Problem 1: Self-Repression with mRNA Dynamics¶

Consider a gene that represses its own production, with explicit mRNA and protein dynamics:

$$\frac{dm}{dt} = \frac{\beta}{1 + p^n} - \gamma m$$

$$\frac{dp}{dt} = km - \alpha p$$

where $m$ is mRNA concentration, $p$ is protein concentration, $k$ is translation rate, $\gamma$ is mRNA degradation rate, and $\alpha$ is protein degradation rate.

a) Find the fixed point $(m^*, p^*)$ and compute the $2\times 2$ Jacobian matrix at this point.

Show hint

At the fixed point, $dm/dt = dp/dt = 0$. The Jacobian is $J_{ij} = \partial(\dot{x}_i)/\partial x_j$ where $x_1 = m$, $x_2 = p$.

b) Calculate the trace and determinant of the Jacobian. Show that $\text{trace}(J) = -\gamma - \alpha$. What does this tell you about the eigenvalues?

Show hint

The trace is the sum of eigenvalues: $\lambda_1 + \lambda_2 = \text{trace}(J)$. If both $\gamma, \alpha > 0$, what can you say about $\text{Re}(\lambda_1) + \text{Re}(\lambda_2)$?

c) Can this system produce sustained oscillations? Explain using the condition that for instability (sustained oscillations), at least one eigenvalue must have positive real part.

Show hint

For a $2\times 2$ matrix, if $\text{trace} < 0$ and $\det > 0$, both eigenvalues have negative real part. Can the trace ever be positive here?

Problem 2: Two Genes - Damped Oscillations¶

Now consider two genes that mutually repress each other:

$$\frac{dx}{dt} = \frac{\beta}{1 + y^n} - \alpha x$$

$$\frac{dy}{dt} = \frac{\beta}{1 + x^n} - \alpha y$$

a) Find the symmetric steady state $(x^*, y^*)$ and compute the Jacobian matrix at this point.

Show hint

At the fixed point, $dx/dt = dy/dt = 0$. By symmetry, $x^* = y^*$. The Jacobian has the form $J_{ij} = \partial(\dot{x}_i)/\partial x_j$.

b) Show that the eigenvalues are $\lambda = -\alpha \pm h'(x^*)$ where $h(u) = \beta/(1+u^n)$. Under what condition do the eigenvalues have imaginary parts?

Show hint

For a $2\times 2$ matrix with structure $J = [[-\alpha, a], [a, -\alpha]]$, the eigenvalues are $\lambda = -\alpha \pm a$. Complex eigenvalues occur when $a$ is imaginary or when... think about what makes $\lambda$ complex.

c) Can the real part of the eigenvalues ever be positive? What type of behavior does this system exhibit and why can't it sustain oscillations?

Show hint

Look at $\text{Re}(\lambda) = -\alpha + \text{Re}(h'(x^*))$. What is the sign of $h'(x^*)$ for a repression function? Can $\text{Re}(\lambda) > 0$?

Problem 3: Frequency of the Three-Gene Repressilator¶

For the three-gene repressilator, the eigenvalues with imaginary parts are:

$$\lambda_{1,2} = -\alpha + h'(x^*) \cdot e^{\pm 2\pi i/3}$$

where $h(u) = \beta/(1+u^n)$ and $x^*$ is the fixed point.

a) Write $\lambda_1$ in the form $\lambda_1 = \sigma + i\omega$ where $\sigma$ is the real part and $\omega$ is the imaginary part. Express both $\sigma$ and $\omega$ in terms of $\alpha$ and $h'(x^*)$.

Show hint

Use $e^{i2\pi/3} = \cos(2\pi/3) + i\sin(2\pi/3) = -1/2 + i\sqrt{3}/2$. Then $\lambda_1 = -\alpha + h'(x^*)\cdot(-1/2 + i\sqrt{3}/2)$.

b) The period of oscillation is $T = 2\pi/|\omega|$. Derive an expression for $T$. How does the period depend on the degradation rate $\alpha$?

Show hint

From part (a), you should have $\omega = (\sqrt{3}/2)h'(x^*)$. Note that $h'(x^*)$ is negative for repression, so use $|\omega|$.

c) Using $h'(x^*)$ from the lecture, express the period in terms of $\alpha$, $\beta$, $n$, and $x^*$. If you want to double the period, what parameters could you adjust and how?

Show hint

$h'(x^*)$ contains $\alpha$ explicitly (through the fixed point condition). Think about which parameters appear in $|h'(x^*)|$ and how $T$ scales with them.

Problem 4: Four Genes Instead of Three?¶

Consider a four-gene cycle where each gene represses the next:

$$\frac{dx_i}{dt} = \frac{\beta}{1 + x_{i-1}^n} - \alpha x_i, \quad i = 1,2,3,4 \text{ (mod 4)}$$

a) The eigenvalues are:

$$\lambda_k = -\alpha + h'(x^*) \cdot e^{i\pi k/2}, \quad k = 0,1,2,3$$

Write out all four eigenvalues explicitly.

Show hint

Evaluate $e^{i\pi k/2}$ for $k = 0, 1, 2, 3$. Remember $e^{i0} = 1$, $e^{i\pi/2} = i$, $e^{i\pi} = -1$, $e^{i3\pi/2} = -i$.

b) Examine the eigenvalue with $k=2$. What is special about it compared to the three-gene case?

Show hint

For $k=2$, you get $\lambda_2 = -\alpha + h'(x^*)\cdot(-1) = -\alpha - h'(x^*)$. This is a real eigenvalue. What's the sign of $h'(x^*)$?

c) Why might even-numbered cycles be problematic for generating sustained oscillations?

Show hint

An even cycle can be "split" into two identical subsystems. Think about what this symmetry implies for stability.