Infinite combinatorics in mathematical biology

Toward understanding the emergence of life: A dual function of the system of nucleotides in the metabolically closed autopoietic organization

General structure of metabolism includes the reproduction of catalysts that govern metabolism. In this structure, the system becomes autopoietic in the sense of Maturana and Varela, and it is closed to efficient causation as defined by Robert Rosen. The autopoietic maintenance and operation of the catalysts takes place via the set of free nucleotides while the synthesis of catalysts occurs via the information encoded by the set of nucleotides arranged in polymers of RNA and DNA. Both energy charge and genetic information use the components of the same pool of nucleoside triphosphates, which is equilibrated by thermodynamic buffering enzymes such as nucleoside diphosphate kinase and adenylate kinase. This occurs in a way that the system becomes internally stable and metabolically closed, which initially could be realized at the level of ribozymes catalyzing basic metabolic reactions as well as own reproduction. The function of ATP, GTP, UTP, and CTP is dual, as these species participate both in the general metabolism as free nucleotides and in the transfer of genetic information via covalent polymerization to nucleic acids. The changes in their pools directly impact both bioenergetic pathways and nucleic acid turnover. Here we outline the concept of metabolic closure of biosystems grounded in the dual function of nucleotide coenzymes that serve both as energetic and informational molecules and through this duality generate the autopoietic performance and the ability for codepoietic evolutionary transformations of living systems starting from the emergence of prebiotic systems.

Overcoming the limits of natural computation in biological evolution toward the maximization of system efficiency

The goal-directedness of biological evolution is realized via the anticipatory achievement of the final state of the system that corresponds to the condition of its perfection in self-maintenance and in adaptability. In the course of individual development, a biological system maximizes its power via synergistic effects and becomes able to perform external work most efficiently. In this state, defined as stasis, robust self-maintaining configurations act as attractors resistant to external and internal perturbations. This corresponds to the local energy–time constraints that most efficiently fit the integral optimization of the whole system. In evolution, major evolutionary transitions that establish new states of stasis are achieved via codepoiesis, a process in which the undecided statements of existing coding systems form the basis for the evolutionary unfolding of the system by assigning new values to them. The genetic fixation of this macroevolutionary process leads to new programmes of individual development representing the process of natural computation. The phenomenon of complexification in evolution represents a metasystem transition that results in maximization of a system’s power and in the ability to increase external work performed by the system.

The hiatus between organism and machine evolution: Contrasting mixed microbial communities with robots

Mixed microbial communities, usually composed of various bacterial and fungal species, are fundamental in a plethora of environments, from soil to human gut and skin. Their evolution is a paradigmatic example of intertwined dynamics, where not just the relations among species plays a role, but also the opportunities - and possible harms - that each species presents to the others. These opportunities are in fact affordances, which can be seized by heritable variations and selection. In this paper, starting from a systemic viewpoint of mixed microbial communities, we focus on the pivotal role of affordances in evolution and we contrast it to the artificial evolution of programs and robots. We maintain that the two realms are neatly separated, in that natural evolution proceeds by extending the space of its possibilities in a completely open way, while the latter is inherently limited by the algorithmic framework in which it is defined. This discrepancy characterizes also an envisioned setting in which robots evolve in the physical world. We present arguments supporting our claim and we propose an experimental setting for assessing our statements. Rather than just discussing the limitations of the artificial evolution of machines, the aim of this contribution is to emphasize the tremendous potential of the evolution of the biosphere, beautifully represented by the evolution of communities of microbes.

The drawbridge of nature: Evolutionary complexification as a generation and novel interpretation of coding systems

The phenomenon of evolutionary complexification corresponds to the generation of new coding systems (defined as а codepoiesis by Marcello Barbieri). The whole process of generating novel coding statements that substantiate organizational complexification leads to an expansion of the system that incorporates externality to support newly generated complex structures. During complexifying evolution, the values are assigned to the previously unproven statements via their encoding by using new codes or rearranging the old ones. In this perspective, living systems during evolution continuously realize the proof of Gödel's theorem. In the real physical world, this realization is grounded in the irreversible reduction of the fundamental uncertainty appearing in the self-referential process of internal measurement performed by living systems. It leads to the formation of reflexive loops that establish novel interrelations between the biosystem and the external world and provide a possibility of active anticipatory transformation of externality. We propose a metamathematical framework that can account for the underlying logic of codepoiesis, outline the basic principles of the generation of new coding systems, and describe main codepoietic events in the course of progressive biological evolution. The evolutionary complexification is viewed as a metasystem transition that results in the increase of external work by the system based on the division of labor between its components.

Mathematics in biological reality: The emergence of natural computation in living systems

Mathematics is a powerful tool to express the computable part of the reality of the physical world. For living systems, mathematical relations emerge internally as an abstracting capacity in the course of development and adaptation to the external world. All living systems possess internal coding structures which represent their embedded description. They are anticipatory in the sense that the embedded description generates deterministic model of their behavior. If the model does not provide a correct result, they can evolve through the acquisition of new statements inside the embedded description that overcome limitations of the existing model. The newly generated statements acquire meaning in and from the changing environment. The growth of complexity, being a consequence of the internal active adaptation to externality performed by the systems, increases the amount of external work and generates the observed patterns of spatiotemporal structures of evolving systems. In living systems, the symbolic memory constraints are dynamic processes in themselves, co-evolving with the other components of biological systems. Separation of the symbolic memory and the dynamic laws (defined as the epistemic cut), required for self-replication of biological systems, forms the basis for their onto-epistemic relation to reality. In this regard, living systems possess their own internal abstracting capacity and invent mathematics. The digital structure of the genetic code is a manifestation of this mathematics.