Gradient Sensing by Template Matching
by
Don Praveen Amarasinghe
Supervisor: Dr. Till Bretschneider
RESUBMISSION
A thesis submitted in partial fulfilment of the requirements for the degree of
Master of Science
Molecular Organisation and Assembly in Cells (MOAC)
Doctoral Training Centre, University of Warwick
July 2013
Contents
Contents
i
Acknowledgements
iii
Abstract
iv
1
Introduction
1
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Aims of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2
3
Model Formulation
6
2.1
Biochemical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.1
One receptor type — competitive model . . . . . . . . . . . . . . . . .
9
2.2.2
Two receptor types — non-competitive model . . . . . . . . . . . . . .
10
2.2.3
Including diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2.4
Comparing the concentrations of A0 and B0 . . . . . . . . . . . . . . .
13
Results
17
3.1
Simulations with instantaneous diffusion of starred chemicals and 1D cells . . .
17
3.1.1
Absence of a chemoattractant gradient . . . . . . . . . . . . . . . . . .
17
3.1.2
Detection of a 2% chemoattractant gradient . . . . . . . . . . . . . . .
19
3.1.3
Detection of chemoattractant gradients of different sizes . . . . . . . .
23
3.1.4
Effects of kA∗ A0 and kB∗ B0 on coefficient values . . . . . . . . . . . . . .
23
Simulations with diffusion and 2D cells . . . . . . . . . . . . . . . . . . . . .
24
3.2
4
Discussion
28
i
Contents
5
Conclusion
30
Bibliography
31
A Appendices
34
A.1 List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
A.2 Code for running the simulations . . . . . . . . . . . . . . . . . . . . . . . . .
35
A.3 Code containing the differential equations for the 1D and 2D Simulations of the
competitive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
A.4 Code containing the differential equations for the 1D and 2D Simulations of the
non-competitive model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
A.5 Simulations studying the effects of no chemoattractant gradient on coefficient
values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
A.5.1 A one-dimensional cell and instantaneous diffusion of starred chemicals
42
A.5.2 A two-dimensional cell and diffusion of primed and starred chemicals .
47
A.6 Simulations studying the effects of chemoattractant concentration gradient size
on coefficient values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
A.7 Simulations studying the effects of kA∗ A0 and kB∗ B0 on coefficient values . . . . .
55
ii
Acknowledgements
Thanks go to family and friends, particularly all of those involved in the MOAC, IBR and
Systems Biology Doctoral Training Centres, for their support throughout the MSc. year. Special
thanks go to Adam Hall and Dave McCormick of MASDOC, one of MOAC’s sister DTCs,
for their guidance in helping me typeset this dissertation with LATEX. I am also grateful to
Robert Lockley for his advice regarding parameter values and the Meinhardt model. Finally,
I am extremely grateful to my supervisor, Dr. Till Bretschneider, for his guidance, support,
MATLAB code, and patience in helping me write this thesis.
Permission for this resubmission was granted by my supervisor and Dr. Hugo van den Berg.
I am grateful to both of them for allowing me a second chance to prove that, academically
speaking, I am not entirely incompetent!
iii
Abstract
Previous studies of gradient sensing in chemotactic eukaryotic cells have involved the use of
a threshold concentration of chemoattractant to elicit a response from the cell. However, this
does not explain the high level of sensitivity to chemoattractant gradients observed in a variety
of different species. This thesis considers a new approach to model gradient sensing based
on the idea of template matching from image processing. The basic premise is that the cell
compares the intracellular pattern of internal components, used to detect and amplify a signal,
with the extracellular diffusion pattern of chemoattractant. The difference in the gradients of
the two patterns will inform the cell of the direction of the source of chemoattractant.
This thesis is a first attempt at using this approach to model gradient sensing. Two theoretical
models are proposed. Each is based on two chemical systems consisting of an initiator chemical,
a local activator and a global inhibitor — one is triggered by the presence of chemoattractant
outside the cell; the other is triggered by an intracellular chemical. Initiation takes place using
receptors in the cell membrane. One model involves these two chemical systems competing for
the same receptors, whereas the other uses separate receptor types for each system of chemicals.
Two coefficients (motivated by the concept of correlation) are proposed to compare the reactiondiffusion patterns that are generated by these systems.
iv
Chapter 1
Introduction
1.1
Background
Various eukaryotic cells are known to detect external signals and direct their motion towards
(or away from) these signals [1, 2]. One way in which these signals present themselves is in
the form of a chemical concentration gradient. Cell movement resulting from the detection and
response to such cues is known as chemotaxis. This feature of eukaryotes plays an important
part in development, immune responses, and the spread of cancer in an organism [3–5]. A
wide variety of species have been studied to analyse the mechanisms of chemotaxis — from
Saccharomyces cerevisiae (the budding yeast) and Dictyostelium discoideum (the chemotactic
social amoeba), to mammalian neutrophils (white blood cells), fibroblasts (connective tissue
cells used for wound repair), and nerve cells [1].
A prominent part of the study of chemotactic behaviour is the development of mathematical
models to simulate cell responses to chemical cues. A large number of these models exists in
the literature and most are based on the standard reaction diffusion system given by:
∂c
= f(c) + D∇2 c,
∂t
(1.1)
where c is a vector of the concentrations of chemicals controlling the process, f is a function
of the chemical concentrations representing the reactions between these chemicals and D∇2 c
is a diffusion term with D being a diagonal matrix of diffusion coefficients [1, 6]. The exact
form for f varies upon the context of the mechanism being modelled. Under relatively simple
1
Chapter 1 Introduction
conditions of an inhibitor diffusing much faster than an activator (known as a ”local excitation,
global inhibition” (LEGI) model), the reaction-diffusion model will give rise to stable pattern
formation [1, 7]. While stable patterns are useful in eliciting sustained responses from a cell, the
formation of permanent patterns does not allow the cell to adapt to changing conditions [1, 5].
One way in which this “locking-in” of patterns can be avoided was proposed by Meinhardt
[7] using a local inhibitor in addition to a local exciter and a global inhibitor. These models
of stable, but not permanent, pattern formation reflect what is observed in cells in real life.
For example, in work by Taniguchi et al. [8] and Gerisch et al. [9], wave patterns of various
intracellular components involved in chemotaxis, including PIP3 (Phosphatidylinositol (3,4,5)triphosphate, a phospholipid synthesised at the front of chemotaxing cells to stimulate actin
polymerisation) and actin. Furthermore, it has been suggested that this is the cause of cell
repolarisation in the absence of external stimuli.
In Iglesias and Levchenko [10] and Devreotes and Janetopoulos [11], it is suggested that chemotaxis can be divided into three distinct activities.
• Motility — Cell movement by periodic extension of self-limited pseudopodia at the cell
anterior and retraction at the rear. Note that no chemotactic gradients are needed to generate pseudopods. For example, Dictyostelium discoideum cells migrate randomly in the
absence of chemotactic cues [12].
• Polarization — Rearrangement of cellular components leading to the development of
separate leading and trailing edges with distinct sensitivities for chemoattractant. This
occurs usually in response to external chemoattractant gradients, but it is also known to
occur in uniform attractant. For example, when stimulated by a uniform dose of chemoattractant, mammalian neutrophils (white blood cells) and Dictyostelium discoideum cells
lacking adenylyl cyclase, ACA, (a chemical which converts adenosine triphosphate, ATP,
to cyclic adenosine monophosphate, cAMP) acquire distinct leading and trailing edges
and begin to migrate at random [13, 14]. Cells need not be polarised at all to respond to
changes in their surroundings [11, 12].
• Gradient sensing — The ability of a cell to detect and amplify spatial gradients, even
when it is immobile. This property is best observed by imaging fluorescently tagged
proteins in cells that have been immobilized by inhibitors of the actin cytoskeleton, such
as Latrunculin, a chemical that prevents actin polymerisation [15].
2
Chapter 1 Introduction
Upon studying the chemotaxis of cells of different species, one observes that there is a number of
ways in which the chemotactic responses between species differ [1]. For example, some models
account for the amplification behaviour of cells, whereby polarisation amplifies the asymmetry
that arises from the concentration gradient present (no matter how small or large). Some models
account for a cell’s sensitivity to new stimuli and response to changes in the overall surrounding
chemoattractant gradient. Other models reflect other behaviour seen in other species, such as
the ability to deal with multiple stimuli and to spontaneuously polarise. It is noted that the
mathematical setup of the model and the species of the cell being studied often mean that the
model constructed displays some, but not all, of these characteristics. For example, reactiondiffusion models are attractive because they account for spontaneous polarisation, high levels of
amplification, and maintain polarisation after stimulus removal. However, they cannot account
for other features seen in some cells, such as non-polar resting states in neutrophils [1].
1.2
Aims of this thesis
A puzzling question is the enormous sensitivity of cells, capable of detecting very small differences in the concentration of a chemoattractant between the cell front and rear — in the case
of Dictyostelium discoideum, a concentration gradient of as little as 1-2% between the “front”
and “rear” is enough [12]. For example, a Dictyostelium discoideum cell of 10 – 20 µm length,
contains around 80,000 cAMP receptors distributed evenly around the cell [16]. For such a cell,
sensing a gradient as shallow as 1-2% relies upon around 130 more receptors on one side of
the cell binding to chemoattractant than on the other side [16, 17]. This is a very small difference in receptor binding and existing models assume that such a small difference can cause
large changes in the behaviour of a cell. Phenomena such as unwanted reactions between intracellular signalling chemicals and other biomolecules may result in noisy intracellular signals.
Various mathematical models for gradient sensing and polarisation attempt to overcome this
issue by providing a mechanism through which minute extracellular gradients can be amplified
to give rise to a strong intracellular response [1]. However, these mechanisms require that cells
are able to adjust the threshold finely for front activation over time, allowing a cell to promote
detection and enhance a response at the top of the chemoattractant gradient. There is also the
existence of the intracellular patterns, as described in Taniguchi et al. [8] and Gerisch et al. [9].
While these are known to occur through the behaviour of intracellular signalling components
3
Chapter 1 Introduction
linked to the actin system as a result of a natural reaction-diffusion process, it is possible that
some cell functions utilise these patterns.
The goal of this thesis, therefore, is to explore a novel approach to gradient sensing for eukaryotic cells using intracellular patterns. Rather than the external gradient promoting frontactivation when above a threshold level, as in Endres and Wingreen [18], it is proposed that the
presence of the gradient alone, no matter how low the absolute concentrations are, is enough
to trigger a directional response. This thesis will consider whether the spontaneously generated
intracellular patterns could be compared with the extracellular gradient, providing a mechanism
similar to that of template matching in digital image processing, where features of an image are
compared with and fitted against a template [19]. If an intracellular gradient is perfectly aligned
with an external gradient, then intracellular chemical reactions that result from this match will
elicit a maximum response from the cell. If there is a slight mismatch in the patterns, these will
reactions will cause a smaller response. This corresponds to taking a convolution of the intracellular and extracellular gradient, a mechanism which essentially would be threshold-free —
no minimum concentration of chemoattractant is required and the response would be generated
by the presence of a gradient, irrespective of the background chemoattractant concentration.
The idea is based on the observation that components signaling to the actin cytoskeleton and
actin itself have been shown to self-organise into dynamic spatial patterns even in the absence
of any extracellular signal gradients, as mentioned previously. It is also based on early work
by Meinhardt [7], which attempts to account for the extraordinary directional sensitivity of
chemotactically sensitive cells. The Meinhardt model assumes that cells have an intrinsic pattern forming system that generates the signals for the extension of pseudopods. An external
signal such as a graded cue is assumed to impose some directional preference onto the pattern
formed. Computer simulations show that the model accounts for the highly dynamic behaviour
of chemotactic cells [7].
With the definition of “gradient sensing” as given by Iglesias and Levchenko [10] in mind,
this thesis is concerned with the pattern formation of chemicals in and around a cell undergoing
chemotaxis — cell movement either as a result of random pseudopod formation or through repolarisation will not be of concern here. First, the basic biochemistry of the model is constructed,
with particular attention to the arithmetic that a living cell can be expected to perform. This
leads on to the development of two mathematical reaction-diffusion systems that will be tested.
4
Chapter 1 Introduction
Finally, the outcomes of simulations of these models will be discussed, along with potential
refinements and other avenues for work.
5
Chapter 2
Model Formulation
In this chapter, a model for gradient sensing using intracellular patterns is developed. First,
details of the underlying biochemistry of the model are proposed using membrane-receptor
binding as a foundation. Then, a mathematical model based upon this biochemical model is
formulated.
2.1
Biochemical Model
The basis for the model proposed here is the binding of chemicals to transmembrane receptors
present in the cell membrane. It will be assumed that the cell has ample membrane receptors to
bind to any relevant chemicals present and that the cell will maintain the number of receptors
present in a given area of the membrane as close to the maximum level as possible. It is also
assumed that receptors will bind a chemoattractant molecule but not release it. This is identical
to basic premise of the “perfectly absorbing sphere” model described by Endres and Wingreen
[18]. The model biochemistry consists of two competing chemical systems, each composed of
three chemicals: the A-system (composed of A, A0 , and A∗ ) and the B-system (composed of B,
B0 , and B∗ ).
Let A represent a chemoattractant outside the cell. It is proposed that the binding of A to
a membrane receptor results in that receptor being unable to bind to any other molecule and
inactive receptors will become activated again after some time. The binding will also trigger
the release of two intracellular chemicals, A0 and A∗ .
6
Chapter 2 Model Formulation
A
Receptors
Membrane
A'
A*
B'
Cytosol
B*
B
Figure 2.1: A summary of the biochemistry of the one-receptor-type, competitive model.
• A0 will trigger a local response to the presence of chemoattractant in that region. This
chemical is considered to be slow-diffusing, to allow it to act locally but not globally, and
fast-acting, to enable a quick response to the presence of chemoattractant.
• A∗ will be produced at a similar rate to the production of A0 . This chemical will react
with A0 to inhibit the cell’s response to chemoattractant. It will be considered slow-acting,
however, so as not to completely diminish the effect of A0 . It is also assumed that it is fastdiffusing, allowing it to travel to any part of the cell membrane and act globally. The main
consequence of this assumption is that A∗ will be considered to be mixed homogenously
in the cell.
Suppose that there is also an intracellular chemical B which is produced centrally and that
this binds, in a similar way, to the same receptors that A binds to. In addition, suppose that
this binding results in the release of two chemicals B0 and B∗ inside the cell that have similar
properties to A0 and A∗ .
These two systems competing for receptor binding, each with an initiation chemical, locally
acting response stimulator (the primed chemical) and a globally acting inhibitor (the starred
chemical). A summary of this biochemical model is given in Figure 2.1. The principle is that
the cell will compare the concentrations of A0 and B0 (with respect to A∗ and B∗ as some form
of “normalisation” — see Section 2.2.4) to monitor changes due to activation by the presence
of chemoattractant and ascertain the direction of an extracellular chemoattractant concentration
gradient. If the distribution pattern of the concentrations of A0 (after normalisation against
A∗ ) matches the pattern of B0 (normalised against B∗ ) along the cell membrane, then the cell
is “pointing in the right direction” (see Figure 2.2). If there are mismatches between these
7
Chapter 2 Model Formulation
Concentration gradient of A (and A')
Low
High
Concentration
gradient of B'
Concentration
gradient of B'
Gradient directions match
Gradient directions do not match
Figure 2.2: A summary of the template matching principle being applied. The orange and blue
regions represent the distribution of A0 and B0 around the cell membrane, with thicker regions
indicating a higher concentration. Note that the concentration gradient of A matches that of A0 .
two distribution patterns, then the cell will use this as an indication that the chemoattractant
concentration gradient and its own internal gradients are not aligned and that it needs to reorient
itself.
The motivation behind this model comes from the structure of G-protein coupled receptors
that are used by Dictyostelium discoideum cells to detect cAMP [20, 21]. Activation of these
receptors results in the attached heterotrimeric G-protein dissociating into two subunits, Gα and
Gβγ [21], and this model retains this feature. However, it is also known that both subunits are
required to stimulate a chemotactic response [22, §14.1]. This model proposes instead that the
two subunits act as a local excitor and a global inhibitor. Furthermore, as mentioned above
in Section 1.1, the formation of wave patterns of F-actin and PIP3 inside cells are known to
occur without the presence of any stimulus [8, 9]. It is envisaged that the B-system represents
a mechanism for this wave formation without stimuli. However, as demonstrated by Meinhardt
[7], the scheme proposed for the B-system will need to incorporate a ”local inhibitor” to ensure
that the intracellular wave patterns that form cannot stabilise [1].
As an extension to this single-receptor-type model, this thesis will also consider the situation
where there are two separate receptor types: one that will bind only to A, producing A0 and
A∗ , and another receptor type that binds only to B, producing B0 and B∗ . This non-competitive
model is motivated by the idea that different receptor types exist to respond to different forms
of extracellular stimuli. For example, Dictyostelium discoideum cells are known to respond to
8
Chapter 2 Model Formulation
Variable
ai
a0i
a∗i
bi
b0i
b∗i
ri
Description
Concentration of A at position i
Concentration of A0 at position i
Concentration of A∗ at position i
Concentration of B at position i
Concentration of B0 at position i
Concentration of B∗ at position i
Concentration of active receptors at position i
Value at start
(See main text)
0
0.1
(See main text)
0
0.1
rtot
Table 2.1: Variables and initial conditions for the single-receptor-type model, given by Equations (2.1)
a mechanical stimulus [23]). These can be coupled to the same intracellular pattern generator.
2.2
2.2.1
Mathematical Formulation
One receptor type — competitive model
For the competitive (single-receptor-type) model, the cell membrane is divided into n sections.
Then, for 1 ≤ i ≤ n, we define seven variables, detailed in Table 2.1. The rates of change of
these variables are governed by a system of non-dimensionalised ordinary differential equations
(ODEs) shown below in Equations (2.1). The parameters used in these equations are listed,
along with their values, in Table 2.2.
da0i
=
dt
da∗i
dt
=
kA+ 0 ai ri
|{z}
kA∗ A0 a0i a∗i
| {z }
−
Binding of A to receptors Natural degradation of A’ Reaction of A’ with A∗
Pn 0
j=1 a j
−
kA∗ A0 a0i a∗i
−
1 + kA− ∗ a∗i
n }
| {z
See main text
db0i
=
dt
kA− 0 a0i
|{z}
−
| {z }
Reaction of A’ with A∗
kB+0 bi ri
|{z}
|
}
−
k ∗ B0 b0i b∗i
|B {z
}
Binding of B to receptors Natural degradation of B’ Reaction of B’ with B∗
Pn 0
j=1 b j
−
kB∗ B0 b0i b∗i
−
1 + kB−∗ b∗i
db∗i
=
dt
n }
| {z
See main text
dri
=
dt
| {z }
Reaction of B’ with B∗
−kA+ 0 ai ri
| {z
}
Receptors binding with A
|
{z
(2.1c)
(2.1d)
}
Natural degradation of B∗
kB+0 bi ri
|{z}
−
(2.1b)
Natural degradation of A∗
kB−0 b0i
|{z}
−
{z
(2.1a)
Receptors binding with B
+ kr (rtot − ri )
| {z }
(2.1e)
Receptor replacement
In Equations (2.1b) and (2.1d), the terms highlighted in blue depict the mean concentrations
9
Chapter 2 Model Formulation
Parameter
rtot
kr
kA+ 0
kA− 0
kA∗ A0
kA− ∗
kB+0
kB−0
kB∗ B0
kB−∗
Value
20
10−5
10−2
10−5
1
0.1
10−2
10−5
1
0.1
Description
Concentration of active and inactive receptors at any position i
Rate of receptor (re-)activation
Rate of binding of A to receptor
Rate of degradation of A0
Rate constant of reaction between A0 and A∗
Rate of degradation of A∗
Rate of binding of B to receptor
Rate of degradation of B0
Rate constant of reaction between B0 and B∗
Rate of degradation of B∗
Table 2.2: Parameters for the single-receptor-type model, given by Equations (2.1)
of A0 and B0 . The inclusion of these averaging terms is motivated by two assumptions. First,
it is assumed that A∗ and B∗ are produced at the same rate as A0 and B0 respectively. Second,
it is also assumed that the diffusion of the starred chemicals is instantaneous. Therefore, the
concentration is assumed to be even along the whole membrane. Furthermore, in the same two
equations, the degradation term takes the form (1 + k− ) × a∗ rather than just k− × a∗ . The reason
for this is that the effect of adding the instantaneous averaging of a0 and b0 is negated, leaving
only contributions specific to a∗ and b∗ , respectively.
2.2.2
Two receptor types — non-competitive model
In the situation of two receptor types, the variables remain unchanged except for ri , which is
replaced with two variables, riA and riB , representing the number of active receptors binding to
A and B, respectively, at position i. The non-dimensionalised ODE system used to model this
10
Chapter 2 Model Formulation
Parameter
A
rtot
B
rtot
krA
krB
Value
10
10
10−5
10−5
Description
Concentration of active and inactive A receptors at any position i
Concentration of active and inactive B receptors at any position i
Rate of A receptor (re-)activation
Rate of B receptor (re-)activation
Table 2.3: Receptor parameters for the single-receptor-type model, given by Equations (2.2)
situation is as follows:
da0i
dt
da∗i
dt
driA
dt
db0i
dt
db∗i
dt
driB
dt
= kA+ 0 ai rA − kA− 0 a0i − kA∗ A0 a0i a∗i
Pn 0
j=1 a j
=
− kA∗ A0 a0i a∗i − 1 + kA− ∗ a∗i
n
A
= −kA+ 0 ai riA + krA rtot
− rAi
= kB+0 bi rB − kB−0 b0i − kB∗ B0 b0i b∗i
Pn 0
j=1 b j
=
− kB∗ B0 b0i b∗i − 1 + kB−∗ b∗i
n
B
= −kB+0 bi riB + krB rtot
− rBi
(2.2a)
(2.2b)
(2.2c)
(2.2d)
(2.2e)
(2.2f)
The reasoning behind the inclusion of the blue averaging terms and (1 + k− ) degradation terms
in Equations (2.2b) and (2.2e) is the same as in Equations (2.1) of the competitive model. The
parameters used in this model are identical to the parameters used in the competitive model, with
the exception of rtot and kr . These are replaced with four parameters, as detailed in Table 2.3.
The initial conditions are the same as those for the competitive model, with those for a and b
A
B
being specified separately, except that riA = rtot
and riB = rtot
for 1 ≤ i ≤ n.
2.2.3
Including diffusion
The two models are now modified to incorporate diffusion into the scheme. In addition to the
inclusion of diffusion terms of the form D∇2 , the major changes are contained in the equations
for the starred chemicals. First, the averaging terms in the rate equations for A∗ and B∗ are
replaced with terms of the form “D∇2 a+k+ ar”. This is because the averaging terms represented
instantaneous and superfast diffusion of A∗ and B∗ and are no longer needed, however the
second term is needed to implement the assumption that these chemicals are produced at the
same rate as A0 and B0 respectively. Second, the replacement of the averaging terms removes
11
Chapter 2 Model Formulation
the need for the degradation terms to take the form (1 + k− ) × a∗ . These now take the form
k− × a∗ .
The dimensionless rate equations for the competitive (single-receptor-type) model with diffusion are given below in Equations (2.3):
∂a0i
∂t
∂a∗i
∂t
∂b0i
∂t
∂b∗i
∂t
∂ri
∂t
= DA0 ∇2 a0i + kA+ 0 ai ri − kA− 0 a0i − kA∗ A0 a0i a∗i
(2.3a)
= DA∗ ∇2 a∗i + kA+ 0 ai ri − kA∗ A0 a0i a∗i − kA− ∗ a∗i
(2.3b)
= DB0 ∇2 b0i + kB+0 bi ri − kB−0 b0i − kB∗ B0 b0i b∗i
(2.3c)
= DB∗ ∇2 b∗i + kA+ 0 ai ri − kB∗ B0 b0i b∗i − kB−∗ b∗i
(2.3d)
= −kA+ 0 ai ri − kB+0 bi ri + kr (rtot − ri )
(2.3e)
The dimensionless rate equations for the non-competitive model with diffusion are given below
in Equations (2.4):
∂a0i
∂t
da∗i
dt
driA
dt
db0i
dt
db∗i
dt
driB
dt
= DA0 ∇2 a0i + kA+ 0 ai rA − kA− 0 a0i − kA∗ A0 a0i a∗i
(2.4a)
= DA∗ ∇2 a∗i + kA+ 0 ai rA − kA∗ A0 a0i a∗i − kA− ∗ a∗i
(2.4b)
A
− riA
= −kA+ 0 ai riA + krA rtot
(2.4c)
= DB0 ∇2 b0i + kB+0 bi rB − kB−0 b0i − kB∗ B0 b0i b∗i
(2.4d)
= DB∗ ∇2 b∗i + kB+0 bi rB − kB∗ B0 b0i b∗i − kB−∗ b∗i
(2.4e)
B
= −kB+0 bi riB + krB rtot
− riB
(2.4f)
The parameter values and initial conditions are as given for the models without diffusion in
Sections 2.2.1 and 2.2.1, with initial conditions for a and b specified separately. The diffusion
constants are given in Table 2.4.
DA0
DA∗
2 × 10−4
2 × 10−3
DB0
DB∗
2 × 10−4
2 × 10−3
Table 2.4: Diffusion coefficients for Equations (2.3) and (2.4)
12
Chapter 2 Model Formulation
2.2.4
Comparing the concentrations of A0 and B0
A key element of the mathematical formulation of the model is the way in which the cell uses
information about the concentrations of A0 and B0 to establish its orientation with respect to a
single chemoattractant gradient. Specifically, what calculations can a cell make at a molecular
level that will allow it to sense a gradient? As mentioned earlier, the method proposed in this
thesis is based on the idea of template matching, as performed in digital image analysis. There
are a number of different ways of implementing a template matching algorithm [19]. One such
method is the use of principal components analysis, by which a computer compares a candidate
image against eigenvectors (or eigenimages) generated from a template image [19, Chapter 8].
This would be useful in gradient sensing, since all the cell would need to do is calculate the angle
between the principal component vectors (or a linear combination of several of the components)
that come from each of the two gradient. This angle would thus be an indicator of how well
the gradients align with each other. However, there is no known biochemical model for a cell
to compute eigenvectors and the angles between them. Furthermore, it is entirely possible that
cells may compute eigenvectors and eigenvalues using a method other than that of finding roots
of a characteristic polynomial and solving simultaneous equations. A discussion of potential
biochemical mechanisms for such calculations is beyond the scope of this thesis.
An alternative, perhaps simpler, method of template matching with computer images is through
convolution (or cross-correlation) [19, Chapter 1]. The template image is moved around the
candidate image and convolutions of the two are calculated — the position with the highest
convolution score represents the best fit of the template to the candidate image [19]. While
the convolution of two functions involves the computation of an integral, a discretised form
of convolution involves simple sums of products [19, Page 10]. However, it is important to
consider the question of whether these calculations can be performed by cells. Addition of the
concentration of two chemicals, X and Y say, could be performed using chemical reactions
that convert both chemicals into the same chemical, Z say. Then the calculation is reduced to
measuring the concentration of chemical Z. Subtraction of concentrations could be done in a
number of different ways. One possible approach would involve the two chemicals reacting
with each other and the cell measuring the concentration of the resulting surplus. Another
possible approach is to monitor the concentration of a chemical whose production is inhibited
by one chemical and promoted by the other. Subtraction may also result in negative numbers
13
Chapter 2 Model Formulation
being produced. To address this issue, a cell could use an equilibrium between two chemicals:
one representing positive numbers, Z+ say, and one representing negative numbers, Z− say. If
there is more X than Y, then the equilibrium will favour Z+ , whereas the reverse would mean
the equilibrium would shift towards the side of Z− .
For multiplication, it could be postulated that molecules can catalyse the speed of a reaction
which produces a chemical that can be used for the final measurement. Suppose that the product of the concentrations of X and Y is to be found, and there is an enzyme that converts a
chemical used for the final multiplication measurement from the inactive form, Z0 say, to the
active form, Z0 say. The binding of X and Y to this enzyme could unlock multiple binding sites
for Z0 . Both X and Y would be needed to fully unlock all binding sites on an enzyme molecule,
whereas as the presence of only one of the two chemical results in fewer sites being unlocked.
Through this mechanism, the speed of Z activation would be used to calculate the product of
the concentrations of X and Y. Division could be performed in a similar way, but with one of X
or Y acting as a non-competitive inhibitor. An alternative approach to multiplication and division could involve converting these calculations into addition and subtraction calculations using
logarithms and exponentials. This could use an equilibria similar to that between Z+ and Z− for
subtraction as described earlier, but there would need to be some means of controlling the ease
with which an equilibrium would shift to one side. This is because, for large values, logarithm
functions are slowly increasing, whereas they are rapidly increasing for small positive values.
A potential problem with the use of these logarithm mechanisms is the ability to control when
a logarithm or an exponential needs to be taken. The various chemical reactions involved need
to be such that additions and subtractions only take place using chemicals that correspond to
logarithms only.
The concept of using a mathematical technique to simplify the arithmetic to be performed can
also be applied to calculating a convolution. The Fourier transform of a convolution is equal
to the product of the Fourier transforms of each of these two functions [24]. If a cell is able to
calculate Fourier transforms, then this result will reduce the problem to calculating a product.
However. this requires a chemical framework for calculating Fourier transforms of patterns and
a discussion of what this could constitute is beyond the scope of this thesis. A simpler approach
would be to consider a correlation coefficient, such as that proposed by Pearson [25]. Assume
that the concentrations of A∗ and B∗ are approximately equal to the mean concentrations of A0
and B0 along the whole membrane (as a consequence of being produced at the same rates but
14
Chapter 2 Model Formulation
diffusing relatively quickly). Then the concentrations of A∗ and B∗ can be subtracted from the
concentrations of A0 and B0 . This gives the deviations from the “mean” concentration along the
cell membrane. These deviations need to be compared on the same scale, but the concentrations
of the A-system chemicals could be on a very different scale to those of the B-system. Therefore, there needs to be a means of normalising these deviations. In Pearson’s coefficient, the
normalising factor is the standard deviation, which involves more arithmetic than computing the
deviations. The concentrations of A∗ and B∗ are also potential candidates, for the same reasons
previously discussed and do not require any calculations to be done. Another alternative would
be to consider the difference in the maximum and minimum values for the concentrations of A0
and B0 . These values would provide some indication of the scale of concentrations, though they
may not remain constant over time.
The essential point to consider is that, while the problem of gradient sensing might be a mathematically trivial problem, the arithmetic that a cell is expected to perform must be backed up
by chemical reaction schemes. With this in mind, two different coefficients for comparing the
concentrations of A0 and B0 are proposed. Both are constructed in such a way that they involve
only the basic operations of addition, subtraction, multiplication, and addition.
• The first involves a subtraction of the normalised concentrations and a maximum calculation.
( 0
)
n
ai b0i
dc1 1 X
=
max ∗ − ∗ , 0 − (1 + kC )c1 ,
dt
n i=1
ai bi
(2.5)
The principle is that the normalised response from the A-system should be greater than
that from the B-system. The calculation of the maximum can be considered as a subtraction where negative numbers are ignored; in the context of the equilibrium between Z+
and Z− described earlier, the concentration of Z− would not be measured. While this is
similar to the use of a threshold response, it is not directly based upon the detection sensitivities of the receptors and the threshold is not uniform across the whole cell membrane.
• The second involves a product of the normalised concentrations of A0 and B0 .
n
dc2 1 X a0i b0i
=
− (1 + kC )c2 ,
dt
n i=1 a∗i b∗i
(2.6)
This coefficient is motivated by Pearson’s coefficient, but a∗i and b∗i are not subtracted
15
Chapter 2 Model Formulation
from a0i and b0i . The reason is connected with existence of two reactions: one between A0
and A∗ and one between B0 and B∗ . These reactions carry out the subtraction, so that the
observed concentrations of A0 and B0 are actually the excesses against the concentrations
of A∗ and B∗ .
Note that both of these coefficients take a single value across the whole cell and not at each
grid point. Both coefficients include a decay term to represent the degradation of the chemical
used by cell to measure the value of the coefficient. In the MATLAB code (see Appendices),
this decay term takes the form (1 + kC )c, where kC is set at 10−3 . That is, the next value of c is
calculated using one of the formulae above, and then (1 + kC ) times the previous value of c is
subtracted. In the simulations of Chapter 3, the coefficient has an initial condition of zero.
16
Chapter 3
Results
Simulations to test the models described in Chapter 2 were performed using MATLAB version
8.1, with ode45 under a finite differences framework. MATLAB codes for these simulations
are provided in the Appendices. All simulations used n = 50 grid points spaced evenly along
a membrane length of 20 units, in addition to the parameter values and the initial conditions
detailed in Chapter 2. Initial conditions for a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) are specified
for each simulation detailed below. The values for a and b were kept constant for the duration
of each simulation.
3.1
Simulations with instantaneous diffusion of starred chemicals and 1D cells
Simulations were run to investigate the behaviour of the two models described in Sections 2.2.1
and 2.2.2, and the two coefficients, c1 and c2 , when diffusion of A∗ and B∗ was assumed to
be instantaneous and the cell was assumed to be one-dimensional. This was done to provide a
testing platform for the model, before diffusion as modelled in Section 2.2.3 was studied.
3.1.1
Absence of a chemoattractant gradient
Control simulations were performed to ascertain the dynamics of c1 and c2 in the absence of
a chemoattractant concentration gradient. Values of ai for 1 ≤ i ≤ n were kept equal to each
17
Chapter 3 Results
other. A variety of initial conditions for b were used, including identical bi values along the
length of the cell and linear gradients starting and ending at the ends of the cell. Further details
of the initial conditions and figures illustrating the results of these simulations are described in
Appendix A.5.1.
In Figures A.1, A.2, A.3, and A.4, the red line on each graph, corresponding to the absence
of chemoattractant A outside the cell, stays at zero for the duration of the simulation. This
is what was expected from all combinations of models and coefficients, since the intracellular
concentrations of A0 and A∗ should not increase under this scenario. The formulation of the
rate equations for both coefficients means that both coefficients remain unchanged from their
initial values of zero. In Figures A.2 and A.4, no response in the c2 coefficient was recorded
when there was an absence of B inside the cell. This is also an expected phenomenon, since
this situation should result in neither B0 nor B∗ being released and, therefore, the value of
dc2
dt
will go to zero. However, Figures A.1 and A.3 show that, when there is a zero concentration of
B and a non-zero concentration of A present, the c1 coefficient does not stay at zero. This result
is explained by the presence of A causing A0 and A∗ to be produced. This causes the sum term
in the expression for
dc2
dt
to increase and dominate the rate equation, until the value of c1 is large
enough for the decay term to counteract this.
Some general patterns emerge when comparing all of the results shown in Figures A.1, A.2, A.3,
and A.4. When the concentration of A around the cell is gradient-free and there is a concentration gradient for B, the responses of both coefficients are unchanged from the situation where
the gradient for B is in the opposite direction. This observation corresponds to the cell detecting
the presence of a chemoattractant in a way that does not change depending on the orientation of
the cell. In addition, when chemoattractant is present, the initial increase in the values of c1 and
c2 under either model occurs after 10 time steps. This feature of the model output is desirable,
as cells have been observed to respond quickly to the presence of chemoattractant in real life
[1]. Furthermore, for each combination of model, coefficient, and b value, the coefficient values
trail off to a background value independent of the the concentration of chemoattractant detected.
This indicates that the cell’s response to chemoattractant is not sustained, allowing the cell to
react to new stimuli.
Finally, recall that the motivation for the models formulated in this thesis is to devise a thresholdfree, gradient sensing mechanism for chemotaxing, eukaryotic cells. This means that a cell’s
18
Chapter 3 Results
response to the presence of chemoattractant should be irrespective of the concentration of
chemoattractant present. Results with the c1 coefficient display both models demonstrating
such behaviour. In Figure A.4, with the exception of the red line representing “no chemoattractant present”, the lines are close together in each graph. Although there are gaps between the
lines in all of these plots, these are small and occur at the start of the simulation. Figure A.2 also
provides another example of the threshold-free sensing behaviour that is desired, although the
separations of the lines at the start of the simulation are more pronounced. By contrast, the results with the c1 coefficient, shown in Figures A.1 and A.1, exhibit the threshold free behaviour,
but only when a concentration gradient for B is present. Furthermore, the gaps between the
lines appear to be more pronounced, but this could be due to the short time span covered in the
graphs.
Figure A.3 also illustrates some anomalous results with the c1 coefficient and the non-competitive
model when a fixed concentration of B is set. For example, when the concentration of B around
the cell is set to 4, the peak responses increase in size with decreasing concentrations of A. The
formula for
dc1
dt
should give a value of zero under the initial conditions and parameter values
used in the simulation, since the sum should equal zero when the concentrations of B-system
chemicals are higher than the concentrations of A-system chemicals. This behaviour is not
observed with the competitive-model, as shown in the graphs in the left column of Figure A.1.
3.1.2
Detection of a 2% chemoattractant gradient
Simulations were carried out to ascertain the responses of the two models and the two coefficients to shallow linear concentration gradients of 2% along the length of the cell. A variety of background concentrations were selected, with a and b concentration gradients fixed in
alignment or in opposite directions to each other. The minimum and maximum values for all
gradients were located at the ends of the cell. The specific values chosen for the endpoints are
given in Tables 3.1 and 3.2. The layout of graphs in each figure corresponds to the layout of
initial conditions in Table 3.2. The legend for the line colours is displayed in Table 3.1.
The dynamics of the c1 coefficient with the competitive model, displayed in 3.1a are dissimilar
to those seen in the simulations discussed in the previous section. At lower concentrations of
B, such as the [0,0.25] gradient, the absolute concentration of A does not have a significant
effect on c1 . However, when both the background concentration and concentration gradient of
19
Chapter 3 Results
400
600
Time
B grad [0,0.5]
800
1
0
200
400
600
Time
B grad [0,1]
800
0.8
0.6
0.4
0.2
0
200
400
600
Time
B grad [0,2]
800
0
200
400
600
Time
B grad [0,4]
800
0.5
0
200
400
600
800
200
400
600
Time
B grad [0.5,0]
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
1
0
600
Time
B grad [1,0]
1
0.5
0
1000
0
0.5
1000
1
0
0
Value of C1
1
0
1
0.5
1000
0.5
0
1.5
1000
Value of C1
200
Value of C1
Value of C1
Value of C1
0
0.5
0
Value of C1
Value of C1
1
0.5
0
Value of C1
B grad [0.25,0]
Value of C1
Value of C1
B grad [0,0.25]
1.5
0.8
0.6
0.4
0.2
0
600
Time
B grad [2,0]
600
Time
B grad [4,0]
1
0.5
0
1000
Time
600
Time
(a) Competitive model with coefficient c1
B grad [0.25,0]
Value of C2
Value of C2
B grad [0,0.25]
3
2
1
0
0
200
400
600
800
3
2
1
0
1000
0
200
400
600
Time
B grad [0.5,0]
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
2
1
0
Value of C2
Value of C2
3
0
200
400
600
Time
B grad [0,1]
800
3
2
1
0
0
200
400
600
800
3
2
1
0
1000
Value of C2
Value of C2
Time
B grad [0,0.5]
3
2
1
0
1000
0
Value of C2
Value of C2
2
0
200
400
600
Time
B grad [0,4]
800
4
2
0
0
200
400
600
800
2
600
Time
B grad [4,0]
4
2
0
1000
Time
600
Time
B grad [2,0]
4
0
1000
Value of C2
Value of C2
Time
B grad [0,2]
4
600
Time
B grad [1,0]
600
Time
(b) Competitive model with coefficient c2
Figure 3.1: Graphs of the values of c1 and c2 against time using the competitive (single-receptortype) model with 2% chemoattractant concentration gradients.
20
Chapter 3 Results
1000
Time
B grad [0,0.5]
1500
1
500
1000
Time
B grad [0,1]
1500
0
500
1000
Time
B grad [0,2]
1500
0.4
0.2
0
500
1000
Time
B grad [0,4]
1500
0.6
0.4
0.2
0
0
500
1000
Time
1500
0.8
0.6
0.4
0.2
0
1000
Time
B grad [0.5,0]
1500
2000
0
500
1000
Time
B grad [1,0]
1500
2000
0
500
1000
Time
B grad [2,0]
1500
2000
0
500
1000
Time
B grad [4,0]
1500
2000
0
500
1000
Time
1500
2000
0.6
0.4
0.2
0
2000
0.8
500
1
0
2000
0
0.5
2000
0.6
0
0
Value of C1
0.8
0.6
0.4
0.2
0
0
1
0.5
2000
Value of C1
500
Value of C1
Value of C1
Value of C1
0
0.5
0
Value of C1
Value of C1
1
0.5
0
Value of C1
B grad [0.25,0]
Value of C1
Value of C1
B grad [0,0.25]
0.8
0.6
0.4
0.2
0
2000
(a) Non-competitive model with coefficient c1
1000
Time
B grad [0,0.5]
1500
1
500
1000
Time
B grad [0,1]
1500
2
1
0
0
500
1000
Time
B grad [0,2]
1500
2
1
0
500
1000
Time
B grad [0,4]
1500
3
2
1
0
0
500
1000
Time
1500
1000
Time
B grad [0.5,0]
1500
2000
0
500
1000
Time
B grad [1,0]
1500
2000
0
500
1000
Time
B grad [2,0]
1500
2000
0
500
1000
Time
B grad [4,0]
1500
2000
0
500
1000
Time
1500
2000
3
2
1
3
2
1
3
2
1
0
2000
500
1
0
2000
0
2
0
2000
3
0
1
0
2000
Value of C2
0
2
0
2000
Value of C2
500
Value of C2
Value of C2
Value of C2
0
2
0
Value of C2
Value of C2
1
0
Value of C2
B grad [0.25,0]
Value of C2
Value of C2
B grad [0,0.25]
2
(b) Non-competitive model with coefficient c2
Figure 3.2: Graphs of the values of c1 and c2 against time using the non-competitive model with
2% chemoattractant concentration gradients.
21
Chapter 3 Results
a1 value
0.255
0.51
1.02
2.04
4.08
Linear gradients of a
an value Line colour on graph
0.25
Red
0.5
Magenta
1
Black
2
Blue
4
Green
Table 3.1: Initial conditions used for a and legend for Figures 3.1 and 3.2.
Linear gradients of b
b1 value bn value b1 value bn value
0
0.25
0.25
0
0
0.5
0.5
0
0
1
1
0
0
2
2
0
0
4
4
0
Table 3.2: Initial conditions used for b in Figures 3.1 and 3.2.
B are increased, small absolute concentrations of A result in a slow rise in the value of c1 over
time, whereas large concentrations of A cause the value of c1 to rise and fall sharply in the first
20 time steps, before decreasing steadily over the rest of the simulation time. Therefore, with
higher concentrations for B, the responses are not uniform across the range of A concentration
gradients that were used, even though all are of the same relative size of 2%. When the c2 coefficient is used, there is a sharp rise and fall at the start of the simulation, with very little difference
in the size and shape of this peak between the use of different background concentrations of A,
as shown in Figure 3.1b. However, with higher concentrations of B, beyond this sharp rise and
fall lies a slower rise and fall in c2 values, with the height of this second peak dependent on the
background concentration of A present.
The non-competitive model results, displayed in Figure 3.2, are very similar to the results with
the same model in the presence of no chemoattractant gradient. The background concentration
of A does not have a significant effect on the responses of the model. However, across all
the simulations, upon swapping the direction of the concentration gradient for B, no change
was observed in the simulation outputs. This is seen in Figures 3.1 and 3.2 by comparing the
left and right columns. This is undesirable, as the basic premise of these models is that a cell
should detect the direction of an external chemoattractant gradient and compare it with its own
intracellular gradient.
22
Chapter 3 Results
3.1.3
Detection of chemoattractant gradients of different sizes
As discussed in the previous section, the models do not appear to distinguish between the situation where a 2% chemoattractant gradient is in the same direction as the intracellular gradient,
and the situation when the gradients are pointing in opposite directions. To investigate whether
the size of the chemoattractant gradient is a factor in the detection of the gradient direction,
simulations were carried out to establish the effects of gradient sizes on the coefficient output.
Further details of the initial conditions for a and b and the results are given in Appendix A.6.
Figures A.9 and A.10 indicate that chemoattractant gradient size is important for determining
the direction. The left-hand column of each figure corresponds to the situations where intracellular and extracellular gradients are facing the same direction, whereas the right-hand column
of each figure corresponds to the situations where they are pointing in opposite directions. In
all combinations of model and coefficient, the outputs from applying 2% or 5% chemoattractant gradients, represented in the figures by the red and magenta lines respectively, are almost
identical between both columns. However, with larger gradient sizes of 10%, 50% and 100%,
represented by the black, blue, and green lines respectively, the differences between the two
columns are clearer.
The observed pattern of responses varies with combination of model and coefficient used. Assume that the responses to the 2% gradient are no different to the responses to no chemoattractant gradient, and denote these to be the “normal” response levels. Figure A.9 shows that both
c1 and c2 values under the competitive model are higher than normal when the A and B concentration gradients were in opposite directions, and lower when the gradients are set in the same
direction. Furthermore, the magnitude of the deviation from the normal level increases as the
size of the A and B concentration gradients increases. However, as shown in Figure A.10, the
non-competitive model, using either c1 or c2 , does not display such a trend, with responses going
above or below the normal response level, depending upon the strength of the chemoattractant
gradient.
3.1.4
Effects of kA∗ A0 and kB∗ B0 on coefficient values
Simulations were performed to study the effects of varying the two parameters representing
the reactions between A0 & A∗ and A0 & A∗ — namely, kA∗ A0 and kB∗ B0 . Details of the initial
23
Chapter 3 Results
conditions and the results of these simulations are detailed in Appendix A.7. The general trend
across all simulations is that increasing both parameters at the same time results in a faster and
higher initial response of the coefficient, but the values attained for the rest of the simulation
time are lower. Thus, the initial response is more acute, but the long term response is suppressed.
The latter is due to the fact that a higher rate of reaction between the activator and inhibitor in
each of the two chemical systems will mean that the activator will be removed very quickly.
One key observation is that the effects of increasing kB∗ B0 appeared to be less pronounced than
when increasing kA∗ A0 . Increasing kB∗ B0 by a factor of 500 appears to roughly double or triple
the values attained by the coefficients in the models, whereas the exponential increases in kA∗ A0
resulted in exponential increases in the maximum value attained by the coefficient.
3.2
Simulations with diffusion and 2D cells
The models were then extended to include periodic boundary conditions and diffusion terms,
as discussed in subsection 2.2.3. The use of periodic boundary conditions enables the cell to
be considered two-dimensional. Although the diffusion of the starred chemicals is assumed to
take place throughout the whole cell, the simulations account for diffusion along the membrane
only. This is because the cytosol is not included in the mathematical formulation of the models.
Control simulations were carried out with these models in the absence of a chemoattractant
gradient, similar to the simulations of the one-dimensional cell in Section 3.1.1. Details of
the initial conditions and the results of the simulations are shown in Appendix A.5.2. With the
competitive model, the absence of chemoattractant results in both coefficients remaining at zero,
irrespective of the concentrations of B, as shown by the red lines in the graphs of Figures A.5
and A.6. This is an outcome that was predicted for the same reasons as the 1D control simulations described earlier. In addition, when the composition of the gradients for b are swapped,
these is no difference in the responses displayed. This result corresponds to the cell having the
same response in gradient-free chemoattractant when rotated 180 degrees. However, between
different concentrations of chemoattractant, the values attained by the coefficients vary significantly from each other. Furthermore, the trend in the response is different to that observed
with the 1D models. Instead of a sharp peak followed by a steady decline, the coefficient values increase rapidly at a rate dependent on the chemoattractant concentration, before remaining
24
Chapter 3 Results
Variable
a
b
Value at endpoints
0.5
0.3
Value at midpoint
0.49
0.4
Table 3.3: Initial conditions used for a and b Figures 3.3 and 3.4.
constant for the remainder of the simulation. This does not conform to the threshold-free property that is desired, as the responses should not vary significantly with different background
levels of chemoattractant. The results obtained for the non-competitive model are similar with
the exception of the c1 coefficient, which decreases slowly in value after peaking earlier in the
simulation.
To investigate how the introduction of a chemoattractant gradient would affect the behaviour of
the models, simulations were carried out using only one pair of initial values for a and b. These
were composed of two linear gradients joined together at the midpoint, with each gradient
covering one half of the cell membrane. Details of the specific values used for the gradients are
given in Table 3.3. Figures 3.3 and 3.4 display the results of these simulations, including plots
of the concentrations of A0 , B0 , A∗ , B∗ , and unbound receptors present along the cell membrane
and the values of both coefficients against time.
Figures 3.3 and 3.4 display the coefficient value rising exponentially over the duration of the
simulation. This rise can be traced to the increasing values of
a0
a∗
and
b0
,
b∗
which control the rate
of change of the coefficients. These are explained by the values of a∗ and b∗ decreasing and the
values of a0 and b0 increasing with time. The large discrepancy between the concentrations of
the primed chemicals and the starred chemicals is due to their respective decay rate parameter
values of 10−5 and 10−1 . Thus the primed chemicals are able to build up in concentration,
whereas the starred chemicals decay rapidly. The receptor concentration in all simulations can
also be seen to decrease over time. Further tests (not shown) with the simulations involved using
higher values for kC in an attempt to maintain unbound receptor levels. However, this did not
yield any changes to this situation.
25
Chapter 3 Results
A bar
0
50
50
time
time
A prime
0
100
150
150
200
0
5
10
position
B prime
15
20
0
50
50
100
150
200
0
5
10
position
15
20
15
20
150
200
B bar
0
time
time
200
100
100
150
0
5
10
position
15
200
20
0
5
10
position
Receptor
0
time
50
100
150
200
0
5
10
position
15
20
C1
C2
8000
Value of C2
Value of C1
1.5
1
0.5
0
0
50
100
Time
150
6000
4000
2000
0
200
0
50
100
Time
Figure 3.3: Plots of a0 , a∗ , b0 , b∗ , concentration of unbound receptors, and coefficients c1 and c2
against time under the competitive (single-receptor-type) reaction-diffusion model.
26
Chapter 3 Results
A star
50
50
50
100
200
time
0
150
100
150
0
5
10
position
15
200
20
100
150
0
5
B prime
10
position
15
200
20
50
50
50
100
150
time
0
100
150
0
5
10
position
5
15
20
200
10
position
15
20
10
position
15
20
150
200
B receptor
0
200
0
B star
0
time
time
A receptor
0
time
time
A prime
0
100
150
0
5
10
position
15
20
200
0
5
C2
C1
3000
10
Value of C2
Value of C1
8
6
4
2000
1000
2
0
0
50
100
Time
150
0
200
0
50
100
Time
Figure 3.4: Plots of a0 , a∗ , b0 , b∗ , concentration of unbound receptors, and coefficients c1 and c2
against time under the non-competitive reaction-diffusion model.
27
Chapter 4
Discussion
While the 1D simulations of the competitive and non-competitive models have illustrated that
gradient sensing through template matching is possible, they also indicate that further refinements are required. For example, simulations with the non-competitive model demonstrate
threshold-free responses to 2% chemoattractant gradients, whereas this is not seen with the
competitive model. However, the competitive model responses to varying gradient sizes follow
a consistent trend with the orientation and sizes of the A and B concentration gradients, whereas
the non-competitive model does not respond consistently to varying gradient sizes. There are
also the anomalous results of the non-competitive model under constant gradient-free A and
B concentrations with the c1 coefficient that need to be investigated further. The models described in Section 2.2.3 that include diffusion terms also require refinement. The effect of the
decay term parameters for the primed and starred chemicals was observed to have affected the
response of the models. Therefore, further work to investigate the effects of parameter values
on the coefficient values obtained needs to be carried out.
Consideration also needs to be given to the biochemical model upon which the competitive and
non-competitive models are based. Although this model was motivated by the wave patterns
of intracellular components observed in Dictyostelium discoideum cells, the models proposed
in this thesis are entirely speculative. While it is possible that the mechanism described in
Section 2.1 might exist in cells, it will be worthwhile to consider existing knowledge of the
biochemical pathways linked to chemotaxis in eukaryotes. Furthermore, if certain features of
the model are considered in a different way (for example, the receptors acting as enzymes), then
this will have an effect on the structure of the mathematical model. In this case, Michaelis28
Chapter 4 Discussion
Menten kinetics could be considered.
One of the main concerns of this thesis has been the formulation of the cellular read-out of a
comparison between the intracellular and extracellular diffusion patterns. Both of the coefficients proposed in this thesis have been based on simple arithmetic operations and both appear
to have flaws when coupled with the models formulated in this thesis. Examples of these flaws
include the higher values of c1 attained under the non-competitive model with smaller A concentrations when the B concentration inside the cell is fixed and gradient-free (see Section 3.1.1),
and the inconsistent dynamics of c2 under the competitive model with diffusion with a B concentration gradient and without a A concentration gradient (see Figure A.6). There may be other,
biologically plausible ways in which the intracellular and extracellular gradients could be compared. Given that these coefficients should be based on a scheme of chemical reactions used to
perform the pattern comparison, rate equations that govern the dynamics of all chemicals used
to compare two gradients should be used, rather than a single numerical read-out representing
the concentration of a single chemical involved in the measurement. This is because the phenomena of decay and diffusion apply to these chemicals as well as to the chemicals directly
involved in generating the patterns.
If it can be assumed that a working model of gradient sensing by template matching can be
formulated, the next step is to consider how the model can be extended. A major step would
be to couple a functioning gradient sensing mechanism with other features of chemotaxis. For
example, if the intracellular and extracellular gradients match well, then the cell will continue to
generate patterns that match the direction of the extracellular pattern, leading to repolarisation
and movement up the concentration gradient of chemoattractant. However, if there is a mismatch of gradients, then the cell could instigate a bias that would cause the next intracellular
pattern that is generated to be a better fit to the extracellular gradient. If cell movement is to
coupled to the gradient sensing model, then perhaps it would be worth using a finite element
method scheme. As an example, in work by Neilson et al. [26], such a scheme is used to study
the Meinhardt model for chemotaxis.
29
Chapter 5
Conclusion
This thesis has considered using a novel template matching approach to model gradient sensing
in chemotactic eukaryotic cells, in an attempt to explain the extraordinary sensitivity of some
eukaryotic cells to shallow chemotactic gradients, whilst avoiding the use of a threshold concentration of chemoattractant required to elicit a response from the cell. The models proposed here
were first attempts to model gradient sensing in this way. Coefficients have also been formulated to describe two possible ways in which a cell could compare intracellular and extracellular
chemical concentration patterns.
The results of simulations using these models and coefficients indicate that neither model behaves completely as desired, though features such as threshold-free responses have been demonstrated. Refinements to the models should be made so that they reflect the real-life biochemistry
of chemotaxing cells. Further work is also required to develop a sensible means for a cell to
calculate differences between intracellular and extracellular concentration gradients, and this
requires a thorough consideration of the ways in which a cell can perform arithmetic using
chemicals and reaction systems.
It is hoped that a working model based on the template matching approach can be developed
and coupled to models of existing biochemical processes in the cell, such as repolarisation to
develop a front and back side to amplify the gradient signal, or pseudopod formation to aid cell
movement towards the source of the chemoattractant.
30
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Polarization of Single Eukaryotic Cells in Response to Guided Cues. PLoS Comput Biol,
7(4):e1001121, April 2011. doi: 10.1371/journal.pcbi.1001121.
[2] Philip K. Maini. On growth and form: spatiotemporal pattern formation in biology, chapter 7 — Some Mathematical Models for Biological Pattern Formation. Wiley Series in
Mathematical and Computational Biology. John Wiley & Sons, Ltd, 1999.
[3] Evanthia T. Roussos, John S. Condeelis, and Antonia Patsialou. Chemotaxis in cancer.
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[5] Fei Wang. The Signaling Mechanisms Underlying Cell Polarity and Chemotaxis. Cold
Spring Harbor Perspectives in Biology, 1(4), 2009. doi: 10.1101/cshperspect.a002980.
[6] J.D. Murray. Mathematical Biology II: Spatial Models and Biomedical Applications, volume 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, third edition, 2003.
[7] H. Meinhardt. Orientation of chemotactic cells and growth cones: models and mechanisms. Journal of Cell Science, 112(17):2867 – 2874, 1999.
[8] Daisuke Taniguchi, Shuji Ishihara, Takehiko Oonuki, Mai Honda-Kitahara, Kunihiko
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[9] Günther Gerisch, Britta Schroth-Diez, Annette Müller-Taubenberger, and Mary Ecke.
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2007.11.011.
[13] Paul W. Kriebel, Valarie A. Barr, and Carole A. Parent. Adenylyl Cyclase Localization
Regulates Streaming during Chemotaxis. Cell, 112(4):549 – 560, February 2003.
[14] Anna Barbara Hauert, Sibylla Martinelli, Camilla Marone, and Verena Niggli. Differentiated HL-60 cells are a valid model system for the analysis of human neutrophil migration
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854, 2002. ISSN 1357-2725. doi: 10.1016/S1357-2725(02)00010-9.
[15] Elena G. Yarmola, Thayumanasamy Somasundaram, Todd A. Boring, Ilan Spector, and
Michael R. Bubb. Actin-Latrunculin A Structure and Function: DIFFERENTIAL MODULATION OF ACTIN-BINDING PROTEIN FUNCTION BY LATRUNCULIN A. Journal
of Biological Chemistry, 275(36):28120 – 28127, 2000. doi: 10.1074/jbc.M004253200.
[16] Masahiro Ueda and Tatsuo Shibata. Stochastic Signal Processing and Transduction in
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ISSN 0006-3495. doi: 10.1529/biophysj.106.100263.
[17] P M Janssens and P J Van Haastert. Molecular basis of transmembrane signal transduction
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0146-0749.
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[19] Roberto Brunelli. Template Matching Techniques in Computer Vision: Theory and Practice. John Wiley & Sons, Ltd, 2009. ISBN 978-0-470-51706-2.
[20] P. S. Klein, T. J. Sun, C. L. 3rd Saxe, A. R. Kimmel, R. L. Johnson, and P. N. Devreotes.
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241(4872):1467 – 1472, September 1988. ISSN 0036-8075.
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[23] Emmanuel Décavé, Didier Rieu, Jérémie Dalous, Sébastien Fache, Yves Bréchet, Bertrand
Fourcade, Michel Satre, and Franz Bruckert. Shear flow-induced motility of Dictyostelium
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[24] Gerald B. Folland. Fourier Analysis and Its Applications. American Mathematical Society, 2009.
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[26] Matthew P. Neilson, John A. Mackenzie, Steven D. Webb, and Robert H. Insall. Modelling
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Stat. Res. Report, 5:1 – 21, 2010.
33
Appendix A
Appendices
A.1
List of Abbreviations
ACA Adenylyl cyclase
ATP Adenosine triphosphate
cAMP Cyclic adenosine monophosphate
LEGI Local excitation, global inhibition
ODE Ordinary differential equation
PDE Partial differential equation
PIP3 Phosphatidylinositol (3,4,5)-triphosphate
34
Chapter A Appendices
A.2
Code for running the simulations
Remark. In the code that follows, references to “a bar” and “b bar” relate to “a star” and “b
star”.
1
clear all
2
hold off
3
clc
4
5
%%% NOTE − Commands that should be commented out to allow this code to run
6
%%% have been uncommented for display in the thesis.
7
8
%time span
9
tfinal=2000;
10
tspan= 0:0.1:tfinal;
11
12
%domain length
13
L=20;%
14
15
%number of grid points
16
P.X=50;
17
18
%space step = Domain length / number of grid points
19
P.deltax = L/P.X;
20
21
%model parameters
22
23
P.Rtotal=20.; %total conc receptor (Competitive (single−receptor−type) ...
model ONLY)
24
25
P.RAtotal=1.; %total conc receptor A (Non−competitive model ONLY)
26
P.RBtotal=1.; %total conc receptor B (Non−competitive model ONLY)
27
28
P.kApp=1e−2; %production of A prime
29
P.kApm=1e−5; %decay of A prime
30
P.kApd=1e0; %depletion of A prime by A bar
31
P.kAbd=1e−1; %decay of A bar
32
35
Chapter A Appendices
33
P.kBpp=1e−2; %production of B prime
34
P.kBpm=1e−5; %decay of B prime
35
P.kBpd=1e0; %depletion of B prime by B bar
36
P.kBbd=1e−1; %decay of B bar
37
38
P.kRp=1e−5; %activation of receptors
39
40
P.kCd=1e−3; %decay of C coefficient
41
42
P.DAp=2e−4; %diffusion of A prime (two−dimensional diffusion models ONLY)
43
P.DAb=2e−3; %diffusion of A bar (two−dimensional diffusion models ONLY)
44
P.DBp=2e−4; %diffusion of B prime (two−dimensional diffusion models ONLY)
45
P.DBb=2e−3; %diffusion of B bar (two−dimensional diffusion models ONLY)
46
47
%initial conditions
48
49
%external gradient (A)
50
%Example for one−dimensional case
51
P.A=linspace(0.05,0.049,P.X)';
52
%Example for two−dimensional case (with diffusion)
53
P.A=vertcat(linspace(0.05,0.049,P.X/2)',linspace(0.049,0.05,P.X/2)');
54
55
%internal gradient (B)
56
%Example for one−dimensional case
57
P.B=linspace(0,1,P.X)';
58
%Example for two−dimensional case (with diffusion)
59
P.B=vertcat(linspace(0.3,0.4,P.X/2)',linspace(0.4,0.3,P.X/2)');
60
61
%other chemicals
62
A prime = zeros(P.X,1);
63
A bar = 1e−1*ones(P.X,1);
64
B prime = zeros(P.X,1);
65
B bar = 1e−1*ones(P.X,1);
66
C=0;
67
R = P.Rtotal*ones(P.X,1); %(Competitive (single−receptor−type) model ONLY)
68
RA=P.RAtotal*ones(P.X,1); %(Non−competitive model ONLY)
69
RB=P.RBtotal*ones(P.X,1); %(Non−competitive model ONLY)
70
71
36
Chapter A Appendices
72
%The following options allow step size to vary
73
%(to ensure conditional stability)
74
%in the two−dimensional case (with diffusion)
75
maxstep=0.1*0.5*P.deltaxˆ2/P.Da bar;
76
options=odeset('MaxStep',maxstep);
77
%Otherwise, the ODE45 options are left empty
78
options=[];
79
80
%In the case of the competitive (single−receptor−type) model
81
[t,y]=ode45(@dydt align1,tspan,[A prime;A bar;B prime;B bar;R;C],options,P);
82
%In the case of the non−competitive model
83
[t,y]=ode45(@dydt align2,tspan,[A prime;A bar;RA;B prime;B bar;RB;C],options,P);
37
Chapter A Appendices
A.3
Code containing the differential equations for the 1D and
2D Simulations of the competitive model
Remark. In the code that follows, references to “a bar” and “b bar” relate to “a star” and “b
star”.
1
function dydt=dydt align1(t,dydt,P)
2
3
A prime = dydt(1:P.X);
4
A bar = dydt(P.X+1:2*P.X);
5
B prime = dydt(2*P.X+1:3*P.X);
6
B bar = dydt(3*P.X+1:4*P.X);
7
R = dydt(4*P.X+1:5*P.X);
8
C = dydt(5*P.X+1);
9
10
%Periodic Boundary Conditions (2D Diffusion case ONLY)
11
P.Xm1=[1,1:P.X−1];
12
P.Xp1=[2:P.X,P.X];
13
14
%%% In the equations below, where there are two equations specified:
15
%%% − the top version is for the 1D simulation
16
%%% − the bottom version is for the 2D simulation with diffusion
17
18
%A prime
19
dydt(1:P.X) = P.kApp*P.A.*R − P.kApm*A prime − P.kApd*A prime.* A bar;
20
dydt(1:P.X) = P.kApp*P.A.*R − P.kApm*A prime − P.kApd*A prime.* A bar + ...
(P.DAp/((P.deltax)ˆ2))*(A prime(P.Xm1)−2*A prime+A prime(P.Xp1));
21
22
%A bar
23
dydt(P.X+1:2*P.X) = sum(A prime)/P.X − P.kApd*A prime.* A bar − (1 + ...
P.kAbd)* A bar;
24
dydt(P.X+1:2*P.X) = ...
(P.DAb/((P.deltax)ˆ2))*(A bar(P.Xm1)−2* A bar+A bar(P.Xp1)) + ...
P.kApp*P.A.*R − P.kApd*A prime.* A bar − (P.kAbd)* A bar;
25
26
%B prime
27
dydt(2*P.X+1:3*P.X) = P.kBpp*P.B.*R − P.kBpm*B prime − ...
38
Chapter A Appendices
P.kBpd*B prime.* B bar;
28
dydt(2*P.X+1:3*P.X) = P.kBpp*P.B.*R − P.kBpm*B prime − ...
P.kBpd*B prime.* B bar + ...
(P.DBp/((P.deltax)ˆ2))*(B prime(P.Xm1)−2*B prime+B prime(P.Xp1));
29
30
%B bar
31
dydt(3*P.X+1:4*P.X) = sum(B prime)/P.X − P.kBpd*B prime.* B bar − (1 + ...
P.kBbd)* B bar;
32
dydt(3*P.X+1:4*P.X) = ...
(P.DBb/((P.deltax)ˆ2))*(B bar(P.Xm1)−2* B bar+B bar(P.Xp1)) + ...
P.kBpp*P.B.*R − P.kBpd*B prime.* B bar − (P.kBbd)* B bar;
33
34
%R
35
dydt(4*P.X+1:5*P.X) = −P.kApp*P.A.*R − P.kBpp*P.B.*R + P.kRp*(P.Rtotal−R);
36
37
%Coefficient C1
38
difference = (A prime./A bar)−(B prime./B bar);
39
dydt(5*P.X+1) = sum(difference(difference>0))/P.X − (1+P.kCd)*C;
40
41
%Coefficient C2
42
A norm = (1./A bar).*(A prime);
43
B norm = (1./B bar).*(B prime);
44
dydt(5*P.X+1) = sum(A norm. * B norm)/P.X − (1+P.kCd)*C;
45
46
end
39
Chapter A Appendices
A.4
Code containing the differential equations for the 1D and
2D Simulations of the non-competitive model
Remark. In the code that follows, references to “a bar” and “b bar” relate to “a star” and “b
star”.
1
function dydt=dydt align2(t,dydt,P)
2
3
A prime = dydt(1:P.X);
4
A bar = dydt(P.X+1:2*P.X);
5
RA = dydt(2*P.X+1:3*P.X);
6
B prime = dydt(3*P.X+1:4*P.X);
7
B bar = dydt(4*P.X+1:5*P.X);
8
RB = dydt(5*P.X+1:6*P.X);
9
C = dydt(6*P.X+1);
10
11
%Periodic Boundary Conditions (2D Diffusion case ONLY)
12
P.Xm1=[1,1:P.X−1];
13
P.Xp1=[2:P.X,P.X];
14
15
%%% In the equations below, where there are two equations specified:
16
%%% − the top version is for the 1D simulation
17
%%% − the bottom version is for the 2D simulation with diffusion
18
19
%A prime
20
dydt(1:P.X) = P.kApp*P.A.*RA − P.kApm* A prime − P.kApd*A prime.* A bar;
21
dydt(1:P.X) = P.kApp*P.A.*RA − P.kApm* A prime − P.kApd*A prime.* A bar + ...
(P.DAp/((P.deltax)ˆ2))*(A prime(P.Xm1)−2*A prime+A prime(P.Xp1));
22
23
%A bar
24
dydt(P.X+1:2*P.X) = sum(A prime)/P.X − P.kApd*A prime.* A bar − (1 + ...
P.kAbd)* A bar;
25
dydt(P.X+1:2*P.X) = ...
(P.DAb/((P.deltax)ˆ2))*(A bar(P.Xm1)−2* A bar+A bar(P.Xp1)) + ...
P.kApp*P.A.*RA − P.kApd*A prime.* A bar − (P.kAbd)* A bar;
26
27
%RA
40
Chapter A Appendices
28
dydt(2*P.X+1:3*P.X) = −P.kApp*P.A.*RA + P.kRAp*(P.RAtotal−RA);
29
30
%B prime
31
dydt(3*P.X+1:4*P.X) = P.kBpp*P.B.*RB − P.kBpm* B prime − ...
P.kBpd*B prime.* B bar;
32
dydt(3*P.X+1:4*P.X) = P.kBpp*P.B.*RB − P.kBpm* B prime − ...
P.kBpd*B prime.* B bar + ...
(P.DBp/((P.deltax)ˆ2))*(B prime(P.Xm1)−2*B prime+B prime(P.Xp1));
33
34
%B bar
35
dydt(4*P.X+1:5*P.X) = sum(B prime)/P.X − P.kBpd*B prime.* B bar − (1 + ...
P.kBbd)* B bar;
36
dydt(4*P.X+1:5*P.X) = ...
(P.DBb/((P.deltax)ˆ2))*(B bar(P.Xm1)−2* B bar+B bar(P.Xp1)) + ...
P.kBpp*P.B.*RB − P.kBpd*B prime.* B bar − (P.kBbd)* B bar;
37
38
%RB
39
dydt(5*P.X+1:6*P.X) = −P.kBpp*P.B.*RB + P.kRBp*(P.RBtotal−RB);
40
41
%Coefficient C1
42
difference = (A prime./A bar)−(B prime./B bar);
43
dydt(6*P.X+1) = sum(difference(difference>0))/P.X − (1+P.kCd)*C;
44
45
%Coefficient C2
46
A norm = (1./A bar).*(A prime);
47
B norm = (1./B bar).*(B prime);
48
dydt(6*P.X+1) = sum(A norm. * B norm)/P.X − (1+P.kCd)*C;
49
50
end
41
Chapter A Appendices
Initial values of ai along the cell
Value of all ai Line colour on graph
0
Red
0.5
Magenta
1
Black
2
Blue
4
Green
Table A.1: Initial conditions used for a and legend for Figures A.1 to A.8.
Value of all bi
0
0.25
0.5
1
2
4
Linear gradients of b
b1 value bn value b1 value bn value
No other graphs in this row
0
0.25
0.25
0
0
0.5
0.5
0
0
1
1
0
0
2
2
0
0
4
4
0
Table A.2: Initial conditions used for b and layout of graphs in Figures A.1, A.2, A.3, and A.4.
A.5
Simulations studying the effects of no chemoattractant
gradient on coefficient values
Control simulations of both the competitive and non-competitive models, with and without
diffusion terms, and using coefficients c1 and c2 , were carried out to assess the effects of a fixed
chemoattractant concentration around the cell.
A.5.1
A one-dimensional cell and instantaneous diffusion of starred chemicals
The models of Sections 2.2.1 and 2.2.2 were tested on a 1D cell. The initial conditions for a
and b are listed in Tables A.1 and A.2. The layout of Table A.2 also corresponds to the graph
layout in Figures A.1, A.2, A.3, and A.4.
42
Chapter A Appendices
1
100
50
Time
B constant 0.5
100
1
0.5
50
Time
B constant 1
100
Value of C1
0.6
0.4
0.2
0
0
0
50
Time
B constant 2
100
Value of C1
0
0.4
0.2
0
0
0
x 10
50
Time
B constant 4
100
−2
−4
0
50
Time
100
0
50
Time
B grad [0,0.5]
100
Value of C1
0
Value of C1
0
1
0.5
0
0
50
Time
B grad [0,1]
100
Value of C1
1
0.5
B grad [0.25,0]
Value of C1
B grad [0,0.25]
1.5
1
0.5
0
1
0.5
0
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0
50
Time
B grad [0,2]
100
Value of C1
50
Time
B constant 0.25
0
50
Time
B grad [0,4]
100
Value of C1
0
Value of C1
0
−3
Value of C1
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
B constant 0
2
0
50
Time
100
1.5
1
0.5
0
0
50
Time
B grad [0.5,0]
100
0
50
Time
B grad [1,0]
100
0
50
Time
B grad [2,0]
100
0
50
Time
B grad [4,0]
100
0
50
Time
100
1
0.5
0
1
0.5
0
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
Figure A.1: Graphs of the value of the c1 coefficient against time using the competitive (singlereceptor-type) model in the absence of a chemoattractant concentration gradient.
43
Chapter A Appendices
x 10
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
B constant 0
−2
−4
100
0
50
Time
B constant 1
100
4
2
0
50
Time
B constant 2
100
Value of C2
0
4
2
0
0
50
Time
B constant 4
100
4
2
0
0
50
Time
100
3
2
1
0
3
2
1
0
Value of C2
0
50
Time
B grad [0,0.5]
100
Value of C2
50
Time
B constant 0.5
B grad [0.25,0]
0
50
Time
B grad [0,1]
100
Value of C2
0
3
2
1
0
0
50
Time
B grad [0,2]
100
Value of C2
Value of C2
B grad [0,0.25]
Value of C2
3
2
1
0
100
Value of C2
3
2
1
0
50
Time
B constant 0.25
Value of C2
Value of C2
Value of C2
Value of C2
Value of C2
Value of C2
0
4
2
0
0
50
Time
B grad [0,4]
100
Value of C2
Value of C2
−3
0
4
2
0
0
50
Time
100
3
2
1
0
3
2
1
0
3
2
1
0
0
50
Time
B grad [0.5,0]
100
0
50
Time
B grad [1,0]
100
0
50
Time
B grad [2,0]
100
0
50
Time
B grad [4,0]
100
0
50
Time
100
4
2
0
4
2
0
Figure A.2: Graphs of the value of the c2 coefficient against time using the competitive (singlereceptor-type) model in the absence of a chemoattractant concentration gradient.
44
Chapter A Appendices
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
2000
0.6
0.4
0.2
0
0.3
0.2
0.1
0
1000 1500
Time
B constant 0.5
2000
Value of C1
500
0
500
1000 1500
Time
B constant 1
2000
Value of C1
0.8
0.6
0.4
0.2
0
0
0
500
1000 1500
Time
B constant 2
2000
Value of C1
0
0
500
1000 1500
Time
B constant 4
2000
0.1
0.05
0
0
500
1000
Time
1500
2000
1
0.5
0
0
500
1000 1500
Time
B grad [0,0.5]
2000
Value of C1
0.5
B grad [0.25,0]
Value of C1
B grad [0,0.25]
1
1
0.5
0
0.8
0.6
0.4
0.2
0
0.6
0.4
0.2
0
0.6
0.4
0.2
0
0
500
1000 1500
Time
B grad [0,1]
2000
Value of C1
1000 1500
Time
B constant 0.25
0
500
1000 1500
Time
B grad [0,2]
2000
Value of C1
500
0
500
1000 1500
Time
B grad [0,4]
2000
Value of C1
0
Value of C1
1.5
1
0.5
0
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
B constant 0
0
500
1000
Time
1500
2000
1
0.5
0
0
500
1000 1500
Time
B grad [0.5,0]
2000
0
500
2000
0
500
2000
0
500
2000
0
500
1
0.5
0
0.8
0.6
0.4
0.2
0
0.6
0.4
0.2
0
0.6
0.4
0.2
0
1000 1500
Time
B grad [1,0]
1000 1500
Time
B grad [2,0]
1000 1500
Time
B grad [4,0]
1000
Time
1500
2000
Figure A.3: Graphs of the value of the c1 coefficient against time using the non-competitive
model in the absence of a chemoattractant concentration gradient.
45
Chapter A Appendices
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
B constant 0
x 10
−2
−4
500
0
100
200 300 400
Time
B constant 0.5
500
Value of C2
0
2
1
3
2
1
0
100
200 300 400
Time
B constant 1
500
Value of C2
3
2
1
0
0
0
100
200 300 400
Time
B constant 2
500
Value of C2
0
0
100
200 300 400
Time
B constant 4
500
4
2
0
0
100
200 300
Time
400
500
2
1
0
0
100
200 300 400
Time
B grad [0,0.5]
500
Value of C2
1
2
1
0
2
1
0
3
2
1
0
3
2
1
0
0
100
200 300 400
Time
B grad [0,1]
500
Value of C2
2
B grad [0.25,0]
Value of C2
B grad [0,0.25]
0
100
200 300 400
Time
B grad [0,2]
500
Value of C2
200 300 400
Time
B constant 0.25
Value of C2
100
Value of C2
Value of C2
Value of C2
Value of C2
Value of C2
Value of C2
0
0
100
200 300 400
Time
B grad [0,4]
500
Value of C2
Value of C2
−3
0
0
100
200 300
Time
400
500
2
1
0
0
100
200 300 400
Time
B grad [0.5,0]
500
0
100
200 300 400
Time
B grad [1,0]
500
0
100
200 300 400
Time
B grad [2,0]
500
0
100
200 300 400
Time
B grad [4,0]
500
0
100
2
1
0
2
1
0
3
2
1
0
3
2
1
0
200 300
Time
400
500
Figure A.4: Graphs of the value of the c2 coefficient against time using the non-competitive
model in the absence of a chemoattractant concentration gradient.
46
Chapter A Appendices
A.5.2
A two-dimensional cell and diffusion of primed and starred chemicals
The models of Section 2.2.3 were tested on a 2D cell using periodic boundary conditions. The
initial conditions for a are as listed previously in Table A.1, whereas the initial conditions for
b are listed in Table A.3. The initial conditions for b were composed of two linear gradients
connected together at the midpoint of the cell. The layout of Table A.3 also corresponds to the
graph layout in Figures A.5, A.6, A.7, and A.8.
47
Chapter A Appendices
Value of all bi
Value at endpoints
(b1 and bn )
0
0.25
0.5
1
2
4
0
0
0
0
0
Linear gradients of b
Value at midpoint Value at endpoints
(b n2 and b 2n +1 )
(b1 and bn )
No other graphs in this row
0.25
0.25
0.5
0.5
1
1
2
2
4
4
Value at midpoint
(b n2 and b 2n +1 )
0
0
0
0
0
Table A.3: Initial conditions used for b and layout of graphs in Figures A.5, A.6, A.7, and A.8.
2000
1000
Time
B constant 1
2000
400
200
0
1000
Time
B constant 2
2000
200
100
0
0
Value of C1
−3
0
x 10
1000
Time
B constant 4
2000
−2
−4
0
1000
Time
2000
0
1000
Time
B grad [0,1,0]
2000
500
0
800
600
400
200
0
600
400
200
0
0
1000
Time
B grad [0,2,0]
1000
0
2000
Value of C1
1000
Time
B grad [0,0.5,0]
1000
Value of C1
0
1500
1000
500
0
0
Value of C1
0
0
2000
Value of C1
1000
Time
B constant 0.5
Value of C1
Value of C1
Value of C1
800
600
400
200
0
0
Value of C1
0
1000
2000
1500
1000
500
0
0
1000
Time
B grad [0,4,0]
0
1000
Time
2000
2000
2000
0
1000
Time
B grad [0.5,0,0.5]
2000
0
1000
Time
B grad [1,0,1]
2000
0
1000
Time
B grad [2,0,2]
2000
0
1000
Time
B grad [4,0,4]
2000
0
1000
Time
2000
1000
Value of C1
500
B grad [0.25,0,0.25]
Value of C1
B grad [0,0.25,0]
2000
Value of C1
1000
Time
B constant 0.25
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
Value of C1
0
1000
Value of C1
Value of C1
Value of C1
B constant 0
15000
10000
5000
0
500
0
800
600
400
200
0
600
400
200
0
Figure A.5: Graphs of the values of the c1 coefficient against time using the competitive (singlereceptor-type) model with diffusion, in the absence of a chemoattractant concentration gradient.
48
Chapter A Appendices
x 10
−4
1
0
8
x 10
1000
Time
B constant 1
2000
1
0
7
x 10
15
10
5
0
0
7
x 10
1000
Time
B constant 2
1000
Time
B constant 4
2000
2000
1000
Time
2000
Value of C2
1000
Time
B grad [0,0.5,0]
2000
2
0
0
x 10
1000
Time
B grad [0,1,0]
2000
2
0
x 10
1000
Time
B grad [0,2,0]
2000
2
0
x 10
1000
Time
B grad [0,4,0]
2000
2
0
1000
Time
0
1000
Time
x 10 B grad [0.5,0,0.5]
2000
0
1000
Time
B grad [1,0,1]
2000
1000
Time
B grad [2,0,2]
2000
1000
Time
B grad [4,0,4]
2000
1000
Time
2000
4
2
0
x 10
4
2
0
0
x 10
4
2
0
0
8
4
0
0
8
4
0
2
8
4
0
4
8
4
8
5
0
x 10
8
10
0
0
8
2
0
0
8
2
0
2
Value of C2
2000
4
Value of C2
x 10
1000
Time
B constant 0.5
8
x 10 B grad [0.25,0,0.25]
B grad [0,0.25,0]
Value of C2
0
Value of C2
0
x 10
Value of C2
1
Value of C2
Value of C2
8
2
8
Value of C2
2000
Value of C2
x 10
1000
Time
B constant 0.25
Value of C2
Value of C2
8
Value of C2
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
−2
0
Value of C2
B constant 0
Value of C2
Value of C2
−3
0
2000
x 10
4
2
0
0
Figure A.6: Graphs of the values of the c2 coefficient against time using the competitive (singlereceptor-type) model with diffusion, in the absence of a chemoattractant concentration gradient.
49
Chapter A Appendices
2000
2000
10000
5000
0
1000
Time
B constant 1
10000
5000
0
1000
Time
B constant 2
10000
5000
0
1000
Time
B constant 4
2000
0
1000
Time
2000
0
1000
Time
B grad [0,1,0]
5000
0
0
1000
Time
B grad [0,2,0]
5000
6000
4000
2000
0
0
1000
Time
B grad [0,4,0]
0
1000
Time
1000
Time
B grad [0.5,0,0.5]
2000
0
1000
Time
B grad [1,0,1]
2000
0
1000
Time
B grad [2,0,2]
2000
0
1000
Time
B grad [4,0,4]
2000
0
1000
Time
2000
5000
0
10000
5000
0
10000
2000
2000
0
10000
2000
10000
0
15000
10000
5000
0
2000
10000
2000
4000
0
0
Value of C1
0
2000
5000
2000
Value of C1
0
1000
Time
B grad [0,0.5,0]
10000
2000
Value of C1
0
0
Value of C1
1000
Time
B constant 0.5
Value of C1
0
Value of C1
0
Value of C1
5000
B grad [0.25,0,0.25]
Value of C1
B grad [0,0.25,0]
15000
10000
5000
0
Value of C1
1000
Time
B constant 0.25
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
Value of C1
0
10000
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
Value of C1
B constant 0
15000
10000
5000
0
5000
0
6000
4000
2000
0
Figure A.7: Graphs of the values of the c1 coefficient against time using the non-competitive
model with diffusion, in the absence of a chemoattractant concentration gradient.
50
Chapter A Appendices
x 10
−2
−4
2000
x 10
1
0
Value of C2
x 10
0
x 10
1000
Time
B constant 2
2000
1
0
0
8
2
x 10
1000
Time
B constant 4
2000
0
1000
Time
0
2
x 10
2000
2000
1000
Time
B grad [0,1,0]
0
0
x 10
1000
Time
B grad [0,2,0]
x 10
1000
Time
B grad [0,4,0]
2000
2000
1000
Time
0
1000
Time
x 10 B grad [0.5,0,0.5]
2000
0
1000
Time
B grad [1,0,1]
2000
1000
Time
B grad [2,0,2]
2000
1000
Time
B grad [4,0,4]
2000
1000
Time
2000
5
0
2
x 10
1
0
0
x 10
2
1
0
0
8
1
0
0
8
2
0
2
8
1
0
7
x 10 B grad [0.25,0,0.25]
10
2000
2
0
4
7
1
8
1
0
0
8
2
1000
Time
B grad [0,0.5,0]
5
8
1
0
0
10
2000
2
8
Value of C2
1000
Time
B constant 1
0
x 10
Value of C2
0
2
7
2
8
Value of C2
2000
Value of C2
Value of C2
8
1000
Time
B constant 0.5
B grad [0,0.25,0]
Value of C2
0
Value of C2
0
x 10
Value of C2
5
4
Value of C2
7
10
Value of C2
x 10
1000
Time
B constant 0.25
Value of C2
7
Value of C2
0
Value of C2
A conc 0
A conc 0.5
A conc 1
A conc 2
A conc 4
B constant 0
Value of C2
Value of C2
−3
0
2000
x 10
2
1
0
0
Figure A.8: Graphs of the values of the c2 coefficient against time using the non-competitive
model with diffusion, in the absence of a chemoattractant concentration gradient.
51
Chapter A Appendices
a1 value
1
1
1
1
1
Linear gradients of a
an value Line colour on graph
1.02
Red
1.05
Magenta
1.1
Black
1.5
Blue
2
Green
Table A.4: Initial conditions used for a and legend for Figures A.9 and A.10.
Linear gradients of b
b1 value bn value b1 value bn value
0
0.25
0.25
0
0
0.5
0.5
0
0
1
1
0
0
2
2
0
0
4
4
0
Table A.5: Initial conditions used for b in Figures A.9 and A.10.
A.6
Simulations studying the effects of chemoattractant concentration gradient size on coefficient values
Simulations were carried out of both the competitive and non-competitive models of Sections 2.2.1 and 2.2.2, to assess the responses of coefficients c1 and c2 to different sizes of
chemoattractant gradients. These simulations were run on a one-dimensional cell. The initial conditions for a and b are listed in Tables A.4 and A.5. Figures A.9 and A.10 describe
the results of the simulations. The graph layout in each figure corresponds to the layout of
Table A.5. The line colour legend is given in Table A.4.
52
Chapter A Appendices
20
40
60
Time
B grad [0,0.5]
80
0.4
0.2
40
60
Time
B grad [0,1]
80
100
Value of C1
20
0.4
0.2
0
0
20
40
60
Time
B grad [0,2]
80
100
0.6
0.4
0.2
0
0
20
40
60
Time
B grad [0,4]
80
100
0.8
0.6
0.4
0.2
0
0
20
40
60
80
0.5
0
Value of C1
0.6
0
1
100
Value of C1
Value of C1
Value of C1
0
0.8
0
Value of C1
Value of C1
0.5
0
Value of C1
B grad [0.25,0]
Value of C1
Value of C1
B grad [0,0.25]
1
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0
20
40
60
Time
B grad [0.5,0]
80
100
0
20
40
60
Time
B grad [1,0]
80
100
0
20
40
60
Time
B grad [2,0]
80
100
0
20
40
60
Time
B grad [4,0]
80
100
0
20
40
80
100
1
0.5
0
100
Time
60
Time
(a) Competitive model with coefficient c1
1
20
40
60
Time
B grad [0,0.5]
80
Value of C2
2
1
0
20
40
60
Time
B grad [0,1]
80
2
1
0
20
40
60
Time
B grad [0,2]
80
2
0
20
40
60
80
20
40
60
Time
B grad [0.5,0]
80
100
0
20
40
60
Time
B grad [1,0]
80
100
0
20
40
60
Time
B grad [2,0]
80
100
0
20
40
60
Time
B grad [4,0]
80
100
0
20
40
80
100
1
2
1
3
2
1
0
100
0
2
0
100
4
0
1
0
100
3
0
2
0
100
Value of C2
Value of C2
0
3
0
Value of C2
Value of C2
2
0
Value of C2
B grad [0.25,0]
Value of C2
Value of C2
B grad [0,0.25]
3
Value of C2
Value of C2
Time
B grad [0,4]
4
2
0
0
20
40
60
80
3
2
1
0
100
Time
60
Time
(b) Competitive model with coefficient c2
Figure A.9: Graphs of the values of the two coefficients against time under different gradients
for a and b using the competitive (single-receptor-type) model.
53
Chapter A Appendices
Value of C1
B grad [0.25,0]
0
200
400
600
Time
B grad [0,0.5]
800
1000
Value of C1
Value of C1
Value of C1
B grad [0,0.25]
0.8
0.6
0.4
0.2
0
0.6
0.4
0.2
0
0
200
400
600
800
0.8
0.6
0.4
0.2
0
0
200
400
600
Time
B grad [0.5,0]
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
0.8
0.6
0.4
0.2
0
1000
Value of C1
Value of C1
Time
B grad [0,1]
0.4
0.2
0
0
200
400
600
800
0.6
0.4
0.2
0
1000
Value of C1
Value of C1
Time
B grad [0,2]
0.6
0.4
0.2
0
0
200
400
600
800
1000
Value of C1
Value of C1
0.4
0.2
0
0
200
400
600
800
1000
600
Time
B grad [2,0]
0.8
0.6
0.4
0.2
0
Time
B grad [0,4]
0.6
600
Time
B grad [1,0]
0.8
0.6
0.4
0.2
0
600
Time
B grad [4,0]
Time
600
Time
(a) Non-competitive model with coefficient c1
400
600
Time
B grad [0,0.5]
800
1
200
400
600
Time
B grad [0,1]
800
2
1
0
0
200
400
600
Time
B grad [0,2]
800
2
1
0
200
400
600
Time
B grad [0,4]
800
3
2
1
0
0
200
400
600
800
Time
400
600
Time
B grad [0.5,0]
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
0
200
400
800
1000
600
Time
B grad [1,0]
2
1
600
Time
B grad [2,0]
2
1
600
Time
B grad [4,0]
3
2
1
0
1000
200
1
0
1000
0
2
0
1000
3
0
1
0
1000
Value of C2
0
2
0
1000
Value of C2
200
Value of C2
Value of C2
Value of C2
0
2
0
Value of C2
Value of C2
1
0
Value of C2
B grad [0.25,0]
Value of C2
Value of C2
B grad [0,0.25]
2
600
Time
(b) Non-competitive model with coefficient c2
Figure A.10: Graphs of the values of the two coefficients against time under different gradients
for a and b using the non-competitive model.
54
Chapter A Appendices
A.7
Simulations studying the effects of kA∗A0 and kB∗B0 on coefficient values
Simulations were carried out to establish the effects of varying the two parameters representing
the reactions between A0 & A∗ and A0 & A∗ — namely, kA∗ A0 and kB∗ B0 . These parameters were
set to values of 0.1, 0.5, 1, 5, 10, 50. The models of Sections 2.2.1 and 2.2.2 were used. As
well as the initial conditions and parameter settings described previously in Section 2.2, the
simulation was run using initial conditions for a and b detailed in Table A.6 and the cell was
assumed to be one-dimensional, as in Section A.5.1. The results of these simulations are shown
in Figures A.11, A.12, A.13, and A.14. The colours of the lines on each of the graphs represent
different initial conditions for a, as indicated in Table A.6. The correspondance between the
choice of parameter values and the resulting graph is explained in Table A.7.
a1 value
0.25
0.5
1
1
1
b1 value
0.25
Linear gradients of a
an value Line colour on graph
1
Red
1
Magenta
1
Black
2
Blue
4
Green
Linear gradient of b
bn value Line colour on graph
1
N/A
Table A.6: Initial conditions used for a & b and legend for Figures A.11 , A.12, A.13 and A.14.
Guide to plot layout with respect to choice of (kA∗ A0 , kB∗ B0 )
(0.1,50) (0.5,50) (1,50) (0.1,50) (10,50) (50,50)
(0.1,10) (0.5,10) (1,10) (0.1,10) (10,10) (50,10)
(0.1,5)
(0.5,5)
(1,5)
(5,5)
(10,5)
(50,5)
(0.1,1)
(0.5,1)
(1,1)
(0.1,1)
(10,1)
(50,1)
(0.1,0.5) (0.5,0.5) (1,0.5) (0.1,0.5) (10,0.5) (50,0.5)
(0.1,0.1) (0.5,0.1) (1,0.1) (0.1,0.1) (10,0.1) (50,0.1)
Table A.7: Table indicating the values used for kA∗ A0 and kB∗ B0 to generate each of the graphs in
Figures A.11, A.12, A.13 and A.14.
55
Value
Value
Value
Value
Value
Value
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
1.5
0
0.5
1
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
3
2
1
0
3
2
1
0
3
2
1
0
3
2
1
0
1
0
3
2
0
1
2
3
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
56
Value
0.4
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Chapter A Appendices
Figure A.11: Graphs of the effect of different values of kA∗ A0 and kB∗ B0 and different a gradients
on the c1 coefficient using the competitive (single-receptor-type) model.
Value
Value
Value
Value
Value
Value
0
1
2
3
2
1
0
3
2
1
0
0
2
4
0
2
4
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
3
2
1
0
2
4
0
2
4
0
2
4
0
2
4
8
6
4
2
0
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
3
2
1
0
0
2
4
0
2
4
0
2
4
0
2
4
6
8
6
4
2
0
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
6
4
2
0
6
4
2
0
6
4
2
0
8
6
4
2
0
8
6
4
2
0
0
5
10
15
Value
Value
Value
Value
Value
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
5
10
0
5
10
0
5
10
0
5
10
15
15
10
5
0
0
10
20
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
60
40
20
0
60
40
20
0
60
40
20
0
80
60
40
20
0
80
60
40
20
0
150
100
50
0
Value
Value
Value
Value
57
Value
6
4
2
0
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Chapter A Appendices
Figure A.12: Graphs of the effect of different values of kA∗ A0 and kB∗ B0 and different a gradients
on the c2 coefficient using the competitive (single-receptor-type) model.
Value
Value
Value
Value
Value
Value
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0.5
1
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
58
Value
1
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
10
20
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Chapter A Appendices
Figure A.13: Graphs of the effect of different values of kA∗ A0 and kB∗ B0 and different a gradients
on the c1 coefficient using the non-competitive model.
Value
0
1
2
0
1
1.5
1
0.5
0
1.5
1
0.5
0
1.5
1
0.5
0
1.5
1
0.5
0
Value
Value
Value
Value
Value
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
3
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
1
2
0
1
2
0
1
2
0
1
2
0
1
2
3
2
1
0
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
Value
0
2
4
0
2
4
0
2
4
0
2
4
0
2
4
6
4
2
0
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
6
4
2
0
6
4
2
0
6
4
2
0
8
6
4
2
0
8
6
4
2
0
0
5
10
Value
Value
Value
Value
Value
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Value
Value
Value
Value
Value
59
Value
2
30
20
10
0
30
20
10
0
30
20
10
0
0
20
40
0
20
40
0
20
40
60
0
0
0
0
0
0
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
1000
Time
2000
2000
2000
2000
2000
2000
Chapter A Appendices
Figure A.14: Graphs of the effect of different values of kA∗ A0 and kB∗ B0 and different a gradients
on the c2 coefficient using the non-competitive model.