Supplementary Materials for · 1 Supplementary Materials We describe briefly the material covered in the first half of the module, and, for the second half of the module, discuss

www.sciencemag.org/cgi/content/full/336/6084/993/DC1

Supplementary Materials for

Learning Biology by Recreating and Extending Mathematical Models

Hillel J. Chiel,* Jeffrey P. Gill, Jeffrey M. McManus, Kendrick M. Shaw

*To whom correspondence should be addressed. E-mail: [email protected]

Published 25 May 2012, Science 336, 993 (2012)

DOI: 10.1126/science.1214192

This PDF file includes:

Materials and Methods References Appendix I

1

Supplementary Materials We describe briefly the material covered in the first half of the module, and, for the second

half of the module, discuss how components of the term paper are submitted by the

students, including the content of the rubrics used to judge paper sections. We also

describe the course wiki that allows students to comment on each other’s term paper

sections and the weekly team evaluations that they submit. We provide more details of the

statistical analysis of the attitude questionnaire (Figure 2 of the main text) and describe the

content and the statistical analysis of the conceptual exam. We describe how students are

kept continuously informed of their progress in the course. Finally, we briefly discuss the

instructional support required for the course. More details about the course are available in

a peer‐reviewed paper published in 2010: http://www.lifescied.org/content/9/3/248.full .

1. Material covered in the first half of the module: See Appendix I for the weekly schedule

for the Spring 2012 semester, which can serve as a guide to the rate at which material is

covered. A link to the wiki that hosts the course materials, including the interactive

textbook, is https://biol300.case.edu/wiki/index.php/Main_Page . If the link is temporarily

not active, please contact the first author directly via email ([email protected]).

In the first half of the module, students work in teams as we introduce students to

key capabilities that they will need from Mathematica (Chapter 2 of the textbook): taking

notes and formatting text; creating notebooks that can organize data, equations, and text;

creating data lists and techniques for manipulating them; exporting and importing data;

plotting data; defining functions; solving equations; numerical integration; programming a

numerical integrator using Mathematica; loading supplementary functions; and creating

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dynamic simulations. Dynamic simulations are one of the most powerful capabilities of

Mathematica that allow students to visualize and manipulate mathematical models [see

http://demonstrations.wolfram.com/ ].

Students are then introduced to the process of modeling by developing a sequence

of increasingly complex models of bacterial growth, starting from a simple discrete‐time

recursive model, then an exponential model, and finally the logistic model and, at each

stage, comparing the model to actual biological data, which motivates the development of

the next, more complex model (Chapter 3 of the textbook). We conclude by pointing out

that this iterative modeling approach is the general process used by anyone who models

biological systems.

Because the students have developed the exponential and logistic models of growth

within a biological context, they understand these models well. In the next section of the

module, we use these two models to introduce fundamental concepts from one‐

dimensional nonlinear dynamical systems theory: state variables, initial conditions, and

parameters; trajectories of individual solutions and the overall flow defined by the

trajectories from all initial conditions; long‐term behavior captured by limit sets (in

particular, equilibrium points); bifurcations (qualitative changes in model behavior) as

parameters are varied; and a brief overview of the overall approach of nonlinear dynamical

systems theory (Chapter 4 of the textbook).

To introduce students to mechanistic biochemical models, we focus on a simple

model of the cell cycle (1), whose equations require students to understand zeroth, first,

and second order kinetics, and Michaelis–Menten enzyme kinetics. We then construct the

model of the cell cycle and allow students to manipulate it (Chapter 5 of the textbook).

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The model of the cell cycle required two state variables, and so students are now

introduced to two‐dimensional nonlinear dynamical systems theory, which readily

generalizes to higher dimensions. In addition to the concepts introduced above,

generalized to two dimensions, students learn about eigenvalues and eigenvectors, the

analysis of stability using a Jacobian matrix, bifurcations in two dimensions, and how to use

these tools to understand the switching between different equilibrium points as the cell

passes different checkpoints during the cell cycle (Chapter 6 of the textbook).

To complete the introduction to the basic tools and concepts of nonlinear dynamical

systems theory, students are then introduced to a two‐dimensional model of an excitable

cell, the Morris–Lecar model (2, 3). This provides the basis for understanding the

relationship between complex eigenvalues and oscillatory behavior. They are also

introduced to bifurcations (such as the Hopf bifurcation) that allow changes in parameter

values to induce switches between equilibrium points and stable oscillations (limit cycles;

Chapter 7 of the textbook). They are also given an exercise to construct the three‐

dimensional Lorenz attractor, and are thus able to explore some of the properties of a

chaotic attractor.

Finally, students are shown a case study of how to read a paper describing an agent‐

based model of schooling behavior in fish (4). They are shown how to analyze the paper’s

equations, how to create a plan for reconstructing a model, and then many of the detailed

steps necessary to reconstruct a working model (Chapter 11 of the textbook).

At the end of each chapter, students are posed problems that not only help them

master the material of each chapter, but steadily increase their ability to construct and

analyze complex models. Thus, by the end of the first half of the module, students are ready

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to tackle their own models. An answer key to the end‐of‐chapter problems is available from

the first author by request.

2. Term paper components and rubrics: During the second half of the module, students

continue to work in teams, and now focus on selecting a recently published paper, and then

using it as the basis for reconstructing a mathematical model of a biological system. (Please

see Appendix II, which provides the rubrics for selecting a published modeling paper to

reproduce, and the rubrics for evaluating components of the term paper, as well as the

criteria for assessing the final term paper). During the second half of the module, we

require students to submit drafts of components of the term paper that each student writes

individually. Each component of the term paper is accompanied by a rubric, that is, a set of

criteria that helps students understand and evaluate the components they submit. Students

must fill out the rubrics and submit them at the time that each term paper component is

submitted, and the instructors return comments to the students based on the criteria in the

rubric. If a component does not meet the criteria stated in the rubric, it is returned to the

student for revision, and late penalties are applied. Students who previously took the class

have given us permission to post exemplars of term paper sections and of entire term

papers, so current students can consult previous excellent examples.

Students submit a potential published paper whose model they propose to

reproduce. Criteria for the selection are whether the equations necessary to reconstruct

the model are present, whether values are given for parameters and initial conditions,

whether the student understands the mathematical formalisms of the model, whether the

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model is just a curve fit (which is not appropriate for the course), and whether there is a

clear comparison between model predictions and actual biological data.

Once the published paper is approved, the term paper proposal is submitted, and the

criteria for its acceptance are that the students describe a clear, testable hypothesis about a

biological system that the mathematical model can be used to test, that they briefly

describe the model that they propose to reconstruct, and they also briefly describe their

plan for reconstructing it.

After the middle of the module, students submit a draft of the Introduction, in which

they describe the significance of the problem, state the central hypothesis that will be

tested by the model, and provide a list of peer‐reviewed references in appropriate

bibliographic format.

A week later, they submit a draft of the Model Description, in which they describe the

state variables, parameters and inputs to the model, provide a term‐by‐term description of

the model components, discuss the assumptions underlying their model, and provide a

description of how they have simulated the model.

Two weeks later, they submit a draft of the Results, in which they describe outputs

from their model, compare them to the outputs of the published model, provide figures

with clear legends, describe how the model addresses the original biological hypothesis,

and describe any discrepancies between their reproduction and the original published

model.

A week later, they submit a draft of the Discussion, in which they address how well

the model supports the original hypothesis, compare their results with those in other

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published papers on the topic, discuss the limitations of their results, discuss discrepancies,

and place their results within the context of the larger scientific literature.

Finally, they submit the final term paper. In addition to the criteria used for each of

the individual sections, they must also describe the extensions to the model, including their

rationale for the extension, the details of how they altered the model, the results they

obtained, and a discussion of the implications of their results, and what they would do next.

In addition, the term paper is judged by how well the model was replicated, the level of

understanding the student demonstrates, the quality of their writing, the difficulty of

extension that was done, and how well it was done.

3. Peer comments on drafts using a wiki: In the current offering of the course, all

submissions must be made to the class wiki. The wiki is run under the open source Linux

operating system Ubuntu (11.10, http://www.ubuntu.com/) using the open source

MediaWiki code (http://www.mediawiki.org/wiki/MediaWiki), and the surveys are done

using the open source Django code (https://www.djangoproject.com/). Ubuntu is run

within the open source virtualizer VirtualBox (https://www.virtualbox.org/) using bridge

networking so that it generates its own unique Internet address. The resulting code is

readily exportable and could be run under Windows, Macintosh or Linux machines. More

details about the technical requirements for setting up and running the wiki are available

from one of the authors (Kendrick M. Shaw) on request.

At the time each submission for a term paper component is due, students must

create a page on the wiki and enter their submission there. Once a wiki page has been

created, other students can then use the Discussion page associated with the new wiki page

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to leave comments for the other students. Because we need to be able to assess each

student’s writing individually, we do not allow students to directly edit one another’s text;

but we strongly encourage students to use the comments that other students have made on

their Discussion page as the basis for editing, correcting and clarifying their own text. The

use of peer comments helps students see the extent to which they have been clear in their

own writing, and how to edit other people’s writing, which are both invaluable skills.

4. Weekly team evaluations: Because instructors circulate and talk to students throughout

the class period as they work on problems or reconstructing models, the instructors can

readily see if the students are working alone or with one another, and how well they are

interacting. To provide an additional evaluation of team function, we provide credit to the

students for filling out a weekly team evaluation survey outside of class (so that their

teammate does not see their evaluation). They rate how well they worked during the past

week, how well their teammate worked, and the overall function of the team on a scale

from 0 to 10. We emphasize that the results are confidential, but we carefully examine

them, and if we see consistently low ratings on any of the questions, we set up meetings

with individual team members, and then with the team as a whole, to see if we can help the

students resolve their problems. These interventions are often sufficient, and we have only

rarely had to reassign teammates.

5. Attitude questionnaire and statistical analysis: At the beginning, middle and end of the

module, students are asked to fill out an attitude questionnaire consisting of the following

six questions:

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(1) How much do you like biology?

(2) How competent are you in biology?

(3) How much do you like mathematics?

(4) How competent are you in mathematics?

(5) How much do you like programming?

(6) How competent are you in programming?

Students may rate themselves on a scale from 0 to 10. Data shown in the main essay

were pooled from answers provided by students in the spring semesters of 2010 and 2011.

The names of students who submitted the data were not revealed to the instructors until

after grades were submitted for each term.

Since the rating scale is ordinal, nonparametric statistics were used to evaluate

differences between majors at the beginning of the semester. Comparing the initial answers

for biology majors to engineering majors (i.e., majors in biomedical or chemical

engineering) using a multivariate Mann Whitney test showed that the patterns of answers

from the two groups were very significantly different (p < 0.00001, N = 25 (biology

majors), 29 (engineering majors)). Applying the nonparametric Kruskal Wallis test to each

sub‐question, and then applying the Bonferroni criterion (i.e., dividing the level of

significance, 0.05, by the 6 comparisons made) showed that biology majors provided very

significantly more positive answers to the questions about biology attitude and

competence (p << 0.00001). Engineers showed a strong trend to report a more positive

attitude towards and sense of competence in computer programming than biology majors

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at the beginning of the semester. These data support the hypothesis that the two

populations significantly differ from one another.

Using students as their own controls, it was possible to compare ratings before and

after the semester for each question, again applying the Kruskal Wallis test to each sub‐

question, and applying the Bonferroni criterion to the level of significance. Engineers

showed a statistically significant improvement in their sense of competence in biology and

in computer programming (p < 0.005, p < 0.006 respectively, N = 29). Biology majors

showed a strong trend towards improvement in their sense of competence in

programming. It is very likely that the lack of significant improvement in the attitudes or

sense of competence in biology on the part of biology majors is a ceiling effect, since

biology majors began the semester rating these both very highly.

6. Conceptual exam and statistical analysis: We wished to assess the “working knowledge”

that students had about concepts that are essential for success in constructing and

analyzing mathematical models of biological systems. Thus, we were not interested in

giving students a highly detailed and complex set of questions, but rather presented them

with questions that could be answered essentially “by inspection” if they knew the relevant

material, but which they would otherwise have little or no idea how to answer.

For example, if shown a plot of a function, could students identify the correct

function from a list of possible functions (e.g., linear, quadratic, exponential, square root)?

Shown a plot of a function, could students identify the correct differential equation whose

solution was that function (e.g., a constant function, a linear function, a growing

exponential function, or a decaying exponential function)? Shown two differential

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equations, could students correctly identify the state variables and distinguish them from

the parameters of the equations? Shown a one‐dimensional differential equation, could

students determine the values of the equilibrium points of the equation? Could they

determine the location and stability of equilibrium points of a one‐dimensional dynamical

system from a plot of the state variable versus the right hand side of the differential

equation? Could they correctly identify the ions that primarily cross the membrane during

the upswing of the action potential in the squid giant axon? For each question, one option

was “Don’t Know”. Students received +1 point for a correct answer, ‐1 point for an

incorrect answer, and 0 points for a “Don’t Know” answer. A total of 17 questions were

asked, so that scores could have ranged from ‐17 (all incorrect answers) to +17 (all correct

answers).

Students filled out the concept exam on the web at the beginning, middle and end of

the module. They were given credit for filling it out. They were not shown the correct

answers. Instructors could examine the answers, and know the major of each student, but

the names were blinded (i.e., answers were not associated with a specific name) until after

grades had been assigned at the end of the semester, so that the score on the conceptual

exam had no influence on the students’ grade. Students were informed of this before taking

the conceptual exam.

Using students as their own controls, it was possible to assess changes in the

number of correct answers from the beginning to the end of the semester. Data analyzed

from the spring semester of 2011 (N = 27 students, which included biology, engineering,

and other majors) was normally distributed, so a paired t‐test was used to analyze the data.

11

The change in scores was highly significant (+4.2 points ± 2.7, p << .00001), suggesting that

the course greatly improved students “working knowledge”.

Appendix III provides the complete set of questions for the Concept Exam, and the

educational objectives that are probed by the questions. Answers are available from the

first author on request.

7. Continuous update on class performance: Students earn points for attendance in class, for

demonstrating understanding of the conceptual content of the benchmark problems based

on their discussions with instructors, for filling out the teamwork, conceptual and attitude

surveys, for submitting components of the term paper as well as the final term paper, and

for commenting on other students’ drafts of term paper components. They may earn extra

credit points by suggesting improvements to the book chapters, by giving an optional

presentation to the whole class, and by providing thoughtful and substantive comments on

other students’ term papers a day before the final term paper is due. Thus, after each class,

it is possible for instructors to update the number of points each student has earned so far

in the semester. To make this transparent to the students, we not only allow them to log in

and see the number of points they have earned out of the total number possible up to that

point, but we also visually convey their progress using a schematic “racetrack” [Figure S1].

The three “runners” shown above the student represent a student in the middle of the A

range (green runner), a student in the middle of the B range (yellow runner), and a student

in the middle of the C range (red runner). The student is represented below the three

“model” runners as a black runner, and so it is immediately clear to the student how well he

or she is currently doing relative to these idealized students.

12

8. Instructional Support: The module is a laboratory, and thus requires as much

instructional support as any other laboratory. To be effective, laboratories need an

instructor to student ratio of about 1 to 12 (1 to 10 or 8 is even better, but is not always

feasible given limitations of personnel). In addition to the primary instructors for the

course who should have some expertise in both biology and modeling, teaching assistants

for the course may be graduate students who have some skill in modeling, or senior

undergraduate students (ideally, those who have previously taken the course, and are

eager to learn how to teach it).

The course is also similar to other laboratory courses in that out‐of‐class time is

necessary for proper preparation. During the both halves of the module, it is valuable to

schedule office hours so that students who need more one‐on‐one instruction to master the

material may be given extra time. It is very important that instructors not “tell the students

what to do”, but give them immediately helpful suggestions that help get them unstuck, and

more general suggestions that help guide their progress through problems, model

reconstruction, and model extensions.

When students begin to select potential published papers for model reconstruction,

it is important that the instructors split the papers among them, and review them to see if

they actually describe a model that can be reconstructed (following the guidelines in the

rubric shown in Appendix II). The instructors should spend additional time before the

second half of the module begins reconstructing components of the models that have been

selected by the students. This often reveals errors in the papers that could completely

frustrate student progress – errors in equations, missing or incorrect parameters, missing

13

initial conditions, or errors in analysis – and students need to be guided to overcome these

problems (or, in some cases, students need to be advised to select a different paper

entirely).

For the continuous progress model to be effective, it is also important for

instructors to spend time after class discussing the progress of each team and each student,

so that if interventions must be made, they can be made early and effectively. Finally,

during the second half of the module, instructors also need to schedule time to read drafts

of the different components of the term paper, as well as student comments on these

components, and provide brief but helpful feedback on how they can be improved. We have

found it very effective to rotate the reading of different components of the paper among

instructors and teaching assistants to ensure that students in the course are assessed

uniformly. We take time to discuss drafts that do not meet the criteria of the relevant

rubric, to ensure that assignment of points is equitable, and that we can provide feedback

to students that will help them bring the section up to the necessary standards.

A rough estimate of two hours of preparation time for each hour of class time is

reasonable, and is typical of the preparation time necessary for other laboratory courses.

Figure S1 – A visual representation of student progress during the semester. The green,

yellow and red “runners” show the points earned in the middle of the A, B, and C ranges,

respectively. The black “runner” shows the current points earned by the student at this

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point in the semester. In the example shown, the student is currently in the high B range,

but with additional effort might still be able to earn an A for his or her term grade.

References

1. Tyson, J. J. and Novak, B. “Regulation of the eukaryotic cell cycle: Molecular antagonism,

hysteresis, and irreversible transitions”, Journal of Theoretical Biology 210:249 – 263,

2001.

2. Morris, C. and Lecar, H. “Voltage oscillations in the barnacle giant muscle fiber”,

Biophysical Journal 35: 193 – 213, 1981.

3. ] Rinzel, J. and Ermentrout, B. “Analysis of neural excitability and oscillations”, in

Methods in Neuronal Modeling, 2nd edition, Koch, C. and Segev, I., eds., The MIT Press,

Cambridge, MA, 1998.

4. Inada, Y. and Kawachi, K., “Order and flexibility in the motion of fish schools”, Journal of

Theoretical Biology 214:371‐387, 2002.

15

Appendix I: Sample Course Syllabus (Spring 2012) January 17, 2012:

Introduction to the Course January 19, 2012:

Mathematica is installed and working on your computer. Hands‐on Start to Mathematica:

http://www.wolfram.com/broadcast/screencasts/handsonstart/ Completely read Chapter 1 before class. Read Chapter 2 through section 2.6 before class.

Begin working through problems in Chapter 2 through section 2.6 Fill out Attitudes Questionnaire. Fill out Conceptual Test.

January 24, 2012:

Read Chapter 2 through section 2.10 before class. Complete exercises through section 2.10, and end of chapter problem 1.

January 26, 2012:

Finish reading all of Chapter 2 before class. Finish all exercises and end of chapter problems. Fill out Team Evaluation after class before 10PM.

January 31, 2012:

Finish reading Chapter 3 through section 3.3 before class. Finish end of chapter problem 1.

February 2, 2012:

Finish reading Chapter 3 through to the end before class. Finish end of chapter problems 2 and 3. Fill out Team Evaluation after class before 10 PM.

February 7, 2012:

Finish reading Chapter 3 through to the end before class. Finish end of chapter problems 4 and 5.

February 9, 2012:

Finish reading Chapter 4 through section 4.4 before class. Finish end of chapter problems 1 and 2. Fill out Team Evaluation after class before 10 PM.

February 14, 2012:


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February 16, 2012: Finish reading Chapter 4 through end of chapter before class. Finish end of chapter problem 4. Fill out Team Evaluation after class before 10 PM.

February 21, 2012:


February 23, 2012:

Finish reading Chapter 5 through end of chapter before class. Finish end of chapter problem 2. Fill out Team Evaluation after class before 10 PM.

February 28, 2012:

Finish reading Chapter 6 through section 6.4 before class. Finish end of chapter problem 1 through section b.

March 1, 2012:

Finish reading Chapter 6 through end of chapter before class. Finish end of chapter problems 1 and 2. Fill out Team Evaluation after class before 10 PM. Select a published model to reproduce by 1 PM before class. Fill out Selected Published Paper Rubric.

March 6, 2012:

Finish reading Chapter 7 through section 7.8 before class. Finish end of chapter problem 1 through section e. Correctly answer all questions in the hypothesis quiz by 1 PM before class. Submit Term paper proposal by 1 PM before class. Fill out Term Paper Proposal Rubric.

March 8, 2012:

Finish reading Chapter 7 through end of chapter before class. Finish end of chapter problems 1 and 2. Fill out Attitudes Questionnaire. Fill out Conceptual Test. Fill out Team Evaluation after class before 10 PM. Provide comments on other students' Term Paper Proposals by 1 PM before class.

March 13, 2012: Spring Break March 15, 2012: Spring Break March 20, 2012:

Finish reading Chapter 11 through end of chapter before class.

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Final deadline for providing helpful extra credit comments on chapters of the textbook.

Clear verbal descriptions of equations and model. Plan and time table for model reconstruction. Fill out Model Plan Rubric by 1 PM before class.

March 22, 2012:

Replicate simple components of model (e.g., single uncoupled equations). Fill out Team Evaluation after class before 10 PM.

March 27, 2012:

Replicate simple components of model. Fill out Term Paper Introduction Rubric. Submit Benchmark I: Introduction by 1 PM before class.

March 29, 2012:

Construction of full model. Fill out Team Evaluation after class by 10 PM.

April 3, 2012:

Construction of full model. Fill out Term Paper Model Description Rubric. Submit Benchmark II: Model Description by 1 PM before class. Comments on other students' Introduction section due by 1 PM before class.

April 5, 2012:

Construction of full model. Fill out Team Evaluation after class by 10 PM.

April 10, 2012:

Model validation: replicate figures or other results from original paper. April 12, 2012:

Model validation: replicate figures or other results from original paper. Fill out Team Evaluation after class by 10 PM.

April 17, 2012:

Model extensions: modify model to test additional hypotheses. Fill out Term Paper Results Rubric. Submit Benchmark III: Results by 1 PM before class. Comments on other students' Model Description section due by 1 PM before class.

April 19, 2012:

Model extensions. Fill out Team Evaluation after class by 10 PM.

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April 24, 2012: Model extensions. Fill out Term Paper Discussion Rubric. Submit Benchmark IV: Discussion by 1 PM before class. Comments on other students' Results sections due by 1 PM before class.

April 26, 2012:

Student Presentations Fill out Final Attitudes Questionnaire. Fill out Conceptual Test. Fill out Team Evaluation. Comments on other students' Discussion sections due by 1 PM (extra credit). Certification that course evaluation form has been filled out (extra credit).

April 29, 2012

Comments on other students' term papers due by 5 PM (extra credit). April 30, 2012:

Submit final Term paper by 5 PM. Submit Final Term Paper Rubric.

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Appendix II: Term Paper Rubrics For all rubrics, choices are radio buttons labeled from 0 (worst) to 10 (best). Selecting a Published Model to Reproduce Equations present? None present (0)

All present (10) Parameters given? None given (0)

All given (10) Initial conditions specified? Not specified or deducible (0) All are specified or deducible (10) Model size? State variable are too many (0) State variables are manageable (10) Math complexity? Do not understand (0) Do understand (10) Only a model fit? Closed form equation fits data (0) Must run simulation (10) Experimentally testable biological predictions?

No testable predictions (0) Testable predictions (10)

_________________________________________________________________________________________________ Term paper proposal rubric Hypothesis proposed?

Missing or vague (0) Specific and falsifiable (10)

Hypothesis verbally highlighted (e.g., "We hypothesize that...")?

Not verbally highlighted (0) Verbally highlighted (10)

Hypothesis highlighted in yellow?

Not highlighted (0) Highlighted (10)

Described number of state variables?

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Insufficient detail (0) Sufficient detail (10)

Described number of parameters?


Described key assumptions?


Key steps to recreate model components?


Tests of model components?


Results from paper that will be reproduced?


Proposed model extensions?


_________________________________________________________________________________________________ Model Plan Rubric Quality of Hypothesis

Vague (0) Clear and readily testable (10)

State variables and parameters

None described (0) All described in detail (10)

What needs to be calculated?

Vague (0) Clearly specified (10)

Realistic intermediate goals

None specified (0) Fully Specified (10)

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Timetable None (0)

Detailed and specific (10) _________________________________________________________________________________________________ Benchmark I: Introduction Significance of problem?

Not explained (0) Well explained (10)

Statement of hypothesis?

No hypothesis (0) Clear and testable hypothesis (10)

List of references?

No references (0) Excellent list of references (10)

Properly formatted references?

No references (0) Fully and correctly formatted (10)

_________________________________________________________________________________________________ Benchmark II: Model Description State variables, parameters and inputs to the model clearly distinguished?

Vague, unclear or incorrect (0) Clear and correct (10)

Term‐by‐term description of model components?

Vague, unclear or incorrect (0) Clear and correct (10)

Have you described model assumptions?

None described (0) All carefully described (10)

Description of equation simulation?

No description (0) Clear and complete description (10)

_________________________________________________________________________________________________ Benchmark III: Results Results described and compared to original paper?

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No results described (0) Clear and complete description (10)

Have you addressed the original biological hypothesis?

No reference to hypothesis (0) Clear connection to hypothesis (10)

Figures and legends to show results?

No figures or legends (0) Complete figures and legends (10)

Discrepancies relative to original model?

No discussion of discrepancies (0) Clear discussion (10)

_________________________________________________________________________________________________ Benchmark IV: Discussion How well does the model support the original hypothesis?

No discussion (0) Clear and complete discussion (10)

Support for hypothesis and assumptions from other data in the literature? No discussion (0) Clear and complete discussion (10) Limitations of results?


Discrepancies and how they affect conclusions?


Relationship to other work in the field?


_________________________________________________________________________________________________ Final Term Paper Rubric How well was the model replicated?

No replication (0) Full replication (10)

Based on the term paper, how well did you understand the material?

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Poor understanding (0) Excellent understanding (10)

How well written is the term paper? Poor writing (0) Excellent writing (10)

How hard was the model extension that you did? No extension (0) Difficult extension (10) How good was the extension? Poor (0) Excellent (10) _________________________________________________________________________________________________ Examples of easier extensions: Changing inputs to a model Exploring some changes in parameters Examples of harder extensions: A detailed bifurcation analysis of the model Mathematical analysis of the model Changing the model form (e.g., adding new terms) In general, an A quality term paper is well‐written, shows an excellent understanding of the material, has a scholarly introduction, clear exposition of the model, excellent results that replicate key figures from the original model and successfully undertakes and describes a challenging extension to the model, and concludes with a scholarly discussion of the strengths and limitations of the model, the results, the extension, and its implications for further study of the biological system. Students are informed at the beginning of the course that if they do not create an extension to the model, they are unlikely to receive a grade of A either in the term paper or in the course.

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Appendix III: Concept Exam Conceptual Exam Question 1

Choose the function that will generate this graph over the range shown: a. f(x) = x2 b. f(x) = x c. f(x) = ex d. f(x) = x e. Don't know ____________________________________________________________________________________ Question 2

Choose the function that will generate this graph over the range shown:

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a. f(x) = x2 b. f(x) = x c. f(x) = ex d. f(x) = x e. Don't know ___________________________________________________________________________________ Question 3

Choose the function that will generate this graph over the range shown: a. f(x) = x2 b. f(x) = x c. f(x) = ex d. f(x) = x e. Don't know __________________________________________________________________________________ Question 4

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Choose the function that will generate this graph over the range shown: a. f(x) = x2 b. f(x) = x c. f(x) = ex d. f(x) = x e. Don't know ___________________________________________________________________________________ Question 5

Choose the differential equation whose solution is shown by the graph above: a. f ' [t] = 0; initial condition f[0] = 1. b. f ' [t] = 3; initial condition f[0] = 1. c. f ' [t] = -f[t]; initial condition f[0] = 1. d. f ' [t] = f[t]; initial condition f[0] = 1. e. Don't know ___________________________________________________________________________________ Question 6

Choose the differential equation whose solution is shown by the graph above: a. f ' [t] = 0; initial condition f[0] = 1.

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b. f ' [t] = 3; initial condition f[0] = 1. c. f ' [t] = -f[t]; initial condition f[0] = 1. d. f ' [t] = f[t]; initial condition f[0] = 1. e. Don't know ___________________________________________________________________________________ Question 7

Choose the differential equation whose solution is shown by the graph above: a. f ' [t] = 0; initial condition f[0] = 1. b. f ' [t] = 3; initial condition f[0] = 1. c. f ' [t] = -f[t]; initial condition f[0] = 1. d. f ' [t] = f[t]; initial condition f[0] = 1. e. Don't know ___________________________________________________________________________________ Question 8

Choose the differential equation whose solution is shown by the graph above: a. f ' [t] = 0; initial condition f[0] = 1. b. f ' [t] = 3; initial condition f[0] = 1. c. f ' [t] = -f[t]; initial condition f[0] = 1. d. f ' [t] = f[t]; initial condition f[0] = 1.

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e. Don't know ___________________________________________________________________________________ Question 9 Consider the following two differential equations:�� (1) x''[t] - α x'[t] + β x[t] y[t] - γ = 0�� (2) y''[t] - δ y'[t] + ε x[t] - η = 0�� The state variables for these two equations are (a) y� (b) x� (c) α and β� (d) γ, δ, ε, and η (e) Answers (a) and (b)� (f) Answers (c) and (d)� (g) Don't know ___________________________________________________________________________________ Question 10 Again consider the same two differential equations: (1) x''[t] - α x'[t] + β x[t] y[t] - γ = 0�� (2) y''[t] - δ y'[t] + ε x[t] - η = 0�� The parameters for these two equations are (a) y� (b) x� (c) α and β� (d) γ, δ, ε, and η (e) Answers (a) and (b)� (f) Answers (c) and (d)� (g) Don't know ___________________________________________________________________________________ Question 11 Consider the following differential equation: x'[t] = x[t]2 – 4 The equilibrium points for this equation are (a) 2

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(b) -2 (c) 0 (d) -1 (e) 1 (f) Answers (a) and (b) (g) Answers (d) and (e) (h) Don't know ___________________________________________________________________________________ Question 12 The right hand side of a differential equation is x[t]3 - 7x[t]2 + 2x[t] + 7. Consider the plot below of the state variable versus the right hand side:

From the graph, as one goes from -2 to +8 along the x-axis, it is clear that there are (a) No equilibrium points (b) Two equilibrium points, one stable and the second unstable (c) Two equilibrium points, one unstable and the second stable (d) Three equilibrium points, one stable, the second unstable, and the third unstable (e) Three equilibrium points, one unstable, one stable, and one unstable (f) Don't know ___________________________________________________________________________________ Question 13 As you vary a model parameter, you note that two equilibrium points approach one another and then disappear. You have just witnessed a (a) Hopf bifurcation (b) Transcritical bifurcation (c) Saddle-node bifurcation (d) Pitchfork bifurcation (e) Don't know ___________________________________________________________________________________

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Question 14 You have analyzed the eigenvalues of an equilibrium point, and have found that they are {-3.2, +7.1, -4.1}. You are looking at (a) A one-dimensional equilibrium point that is stable (b) A two-dimensional equilibrium point that is unstable (c) A two-dimensional equilibrium point that is a saddle (d) A three-dimensional equilibrium point that is a saddle (e) A three-dimensional equilibrium point that is unstable (f) Don't know ___________________________________________________________________________________ Question 15 You have analyzed the eigenvalues of an equilibrium point, and have found that they are {0.5 + 0.3 I, 0.5 - 0.3 I}, where I = −1 . You are looking at (a) A one-dimensional equilibrium point that is half stable (b) A two-dimensional equilibrium point that is an unstable spiral (c) A two-dimensional equilibrium point that is a stable node (d) A three-dimensional equilibrium point that is an unstable spiral (e) A three-dimensional equilibrium point that is a stable node (f) Don't know ___________________________________________________________________________________ Question 16 You have inoculated a flask with bacteria, and find after several hours that the number is far greater than the initial number you started with, but that the population size is no longer changing. You are looking at (a) The lag phase (b) The log phase (c) The stationary phase (d) The death phase (e) Don't know ___________________________________________________________________________________ Question 17 You are studying the squid giant axon, and using radioactive isotopes to trace the movement of ions across the membrane. During the upswing of the action potential, you primarily observe (a) Influx of calcium ions (b) Influx of potassium ions (c) Efflux of sodium ions (d) Influx of sodium ions

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(e) Efflux of potassium ions (f) Don't know ___________________________________________________________________________________ Educational objectives: (1) Recognition of mathematical functions (Questions 1 – 4) (2) Recognition of simple solutions of differential equations (Questions 5 – 8) (3) Distinguishing state variables and parameters (Questions 9 – 10) (4) Ability to apply basic concepts from nonlinear dynamics systems theory – equilibrium, stability, bifurcations (Questions 11 – 15) (5) Recall of basic concepts from microbiology, which were presented in Chapter 3 (Question 16) (6) Recall of basic concepts from neurophysiology, which were presented in Chapter 7 (Question 17)

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Supplementary Materials for · 1 Supplementary Materials We describe briefly the material covered in the first half of the module, and, for the second half of the module, discuss