Randomness increases biological organization: a mathematical understanding of Gould’s critique of evolutionary progress

In this text, we informally expose the mathematical analysis in [BL09] of Gould’s ideas on the increase of “complexity” along biological evolution as a result of random paths. We also revisit some related theoretical investigations on randomness (contingency) and symmetry breakings in biology, following [LM12]. Gould made several fundamental observations on how phenotypic complexity increases on average, in a random evolution, without a bias towards an increase. Technically, we understand complexity as anti-entropy, a proper biological observable. Its increase, by symmetry changes, involves a strong form of randomness. In more usual biological terms, an increase of complexity involves new elements of biological organization and the latter are “contingent” because they are not determined by the current state of the life dynamics. 1 Randomness and Complexification in Evolution. Available energy consumption and transformation, thus entropy production, are the unavoidable physical processes underlying all biological activities, including reproduction with variation. At the origin of life, bacterial exponential proliferation was (relatively) free, as other forms of life did not contrast it. Diversity, even in bacteria, by random differentiation, produced competition and a slow down of the exponential growth (see diagram 3). Simultaneously, though, this started the early variety of live, a process never to stop. Gould, in several papers and in two books [Gou89, Gou97], uses this idea of random diversification in order to understand a blatant but too often denied fact: the “complexification” of life. The increasing complexity of biological structures has been often denied in order to oppose finalistic and anthropocentric perspectives, which viewed life as aiming at Homo sapiens as the “highest” result of the (possibly intelligent) evolutionary path (or design). Yet, it is a fact that, under many reasonable measures, an eukaryotic cell is more “complex” than a bacterium; a metazoan, with its differentiated cells, tissues and organs, is more “complex” than a cell ... and that, by counting neurons and their connections, cell networks in mammals are more complex that in early triploblast (which have three tissues layers) and these have more complex networks of all sorts than diplobasts (like jellyfish, a very ancient life form). This non-linear increase can be quantified by counting tissue differentiations, networks and more, as hinted by Gould and more precisely proposed in [BL09]. We will first summarize and comment this latter paper here. The point is: how are we to understand this change towards complexity without invoking global aims? Gould provides a remarkable answer based on the analysis of the asymmetric random diffusion of life. Asymmetric because, by principle, life cannot be less complex than bacterial life. So, reproduction with variability, along evolutionary time and in an abstract space, randomly produces more complex 1 Invited lecture, to appear in the proceedings of the Conference, Stephen J. Gould heritage: Nature, History, Society, Venice (It.), May 10 12 , 2012 2 Centre Cavaillès, CIRPHLES, CNRS et Ecole Normale Sup., Paris http://www.di.ens.fr/users/longo/ 3 Tufts University, Department of Anatomy and Cellular Biology, Boston 4 Some may prefer to consider viruses as the least form of life. The issue is controversial, but it would not change at all Gould’s and our perspective: we only need a minimum biological complexity which differs from inert matter individuals just as possible paths. Some happen to be compatible with the environment, resist and proliferate (a few even very successfully) and keep going further and randomly producing also more complex forms of life. Also, since the random exploration of possibilities may, of course, decrease the complexity, no matter how this is measured. Yet, there is a key general principle by which we will understand Gould’s analysis: Any asymmetric random diffusion propagates, by local interactions, the original symmetry breaking along the diffusion. Typically, in a liquid, a drop of dye against a (left) wall, diffuses (towards the right) by local bumps of the particles against each other. That is, particles transitively inherit the original (left) wall asymmetry and propagate it globally by local random interactions. Thus there is no need for a global design or aim: the random paths that compose any diffusion, also in this case help to understand a random growth of complexity, on average. On average, as, of course, there may be local inversion in complexity; yet, the asymmetry randomly forces to “higher complexity” (a notion yet to be defined). This is beautifully made visible by figure 1, in [Gou89], page 205. The image explains the difference between a random, but oriented development (the right figure, 1b, along time, the abscissa axis), and the non-biased, purely random diffusive bouncing of life expansion on the lower wall of complexity, which is “quantified” on the ordinate axis, the left figure 1a. (a) Passive trend, there are more trajectories near 0. (b) Driven trend, the trajectories have a drift towards an increased mean. Figure 1: Passive and driven trends. In one case, the boundary condition, materialized by a lower wall for complexity, is the only reason why the mean increases over time. As a result, this increase is slow. However, in the case of a driven trend or biased evolution, it is the rule of the random walk that leads to an increase of the mean over time (there would be an intrinsic trend in evolution), and the increase of the mean is linear as a function of time. Gould’s and our approach are based on passive trends, which means that we do not need any intrinsic bias for increasing complexity in the process of evolution. Our work has been to justify and frame mathematically Gould’s beautiful intuition, in short here, extensively in [BL09]. Of course, time runs on the horizontal axis, but ... what is in the vertical one? Anything or, more precisely, anywhere the random diffusion takes place or the intended phenomenon “diffuses in”. In particular, the vertical axis may quantify “biological complexity” whatever this may mean. The point Gould wants to clarify is the difference between a fully random vs. a random and biased evolution. The biased right image does not apply to evolution: bacteria are still on Earth and very successfully. Any finalistic bias would instead separate the average random complexification from the lower wall. In either cases, as we said, complexity may locally decrease: tetrapodes may go back to the sea and lose their podia (the number of folding decreases, the overall body structure simplifies). Some cavern fishes may loose their eyes, in their new dark habitat; others, may lose their red blood cells [Ruu54]. Thus, the local propagation of the original asymmetry may be biologically understood as follows: on average, variation by simplification may lead towards a biological niches that has more chances to be already occupied. Thus, global complexity increases as a purely random consequence of variability and on the grounds of local effects: the greater chances, for a “simpler” organism, to bump against an already occupied niche. Thus, more complex variants have just slightly more chances to survive and reproduce — but this slight difference is enough to produce, in the long run, very complex biological organisms. Note that, if variability and, thus, diversity are grounded on randomness, then randomness contributes to structural stability, in biology: diversity is a component of the stability of a species, a population, even an organism (e. g. the irregularities in the fractal structure of an organ may contribute to the organism adaptivity, a concept to be developed elsewhere). That is, by a sound analysis of randomness, as “contingency” in Gould’s sense, complexity and diversity increase with no need for finalism nor a priori “global aim” nor “design” at all. They increase just a consequence of an original symmetry breaking in a random diffusion on a very peculiar phase space: bio-mass × complexity × time (see figure 3 for a complete diagram). Consider now that both in embryogenesis and in evolution, increasing complexity is a form of local reversal of entropy. The global entropy of the Universe, as energy dispersal (or disorder, in biology) increases (or does not decrease). However, locally, by using energy of course, life inverses the entropic trend and creates (new) organization or increases complexity. Of course, embryogenesis is a more canalized process, while evolution seems to explore a diversity of “possible” paths, within the ecosystemto-be. Most turn out to be incompatible with the environment, thus they are eliminated by selection. In embryogenesis increasing complexity seems to follow an expected path and it is partly so. But only in part as failures, in mammals say, reach 50% or more: the constraints imposed, at least, by the inherited DNA and zygote (and by the ecosystem as well), limit the random exploration due to cell proliferation. Yet, their variability, jointly to the many (variable) constraints present in development (first, a major one: DNA), is an essential component of cell differentiation. Tissue differentiation is, from our point of view, a form of (strongly) regulated/canalized variability along cell reproduction. In conclusion, by different but correlated effects, complexity increases, on average, and reverts entropy locally. In [BL09], we called anti-entropy this observable opposing entropy, both in evolution and embryogenesis; its peculiar nature is based on reproduction with random variation, submitted to constraints. As observed in [LM12], anti-entropy differs from negentropy, which is just entropy with a negative sign. First when anti-entropy is added to entropy, the sum never gives 0. Moreover, anti-entropy is realized in a very peculiar singularity (different from 0): a non-null interval of extended criticality [BL11, LM11]. In the next section, we will use this notion to provide a mathematical frame for a further insight by Gould. 2 (Anti-)Entropy in Evolution. 2.1 The diffusion of Bio-mass over Complexity. In yet another apparently naive drawing, Gould proposes a further visualization of the increasing complexity of organisms along evolution. It is just a qualitative image that the paleontologist draws on the 5 This approach does not exclude adaptive effects, which may lead both to greater or even lower complexity, but are locally successful. grounds of his experience. It contains though a further remarkable idea: it suggests the “phase space” (the space of description: observables and parameters) where one can analyse complexification. It is bio-mass density that diffuses over complexity, that is, figure 2 qualitatively describes the diffusion of the frequency of occurrences of individual organisms per unity of complexity. Figure 2: Evolution of complexity as understood by Gould. This illustration is taken from [Gou97], page 171. This is just a mathematically naive, global drawing of the paleontologist on the basis of data. Yet, it poses a conceptual challenge. The diffusion, here, is not along a spatial dimension. Physical observables usually diffuse over space in time; or, within other physical matter (which also amounts to diffusing in space). Here, the diffusion of bio-mass density (or number of occurrences of individual organisms weighted by their individual mass, the ordinate axis) takes place over an abstract dimension, “complexity”, on the abscissa axis. But what does biological complexity exactly mean? Hints are given in [Gou97]: the addition of a cellular nucleus (from bacteria to eukaryotes), the formation of metazoa, the increase in body size, the formation of fractal structures (usually — new — organs) and a few more.... In a sense, any added novelty, provided by random “bricolage” and Gould’s “exaptation” along evolution contributes to complexity when they are at least for some time compatible with the environment. Only a few organisms become more complex over time, but, because of the original symmetry breaking mentioned above (and now represented by the left wall of complexity, where bacteria are), this is enough to increase the global complexity. Of course, the figure above is highly unsatisfactory. It gives two slices over time where the second one is somewhat inconsistent: where are dinosaurs at present time? It is just a sketch, but an audacious one if analyzed closely. Mathematics may help us to consistently add the third missing dimension: time. A simple form of diffusion equation of q in time t over space x is: