Derivation

We use the following notations for the wiki: [X] denotes the concentration of substance X

0.1 ODEs under transcription equilibrium assumption

We first derive the ordinary differential equations for protein concentrations under two scenarios: constitutive expression and repressed expression. In deriving these equations, we use the assumption that mRNA concentration is always at equilibrium, as transcription is many orders of magnitude faster than translation.

For the first scenario, the gene is constitutively expressed. Transcription and translation are constantly active, and there are no repressors in the system (see ^[1]). In this case, the equations are:

\begin{eqnarray} \dot{m} = k_1 - d_1 m \\ \dot{p} = k_2 m - d_2 p \end{eqnarray}

where m = [mRNA], p = [Protein], k₁ = constitutive transcription rate (constant), d₁ = mRNA degradation rate, k₂ = translation rate, d₂ = protein degradation rate^[1].

According to the transcription equilibrium assumption,

\begin{eqnarray} \dot{m} = 0 \Rightarrow m = \frac{k_1}{d_1} \end{eqnarray}

Substituting this value of m into the equation for p, we get:

\begin{equation} \dot{p} = \frac{k_1 k_2}{d_1} \frac{K^n}{K^n + R^n} - d_2 p \end{equation}

Summary

Our project this year is to quantitatively model out lacILov system with ODE’s. It follows the methodology applied in Timoth S gardners paper “A genetic toggle switch in ecoli” 2005. We began by abstracting away the details of specific promoters and repressors (figure 1) to get a simplified view of the interactions of our system. Afterwards we modeled the interactions through a set of first order ordinary differential equations. Using various assumptions to reduce the number of equations and parameters, along with the application of nondimensionalization we obtained our final result:

Equations 1, 2, 3

\begin{eqnarray} \frac{dx_2}{d\tau} = \psi_1 - \gamma_2 x_2 \\ \frac{d\theta}{d\tau} = k\psi_1 - \gamma_\theta \theta \\ \frac{d\lambda}{d\tau} = \frac{\alpha_\gamma}{1+x_2^n} - \gamma_\lambda \lambda \end{eqnarray}

These 3 equations encapsulate the core nature of our system. Note that all the parameters and variables have no dimensions, so our results may be generalized to other light activated systems of the same structure. Mapping our abstracted variables back to our system we see that:

Equation 1

Represents the rate of change of the CI repressor, whose activation depends on whether or not light is on and exhibits linear scaling with respect to its promoter strength.

Equation 2

Is the rate of change of sgrna and it is important to note that from the equations, its expression is indirectly linked to the CI repressor via the psi term.

Equation 3

Is the key result of our system, it represents the rate of change of Anti-CRISPR. Our model confirms that the nature of Anti-CRISPR activation is inversely proportional to LacILov activation.