Transcript How biology is changing mathematics and why all math is useful.

How biology is changing mathematics
and why all math is useful.
Paul J. Hurtado
Mathematical Biosciences Institute
Aquatic Ecology Laboratory
Department of Evolution, Ecology & Organismal Biology,
The Ohio State University
Overview
• Computing & biology impacting mathematics?
– Modern biology is a “hard science”
– Computing, science and mathematics
• All math is useful!
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Geometry & Topology: Mariel Vazquez
Algebra: Marisa Eisenberg, Marty Golubitsky
Combinatorics, Graph Theory: Reka Albert
Core applied math: Probability, Linear Algebra,
Analysis, Dynamics & Bifurcation Theory, Statistics.
Mathematics in Biology
Open letter to [Bio Department] faculty
I am finishing my fifth year as a PhD student in the [Bio Department]…. While I have many good things
to say about [this university] and about our Department, I am aware of one severe deficiency in our
graduate program: we do not provide adequate training to our graduate students in quantitative
biology.
To be very explicit about what I mean…
(i) Understanding statistical models. …
(ii) Computer programming. It is hard to overstate the importance of this for any biologist…
(iii) A basic understanding and feeling for the role of deterministic/stochastic modeling in biology...
… During the past few months, I have given seminars at a number of high-caliber universities… UC
Berkeley, Harvard, UC Davis, the University of Texas, and perhaps several others. But I am consistently
shocked by what seems to me a substantial deficiency in our training relative to those other programs.
I have found that interacting with my peer group from those other schools (and not just students from
any single lab) is often an eye-opening experience. It isn’t that these folks are theoreticians: many
are every bit as empirical as the average [Bio Department] graduate student. But I consistently come
away from these interactions with a sense that the ‘dialogue’ about issues involving all things
quantitative in [my field] is on a distinctly higher level from what we typically experience here.
… This does not reflect a difference in quality of graduate students. It does, however, reflect a
difference in the amount of quantitative training and the emphasis on quantitative approaches in our
respective programs. In my experience, the ability to utilize quantitative methods is often the
greatest determinant of productivity in graduate school – I have no data here, but…
... I have seen the CVs of many graduate students from those previously mentioned schools against
whom I will be competing for faculty positions in the next few years and I know what distinguishes
their competitive publication records from those of graduate students who have published only one or
several papers by the time they complete their PhDs. Without strong quantitative skills, our
students are at a distinct disadvantage relative to students in those other programs.
It is almost ironic that perhaps the most important skill I have acquired as a graduate student was not
obtained [here]. I learned the fundamentals of computer programming through a simple 2-hour per
week, single semester seminar [elsewhere]. This skill has fundamentally changed everything about
the way I do science and transcends “programming” per se. It has opened the door to thinking
quantitatively about challenging problems in [my field of biology], has led me to question ‘black
box’ solutions to data analysis, and has given me the confidence to pursue my own approaches
when the ‘black box’ solutions are simply inadequate for dealing with real-world data…
I don’t think that we necessarily need more [courses] (with the exception of [bio]-oriented computer
programming…). I contend that the entire culture surrounding quantitative approaches in [my field
of biology in this department] … needs to be changed, and I can think of one good way to do this:
[hire] a high-profile quantitative [biologist] … to attract a core group of quantitative/computational
postdocs and graduate students who are trained to think about similar intellectual problems as [the
biologists in] our department. When I have surveyed programs in [my field] at some of the schools
mentioned earlier, it is very obvious to me that what they have – and what we lack – is precisely this
sort of expertise.
…I can see no other hole with such far-reaching consequences for graduate training as the deficit in
quantitative … biology.
To summarize…
• Quantitative skills are a hot commodity in
biology! These include math, stats, computing.
• Thinking quantitatively = a very good thing!
• Biologists are taking ownership of developing,
teaching and learning quantitative tools.
• There is “no other hole with such far-reaching
consequences for graduate training as the
deficit in quantitative … biology.”
Computing is part of the story…
Biology Applications yield New Math!
“Math to Bio, Bio to Math”
• Biology is drawing heavily from mathematics,
statistics and scientific computing as it
becomes a more quantitative science.
• New applications can lead to new
mathematical questions and techniques.
• This back and forth requires some
mathematicians to become biologists, and
some biologists to become mathematicians.
Overview
• Computing & biology impacting mathematics?
– Modern biology is a “hard science”
– Computing, science and mathematics
• All math is useful!
–
–
–
–
Geometry & Topology: Mariel Vazquez
Algebra: Marisa Eisenberg, Marty Golubitsky
Combinatorics, Graph Theory: Reka Albert
Core applied math: Probability, Linear Algebra,
Analysis, Dynamics & Bifurcation Theory, Statistics.
Traditional Applied Math
• Analysis, Dynamical Systems
– Differential equations models
• Probability
– Stats fundamentals, Stochastic processes
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Statistics
Linear Algebra
Calculus
…
(Simplified) Model
n
x
y
Why Bistability?
First, transform the direct transmission (3D) model by
(n, x, y) –> (n, p = x+y, i = y/p)
This gives,
Host-Resource Dynamics
Host-Resource Dynamics
Host-Resource Dynamics
Host-Resource Dynamics
Host “Hydra Effect”
Hydra Effect +
Density Dependent Transmission
Singular Hopf
1. Hopf bifurcation in the fast subsystem:
Implications for other kinds of dynamics?
2. Averaging:
How valid is this
approximation away
from the limit?
Co-author: Chris Schepper
The Model
n
x
y
Consumer/Host
Parasite/Disease
Basic Reproduction Number
Basic Reproduction Number
Three-species Dynamics
Traditional Applied Math
• Analysis, Dynamical Systems
– Differential equations models
• Probability
– Stats fundamentals, Stochastic processes
•
•
•
•
Statistics
Linear Algebra
Calculus
…
(Part II)
Ultimate Goal
Model
We model the Central Basin of Lake Erie, ignore horizontal space, dividing
the water column into 24 “patches” each roughly 1 meter deep.
N1
N1
N2
N2
N3
N…
N24
∑mξiNi
mфi(N)
N3
N…
N24
Model
We model the Central Basin of Lake Erie, ignore horizontal space, dividing
the water column into 24 “patches” each roughly 1 meter deep.
Population Dynamics
Population Dynamics
Problem: Quality Index ri
• Movement based on r = G/μ can lead to
tolerance of terribly high mortality rates!
• Solution? Stimulus ≠ Response! r = g(G)f(1/μ)
f(1/μ)
1/μ
1/μ
1/μ
Problem: Quality Index ri
• Movement based on r = G/μ can lead to
tolerance of terribly high mortality rates!
• Solution? Stimulus ≠ Response! r = g(G)f(1/μ)
• What about g(G)?
– Predator encounters inhibit foraging.
– Solution: discount ideal growth rate G accordingly.
– Probability model gives g(G, μ) = G exp(-λh) where
λ(μ) = predator attack rate, h = time displaced.
Virulence Evolution?
Model
Individual Variation
Population Level Consequences?
• High individual variation.
• Mortality is environmentally driven, random.
Pathogen Fitness
Host Fitness
Quantify Sensitivity:
Transmission-Virulence Tradeoff
Closing Remarks
• Computing & biology are creating new math!
– Computing allows easy access to quantitative thinking.
– Biology problems are pervasive in society, complex!
• All math is useful!
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Follow your passion!
Learn some programming, stats, applied math.
Collaborate
Broaden yourself mathematically; become a scientist!
Questions?