Before we go into pattern formation let's quickly review the key concepts of Reaction-Diffusion equations:
As we discussed in the introduction, one of the active areas of research in developmental biology (specifically embryology) is how living things develop from a single fertilized egg to complex organisms. The main theory is that in the early embryonic stages, chemical gradients are established that act like a GPS system that give cells information about where they are in the cell. Based on this positional information first proposed by Lewis wolpert [(wolpert_1969)], the cells will differentiate and become parts of different organs in a spatially organized way. (Note that specifying positional information and the actual differentialtion are distinct processes, and we are focusing on how this “GPS system” is set up). How this is done is still an open question, and it is an active area of research, but one of the main hypotheses of biological pattern formation is a reaction-diffusion model first proposed by Alan Turing.
In 1952, Turing published a paper titled The chemical basis of morphogenesis [(turing_1952)], where he proposed a reaction-diffusion model for pattern formation, in which diffusion was the source of the instability that caused patterns to form. This is somewhat counterintuitive, because so far we have studied diffusion as a stabilizing force. However, we will see that the key is that this instability comes from the balance between the reactive and diffusive terms that govern interacting chemical species that are diffusing within some spatial domain.
Consider the system of equations: \begin{alignat}{3} \frac{\partial u}{\partial t} &= D_u\frac{\partial^2u}{\partial x^2} &&+ f(u,v), \label{difrxn1} \\ \frac{\partial v}{\partial t} &= D_v\frac{\partial^2 v}{\partial x^2} &&+ g(u,v). \label{difrxn2} \end{alignat}
where $D_u$ and $D_v$ are diffusion constants, $u$ and $v$ are functions of position and time $u(\vec{x}, t)$, $v(\vec{x}, t)$, and $f(u,v)$ and $g(u,v)$ describe how $u$ and $v$ interact.
We introduce a couple of definitions:
Definition 1: Patterns are stable, time-independent, spatially heterogeneous solutions of Equations \eqref{difrxn1} and \eqref{difrxn2}.
Definition 2: A diffusion-driven instability, or Turing instability, occurs when a steady state, stable in the absence of diffusion, becomes unstable when diffusion is present.
When will we get patterns? In the next section of the notes we derive the conditions that are needed for a system to have Turing instabilities (i.e. to have patterns). Keep in mind that our goal is to have:
In this section we will take the general equations \eqref{difrxn1} and \eqref{difrxn2} and analyze the conditions for Turing Instabilities to occur. The main goal in this section is to find what the system must satisfy to be stable without diffusion (time-independent) and to become unstable when the diffusion terms are added in (spatially heterogeneous).
We assume a stationary uniform state $(u_0, v_0)$ exists (i.e. $f(u_0, v_0) = g(u_0,v_0) = 0$).
Let $u(x,t) = u_0+\tilde{u}$ and $v(x,t) = v_0+\tilde{v}$, where $\tilde{u}$ and $\tilde{v}$ are small. Note that when we do a Taylor Expansion about the fixed points, we get:
\begin{align*} f(u,v) &= f(u_0, v_0) + \tilde{u}\frac{\partial f(u_0,v_0)}{\partial u} + \tilde{v}\frac{\partial f(u_0,v_0)}{\partial v} + ... ,\\ g(u,v) &= g(u_0, v_0) + \tilde{u}\frac{\partial g(u_0,v_0)}{\partial u} + \tilde{v}\frac{\partial g(u_0,v_0)}{\partial v} + ... \end{align*}
so when we linearize Equations \ref{difrxn1} and \ref{difrxn2} about $(u_0, v_0)$ we get, \begin{gather} \frac{\partial \tilde{u}}{\partial t} = f_u + f_v + D_u\frac{\partial^2 \tilde{u}}{\partial x^2} + ... \label{eq:LinTura}\\ \frac{\partial \tilde{v}}{\partial t} = g_u + g_v + D_v\frac{\partial^2 \tilde{v}}{\partial x^2} + ... .\label{eq:LinTurb} \end{gather} For simplicity we can rewrite this in matrix notation as, \begin{equation}\label{eq:vecForm} \frac{\partial}{\partial t} \binom{\tilde{u}}{ \tilde{v}} = \left(D\frac{\partial}{\partial x^2} + J_1 \right) \binom{\tilde{u}}{\tilde{v}},\end{equation}
where, \begin{equation} \label{eq:vecForm2} J_1 = \left( \begin{array}{cc} f_u & f_v \\ g_u & g_v \end{array} \right), \end{equation}
and, \begin{equation} \label{eq:vecForm3} D = \left( \begin{array}{cc} D_u & 0 \\ 0 & D_v \end{array} \right). \end{equation}
When diffusion is absent, we want our system to be stable. What needs to hold in order for this to be true?
The diffusion-less linearized system looks like: \begin{equation} \label{eq:noDiff} \frac{\partial}{\partial t} \binom{\tilde{u}}{\tilde{v}} = \left( \begin{array}{cc} f_u & f_v \\ g_u & g_v \end{array} \right) \binom{\delta u}{\delta v}. \end{equation}
In seeking diffusion-driven instabilities (see Definition 2), we are thus looking for steady-state solutions which are asymptotically stable. This requires $Re(\lambda_{1,2}) <0$, where $\lambda_{1,2}$ are the eigenvalues of $J_1$.
Recall from linear algebra that in a $2\times 2$ matrix, the trace of the Jacobian $\tau = \lambda_1 + \lambda_2$ and the determinant $\Delta = \lambda_1\cdot \lambda_2$. Taken together with the fact that $Re(\lambda_{1,2}) <0$, we get the following two conditions for the stability of \eqref{eq:noDiff}:
\begin{align} \tau &= f_u + g_v < 0, \label{eqn:condition1noDiff}\\ \Delta &= f_ug_v - f_vg_u > 0. \label{eqn:condition2noDiff} \end{align}
Now, according to Def. 1, patterns are time-independent and spatially heterogeneous solutions to Eq. \eqref{difrxn1} and \eqref{difrxn2}. We assume that the solution is separable, so set: \begin{gather} \delta \tilde{u}(x,t) = A(t) e^{iqx} \text{, and}\\ \delta \tilde{v}(x,t) =B(t) e^{iqx}, \end{gather} where each q is the wave-number of a Fourier mode.
Then, the diffusion terms become: \begin{align} D_u \frac{\partial^2}{\partial x^2}A(t) e^{iqx} &= -q^2D_uA(t)e^{iqx}, \label{perturbU} \\ D_v \frac{\partial^2}{\partial x^2}B(t) e^{iqx} &= -q^2D_vB(t)e^{iqx}, \label{perturbV} \end{align}
After we perturb the system by find that the Jacobian for this system is:
\begin{equation} \label{JacobianPert} \frac{\partial}{\partial t} \binom{\delta\tilde{u}}{\delta \tilde{v}} = \left( \begin{array}{cc} f_u - q^2D_u & f_v \\ g_u & g_v-q^2D_v \end{array} \right) \binom{\delta\tilde{u}}{\delta\tilde{v}} \end{equation}
Equation \eqref{JacobianPert} is stable when: \begin{align} \tau &= f_u + g_v - q^2(D_u+D_v) < 0,\label{eqn:condition1Diff}\\ \Delta &= (f_u-q^2D_u)(g_v-q^2D_v) - f_vg_u > 0. \label{eqn:condition2Diff} \end{align}
Notice that the first condition \eqref{eqn:condition1Diff} is always true since $D_u, D_v \in \mathbb{R}^+$ and by \eqref{eqn:condition1noDiff}.
Thus, if we want the system to become unstable we need \eqref{eqn:condition2Diff} to be false.
We look for $q>q_{min}$ where $q_{min}$ is the first mode that can cause an instability, i.e. satisfy, \begin{equation} \label{eqn:condQmin} H(q^2)=(f_u-q^2D_u)(g_v-q^2D_v) - f_vg_u > 0. \end{equation}
Notice that \eqref{eqn:condQmin} is a quadratic with respect to $q^2$, so \begin{equation} \label{eqn:detqsq} q_{min}^2 = \frac{D_ug_v + D_vf_u}{2D_uD_v}, \end{equation} and if we look at where the determinant of the quadratic is positive, we find that:
\begin{equation} \label{eqn:conditiondetpos} D_ug_v + D_vf_u > 2\sqrt{D_uD_v(f_ug_v- g_uf_v)} \end{equation}
This may seem a bit too abstract, so in the next section of the notes we will go through an Activator-Inhibitor model known as the Gierer-Meinhardt model.
You can now proceed to the next chapter: Turing Instabilities II
[(:turing_1952» author: Turing, A. ref-author: Turing title: The chemical basis of morphogenesis journal: Philos. Trans. R. Soc. volume: 237 year: 1952 )]
[(:wolpert_1969» author: Wolpert L. ref-author: Wolpert title: Positional information and the spatial pattern of cellular differentiation journal: J. of Theoretical Biology volume: 25 number: 1 year: 1969 )]