{"cells": [{"cell_type": "markdown", "metadata": {}, "source": ["# Homework 1.2: Autoactivation, bistability, and the importance of leakage (30 pts)\n", "\n", "
"]}, {"cell_type": "markdown", "metadata": {}, "source": ["In this problem, we will explore a very simple circuit: autoactivation of a single gene, shown in the figure below. We will assume a separation of time scales in mRNA and protein degradation/dilution such that we only need to consider the dynamics of the protein, and we will call its concentration $x$. We will further assume that the autoactivation effect can be modeled as a Hill function with Hill coefficient $n$.\n", "\n", "\n", "\n", "\n", "\n", "
\n", "\n", "**a)** Write down an ODE describing the dynamics of $x$. Include leakage parametrized by $\\alpha$.\n", "\n", "**b)** Show that for $n = 1$, only one steady state exists for $\\alpha > 0$. Show further that when $\\alpha = 0$, two steady states may exist, but only one is stable.\n", "\n", "**c)** In part (b), we considered the case where activation is not cooperative. We now consider the other extreme, where activation is infinitely cooperative. Derive the condition for bistability when $n\\to\\infty$. In other words, for what parameter values does the circuit exhibit two stable steady states? Discuss this condition as it relates to design principles for bistability."]}, {"cell_type": "markdown", "metadata": {}, "source": ["
"]}], "metadata": {"anaconda-cloud": {}, "kernelspec": {"display_name": "Python 3", "language": "python", "name": "python3"}, "language_info": {"codemirror_mode": {"name": "ipython", "version": 3}, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.7"}}, "nbformat": 4, "nbformat_minor": 4}