In 2000, Gillespie rehabilitated the chemical Langevin equation (CLE) by describing two conditions that must be satisfied for it yield a valid approximation of the chemical master equation (CME). In this work, we construct an original path integral d
escription of the CME, and show how applying Gillespies two conditions to it directly leads to a path integral equivalent to the CLE. We compare this approach to the path integral equivalent of a large system size derivation, and show that they are qualitatively different. In particular, both approaches involve converting many sums into many integrals, and the difference between the two methods is essentially the difference between using the Euler-Maclaurin formula and using Riemann sums. Our results shed light on how path integrals can be used to conceptualize coarse-graining biochemical systems, and are readily generalizable.
Stochastic mechanics---the study of classical stochastic systems governed by things like master equations and Fokker-Planck equations---exhibits striking mathematical parallels to quantum mechanics. In this article, we make those parallels more trans
parent by presenting a quantum mechanics-like formalism for deriving a path integral description of systems described by stochastic differential equations. Our formalism expediently recovers the usual path integrals (the Martin-Siggia-Rose-Janssen-De Dominicis and Onsager-Machlup forms) and is flexible enough to account for different variable domains (e.g. real line versus compact interval), stochastic interpretations, arbitrary numbers of variables, explicit time-dependence, dimensionful control parameters, and more. We discuss the implications of our formalism for stochastic biology.
Nature is in constant flux, so animals must account for changes in their environment when making decisions. How animals learn the timescale of such changes and adapt their decision strategies accordingly is not well understood. Recent psychophysical
experiments have shown humans and other animals can achieve near-optimal performance at two alternative forced choice (2AFC) tasks in dynamically changing environments. Characterization of performance requires the derivation and analysis of computational models of optimal decision-making policies on such tasks. We review recent theoretical work in this area, and discuss how models compare with subjects behavior in tasks where the correct choice or evidence quality changes in dynamic, but predictable, ways.
Training individuals to make accurate decisions from medical images is a critical component of education in diagnostic pathology. We describe a joint experimental and computational modeling approach to examine the similarities and differences in the
cognitive processes of novice participants and experienced participants (pathology residents and pathology faculty) in cancer cell image identification. For this study we collected a bank of hundreds of digital images that were identified by cell type and classified by difficulty by a panel of expert hematopathologists. The key manipulations in our study included examining the speed-accuracy tradeoff as well as the impact of prior expectations on decisions. In addition, our study examined individual differences in decision-making by comparing task performance to domain general visual ability (as measured using the Novel Object Memory Test (NOMT) (Richler et al., 2017). Using Signal Detection Theory (SDT) and the Diffusion Decision Model (DDM), we found many similarities between expert and novices in our task. While experts tended to have better discriminability, the two groups responded similarly to time pressure (i.e., reduced caution under speed instructions in the DDM) and to the introduction of a probabilistic cue (i.e., increased response bias in the DDM). These results have important implications for training in this area as well as using novice participants in research on medical image perception and decision-making.
Cell polarization and directional cell migration can display random, persistent and oscillatory dynamic patterns. However, it is not clear if these polarity patterns can be explained by the same underlying regulatory mechanism. Here, we show that ran
dom, persistent and oscillatory migration accompanied by polarization can simultaneously occur in populations of melanoma cells derived from tumors with different degrees of aggressiveness. We demonstrate that all these patterns and the probabilities of their occurrence are quantitatively accounted for by a simple mechanism involving a spatially distributed, mechano-chemical feedback coupling the dynamically changing extracellular matrix (ECM)-cell contacts to the activation of signaling downstream of the Rho-family small GTPases. This mechanism is supported by a predictive mathematical model and extensive experimental validation, and can explain previously reported results for diverse cell types. In melanoma, this mechanism also accounts for the effects of genetic and environmental perturbations, including mutations linked to invasive cell spread. The resulting mechanistic understanding of cell polarity quantitatively captures the relationship between population variability and phenotypic plasticity, with the potential to account for a wide variety of cell migration states in diverse pathological and physiological conditions.
Cells crawling through tissues migrate inside a complex fibrous environment called the extracellular matrix (ECM), which provides signals regulating motility. Here we investigate one such well-known pathway, involving mutually antagonistic signalling
molecules (small GTPases Rac and Rho) that control the protrusion and contraction of the cell edges (lamellipodia). Invasive melanoma cells were observed migrating on surfaces with topography (array of posts), coated with adhesive molecules (fibronectin, FN) by Park et al., 2016. Several distinct qualitative behaviors they observed included persistent polarity, oscillation between the cell front and back, and random dynamics. To gain insight into the link between intracellular and ECM signaling, we compared experimental observations to a sequence of mathematical models encoding distinct hypotheses. The successful model required several critical factors. (1) Competition of lamellipodia for limited pools of GTPases. (2) Protrusion / contraction of lamellipodia influence ECM signaling. (3) ECM-mediated activation of Rho. A model combining these elements explains all three cellular behaviors and correctly predicts the results of experimental perturbations. This study yields new insight into how the dynamic interactions between intracellular signaling and the cells environment influence cell behavior.
Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I prese
nt a relatively simple and efficient, non-linear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and non-linear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have non-linear patterning regimes not previously detected. Analysis of a protein regulatory network related to chemotaxis shows its application in a more complex setting where other non-linear methods become intractable. Predictions of this method are verified against results of numerical simulation, linear stability, and full PDE bifurcation analyses.
Migrating cells possess intracellular gradients of Rho GTPases, but it is unknown whether these shallow gradients themselves can induce motility. Here we describe a new method to present cells with induced linear gradients of active, endogenous Rac w
ithout receptor activation. Gradients as low as 15% were sufficient to not only trigger cell migration up the synthetic gradient, but also to induce both cell polarization and repolarization. Response kinetics were inversely proportional to Rac gradient values, in agreement with a new mathematical model, suggesting a role for natural input gradient amplification upstream of Rac. Increases in Rac levels beyond a well-defined threshold dramatically augmented polarization and decreased sensitivity to the gradient value. The threshold was governed by initial cell polarity and PI3K activity, supporting a role for both in defining responsiveness to natural or synthetic Rac activation. Our methodology suggests a general way to investigate processes regulated by intracellular signaling gradients.
