Problem 1: Nullcline Analysis and Bistability¶

The toggle switch consists of two mutually repressing genes. Here's a normalized model:

$$\frac{dx}{dt} = \frac{\beta_x}{1+y^n} - x, \quad \frac{dy}{dt} = \frac{\beta_y}{1+x^n} - y$$

For symmetric parameters $\beta_x = \beta_y = \beta$, derive the condition on $\beta$ and $n$ for bistability to exist.

The nullclines are $x = \beta/(1+y^n)$ and $y = \beta/(1+x^n)$. For what parameter values do these curves intersect three times? Express your answer as a relationship between $\beta$ and $n$.

Problem 2: Modeling Inducers in a Toggle Switch¶

In our analysis of the Gardner and Collins synthetic toggle switch, we did not include a way to represent the addition of the inducer signals. Such signals typically function by decreasing a receptor's affinity for its target binding site, in other words by increasing the value of the Hill repression constant $k$.

Create a plot representing the following scenario:

The circuit is built with parameters $\beta$, $\gamma$, $n$ fixed to be $\beta = 10$, $\gamma = 1$, $n=10$.
The plot has sliders for $k_x$ and $k_y$, which represent the presence of inducer signal to inactivate the repressors $X$ and $Y$, respectively. (Note that this representation assumes that the inducer's action is linearly proportional to its concentration, which is typically not true but gives the right qualitative result).
The plot shows how the nullclines, change as you add and remove the inducer signals.

What insights did you gain from the visualization?

Problem 3: Stochastic Gene Expression¶

Part (a): Transcriptional Bursting¶

Consider a two-state promoter model:

$$\text{Gene}_{\text{off}} \underset{k_{off}}{\stackrel{k_{on}}{\rightleftharpoons}} \text{Gene}_{\text{on}} \xrightarrow{\alpha} \text{Gene}_{\text{on}} + \text{mRNA}$$ $$\text{mRNA} \xrightarrow{\gamma_m} \emptyset$$

The steady-state statistics are characterized by:

Mean: $\langle \mathrm{mRNA}\rangle = \alpha\cdot k_\mathrm{on}/(\gamma_m(k_\mathrm{on} + k_\mathrm{off}))$
Fano factor: $F = 1 + b/(1+b)$, where $b = \alpha/k_\mathrm{off}$ is the burst size

For the lac operon, suppose $k_\mathrm{on}$ = 0.1 min⁻¹, $k_\mathrm{off}$ = 0.2 min⁻¹, $\alpha$ = 6 min⁻¹ when ON, $γ_m$ = 0.33 min⁻¹.

Calculate the mean, variance, and coefficient of variation (CV) of mRNA copy number. Compare this to a Poisson process with the same mean. What is the physical interpretation of the burst size $b$?

Part (b): Translational Amplification of Noise¶

For the two-stage process DNA → mRNA → Protein, theoretical analysis shows:

$$F_{\text{protein}} \approx 1 + \frac{\beta}{\gamma_m}$$

where $\beta/\gamma_m$ represents the translational burst size. Given $\beta$ = 20 proteins·mRNA⁻¹·min⁻¹ and $\gamma_m$ = 0.33 min⁻¹:

Calculate the protein Fano factor and CV. If steady-state protein levels are ~1000 molecules/cell, what is the standard deviation? Why is protein noise super-Poissonian ($F > 1$) despite proteins being produced from many independent mRNA?

Part (c): Experimental Predictions¶

In your homework, you'll compare deterministic and stochastic simulations. Based on your analysis above, predict:

For which species (mRNA or protein) will relative fluctuations (CV) be larger?
During a transition from repressed to induced state, will cell-to-cell variability be higher during the transient or at steady state?
What summary statistics beyond the mean should you report?