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TOPIC: Assessing the effectiveness of drug treatments using mathematical modeling - Application of Mathematical Models Cell Application TUTOR GUIDE MODULE CONTENT: This module contains simple exercises for biology majors to begin applying and interpreting mathematical models of biological systems. TABLE OF CONTENTS Alignment to HHMI Competencies for Entering Medical Students………………...1 Outline of concepts covered, module activities, and implementation……..……....2 Module: Worksheet for completion in class........................................................3-9 Pre-laboratory Review Questions (optional)...…….…...……………….………10-11 Suggested Questions for Assessment.................................................................12 Guidelines for Implementation……………………………...............…..............12-13 Contact Information for Module Developers........................................................14 Alignment to HHMI Competencies for Entering Medical Students: Competency E1. Apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world. Learning Objective E1.2. Interpret data sets and communicate those interpretations using visual and other appropriate tools. Activity 1 2. E1.3. Make statistical inferences from data sets (evaluating best fit linear relationships based on calculating error sums of squares) E1.5. Make inferences about natural phenomena using mathematical models 3. 1 3. Mathematical Concepts covered: - mathematical modeling in a biological context - linear models - regression models In class activities: - group discussion - graphing and interpreting data - construction of linear models - using regression approach calculations for determining “best fit” relationships between two variables. Components of module: - preparatory assignment to complete and turn in as homework before class - in class worksheet: - discussion questions - plotting and interpreting data - calculations of sums of squared error values to quantitatively assess the goodness of fit of lines to observed data. - suggested assessment questions - guidelines for implementation Estimated time to complete in class worksheet - 60 minutes Targeted students: - first year-biology majors in introductory biology course covering cell and molecular biology Quantitative Skills Required: - Basic arithmetic - Logical reasoning - Interpreting data from tables - Graph/Data Interpretation 2 Worksheet: Introduction to Mathematical Modeling in Biology Biological processes, such as the conversion of sunlight to plant biomass, the transcription of DNA to RNA, the growth of cells in an organism and the rate of growth of populations, are influenced by many factors (variables). Although the scientific fields of chemistry and physics have long relied on the use of mathematical models, the use of mathematical models in biology has been much less extensive. This is changing, however, and all practicing biologists in the future are going to need to be skilled in the use of mathematical and statistical models to understand the biological processes they are studying. Our ability to gather more sophisticated data on more variables regulating biological systems requires that we find ways of integrating, organizing, and evaluating this information. Mathematical models provide tools for doing just that, as they precisely describe relationships among variables that drive biological systems and provide us with a means to test our understanding of how these systems work. In this module we will begin to introduce you the use of mathematical models in biology. Discussion Questions To begin, form groups of 3 students. Discuss and write down the answers to the following questions: 1. What is a mathematical model? What is a mathematical model of a biological system? 2. How does a scientist know if a model of a biological system is a “good” model? Lab Exercises – Part I Students: To receive credit for this exercise, provide short answers and supporting graphs and calculations for all questions below. Data Set – plotting data, model evaluation 3 The drug Angioblock was developed to treat some forms of cancer. Angioblock acts by inhibiting the formation of blood vessels; cancer cells require the formation of many new blood vessels in order to form tumors, so blocking the formation of new vessels preferentially harms cancer cells. To see if Angioblock has an effect on tumor cell division and if the effect depends on tumor size, colorectal tumors of different sizes were treated for one week with Angioblock and then the number of tumor cells that were dividing (the proliferation index) was measured in each tumor. Consider the following experimental data: Observation (i) Tumor (mm) 1 2 3 4 5 6 7 8 9 10 8 11 13 15 19 23 27 33 35 44 Diameter Proliferation Index without Angioblock 375 455 722 318 216 161 232 245 223 154 Proliferation Index with Angioblock 325 380 337 248 147 103 152 175 153 79 Activities and Questions: 1. In your groups, first outline two hypotheses that you might test based on the description above. The first step to address these hypotheses is to plot the data. In your groups, decide what types of plots could be made. In each case, what is the dependent variable of interest (what will you plot on the y-axis)? What is the independent variable (to be plotted on x-axis)? Why did you choose one variable for the y axis and the other one for the x axis (in other words, what is the question the researchers are trying to answer and how could these data be used to provide them with an answer)? 4 2. a. Each individual in the group should now take a piece of graph paper and plot all the data on one graph and use different symbols to distinguish data from the tumors that are treated with the drug from those that were not (e.g., use diamonds for controls, squares for angioblock treatment). b. Based on the graphs produced, do you think the effect of Angioblock depended on the tumor size? Why or why not? c. Now, looking at the graphs, draw a line through each data set that you feel best fits the data points (one line through the proliferation index data without Angioblock and the other for the data that came from the Angioblock treatment). Now use the formula for a line (y=mx+b) to come up with a linear model for each line, where y is the dependent variable, b is the value of y when x = 0, and m = the slope of the line (change in y / change in x). Write a sentence explaining in words exactly what the relationship between tumor diameter and proliferation index is for each treatment group. 5 Lab Exercises Part 2 3. We will now examine how well the line you drew fit the actual data for the Angioblock treatment group (your groups' line will be compared to ones drawn by other groups to see who drew the line that best fit the data using the Error Sum of Squares calculation described below). Each group must first decide which group member drew the line that best fit the data take a moment now to do this. The formula below (the error sums of squares formula) is a procedure that is used in statistical analyses to find the line that best fits the data in experiments of this nature. Using the formula below calculate the error sum of squares (calculation shown below) for the line that was drawn through the data from the Angioblock treatment group from the Data set above. To do this calculation, find the value of each value of Y (the proliferation index) that corresponds to a given value of X (tumor diameter) for each observation i (using the data from Data set). Now for every observation i (the measured proliferation index at each tumor diameter measured) determine the difference between every observed value of Y at each X value and the value of Y that passes through the line that you drew at that X value. Calculate a total Error Sum of Squares separately for each model (A and B) In the end you should have a sum of squares error value for each line. When your group is finished raise your hand. When all groups are finished one member of each group will give the formula for their line and the error sum of square value for that line (please include this information on the sheet you will hand in for this question, include your calculations). The winning group will have the lowest error sums of squares value (and will so be the line that best fits the data). 6 MODULE FEEDBACK - Each year we work to improve the modules in the active learning "discussion" sections. Please answer the following question with regard to this module on this sheet and turn in your answer to the TA. You can do this anonymously if you like by turning in this sheet separately from your module answers. How helpful was this module in helping you understand the fundamental concepts in mathematical modeling of biological data? A = Extremely helpful B= Very helpful C= Moderately helpful D= A little bit helpful E = Not helpful at all Module Rating ____________ Thank you! 7 Pre-laboratory Exercise: To be completed before you come to class and handed in at the beginning of class. This homework is designed to review lines and linear models and to prepare you for the upcoming module on mathematical modeling and subsequent models in Ecology that you will have to work with. If you encounter an unfamiliar term, please refer to a textbook for high school algebra or pre-calculus. On her way to her volleyball game, Joanne stops by the grocery store for a healthy snack that will give her energy for the game. She decides to get a jar of peanut butter and some bananas. She notices that the peanut butter costs $3.99 for a jar and that the bananas cost $0.49 per pound of bananas. 1. How much will one pound of bananas and a jar of peanut butter cost? 2. Joanne decides to bring a snack for each of the 6 girls on the volleyball team. How much will it cost for six pounds of bananas and a jar of peanut butter? 3. On the grid provided below, draw a Cartesian coordinate system with number of pounds of bananas on the x-axis and the total cost of the peanut butter and bananas on the y-axis. Plot the two points that you found for the previous two questions. 8 7 6 5 4 3 2 1 0 4. Plot the line on the grid above through the two points that you found. Now compute the slope of the line that goes through these two points (remember, “rise over run”: (y2-y1) = m (x2-x1) where m is the slope of the line). 8 Now write the equation of the line in slope-intercept form, y= mx + b, where b is the y-intercept (the place where the line crosses the y axis). Congratulations! You have just made a mathematical model of peanut butter and bananas! 5. What does the slope correspond to in terms of the scenario above? What is the y-intercept of the line? What does it correspond to in the scenario above? 6. What is the independent variable? What is the dependent variable? Respond in symbols as well as in words. 7. How would the line change if the price of bananas increased or decreased? How would the line change if the cost of the jar of peanut butter increased or decreased? 9 Suggested Questions for Assessment E1. Apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world. E1.2. Interpret data sets and communicate those interpretations using visual and other appropriate tools. 1 2.a. – c. E1.5. Make inferences about natural phenomena using mathematical models 3. Guide for implementation: Discussion Have students break up into groups of 3 to discuss and come up with answers to the questions. Groups get together to talk about each question – work pauses in after 10 minutes or as soon as you think each group has come up with something for all questions (no longer than 15 minutes). The TA should then pick a person from 3 groups chosen at random to share with the class their group’s answer to one of the questions. Tell them that everyone in the group should be prepared to share the answer with the class, as you will chose who speaks randomly among the group. Suggested ideas that should emerge from each question are listed below. An alternative way to run this discussion section of the class would be to run it “Question Time” style where you let them discuss each question in turn for three minutes, then ring a bell or use another method to cut off discussion, then have the whole group report their answer. Then move on to the next question. 1. What is a mathematical model? A mathematical description of a system. A model of a biological system is a description of a biological process using mathematics. Mathematical models typically involve formulas that describe relationships among variables and also provide information on the relative strength of these variables on a given biological observation (population growth rate, mutation rates of DNA). Models are typically organized into dependent and independent variables (but not always). TAs should guide discussion toward mathematical models as ways to evaluate 1) how well we understand the biotic or abiotic factors that influence biological systems 2) our ability to predict how changes in certain variables influence biological systems. 2. How does a scientist know if a model is good? - scientists can evaluate how well the model explains the observations it was based on – how well does the model fit the observations. - scientists can evaluate how accurate the model is in predicting future events (either through new experiments or through new observations that were not used to construct the original model). 10 Guide for Implementation: Lab Activity 1) TA hands out a copy of the first data sheet and a piece of graph paper to each member of a group. Each individual is instructed to plot the data points and construct lines through the data points based on the model equations. Each individual will then decide which model (line) more accurately “fits” the data and provide rationale. One or two groups share their decision with the class. a) The graph paper should have x and y axes drawn in lower left when held in landscape orientation. b) If possible, project the data plot & regression lines for discussion –maybe via instructor’s computer & projector or overhead projector. c) Discussion of possible data outliers may come up – explain why the data may not fall exactly along the best fit line. 2) Instructor provides brief explanation of least squares method of evaluating regression lines, writing the equation for this calculation on the board. Students then perform these calculations in their groups (can divide up who does computation for which model). a) Sample explanation: By plotting the data it looks like y varies with x. The models describe a linear relationship where y varies with x. But the data points don’t actually all fall on the line. The distance a point falls from the line is a form of error. The model that best explains, or fits, the data is the one that the data points are closest to, or has the least error (the smallest error sum of squares). n ErrorSumOfSquares = å (Yi - Yˆi ) 2 i=1 Yi is a data point at Xi, and Yhat is the value of Y that falls on the line at Xi as defined by the model. 3) Instructor-led discussion (as a class) about which group’s model is the “best fit”. a) The actual best fit lines are shown in the answer key. This information may or may not be shared and is up to the instructor. b) Instructor may foreshadow whether relationships may be linear or not, and whether the regression is valid beyond the range of the data. 11 Module Developers: Please contact us if you have comments/suggestions/corrections Kathleen Hoffman Department of Mathematics and Statistics University of Maryland Baltimore County [email protected] Jeff Leips Department of Biological Sciences University of Maryland Baltimore County [email protected] Sarah Leupen Department of Biological Sciences University of Maryland Baltimore County [email protected] Acknowledgments: This module was developed as part of the National Experiment in Undergraduate Science Education (NEXUS) through Grant No. 52007126 to the University of Maryland, Baltimore County (UMBC) from the Howard Hughes Medical Institute. 12