Math 4600, Homework 4
1. (Computing) Consider a generalized version of the 1D (non-dimensionalized) model for gene activation
from class:
dx
xn
= s − rx +
dt
1 + xn
Use 1D phase portraits to answer the following questions. (Restrict your investigations to r ∈ (0, 1); try
starting with r = 0.4)
a. For n = 2, what happens as r varies? Is there a bifurcation? Consider cases where s > 0 and s = 0.
b. For n = 1, describe the behavior of the system for s = 0; r = 0.4. What happens as you vary s? As
you vary r?
c. Let s = 0 and investigate the system by varying r for n = 3;4; 5. Are there any qualitative differences
in the systems behavior for these values of n?
d. What do your investigations say about cooperativity’s importance (or lack thereof) in determining
the behavior of the gene activation model? Does the specific form of cooperativity matter?
2. (Computing) This problem refers to the original gene activation model we looked at it in class. Suppose
that the hormone signal level (previously called S0 ) is not constant; instead, the cell experiences a pulse
of hormone signal S(t) described by

t < Ton
 −δs S
dS
− δs S Ton < t < Tof f
=

dt
−δs S
Tof f < t
dg
g2
= k1 S − k2 g + k3 2
dt
k4 + g 2
Note that δs describes the degradation of hormone signal, and is the constant rate at which hormone
is delivered into the cell between times Ton and Tof f . For parameter values, choose k1 = 1, k2 = 3, k3 =
20, k4 = 2, ds = 1, Ton = 10, Tof f = 15, and initial conditions S(0) = 0, g(0) = 0.
a. Simulate this system in R, and allow it to run long enough for equilibrium to be reached. Investigate
the effect of varying within the parameter range of (.1, 1). Pick two values of that give very
different end levels of gene product (caused by a saddle-node bifurcation in the model). Produce a
time series plot of each case.
b. Make a bifurcation diagram of equilibrium values g ∗ vs S, thinking of S as a parameter in your model.
On your bifurcation diagram, plot a trajectory of (g(t), S(t)) for the case in part (a) in which the
value of g increases dramatically.
3. Imagine that you have a cell with capacitance C = 1, and only one ionic current IL with constant
conductance gL = 30 and the reversal potential VL = −60 mV.
a. Write the current-balance equation. Find steady state voltage. Solve the equation analytically with
initial condition V0 = −70 mV. Plot the solution. What is the behavior of voltage as time goes on?
b. Let’s say we have expressed another type of channel, which produces a currrent IM with a constant
conductance gM and the reversal potential VM = 0 mV. Find the steady state voltage in this model
with two currents for gM =.1, 30 and 100. Which current dominates in each of the cases?
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