  Markov state models (MSMs) and Master equation models are popular approaches
to approximate molecular kinetics, equilibria, metastable states, and reaction
coordinates in terms of a state space discretization usually obtained by
clustering. Recently, a powerful generalization of MSMs has been introduced,
the variational approach (VA) of molecular kinetics and its special case the
time-lagged independent component analysis (TICA), which allow us to
approximate slow collective variables and molecular kinetics by linear
combinations of smooth basis functions or order parameters. While it is known
how to estimate MSMs from trajectories whose starting points are not sampled
from an equilibrium ensemble, this has not yet been the case for TICA and the
VA. Previous estimates from short trajectories, have been strongly biased and
thus not variationally optimal. Here, we employ Koopman operator theory and
ideas from dynamic mode decomposition (DMD) to extend the VA and TICA to
non-equilibrium data. The main insight is that the VA and TICA provide a
coefficient matrix that we call Koopman model, as it approximates the
underlying dynamical (Koopman) operator in conjunction with the basis set used.
This Koopman model can be used to compute a stationary vector to reweight the
data to equilibrium. From such a Koopman-reweighted sample, equilibrium
expectation values and variationally optimal reversible Koopman models can be
constructed even with short simulations. The Koopman model can be used to
propagate densities, and its eigenvalue decomposition provide estimates of
relaxation timescales and slow collective variables for dimension reduction.
Koopman models are generalizations of Markov state models, TICA and the linear
VA and allow molecular kinetics to be described without a cluster
discretization.
