MathWorks Simulations

Using the previously derived expressions from our ODEs, we simulated our equations for cI Protein, sgRNA and anti-CRISPR, shown in Figure 1.

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_\lambda}{1+x_2^n} - \gamma_\lambda \lambda \end{eqnarray}

We then used the Mathworks Simulink package to derive solutions to our system and model our system for a range of parameters.

ODE Solution

Solving:

\begin{eqnarray} \frac{x_2}{dt} = \alpha - \gamma x_2 \\ \frac{x_2}{dt} + \gamma x_2 = \alpha \end{eqnarray}

Integrating Factor:

\begin{eqnarray} e^{\int \gamma dt} = e^{\gamma t} \end{eqnarray}

Multiplying both sides by our integrating factor:

\begin{eqnarray} (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \alpha e^{\gamma t}\\ \int (\frac{x_2}{dt} + \gamma x_2)e^{\gamma t} = \int \alpha e^{\gamma t} \\ x_2 = \frac{\alpha}{\gamma} + ce^{-\gamma t} \end{eqnarray}

R plots

Our GitHub repository contains all our code for the following R plots and R analysis, as well as for generating the above simulations.

R Analysis

Analyzed in R for this model, and got the following values with adjusted R-squared and p-value:

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)
(Intercept)          2.87199    0.21773   13.19 1.47e-15 ***
c(time, time, time)  0.15267    0.01142   13.37 9.74e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2935 on 37 degrees of freedom
Multiple R-squared:  0.8285,	Adjusted R-squared:  0.8238
F-statistic: 178.7 on 1 and 37 DF,  p-value: 9.741e-16

Intercept represents the equilibrium value of LacILov, and thus our intercept:

\begin{eqnarray} 2.879199 \pm (0.21773)(2.026) \\ 2.879199 \pm 0.44112098 \end{eqnarray}