Remarks on a population-level model of chemotaxis: advection-diffusion approximation and simulations

This note works out an advection-diffusion approximation to the density of a population of E. coli bacteria undergoing chemotaxis in a one-dimensional space. Simulations show the high quality of predictions under a shallow-gradient regime

A Characterization of Scale Invariant Responses in Enzymatic Networks

An ubiquitous property of biological sensory systems is adaptation: a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. Recently, it was shown theoretically and experimentally that many adapting systems, both at the organism and single-cell level, enjoy a remarkable additional feature: scale invariance, meaning that the initial, transient behavior remains (approximately) the same even when the background signal level is scaled. In this work, we set out to investigate under what conditions a broadly used model of biochemical enzymatic networks will exhibit scale-invariant behavior. An exhaustive computational study led us to discover a new property of surprising simplicity and generality, uniform linearizations with fast output (ULFO), whose validity we show is both necessary and sufficient for scale invariance of enzymatic networks. Based on this study, we go on to develop a mathematical explanation of how ULFO results in scale invariance. Our work provides a surprisingly consistent, simple, and general framework for understanding this phenomenon, and results in concrete experimental predictions

An observability result related to active sensing

For a general class of translationally invariant systems with a specific category of nonlinearity in the output, this paper presents necessary and sufficient conditions for global observability. Critically, this class of systems cannot be stabilized to an isolated equilibrium point by dynamic output feedback. These analyses may help explain the active sensing movements made by animals when they perform certain motor behaviors, despite the fact that these active sensing movements appear to run counter to the primary motor goals. The findings presented here establish that active sensing underlies the maintenance of observability for such biological systems, which are inherently nonlinear due to the presence of the high-pass sensor dynamics

Scale-invariant systems realize nonlinear differential operators

In recent years, several biomolecular systems have been shown to be scale-invariant (SI), i.e. to show the same output dynamics when exposed to geometrically scaled input signals (u → pu, p > 0) after pre-adaptation to accordingly scaled constant inputs. In this article, we show that SI systems-as well as systems invariant with respect to other input transformations-can realize nonlinear differential operators: when excited by inputs obeying functional forms characteristic for a given class of invariant systems, the systems' outputs converge to constant values directly quantifying the speed of the input

Fold-change detection in a whole-pathway model of Escherichia coli chemotaxis

There has been recent interest in sensory systems that are able to display a response which is proportional to a fold change in stimulus concentration, a feature referred to as fold-change detection (FCD). Here, we demonstrate FCD in a recent whole-pathway mathematical model of Escherichia coli chemotaxis. FCD is shown to hold for each protein in the signalling cascade and to be robust to kinetic rate and protein concentration variation. Using a sensitivity analysis, we find that only variations in the number of receptors within a signalling team lead to the model not exhibiting FCD. We also discuss the ability of a cell with multiple receptor types to display FCD and explain how a particular receptor configuration may be used to elucidate the two experimentally determined regimes of FCD behaviour. All findings are discussed in respect of the experimental literature

Zeros of nonlinear systems with input invariances

A nonlinear system possesses an invariance with respect to a set of transformations if its output dynamics remain invariant when transforming the input, and adjusting the initial condition accordingly. Most research has focused on invariances with respect to time-independent pointwise transformations like translational-invariance (u(t) -> u(t) + p, p in R) or scale-invariance (u(t) -> pu(t), p in R>0). In this article, we introduce the concept of s0-invariances with respect to continuous input transformations exponentially growing/decaying over time. We show that s0-invariant systems not only encompass linear time-invariant (LTI) systems with transfer functions having an irreducible zero at s0 in R, but also that the input/output relationship of nonlinear s0-invariant systems possesses properties well known from their linear counterparts. Furthermore, we extend the concept of s0-invariances to second- and higher-order s0-invariances, corresponding to invariances with respect to transformations of the time-derivatives of the input, and encompassing LTI systems with zeros of multiplicity two or higher. Finally, we show that nth-order 0-invariant systems realize – under mild conditions – nth-order nonlinear differential operators: when excited by an input of a characteristic functional form, the system’s output converges to a constant value only depending on the nth (nonlinear) derivative of the input