====== Reaction-Diffusion Equations 2 ======
===== Why do we need Diffusion AND Reactions? =====
What benefit do diffusion-reaction models have over just reaction or just diffusion models?
As discussed in the introduction, one of the big questions in developmental
biology is how complex organisms emerge from a single fertilized egg.
The answer has long been thought to be chemical gradients. If you have different
chemicals at different concentrations throughout the embryo, and you apply a
threshold function so that the cell differentiates depending on the chemicals
immediately surrounding it, then you get a spatially organized differentiation of
cells that will later develop into different organs and body parts.
The question of how these gradients are set up were subject of great debate, and
at the beginning, diffusive models were discarded because it was assumed to be
too slow to establish a stable gradient.
In 1970, Francis Crick (who would later win the Nobel Prize for discovering DNA)
proposed that diffusion was a plausible mechanism since it was
fast enough. [(crick_1970)]
He modeled this as a one-dimensional diffusion problem much like the example above.
In other words, he imagined an embryo as a line of cells.
His equation was,
\begin{equation} \label{eqn:Crick1} \frac{\partial C(x,t)}{\partial t} = D \frac{ \partial^2C(x,t)}{\partial x^2}, \end{equation}
where $C(x,t)$ is the concentration of the chemical at position x and time t.
Furthermore, he placed boundary conditions $C(0,t) = C_0$ and $C(L,t) = 0$, where
L is the length of the embryo.
Crick wanted a stable gradient (in time), so he solved \eqref{eqn:Crick1} by setting
the LHS to zero. The system becomes
\begin{equation}\label{eqn:Crick2} \frac{\partial^2 C(x,t)}{\partial x^2} = 0, \end{equation}
which implies that the concentration gradient will be a straight line when it is stable.
He then verified that the stable linear gradient would be set up fast enough.
Diffusion is a random walk process and our derivation above showed that the
diffusion constant has dimensions $L^2t^{-1}$. The time it takes to set up the gradient
is,
\begin{equation} \label{eqn:Crick3} t - \frac{A(nl)^2}{D}, \end{equation}
where $t$ = time in seconds, $n$ = number of cells in the embryo, $l$ = length of each
cell in cm, and $D$ = diffusion constant in $cm^2s^{-1}$. $A$ is a numerical constant
that is fit from data. Assuming that the time it takes a real embryo to set up
the chemical gradient is around three hours, Crick found that diffusion would be
fast enough if L was on the order of milimeters, which is the case with fruit flies.
Crick did not have the tools to check whether the concentration gradient of
bicoid in fruit fly embryos was linear, so he was satisfied with using just diffusion.
He also used boundary conditions without stating what could biologically explain
the $u(L,t)=0$ condition.
New imaging technologies have been developed that allow scientists to
measure the concentration of chemicals in embryos.
One very important chemical is called bicoid and it is important in establishing
an asymmetry in the anterior-posterior axis (i.e. it determines what cells become
part of the head and which ones become part of the body).
The bicoid gradient can be approximated by an exponential curve.
Instead of relying solely on boundary conditions like Crick, models of bicoid have
introduced a reaction term. Bicoid will take time both to get produced, and once
it is created it degrades naturally.
This new model looks like:
\begin{equation} \label{eqn:revMod1} \frac{\partial C(x,t)}{\partial t} =
D(t) \frac{ \partial^2C(x,t)}{\partial x^2} - \frac{1}{\tau}C(x,t) + \rho(x,t), \end{equation}
where $D$ is the diffusion constant, $\tau$ is the degradation rate, and
$\rho$ is a synthesis rate [(little_2011)]
Consider a simpler version of \eqref{eqn:revMod1},
\begin{equation} \label{eqn:revMod2} \frac{\partial C(x,t)}{\partial t} =
D \frac{ \partial^2C(x,t)}{\partial x^2} - \frac{1}{\tau}C(x,t), \end{equation}
which we can solve for a stable gradient analytically,
\begin{equation} \label{eqn:revMod3} C(x,t) = C_0 e^{-x/\lambda},\quad
\lambda = \sqrt{D\tau} \end{equation}
{{dynsys:reactdiffusetwo:audio_bicoid_gif.mp3}}
{{ dynsys:reactdiffusetwo:bicoid2.gif }}
This example shows why sometimes reaction and diffusion are needed to make realistic
biological models. Diffusion can model spatial phenomena, but often, like in the case
of bicoid, we have local reactions that can only be included in the model with
equations like the Reaction-Diffusion PDE in \eqref{eqn:revMod1}.
===== Reaction-Diffusion equations and spatial domains =====
So far we have seen how reaction and diffusion can work together to create biological models,
and how diffusion is really fast in short length scales. However, we have not studied
the spatial properties of Reaction-Diffusion equations.
Let us consider one last example of a one-dimensional reaction diffusion equation
to study the effect of the spatial domain on the structure that emerges [(kierstead_1953)].
/* Following Kierstead and Slobodkin */
Imagine we have a phytoplankton population and it can only survive in some water
mass that has the adequate temperature and dissolved nutrients. If the water mass
is isolated (i.e. surrounded by water where the phytoplankton will die),
is there a minimum water mass size so that the phytoplankton population can increase?
In the ocean, this water mass would be three-dimensional, but let us take
a simpler one-dimensional approach in which we consider a mass of water
that has been stretched out into a very thin tube. We impose boundary conditions
on the concentration of phytoplankton $c$ such that any phytoplankton at the edges are automatically destroyed, and also that the concentration is constant at $t = 0$. That is, we have
\begin{gather}
c(0,t) = 0 = c(L,t) \label{bcplank} \\
c(x, 0) = c_0. \label{icplank}
\end{gather}
Phytoplankton cannot swim against the current of the ocean, so its motion
can be described by diffusion. Therefore, if the phytoplankton population does not
grow or decrease, its concentration will be the solution to the diffusion equation,
\begin{equation} \label{eqn:Plank1} \frac{\partial c}{\partial t} = D \frac{\partial^2c}{\partial x^2}. \end{equation}
However, phytoplankton are living organisms, so they will reproduce at a
rate that is proportional to their concentration.
Therefore we add a constant term to Equation \eqref{eqn:Plank1},
\begin{equation} \label{eqn:Plank2} \frac{\partial c}{\partial t} = D \frac{\partial^2c}{\partial x^2} +Kc, \end{equation}
where K is a growth constant.
Before we solve this via separation of variables, we can simplify our problem by scaling out the diffusion-less exponential growth,
\begin{equation} \label{eqn:Plank3}
c(x, t) = f(x, t)e^{Kt},
\end{equation}
and substituting Equation \eqref{eqn:Plank3} into Equation \eqref{eqn:Plank2}, we find
that f must satisfy the standard diffusion (or heat) equation of the previous part,
\begin{equation} \label{eqn:Plank4}
\frac{\partial f}{\partial t} = D \frac{\partial^2f}{\partial x^2}.
\end{equation}
By the standard techniques of Fourier series, and using the boundary conditions of $c = 0$ and $x = 0, L$, we find that
\begin{equation} \label{eqn:Plank5}
f = \sum_{n=1}^\infty B_n \sin \left(\frac{n\pi x}{L}\right) e^{-n^2\pi^2D/L^2t},
\end{equation}
where $B_n$ are the Fourier sine coefficients given by,
\begin{equation} \label{eqn:Plank6}
B_n = \frac{2}{L} \int_0^L c_0 \sin \left(\frac{n\pi x}{L}\right) \, \de{x},
\end{equation}
for $n = 1, 2, \ldots$, which are then computed for given initial concentration, $c_0$.
Substituting \eqref{eqn:Plank4} into \eqref{eqn:Plank5} we get the concentration,
\begin{equation} \label{eqn:Plank7} c(x,t) = \sum_{n=1}^\infty B_n \sin \left(\frac{n\pi x}{L}\right) e^{(K-n^2\pi^2D/L^2)t}. \end{equation}
==== What will be the steady state of the phytoplankton population? ====
They key is to note from \eqref{eqn:Plank7} that because the Fourier coefficients are bounded and decreasing as $n \to \infty$, and since the sinusoidals are well behaved, the long term behavior of the system will be controlled by the time term in Equation \eqref{eqn:Plank7},
\begin{equation} \label{eqn:Planktime}
e^{(K-n^2\pi^2D/L^2)t}.
\end{equation}
The sign of the argument of the exponential will determine whether the
population of the plankton will grow, stay the same or decay as time goes to infinity. Thus, we obtain three cases related to the argument of \eqref{eqn:Planktime}:
* If the argument is exactly zero, then the population will stay the same.
* If the argument is negative, then the population will decay over time.
* If the argument is positive, then the population will grow over time.
Moreover, if these conditions hold for the $n=1$ mode, then the $n = 2, 3,$... higher modes don't change the steady-state behaviour. For $n = 1$, the //bifurcation// (the point at which the behavior changes from decay to growth) is found at the length, $L$ such that
\begin{equation} \label{eqn:Plank8}
K - \pi^2\frac{D}{L^2} = 0,
\end{equation}
and thus, we can obtain a //critical length// given by
\begin{equation} \label{eqn:Plank10}
L_c = \pi\sqrt{\frac{D}{K}}.
\end{equation}
In summary, we have found the critical length of the domain such that
* If $L= L_c$ the population will stay the same.
* If $L>L_c$ the population will increase.
* If $LL_c$ then the reactive term $Kc$ will dominate the long-term behavior,
but when $L>
author: Crick, F.
ref-author: Crick
title: Diffusion in Embryogenesis
journal: Nature
volume: 225
year: 1970
)]
[(:einstein_1956>>
author: Einstein, A.
ref-author: Einstein
publisher: Dover Publications
title : Investigations on the Theory of the Brownian Movement
year : 1956
)]
[(:mehrer_2009>>
author: Mehrer, H. and Stolwijk, N.A.
ref-author: Mehrer and Stolwijk
title: Heroes and Highlights in the History of Diffusion
journal: Diffusion Fundamentals
volume: 11
number: 1
year: 2009
)]
[(:nelson_1967>>
author : Nelson, E.
ref-author: Nelson
publisher : Princeton University Press
title : Dynamical theories of Brownian motion
year : 1967
)]
[(:little_2011>>
author: Little, SC, Tkacik G, Kneeland TB, Wiechaus EF, Gregor T
ref-author: Little et al
title: The Formation of the Bicoid Morphogen Gradient Requires Movement from
Anteriorly Localized mRNA
journal: PLoS Biology
volume: 9
number: 3
year: 2011
)]
[(:kierstead_1953>>
author: Kierstead, H, and Slobodkin LB.
ref-author: Kierstead and Slobodkin
title: The size of water masses containing plankton blooms
journal: J. Mar. Res
volume: 12
number: 1
year: 1953
)]
[(:murraySperb_1983>>
author: Murray J.D. and Sperb R.P.
ref-author: Murray and Sperb
title: Minimum domains for spatial patterns in a class of reaction diffusion equations
journal: J. Math. Biology
volume: 18
year: 1983
)]