Рассмотрены две пространственные модели реакции–диффузии, найдены параметрические зоны диффузионной неустойчивости. Исследованы сценарии детерминированной и стохастической генерации паттернов. Показано, что обе модели мультистабильны с большим количеством сосуществующих аттракторов–паттернов. Под воздействием шума возможны переходы между стационарными неоднородными состояниями, а также генерация паттернов в зоне диффузионной устойчивости. / Two spatial reaction-diffusion models are considered, and parametric zones of diffusion instability are found. Scenarios of deterministic and stochastic pattern formation are investigated. It is shown that both models are multistable with a large number of coexisting attractor-patterns. Under the influence of noise, transitions between stationary inhomogeneous states are possible, as well as the generation of patterns in the zone of diffusion stability.

This dissertation is concerned with the potential multistability of protein concentrations in the cell that can arise in biochemical networks. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. Models of multisite protein phosphorylation have shown that this mechanism is able to exhibit unlimited multistability. Nevertheless, these models have not considered enzyme docking, the binding of the enzymes to one or more substrate docking sites, which are separate from the motif that is chemically modified. Enzyme docking is, however, increasingly being recognised as a method to achieve specificity in protein phosphorylation and dephosphorylation cycles. Most models in the literature for these systems are deterministic i.e. based on Ordinary Differential Equations, despite the fact that these are accurate only in the limit of large molecule numbers. For small molecule numbers, a discrete probabilistic, stochastic, approach is more suitable. However, when compared to the tools available in the deterministic framework, the tools available for stochastic analysis offer inadequate visualisation and intuition. We firstly try to bridge that gap, by developing three tools: a) a discrete `nullclines' construct applicable to stochastic systems - an analogue to the ODE nullcines, b) a stochastic tool based on a Weakly Chained Diagonally Dominant M-matrix formulation of the Chemical Master Equation and c) an algorithm that is able to construct non-reversible Markov chains with desired stationary probability distributions. We subsequently prove that, for multisite protein phosphorylation and similar models, in the deterministic domain, enzyme docking and the consequent substrate enzyme-sequestration must inevitably limit the extent of multistability, ultimately to one steady state. In contrast, bimodality can be obtained in the stochastic domain even in situations where bistability is not possible for large molecule numbers. We finally extend our results to cases where we have an autophosphorylating kinase, as for example is the case with $Ca^{2+}$/calmodulin-dependent protein kinase II (CaMKII), a key enzyme in synaptic plasticity.

In this thesis, I investigate dynamic and computational properties of prokaryotic signalling architectures commonly known as the Two Component Signalling networks and phosphorelays. The aim of this study is to understand the information processing capabilities of different prokaryotic signalling architectures by examining the dynamics they exhibit. I present original investigations into the dynamics of different phosphorelay architectures and identify network architectures that include a commonly found four step phosphorelay architecture with a capacity for tuning its steady state output to implement different signal-response behaviours viz. sigmoidal and hyperbolic response. Biologically, this tuning can be implemented through physiological processes like regulating total protein concentrations (e.g. via transcriptional regulation or feedback), altering reaction rate constants through binding of auxiliary proteins on relay components, or by regulating bi-functional activity in relays which are mediated by bifunctional histidine kinases. This study explores the importance of different biochemical arrangements of signalling networks and their corresponding response dynamics. Following investigations into the significance of various biochemical reactions and topological variants of a four step relay architecture, I explore the effects of having different types of proteins in signalling networks. I show how multi-domain proteins in a phosphorelay architecture with multiple phosphotransfer steps occurring on the same protein can exhibit multistability through a combination of double negative and positive feedback loops. I derive a minimal multistable (core) architecture and show how component sharing amongst networks containing this multistable core can implement computational logic (like AND, OR and ADDER functions) that allows cells to integrate multiple inputs and compute an appropriate response. I examine the genomic distribution of single and multi domain kinases and annotate their partner response regulator proteins across prokaryotic genomes to find the biological significance of dynamics that these networks embed and the processes they regulate in a cell. I extract data from a prokaryotic two component protein database and take a sequence based functional annotation approach to identify the process, function and localisation of different response regulators as signalling partners in these networks. In summary, work presented in this thesis explores the dynamic and computational properties of different prokaryotic signalling networks and uses them to draw an insight into the biological significance of multidomain sensor kinases in living cells. The thesis concludes with a discussion on how this understanding of the dynamic and computational properties of prokaryotic signalling networks can be used to design synthetic circuits involving different proteins comprising two component and phosphorelay architectures.

Microelectromechanical systems (MEMS) depend on mechanical deformation to sense their environment, enhance electrical circuitry, or store data. Nonlinear forces arising from multiphysics phenomena at the micro- and nanoscale -- van der Waals forces, electrostatic fields, dielectric charging, capillary forces, surface roughness, asperity interactions -- lead to challenging problems for analysis, simulation, and measurement of the deforming device elements. Herein, a foundation for the study of mechanical deformation is provided through computational and experimental studies of MEMS microcantilever capacitive switches. Numerical techniques are built to capture deformation equilibria expediently. A compact analytical model is developed from principle multiphysics governing operation. Experimental measurements support the phenomena predicted by the analytical model, and finite element method (FEM) simulations confirm device-specific performance. Altogether, the static multistability and quasistatic performance of the electrostatically-actuated switches are confirmed across analysis, simulation, and experimentation. <p><br></p> <p>The nonlinear multiphysics forces present in the devices are critical to the switching behavior exploited for novel applications, but are also a culprit in a common failure mode when the attractive forces overcome the restorative and repulsive forces to result in two elements sticking together. Quasistatic operation is functional for switching between multistable states during normal conditions, but is insufficient under such stiction-failure. Exploration of dynamic methods for stiction release is often the only option for many system configurations. But how and when is release achieved? To investigate the fundamental mechanism of dynamic release, an atomic force microscopy (AFM) system -- a microcantilever with a motion-controlled base and a single-asperity probe tip, measured and actuated via lasers -- is configured to replicate elements of a stiction-failed MEMS device. Through this surrogate, observable dynamic signatures of microcantilever deflection indicate the onset of detachment between the probe and a sample.</p>

There is an increasing concern for the impact of humans on the environment. Traditionally, ecological models consider human influence as a constant or linearly varying parameter, whereas socioeconomic models and frameworks tend to oversimplify the ecological system. But tackling complex environmental challenges faced by our societies requires interdisciplinary approaches due to the intricate feedbacks between the socioeconomic and ecological systems involved. Thus, models of social-ecological systems couple an ecological system with a socioeconomic system to investigate their interaction in the integrated dynamical system. We define this coupling formally and apply the social-ecological approach to three ecological cases. Indeed, we focus on eutrophication in shallow freshwater lakes, which is a well-known system showing bistability between a clear water state and a turbid polluted state. We also study a model accounting for an aquifer (water stock) and a model accounting for a biotic population exhibiting bistability through an Allee effect. The socioeconomic dynamics is driven by the incentive that agents feel to act in a desirable or undesirable way. This incentive can be represented by a difference in utility, or in payoff, between two strategies that each agent can adopt: agents can cooperate and act in an environment-friendly way, or they can defect and act in an ecologically undesirable way. The agents' motivation includes such factors as the economic cost of their choice, the concern they feel for the environment and conformism to the collective attitude of the human group. Thus, the incentive to cooperate responds to the state of the ecological system and to the agents' collective opinion, and this response can be linear, nonlinear and monotonic, or non-monotonic. When investigating the mathematical form of this response, we find that monotonic non-linear responses may result in additional equilibria, cycles and basins of attraction compared to the linear case. Non-monotonic responses, such as resignation effects, may produce much more complicated nullclines such as a closed nullcline and weaken our ability to anticipate the dynamics of a social-ecological system. Regarding the modelling of the socioeconomic subsystem, the replicator dynamics and the logit best-response dynamics are widely used mathematical formulations from evolutionary game theory. There seems to be little awareness about the impact of choosing one or the other. The replicator dynamics assumes that the socioeconomic subsystem is stationary when all agents adopt the same behaviour, whereas the best-response dynamics assumes that this situation is not stationary. The replicator dynamics has formal game theoretical foundations, whereas best-response dynamics comes from psychology. Recent experiments found that the best-response dynamics explains empirical data better. We find that the two dynamics can produce a different number of equilibria as well as differences in their stability. The replicator dynamics is a limit case of the logit best-response dynamics when agents have an infinite rationality. We show that even generic social-ecological models can show multistability. In many cases, multistability allows for counterintuitive equilibria to emerge, where ecological desirability and socioeconomic desirability are not correlated. This makes generic management recommendations difficult to find and several policies with and without socioeconomic impact should be considered. Even in cases where there is a unique equilibrium, it can lose stability and give rise to sustained oscillations. We can interpret these oscillations in a way similar to the cycles found in classical predator-prey systems. In the lake pollution social-ecological model for instance, the agents' defection increases the lake pollution, which makes agents feel concerned and convince the majority to cooperate. Then, the ecological concern decreases because the lake is not polluted and the incentive to cooperate plummets, so that it becomes more advantageous for the agents to defect again. We show that the oscillations obtained when using the replicator dynamics tend to produce a make-or-break dynamics, where a random perturbation could shift the system to either full cooperation or full defection depending on its timing along the cycle. Management measures may shift the location of the social-ecological system at equilibrium, but also make attractors appear or disappear in the phase plane or change the resilience of stable steady states. The resilience of equilibria relates to basins of attraction and is especially important in the face of potential regime shifts. Sources of uncertainty that should be taken into account for the management of social-ecological systems include multistability and the possibility of counterintuitive equilibria, the wide range of possible policy measures with or without socioeconomic interventions, and the behaviour of human collectives involved, which may be described by different dynamics. Yet, uncertainty coming from the collective behaviour of agents is mitigated if they do not give up or rely on the other agents' efforts, which allows modelling to better inform decision makers.

<p dir="ltr">Origami has emerged as a design paradigm to realize morphing structures with rich kinematic and mechanical properties. Biological examples augment the potential folding design space by suggesting intriguing routes for achieving and expanding crease patterns which traditional origami laws are unable to capture. Specifically, spring origami theory exploits the material system architecture and energy storage mechanism of the earwig wing featuring one of the highest folding ratios in the animal kingdom (1:18), minimal energy required for deployment and collapse of the wing, and bistability locking the wing in closed, and open configurations for crawling through tunnels, and flight, respectively. The central mechanism responsible for bistability in the wing features a non-developable crease pattern with a non-zero Gaussian curvature. Reconfiguring, or even flattening a structure with such an intrinsic property requires stretching or tearing; soft, rubbery material found in the creases of the central mechanism allows for stretching enabling shape transformations between open and closed states without tearing. In the first part of this thesis, such characteristics are transferred to a synthetic bistable soft robotic gripper leveraging the shape adaptability and conformability exhibited by the biological organism to minimize actuation energy. This is achieved by integrating soft, flexible material in the bioinspired gripper that allows kinematically driven geometries to grasp and manipulate objects without continuous actuation. Secondly, the stiffening effect from spring origami is utilized in a bioinspired wing for an aerial--aquatic robot. Transitions between air and sea in multimodal robots is challenging, however, a structurally efficient and multifunctional membrane is developed to increase locomotive capabilities and longer flights. This is motivated by the flying fish's locomotive modules and origami design principles for deployment and folding. Additionally, to keep the wing in a stiff state while gliding, spring origami bistable units are integrated into the membrane inducing self-stiffening and a global curvature reducing energy expenditure while generating lift. While the previous examples present solutions to adaptive manipulation and membrane multifunctionality, once programmed, their shapes are fixed. In the third application, a class of multistable self-folding origami architectures that are reprogrammable post fabrication are presented. This is achieved by encoding prestrain in bilayer creases with anisotropic shrinkage that change shape and induce a local curvature in the creases in response to external stimuli. The topology of the energy landscapes can thus be tuned as a function of the stimulation time and adaptable post fabrication. The proposed method and model allows for converting flat sheets with arranged facets and prestrained mountain-valley creases into self-folding multistable structures. Lasty, encoding crease prestrain is leveraged to manufacture a biomimetic earwig wing featuring the complex crease pattern, structural stability, and rapid closure of the biological counterpart. The presented method provides a route for encoding prestrain in self-folding origami, the multistability of which is adaptable after fabrication.</p>