Breaking symmetry underpins a variety of natural phenomena, including animal patterning, and enables nature to transcend beyond simplicity. Photo credit: Nick Fewings via Unsplash
…many phenomena in the natural world arise from underlying dynamics rife with asymmetries…
Nature is often characterised by beautiful and emergent symmetry such as the intricate geometry of snowflakes, the patterns found on the skins of animals, and the equipoise of ecosystems. The ungoverned self-organisation of biological processes can leave one awestruck at the balance, stability, and harmony of the natural world, especially that which is found within, or made up of, living organisms. Despite this, many of the physical laws separating living systems from inanimate matter are based upon hidden violations of symmetry in the underlying physics. From animal patterning to the stability of ecosystems, many phenomena in the natural world arise from underlying dynamics rife with asymmetries that tip the scales between opposing forces and lead to the emergence of living systems which are constantly evolving.
How the leopard got its spots
Despite being renowned for his pioneering work in computing, Alan Turing’s contributions to biology are often forgotten by popular science. In his seminal 1952 paper, Turing proposed a simple mechanism by which patterns could form from the interactions of two chemical substances. This simple model represented the first plausible theory explaining the formation of patterns in a host of biological systems spanning an enormous range of scales. Whilst the original theory focused on “morphogenesis” — the process by which cells, tissues, and organisms develop their shapes — it has since found applications in the modelling of a vast array of phenomena, including the formation of patterns in the coats of animals.
Turing’s theory begins by considering a two-species model of interacting chemicals, composed of an activator, which promotes the production of both species, and an inhibitor, which prevents the production of the activator. Whilst this asymmetry alone can produce concentration gradients that lead to the formation of transient patterns, they ultimately decay and disappear. To explain the formation of the persistent patterns all around us, Turing introduced diffusion into the model, the spreading of the chemical species in space. Crucially, given an asymmetry in the diffusion rates of the two species, Turing’s model can give rise to a plethora of persistent and emergent patterns, including those consistent with the easily recognisable patterns of animal skins. Contrary to intuition, many of the most ubiquitous symmetrical patterns in nature find their roots in this inceptive asymmetry in rates of diffusion. In a manner that was unforeseen, this simple model, with slight variation in the value of its parameters, can reproduce the vast array of natural patterns, stemming entirely from a seemingly innocuous linear instability and broken diffusive symmetry.
Time’s arrow and the thermodynamics of life
Discussions of symmetry often focus on the spatial domain such as the visual patterns discussed earlier. Nevertheless, the most pervasive asymmetry in nature is that of time-reversal asymmetry — namely the emergence of the so-called arrow of time. The definiteness with which time flows in a single direction is a direct consequence of the famed second law of thermodynamics, which states that a system will tend to become more disordered over time. The disorder of a system is measured by its entropy which must therefore increase as a system progresses towards thermodynamic equilibrium, the state in which it is no longer exchanging energy with its environment. Thus, a system, if left alone, will dissipate heat to its environment until it reaches thermodynamic equilibrium. A classic example of this progression is the melting of an ice-cube kept at room temperature. Once the ice, now water, has reached the temperature of the room, there is no more heat to be dissipated as the system is at thermodynamic equilibrium with its environment. Crucially, the restriction that entropy can only increase in time yields the melting process irreversible, thus breaking time-reversal symmetry, and outlaws possibilities such as a puddle of water spontaneously freezing. But what would happen if time were to be reversed and entropy decreased? In an 1867 letter to Peter Tait, James Clerk Maxwell outlined a paradox that would later become known as Maxwell’s demon. He suggested a hypothetical experiment in which an intelligent agent handled a container with a mixture of hot, fast-moving particles and colder, slow-moving particles. By manipulating a sliding partition, the agent would then sort the hot and cold particles into two halves of the container, which would naturally decrease the entropy of the system without dissipating energy, seemingly in violation of the second law. The paradox cast a shadow of doubt over the famed law until 1929 when Leo Szilard suggested that the act of acquiring information, in this case about the speed of the particles, was not energetically free. He proposed that the act of acquiring and then storing information would result in the dissipation of energy that appeared to be missing from the experiment. His work was the first clue towards a link between information theory and thermodynamics, which has gone on to occupy statistical physicists up to the present day.
…the most pervasive asymmetry in nature is that of time-reversal asymmetry — namely the emergence of the so-called arrow of time.
Whilst for water, thermodynamic equilibrium may seem an innocuous fate, it implies “heat-death” for living systems. When the exchange of energy and information between an organism and its environment ceases the organism dies, decays and, ultimately, ceases to exist. In his 1944 book What is Life? Erwin Schrödinger presented his thesis on the role of physics in the emergence and functioning of living systems. His treatise ranged from theories that ultimately inspired the discovery of DNA to personal musing on consciousness and the nature of free will. Nevertheless, the book’s most direct contribution was to posit nonequilibrium thermodynamics as the central physical theory that could explain the emergence of life. In particular, Schrödinger argued that living systems evade this tendency to decay by consuming energy and dissipating entropy as heat to their environments, thus staving off heat death. Importantly, entropy-producing systems must violate time-reversal symmetry, thus the arrow of time has become a defining feature in the dynamics of living organisms.
Kings and pawns, predators and prey
‘We are all either kings or pawns; emperors and fools’, words that reflect the unavoidable hierarchy of human society, often attributed to Napoleon Bonaparte. However, such hierarchical structure is not limited to human societies, but emerges in a plethora of animal communities, both within and between species. Whilst the celebrated position of alpha-males in wolf packs has been called into question due to faulty assumptions about the qualities that result in high-status amongst animal populations, somewhat unexpected communities have shown the emergence of leader-follower organisation, such flocks of flying pigeons. Importantly, the emergence of these hierarchical structures does not require central organisation or a multi-body interaction, but instead can arise from subtle asymmetries in local relationships between pairs of agents, in this case birds. These individual interactions often follow globally consistent patterns such as grandmother leads mother leads daughter, in contrast to relatively uncommon ‘loop-like’ relationships such as lion leads lioness leads lion. The result of this is a system that self-organises into a hierarchy of leader and follower agents, such as those found in flocks and schools of animals.
Contrary to the subtle asymmetries in same-species interactions, the most blatantly asymmetric ecological interaction of all is the predator-prey relationship. Additionally, the stability of an ecosystem to perturbations such as human intervention, natural disaster, or genetic mutation hinges on the organisation of such relationships. In his seminal 1972 paper, Robert May showed, rather counterintuitively, that as a complex system of interacting agents grows, so does the likelihood that it is unstable. May’s argument considered the stability of a random interaction network of animal populations, showing that in the limit, such a system was almost surely unstable. This prompted the emergence of the ‘diversity-stability debate’, a range of theories attempting to explain the diversity of real-world ecosystems in spite of this apparent paradox. In particular, the validity of May’s random interaction model was called into question. Most significantly, more plausible models of ecological interactions actually showed a marked increase in stability over May’s random model. These results suggested that, whilst entirely asymmetrical, the hierarchy of predator-prey interactions within the food-chain appears carefully organised into a stable yet imbalanced structure. Such a structure differs crucially from a random assortment, a testament to the self-organising, harmonising features of the evolutionary mechanism, even in the presence of major violations of symmetry and balance.
…the most blatantly asymmetric ecological interaction of all is the predator-prey relationship.
Symmetry and simplicity, asymmetry and complexity
In his 1938 book Symmetry, the celebrated mathematician and physicist Hermann Weyl argued that the study of physical sciences ultimately boiled down to the study of symmetries. Weyl’s position was consistent with the history of physics and mathematics where principles of symmetry had both birthed and bridged disparate theories for centuries. However, such a stance was called into question by Nobel laureate Philip Anderson in his 1972 article More is different, where he argued against a reductionist approach to science that abstracted away microscopic, and often asymmetric, interactions between units with assumptions of symmetry and homogeneity. His work proved prophetic as it anticipated the development of complexity science as a novel philosophy on the physics of systems with so-called emergent properties, where the behaviour of the whole system cannot be explained simply by its individual parts. Many prominent figures in classical physics such as Schrödinger, Prigogine and Gell-Mann turned to the study of complex systems towards the end of their careers, often in an attempt to develop a more comprehensive physics of biology and life. The pinnacle of such efforts came in the form of the 2021 Nobel prize in Physics, a first in the study of complex systems, awarded to Giorgio Parisi, in part for his analysis of the Sherrington-Kirkpatrick model, a model of interacting magnetic spins, or binary neurons, with asymmetric coupling strengths.
The emergent complexity of life calls for a new physics for biology, one that is capable of explaining the beauty and harmony of the natural world and its governing dynamics. Nevertheless, such a physics appears to hinge on both subtle and flagrant violations of symmetry and balance that, in turn, explain the dynamism, vibrance, and complexity of life.
