Evolution and non-equilibrium physics. A study of the Tangled Nature Model

We argue that the stochastic dynamics of interacting agents which replicate, mutate and die constitutes a non-equilibrium physical process akin to aging in complex materials. Specifically, our study uses extensive computer simulations of the Tangled Nature Model (TNM) of biological evolution to show that punctuated equilibria successively generated by the model’s dynamics have increasing entropy and are separated by increasing entropic barriers. We further show that these states are organized in a hierarchy and that limiting the values of possible interactions to a finite interval leads to stationary fluctuations within a component of the latter. A coarse-grained description based on the temporal statistics of quakes, the events leading from one component of the hierarchy to the next, accounts for the logarithmic growth of the population and the decaying rate of change of macroscopic variables. Finally, we question the role of fitness in large scale evolution models and speculate on the possible evolutionary role of rejuvenation and memory effects. Introduction. Initially perceived as a challenge to gradualism, punctuated equilibria are now widely accepted [1,2] as key features of large scale darwinian evolution. Their striking similarity to intermittency in ‘aging’ [3–6] complex materials is not well understood, but may hold clues on how life evolves from matter [7]. The origin of this similarity is addressed below by analyzing the Tangled Nature Model (TNM) dynamics [8,9] as a non-equilibrium physical process. While physics ideas are common in evolution models [10, 11], evolution itself has not previously been modeled as a physical process, bar attempts [12–14] inspired by Self Organized Criticality (SOC) [15], according to which punctuations are the manifestation of stationary fluctuations. We see them instead as the manifestation of a spontaneous physical process. But how can a pertinent free energy then be defined and why does the process decelerate over time [16,17]? In spite of its simplicity, the TNM, an individual based stochastic model of ecosystem dynamics, captures key aspects of co-evolution, e.g. its decelerating nature [18], its log-normal species abundance distribution [19] and, in a version including spatial migration, the area law [20]. Punctuations, here called quakes, irreversibly disrupt quasi-Evolutionary Stable Strategies (qESS), periods of metastability where population and the number of extant species, or diversity, fluctuate reversibly. Statistical physics is used to connect microscopic interactions, defined in darwinian terms at the level of individuals, to macroscopic properties, e.g. population and diversity. Along the way, we introduce the concepts of core and cloud species and implement an adaptation of the lid method [21] originally developed to map out complex energy landscapes. We find that: i) The growing duration of qESS reflects an entrenchment into metastable configuration space components of increasing entropy; ii) The decreasing rate of evolution stems from a logarithmic time growth of the entropic barriers separating successive qESS; iii) Rare fluctuations in a time series of positive couplings extending from the core to the cloud trigger the quakes. The physical picture emerging highlights the similarity of evolution and physical aging of complex materials. The ubiquitous role of hierarchies in complex dynamics [22, 23] suggests that similar conclusions might hold beyond the TNM. Background. Our results are based on simulations performed at the SDU horseshoe cluster, using C code dep-1 ar X iv :1 30 9. 18 37 v2 [ qbi o. PE ] 1 8 A ug 2 01 4 Nikolaj Becker and Paolo Sibani veloped from scratch. Detailed information on the model parameters, the initial conditions, and how to generate the couplings can be found in Ref. [9], which should be consulted for further details. For convenience, some definitions and known properties are given below. The TNM’s variables are binary strings of length K, i.e. points of the K dimensional hypercube. Variously called species or sites, these are populated by agents or individuals, which reproduce asexually in a way occasionally affected by random mutations. Only a tiny fraction of the possible species ever becomes populated during simulations lasting up to one million generations. The extant species, i.e. those with non-zero populations at a given time, are collectively referred to as ecosystem, and their number as diversity. With probability θ, a pair (a, b) of species has non-zero couplings, (Jab, Jba), describing how b affects the reproductive ability of a and vice-versa. Empirically, the distribution of the generated couplings is well described by the Laplace double exponential density p(x) = 1 2ae −|x−x|/a. The parameters x and a are estimated to −0.0019 and 0.0111, respectively. Extant species cluster together, their closeness expressed by the Hamming distance, the number of bits by which their strings differ. Let S, Nb(t) and N denote the ecosystem, the population size of species b, and the total population N(t) = ∑ bNb(t). An individual of type a is chosen for reproduction with probability na = Na/N , and succeeds with probability poff(a) = 1/(1 + e −Ha), where