Glenn LedderDepartment of MathematicsUniversity of Nebraska-Lincolngledder@math.unl.edu

funded by NSF grant DUE 0536508

A Terminal Post-Calculus-I Mathematics Course for

Biology Students

My Students• From Calculus I:

– Biochemistry majors– Pre-medicine majors– Biology majors

• From Business Calculus:– Natural Resources majors

• Took Calculus I in a past life:– Biology and Agronomy graduate students

My Course Format

• 15 weeks

• 5 x 50-minute periods each week

• Computer lab access as needed– We use the lab an average of 2 x per week– I use R, which is popular among biologists

Formatting Note

The rest of the talk is lists of topics, with comments and examples as needed:

Topics in blue are elaborated on 1 or more additional slides.

Topics in black aren’t. (I have little to add to what is readily available elsewhere.)

Outline of Topics

1. Mathematical Modeling (2-3 weeks)

2. “Review” of Calculus (1 week)

3. Probability (4-5 weeks)

4. Dynamical Systems (5 weeks)

5. Student Presentations (1 week)

Unexpected Difficulties (1 week)

1. MATHEMATICAL MODELING

• Functions with Parameters

• Concepts of Modeling

• Fitting Models to Data

• Empirical/Statistical Modeling

• Mechanistic Modeling

1. MATHEMATICAL MODELING

Functions with Parameters• Parameter: a quantity in a mathematical

model that can vary over some range, but takes a specific value in any instance of the model

• Perform algebraic manipulations on functions with parameters.

• Identify the mathematical significance of a parameter.

• Graph functions with parameters.

Functions with Parameters

y = e-kt y = x3 − 2x2 + bx

The half-life is ½ = e-kT,

or kT = ln 2

Parameters can change the

qualitative behavior.

Concepts of Modeling• The best models are valid or useful, not

correct or true.

• Mathematics can determine the properties of models, but not the validity. (data)

• Models can be analyzed in general; simulations illustrate instances of a model.

• The same model can take different symbolic forms (ex: dimensionless forms).

1. MATHEMATICAL MODELING

Fitting Models to Data• Fit the models

Y = mX, y = b + mx, z = Ae-kt using linear least squares.

• In what sense are the results “best”?

Fitting Models to Data

• The least squares fit for m in Y = mX is the vertex of the quadratic function

F(m) = (∑X2) m2 − 2 (∑XY) m + (∑Y2) .

• The least squares fit for b and m in y = b + mx comes from fitting Y = mX to

X = x – x, Y = y - y(We assume the best line goes through the mean

of the data.)

1. MATHEMATICAL MODELING

Empirical/Statistical Modeling• Explain where empirical models come

from. (looking at graphs of data)

• Use AICc (corrected Akaike Information Criterion) to compare statistical validity of models.

Empirical/Statistical Modeling

The odd-numbered points were used to fit a line and a quartic polynomial (with 0 error). But the even-numbered points don’t fit the quartic at all.• Measured data comprise only 0% of the points on a curve. Complex models are unforgiving of small measuring errors.

1. MATHEMATICAL MODELING

Mechanistic Modeling• Discuss the relationship between real

biology, a conceptual model, and a mathematical model. (Ledder, PRIMUS 2008)

• Derive the Monod growth function (Holling II).

• Use linear least squares to approximately fit models of form y = m f ( x; p) to data from BUGBOX-predator.

Mechanistic Modeling

Fitting y = m f ( x; p):•Let ti = f (xi; p) for any given p.

•Then y = mt with data for t and y.

•Define G(p) by

•Best p is the minimum of G.

2)(min)( iimmtypG

2. “REVIEW” OF CALCULUS• The derivative as the slope of the graph.• The definite integral as accumulation in

time, space, or “structure.”• Calculating derivatives.• Calculating elementary definite integrals by

the fundamental theorem (and substitution).• Approximating definite integrals.• Finding local and global extrema.

• Everything with parameters!

Demographics / Population Growth

Let l(x) be the probability of survival to age x.Let m(x) be the rate of production of offspring for

parents of age x.Let r be the population growth rate.Let B(t) be the total birth rate.How do l and m determine B (and r)?• The birth rate should increase exponentially with

rate r. (it has to grow like the population)• The birth rate can be computed by adding up the

births to parents of different ages.

Demographics / Population Growth

Population of age x if no deaths:Actual population of age x:Birth rate for parents of age x:Total birth rate at time t:

Total birth rate at time t:

Euler equation:

0

)()()()( dxxmxlxtBtB

dxxmxlxtB )()()(

dxxlxtB )()(

dxxtB )(

rteBtB )0()(

0)()(1 dxxmxle rx

3. PROBABILITY

• Characterizing Data• Basic Concepts• Discrete Distributions• Continuous Distributions• Distributions of Sample Means• Estimating Parameters• Conditional Probability

Distributions of Sample Means

Frequency histograms for sample means from a geometric distribution (p=0.25), with n = 4, 16, 64, and ∞

4. DYNAMICAL VARIABLES• Discrete Population Models

Example: Genetics and Evolution

• Continuous Population Models

Example: Resource Management

• Cobweb Plots

• The Phase Line

• Stability Analysis

Genetics and EvolutionSickle cell anemia biology:

• Everyone has a pair of genes (each either A or a) at the sickle cell locus:– AA: vulnerable to malaria– Aa: protected from malaria– aa: sickle cell anemia

• Babies get A from an AA parent and either A or a from an Aa parent.

Let p by the prevalence of A.Let q=1-p be the prevalence of a.Let m be the malaria mortality.

Genotype AA Aa aaFrequency p2 2pq q2

Fitness 1-m 1 0Next Generation (1-m) p2 2pq 0

The next generation has 2 pq of a and

2(1-m) p2 + 2 pq of A:

tt

t

ttt

ttt qqm

qqppm

qpq2)1)(1(4)1(2

221

Resource Management

Let X be the biomass of resources.Let K be the environmental capacity.Let C be the number of consumers.Let G(X) be the consumption per consumer.

)(1 XGCKXXR

dTdX

• Holling type 3 consumption– Saturation and alternative resource

22

2

)(XA

QXXG

0 A 2A 3A 4A0

0.25Q

0.5Q

0.75Q

Q

X

G

Dimensionless Version

k represents the environmental capacity.c represents the number of consumers.

2111

xx

kx

ccx

dtdx

RACQc

AKk

RtTAxX ,,,

4. DISCRETE DYNAMICAL SYSTEMS

• Discrete Linear Models

Example: Structured Population Dynamics

• Matrix Algebra Primer

• Eigenvalues and Eigenvectors

• Theoretical Results

Presenting Bugbox-population, a real biology lab for a virtual world.

http://www.math.unl.edu/~gledder1/BUGBOX/

Boxbugs are simpler than real insects:– They don’t move.– Development rate is chosen by the experimenter.– Each life stage has a distinctive appearance.

larva pupa adult

• Boxbugs progress from larva to pupa to adult.• All boxbugs are female.• Larva are born adjacent to their mother.

Structured Population Dynamics

The final “bugbox” model:

Let Lt be the number of larvae at time t.

Let Pt be the number of juveniles at time t.

Let At be the number of adults at time t.

Lt+1 = s Lt + f At

Pt+1 = p Lt

At+1 = Pt + a At

Computer Simulation Results

A plot of Xt/Xt-1 shows that all variables tend to a constant growth rate λ

The ratios Lt:At and Pt:At tend to constant values.

4. CONTINUOUS DYNAMICAL SYSTEMS

• Continuous Models

Example: PharmacokineticsExample: Michaelis-Menten Kinetics

• The Phase Plane

• Stability for Linear Systems

• Stability for Nonlinear Systems

Pharmacokinetics

x′ = Q(t) – (k1+r) x + k2 y

y′ = k1 x – k2 y

Q(t)

r x

k1 x

k2 yx(t) y(t)

blood tissues

References• PRIMUS 18(1), 2008

– R.H. Lock and P.F. Lock, Introducing statistical inference to biology students through bootstrapping and randomization

• Teaching statistics through discovery– T.D. Comar, The integration of biology into calculus courses

• Demographics, genetics– L.J. Heyer, A mathematical optimization problem in

bioinformatics• Excellent introductory problem in sequence alignment

– G. Ledder, An experimental approach to mathematical modeling in biology

• Modeling, theory and pedagogy

• Britton (Springer)• Cobweb plots

• Brauer and Castillo-Chavez (Springer)• Resource management