Competance in Bacteria

Competance is a state in which bacteria can take in DNA from the environment and add it to their own. It is a desperate action in attempt to speed up evolution in order to survive.

We will look at an alternative circuit that can replicate the behavior of the native competance circuit.

Model

\[x \equiv \text{ComK} \ , \ \ y \equiv \text{MecA}\]

The reactions are the following:

\[\begin{align*} x &\rightarrow x \\ x &\rightarrow y\\ y &\dashv \ x \\ \end{align*}\] \[\begin{align*} \dot{x} &= f(x) - rxy \\ \dot{y} &= f(x) - y \end{align*}\]

Like we did in the previous lecture, we want to work with a non-dimensional system. Take the variables \(x\) and \(y\) to already be scaled and dimensionless.

\[f(x) = \alpha_0 + \frac{\alpha x^n}{1+x^n}\]

Again, let’s do fixed point analysis on our system.

\[0 = f(x^*)-rx^* y^* \ , \ \ 0 = f(x^*)-y^*\] \[\Rightarrow y^* = \frac{1}{r} \frac{f(x^*)}{x^*} \ , \ \ y^* = f(x^*)\]

These two equations are known as nullclines. The intersection points of nullcilnes are the fixed points of the system.

\[x^* = \frac{1}{r} \ , \ \ y^* = f \left( \frac{1}{r} \right)\]

Are the fixed points stable?

Let’s linearize the system of equations.

\[x = x^* + \delta x \ , \ \ y = y^* + \delta y\] \[\partial_t(x^*+\delta x) = f(x^*+\delta x) - r(x^*+\delta x)(y^*+\delta y) \\ \delta\dot{x} = f(x^*) + \delta xf'(x^*)-rx^*y^*-ry^*\delta x - rx^*\delta y \\ \Rightarrow \delta\dot{x} = \delta x \left[ f'(x^*)-ry^* \right]+\delta y \underbrace{\left[-rx^* \right]}_{=-1}\]

We get a similar equation for $y$.

\[\delta\dot{y} = \delta x \left[f'(x^*)\right] - \delta y \\ \text{let} \ a \equiv y^* \ , \ \ b \equiv f'(x^*)\] \[\dot{ \begin{bmatrix} \delta x\\ \delta y \end{bmatrix} } = \underbrace{ \begin{bmatrix} b-ra & -1 \\ b & -1 \end{bmatrix} }_J \ \begin{bmatrix} \delta x\\ \delta y \end{bmatrix}\] \[\det(J-\lambda I) = 0 \\ \Rightarrow \lambda_{\pm} = \frac{1}{2} \left[b-ra-1\pm\sqrt{(b-ra-1)^2-4ra} \ \right]\]

We can figure out the stability by finding the sign of the eigenvalues. Specifically, let’s examine some limiting cases.

1. Weak repression \(\ r<<1\).
2. Strong repression \(\ r>>1\).
3. Intermediate repression \(\ r\sim 1\).

1. Weak Repression

\[b = f'(x^*) \to 0 \\ ra \to \epsilon \\ \lambda_{\pm} = \frac{1}{2} \left[b-1-\epsilon\pm\sqrt{(b-1-\epsilon)^2-4\epsilon} \ \right] \\ \lambda_{\pm} = \frac{1}{2} (b-1-\epsilon) \left[1\pm\sqrt{1-\frac{4\epsilon}{(b-1-\epsilon)^2}} \ \right] \\ \lambda_{\pm} = \frac{1}{2} (b-1-\epsilon) \left[1\pm\sqrt{1-\epsilon'} \right]\]

So we can say the following about each term..

\[(b-1-\epsilon) \approx -1 \ , \ \ \left[1\pm\sqrt{1-\epsilon'}\right]>0 \\ \therefore \lambda_{\pm} < 0\]

Therefore in the case of weak repression the concentrations \(x\) and \(y\) are large and stable.

2. Strong Repression

\[\alpha >> 1 \ , \ \ b<<1 \ , \ \ b \sim \epsilon\] \[\lambda_{\pm} \approx \frac{1}{2}\left[-ra-1\pm\sqrt{(ra)^2+2ra-4ra} \ \right] \\ \lambda_{\pm} = -\frac{1}{2} \left[ra+1\pm(ra-1)\right] < 0\]

Therefore in the case of strong repression the concentrations \(x\) and \(y\) are small and stable.

3. Intermediate Repression

\[a = f(x^*) = f(1) = \frac{\alpha}{2} \ \text{ with }\alpha_0 \to 0 \\ \begin{align*} b &= f'(x^*) = \left. \frac{\alpha nx^{-n-1}}{(x^{-n}+1)^2} \right|_{x=x^*} = f(x) \left. \frac{n}{x(1+x^n)}\right|_{x=x^*} \\ b &= \frac{\alpha n}{4} \end{align*} \\ \lambda_{\pm} = \frac{1}{2} \left[\frac{\alpha n}{4}-\frac{\alpha}{2}-1\pm\sqrt{\text{...}} \right] \\\]

Note \(\ \frac{\alpha}{2} \left(\frac{n}{2}-1\right) > 1\).

\[\begin{align*} n > 2 &\Rightarrow \text{Cooperativity} \\ \alpha >> 1 &\Rightarrow \text{Strong Activation} \end{align*}\]

In this case of intermediate repression, \(r\sim1\), the system is unstable.

In summary, in the cases of large and small \(r\) there is stability and when $r\sim1$ there is instability.

Note, that we have not considered noise in the system. With noise the stable system may be kicked into an unstable period of time and then fall back to a fixed point. For $r»1$ the stable point can succumb to noise more easily since the species population is so small, and so noise has a much bigger effect.

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