Models of developmental plasticity
and cell-growth
Graeme Wake
Massey University @ Auckland
New Zealand
Workshop on Evolution, Barcelona, Spain
June/July 2014
1
Outline
0.Introduction to Epigenetics
1. Model of evolution- Phenotype change
(a)inducible secondary phenotypes –
a stochastic model
(b)when the response to an environmental
cue is initiated after a period of waiting
(c) continuous phenotype spaces- where we
quantify the phenotype state
2. Cell-growth of a population
-steady-size distribution of an evolving cohort of
cells
2
• Epigenetics is a relatively new scientific
field; research only began in earnest in the
mid Nineties, and has only found traction in
the wider scientific community in the last
decade or so. We
have such a group at Gravida.
• The epigenetic “interpretation” of your inherited
DNA is that it is not fixed – it alters dramatically
(called a phenotype). And not just during big life
changes, like puberty or pregnancy. Now research
has found it can also change due to environmental
factors, such as our stress levels, if we smoke etc.
3
Plasticity. Means by which living
organisms mutate to optimise
their fitness.
Methylation. Chemical Process by
which genes can be switched offthereby changing our phenotype.
There is also increasing evidence
that certain cancers are caused by
misplaced epigenetic tags.
4
1 Outline.
(a) Outline
•
•
•
•
•
•
1. Introduction to the biology
2. Nishimura’s deterministic model
3. A stochastic model of Plasticity
4 Results
5. Discussion of the stochastic model
6. Generalisations: Developmental time
Under Development: deterministic, see (b)
5
1. Introduction
• An organism may express different
phenotypes (gene expression) in response to
changes in the environment. This
phenomenon is called plasticity and appears
to be an important mechanism enhancing an
organism’s ability to survive and reproduce.
• That is, plasticity induces a fitness advantage
for the organism.
6
• Nishimura (2006)* proposed that an organism faced
with a change in environmental circumstances would
seek to minimise the energy cost of adapting to the
new environment and the energy cost of
reproduction. To illustrate these concepts Nishimura
(2006) constructed a simple model of a “prey” in an
environment which changes to favour “predation”.
The prey may elicit a plastic response to reduce
predation, but this response will require extra energy
to induce.
*
Nishimura,K. “Inducible plasticity; optimal waiting time for the development of an
inducible phenotype”, Evolutional Ecology Research, Vol 8, 2006, pp 553-559.
7
• If the predation rate increases it destroys
the population fitness;
• Changes in the characteristics are
initiated with consequential changes in
the Energy content.
8
• In epidemiology, a popular theory is that the
rising incidences of coronary heart disease
and Type II diabetes in human populations
undergoing industrialization is due to a
mismatch between a metabolic phenotype
determined in development and the
nutritional environment during development,
to which an individual is subsequently
exposed. This is known as the
'Thrifty phenotype' hypothesis.
9
2. Nishimura’s deterministic model
• Assume a prior specific death rate of μ1
for the “prey” and, when the environment
changes, it has a death rate of μ2 (with
μ1 >μ2) and that the development of the
plastic response incurs an energy cost from a
base of c0 to c1 (with c0 < c1).
10
•
A fitness function W(t) for the suitable fitness currency as a function of time t at
which the plasticity is expressed - linking the predation rate to the energy cost of
the plastic response:
W (t )  e
 ( 1t  2 (T t ))
[ E  c0t  c1 (T  t )]
(1)
The first term = the survival probability in the predator environment;
The second term = the amount of remaining energy;
where T = total time, t = the switching time,
E = total energy budget, W = “plasticity”,
ci = the Energy consumption rate in the respective modes i= 0,1.
11
co is the default baseline energy cost to the prey
individual per time, and
c1 includes the additional costs of building and
maintaining the defensive phenotype per unit time,
c1 > co.
W (t )  e
 ( 1t  2 (T t ))
[ E  c0t  c1 (T  t )]
This is a simple counting exercise.
• Now differentiate w.r.t t …….
12
• Nishimura uses equation (1) to find the maximum
fitness response for the time at which plasticity is
expressed following the change in the environmental
conditions which most advantages predation. This is
found to be trivially, W`(t) = 0, at which time the
plasticity is (actually) maximised, which gives:
(2)
E

c
T
1
*
1
t 
1   2 c1  c0
• But this assumes a deterministic approach in cohort
where there is no variability,
So……. Introduce variability…
13
3. A Stochastic Model of Plasticity
• Suppose that the fitness of the organism at
time t given by equation (1) is now a
stochastic variable. In terms of pressure for
the development of plasticity it is the
expected value of the fitness that is of
interest. That is, the time of plastic response t
that maximises the expected value of the
fitness is of interest in determining the
evolutionary path.
14
• Let α = μ1 - μ2 > 0
•
A = E – c1T
•
c = c1 – co > 0
Then
W (t )  e
t  2T
(2)
[ A  ct ]
Let the time of the plastic response be a
normally distributed random variable with
2
mean t and variance  t . Note this is not
assuming W is Gaussian, only its components.
Q. Should this be a Beta distribution????
15
• Then the expected value of the fitness of the
organism is:
E[ w(t )]  E[e
t  2T
e
 2T
e
 2T
[ A  ct ]]
t
E[e ( A  ct )]
t
AE [e ]  ce
 2T
E[te
t
]
16
Noting that
t
E[te ]  e
1 2 2
t    t
2
(t  2 )
2
t
17
Then we get
E[ w(t )]  e
1 2 2
  t t   2T
2
A  c(t  2 
2
t
•
(3)
• Note that equation (3) is similar to the deterministic
case expressed by equation (1) with t →
- α
2
:
t
t
W (t )  e
 ( 1t  2 (T t ))
[ E  c0t  c1 (T  t )]
18
• Equation (3) has 2 variables associated with
the plasticity, the time of the response in
relation to the environmental stimulus and
the variance of this response in the population
of interest . Finding the maximum fitness by
differentiating with respect to each of these
variables and setting to zero gives the two
inconsistent equations:
ct   t2   c  A
ct   t2   2c  A
(4)
19
• The fitness of the organism is maximised when
plasticity acts at the time from the environmental
stimulus given by:
1 A
2
t     t
 c
(5)
E  c 2T
1
2


 ( 1   2 ) t
1   2 c2  c1
*
t
This uses the first equation
∂/∂ t = 0.
20
• This shows that when the time of plasticity is
stochastic then the mean time of plasticity
optimising the expected value of the fitness is
extended by an amount given by . In this
respect
 t2 variation in the response of plasticity
in the population might be seen as an
advantage in terms of providing extra time for
the average response in the population to be
most effective.
21
• However, in terms of Life History analysis the
variance of the response to plasticity might be
regarded as the variable of interest. That is, an
organism might seek to optimise fitness by
manipulating the variance of the response, perhaps
by collecting a variety of alleles or by developing
epigenetic modes to express such variation. In this
case the optimum variance of plasticity to
maximise the fitness is given by:
1
1 A
t  t   
  c
2
•
1 
1
E  c2T 
 t 



1  2  1  2 c2  c1 
•
22
4. Results
• However, the results show that the maximum
expected value of fitness is not given by an unique
pair of numbers for the average time of plasticity and
the variance of plasticity, but rather by a set of
numbers lying on a straight line in the plane of the
average time and the variance of plasticity.
Repeating this here…..New term
A
  t2
 c
E  c 2T
1


 ( 1   2 ) t2
1   2 c 2  c1
t* 
1

• This is shown on the following diagram:
23
24
120
Frequency
80
40
0
0
25
50
Time to plasticity
75
• Figure 2: TheL1frequency distributions for the time to plasticity
L2
for 2 populations
each having the same optimal expected
value for fitness given by equation (3).
25
• A population might not be on the optimal
mean time – variance of plasticity line for a
number of reasons. For example, the average
energy available E might change, resulting in a
parallel shift of the optimum mean time
variance line, or the predation rates might
change altering the slope.
26
• If the population ends up off the optimal line
the question arises of the expected path in
the mean time of plasticity – variance of
plasticity space that it will take to get to the
new optimal line.
• To calculate this path we assume that the
probability distribution of time t to switch
states (the plasticity factor for the population)
remains Gaussian. This means that the new
environment induces a similar average change
in t across the population, preserving the
Gaussian character of its probability density.
27
Transition to a new optimal state:
• Assuming that the path of the population in mean
time of plasticity – variance of plasticity space
proceeds in a direction most favourable to the
population this means
2

t
,

that the path from a point
0
0 ,t 
to a point t1 ,  12,on
t  the new optimal line .
That is, the population will follow a path that is
perpendicular to the level curves of the expected
value of the fitness function (3).
28
• Let x = variance,
y = mean time
w1 = maximum of the expected value of
fitness, with
available energy is E1; w1 = F(x,y)
Then E1 → E2
The path to w2 = maximum of the expected value
of fitness, with
available energy E2 will follow the o.d.e
dy = ∂F/ ∂y
dx
∂F/ ∂x
which is perpendicular to the level curves, that is the steepest
path. This is shown in the following diagram…it is obtainable
analytically!!!!
29
•
Figure 1. The trade – off between the expected value of the time for plasticity and
the variance of the time for plasticity in a population subjected to 2 different
energy availabilities, and the optimal path of time to plasticity mean and variance
when the energy available changes from low to high.
30
Example
Consider a population with
T = 100 days,
an average energy level of 100 units,
μ1 = 0.2, μ2 = 0.1 days-1
c1 = 0.5, c2 = 2 units. days-1
standard deviation σ = 8 days.
This will have an optimal average time to optimal
plasticity of 84.1 days. (The deterministic
calculation is 77.7 days, that is, 6.4 days less).
If the energy level available from the environment
changes to 105 units then the equations imply the
population will move to an optimal mean time for
plasticity of 81.8 days with a standard deviation of
7.2 days.
31
Example continued
The extra energy available results in a shorter
time to implement plasticity, because the
population has more energy to spare on
maintaining plasticity for a longer time.
Energy Mean time
Std Deviation
100
84.1 days
8 days
105
81.8 days
7.2 days
Outcomes in Red
32
4. Discussion
• The analysis presented here shows a relationship
between the average time for plasticity and the
variance of the time for plasticity.
• The higher the variance the longer the average time
for plasticity that maximises the organism’s fitness.
• If there was a constraint which acted to ensure that
plasticity required some minimum time, the organism
might adjust the variance of the time for plasticity to
maximise fitness.
33
• The smaller the difference between the death
rates the greater the variance which ensures
maximum fitness (for a given average time for
plasticity).
• Alternatively, the greater the difference
between death rates the smaller the variance
for maximum fitness.
• Similarly for the energy cost of implementing
plasticity.
• The moments (1st and 2nd) of the optimal
distribution are obtained by a shift in available
energy easily by solving a first order o.d.e.
34
(b). Deterministic Generalisation
• After an environmental cue is detected, the organism waits for
a period t before initiating development of the induced
phenotype, which takes time τ. During this waiting time the
constant per unit-time cost of the yet to be induced phenotype
is c1, and the death rate μ1 is also constant. During
development, these parameters change to cd and μd,
respectively, and after development
is complete they become c2 and μ2. Although we do not
assume so, biological considerations imply that
cd > c2 > c1.
Similarly, it seems reasonable to infer that μ1 > μd,> μ2.
0
t
t+τ
T
│
│
│
│
waiting developmental afterwards
time
35
• We investigate various different realistic forms of cd :
(a) first, that the product cdτ is a constant, D, which can be
thought of as the (fixed) cost of development;
(b) second, that cd is a constant, and so the cost of
development is a linearly increasing function of development
time, τ, that is,
and, finally,
(c) we put
D    cd
D    K0  K d e r
where Ko, Kd and r are constants, which implies that rapid
development is more expensive: the cost of development, , is
now a decreasing function of the development time. Fitness
is evaluated at a time T, measured from the onset of the
environmental cue.
36
• Again following, Nishimura (2006), we assume
that the total initial energy budget is a constant,
E, and the background cost of plasticity reduces
this energy by a constant amount c0. If the
environment does not change (and there is no
cue), therefore, the total cost to the organism is
c0 + c1T. If the organism detects the cue and
develops the induced phenotype, the cost is
c0  c1 t     c2 T  t   
• . The proportion of the population which
survives is, following Nishimura (2006),
exp[  1t  d  2 T  t    ]
37
• and hence the fitness function is
W  t ,   e 1t e d e
 2 T t  
 E  c0  c1t  cd  c2 T  t   
• Nishimura’s (2006) Equation (1) can be
recovered by substituting cd = c1 and μd = μ1.
•
This formulation assumes the cue is always
detected and is perfectly reliable. It is
important to realize that the evolution of
plasticity may be favored even when these
two assumptions are violated (Moran 1992;
Sultan & Spencer 2002).
38
References
• Moran, N.A. 1992. The evolutionary maintenance of
alternative phenotypes. American Naturalist 139:
971-989.
• Sultan, S.E. and H.G. Spencer. 2002. Metapopulation structure favors plasticity over local
adaptation. American Naturalist 160:271-283.
• Wake, G.C., A.B. Pleasants, A. Beedle and P.D.
Gluckman. 2008. The optimal waiting time for the
induction of plasticity for a phenotype in a
population subject to random environmental effects.
(M.B.E. Journal).
39
1(b) Plastic Adaptive response
A deterministic model.
• Predictive adaptive responses (PARs) are a form
of developmental plasticity in which the
phenotypic response to an environmental cue is
delayed until some time after the cue and the
induced phenotype may anticipate (i.e., be
completely developed) before exposure to the
eventual environmental state forecast by the cue.
• We model this sequence of events to discover,
under various assumptions concerning the cost of
development, what lengths of delay,
developmental time and anticipation are optimal,
that is, the effect for the host is most
advantageous.
40
Procedure for these modifications
• Proceed as before, optimise the fitness
function on the region
{(t, τ ) : 0 < t + τ < T }.
Note: The maximum may, for some
parameter values not be an internal
maximum.
This is trivial calculus really with big
impact.
41
Figure 1: Life history of the model organism. The dashed
yellow line shows the death rate for an organism that fails to
respond
to
the
environmental
cue.
42
Summary
• Phenotypic plasticity is ubiquitous in the
biological realm and it has fundamental
consequences for our understanding of
evolution.
• The realization that the induced response
need not begin immediately after the
environmental cue has generally been underappreciated by evolutionary biologists.
43
• Such a delay may have important
consequences for fitness and, indeed, some
recent authors have argued that a number of
phenotypic changes observed in adult humans
and other mammals that have deleterious
health effects (e.g., diabetes, hypertension),
are in response to inappropriate
environmental cues detected as a fetus or
newborn .
44
• It is further claimed that the physiological
mechanisms underlying many of these
delayed responses have been selected and we
label them
“predictive adaptive responses.”
PAR is an in-topic right now amongst
evolutionists.
45
Outcomes
• Our modelling shows that, for a variety of formulations
for the costs of development, predictive adaptive
responses (PARs) maximize fitness when there is no
anticipation (the time between the completion of the
development of the induced phenotype and the onset
of the eventual environment).
• In addition, when rapid development is costly, fitness is
maximized when there is no delay between the
environmental cue and the initiating of development,
and hence all available time is allotted to development.
• Developmental costs appear to be privileged, fitness
being maximized whenever the additional costs of
development are minimized. Second in the hierarchy is
the delay time, with fitness maximized when any time
not required for development is assigned to the delay.
46
• Here we have extended Nishimura’s model in four
ways:
(1) we introduce a cohort and focus on the probability
distribution of development time;
(2) we allow the parameters during development to be
different, albeit constant, from those during the
waiting time which we introduced;
(3) we permit development time to be a variable and
optimize it simultaneously with waiting time;
(4) we assume that the cost of developing the induced
phenotype is an exponential function of
development time (details omitted).
This has enormous effect on phenotype plasticity and
organism development especially for a fetus.
47
Any Questions??
48
1.(c) A simple 2-dimensional model
• Let u(t), and v(t) be concentrations of healthy and
infected target cells respectively.
• Healthy target cells are generated at a constant rate b and
die at the per capita rate μ.
• Virus infect healthy cells at a rate α u*v “mass kinetics”;
• The virus-infected particles die at the per capita rate m.
• In this model framework, a virus is described by the
constants β and m only.
du
 b  u (t )v(t )  u (t ),
dt
dv
 u (t )v(t )  mv(t )
dt
49
Outcomes
Steady states (U,V)
USE Ro
Qo (U,V) = (b/μ, 0): a disease-free equilibrium
Q* (U,V) = (m/β, b/m - μ/β) Endemic equilibrium
= (b/μRo, μ(Ro – 1) /β),
where Ro = b β/ μm serves as
- the strain’s basic reproduction number
- a measure of its Darwinian fitness
50
Clearly
Qo is globally asymptotically
stable/unstable if Ro </> 1;
Q* is globally asymptotically
stable/unstable if Ro >/<1 ;
51
V
Bifurcation
Diagram:
-One globally
asymptotically
stable state
Ro
Ro
Arrow of
time
52
(b) Strain Space and Random Mutations
•
•
•
•
•
•
•
•
•
Let us assume that several different strains are present at
the same time;
Each strain is characterized by β and m only;
Let us introduce a one-dimensional strain space S = R+ =
{s>0};
Each strain corresponds to a point s of this space;
Each infected cell is now infected with a particular strain;
Hence β becomes a function of s;
. . . while m, for the sake of simplicity, is the same for all
strains;
For simplicity we use a linear function β = a * s;
Now we can model random mutations by dispersion in the
strain space.
53
New Model
This is an integro-differential system,
with a non-local term. Solve!!

du
 b  au (t )  sv(t , s)ds  u (t )
0
dt
2
v
 v(t.s)
 asu(t )v(t , s)  mv(t , s)  D
2
t
s
Here D is the drift dispersion coefficient,
D = (variance)/2
54
Here
 2v
- dispersion D s 2 describes
random mutations;
- v(t; s) is the density distribution of
the infected cells in S,
[u, v] = cells.mm-3; and
- VT(t) = ∫0∞v(t, s)ds is the total
concentration of infected cells.
55
The domain of the system is
(t,s) ϵ R+2;
The natural boundary condition at s = 0
is the no-flux condition
v(t , s  0)
0
s
The system should be complemented
by non-negative initial conditions
u(0), v(0,s) given and appropriate
boundedness conditions as s →∞ .
56
3. Mathematical analysis
•
Some computation in Korobeinikov and
Dempsey, MBE (in press).
D = 10-8 day-1
D= 10-6 day-1
•
•
The distribution of infected cells moves to higher fitness.
• The speed of the evolution depends on the available resources.
57
Numerical Simulations
• Travelling waves time in years
moving toward
higher fitness.
• Velocity of the
wave is variable
and depends on
the abundance of
the resources.
D = 10-6 days-1
58
Steady-states (drives the dynamics)
• Notice these computed solutions suggest
v(t,s) → 0 as t →∞.
• These will be (U,V(s)):

U ( a  sV (s)ds)  b, DV " (s)  (asU  m)V ( s)  0.
0
V ’(0) = V(∞) =0.
59
These uncouple to give the nonlocal
equations
abs
DV ' ' ( s)  (
 m)V ( s)  0, s  0

 a  sV ( s)ds
0
V ‘(0) = V(∞) =0.
Solve this? Are there any non=trivial solutions?
Then,
U
b

  a  sV ( s )ds
0
60
Algorithm used
(a) Assume we know the definite integral for V(s)
in
abs
DV ' ' ( s)  (
 m)V ( s)  0, s  0

 a  sV ( s)ds
0
Call it K,
abs
So DV " ( s) (
 m)V ( s)  0,V ' (0)  V ()  0.
 aK
61
V(s)≡ 0 or V(s) is an Airy function.
(b) Airy functions are solutions of
y”(x) – xy(x) =0, x in R, y = Ai(x), Bi(x)
62
(c) Get V(s,K), and solve the
transcendental
equation for K

 sV (s, K )ds  K
0
K having a non-trivial positive solution gives an
equivalent condition for “fitness” on the
parameters – it will effectively give us Ro in
terms of a, b, m, μ, and D, and we write it as
the Nominal condition Ro > 1. This work is in
progress.
63
Toy non-local problem
Consider the non-linear non-local eigenvalue problem
 y ' ' ( s )  sy ' ( s )  
2
y ( s )
 0, s  0; y ' (0)  y ()  0.
 y(s)ds
0
This has non-trivial solutions
y = A exp(-x2/2σ2), when
λ = A√(π/2) σ ~ A
64
So “λo“ = 0.
65
4. Conclusions
• Phenotype expression can be modelled using
structured population models;
• These involve challenging nonlocal nonlinear
eigenvalue problems, with thresholds able to
be quantified.
• Algorithms are available…..to be continued.
66
Tumour cell biology and some new
non-local calculus: Outline
• Elementary Maths
• Biology background
• A new generic model
• Key mathematical challenges
67
This is all about “Non-local
Calculus” where cause and
effect are separated by time,
distance, age, size, etc, Often
called Functional Calculus.
Arose for me in:
Population models(discrete time delay)
Harvesting models(continuous time delay)
Tumour Cell-growth (delays/size/division)
Ignition –in mines (space - action at a distance)
Useful results: New Maths!!!!
68
0. Time delay: Arise in poulation
models.
First example; “R18 Calculus”.
x‘(t) = x(t-T), t ≥ 0;
x(t) = xo(t), -T < t ≤ 0.
Questions:
Questions:
1.
1. Well-posed?
Well-posed?
2.
2. What
What space
space isis yx in?
in?
3.
3. Solution
Solution ++ ??
?? PRIZE!!!!!
69
Solution of Malthusian-delay problem with xo(t) = 1
70
Questions:
1. Well-posed? Yes
2.What space is x in? Depends on xo(t).
3.Solution + ??
x(t) = ∑∞ cn exp(λnt), where the λn are in the
n=0
set {λ : λ = exp(- λ T) } ~ N
λ o ≈ 0.56 (when T = 1 );
Re(λn ) < 0, n >0 and occur in
complex conjugate pairs:
messy calculation.
:
71
Transcendental equation
λ = exp(- λ), when T = 1 ;
λ1 = Re(λ), λ2 = Im(λ) satisfy
λ1 = exp(- λ1 ) cos(λ2 ) (i)
and λ2 = - exp(- λ1 ) sin(λ2 ). (ii)
Only one real solution.
Infinite number of complex conjugate
solution pairs.
72
Solutions of real part (red (i)) and imaginary (blue (ii)) part of exp(-λ) – λ= 0
73
1. Biology: Cell- Cycles
74
1. Biology background
• The ability of cells to divide asymmetrically is
essential for generating diverse cell types during
development.
• The past 10 years have seen tremendous
progress in our understanding of this important
biological process.
• The relevance of asymmetric cell division for stem
cell biology has added a new dimension to the
field, and exciting connections between
asymmetric cell division and tumour genesis have
begun to emerge.
75
• Cell divisions producing two daughter cells that
adopt distinct fates are defined as asymmetric.
• In all organisms, ranging from bacteria to
mammals, in which development has been
studied extensively, asymmetric cell divisions
generate cell diversity.
• Asymmetric cell divisions can be achieved by
either intrinsic or extrinsic mechanisms, Intrinsic
mechanisms involve the preferential segregation
of cell fate (read “size”) determinants to one of
two daughter cells during mitosis.
• Asymmetrically segregated factors that bind cell
fate determinants and orient the mitotic spindle
may also be necessary to ensure the faithful
segregation of determinants into only one
daughter cell.
76
Intrinsic and extrinsic mechanisms in asymmetric cell divisions.
Hawkins N , Garriga G Genes Dev.
1998;12:3625-3638
77
©1998 by Cold Spring Harbor Laboratory Press
Intrinsic and extrinsic mechanisms in asymmetric
cell divisions.

Intrinsic factors indicated in red are distributed
asymmetrically in the progenitor cell and inherited
by one daughter cell (Z).

These factors can be cell fate determinants that
specify the fate of the daughter cell that inherits
them, factors that distribute cell-fate
determinants, or factors that orient the spindle
.
 Extrinsic mechanisms require cell signalling that
can occur (1) between daughter cells, (2)
between a daughter cell and surrounding cells, or
(3) between the precursor cell and surrounding
78
cells.

Taxol effect: 6 weeks “in vivo”
79
Our Objective
.
SSD behaviour: n( ,t) evolves like:
Cell Count
SSD Behaviour
10
9
8
7
6
5
4
3
2
1
0
t2
t1
0
1
Cell Size
2
3
4
80
Melanoma Cells in Vitro
NZM13 DNA profile 0 hours after the addition of Taxol
800
Model
700
Data
Cell Count
600
Doubling Time:
500
Model: 72 hours
400
Data: 76 hours
300
200
100
0
0
0.5
1
1.5
2
2.5
x (Relative DNA Content)
3
3.5
81
2. Notation and Our Model
• Independent variables
t time in hours : [t] = T
x “size” (=fate) in ngrams: [x] = M
 Dependent variable
-1
n(x,t) bnumber density
 n( x, t )dxof cells w.r.t x [n] = M
b
M(t) =
 n( x, t )dx
a
= # cells in a ≤ x ≤ b.
Size
a
|
0
τ
||
|
x
ξ
Transfer function W(x,ξ) = # cells of size x from
division of one cell of size ξ : [W] = M-1 .
x
82
More notation
Transfer function W(x,ξ) = # cells of size x from
division of one cell of size ξ : [W] = M-1
Frequency at which cells divide into two or
more daughter cells b(x) : [ b ] = T-1
Per capita death rate µ(x) : [µ] = T-1
Per capita growth rate g(x) : [g] = M T-1
83
The equation.
Analogy
• Think of n as a “concentration”, as a
function of “position” x;
• g as a velocity;
• Do a balance on n…….The ultimate non-local p.d.e
0
τ
x
ξ
(2)
•B.c, i. c:
n(0,t) = n(∞,t) =0; n(x,0) = no(x) (arbitrary)
n


( gn) 
t
x
x

 b( )W ( x,  )n( , t )d (credits (1))
x
 (b( x )  W ( , x )
0
(1)

x
d ( 2)   ( deaths )) n( x, t )( debits )
84
The splitting function W(x,ξ)
• What sort of W(x,ξ)’s are feasible???
• Answer….
If a cell of size x splits to give many cells of size τ,
0 < τ < x, conservation of mass demands
x
x   W ( , x)d
(*)
0
This is an ill-posed partial Volterra equation.
85
A simplification to The Equation
Using (*) we get The Equation simplified as

n ( gn)

  b( )W ( x,  )n( , t )d  (b( x)   )n( x, t )
t
x
x
This satisfies “mass” conservation, and has
to be solved with the b.c and i.c.
n(0,t) = n(∞,t) = 0, n(x,0) = no(x)
and with admissible W(x,ξ) satisfying (*).
86
Examples of W(x,ξ) to date…
1. W(x,ξ) = 2 δ(ξ/2 – x) : δ = the Dirac-delta
function, the symmetrical special case: two
daughter cells of equal size.
2. W(x,ξ) = W(x – ξ) = δ(x – ξ) , not real in the
sense that the cells are not really dividing,
but it satisfies (*)!!!!
3. W(x,ξ) = U(x)/V(ξ), the separable case:
(noting U,V have different dimensions)
x
then (*) gives:
1
V ( x)   U ( )d

x0
87
4. The most realistic case
binary/asymmetric: α > β >1 constants
І
І
І
І
І
0
z =x/α z = x/β [ x ] ξ = βx :ξ = αx
out
in
Thus
W(x, ξ) = δ(ξ/α – x) + δ(ξ/β – x)
The mass conservation equation
1
x
x   W ( , x)d
implies
0
α
+
= 1, or β =
α
β
α−1
1
1
88
3. Key mathematical challenges
• Solve THE equation for various W;
• Look for separable solutions
n(x,t) = e -λt y(x)
• Get a non-local first-order singular eigenvalue
problem,
1. With W(x, ξ) = 2 δ(ξ/2 – x), g, b constant
(gy)’(x) = 4b y(2x) – (b + μ)y(x) + λ y(x);
y(0) = y(∞) = 0 ; y ≠ 0.
89
4. The binary/asymmetric case:
With W(x, ξ) = δ(ξ/α – x) + δ(ξ/β – x),
g, b constant and 1/α + 1/β = 1 we get
(gy)’(x) = b α y(α x) + b β y(βx) –
-(b + μ)y(x) + λ y(x);
y(0) = y(∞) = 0 ; y ≠ 0.
This is a two term “pantograph equation”!!!
This is an eigenvalue problem.
What are the
eigenvaues/eigenfunctions??
90
Conjectures, for some cases, yet to
be fully understood.
Spectrum = {λn : n ε N} ;
Eigenfunctions {yn(x) : n ε N } “complete” in
some sense;
n(x,t) = Σ cn exp(−λnt) yn(x)
nεN
Note: there maybe some non-discrete/continuous spectra, as in
y’’(x) + λy(x) = 0, 0 < x <∞; y(0) = 0, │ y(∞ )│ < ∞.
Then SSD- behaviour is n(x,t) ~ exp(−λmt) ym(x),
where λm = the eigenvalue with smallest real part.
If Re (λm ) > 0, cohort decreasing;
Re (λm ) < 0, cohort increasing:
Both asymptotically of constant shape. This is seen experimentally.
91
Future Extensions
• Let growth be stochastic, using the stochastic
growth given by
dx = gdt + σdX,
(A Langevin equation, where X is a Wiener
process, with variance σ2 ) and using the
Fokker-Planck equivalence, interpreting n(x,t)
2
as a p.d.f, we add to the LHS
 ( Dn)

x 2
where D = σ2/2, making The Equation parabolic,
not hyperbolic. The same challenges will arise.
.
Questions/Comments
Acknowledgments:
• Professor Helen Byrne (Oxford) for pointing
out the need for mass conservation on the
splitting function W, thus giving the
x
condition
(*)
x   W ( , x)d
0
• Support from
93
Questions/Comments
Epigenetics: How to alter your genes
We’ve long been told our genes are our destiny.
But it’s now thought they can be changed by
habit, lifestyle, even finances. What does this
______mean for our children?_______
Thank you to Gravida for financial support
94