Elowitz & Leibler 2000 - Repressilator
BiologicalOscillations.elowitz_2000 — Constantelowitz_2000Generalized Repressilator model based on Elowitz and Leibler 2000 Nature article.
In contrast to the original model, this implementation has independent parameters for each mRNA and protein pair:
\[ \begin{align*} \frac{\mathrm{d} m_1\left( t \right)}{\mathrm{d}t} =& \gamma_1 - \delta_1 m_1\left( t \right) + \mathrm{hillr}\left( P_3\left( t \right), \alpha_3, K_3, n_3 \right) \\ \frac{\mathrm{d} m_2\left( t \right)}{\mathrm{d}t} =& \gamma_2 - \delta_2 m_2\left( t \right) + \mathrm{hillr}\left( P_1\left( t \right), \alpha_1, K_1, n_1 \right) \\ \frac{\mathrm{d} m_3\left( t \right)}{\mathrm{d}t} =& \gamma_3 - \delta_3 m_3\left( t \right) + \mathrm{hillr}\left( P_2\left( t \right), \alpha_2, K_2, n_2 \right) \\ \frac{\mathrm{d} P_1\left( t \right)}{\mathrm{d}t} =& \beta_1 m_1\left( t \right) - \mu_1 P_1\left( t \right) \\ \frac{\mathrm{d} P_2\left( t \right)}{\mathrm{d}t} =& \beta_2 m_2\left( t \right) - \mu_2 P_2\left( t \right) \\ \frac{\mathrm{d} P_3\left( t \right)}{\mathrm{d}t} =& \beta_3 m_3\left( t \right) - \mu_3 P_3\left( t \right) \end{align*} \]
where
\[\text{hillr}(P(t), \alpha, K, n) = \alpha \frac{K^n}{P(t)^n + K^n}\]
We can obtain an oscillatory solution by using the following parameters
| $\alpha_1, \alpha_2, \alpha_3$ | 0.5 | $K_1, K_2, K_3$ | 40 | $\beta_1, \beta_2, \beta_3$ | 0.05 | $n_1, n_2, n_3$ | 2 |
| $\gamma_1, \gamma_2, \gamma_3$ | 0.005 | $\mu_1, \mu_2, \mu_3$ | 0.005 | $\delta_1, \delta_2, \delta_3$ | 0.0025 |
and the initial condition
| $m_1(0), m_2(0), m_3(0)$ | 0.0 | $P_1(0)$ | 20.0 | $P_2(0), P_3(0)$ | 0.0 |