A Maximum Caliber analysis of the Foldon Hypothesis.

Different models of protein folding favor different mechanisms. Some models rely on a defined pathway, while other models rely on a more heterogeneous set of pathways. The Foldon Hypothesis is based on the concept of a defined pathway. According to the Foldon Hypothesis, protein folding is characterized by the stepwise assembly of small, cooperative units called foldons. The Maximum Caliber (Max Cal) method provides an opportunity to test this model. The Max Cal method gives the probabilities for dynamical trajectories, in much the same way that the maximum entropy principle gives the probabilities for equilibrium states. In this study, the Max Cal method was applied to folding data for the protein cytochrome c (cyt c; Hu W, Kan ZY, Mayne L, Englander SW. Proc Natl Acad Sci U S A. 2016;113:3809-3814). The overall picture to emerge from this analysis is that the data do not require a dominant, defined pathway. The folding of cytochrome c is likely a heterogeneous process that includes multiple pathways.

Critical Comparison of MaxCal and Other Stochastic Modeling Approaches in Analysis of Gene Networks.

Learning the underlying details of a gene network with feedback is critical in designing new synthetic circuits. Yet, quantitative characterization of these circuits remains limited. This is due to the fact that experiments can only measure partial information from which the details of the circuit must be inferred. One potentially useful avenue is to harness hidden information from single-cell stochastic gene expression time trajectories measured for long periods of time-recorded at frequent intervals-over multiple cells. This raises the feasibility vs. accuracy dilemma while deciding between different models of mining these stochastic trajectories. We demonstrate that inference based on the Maximum Caliber (MaxCal) principle is the method of choice by critically evaluating its computational efficiency and accuracy against two other typical modeling approaches: (i) a detailed model (DM) with explicit consideration of multiple molecules including protein-promoter interaction, and (ii) a coarse-grain model (CGM) using Hill type functions to model feedback. MaxCal provides a reasonably accurate model while being significantly more computationally efficient than DM and CGM. Furthermore, MaxCal requires minimal assumptions since it is a top-down approach and allows systematic model improvement by including constraints of higher order, in contrast to traditional bottom-up approaches that require more parameters or ad hoc assumptions. Thus, based on efficiency, accuracy, and ability to build minimal models, we propose MaxCal as a superior alternative to traditional approaches (DM, CGM) when inferring underlying details of gene circuits with feedback from limited data.

Solving Equations of Motion by Using Monte Carlo Metropolis: Novel Method Via Random Paths Sampling and the Maximum Caliber Principle.

A permanent challenge in physics and other disciplines is to solve Euler-Lagrange equations. Thereby, a beneficial investigation is to continue searching for new procedures to perform this task. A novel Monte Carlo Metropolis framework is presented for solving the equations of motion in Lagrangian systems. The implementation lies in sampling the path space with a probability functional obtained by using the maximum caliber principle. Free particle and harmonic oscillator problems are numerically implemented by sampling the path space for a given action by using Monte Carlo simulations. The average path converges to the solution of the equation of motion from classical mechanics, analogously as a canonical system is sampled for a given energy by computing the average state, finding the least energy state. Thus, this procedure can be general enough to solve other differential equations in physics and a useful tool to calculate the time-dependent properties of dynamical systems in order to understand the non-equilibrium behavior of statistical mechanical systems.

Cell responses only partially shape cell-to-cell variations in protein abundances in Escherichia coli chemotaxis.

Cell-to-cell variations in protein abundance in clonal cell populations are ubiquitous in living systems. Because protein composition determines responses in individual cells, it stands to reason that the variations themselves are subject to selective pressures. However, the functional role of these cell-to-cell differences is not well understood. One way to tackle questions regarding relationships between form and function is to perturb the form (e.g., change the protein abundances) and observe the resulting changes in some function. Here, we take on the form-function relationship from the inverse perspective, asking instead what specific constraints on cell-to-cell variations in protein abundance are imposed by a given functional phenotype. We develop a maximum entropy-based approach to posing questions of this type and illustrate the method by application to the well-characterized chemotactic response in Escherichia coli. We find that full determination of observed cell-to-cell variations in protein abundances is not inherent in chemotaxis itself but, in fact, appears to be jointly imposed by the chemotaxis program in conjunction with other factors (e.g., the protein synthesis machinery and/or additional nonchemotactic cell functions, such as cell metabolism). These results illustrate the power of maximum entropy as a tool for the investigation of relationships between biological form and function.