Team:Stockholm/model-simulation

<!doctype html>iGEM Stockholm Model Simulation

Model Simulation

P2 genetic switch

We first simulated C protein concentration dynamics in P2 genetic switch system over time using the final derived deterministic model:

$$\frac{dx}{dt\prime}=\frac{1+\tau u x^2}{1+ux^2+\sigma u^2x^4}-\gamma x=f(x)$$

We selected a time step of dt=0.01, as well as arbitrary parameters σ=2, u=1, γ=3, τ=2 and initial concentration of x=0.8. The following time series graph shows the stabilization of the protein concentration to a steady state.

Figure 1. C protein concentration over time in phage P2 switch system, deterministic model. σ=2, u=1, γ=3, τ=2, x0=0.8.

In order to determine the steady state concentration, we examine the f(x) versus x graph, from which we can tell the point where the graph crosses horizontal axis, corresponding to dx/dt=0, i.e. the steady state concentration. We determine this concentration being ~ 0.371.

Figure 2. f(x) versus x graph of deterministic P2 switch model. σ=2, u=1, γ=3, τ=2, x0=0.8.

Simulating the model adapted to a stochastic setting results in a similar graph, which, however, includes more fluctuations due to random noise, introduced into the model.

Figure 3. C protein concentration over time in phage P2 switch system, stochastic model. σ=2, u=1, γ=3, τ=2, x0=0.8.

To our knowledge, this is the first attempt to derive and simulate a model for P2 bacteriophage genetic switch.

Model Plasmid

We first simulated C and Cox protein concentration dynamics in Model Plasmid system over time using the final derived deterministic model:

$$\frac{dx}{dt}=\frac{K_{t1}K_1[arabinose]}{1+arabinose}-αx$$

$$\frac{dm}{dt}=\frac{arabinose}{1+arabinose}-βm$$

We selected a time step of dt=0.01, as well as arbitrary parameters σ=2, u=1, γ=3, τ=2 and initial concentration of x=0.5. The following time series graph shows the stabilization of all C and Cox protein concentrations to a steady state.

Figure 4. C and Cox protein dynamics over time in the Model Plasmid.

It depicts that C protein increases exponentially up to a certain concentration (x=1.75) and stabilizes after a certain time point, in this case t= 1 unit. Cox protein seem to be almost stabilised throughout by t=2 .

In order to determine the steady state concentration, we examine the f(x) versus x [dx/dt v/s protein concentration] graph, from which we can tell the point where the graph crosses horizontal axis, corresponding to dx/dt=0, i.e. the steady state concentration.

Figure 5. f(x) vs x graph of Model Plasmid.

We determine this concentration for C protein= 1.1 and Cox protein = 0.9

Simulating the model adapted to a stochastic setting results in a similar graph, which, however, includes more fluctuations due to random noise, introduced into the model.

Figure 6. C and Cox protein dynamics in Model Plasmid over time, stochastic model.

Switch Plasmid

We first simulated C, Cox and TetR protein concentration dynamics in Switch Plasmid system over time using the final derived deterministic model:

$$\frac{dx}{dt}=\frac{k_{t1}+k_{t3}K_1K_Dr^2}{1+K_3[arabinose]+K_1K_Dr^2+K_D^2K_1K_2r^4}-αx$$

$$\frac{dm}{dt}=\frac{k_{t2}K_3[arabinose]}{1+K_3[arabinose]+K_1K_Dr^2+K_D^2K_1K_2r^4}-βx$$

$$\frac{dr}{dt}=\frac{k_{t2}K_3[arabinose]}{1+K_3[arabinose]+K_1K_Dr^2+K_D^2K_1K_2r^4}-γx$$

We selected a time step of dt=0.01, as well as arbitrary parameters σ=2, u=1, γ=3, τ=2 and initial concentration of x=0.5. The following time series graph shows the stabilization of all the three protein concentrations to a steady state.

Figure 7. C, Cox, TetR protein dynamics over time in the Switch Plasmid.

It depicts that C protein increases exponentially up to a certain concentration (x=0.8) and stabilizes after a certain time point, in this case t= 2 units. Cox and TetR protein seem to decrease in concentration up to a certain time point (t=0.5) and stabilizes at t=1.

In order to determine the steady state concentration, we examine the f(x) versus x [dx/dt v/s protein concentration] graph, from which we can tell the point where the graph crosses horizontal axis, corresponding to dx/dt=0, i.e. the steady state concentration.

Figure 8. f(x) vs x graph of Switch Plasmid.

We determine this concentration for C protein= 0.8 , Cox protein = 0.3, TetR protein=0.2

Simulating the model adapted to a stochastic setting results in a similar graph, which, however, includes more fluctuations due to random noise, introduced into the model.

Figure 9. C, Cox, TetR protein dynamics in Switch Plasmid over time, stochastic model.

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