Why Is MathematicalBiology So Hard?Michael C. Reed

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Although there is a long history of the applica-tions of mathematics to biology, only recently hasmathematical biology become an accepted branchof applied mathematics. Undergraduates are doingresearch projects and graduate students are writ-ing Ph.D. dissertations in mathematical biology,and departments are trying to hire them. But whatshould the Ph.D. training consist of? How shoulddepartments judge work in mathematical biology?Such policy questions are always important andcontroversial, but they are particularly difficulthere because mathematical biology is very differ-ent from the traditional applications of mathe-matics in physics. Ill begin by discussing the nature of the field itself and then return to the policy questions.

Wheres Newtons Law? The phenomena thatmathematical biology seeks to understand and pre-dict are very rich and diverse and not derived froma few simple principles. Consider, in comparison,classical mechanics and continuum mechanics.Newtons Law of Motion is not just a central ex-planatory principle; it also gives an immediate wayto write down equations governing the importantvariables in a real or hypothetical physical situa-tion. Since the Navier-Stokes equations expressNewtons Law for fluids, they are fundamental andhave embedded in them both the fundamentalprinciple and the complexity of the fluid phe-nomena that we see. Thus a pure mathematicianwho proves a theorem about the Navier-Stokesequations and an applied mathematician who develops new numerical tools knows that he or shehas really contributed something. Alas, there areno such central fundamental principles in biology.There are principles of coursesome would say

dogmassuch as evolution by natural selection,no inheritance of acquired characteristics, orDNA RNA proteins. But these are not trans-latable into mathematical equations or other struc-tures without hosts of additional facts and as-sumptions that are context-dependent. This meansthat mathematical biology is very unsatisfying forpure mathematicians, who usually are interestedin discovering fundamental and universal structuralrelationships. It also means that there is no math-ematics of biology in the same way that ordinarydifferential equations is the mathematics of clas-sical mechanics and partial differential equationsis the mathematics of continuum mechanics.

Diverse, yet special. Because of evolution, bio-logical systems are exceptionally diverse, complex,and special at the same time, and this presents sev-eral difficulties to a mathematician. The first ischoosing what to work on. Theres too much biol-ogy! How do changes in the physics or chemistryof a particular environment affect the species thatlive in the environment (ecology)? How do diseasesspread within a population (epidemiology)? How do the organ systems of the human body work(physiology)? How do the neurons in our brainwork together to allow us to think and feel and calculate and read (neurobiology)? How does ourimmune system protect us, and what are the dynamical changes that occur when we are underattack by pathogens (immunology)? How do cellsuse physics and chemistry to accomplish funda-mental tasks (cell biology, biochemistry)? How doesthe genetic code, inscribed in a cells DNA, give rise to a cells biochemical functioning (molecularbiology and biochemistry)? How do DNA sequencesevolve due to environmental pressures and randomevents (genomics and genetics)?

The second difficulty is that a priori reasoningis frequently misleading. By a priori reasoning I

Michael C. Reed is professor of mathematics at Duke University. His email address is reed@math.duke.edu.

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mean thinking how we would design a mechanismto accomplish a particular task. As a simple example, both birds and planes burn something to create energy that can be turned into potentialenergy, and both have to use the properties of fluids that are implicit in the Navier-Stokes equa-tions. But that doesnt mean that one understandsbirds if one understands planes. To understand howa bird flies, one has to study the bird. Modelers aresometimes satisfied that they have created a mathematical model that captures the biologicalbehavior. But that is not enough. Our purpose isto understand how the biological mechanisms giverise to the biological behavior. Since these biolog-ical mechanisms have been designed by evolution,they are often complicated, subtle, and very spe-cial or unusual. To understand them, one mustimmerse oneself in the messy, complex details ofthe biology; that is, you must work directly with biologists.

Thirdly, different species (or different tissues ordifferent cells) may accomplish the same task bydifferent mechanisms. An astounding array of spe-cial mechanisms allows animals to exploit specialniches in their environments. For example, a diverseset of locomotory mechanisms are used at differ-ent size scales. Thus, when you have understoodbird flight completely, you have not even startedon the butterfly or the fruit fly. So, even when oneis successful, one may have provided understand-ing only in particular cases.

We can already draw some conclusions. Dont domathematical biology to satisfy a desire to find uni-versal structural relationships; youll be disap-pointed. Dont waste time developing methods ofmathematical biology; the problems are toodiverse for central methods. Whats left is the bi-ology. You should do mathematical biology only ifyou are deeply interested in the science itself. If youare, theres lots of good news. We mathematiciansare experts at thinking through complex relation-ships and formulating scientific questions as math-ematical questions. Some of these mathematicalquestions are deep and interesting problems inpure mathematics. And most biologists know thatthe scientific questions are difficult and compli-cated, so they want our help. Theres some bad newstoo; there are three more reasons why the field isso hard.

The problem of levels. In many biological prob-lems one is trying to understand how the behaviorof the system at one level arises from structures andmechanisms at lower levels. How does the coordi-nated firing of neurons give rise to the graceful motion of an arm? How does the genetic code inDNA create, maintain, and adjust a cells bio-chemistry? How does the biochemistry of a cell allow it to receive signals, process them, and sendsignals to other cells? How does the behavior of

groups of cells in the immune system give rise tothe overall immune response? How do the proper-ties of individual bees give rise to the behavior ofthe hive? How do the cells in a leaf cooperate toturn the leaf towards the sun? How does the varied behavior of individuals contribute to thespread of epidemics?

We are familiar with these types of questionsfrom physics. What are the right variables to describe the behavior of a gas, and how do the values of these variables arise from the classical mechanics of the molecules making up the gas? The behavior at the higher level is relatively sim-ple, and Newtons law suggests the few importantvariables at the lower level; even so, the proofs arenot easy. In the case of biological systems thesequestions are even more difficult, because the objects at the lower level have been designed byevolution (or trained by feedback control; see below)to have just the right special properties to giverise to the (often complicated) behavior at thehigher level. And it is usually not easy to decidewhat the important variables are at the lower level.If your model has too few, you will not be study-ing the real biological mechanism. If your modelhas too many, it may be so complicated that a lifetime of computer simulations will not give newbiological understanding. You need ideas, guess-work, experience, and luck. You need to be able todeduce the consequences from the assumptions.That is what mathematicians are good at.

The difficulty of experimentation. We mathe-maticians often have an overly simple view of ex-periments and the role they play, probably becausewe dont conduct them ourselves. A theory is testedby deciding on a few crucial variables and design-ing the right experimental setup. For example, onemeasures how fast metal beads and feathers fall in a vacuum or the angle subtended by two stars.However, the complicated histories of interactionbetween theory and experiment in quantum me-chanics, nuclear physics, and elementary particlephysics in the twentieth century show that this simple view is naive. And for several reasons the experimental situation is even more difficult in biology.

First, one is often interested in how the behav-ior at one level arises from lower levels. Typically,this emergent behavior cannot be seen in any ofthe parts at the lower level but arises because ofcomplex interactions among the parts. Unfortu-nately, it can be misleading to study the parts inisolation. For example, I try to understand howcertain biochemical networks in mammalian cellsfunction. The networks give rise to systems ofODEs in which the nonlinear terms depend on theenzyme kinetics for each separate enzyme. Theenzymes can be isolated and their reaction kinet-ics studied in vitro in experiments that combine

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pure enzyme with pure substrate. But in the soup,in the real cell, the enzymes and substrates are bind-ing to or being affected by other chemicals too, soone is unsure whether the in vitro experimentsreflect the true in vivo kinetics. That is, each ofthe parts at the lower level behaves differently inisolation than when it is connected to the otherparts.

Second, chance plays a role, not only in experi-mentation, but perhaps in explanation. Why aretwo neighboring fields dominated by black-eyedSusans and poppies respectively? Are the soils different? Are the local plant and animal species different? Or perhaps the explanation is a chanceevent in the past (or all of the above).

Third, individuals (whether cells or flowers orpeople) are both similar and different. How doesone know whether data collected is special ortypical? How does one assure oneself that rat datatells us something about humans or, indeed, some-thing about other rats?

Finally, it is characteristic of living systems thatthe parts themselves are not fixed but ever chang-ing, sometimes even affected by the behavior of thewhole (see feedback control, below). A simple truestory illustrates this point. An experimenter (#1)who used rats in his experiments was getting veryunusual results, and the results were not repeat-able week to week. After six months of this he hadto stop his experiments and investigate the rats.His rats were housed in his universitys vivarium.It turned out that another experimenter (#2) hada mean technician, and when the other experi-menters technician came to get #2s rats, #2s ratswould cry out, upsetting the rats belonging to #1.The behavior of #1s rats in experiments dependedon whether #2 was doing an experiment the sameday! Whew, Im glad I became a mathematician.

For all these reasons, biological data must be approached cautiously and critically. Since bio-logical systems are so diverse and everything seems to interact with everything else, there aremany possible measurements, and enormousamounts of data can be produced. But data itselfis not understanding. Understanding requires aconceptual framework (that is, a theory) that identifies the fundamental variables and theircausative influences on each other. In messy bio-logical problems, without simple fundamental principles like Newtons Law, useful conceptualframeworks are not easy to propose or validate. Onemust have ideas about how the structure of (or behavior of) the whole is related to the assumptionsabout the parts. Thinking through such ideas and proving the consequences of the assumptionsare important ways that we mathematicians canmake contributions.

The problem of feedback control. It is com-mon to think of biological systems as fragile.

However, most are very stable, and it is almost atautology to say so, because they must all operatein the face of changing and fluctuating environ-mental parameters; so if they werent stable, theywouldnt be here. We are familiar from engineer-ing with the concept of feedback control, wherebyvariables are sensed and parameters are then resetto change the behavior of the system. The nephronsin the kidney sense NaCl concentration in the bloodand adjust filtration rate to regulate salt and waterbalance in the body. The baroreceptor loop regu-lates blood pressure, heart rate, and peripheral resistance to adjust the circulation to differentchallenges. Numerous such control systems areknown and studied in animal and plant physiology.

There is another kind of feedback that operatesbetween levels that poses special problems. Hereare two examples. In the auditory system sensoryinformation is transformed in the cochlea to elec-trical information that proceeds up the VIIIth nerveto the cochlear nucleus and from there to variousother nuclei in the brain stem (a nucleus is a largeanatomically distinct group of cells) and on to themidbrain and the cortex. Surprisingly, there are alsoneural projections from the cortex that influencethe sensitivity of the cochlea. Second, the dogmaDNA RNA proteins function has turnedout to be a naive fiction. Genes (pieces of DNA) dontturn themselves on or off but are activated or in-hibited by proteins. That is, proteins affect thegenes, adding a reverse loop to the simple picture,implying of course that the genes affect each otherthrough the proteins. The fundamental objects,that is, the objects most closely related to func-tion, may not be genes or proteins but small networks involving both genes and proteins thatrespond in certain ways to changes in the cells environment. These kinds of examples show thatthe nineteenth-century picture of a machine withparts is a very inappropriate metaphor for (at leastsome) biological systems. When there is feedbackbetween levels, it is hard to say which are the parts!In fact, it may be hard to say which are the levels,and therefore our traditional scientific researchparadigm of breaking things into smaller parts(lower levels) may not be successful.

This is not just a philosophical point but a fun-damental research issue that deepens the impactof the previous four difficulties. Take, for example,the question of dendritic geometry. Its been onehundred years since Ramon y Cajal made beauti-ful drawings of complicated dendritic arbors onnerve cells. Is the geometry important? Surely itmust be, we feel, since cells in the same brain nu-cleus in different individuals seem to have roughlysimilar dendritic arborization. And, indeed, thereare examples where it is understood how specificdendritic geometry creates specific neuron-firingproperties and presumably specific cell function,

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though it is not always clear what function meansfor a single cell embedded in layers and layers ofa large neural network. On the other hand, supposea cell is part of a large neural network whose job it is to transform its pattern of inputs into a corresponding pattern of outputs. This neural network may have been trained to do this job byfeedback control from a higher level, in which casethe details of the dendritic geometry (and even thedetails of the neural connections) may not be im-portant at all. The details arose from the training,and they are whatever they need to be to give thebehavior at the higher level. Furthermore, for largenetworks there may be many choices of detailsthat give the same network behavior, in which caseit will be hard to infer the behavior of the wholeby studying the properties of the parts.

I now want to turn to the policy questions that I mentioned at the beginning. I have been using the term mathematical biology to refer in the broadest way to quantitative methods in the bio-logical and medical sciences. Physicists, chemists,computer scientists, and biological and medical researchers with some mathematical training canand do contribute to the field I have been referringto as mathematical biology. But let us now narrowthe focus to mathematics education, both under-graduate and graduate, and the mathematics jobmarket.

Undergraduate education. Mathematical biologyis an extremely appealing subject to undergradu-ate students with good training in freshman andsophomore mathematics. Many are naturally in-terested in biology, and all know that we are in themidst of a revolution in the biological sciences.They are usually amazed and delighted that themathematical techniques that they have learned canbe used to help understand how biological sys-tems work. Further, mathematical biology is a per-fect subject for undergraduate research projects.Biology is so diverse and so little quantitative mod-eling has been done that it is relatively easy to findprojects that use undergraduate mathematics innew biological applications. The students find suchprojects to be very rewarding. They know that theundergraduate major consists mostly of nineteenth-century mathematics; of course they are excited bytwenty-first-century applications. Here at Duke wehave found that the availability of projects inmathematical biology has attracted many studentsto the mathematics major. Of course, it helps tohave a mathematical biologist on the faculty, butit is not necessary. Any mathematician can createand supervise such projects by working coopera-tively with local biologists. It requires only the effort to make the connections and tolerance forappearing nonexpert to the students (somethingwe are not used to!).

Graduate education. There is quite a bit of dis-agreement about the proper mathematics graduatetraining of a student who wants to be a mathe-matical biologist. Ill simplify the discussion into two(extreme) positions. The first position emphasizesmaximal contact with biology and biologists aspart of graduate training. Graduate students shouldtake biology courses (including labs) and should par-ticipate in or even initiate collaborative modelingprojects. This way they learn a lot of biology, and,even more importantly, they learn modeling andhow to communicate with biologists. By doing thisthey study less mathematics of course, but they canlearn what they need later when they need it. Thesecond position emphasizes training as a mathe-matician first. Graduate students should receive thetraditional training in analysis or applied mathe-matics (or other subjects), and (ideally) the thesisshould contain some applications to biologicalproblems. But the graduate student should notspend too much time slogging around in the bio-logical details or working on collaborative projects.It is the job of the thesis advisor to be the interfacebetween the graduate student and the biology.Later, after the Ph.D., when the mathematician is es-tablished, he or she can choose to become involvedin collaborations and learn more biology.

I guess that most mathematical biologists sup-port the first position. I support the second, perhaps because that is the route that I followedmyself. Mathematical biology is really a very hardsubject (I hope I have convinced you of that), anda great many ideas and techniques from differentbranches of mathematics have proven useful. Somathematical biologists need broad training inmathematics. Secondly, I believe that only deepand rigorous graduate training creates mathe-maticians who can not only learn new mathemat-ics when they need it, but who can also recognizewhat they need to learn.

Hiring issues. Hiring a mathematical biologistposes special challenges for departments. Most math-ematicians have no idea how large the field biologyis or how large the research communities are. Two examples illustrate this. Here at Duke, Arts and Sci-ences has 469 tenure-track faculty members, and the Medical Center has 767 (not counting clinical faculty). My colleague Harold Layton is a mathe-matician who works on the kidney, so of course hegoes to the annual meeting of the American Societyfor Nephrology, where the typical registration num-ber, 12,000, completely dwarfs the registration at the Joint Mathematics Meetings. And those consti-tute just a subset of the researchers who work on thekidney! So, the first issue is thinking about what kindof mathematical biologist you want. It is a good ideato involve local biologists and medical researchersin preliminary discussions, both to educate depart-ment faculty and to understand the local context.

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The question of how best to judge job candidatesis particularly difficult in mathematical biology.First, most mathematicians know less biology thantheir tenth-grade daughters. Thats just the way itis; biology was not a mathematically related disci-pline when we were growing up. But, more impor-tantly, mathematical biology really isnt a field ofmathematics with a coherent community that cantestify meaningfully about young people. Mathe-matical biology is fragmented because biology itself is so diverse. A mathematician working on thelung might want to talk to pulmonary physiologistsor geometric analysts who are experts on fractals,but why would he or she want to talk to mathe-matical biologists working on the kidney, the neu-robiology of hearing, or the epidemiology of AIDS?In each area of specialization there is a tremendousamount of biology to learn, and if one doesnt havethe background, its hard to judge the strength ofan individuals contributions. This is true even forus, the mathematical biologists, the experts youexpect to consult. So hiring in mathematical biol-ogy necessarily involves intuition and high risks aswell as high potential payoffs for the departmentand the college.

The first step. The best way for departments toovercome these difficulties is to encourage seniorfaculty to become involved in bringing biological applications and student projects into the under-graduate curriculum. This should be done by working cooperatively with local biologists to create examples and projects related to their own specialties. All mathematicians can do this, and they do not have to give up their own researchagendas or become mathematical biologists; it only requires effort.

This strategy for engagement with biology hasgreat benefits both for departments and individu-als. The faculty as a whole will become educated inbiology and thus better able to judge job candidatesin mathematical biology, and the undergraduate curriculum will be more attractive. More importantly,departments and individuals will be participatingintellectually in the biological revolution, the great-est scientific revolution of our times, perhaps of alltimes. The task is to understand how life, in all its diversity and detail, works. This includes how weact, think, and feel, and how we influence and areinfluenced by other forms of life. We mathemati-cians have the technical and intellectual tools tomake enormous contributions. So, surely, this is ourresponsibility and our opportunity.