NonAutonomous Reaction Diffusion Equations: Biological Pattern Formation

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Reaction Diffusion Models of Biological Pattern
FormationThe Effects of Domain Growth and Time
Delays
EA Gaffney Collaborators NAM Monk, E Crampin,
PK Maini
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Summary of Background

• In 1952 Turing proposed that
• Pattern Formation during morphogenesis might
  arise through
• an instability in systems of reacting
  chemicals (morphogens),
• which is driven by diffusion.
• Heterogeneous concentrations of these chemicals
  form a pre-pattern.
• Subsequent differentiation of tissue/cell
  type is in response to whether
• or not one of these morphogens exceeds some
  threshold locally.
• The equations describing this for two reacting
  constituents on a
• stationary domain are of the form

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In the simplest setting there is a unique
homogeneous steady state at (a0, b0), given
by solutions of The Jacobean
at the stationary point is of the form

The kinetics are always chosen such that, in the
absence of diffusion, the homogeneous steady
state is stable (and thus the instability is
diffusion driven) In the presence of
diffusion, for a sufficiently large domain, and
quite reasonably assuming the components of the
Jacobean are O(1) compared to the scales e, 1/e
the homogeneous steady state is unstable if
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• For
• a domain larger than the critical size, again
  assuming the components of the Jacobean are
  O(1) compared to the scales e , 1/e
• e ltlt1 sufficiently small
• the rate of growth of the fastest growing mode,
  µ, is given by
• where

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• A key feature of all Turing-Pair kinetics is
• Morphogen induced production of morphogen
• Short range activation, long range inhibition.

A specific example Schnakenberg Kinetics
p 0.9, q 0.1, e ltlt 1. 1/T0 ? is
the decay rate of the activator, b
In particular the activator production is
equivalent to a law of mass action rule
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• Potential Examples
• Avian feather bud formation
• HS Jung, , L Wolpert, ... et al, Developmental
  Biology 1998
• Vertebrate limb formation
• CM Leonard et al, Developmental Biology, 1991.
  
• TGF-ß2 possible activator inhibitor
  undetermined
• Zebrafish mesendoderm induction
• L Solnica-Kreznel, Current Biology, 2003
• Nodal (Squint) gene product is an activator.
• Lefty gene product is an inhibitor
• Evidence that range of Lefty's influence
  exceeds Nodal's.
• Molecular details remain to be uncovered of
  their interaction (though progress is rapid on
  this point)
• Molecular details remain to be determined on
  the differential in

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Difficulties

• Reaction Diffusion Patterns can be observed to be
  very sensitive to
• Noise (in initial conditions and generally).
• Perturbations of the domain shape.
• Turing Morphogens are hard to find.
• However, more and more molecular data is being
  produced in developmental studies. These indicate
  that a possible Turing Pair are Nodal and Lefty
  in Zebrafish mesendodermal induction.
• The molecular data also indicates that Nodal and
  Lefty and other putative Turing pairs induce
  each others' production by signal transduction
  and gene expression.
• For example, in situ hybridisation reveals mRNA
  transcripts of the proteins speculated to be
  Turing pairs.
• The extracellular domain is complicated and
  tortuous.
• The precise details of the kinetic functions are
  only ever speculated.

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Formulating a model
Complicated extracellular domain
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Robustness and Reaction diffusion on growing
domains
The pattern produced by an RD system can be
sensitive to the details of the initial
conditions
0.3
frequency
0.2
0.1
Mode Number
5
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20
15
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• Numerical Simulations of Kondo, Asai, Goodwin and
  others
• indicate that
• domain growth can lead to robust pattern
  formation, ie. an
• insensitivity to noise and randomness in the
  initial conditions.
• This has previously motivated a detailed
  investigation of
• the stabilising influence of domain growth
• the mechanisms by which it produces robustness
• conditions for which one may expect robust
  pattern formation

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Model Formulation Incorporating uniform domain
growth
Uniform growth
Rescaling
gives
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Activator Exponential Growth Schnakenberg
Kinetics
Exponential Domain Growth Self similarity
arguments indicate this behaviour will continue
indefinitely in time. The pattern is
insensitive to details of the initial
conditions These observations hold over five -
six orders of magnitude of domain growth. The
robustness is insensitive to the details of the
kinetics (providing pattern initially forms).
Linear Domain growth Frequency doubling
behaviour breaks down more readily. No self -
similarity arguments.
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Space
Space
30
10
4000
500
Time
Activator Linear Growth Schnakenberg Kinetics
30
Space
10
Time
2
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• Pattern formation for logistic growth
  (Schnakenberg Kinetics).
• For the exponential phase of logistic growth,
  robustness is observed.
• However, there is the possibility of a loss of
  robustness as the domain growth saturates, as
  above for the centre plot.
• However, the saturation domain size increases
  by a factor of 1.0015 on moving from left to
  right.
• This indicates an extreme fine tuning of
  parameters is required to loose robustness for
  logistic growth.

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Conclusions Robustness and Domain Growth

• Including GROWTH in 1D reaction diffusion models
  leads to
• Robustness to noise in the initial conditions
  over 4-5 orders
• of magnitude of the growth rate for
  exponential growth
• Semi-scale invariance. No need for parameter
  fine tuning or
• feedback between the domain size and the
  kinetics.
• Persistence of robustness for logistic
  saturating growth.
• Robustness independent of the exact details of
  the model
• providing pattern initially forms.

Dismissal of Reaction Diffusion as a Pattern
Formation mechanism on the grounds of robustness
not necessarily founded if slow domain growth is
relevant.
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Time Delays
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Question How are the effects of signal
transduction and gene expression time delays
incorporated into a model?
Signal
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In more detail ...
Transcription and Translation delay dependent
on size on protein. It is at least 10 minutes
and can be several hours. Signal transduction
serves to only increase this delay
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For a suitable non-dimensionalisation, the
Schnakenberg reaction diffusion equations, in the
presence of domain growth and time delays, can be
written in the form
A, 2B lost from reaction at time t
3B gained from reaction at time t-t
u is the velocity field of the domain growth
y takes values in 0, L(t), where L(t) is
the (non-dimensionalised) domain length
t is the gene expression time delay
where
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Model Investigations
Linear Analysis.
(a, b) is the homogeneous steady state solution
Substitute
into the model equations to obtain (on neglecting
O(?2))

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• In the absence of a time delay, with or without
  domain growth, substitute
• The critical value of the domain length is
  given via
• There are no oscillations at this critical
  point, nor for the fastest growing mode (at
  least providing e2 is sufficiently small).
• The rate of growth of the fastest growing mode,
  µ, is again given by

into
Schnakenberg Kinetics

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In the absence of domain growth, substitute
into

to obtain

• For e ltlt1 sufficiently small and q11 q12
  0, q22 gt 0, p22q22 gt 0, as with Schnakenberg
  kinetics, one has
• There are no oscillations at the critical
  domain length
• The critical length for the onset of
  instability is unchanged by the time
  delay

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In the absence of domain growth, substitute
into

to obtain

• For e ltlt1 sufficiently small and q11 q12
  0, q22 gt 0, p22q22 gt 0, as with Schnakenberg
  kinetics, one has
• Writing ? µ i?, the rate of growth of
  the fastest growing mode, µ, is given
  implicitly by

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We have the fastest growing mode has a growth
rate given by while in the absence of domain
growth the fastest growth rate is given by the
above expression with t 0.
1
We focus on non-oscillatory modes (? 0), as
oscillations are not observed in Schnakenberg
kinetics One can solve the above in terms of
the LambertW functions on neglected the O(e2)
terms. On the left is a plot of the ratio
µ(t, ? 0) / µ(t 0, ? 0) as a function of
tq22, t(p22-e2p2/?)
0.8
0.6
0.4
0.2
0
-1
t(p22-e2p2/?)
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We have the fastest growing mode has a growth
rate given by while in the absence of domain
growth the fastest growth rate is given by the
above expression with t 0.
1
We focus on non-oscillatory modes (? 0), as
oscillations are not observed in Schnakenberg
kinetics One can solve the above in terms of
the LambertW functions on neglected the O(e2)
terms. On the left is a plot of the ratio
µ(t, ? 0) / µ(t 0, ? 0) as a function of
tq22, t(p22-e2p2/?)
0.8
0.6
0.4
0.2
0
-1
t(p22-e2p2/?)
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Explicitly Solving the Linear Equations to
obtain insight for when there are both Time
delays and Domain Growth
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We can see that time delays do greatly increase
the time it takes to leave the homogeneous steady
state, as indicated by analysis
A sufficiently large value of td 2ln(2)
? Time Delay/Doubling Time can result in the
large time asymptote of the linear theory
decaying to zero.
These results appear to be completely general.
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t/t0
t/t0
td/t0
td/t0
The minimum of, say, 50 and the large time
asymptotic value of the components of An1 are
plotted against d, t/t0, td/t0 for various
parameters. Note that the large time asymptote
is always small for sufficiently large td.
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Thus one cannot rely on a naive linear analysis
predicting an instability via growth away from
the homogeneous steady state.
The large time asymptote may decay to zero for
sufficiently large td. Whether the
intermediate behaviour triggers pattern formation
depends on the non-linear dynamics
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Conclusions from the linearised equations

• Domain Growth, No Time Delay. In the absence of
  time delays, domain growth does not have
  much effect on the linear analysis
• Time delays, No Domain Growth. There will
  typically be a substantial patterning lag
• The location of the onset of the instability is
  independent of the time delay
• The ratio of the fastest growing modes in the
  presence of the time and in the absence of the
  time delay will typically be large.

• Domain Growth Time Delays.
• There will typically be a substantial
  patterning lag
• A naive linear analysis is conceptually flawed
  for the prediction of instability. Whether the
  intermediate behaviour triggers pattern
  formation depends on the non-linear dynamics
• The behaviour of the large asymptote is
  governed by the parameter
• td 2ln(2) ? Time
  Delay/Doubling Time

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Numerical Simulations of the Nonlinear equations
with Schankenberg kinetics and domain growth
and time delays
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Initial Conditions for t in the domain 0,t
a
b
x
x
The initial conditions are typically given by the
solid lines above (IC1) and the dashed lines
(IC2). We also consider multiplying these
initial conditions by time dependent factors.
One example is 1 0.0025 cos (px) cos
(pt/(2t)) All the behaviour observed below is
representative of the numerous initial conditions
considered. Similarly for an order of magnitude
variation of the parameters t, e, d.
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t 0, IC 1
t 0, IC 2
Stationary Domain Gray Scale plots of the
activator (Schnakenberg) There are no
oscillations. The final pattern is sensitive to
the details of the initial conditons. t0
corresponds to a delay of 12 minutes in the
dimensional model A delay of t0 induces a
patterning lag of about 60t0 A delay of 4t0
induces a patterning lag of about 240t0
t t0, IC 1
t t0, IC 2
t 4t0, IC 1
t 4t0, IC 2
x
x
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t 0, IC 1
t 0, IC 2
?103
?103
Growing Domain Gray Scale plots of the
activator (Schnakenberg) Domain doubling time
of 2 days There are no oscillations. t0
corresponds to a delay of 12 minutes in the
dimensional model Time delays delay the onset of
peak doubling
?/?(t 0)
t t0, IC 1
t t0, IC 2
?103
?103
?/?(t 0)
t 4t0, IC 1
t 4t0, IC 2
?103
?103
?/?(t 0)
x
x
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t 0, IC 1
t 0, IC 2
?103
?103
Growing Domain Gray Scale plots of the
activator (Schnakenberg) Domain doubling time
of 2 days There are no oscillations. t0
corresponds to a delay of 12 minutes in the
dimensional model Time delays delay the onset of
peak doubling Larger time delays result in the
absence of the Turing instability
?/?(t 0)
t t0, IC 1
t t0, IC 2
?103
?103
?/?(t 0)
t 8t0, IC 1
t 8t0, IC 2
?103
?103
?/?(t 0)
x
x
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t t0, IC 1
t 0, IC 1
?/?(t 0)
t 2t0, IC 1
t 4t0, IC 1
?/?(t 0)
?/?(t 0)
x
x

• Growing Domain
• The time delay induces a delay to patterning
• A time delay of t0, i.e. 12 minutes induces a
  lag of a domain doubling time, i.e. 2 days

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t 4t0, IC 1
t 4t0, IC 1
?103
?103
Domain Doubling Time 2 days
?/?(t 0)
x
x
t t0, IC 2
t 4t0, IC 1
?103
t t0, IC 1
?103
?103
5
Domain Doubling Time 12 hours
?/?(t 0)
x

• Growing Domain
• The behaviour of the system appears to be
  governed by the parameter grouping
• td 2ln(2) ?Time
  Delay/Doubling Time

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t 4t0, IC 1
t 4t0, IC 1
?103
?103
Domain Doubling Time 2 days
?/?(t 0)
t t0, IC 2
t 4t0, IC 1
?103
t t0, IC 1
?103
?103
5
Domain Doubling Time 12 hours
?/?(t 0)
x
x
x

• Growing Domain
• The behaviour of the system appears to be
  governed by the parameter grouping
• td 2ln(2) ?Time
  Delay/Doubling Time

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t 4t0, IC 1
t 4t0, IC 1
?103
?103
Domain Doubling Time 2 days
?/?(t 0)
t 16t0, IC 1
t 16t0, IC 2
t 4t0, IC 1
?103
?103
?/?(t 0)
Domain Doubling Time 8 days
?/?(t 0)
x
x
x
x

• Growing Domain
• The behaviour of the system appears to be
  governed by the parameter grouping
• td 2ln(2) ?Time
  Delay/Doubling Time

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x
x
x
x
40
Irregular behaviour also possible, along with
oscillations, before the failure of the Turing
instability as one increases the time delay.
x
x
x
x
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• Schakenberg Model. Results. Summary
• Stationary Domain
• No oscillations generally
• Pattern is sensitive to the initial conditions
• Time delays can induce a large patterning lag
• Growing Domain Results. Summary
• No oscillations generally
• Time delays can induce a large patterning lag
• Time delays can induce irregular behaviour and
  a failure of the
• Turing instability
• The behaviour of the system is governed by the
  parameter
• grouping td 2ln(2) ?Time Delay/Doubling
  Time

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• Conclusions. Time Delays and Biological RD
  Systems
• We have
• motivated the biophysical need for the
  inclusion of
• signal transduction and gene expression
  time delays in models
• of biological pattern formation
• We have demonstrated how these delays can be
  included in
• one of the simplest "long range
  activation-short range inhbition"
• pattern forming reaction diffusion models.
• While we have not typically found oscillations,
  we have found that
• Time delays can make a large difference to the
  patterns emerging
• from the models, especially with regard to
  patterning lags and,
• for growing domains, the failure of the
  Turing instability
• Thus

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• The above observations do not rule out reaction
  diffusion as a
• putative pattern formation mechanism, whether on
  a stationary
• or uniformly growing spatial domain.
• However, when considering patterning events
  especially those for
• which rapid establishment of pattern is critical,
  such as in the tissues
• of developing embryos, our results show that
• any putative time delays cannot be neglected in
  general without
• careful justification.
• our finding that time delays can dramatically
  increase the
• time taken for the reaction diffusion system
  to initiate patterns
• imposes potentially severe constraints on the
  potential
• molecular details of any Turing system that
  might operate
• during developmental patterning.

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• Future Work/Further Questions
• Continue investigating the extent to which the
  results are general, especially for
  models with "short range activation, long range
  inhibition"
• Kinetics with a negative feedback loop,e.g.
  Gierer Meinhardt
• Kinetics with more than two componenents and
  multiple time delays. Are the patterning lags
  cumalative?
• Other biological pattern formation mechanisms
  e.g. the mechanochemical models
• If the results are general
• We have, in general, potentially severe
  constraints on the reaction diffusion mechanism
  and other mechanisms of biological pattern
  formation
• If the results are not general
• there is a clear distinction between the
  patterning forming behaviour of "short range
  activation, long range inhibition" on the
  inclusion of time delays. This would have a
  substantial impact in that the choice of the
  kinetics really does matter!

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Complicated extracellular domain. An exercise in
homogeneisation theory
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The End
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Caveats

• Reaction Diffusion Patterns are observed to be
  very sensitive to
• Perturbations of the domain shape.
• Noise (in initial conditions and generally).

53
Background

• In 1952 Turing proposed that
• Pattern Formation during morphogenesis might
  arise
• through an instability in systems of reacting
  chemicals
• (morphogens), which is driven by diffusion.
• Heterogenous concentrations of these chemicals
  form a
• pre-pattern. Subsequent differentiation of
  tissue/cell
• type is in response to whether or not one of
  these
• morphogens exceeds some threshold locally.

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Typical Patterns Formed by This Mechanism

• Such a mechanism can often
• produce plausible looking results
• Mammal coat patterns
• (Murray)

• And interesting, non-trivial, predictions
• You shouldnt be able to find a
• mammal with a striped body and
• a spotted tail.

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Domain Growth

• Numerical Simulations of Kondo, Asai, Goodwin and
  others
• indicate that
• domain growth can lead to robust pattern
  formation, ie.
• insensitivity to noise and randomness in the
  initial conditions.
• motivates a detailed investigation of
• the stabilising influence of domain growth
• the mechanisms by which it produces robustness
• conditions for which one may expect robust
  pattern formation

56
Arcuri Murray Conclusions

• By observations of numerical simulations
• domain growth yields a tendency towards
  robustness, but
• this is far from universal.
• This had led some theoretical biologists to
  dismiss Reaction Diffusion
• Even as a possible/plausible pattern formation
  mechanism (eg Saunders
• Ho, Bull Math Bio, 1995).
• BUT
• One should derive the Reaction Diffusion
  Equations with domain
• growth from conservation laws, not crude
  scaling arguments.

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Lock in
Thus we have frequency doubling cascades of
self-similar patterns.
Of course, one cannot expect to hold exactly
in practice so a form of start-near, stay near
stability is also required for a frequency
doubling cascade.
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Domain Growth Initial Conclusions

• Incorporation of exponential domain growth
  initially appears to give
• Robust pattern Formation, ie. No dependence on
  details
• of initial conditions
• No need to fine tune the parameters
• No need to fine tune the model.

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Breakdown of Mode Doubling

• Mode Doubling breaks for high and low growth
  rates.
• Exponential Growth, Schnakenburg Kinetics

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Breakdown of Mode Doubling

• Breakdown at High Growth rates appears to be due
  to the fact
• Further peak reorganisation occurs before
  splitting peak
• reaches quasi-steady state.
• This in turn prevents Lock in, ie a point where
• As we get a total breakdown of
  pattern.
• For Low growth rates we also get breakdown

61
Low Growth Rate Mode Doubling Breakdown

• Low Growth rate frequency doubling breakdown is
• noise dependent
• occurs after lock in

62
General Conclusion
Robust Pattern Formation occurs for exponential
growth with Schnakenburg kinetics over 4-5
orders of magnitude of the growth rate.
63

• Do not expect self similar cascade to proceed
  indefinitely
• for linear growth
• Given Stability assumptions, if there is a
  such that
• or equivalently
• then these coincide for
• If a sequence with a linear growth rate
  undergoes M
• frequency doublings before breakdown then
• A sequence with a linear growth rate of
  will undergo

Prediction