Mathematics, Biology, and Physics: Interactions and Interdependence Michael C. Mackey∗ Mois´es Santill´an† October 24, 2018 Introduction Centuries, scientific disciplines started to be hierarchically classified. This classification Modern science means that which has a solid works well in some instances, and without conceptual framework, and which considers it dealing with the rapidly growing body of experimental results as the ultimate litmus knowledge of the past 200 years would have test against which to validate any theoretical been difficult. However, it fails to fairly rep- construct. Its birth can be traced back to the resent the interdisciplinary work which has 16th and 17th Centuries. The work of people been, and continues to be, highly important. like Nicholaus Copernicus, Galileo Galilei, Jo- In this paper we give a taste of the rich histor- hannes Kepler, William Harvey, Vesalius, and ical relation between physics, mathematics, others was seminal to this development. Be- and the biological sciences. We argue that fore the so-called Scientific Revolution, natu- this will continue to play a very important ral philosophers (the forefathers of scientists role in the future, based on historical exam- as we know them today) did not perform ex- ples and on a brief review of the current sit- periments as manual labor was considered uation. a lower class activity. This attitude, inher- ited from the Greeks, changed between the 16th and the 18th Centuries, as merchants The 18th and 19th Cen- and craftsmen gained economic and political turies power. As a result economics, politics, and science went through significant changes. In Electrophysiology is the science that studies that period democracy, capitalism, and mod- the interaction between electromagnetic fields ern science were founded and emerged as the arXiv:q-bio/0409001v1 [q-bio.OT] 1 Sep 2004 and biological tissues. This includes the gen- cornerstones of a new era. eration of electric or magnetic fields and elec- During the Enlightenment, in the latter tric currents in some specialized organs, the part of the 18th and early part of the 19th intrinsic electric and magnetic properties of ∗e-mail: [email protected], Departments of tissue, the response of specialized cells (like Physiology, Physics & Mathematics and Centre for neurons and muscle cells) to stimulation, etc. Nonlinear Dynamics, McGill University, 3655 Prom- Up to the middle of the 19th Century, the his- enade Sir William Osler, H3G 1Y6 Montreal, QC, torical development of electrophysiology par- CANADA †e-mail: [email protected], Permanent address: alleled that of electromagnetism. The first Depto. de F´ısica, Esc. Sup. de F´ısica y Matem´aticas, electric generating machines and the Leyden Inst. Polit´ecnico Nal. 07738 M´exico D. F., MEXICO´ jar were constructed to produce static elec- 1 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.