Modeling - Math 056 - Homework 7

1. Read one of the derivations of Michaelis-Menten kinetics. These are located on my webpage. One of the techniques used is establishing that a variable is at 'quasi steady-state'. Explain briefly what this is and how it is used in the derivation. Specifically identify the 'fast' variable and the 'slow' variable.

2. Suppose that mass-action kinetics is used instead of Michaelis-Menten kinetics in a chemostat model. The model is therefore,

dH/dt = µ₁SH - DH,                                                       (1)

dS/dt = D(S_i - S) - µ_i/Y₁ SH.                                          (2)

Using the following rescales,

x = S/S_i, y = H/Y₁S_i, t = DT,

the model is reduced to,

dx/dt = 1 - x - Axy,                                                       (3)

dy/dt = Axy - y,                                                            (4)

where A = µ₁S_i/D.

(a) Identify all of the variables and parameters in equations (1)-(2).

(b) Add equations (3)-(4). Then treat x(t) + y(t) as a single variable and solve.

(c) Show that lim_t→∞ x(t) + y(t) = 1.

(d) Use x(t) + y(t) = 1 to write (4) only in terms of y. Show that this can be written as,

dy/dt = (A - 1)y (1 - (y/(A-1/A))).

(e) What equation is this? What is lim_t→∞y(t) (you should not have to calculate this). Biologically interpret this result using the original parameters.

3. Variants of the chemostat. Include any assumptions that you make.

(a) Suppose that at high densities bacteria start secreting a chemical that inhibits their own growth. How would you model this situation?

(b) In certain cases, two or more bacteria species are kept in the same chemostat and compete for a common nutrient. Suggest a model for such competition experiments.

4. An organic pollutant enters a lake at a constant rate. Bacterial action metabolizes the pollutant at a rate proportional to its mass. In doing so, the dissolved oxygen in the lake waters is used up at the same rate that the pollutant decomposes. However, oxygen in the air re-enters the lake through surface-to-air contact (this is call re-aeration) at a rate proportional to the difference between the maximum dissolved oxygen level that the lake can support and its current actual value. Let x₁(t) be the mass of pollutant and x₂(t) be the level of oxygen in the water at time t. Furthermore, let x_m be the maximum (saturation) level of oxygen in the water.

(a) Define the remaining necessary parameters and write a linear system of differential equations to model the behavior.

(b) For the students who have taken differential equations, find solutions for x₁(t), x₂(t). Hint: Since the organic pollutant enters at a constant rate, it is a nonhomogenous differential equation. Use the variation of parameters.

(c) Find lim_t→∞ x₁(t), x₂(t).

5. A biochemical reaction involving a substrate, S, and an enzyme, E, and yielding a complex, C, is quantified,

E' = -k₊ES + k_-C + kC = -C',                                        (5)

and can be simplified to

C' = k₊(E₀ - C)(S₀ - C) - (k + k_-)C = f(C),                     (6)

where S₀ is the total amount of substrate that is available at any time, either bound up in C or totally unbound, and E₀ is the amount of E available at t = 0.

(a) Identify the relationships that are used to simplify (5) to (6).

(b) Show that f(C) has two real and positive roots. What is the biological meaning of the two roots?

(c) Find the explicit solution for C(t).

(d) What is the lim_t→∞C(t)?

6. A semitoxic chemical is ingested by an animal and enters its bloodstream at the constant rate D. It distributes within the body, passing from blood to tissue to bones at constant rates. It is excreted in urine (from the blood) and sweat (from the tissues) at rates u and s respectively. Let x₁, x₂, and x₃ represent concentrations of the chemical in the three areas of the body.

(a) Assuming linear exchange between the three compartments, write equations for dx_i/dt.

(b) Find the steady-state values. Simplify notation by using your own symbols for ratios of rate constants that appear in the expressions.

(c) How would you investigate whether this steady state is stable? Note: you do not have to do it.

7. For the following: sketch the directional field, dy/dy vs. y, and the 1-D phase portrait to determine the behavior. Verify with a dfield plot.

(a) dy/dt = y²

(b) dy/dt = 1 - y/1+y

(c) dy/dt = ye^y-1

8. Using the non-dimensionalized chemostat model that we discussed in class,

(a) Sketch a phase portrait of the two different biological outcomes (you might want to select relevant parameter values).

(b) Use pplane to confirm your results (you will definitely have to select relevant parameter values).

(c) Describe in words what would happen if we set up the chemostat to contain the following:

i. A small number of bacteria with excess nutrient in the growth chamber.

ii. A large number of bacteria with very little nutrient in the growth chamber.

9. A model for the effect of a spruce budworm on a forest is given by

dS/dt = r_SS(1 - (SK_E/K_SE)),

dE/dt = r_EE(1 - (E/K_E)) - P(B/S),

where

S(t) = total surface area of trees,

E(t) = energy reserve of trees.

(a) Interpret possible meanings of these equations.

(b) Sketch nullclines and determine how many steady states exist.

(c) Draw a phase-plane portrait of the system. Show that the outcomes differ qualitatively depending on whether B is small or large.

(d) Interpret what this might imply biologically.

(e) Interested in this model? You can read the scientific article posted on my website.