Using the Scientific Method to Motivate Biology Students to Study Precalculus

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Using the Scientific Method toMotivate Biology Students toStudy PrecalculusJames P. Fulton & Linda SabatinoPublished online: 22 Jan 2008.

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PRIMUS, XVIII(1): 5–21, 2008Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970701745007

Using the Scientific Method to Motivate BiologyStudents to Study Precalculus

James P. Fulton and Linda Sabatino

Abstract: During the last two years we have developed a precalculus coursecustomized around biology by using the scientific method as a framework to engageand motivate biology students. Historically, the precalculus and calculus coursesrequired for the Suffolk County Community College biology curriculum were designedusing examples from the physical sciences with little direct relation to biology. Thebiology-related mathematics in the curriculum was limited to that presented in biologycourses. Favorable results from the biology department spreadsheet and web pageLICIL project to increase math awareness led us to our current NSF CCLI projectto develop an alternative approach to teaching mathematics to biology students. Weselected the scientific method because it is taught at the beginning of every intro-ductory biology course and is also the first fundamental level at which mathematicsand biology are connected.

In this article, we describe the scientific method approach to introducing precal-culus, the sequence of topics covered in precalculus and the introductory biologycourse, and the challenges and successes of implementing this new course wheremathematics topics are presented from the standpoint of a scientific experiment.Examples of the computer lab component of the course where Excel is used for dataorganization and analysis are provided along with how the scientific method can beused in precalculus to make predictive models that have direct relevance to the biologycurriculum.

Keywords: Precalculus, biology, scientific method, excel.

INTRODUCTION

From the mathematician’s perspective, the primary goal of a precalculuscourse is to lay the foundation for the study of functions, which is needed in the

Address correspondence to James P. Fulton, Suffolk County Community College,Mathematics Department, 533 College Road, Selden, NY 11784, USA. E-mail:[email protected]

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6 Fulton and Sabatino

subsequent series of calculus courses. The precalculus simply bridges the gapbetween algebra and calculus, by providing useful concepts and techniquesfor calculus. As mathematicians, we teach the material in a hierarchicalfashion, and introduce topics in an order and sequence more conducive formathematical expediency, rather than in ways that might engage and motivatethe students. Furthermore, just as calculus has it roots in the physical sciences,the precalculus course also follows this lead and approaches most topicsfrom a physical sciences perspective. The examples and motivating problemspredominantly come from the physical sciences, which often leave biologystudents wondering why they are studying this material, since there appearsto be little or no connection to biology.

A fresh perspective on the relation of mathematics and biology has begunto emerge in both the mathematics and biology communities, and this canand should impact the teaching of mathematics to biologists. Some instructorshave adapted to this new perspective and have reorganized their courses toinclude some life science examples or modules to show the relevance ofmathematics to biology, yet they still follow the same pedagogical approachthey always used. Instead of fitting the biology into the mathematics viaunconnected examples, we believe that the mathematics can be customizedaround biology. A major premise of this article is that just as calculus wascreated around and motivated by physics, so too can the study of precalculus,and for that matter calculus, be built around the study of biology. Rather thancreating specialized “math-bio” modules, or various biological examples touse in introducing the different mathematical concepts, we adopt a differentapproach. We want to introduce the mathematics to motivate and enhancethe biology and not the other way around. The precalculus will be presentedas an integral and necessary part of biology, just as it is for physics. Themathematics needs to be shown to be a necessary consequence of studyingand understanding biology.

BACKGROUND

There is a growing need to increase the mathematics level of biology students[7, 3], at a time when the mathematics educational level of all US studentsgraduating from high school is on the decline.

“In the fourth grade, our schoolchildren performed quite well on thepaper-and-pencil test in science; they were outperformed by only onecountry and were above the international average in mathematics. Yetthe eighth-grade U.S. students fell substantially behind their interna-tional peers. These students performed below the international averagein mathematics and just above the average in the written scienceachievement tests.” [9]

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Using the Scientific Method to Motivate 7

This decline continues unabated through the 12th grade. More recently, theAmerican College Testing (ACT) Organization did a survey in which theylooked at the:

“Differences between what high schools are teaching and what collegeswant incoming students to know

• In mathematics, high school teachers tend to give advanced contentgreater importance than do college instructors. College instructorsrate a rigorous understanding of mathematics fundamentals as beingmore important than brief exposure to advanced content.

• In science, high school teachers consistently rate knowledge ofcontent (specific facts and information) as more important thanan understanding of science process and inquiry skills. Collegeinstructors, in contrast, rate these skills in the opposite way-scienceprocess skills are more important for students to possess when theyenter college, they say, than knowledge of specific content.” [4, 5]

Add to this the large disconnect between science and mathematics coursesand you have a pretty good picture of what is happening. While biologystudents are now required to have a substantial preparation in mathematics,the educational system is actually making them less prepared. This problem isoften magnified at the community college level, which has a greater diversityin students’ skill levels.

Our college, Suffolk County Community College (SCCC), is the largestcommunity college in the State University of New York (SUNY) system,serving approximately 22,000 students on three campuses. Less than 1%of our students have a high school average of 90% or better with themajority (50%) having averages ranging from 70 – 79. In addition, 50 – 60%of entering students must take remedial courses in Mathematics (Arithmeticand Algebra I) and/or English. The typical student is under-prepared inmathematics with little or no algebraic manipulation skills and has littlemotivation towards learning mathematics. Many have not been successful intheir mathematics courses in the past, and fail to see its relevance to theirfuture.

Biology majors at SCCC are required to finish two semesters of calculus.Most of the students that start the program are not prepared to start calculusand a number must begin by taking an intermediate algebra class beforethey can go on to take the two required precalculus courses leading to thecalculus. This means that biology majors may have to take up to 20 credits ofmathematics starting from Algebra up to Calculus II. As traditionally taught,none of these mathematics courses have any direct relation to their biologycourses. There are not even appropriate biology-related examples presented in

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the classes. Students are ultimately left with the impression that mathematicsis not important for a biologist.

Under a previous NSF grant [6], one of the authors tried to rectify thisproblem and enhance the mathematics received by biology majors, as part ofa Long Island Consortium of Integrate Learning (LICIL) (an NSF sponsoredinitiative) project, an “Interdisciplinary Approach to Increase Math Awarenessin Biology by Publishing Data on the Web.” Under this project, facultyfrom the SCCC biology department developed spreadsheet and web pagetechnology to increase mathematics awareness in their classes. In addition tousing Excel for data analysis of class projects, faculty published data on theWeb to be used by other sections of the same course, other disciplines andcolleges as applications of mathematics in biology.

Results of the LICIL project indicated that using Excel for rapid statisticalanalysis and conversion of data collected from biology class projects to graphsenhanced learning for students in introductory biology courses. Excel wasused for analysis of data from class exercises, such as Effect of Temperatureand pH on Enzymatic Function, Cellular Respiration, Population Genetics,Plant Competition, Predator Prey Relationships, and Effect of Exercise onHeart Rate and Cardiac Output. By posting the results on the Web, the datacould be used by all of the students in multiple section courses for statisticalanalysis, or by biology faculty to create simulations and develop “what if ”analyses and to provide mathematics faculty with access to real life examples.Students were exposed to the quantitative analysis that they were not gettingin their requisite mathematics courses.

As part of an NSF grant obtained jointly with Farmingdale StateUniversity [1], a new methodology to teach mathematics to biologists wasdeveloped. To show the necessary connection between mathematics andbiology, we organized the precalculus course around the scientific method,and included a heavy dose of biological applications, and mathematics labora-tories in biology.

APPROACH

Biologists establish which general principles are true amongst all thepossible alternatives by systematically testing various scenarios to see whichhypothesis or pattern best represents the outcome of the phenomenon underinvestigation. The traditional scientific method is roughly comprised of fourdistinct phases:

1. Characterization Stage: Observe and describe a phenomenon or group ofphenomena.

2. Hypothesis Stage: Formulate a hypothesis to explain the phenomena.[causal mechanism or mathematical relation (model)]

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3. Prediction Stage: Use the hypothesis to predict the existence or occur-rence of the phenomena, or to quantitatively predict the results of newobservations

4. Experiment/Validate Stage: Perform experiments that either validate ordisprove the hypothesis.

Other more flexible variations of the scientific method have beenproposed [2]; however, in this article we follow the more traditionaldefinition.

We organized the precalculus around the first three phases of the scientificmethod. The Characterization stage begins with measurements, and sincemeasurements involve numerical data, our study begins with numbers. Thetopics include scientific notation, as well as an introduction to units and unitsystems. A thorough understanding of real numbers, although critical fora formal study of calculus, is not pursued. Understanding the infinite andinfinitesimal at this level was considered to be beyond the scope of what wewanted to accomplish. Although these two concepts exist mathematically, wehave no evidence of their existence otherwise, and invariably either idea leadsto deep philosophical arguments or to total student confusion at this level.

We next move on to working with formulae. These are the “proven”models (laws) obtained from past investigations of other scientists. Studentsneed to be able to work with them and understand how to evaluate andmanipulate them. We begin to classify these formulae, informally at first, aseither linear, polynomial, rational, exponential, or logarithmic, (trigonometriccases are treated later, but could also be introduced if appropriate models arefound). At this point we do not make the graphical connection. Wheneverpossible, we work with formulae and calculations that are tied to the biologicalsubject currently being covered in the biology course. In Table 1, we showsome common formulae used by biologists that we present and discuss at thispoint in the course.

We also introduce properties useful for manipulating the formulae andsolving for other variables. The concepts of dependent and independentvariables are introduced and discussed from the standpoint of a scientificexperiment. Also, we discuss the various laws/models with examples. Thegoal is to get students familiar and comfortable with the algebraic represen-tation and manipulation of such functions, without introducing them as such.Exponentials and logarithms are connected via “undoing” operations froma purely algebraic manipulation perspective. The emphasis is on biologicalexamples and asking for different results/predictions of the results.

We move on to make the graphical connection as part of the Hypothesisstage. This is where we emphasize that we are trying to find and fit a patternto. We start with collected data that follow a linear or nearly linear pattern,as this is the easiest model to introduce and create. We also talk aboutmeasurements and errors in measurements and plot these points on a Cartesian

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Table 1. Some Common Formulae in Biology

Phenomenon/Law Formula

pH pH= – log[H+]Half life t1/2 = In�2�

r ≈ 0�693r

Hardy-Weinberg Law (p+q)2 = p2 +2pq+q2 =1p+q = 1

Michaelis-Menton Equation(Enzyme Kinetics)

V = VMax�S�Km + �S�

MacArthur-Wilson Species-Arealaw (allometry)

S = cAz

General scaling law Y = Y0Mb

Haldane Function (Geneticmapping)

CAB = 12

(1− e−2XAB

)

grid. We reintroduce the notion of dependent and independent variables andgo from these discrete data to trying to come up with our own “continuous”formula/model/law that shows how our data are related. We find the “best”fitting line by hand and then show how to enter data and graph the pointsusing Excel.

An excellent example for introducing these concepts is the standard curve(see Appendix), which is a common quantitative research tool in biology.It is a method of plotting known data to determine the concentration ofan unknown substance, particularly proteins and DNA. The analysis is firstperformed with various known concentrations of a substance similar to thatbeing measured. A new standard curve is constructed each time a new seriesof tests is set up, since one must account for variations in experimentalconditions (solution preparation, spectrophotometer sensitivity, temperature,etc.) The particular analysis procedure used may measure absorbance, opticaldensity, luminescence, fluorescence, radioactivity, Rf (retention factor) orsomething else.

The data from known samples is then used to make a standard curveby plotting the concentration on the y-axis and assay measurement on thex-axis. If the data exhibits a linear relationship, the standard curve will be astraight line. The same analysis is then performed with samples of unknownconcentration. To analyze the data, one locates the measurement on the x-axisthat corresponds to the assay measurement of the unknown substance andfollows a line to intersect the standard curve. The corresponding value onthe y-axis is the concentration of substance in the unknown sample. Again,the focus is on using biological examples and, if possible, data from thestudents’ biology labs to introduce and reinforce the mathematical concepts.This exercise can also lead to a good discussion of dependent and independent

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variables. In this example, there is no clear distinction from a mathematicalperspective, but scientifically we usually choose the unknown quantity as thedependent variable.

At this point we can also show how Excel has a built-in feature thatproduces a linear model and we compare this result to that which we did byhand. We then introduce the idea of linear regression as the “best” approachto model this data. A brief discussion follows, as to why this is the bestapproach in regards to “minimizing the error.” The discussion is intuitiverather than rigorous.

Next we introduce the notion of relations and functions and how toidentify each. We stress that, as scientists, it is our goal to begin to identifyand organize these functional patterns and from this we are lead to the variousproperties of these different functions. We make the connection that differenttypes of functions can have different types of graphs and similar functionshave similar graphs. This leads to the study of the various “families offunctions,” as given by both their algebraic and graphical representations. Wealso study the connection between the various functional families and changesin their graphs. This is more of an investigative approach, ideally suited forthe computer lab component of the course. The abstract concept of functionalnotation and how it is useful in describing simple changes in patterns such as atranslation, shift, or inversion is then introduced. We then formally distinguishthe families of functions and identify and list their properties.

We try to emphasize that since these patterns occur in nature weneed to be proficient in seeing, understanding, and working with them.Functions are presented as models that characterize unique relationshipsbetween measurable phenomena. Again the emphasis is on using examplesfrom biology to introduce and motivate the mathematics.

A discussion of the different approaches for coming up with a function,model, or law follows. We show that a model can be obtained either by fittingdata to a particular function type (regression: linear and non-linear) or byarguing from first principles based upon the known behavior of the data. HereExcel provides many useful tools. At this point the emphasis is not on fullyunderstanding regression, but only on an intuitive idea of what is happeningand why it works.

Once we have these functional models, we now use them to make predic-tions. This frequently involves the process of finding zeros or regions ofincreasing, decreasing, or unchanging trends. The focus is on seeing if ourhypothetical model makes reasonable predictions. We emphasize that thiswould often require an appropriate experiment to validate our model. We thendiscuss this in terms of the scientific method. This is where we develop thenecessary analysis tools and techniques in the precalculus for analyzing andvalidating our model.

Two different models are created via the scientific method: one basedupon measured data and the other based upon first principles. This is where we

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introduce difference equations and how they can be used to make predictivemodels. We study populations using this approach and show how the graphof the exponential function seems to arise out of a simple difference equation.This provides a good introduction to and comparison between arithmetic andgeometric growth.

The study of difference equations works well at this level to introducenew patterns and analysis techniques. Advancing to a study of more thanone population, as in predator-prey problems, can give students exposure tomany new and important concepts such as limit cycles, limit points, andeven chaos if desired. Care must be taken here, as the mathematical maturityand sophistication level of the students may not be enough to do more thanjust illustrate what is happening and how the mathematics reproduces thesepatterns in nature. More importantly, however, is how potential models canbe constructed from first principles as opposed to fitting measured data. Itis through both approaches that a better understanding of the underlyingphenomena is achieved. A summary of the scientific method approach tointroducing precalculus along with the associated biology topics is shown inTable 2 below.

STUDENT ANALYSIS/FEEDBACK

In addition to developing the course we also analyzed the impact of thisapproach on both the student retention rates and final grades [8]. In the twosections of the course that ran during different semesters, the retention rate was100%. In the traditional precalculus course the retention rate is consistently80-82.5%, where the retention rate is simply defined as the number of studentsthat complete the course. There was no significant improvement in the finalstudent grades, but this metric could have been biased by the greater retentionof marginal students.

A class survey was performed and analyzed by the SCCC Office ofInstitutional Research and compared against other mathematics courses at thecollege. Relevant statements and responses are provided in Table 3 (Note: aresponse of 1 means a student strongly disagrees, and a 5 means a studentstrongly agrees with the statement.)

The above results clearly show a measurable improvement in how thestudents reacted to the different approach. In addition to the numerical rating,the students were also able to provide a written response to the followingquestion: Aside from the quality of the instruction, what is particularly goodor bad about the course? Sample student comments are as follows: “verygood preparation for the (biology) class following this one,” “it is goodbecause it applies to my major,” “ I enjoyed the computer involvement,” “Iam in my third attempt at this course and I am finally passing,” “this courseis different, in that it allows for further understanding of material and not

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Table 2. Summary of precalculus course material

Modern Biology I Precalculus Math Labs

Tools & Techniques Characterize Scientific Method, Measurement Systems, Formulas in BiologyBiomolecules I Excel, Standard Curves (Linear Regression) Standard CurveBiomolecules II Hypothesize Graphs, Functions & ModelsBiomolecules III Families of Functions & Transformations (Polynomials,

Rationals, Power, Exponential, Logarithmic, Trigonometric)Enzyme Activity Families of Functions & Transformations (Polynomials,

Rationals, Power, Exponential, Logarithmic, Trigonometric)Enzyme

Cell Structure &Function

Families of Functions & Transformations (Polynomials,Rationals, Power, Exponential, Logarithmic, Trigonometric)

Photosynthesis Finding Patterns: Data Fitting Using Nonlinear Regression Photo-synthesisCellular Respiration Finding Patterns: First Principles & Difference EquationsCell Division Finding Patterns: First Principles & Difference EquationsGenetics I Finding Patterns: First Principles & Difference Equations Difference

EquationGenetics II DNA Predict Working with models (functions). Interpreting graphs, solving

for zeros, looking for trends, simplifying expressions,operations on functions.

Genetics III Expression& Tech

Working with models (functions). Interpreting graphs, solvingfor zeros, looking for trends, simplifying expressions,operations on functions.

Evolution Working with models (functions). Interpreting graphs, solvingfor zeros, looking for trends, simplifying expressions,operations on functions.13

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Table 3. Comparative responses to class survey

Traditional New

Statement RatingStandardDeviation Rating

StandardDeviation

My knowledge of thesubject increased.

4.027 0.558 4.64 0.50

This class met myexpectations.

4.024 0.544 4.55 0.69

This course helped meto become a betterstudent.

3.872 0.651 4.55 0.69

just a memorization of formulas,” “I really don’t like math, but I do likescience. Bringing them together allows me to learn and enjoy the class,”“The course has everything for math and biology.” Overall these commentsshow that the students are clearly responding more favorably to the newapproach. In addition, they are beginning to see the relevance of mathematicsto biology, and this has helped to motivate them to remain and succeedin class.

ISSUES

In creating and implementing this new course, a few issues surfaced that needto be addressed before the course can be fully successful. Some of the issuesare logistical, while others required innovation in course material as well asclose cooperation between biology and mathematics faculty if the course wasto be successful. Here are the major issues we found in trying to implementthis new course:

1. Only a small percentage of students taking Modern Biology are biologymajors. (Many are health and pre-med majors) The original intent wasto focus on biology majors, but this was broadened to include all healthscience majors, since the pool of students was so small.

2. Data posting and analysis using Excel are not an integral part of all biologycourses.

3. The prerequisite for Modern Biology (in fact, all laboratory sciencesclasses) is only Beginning Algebra (non-college level) while the prereq-uisite for Precalculus is Intermediate Algebra. The prerequisite for bothshould be the same, or there should be a corequisite of precalculus forModern Biology.

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4. More time for the Mathematics labs would be helpful, as this will enablestudents to more fully understand what they are doing.

5. Better coordination between the Precalculus course and the ModernBiology course is needed. We need to correlate the precalculus topics withthe particular biology topic. Also more interactions between faculty fromthe two departments would be extremely helpful.

6. The order, type, and depth of each precalculus topic, needs furtherattention. At times the material did not flow as well as we would have liked.

7. Most students seem to be passive learners. They do not fully participatein their learning. They simply believe that by sitting in class they will“get” whatever they need. This means that new concepts do not go overwell, since concepts require an actively engaged mind to comprehend.This was especially apparent when we were working with transformationsof functions.

CONCLUSIONS AND FUTURE DIRECTIONS

The students responded better to the new approach to precalculus, asevidenced by their course evaluation responses. The statistics show that theyare more likely to complete the course and their written responses show thatthey are beginning to appreciate the connections between mathematics andbiology and are actually starting to enjoy it. The scientific method has provento be an excellent unifying platform in which to introduce and “tie-together”precalculus and biology. The students can finally see how mathematics appliesto biology, and the accompanying computer labs serve to further reinforcethis investigative approach.

Using Excel in the first semester of a sequence course enables a moresophisticated use of the technology in subsequent courses and also showsa direct correlation to their mathematics course. Excel is also a great toolfor analyzing patterns in data. It makes some of the laborious and tedioustasks more dynamic and compelling, since results can be presented quicklyand compactly.

It is important for biology majors to see connections with mathematicsin their sequence of biology courses. Thus, in addition to having biologyexamples/applications in mathematics courses, biology courses need a moreactive approach utilizing the Scientific Method in the laboratories to providecontinuity of ideas, as well as a direct link to what is being taught in precal-culus. At minimum, biology students need to know how to properly graphdata from their experiments (e.g., Effect of Temp on Enzymatic Functions infirst semester courses such as Anatomy & Physiology I); as they continue on,they have to learn how to analyze these graphs as well as more complex data.For example, when blood vessel diameter and blood flow (cardiac output)is addressed in a mathematics module, biology faculty should bring this

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information into their classes allowing students to predict “what-ifs” basedon the model. Similar scenarios can result from plant competition/nutritionexperiments, too. Biologists need to become proficient at unveiling thequantitative patterns that may be buried in their data, and mathematics isthe key.

Although this is just the “first” course in a biology majors’ mathematicssequence, we recommend a complete mathematics curriculum, including bothstatistics and calculus, following this approach. A strategy where both themathematics and biology departments work together to create this shouldbe encouraged, and joint professional development workshops would be anessential tool in making this happen.

Overall the course was very successful, but what is now needed is a groupof faculty from both departments to revise and coordinate the entire sequenceof topics in all the courses. The courses need to reflect this interdisciplinaryapproach to stress that mathematics is an integral component of every biologycurriculum.

REFERENCES

1. Gordon, Winn, Sabatino, and Fulton. 2004. NSF DUE Award #0442160,Developing a Mathematics Curriculum to Serve the Biosciences: The FirstStep. Washington, DC: NSF.

2. Hillborn, R. and M. Mangel. 1997. The Ecological Detective. ConfrontingModels with Data. Princeton, N. J.: Princeton University Press.

3. National Research Council. 2003. BIO2010: Transforming Under-graduate Education for Future Research Biologists. Washington, DC:NRC.

4. Olson, L. 2007. High School, College Standards Out of Sync, SurveyFinds. Education Week.

5. Olson, L. 2007. Views Differ on Defining College Prep: How to gaugereadiness sparks vexing questions. Education Week.

6. Sabatino, L. 1998. SUNY Suffolk Community College. LICIL Award.Interdisciplinary Approach to Increase Math Awareness in Biology byPublishing Data on the Web.

7. Salem, A. and J. Dilts. 2005. The Curriculum Foundation Workshop,CRAFTY.

8. SCCC Office of Institutional Research. 2007. Suffolk County CommunityCollege, 533 College Rd, Selden NY, 11784.

9. Valverde, J. and W. Schmidt. 1997. Refocusing U.S. Math and ScienceEducation, Science and Technology Magazine, (Winter, 1997–98),http://ustimss.msu.edu/.

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APPENDIX: EXAMPLES OF LAB ASSIGNMENTS

Standard Curve for Molecular Weight Measurements

Determination of molecular weight is sometimes accomplished by using arelative mobility measurement technique. A common abbreviation for relativemobility is Rf (the f is a subscript), for “retention factor.” This procedure iscalled electrophoresis and involves the relative mobility (the distance migratedby a band divided by the distance migrated by the dye front) of a substancebeing tested. Relative mobility of a polypeptide band is related to its molecularmass. By running a set of protein standards of known molecular mass, astandard curve of mass versus relative mobility can be produced, from whichestimates of apparent mass can be made for unknown proteins. The Rf iscalculated as the ratio of the distance migrated by the molecule to that migratedby a marker dye-front.

A simple way of determining relative molecular weight by electrophoresisis to plot a standard curve of Rf vs. log(Mw) for known samples, and readoff the log(Mw) of the sample after measuring distance migrated on the samegel. After introducing the sample data for Rf, the sample Mw is calculated.

A linear relationship exists between the logarithm of the molecular weight(Mw) of the nucleic acid, and its Rf.

Log(Mw) = m Rf + b

This is an example of a log-linear relation. The relation is linear in onevariable (Rf) and logarithmic in another (Mw). In this section we show howto work with these relations in two different ways.

A series of tests were performed and the following Rf measurementswere obtained:

Molecular Wt. Log (Mw) Rf

15848.93 4.2 0.3739810.72 4.6 0.2763095.73 4.8 0.24

125892.50 5.1 0.16

TASKS:

1. Enter this data into Excel.2. Create two graphs of the data that are appropriately titled and labeled

(Include your name on the graphs) One graph should use the actual

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molecular weight as the independent variable while the other graph shoulduse the logarithm of the molecular weight as the independent variable.

3. Print out both graphs.4. Use a straight edge to sketch a line that “best” approximates the trend of

these data.5. Pick two points on each line that you’ve drawn [(x1,y1) and (x2,y2)] and

use these points to find the equations of these lines using the two-pointformula for a line:

y−y1 =(

y2 −y1

x2 −x1

)�x −x1�

Solve the equation for y.6. Now use the Trendline feature of Excel to find the “best” fit lines to the data.7. Display the equations and the associated R2 values of the trendlines on

the graphs and print out the graphs.8. Using the graph that displays the actual molecular weight, change the

axis to display logarithmic values instead.9. Print this graph out.

10. Use the Excel trendline feature and construct a logarithmic trendline andprint out both the equation and the R2 value and print the resulting graph.

11. Compare your equations to the Excel generated equations.12. Discuss the difference between the trendline from the graph with the

actual molecular data displayed on a logarithmic scale with that of thegraph using the logarithm of the data.

13. Use one of the Excel equations (your choice) to determine the molecularweights of the following unknown Rf samples:

Rf

0.430.310.170.06

Rational Functions in Biology

Basics of Enzyme Kinetics

Enzymes are organic catalysts that control the rate of chemical reactionsin cells. In general, enzymes speed up the rate of reaction by lowering the

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activation energy required to start reactions. Enzymes are extremely efficient.They can catalyze reactions at rates up to 10 billion times higher than compa-rable non-catalyzed reactions.

The chemistry in biological organisms, however, is fundamentallydifferent from that in test tubes and in chemical engineering largely due tothe unique roles played by enzymes. Living systems depend on chemicalreactions, which on their own, would occur at extremely slow rates. Enzymesare catalysts, which reduce the needed activation energy so these reactionsproceed at rates that are useful to the cell. With the presence of enzymes asspecific catalysts, biochemical reactions are greatly accelerated. The uniquerole of enzymes also makes the kinetic of such reactions significantly differentfrom that of conventional chemical kinetics. The key difference between theenzyme kinetics and the conventional bimolecular reactions is the large excessof substrate concentration with respect to the enzyme concentration.

In typical enzyme-catalyzed reactions, reactant and product concentra-tions are usually hundreds or thousands of times greater than the enzymeconcentration. Consequently, each enzyme molecule catalyzes the conversionto product, of many reactant molecules. The kinetics of simple reactions inthe body, were first characterized by biochemists Michaelis and Menten. Theconcepts underlying their analysis of enzyme kinetics continue to provide thecornerstone for understanding metabolism today, and for the development andclinical use of drugs aimed at selectively altering rate constants and interferingwith the progress of disease states.

The Michaelis-Menten equation,

V = VMax �S�

�S�+Km

is a quantitative description of the relationship among the rate of an enzyme– catalyzed reaction [V], the concentration of substrate [S] and two constants,VMax and Km (which are constant for the particular equation). The symbolsused in the Michaelis-Menton equation refer to the reaction rate [V], maximumreaction rate (VMax), substrate concentration [S] and the Michaelis-Mentonconstant (Km).

Now this is not the most useful form of the equation. We often definev = V/Vmax, to make things easier to understand and work with. This is calledscaling a variable. Then the Michaelis-Menten equation becomes:

v = �S�

�S�+Km

The Michaelis-Menten equation can also be used to demonstrate that at thesubstrate concentration that produces exactly half of the maximum reactionrate, i.e., 1/2 VMax, the substrate concentration is numerically equal to Km.This fact provides a simple yet powerful bioanalytical tool that has been used

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pH

Enz

yme

Act

ivity

Enz

yme

Act

ivity

5 7 9 °C10 30 50

to characterize both normal and altered enzymes, such as those that producethe symptoms of genetic diseases. Furthermore, the activity of enzymes isstrongly affected by changes in pH and temperature. Each enzyme works bestat a certain pH and temperature (see graphs below), its activity decreasing atvalues above and below that point.

A. Use the Excel spreadsheet provided to obtain graphs of the scaledMichaelis-Menten Equation:

v = �S�

�S�+Km

Note: v will range from 0 to 1. A value of 1 corresponds to the maximumreaction rate, Vmax.

Let S range from 0 to 50 mMVary Km as: 1, 2, 4, and 8mM

Answer the following:

1. Describe what happens to v as you change Km.2. For what value of S is v equal to half the maximum reaction rate?3. What happens as S → �?

B. We can often combine the various base functions listed above to makenew functions. For example, say we wished to find functions that givethe same curves illustrated for the reaction rate dependence on both pHand temperature given above.

For temperature dependence (graph) shown above consider the function:f�x� = a

(xb

)me−� x

b �n

, where x represents temperature in degrees Celsiusand f(x) is the reaction rate.

1. Use the Excel spreadsheet provided to graph this function for a=5,b=60, m=0.6 and n=12. Let x range from 0 to 80.

2. For what temperature (x value) is the reaction rate the largest?

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3. Why did we choose these values for x (recall the physical meaningof x)?

4. Compare the results to the graph given above.5. Vary a, b, m, and n to see how the graph changes.

C. For the pH dependence (graph) shown above, consider the function:

f�x� = ae−� x−bc �

2

, where x is the pH and f(x) is the reaction rate.

1. Use the Excel spreadsheet provided to graph this function for a=4,b=8, and c=3. Let x range from 0 to 14.

2. For what pH (x value) is the reaction rate the largest?3. Why did we choose these values for x (recall the physical meaning

of x)?4. Compare the results to the graph given above.5. Vary a, b m and n to see how the graph changes.

BIOGRAPHICAL SKETCHES

James Fulton received his PhD in Applied Mathematics from the Universityof Virginia. He has been at Suffolk County Community College for 5 years.Prior to coming to Suffolk he has worked for GE’s Research and DevelopmentCenter, and as a Research Scientist at NASA Langley Research Center.His expertise is in mathematical modeling and he has created and appliedmathematical modeling to a variety of applications in physics, engineeringand biology.

Linda Sabatino received her MA in biology from Hofstra University. She hasbeen on the faculty at Suffolk County Community College for 32 years and isa recipient of the SUNY Chancellor’s award for Excellence in Teaching. Shehas received NSF grants to develop active learning web modules in biologythat use an investigative-based approach to data collection and analysis instudent designed experiments including digital imaging and morphologicalmeasurements for statistical and graphical analysis. As former Coordinatorof SCCC’s Teaching and Learning Center and current member of the TLCAdvisory Board she is active in professional development projects and isCo-PI for SCCC’s STEM award.

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Documents

Using the Scientific Method to Motivate Biology Students to Study Precalculus