Mathematical Modelling of Integrated Signalling Networks in Stomatal Guard Cells
Author(s)
Beguerisse Diaz, Mariano
Type
Thesis
Abstract
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.
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.
Date Issued
2012-03
Date Awarded
2012-05
Advisor
Desikan, Radhika
Barahona, Mauricio
Stark, Jaroslav
Sponsor
Biotechnology and Biological Sciences Research Council (Great Britain) ; Microsoft Research
Publisher Department
Cell and Molecular Biology
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)