In the preceding sections, we have seen how the twin phenomena of self-organization and emergence create a wide variety of intricate and beautiful patterns in the molecular realms studied by physicists and biochemists – from microscopic micelles, folded proteins, and the DNA double helix to the easily visible tessellations and spirals of B´enard cells and chemical clocks.
We shall now turn our attention to another striking and widespread principle of organi- zation and order in nature, known as chirality, or “handedness” (from the Greek chiros – “hand”). We are, of course, quite familiar with handedness from our bodily experience. Our two hands (and feet) are not identical but are mirror images of each other. In the language of mathematics (seeSection 8.4.3below), such mirror images are said to be asymmetric under reflection, or “chiral.”
To see how chirality is related to self-organization and order in nature, we need to reconsider the chemistry of life, and in particular the chemistry of the carbon atom, which
8.4 Mathematical patterns in the living world 169
Figure 8.14 Two different α-amino acids that are chiral objects; the two mirror images are not superimposable.
is the chemical backbone of all important biomolecules, from the amino acids to the sugars, and from the lipids to the nucleic bases. The normal carbon atom, in the language of chemistry, is tetravalent, which means that it can be bound to four chemical groups. For example, in methane (CH4), carbon is bound to four hydrogen atoms; in chloroform
(CHCl3) to one hydrogen and three chlorine atoms.
Something interesting happens when carbon is bound to four different groups, as is the case of the naturalα-amino acids (seeFigure 8.14). When this happens, the resulting molecule can exist in two different forms – that is, as two different molecules that are mirror images of each other: the two forms are chiral; they are not superimposable on each other. Of the two forms ofα-amino acid, intriguingly, only the left-handed version (the so-called l-form) is present in nature, with negligibly few exceptions.
In the language of chemistry, compounds with the same molecular formula but different structures are known as isomers, and isomers differing in the spatial order are generally called stereoisomers, while the two chiral mirror images are called “enantiomers,” or “optical isomers.” The last term refers to their distinctive optical property of rotating plane polarized light in different directions – clockwise or counterclockwise. Two distinct chiral isomers are indistinguishable in terms of all other physical properties.
When chemists synthesizeα-amino acids (or any other molecules containing asymmetric carbon atoms) by normal laboratory procedures, they produce automatically a 50:50 mixture of the two enantiomers. (The synthesis of one pure enantiomer is possible but extremely difficult – e.g., by using asymmetric catalysts – and the same can be said about the separation of one enantiomer from the other.) However, in nature (with very few exceptions) only the l-form of theα-amino acid is present, and the biochemical reactions in living organisms only produce the l-form of theα-amino acid. Biochemists speak of “homochirality” to indicate situations in which all compounds exhibit the same type of chirality. All our
170 Order and complexity in the living world
proteins are homochiral, being constituted solely by l-amino acids, and this intriguing asymmetry in nature is not restricted to this class of compounds. All natural sugars and their polymers are chiral, asymmetric molecules existing in nature only as one type of enantiomer (seeBox 9.4for definitions of polymers, peptides, and proteins).
It seems, then, that nature is intrinsically asymmetric, and the question is: why? What is the evolutionary advantage of this asymmetry? To answer this question, let us consider the example of a polypeptide hormone made of a linear sequence of, say, ten, amino acids. Biologically, this hormone works because it has a very specific interaction with a biological “receptor,” generally a membrane protein that recognizes the hormone’s structure and spatial form. Now, suppose that both l- and d-forms (d for dextrorotatory, the mirror image of the l-form) of amino acids were present in nature, being synthesized and incorporated into a growing chain with the same probability. The polypeptide would then exist in 210,
or about 1,000, possible forms. It is obvious that the specificity of interaction with the receptor would be extremely difficult, if not impossible. Extrapolating this calculation to a protein with 50 amino-acid residues, we would obtain the astronomical number of 250 protein isomers, approximately 1015, or 1,000 trillion.
However, these astronomical numbers of hypothetical proteins with combinations of l- and d-amino acids are reduced to a single protein simply by having only one optical isomer – that is, by using only the l-amino acids. What a trick! It is evident that this amounts to a huge evolutionary advantage. There is no doubt, then, that homochirality is an extremely powerful principle to bring order and simplicity into the structures of life. Without homochirality, life as we know it would be impossible; and this consideration also suggests that, most probably, this molecular asymmetry was present in the very first steps of the origin of life.
This brings us to the important question: what is the origin of chirality in nature? In other words, what induced the symmetry breaking that favored one kind of chiral molecule over the other? We will discuss this question later on (inSection 8.4.6) within the broader context of symmetry and symmetry breaking. Here we just want to add the interesting observation that in nature the basic chiral asymmetry at the molecular level is generally attended by a high degree of symmetry at the macroscopic level.
The splendid symmetrical patterns exhibited by flowers, insects, and higher organisms, including the bilateral symmetry of mammals, are well known. The relation between symmetry and order is quite apparent, and it is also evident that symmetry in our living world corresponds to an economical strategy of nature: to make a flower with several identical petals, or a butterfly with identical wings, the organism needs just one set of genes, repeated several times. Indeed, the relation between molecular asymmetry and macroscopic symmetry is a fascinating aspect of order in nature.
Of course, symmetry in nature also has an evolutionary value. In many animals it is directly related to beauty as a mating attractor – think, for example, of the spectacular display of the peacock’s feathers – and, more generally, it can serve as a recognition pattern, also among different species. In human civilization, symmetry is highly valued in all forms of art, from the architecture of the most primitive temples to modern painting and
8.4 Mathematical patterns in the living world 171
computer design. We shall return to the intriguing role of symmetry and beauty in evolution in a subsequent chapter when we discuss the basic characteristics of human nature (see
Section 11.3.3).
8.4.2 “Biomathematics” – a new mathematical frontier
Asymmetry in nature, however, is not restricted to the molecular level but is conspicuous also in the macroscopic world. A spiral, or a helix, for example, can be right-handed or left-handed. Spirals, in particular, seem to be ubiquitous in nature, appearing in the growth patterns of many plants and animals, as well as in the vortices of turbulent flows of water and air, and the accumulations of stars in giant spiral galaxies.
Indeed, spiral patterns in the growth of leaves and flower petals, as well as in the pig- ments of seashells and other animals, have long been known by botanists and zoologists; and it is not surprising that mathematicians, too, became fascinated with these extraordi- nary markings on the skins and exoskeletons of animals, and tried to find mathematical explanations.
One of the first to do so was the Scottish mathematician and biologist D’Arcy Thompson in the nineteenth century. In his pioneering book On Growth and Form, Thompson (1917) took his inspiration from the successful use of mathematics to understand nature’s patterns in the physical sciences, and advocated a similar approach in biology. He identified numerous mathematical patterns in the living world – the spiral shapes of shells, stripes of zebras, and numerical regularities of plant growth – and he tried to explain them in terms of underlying abstract principles. He failed to do so, however, because (as we know today) the mathematics of life is much more subtle and hidden than that of the nonliving world, and thus Thompson’s book, although widely regarded as a classic today, had no significant influence on mainstream biology.
With the advent of complexity theory (seeChapter 6), which is essentially a mathematics of patterns, the situation changed dramatically. The techniques of nonlinear dynamics opened up exciting possibilities of modeling and explaining many details in the emergence of biological forms and revealed a variety of new connections between mathematics and biology. Indeed, in the 1990s, the mathematician Ian Stewart (1998, p. xii) argued forcefully that “biomathematics” would be the new mathematical frontier in the twenty-first century:
I predict – and I am by no means alone – that one of the most exciting growth areas of twenty- first-century science will be biomathematics. The next century will witness an explosion of new mathematical concepts, of new kinds of mathematics, brought into being by the need to understand the patterns of the living world.
The methods used in this new discipline include those of nonlinear dynamics, group theory, and topology – even knot theory. What they all have in common is that they are qualitative approaches, dealing with patterns, order, and complexity. In this section we shall discuss only one mathematical concept, which is of central importance in contemporary physics and is now being used increasingly also in biology: the concept of symmetry.
172 Order and complexity in the living world