# Turing Patterns

Have you ever observed similar patterns in seemingly unrelated objects or places? Or wondered how these common patterns form to begin with? In 1952, Alan Turing published his paper The Chemical Basis of Morphogenesis describing a mechanism for the spontaneous formation of spots and stripes. Over the years, it has been shown that Turing systems are capable of reproducing a great variety of patterns. We were inspired by a recent paper by Karig et al. which showed that Turing patterns could form in a biological system outside of the strict conditions specified in the original paper. We hoped to use what we learned from modeling the light operon to introduce another means to control pattern formation.

Turing's paper stipulates that under certain conditions small perturbations in a homogeneous system can be destabilized by diffusion, leading to the growth of large scale patterns. This is a rather counterintuitive result because we typically witness diffusion as a stabilizing force, like dropping an ice cube into a hot coffee. See the animation below for a visualization of stabilizing diffusion.

## The Activator-Inhibitor Model

Over time models were developed for specific systems that are optimized for producing Turing patterns. One such model is the activator-inhibitor model.

The crux of the activator-inhibitor model (and more generally any Turing system) is the set of two diffusible signal molecules which react and diffuse over time. The activator drives the expression of itself and the inhibitor. The inhibitor represses the activator, slashing its own expression. Another feature- or rather, restriction- of this model is that the inhibitor must diffuse at a much faster rate than the activator. The resulting behavior is known as local activation and lateral inhibition. The activator’s self-inducing behavior causes local peaks to form while the inhibitor reduces the spread of those peaks.

## Reaction-Diffusion Equations

How are inhibitor and activator concentrations represented mathematically? By using a class of partial differential equations called Reaction-Diffusion equations. A partial differential equation is defined the same way an ordinary differential equation is, except it has more than one independent variable. See our Modeling 101 page to learn about ODE's. The following is the general form of a pair of reaction-diffusion equations:

$$\frac{\partial u}{\partial t}=F(u,v)+D_u\Delta u$$
$$\frac{\partial v}{\partial t}=G(u,v)+D_v\Delta v$$

Where
• We define $u$ as a real-valued function where $$u(\vec{x},t)$$ represents activator concentration at position $$\vec{x}=(x,y)$$ and time $$t$$
• We define $$v$$ as a real-valued function where $$v(\vec{x},t)$$ represents inhibitor concentration at position $$\vec{x}=(x,y)$$ and time $$t$$
• $$F(u,v), G(u,v)$$ = reaction terms (production, degradation, etc.) of $$u$$ and $$v$$.
• $$D_u, D_v$$ - diffusion coefficients
• $$\Delta = \nabla ^2$$ is the Laplacian operator (note: it's taken with respect to $$\vec{x}$$)

In other words, the rate of change of concentration over time is dependent upon how much of a molecule is being produced and degraded and how fast it spreads out. To fully flesh out this problem, we also have to specify the boundary conditions. In contrast to an initial condition, a boundary condition commonly specifies the value of a function at the boundary points (Dirichlet) or its normal derivative along a surface (Neumann). In our case, the boundary represents the impermeable walls of a container such as a petri dish. We impose the following Neumann boundary condition on our equations:

$$0=\frac{\partial y}{\partial n}(x)=\nabla y(x) \cdot \hat{n}(x)$$

## Modulating Spot Size

In the Stochastic Turing paper by Karig et al., it was experimentally demonstrated that IPTG could be used to relieve LacI repression of CI, ultimately increasing inhibition of the activator and inhibitor, decreasing spot size, and causing GFP to be more strongly expressed. We hypothesized that the pDawn promoter, which expresses CI under light, could perform the same function of the LacI/IPTG module while doubly offering a better measure of control due to the abrupt nature of light activation/deactivation. Below we modeled the expression of CI under both schemes at a starting concentration of 10 $$\mu mol$$. The LacI/IPTG model shows minimal leftward shift even with extreme amounts of IPTG being applied. On the contrary, CI under the light operon displays a noticeable downward shift under increasing light.