed to reproduce experimental or (quantum mechanically) computed observables. However, even in such empirical MD simulations timescales can be quite long, caused by the presence of dynamical bottlenecks between metastable states.  Exploring these transitions by direct MD is extremely costly, for example in the case of protein folding or nucleation.  Further coarse graining towards simpler force fields could solve this problem, but then important molecular details might get lost. A better option is to coarse-grain in time and make use of the emergent Markovianity of the dynamics, which leads to a probabilistic Markov state description. Still, this requires knowledge of the state-to-state transition rates that are hard to obtain by direct MD as these transitions occur very rarely on the timescale of the simulation. This rare event problem can be solved by computing the probability to find the system at the dynamical bottleneck, the transition state, by slowly restraining the system towards it. This in turn requires a way to identify this transition state by a collective variable or reaction coordinate (sometimes also called the importance function). This very important reaction coordinate (RC) is ideally  a low dimensional representation of the high-dimensional space. Finding it requires a dimensionality reduction of the phase space, but is often elusive. In fact, it is kind of a chicken and egg problem: To study the rare event, one needs the RC, but to find the RC one needs to observe the rare event first.  

A solution to this problem is to use the concept of trajectory sampling. One can sample trajectories between easily identifiable metastable states, and extract the RC from the resulting path ensemble by a dimensionality reduction, e.g. using machine learning.  The thus obtained knowledge can then be used to better describe the kinetics and dynamics in complex molecular systems.   

Of course, this approach still relies on accurate force fields. Recent work demonstrated that it is possible to correct for force field inaccuracies by incorporating experimental kinetic constraints into the computed trajectory ensemble, through the use of the maximum (path) entropy principle, even further improving our understanding of rare events in complex systems.