arXiv:q-bio/0409001v1 [q-bio.OT] 1 Sep 2004 nt oi´cioNl 73 ´xc .F,M F., M´exico D. 07738 Polit´ecnico Matem´aticas, F´ısica Nal. y de Inst. Sup. F´ısica, Esc. de Depto. nd i ila se,HG16Mnra,QC, Montreal, 1Y6 H3G Osler, CANADA William Sir Prom- for 3655 enade University, Centre McGill and Dynamics, Mathematics Nonlinear & Physics Physiology, ato h 8haderypr fte19th the of part early and 18th the of part era. new the a as of emerged and cornerstones founded were science mod- ern and capitalism, In democracy, and period changes. that politics, significant economics, through went result science a As political merchants and power. as economic gained Centuries, craftsmen the 18th and between the changed and inher- Greeks, 16th attitude, the This from considered activity. ited was class lower labor ex- a manual perform not as did today) periments them scientists know we of as forefathers (the philosophers natu- ral Revolution, Be- Scientific so-called development. the this fore to seminal was others and Vesalius, Harvey, William Kepler, hannes Jo- people Galilei, Galileo of Copernicus, work Nicholaus The like the Centuries. to back 17th traced and be 16th can birth theoretical Its any construct. litmus validate to ultimate which against the test as considers results which experimental and solid a framework, has which conceptual that means science Modern Introduction uigteElgtnet ntelatter the in Enlightenment, the During ∗ † ahmtc,Booy n hsc:Itrcin and Interactions Physics: and Biology, Mathematics, -al ooef.p.x emnn address: Permanent [email protected], e-mail: -al akycdmgl.a eatet of Departments [email protected], e-mail: ihe .Mackey C. Michael Interdependence coe 4 2018 24, October EXICO ´ ∗ 1 a eecntutdt rdc ttcelec- static produce to constructed Leyden were first the jar The and machines generating electromagnetism. electric of that alleled par- electrophysiology his- the of Century, development 19th torical the of middle etc. the to stimulation, (like Up to cells cells) muscle specialized and of neurons of response properties the magnetic tissue, the and organs, electric specialized intrinsic some elec- in and currents fields tric magnetic or gen- electric the of includes eration This tissues. biological and fields studies electromagnetic between that interaction the science the is Cen- Electrophysiology 19th and turies 18th The sit- current the of review exam- brief uation. a historical on important on and based very ples future, a the that play in argue to role We continue will sciences. this biological mathematics, the physics, and histor- between rich the relation of taste ical a give important. we paper highly this be, In has to continues which and work rep- been, fairly interdisciplinary to the fails have resent it would However, years of difficult. 200 body been past growing the without rapidly of and the knowledge with instances, dealing some it in classification well This works be classified. to hierarchically started disciplines scientific Centuries, Mois´es Santill´an † tricity for a specific purpose: to “electrify” tions occurred. (It has been suggested that and to stimulate humans. The Voltaic pile the assistant was Galvani’s wife Lucia, who was developed with the idea of galvanic (i.e. is known to have helped him with his experi- direct current, as opposed to faradic or alter- ments). This is cited as the first documented nating current) stimulation. Bioelectric and experiment in neuromuscular electric stimu- biomagnetic measurements were the incentive lation. for the development of sensitive measurement Galvani continued the stimulation studies instruments, like the galvanometer and the with atmospheric electricity on a prepared capillary electrometer. Thus, it is no sur- frog leg. He connected an electric conductor prise that some scientists of the time made between the side of the house and the nerve important contributions to the development innervating the frog leg. Then he grounded of both the biological and the physical sci- the muscle with another conductor in an ad- ences. In the following paragraphs we present jacent well. Contractions were obtained si- a brief review of the work of some of these in- multaneous with the occurrence of lightning terdisciplinary workers. We do not attempt flashes. In September 1786, Galvani was try- to present a detailed review of the history of ing to obtain contractions from atmospheric electrophysiology, as our purpose is only to electricity during calm weather. He sus- exemplify the rich interdisciplinary interac- pended frog preparations from an iron railing tions of the 18th and 19th Centuries. in his garden by brass hooks inserted through The essential invention necessary for the the spinal cord. Galvani happened to press application of a stimulating electric current the hook against the railing when the leg was was the Leyden jar (a capacitor formed by also in contact with it. Observing frequent a glass bottle covered with metal foil on the contractions, he repeated the experiment in a inner and outer surfaces), independently in- closed room. He placed the frog leg on an iron vented in Germany (1745) and The Nether- plate and pressed the brass hook against the lands (1746). With it, Benjamin Franklin’s plate, and muscular contractions occurred. experiments allowed him to deduce the con- Systematically continuing these experiments, cept of positive and negative electricity in Galvani found that when the nerve and the 1747. Franklin also studied atmospheric elec- muscle of a frog were simultaneously touched tricity with his famous kite experiment in with a bimetallic strip of copper and zinc, a 1752 (many American school children have contraction of the muscle was produced. This heard the apocryphal stories of Franklin fly- experiment is often cited as the classic study ing kites during thunderstorms strings soaked to demonstrate the existence of animal elec- in salt water). tricity. Galvani did not understand the mech- The most famous experiments in neu- anism of the stimulation with the bimetallic romuscular stimulation of the time were strip. His explanation for this phenomenon performed by Luigi Galvani, professor of was that the bimetallic strip was discharging anatomy at the University of Bologna. His the animal electricity existing in the body. first important finding is dated January 26, Galvani’s investigations intrigued his friend 1781. A dissected and prepared frog was ly- and colleague Alessandro Volta (professor of ing on the same table as an electric machine. physics in Pavia), who eventually came up When his assistant touched the femoral nerve with a totally different (and correct) expla- of the frog with a scalpel, sparks were si- nation for the phenomena that Galvani was multaneously discharged in the nearby elec- trying to explain. In the process, Galvani and tric machine, and violent muscular contrac- Volta maintained their friendship (in spite

2 of their differences of scientific opinion), and The 20th Century Volta developed the ideas that eventually led to the invention of the Voltaic pile in 1800 In the 18th and 19th Centuries interdisci- (forerunner of the modern battery), a battery plinary research bridging physics, mathemat- that could produce continuous electric cur- ics and biology was carried out by scien- rent. Incidentally, Volta only completed the tists educated as physicians. The 20th Cen- equivalent of his doctoral dissertation when tury witnessed a reversal of this trend with he was 50 years old! major contributions to biology from people with solid backgrounds in physics and math- ematics. There are two of these disciplines in which the contributions by physicists and All of these contributions to electrophysiol- mathematicians were particularly important: ogy were experimental. The first significant electrophysiology (following the tradition of theoretical contributions were made by the Galvani, Volta, Helmholtz, etc.) and molec- German scientist and philosopher Hermann ular biology. Ludwig Ferdinand von Helmholtz. A physi- cian by education and, in 1849, appointed Electrophysiology professor of physiology at K¨onigsberg, he moved to the chair of physiology at Bonn The growth of biophysics owes much to A. V. in 1855 and, in 1871, was awarded the Hill, whose work on muscle calorimetry was chair of physics at the University of Berlin. essential to our understanding of the physi- Helmholtz’s fundamental experimental and ology of muscle contraction. Hill received an theoretical scientific contributions in the field undergraduate degree in physics and math- of electrophysiology included the demonstra- ematics, and a doctorate in physiology, all tion that axons are extensions of the nerve from Cambridge. Besides his work on mus- cell body, the establishment of the law of cle contraction, Hill also addressed problems conservation of energy (the First Law of related to the propagation of the nervous im- Thermodynamics), the invention of the myo- pulse, the binding of oxygen by hemoglobin, graph, and the first measurement of the ac- and on calorimetry of animals. He discov- tion potential conduction velocity in a mo- ered that heat is produced during the nerve tor nerve axon. Besides these, the contribu- impulse. Hill’s original papers reveal an el- tions of Helmholtz to other fields of science in- egant mixture of biological concepts and ex- clude fundamental work in physiology, acous- periments together with physical and math- tics, optics, electrodynamics, thermodynam- ematical theory and insight. His discoveries ics, and meteorology. He invented the oph- concerning the production of heat in muscle thalmoscope and was the author of the the- earned him the Nobel Prize in 1922, and his ory of hearing from which all modern theories research gave rise to an enthusiastic follow- of resonance are derived. Another important ing in the field of biophysics. He was in- contribution to the development of biophysics strumental in establishing an extremely suc- was Helmholtz’s philosophical position in fa- cessful interdisciplinary school in Cambridge, vor of founding physiology completely on the whose investigators received a number of No- principles of physics and chemistry at a time bel prizes. when physiological explanations were based A few years later Bernard Katz, working on vital forces which were not physical in na- at University College London with his stu- ture. dent Paul Fatt, made a major advance in

3 our understanding of the chemical and quan- were A. L. Hodgkin and A. F. Huxley. Both tal nature of synaptic transmission in the pa- studied physics, mathematics, and physiol- pers “An analysis of the end-plate potential ogy at Trinity College, Cambridge, whose recorded with an intra-cellular electrode” and high table included, at that time, an aston- “Spontaneous subthreshold activity at motor ishing array of scientific talent with people nerve endings”, which were marvels of exper- like J. J. Thomson, Lord Rutherford, F.W. imental investigation combined with math- Aston, A.S. Eddington, F.G. Hopkins, G. ematical modelling of stochastic processes. H. Hardy, F.J.W. Roughton, W.A.H. Rush- Katz was one of the recipients of the 1970 ton, A.V. Hill, and E.D. Adrian. Hodgkin Nobel Prize for “discoveries concerning the and Huxley developed a long-lasting collab- humoral transmittors in the nerve terminals oration, interrupted only by the outbreak of and the mechanism for their storage, release World War II. and inactivation.” In 1938 Hodgkin spent the summer with Jumping back a few decades, the German Cole at Woods Hole, and they demonstrated physical chemist Walter Nernst was inter- the overshoot of the action potential which ested in the transport of electrical charge in had significant implications in terms of po- electrolyte solutions. His work intrigued an- tential ionic mechanisms. It seems reasonable other physicist, Max Planck, one of the fa- to suppose that Cole and Hodgkin discussed thers of modern quantum theory, who ex- the possible meanings of these discoveries and tended Nernst’s experimental and theoreti- what types of experiments were needed to de- cal work, eventually writing down a trans- termine exactly what was going on. Because port equation describing the current flow in of their training they would have seen that an electrolyte under the combined action of some means must be found to bring under ex- an electric field and a concentration gradi- perimental control the variable (either mem- ent. This work lay largely forgotten until the brane current or membrane voltage) that is 1930s, when it was picked up by the physi- responsible for the all-or-nothing behavior of cist Kenneth S. Cole at Columbia Univer- the action potential. Hence taming the ac- sity and his graduate student David Goldman tion potential required either controlling the (originally trained in physics). They realized current or the voltage. They undoubtedly that the work of Nernst and Planck (in the realized that space clamping was necessary form of the Nernst-Planck equation) could be for both current and voltage clamping and used to describe ion transport through bio- since both knew cable theory, they knew that logical membranes and did so with great ef- space clamping was best done by drastically fect. Their work resulted in the development reducing internal resistance (space clamping) of the Goldman equation, which describes the so the space constant was much longer than membrane equilibrium potential in terms of the length of axon under study. intra- and extracellular ionic concentrations The Second World War interrupted these and ionic permeabilities. This background investigations and Cole, like hundreds of theoretical work of Nernst and Planck was other scientists, was caught up in the war also instrumental in helping Cole to experi- effort. Cole moved from Columbia to the mentally demonstrate that there was a mas- Manhattan Project in Chicago and worked sive increase in membrane conductance dur- on radiation dosimetry and radiation dam- ing an action potential. age in tissues during the war. After the war Two of the most distinguished alumni he was at the University of Chicago for a of Hill’s Cambridge interdisciplinary school few years. When the war was over, one of

4 the positive outcomes was the existence of of the techniques that he had learned from the high-input impedance vacuum tubes that had Russian applied mathematics literature to an been developed for the amplifiers in radar analysis of the Hodgkin-Huxley equations. receivers. Cole, working with Marmont in That reduction of the Hodgkin-Huxley equa- Chicago, used these new electronic advances tions later became known as the FitzHugh- to build a feedback circuit that allowed them Nagumo model and has given us great in- to space clamp axons. These axons devel- sight into the mathematical and physiolog- oped an all-or-none action potential when suf- ical complexities of the excitability process. ficiently depolarized, and the implication was Another consequence of the Hodgkin-Huxley that voltage clamping was necessary to tame model, taken to its interpretational extreme, the axon to measure the dependence of mem- was the implication that there were micro- brane current on membrane voltage. scopic “channels” in the membrane through Shortly after the war (1948), Hodgkin vis- which ions would flow and which were con- ited the United States and Cole’s laboratory trolled by membrane potential. There were in Chicago, and realized that the results of strong experimental data also leading to the the space clamp experiments meant that volt- same conclusion including the binding of age clamping was the way to go. On his re- tetrodotoxin (TTX) to nerve membranes to turn to England he teamed up with Huxley to block sodium currents, titration studies indi- really measure what was going on during the cating that there were about 20 TTX binding generation of an action potential in the squid sites per square micrometer, and membrane giant axon. This work was published in a noise measurements. However, it was left to brilliant series of five papers in the Journal of the German physicist Erwin Neher, in con- Physiology in 1952. The final one is an intel- junction with the physiologist Bert Sakmann, lectual tour de force combining both experi- to develop the technology and techniques that mental data analysis and mathematical mod- eventually allowed them to demonstrate the elling (the Hodgkin-Huxley equations) that existence of these ion channels. They were eventually won Hodgkin and Huxley the No- awarded the Nobel Prize in 1991 for this bel Prize in 1963, along with J.C. Eccles, “for work. Modifications of the Hodgkin-Huxley their discoveries concerning the ionic mech- equations were soon proposed for cardiac tis- anisms involved in excitation and inhibition sue as well as a myriad of other excitable cells. in the peripheral and central portions of the Extensions of the work of Hodgkin and nerve cell membrane.” Huxley the mathe- Huxley soon followed. For example, J. W. matician/physiologist was not content to stop Woodbury (a physicist turned physiologist) there, however, and went on to publish his and his student W. E. Crill found that current celebrated review of muscle contraction data injected into one cell in a sheet of heart mus- and its synthesis into the mathematically for- cle changed the membrane voltage in nearby mulated cross bridge theory in 1957, a theory cells in an anisotropic manner. This showed that still stands in its essential ingredients to- that there must be low resistance connections day. between abutting cells in heart tissue and The Hodgkin-Huxley model for excitabil- paved the way for the discovery and charac- ity in the membrane of the squid giant axon terization of gap junctions between the cells is complicated and consists of one nonlin- (in a variety of tissues such as epithelia). ear partial differential equation coupled to Woodbury also showed that Eyring reaction three ordinary differential equations. In the rate theory, learned from his famous foster early 1960s Richard FitzHugh applied some Doctorvater Henry Eyring, can be used to ex-

5 plain the linear current-voltage relationship pact is so broad and pervasive. The Notices of open sodium channels by choosing the ap- of the American Mathematical Society (De- propriate electrochemical potential profile en- cember 1999) has a very nice article by Nancy countered by a sodium ion while traversing a Kopell with some of the mathematical side of Na ion channel. This, together with other the story, and Nature Neuroscience (Novem- lines of types of experimental evidence men- ber 2000) featured some of this from a bio- tioned above, established the feasibility of the logical perspective in an interesting and lively ion channel concept before single channel con- series of survey articles. ductances were directly measured by Neher. One of the most remarkable individuals Molecular biology interested in the dynamic behavior of sim- ple nervous systems was H. K. Hartline of Genetics started in 1866, when Gregor the John Hopkins University. Hartline was Mendel first deduced the basic laws of inher- trained as a physiologist, and following re- itance. However, modern genetics, with its ceipt of his M.D. spent an additional two capacity to manipulate the very essence of liv- years at Hopkins taking mathematics and ing things, only came into being with the rise physics courses. For some unaccountable of molecular investigations culminating in the reason he was still not satisfied with his breakthrough discovery of the structure of training and obtained funding to study for DNA; for which Francis Crick, James D. Wat- a further year in Leipzig with the physicist son, and Maurice Wilkins received the Nobel Werner Heisenberg and a second year in Mu- prize in 1962. The contribution of physics nich with Arthur Sommerfeld. Armed with and physicists to this, what Watson calls Act this rather formidable training in the bio- 1 of molecular biology’s great drama, was logical, mathematical, and physical sciences seminal. Here we review the work of some he then devoted the majority of his profes- of the physicists who helped shape molecular sional life at Hopkins to the experimental biology into the exciting science it currently study of the physiology of the retina of the is. horseshoe crab Limulus. His papers are a Max Delbr¨uck received his doctorate in marvel of beautiful experimental work com- theoretical physics from the University of bined with mathematical modelling designed G¨ottingen, and then spent three postdoctoral to explain and codify his findings, and his years in England, Switzerland, and Denmark. life work justly earned him the Nobel Prize His interest in biology was aroused during in 1967 (with George Wald) “for his dis- his stay in Denmark by Niels Bohr’s specu- coveries concerning the primary physiological lation that the complementarity principle of and chemical visual processes in the eye.” As quantum mechanics might have wide applica- an aside we should point out that FitzHugh tions to other scientific fields, and especially (of the FitzHugh-Nagumo reduction of the to the relation between physics and biology. Hodgkin-Huxley model) received his Ph.D. Back in Berlin, Delbr¨uck initiated an interdis- in biophysics (where he learned mathemat- ciplinary collaboration with Nikolai W. Tim- ics, physics, and chemistry) under Hartline ofeeff and Karl G. Zimmer on biologically in- after completing his biological studies at the spired problems. Based on X-ray induced University of Colorado. mutagenesis experiments and applying con- One can hardly underestimate the impact cepts from quantum mechanics, they sug- that this work in excitable cell physiology has gested that chromosomes are nothing more had on the biological sciences since the im- than large molecules and that mutations can

6 be viewed as ionization processes. These re- on the Italian front. For a variety of reasons, sults were published in 1935. Schr¨odinger’s Schr¨odinger moved constantly, holding posi- little book “What is Life?” (1944) was in part tions in Austria, Switzerland, Germany, Eng- inspired by this paper. land, and then Austria again. Soon after he In 1937, Delbr¨uck moved from Germany to took up this last position in Graz, Austria fell the United States, and decided to remain af- into the hands of the Nazis, and Schr¨odinger ter the start of World War II. At that time escaped to Ireland since his initial departure he initiated a fruitful collaboration with Sal- from Berlin when the National Socialists took vador Luria on the genetic structure of bac- power was considered an unfriendly act. teriophage (bacteria-infecting viruses) and on In Ireland, Schr¨odinger joined the Institute the genetic mechanism of DNA replication. for Advanced Studies in Dublin. His con- After the outbreak of the war, Delbr¨uck and tract required him to give a yearly series of Luria were classified as ’enemy aliens’ by public lectures. In 1943, he elected to dis- the American government despite their open cuss whether the events in space and time opposition to the Nazi and Fascist regimes. which take place within the spatial bound- This classification fortuitously allowed them ary of a living organism can be accounted for to pursue their own investigations without by physics and chemistry in light of the most having to join any military project. For “their recent developments in quantum mechanics discoveries concerning the replication mecha- and its application to genetics. These lec- nism and the genetic structure of viruses,” tures were published in book form in 1944 Delbr¨uck and Luria were awarded the Nobel under the title “What is Life?” After dis- Prize in 1969, along with Alfred D. Hershey. cussing how thermodynamics plays a role in In the early 1950’s Delbr¨uck’s research inter- the processes of life and reviewing the not- ests shifted from molecular genetics to sen- so-recent results on mutagenesis by Delbr¨uck sory physiology, with the goal of clarifying the et al., Schr¨odinger argued in “What is Life?” molecular nature of the primary transduction that life could be thought of in terms of stor- processes of sense organs. Delbr¨uck was also ing and transmitting information. Chromo- involved in setting up an institute of molecu- somes were thus simply bearers of informa- lar genetics at the University of Cologne. It tion. Because so much information had to was formally dedicated on June 22nd, 1962, be packed into every cell, Schr¨odinger argued with Niels Bohr as the principal speaker. His it must be compressed into what he called lecture entitled “Light and Life Revisited” a ‘hereditary code-script’ embedded in the commented on his original one of 1933, which molecular fabric of chromosomes. To under- had been the starting point of Delbr¨uck’s in- stand life, then, it was necessary to iden- terest in biology. It was to be Bohr’s last for- tify these molecules, and crack their code. mal lecture. He died before completing the Schr¨odinger’s book had the very positive ef- manuscript of this lecture for publication. fect of popularizing the Delbr¨uck paper and Erwin Schr¨odinger is regarded as one of the of rephrasing some important questions de- fathers of quantum mechanics. However, his rived from it in a language accessible to the interests went far beyond physics. He was non-expert. The book’s publication could not particularly interested in philosophy and biol- have been better timed, and it was tremen- ogy. Early in his career, he made substantial dously influential. Many of those who would contributions to the theory of color vision. play major roles in the development of molec- Schr¨odinger’s personal life was tumultuous. ular biology were drawn to this field after He participated as an officer in World War I reading “What is is Life?” Schr¨odinger’s re-

7 cruits included Francis Crick, James D. Wat- come a painter in Paris. However, he too son, Maurice Wilkins, Seymour Benzer, and had read Schr¨odinger’s book and biology in- Fran¸cois Jacob. tervened. Franklin, working in Wilkins’ lab, Francis Crick studied physics at Univer- recorded the DNA X-Ray diffraction patterns sity College, London. After graduating, he that allowed Watson and Crick to beat Paul- started research for a doctorate, but this was ing in the race to determine the structure of interrupted by the outbreak of World War DNA. Crick, Watson, and Wilkins received II. During the war he worked as a scientist the Nobel Prize in 1962 “for their discov- for the British Admiralty, mainly on mag- eries concerning the molecular structure of netic and acoustic mines. When the war nuclear acids and its significance for infor- ended, Crick had planned to stay in mili- mation transfer in living material.” Rosalind tary research but, on reading Schr¨odinger’s Franklin had died at an early age a few years book, he joined the Medical Research Council before, and was not recognized for her essen- Unit in Cambridge to study biology. In 1951, tial contributions. Crick started a collaboration with James D. Knowing the structure of DNA was only Watson, who came to Cambridge as a post- the start. Next it was necessary to find the doctoral fellow. Watson had originally con- sequence of genes and chromosomes, to un- sidered being a naturalist, but he was also derstand the molecular machinery used to hooked on gene research by Schr¨odinger’s read the messages in DNA, and to understand book. Linus Pauling had discovered the alpha the regulatory mechanisms through which the helix protein structure by making scale mod- genes are controlled. These questions were els of the different parts of the molecule, and answered by a second generation of molecular working out possible 3-dimensional schemes biologists like Seymour Benzer, Sydney Bren- to infer which type of helical fold would ner, Fran¸cois Jacob, Jacques Monod, and be compatible with the underlaying chemi- Walter Gilbert. Seymour Benzer and Walter cal features of the polypeptide (amino acid) Gilbert had also been educated as physicists, chain. Following Pauling’s approach, Watson but were attracted to the excitement of the and Crick started to look for the structure new science. Seymour Benzer also heeded the of DNA, which in 1944 had been discovered clarion call of the Schr¨odinger book. He was to be the substance making up the chromo- a pioneer of gene sequencing. Among other somes. They finally succeeded in the Spring things, Benzer was the first to produce a map of 1953. Not only did they determine the of a single bacteriophage gene, rII, showing structure of DNA, but they also proposed a how a series of mutations (all errors in the scheme for its replication. gene script) were laid out linearly along the Essential for the work of Watson and Crick viral DNA. were the experimental results of Rosalind Walter Gilbert received his doctorate in Franklin and Maurice Wilkins. Franklin had theoretical physics and, after becoming pro- a background in chemistry while Wilkins was fessor at Harvard, worked on particle physics a physicist. During World War II, Wilkins and quantum field theory for a number of worked in the Manhattan Project. For him, years. Then his interests shifted. In 1960 as for many other of the scientists involved, Gilbert joined James Watson and Fran¸cois the actual deployment of the bombs in Hi- Gros in a project to identify messenger roshima and Nagasaki, the culmination of all RNA. After a year of work on this prob- their work, was profoundly disillusioning. He lem, Gilbert returned to physics only to re- considered forsaking science altogether to be- return to molecular biology shortly after-

8 wards. Some of the more important contri- problems involving biomathematics (Septem- butions of Gilbert and his collaborators to ber, 1995) or molecular biology (April and this field are: the discovery that a single May, 2002). messenger molecule can service many ribo- Many major universities in the world have somes at once and that the growing proteinic at least one research group working in these chain always remains attached to a transfer fields. However, listing them all is be- RNA molecule; the isolation of the lactose yond the scope or the intent of this arti- repressor, the first example of a genetic con- cle. Our purpose has only been to illustrate trol element; the invention of the rolling circle how widespread and important biophysics model, which describes one of the two ways and biomathematics have been in the past DNA molecules duplicate themselves; the iso- few centuries and the increase in their impor- lation of the DNA fragment to which the lac tance in the past few decades. repressor binds; and the development of rapid Darwin’s theory states that, given the en- chemical DNA sequencing and of recombi- vironmental conditions, the fittest individu- nant DNA techniques. Walter Gilbert and als are the ones that survive and reproduce. Frederick Sanger received the Nobel Prize in However, it is impossible to identify the cur- 1980, “for their contributions concerning the rent fittest individuals whose genes are go- determination of base sequences in nucleic ing to pass to the next generation. They can acids.” be pinpointed only after they have survived. Thus, according to some, Darwinism is tau- tological since it only predicts the survival of Present and future per- the survivors. In trying to foresee the future of science, we face the same problem. It is not spectives possible to identify the current areas of scien- tific research that will play a relevant role in What we have described so far have been a the development of science and technology. few of the significant advances made in the Even though we acknowledge this problem, study of systems in which there was a cer- it is our belief that, given the fruitful his- tain clear and obvious physics and mathe- torical relation and the present blooming of matics component to the research being car- biological, physical, and mathematical inter- ried out. The advances made in the biological disciplinary sciences, they are going to be so understanding were often quite dependent on important in the near future that the avant the application of physical and mathematical garde biological scientists will be those with a principles, or the development of the physics strong background in both the biological and and the mathematics was clearly driven by the physical-mathematical sciences. observations in biology. This strong interde- The mathematical and computational pendence is mirrored in the highlighting of modelling of biological systems is a subject biologically oriented problems in the new mil- of increasingly intense interest. The acceler- lennium (January, 2000) issues of Physics To- ating growth of biological knowledge, in con- day and the Notices of the American Mathe- cert with a growing appreciation of the spa- matical Society as well as the special Novem- tial and temporal complexity of events within ber, 2000 Nature Neuroscience issue “Com- cells, tissues, organs, and populations, threat- putational Approaches to Brain Function.” ens to overwhelm our capacity to integrate, The Notices of the American Mathematical understand, and reason about biology and Society have on several occasions focussed on biological function. The construction, anal-

9 ysis, and simulation of formal mathematical still to be developed. This has opened models is a useful way to manage such prob- a whole new area of research known as lems. Metabolism, signal transduction, ge- bio-informatics, which is rapidly growing netic regulation, circadian rhythms, and var- and, presumably, will keep on growing ious aspects of neurobiology are just a subset at an accelerated pace in the next few of the phenomena that have been successfully years. However, we are of the opinion treated by mathematical modelling. What that the sequence analysis component are the likely areas of advancement for the of bio-informatics will quickly evolve to future? Predicting the future has fascinated become a mere tool widely and easily and confounded man for centuries, probably used by scientific practitioners (in anal- for as long as he has been able to articulate ogy with the transition from scientific the concept of the future. For example, some computing being done on large main- relatively recent predictions were: frame computers a few decades ago, and now being almost exclusively carried out Physics is finished, young man. It’s on inexpensive workstations). a dead-end street. -Unknown teacher of Max Planck, • The classification aspects of bio- late 19th century informatics will be rapidly replaced by efforts to understand the regulation of I believe that the motion picture is gene networks using established and new destined to revolutionize our educa- techniques from non-linear dynamics. tional system and that in a few years Mathematical modelling and analysis it will supplant largely, if not en- of the mechanisms of gene regulation tirely, the use of textbooks. will continue at an ever accelerating -Thomas Edison, 1922 pace. This, in conjunction with the already established ability to produce It is probable that television drama “designer” molecular circuits, will be of high caliber and produced by instrumental in the targeted treatment first-rate artists will materially raise of disease through gene therapy. the level of dramatic taste of the na- tion. • Attempts to understand the noisy inter- -David Sarnoff, 1939 actions in gene regulation and expression at the single cell level will lead to the Being aware of the almost certain folly of development of new mathematical tech- trying to predict the future, as illustrated niques for dealing with chemical reac- by these quotations, we nevertheless take the tions in which the law of large numbers leap and mention several areas in which we cannot be invoked. feel that significant advances are likely to take place over the present century. • The Herculean efforts of countless neu- robiologists over the past century have • The sequencing of human and other given us much insight into the func- genomes has provided a spectacular tioning of single neurons as well as the amount of data which needs to be or- behavior of simple neural circuits and ganized and analyzed before its signif- some extremely simple sensory and mo- icance becomes clear. The mathemat- tor systems. This progress will continue ical techniques necessary to do so are and lead to the efficient treatment of

10 many neuron-related diseases, to a bet- systems will improve the design of ar- ter design of protheses, and perhaps, to tificial organs, protheses, and implants a deeper understanding of the relation through the development of hybrid between brain and mind. Shall we, at animate-inanimate devices. some time, be able to really understand phenomena like cognition and memory? • Epidemiological research aided by math- Maybe, maybe not. Perhaps, as some ematical modelling and statistical anal- philosophers maintain, the human mind ysis will help us understand the dynam- is unable to understand itself. However, ics of disease transmission and to design we firmly believe that the neurophysio- more efficacious treatment and vaccina- logical sciences will thrive in the near tion strategies. future, with physics and mathematics playing a central role in such progress. • The difficulty in collecting high reso- Examples are the use of vagal stimula- lution temporal and spatial data from tion to abort epileptic seizures, and deep ecological and meteorological systems brain stimulation to control the tremor has limited the success of mathemati- of Parkinson disease. cal modelling approaches in these fields. The availability of more sophisticated ge- • Biophysical advances in determining the ographic information systems and mas- structure and dynamic properties of sive parallel computational power will al- membrane channels and receptors have leviate these problems. proceeded at a rapid pace over the past decade. There is every reason to antici- pate that this will only accelerate in the Summary future. The accumulated knowledge, in conjunction with modelling and produc- There has been a long and rich tradition tion of designer molecules, will enable of fruitful interdisciplinary interplay between the efficient development and production the physical and biological sciences extend- of drugs specifically targeted to the elim- ing over several centuries, as we have illus- ination of disease symptoms if not the trated with a few examples. Many other ex- disease itself. amples could have been offered to illustrate the point, and would simply serve to highlight • The accelerated rhythm at which tech- the rich interactions between apparently dis- nology is progressing makes us believe parate branches of science. We expect that that, in the near future, it will be pos- these interactions and interdependence will sible to combine knowledge and tech- continue and become even stronger in the fu- niques from biology, chemistry, biochem- ture. istry, computer science, engineering and physics to engineer designer molecules for specific medical and industrial pur- Acknowledgments poses. We are grateful to N. Anderson-Mackey, R. • Interdisciplinary work focused in the FitzHugh and J. Walter Woodbury for ex- development of bio-materials, bio- tensive comments on this article, which is electronic devices, and bio-mechanical largely based on a lecture with the same title

11 given by MCM 4 May, 2001, at the Univer- • Scr¨odinger, E. What is Life?: The Phys- sity of Oxford as the Leverhulme Professor ical Aspects of the Living Cell. Cam- of Mathematical Biology for the 2001 aca- bridge: University Press, 1944 demic year. This work was supported by COFAA-IPN (M´exico), EDI-IPN (M´exico), • Watson, J. DNA: The Secret of Life. MITACS (Canada), the Natural Sciences and New York: Alfred A. Knopf, 2003 Engineering Research Council (NSERC grant OGP-0036920, Canada), and Le Fonds pour la Formation de Chercheurs et l’Aide la Recherche (FCAR grant 98ER1057, Qu´ebec).

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