Chick or Egg, which came first - a debate no more?

This famous question, whose origin can be traced far back into the Greek philosopher Plutarch's Moralia at the beginning of the first millennium, has been much used and abused over the course of the history since then. Finally, in a debate organized by Disney to promote the DVD launch of its film "Chicken Little", a team comprising a geneticist, a philosopher and a chicken farmer appears to have cracked the puzzle.

First, to clarify the debate itself - the egg of course refers to a chicken egg, and not any other egg such as an ostrich egg, else there is no debate to begin with. But what is the definition of a chicken egg? There are in fact three definitions. First, an egg from which a chicken is hatched. Second, an egg laid by a mother hen. And third, an egg laid by a hen from which a chick is born. There seems to be no doubt about the first definition. And if one also assumes the second, then the third definition is redundant. The first two definitions toegther, however, gives rise to the circularity, and hence, the debate on which came first. So, in order to break the circularity, one of them must have been wrong when the first chick or the first egg appeared. But which one?

Therein comes Darwin's evolution (what else?) to the rescue, and a simple fact of genetics. The answer seems to be almost straightforward in hindsight. Evolution teaches us that all species, including our chicken, derived (or speciated, using technical jargon) from an ancestral species. And genetics tell us that an individual's genetic makeup does not change during its lifetime. Therefore, the ancestral "pre-chicken" individual could not have metamorphosed into a modern hen as it was growing up, and instead the changes must have happened within the egg during the earliest stages of embryonic development, when rapid chromosomal mixing was taking place. To put it simply, the first chicken egg was laid, not by a hen, but by a "pre-hen". And therefore, egg came first, and the chicken followed.

The argument might still appear preposterous to many were it not for a sobering fact of evolution, which is the underlying continuity of changes as a species moves up its evolutionary ladder. (There are glaring exceptions, as the huge gaps in the fossil records stretching back to a few billion years bear testimony to, but that is a whole different story). We humans did not arrive overnight from tree-hopping monkeys, but progressed through a series of intermediate species over millions of years. Likewise, the ancestor of the hen might have been quite like the hen itself, and probably did not even spot the odd chick as it mingled happily with the rest of its nestlings.

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The Reasonable Ineffectiveness of Mathematics in the Biological Sciences

To many the title presents a poor parody of the famous quote "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", made by the 20th century physicist Eugene Wigner. In Wigner's era physics ruled the domain of natural sciences riding on the twin triumph of relativity and quantum physics, and what he really meant was the unique ability of mathematics in advancing the physical understanding of the Universe. Examples abound: Newton invented calculus (years before Leibniz, who was a mathematician) to develop his laws, and Einstein learnt Minkowski and Reimann geometry to develop his relativity theory. Likewise the eventual success of quantum physics depended heavily on Schroedinger equation. In subsequent decades biological sciences have made rapid strides (ignoring Darwin's evolution theory, the biggest stride of them all, which occurred in mid 19th century), with amazing progress in genetics, molecular biology and ecology, and can today rightfully claim to be an equal partner in describing the science of the nature. This has brought up the question, particularly after Wigner made his poignant observation, about the comparative lameness of mathematics when it comes to biology. I have myself mused on it for some time, and finally decided to put my thoughts in writing after reading a recent discussion on NECSI listserv.

Of course, while flipping through a biology textbook today, one would come across quite a few mathematical notations and equations. It is also true for the numerous research papers published in various peer-reviewed biology journals. This is all because of the development of an entire new field of mathematical biology, with the increasing realization that mathematics can make the job of describing many biological phenomena relatively simple. But Wigner did not use the word "effective" to mean mathematics merely as a convenient tool or a language to describe physics; to him discovery of the physical laws would be at best indefinitely delayed, and at worst impossible, without fundamental contribution from mathematics. (I am skating on thin ice here: there are biologists with considerably more authority than me who can point to a number of similar contributions of mathematics to biology, such as Mendelean laws of inheritence, Hardy-Weinberg law of population genetics, Trophic-Dynamic concept of energy transfer, and so on. But the argument against these examples again boils down to whether mathematics was really needed for their discovery, or merely served as a suitable language to present them.)

The reason for this discrepancy clearly lies in the difference in the very structure of the two sciences. Physics is a science of universal phenomena, where each physical law is reproducible anytime anywhere in the Universe given identical conditions (water always boils at 100^oC and freezes at 0^oC under the same ambient temperature and pressure). Biology, by contrast, is a science of here-and-now; that is, each and every biological entity and event is unique and distinct from any other. No two human beings are identical, even when from the same location, race or family. Same holds for individuals belonging to any non-human species. The rules and conditions governing the persistence (or extinction) of a species in two habitats are never same, because the detailed biotic and abiotic environments in the two habitats always differ to some degree. However, at the very basic level all biological systems are composed of electrons, protons, neutrons etc., and as such they must abide by physical laws at this elementary level. Also, universality holds for many biological principles: for instance, Gause's law of competitive exclusion, which states that one species will always outcompete all others when vying for identical resources (the biggest tree always wins against smaller plants when they compete for soil nutrition and sunlight on the same habitat patch). It is easy to see that given identical environmental conditions, this principle should hold everywhere. The catch is in the detail: environmental conditions are never identical in any two places, and therefore the notion of universality, while still valid in biology, is irrelevant in addressing pressing problems of species conservation in as varied places as, for example, Alaska and Hawaii. Thus, biologists often must forego unifying approaches, and study their systems on case-by-case basis. As someone in NECSI forum pointed out, "Simpson doctrine" says it best: Physicists study principles which apply to all phenomena; biologists study phenomena to which all principles apply.

Where does mathematics fit in to this absence of universality in biological sciences? Before we get into that, I should qualify the kind of mathematics that was prevalent in Wigner's time, which was the so called traditional, or continuous, mathematics such as differential and integral calculus with the implicit assumption of continuous (and infinite) time and space. Walter M. Elsasser proposed that it is this property (or limitation) of the traditional mathematics that makes it so effective in physical sciences and the same time ineffective in biological sciences. He delineated sciences into studying two broad classes of objects: the homogeneous (and infinite) class which is characterized by a small number of kinds of objects with a large (almost infinite) number of exact copies of each kind, and the heterogeneous (and finite) class that concerns with objects whose variety far exceeds the number of copies of each kind. Physics, with its requirement of universality and reproducibility of the laws under identical conditions, makes an assumption of homogeneity of space and time as well as the existence of infinite copies, and therefore deals with homogeneous class of objects. Biology, by contrast, because of the sheer variety of the subjects with almost no two identical copies in nature, is said to study objects of the heterogeneous class. It was Elsasser's contention that the continuous mathematics is historically well adapted for the homogeneous class and thus for physical sciences, but not the heterogeneous class for which he claimed that a new kind of logic is needed.

With the advent of computer, many believe such a logic can now be built on computer-based discrete mathematics (including information theory, combinatorics, algorithmics etc.), which has the potential to make fundamental contributions to biological sciences. To give an example, it is possible to develop theoretical models of oceanic plankton dynamics with differential equations representing the population size as a continuous variable, because a typical plankton colony in bloomtime contains upto trillions of individuals. But a population of the bighorn sheep, a threatened species with urgent conservation concern, typically numbers only in 50's and 100's of individuals, and modeling the dynamics of such a population would involve finite mathematics in which each individual needs to be tracked separately. Such an endeavor is possible only in computer-based models similar to the cellular automata type systems, which was developed by Stephen Wolfram and is strongly advocated in his somewhat ostentatiously titled book "A New Kind of Science". (I remember listening to one of his promotional lectures in the University of Michigan few years ago, and was unimpressed by his arrogance.) Another type of mathematics that has become particularly relevant to population biology and ecology is the stochastic mathematics, which is often used to introduce variations into population parameters of the computer models, in order to capture the uniqueness of the individuals making up the population. Thus, it seems we are not far from experiencing a sense of wonder at the effectiveness of the "new mathematics" in biology, similar to what Wigner felt in 1960.

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India's ODI loss to WI

Finally India's astonishing run of successful ODI chases has ended, after a record 17 wins on the trot. And the manner of the loss itself, to WI yesterday by only 1 run in the last over, speaks of the competitive mindset that the present Indian team possesses. All good things come to end some day, and it was great while it lasted. The team should be pleased that this achievement has come by playing against every big opposition in all conditions both home and away. India has come a long way from the days when it replaced SA for the ultimate "choker" tag, even though it still has a long way to go (for example, winning a significant series abroad). Since the beginning of Dravid-Chappell era, multiple factors have contributed to this success, not least of which is the resurgence of our middle order bat, which more than compensated for Shewag's form slump and the resulting rickety top order. Often it is the top order that sets up a game, but middle order finishes it (discounting those matches that are decided by bowlers). India's top order wobble is also the reason for its indifferent performence in defending scores. In any case, let us keep fingers crossed until the World Cup.

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"Extinction Vortex"

I read an interesting paper in a biology journal last week, on the processes of extinction of biological species. First, couple of definitions. A "species", using an intuitive working definition that should suffice for this post, is one that does not naturally interbreed with other species. (There are rare exceptions: CNN News reported last week the discovery of a hybrid bear born from a polar mother and a grizzly father.) The "extinction" of a species usually means the death of the last known individual of that species on the earth. This is also known as a "global extinction", which signifies the permanent and irreversible loss of a species from the planet. Biologists use another and more easily tractable definition, the so called "local extinction", which implies the disappearance of a population of the species in a certain region, while it may still persist elsewhere.

Now, back to the paper, which appeared January this year in the acclaimed journal Ecology Letters (a subscription is required to read it). The main objective of the paper is to validate an important theoretical prediction: as a population - defined as a collection of individuals of a particular species within a region - declines in numbers and approaches (local) extinction, the factors causing the decline intensify, driving the population faster and faster towards extinction. This is known as the "extinction vortex". (Mathematically, the rate of population decline increases with decreasing population size.) The paper offers the first empirical verification of this prediction for a wide array of species, from mammals to birds, turtles and even one fish population. Such empirical success of fundamental theoretical precepts is rare in biology, and is a source of excitement for praciticing scientists.

Let the biologists gloat over their success: what has it got to do with the general public? Extinction, be it an inconsequential insect population from the backyard or the gorgeous ivory-billed woodpecker (see the picture above) from the entire planet, is first and foremost a loss, and not merely from moral or aesthetic point of view (even though they are significant). Scientists are still figuring out the intricate ways each and every species contributes to sustaining the balance of the ecosystem around us, which itself provides important services to humankind. The loss of the insect population in the backyard can trigger a precipitous decline of the still larger insects that eat them, which in turn can starve the local bird population that feeds on these larger insects. The depletion of the birds, either by death or emigration, deprives the plant specices of an important mode of dispersal - the seeds and pollen carried along by the birds to far away places. This is not a mere theoretical construct, examples of such keystone species are many in nature. While a global extinction cannot be reversed after it happens, extinction can be prevented from happening in the first place. Scientists and non-scientists alike have a role to play in this endeavor, and the first step is to develop an awareness of the key processes involved as a species hurtles towards fateful oblivion.

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Must we slay the golden geese?

So the debate over player burn-out continues, fueled by the latest comments from Shewag today. This is getting interesting - Australians were at it for quite some time, so were England and Pakistan players, and now joined publicly by Indians. Gavaskar, for all his wisdom, recently stirred up some unwelcome emotion by making that corny remark about cricketers not showcasing 365-day commitment. Do not get me wrong - I am with the people who demands action in return for parting with substantial chunks of their poor middle class paycheck just to catch a glimpse of their heroes, besides pampering them with cult-like devotion. But I believe our interest is better served if a cricketer's playing career is prolonged - who wouldn't enjoy watching Tendulkar swing away for another few years! And to do that, instead of killing the golden geese now, it makes sense to allow them enough rest in between games to avoid potential burn-out.

Let us bring up some numbers. Gavaskar, during his 16-yr international career, played 125 Tests and 108 ODIs - that is one day out in the field for a week in between (this assumes all Tests ran the full course of 5 days). Tendulkar, by comparison, over the same period played 130 Tests and 350 ODIs - each day of play with 5 days in between. The difference, while considerable in statistical terms (who wouldn't like an extra 2-day rest!), does not quite cause eyebrows to shoot up, right? Now factor in the days of warm-up matches to recondition players to new place, net practice before the actual contest, and the travel before and after every match - cricketers today literally live out of suitcases. But then, of course, players in Gavaskar era did not earn the kind of money they do now.

I guess to many the equation is simple - divide the annual earning by the days of play in that year, and this number should stay within similar range across years (after adjusting for inflation). So, for example, if you earn twice today, you should put in twice as many days on the ground. Two factors have complicated this equation over the 90's decade. First, BCCI has become the richest body in world cricket (through lucrative sponsorship and telecasting deals with big industries), resulting in vastly inflated match fees of the contracted players. Second, the players themselves make unprecedented money through corporate endorsements (a trend started with none other than the brand "Tendulkar" himself). I must add, however, that it was Gavaskar's team that, led by Kapil Dev, brought home the 1983 World Cup, which overnight made cricket the prima donna of Indian sport and turned industries' attention to it in the first place. (I was a slip of a young lad that time, and still vividly remember the fireworks lighting up the night sky as a beaming Kapil held aloft the Prudential Cup at Lords.)

So, we cannot really expect players to put in that many days each year in the field to justify their current income, can we? Or, if one wants to be methodical, why not add up all the days a player actually spends to earn his money - that should include travel times plus practice sessions and matches plus press appearances and interviews, and add to that the days of rehearsals and shootings to appear in a TV commercial. Anyway, this post has gone on far too long already, I must sign off now. To conclude, I'll happily pay to watch a Tendulkar, a Shewag, an Afridi, or a Shoaib, in full glory fewer days a year but over many years. And I believe many will agree.

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Random Walk on a Mobius Band

As the first post of this blog, I should say something about the title "Random Walk", and more importantly, the little horizontal "8" following it (my poor attempt at HTML coding of a Mobius Band). Why Mobius Band (or "Mobius Strip")? And, what IS a Mobius Band? According to Wikipedia.org, it is a one-sided surface, in the sense that you can walk on both sides of it without ever meeting an edge. Imagine an ant on a flat piece of paper - to go from one side of the paper to the other side it must make a U-turn over an edge. So, how is a one-sided surface at all possible? Cut a paper strip 10-inch long and 1-inch across, give it a half-twist and attach the two ends together - you have a Mobius Band! The ant can now go on and on over both sides of the strip and never come across an edge (see Escher's famous depiction above).

As to "Why Mobius Band" - I don't know about you, but happens many times with me that, after my thought process has taken off on a random excursion, it goes over a wide range of different issues and often many sides of the same issue, but eventually comes back to the original topic that triggered the thought in the first place. However, even though the trajectory is zigzag and the issues are unrelated to one another, yet there is the continuity of an underlying thread that binds one thought to its immediate predecessor (much like the thought of a fishing trip triggering another thought on today's lunch menu). A statistical physicist would probably call it an example of a "Markov process", in which the current state is determined solely by the preceding state, and not the ones before that. But I digress: my point is, this underlying continuity resembles to me the absence of an abrupt "edge", and your thought can wander away and flip around and still return to the starting point without ever making a U-turn, as if it is walking on a Mobius Band of its own.

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