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A Terminal Post-Calculus-I Mathematics Course for Biology Students Glenn Ledder Department of Mathematics University of Nebraska-Lincoln [email protected] funded by NSF grant DUE 0536508 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. “Review” of Calculus (2-3 weeks) (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=ln2 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 – x̄, Y = y - ȳ (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 = mf ( x; p) to data from BUGBOX-predator. Mechanistic Modeling Fitting y = mf ( x; p): 1. Let ti = f (xi; p) for any given p. 2. Then y = mt with data for t and y. 3. Define G(p) by G ( p) min m ( y mt ) 2 i 4. Best p is the minimum of G. i 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)? 1. The birth rate should increase exponentially with rate r. (it has to grow like the population) 2. The birth rate can be computed by adding up the births to parents of different ages. Demographics / Population Growth B(t x) dx Population of age x if no deaths: B(t x)l ( x) dx Actual population of age x: Birth rate for parents of age x: B(t x)l ( x)m( x) dx Total birth rate at time t: B(t ) B(t x)l ( x)m( x) dx 0 B(t ) B(0) ert Total birth rate at time t: Euler equation: 1 e 0 rx l ( x)m( x) dx 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 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. 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 aa Frequency p2 2pq q2 1-m 1 0 (1-m) p2 2pq 0 Fitness Next Generation The next generation has 2 pq of a and 2(1-m) p2 + 2 pq of A: 2 pt qt qt qt 1 2 2(1 m) pt 4 pt qt (1 m)(1 qt ) 2qt 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. dX X R X 1 C G( X ) dT K • Holling type 3 consumption – Saturation and alternative resource 2 QX G( X ) 2 A X2 Q 0.75Q G 0.5Q 0.25Q 0 0 A 2A X 3A 4A Dimensionless Version t K CQ X Ax , T , k , c R A RA 1 x dx x cx 1 2 dt c k 1 x k represents the environmental capacity. c represents the number of consumers. 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 = sLt + fAt Pt+1 = pLt At+1 = Pt + aAt 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: Pharmacokinetics Example: Michaelis-Menten Kinetics • The Phase Plane • Stability for Linear Systems • Stability for Nonlinear Systems Pharmacokinetics Q(t) blood x(t) k1 x k2 y tissues y(t) rx x′ = Q(t) – (k1+r) x + k2 y y′ = k1 x – k2 y 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