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! 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… Understanding statistical models. … 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 Statistics Linear Algebra Calculus …

(Simplified) Model y n x B propto sigma*beta. Important later, B captures ‘net effect’ of resource dependence.

(n, x, y) –> (n, p = x+y, i = y/p) 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 Hopf bifurcation in the fast subsystem: Implications for other kinds of dynamics? Averaging: How valid is this approximation away from the limit? Co-author: Chris Schepper

The Model y n x Consumer/Host Parasite/Disease Basic Reproduction Number Basic Reproduction Number

Three-species Dynamics

Routh-Hurwitz Stability Criteria Which has a characteristic polynomial of the form The equilibrium (n,x,y) is stable when each of the following hold Bonus: A Hopf Bifurcation occurs where the third criterion fails.

Hopf Bifurcation Think of region 1, EQ2 stable. We’d like to predict stability consequences of resource dependent disease parameters. Turns out slope tells us a lot. Two slopes though, so need relative slope aka relative sensitivity to assess their combined effect. Formally … dH positive ==> increase size of region where EQ2 stable. Negative, decrease.

Traditional Applied Math Analysis, Dynamical Systems Differential equations models Probability Stats fundamentals, Stochastic processes Statistics Linear Algebra Calculus …

(Part II) Ultimate Goal

Model ∑mξiNi N1 N1 N2 N2 mфi(N) N3 N3 N… N… N24 N24 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 ∑mξiNi mфi(N) N3 N3 N… N… N24 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) 1/μ f(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 Crude SA, top 3 params that give variation. Growth rate and details of non-specific response.

Population Level Consequences? High individual variation. Mortality is environmentally driven, random. Pathogen Fitness Host Fitness Stochastic process, need to know some probability! Quantify Sensitivity:

Transmission-Virulence Tradeoff Tradeoff curve = 

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! Follow your passion! Learn some programming, stats, applied math. Collaborate Broaden yourself mathematically; become a scientist!

Questions?