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Starting Calculus for Biologists

MTHS 1060-Biology: A Freshman BasedApproach to the Integration of Mathematics,Science and Computation Within a Biological

Science DepartmentIntroductory Remarks

James K. Peterson

Department of Biological Sciences and Department of Mathematical SciencesClemson University

August 16, 2013

Outline

1 A Calculus Course For BiologistsOur Point of ViewMathematics, Science and Computer Science NeededThe Implementation

2 The Syllabus

Abstract

This lecture introduces the “Starting Calculus For Biologists:Derivatives, Integration and Modeling” course we are offeringat Clemson University. The notes for this course are printed usingprint on demand technologies and are available atwww.lulu.com/spotlight/gneuralgnome. The biology majorrequires a single first semester calculus course and one semester ofstatistics at the present time, but we feel that a biology major atthis time of change in the sciences should be exposed to acombination of science, mathematics and the use of computationaltools to solve problems that can not be solved by hand. We willstart you on this journey in this course, but, naturally, we wouldlike to encourage you to learn more.

There is a followup course called “Calculus Two ForBiologists: A Beginning Getting Ready For Models andTheir Analysis” you can take which will give you moreconcepts. This is called MTHS1110 and you could take itright after this one.

There is then another course called “Calculus Three ForBiologists: Partial Differential Equation Models” whichcontinues with more advanced material also. It is currentlytaught as MTHS450-Section 101 which introduces you topartial differential equation models and builds a nice model ofan excitable neuron.

If you are interested in these other course, just ask!

There is a followup course called “Calculus Two ForBiologists: A Beginning Getting Ready For Models andTheir Analysis” you can take which will give you moreconcepts. This is called MTHS1110 and you could take itright after this one.

There is then another course called “Calculus Three ForBiologists: Partial Differential Equation Models” whichcontinues with more advanced material also. It is currentlytaught as MTHS450-Section 101 which introduces you topartial differential equation models and builds a nice model ofan excitable neuron.

If you are interested in these other course, just ask!

A Calculus Course For Biologists

Rethinking The Presentation of Calculus

This course is an integration of biology, mathematics andcomputational tools and is designed to help students be preparedfor work in modern biological sciences areas. For convenience, letBMC denote this interdisciplinary triad. We have a traditionalcollege entity at Clemson University having the following structure

80% or more tenured faculty having little interest in the use ofthe triad of mathematics, computation and science in theirown course development and teaching.Incoming majors having uneven training in mathematics andcomputer science and may even be biased againstmathematics and computer science as part of the discipline ofbiology.Prior to this course, we had an existing mathematicsrequirement of one semester of engineering based calculus butthe material was not combined with interesting biology and somany students did not like it for that reason.

Our Point of View

We have the following point of view:

Mathematics is subordinate to the biology: the course buildsmathematical knowledge needed to study interesting biologicalmodels.

Start slow with models that use algebra only and introducecalculus ideas as tools that can aid us to understand ourmodels better.

More interesting biology requires more difficult mathematicsand concomitant intellectual resources.

We can easily build models and fully understand them eventhough we can’t solve them by hand – so we must start usingcomputational techniques.

Our Point of View

Mathematics, Science and Computer Science Needed

Combining Threads from Mathematics Courses

In outline, these are the topics we have chosen to use:

We ease you into the use of algebra by studying a simpleevolutionary model called Viability Selection.

At this point we introduce you to using Matlab as ourcomputational tool.We next introduce you to the ideas of limits. We show youright up front the limits which lead to the idea of continuity,differentiation and integration using as much biology as wecan. These are the basic ideas we will flesh out throughoutthe course.With the introductions done, we work through the details forcontinuity and differentiation and even talk about the greatsin and cos functions!Then we add the inverse of taking a derivative – theantiderivative.Substitution ideas for finding antiderivatives.

We ease you into the use of algebra by studying a simpleevolutionary model called Viability Selection.At this point we introduce you to using Matlab as ourcomputational tool.

We next introduce you to the ideas of limits. We show youright up front the limits which lead to the idea of continuity,differentiation and integration using as much biology as wecan. These are the basic ideas we will flesh out throughoutthe course.With the introductions done, we work through the details forcontinuity and differentiation and even talk about the greatsin and cos functions!Then we add the inverse of taking a derivative – theantiderivative.Substitution ideas for finding antiderivatives.

We ease you into the use of algebra by studying a simpleevolutionary model called Viability Selection.At this point we introduce you to using Matlab as ourcomputational tool.We next introduce you to the ideas of limits. We show youright up front the limits which lead to the idea of continuity,differentiation and integration using as much biology as wecan. These are the basic ideas we will flesh out throughoutthe course.

With the introductions done, we work through the details forcontinuity and differentiation and even talk about the greatsin and cos functions!Then we add the inverse of taking a derivative – theantiderivative.Substitution ideas for finding antiderivatives.

We ease you into the use of algebra by studying a simpleevolutionary model called Viability Selection.At this point we introduce you to using Matlab as ourcomputational tool.We next introduce you to the ideas of limits. We show youright up front the limits which lead to the idea of continuity,differentiation and integration using as much biology as wecan. These are the basic ideas we will flesh out throughoutthe course.With the introductions done, we work through the details forcontinuity and differentiation and even talk about the greatsin and cos functions!

Then we add the inverse of taking a derivative – theantiderivative.Substitution ideas for finding antiderivatives.

We ease you into the use of algebra by studying a simpleevolutionary model called Viability Selection.At this point we introduce you to using Matlab as ourcomputational tool.We next introduce you to the ideas of limits. We show youright up front the limits which lead to the idea of continuity,differentiation and integration using as much biology as wecan. These are the basic ideas we will flesh out throughoutthe course.With the introductions done, we work through the details forcontinuity and differentiation and even talk about the greatsin and cos functions!Then we add the inverse of taking a derivative – theantiderivative.

Substitution ideas for finding antiderivatives.

We ease you into the use of algebra by studying a simpleevolutionary model called Viability Selection.At this point we introduce you to using Matlab as ourcomputational tool.We next introduce you to the ideas of limits. We show youright up front the limits which lead to the idea of continuity,differentiation and integration using as much biology as wecan. These are the basic ideas we will flesh out throughoutthe course.With the introductions done, we work through the details forcontinuity and differentiation and even talk about the greatsin and cos functions!Then we add the inverse of taking a derivative – theantiderivative.Substitution ideas for finding antiderivatives.

Combining Threads from Mathematics CoursesContinued

Riemann integration and the Fundamental Theorem ofCalculus. With this under our belts, we can start solvingmodels.

The natural logarithm and exponential function. We use theFundamental Theorem of Calculus to set these up.

Now we can do simple rate models – our first differentialequation model.

Next, we can study a simple model of protein transcription.

Our first nonlinear model is the logistics model.

We then learn how to approximate functions using calculus:tangent lines and all that.

The idea of the minimum and maximum of a function andhow we might find them.

How to solve our models using Matlab using Euler’s method.

A more advanced Protein transcription model – we can solvethese better with Euler’s method. We add in a bit morebiology here such as binding time estimates, bound fractionsand so forth.

Many models need to have more than one independentvariable and so we will need tools to handle that. We startwith matrices and vectors.

We then stop and explore a portion of a tumor suppressorgene model of cancer. There are three variables here and wecan use our ideas of approximation and simple rate models alltogether.

We then introduce how to draw surfaces using Matlab.

Partial derivatives and tangent planes.

The chain rule for partial derivatives.

The error we make with a tangent plane approximation to afunction of two variables.

A nice derivation of how to setup a regression line.

We finish with a discussion of Hamilton’s rule which is amodel of how altruistic genes might become prevalent in apopulation. Most of this discussion using only algebra butwith a lot of thinking behind it!

The important parameter in Hamilton’s rule is r which iscalled the coefficient of relatedness and it is hard to explain.We offer several different ways to understand it ranging fromthe slope of a regression line to a term in a chain ruleexpansion.

The Implementation

How Do We Do The Implementation?

We interweave the calculus, differential equation models, MatLaband numerical methods carefully.

Mathematics is introduced as needed for the modeling.

Graders help with the daily busywork.

The lectures and textbook use many examples with dailycomments and anecdotes on how this stuff relates to biology.

The Implementation

The freshman and sophomore students need to learn how to study.

Daily homework with lots of algebra and manipulation is used.Doing 1 or 2 problems encourages mimicry, while doing 5 − 6of each type builds mastery.

We discuss the proper way to do homework: following notesfor the first 2 problems is fine, but they need to close theirbooks and notes and see if they can do the problem withoutlooking at their notes.

We discuss how to study in general and the discipline neededto study for all their classes in college so that they avoid thefeast and famine cycle.

We provide study guides and sample exams.

The Implementation

The Followup Mathematics Course

Our goals are to

Add more mathematical tools that lead the students intomodeling realms involving one and two variables.

Discuss models which require abstraction of biological detailto gain insight.

Develop computer modeling skills via the use of MatLab.

Always remember that the mathematical content issubservient to the biology.

All parts of the course must be integrated, so we don’t wantto do mathematics, science or computer approaches for theirown intrinsic value. Experts in these separate fields must workhard to avoid this. This is the time to be generalists andalways look for connective approaches.

The Implementation

Our goals are to

The Implementation

Our goals are to

The Implementation

Our goals are to

The Implementation

Our goals are to

The Implementation

More Calculus for Biologists Design Philosophy

Our design philosophy is as follows:

Models are carefully chosen to illustrate the basic idea that weknow far too much detail about virtually any biologicallybased system we can think of.

We must learn to throw away information in the search of theappropriate abstraction. The resulting ideas can then bephrased in terms of mathematics and simulated or solved withcomputer based tools. However, the results are not useful, andmust be discarded and the model changed, if the predictionsand illuminating insights we gain from the model are incorrect.

The Implementation

Models are carefully chosen to illustrate the basic idea that weknow far too much detail about virtually any biologicallybased system we can think of.

We must learn to throw away information in the search of theappropriate abstraction. The resulting ideas can then bephrased in terms of mathematics and simulated or solved withcomputer based tools. However, the results are not useful, andmust be discarded and the model changed, if the predictionsand illuminating insights we gain from the model are incorrect.

The Implementation

We must always remember that throwing away informationallows for the possibility of mistakes. This is a hard lesson tolearn, but important.

Models from population biology, genetics, cognitivedysfunction, regulatory gene circuits and many others aregood examples to work with. All require massive amounts ofabstraction and data pruning to get anywhere, but theillumination payoffs are potentially quite large.

The Implementation

We must always remember that throwing away informationallows for the possibility of mistakes. This is a hard lesson tolearn, but important.

Models from population biology, genetics, cognitivedysfunction, regulatory gene circuits and many others aregood examples to work with. All require massive amounts ofabstraction and data pruning to get anywhere, but theillumination payoffs are potentially quite large.

The Syllabus

Homework Policies

Complete all homework assignments on time. Late homeworkis not accepted without my ok. Suitable excuses includeillness, university excused absence and family emergencies, butyou must check with me before you hand in anything in thissituation.

Each homework exercise should be clearly stated before yougive your solution even if that means you copy a lengthystatement.

Staple your homework.

Do not use the back of the paper.

The Syllabus

MTHSC 106 Biology Syllabus Part 1

DATE Topic

Lecture 1 W August 21 Algebra and Viability SelectionLecture 2 F August 23 Viability update equationsLecture 3 M August 26 Introducing MatLab for graphsLecture 4 T August 27 Limiting behavior: roots and polysLecture 5 W August 28 Limiting behavior: continuity

Limiting behavior: derivativesLecture 6 F August 30 Limiting behavior: integrationLecture 7 M September 2 ContinuityLecture 8 T September 3 Derivatives and the Product RuleLecture 9 W September 4 The Quotient and Chain rulesLecture 10 F September 6 Sin and cos stuff

The Syllabus

DATE Topic

Lecture 11 M September 9 AntiderivativesLecture 12 T September 10 Trig Antiderivatives and Substitution ILecture 13 W September 11 Substitution IILecture 14 F September 13 Riemann sumsLecture 15 M September 16 Riemann integrals and FToC

T September 17 Exam 1Lecture 16 W September 18 FToC again, CFToCLecture 17 F September 20 Substitution IIILecture 18 M September 23 ln and expLecture 19 T September 24 ln and exp Derivatives

ln and exp AntiderivativesLecture 20 W September 25 ln properties

The Syllabus

DATE Topic

Lecture 21 F September 27 exp propertiesLecture 22 M September 30 Simple Rate modelsLecture 23 T October 1 Half lives, doubling timesLecture 24 W October 2 Protein SynthesisLecture 25 F October 4 Signalling scenariosLecture 26 M October 7 Transcription Error ILecture 27 T October 8 Transcription Error IILecture 28 W October 9 Tangent Lines: func approxLecture 29 F October 11 Rolle’s Theorem

The Mean Value TheoremM October 14 Fall BreakT October 15 Fall Break

Lecture 30 W October 16 Quadratic approximation

The Syllabus

DATE Topic

Lecture 31 F October 18 Differences in ExponentialsLecture 32 M October 21 ExtremalsLecture 33 T October 22 Cooling ModelsLecture 34 W October 23 Cancer Model ILecture 35 F October 25 Cancer Model IILecture 36 M October 28 Euler’s Method I

T October 29 Exam 2Lecture 37 W October 30 Euler’s Method IILecture 38 F November 1 Logistic modelsLecture 39 M November 4 Advanced protein models ILecture 40 T November 5 Advanced protein models II

The Syllabus

DATE Topic

Lecture 41 W November 6 Runga-Kutte methodsLecture 42 F November 8 Matrices, Vectors etcLecture 43 M November 11 Drawing SurfacesLecture 44 T November 12 Partial DerivativesLecture 45 W November 13 Tangent PlanesLecture 46 F November 15 Second order partialsLecture 47 M November 18 Tangent Plane errorLecture 48 T November 19 ExtremalsLecture 49 W November 20 Regression linesLecture 50 F November 22 Shared Common Good models

The Syllabus

DATE Topic

Lecture 51 M November 25 Hamilton’s ruleT November 26 Exam 3W November 27 Thanksgiving BreakR November 28 Thanksgiving BreakF November 29 Thanksgiving Break

Lecture 52 M December 2 Gene SurvivalLecture 53 T December 3 Additive Fitness

Additive GeneticsLecture 54 W December 4 An optimization approach

F December 6 Reading Day: No ClassDecember 9 - 13 Finals

The Syllabus

MTHSC 106-Biology Exams

DATE Value

Exam 1: September 17 100 PointsExam 2: October 29 100 PointsExam 3: November 26 100 PointsFinal: Sometime in December 9 - December 13 200 Points

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