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%\markboth{Rohan Attele and Dan Hrozencik}{Bio Math and Computational Algebra}
\title{\bf Mathematical Biology and Computational Algebra at the Sophomore Level}
\author{Rohan Attele\footnote{kattele@csu.edu} and
Dan Hrozencik\footnote{dhrozenc@csu.edu},\\ Department of Mathematics, Chicago State University}

%\address{Department of Mathematics\\ Chicago State University\\ Chicago, IL 60628.}
%\email{kattele@csu.edu \\ dhrozenc@csu.edu}
\date{}

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\begin{center}
 \begin{tabular}{|p{0.18\textwidth}|p{.78\textwidth}|}\hline
 \multicolumn{2}{|c|}{{\bf Name of the Institution:} Chicago State University}\\ \hline
 {\bf Size} & about 7500 students.\\\hline
 {\bf Institution Type} & Chicago State University is a public, comprehensive university that provides access to higher education for students of diverse backgrounds and educational needs. \\\hline
\parbox{1in}{\bf Student \\ Demographic}& Mathematics majors with prerequisites of linear algebra, probability and statistics course, and two specific freshman biology courses.   Biology majors with prerequisites of a sophomore botany course and a sophomore zoology course.  \\\hline
{\bf Department  Structure}& Mathematics and Biology are two MS degree granting departments  in the College of Arts and Sciences.\\\hline
 \end{tabular}
 \end{center}


\section{Abstract}
Biology provides a rich source of problems that can be solved by non-infinitesimal algebraic techniques.   This article discusses the content of an  interdisciplinary research-oriented sophomore course in computational algebra and  biology created at Chicago State University.   The course is  part of a an attempt to teach abstract algebra on a foundation of computational algebra done at the sophomore level.

\section{Course Structure}
\begin{description}
\item[Weeks per term:] 16 weeks.
\item[Classes per week:] Two 100 minutes sessions.
\item[Labs per week:] No specific time set aside for labs; computer or experiment assignments done as needed.
\item[Average class size:] Three in the pilot; 15-20 can be accommodated.
\item[Faculty/dept per class, TAs:] Team-taught by one mathematics instructor and one biology instructor.  No TAs.
\item[Next course:] Not yet  approved.  It will be a course in computational algebra.
\end{description}

\section{Chicago State University, CSU}

Chicago State University has consistently led Illinois public universities in the conferring of baccalaureate and master’s degrees to African American students.  CSU graduates more African Americans with a master’s degree in mathematics than all other Illinois private and public colleges and universities combined, and is a national leader in graduating African Americans with a masters' degree in mathematics.

\subsection{Department of Mathematics and Computer Science}
The department has nineteen faculty members whose active research areas include mathematical biology, geometric algebras, logic, network security, software engineering, programming languages, and analysis (real, complex, and numerical, ODE, PDE). It has shifted focus from being service oriented to offering a  research oriented curriculum that prepares students to work in academic and industrial settings.

The following table summarizes the distribution of students in the undergraduate majors and graduate programs.

\begin{center}
\begin{tabular}{cccccccc}
{\sc  {major}} & 02-03 &03-04 & 04-05 &05-06 & 06-07&07-08&08-09\\
{Math UG }   & 73     & 79&76&70&53&35&46\\
{Math GR}    & 27&30&21&25&22&22&19\\ 
{Computer science UG } & 204 & 184&134&119&94& 87&82 \\
{Computer science GR} &4&9&15&14&12&13  &8\\
\end{tabular}
\end{center}


\section{Background and Motivation}
This paper describes a part of on-going attempts to develop a computational algebra curriculum at the sophomore level that will lay a foundation for abstract algebra done  at the junior/senior level.   Genetics provides a source of real-world problems for computational algebra.
At the MAA  MathFest 07 the authors made a presentation on the project.

In private industry, mathematicians are most often part of interdisciplinary teams working closely with biologists, physicists, economists, etc.,  to solve complex problems (U.S. Department of Labor, 2010). Where in the undergraduate curriculum do students get an experience that will help them to see how mathematicians work, and to decide if that is right for them? Can we get it to them early enough so that it can be useful to their career planning?

In particular, mathematics curricula should prepare majors to have the skills needed to work in  teams and bring their mathematics expertise to bear upon interdisciplinary problems.  Thus mathematics majors should gain experience in solving  a significant real-world problem and in working with individuals having other majors.

We designed \emph{Introduction to Research in Mathematical Biology I}, Math 2180, to provide such an experience. In particular, it provides:

\begin{itemize}
\item   An early opportunity for students to work in interdisciplinary teams on a real-world research problem.
\item   An opportunity for students to be the experts in their content areas.
\item   An opportunity to use relatively elementary mathematics to work on research problems.
\item   An opportunity to work on a research problem that could continue throughout their undergraduate careers.
\end{itemize}

\section{Description of the Course}

The course was open to both mathematics   and biology  majors as an elective.  It was  team-taught by a mathematics  and biology professor,  and was offered concurrently by both departments to ensure a mix of both majors.

A key idea was to build teams of students with differing abilities.  Biology and mathematics majors brought different experience and expertise to the class, and then worked together to develop and analyze mathematical models. Both mathematics and biology majors needed to know enough about the other area so that they could combine their ideas. For that reason, the prerequisites for biology majors and mathematics majors were different.  A mathematics major needed to have a first course in linear algebra, a  probability and statistics course, and two freshman biology courses: A Survey of the Plant Kingdom, and A Survey of the Animal Kingdom.   A biology major needed to have College Algebra, a sophomore botany course (Biology of Algae, Plants and Fungi), and a sophomore zoology course (Biology of Animals).

The initial class (Spring 2007) had only four students (one biology major and three  mathematics majors, one of whom also was a biology minor).  One of the mathematics students attended the summer 2007 mathematical biology workshop at the Mathematical Biology Institute of The Ohio State University.  The low had two causes: the advisors were not well aware of the course and the course was an elective.   The department has asked that the course satisfy the college interdisciplinary requirement beginning with Fall 11.

The course met twice a week in a computer lab or a biology lab.    The beginning of the course had a heavier emphasis on mathematics instruction.  Students used Excel spreadsheets to study difference equations, Excel add-ins (PopTools) for analyzing population demography data, and MATLAB\footnote{MATLAB was acquired with an internal grant from the Research Development Office of the College of Arts and Sciences.} in matrix modeling of population dynamics. Students learned structured population models, transition matrices, methods for finding eigenvalues and eigenvectors and their interpretations in terms of stability. We covered most of Chapter 4 of Caswell's \emph{Matrix Population Models} (2006) with introductory material from Allman and Rhodes' \emph{Mathematical Models in Biology} (2004).   Students learned to compute powers of matrices using eigenvalues and to approximate eigenvalues using the power method.

Once the students have enough mathematics background, they move on to the more biological aspects of the course. They study models of molecular evolution and phylogenetics, using Excel to explore the evolution of an invasive plant over time and space. This is Chapters 4 and 5 of the Allman and Roads text, omitting the mathematical background material. Also discussed were the Jukes-Cantor and Kimura models, corresponding phylogenetic distances, and log-det distance. The students constructed phylogenetic trees and learned neighbor joining and maximum parsimony methods.

We wanted to have mathematics majors and biology majors work in pairs (or small groups) on a research project, but the small initial class size did not permit the full realization of this goal.    Students collected population demography data using plants in the university's  Research and Teaching Prairie Garden and used long-term population demography data of endangered plants collected by the biology professor. The students then used matrix population models learned in the course  to project population persistence.

The course was offered at the sophomore level in order for students to do a long-term research project.   One of the mathematics undergraduate students ultimately joined the graduate program in mathematics and took two classes in genetics.  As an extension of her work in population models, she is now  finishing her masters' research project to model endangered plant populations incorporating inbreeding with data gathered from Illinois and Indiana state parks.

The course was again offered in Fall 10 but only for two senior graduating mathematics majors needing an elective.   One of them became interested in mathematical biology and is now in the process of applying for a NSF  Graduate Fellowship  Program (GRFP).    The student's research plan for the GRFP is based on Chicago State's aquaponics program, and is a theoretical  extension of her research done for the course.  She is gathering data on the length, relative positions, and velocities of fish, the concentration of nitrites in fish effluent before and after being absorbed by the plants, and the total weight of supported plants.   The data will be used for computations on the angle between velocities of different fish, minimal distance between fish, plant absorption rates of nitrites, and estimating the total weight of the fish. This research will span at least two semesters.

\section{Conclusion}

Promoting the use of relatively elementary non-infinitesimal modeling techniques and reliance on technology is a hallmark of this project. It demonstrates that even at the sophomore level, mathematics and biology students can interact significantly.   Students inclined toward applied mathematics  can meaningfully experience mathematics in a biological setting.

\section*{References}

Allman, E. S. and J. A. Rhodes, 2004: \emph{Mathematical Models in Biology}. Cambridge University Press,  \\370 pp.

\medskip
\noindent
Caswell, H., 2006: \emph{Matrix Population Models}. Updated 2nd ed. Sinauer Associates, Inc., 722 pp.

\medskip
\noindent
U.S. Department of Labor,  Bureau of Labor Statistics, 2010--2011 ed: Occupational Outlook Handbook: Mathematicians. [Available online at http://stats.bls.gov/oco/ocos043.htm.]

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