Mathematical Modelling of Integrated Signalling Networks in Stomatal Guard Cells
Creators: name: Beguerisse Diaz, Mariano
Publisher: name: Imperial College London
Year: 2012

description: Stomata are tiny pores in plant leaves that regulate gas and water exchange between plants and their environment. Abscisic acid and ethylene are two well-known elicitors of stomatal closure. Yet when stomata are presented with a combination of both signals, they fail to close; these observations are hard to reconcile biologically and their explanation is not easily obtained by experimental means alone. To shed light on this unexplained behaviour, a combination of mathematical, computational, and experimental techniques are used. A differential equation model of stomatal closure is constructed from known biochemical interactions; this modelling process has motivated the collection of experimental measurements of components in the pathway, at time points beyond what is usually found in the literature. The experimental observations include stomatal aperture and hydrogen peroxide production in Arabidopsis thaliana guard cells treated with abscisic acid, ethylene, and a combination of both. These measurements show that sustained high levels of hydrogen peroxide are required to achieve stomatal closure and that guard cells exhibit increased antioxidant activity when treated with a combined dose of abscisic acid and ethylene. Additionally, the experimental observations and modelling suggest a distinct role for two antioxidant mechanisms during stomatal closure: 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 present. The model indicates that the presence of this ‘and’ mechanism in the antioxidant response is crucial to explain the lack of closure under a combined stimulus. Estimating parameters from data is a key stage of the modelling process, particularly in models of biological systems many parameters need to be estimated from sparse and noisy data sets, such is the case of the stomatal closure model presented here. Over the years, a variety of heuristics have been proposed to solve this complex optimisation problem, with good results in some cases yet with limitations in the biological setting. In this thesis, an algorithm for model parameter fitting is developed combining ideas from evolutionary algorithms, sequential Monte Carlo methods and direct search optimisation. The method is shown to perform well even when the order of magnitude and/or the range of the parameters is unknown. The method refines iteratively a sequence of parameter 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 this thesis. In a special but important case the output of an entire cascade can be represented 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 results show how one can reduce the number of equations and parameters in ODE models of cell signalling cascades, and how delay differential equation models can sometimes be approximated through the use of simple expressions involving the incomplete gamma function. The expressions with the lower incomplete gamma functions are used in the construction of the model of stomatal closure to represent the activation dynamics of the antioxidant mechanisms in guard cells. Fitting the stomatal closure model parameters to the data indicates that the antioxidant responses should have two different timescales, which can explain the lack of closure under a combined ABA and ethylene stimulus.