Why This Class? James K. Peterson. August 22, Department of Biological Sciences and Department of Mathematical Sciences Clemson University

Transcription

1 Why This Class? James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University August 22, 2013

2 Outline 1 Our Point of View Mathematics, Science and Computer Science Needed The Implementation 2 A Typical Syllabus

3 Abstract This lecture introduces the Calculus For Biologists course we are offering at Clemson University as MTHSC 111. The notes for this course have evolved from handwritten lectures to a printed book in its third revision with a fourth revision being released the summer of The text notes are printed using print on demand technologies and are available at This course was designed, implemented and added to the curriculum as the new mathematics course MTHSC 111 and replaced the previous requirement of second semester engineering calculus. It has been taught to about 700 students since 2006.

4 Rethinking The Presentation of Calculus We have developed this course to be an integration of biology, mathematics and computational tools for placement within a typical existing department of biological sciences. We have a traditional college entity at Clemson University having the following structure 80% or more tenured faculty having little interest in the use of the triad of mathematics, computation and science in their own course development and teaching. Incoming majors having uneven training in mathematics and computer science and they may even be biased against mathematics and computer science as part of the discipline of biology. Prior to this course, we have an existing mathematics requirement of two semesters of engineering based calculus but the material was not combined with interesting biology and so many students did not like it for that reason.

5 Our Basic Plan Here is the basic plan: We are replacing the first semester calculus course with a new starting calculus for biologists course in Fall It is an experiment that should work out fine and it currently has about 60 students in two sections., here MTHSC 106. We have eplaced the existing Second Semester Calculus Course which is called Math-Two-Engineering (here MTHSC 108) with the course you are taking which is specifically designed for Biology majors called Math-Two-Biology here MTHS 111 Add model based points of view to the first year fundamental biology sequence Biology-One and Biology-Two, here BIOL 110 and 111.

6 Our Point of View Rethinking The Presentation of Calculus Two We have the following point of view: Mathematics is subordinate to the biology: the course builds mathematical knowledge needed to study interesting nonlinear biological models.

7 Our Point of View Rethinking The Presentation of Calculus Two We have the following point of view: Mathematics is subordinate to the biology: the course builds mathematical knowledge needed to study interesting nonlinear biological models. More interesting biology requires more difficult mathematics and concomitant intellectual resources.

8 Our Point of View Rethinking The Presentation of Calculus Two We have the following point of view: Mathematics is subordinate to the biology: the course builds mathematical knowledge needed to study interesting nonlinear biological models. More interesting biology requires more difficult mathematics and concomitant intellectual resources. It is better to choose a few nonlinear models and discuss them very carefully; e.g., logistics, Predator - Prey and disease models.

9 Our Point of View Rethinking The Presentation of Calculus Continued Since graphical interfaces lose value with more variables, we explore how to obtain insight from a six variable linear Cancer model.

10 Our Point of View Rethinking The Presentation of Calculus Continued Since graphical interfaces lose value with more variables, we explore how to obtain insight from a six variable linear Cancer model. We always stress how we must abstract out of biological complexity the variables necessary to build models and how we can be wrong.

11 Our Point of View Rethinking The Presentation of Calculus Continued Since graphical interfaces lose value with more variables, we explore how to obtain insight from a six variable linear Cancer model. We always stress how we must abstract out of biological complexity the variables necessary to build models and how we can be wrong. We always stress that the BMC triad is a very useful paradigm for developing insight in science.

12 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses In outline, these are the topics we have chosen to use: From Calculus 2: Simple substitution. The Fundamental Theorem of Calculus. The Logarithm and Exponential Function Integration by Parts and Partial Fraction Decomposition Methods. Approximating functions by tangent lines and the error that is made.

13 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses In outline, these are the topics we have chosen to use: From Calculus 2: Simple substitution. The Fundamental Theorem of Calculus. The Logarithm and Exponential Function Integration by Parts and Partial Fraction Decomposition Methods. Approximating functions by tangent lines and the error that is made. From Linear Algebra: Here MTHSC 311 Vectors and matrices. Linear systems of equations. Determinants and what they mean. Eigenvalues and Eigenvectors for 2 2 matrices.

14 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Linear Ordinary Differential Equations: here MTHSC 208 Simple Rate Equations and their applications. The idea of an integrating factor. Protein Synthesis models. Building a Newton s Cooling model from data. The logistics model. Complex numbers and complex functions. Second order linear homogeneous equations. Systems of two linear differential homogeneous equations. Graphical analysis of systems of two linear differential homogeneous equations.

15 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Nonlinear Ordinary Differential Equations: here MTHSC 208 level material but not usually covered. The Predator - Prey model. Conservation law. Restriction of positive solutions. Periodicity of solutions. Graphical analysis. Usefulness in explaining some data.

16 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Nonlinear Ordinary Differential Equations: here MTHSC 208 level material but not usually covered. The Predator - Prey model. Conservation law. Restriction of positive solutions. Periodicity of solutions. Graphical analysis. Usefulness in explaining some data. The Predator - Prey model with self-interaction. Restriction of positive solutions. periodicity of solutions. Graphical analysis. Failure to explain data and what that means for modeling.

17 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued A disease model. Restriction of positive solutions. Graphical analysis. Usefulness in explaining epidemics.

18 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued A disease model. Restriction of positive solutions. Graphical analysis. Usefulness in explaining epidemics. Higher Dimensional Differential Equation Systems: here MTHSC 208 A six variable linear cancer model. Model used to find important relationships between factors that cause mutations in tumor suppressor genes.

19 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued A disease model. Restriction of positive solutions. Graphical analysis. Usefulness in explaining epidemics. Higher Dimensional Differential Equation Systems: here MTHSC 208 A six variable linear cancer model. Model used to find important relationships between factors that cause mutations in tumor suppressor genes. Asymptotic behavior not useful. Behavior over typical human lifetime is what is important.

20 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued A disease model. Restriction of positive solutions. Graphical analysis. Usefulness in explaining epidemics. Higher Dimensional Differential Equation Systems: here MTHSC 208 A six variable linear cancer model. Model used to find important relationships between factors that cause mutations in tumor suppressor genes. Asymptotic behavior not useful. Behavior over typical human lifetime is what is important. Graphical analysis not useful.

21 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued A disease model. Restriction of positive solutions. Graphical analysis. Usefulness in explaining epidemics. Higher Dimensional Differential Equation Systems: here MTHSC 208 A six variable linear cancer model. Model used to find important relationships between factors that cause mutations in tumor suppressor genes. Asymptotic behavior not useful. Behavior over typical human lifetime is what is important. Graphical analysis not useful. Numerical methods not useful.

22 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued A disease model. Restriction of positive solutions. Graphical analysis. Usefulness in explaining epidemics. Higher Dimensional Differential Equation Systems: here MTHSC 208 A six variable linear cancer model. Model used to find important relationships between factors that cause mutations in tumor suppressor genes. Asymptotic behavior not useful. Behavior over typical human lifetime is what is important. Graphical analysis not useful. Numerical methods not useful. Determining validity of the model is the most important thing. Errors made in approximations determine this.

23 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Numerical Methods for Differential Equations: here MTHSC 208 and MTHSC 360. We use MatLab, but we could easily switch to the open source octave if cost was a factor for the site license. Euler s Method and its error analysis.

24 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Numerical Methods for Differential Equations: here MTHSC 208 and MTHSC 360. We use MatLab, but we could easily switch to the open source octave if cost was a factor for the site license. Euler s Method and its error analysis. Using MatLab to solve first order differential equations with Euler s method using scripts.

25 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Numerical Methods for Differential Equations: here MTHSC 208 and MTHSC 360. We use MatLab, but we could easily switch to the open source octave if cost was a factor for the site license. Euler s Method and its error analysis. Using MatLab to solve first order differential equations with Euler s method using scripts. Learning how to plot results.

26 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Numerical Methods for Differential Equations: here MTHSC 208 and MTHSC 360. We use MatLab, but we could easily switch to the open source octave if cost was a factor for the site license. Euler s Method and its error analysis. Using MatLab to solve first order differential equations with Euler s method using scripts. Learning how to plot results. Discussion of Runga-Kutte methods and their use via MatLab scripts.

27 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued Numerical Methods for Differential Equations: here MTHSC 208 and MTHSC 360. We use MatLab, but we could easily switch to the open source octave if cost was a factor for the site license. Euler s Method and its error analysis. Using MatLab to solve first order differential equations with Euler s method using scripts. Learning how to plot results. Discussion of Runga-Kutte methods and their use via MatLab scripts. MatLab scripts for the numerical solution of systems of differential equations.

28 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued MatLab scripts for the numerical solution of Predator - Prey models.

29 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued MatLab scripts for the numerical solution of Predator - Prey models. MatLab scripts for the numerical solution of disease models.

30 Mathematics, Science and Computer Science Needed Combining Threads from Mathematics Courses Continued MatLab scripts for the numerical solution of Predator - Prey models. MatLab scripts for the numerical solution of disease models. MatLab scripts for plotting multiple phase plane portraits simultaneously.

31 The Implementation How Do We Do The Implementation? We interweave the calculus, differential equations, MatLab and numerical methods, linear algebra and modeling material carefully. Mathematics is introduced as needed for the modeling. The first part of the course must introduce specific mathematical material and they are not quite ready for models. So they find this more challenging.

32 The Implementation How Do We Do The Implementation? We interweave the calculus, differential equations, MatLab and numerical methods, linear algebra and modeling material carefully. Mathematics is introduced as needed for the modeling. The first part of the course must introduce specific mathematical material and they are not quite ready for models. So they find this more challenging. Graders help with the daily busywork.

33 The Implementation How Do We Do The Implementation? We interweave the calculus, differential equations, MatLab and numerical methods, linear algebra and modeling material carefully. Mathematics is introduced as needed for the modeling. The first part of the course must introduce specific mathematical material and they are not quite ready for models. So they find this more challenging. Graders help with the daily busywork. The lectures and textbook use many examples with daily comments and anecdotes on how this stuff relates to biology.

34 The Implementation How Do We Do 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 6 of each type builds mastery.

35 The Implementation How Do We Do 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 6 of each type builds mastery. We discuss the proper way to do homework: following notes for the first 2 problems is fine, but they need to close their books and notes and see if they can do the problem without looking at their notes.

36 The Implementation How Do We Do 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 6 of each type builds mastery. We discuss the proper way to do homework: following notes for the first 2 problems is fine, but they need to close their books and notes and see if they can do the problem without looking at their notes. We discuss how to study in general and the discipline needed to study for all their classes in college so that they avoid the feast and famine cycle.

37 The Implementation How Do We Do 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 6 of each type builds mastery. We discuss the proper way to do homework: following notes for the first 2 problems is fine, but they need to close their books and notes and see if they can do the problem without looking at their notes. We discuss how to study in general and the discipline needed to study for all their classes in college so that they avoid the feast and famine cycle. We provide study guides and sample exams.

38 The Implementation Our Four Year Quantitative Emphasis Idea To build a more cohesive plan for the development of these points of view within Biological Sciences, we would like to have students take this kind of course each year in their biology major. Our goals are to Add more mathematical tools that lead the students into modeling realms involving partial differential equations.

39 The Implementation Our Four Year Quantitative Emphasis Idea To build a more cohesive plan for the development of these points of view within Biological Sciences, we would like to have students take this kind of course each year in their biology major. Our goals are to Add more mathematical tools that lead the students into modeling realms involving partial differential equations. Discuss models which require abstraction of biological detail to gain insight.

40 The Implementation Our Four Year Quantitative Emphasis Idea To build a more cohesive plan for the development of these points of view within Biological Sciences, we would like to have students take this kind of course each year in their biology major. Our goals are to Add more mathematical tools that lead the students into modeling realms involving partial differential equations. Discuss models which require abstraction of biological detail to gain insight. Continue to develop computer modeling skills via an increased use of MatLab.

41 The Implementation Our Four Year Quantitative Emphasis Idea To build a more cohesive plan for the development of these points of view within Biological Sciences, we would like to have students take this kind of course each year in their biology major. Our goals are to Add more mathematical tools that lead the students into modeling realms involving partial differential equations. Discuss models which require abstraction of biological detail to gain insight. Continue to develop computer modeling skills via an increased use of MatLab. Always remember that the mathematical content is subservient to the biology.

42 The Implementation More Calculus for Biologists Design Philosophy Our design philosophy is as follows: All parts of the course must be integrated, so we don t want to do mathematics, science or computer approaches for their own intrinsic value.

43 The Implementation More Calculus for Biologists Design Philosophy Our design philosophy is as follows: All parts of the course must be integrated, so we don t want to do mathematics, science or computer approaches for their own intrinsic value. Models are carefully chosen to illustrate the basic idea that we know far too much detail about virtually any biologically based system we can think of. We must always remember that throwing away information allows for the possibility of mistakes. This is a hard lesson to learn, but important.

44 The Implementation More Calculus for Biologists Design Philosophy Our design philosophy is as follows: All parts of the course must be integrated, so we don t want to do mathematics, science or computer approaches for their own intrinsic value. Models are carefully chosen to illustrate the basic idea that we know far too much detail about virtually any biologically based system we can think of. We must always remember that throwing away information allows for the possibility of mistakes. This is a hard lesson to learn, but important. Models from population biology, genetics, cognitive dysfunction, regulatory gene circuits and many others are good examples to work with. All require massive amounts of abstraction and data pruning to get anywhere, but the illumination payoffs are potentially quite large.

45 The Implementation More Calculus For Biologists: We have decided to focus on these topics for the followup course. This is currently taught as MTHSC 450 Section 101 and has been offered to 1-3 students per semester for about 5 years. Analysis of predator-prey dynamics using both linear and nonlinear theory.

46 The Implementation More Calculus For Biologists: We have decided to focus on these topics for the followup course. This is currently taught as MTHSC 450 Section 101 and has been offered to 1-3 students per semester for about 5 years. Analysis of predator-prey dynamics using both linear and nonlinear theory. General ideas from nonlinear differential equation systems.

47 The Implementation More Calculus For Biologists: We have decided to focus on these topics for the followup course. This is currently taught as MTHSC 450 Section 101 and has been offered to 1-3 students per semester for about 5 years. Analysis of predator-prey dynamics using both linear and nonlinear theory. General ideas from nonlinear differential equation systems. General ideas for linear partial differential equations Separation of variable and Fourier series etc.

48 The Implementation More Calculus For Biologists: We have decided to focus on these topics for the followup course. This is currently taught as MTHSC 450 Section 101 and has been offered to 1-3 students per semester for about 5 years. Analysis of predator-prey dynamics using both linear and nonlinear theory. General ideas from nonlinear differential equation systems. General ideas for linear partial differential equations Separation of variable and Fourier series etc. The cable equation (used to study transmission of depolarization in neurons).

49 The Implementation More Calculus For Biologists: We have decided to focus on these topics for the followup course. This is currently taught as MTHSC 450 Section 101 and has been offered to 1-3 students per semester for about 5 years. Analysis of predator-prey dynamics using both linear and nonlinear theory. General ideas from nonlinear differential equation systems. General ideas for linear partial differential equations Separation of variable and Fourier series etc. The cable equation (used to study transmission of depolarization in neurons). Solution of the cable equation.

50 The Implementation More Calculus For Biologists: We have decided to focus on these topics for the followup course. This is currently taught as MTHSC 450 Section 101 and has been offered to 1-3 students per semester for about 5 years. Analysis of predator-prey dynamics using both linear and nonlinear theory. General ideas from nonlinear differential equation systems. General ideas for linear partial differential equations Separation of variable and Fourier series etc. The cable equation (used to study transmission of depolarization in neurons). Solution of the cable equation. The Hodgkin-Huxley models of action potentials.

51 The Implementation Why These Topics? Why these topics? The students have not yet seen partial derivatives as they are not taking the traditional Calculus 3 for engineers, so they need to be exposed to that right away.

52 The Implementation Why These Topics? Why these topics? The students have not yet seen partial derivatives as they are not taking the traditional Calculus 3 for engineers, so they need to be exposed to that right away. The analysis of Predator - Prey models (using graphical tools and conservation laws as done in the first course) can naturally be augmented by looking at the eigenvalues of the appropriate Jacobian matrix.

53 The Implementation Why These Topics? Why these topics? The students have not yet seen partial derivatives as they are not taking the traditional Calculus 3 for engineers, so they need to be exposed to that right away. The analysis of Predator - Prey models (using graphical tools and conservation laws as done in the first course) can naturally be augmented by looking at the eigenvalues of the appropriate Jacobian matrix. Deriving the cable equation from Biophysical principles ties together many ideas from mathematics and science.

54 The Implementation Why These Topics? Why these topics? The students have not yet seen partial derivatives as they are not taking the traditional Calculus 3 for engineers, so they need to be exposed to that right away. The analysis of Predator - Prey models (using graphical tools and conservation laws as done in the first course) can naturally be augmented by looking at the eigenvalues of the appropriate Jacobian matrix. Deriving the cable equation from Biophysical principles ties together many ideas from mathematics and science. The separation of variables technique for PDE exposes the student to many new interesting mathematical things which enable additional models to be attacked.

55 The Implementation Why These Topics? Why these topics? The students have not yet seen partial derivatives as they are not taking the traditional Calculus 3 for engineers, so they need to be exposed to that right away. The analysis of Predator - Prey models (using graphical tools and conservation laws as done in the first course) can naturally be augmented by looking at the eigenvalues of the appropriate Jacobian matrix. Deriving the cable equation from Biophysical principles ties together many ideas from mathematics and science. The separation of variables technique for PDE exposes the student to many new interesting mathematical things which enable additional models to be attacked. The cable equation has many applications and so is a good vehicle for further modeling.

56 A Typical Syllabus MTHSC 111 DATE Lecture 1 W August 21 Lecture 2 F August 23 Lecture 3 M August 26 Lecture 4 T August 27 Lecture 5 W August 28 Lecture 6 F August 30 Lecture 7 M September 2 Syllabus Part I TOPIC Riemann Sums More Riemann Sums Introduction to Matlab FTOC Subsitutions Reviewed CFToC and Substitutions More FToC, defining ln and e

57 A Typical Syllabus MTHSC 111 DATE Lecture 8 T September 3 Lecture 9 W September 4 Lecture 10 F September 6 Lecture 11 M September 9 Lecture 12 T September 10 Lecture 13 W September 11 Lecture 14 F September 13 Syllabus Part II TOPIC log and exp properties log and exp derivs, integrals Rate Differential Equations Continued Application: Gene Transcription Partial Fraction Decompositions Integration By Parts

58 A Typical Syllabus MTHSC 111 Syllabus Part II DATE TOPIC Lecture 15 M September 16 Continued: Lecture 16 W September 18 The Logistic Type Model T September 17 Exam 1 Lecture 16 W September 18 Lecture 17 F September 20 Continued I Lecture 18 M September 23 Complex numbers, vectors and matrices Lecture 19 T September 24 Taylor Polynomials Lecture 20 W September 25 Euler s Method Lecture 21 F September 27 Matlab for models

59 A Typical Syllabus MTHSC 111 DATE Lecture 22 M September 30 Lecture 23 T October 1 Lecture 24 W October 2 Lecture 25 F October 4 Lecture 26 M October 7 Lecture 27 T October 8 Lecture 28 W October 9 Lecture 29 F October 11 Syllabus Part III TOPIC Second Order Linear ODE Characteristic Equation Case: Real, distinct roots Case: complex roots: Matlab Matlab for second order models Linear Systems, Char. Eqn. Linear System IVPs Eigenvalues and Eigenvectors

60 A Typical Syllabus MTHSC 111 Syllabus Part IV DATE TOPIC M October 14 Fall Break T October 15 Fall Break Lecture 30 W October 16 Continued Lecture 31 F October 18 Nullclines Lecture 32 M October 21 Phase Plane Analysis Lecture 33 T October 22 Continued Lecture 34 W October 23 Runga-Kutte methods for model systems Lecture 35 F October 25 Continued Lecture 36 M October 28 Linear systems: external inputs T October 29 Exam 2 Lecture 37 W October 30 Proteins and Climate Models Lecture 38 F November 1 Predator Prey Models

61 A Typical Syllabus MTHSC 111 DATE Lecture 39 M November 4 Lecture 40 T November 5 Lecture 41 W November 6 Lecture 42 F November 8 Lecture 43 M November 11 Lecture 44 T November 12 Lecture 45 W November 13 Lecture 46 F November 15 Lecture 47 M November 18 Lecture 48 T November 19 Syllabus Part IV TOPIC PP nullclines PP NLCL Bounded Trajectories Trajectories are periodic Continued Adding Fishing Adding Self Interaction Assembling nullclines SIR Disease Models, nullclines SIR Phase Plane Analysis

62 A Typical Syllabus MTHSC 111 DATE Syllabus Part IV TOPIC Lecture 49 W November 20 The Cancer Model Lecture 50 F November 22 Top Pathway Exact Solutions Lecture 51 M November 25 Top Pathway approxs. T November 26 Exam 3 W November 27 Thanksgiving Break R November 28 Thanksgiving Break F November 29 Thanksgiving Break Lecture 52 M December 2 Continued Bottom Pathway Approx Lecture 53 T December 3 Bottom Pathway Approximations Lecture 53 W December 4 Which pathway is dominant? F December 6 Reading Day: No Class R December 12 11:30 am - 2:00 pm Final

63 A Typical Syllabus MTHSC 111 DATE Exams Value Exam 1: 9/ Points Exam 2: 10/ Points Exam 3: 11/ Points Final: 12/12 11:30 am - 2:00 pm 200 Points