Homework 3

(c) 2019 Justin Bois and Michael Elowitz. With the exception of pasted graphics, where the source is noted, this work is licensed under a Creative Commons Attribution License CC-BY 4.0. All code contained herein is licensed under an MIT license.

This document was prepared at Caltech with financial support from the Donna and Benjamin M. Rosen Bioengineering Center.

This homework was generated from a Jupyter notebook. You can download the notebook here.


3.1: Co-substrate compensation, 50 pts

This problem derived from discussions with H. Y. Kueh, and is based on his paper (Biophys J., 104, 1338--1348, 2013).

To put this problem in context, let us think about an intriguing question. Oxygen is delivered to the cells of your body through your blood. Naturally, cells are situated at difference distances from blood vessels. As oxygen diffuses away from the blood vessels, different cells experience different local oxygen concentrations, often very different concentrations. How, then, do cells respire at the same rate, despite the large difference in oxygen concentration?

This is a question of robustness. Is rate of respiration robust to differences in oxygen concentration over a physiologically relevant operating regime? To address this question consider a simple model of the mitochondrial electron transport chain (ETC) consisting of a single electron carrier and two reactions that catalyze its oxidation and reduction, shown in below.

a) Write down a system of ODEs describing the dynamics of this network. Assume that the rate of oxygen reduction by enzyme $\mathrm{E}_0$ is hyperbolic with respect to oxygen concentration with a half-maximal value $K_0$, i.e., $v_0 \propto [\mathrm{O}_2]/([\mathrm{O}_2]+ K_0)$. For simplicity, also assume that the rate of carrier reduction (by enzyme $\mathrm{E}_1$) is first order with respect to the carrier concentration. Finally, assume that the total carrier concentration is conserved, i.e., $C_T = [C_0] + [C_1] = \text{constant}$.

b) Compute the response of the system to a 5-fold drop in oxygen concentration from a value greater than $K_0$ to one lower than $K_0$, for one set of parameters of your choosing. Does the system maintain a constant oxygen consumption rate after the drop in oxygen levels? If not, repeat these simulations for a different set of parameters to identify a set of parameters where rate constancy is maintained.

c) Solve for the steady-state oxygen consumption rate as a function of oxygen concentration. Using this expression, derive an analytical expression for the oxygen concentration at which oxygen consumption rate is half-maximal. We will call this concentration $K_m$.

d) Now, consider the regime where the maximal rate of enzyme $E_0$ is much greater than that of $E_1$. Show that, in this regime:

  1. The model recapitulates the Chance relationship (B. Chance, \textit{J. Gen. Phys.}, 1965), which states that the $K_m$ scales linearly with the maximal rate of electron transfer, and inversely with the reaction rate constant for oxygen reduction.
  2. The system implements integral feedback. Specifically, show that the time integral of the difference between an enzyme's operational velocity and its steady-state velocity is conveyed to the enzyme by the levels of reduced carrier.

e) What are the main conclusions that you were able to reach from this simple toy model of the ETC?


3.2 Kinetic proofreading in the immune system, 50 pts

In class, we talked about kinetic proofreading in the context of translation. In this problem, we will consider a model for kinetic proofreading in the immune system. T cells recognize specific agonist major histocompatibility complexes (MHC) on an antigen presenting cell (APC). The APCs also have many endogenous MHCs that should not trigger T cell activation. It has been proposed that T cells are able to distinguish agonist MHCs (which should trigger T cell activation) and endogenous MHCs (which should not trigger T cell activation) via a kinetic proofreading mechanism. Such a mechanism is shown in below. In the figure, M = MHC and T = T cell receptor (TCR). The complexes $\mathrm{C}_i$ denote MHC-TCR complexes with various stages of phosphorylation. In this model, all complexes may dissociate and "reset" to an unbound MHC and TCR. The rate of activation of T cells is proportional to the concentration of complex $\mathrm{C}_n$. We say that $\mathrm{C}_n$ is the "active form."



In this problem, we will provide a bit less guidance than usual in the analysis and leave you to think carefully about what plots and what limits are useful for analyzing the system and making conclusions about it. This is meant to help train you think carefully and creatively about new circuits you encounter. Specifically, you will think about the competing needs of the T cell activation system:

  • The system must be selective. That is, T cells should only be activated upon interaction with an agonist MHC, and not with endogenous MHCs.
  • The system must be sensitive. That means it should get considerable T cell activation when the agonist MHC is present.

a) Assuming mass action kinetics, write down a system of ODEs for the concentrations of all chemical species. Consider only a single MHC species.

b) Until part (e), assume that $k_{-n} = k_{-1}$. Show that at steady state, the fraction of complexes that are in the active form is $\alpha^n$, where

\begin{align} \alpha = \frac{1}{1 + k_{-1}/k_p}. \end{align}

Here we again consider only a single MHC species. Hint: Recall the sum of the first $n$ terms of a geometric series,

\begin{align} \sum_{k=0}^{n-1} x^k = \frac{1 - x^n}{1-x}, \end{align}

provided $0 < x < 1$.

c) Compute the fraction of MHC that is bound to active TCR complexes, again considering only a single MHC species.

d) Discuss, with plots, limits, and/or words, how these results relate to the selectivity and sensitivity of the system. You can compute other quantities in our arguments if you like, such as the ratio of active complexes that are bound to agonist and endogenous MHCs.

e) It has been argued that $k_{-n}$ can be tuned to get better performance of the system in terms of sensitivity without a comparable sacrifice in selectivity. Evaluate this claim.

f) If you are feeling curious and motivated, investigate other modifications to the kinetic proofreading mechanism. You might want to postulate about how the fact that the interactions between MHCs and TCRs are happening on cell membranes. This part of the problem has no points associated with it, but is meant to provide something to think about and possibly to spark conversations among students and the course staff.