HIGHLIGHTS• A push-pull enzyme-substrate system is ultrasensitive under enzyme saturation.• Deterministic chemical rate equations are inadequate for small substrate populations.• We adopt a probabilistic approach, starting from master equation.• Fluctuations are estimated within the linear noise approximation.• Analytical results are supported by stochastic simulations.ABSTRACT Zero-order ultrasensitivity (ZOU) is a long known and interesting phenomenon in enzyme networks. Here, a substrate is reversibly modified by two antagonistic enzymes (a 'pushpull' system) and the fraction in modified state undergoes a sharp switching from near-zero to near-unity at a critical value of the ratio of the enzyme concentrations, under saturation conditions. ZOU and its extensions have been studied for several decades now, ever since the seminal paper of Goldbeter and Koshland (1981); however, a complete probabilistic treatment, important for the study of fluctuations in finite populations, is still lacking. In this paper, we study ZOU using a modular approach, akin to the total quasi-steady state approximation (tQSSA). This approach leads to a set of Fokker-Planck (drift-diffusion) equations for the probability distributions of the intermediate enzyme-bound complexes, as well as the modified/unmodified fractions of substrate molecules. We obtain explicit expressions for various average fractions and their fluctuations in the linear noise approximation (LNA). The emergence of a 'critical point' for the switching transition is rigorously established. New analytical results are derived for the average and variance of the fractional substrate concentration in various chemical states in the near-critical regime. For the total fraction in the modified state, the variance is shown to be a maximum near the critical point and decays algebraically away from it, similar to a second-order phase transition. The new analytical results are compared with existing ones as well as detailed numerical simulations using a Gillespie algorithm.

Bacteria such as Escherichia coli move about in a series of runs and tumbles: while a run state (straight motion) entails all the flagellar motors spinning in counterclockwise mode, a tumble is caused by a shift in the state of one or more motors to clockwise spinning mode. In the presence of an attractant gradient in the environment, runs in the favourable direction are extended, and this results in a net drift of the organism in the direction of the gradient. The underlying signal transduction mechanism produces directed motion through a bi-lobed response function which relates the clockwise bias of the flagellar motor to temporal changes in the attractant concentration. The two lobes (positive and negative) of the response function are separated by a time interval of ∼ 1s, such that the bacterium effectively compares the concentration at two different positions in space and responds accordingly. We present here a novel path-integral method which allows us to address this problem in the most general way possible, including multi-step CW-CCW transitions, directional persistence and power-law waiting time distributions. The method allows us to calculate quantities such as the effective diffusion coefficient and drift velocity, in a power series expansion in the attractant gradient. Explicit results in the lowest order in the expansion are presented for specific models, which, wherever applicable, agree with the known results. New results for gamma-distributed run interval distributions are also presented.

Signal propagation over long distances is a ubiquitous feature of multicellular communities, but cell-to-cell variability can cause propagation to be highly heterogeneous. Simple models of signal propagation in heterogenous media, such as percolation theory, can potentially provide a quantitative understanding of these processes, but it is unclear whether these simple models properly capture the complexities of multicellular systems. We recently discovered that in biofilms of the bacterium Bacillus subtilis, the propagation of an electrical signal is statistically consistent with percolation theory, and yet it is reasonable to suspect that key features of this system go beyond the simple assumptions of basic percolation theory. Indeed, we find here that the probability for a cell to signal is not independent from other cells as assumed in percolation theory, but instead is correlated with its nearby neighbors. We develop a mechanistic model, in which correlated signaling emerges from cell division, phenotypic inheritance, and cell displacement, that reproduces the experimentally observed correlations. We find that the correlations do not significantly affect the spatial statistics, which we rationalize using a renormalization argument. Moreover, the fraction of signaling cells is not constant in space, as assumed in percolation theory, but instead varies within and across biofilms. We find that this feature lowers the fraction of signaling cells at which one observes the characteristic power-law statistics of cluster sizes, consistent with our experimental results. We validate the model using a mutant biofilm whose signaling probability decays along the propagation direction. Our results reveal key statistical features of a correlated signaling process in a multicellular community. More broadly, our results identify extensions to percolation theory that do or do not alter its predictions and may be more appropriate for biological systems.

Spatiotemporal pattern formation plays a key role in various biological phenomena including embryogenesis and neural network formation. Though the reaction−diffusion systems enabling pattern formation have been studied phenomenologically, the biomolecular mechanisms behind these processes have not been modeled in detail. Here, we study the emergence of spatiotemporal patterns due to simple, synthetic and commonly observed two-and three-node gene regulatory network motifs coupled with their molecular diffusion in one-and two-dimensional space. We investigate the patterns formed due to the coupling of inherent multistable and oscillatory behavior of the toggle switch, toggle switch with double self-activation, toggle triad, and repressilator with the effect of spatial diffusion of these molecules. We probe multiple parameter regimes corresponding to different regions of stability (monostable, multistable, oscillatory) and assess the impact of varying diffusion coefficients. This analysis offers valuable insights into the design principles of pattern formation facilitated by these network motifs, and it suggests the mechanistic underpinnings of biological pattern formation.

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