Calculus For Biologists: A Beginning – Getting Ready For Models and Analyzing Models The ...

541 Pages·2008·3.04 MB·English

Calculus For Biologists: A Beginning – Getting Ready For Models
and Analyzing Models
The seadragons were intrigued by Calculus and flocked to the
teacher.
James K. Peterson
Department of Biological Sciences
Department of Mathematical Sciences
Clemson University
email: [email protected]
(cid:13)c James K. Peterson Version July 31, 2008
Gneural Gnome Press
August 25, 2008

Dedication
I dedicate this work to the students who have learned this material in its various preliminary
versions, the practicing biologists and immunologists who have helped an outsider think more
biologically and to my family who have listened to my ideas in the living room and over dinner
for many years. I hope that this text helps inspire all my students to consider mathematics and
computer science as indispensable tools in their own work in the biological sciences.
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Abstract
Thisbooktriestoshowbeginningbiologymajorshowmathematics,computerscienceandbiology
can be usefully and pleasurably intertwined. After the necessary start up costs to develop some
essential calculus tools, we use a few select models to illustrate how these three fields influence
each other in interesting and useful ways. Indeed, we believe that the three must be considered
as part of a larger whole and that the use of these ideas is perhaps best described as an emergent
phenomena! And we must always be careful to make sure our use of mathematics gives us insight.
These cautionary words about the modeling of the physics of stars from 1938 should be taken to
heart:
Technical journals are filled with elaborate papers on conditions in the interiors of model
gaseous spheres, but these discussions have, for the most part, the character of exercises in
mathematical physics rather than astronomical investigations, and it is difficult to judge the
degree of resemblance between the models and actual stars. Differential equations are like
servants in livery: it is honourable to be able to command them, but they are ”yes” men,
loyally giving support and amplification to the ideas entrusted to them by their master. –
Paul W. Merrill, The Nature of Variable Stars, 1938, quoted in Arthur I. Miller Empire of the
Stars: Obsession, Friendship, and Betrayal in the Quest for Black Holes, 2005”
The relevance of this quotation to our pursuits is clear. It is easy to develop sophisticated
mathematical models that abstract from biological complexity something we can then analyze
with mathematics or computational tools in an attempt to gain insight. But as Merrill says,
it is difficult to judge the degree of resemblance between the models and actual [biology]
where we have taken the liberty to replace physics with our domain here of biology. We should
never forget the last line
Differential equations are like servants in livery: it is honourable to be able to command
them, but they are ”yes” men, loyally giving support and amplification to the ideas entrusted
to them by their master.
We must always take our modeling results and go back to the scientists to make sure they retain
relevance.
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Acknowledgements
We would like to thank all the students who have used the various iterations of these notes as
they have evolved from handwritten to the fully typed version here. We particularly appreciate
your interest as this course is required and uses mathematics; a combination that causes fear in
many biological science majors. We have been pleased by the enthusiasm you have brought to
this interesting combination of ideas from many disciplines.
We also thank our colleagues Jo Hoffacker, Hugh MacMillan and Tim Teitloff for their help
in teaching various versions of this material. We also appreciate all the help Jim Colachino in
Biology has given us. Finally, we gratefully acknowledge the support of Hap Wheeler in the
Department of Biological Sciences for believing that this course would be useful to his students.
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History
Based On:
Notes On MTHSC 108 for Biologists developed during the
Spring 2006,
Fall 2006,
Spring 2007,
Fall 2007 and
Spring 2008 courses.
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