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Description: We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.

Description: Spreading processes are important drivers of change in social systems. To understand the mechanisms of spreading it is fundamental to have information about the underlying contact network and the dynamical parameters of the process. However, in many real-wold examples, this information is not known and needs to be inferred from data. State-of-the-art spreading inference methods have mostly been applied to modern social systems, as they rely on availability of very detailed data. In this paper we study the inference challenges for historical spreading processes, for which only very fragmented information is available. To cope with this problem, we extend existing network models by formulating a model on a mesoscale with temporal spreading rate. Furthermore, we formulate the respective parameter inference problem for the extended model. We apply our approach to the romanization process of Northern Tunisia, a scarce dataset, and study properties of the inferred time-evolving interregional networks. As a result, we show that (1) optimal solutions consist of very different network structures and spreading rate functions; and that (2) these diverse solutions produce very similar spreading patterns. Finally, we discuss how inferred dominant interregional connections are related to available archaeological traces. Historical networks resulting from our approach can help understanding complex processes of cultural change in ancient times.

Description: Molecular simulations of ligand-receptor interactions are a computational challenge, especially when their association- (``on''-rate) and dissociation- (``off''-rate) mechanisms are working on vastly differing timescales. In addition, the timescale of the simulations themselves is, in practice, orders of magnitudes smaller than that of the mechanisms; which further adds to the complexity of observing these mechanisms, and of drawing meaningful and significant biological insights from the simulation. One way of tackling this multiscale problem is to compute the free-energy landscapes, where molecular dynamics (MD) trajectories are used to only produce certain statistical ensembles. The approach allows for deriving the transition rates between energy states as a function of the height of the activation-energy barriers. In this article, we derive the association rates of the opioids fentanyl and N-(3-fluoro-1-phenethylpiperidin-4-yl)- N-phenyl propionamide (NFEPP) in a $\mu$-opioid receptor by combining the free-energy landscape approach with the square-root-approximation method (SQRA), which is a particularly robust version of Markov modelling. The novelty of this work is that we derive the association rates as a function of the pH level using only an ensemble of MD simulations. We also verify our MD-derived insights by reproducing the in vitro study performed by the Stein Lab, who investigated the influence of pH on the inhibitory constant of fentanyl and NFEPP (Spahn et al. 2017). MD simulations are far more accessible and cost-effective than in vitro and in vivo studies. Especially in the context of the current opioid crisis, MD simulations can aid in unravelling molecular functionality and assist in clinical decision-making; the approaches presented in this paper are a pertinent step forward in this direction.