Math 4428: Mathematical Modeling via Mathematical Biology

    Spring, 2004,  12:20pm - 01:10pm, MWF,  Vincent Hall 206
 [Last year's friendly Mission Statement  here]

  • Instructor:  Jackie Shen, Vincent Hall 539,  625-3570; jhshen@math.umn.edu;  www.math.umn.edu/~jhshen.
  • Office Hours:  MW 2:30pm - 3:30pm, or by appointments.
  • Prerequisites:  Math 2243 (Linear Algebra & Differential Equations)  or equivalent. Some elementary Probability and Multivariable Calculus (Math 2263) are recommended. Thus the ideal audience would mainly consist of juniors, seniors, and graduate students.
  • Textbook: Modeling Differential Equations in Biology, by Professor Clifford H. Taubes, Harvard University, published by Prentice Hall, 2001.  ISBN: 0-13-017325-8. (Extra topics will be inserted from time to time. Examples: Mathematical Physiology by James Keener et. al, Springer, Best Title in 1998; Neurobiology and Neural Networks; and Vision and Learning Theory.)
  • Computation: Software MATLAB (by The MathWorks, Inc.) will be ideal and enough.  
  • Workload:   Five Problem Sets[NO late homework will be accepted without good excuses], aone-hour Midterm Exam[ NO books, notes, traditional one-page "cheating" sheet, or Laptops. Standard calculator is OK . NO makeup tests without rigorous emergency reasons. Athletes please present Proofs of Activities in advance ], plus a one-hour Final  Exam [same policies as the Midterm].
    • Homework: 30 %. 
    • Midterm Exam (at the regular lecture time, Friday, March 12, 2004 ): 30 %. 
    • Final Exam (at the regular lecture time, Friday, May 7, 2004 ): 40 %. 
  • Statement on Scholastic Conduct: Each student should read the college bulletin for the definitions and possible penalties for scholastic dishonesty. Students suspected of cheating will be reported to the Scholastic Conduct Committee.

Initially created on January 19, 2004;  Last updated on January 19, 2004.


Problem Set 1 [Assigned on Monday, Feb 1; Due on Friday, Feb 13]:

Chapters/Themes

Pages/Applications

Problems (© Jackie Shen, 2004)

2

43

1(d), 2(b)[following 1(d) only]

3

62

1(b and c), 2 (b and c)

4

71

1

Extra

481

1

Exponential Law

Carbon dating

A piece of fish fossil was just unearthed somewhere in Africa. Measurement shows that its concentration of C14 is about 12% of that of the contemporary environment. Assume that the concentration in the environment has remained stable in history, and any life maintains the same concentration level of C14 as the environment before its death. Determine the age of this fish fossil. (The half-life time of carbon 14 is t1/2=5730 years.) Clearly explain the model you use and its associated data.

Exponential Law with Constant Supply

Gandalf's checking account

UMN freshman Gandalf just opened a checking account at the US Bank with initial deposit $5000. The generous bank manager gave him a fixed monthly interest rate of 0.1%. To pay for tips in restaurants (he often dines out), Gandalf plans to withdraw B dollars each month from the account on average.
(1) Suppose B=4 dollars/month. What is the balance exactly four years later when Gandalf finishes his B.S.;   (2) Determine the maximum value of Bmax, so that any more aggressive withdrawal B above it will eventually empty the account.

Logistic Growth Law

Aragorn's fish pond

Out of the 10000 lakes that Minnesota claims, our UMN junior Aragorn owns one of them with current fish population p0=8 (units). After spending 4 years in measuring and modeling the lake, Aragon's biology senior friend Gimli has concluded that the Logistic Model: dp/dt = rp(m-p) works quite accurately for the fish population p(t) without any human interference, with r=0.01, and m=20 (units). The time (i.e. t) unit is in years.
(1) In such a natural environment, what are the precise fish populations one year later and 10 years later?   (2) Suppose that from now on, Aragorn decides to sell C (units) of fish annually out of his lake. Help Aragorn to decide the maximum level Cmax, beyond which the fish population is put at the risk of eventual depletion.

Problem Set 2 [Assigned on Wed, Feb 18, Due on Mon, March 1, 2004]:

Chapters/Themes

Pages/Topics

Problems (Population Dynamics of Two Species)

Generic Equilibria

Local Behavior

(a) Sketch the local dynamics for each generic equilibrium type: a stable node, an unstable node, a focusing (stable) spiral, a defocusing (unstable) spiral, and a saddle point.
(b) For each generic equilibrium type, describe qualitatively the behavior of the associated two eigenvalues (i.e. real or complex, and their signs).

Eigenvalues

Real or Complex

Find the eigenvalues for each of the 2-by-2 matrices:

A = [4, -1; -1, 4]     B = [1, -2; 2, 1].

Here the semicolons denote row breaks.

Characteristic Polynomials

A Formula to Impress Jerry

(a) Express the characteristic polynomial of a 2-by-2 matrix A in terms of its trace tr(A) and determinant det(A).
(b) Suppose via his cellular phone, Jerry transmits to you the following information tr(A)= 7, det(A)=12 of a 2-by-2 matrix A. But leaving in a rush for the movie The Matrix , he forgets to transmit to you the entire matrix A. Can you impress him by sending him the precise information of the two eigenvalues, even without knowing A?

Linear Systems

Exact Solutions

Consider the linear system

p'(t)=3p-q;   q'(t)=-p + 3q;     with initial conditions p0=1, q0=2.

Find the exact solution p(t) and q(t). Is (0, 0) a node, spiral, or a saddle point?

6/Null Clines

103/Nonlinear System

6

8/Null Clines

139/Linear System

5

5/Lotka-Volterra

81/Stability

Find ALL the equilibrium states and their stability types (i.e. stable or unstable) for the Lotka-Volterra equation (10) with a=3, b=2, c=1, d=2, e=2

 

Problem Set 3 [Assigned on: Friday, March 5;  Due on Wed, March 24 (after the spring break)]

Chapters 13, 15

Problems (© Jackie Shen, 2004) [ Prediction is the Power -Jackie ]

Advection: Drug  in Blood

[Drug Absorption in an Idealized Blood Vessel] Consider an idealized 1-D blood vessel (from x=-œ to x=+œ) with constant flow speed c=2. A patient takes a certain type of drug to cure his disease. Suppose initially the concentration profile of the drug chemical is f(x)= exp(- x 4 ), and it is absorbed from the vessel to the ambient cellular environment at a rate of r=1. Assume that the system could be well modeled by the advection equation: ut = -c u x - ru, with u =u (x, t) denoting the concentration at time t. Predict the drug concentration at x=1 and time t =1.

Advection: Wind and Rain

[Are You a Weatherman ?] Idealize the US geography by thinking Minneapolis, Buffalo, and Boston as three spots in a 1-dimensional world (from x= -œ to x=+œ): x0=0, x1=2, and x 2=5. Today (t=0), the cloud concentration is measured to be f(x)=1 + cos (p x).  Suppose that (i). clouds are carried to the east (i.e. x>0 direction) by a steady wind of speed c=7 (units); and (ii) due to heavy rain releasing, the cloud concentration decays at a rate of r =2. Assume that this idealized weather system is well approximated by the advection equation in the previous problem. Predict tomorrow's (t=1) cloud concentrations at Boston.

Advection with Variable Speed

Consider the general advection model u t +q x= - k. Suppose that k=0 (i.e. lossless) and q(x,t)=c u(x,t), with c=c(x)=2x depending on the position. (1) Where will a particle initially at x0 =4 be at time t=ln 2 (natural logarithm)? (2) What is the initial position of a particle being observed at x=1 at time t=ln 2? (3) Assume at time t=0, two particles sit at x0 =1 and x0 =2 separately with initial distance 1. Predict their distance at time t=ln 2. Explain intuitively why the distance expands. [Hint keywords: slower and faster.]

Advection: Lossy or Lossless?

Suppose q(x,t)=c(x) u(x,t) and k(x,t)=r u(x,t). If the overall lossy advection equation reads as:  u t + 2 x u x =0, what is the loss rate constant r? [Many people would think that there is no loss involved in this equation!]

Advection: Total Population

Consider a lossy advection system with an unknown c(x) but known oscillatory loss rate r=r(t)=cos(t), which is time dependent.  Let

Q(t) = \integral u(x, t) dx     from –infinity to +infinity

denote the total population at each time t. Suppose initially Q(0)=1 (unit). What is the total population at time t= p ?

Attention:   Midterm on Friday, March 12 will cover up to PS 3 (inclusive).   Wed, March 10 will be a review lecture for the midterm.  Extra Office Hours:  10:00am-12:00pm, Fri, March 12.

 

Problem 4, Monday,  March 29.               Due on: Wednesday, April 14, 2004.

 

Subjects

Problems (© Jackie Shen, 2004) [ Life has randomness, and randomness gives life ]

Conditional Probability: Car accidents in Minneapolis

1. City government gives the following conditional probabilities of daily # of car accidents under different weather conditions during March in Minneapolis:  

N=n

n=1

n=2

n=3

n=4

p(N=n | rain)

0.3

0.3

0.3

0.1

p(N=n | snow)

0.1

0.4

0.3

0.2

p(N=n | sunny)

0.6

0.3

0.1

0

The chances for each condition in a normal March are:

Prob(rain) = 0.3,    Prob(snow) =0.3,  and   Prob(sunny) = 0.4.

(1.1) What is the chance p(2) for a given day in March to have n=2 accidents?

(1.2)  On average in March, how many daily accidents could be expected?

Binomial Diffusion in Polarized Tissues

2. Let us model a piece of  thin skin tissue by a 2-D lattice

{(n, m) :  n, m =0, 1, 2, ...) }.

Suppose initially 1 million drug molecules are injected at (0, 0) for a skin test, and that each molecule moves independently by the following rules (within a unit time):
 (a) with p=75% chance it moves to the right neighboring lattice point (i.e. n++);
 (b) with q=25% chance it moves to the upper neighboring lattice point (i.e. m++);
Predict at time t=6, how many molecules could be observed at site (4, 2).

1-800-Service Station:  Be a smart manager

3. You are just appointed as the manager for a 1-800 customer service call station which currently has 7 people answering incoming calls. Your research finds:

(a)    #incoming calls can be well modeled by Poison P(100) (in one hour);

(b)   on average,  each incoming call takes 6 minutes to resolve its issue.

Should you consider hiring more people? How many openings in an optimal way ? 

Exponential Neuron Firing

4. Suppose the time interval T for an idealized neuron to fire a new spike is well modeled by the exponential type E(10).
(4.1) What is the average firing rate per unit time?
(4.2) On average, how many firings could be expected within 10 units of time?
(4.3) What is the average waiting time for a new firing to occur?

Poisson Signaling

5. Two cells independently emit a same type of signaling chemicals (i.e. molecules) into the extra-cellular environment. Suppose they are well modeled by Poisson's P(10) and P(20) (within a unit time).
(5.1) On average, how long does it take for the combined system to emit 6000 signaling molecules?
(5.2) What is the probability type of the combined system: Gaussian, exponential, or still Poisson?
(5.3) What is the average waiting time for the combined system.

Sense and Sensibility: Selective firing

6. Suppose that the thumb's fingertip has n=10 temperature sensors, and when it touches a material surface (say wood, iron, etc): 
 (a) the number of sensors activated is subject to the binomial B(n=10, p=1/2);
 (b) each sensor, when activated, starts to fire spikes independently according to  

      Poisson P(5) (within a unit time).
All the spikes are then transmitted to the brain and counted by the central nerve system, allowing us to feel the relative temperature of the material. What is the probability distribution for the total number M of spikes (within a unit time) counted by the central nerve system?

 

Problem 5  [on Markov Chains]    Assigned on: Friday, April 16,   Due:  Friday, April 30, 2004]

Subjects

Problems (© Jackie Shen, 2004) [ Out of Randomness Rise Structures and Predictability ]

Markov Chains: Trinity’s Matrix:

Neo Needs Help!

(1) State the Markov Transport Theorem, (2) from which derive the Markov Equilibrium Equation;   Using her cellular phone, Trinity transfers to Neo the following 3 by 3 Markov transition matrix:  M=[ 0.3,  0.2,  0.1 (1st row); 0.2, 0.4, b (2nd row); 0.5, a, 0.5 (3rd row)] . But the entries a and b are lost due to Agent Smith’s interference. (3) Help Neo recover them.  (4) Draw the three-state Markov transition graph associated to this matrix.

Markov Equilibrium of a Simple Organism

Consider a 4-cell organism with all cells linearly sitting at positions (0, 1, 2, 3) along x-axis. A certain type of nutrition particles randomly (and independently) move through the cell membranes. Suppose that  P(i+1 | i) = p,  P(i-1 | i)=q, and P(i | i)=1-p-q, are the transfer probabilities for each individual particle, i=0:3, assuming that P(4|3)=0 (since no 4 exists) and P(-1 | 0)=0. Suppose under a laser beam, initially the cell at 0 produces 10 million particles. Afterwards the laser beam is removed and no particles are produced or destroyed. Predict the long-term distribution when (case 1): p=q=1/3; and (case 2) p =2q=1/2. 

Capturing and Waiting Time:

A Lazy Immune Cell vs. a Virus:

Consider a 1-D tiny cellular environment inside human body, which can be modeled by four sites along the x-axis: 0, 1, 2, and 3. Suppose a lazy immune cell is immobile and always stays at site 0, while a virus is energetic and moves around. Whenever the virus moves to site 0, it will be captured and eaten up by the immune cell. Suppose the virus’ random walk is described by : P(i-1 | i)= P(i+1 | i)=1/3, i=1, 2, 3, assuming that P(4|3)=0. Also notice that P(1 | 0)=0. Suppose the virus is initially at site 3.  (1) What is the probability that the bacteria will be eaten up at some finite time T?  (2) What is the average waiting time t3=E[T3] for the virus to be destroyed by the immune cell?

Lazy but Clever Immune Cells

Following the preceding problem, suppose in addition that our immune cell is smart and emits certain type of chemical signal, so that the transition probabilities of the virus is altered to:  P(i-1 | i)=3/4  and P(i+1 | i)=1/4 (i.e. causing the virus to prefer the motion toward the immune cell).  (1, 2) Answer the same two questions in the preceding problem. (3) Is the average waiting time t3=E[T3] indeed shortened?

Gene Regulated Network: Hi, Mr. Gene!!!!!! (Have you ever wondered how structures (limbs/head, etc.) could arise from a single cell?)

Ever wondered how Mr. Gene controls growth and creates body differences? Consider a simple linear cell divided into four sites, labeled by 0, 1, 2, and 3.  Suppose its head is mainly made of protein S, while the distribution of S is controlled by Mr. Gene G.

Suppose gene G has a  fixed  concentration distribution inside the cell:  (G(0)=7, G(1)= 4, G(2)=2, G(3)=1) in certain units, meaning that, for example, 4 units of gene G are concentrated at site 1.

Suppose initially protein S is distributed by (S(0)=2, S(1)=4, S(2)=4, S(3)=2), in millions, and that each protein molecule moves independently according to Markov transition law: P(i-1 | i) =(G(i-1)-G(i)) / (G(i-1)-G(i+1));   P(i+1 | i) = (G(i) - G(i+1)) / (G(i-1)-G(i+1)),   i=1, 2, 3,  assuming that  G(4)=G(3) to use this formula for boundary site i=3, and P(1 | 0)=1/3,  P(0|0)=2/3.  Now you see how Mr. G has regulated the distribution of protein S !

What is the long-term eventual distribution of protein S?  Is it uniform (i.e. same amount at each site)? Notice that biologically the head will develop where S is highly concentrated.

Note

Last problem set of Spring, 2004.  Thank you all for making the course a fun and pleasure!