MATH 437/537–Assignment 2 Due Feb 9, 2023

Prof. A. Khadra

1. Consider the following physical quantities and their associated units

Variables Radius r Energy E Time t

Air density ρ Air pressure P

Units (Dimensions)

L ML2T−2 T

ML−3 ML−1T−2

     Table 1: M represents the unit of mass.

(a) Use Buckingham π Theorem to find dimensionless variables. How many did you find?

(b) If 0 < r ≪ 1, can you predict what will happen to air pressure P in this case? What kind of

system can exhibit such a behaviour?

2. In modeling the phagocytic ability of macrophages, we developed three models, one of which (Model 2) assumed that naïve macrophages, i.e. class of macrophages with no apoptotic cells inside them M0, go through an activation step upon apoptotic-cell uptake (see lecture notes).

According to this new formalism, find estimates for the rate constants ka and ke by considering the time evolution of the model in the initial stages, then use QSS approximation to show that the percent phagocytosis φ and phagocytic index Iφ are approximately given by

and

Iφ ≈ 100 where λa = kaA/kd and λ = keA/kd.

Based on these two estimates, deduce that

λa , (1 − λ + λa)(1 − λ)

φ ≈ 100 λa

1 − λ + λa

 ke ≈ ka(Iφ − φ)(100 − φ). φ2

3. We saw during lectures that the remission-relapse (flare-up) model describing the destruction of a specific tissue Te by effector immune cells Ee, is given by the following system of equations

(1)

dTe dt

= re Te − ek Te Ee

dEe Te Ee2 e

dt = pek1+Te+sek2+Ee2−k3E 1

where re is the recovery rate of the tissue, ek is the destruction rate of the tissue by effector immune cells, pe is the stimulation rate of effector immune cells by target tissue, se is the rate of cooperation of effector immune cells, k1 and k2 are half-maximum stimulation and cooperation, respectively, and k3 is the turnover rate of effector immune cells.

By applying the appropriate substitution, the model can be non-dimensionalized into

dT dτ

= rT−kTE

dE T E2

(2)

dτ = p1+T +s1+E2 −E.

(a) Apply the necessary substitutions to non-dimensionalize system (1) into system (2). Determine

how the parameters r,k,p and s depend on re,ek,pe,se,k1,k2 and k3.

(b) Let r = 1, k = 2, p = 1 and s = 1 (which all happen to be dimensionless parameters based on

part (a)). Use XPPAUT to plot the one-parameter bifurcation diagram of E with respect to s.

(c) Use XPPAUT and part (b) to plot the two-parameter bifurcation diagram of this model with

respect to s and p. In the latter, identify the “non-physiological” regime by coloring it gray.

(d) Use XPP to plot the nullclines of the model in the T,E-plane along with the stable/unstable

manifolds (invariant sets) of the key fixed points when s = 1, s = 1.8 and s = 2.

4. We already know that normal cells in healthy tissue express self-antigens on their surface in the form of pMHC complexes that identify them as belonging to their original host. However, genetic alteration characterizing tumours often generates new surface proteins in cancer cells that are not expressed by normal cells. These so-called tumour antigens can be particularly effective in their ability to stimulate an immune response, or they can be weak and fail to do so. One would be interested to find out if the immune response would be able to produce alternating phases of tumour progression and regression.

Let P be the population of tumour (target) cells, e0 be the population of circulating immune cytotoxic cells and e1 be the population of immune cells bound to the pMHC on target cells via immune synapses. The kinetic equations describing the temporal evolution of the cellular densities of these three cell populations are given by

dP = rP(1−P)−be0P dt K

de0 dt

de1 dt

where rP(1−P/K) represents the logistic term describing the evolution of target cells in the absence of immune cells, s is the thymus input (source of immune cells), be0P is the binding of immune cells with host cells and their turnover, a0e0 is the natural turnover of unbound immune cells, and ke1 the transition of bound immune cells into unbound cells.

(a) Non-dimensionalize the model by applying the substitutions τ = rt, x = P/K, yi = a0ei/s, i = 0, 1. Simplify your equations by setting the resulting parameter combinations to the following: β = bs/(a0r), α = a0/k, η = k/r and κ = bK/k.

(b) Show that when β < 1, the non-dimensionalized (scaled) model has two steady states: S1 (with no elevation in the number of bound immune cells y1) and S2 (with elevated level of y1).

 = s−a0e0 −be0P +ke1 (3) = be0P−ke1,

2

(c) Show that the steady state S1 is always unstable in the interval β ∈ [0, 1). Explain what happens at β = 1 and identify the type of bifurcation point that occurs at that value.

(d) Consider the simple scenario in which κη = αη = 1 and η = 1 − β. Write down the Jacobian matrix at S2 and its characteristic equation written in the form p(λ) = λ3 + d1λ2 + d2λ + d3. Use the Routh Hurwitz criterion to determine the stability of the steady state S2.

5. STUDENTS IN MATH 537 ARE EXPECTED TO DO THIS PROBLEM; STUDENTS IN MATH 437 CAN SOLVE THIS FOR A BONUS

Consider Fisher’s equation, given by

∂p = D ∂2p + αp(1 − p), ∂t ∂x2

which describes the dynamics of a population density p of a given species, such as immune cells, diffusing in one dimension at a rate D.

(a) Show that traveling wave solutions to Fisher’s equation, given by P (z) = p(x, t), where z = x−vt, must satisfy

(4)

What happens if the condition in part (c) is not met? [Sketch the resulting phase-plane diagram and discuss why one cannot obtain realistic traveling wave.]

(d) Conclude what the minimum wave speed is.

dP dz

=S

dS = −αP(1−P)−vS,

(b) Find the steady states of system (4) and determine the Jacobian matrices at these points.

(c) Show that one of the steady states is stable and the other is a saddle, provided that

?v?2 α

−D −4D >0.

dz D D

3