The Introductory Life Science Mathematics Sequencewikifuse.pbworks.com/f/LifeSciMath.pdf · 4...

The Introductory Life Science Mathematics Sequence

The Mathematics & Computer Science Department, Emory Universityin collaboration with the Center for Science Education

and the Department of Biology

Emory UniversityAtlanta GA 30322

Quantitative Biology Curriculum Planning WorkshopETSU 18–20 July 2007

Math & CS (Emory) Life Science Math QBCPW/ETSU /July 2007 1 / 13

Outline

1 The Emory Setting, Objectives, Courses & ChallengesThe ObjectivesThe CoursesThe Challenges

2 Examples of the Models and Methods in the CoursesDiscrete Models – Population Dynamics & CobwebbingDiscrete Models – Epidemics & Difference EquationsContinuous Models – Predator-Prey & Differential EquationsProbability and Statistics – The CLT & Sample Means

3 References and ContactsReferencesContacts


The Setting The Objectives

Objectives of the Courses

Mathematical, computational and statistical methods are of ever-growingimportance in the life sciences. We introduce the basic quantitative toolsrequired in modern life science research, adhering to the recommendations ofBIO 2010 [NRC] and Math & Bio 2010 [MAA]. Courses emphasize

modeling change in biological systems via discrete dynamics,continuous differential equations, and stochastic processes, andorganizing and analyzing data in information-intensive application areassuch as molecular evolution and genetics.

Researchers in the biological sciences and health science professionals mustbe discerning readers of the research literature. In particular, criticalevaluation of the construction of, and conclusions drawn from, statisticalstudies is indispensable. This requires

an understanding of probability theory, as the underpinning ofinferential statistics, andexposure to a variety of statistical methods used to establish confidenceand test hypotheses.


The Setting The Courses

Outline of Math 115 – Life Science Mathematics 1

Motivate the use of mathematical methods by constructing a series ofmodels for population dynamics based on difference equations.

Introduce differential calculus in order to describe continuous change.Include: differentiation of polynomial, exponential, logarthmic, andtrigonometric functions; graphing, optimization, and applications to growthand decay, and to periodic phenomena.

Develop discrete and continuous models of populations, infectious diseaseand predator-prey systems. Include analysis via cobwebbing and phase lines,and characterization of equilibrium stability with the derivative.

Introduce antidifferentiation, including the substitution method andintegration by parts, and pure-time differential equations.

Show how autonomous differential equations arise in modeling processesalready seen in population growth, epidemics and predator-prey systems.

Develop the definite integral, linked to antidifferentiation by the

Fundamental Theorem of Calculus, and include applications to area and

averaging.


The Setting The Courses

Outline of Math 116 – Life Science Mathematics 2

Examine systems of differential equations, particularly predator-prey systemsand the SIR model of infectious disease. Use slope fields and phase planediagrams to describe the qualitative behavior of solutions.

Use genetics and molecular evolution to motivate the introduction ofprobability theory and statistics.

The introduction to probability theory includes: discrete spaces,conditional probability, rare disease examples, binomial and geometricdistributions, random variables, expectation, continuous randomvariables, probability density functions, the normal distribution, and theCentral Limit Theorem.The introduction to statistical methods includes: descriptive andinferential statistics, with emphasis on hypothesis testing, samplingdistributions and analysis of variance.

Additional material includes probabilistic topics such as random walksapplied to diffusion and genetic drift, and statistical tools such as theχ2-test, contingency tables, goodness-of-fit, and nonparametric tests.


The Setting The Challenges

Diverse Student Backgrounds

The sequence is now the main mathematics requirement for the Biology major.

Consequently student math backgrounds and aptitude vary widely. We have

instituted a mandatory additional contact hour that will allow review of material,

question-and-answer, and more frequent testing. An available resource at Emory

College: supplemental instruction program based on peer tutoring.

Integration of Life Science Topics

Intriguing problems rooted in biological phenomena are key to establishing

student interest. Presentations from Biology faculty can be used for particular

topics [eg, modeling epidemics or molecular evolution]. Use of real experimental

data is an important aspect in statistical testing and experimental design.

Coordination of Several Sources

We have not found one text adequate for the current content of the sequence.

One provides the life science motiviation [6], another, the calculus basics [2]; web

materials, the DE topics [3]; and two texts, the probability and statistics [4, 6].

Care must be taken to maintain clarity of sources and our expectations.


The Setting The Challenges

Assessment

Anecdotal evidence from the Biology Department indicates that students takingMath 115/116 rather than standard calculus have a better grasp of the math andstatistics involved in studying biology. However, we have no systematicallyobtained evidence.

Goal: Formulate a specific and quantifiable set of expectations of the sequence

and design assessment tools in collaboration with the Center for Science

Education and the Biology Department.

Integration of Courses

There are strong indications that integrated math and science curriculasignificantly reinforce topics separately treated in different departmental offerings.

Goal: Cooperate with Biology, the CSE, and other science departments in

coordinating treatment of common topics, such as population dynamics,

molecular evolution, and genetics. Similarly, common methods in Math & CS and

science department courses, such as modeling and hypothesis testing, could be

mutually reinforcing.


Examples Discrete Models – Population Dynamics & Cobwebbing

Population Dynamics & Cobwebbing

On the left is the updating function for population change: pt+1 = pt + 1.8pt(1− 0.1pt).Cobwebbing shows convergence to an equilibrium. On the right is the data displaying asolution, that is, pt as a function of t.

Does the initial condition determine the long term behavior of the population? Can yousolve explicitly for pt as a function of t, given p0 and the updating function?


Examples Discrete Models – Epidemics & Difference Equations

Epidemics & Difference Equations

These are two examples of S − I − R models based on the difference equations:

St+1 = St − αSt It and It+1 = It + αSt It − γIt .

St (susceptibles) vs t in yellow; It (infecteds) vs t in green; Rt (removeds) vs t in blue.

Left: α = 0.001, γ = 0.07. Right: α = 0.001, γ = 0.5. Describe the differences in theoutcomes. Does the model describe observable, that is, biologically reasonable,behavior?


Examples Continuous Models – Predator-Prey & Differential Equations

Predator-Prey Systems & Differential Equations

Let p = p(t) and q = q(t) be two interacting populations with change described by:dp

dt= 0.4p(1− 0.1p)− 0.05pq ;

dq

dt= 0.8q + 0.08pq .

Left: Phase plane trajectory for p and q. Right: Solutions for p(t) and q(t). Is this apredator-prey system? If yes, which species is prey?


Examples Probability and Statistics – The CLT & Sample Means

The Central Limit Theorem & Sample Means

The CLT asserts that if sufficiently large samples are drawn from a population [in thiscase one with exponential distribution with parameter λ = 0.2 and, hence, mean µ = 5.0and variance σ2 = 25.0], the distribution of sample means is approximately normal.

Left: Sample size n = 3. Right: n = 30, with a very close fit to the normal curvepredicted by the CLT. Can you approximate the variances for each? Can you account forthe difference in variances?


References and Contacts References

References

1 BIO 2010 – Transforming Undergraduate Education for Future Research Biologists,National Research Council, The National Academies Press [Washington DC, 2003]– specifies mathematical, statistical and computational needs, see Appendix F

2 Calculus Lite (3rd), F. Morgan, AK Peters [Wellesley, MA, 2001] – provides thedifferential and integral calculus materials for Math 115 – Life ScienceMathematics 1

3 CCP Materials, L. Moore and D. Smith,www.math.duke.edu/education/modules2/materials – supports DE modelingtopics

4 Chance in Biology – Using Probability to Explore Nature, M. Denny and S. Gaines,Princeton University Press [ Princeton, 2000] – provides introductory material onprobability and the stochastic nature of nature

5 Math & Bio 2010, L. Steen [ed], Mathematical Association of America [2005] –essays on the interrelations among biology, computing and mathematics forundergraduates

6 Mathematical Models in Biology, E. Allman and J. Rhodes, Cambridge University

Press [Cambridge 2004] – the main source of life science topics throughout the

sequence


References and Contacts Contacts

Emory Contacts

Mathematics & Computer Science Department:

Department website:http://www.mathcs.emory.eduUndergraduate program director: [email protected] 115-116 course coordinator [2007-08]: Dwight Duffus –[email protected] and handout available at:http://www.mathcs.emory.edu/math4bio

Center for Science Education:

CSE website:http://www.cse.emory.edu


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