# Modelling Reaction-Diffusion Systems

Pulsating spots

Mathematically, reaction-diffusion systems are coupled nonlinear differential equations that can be solved numerically. Our first step will be to implement/model a simple two-component system originally proposed by Turing. After validating our numerical method and computating our first patterns (img to the right), we will be thinking about a more realistic system describing the behaviour of the activator and inhibitor system that we intend to engineer with B.subtilis. This will include an analysis of enzyme kinetics and we hope to deduce the parameter ranges, in which our B.subtilis construct will be able to form patterns, thus feeding back into our design decisions (promoter strength, rbs choice etc.).

We will also take an investigative approach and ask whether Turing-like patterns can originate from a system that is simpler than the one envisioned, consisting of fewer components, e.g. depending on a single signalling molecule only.

## Introduction

How does pattern formation occur? Turing considered diffusion to be the crucial component. He showed that pattern formation occurs of the system has a stable steady state in the absence of diffusion, but allows unstable states to develop when the diffusion term is added. This results in divergence (or, in a biological context, gene expression/cell differentiation) and subsequent pattern formation. In the following sections, we shall deduce conditions for pattern formation for this simple system.

We will work with a two-component system and we non-dimensionalise the variables. The resulting general form is:

$\frac{\partial A}{\partial t} = \gamma f(A,B) + \nabla^2 A$
$\frac{\partial B}{\partial t} = \gamma g(A,B) + d \nabla^2 B$

where γ (determining the scale) is a constant and d stands for the diffusion ratio. We note that non-dimensionalisation has the added advantage that we can now map a specific pattern onto a wide range of biological parameters, the easiest example being that two different pairs of diffusion rates will result in the same pattern if the respective diffusion ratios remain unchanged.

## Turing System

(also known as Schnakenberg reaction)

### Case: Without diffusion, need stable steady state

Moving flame fronts

Consider the following system without diffusion terms:

$\frac{\partial A}{\partial t} = \gamma (\alpha - A + A^2B) = \gamma f(A,B)$
$\frac{\partial B}{\partial t} = \gamma (\beta - A^2B) = \gamma g(A,B)$

We determine the steady state solution and add a small pertubation $\tilde{A}, \tilde{B}$ to linearise the system about the steady state in order to determine its stability. In Matrix notation, the linearised system can be written as:

$\begin{pmatrix} \tilde{A_t} \\ \tilde{B_t} \\ \end{pmatrix} = \gamma \begin{pmatrix} f_A & f_B \\ g_A & g_B \\ \end{pmatrix} . \begin{pmatrix} \tilde{A} \\ \tilde{B} \\ \end{pmatrix}$

where the Jacobian is evaluated at the steady states of A and B. This is a set of coupled first-order ODEs and solutions are proportional to expt). In order to have a stable steady state, we need the real part of λ to be negative.

This requires fA + gB < 0 (need -ve trace) and fAgBfBgA > 0 (need +ve determinant), which we note as our first two conditions for pattern formation.

### Case: With diffusion, need unstable steady states

Now we add diffusion to our system and require the resulting steady state to be unstable. After linearisation, the equation can be expressed as follows:

\begin{align} \begin{pmatrix} \tilde{A_t} \\ \tilde{B_t} \\ \end{pmatrix} &= \gamma \begin{pmatrix} f_A & f_B \\ g_A & g_B \\ \end{pmatrix} . \begin{pmatrix} \tilde{A} \\ \tilde{B} \\ \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ 0 & d \\ \end{pmatrix} . \nabla^2 \begin{pmatrix} \tilde{A} \\ \tilde{B} \\ \end{pmatrix} \\ & \equiv \gamma \mathbf{M}.\mathbf{w} + \mathbf{D}.\nabla^2\mathbf{w} \end{align}

In the following steps, we consider solutions of this particular form $\mathbf{w}(\mathbf{x},t) = \sum_{k} c_{k} e^{\lambda t} \mathbf{v}_k(\mathbf{x})$

with $\mathbf{v}_k$ satisfying $\nabla^2 \mathbf{v}_k + k^2\mathbf{v}_k = 0$.

After substituting into the linearised system, we can see that for each index k, we require:

$det(\lambda \mathbf{I} - \gamma \mathbf{M} + k^2\mathbf{D}.\mathbf{v}_k) = 0$.

We now need to determine the conditions for unstable steady states, i.e. conditions on the parameters for which Re(λ)>0 hold. The calculations will be standard and similar to the previous case: We need the trace and the determinant of the matrix $\gamma \mathbf{M}-k^2\mathbf{D.v}_k$ to infer stability properties. Thus, we arrive at a set of inequalities that we can simplify to give us the following conditions:

\begin{align} 0 < \beta - \alpha &< (\alpha+\beta)^3 \\ d(\beta-\alpha) &> (\alpha+\beta)^3 \\ (d(\beta-\alpha)-(\alpha+\beta)^3)^2 &> 4d(\alpha+\beta)^4\\ \end{align}

These are a set of conditions that our parameters must satisfy in order for this particular system to develop patterns.

Given a (two-component) system, we can now infer which parameters to choose in order for the system to develop patterns. This saves us a lot of time as reaction diffusion systems are extremely sensitive to parameter changes and simply trying out sets of parameters will usually result in no patterning at all.

# Bacillus and its signalling systems

We will first look at the specific components of our construct and models describing the behaviour of the agr and lux quorum-sensing systems will be introduced. We will then model our envisioned activator-inhibitor system and deduce under which conditions pattern formation will occur.

## Peptide-signalling system

We have created a simple model describing the agr-receiver device and it exhibits a two-state behaviour, akin to a switch. This is typical for quorum-sensing systems: once concentrations of AIP (our signalling molecule) pass a certain threshold value, the agr-receiver will jump from its previously "off-"state (negligible levels of PoPS from the P2 promoter) into an "on-"state (significant levels of PoPS). In nature, once the bacterial density (and thus AIP concentration) surpasses a critical value, the bacteria will e.g. sporulate, become virulent or fluoresce - behaviour that is biologically expensive (however, critical) and only advantageous at high cell densities.

The equations of this model read as follows:

\begin{align} \frac{\partial [A]}{\partial t} &= p_{2} \frac{[A^{+}]^n}{1+[A^{+}]^n} + \beta - p_{+} [CP] [A] + p_{-} [A^{+}] - \delta [A] \\ \frac{\partial [A^{+}]}{\partial t} &= p_{+} [CP] [A] - p_{-} [A^{+}] - \delta [A^{+}] \\ \frac{\partial [C]}{\partial t} &= p_{2} \frac{[A^{+}]^n}{1+[A^{+}]^n} + \beta - c_{+} [C] [P] + c_{-} [CP] [P] - \delta [C] \\ \frac{\partial [P]}{\partial t} &= - c_{+} [C] [P] + c_{-} [CP] - \delta [P] \\ \frac{\partial [CP]}{\partial t} &= c_{+} [C] [P] - c_{-} [CP] - \delta [CP] \\ \end{align}

A rigorous justification (i.e. all the steps leading to this set of equations) should probably be on here as well.

### Sender

Interesting note: There is experimental evidence that AIP production rates are not depended on the basal expression level of agrB and agrD.

## AHL-signalling system

According to our experiments, B.subtilis does not degrade AHL.

Model behaviour, look at Bangalore 2007 & Canton et al. (2008), assay lux-system in B.subtilis.

## Activator-inhibitor system

Local activation, lateral inhibition.

# Beyond Turing

Pattern formation with a single signalling molecule only?