The hidden Markov model (HMM) has been a workhorse of single molecule data analysis and is now commonly used as a standalone tool in time series analysis or in conjunction with other analyses methods such as tracking. Here we provide a conceptual introduction to an important generalization of the HMM which is poised to have a deep impact across Biophysics: the infinite hidden Markov model (iHMM). As a modeling tool, iHMMs can analyze sequential data without a priori setting a specific number of states as required for the traditional (finite) HMM. While the current literature on the iHMM is primarily intended for audiences in Statistics, the idea is powerful and the iHMM's breadth in applicability outside Machine Learning and Data Science warrants a careful exposition. Here we explain the key ideas underlying the iHMM with a special emphasis on implementation and provide a description of a code we are making freely available. In a companion article, we provide an important extension of the iHMM to accommodate complications such as drift.

A unified stochastic formulation of dissipative quantum dynamics. I. Generalized hierarchical equations The Journal of Chemical Physics 148, 014103 (2018) We review here Maximum Caliber (Max Cal), a general variational principle for inferring distributions of paths in dynamical processes and networks. Max Cal is to dynamical trajectories what the principle of maximum entropy is to equilibrium states or stationary populations. In Max Cal, you maximize a path entropy over all possible pathways, subject to dynamical constraints, in order to predict relative path weights. Many well-known relationships of non-equilibrium statistical physics-such as the Green-Kubo fluctuation-dissipation relations, Onsager's reciprocal relations, and Prigogine's minimum entropy production-are limited to near-equilibrium processes. Max Cal is more general. While it can readily derive these results under those limits, Max Cal is also applicable far from equilibrium. We give examples of Max Cal as a method of inference about trajectory distributions from limited data, finding reaction coordinates in bio-molecular simulations, and modeling the complex dynamics of non-thermal systems such as gene regulatory networks or the collective firing of neurons. We also survey its basis in principle and some limitations.

Single-molecule localization microscopy has the ability to measure spatial proximity between individual molecules with tens of nanometers precision. Extracting meaningful biological results, however, requires fully characterizing the distribution of molecular behaviors, which in turn, necessitates analyzing large numbers of individual measurements. Making large numbers of replicate measurements in a single imaging session has been made possible in recent years by large area detectors that afford an ultrawide field-of-view as well as fast frame rates. A remaining barrier to ultrawide-field imaging is that optical aberrations become pronounced when imaging far away from the central optical axis, which can compromise the precision and accuracy of point-spreadfunction (PSF) fitting across the field-of-view. Here, we present a computational phase retrieval routine based on vectorial PSF models to account for the spatially-variant aberrations in two color channels of a 3D singlemolecule localization microscope. By computationally correcting the aberrations during data post-processing, we are able to localize emitters in an ultrawide filed-of-view with improved precision and accuracy compared to approaches based on analytical PSF models. The use of a spatially-variant PSF model enables accurate emitter localization in x, y and z over the entire field-of-view, so that the reconstructed super-resolution images and singlemolecule trajectories accurately reproduce the relative spatial arrangement among all localized emitters.

We present a unified conceptual framework and associated software package for single molecule Förster Resonance Energy Transfer (smFRET) analysis from single photon arrivals leveraging Bayesian nonparametrics, BNP-FRET. This unified framework addresses the following key physical complexities of an smFRET experiment, including: 1) fluorophore photophysics; 2) continuous time dynamics of the labeled system with large timescale separations between photophysical phenomena such as excited photophysical state lifetimes and events such as transition between system states; 3) unavoidable detector artefacts; 4) background emissions; 5) unknown number of system states; and 6) both continuous and pulsed illumination. These physical features necessarily demand a novel framework that extends beyond existing tools. In particular, we propose a second order hidden Markov model (HMM) and Bayesian nonparametrics (BNP) on account of items 1, 2 and 5 on the list, respectively. In companion manuscripts II and III, we discuss the direct effects of these key complexities on the inference of parameters for continuous and pulsed illumination, respectively.

Complex feedback systems are ubiquitous in biology. Modeling such systems with mass action laws or master equations requires information rarely measured directly. Thus rates and reaction topologies are often treated as adjustable parameters. Here we present a general stochastic modeling method for small chemical and biochemical systems with emphasis on feedback systems. The method, Maximum Caliber, is more parsimonious than others in constructing dynamical models requiring fewer model assumptions and parameters to capture the effects of feedback. Maximum Caliber is the dynamical analog of Maximum Entropy. It uses average rate quantities and correlations obtained from short experimental trajectories to construct dynamical models. We illustrate the method on the bistable genetic toggle switch. To test our method, we generate synthetic data from an underlying stochastic model. MaxCal reliably infers the statistics of the stochastic bistability and other full dynamical distributions of the simulated data, without having to invoke complex reaction schemes. The method should be broadly applicable to other systems.