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Markov state models (MSMs) and Master equation models are popular approachesto approximate molecular kinetics, equilibria, metastable states, and reactioncoordinates in terms of a state space discretization usually obtained byclustering. Recently, a powerful generalization of MSMs has been introduced,the variational approach (VA) of molecular kinetics and its special case thetime-lagged independent component analysis (TICA), which allow us toapproximate slow collective variables and molecular kinetics by linearcombinations of smooth basis functions or order parameters. While it is knownhow to estimate MSMs from trajectories whose starting points are not sampledfrom an equilibrium ensemble, this has not yet been the case for TICA and theVA. Previous estimates from short trajectories, have been strongly biased andthus not variationally optimal. Here, we employ Koopman operator theory andideas from dynamic mode decomposition (DMD) to extend the VA and TICA tonon-equilibrium data. The main insight is that the VA and TICA provide acoefficient matrix that we call Koopman model, as it approximates theunderlying dynamical (Koopman) operator in conjunction with the basis set used.This Koopman model can be used to compute a stationary vector to reweight thedata to equilibrium. From such a Koopman-reweighted sample, equilibriumexpectation values and variationally optimal reversible Koopman models can beconstructed even with short simulations. The Koopman model can be used topropagate densities, and its eigenvalue decomposition provide estimates ofrelaxation timescales and slow collective variables for dimension reduction.Koopman models are generalizations of Markov state models, TICA and the linearVA and allow molecular kinetics to be described without a clusterdiscretization.

Variational Koopman models: slow collective variables and molecular kinetics from short off-equilibrium simulations