Friday, February 27, 2015

Alan Turing, Biological Physicist

Recently, my wife and I went to the theater to see The Imitation Game, about Alan Turing and the breaking of the enigma code during World War II. It is a fascinating movie. I’m a big fan of Benedict Cumberbatch, who played Turing (I particularly enjoy his portrayal of Sherlock Holmes in the TV series Sherlock), and I always enjoy performances by Keira Knightly.

Turing was primarily a mathematician, but he did publish one paper that straddled the disciplines of mathematical biology and biological physics: A. M. Turing, 1952, “Chemical Basis of Morphogenesis.” Philosophical Transactions of the Royal Society of London. Series B, Volume 237, Pages 37–72. The abstract is reproduced below.
It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.
Mathematical Biology, by James Murray, superimposed on Intermediate Physics for Medicine and Biology.
Mathematical Biology,
by James Murray.
You can learn more about Turing’s theory in James Murray’s book Mathematical Biology (I am basing my comments on the edition in the Oakland University library: the Second, Corrected Edition, 1993). Murray writes
Turing’s (1952) idea is a simple but profound one. He said that if, in the absence of diffusion….[two chemicals] A and B tend to a linearly stable uniform steady state then, under certain conditions, which we shall derive, spatially inhomogeneous patterns can evolve by diffusion driven instability… Diffusion is usually considered a stabalising process which is why this was such a novel concept. To see intuitively how diffusion can be destabilizing consider the following, albeit unrealistic, but informative analogy.

Consider a field of dry grass in which there is a large number of grasshoppers…
I don’t know about you, but I gotta love someone who explains mathematics using dry grass and grasshoppers.

Diffusion is a key concept underlying Turing’s work. Russ Hobbie and I discussion diffusion in Chapter 4 of the 4th edition of Intermediate Physics for Medicine and Biology, and it is one of the central ideas in all of biological physics. Diffusion-driven instabilities play a role when analyzing the Belousov-Zhabotinsky oscillating chemical reaction, and are relevant to explaining how leopards get their spots (a spotted leopard graces the cover of Murray’s book; whenever I search for his book in the stacks of the OU library, I just look for the leopard).

Murray continues
A reaction diffusion system exhibits diffusion-driven instability or Turing instability if the homogeneous stead state is stable to small perturbations in the absence of diffusion but unstable to small spatial perturbations when diffusion is present. The usual concept of instability in biology is in the context of ecology, where a uniform steady state becomes unstable to small perturbations and the populations typically exhibit some temporal oscillatory behaviour. The instability we are concerned with here is of a quite different kind. The mechanism driving the spatially inhomogeneous instability is diffusion: the mechanism determines the spatial pattern that evolves. How the pattern or mode is selected is an important aspect of the analysis.
Not only did Turing make a monumental contribution to deciphering the enigma code, but also he helped to develop the field of mathematical biology. In my book, that makes him a biological physicist.

Friday, February 20, 2015

The Ugliest Equation

The 4th edition of Intermediate Physics for Medicine and Biology contains thousands of equations. One of them, Equation 15.22, gives the cross section for energy transferred to electrons during Compton scattering, as a function of photon energy (x = hν/mc2). Of all the equations in the book, it is the ugliest.

Equation 15.22 in Intermediate Physics for Medicine and Biology, giving the cross section for energy transferred to electrons during Compton scattering, as a function of photon energy. It is the ugliest equation in the book.
Eq. 15.22 in IPMB; the ugliest equation.

Russ Hobbie and I write
Equation 15.22 is a rather nasty equation to evaluate, particularly at low energies, because many of the terms nearly cancel.
Examining the behavior of the expression in brackets at small x should be easy: just take its Taylor’s series (to review Taylor’s series, see Appendix D of IPMB). In order to get the correct answer, however, you need to keep not two terms in the expansion, or three, but four! The Taylor’s series you need are

Taylor series needed to analyze Equation 15.22 in Intermediate Physics for Medicine and Biology.
Three Taylor's series needed to analyze Eq. 15.22.

Oddly, the expansion for the fourth term 4x2/3(1+2x)3 doesn’t even need one term in its expansion; it is small and doesn’t contribute to the limiting behavior. Plug these all in, and you find that the terms in x−2, x−1, and x0 all vanish. The first nonzero term is linear: 4x/3.

Out of curiosity, I evaluated each of the five terms in the expression using x = 0.01. I got

20001.9608 - 0.9900 + 9900.0192 - 0.0001 - 29900.9771  =  0.0128

The terms really do “nearly cancel.”

Friday, February 13, 2015

Willem Einthoven, Biological Physicist

In Chapter 7 of the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I mention Einthoven’s triangle. This triangle is formed by three electrodes used to measure the electrocardiogram: one on the right arm, one on the left arm, and one on the left leg. Who is this Einthoven of Einthoven’s triangle? He is an excellent example of a scientist well versed in both physics and physiology.

Asimov’s Biographical Encyclopedia of Science and Technology, superimposed on the cover of Intermediate Physics for Medicine and Biology.
Asimov's Biographical
Asimov’s Biographical Encyclopedia of Science and Technology describes Einthoven in this way:
Einthoven, Willem (eyent’-hoh-ven)
Dutch physiologist
Born: Semarang, Java (now part of Indonesia), May 22, 1860
Died: Leiden, September 29, 1927

Einthoven’s father was a practicing physician serving in the East Indies, which was then a Dutch colony. The father died in 1866, and in 1870 the family returned to the Netherlands and settled in Utrecht. In 1878 Einthoven entered the University of Utrecht and began the study of medicine, although always with considerable interest in physics. He obtained his medical degree in 1885 and was at once appointed to a professorship of physiology at the University of Leiden, serving there the remainder of his life.

The physicist in him provoked his interest in the tiny electric potentials produced in the human body…. In 1903 Einthoven developed the first string galvanometer. This consisted of a delicate conducting thread stretched across a magnetic field. A current flowing through the thread would cause it to deviate at right angles to the direction of the magnetic field lines of force, the extent of the deviation being proportional to the strength of the current. The delicacy of the device was sufficient to make it possible to record the varying electrical potentials of the heart.

Einthoven continually improved his device and worked out the significance of the rises and falls in potential. By 1906 he was correlating the recordings of these peaks and troughs (the result being what he called the electrocardiogram) with various types of heart disorders….For the development of electrocardiography Einthoven was awarded the 1924 Nobel Prize in medicine and physiology.
Willem Einthoven (1860-1927): Father of Electrocardiology.
Willem Einthoven (1860-1927):
Father of Electrocardiology.
I became intrigued by Einthoven’s skill at both mathematics and medicine, so I decided to explore deeper into how he straddled these two fields. The book Willem Einthoven (1860-1927): Father of Electrocardiography, by H. A. Snellen, provided these insights:
[Einthoven’s work] demanded more knowledge of mathematics than Einthoven’s high school and medical training had provided. He supplemented this mainly through self-study; learning differential and integral calculus from Lorentz’ book on the subject in the early 1890’s.

30 Years later he presented a copy of this book to Frank Wilson with the words: “May I send you the excellent book of Lorentz’ Differential- und Integralrechnung? I have learned my mathematics from it after my nomination as a professor in this University and I hope you will have as much pleasure and profit by it as I have had myself.”

In physical matters he was aided by his correspondence and talks with his friend (and later brother-in-law) Julius, who became extra-ordinary professor of physics at Amsterdam and subsequently full professor at Utrecht, where they had studied together.

Einthoven profited also from written and personal contact with the somewhat older and already famous Lorentz, professor of theoretical physics at Leiden….

Einthoven the physiologist with a marked general concern about patients and general medicine was at heart a physicist though not by training and office…

Most of the important topics in the correspondence [between Einthoven and English physiologist A. V. Hill] are reflected in Hill’s obituary of Einthoven in Nature. I quote a few lines, which bear testimony to Hill’s keen observation and his sincere admiration of Einthoven: “Einthoven’s investigations cover a wide range, but they are all notable for the same characteristic—the mastery of physical technique which they show. Einthoven, in spite of his medical training and his office, was essentially a physicist, and the extraordinary value of his contributions to physiology, and therewith indirectly to medicine, emphasizes the way in which an aptitude—in Einthoven’s case a genius—for physical methods can aid in the solution of physiological problems.”
Many scientists have made the leap from physics to biology (see my blog entry of a few weeks ago for examples). Einthoven did the opposite: going from biology to physics. I’ve always suspected this is the more difficult path, and it certainly seems to be the less common one. Yet, he appears to have made the journey successfully. Snellen’s book provides no anecdotes about how Einthoven picked up his mathematics and physics, but I imagine he must of spent many a night slogging through Lorentz’s book, painstakingly teaching himself the subject.

I suspect IPMB can aid physicists moving into biology and medicine. I wonder how useful it is for someone like Einthoven, travelling in the other direction?

Friday, February 6, 2015

The Sinc Function

In the 4th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss many mathematical functions, from common ones like the sine function and the exponential function to less familiar ones like Bessel functions and the error function. A simple but important example is the sinc function.

Sinc(x) is defined as sin(x)/x. It is zero wherever sin(x) is zero (where x is a multiple of π), except at x = 0, where sinc is one. The shape of the sinc function is a central peak surrounded by oscillations with decaying amplitude.

A plot of sinc(x) = sin(x)/x as a function of x.
The sinc function.

The most important property of the sinc function is that it is the Fourier transform of a square pulse. In Chapter 18 about magnetic resonance imaging, a slice of a sample is selected by turning on a magnetic field gradient, so the Larmor frequencies of the hydrogen atoms depend on location. To select a uniform slice, you need to excite hydrogen atoms with a uniform range of Larmor frequencies. The radio-frequency pulse you must apply is specified by its Fourier transform. It is an oscillation at the central Larmor frequency, with an amplitude modulated by a sinc function.

When you integrate sinc(x), you get a new special function that Russ and I never discuss: the sine integral function, Si(x)

A plot of the sine integral function, Si(x), versus x.
The sine integral function, Si(x).
This function looks like a step function, but with oscillations. As x goes to infinity the sine integral approaches π/2. It is odd, so as x goes to minus infinity it approaches –π/2.

The sinc function and the sine integral function resemble the Dirac delta function and the Heaviside step function. In fact, sinc(x/a)/a gets taller and taller, and the side lobes fall off faster and faster, as a approaches zero; it becomes the delta function.Similarly, the sine integral function becomes—to within a constant term, π/2—the step function.

Special functions often have interesting and beautiful properties. As I noted earlier, if you integrate sinc(x) from zero to infinity you get π/2. However, if you integrate the square of sinc(x) from zero to infinity you get the same result: π/2. These two functions are different: sinc(x) oscillates between negative and positive values, so its integral oscillates from above π/2 to below π/2, as shown above; sinc2(x) is always positive, so its integral grows monotonically to its asymptotic value. But as you extend the integral to infinity, the area under these two curves is exactly the same! I’m not sure there is any physical significance to this property, but it is certainly a fun fact to know.