The variational principles called maximum entropy (MaxEnt) and maximum caliber (MaxCal) are reviewed. MaxEnt originated in the statistical physics of Boltzmann and Gibbs, as a theoretical tool for predicting the equilibrium states of thermal systems. Later, entropy maximization was also applied to matters of information, signal transmission, and image reconstruction. Recently, since the work of Shore and Johnson, MaxEnt has been regarded as a principle that is broader than either physics or information alone. MaxEnt is a procedure that ensures that inferences drawn from stochastic data satisfy basic self-consistency requirements. The different historical justifications for the entropy S ¼ À P i p i logp i and its corresponding variational principles are reviewed. As an illustration of the broadening purview of maximum entropy principles, maximum caliber, which is path entropy maximization applied to the trajectories of dynamical systems, is also reviewed. Examples are given in which maximum caliber is used to interpret dynamical fluctuations in biology and on the nanoscale, in single-molecule and few-particle systems such as molecular motors, chemical reactions, biological feedback circuits, and diffusion in microfluidics devices.

Recent studies have shown that the diffusivity of enzymes increases in a substrate-dependent manner during catalysis1,2. Although this observation has been reported and characterized for several different systems3–10, the precise origin of this phenomenon is unknown. Calorimetric methods are often used to determine enthalpies from enzyme-catalysed reactions and can therefore provide important insight into their reaction mechanisms11,12. The ensemble averages involved in traditional bulk calorimetry cannot probe the transient effects that the energy exchanged in a reaction may have on the catalyst. Here we obtain single-molecule fluorescence correlation spectroscopy data and analyse them within the framework of a stochastic theory to demonstrate a mechanistic link between the enhanced diffusion of a single enzyme molecule and the heat released in the reaction. We propose that the heat released during catalysis generates an asymmetric pressure wave that results in a differential stress at the protein–solvent interface that transiently displaces the centre-of-mass of the enzyme (chemoacoustic effect). This novel perspective on how enzymes respond to the energy released during catalysis suggests a possible effect of the heat of reaction on the structural integrity and internal degrees of freedom of the enzyme.

Summary ATP-dependent proteases are vital to maintain cellular protein homeostasis, but the mechanisms by which these machines couple ATP hydrolysis to mechanical protein unfolding and translocation remain unclear. Here, we study the mechanisms of force generation and inter-subunit coordination in the ClpXP protease from E. coli. We identify phosphate release as the force-generating step of the ATPase cycle and reveal that ClpXP translocates substrate polypeptides by highly coordinated conformational changes in up to four ATPase subunits. To unfold stable substrates like GFP, ClpXP must use this maximum successive firing capacity. The dwell duration between individual bursts of translocation is constant and governed by an internal clock, regardless of the number of translocating subunits, implying that ClpXP operates with constant “rpm” but different “gears”.

Super-resolution microscopy provides direct insight into fundamental biological processes occurring at length scales smaller than light’s diffraction limit. The analysis of data at such scales has brought statistical and machine learning methods into the mainstream. Here we provide a survey of data analysis methods starting from an overview of basic statistical techniques underlying the analysis of super-resolution and, more broadly, imaging data. We subsequently break down the analysis of super-resolution data into four problems: the localization problem, the counting problem, the linking problem, and what we’ve termed the interpretation problem.