Mathematical modelling of integrated signalling networks in stomatal guard cells

Stomata are tiny pores in plant leaves that regulate gas and w ater exchange between plants and their environment. Abscisic acid and ethylene ar e two well-known elicitors of stomatal closure. Yet when stomata are presented with a co mbination of both signals, they fail to close; these observations are hard to r ec ncile biologically and their explanation is not easily obtained by experimental means al one. To shed light on this unexplained behaviour, a combination of mathematical, com putational, and experimental techniques are used. A differential equation model of stoma al closure is constructed from known biochemical interactions; this modelling process ha motivated the collection of experimental measurements of components in the pathway, at time points beyond what is usually found in the literature. The experimental observ ations include stomatal aperture and hydrogen peroxide production in Arabidopsis thalianaguard cells treated with abscisic acid, ethylene, and a combination of both. These measuremen ts show that sustained high levels of hydrogen peroxide are required to achieve stomata l closure and that guard cells exhibit increased antioxidant activity when treated with a combined dose of abscisic acid and ethylene. Additionally, the experimental observation s a d modelling suggest a distinct role for two antioxidant mechanisms during stomatal closur e: a slower, delayed response that is activated by a single stimulus (abscisic acid or ethylene) and another more rapid mechanism that is only activated when both stimuli are prese nt. The model indicates that the presence of this ‘and’ mechanism in the antioxidant resp on e is crucial to explain the lack of closure under a combined stimulus. Estimating parameters from data is a key stage of the modelli ng process, particularly in models of biological systems many parameters need to be esti mated from sparse and noisy data sets, such is the case of the stomatal closure model pres nted here. Over the years, a variety of heuristics have been proposed to solve this compl ex optimisation problem, with good results in some cases yet with limitations in the biolog ical setting. In this thesis, an algorithm for model parameter fitting is developed combin ing ideas from evolutionary algorithms, sequential Monte Carlo methods and direct sear ch optimisation. The method is shown to perform well even when the order of magnitude and/or the ange of the parameters is unknown. The method refines iteratively a sequence of para meter distributions through local optimisation combined with partial resampling from a historical prior defined over the support of all previous iterations. The method is tested on biological models using both simulated and real experimental data, and it estimates the parameters efficiently even in the absence of a priori knowledge about the parameters. Then, this method is used to find the parameter values of the much larger stomatal closure model from experimental observations. A classic model of linear activation cascades is studied in t h s thesis. In a special but important case the output of an entire cascade can be repr sented analytically as a function of the input and a lower incomplete gamma function. If the inactivation rate of any component is altered, the change induced at the output is independent of the position in the cascade of the modified component. These analytical resu lts show how one can reduce the number of equations and parameters in ODE models of cell s ignalling cascades, and how delay differential equation models can sometimes be app roximated through the use of simple expressions involving the incomplete gamma funct io . The expressions with the lower incomplete gamma functions are used in the constructi on of the model of stomatal closure to represent the activation dynamics of the antioxi dant mechanisms in guard cells. Fitting the stomatal closure model parameters to the data in dicates that the antioxidant responses should have two different timescales, which can e xplain the lack of closure under a combined ABA and ethylene stimulus.