Homework 7.1: Cytokines and homeostasis (50 pts)

This problem is based on analysis inHart, et al., 2014.

Cytokines are small proteins that are widely used in cellular signaling. Importantly, they orchestrate cell growth and differentiation in contexts such as hematopoietic stem cell differentiation. In this problem, we will investigate the control of cell proliferation by cytokines in paradoxical regulation.

Below is a diagram of a circuit for paradoxical regulation of cell proliferation by cytokines.

diagram of paradoxical cytokine circuit

The cytokine, m, has a basal production rate of \(I_0\) from other cells. The cells, schematically shown as C in the diagram, also secrete cytokines at a rate \(\beta_1\). The cytokine regulates the proliferation of cells, described by \(\beta(m)\). The cells die with rate \(\gamma_c(m)\), also regulated by the cytokine. The cytokine degrades with a rate \(\gamma_m\). Finally, the cell takes in cytokine (thereby clearing it) at a rate \(f(m)\).

We can write differential equations to describe the dynamics of the concentration of cells, \(c\), and of cytokine, \(m\).

\begin{align} \frac{\mathrm{d}C}{\mathrm{d}t} &= \beta_c(m)\,c - \gamma_c(m)\,c, \\[1em] \frac{\mathrm{d}m}{\mathrm{d}t} &= I_0 + \beta_1 c - \alpha_0 c \,f(m) - \gamma_m(m). \end{align}

For simplicity, we will take \(\gamma_m(m) = \gamma_m m\). We will take

\begin{align} f(m) = \frac{(m/k)^{n}}{1 + (m/k)^{n}}, \end{align}

a Hill function. For simplicity, we will take \(n = 2\), approximately the value obtained experimentally in the Hart, et al. paper. Also in accordance with experimental measurement, we take \(\gamma_c(m)\) to be a linear function of \(m\),

\begin{align} \gamma_c(m) \to \gamma_c + \alpha_c m. \end{align}

Finally, we also write \(\beta_c(m)\) as a Hill function,

\begin{align} \beta_c(m) = \beta_0\,f(m). \end{align}

It has the same form as \(f(m)\) since the \(\alpha_0 c\, f(m)\) term in the \(m\)-dynamics describes uptake of \(m\) by cells, and the \(\beta_c(m)\) term in the \(c\)-dynamics described cytokine mediated growth, dependent on the uptake rate. Thus, our system of ODEs is

\begin{align} \frac{\mathrm{d}c}{\mathrm{d}t} &= \beta_0\,\frac{c\,(m/k)^2}{1 + (m/k)^2} - \gamma_c c - \alpha_c m c,\\ \frac{\mathrm{d}m}{\mathrm{d}t} &= I_0 + \beta_1 c - \alpha_0 \,\frac{c\,(m/k)^2}{1 + (m/k)^2} - \gamma_m m. \end{align}

This is the dynamical system we will analyze in the problem.

a) Show that the equations can be nondimensionalized to go from eight parameters to five. Specifically, show that the equations may be written, with appropriate re-definition of variables and parameters, as

\begin{align} \frac{\mathrm{d}c}{\mathrm{d}t} &= \beta_0\,\frac{m^2 c}{1 + m^2} - \gamma c - \alpha_c m c\\[1em] \frac{\mathrm{d}m}{\mathrm{d}t} &= I_0 + c - \alpha_0\,\frac{m^2 c}{1 + m^2} - m. \end{align}

Henceforth, we will be working with these dimensionless variables and parameters, e.g., \(m\) refers to the redefined dimensionless cytokine concentration.

b) We will now find the nullclines corresponding to \(\mathrm{d}c/\mathrm{d}t = 0\).

  1. Show that there is always a steady state with \(c = 0\). Discuss why this nullcline is a vertical line in the \(c\)-\(m\) plane.

  2. For arbitrary nonzero \(c\), \(\mathrm{d}c/\mathrm{d}t\) vanishes for particular values of \(m\). Show that there are either zero or two such values. Demonstrate that a necessary condition for existence of nonzero values of \(m\) for which \(\mathrm{d}c/\mathrm{d}t = 0\) is \(\beta_0 > \gamma\). Hint: It may help to write the equation for \(\mathrm{d}c/\mathrm{d}t = 0\) as a cubic polynomial. What does this mean physically? Describe why the nullclines for nonzero \(c\) are horizontal lines in the \(c\)-\(m\) plane.

If you are feeling ambitious, you may show that the necessary and sufficient condition for having two nonzero values of \(m\) for which \(\mathrm{d}c/\mathrm{d}t = 0\) is \(\alpha_c^2\left(\beta_0^2 - 20\beta_0\gamma - 8\gamma^2\right) + 4\gamma(\beta_0 - \gamma)^3 - 4\alpha_c^2 > 0\), but this is not required for this problem.

c) Now we will find the nullcline corresponding to \(\mathrm{d}m/\mathrm{d}t = 0\). The nullcline may be written as \(c(m)\), that is, \(c\) as a function of \(m\). Derive this function. For what values of \(m\) is the function defined (bearing in mind that \(m,c\ge 0\))?

d) Now that we have the nullclines, we can compute the fixed points (a.k.a. steady states), which is where the nullclines cross.

  1. The first fixed point is the one with \(c = 0\). What is the value of \(m\) at this fixed point?

  2. For the other (possibly) two fixed points, analytical solutions exist. However, the analytical solution is a mess, and it is easier to consider numerical solutions. The dimensionless parameters used in Hart, et al. paper, which are based on experimental measurements, are: \(\beta_0 = 9.78\), \(\gamma = 2.14\), \(I_0 = 24.67\), \(\alpha_c = 1.06\), and \(\alpha_0 = 11.21\). For this parameter set, find the other two fixed points.

  3. Plot all of the nullclines for the given set of parameters. Overlay the fixed points as dots.

  4. In part b2, you already discussed conditions on \(\beta_0\) and \(\gamma\) that are necessary to get bistability. From your analytical analysis, can you comment on the magnitudes of \(I_0\) and \(\alpha_0\) are are needed to get bistability? What does this mean physically?

e) Plot the separatrix.

f) Finally, overlay a phase portrait for the system.

g) Solve the system of ODEs for the given parameter set starting with an initial condition of \(c\) and \(m\) just above the separatrix. Do this again with an initial condition just below the separatrix. Plot the temporal profiles of \(c\) and \(m\) and comment on the results.