EDUCATIONFORUM
We do not argue against the mathematical
courses included in current undergraduate
biology curricula. But we believe that these
courses should be revised and extended.
Many key computational ideas can be better communicated and absorbed by biology
undergraduates with few prerequisites, in a
way that will make the students excited about
bioinformatics as a scientific discipline and
more creative when they employ bioinformatics methods and ideas in the future. We
feel that the best way to engage biology
undergraduates in bioinformatics is to appeal
to their innate intuition and common sense
and to avoid mathematical formalism as
much as possible. The proposed course may
become a first step toward building the new
computational curriculum for biologists.
References and Notes
1. National Research Council, BIO2010, Transforming
Undergraduate Education of Future Research Biologists
(National Academies Press, Washington, DC, 2003).
2. W. Bialek, D. Botstein, Science 303, 788 (2004).
3. P. A. Pevzner, Bioinformatics 20, 2159 (2004).
4. L. M. Iyer et al., Genome Biol. 2(12), RESEARCH0051.1
(2001).
5. P. Grindrod et al., Math. Today 44, 80 (2008).
6. N. Wingreen, D. Botstein, Nat. Rev. Mol. Cell Biol. 7, 829
(2006).
7. L. J. van ’t Veer et al., Nature 415, 530 (2002).
8. L. J. Gross, Cell Biol. Educ. 3, 85 (summer 2004).
9. R. Brent, Cell Biol. Educ. 3, 88 (summer 2004).
10. We are grateful to all participants of RECOMB Bioinformatics Education conference (La Jolla, 14 and 15 March 2009)
for many comments on various aspects of bioinformatics
education. We are also grateful to V. Bafna, N. Bandeira,
A. Tanay, and G. Tesler for many interesting discussions
and suggestions. The conference was supported by the
Howard Hughes Medical Institute Professors Program.
10.1126/science.1173876
EDUCATION
Mathematical Biology Education:
Beyond Calculus
Training in developing algebraic models
is often overlooked but can be valuable
to biologists and mathematicians.
Raina Robeva1* and Reinhard Laubenbacher2
I
n 2003, the National Research
Council’s BIO2010 report
M
recommended aggressive
curriculum restructuring to
educate the “quantitative biol- Ge
E
ogists” of the future (1). The
number of undergraduate and
L
graduate programs in mathematical and computational
biology has since increased,
and some institutions have added
courses in mathematical biology related to
biomedical research (2, 3). The National
Science Foundation (NSF) and the National
Institutes of Health are funding development
workshops and discussion forums for faculty (4, 5), research-related experiences (6,
7), and specialized research conferences in
mathematical biology for students (8, 9).
This new generation of biologists will
routinely use mathematical models and computational approaches to frame hypotheses,
design experiments, and analyze results. To
accomplish this, a toolbox of diverse mathematical approaches will be needed.
Nowhere is this trend more evident than in
Department of Mathematical Sciences, Sweet Briar College, Sweet Briar, VA 24595, USA. 2Virginia Bioinformatics
Institute, Virginia Tech, Blacksburg, VA 24061, USA.
1
*To whom correspondence should be addressed. E-mail:
[email protected].
542
)
Le
DE and Boolean models of the lac operon mechanism. Each component of the shaded part of the wiring
diagram is a variable in the model, and the compartments outside of the shaded region are parameters.
Directed links represent influences between the variables: A positive influence is indicated by an arrow; a
negative influence is depicted by a circle.
systems biology. At the molecular level, this
involves understanding a complex network of
interacting molecular species that incorporates gene regulation, protein-protein interactions, and metabolism. Two types of models have been used successfully to organize
insights of molecular biology and to capture
network structure and dynamics: (i) discreteand continuous-time models built from difference equations or differential equations
(DE) models, which focus on the kinetics
of biochemical reactions; and (ii) discretetime algebraic models built from functions
of finite-state variables (in particular Boolean
networks), which focus on the logic of the
network variables’ interconnections.
Algebraic models were introduced in
1969 to study dynamic properties of gene
regulatory networks (10). They have proven
useful in cases where network dynamics
are determined by the logic of interactions
rather than finely tuned kinetics, which often
are not known. Published algebraic models
include the metabolic network in Escherichia coli (11) and the abscisic acid signaling
pathway (12).
The use of algebraic methods is extending beyond systems biology. Methods
from algebraic geometry have been used
in evolutionary biology to develop new
approaches to sequence alignment (13), and
new modeling of viral capsid assembly has
been developed using geometric constraint
theory (14). Algorithms based on algebraic
combinatorics have been used to study RNA
secondary structures (15).
25 JULY 2009 VOL 325 SCIENCE www.sciencemag.org
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tional ideas (hierarchical clustering, pattern classification, and feature selection) are
introduced as part of the “story” to promote
ease of understanding by students. Such
examples may also instill in students the
real-life impact of computational methods.
This research, for example, led to the first
cancer diagnostic chip to be approved by
the U.S. Food and Drug Administration; it is
currently used to determine which patients
may benefit from additional chemotherapy.
The question of whether similar innovative courses can be implemented at the
undergraduate level at many universities
remains subject to debate (8, 9). Nevertheless, such courses are pioneering steps
toward developing a new computational
biology curriculum.
EDUCATIONFORUM
DE Versus Algebraic Models
Many institutions now offer DE-focused
courses that include problems from the life
sciences, rather than just the traditional
linkages with physics and engineering. For
example, in bio-calculus and precalculuslevel courses, differential and difference
equations are used to model system dynamics. Textbooks emphasize development of
DE models for various biological systems
related to, e.g., population dynamics and the
spread of an epidemic (16, 17). In contrast,
few curricular materials linking biology
with modern algebra have been created. The
courses “Mathematics for Computational
Biology” (18) and “Algebraic Statistics for
Computational Biology” (19) are two such
examples targeting advanced undergraduate mathematics and biology students. A
NSF curriculum development pilot project
to produce algebraic educational modules is
under way (20), as is a project to create an
integrated biology curriculum, incorporating mathematics, statistics, and computational methods beyond calculus (21).
This relative lack of interest and investment in algebraic approaches compared
with DE is due primarily to a lack of awareness of the importance of algebraic models in modern biology and mathematics
research, as well as to a lack of appropriate
educational materials on algebraic models,
rather than to inaccessibility of the essential
underlying mathematics.
Variables in a DE model can span a continuous range of biologically feasible values. Modelers need detailed knowledge of
interactions between variables, for example, specifics of control mechanisms, rates
of production and degradation, or highest
and lowest biologically relevant concentrations. But in an algebraic model, only values from a finite set are allowed. The special
case of a Boolean network allows only two
states, for example, 0 and 1, representing
the absence or presence of gene products in
a model of gene regulation. In contrast to
DE, the information necessary to construct
a Boolean model requires only conceptual
understanding of the causal links of dependency. Thus, continuous models are quanti-
tative while Boolean models are qualitative
in nature.
To illustrate, we use one of the simplest and best-understood mechanisms of
gene regulation: the lactose (lac) operon
that controls the transport and metabolism
of lactose in E. coli (fig. S1). A wiring diagram is constructed to represent major components and causal interactions of the system (see figure, page 542).
The wiring diagram on the left of the figure depicts both DE and Boolean models of
the lac operon. Although it involves only three
variables, the DE model is complex, involving additional parameters reflecting the reaction kinetics. The Boolean model is simpler,
reflecting only basic interactions depicted in
the wiring diagram. Despite its simplicity, the
Boolean model exhibits the same qualitative
behavior as the DE model. Both are capable
of reflecting the key feature of bistability of
the operon (22).
In contrast with the high level of detail
needed for the construction of the DE model,
the Boolean model is relatively intuitive and
was essentially obtained by translating a
prose description of the molecular pathways
depicted in fig. S1 and formalized by the
wiring diagram into logical statements. This
feature of algebraic models makes them particularly appealing as an entry into mathematical modeling.
Constructing algebraic models of biological networks requires only a modest amount
of mathematical background, some of which
is typically included in a college algebra
course. This provides a quick path to mathematical modeling for students (and researchers, for that matter) in the life sciences. For
mathematics students, algebraic models of
biological networks provide a meaningful
way to introduce many of the concepts in the
discrete mathematics curriculum. And they
are easily connected to more advanced topics in abstract algebra (23).
A Call for Change
Algebraic models should be considered
critical for the professional development of
biologists. Mathematics and biology educators must work to determine the best way
of including these in undergraduate curricula. Because the mathematics involved is
more easily accessible to faculty in the life
sciences, relative to DE, this could lead to
increased engagement on their part. Algebraic models will introduce problems from
modern biology into the traditional mathematics courses, bringing life to the primarily theoretical abstract algebra curricula.
Concurrent or subsequent introduction to
DE models in the courses already in place
will only reinforce the conceptual framework and further elevate students’ mathematical sophistication.
References and Notes
1. National Research Council, BIO2010: Transforming
Undergraduate Education for Future Research Biologists
(National Academies Press, Washington, DC, 2003).
2. N. Wingreen, D. Botstein, Nat. Rev. Mol. Cell Biol. 7, 829
(2006).
3. R. Robeva, Methods Enzymol. 454, 305 (2009).
4. Over the Fence: Mathematicians and Biologists Talk
About Bridging the Curricular Divide, Mathematical
Biosciences Institute workshop, Columbus, OH, 1 and 2
June 2007.
5. Computational and Mathematical Biology, Mathematical
Association of America, Professional Enhancement
Programs, 2003, 2004, 2005, 2006, and 2007.
6. Expanding Biomedical Research Opportunities at the
Undergraduate Level, in Recovery Act Limited Competition: NIH Challenge Grants in Health and Science
Research (RC1) (National Institutes of Health Program
Solicitation 12-DK-102, Challenge Grant Applications,
2009); http://grants.nih.gov/grants/funding/challenge_
award/Omnibus.pdf.
7. Interdisciplinary Training for Undergraduates in Biological and Mathematical Sciences, (National Science
Foundation Program Solicitation, NSF 08-510, 2008);
www.nsf.gov/pubs/2008/nsf08510/nsf08510.htm.
8. Undergraduate Conference at the Interface between
Mathematics and Biology, National Institute for Mathematical and Biological Synthesis (NimBioS), Knoxville,
TN, 23 and 24 October 2009.
9. Undergraduate Research Conference in Quantitative
Biology, Appalachian State University, Boone, NC, 2 and
3 November 2007 and 14 and 15 November 2008.
10. S. A. Kauffman, J. Theor. Biol. 22, 437 (1969).
11. A. Samal, S. Jain, BMC Syst. Biol. 2, 21 (2008).
12. R. Zhang et al., Proc. Natl. Acad. Sci. U.S.A. 105, 16308
(2008).
13. L. Pachter, B. Sturmfels, Proc. Natl. Acad. Sci. U.S.A. 101,
16132 (2004).
14. M. Sitharam, M. Agbandje-Mckenna, J. Comput. Biol. 13,
1232 (2006).
15. A. Apostolico et al., Nucleic Acids Res. 37, e29 (2009).
16. C. Neuhauser, Calculus for Biology and Medicine
(Prentice-Hall, Upper Saddle River, NJ, ed. 2, 2003).
17. F. Adler, Modeling the Dynamics of Life: Calculus and
Probability for Life Scientists (Thompson, Belmont, CA,
ed. 2, 2005).
18. “MATH 127—Mathematics for Computational Biology”
(University of California, Berkeley, CA, 2007);
http://math.berkeley.edu/~bernd/math127.html.
19. “MATH, 295—Algebraic Statistics for Computational
Biology” (Duke University, Durham, NC, 2005); www.
math.duke.edu/graduate/courses/fall05/math295.html.
20. R. Robeva et al., National Science Foundation DUE award
0737467, 2008; www.nsf.gov:80/awardsearch/
showAward.do?AwardNumber=0737467.
21. Biology Through Numbers Howard Hughes Medical
Institute Award, 2006; www.hhmi.org/research/
professors/neuhauser.html.
22. E. M. Ozbudak, M. Thattai, H. N. Lim, B. I. Shraiman,
A. van Oudenaarden, Nature 427, 737 (2004).
23. R. Laubenbacher, B. Sturmfels, Am. Math. Month.
arXiv:0712.4248v2, in press.
24. The authors thank R. Davies from Sweet Briar College
for her review of the biological content. Support by the
NSF under the Division of Undergraduate Education; NSF
Course, Curriculum, and Laboratory Improvement (CCLI)
award 0737467 is also gratefully acknowledged.
Supporting Online Material
www.sciencemag.org/cgi/content/full/325/5940/542/DC1
www.sciencemag.org SCIENCE VOL 325 31 JULY 2009
Published by AAAS
Downloaded from www.sciencemag.org on October 5, 2009
Because of the success of DE methods
for modeling biochemical networks, they
have been the primary focus of curriculum
reform efforts, whereas algebraic models
have received less attention. But algebraic
models have many features needed to integrate mathematics and biology into the training of students at all levels; thus, their inclusion in curricula is necessary and timely.
10.1126/science.1176016
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