A Sense of Place-A Model of Synaptic Plasticity in the Hippocampus
Colette_Parker_Final_Copy
1. Models of Cutaneous Mosaicism:
Congenital Melanocytic Naevi
G14MBD
MSc Dissertation in
Mathematical Medicine and Biology
2014/15
School of Mathematical Sciences
University of Nottingham
Colette Parker
Supervisor: Prof. Stephen Coombes & Prof. Markus Owen
With thanks to Dr. Veronica Kinsler, Dept. Dermatology, Great Ormond
Street Hospital for Children NHS Trust
I have read and understood the School and University guidelines on plagiarism. I confirm
that this work is my own, apart from the acknowledged references.
2. Abstract
Mosaicism of the skin depicts two or more differing populations of cells with non-
identical genotypes, which is sometimes evident through a variation in pigmentation.
There are many different types of skin mosaics. Congenital melanocytic naevi are clusters
of benign immature melanocytes (a melanin producing cell). The naevi originate from
mutations that occur in embryonic development. This report aims to conduct a review of
previous research into the origins of the naevi (both experimental and computational), and
to construct a mathematical model to represent the proliferation and migration of mutated
melanoblasts (a cell that is a precursor of a melanocyte) in a developing embryo. The
images and statistics produced show many of the features and common traits displayed
and found in the naevi studied clinically.
5. 1 Introduction
Mosaicism is a term used to described a genetic state whereby an animal (in our case
human) exists with two genetically differing populations of cells. In this dissertation, we
aim to present the reader with a review of literature relevant to different types of cutaneous
(skin) mosaicism, and to give an overall understanding of the condition (section 2). We will
critique models that have been presented to try and demonstrate how the pigmentation
patterning in can skin occur in this genetic disorder (section 3). The remainder of the
report will introduce a new model, highlighting the methods used to develop it as well as
the limitations it holds. (section 4). It is hoped that with further technological advances,
as well as dermatological and embryological discoveries, this model could be enriched and
expanded in order to better understand how certain patterns of mosaicism develop in
embryos.
2 Literature Review
Cutaneous mosaicism has been studied by dermatologists and geneticists for many years
(from the early 1900s to present day), but many other areas of research such as embry-
ology, genetics, dermatology and oncology have been vital to understanding it further.
For example, through embryological research it has been possible to understand how pat-
terning originates in early stage embryos [40]. In this section, a summary of the different
types of mosaicism as well as a brief history of previous research, will be given. Not all
medical opinions are in line with each other, so the following gives a review of a collection
of different thought processes. Research in this area is ongoing.
2.1 Different types of Cutaneous Mosaicism
There are many different types of cutaneous mosaicism classified by certain phenotypical
characteristics. This section aims to outline the differences between the diagnoses and
highlight relevant research for each.
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6. Blaschko’s Lines The lines of Blaschko were originally discussed by Alfred Blaschko
in 1901 [5]. It is widely believed that Blaschko’s lines reflect patterns of ectodermal
cell development, which only become visible when mosaicism exists. The early embryo
consists of three primary layers, ectoderm, mesoderm and endoderm; the ectoderm is the
layer from which epidermal skin cells and pigment cells are formed. Linear lines are seen
on the extremities, whilst lines on the anterior trunk and back are S-shaped and V-shaped
respectively [2]. Blaschko’s lines are found in many cases of mosaicism, for example Type
1a: Incontinentia pigmenti (IP).
Figure 1: Figure 1a: Blaschko’s Lines [5]; here the shape of the lines is visible; linear on
extremities, S-shaped on the anterior trunk and V-shaped on the back. These lines exist
in all humans due to the development process, however they are visible due to variation
in pigmentation in patients with congenital mosaicism. Figure 1b: A clinical example of
Blaschko’s lines on the back; note the similarity with the diagram on the left [41].
2.1.1 Type 1a: Incontinentia pigmenti
Incontinentia pigmenti (IP) are known colloquially as Blaschko’s lines, and are charac-
terised by narrow bands of pigmentation which are darker or lighter than the surrounding
skin. (McCune-Albright syndrome also expresses Blaschko’s lines, but these are much
broader) IP is also referred to as Bloch-Sulzberger syndrome, Bloch-Siemens syndrome,
melanoblastosis cutis linearis, and pigmented dermatosis-Siemens-Bloch type [2].
IP is a form of cutaneous mosaicism resulting from a single specific mutation of the
gene IKBKG (Inhibitor of Kappa B kinase Gamma). The prevalence of IP is about 1 in
every 500,000 [3]. Both skin changes and other organ abnormalities are exhibited as a
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7. result of X-linked genodermatosis (an inherited genetic condition of the skin, linked to the
X chromosome). IP patients are predominantly female, as the mutation is usually lethal
in male embryos. The variation of phenotypic expression (how the genetic condition is
visibly apparent in the skin or other organs) is very large; individuals may experience skin
blistering (in the first few weeks of life), followed by a hypertrophic rash (a raising of the
skin in a lump form due to the over-production of collagen by the body) which manifests
itself longer term as hyper-pigmentation. The mutation’s effect on other organs is a
characteristic often used in differential diagnosis of IP. Patients may experience alopecia
of the hair (both on the scalp and on affected skin areas). Following tooth eruption during
infancy, children may experience abnormalities in the shape, texture and structure of their
teeth and/or palate. Later in life, some individuals experience unevenness, dystrophy
(degeneration/death) or periungual tumours (commonly known as warts) of the nails;
this is most frequently seen on the first 3 digits of the hands. A serious consequence of IP
is that it may affect the Central Nervous System (CNS); this can cause seizures, motor
impairment and/or mental retardation.
2.1.2 Type 1b: McCune-Albright syndrome
McCune-Albright syndrome (MAS) is another type of mosaicism that expresses Blaschko’s
lines. The band width of the lines is broader than Incontinentia Pigmenti. MAS is caused
by a post zygotic activating mutation (a mutation that occurs after birth) of the GNAS
gene which codes for G protein subunit Gs alpha in the affected cells. The prevalence
of the condition ranges from 1 in 100,000 to 1 in 1,000,000 and roughly twice as many
females are affected than males [4]. The phenotypic expression falls into the following
categories [1];
1. Abnormal Skin Pigmentation-Similarly to IP, patchy skin pigmentation in MAS
may follow Blaschko’s lines. The abnormal skin tissue is described as caf´e-au-lait
due to it’s ‘coffee with milk’ colouring and affects roughly 60% of patients.
2. Bone abnormalities Polyostotic fibrous dysplasia can be a symptom; this is where
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8. normal bone tissue is replaced by fibrous tissue, causing weakened limbs and a higher
probability of fracture. The pelvis and femur are the most frequently affected bones.
MAS can also cause less common symptoms such as scoliosis and facial asymmetry.
3. Endocrine abnormalities Over-active hormones can lead to precocious puberty;
seen in a ratio of 9:1 female to male.
4. Other Developmental delay, liver disease, abnormal heart rhythms and/or high
blood pressure.
2.1.3 Type 2: Becker naevus / vascular malformation
The Becker naevus/vascular malformation types of cutaneous mosaicism display a che-
querboard style pattern.
ˆ Becker naevus This is an overgrowth of the epidermis, melanocytes and hair
follicles [5] and mainly affects the upper trunk or shoulders, primarily in males. The
cause of this has not yet been understood or identified.
ˆ Vascular Malformation There are two common types of vascular malformation.
The salmon patch (naevus simplex) affects 2 in 5 newborn children and is a light pink
colouring which disappears about a year after birth. The port wine stain (naevus
flammus) variety of vascular malformation is more prominent in colour and lasts a
lifetime, with a growth rate proportional to the growth of the individual. This is
seen in 3 in every 100 newborns. Vascular malformation is caused by a mutation
of the GNAQ gene [6]. The term ‘Vascular’ derives from the abnormal clusters of
blood vessels that occur during foetal development, causing the unusual colouring.
2.1.4 Type 3: Mosaic trisomy13 (Phylloid pattern )
Mosaic trisomy13 (Pataus Syndrome) displays leaf-like patterns in pigmentation. It is a
chromosomal disorder which severely affects intellectual capabilities. It is caused when
cells have two normal copies of chromosome 13 plus an extra copy attached to one of the
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9. original chromosomes. The mosaicism effect results from the difference between normal
cells with only 2 copies of chromosome 13, and those with the extra chromosome [7]. Com-
mon symptoms include; a third fontanelle (soft spot on skull) in infants, polydactylism
(appearance of extra digits), flat nasal bridge and clenched fists [8]. Other less evident
expressions of this syndrome may be seen in heart abnormalities or seizures (suggesting
an influence on the CNS). The prevalence of this condition is 1 in every 200,000 births.
The severity of this condition is proportional to the fraction of cells containing extra
copies of chromosome 13; the higher the ratio of abnormal (mutated) cells to normal cells
(containing only one chromosome 13), the more severe the condition will appear. This
ratio will depend on when the first mutation occurs in the developing embryo; if it is early
on, the fraction of cells that are affected is likely to be higher, whereas if the mutation
occurs near birth, the abnormal cells will have had less time to migrate and proliferate.
2.1.5 Type 4: Large congenital melanocytic naevi
Congenital melanocytic naevi (CMN) appear as large patches on the skin without any
mid-line separation (a partition of the different skin types following a central line along
the body). The cause of CMN is thought to result from the migration and proliferation
of melanoblasts (which we aim to model later) from the neural crest to the dermis of the
embryo.
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10. Figure 2: Diagram showing the structure of an early embryo; the melanoblasts migrate
from the neural crest to the ectoderm of the embryo [44].
These precursor cells migrate further to the epidermis where they differentiate to form
melanocytes which can be present in many different organs, but are visible to the naked
eye on the patches of skin [10]. CMN are classified in relation to their predicted maximum
diameter in adulthood. Small patches (less than 1.5cm) are the most prevalent, with 1
in 100 newborns exhibiting them. Both medium patches (with a diameter between 1.5
and 20cm) and small patches tend to be round or oval in shape and light-dark brown
in colour. They often have mammilliated surfaces (small protuberances or ‘nipple’-like
structures) with hypertrichosis (an abnormal amount of hair). The rarer large and giant
CMN (over 20cm and 50cm respectively) are more likely to be asymmetrical with irregular
borders and variation of pigmentation. Giant naevi will, in most cases, be surrounded by
a scattering of much smaller ‘satellite naevi’ [5]. The severity of the complications caused
by the condition is proportional to the size of the area affected. Congenital naevi can
also be classified into subtypes based on the phenotypic variation that can occur. The
subtypes are listed below:
ˆ Caf´e-au-lait macule A macule is an area of skin discolouration. Caf´e-au-lait mac-
ules (CALMs) fall into the small or medium categories, with lesions typically being
under 10cm and oval in shape. The name describes the colour as being ‘coffee with
milk’. CALMs associated with the NF1 gene on chromosome 17 result from an
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11. inherited dominant disorder [9] (a ‘dominant’ genetic disorder will be expressed in
an individual with one or more copy(s) of the defective gene; this is different to a
‘recessive’ disorder which requires 2 copies of the abnormal gene in order for expres-
sion to occur, with just 1 gene a ‘recessive’ disorder will have dormant symptoms).
Note that McCune-Albright syndrome is one large caf´e-au-lait macule.
ˆ Speckled lentiginous naevus Also called naevus spilus, these naevi tend to be
darker spots on a flat tan background.
ˆ Satellite lesions Spotting of skin appear on the periphery of larger melanocytic
naevi.
ˆ Tardive naevi Not present at birth, these naevi develop afterwards. It is assumed
to result that a lower level of synthesis of melanin is responsible for the longer growth
period before the naevi appear.
ˆ Garment naevi reflect the image of an individual wearing a bathing costume, coat
sleeve or cape, as the naevi are present in the central areas or arm and shoulders
respectively.
ˆ Halo This is apparent when skin surrounding a present CMN is lighter, form-
ing a ‘halo’-like impression. Experiments have shown this is due to destruction of
melanocytes by the immune system; T cells (a subtype of white blood cells) and
other antibodies are believed to be responsible for this [45].
Complications of CMN The main complication of CMN is neurological involvement,
where affected patients can be born with melanin-producing cells within the brain, or
brain tumours, or other abnormalities of brain development. An important post-natal
complication is the increased susceptibility to melanoma, which can occur either within
the brain or spinal cord, or within the skin [12]. Large congenital melanocytic naevi will
be the focus for the remainder of this report.
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12. 2.1.6 Type 5: CHILD syndrome
Congenital Hemidysplasia with Ichthyosiform naevus and Limb Defects (CHILD) syn-
drome is an X-linked dominant disorder with fatality for male embryos [11] caused by
a mutation of the gene NSDHL which codes for a steroid-like protein. This mutation
affects the cholesterol biosynthetic pathway and can lead to blistering and dermatologi-
cal patterns in the skin. These follow a strict mid-line demarcation (a separation along
the mid-line of the abdomen) and are likely to be found in body folds such as armpits,
groin areas and between digits. CHILD syndrome is commonly known as lateralisation
mosaicism.
2.2 History of research
Since the early 1900s cutaneous mosaicism has been an area of interest for dermatolo-
gists, neurologists and other scientific professionals [2]. The complicated nature of the
development of these conditions has relied heavily on discoveries in other areas of science;
specifically embryonic development and neuroscience. This section aims to give a broad
review of the history of research conducted into cutaneous mosaicism, with a strong focus
on Type 4: Congenital Melanocytic Naevi.
2.2.1 Analysing 16 years of results
Veronica Kinsler et al. from the Department of Dermatology, Great Ormond Street
Hospital for Children NHS Trust, conducted a review of 16 years worth of patient data
(1991-2007) [12]. The research focused on acknowledging complications and links between
different phenotypic expressions. Their findings, based on 224 patients, showed new trends
as well as confirming and disputing hypotheses. An association between male patients
and a likelihood for neurological complications (identified using MRI of the CNS) was
discovered. The data also revealed a proportionality relation between the size of the
CMN site and the presence of satellite lesions, and neurological complications. Further
to this, the review revealed that the previous hypothesis suggesting that the location of
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13. the CMN site was related to the presence of neurological complications, was statistically
unjustified.
2.2.2 Timeline
In 1901, Alfred Blaschko, a German dermatologist first described the phenomenon of
differences in skin pigmentations following certain paths; this became commonly known
as ‘Blaschko’s lines’. In his paper [13], he discussed his findings which were based on 140+
patients who had skin lesions with liver complications. It was Blaschko himself who first
proposed that these lines originated from germ-line development(this is the development
of the cell lineage), but he did not quantify his hypotheses. Prior to this, skin lesions
were removed by cryosurgery beginning in 1899, with electro surgery often replacing this
from 1909. Cryosurgery involves the application of extremely cold substances to localised
cancerous tissues [28]. Electro surgery uses electricity for a variety of purposes, including
the cutting and dessication of unwanted skin [29]. The invention of Magnetic Resonance
Imaging in 1977, followed by its commercial use in 1980 enabled more research to be done
into the implications of genetic mutations [30]. 1994 brought the discovery of Keratin
10 mutations in lesional skin; this correlates with keratinocyte genetic mosaicism [31]. In
1998, Mary Lyon [32] revealed that LINE-1 (long interspersed nuclear elements) promote
the inactivation of the X chromosome. This finding has led to greater understanding of
why some types of mosaicism are more common in certain sexes [32] as females have 2 X
chromosomes, whilst males have 1 X and 1 Y chromosome, so are less likely to exhibit
conditions which are linked to the X chromosome.
2.3 Recent developments
Melanomas : Dermatologists are keen to monitor naevi to ensure any melanomas (can-
cerous cells) are caught early as this ensures the patient the best chance of full recovery.
Melanomas often appear in pigmented moles, as they can be a cancer of the melanocytes
(pigment cell). 5% (1 in 20) children born with a large or thick naevi develop a melanoma
11
14. in their life [21]. This is higher than the general average of 1 in 55 people developing skin
cancer in their lifetime.
Classification : In 2012, Sven Krengel proposed a new classification system for CMN,
based on predicted adult size (calculated from infant size), localization (where the naevi
is visible), description (colouring, texture) and concentration of satellite naevi. Krengel
devised this new system to provide an international classification tool for CMN, to bring
together different institutions [22].
2.4 Treatment and Procedures
Although there is a limited amount that can be done to treat the naevi, screening is
performed for neurological abnormalities, and for melanoma development. It is vital that
research is done to further understand the causes and implications in order to improve
further treatment. In 2010, a tissue bank of naevi biopsies was collated from various
sources across the USA. This will enable scientific research as a large collection of such
tissues will build a database of information about certain genetic traits and phenotypes
which could point towards further understanding of the origin of these conditions [18].
Total Body Photography (TBP) TBP is an important tool used to detect melanomas
early on. The original image taken roughly at the time of diagnosis can be used as a base-
line and referred to in the future to highlight any changes. This is particularly useful in
spotting growth which could indicate melanomas [33].
Blood and DNA sampling Easier to obtain than skin, blood samples can give re-
searchers important clues as to what causes CMN to occur and how [19].
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15. 3 Related Models in literature
Before developing a new model, we will discuss models available in literature that consider
migration and proliferation of precursor cells. Initially, most models consider a discrete
system which under certain assumptions can be extended to a continuous system at a
limit. The aim of these models is to understand and predict the spread of cells over time,
allowing doctors to predict how a patient’s affected skin may change throughout their
life. By varying certain parameters that may depend on the genetics of an individual
(for example their race, skin type or genetic history of mosaicism), dermatologists hope
to understand the roots of the different phenotypic expressions. Benefits of producing
an accurate model could include advances in treatment of conditions, or even prevention
with genetic manipulation.
3.1 Models of Migration
There are many factors which contribute to migration. Cells may migrate in response
to gradients in concentration of different substances. Diffusion of molecules (cells or
other mediums) from areas of high concentration to low concentration accounts for a
large majority of movement. Cells may also move in response to signals; this is generally
termed ‘taxis’ and may involve a response to chemical gradients.
Cellular Automata A Cellular Automaton is a grid (or lattice) structure which can
hold information about the ‘state’ of the cells. The state a cell is in will be one of a
finite possible number of states. The grid is then monitored for a certain number of time
increments where cells may change their ‘state’ (examples of this ‘state’ include an on/off
function, a subtype of cell or a differing marking).Cellular Automata have been used
to replicate many patterns in nature, for example seashell patterning [26]. In mosaicism,
cellular automata are useful as the ‘state’ held could represent cells which express different
pigmentation, or of migrating vs. proliferating cells. This will be discussed further in
section 4.
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16. 3D Model of Migration In the model we will be proposing a major simplification is
made; the model is only two-dimensional. This is a realistic assumption due to the fact
the organism we are considering is in the very early stages of embryonic development,
where the size and shape of the organism mirrors that of a small flat disc.
Figure 3: Early Human Embryo development; displaying flat disc properties. [39]
In a paper originally published in January 2014 entitled “Implementing a Numerical
Package to Model Collective Cell Migration” by Stonko et al. [23], a three-dimensional
model of migration of cells in the fruit fly, Drosophila melanogaster is presented. An
ODE system is introduced, with each cell having it’s own ODE that tracks it’s position
with time. This system actually considers only the [x,y] coordinate of each cell. However,
further extension by Stonko et al. using MATLAB® [34] allows for a radius to be added
to each cell, then multiple ‘slices’ of 2D [x,y] plots are collected and plotted together
to form a 3rd dimension. This paper’s results focused on the sufficiency of four ‘funda-
mental forces’; adhesion, repulsion, migration and stochasticity to accurately portray the
biological system.
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17. 3.2 Models of Proliferation
Mitosis is the process of a single cell dividing into two identical daughter cells. Prolifera-
tion of cells is the production of new cells as a result of mitosis. The rate of proliferation
may be controlled by a variety of things, including characteristics of the embryonic envi-
ronment. There is clearly a heavy dependence on space and time; space is required for cell
populations to grow, and this process takes time. Proliferation in a model will be coupled
with cell death; clearly the higher the rate that cells are being produced, the higher the
death rate as the cells compete for space and nutrients [42].
3.2.1 Modelling Chimæra experiments
Mosaicism can be modelled through the observation of Chimæric animals (animals with
4 or more parents). Chimæras are created through the fusion of early embryos, mutation,
grafting or other genetic techniques. By observation of the patterns created through the
presence of multiple sub populations of cells, it has been possible to gain a greater un-
derstanding of how mosaicism is established [20]. In the paper “Cellular automata and
integrodifferential equation models for cell renewal in mosaic tissues.” by Bloomfield et
al. [20], both discrete and continuous models are developed which represent the prolifer-
ation of different populations of cells. A simple example of the progression from discrete
to continuous involves a 2D lattice structure, with 2 different cell subtypes; A and B.
Bloomfield et al. consider cell birth and death; cells are born to replace those lost. If
a(x, t) represents the proportional concentration of A (and so 1 − a(x, t) is the equivalent
for B), then under certain assumptions, a continuous model can be created;
∂a
∂t
= α(f(Ia) − a)
Here, α is the cell replacement rate, Ia is the normalised integral over the area of the
concentration, and we require f(0) = 0, f(1) = 1 to ensure steady states at 0 and 1.
f(Ia) is chosen to either be linear or a smooth approximation to the step function. Linear
15
18. stability analysis gives a good idea of the long term behaviour of the model. Bloomfield
et al. used a numerical method in which they discretised the domain and integrated over
the grids contained in the domain (this was a justified approximation as ‘...with a fine
lattice, it is reasonably accurate; note that simulations with a reduced lattice give the
same qualitative behaviour.’. Simulations were run with differing initial distributions of
cell types. In the long term runs, graphs show how pattering can occur as the two cell
types (see below).
Figure 4: Graphs taken from the paper by Bloomfield et al. [20]. White areas show cells
of type A, black represent type B
3.3 Combined Models of Migration and Proliferation
In a paper entitled ‘Simulating invasion with cellular automata: connecting cell-scale and
population-scale properties.’ by Simpson et al. [24], a cellular automaton (CA) was used
to model motility and proliferation of cells. The figure below shows the rules that are
chosen to simulate these processes
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19. Figure 5: Diagram showing the two CA rules. Grids containing no cells are black, and
cells can be located in red or blue. ‘If a cell is motile it can transition to any of the
nearest four neighbouring sites with equal probability if the site is empty and that the
local density around the cell in the new location is not above carrying capacity. If a cell is
proliferative, it can place a daughter into one of four directions ‘so that the two daughter
cells move in opposing directions.’[24]
The motility rules here are limited to only 4 directions; in our model proposed in
section 4, we aim to extend this to 8 directions (with diagonals as the extra possible
moves); this should give a closer match to the real biological continuous system we are
modelling. This paper also explains the link to Fisher’s equation (as presented in Murray’s
1989 book, Mathematical Biology) [25]. Fisher’s equation:
∂u
∂t
= D 2
u + su(1 − u)
such that u = u(r, t) represents the concentration of different cells, D 2
u represents the
diffusion terms (motility) and su(1 − u) is logistic growth (proliferation). D & s are the
diffusion coefficients and the rate of logistic growth respectively. r ∈ RN
, where N is the
spatial dimension being considered.
3.4 Other Related Models
3.4.1 Turing pattern formation
In 1952, Alan Turing published a paper entitled ‘The Chemical Basis of Morphogenisis’
[15]. In this work, he proposed a model for embryonic growth and showed how certain
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20. conditions could lead to pattern formation. Initially considering cells as geometric points,
and further extending this to a continuous distribution, Turing derived the following set
of equations;
∂(u)
∂t
= D 2
u + f(u)
where u is the vector of chemical concentrations, D is a matrix of constant diffusion
coefficients (assuming that for each medium the diffusion coefficients are constant in
space and time), and f(u) is a function representing the reaction kinetics. The state of
the system may change with time, position of cells, diffusion of substances and further
chemical reactions. In addition to this model, Turing imposed certain initial and boundary
conditions. Turing posed the question; when does instability occur as a result of diffusion
(‘Diffusion Driven Instability (DDI)’)? In order to find the answer to this, one should
consider the steady state in the absence of diffusion, linearise about this value and find
conditions on the system for instability to occur [14]. It is straightforward to look at an
example in 2D; r ∈ R2
u = (u, v), f(u) = (f(u, v), g(u, v))T
and
D =
Du
0
0 Dv
The following give the conditions for DDI:
fugv − fvgu > 0, Du
gv + Dv
fu > 2
√
DuDv fugv − fvgu
The reaction kinetics described by f(u) vary depending on the system we are considering.
Turing’s findings can be applied to many different patterning examples seen in nature
including animal coat patterns, shell markings and pigmentation variation.
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21. Relevance to our model In the model presented later in the paper, the reaction
kinetics considered involve migration and proliferation of cells. If we were to structure
our model in the continuous way that Turing suggests then our f(u) would look something
like:
f(u) = su(1 − u)
where s is a birth rate constant and f(u) represents logistic growth of the cell population.
The migration would be incorporated into the diffusion term. As we will consider only
2 dimensions, D 2
u = ∂2
(u)/∂2
x + ∂2
(u)/∂2
y. Our diffusion constants Du
and Dv
will
depend on the properties of the substance that the cells are moving through; how viscous
it is, the population density etc.
Uses of Turing Analysis: Melanocyte differentiation in Zebrafish The phe-
nomenon of Turing pattern formation can be seen in Zebrafish; this occurs as a result
of different types of pigment cells interacting [16]. Zebrafish are highly beneficial to ge-
netic research as they ‘are vertebrates and therefore share a high degree of sequence and
functional homology with mammals, including humans. Due to the conservation of cell
biological and developmental processes across all vertebrates, studies in fish can give great
insight into human disease processes.’[17].
4 Proposed Model
In this section, a proposed method of modelling the development of Congenital Melanocytic
Naevi (CMN) will be presented. Attention will be given to any assumptions or simplifica-
tions made in the process. We will then discuss the model and produce results which can
be evaluated in order to qualitatively draw correlations with clinical examples of the con-
dition, as well as performing statistical analysis which may or may not show consistency
with clinical data collected previously by various sources (see section 2.2.)
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22. 4.1 Objective
The aim of this project is to produce a model for the migration and proliferation of
melanocytes in embryos, which can be then used to understand how certain patterns
occur in patients who express CMN . This project was motivated by the research of Dr.
Veronica Kinsler, Dept. Dermatology, Great Ormond Street Hospital [35]. For many
years, her team have been treating patients that exhibit varieties of this condition.
4.2 Assumptions made
As with all mathematical models, we are required to make some assumptions and simpli-
fications in order to implement the processes to produce results. In addition to this, as
research in this area is ongoing (particularly in relation to early embryonic development),
some factors and characteristics that contribute to the model have not been confirmed
scientifically, but are commonly agreed upon. For example, the exact time frame of devel-
opment is not definitive but embryologists make assumptions based on previous research
data. In the creation of this mathematical model for Cutaneous Mosaicism, Dr Veronica
Kinsler has been consulted as to her informed opinion on the mechanisms underlying the
condition; her input enabled us to make calculated decisions regarding unknown param-
eters [43].
1. 2D discrete domain - In the proposed model we consider a lattice; this may
appear to be a simplification of the embryo as a continuous medium, however we
are modelling a discrete finite number of cells, so it is appropriate. Later, this
report will look at the link to continuous models, which is done for mathematical
convenience. It is valid to ‘drop’ a dimension as at such early development, embryos
do appear as a simple flat disc. The discussion will look at how this model could be
extended to 3D. We consider a lattice structure, where cells, although intended to lie
within a 2D domain, will inevitably move and divide to be on top of one another. It
was considered that the model might involve a loop which ensures that cells cannot
move to an already occupied grid on the lattice. However, this was computationally
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23. expensive and would have been less efficient. As the number of cells and time scale
we are considering are low, this idea was neglected. Besides, ultimately the embryo
is 3-dimensional and so in many ways this is a realistic option.
2. Senescence & Cell death are negligible - The parameters determining the
rate of mitosis have been calculated using data which shows the net growth in cell
numbers over time. Therefore, our birth rate incorporates the death rate, so we do
not need to address this individually. Senescence (the cessation of division) has also
been included in this calculation.
3. Elongation of patterns - Previous research has shown that the cell cluster is
longer than it is wide. This ‘stretched’ appearance correlates with the cranio-caudal
extension of the embryo. Although not much is yet known as to why this happens,
it is clearly a defining feature essential in forming the patterns exhibited by patients.
Different methods have been considered to implement this model, and all will be
discussed later in this section.
4. Motile cells become Proliferating cell - A parameter c = 6 (hours) is introduced
to represent the age at which a motile cell stops migrating and starts the process of
proliferation.
5. Non-dimensionalisation - Before we use our model to generate results, we make
the simplification of non-dimensionalisation. This involves considering the units
relevant to each variable or measure in the model, and scaling with appropriate
known constants. In our model, the spatial units are likely to be in micrometers,
and we discretise time into hour slots.
6. Other cellular processes - Cell signalling and environmental factors are largely
ignored in the model we present. They are considered negligible, and our focus is
on finding the spread of the patterns at certain times.
ˆ Nutrient Transport - The transport and production of nutrients will heavily
21
24. depend on the location and number of cells. The larger the population, the
higher the demand for nutrients. We will assume a steady state has been
reached for this.
ˆ Extracellular Matrix - Extracellular matrix proteins may be produced by cells.
This could affect the overall distribution, however we have chosen to neglect
to model this as we suppose it’s effect would be negligible at this early stage.
ˆ Differentiation - As this model deals with embryonic stem cells, it is important
that the process of differentiation is considered. However, the patterns are
formed by just one cell type-melanocytes; cells which secrete melanin (a dark
pigment that shows as the naevus).
4.3 Definitions of processes involved
ˆ Random Walk- A random walk is a stochastic process which involves a sequence
of randomly generated steps that create a path for a particle. More formally, if
X(t) is a particle trajectory that begins at position X(0) = X0. The random walk
is modelled by the following expression: X(t + τ) = X(t) + φ(τ), where φ is the
random variable that describes the probabilistic rule for taking a subsequent step
and τ is the time interval between steps [27].
4.4 Progression of development of model
With mathematical models, it is often simplest to start from a very basic model and add in
certain more complex aspects to develop it over time. In creating a model that represents
the migration and proliferation of melanocytes, this was particularly important as it
enabled us to expand and improve the model and also show how variation of parameters
or behaviours affects the overall outcome.
22
25. 4.4.1 Basic Structure
As a basis, two models were created, representing migration and proliferation of one cell
respectively. To model the movement of a cell, a random walk was considered. Initially,
the cell could only move in 4 possible directions on the 2D lattice; North, South, East or
West. This was later expanded to 8 directions. The below figure shows an example plot
of a random walk of a single cell under certain conditions.
Figure 6: Random walk on 2D lattice with equal probability of moving in x or y direction,
shown for 10000 time steps.
When considering the proliferation of the cell, questions arose as to how often the cell
would divide and where the daughter cell would be placed. The combination of these two
separate models raised further questions including:
ˆ Do cells participate in cycles of proliferation/migration, and how long do these last?
ˆ Can cells move on top of each other in the lattice?
ˆ Are there different subtypes of cells; e.g do some cells only divide, whilst others can
migrate and/or divide?
23
26. The figure below shows a simple combination of one cell dividing followed by the migration
of both daughter and parent cell.
Figure 7: Division followed by migration on 2D lattice with equal probability of moving
in each direction, shown for 10000 time steps.
4.4.2 Initial Model 1
An isolated cell is placed at the centre (the origin). It is then allowed to divide for a
certain amount of ‘divisions’; each daughter cell can be placed in one of the 8 neighbouring
positions on the lattice. The parameters can be altered to change the probability of placing
in certain locations or change the likelihood of division occurring at all. We track the
division of the first cell to produce one daughter, followed by the division of that daughter
and so on until the total time (N = 4 weeks = 28 days = 672 hours) is up. This produces
a ‘path’ on the lattice following the placement of daughters. When the first daughter cell
divides, it is of course possible for the original parent cell to divide again. Thus another
path is created with ‘divisions-1’ daughter cells. This happens repeatedly and each cell’s
divisions are tracked up to the time limit (N).
Once we have tracked all the cells undergoing proliferation, we allow some time for
24
27. migration. This is modelled using a 2D random walk as mentioned; the probability of
moving in a certain direction can be changed in the parameters. It is also possible to stay
in the same location in a time step.
After a certain number of steps, if the cell is located on the longitudinal axis (x = 0),
it can only move up or down that axis, and not away from it (this accounts for a vertical
bias in the system). It is important that this is only inflicted after 3+ time steps (1 time
step = 1 hour), as otherwise our model would just show cells in a straight 1D line on the
y-axis. A boundary of a 20 × 20 grid (i.e. each axis ranges from -10 to 10) is inflicted;
if a cell is on a boundary, it cannot move outside of the boundary, and so it is reflected
off the boundary or stays still for that time step. It was decided that this model did not
reflect the system closely enough, and computational time/expense would be large.
4.4.3 Initial Model 2
In the following model, we consider that each cell may divide or migrate in a single time
step. The flow chart displays the process behind the decisions.
25
28. Figure 8: Flow chart showing the migration or proliferation of cells.
26
29. The main difference to note here is that on each loop, the model scans for all cells
present and tracks what happens to them at that time step; they may move or divide.
Once all cells have been checked and their actions recorded, the time is increased and
we consider what happens to each cell in the next interval. This is repeated until we
have reached the end of the allocated total time period (N). The below diagram shows
pictorially an example of this model.
Figure 9: Diagram showing short time lapse; f is the parameter given to the frequency
of segments being added. The dark blue circles represent the ‘original’ parent cell in each
segment; the light blue cells are it’s ‘daughters’ etc.
4.4.4 Improvements made
After consultation with Dr. Veronica Kinsler, [36], we were able to further improve the
model and answer some of the above questions. One of the main issues raised which had
not been addressed in either of the initial models was that when a cell divides, only the
new daughter cell migrates. The original cell should stay put, and divide again at the
appropriate time.
27
30. The notion of a North-South bias in migration was disputed and it was suggested that
the organism produces increasing numbers of longitudinally orientated segments. These
segments would be exact replicas of the original single cell model, and again that original
cell per segment would not migrate.
Further to this, it was decided that there would be a vague idea of cycles of proliferation
and migration; cells which divide then have a stationary daughter cell and a migratory
daughter cell, and the migratory cell after time finishes migrating and becomes a dividing
cell, with the same outcome as the original parent cell (it stays stationary, its motile
daughter migrates). Thus each cell has it’s own independent cycle; a period of migration
followed by division (the ‘daughter’ cells) or simply a life cycle of division (the ‘parent’
cells).
4.4.5 Segmentation: How is best to implement this?
There is some suggestion in literature that the organism starts to grow extra segments after
3 weeks of development [36]. Although there are 33 segments in a human (corresponding
to the vertebrae), in a developed foetus there are 40-44. These are all present by the 5th
week of development. Choosing how to model this posed the following questions:
ˆ Does the addition of new segments occur at regular time intervals from the end of
Week 3 to the end of Week 5 or is this random?
ˆ If the creation of segments is random, is it reasonable for us to model it regularly?
ˆ Where should the new segments be placed? In order to fulfil the criteria of a
lengthened organism, we know they are placed on the y-axis, but how far above (or
below) the current cluster of cells?
ˆ Although not biologically accurate, would it be more efficient to model the segmen-
tation of the organism as happening over the whole 5 week period from day 1? This
would mean a greater time between new segments being created, and potentially a
more even spread of cells.
28
31. One proposed segmentation model is depicted below in a pictorial format.
Figure 10: Diagram showing time lapse as new segments are added at a distance from
present cluster of cells. Again, the dark blue circles represent the ‘original’ parent cell in
each segment; the light blue cells are it’s ‘daughters’ etc.
4.4.6 Definition of the state matrix
The state matrix holds information about all cells; there is one row for every cell present
(hence the matrix grows in row number over time), and 4 columns give information about
that cell. The first column gives the x-coordinate of the cell, and similarly the second
column gives the y-coordinate of the cell. The third column holds the time t∗ (hours)
since the cell last divided (if it is a proliferating cell), or the age since it was born (if it is
currently a motile cell). Recall, this is reset to zero for 2 reasons;
1. If the cell changes from a migrating cell to a non-motile dividing cell (at t∗ = c), or
2. If the cell is a dividing cell and divides into two daughter cells (at t∗ = 24).
The fourth and final column gives the type of cell; whether it is a motile cell, in which
case a 1 is placed in this column, or a parent cell of type 0. Note that is is possible to
go from cell type 1 to cell type 0 at the appropriate value of t∗. At every time step the
state matrix changes; some cells will simply ‘move’ and so their coordinates (columns 1
or 2) will change. Other cells will change from a migrating cell to a dividing cell (column
4 changes). At each time step, t∗ is increased by 1 hour unless we are at a changing point
(given by points (1) and (2) above, in which case t∗ is set to 0. When a daughter cell is
created, a new row is added to the end of the matrix to contain the information for it; it
will be of type 1, with age t∗ = 0, and it’s coordinates will be one of the lattice points
neighbouring the parent cell’s coordinates; this is random.
29
32. 4.4.7 Speeding up the computation time
The main issue with the models presented above is the time it takes to produce results.
After consideration of how to reduce this, it was decided that rather than visiting each
individual cell and applying the rules to it for that time step, the model would scan over
all the cells, locating the different types (e.g. motile cells, motile cells old enough to
become dividing cells, dividing cells) and apply suitable vectors to the state matrix. This
would have different values in each position. The below shows the steps followed:
1. Cell Type- Matlab locates motile cells (type 1) in the matrix which are old enough
to become a dividing cell (t∗ > 6); it then creates a vector the size of the state
matrix which is added to the ’type’ column, changing the appropriate cells to type
0 (dividing cells). The time of these cells is reset to zero. Where the cells are not
of this type, it simply adds ‘0’ to the column, therefore not altering it. Here the
vector added will take the form of 1 column, and a row for each cell. An example
of this is;
-1
0
0
When this vector is added to the fourth column of our state matrix, the 1st cell
changes from a motile cell to a dividing cell, and the remaining cells stay in their
original state.
2. Migration- A random 2 column matrix is generated which holds a pair of numbers
for each cell (each one is -1, 0 or 1); this is then multiplied element by element
with the 4th column of the state matrix (the ‘type’ column; to ensure it only affects
motile cells of type 1) and added to the location columns (columns 1 and 2) of the
30
33. state matrix. For example, if the matrix is
1 0
-1 -1
0 0
the first cell will move to the East (right), and the second will move South West
(down and left) whilst the third will not move at all.
3. Proliferation- The model locates parent cells (type 0 in column 4) in the matrix
which are old enough to divide (their time in the 3rd column is t∗ = 24). Using a
method similar to that for migration, a random matrix of numbers is created which
will determine where the daughters produced by these cells will be placed. This is
applied to the relevant cells and stored separately, then added as additional rows
to the end of the state matrix (representing daughter cells (type 1), in a location
one grid away from their parent). The age of both the parent cell and daughter
cell is reset to zero. For example, if our in our state matrix we have 3 cells; one is
a motile cell, and two are dividing cells. Of the dividing cells, one has an age of
t∗ = 24, so it is ready to divide. Therefore the model generates a random number
which determines where the daughter is to be placed. Let us suppose that it is to be
placed one place to the right of the parent i.e in [−4, 9]. Then the matrix calculation
becomes:
13 −4 4 1
−5 9 24 0
1 2 5 0
+
0 0 1 0
0 0 −24 0
0 0 1 0
The output is the result of the above calculation, plus the extra row added to
31
34. represent the daughter cell:
13 −4 5 1
−5 9 0 0
1 2 6 0
−4 9 0 1
These new rules for tracking the state of cells is faster than the previous methods presented
as it does not require a ‘for t*=0..N’ loop. In each time step, the previous method would
look at each cell and ask the questions ‘Is it a migrating cell ready to become a dividing
cell?’, ‘Is it a migrating cell ready to move?’, ‘Is it a dividing cell ready to divide?’ one
by one. This was very time consuming in the later stages due to the large number of
cells. The new rules locate all cells that answer ‘Yes’ to a certain one of the questions,
and applies the change to all of them at once. Therefore only 3 actions are taken. This
maximises the use of time as ‘for’ and ‘if’ loops are slow in Matlab.
4.5 The Final Model
The flow chart depicted below shows the progression of decisions made in the final model
proposed . The changes discussed above in 4.4.4, 4.4.5 & 4.4.7 have been implemented.
A new parameter ‘f’ is introduced; this represents the frequency at which the organism
expands longitudinally. For the first 10 days, if N (the total time in hours) is a multiple
of 6 hours, the cluster of cells grows by adding an extra cell above (or below) the first;
this is a dividing cell at the origin of this new segment that is produced every 6 hours.
This segment then follows the same rules as the first, and the cells contained in the whole
organism are tracked simultaneously. The timing was worked out to ensure segmentation
was complete by day 10.
32
35. Figure 11: Flow chart for final model
The pictorial description below describes the above:
33
36. Figure 12: Diagram showing time lapse as new segments are added at a distance from
present cluster of cells.
4.5.1 Time line of Events
The table below outlines the time taken for different stages in development.
Table 1: Time Line of Events
Week Events Description
1 No melanocytes yet present Simple ‘empty’ lattice structure
2 Mitosis and Proliferation start Small cluster of cells
3 New segment every 6 hours; placed on longitudinal axis. Larger stretched cluster
4 Production of new cells ends on day 31 Expanding oval like structure
5 Continuation Results appear to match patterns
There are many parameters used in this model and it is important to note the relevance of
each of them and to understand why the default value is chosen. The table below outlines
the parameters used and their properties.
Table 2: Parameters used in final model
Parameter Default Value Valid range Description
U 100 U>0 Upper x boundary
T -100 T<0 Lower x boundary
W 100 W>0 Upper y boundary
Q -100 Q<0 Lower y boundary
w 4 n/a Number of weeks model is run for
Dt 1 Dt>0 Time increment (hours)
N 24 × 7 × w = 672 N>0 Number of hours (time increments)
F 24 F>0 Division time (hours)
f 6 f>0 Frequency of new segment production
(every 6 hours for 10 days)
c 6 c> 0 Age that motile cell becomes stationary
and undergoing mitosis (hours)
px, (= py) 0.25 0 < px, py < 1 Probability of moving in horizontal (vertical) direction,
also used as probability of placement of
daughter cell in certain neighbour
34
37. From the table, it can be seen that the probability of migration is equal for all directions.
The nature of a random walk model allows the option to add a bias to certain directions
of movement. For example, it may be more likely that the cell moves to the East, so we
can assign a higher probability to this than to the other directions.
4.6 Limitations
This section will highlight the limitations that the model holds and explain why these are
justified or how future improvements could reduce their effect.
4.6.1 Placement of Daughter Cell
In the development of this model, it was considered that a ‘parent’ cell may place it’s
‘daughter’ cell in any neighbouring location on the lattice. However, this extra element of
chance increases computational time dramatically and so it may be decided when running
for longer or with a larger number of cells that ‘daughter’ cells would be allocated the
same location as their parent. This is a good approximation as the ‘daughter’ cell will
likely move in the next time step anyway.
4.6.2 Growth of Organism
To replace the notion of a North-South bias, and to fulfil the requirement that the cluster
of cells appears to grow longitudinally, the final model introduced the growth of a new
segment at a certain frequency. Although this implementation shows the results we would
expect, it is an ambitious approximation to make for many reasons:
1. Most obviously, it is not apparent as to what would trigger the organism to grow
an extra segment so suddenly.
2. The ‘frequency’ we assign is a calculated not observed measure; we arrived at this
number by looking at the overall growth of the organism over the time allocated
and assumed growth occurs at regular intervals.
35
38. 3. Cells can move on top of each other; this doesn’t obey the idea that at very early
stages of embryonic development only a single lattice layer is present.
4.7 Simulations
In this section we will examine some examples of the patterns produced by running this
model and we will analyse any similarities and/or differences between them and real
pigmentation visible on patients with CMN.
4.7.1 Distribution of Cells
The picture below shows how the distribution of cells changes with time; they spread out
and by the end of the 5th week we can see some minor scattering at the edges which
shows how satellite naevi may form;
Figure 13: Distribution Plot of cells at different times in development.
This progression shows that by the end of the third week (in particular by Day 17),
all segments are in place along the x axis, and cells are migrating and proliferating from
there. Clearly, as time elapses, the spread of cells widens and lengthens.
36
39. 4.7.2 Density of Cells
The plots shown above give us an idea of where the cells fall on the lattice. However, it
is not clear how many cells are present in any given location on lattice. To study this, a
plot of the density of cells was produced.
Figure 14: Density Plot taken at different times in development; red is very high density,
dark blue is low density.
Note, the above figure shows the density of the new cells produced during the time
frame stated above the figure. The central areas are densely populated by cells as this is
where the cells originated from. If we take a cross section at different points and view the
densities here, we get the figure below:
37
40. Figure 15: Cross sectional views of density at y=0 and y=30 respectively. Green; 3 weeks
after fertilisation, Blue: 4 weeks after fertilisation, Red: 5 weeks after fertilisation.
As expected, the edges of the cluster of cells are not smooth, but instead we see some
scattering of cells if we look closely; this will be discussed in relation to satellite lesions
in the analysis section 4.8.
4.8 Analysis
4.8.1 Parameter Variation
To see whether a further spread of cells can be achieved in the same time period, certain
parameters were changed. Firstly, c (the age at which a motile cell becomes a dividing
cell) was increased from 6 to 10. In addition, the probability of moving in a certain
direction px and py were raised to 0.5. This meant that at every time step, each motile
cell would definitely move ±1 in both the x and y direction. This increased mobility and
the results can be seen in the final diagrams:
38
41. Figure 16: Distribution of cells with c = 10, px = py = 0.5.
If we again generate the relative density of the distribution at different points, we can
analyse how the spread of cells differs with these new altered parameters:
Figure 17: Relative density of cells for c = 10, px = py = 0.5.
The cells reach a much higher distance from the origin; the edges of the cluster with
these parameters reach to x = ±22 and y = ±57, whereas previously it was x = ±16 and
y = ±52. These boundaries have increased by 37.5% and 9.6% respectively. The smaller
increase in spreading longitudinally is expected as most of the spread in this direction is
accounted for by the creation of new segments along the y axis during the first 10 days
of melanocyte development.
39
42. 4.8.2 Satellite Lesions
By running the model numerous times, it is possible to see how likely we are to get
so called ‘Satellite Lesions’. The picture below shows a close up view of the edge of a
simulation (run for 5 weeks with the parameters described in ‘Parameter Change 1’).
Figure 18: Close up of density plot after 5 weeks (parameters as in ‘Parameter Variation
1’); the red ellipse circles a couple of cells which are separate from the main cluster, and
could therefore account for satellite lesions.
40
43. 4.8.3 Comparison to Clinical examples
The nature of this condition is that there is a level of randomness; not all clinical examples
will match the simulations our model produces. However, it is possible to spot correlations
between them. Some examples will be shown here.
Figure 19: Comparison of a clinical example of CMN with a computer simulation; note
we have a similar shape. This simulation has been run for only 2 weeks (336 hours), and
we have assumed that this occurred after the opportunity for segmentation; this is why
there is no obvious bias of cells spread in the vertical direction. The short time period
allocated to the model also accounts for the smaller size of the naevi [37]
Figure 20: Comparison of a clinical example of CMN on the abdomen of a 12 year old boy
[38], with a computer simulation; note the similar speckled edging and relative sizes of the
length to width. This simulation was run for the full 4 weeks (N = 672 hours), taking the
time passed since fertilisation to 5 weeks. Unlike the previous example, segmentation was
introduced here in the way explained in section 4.5. This accounts for the longitudinal
spread of cells.
41
44. Figure 21: Comparison of a clinical example of CMN on the back and top of legs of a
young girl [37], with a computer simulation; note the similarity in the peak; this was
achieved in the simulation by halting the segmentation after 8 days and continuing it
later in the process (in the 5th week since fertilisation). There is also a correlation in
how far down the left leg that the naevi reaches; this occurred by chance after multiple
simulations.
The above pictures and simulations prove that this model can produce accurate ex-
amples of CMN distributions. The differences between them are due to changes in the
parameters used, and more importantly the length of time they are allocated, and when
this occurs in relation to fertilisation (it appears the greater the time between fertilisation
and melanocyte development, the smaller the naevi). The nature of CMN is that they
effectively wrap around the body, with the mid-line matching the cranio-cordial axis of
the human- this can be seen in these pictures here- if we were to map the simulation onto
the clinical photo, this would be evident.
4.9 Extension to Continuous medium
The model we use is based on a 2 dimensional random walk. It is possible to extend this
to a continuous time random walk (CTRW) (also in 2D). CTRWs are Markov processes.
This section will highlight how our model could be formulated in continuous time. For
simplicity, let us assume that the probability of moving in any given direction on lattice is
equal and called ‘p’ (p = 1/8 as there are 8 neighbouring lattice locations we could move
42
45. to). Suppose that time no longer has the restriction of being discrete; it must simply
belong to the real positive number group. If the distance between lattice sites is called
‘d’ (in our model, we assumed a dimensionless value of 1), we seek what happens in the
limit as d → 0; this represents the idea of a continuous medium. Further, we consider an
isotropic medium; the conditions are invariant with respect to direction; this means that
the transition of a cell from any two lattice sites has equal probability. In a random walk
we may consider:
P([x(t), y(t)] = [x , y ]|[x(0), y(0)] = [x0, y0])
(the probability that at time t, [x, y] = [x , y ], given we start at [x0, y0]. This can be
expressed as a sum of possible movements; expanding this and neglecting high order
terms (as we assume they are small)leaves us with:
∂(P)
∂t
= D 2
P
which we recognise as Turing’s theorem from 3.4.1.
5 Discussion
This section will discuss the work presented above and highlight areas that could be
extended or improved.
5.1 Comparison to previous models and discussion of assump-
tions made
Clearly, our model uses a 2D random walk as its basis, with certain rules and altered
parameters which obey what we know or assume about the embryonic development.
Therefore, it has drawn on aspects of other models, whilst being unique. One of the
main assumptions made was that we only consider 2 dimensions. The reasoning behind
this was that we felt each lattice site could hold more than one cell at any given time.
43
46. This allows cells to move in any direction no matter what is already in the lattice sites.
However, it is important to consider that there might be a need for a cap on the amount
of cells that can be present in a lattice site. Potentially the model could be improved by
researching the number of cells present in embryos at certain time periods and the size
of the embryo. This would allow a calculation of the number of cells that can fit into
each lattice site (which will depend on the size of the site before non-dimensionalisation).
Another way to address this issue would be to introduce a third dimension into the model
at an appropriate point. The time to do this would be when the embryo’s appearance is
no longer that of a flat disc, but of a more complicated 3D object.
5.2 Uses for this Model
This model will be useful to researchers to understand more about the following charac-
teristics and questions relating to Congenital Melanocytic Naevi (CMN);
1. Does the time that melanocytes start development in the embryo affect the size/dis-
tribution of the CMN?
2. Why is there a correlation between the size of the CMN and the presence of satellite
lesions?
3. Can we predict the shape of the CMN by using parameters engineered to match
an individuals skin type?
4. It is useful to be able to map clinical CMN examples to computerised models to
see correlations; this could enable doctors to understand why certain areas of skin
may be textured differently, and to potentially predict future growth/changes in the
CMN.
5. This model could also be used with different types of cells interacting; for example
we could introduce a cancerous cell and track it’s movement and interaction with
melanocytes. This could increase understanding of the timing and other character-
istics of melanoma development.
44
47. 5.3 Extension Possibilities
There are many ways in which this project could be extended to better present the
biology behind this condition. This section will discuss a few of them. Clearly, a lot of
the development of this model relied on assumptions of parameters, or ‘educated guesses’
as a result of previous research. As more and more discoveries are made in embryology
and other defining areas, it will be possible to fine tune this model to produce results that
closer match what is seen in nature. This in turn will improve the reliability of statistical
data produced from the model. As discussed previously, a limit on the number of cells
allowed in each lattice site would increase the similarity to real life. This is a large extra
condition on the model, so is likely to be expensive and time-consuming, which could
render it computationally inefficient at this stage. Making use of faster technology such
as a the Graphics Processor Unit (GPU) on most modern computers will alleviate for
this.
6 Conclusion
This project has presented a synopsis of the different forms of skin mosaicism, and given
examples of how these have previously been modelled. The development of a new model
to produce plots of cells distributed in different ways has allowed for comparison to clin-
ical examples of the Congenital Melanocytic Naevi. Analysis has been done that shows
how the spread of cells changes with time. The limitations and assumptions discussed
throughout this report arise mainly from a lack of data on certain time scales or other
parameters. With further research, it is hoped that scientists will have a greater under-
standing of the mechanisms involved in the development of CMN in embryos, which can
then be implemented into the model to improve the relevance and accuracy.
45
48. A Raw data
The below list displays snippets of code used to generate results above;
1. Code for migration, proliferation and segmentation in weeks 2-3.
%Global Parameters
global N p x p y no x no y Dt F w d U T W Q c
w=2; %Number of weeks
d=7*w; %Number of days
N=24*d; %Total length of time to run for (Number of hours)
Dt= 1; %Time increment
F=24/Dt; %Division time- every 'F' hours in the life of a cell, ...
a division occurs
c=6; %Age to become a parent cell
clf;
% Boundary
U = 100; % upper x boundary
T= -100; % lower x boundary
W = 100; % upper y boundary
Q = -100; % lower y boundary
% Initial Conditions; t=0, we have one daughter and one parent ...
cell at (0,0):
state=[0 0 0 0
0 0 0 1];
%%Type of cell
%0; parent
%1; daughter
clf
nFrames=N;
cstring='rgbcmky'; %String of colours to identify paths of ...
different cells
n=0;
%Probabilities
p x= 0.25; %Probability move +/- 1 in x-direction
p y= 0.25; %Probability move +/- 1 in y-direction
no x=1-2*p x; %Probability of not moving in x-direction
no y=1-2*p y; %Probability of not moving in y-direction
work=[0 0 0 0];
for t=1:N;
if t<11*24 && mod(t,6)<1 %If t<11 days (ie up to 10 days) ...
and is a multiple of 6; this gives us a new segment ...
every 6 hours for 10 days
y max=max(state(:,2));%finds maximum y value
y min=min(state(:,2));%finds minimum y value
y tot=max(abs(y max),abs(y min));
state(end+1,:)= [0,((-1)ˆn)*(y tot+1),0,0]; %Creates a ...
new cell above or below the furthest out cell
newcell=size(state,1);
n=n+1;
end
%% Migration
% Does the motile cell become a dividing cell?
46
49. f=state(:,3)==c; %Locates cells with an age of 6
e=state(:,4)==1; %Locates cells with type '1'
typechange=f.*e;
state(:,4)=state(:,4)-typechange; %Changes Migratory cell to ...
a dividing cell
state(:,3)=state(:,3).*(¬typechange); %Changes age of cells ...
just affected to zero
% Migration
length=size(state,1); %Calculates how many cells to scan over
r x=rand(length,1); %Generates a vector of random numbers ...
between 0 and 1
r y=rand(length,1); %Generates a vector of random numbers ...
between 0 and 1
x jump=[(r x<p x)-((r x>p x)&(r x<2*p x))].*state(:,4); ...
%Calulates where random numbers fall and generates +1, -1 ...
or 0 for each
y jump=[(r y<p y)-((r y>p y)&(r y<2*p y))].*state(:,4);
state(:,1)=state(:,1)+x jump;
state(:,2)=state(:,2)+y jump; %Applies vectors to states
%% Proliferation of parent cells
a=state(:,3)==24; %Locates cells old enough to divide
b=state(:,4)==0; %Locates cells of type '0' ie parent cells ...
who are ready to divide
div=a.*b; %Gives cells which satisfy both of the above
numcells=size(state,1); %Calculates how many cells to scan over
s x=rand(numcells,1); %Generates a vector of random ...
numbers between 0 and 1
s y=rand(numcells,1); %Generates a vector of random numbers ...
between 0 and 1
state(:,3)=state(:,3)-(24.*div); %Resets time count of ...
relevant cells to zero
x place=[(s x<p x)-((s x>p x)&(s x<2*p x))].*div; %Calulates ...
where random numbers fall and generates +1, -1 or 0 for each
y place=[(s y<p y)-((s y>p y)&(s y<2*p y))].*div;
work=[state(:,1)+x place,state(:,2)+y place,state(:,3),state(:,4)];
L=[div,div,div,div]; %4x4 matrix
daughters=work.*L; %Creates new matrix of daughter cells
daughters( ¬any(daughters,2), : ) = []; %Deletes zero rows
if size(daughters,1)>0
state=[state;daughters]; %Adds any daughter cells ...
created to end of state matrix
end
state(:,3)=state(:,3)+Dt; %Tracks time steps
figure(1)
clf
scatter(state(:,1),state(:,2),cstring(mod(t,7)+1)); %Plots ...
scatter diagram of cells at time point
axis([T U Q W]);
axis([-100 100 -100 100])
xlabel('X Displacement');
ylabel('Y Displacement');
title('Distribution of cells 3 Weeks after Fertilization')
end
fid = fopen('StateAfter3Weeks.txt', 'wt'); % Open for writing
47
50. for a=1:size(state,1)
fprintf(fid, '%d ', state(a,:)); %Records state matrix in a ...
file for use running in week 4
fprintf(fid, 'n');
end
2. Code for creating a density plot of the distribution in weeks 2-3.
function out = WeekFivePrint(x,y,method,radius,N,n,po,ms)
state = importdata('FinalState.txt'); x=state(:,1);y=state(:,2);
if nargin==0
scatplotdemo; return
end
if nargin<3 | isempty(method)
method = 'vo';
end
if isnumeric(method)
gsp(x,y,method,2); return
else
method = method(1:2);
end
if nargin<4 | isempty(n)
n = 5; %number of filter coefficients
end
if nargin<5 | isempty(radius)
radius = sqrt((range(x)/30)ˆ2 + (range(y)/30)ˆ2);
end
if nargin<6 | isempty(po)
po = 1; %plot option
end
if nargin<7 | isempty(ms)
ms = 4; %markersize
end
if nargin<8 | isempty(N)
N = 100; %length of grid
end
%Correct data if necessary
x = state(:,1); y = state(:,2);
%Asuming x and y match
idat = isfinite(x); x = x(idat); y = y(idat); holdstate = ishold;
if holdstate==0
cla; end
hold on
dd = datadensity(x,y,method,radius); %Caclulate data density
xi = repmat(linspace(min(x),max(x),N),N,1);%Gridding
yi = repmat(linspace(min(y),max(y),N)',1,N);
zi = griddata(x,y,dd,xi,yi);
zi(isnan(zi)) = 0;% Bidimensional running mean filter
coef = ones(n(1),1)/n(1); zif = conv2(coef,coef,zi,'same');
if length(n)>1
for k=1:n(2)
zif = conv2(coef,coef,zif,'same');
end end
48
51. ddf = griddata(xi,yi,zif,x,y);%New Filtered data densities
switch po
case {1,2}
if po==2
[c,h] = contour(xi,yi,zif);
out.c = c; out.h = h;
end %if
hs = gsp(x,y,ddf,ms); out.hs = hs; colorbar
case {3,4}
if po>3
[c,h] = contour(xi,yi,zi); out.c = c;
end %if
hs = gsp(x,y,dd,ms);out.hs = hs;colorbar
end
dd(idat) = dd; %Relocate variables and place NaN's
dd(¬idat) = NaN; ddf(idat) = ddf; ddf(¬idat) = NaN;
out.dd = dd;%Collect variables
out.ddf = ddf; out.radius = radius; out.xi = xi; out.yi = yi;
out.zi = zi; out.zif = zif;
if ¬holdstate; hold off
end return
function scatplotdemo
po = 2; method = 'squares';radius = []; N = []; n = [];
ms = 5; x = randn(1000,1); y = randn(1000,1);
out = WeekFivePrint(x,y,method,radius,N,n,po,ms)
return
function dd = datadensity(x,y,method,r)% Data Density
Ld = length(x);%Computes the data density (points/area) of ...
scattered points
dd = zeros(Ld,1);
switch method %Calculate Data Density
case 'sq' %---- Using squares ----
for k=1:Ld;dd(k) = sum( x>(x(k)-r) & x<(x(k)+r) & ...
y>(y(k)-r) & y<(y(k)+r) );
end %for
area = (2*r)ˆ2; dd = dd/area;
case 'ci'
for k=1:Ld;dd(k) = sum( sqrt((x-x(k)).ˆ2 + (y-y(k)).ˆ2) < ...
r );
end
area = pi*rˆ2; dd = dd/area;
case 'vo' %----- Using voronoi cells ------
[v,c] = voronoin([x,y]);
for k=1:length(c)
if all(c{k}>1)
a = polyarea(v(c{k},1),v(c{k},2)); dd(k) = 1/a;
end end end
return
function varargout = gsp(x,y,c,ms)%Graf Scatter Plot
map = colormap;%Graphs scattered poits
ind = fix((c-min(c))/(max(c)-min(c))*(size(map,1)-1))+1; h = [];
for k=1:size(map,1)
if any(ind==k)
h(end+1) = line('Xdata',x(ind==k),'Ydata',y(ind==k), ...
'LineStyle','none','Color',map(k,:), ...
49
52. 'Marker','.','MarkerSize',ms); end
end
if nargout==1; varargout{1} = h;
end
figure(1);axis([-100 100 -100 100])
xlabel('X distribution');ylabel('Y distribution');
title('Density of Distribution of Cells 5 weeks after fertilisation');
return
3. Histogram showing density distribution at Y=0.
%% Plot density as fn of position-Histogram
a = importdata('StateAfter3Weeks.txt'); b = ...
importdata('StateAfter4Weeks.txt'); c = ...
importdata('FinalState.txt');
%% After 3 weeks
p a=-1<a(:,2)<1; %Probability that y=0
H a=[p a p a p a p a]; %Repeated vectors
l a=a.*H a; %New matrix- component multiplication
nbins=40; %Bins for histogram
G a=l a(:,1); %Locates x coordinate we are interested in
%% After 4 weeks
p b=-1<b(:,2)<1 ; %Probability that y=0
H b=[p b p b p b p b]; %Repeated vectors
l b=b.*H b; %New matrix- component multiplication
nbins=40; %Bins for histogram
G b=l b(:,1); %Locates x coordinate we are interested in
%% After 5 weeks
p c=-1<c(:,2)<1; %Probability that y=0
H c=[p c p c p c p c]; %Repeated vectors
l c=c.*H c; %New matrix- component multiplication
nbins=40; %Bins for histogram
G c=l c(:,1); %Locates x coordinate we are interested in
%% Plot Histogram
figure(1); hist(G a,nbins); hold on
hist(G b,nbins); hist(G c,nbins); h = findobj(gca,'Type','patch');
set(h(1),'FaceColor','r','EdgeColor','r','facealpha',0.15); ...
set(h(2),'FaceColor','b','EdgeColor','b','facealpha',0.15); ...
set(h(3),'FaceColor','g','EdgeColor','g','facealpha',0.15);
axis([-12 12 0 2.7*10ˆ5]); xlabel('x value'); ylabel('Density of ...
cells'); legend('After 3 weeks', 'After 4 weeks', 'After 5 weeks');
title('Density of cells at y=0') hold off
50
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