TOPIC: Animal Physiology: Size and Surface Area in Animal Physiology

TUTOR GUIDE

MODULE CONTENT: The major goals of the module are for students to a) be exposed to the overwhelming importance of size in an animal’s life; b) understand the relationships between surface area, volume, and size; and c) see how the relationship between surface area and its volume is fundamental to the operation of many animal systems. The module includes simple calculations of surface area, an introduction to the mathematical relationship between size and heat loss/metabolic rate, and a series of questions exploring the relationship between surface area and organ function. There is also an opportunity for students to extend this understanding to the cellular level in optional additional exercises.

The module is designed to be implemented in a 50-minute classroom session with a preparatory assignment for students to complete and turn in at the beginning of the session and optional follow-up homework questions. The module is designed for first-year biology majors in an introductory biology course. The role of size and especially surface area to volume ratio are critical to nearly all animal systems as well as at the cellular level, but are usually not dealt with directly in lecture, so this module provides an opportunity for students to connect information from different systems using this theme, as well as develop the module-specific skills and extend the skills they developed in the Introduction to Mathematical Modeling in Biology Module, if that module is used previous to this one.

TABLE OF CONTENTS

Alignment to HHMI Competencies for Entering Medical Students………………...2Outline of concepts covered, module activities, and implementation……..……....3 Module: Worksheet for completion in class......................................................4 - 8Pre-laboratory Exercises................................................................................9 - 11Suggested Take home Questions................................................................12 - 13Suggested Questions for Assessment.........................................................14 - 15Guidelines for Implementation……………………………...............…...................16Contact Information for Module Developers........................................................17

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Alignment to HHMI Competencies for Entering Medical Students:

Competency Learning Objective ActivityE1. Apply quantitative reasoning and appropriate mathematics to describe or explain phenomena in the natural world.

E1.1. Demonstrate quantitative numeracy and facility with the language of mathematics.

4a-c; preparatory 1,3a

E1.2. Interpret data sets and communicate those interpretations using visual and other appropriate tools.

3a,b; preparatory 2

E1.5. Make inferences about natural phenomena using mathematical models.

3c; preparatory 3b; take home 1,2

E1.7. Quantify and interpret changes in dynamical systems.

5; take home 3

E7: Explain how organisms sense and control their internal environment and how they respond to external change.

E7.1. Explain maintenance of homeostasis in living organisms by using principles of mass transport, heat transfer, energy balance, and feedback and control systems.

1,2,6; take home 4,5

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Accompanying material- A Powerpoint file is included (Size Matters Slideshow.ppt), though questions and other content could be written on a board and/or given verbally, or transferred to a worksheet.

Mathematical Concepts covered:- basic arithmetic- power functions

Components of module:- preparatory assignment - in class activities (preferably no worksheet - see guidelines for implementation):- 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

Quantitative Skills Required: - Basic arithmetic - Logical reasoning - Graph/Data Interpretation

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WORKSHEET: Size and Surface Area in Animal Physiology

In groups of 4-5, discuss and answer the following questions:

1. Why are mittens warmer than gloves?

2. Why do muffins cook faster than bread, made from the same batter?

3. As you saw in the homework, small animals lose heat faster, for their size, than large animals. Partly because of this, in order to survive, the ancestors of small animals have had to obtain a lot of chemical energy and therefore to eat more, for their size, than large animals. This shows how much a vole and a rhino have to eat in a week compared to their body size (note, the rhino still eats far more total!):

How much energy animals must obtain from food depends on how fast they are using up energy—their metabolic rate. Here is the equation for figuring out an animal’s total metabolic rate (not the metabolic rate per gram):

M=Wb

where M= total metabolic rate in joules/hour, W= mass of the animal in grams, and b determines how (how quickly and in what direction) metabolic rate changes with size. A simple hypothesis about the relationship between metabolic rate and size might be that an animal that was twice as large as another animal would have twice as large a metabolic rate (need twice as much energy from food). In other words, metabolic rate varies linearly with size, that is, b=1.

A mouse weighs 20g and a small elephant weighs 2,000,000g.

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a) Using a value of 1 for b (so, simply, M=W), plot the mouse and the elephant on a graph with size on the x axis and total metabolic rate on the y axis. Based on this value for b, an elephant’s metabolic rate would be _________ times higher than a mouse’s.

b) Scientists have now collected metabolic rate data from many animals of different sizes. These data are best fit with the line M=W0.75 (instead of M=W1). Using this new value for b, calculate the metabolic rate for the mouse and elephant and plot them on your graph. Based on your calculations:an elephant’s metabolic rate is _________ times higher than a mouse’s.

c) For both b=1 and b=0.75, plot these three animals on a graph: a bat (10g), a sea otter (50 kg), and a camel (500kg); there are 1000g in a kg. For which animal is the difference between the two points the largest? Considering for comparison that a person must obtain about 85 calories (355 joules) per hour, on average, from food, which of these animals probably could not have evolved if metabolic rate scaled linearly with size (if b were 1)?

4. Mammals have extraordinarily high metabolic rates, so they benefit from having a lot of surface area to provide oxygen and void carbon dioxide (reactants and products of aerobic respiration). Mammal lungs branch many times, like a tree, and end in alveolar sacs, where oxygen diffuses from the air to the blood and carbon dioxide from the blood to the air. If the lungs did not branch at all, there would be one alveolar sac. If they branched once, there would be two alveolar sacs.

a) How many alveoli would there be if the lungs branched twice? b) What if the lungs branched 8 times? c) What if—as is actually the case— the lungs branched 23 times?

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5. Each alveolar sac contains 40 gas-exchange spheres, or alveoli, and each alveolus is 0.07mm2 in area. What is the total alveolar (surface) area and the surface (alveolar) area to volume ratio (assume a lung volume of 2L, or 2,000,000 mm3) of the lung of a person? How does this SAVR compare to that of the blocks and animals you calculated in the homework?

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6. Instead of having lungs that transfer oxygen into the bloodstream, insects use diffusion to move air through tubes directly to each part of their bodies:

Given this system, and remembering that the rate of diffusion decreases with distance squared, why are there no flies as big as mice—but stick insects can be as long as half a meter (almost two feet)?

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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 a fundamental concept in animal physiology? A = Extremely helpfulB= Very helpfulC= Moderately helpfulD= A little bit helpfulE = Not helpful at all

Module Rating ____________

Thank you!

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Pre-laboratory Exercises: Size and Surface Area ModuleTo be handed in at the beginning of class (or submitted electronically before class).

Name: ________________________

Surface Area to Volume Ratios Worksheet

1. Fill out the table below by calculating surface area, volume, and surface area to volume ratio (SVR) for each of the cubes.

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Cube Surface Area Volume SVR

1

1

2

2

3

3

4

4

2. Describe the relationships between surface area, volume, and SVR (y-axis) to increasing cube size (x-axis) in a graph and in a few sentences.

3. a. Animals can be thought of as simple three-dimensional shapes. Suppose a black-tailed prairie dog is represented as a rectangular prism with units 2x1x1 (length x width x height) and an American bison is represented as a rectangular prism with units 24x6x14. Fill out the table below by calculating surface area, volume, and SVR for the bison and prairie dog.

Animal Surface Area Volume SVR

b. Mammals maintain a relatively constant body temperature. This is an energy-intensive task especially in the winter, when animals lose heat to the cold air. Because an animal can only lose heat across its surface, the rate of heat loss is proportional to surface area. Assuming that 10 joules of heat are lost per hour for every unit of surface area:

What is the total amount of heat lost per hour in the prairie dog? The bison?

What is the heat loss per unit size (volume) in the prairie dog? The bison?

An animal’s total amount of heat is just its volume times the amount of heat in each unit volume. Assuming bison and prairie dogs maintain the same body temperature, which one is cooling down faster? How much faster?

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Suggested Take Home Questions:

(Alternatively, these could be done in class, if time allows.)

1. Essential molecules pass into the cell via the cell membrane. What parameter of a cell determines how much (total) a cell can absorb per unit time: surface area, volume, or mass?

2. The cell must obtain essential molecules so that the organelles of the cell can function. What parameter determines how many organelles the cell is providing for: surface area, volume, or mass?

[Table from the homework repeated here to be used on the next question:]

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Cube Surface Area Volume SVR

1

1

2

2

3

3

4

4

3. Assume that to provide enough essential molecules for survival, a cell requires 2.5 units of surface area for every 1 unit of volume. Cells of which of the sizes shown above could survive?

4. Why are single cells limited to such a small size?

5. [Alternatively, this could be an assessment question] The cells lining the small intestine each have numerous microvilli projecting from their cell membranes. What is the specific survival value of this arrangement? What must have happened in the history of vertebrate animals to bring about this arrangement over time?

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Suggested Questions for Assessment

Optional Additional Assessment Questions:

1. Why do you unload the forks first—before the knives and spoons-- from the silverware rack of a just-opened hot dishwasher?

2. Amphibians are able to breathe across their skin as well as in their lungs. During the mating season, when energy demands from calling are very high, male Hairy Frogs grow filamentous projections on their legs and sides. What could be the survival value of these projections? Why might males who produced these projections during the breeding season be selected for in evolution?

3. You have two dogs, one that weighs 30 pounds and one that weighs 60 pounds. How much does the big dog eat compared to the small dog?a) it eats the same amountb) it eats lessc) it eats more, but not twice as muchd) it eats twice as muche) it eats more than twice as much

4. In discussion section you calculated the surface area to volume ratio (SAVR) of a human lung. Given that humans maintain a tightly regulated body temperature and reptiles do not, predict the SAVR of a reptile’s lung. What evolutionary pressures might have produced these differences?

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Learning Objective ActivityE1.5. Make inferences about natural phenomena using mathematical models.

3c; preparatory 3b; take home 1,2

E7.1. Explain maintenance of homeostasis in living organisms by using principles of mass transport, heat transfer, energy balance, and feedback and control systems.

1,2,6

5. Consider the equation for metabolic rate, M=Wb, where M= total metabolic rate in joules/hour, W= mass of the animal in grams, and b determines how (how quickly and in what direction) metabolic rate changes with size. Complete the following table:

With b=1: Animal W (mass in grams) M (total metabolic rate)Star-Nosed Mole 50Virginia Opossum 4000Hippopotamus 1,500,000

With b=0.75:Animal W (mass in grams) M (total metabolic rate)Star-Nosed Mole 50Virginia Opossum 4000Hippopotamus 1,500,000

Plot the data for each table on graph paper or in Excel. Where does a Delmarva Squirrel (1500 g) fall on both graphs? What about a Sumatran tiger (110,000g)? Explain why b=0.75 is conducive to the existence of large animals.

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Guidelines for Implementation:

Collect homework. Have students break up into groups, ideally of 4-5 students each, and give each student group a simple dry-erase board (available for about $3 apiece at office-supply stores), mini-chalkboard, or large piece of paper, and a marker. For this module we recommend that NO handout be given; questions and concepts are projected as a slideshow or written on the board by the instructor. A .ppt file is provided as part of the module: contact leupen@umbc.edu. A key is available: email leupen@umbc.edu. Students get in groups of 4-5, and are shown a question in a slide. Each student group writes down their answer on their white board or large paper, and at instructor’s signal, they all hold up their boards/papers. Based on the answers on the papers, the TA moves forward or leads a discussion. The first two questions are essentially warm-up questions to help students become comfortable with the material.

Give the students a few minutes with each question or sub-question—not a long time, no more than 3-4 minutes per question-- and some questions need only a minute, such as the first two questions. The shorter the interval, the higher the level of energy and interest in the room. As the students work, circulate and assist them (without giving them the answer, of course). At the end of the time period for the question, announce that there are 10 seconds remaining, then ring a bell or use some other pre-agreed signal, and at the signal, all student groups hold up their white boards with their answers. Use the boards as a basis for discussion if answers differ. If most student groups have the right answer, move on quickly to the next question. The Powerpoint file is provided as part of the module and has answers in the “notes” section of each slide.

Alternatively, the questions can all be given together to each student group as a worksheet. This sounds like less work and stress for the TAs/instructors, but the one-at-a-time-method keeps everyone on track, energized and having fun. Try it!

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Module Developers:Please contact us if you have comments/suggestions/corrections

Kathleen HoffmanDepartment of Mathematics and StatisticsUniversity of Maryland Baltimore Countykhoffman@math.umbc.edu

Jeff LeipsDepartment of Biological SciencesUniversity of Maryland Baltimore Countyleips@umbc.edu

Sarah LeupenDepartment of Biological SciencesUniversity of Maryland Baltimore Countyleupen@umbc.edu

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.

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