A Terminal Post-Calculus-I Mathematics Course for Biology Students

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A Terminal Post-Calculus-I Mathematics Course for
Biology Students

Glenn Ledder Department of Mathematics University
of Nebraska-Lincoln gledder_at_math.unl.edu funded
by NSF grant DUE 0536508
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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

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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

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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 arent. (I have little to add to
  what is readily available elsewhere.)

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Outline of Topics

• Mathematical Modeling (2-3 weeks)
• Review of Calculus (1 week)
• Probability (4-5 weeks)
• Dynamical Systems (5 weeks)
• Student Presentations (1 week)
• Unexpected Difficulties (1 week)

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1. Mathematical Modeling

• Functions with Parameters
• Concepts of Modeling
• Fitting Models to Data
• Empirical/Statistical Modeling
• Mechanistic Modeling

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1. Mathematical ModelingFunctions 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.

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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.
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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).

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1. Mathematical ModelingFitting 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?

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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.)

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1. Mathematical ModelingEmpirical/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.

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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 dont fit the quartic at
  all.
• Measured data comprise only 0 of the points on
  a curve. Complex models are unforgiving of small
  measuring errors.

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1. Mathematical ModelingMechanistic 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.

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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.

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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!

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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.

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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

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3. Probability

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

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Distributions of Sample Means
Frequency histograms for sample means from a
geometric distribution (p0.25), with n 4, 16,
64, and 8
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4. Dynamical Variables

• Discrete Population Models
• Example Genetics and Evolution
• Continuous Population Models
• Example Resource Management
• Cobweb Plots
• The Phase Line
• Stability Analysis

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Genetics and Evolution

• Sickle 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.

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• Let p by the prevalence of A.
• Let q1-p be the prevalence of a.
• Let m be the malaria mortality.

Genotype AA Aa aa
Frequency p2 2pq q2
Fitness 1-m 1 0
Next Generation (1-m) p2 2pq 0
The next generation has 2 pq of a and
2(1-m) p2 2 pq of A
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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.

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• Holling type 3 consumption
• Saturation and alternative resource

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Dimensionless Version

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

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4. Discrete Dynamical Systems

• Discrete Linear Models
• Example Structured Population Dynamics
• Matrix Algebra Primer
• Eigenvalues and Eigenvectors
• Theoretical Results

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• 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 dont 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.

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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.

Lt1 s Lt f At
Pt1 p Lt
At1 Pt a At
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Computer Simulation Results
A plot of Xt/Xt-1 shows that all variables tend
to a constant growth rate ?
The ratios LtAt and PtAt tend to constant
values.
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4. Continuous Dynamical Systems

• Continuous Models
• Example Pharmacokinetics
• Example Michaelis-Menten Kinetics
• The Phase Plane
• Stability for Linear Systems
• Stability for Nonlinear Systems

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Pharmacokinetics
Q(t)
k1 x
blood
tissues
k2 y
x(t)
y(t)
r x

• x' Q(t) (k1r) x k2 y
• y' k1 x k2 y

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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