Yes, I love math, chemistry, and physics for insight into biology
I’ve enjoyed hugely applying the concepts and methods of physics and chemistry, the fields in which I started, to biological systems. It is a bit unusual to have published scientific articles in plant physiology and ecology that included integral equations, Lagrange multipliers, root-searching algorithms, differential equations, adjoint equations, linear algebra, spherical trigonometry, and such. (I tip my hat to Mark Denny, whose command of math in biology is striking and so insightful – check out his Air and Water book!)
Why use mathematical models in biology? In a phrase, they tell us much more than “conceptual models,” verbal models with linkages but nothing quantitative. In fact, we unconsciously use models at every turn in science, or in real life. We do a titration and assume the model of chemical equilibria, maybe with some explicit parameters such as pKa and pKb. We have in our minds a model of how plants grow, even if we don’t quantify how much nutrient (N, P, …) is in the soil or what genetic capabilities a plant has for branching and leafing patterns, or how it intercepts sunlight. We do know that nutrients, developmental patterns, and light interception all matter, in a “soft” way. We have to have a model of the world, even to walk on a hill. Some of us go deeper, into “first principles,” looking at how properties of enzymes and environmental variables quantify photosynthesis, or how electronic energy levels in metals predict conductivity. (We don’t deliberately go too far, using, say, quantum chromodynamics and quantum electrodynamics to explain molecular bonding!)
Putting it together: how and why I do modeling
In 2009 I co-organized a session of the Ecological Society of America on modeling, along with Lou Gross of the University of Tennessee and 4 others. I established a wiki and posted some of the gleanings from my career:
- My philosophy of modeling, an htm file. It includes links that are mostly inactive, except for two:
- The syllabus I used in my biological modeling class – rather exhaustive at 14 pages, covering modeling goals, math, methods, etc.
- An example of a model, never published, on optimizing deficit irrigation of pecan trees. This was built around the time that I published a number of papers on pecan water use and nutrition with co-authors, and before I co-authored a paper analyzing many fluxes out of and into pecan orchards (water vapor, CO2, heat, solar radiation, thermal infrared radiation), using comprehensive models to analyze field data from an eddy-covariance system. That analysis gave the striking conclusion that pecans, very anomalously, lack a water-preserving response to atmospheric humidity. I link that paper here, too.
- Types of models, another htm file, covering mathematical and conceptual structure. Two of the links are active and relevant:
- Experimental data and a novel model of functional balance, illuminating the physiological acclimations of plants to grow at extremely low concentrations of N and P. My wife, Dr. Lou Ellen Kay, and I built and ran a monster apparatus (at least when you consider two people had to do it, day and night).
- A model, based on generic data, of how plants may change in performance as atmospheric CO2 rises. Changes in water- and nitrogen-use efficiency, nitrogen content, and CO2 assimilation rate vary among species; plausible interspecies variations portend significant changes in species distributions and abundance.
- Models discriminated by purpose: optimization, simplification, developing testable hypotheses, synthesizing multiple concepts, teaching concepts, and more.
- Interesting uses of inverse models, using models to estimate inputs (parameters that describe the dynamical system).
- A flier of sorts, attempting to interest ecologists working on the Jornada Long-Term Ecological Research site in developing truly quantitative models – getting a clear understanding of state variables, initial conditions, boundary conditions, parameters, equations of motion.
- My presentation on possible roles of models of plant physiology, structure, and ecological interactions, at a symposium I was tasked with organizing at the Mathematical Biology Institute at the Ohio State University. The international group of 40 participants was a joy to be with.
A caveat, or many: limits of optimization models
One thread in a number of my publications is optimization. My first application was to try to understand the pattern of plants’ investment in photosynthetic capacity as it varies with optical depth in the canopy. A qualitative argument is plausible: invest more where the return is best. To get quantitative, I worked with Dutch physicist Frits Wiegel, using the calculus of variation and Lagrange mutlipliers to put numbers to it (American Naturalist 132 (1988): 67-86). Of further interest, we noted that plants show the pattern we derived, but the average leaf mass per area seems to have evolved to be lower than we predicted – leaves are thinner, and, for the same total mass, cover more ground area than may be optimal for all the conspecific plants in a stand. We proposed that the extra shading of competitors pays off; this was tested in 1999 by Schieving and Poorter (New Phytologist 143: 201-211_! In a monograph (Functional Biology of Crop Plants, Croom Helm, London/Timber Press, Beaverton, OR, 1987) I worked in a number of other cases. I touched on optimization in additional publications.
Optimization seemed to offer insight into how plants work, physiologicall and ecologically. Others applied optimization to complex problems as well, notably Tim Buckley, now at the University of Sydney, and Ülo Niinimets of the Estonian University of Life Sciences. Of course, we are increasingly aware that we don’t know the objective function for plants – what are they trying to maximize or minimize? Total growth is not a good answer, though agronomists would like it so; there are risks of herbivory for plants of high photosynthetic competence, and risks of depleting soil nutrients too early; there are phylogenetic constraints – sorry, guys, we don’t have the genetic tools in our species; there are physicochemical constraints – Rubisco, the key enzyme for CO2 fixation in the photosynthesis of all green plants, can’t help incorporating O2 into nascent sugars and reversing the work of the past. Finally, in 2015, Tim Buckley and I exchanged emails sending up a white flag about optimization.
Rewarding: predictive models
In 1984, I published two articles in Photosynthetica (18: 549-568 and 569-595; yes, long and detailed). I presented a detailed model of plant leaves, resolving their optical and photosynthetic structure as well as control systems. I parametrized the models, including the examination of the effect of changing total investment in chlorophyll. Surprisingly, at first, when put into a whole-canopy model with light interception, the model predicted that leaves with half-normal chlorophyll content, as in well-known mutant peas, would confer a gain of about 8% in seasonal biomass yield! Basically, more light was shared with leaves deeper in the canopy, increasing total light utilization. Remarkably, John Hesketh’s group at the University of Illinois field-tested this and found a gain of 8%! (Crop Science 29: 1025-1029, 1989.) Why do agronomists have no interest? It’s not because the only “pale mutants” are heterozygous, planting out as 1/4 full green, 1/2 pale, and 1/4 albino = doomed. There are homozygous pale mutants…but pale is the word that puts farmers off their feed; the plants don’t look good. These plants work better in monocultures than do wild-types…but not in competition in natural selection. Nor, it seems, in artificial selection if breeders don’t want to risk proposing odd plants to farmers.