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

Biological thermodynamics: Ervin Bauer and the unification of life sciences and physics

Biological systems operate toward the maximization of their self-maintenance and adaptability. This is achieved through the establishment of robust self-maintaining configurations acting as attractors resistant to external and internal perturbations. Ervin Bauer (1890-1938) was the first who formulated this essential thermodynamic constraint in the operation of biological systems, which he defined as the stable non-equilibrium state. The latter appears as the basic attractor relative to which biological organization is established. The stable non-equilibrium state represents a generalized cell energy status corresponding to efficient spatiotemporal organization of the fluxes of matter and energy and constantly reproducing the conditions of self-maintenance of metabolism and controlling the rates of major metabolic fluxes that follow thermodynamically and kinetically defined computational principles. This state is realized in the autopoietic structures having closed loops of causation based on the operation of biological codes. The principle of thermodynamic buffering determines the conditions for optimization of the fluxes of load and consumption in metabolism establishing the conditions of metabolic stable non-equilibrium. In developing and evolving biological systems, the principle of stable non-equilibrium is transformed into the principle of increasing external work, which is grounded in the hyper-restorative non-equilibrium dynamics. Bauer's concept of the stable non-equilibrium state puts thermodynamics into the frames of the internal biological causality governing self-maintenance and development of living systems. It can be defined as a relational theory of biological thermodynamics since the standard to which it refers represents the actual biological function rather than the abstract state of thermodynamic equilibrium.

Toward the Relational Formulation of Biological Thermodynamics

Classical thermodynamics employs the state of thermodynamic equilibrium, characterized by maximal disorder of the constituent particles, as the reference frame from which the Second Law is formulated and the definition of entropy is derived. Non-equilibrium thermodynamics analyzes the fluxes of matter and energy that are generated in the course of the general tendency to achieve equilibrium. The systems described by classical and non-equilibrium thermodynamics may be heuristically useful within certain limits, but epistemologically, they have fundamental problems in the application to autopoietic living systems. We discuss here the paradigm defined as a relational biological thermodynamics. The standard to which this refers relates to the biological function operating within the context of particular environment and not to the abstract state of thermodynamic equilibrium. This is defined as the stable non-equilibrium state, following Ervin Bauer. Similar to physics, where abandoning the absolute space-time resulted in the application of non-Euclidean geometry, relational biological thermodynamics leads to revealing the basic iterative structures that are formed as a consequence of the search for an optimal coordinate system by living organisms to maintain stable non-equilibrium. Through this search, the developing system achieves the condition of maximization of its power via synergistic effects.

Macroevolution, differentiation trees, and the growth of coding systems

An open process of evolution of multicellular organisms is based on the rearrangement and growth of the program of differentiation that underlies biological morphogenesis. The maintenance of the final (adult) stable non-equilibrium state (stasis) of a developmental system determines the direction of the evolutionary process. This state is achieved via the sequence of differentiation events representable as differentiation trees. A special type of morphogenetic code, acting as a metacode governing gene expression, may include electromechanical signals appearing as differentiation waves. The excessive energy due to the incorporation of mitochondria in eukaryotic cells resulted not only in more active metabolism but also in establishing the differentiation code for interconnecting cells and forming tissues, which fueled the evolutionary process. The "invention" of "continuing differentiation" distinguishes multicellular eukaryotes from other organisms. The Janus-faced control, involving both top-down control by differentiation waves and bottom-up control via the mechanical consequences of cell differentiations, underlies the process of morphogenesis and results in the achievement of functional stable final states. Duplications of branches of the differentiation tree may be the basis for continuing differentiation and macroevolution, analogous to gene duplication permitting divergence of genes. Metamorphoses, if they are proven to be fusions of disparate species, may be classified according to the topology of fusions of two differentiation trees. In the process of unfolding of morphogenetic structures, microevolution can be defined as changes of the differentiation tree that preserve topology of the tree, while macroevolution represents any change that alters the topology of the differentiation tree.

Reflexive structure of the conscious subject and the origin of language codes

The code paradigm in biological and social sciences arises to Aristotle. For conscious activity, Aristotle introduced the notion of reflexive self-awareness in sense perception. This reflexive process generates the codes that signify sensual perceptive events and constrain human behavior. Coding systems grow via the generation of hypertextual statements reflecting new meanings in the process defined by Marcello Barbieri as a codepoiesis. It results in the establishment of higher-level codes (metacodes) forming the semiotic screen that has a nature of the set of perceived objects internalized by the conscious subject in encoding the symbolic actions. The characteristic feature of the semiotic screen consists in its property of being shared between the communicating agents. A sufficient complexity of nervous system, through the appearance of mirror neurons that are fired both when a subject executes certain action and when he observes another subject performing a similar action, represents a prerequisite for the emergence of reflexive codes in evolution. The codes appearing as a result of reflexive awareness and establishing different sociotypes, span from the symbolic systems of art and music through the common language to the formal language of logic and mathematics. Social dynamics is based on the implementation of reflexive coding activity and results in the growth and decay of social systems and civilizations.

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.

Quantum-like model for unconscious-conscious interaction and emotional coloring of perceptions and other conscious experiences

Quantum measurement theory is applied to quantum-like modeling of coherent generation of perceptions and emotions and generally for emotional coloring of conscious experiences. In quantum theory, a system should be separated from an observer. The brain performs self-measurements. To model them, we split the brain into two subsystems, unconsciousness and consciousness. They correspond to a system and an observer. The states of perceptions and emotions are described through the tensor product decomposition of the unconscious state space; similarly, there are two classes of observables, for conscious experiencing of perceptions and emotions, respectively. Emotional coloring is coupled to quantum contextuality: emotional observables determine contexts. Such contextualization reduces degeneration of unconscious states. The quantum-like approach should be distinguished from consideration of the genuine quantum physical processes in the brain (cf. Penrose and Hameroff). In our approach the brain is a macroscopic system which information processing can be described by the formalism of quantum theory. The paper is concluded with experimental test of contextual emotional coloring of conscious experiences based on Bell type inequalities which are treated in the contextual framework.

Biological computation running on quantum computation

Biological computation supporting biological phenomena functionally practices the underlying quantum computation indexically, rather than symbolically. An advantage of the indexical operation of quantum computation rests upon a significant reduction of the computational complexity compared with the corresponding classical counterpart running exclusively upon the symbol manipulation. The reduction of the complexity is sought in allowing for the participation of multiple processors running concurrently in a parallel manner. The concurrent distribution of multiple processors operating mutually in an inseparable manner lets each processor regard the rest of the distribution as its own environment. The environment thus formed and detected by each processor may differ from the similar ones appropriated to the other individual participants nearby. Both the individual processor and the corresponding environment turn out to be agential. Quantum computation practiced indexically may serve as a precursor agency apt for both forming Jakob von Uexküll's umwelt towards the environment and making use of James J. Gibson's affordance from the environment. The individual environment to each material participant there is already indexically agential in pulling that participant in.

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.

Voevodsky's unfinished project: Filling the gap between pure and applied mathematics

In a series of lectures given in 2003, soon after receiving the Fields Medal for his results in the Algebraic Geometry, Vladimir Voevodsky (1966-2017) identifies two strategic goals for mathematics, which he plans to pursue in his further research. The first goal is to develop a ''computerised library of mathematical knowledge,'' which supports an automated proof-verification. The second goal is to ''bridge pure and applied mathematics.'' Voevodsky's research towards the first goal brought about the new Univalent foundations of mathematics. In view of the second goal Voevodsky in 2004 started to develop a mathematical theory of Population Dynamics, which involved the Categorical Probability theory. This latter project did not bring published results and was abandoned by Voevodsky in 2009 when he decided to focus his efforts on the Univalent foundations and closely related topics. In the present paper, which is based on Voevodsky's archival sources, I present Voevodsky's views of mathematics and its relationships with natural sciences, critically discuss these views, and suggest how Voevodsky's ideas and approaches in the applied mathematics can be further developed and pursued. A special attention is given to Voevodsky's original strategy to bridge the persisting gap between pure and applied mathematics where computers and the computer-assisted mathematics play a major role.