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The ability of cells to react to changes in their environment is critical to their survival. Effective decision making strategies leading to the activation of specific intracellular pathways hinge on cells sensing and processing extracellular variation. We will only be able to understand and manipulate how cells make decisions if we understand the “design” of the mechanisms that enable them to make such decisions, in terms of how they function, and in terms of their limitations and architecture. In this thesis, using mathematical modelling, I revisited classical signal sensing and transduction mechanisms in light of recent developments in methodological approaches and data collection. I studied the sensing characteristics of one of the simplest of sensors, the allosteric sensor, to understand the limits and effectiveness of its “design”. Using the classical Monod-Wyman-Changeux model of allostery, I defined and evaluated six engineering-inspired characteristics as a function of the parameters and number of sensors. I found that specifying one characteristic strongly constrains others and I determined the trade-offs that follow from these constraints. I also calculated the probability distribution of the number of input molecules that maximizes information transfer and, as a consequence, the number of environmental states a given population of sensors can discriminate between. Next, I proposed a new general model of phosphorylation cycles that can account for experimental reports of ultrasensitivity occurring in regimes that are far away from Goldbeter and Koshland’s zero-order saturation, the classical ultrasensitivity-generating mechanism. The new model exhibits robust ultrasensitivity in “anti-zero-order” regimes. The degree of ultrasensitivity, its limits, and its dependence on the parameters of the system are analytically tractable. The model can, additionally, explain in an intuitive way some puzzling experimental observations. Finally, I addressed the problem of integrating different types of signals from multiple sources, and performed some preliminary exploration of how cells can “learn” to associate and dissociate correlated signals in non-evolutionary time-scales. This work provides insights into the function and robustness of signal sensing and transduction mechanisms and as such is applicable to both the study of endogenous systems and the design of synthetic ones.

Entry into mitosis in eukaryotes depends on the activation of the Cyclin-dependent kinase 1 (Cdk1), which phosphorylates many mitotic protein substrates. Activation of Cdk1 requires formation of a complex with Cyclin B (CycB), which gradually rises in concentration during interphase. However, in most organisms Cdk1 activation is not gradual but switch-like, because phosphorylation of the Cdk1-CycB complex by the Wee1 kinase normally keeps Cdk1-CycB inactive during interphase. Mitotic entry is induced when rapid dephosphorylation of Cdk1-CycB by the Cdc25 phosphatase causes abrupt activation of Cdk1-CycB. Cdk1-CycB in turn phosphorylates both Wee1 and Cdc25 leading to Cdc25 activation and Wee1 inhibition. This regulation creates both a positive and a double-negative feedback loop in the system, which are thought to generate a sharp, bistable switch that controls mitotic entry. Bistability is known to require positive feedback and ultrasensitivity, however, how ultrasensitivity arises in the mitotic switch is subject to extensive research efforts both experimentally and theoretically. In this thesis I explore several possible sources of ultrasensitivity in the mitotic switch through mathematical modelling. Based on theoretical considerations and experimental evidence, I show that the existence of multiple positive feedback loops, multisite phosphorylation, and Cdk1-CycB-dependent regulation of Cdk1-counteracting phosphatase activity can all contribute to ultrasensitivity and bistability in the mitotic switch. I analyse models of the mitotic switch including these bistability-generating mechanisms, to simulate and explain experimental data and make testable predictions. I argue that it is unlikely that a single mechanism is responsible for ultrasensitivity in this system, and that bistability requires a combination of different sources, including the ones studied here and others such as enzyme saturation and sequestration effects. I also highlight the importance of network architecture and coherent regulation of opposing reactions in generating efficient biochemical switches. Finally, I draw on recent experimental evidence and ideas derived from this analysis to propose a revised network of the mitotic switch.

The phosphorelay is a ubiquitous biological module that plays a fundamental role in signal transduction and stress response coordination in organisms ranging from bacteria to plants. Despite their central role, the manner in which they integrate information is poorly understood. Furthermore, naturally occurring systems have a number of key architectural variations whose purpose remains mysterious. These variations include the number of stages in the relay, the number of proteins from which the relay is built, and the set of stages which are targeted by phosphatases and kinases. In this work, I create a unified framework for understanding the function of phosphorelays and their architectural variations. Central to this investigation are a pair of models for the phosphorelay, one of which is an Ordinary Differential Equations based model, and the other of which is a Chemical Master Equation based model. The ODE based model is used to rigorously demonstrate that a phosphorelay is monotone, and thus converges to some steady state. In turn, the steady state output is elucidated in terms of the parameters that determine the net influx and net efflux at each stage as well as the growth rate. This steady state output function provides an elaboration on a prior hypothesis which suggests that a long phosphorelay provides additional phosphoregulation targets. Specifically, we find that the output of a phosphorelay is proportional to the net influx rate divided by the sum of various products of efflux signals. In the large efflux signal limit, effluxes are effectively multiplied to generate the final output. In this way, the activity of phosphatases which act on multiple stages in the relay are multiplied, allowing the phosphorelay to act as an analog computation device for this specific function. Growth is shown to have an unexpectedly powerful effect on relay output. In the most extreme cases, the phosphorelay output is shown to obey a power law with respect to growth, with an exponent which can be as large as the length of the relay, and which is also mediated by other key architectural variations. Thus, the phosphorelay can be utilized as a device which allows an organism to select behavior by comparing its growth rate to a threshold, where the level and sharpness of this threshold can be controlled by architectural and parametric changes. These results also provide design laws for building phosphorelays which are robust to growth rate variation. Phosphorelays are also known to have a substantial effect on growth rate, and thus the relay and growth rate form a cross inhibitory loop. We show that under reasonable parametric conditions, a phosphorelay can thus be used as a hysteretic growth switch, and discuss the implications of this idea. Many of these ideas are then supported through numerical simulation of the ODE outside of the parametric conditions which allowed these ideas to be derived. They are further supported through investigation of a CME based model, and biological experiments are suggested that could validate these ideas. Finally, we discuss these results in the context of the Bacillus subtilis phosphorelay, whose output is known to affect the production of every relay protein but one. We hypothesize that this single feedback is missing in order to preserve the analog computation function.