Computational power and generative capacity of genetic systems

Accepted Manuscript Title: Computational power and generative capacity of genetic systems Author: Abir U. Igamberdiev Nikita E. Shklovskiy-Kordi PII: DOI: Reference:

S0303-2647(16)30003-X http://dx.doi.org/doi:10.1016/j.biosystems.2016.01.003 BIO 3632

To appear in:

BioSystems

Received date: Revised date: Accepted date:

8-12-2015 25-1-2016 27-1-2016

Please cite this article as: Igamberdiev, A.U., Shklovskiy-Kordi, N.E.,Computational power and generative capacity of genetic systems, BioSystems (2016), http://dx.doi.org/10.1016/j.biosystems.2016.01.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Computational power and generative capacity of genetic systems Abir U. Igamberdiev1, Nikita E. Shklovskiy-Kordi2 1

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Email addresses: [email protected]; [email protected]

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Department of Biology, Memorial University of Newfoundland, St. John’s, NL, A1B 3X9, Canada; 2National Research Center for Hematology, Moscow, 125167, Russian Federation

Abstract

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Semiotic characteristics of genetic sequences are based on the general principles of linguistics formulated by Ferdinand de Saussure, such as the arbitrariness of sign and the linear nature of the signifier. Besides these semiotic features that are attributable to the basic structure of the genetic code, the principle of generativity of genetic language is important for understanding biological transformations. The problem of generativity in genetic systems arises to a possibility of different interpretations of genetic texts, and corresponds to what Alexander von Humboldt called “the infinite use of finite means”. These interpretations appear in the individual development as the spatiotemporal sequences of realizations of different textual meanings, as well as the emergence of hyper-textual statements about the text itself, which underlies the process of biological evolution. These interpretations are accomplished at the level of the readout of genetic texts by the structures defined by Efim Liberman as “the molecular computer of cell”, which includes DNA, RNA and the corresponding enzymes operating with molecular addresses. The molecular computer performs physically manifested mathematical operations and possesses both reading and writing capacities. Generativity paradoxically resides in the biological computational system as a possibility to incorporate meta-statements about the system, and thus establishes the internal capacity for its evolution. Keywords: biological evolution; generativity; genetic code; language; molecular computation; physical carriers of mathematical operations

1. Introduction. Genetic language and its operation It was considered throughout the human history that all known texts were written by humans. However in 1953 it was established that the hereditary information in cells is presented as a DNA code sequence of four letters-nucleotides. The matrix principle of heredity was formulated as a scientific hypothesis which included the double helix idea by Nikolai Koltsov (1927, in more detail 1936), while the first model of the genetic code was suggested by George Gamow (1954). The information in DNA, similarly as in human texts, is presented as a linear order of letters (symbols). Like in books where some information is attributed to pictures and to book 1

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shape, the secondary and higher structures of DNA also bear certain information; however for simplicity we usually consider primarily the linear information encoded by four nucleotides. For the reading device, all same signs (letters) are identical.

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The genetic language is based on the rigid structure of genetic code which is universal for all organisms on Earth except of minor deviations in mitochondria and several lower organisms. The genetic code possesses all basic characteristics of the linguistic semiotic system that were formulated by Saussure (1911). One of such characteristics is the arbitrariness of sign, expressed in the fact that the correspondence of triplets of the genetic code to amino acids is not based on any physical similarity and could be different (which is observed at a limited degree in the mitochondrial code). Another important characteristic feature is the linear nature of the signifier, which fully corresponds to the linearity of human alphabetic script. Any language has the syntactic, pragmatic and semantic aspects. In the genetic language, its syntax is represented by the combinatorial rules of interactions between nucleotides; the pragmatics is realized via the context-dependent transcription, while the semantics appears as the function of the transcription products (Witzany, 2016).

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For language operation, a device is needed which can read it, and a subject (“self”) which can perceive it. Originally the concept of such reading device in biological cells was introduced by Efim Liberman (1972) and developed in his subsequent papers (Vaintsvaig and Liberman, 1973; Liberman, 1978, 1979, 1983, 1989). According to this concept, DNA is not a compendium of genes but a molecular text representing a program for the molecular computer of the cell (MCC). The necessity of MCC arises from the fact that that no natural code codes itself but needs some competent agents that act on this code (Witzany, 2016). These agents form a machine-like structure which function is to decode genetic statements. MCC is a system consisting of DNA, RNA and proteins addressing them. Its operators cut and crosslink the molecules at certain places determined by the program written on DNA. The enzymatic activity necessary for this function is associated with corresponding protein enzymes and with the enzymatic activity of RNA. The scheme of MCC operation and of its transformational and evolutionary consequences is presented on Figure 1. It shows that the process of molecular computation is based on reading and interpreting of genetic texts in the expense of energy resulting in the coordination of biosystem’s functions with the events of the observable world. This coordination possesses generative properties and results in the open process of evolution via writing new genetic texts. The process of natural calculation performed by MCC uses the Brownian movement of molecular structures. Multiple searches of addressed molecular operations of words-molecules are possible due to heat motion (Liberman, 1972). The certainty of interpretation of the sequence and its time-irreversible reading are defined, according to Liberman, by the loss of energy for calculation which represents the price of action and determines the physical limitation of computation (counting). For the meaning of signs, their chemical matter is important only for the 2

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process of readout by MCC, while the information by itself is static and independent from its carrier. In the physical reality of the cell, the genetic information receives its interpretation, and its material carriers (nucleotides) exhibit their real physical properties, in particular in the secondary, tertiary and quaternary DNA structures up to chromosomes that possess the speciesspecific architecture.

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The genetic language expresses the living self (corresponding to the “quantum regulator” in Liberman’s concept, see Liberman 1983), while the human language expresses the conscious self. It is widely accepted that the process of computation as well as the process of fixation of the result of measurement are realized by the conscious observer. However this statement may not be valid even at the level of human brain. The experiments of Benjamin Libet (1985) show that consciousness appears as an epiphenomenon of brain states, and the reduction of uncertainty in volition actions takes place at the level of the unconscious before any realization of the awareness. The important examples of the unconscious choice have been established for animal behaviour in the studies of Gunji group (Fukano et al., 2004; Migita et al., 2005).

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Gamkrelidze (1989) analyzed the structural isomorphism between the two codes (genetic and linguistic) in relation to the existence of two approaches to explain such isomorphism, one arising to Jacob (1977) and the other to Jakobson (1971). According to Francois Jacob, this isomorphism appeared as a result of the structural coincidence between the two systems bearing similar information functions, while Roman Jakobson derived this isomorphism from the phylogenetic construction of the linguistic code on the basis of the structural principles of the genetic code. Such opposition can be resolved via understanding of the universal principles that are reproduced independently for different digital semiotic systems and arising to the combinatorial rules that were initially formulated in the Chinese “I Ching” book (Petoukhov, 2006, 2016). Both the genetic and the human alphabetic languages are based on the linear representation, while the interactions based on the secondary and tertiary structure of nucleic acids and proteins resemble the Chinese and Egyptian hieroglyphics (Doerfler, 1982). Ratner (1993) analyzed the genetic language as a collection of rules and regularities of encoding the genetic information in the course of operation of the genetic texts. The genetic language possesses the alphabet, the grammar, the system of punctuation, and semantics. Searls (1997, 2002) pointed the universality of resemblance between the genetic and the human language and emphasized the linguistic basis of the basic methods and approaches of bioinformatics. As Eigen and Winkler (1981, p. 282) stated, “Nature’s two great evolutionary processes – the development of all forms of life and the evolution of the intellect – both depended on the existence of language”. 2. Generativity and evolution A key feature of language is generativity, which means a possibility to create an infinite number of meaningful sentences from the fixed and finite number of basic elements (sounds, letters, and 3

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words). Alexander von Humboldt (1792, cited by Chomsky, 1966b) characterized this feature as “the infinite use of finite means”. Generativity distinguishes the human language from the communications of nonhuman species. It is based on the vocabulary of units and rules for combining them that makes possible the formation of novel but recognizable structures. Some of the generative aspects of human language are captured by generative grammars that use recursion (Corballis, 1992); however the current versions of the generative grammar possess limited universality (see below).

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The understanding that genericity is the basic property of living systems that drives their evolution, was evident in the works of several authors, however the notion of genericity initially was not incorporated into the scientific paradigm. The “complexifying force” of Lamarck may be considered as a very general definition of the evolutionary generativity. Contrary to the widespread views, the direct adaptation is not the main principle of the Lamarckian evolution because, according to Lamarck, it can operate within one gradation (i.e., it is limited within the single level of complexity). To transform for another level of complexity, the complexifying force was postulated (Lamarck, 1809). The notion of complexifying force can be incorporated in the scientific paradigm via the semiotic principle of metasystem transition (Turchin, 1977). In modern terms, the growth of complexity can be the primary consequence of quantum measurement when the system measures itself in the environment generating infinite recursion (Igamberdiev, 2014). Both measurement and evolution are open processes that cannot be adequately formalized in terms of the first order logic. Evolution becomes possible when the closed set of energy-degenerate, rate-independent genetic symbols (appearing as a hereditary memory) is distinguished from the rate-dependent open dynamics of construction it controls (Pattee, 2001). This set possesses a complexifying capacity which is non-computable by itself but can result in the formation of more complex systems possessing the property of decidability via the process of internal computation that halts in the end. Understanding genericity is the most important challenge for modern science. The division of all complex systems into non-generic and generic was clearly outlined by Robert Rosen who defined living systems as “closed to efficient causation” and claimed that their genericity is the main difference from the physical automata amenable by the Newtonian models (Rosen, 1999). According to Rosen, the “functional component” of living systems that makes them generic, is context-dependent and has no meaning outside its role in the system. In accordance with the closeness to efficient causation, it has a self-referential dimension. Information as a fixed choice exists only in generic systems. Analyzing the physical model of the Universe, Robert Rosen (1999, p. 44) wrote: “The conventions on which contemporary physics rest amount to asserting that the world of material nature, the world of causal entailment, is a predicative world. It is a world of context-independent elements, with a few finitary operations, in which impredicativities cannot arise. This is what makes it objective. But this objectivity is bought very dearly: its cost is a profound non-genericity of that world, an impoverishment of what can be entailed in it. Most profoundly, it is a world in which life does not exist. But life does exist.” 4

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Understanding evolution as a generic phenomenon, as we mentioned, arises to the complexifying principle of Lamarck. The understanding of evolution in Darwin’s model is also generic, but this genericity is firmly restricted by the limitations of the process of random and non-directional variation that is rather chaotic in the original Darwinian model. Generativity of the evolutionary process as a philosophical concept was formulated by Henri Bergson (1917). For its implementation by using strictly scientific terms, the concept of autocatalysis became important. It was introduced as the concept of hypercycle by Eigen and Schuster (1978) and as the theory of autocatalytic sets by Kauffman (1986, 1993). In his most recent works, Kauffman (2014) claims that biological evolution can be understood only beyond the Newtonian paradigm and must include the understanding of genericity. The problem of origin of life can be interpreted in a way of how the simplest generic system appeared. The generic system here is a self-referential system that can memorize its generic choice, i.e., possesses its own embedded semiotic description with a digital language. Thus the appearance of the genetic language opened a possibility of potentially infinite evolutionary development of life.

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Language generativity can be described as the ability to produce sentences never before said, and to understand sentences never before heard. The symbols of a language can be combined in unusual and new ways and still convey meaning. Generativity was defined as the unbounded scope of reference and freedom from control by identifiable stimuli (Chomsky, 1966). The properties of multivalence and complexity are implemented in the genetic language (Trifonov, 1989) by other means than in human language. Popov et al. (1996) emphasized that the genetic text can be read by different ways, each time using the alternative choice of signs and ignoring other signs. This becomes possible, e.g., via the overlap of sequences which makes genetic texts more complicated as compared to the texts of human language. Collado-Vides (1989, 1992) made an attempt to define generative principles of the genetic and human languages and to formulate the principles of transformation of genetic texts. In this framework, the genetic syntactic structure is considered as an equivalent of sentence, and sign iteration appears as an intermediation between potentialities that are actualized and actualities that are potentiated (Andrade, 2014). In the transformations that ensure generative properties of the genetic language, an important role belongs to the non-coding DNA sequences, which provide the redundancy of genetic texts. This redundancy is necessary for the realization of transformational possibilities of the genetic system and its ability for realignment and evolution (Mantegna et al., 1994). Ji (1997), by analyzing the isomorphism between the genetic and human languages, showed that both types of language share 10 out of 13 properties of human language mentioned by Hockett (1960). According to Ji (1997), the structural genes correspond to lexicon, while the “spatiotemporal” genes correspond to the grammar of the genetic language. This grammar is identified as a mapping of nucleotide DNA sequences into its four-dimensional patterns of folding that control gene expression. This mapping is considered as “the second genetic code”. The genetic language

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introduces in biology the principle of “rule-governed creativity”, which substantiates a potentially infinite evolutionary development of living organisms.

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Later Ji (1999) developed the model of cell computer that has an essential similarity to the earlier model of Efim Liberman (1972, 1989, see also Liberman et al., 1989). Following the logic of the work of Adleman (1994), who demonstrated experimentally that DNA molecules can be manipulated in vitro in a way that they realize certain type of mathematical calculation, Ji (1999) formulated the principles of molecular semiotics which describe the operation of cell computer. The generative principles that follow from this concept on the basis of change of spatial forms were characterized in more detail in the paper of Ji and Ciobanu (2003). Conrad (1999) developed the concept of molecular computation which is similar to the concept of Liberman but lacks the idea of the internal quantum regulator. In other words, Conrad considered biological systems from the point of third person descriptions, while Liberman emphasized the importance of the first and second person descriptions of the biological system raising the issue of how it transforms and makes internal decisions itself as measuring and interacting with the others (Matsuno, 2003).

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The similarities and differences in the approaches of these two scientists were outlined in their joint paper (Conrad and Liberman, 1982). The patterns of recognition by proteins and nucleic acids on the basis of form complementarity, according to Conrad, result in the multiplicity of solutions having different degrees of context dependence. The self-organizing principle of computation on the basis of DNA was analyzed in the work of Conrad and Zauner (1998). The work of Adleman (1994), in which the computational properties of DNA were described, has a great scientific significance as it formulates the formal basis of molecular computation. However we should remember that the concept of DNA-based computation was first introduced by Liberman in 1972, and in the next year the equivalency of the molecular cell computer to the universal computational machine (Markov’s algorithms) was demonstrated (Vaintsvaig and Liberman, 1973). Liberman’s contributions also preceded the important works of Deutsch (1985) on quantum computation. Generativity corresponds to the “Self-growing Logos” of Heraclitus and can be defined as a consequence of the ability of text to generate a hypertext, which signifies the text itself so that some statements acquire the property to reflect the whole system. In formal logic this corresponds to the postulation of incompleteness of the formal system and to its relative resolution by the means of Gödel numbering in which certain statements encode the whole system. According to the approach arising to Darwin, generativity in evolution is a consequence of casual mistakes. However to become generative, the casual mistake should be internalized within the complex formal system. John von Neumann (1966) proved the theorem claiming that for reproduction of finite automata, a sufficiently high level of complexity is needed. Such level of complexity can correspond to the ability of formal systems to generate Gödel numbers (corresponding to the hypertextual statements in linguistics), in other words, to be generative and to build sufficiently advanced hypertext which allows reproducing the given system and make it 6

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even more complex after reaching a certain threshold. This can also be defined as an abstracting capacity, in other words, an organism is able of abstracting its own durability as a class property out of different individual events (Matsuno, 2014).

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3. Paradox of computation versus generativity

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Biological evolution should be considered as a consequence of generativity of the genetic language, while the social evolution is a consequence of generativity of human language. The systems of animal communication are not generative. They use limited sets of symbols and can change only in parallel with the changes driven by biological evolution. This may be explained by the principle that only sufficiently complex systems can generate hypertexts, while the communicative systems of animals are not sufficiently rich to produce the hypertextual statements.

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In the framework of cybernetics, computation and generativity seem to be the antithetical concepts. In order to save its decidability, computation must halt in the end. In contrast, generativity and open-endedness are the main presumptions of biological evolution (Pattee 1989, 2001). Resolution of the paradox of computation and generativity in the classical evolutionary approach is attributed to casual mistakes in computational programs which occasionally can be generative, and this can be proved by natural selection. How a casual mistake is internalized within the complex formal system to become generative is left beyond the framework of this approach. Conrad and Liberman (1982) were probably the first researchers who formulated the crucial question how to figure out the open-endedness from computation. The open-endedness, according to their views, comes from recombination of biological macromolecules in the course of realization of the genetic program, i.e., during the performance of a certain type of mathematical calculation. Generativity can paradoxically reside in the computational system not as a property of “incomplete computation” but rather of the self-expanding computation which is able to internalize meta-statements about the formal system in an expanded system that incorporates these meta-statements. Generativity corresponds to the recursivity of embedding propositions within sentences, and the system perpetually resolves the paradox of computation and generativity via the process of recursive embedding and thus possesses the internal capacity for evolution. When we consider genetic program as a case of natural computation, we have to assume the direct isomorphism between computer programs and mathematical proofs (Curry–Howard isomorphism). In molecular biology, its concrete observations are decidable as revealed in many concrete examples of the buildup of DNA, RNA and proteins. The transitions from one molecular structure to another can be viewed as deductive processes. The genetic system represents a natural deduction system possessing the Curry-Howard isomorphism (Boniolo et al., 2015). In this isomorphism, the programs appear as mathematical proofs and the propositions as 7

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interpretation types. To establish the correspondence between proofs and propositions, the second-order logic can be applied. In fact, the Curry–Howard isomorphism links the genetic system to the intuitionistic second-order propositional logic, which formulae are constructed as propositional variables via implication and universal quantification of the statements. It can be expressed as a λ-calculus based on the function abstraction and the application using variable binding and substitution.

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The link between logic and computation that is attributed to Curry and Howard is related to the operational interpretation of intuitionistic logic (Brouwer–Heyting–Kolmogorov interpretation). What appears in the first-order logic as incompleteness in which the capacity for generativity is grounded, the second-order logic resolves in much more advanced way (Rossberg, 2004). Kolmogorov complexity (see Kolmogorov, 1965) is non-computable as well as the complexification process itself (appearing as a non-computable function), however the duality of the first-order and the second-order logics in biological evolution can bring the understanding of evolution as an intuitionistic construction in which meta-statements are incorporated making the newly generated system complete and deductive.

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While the first-order logic quantifies the variables that range over individuals, the second-order logic also quantifies over sets. The logics of third and higher orders quantify over sets of sets and their semantics are more complex. The second order logic represents the functional relation as a domain of discourse, so a predicate acquires the property of universal quantifier (corresponding to the Platonic eidos). In molecular biology these quantifiers would correspond, in particular, to Hox genes, i.e., the categories of transcriptional factors that can initiate the programs of specific differentiation patterns (e.g. PAX6 gene for eye development). The CRISPR-Cas system that confers resistance to foreign genetic elements and provides a form of acquired immunity via targeted genome editing is another category of the universal transcriptional quantifier representing a meta-statement that introduces the acquired memory in the genome (Koonin and Makarova, 2013). 4. Generativity of genetic language

The problem of generativity of genetic language has not yet received a significant development in genetics and is usually resolved on the basis of the concept arising to Charles Darwin that casual mistakes (defined in genetics as mutations) by themselves are generative and they determine the evolutionary process by means of natural selection. The question what makes casual mistakes generative, and what structure permits the internalization of “mistakes” and its own generative transformation, was not properly set at the time of Darwin and later in classical genetics. In a very general approximation, we can state that the logical basis of evolution is the incompleteness of formal system of the genetic language (Igamberdiev, 2007), while the physical basis is the principle of uncertainty of Heisenberg (Matsuno 1992a, Igamberdiev, 1993, 1998, 2004, 2007). Both the logical incompleteness and the physical uncertainty are present in the structure of quantum measurement, in which the system measures itself in the environment 8

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(Igamberdiev, 2014). The internal measuring device measures, among other elements of the measured system, itself in the state of measuring, which limits the possible accuracy of quantum measurement and manifests itself as the uncertainty principle (Matsuno, 1992a). This means that the quantum measurement is an elementary generative structure underlying all events in the Universe. How does this principle relate to generativity of genetic texts?

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The answer can be found in the concept of generativity of genetic texts that arises to Liberman. We can interpret the ideas of Liberman in a way that generativity is the basic property of the system of readout of genetic text which is defined in his works as “the molecular cell computer” (MCC) and represents a part of the cell that directly operates with the hereditary information. This MCC is equivalent to the universal computer using Markov’s algorithms (Vaintsvaig and Liberman, 1973). Alternative readouts in this computer correspond to the processes of recombination, splicing, actions of small RNAs. These processes were predicted by Liberman (1972) several years before their discovery.

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A potential set of alternative readouts of the genetic text is realized in the process of individual development and adaptation of organisms. A possibility of new realizations non-determined by the system and appearing due to the incompleteness of the formal system unlocks the new level of generativity and determines biological evolution. The “generative grammar” of the genetic system is defined by the rules of recombination of the genetic material according to molecular addresses. The simplest prerequisites of generativity in the genetic language are based on the operations related to the large-scale perturbations in genomes (deletion, inversion, transposition, duplication). These perturbations possess limited generative capacity, closure properties, decidability (Dassow et al., 1997). There was even an exercise of development of the evolutionary theory based on the single principle of gene duplications (Ohno, 1970). The expansion of generative capacity can be achieved via the splicing procedures in particular or as the introduction of the combinatorial power of RNA to the system “genome – proteome” in general. The effect of splicing rules applied to finite multisets of words using the sequential types of parallel derivation strategies has been analyzed in the work of Dassow and Vaszil (2004) and a huge expansion of the generative capacity in such systems was revealed. If the MCC operates with molecular addresses and represents the set of possible realizations, the choice of these realizations can be attributed to the agency that governs MCC. Liberman considered living cell as a system with the internal point of view, thus possessing an internal self (Conrad and Liberman, 1982; Liberman, 1983; Minina and Liberman, 1990; Liberman and Minina, 1996). This cellular self was defined by Liberman as a “quantum regulator”, later it was described as a coherent internal quantum state (IQS) (Igamberdiev, 2004), sending the commands of decoherence implemented by means of MCC, which results in adaptive reactions, spatial movements and morphogenetic changes of biosystems (Igamberdiev, 2012a). The ideas of Liberman on calculating operations generating the physical space-time and irreversibility connect the genetic language with the space-time structure which appears as a result of

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calculation, while irreversibility is introduced due to the loss of energy for calculating procedures.

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Figure 2 depicts the abstract scheme of the biological system that includes the internal quantum state (IQS) shielded from thermal fluctuations, with its most coherent part corresponding to perception and conscious activity, the MCC corresponding to the genetic system operating with molecular addresses, and the heat machine of the organism. The system exists in the environment immersed in the cosmic microwave background. The temperatures are calculated for IQS from the rate of emission and energy of coherent quanta (Matsuno and Paton, 2000, Igamberdiev 2004, 2007). The picture shows that the long-living quantum coherent state is embodied into a robust quantum heat engine feeding on quantum decoherence that supports the coherent state (Matsuno, 2006), while the heat engine of the organism operates in the external ambient environment that is in turn embedded into the cosmic microwave environment of the Universe.

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Although the rules of the genetic code are strictly fixed, so Nalimov (1981) was able to define them as “the rigid language of the genetic code”, even within the code certain variations in interpretation exist, and the common phenomenon is editing of the transcribed text, especially in mitochondria. The basic structure of the genetic language, which is the genetic code, is more rigid than the human language and corresponds more to the unambiguous rules of formal logic. But an open area of freedom is expanded at the level of hypertexts, which provides the flexibility in the individual development and adaptation of organisms and opens potentially infinite possibilities for evolution.

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Genome is characterized by the complementarity between linear texts and their superposition. This complementarity assumes that the text and the hypertext should be separated by a time interval which divides the system and its embedding. Overlapping genes, alternatively spliced sequences, DNA and RNA editing, introns, recombination in accordance with molecular addresses all form a complex hypertext generating potentially infinite number of language games. Genome as a complete semiotic structure exists as a complementary set of its alternative combinations that unavoidably include logical paradoxes determining its temporal dynamics (Isalan, 2009). The complete genome structure is a superposition of alternative realizations which generates one single realization in a concrete moment of time. The combinatorial possibilities of genome are increased enormously via the pool of mobile genetic elements including viruses, transposons and other agents of the horizontal transfer of genetic information. 5. The role of RNA in generativity of genetic language Recursion is either impossible or strictly limited in the dual system “gene – protein”. Generativity is such system can be limited by the evolution by means of mutations and their selection. However in the system with an intermediate component (RNA), a huge flexibility appears and its evolution can take place as a complex language game, in which RNA becomes a part of the temporal component, mediating incomplete identification and increasing the plasticity 10

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between genes and their realized functions. Liberman (1972, 1989) defined the primary role of RNA in operation of the molecular readout device and its evolution. Even the role of accidental choice is increased significantly in the system that includes RNA. Therefore the RNA pool can be considered as a realizer of a powerful language game, in which the Wittgensteinian family resemblances (Familienähnlichkeiten) are formed, with the selection of pragmatically valid realizations. The RNA pool thus represents a powerful source of the “formal superposition” in the sense of Bordonaro and Ogryzko (2013) which represents the mechanism driving evolutionary adaptation.

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The casual rearrangement of nucleotides in RNA-based multiplication of generativity can be provided via the action of the enzyme polynucleotide phosphorylase operating without energy cost. It is the main buffering enzyme for many types of genetic recombination. A new sequence generated via such equilibrium rearrangement can potentially acquire meaning and direct the macroscopic movement of the components of MCC.

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A special type of generativity exists in ciliates, which enables to define the generative role of RNA. These unicellular organisms possess two nuclei, a small diploid micronucleus and big polyploid macronucleus. The latter is formed through the amplification of micronucleus genome and its heavy editing. Micronucleus does not express genes, it is a kind of germplasm in the sense of August Weismann, while macronucleus produces RNA for vegetative growth. The templates of maternal RNA organize DNA rearrangements and transfer spontaneous mutations appearing during the life cycle to next generations (Nowacki and Landweber, 2009). This means that the ciliate genome is an epigenome assembled from the templates formed during previous generations. Ciliates are characterized by the spatial separation (between the two nuclei) of the edited and non-edited genome, while in other eukaryotes this separation is rather temporal. Ciliates are therefore unique for the study of generative editing of the genome that provides the evolutionary flexibility (Landweber and Kari, 1999). The parental non-coding DNA molecules play a role in instructing whole genome reorganization. The latter includes the removal of practically all non-coding DNA and sorting of remaining fragments, which results in the formation of generative somatic genomes (Nowacki et al., 2010). Nowacki et al. (2011) define the biological role of DNA in sculpting of genetic information in cells. In the classical scheme of the central dogma of molecular biology, the general role of all RNA forms is not clearly defined. In our views, the most general role of RNA consists in the generation of possibilities of new Gödelian (hypertextual) statements which can be memorized in genome. Different types of RNA fulfil a generative function that consists in providing dynamical flexibility in the system and enabling potentially infinite unfolding of the genetic language. In this anticipatory language game, RNA moderates an incomplete identification between genes and functions and increases the plasticity of this interaction (Igamberdiev, 2015). Shapiro (2013, 2016) has suggested to reconsider our understanding of genome as a Read-Only Memory, changing via copying of casual errors, to a sophisticatedly formatted system of data 11

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storage Read-Write (RW), perpetually changing according to cellular modifications and inscriptions. These inscriptions occur at three temporal levels: of cellular reproduction, of multicellular development, and of evolutionary transformations, and include the diversity of processes characteristic for each level such as the formation of nucleoprotein complexes, epigenetic formatting, and structural transformations of nucleotide sequences. The writing function of genome directly depends on the generation of new “words” and loading them by meaning. Therefore it is not a direct writing and not a directly memorized adaptation in the sense of Lamarck, but it is the process mediated logically by generative properties of the system. Witzany (2014, 2016) considers different ways of the appearance of new evolutionary features on the basis of genome editing with participation of different types of RNA molecules.

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Operation of MCC, according to Liberman (1989), is not limited by the operations on DNA and RNA but also includes post-translational modifications of proteins. This level of genetic language operates with three-dimensional objects. The important role belongs here to the allosteric regulation of proteins-operators. According to the central dogma, the transfer of information is possible from DNA to RNA and then to proteins. The transfer of information from RNA to DNA corresponds to the reverse transcription. The arrow from proteins to RNA is absent (the reverse translation is impossible); however the changes in DNA are mediated by the proteinbased MCC via the pool of RNA and its combinatorial rearrangements, and correspond to the evolutionary adaptation of organism in the changing environment via writing new genetic texts.

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Figure 3 depicts the general scheme of generation and transfer of information in biological systems which incorporates the central dogma of molecular biology into the structure of (M,R) system of Robert Rosen (1991) in its modification presented in the book of Igamberdiev (2012b, p. 140). In the current scheme, which is adapted to reflect evolutionary transformations, M corresponds to maintenance and R to evolutionary reconstruction of biosystem. The total pool of RNA is divided in the translational and combinatorial pools; the latter is formed through generative dynamics and can be memorized in DNA via reverse transcription. The scheme postulates a possibility of translation of elements of the combinatorial RNA pool into protein molecules for testing their functionality (arrow 7), and a subsequent selection of useful combinations to be memorized via reverse translation (arrow 6). This process may take place, in particular, during the immune response. The mechanism of alternative splicing possibly appeared as a consequence of this general function, which establishes a new invariance in the generatively transformed system. 6. Generativity of human language The problem of generativity of human language was not explicitly stated and formulated until the second half of the XX century. The reason for this was, in part, the absence in linguistics of the general theory of language evolution like the theories of Jean Baptist Lamarck, Charles Darwin or Leo Berg in biology, despite of wide discussions of language evolution also by biologists

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(e.g., Berg, 1922), and of the similarity of the basic principles of language evolution and biological evolution.

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Chomsky (1965) in “Aspects of the Theory of Syntax” formulated the idea that the unlimited expansion of any natural language is possible due to the presence of the recursive mechanism of embedding of statements within sentences. The universal generative grammar was developed by Chomsky in attempt to explain the recursivity of human language. This theory was elaborated on the basis of limited number of languages which poses certain restrictions on its universality (Ivanov, 2004). The enormous popularity of the theory of generative grammar became the consequence of the fact that this theory captured probably the most important property of language and introduced formally defined rules of generative transformations in linguistics. This is despite of the fact that the generative grammars as they were formulated by Chomsky are rather the consequences than the causes of language recursivity. Also these grammars remain the sets of speculative formal rules that are unlikely materialized in real physical (neuronal) structures like the genetic code is materialized in DNA.

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In its basis, language recursivity is grounded in the incompleteness of formal system. The superficial rules postulated by Chomsky are the post-hoc observations rather than the predictions of possible realizations in the language (Sampson, 1979). More advanced (but vaguely formulated) in the understanding of generativity could be the concept of Nalimov (1979) claiming that the operation of human language is based on the application of a selective filter associated with the cognitive activity, which selects meanings from the potential field. This selection is context-dependent and can be analyzed via the approach to understanding complexity developed by Kolmogorov (1965). Matsuno (1992b) suggested the idea that the natural language processor of brain realizes “non-programmable computation” in which the average number of different lexical meanings of one word is a quantitative characteristic of the price of action in such metamathematical operation. This concept has a similarity with the concept of Liberman because the postulated processor of brain should act similarly to the MCC in Liberman’s theory. The concrete mechanisms of its operation are much more obscure than the work of genetic system but they both bear similar principles of linguistic generativity. Recursion in linguistics makes possible the discrete infinity of unfolding the language via the process of embedding of phrases inside the phrases of the same type within the hierarchical structure. Generativity of human language from the time of its origin unlocked the open process of social evolution, which arose as essentially novel evolutionary phenomenon. Recursivity is based on the deterministic ambiguity which is selectively reduced from the potential field of meanings rather than on some fixed rules of the generative grammar. Simplification of the selective filter can lead to a loss of recursivity which is probably observed in the Pirahã language of the aboriginal people of the Amazon rainforest (Everett, 2013). 7. Conclusion: Phenomenon of life and phenomenon of man

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In living cells, new DNA texts appear in the processes following the meiotic cell division, as well as during the horizontal gene transfer, via the application of various biochemical tools necessary for cutting, splicing, polymerizing and modifying DNA. Only humans acquired the ability to build the new texts which are independent from the genetic information and having the structure based on the linear sequence of symbols. The phenomenon of life, in accordance with the views of Efim Liberman discussed in this paper, can be defined as an internal process of calculation by means of the physical system with the embedded mathematical operations. The strategy of ontogenetic and evolutionary adaptation rises to the correctness of calculations performed by the MCC. Life exists in the unique natural form of living cells having MCC and operating according to the program written on DNA. The phenomenon of man is based on the freedom of operation with the symbols of human language and of building the new texts. The notion of biosphere corresponds to the total set of all existing genetic texts while the noosphere acquires the meaning of the total set of texts written by man.

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Acknowledgment

The paper is dedicated to the memory of Professor Efim A. Liberman (1925-2011) whose ideas shaped the conceptual basis of our understanding of molecular computation in living cells.

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The authors thank Professor Vyacheslav V. Ivanov for his creative inspiration that promoted this publication and for his valuable support of the manuscript. This work was supported by the grants 13-07-00967, 13-07-00847 and 14-07-00904 of the Russian Foundation for Basic Research (to N.E.S.-K.) and by the grant 355753-2008 of the Natural Sciences and Engineering Research Council of Canada (to A.U.I.).

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Legends to the figures Figure 1. General scheme showing the process of molecular computation, which is based on reading and interpreting of genetic texts in the expense of energy resulting in the coordination of biosystem’s functions in the observable world and in the evolution via writing new genetic texts.

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Figure 2. Representation of biological system that includes the internal quantum state (IQS) shielded from thermal fluctuations, with its most coherent part corresponding to perception and conscious activity, the molecular computer of the cell (MCC) corresponding to the genetic system operating with molecular addresses within the heat machine of the organism. The system exists in the environment immersed in the cosmic microwave background. Temperatures (K) are calculated for IQS from the emission of coherent quanta (Matsuno and Paton, 2000).

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Figure 3. Generation and transfer of information in biological system. The scheme incorporates the central dogma of molecular biology into the modified structure of (M,R) system of Robert Rosen. In the current scheme M corresponds to maintenance, and R to evolutionary reconstruction of biosystem. 1 – replication; 2 – transcription; 3 – translation; 4 – regulation of DNA readout via transcriptional factors and other functions of transcriptional machinery; 5 – combinatorial RNA rearrangement; 6 – appending DNA texts (writing capacity via reverse transcription); 7 –apparent translation of RNA combinations for testing functionality of novel proteins. Arrows 1, 2, 3 correspond to the reading function of genome (the initial structure of the central dogma), arrow 4 to the protein machinery for transcription, arrows 5 and 7 to the combinatorial dynamics of information and its testing, arrow 6 to the writing capacity of genome. For simplification, RNA replication, combinatorial dynamics of DNA, and protein machinery for translation are not shown.

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