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Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition operator are useful for visualization, and they can provide an efficient basis for computing statistics, such as the likelihood and average time of events (predictions). Here, we develop inexact iterative linear algebra methods for computing these eigenfunctions (spectral estimation) and making predictions from a dataset of short trajectories sampled at finite intervals. We demonstrate the methods on a low-dimensional model that facilitates visualization and a high-dimensional model of a biomolecular system. Implications for the prediction problem in reinforcement learning are discussed. 

Transition path theory provides a statistical description of the dynamics of a reaction in terms of local spatial quantities. In its original formulation, it is limited to reactions that consist of trajectories flowing from a reactant set A to a product set B. We extend the basic concepts and principles of transition path theory to reactions in which trajectories exhibit a specified sequence of events and illustrate the utility of this generalization on examples. 

Cells dynamically control their material properties through remodeling of the actin cytoskeleton, an assembly of cross-linked networks and bundles formed from the biopolymer actin. We recently found that cross-linked networks of actin filaments reconstituted in vitro can exhibit adaptive behavior and thus serve as a model system to understand the underlying mechanisms of mechanical adaptation of the cytoskeleton. In these networks, training, in the form of applied shear stress, can induce asymmetry in the nonlinear elasticity. Here, we explore control over this mechanical hysteresis by tuning the concentration and mechanical properties of cross-linking proteins in both experimental and simulated networks. We find that this effect depends on two conditions: the initial network must exhibit nonlinear strain stiffening, and filaments in the network must be able to reorient during training. Hysteresis depends strongly and non-monotonically on cross-linker concentration, with a peak at moderate concentrations. In contrast, at low concentrations, where the network does not strain stiffen, or at high concentrations, where filaments are less able to rearrange, there is little response to training. Additionally, we investigate the effect of changing cross-linker properties and find that longer or more flexible cross-linkers enhance hysteresis. Remarkably plotting hysteresis against alignment after training yields a single curve regardless of the physical properties or concentration of the cross-linkers. 

The proteins that make up the actin cytoskeleton can self-assemble into a variety of structures. In vitro experiments and coarse-grained simulations have shown that the actin crosslinking proteins α-actinin and fascin segregate into distinct domains in single actin bundles with a molecular size-dependent competition-based mechanism. Here, by encapsulating actin, α-actinin, and fascin in giant unilamellar vesicles (GUVs), we show that physical confinement can cause these proteins to form much more complex structures, including rings and asters at GUV peripheries and centers; the prevalence of different structures depends on GUV size. Strikingly, we found that α-actinin and fascin self-sort into separate domains in the aster structures with actin bundles whose apparent stiffness depends on the ratio of the relative concentrations of α-actinin and fascin. The observed boundary-imposed effect on protein sorting may be a general mechanism for creating emergent structures in biopolymer networks with multiple crosslinkers.