Studies on evolutionary and selective properties of hypercycles using a Monte Carlo method

Nonequilibrium Entropic Bounds for Darwinian Replicators

Life evolved on our planet by means of a combination of Darwinian selection and innovations leading to higher levels of complexity. The emergence and selection of replicating entities is a central problem in prebiotic evolution. Theoretical models have shown how populations of different types of replicating entities exclude or coexist with other classes of replicators. Models are typically kinetic, based on standard replicator equations. On the other hand, the presence of thermodynamical constraints for these systems remain an open question. This is largely due to the lack of a general theory of statistical methods for systems far from equilibrium. Nonetheless, a first approach to this problem has been put forward in a series of novel developements falling under the rubric of the extended second law of thermodynamics. The work presented here is twofold: firstly, we review this theoretical framework and provide a brief description of the three fundamental replicator types in prebiotic evolution: parabolic, malthusian and hyperbolic. Secondly, we employ these previously mentioned techinques to explore how replicators are constrained by thermodynamics. Finally, we comment and discuss where further research should be focused on.

Some mechanistic requirements for major transitions

Major transitions in nature and human society are accompanied by a substantial change towards higher complexity in the core of the evolving system. New features are established, novel hierarchies emerge, new regulatory mechanisms are required and so on. An obvious way to achieve higher complexity is integration of autonomous elements into new organized systems whereby the previously independent units give up their autonomy at least in part. In this contribution, we reconsider the more than 40 years old hypercycle model and analyse it by the tools of stochastic chemical kinetics. An open system is implemented in the form of a flow reactor. The formation of new dynamically organized units through integration of competitors is identified with transcritical bifurcations. In the stochastic model, the fully organized state is quasi-stationary whereas the unorganized state corresponds to a population with natural selection. The stability of the organized state depends strongly on the number of individual subspecies, n, that have to be integrated: two and three classes of individuals, [Formula: see text] and [Formula: see text], readily form quasi-stationary states. The four-membered deterministic dynamical system, [Formula: see text], is stable but in the stochastic approach self-enhancing fluctuations drive it into extinction. In systems with five and more classes of individuals, [Formula: see text], the state of cooperation is unstable and the solutions of the deterministic ODEs exhibit large amplitude oscillations. In the stochastic system self-enhancing fluctuations lead to extinction as observed with [Formula: see text] Interestingly, cooperative systems in nature are commonly two-membered as shown by numerous examples of binary symbiosis. A few cases of symbiosis of three partners, called three-way symbiosis, have been found and were analysed within the past decade. Four-way symbiosis is rather rare but was reported to occur in fungus-growing ants. The model reported here can be used to illustrate the interplay between competition and cooperation whereby we obtain a hint on the role that resources play in major transitions. Abundance of resources seems to be an indispensable prerequisite of radical innovation that apparently needs substantial investments. Economists often claim that scarcity is driving innovation. Our model sheds some light on this apparent contradiction. In a nutshell, the answer is: scarcity drives optimization and increase in efficiency but abundance is required for radical novelty and the development of new features.This article is part of the themed issue 'The major synthetic evolutionary transitions'.

Emergence of diversity in a model ecosystem

The biological requirements for an ecosystem to develop and maintain species diversity are in general unknown. Here we consider a model ecosystem of sessile and mutually excluding organisms competing for space [Mathiesen et al. Phys. Rev. Lett. 107, 188101 (2011)]. Competition is controlled by an interaction network with fixed links chosen by a Bernoulli process. New species are introduced in the system at a predefined rate. In the limit of small introduction rates, the system becomes bistable and can undergo a phase transition from a state of low diversity to high diversity. We suggest that isolated patches of metapopulations formed by the collapse of cyclic relations are essential for the transition to the state of high diversity.

Ecosystems with mutually exclusive interactions self-organize to a state of high diversity

Ecological systems comprise an astonishing diversity of species that cooperate or compete with each other forming complex mutual dependencies. The minimum requirements to maintain a large species diversity on long time scales are in general unknown. Using lichen communities as an example, we propose a model for the evolution of mutually excluding organisms that compete for space. We suggest that chainlike or cyclic invasions open for creation of spatially separated subpopulations that subsequently can lead to increased diversity. In contrast to its nonspatial counterpart, our model predicts robust coexistence of a large number of species. It is demonstrated that large species diversity can be obtained on evolutionary time scales, provided that interactions between species have spatial constraints. In particular, a phase transition to a sustainable state of high diversity is identified.

Error propagation in the hypercycle

We study analytically the steady-state regime of a network of n error-prone self-replicating templates forming an asymmetric hypercycle and its error tail. We show that the existence of a master template with a higher noncatalyzed self-replicative productivity a than the error tail ensures the stability of chains in which m < n-1 templates coexist with the master species. The stability of these chains against the error tail is guaranteed for catalytic coupling strengths K on the order of a. We find that the hypercycle becomes more stable than the chains only if K is on the order of a2. Furthermore, we show that the minimal replication accuracy per template needed to maintain the hypercycle, the so-called error threshold, vanishes as square root of n/K for large K and N < or = 4.

Viral sex, levels of selection, and the origin of life

Several issues in Chao's related paper J. theor. Biol. (1991, 153, 229-246) are revisited. It is argued that mixes of segments from different viral coinfection groups cannot be regarded as sex, unless one is willing to accept that these groups are replicators and individuals. But, because selection in coinfection groups is dynamically analogous to that in trait groups in structured demes, one should also regard these latter groups as replicators. This approach is unacceptable since the groups in question have irregular ploidies, an unfixed number of parents, and no rules analogous to those of meiosis. It is emphasized, however, that the effective presence of neighbour-modulated fitness can ensure dynamical coexistence of covirus segments, even if the equal net reproduction rate within groups is not warranted. It seems that during the origin of coviruses from complete viruses, a higher-level evolutionary unit has become disintegrated, whereas during the origin of life a higher-level unit, the protocell, has emerged from lower-level ones, i.e. unlinked, replicating genes. These two gene-level systems are not homologous, but analogous. Although it is true that the resistance to parasites and the need to avoid a mutational collapse of the genome are likely to have called for some compartmentation in precellular stages of evolution, no clear demonstration, that the proposed mechanisms (the compartmentalized hypercycle and the stochastic corrector model) do in fact solve the error threshold problem, exists. Neither has a plausible mode of protocellular sex been suggested.

Study of an error-prone hypercycle formed from two kinetically distinguishable species

Asymmetry within both the amplification factor values (Ak) and cross-catalytic hypercyclic constant (Kjk) and its influence on the stability of a two-membered error-prone hypercycle has been exhaustively studied from a deterministic point of view and the bifurcation diagram as a function of the quality factor (Q) has been obtained. In the more general case, several Q critical values appear, changing their relative position in the diagram depending on the Ak and Kjk values. The order of the Q critical values affects both the general properties of the system and the stability of the hypercyclic organization. The importance of this asymmetry in the selective and evolutionary properties of the hypercycle is also discussed.

Influence of the hypercycle on the error threshold: a stochastic approach

The role of fluctuations on the error threshold of the hypercycle has been studied by a stochastic approach on a very simplified model. For this model, the master equation was derived and its unique steady state calculated. This state implies the extinction of the system. But the actual time necessary to reach the steady state may be astronomically long whereas for times of experimental interest the system could be near some quasi-stationary states. In order to explore this possibility a Gillespie simulation of the stochastic process has been carried out. These quasi-stationary states correspond to the deterministic steady states of the system. The error threshold shifts towards higher values of the quality factor Q. Moreover, information about the fluctuations around the quasi-stationary states is obtained. The results are discussed in relation to the deterministic states.