Algebraic and Discrete
Biological Models for
Undergraduate Courses
Algebraic Methods? Why?
Raina Robeva
Sweet Briar College
June18, 2014
NIMBioS
Knoxville, TN
Science Advances
A Changing Landscape
Molecular Biology and Genetics
undergoing fundamental changes
Biomathematics
Health-related breakthroughs
Technological and Computational advances
Acceptance of mathematical biology
as a field of applied mathematics
Opportunities for Professional Development
Funding
Establishing Policy
2009
Food, environment, energy and health
“…The essence of the New Biology is integration – re-integration of the
many subdisciplines of biology, and the integration into biology of physicists,
chemists, computer scientists, engineers, and mathematicians to create a
research community with the capacity to tackle a broad range of scientific
and societal problems.
A New Biology for the 21st Century, NRC 2009
2010
…it is becoming increasingly clear that there is much untapped potential in mathematical fields
that are not traditionally considered as applied. Good examples are recent applications of
algebraic geometry to biological problems and the use of methods from algebraic topology for
SIAM’s response
high dimensional data analysis.
to the New Biology… report
Establishing Policy
2010
“All four challenges [of the New Biology] require the construction and analysis of predictive
mathematical models of large, nonlinear dynamic networks that span several spatial and
temporal scales. Understanding and manipulating these systems will require large, multiscale, nonlinear, and hybrid models. Existing simulation and analysis tools for such models
are in their infancy, or nonexistent in some cases. For instance, an increasingly popular
modeling paradigm for complex networks in fields ranging from molecular biology to ecology
is agent-based modeling, which captures the important feature of many complex systems
that global behavior emerges from local interactions. Very few analysis tools exist for such
models.”
SIAM’s response
to the New Biology… report
ABM - Flocking
ABM - Flocking
Run Example
Local Rules (apply to each agent)
Each bird follow three rules: "alignment", "separation",
and "cohesion".
Alignment - a bird tends to turn so that it is moving in
the same direction that nearby birds are moving
Separation - a bird will turn to avoid another bird which
gets too close
Cohesion - a bird will move towards other nearby birds
The Role of Mathematics
Systems Biology: …a field that examines how “… large number of
functionally diverse, and frequently multifunctional, sets of elements
interact selectively and non-linearly to produce coherent behavior”.
Kitano, H. (2002) Computational Systems Biology. Nature, 420, 206-210.
A Good Mathematical Models Should
Be Able To…
Capture the basic biological behaviors of the
system
Simulate the system’s responses to stimuli
and robustness to noise
Reproduce system’s key properties such as
long-term behavior, equilibrium states, limit
cycles, stability, robustness…
Types of Mathematical Models
Difference Equations Models
xn1  axn (1  xn )
Differential Equations Models
What type of
model would be
best?
Boolean Networks Models
dS
 rS
dt f  A
M
f A  A  (L  M )
f L  M  ( L ~ M )
Finite Dynamical Systems Models
X  finite set of values
V  X n  {x  ( x1 , x2 ,, xn ) | xi X }
X  Z p , p  prime
E  {( x, f ( x)) | x  V }
Probability (Stochastic) Models
t
t
X t  X 0    (t , X t )dt    (t , X t ) dWt
Many, many, more…
0
0
Models can be broadly divide into…
Quantitative – giving detailed quantitative output
and requiring detailed specific knowledge
(assumptions) about the interaction between the
system’s components.
Qualitative – generating qualitative output for the
systems in terms of convergence to equilibrium,
oscillations, long term behavior, and steady states
and requiring only information about the network
of interactions between the components.
Models can be broadly divide into…
Continuous (space continuous) – each variable
can take on values from a continuum of values.
Require detailed specific knowledge about the
fine kinetics of the system’s components
interaction.
Discrete (space discrete) – each variable can take
any out a fixed finite number of values. Require
information about the network of interactions
between the components and information about
how the “states” are different.
Major Research Goals
To understand system’ structure,
functionality and interactions and to develop
mathematical methods to describe the system in
those terms
Develop ways to simulate the system’s
behavior, evolution, and dynamics
Develop methods to control systems’ design
properties or real-time behavior of the system
Education Policy: A New Biology
2009
“Rather than teaching each level of biological organization separately-from
molecules to cells to organs, etc., and on to ecosystems (if time allows)-a
New Biology curriculum would emphasize the interconnections among
those levels to understand system-level phenomena… Students and
teachers alike will recognize that memorization of observations and facts do
not allow one to understand or predict how complicated biological systems
behave—and without that ability one will not be able to solve problems.”
A New Biology for the 21st Century, NRC 2009
2010
“…Mathematicians would benefit from a new program that supported university efforts to
develop curricula and programs focused on horizontal integration of mathematics training with
other disciplines.”
SIAM’s response
to the A New Biology… report
Education Policy: BIO 2010
2003
2005
Most biology majors take no more than one year of calculus, although some also take an
additional semester of statistics. Very few are exposed to discrete mathematics, linear
algebra, probability, and modeling topics, which could greatly enhance their future research
careers. These are often considered advanced courses; however, many aspects of discrete
math or linear algebra that would be relevant to biology students do not require calculus as a
prerequisite. While calculus remains an important topic for future biologists, the committee
does not believe biology students should study calculus to the exclusion of other types of
mathematics. Newly design courses in mathematics that cover some calculus as well as the
other types of math mentioned above would be suitable for biology majors and would also
prove useful to students enrolled in many other undergraduate majors.
Executive Summary, BIO 2010
Establishing Policy
2011
“Biology Education for all students”
A Vision and Change
report of the AAAS, 2011
The report contains, among many other things, a list “National
Calls for Improving Undergraduate Science Education” of the most
important national initiative undertaken in the last 25 years.
2009
Competency E1
“Apply quantitative reasoning and appropriate mathematics to
describe or explain phenomena in the natural world.”
Scientific Foundations for Future Physicians
report of the AAMC &HHMI 2009
Enforcing Change
MCAT 2015 will include a “Critical Analysis and
Reasoning Skills” section
2012
Enforcing Change
2011
2009
Science 25 September 2009: Vol. 325 no. 5948 pp. 1627-1628
DOI: 10.1126/science.1180821
Enforcing Change
The average score on the new grid-in questions that require students to use mathematics to
Science 25 September 2009: Vol. 325 no. 5948 pp. 1627-1628
solve biological problems
was
36% (for 1200 students who took the exam).
DOI:
10.1126/science.1180821
http://www.totalregistration.net
http://apcentral.collegeboard.com/apc/public/repository/ap08_Biology_G
radeDistributions.pdf
http://apcentral.collegeboard.com/apc/public/repository/ap09_Biology_G
radeDistributions.pdf
http://apcentral.collegeboard.com/apc/public/repository/2010_Biology_S
core_Dist.pdf
http://www.totalregistration.net/index.php?option=com_content&view=a
rticle&id=487:2013-ap-exam-score-distributions&catid=18:misc
Success Story: Bio-Calculus
The most popular choice in the college
curricula linking mathematics and biology
Material builds on a long and successful “track
record” of the importance of those methods
Calculus and ODE are the foundation of the
kind of mathematics biologists know best
2013
2014
2015
TOC
But…
Course material does not always capture the
energy and excitement of the new interdisciplinary landscapes.
http://scienceblogs.com/intersection/2008/01/29/crackingventers-genetic-code/
E Pennisi Science 2014;343:1426-1429
Mathematical methods other than Calculus such
as various discrete and algebraic approaches have
also proved to be critically important
Calculus is a difficult first step to modeling
Science 31 July 2009: Vol. 325 no. 5940 pp. 542-543 ; DOI: 10.1126/science.1176016
Calculus is just one tool…
Nature, Volume 7, November 2006
Recent Textbooks on Systems Biology
2012
There are many others!
2013
2013
More Books and Resources
2003
2014
2013
2008
2010
2008
More Books and Resources
2013
2014 (coming in mid-August)
2015 (coming out in March)
Algebraic and
Discrete
Mathematical
Methods for Modern
Biology
Edited by
Raina Robeva
Academic Press
2013
Past Workshops
2010, 2011
2013
2010
Current and Upcoming Workshops
June 18-20, 2014
July 23-25, 2014
August 6-7, 2014
August 7-8, 2014
You are not alone
“BIO SIGMAA was created
to strengthen the ties
between research and
education and to provide a
venue for sharing ideas.”
Erick Marland
Founding Board Member of Bio
SIGMAA
“The focus of BIO
SIGMAA is on the
pedagogy of
mathematical and
computational biology
at the undergraduate
level.”
From the
BIO SIGMAA Charter
Our Tutorial
Biology Topics:
Gene Regulation
Phylogenetic Trees
CpG Islands
RNA Secondary Structures
Metabolic Pathways
Tutorial Structure
Level I lectures on all topics
Concurrent Level II lectures on Friday
Each lecture followed by a 90-min Lab
All materials available from the Tutorial’s
WordPress site:
http://www.nimbios.org/wordpress-training/mathbio/
Get Ready for a Great Tutorial!