Mechanisms of directed evolution of morphological ...

BioSystems 168 (2018) 26–44

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Review article

Mechanisms of directed evolution of morphological structures and the problems of morphogenesis

T

⁎

Alexey V. Melkikha, , Andrei Khrennikovb,c a

Ural Federal University, Yekaterinburg, 620002, Mira str. 19, Russia International Center for Mathematical Modelling in Physics and Cognitive Sciences, Linnaeus University, Växjö, S-35195, Sweden c National Research University of Information Technologies, Mechanics and Optics (ITMO), St. Petersburg 197101, Russia b

A R T I C LE I N FO

A B S T R A C T

Keywords: Morphogenesis topology Complexity Partially-directed evolution Non-Archimedean analysis Quantum models of interaction Active information Bohmian-like dynamics

Morphogenesis mechanisms are considered from the point of view of complexity. It has been shown that the presence of long-range interactions between biologically important molecules is a necessary condition for the formation and stable operation of morphological structures. A quantum model of morphogenesis based on nonArchimedean analysis and the presence of long-range interactions between biologically important molecules has been constructed. This model shows that the evolution of morphological structures essentially depends on the availability of a priori information on these structures. Critical steps in evolution related to the most important morphological and behavioral findings have been analyzed; the results have shown that the implementation of such steps can only be explained within the framework of a partially directed evolution. Thus, the previously proposed model for a partially directed evolution is established for modeling the evolution of morphological structures.

1. Introduction Mechanisms for the emergence of complex living systems are still a matter of debate. In particular, morphogenesis mechanisms play an important role in evolution because only a full-fledged organism that emerged based on its genes is able to reproduce and be effective. In this regard, it is necessary to assess the real complexity of living systems and the mechanisms of their occurrence. In earlier works (Melkikh, 2014a, 2015a; Melkikh and Khrennikov, 2017a, 2017b), we proposed a partially directed evolution model, the most significant point of which is the statement regarding the directivity (controllability) of evolutionary processes. Quantum mechanisms of changes in the genome, based on the long-range interaction potential between biologically important molecules, are proposed as one of possible explanation of such directivity. The following are the main provisions of the model for partially directed evolution (Melkikh, 2014a): A. Evolution is partially directed, i.e., there exists a priori information in accordance with which a directional change in the genome occurs. Only the existence of a priori directivity of evolution allows us to explain its characteristic rate. Because the number of variants of information molecules (DNA) increases exponentially with increasing DNA length, when the number of nucleotides exceeds 102-103, the

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enumeration of all variants of information sequence becomes impossible during the lifetime of the universe. At the same time, it is shown that any means of accelerating evolution other than an exhaustive search of all possible nucleotide sequences (e.g., block coding, molecular exaptation and cumulative selection) implicitly assumes the existence of such a priori information. B. Randomness in evolution is the result of uncertainty in the environment. An accident can be a natural consequence of uncertainty in the environment and in the organism itself. In particular, randomness can play an important role in molecular recognition of the environment. C. Quantum mechanics plays an important role in organization of directed mutational fluxes. Mathematically, partially directed evolution may be expressed (Melkikh and Khrennikov, 2017b) in the fact that the state of the system at time (t +1) is determined by the limitations existing a priori, i.e., at time t. That is, the properties of the next generation are determined not only by which descendants survive under the current circumstances but also by the fact that the properties of the descendants themselves partly existed before they encountered the new environment. It is clear that this is an additional mechanism of evolution that is not associated with random mutations or with selection. Note that in automata theory the existence of a priori constraints is natural and is one of their most

Corresponding author. E-mail addresses: [email protected] (A.V. Melkikh), [email protected] (A. Khrennikov).

https://doi.org/10.1016/j.biosystems.2018.04.004 Received 20 December 2017; Received in revised form 19 April 2018; Accepted 24 April 2018 0303-2647/ © 2018 Elsevier B.V. All rights reserved.

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averaging. Morphogen gradients are realized in space between the cells, and these representations underlie the Turing model (1). However, experiments do not always confirm the Turing model (Maini and Baker, 2012). A number of morphogenesis models (Maini and Baker, 2012) are based on the long-range inhibition – short-range activation scheme, which is a generalization of the initial Turing’s model. As demonstrated by protein folding models and models of interactions between biologically important molecules (Melkikh, 2015a; Melkikh and Meijer, 2018), long-range interactions are fundamentally important at different levels of organization in living systems, in particular at the level of molecules. Isaeva et al. (2012) considered the topological features of morphogenesis. During morphogenesis, there is a constant violation of symmetry, and global and local order is important. The authors introduced the term “topological positional information”, which reflects the fact that information about a certain structure depends on the place where it is located. Similar to other authors, Isaeva et al. assign a large role in morphogenesis to the cytoskeleton. The authors consider the analogy of the cytoskeleton with a liquid crystal. In particular, centrioles play the role of the center for microtubules in the animal cell. This is similar to a drop of a nematic liquid crystal on the surface of a solid − in both cases there is a singular point that determines the astro-like structure of the system. The polarization of the egg during the initial stages of its development is largely determined by the cortical peripheral layer. This layer largely influences morphogenesis. In this case, the ion flux through the membrane and the change in the resting potential it creates can play the role of a switch in the polarization process (Isaeva et al., 2012). The authors considered the singularities of morphogenetic fields, noting the role of mechanical stresses in morphogenesis. Mechanical stresses affect not only the shape of the cell, from which organ or tissue consists, but also the rate of the expression of certain genes. The authors paid considerable attention to fractal morphological structures (neurons, blood vessels, and jellyfish forms) with fractional dimensions. In conclusion, the authors note that understanding topological patterns can help explain morphogenesis. The classical Turing model for animal skin patterns was considered in this work (Murray, 2012). Trifurcations, situations in which a system can choose between three possible states, were considered. A number of morphogenesis models are based on mechano-chemical concepts (Oster et al., 1983; Maini, 1997). In the mechanical model proposed by Oster et al. (1983), the following three variables are considered: cell density n(x,t), matrix density ρ(x,t) and matrix displacement u(x,t) at position x and time t. These variables can be written into the following equations:

common features. In particular, in the frame of the model of partially directed evolution it is possible to consistently explain many different evolutionary phenomena, such as the finite lifespan of organisms, the existence of the sexes, the genetic diversity of populations, the effect of the Red Queen, and phenotypic plasticity (Melkikh and Khrennikov, 2017a). It is necessary to further develop this model to include mechanisms for the growth and evolution of complex morphological systems. One of the tasks associated with the evolutionary theory is to assess the complexity of genetic enumeration tasks. This task is how to get a new species of animals (plants) by shuffling nucleotides (genes). The complexity of such a task essentially depends on the availability of a priori information on the evolving system and on the explicit consideration of information in the evolutionary theory. However, the complexity of the organs and not the genes on which they are based is a separate problem. On the other hand, the mechanisms of morphogenesis remain largely unclear. How does a cell in a part of the body know when it needs and does not need to divide? How is the shape of the cell controlled when it interacts with neighboring cells? In this paper, we consider mechanisms for morphogenesis (ontogenesis) from the point of view of complexity, and mechanisms of directed evolution of complex morphological structures. 2. Unsolved problems associated with morphogenesis 2.1. Models for morphogenesis and their problems The main directions in the modeling of morphogenesis at present are the Turing model of the “reaction-diffusion” type and the mechanochemical model. The latter is based on the equations of mechanics for totality of cells, which is considered to be a continuous medium. In this case, topological considerations play an important role because qualitative changes in the topology of a system can occur repeatedly in the development process. Note that topological restrictions should exist not only at the level of organs and tissues, but as will be shown below, they play a very important role at the level of biologically important molecules. The famous Turing model (Turing, 1952) is a system that includes the following equations: ∂u ∂t ∂v ∂t

= γf (u, v ) + Du Δu, = γg (u, v ) + Dv Δv,

(1)

where the functions f (u, v) and g (u, v) are responsible for the kinetics of the reactions between substances u (activator) and v (inhibitor), Di are the diffusion coefficients of the substances, and γ is the scale factor. We note that if these equations (reaction-diffusion type) are natural for many chemical reactions, then a very important question arises about the complexity of reacting substances that are biologically important molecules. Is it possible to achieve all possible states during the reactions between substances? The term “morphogenetic field” can be considered a region of cells from which a certain organ is formed. In this case, each cell contains all the information about the organ. Transplantation and regeneration experiments (for example, see Morozova and Shubin, 2013) show that cells move according to a certain plan. The behavior and development of cells is predetermined by both the organ from where they originate and by the organ to where they migrate. A number of morphogenesis models are based on the idea that a minimum free energy corresponds to a specific tissue cell arrangement. For example, this is the purse-string model for tissue folding (Maini and Baker, 2012). We note that this problem is similar to the problem of protein folding, which will be considered below. As experiments show, substances diffuse within an embryo, controlling its growth. In this case, protection from noise (random processes) is performed by

∂n ∂t

= −∇J + rn (N − n), ∂ρ ∂t

( ) ∂u

= −∇ ρ ∂t .

These equations are based on the known conservation laws and the theory of elasticity. The question is how long are these equations applicable in this situation. By default, it is assumed that a local equilibrium exists at the single cell level and that its physics is clear, which is why cells are referred to as a collective, such as their density, etc. Thus, can equations be written for the intracellular structures (cytockeleton, organelles etc)? Even though morphogenetic models qualitatively and correctly predict the formation of structures in living systems, the main problem is the substantiation of these models at the molecular level. As will be shown below, the interaction of complex biologically important molecules is an unresolved problem. In particular, the states of these molecules cannot be enumerated by the presence of only short-range interactions, meaning that a local equilibrium for this system cannot be 27

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fraction of a second in the cell. As shown in (Melkikh and Meijer, 2018), topological problems make the paradox unsolvable for short-range potentials (that is, potential energies of interaction between individual atoms of biologically important molecules). Short-range potentials mean that the forces of interaction between particles decrease rapidly with distance. This means that the largest contribution to the interaction between the particles is made only by the nearest neighbors, and interaction with the more distant particles can be neglected. The longrange forces slowly (according to a power-law) decrease with distance. In this case, interaction with distant particles, and not only with nearest neighbors, becomes important for understanding the dynamics of the system. The reactions between proteins and ligands and between proteins and DNA are the most important components for any cell to work. Thus, if the generalized Levinthal’s paradox remains unresolved, it is not clear why the cell works at all. This, of course, also applies to morphogenesis and its models; there can be no question regarding the exact construction of organs and tissues if the biologically important molecules are unstable. Instability in this case means that even if biologically important molecules are present in the medium, small perturbations caused by thermal noise are likely to lead to the fact that biologically useful structures (molecular machines, cellular and supracellular structures, etc.) will not arise from these molecules. As noted above, instability arises in the case of the action of shortrange potentials alone between atoms of biologically important molecules. As we know, in reality, biological molecules are stable. This can be achieved by the existence of long-range forces (Melkikh, 2014b; Melkikh and Meijer, 2018).

introduced. 2.2. Morphogenesis at the molecular level. Why a cell works? 2.2.1. Generalized Levinthal’s paradox Analysis of experimental data as well as morphogenesis models shows that reactions between various substances (morphogens) and mechano-chemical processes involving the cytoskeleton play key roles in morphogenesis accuracy. As shown in a previous paper (Melkikh and Meijer, 2018), the problem associated with the accuracy of reactions between biologically important molecules (key-lock or hand-glove) has been unsolved. Consequently, the morphogenesis accuracy problem has also been unsolved. One key to understanding cell functioning is the protein folding problem. This problem has been considered repeatedly, and (according to (Melkikh and Meijer, 2018)) the following three main approaches for the problem of folding and interactions between complex biologically important molecules can be distinguished: – biophysical approach, – geometrical approach, and – molecular docking. In the biophysical approach (for example, see Bryngelson and Wolynes, 1987; Onuchic et al., 1997; Wolynes, 2015; Grosberg and Khokhlov, 2010; Finkelstein and Garbuzinsky, 2013; Martinez, 2014; Ben-Naim, 2013; or Finkelstein et al., 2017), protein folding is considered from a statistical physics point of view. At the same time, most researchers consider the issue of folding speed (the Levinthal’s paradox) to be solved based on the funnel-like landscape paradigm (see, for example, Onuchic et al., 1997; Wolynes, 2015). The geometrical approach (for example, see Berger and Leighton, 1998; Bern and Hayes, 2011; Crescenzi et al., 1998; Jiang and Zhu, 2005; Shaw et al., 2014; Guyeux et al., 2014; or Kauffman, 2015) does not focus on statistical physics but mainly addresses topological problems associated with folding. NP-completeness for both two-dimensional and three-dimensional folding has been proven (Berger and Leighton, 1998; Crescenzi et al., 1998). Molecular docking and molecular recognition (see, e.g., Mobley and Dill, 2009; Kahraman et al., 2007; Wang and Pang, 2007) are separate areas in which the configuration of the protein and ligand is calculated. Some simplifying assumptions are used to smooth the energy landscape. According to Melkikh and Meijer (2018), only the geometrical approach can be considered as an approach based on first principles, as it is the only approach that takes into account the specific topology of biologically important molecules. In many other cases, it is either ignored or the energy landscape is simply smoothed; however, the task becomes substantially different and ceases to correspond to what is occurring in nature. Because the folding and reaction problem is NP-complete, i.e., it requires an exponentially large number of steps, this leads to a contradiction with the characteristic velocities and the accuracies of these processes. As was shown in Melkikh, 2014a, 2015a, the basic processes in cells will simply not occur. Proteins will not have time to achieve their native conformations, biologically important molecules will become entangled (in the classical sense) and form inefficient complexes, and genome control will be impossible. Will the cell work at all? Based on the consideration of the protein folding problem as well as reactions between biologically important molecules in the paper (Melkikh and Meijer, 2018), the generalized Levinthal’s paradox was formulated. The paradox reflects the discrepancies between the exponentially large times required for a protein to reach its native conformation, the achievement by reacting complex molecules of their final correct configurations, and times of folding that are equal to a

2.2.2. Morphogenes and boolean satisfiability problem miRNAs, ncRNAs, transcription factors (HOX genes), short secretory proteins, usually act as morphogens (for example, see Sagner and Briscoe, 2017; Inui et al., 2012). Morphogen concentration is important in morphogenesis; however, morphogen conformational degrees of freedom are not usually considered. Most importantly, the “morphogen + receptor” system is a complex system. In many respects, this reaction is similar to “proteinprotein”, “protein-ligand” and other reactions. The complexity of these reactions was considered in (Melkikh and Meijer, 2018). As shown in the paper (Melkikh and Meijer, 2018), the problem of recognizing morphogens can be reduced to a classical boolean satisfiability problem. As is known, the boolean satisfiability problem (SAT) is NP-complete (Cook, 1971). NP-complete means that no fast solution of this problem is known and solution and time required to solve the problem increases exponentially as the size of the problem grows. The problem is the following: when considering a Boolean formula consisting only of the names of variables, brackets and OR, AND, and NOT operations, can we assign to all variables that are in the formula values FALSE or TRUE, so that the formula becomes true? In a previous work (Kauffman, 2015), it was proposed to use a mathematical logic language to describe protein folding processes. The names of binding sites (protein domains and individual atoms) would be the variables and any structures (parts of molecules) can interact with one another. These structures can be numbered in one way or another. The logical variables are TRUE or FALSE with respect to the correct or incorrect connection of sites. Brackets order the binding, i.e. specify the order of the reaction (folding). The true formula is expressed by the fact that the configuration resulting from the reaction between two (or more) molecules or their folding is native (true). When two macromolecules come together, the funnel-like landscape is not formed due to the presence of “forks”. This inevitably leads to the formation of a large number of metastable long-lived states that do not correspond to any effective structures. The efficiency of such a structure, in view of the exponentially large number of its states, is not obvious, and must be proved separately. Such evidence is not available 28

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Fig. 1. Two molecules just before the interaction. Squares of different colors designate different amino acid residues.

in the literature. Let two molecules have the following characteristics before interaction (see, Fig.1 (of this article) and Melkikh and Meijer, 2018): After an interaction, both the specific binding sites and the shape of the resulting molecule will change (Fig. 2): As a result of the interaction between the two logical structures, a new logical structure will be obtained. Will the actual structure of each molecule change after interaction? Will this new structure be true? The classical boolean satisfiability problem (SAT) can be reformulated within the framework of quantum mechanics (for example, de Araujo and Finger, 2011). The authors considered the logical relationship between the SAT and QSAT (quantum-SAT) problems. The relationship between these problems is only superficial and does not allow for direct comparisons between the NP and QMA (QuantumMerlin-Arthur) classes (the quantum version of the MA (Merlin-Arthur) class, which in turn is a classical probabilistic generalization of NP). Thus, this issue remains open.

Fig. 2. Two molecules just after the interaction.

The cytoskeleton consists of various proteins (actino-myosin, keratin, tubulin-dynein, etc.), which form structures such as microtubules and microfilaments. The cytoskeleton plays a crucial role in maintaining the shape of the cell. The number of degrees of freedom associated with the cytoskeleton is very large, which allow the cell and its organelles (mitochondria, vacuoles, chloroplasts, magnetosomes, Golgi complex, nucleus, etc.) to have an almost infinite number of different configurations. Another important role of the cytoskeleton is the propagation of differentiation waves (see, for example, Gordon, 1999; Gordon and Gordon, 2016a, 2016b) in embryos. According to the authors, the forces arising in the actin ring change the shape of the cell. If there are connections between the cells, the contraction of the cell also causes the contraction of the neighboring cell and so on. On the other hand, mechanical stresses are now well known to change gene expression. All of the signal transduction pathways, whether biochemical or mechanical or some combination of both, eventually converge to the level of the DNA. The signal eventually activates or deactivates a specific set of proteins which bind to the DNA at specific regulatory sequences. All of this leads to the transcriptional activation or repression of specific

2.2.3. Mechano-chemical processes at the molecular level and their stability. Cytoskeleton and extracellular matrix The cytoskeleton is present in all eukaryotic cells; its functions include maintaining and adapting the cell's shape to external influences, exo- and endocytosis, ensuring cell movement as a whole, active intracellular transport and cell division. 29

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differ from intracellular molecules, as they also have a large number of degrees of freedom. Thus, the “reaction-folding” paradox (generalized Levinthal’s paradox, Melkikh and Meijer, 2018) is fully applicable to these molecules. These paradoxes regarding the interaction of the cytoskeleton and extracellular matrix as well as the whole entire system that controls division and cell form, suggest that there must be some more subtle mechanism controlling these processes. Such a mechanism can be part of some cybernetic system. In work (Gordon and Stone, 2016) it is suggested to consider the development of the embryo as a hierarchical construction of cybernetic systems. In such a system, the waves of differentiation could be mechanisms for changing the state of cells. However, such a mechanism needs to be justified at the molecular level. In particular, the key issue remains the presence (absence) of longrange forces in such a system.

regulons. For the cell that starts a differentiation wave, authors (Gordon, 1999; Gordon and Gordon, 2016a, 2016b) hypothesized that the initial signal could be purely mechanical. An estimate of the possible number of such configurations can be determined by estimating of the number of possible DNA conformations. In previous studies (Melkikh and Khrennikov, 2017b; Melkikh and Meijer, 2018), it was proposed that a DNA chain on the characteristic persistence length (characteristic length at which a polymer molecule curves substantially) can take at least two different states. As a result, the following formula was obtained for the total number of possible DNA spatial states:

2L / Lpers. This is estimation from below, but with enzyme action this length can be made much smaller. However, even this estimation gives rise to approximately significant number of variants:

2.3. Quantum model of morphogenesis, non-Archimedean space and bohmian-like dynamics

7

22 × 10 . This number of variants is so large that it is impossible to enumerate them during the lifetime of the universe with parallel operation of all living beings that ever lived on Earth. This means that DNA during folding has become entangled (in classical sense) in any of the exponentially large number of amorphous states. This also applies fully to the cytoskeleton, as it has a comparable (or even greater) number of possible conformations. This means that the cell form in multicellular organisms is practically infinitely diverse. The extracellular matrix can in some sense be regarded as a continuation of the cytoskeleton. Extracellular matrix molecules are produced by the cell and ensure the structural integrity of tissue. The composition of the extracellular matrix is somewhat different in different multicellular organisms. The gel of polysaccharides and fibroid proteins plays the role of a buffer in the event of external stresses in the tissue (for example, Humphrey et al., 2014). In plant cells the flows of matter from one plant cell to another are regulated by plasmodesma. However, recognition mechanisms for transportable molecules and the transport mechanisms themselves remain unclear (for example, see Gallaher and Benfey, 2005). In recent decades, accumulated experimental data (for example, see Piccolo et al., 2014) has indicated that the behavior of the cell (such as division) is due in large part to the mechanical effects on the cell by its neighbors. This contradicts the previously accepted paradigm, according to which genes dictate a cell, how to behave. Cells divide until they begin to make contacts another cells. In this case, the contactinduced inhibition of division occurs, whereby special YAP and TAZ proteins function as a molecular gene switch (Piccolo et al., 2014). The cytoskeleton communicates with the environment through these adhesive proteins that permeate the membrane, which are also attached to the extracellular matrix filament network. The cytoskeleton compensates for the stresses arising in the matrix, which is one of the manifestations of homeostasis. Usually, these proteins are in the cytoplasm, but when the cytoskeleton is stretched, they move into the nucleus and bind to target DNA sites to activate specific genes that trigger proliferation. If the space around the cell is limited, these proteins do not migrate to the nucleus when the cytoskeleton is stretched, and there is no gene activity. It is believed (for example, see Piccolo et al., 2014) that the YAR/TAZ system reacts to changes in the three-dimensional system design (the organ as a whole). Feedback in the form of collagen → integrin → cytoskeleton gives the cell information about when it needs to divide and where the organ will grow. The main question that arises when considering the extracellular matrix is how its components are controlled? In particular, how can a matrix control the movement of a cell? As already noted above, the cell itself has an exponentially large number of degrees of freedom. Accordingly, the control of such a structure (and its movement) requires comparable complexity. In this sense, intercellular matrix molecules (collagens, proteins, proteoglycans, etc.) do not fundamentally

Taking into account what was said above about the generalized Levinthal’s paradox with respect to intracellular structures, it becomes obvious that our ideas about the work of biologically important molecules based on classical physics are largely untrue. In our opinion, the quantum models make it possible to solve the generalized paradox, as well as to explain morphogenesis at the molecular level. The need for quantum models for morphogenesis was previously expressed in works by Igamberdiev (2012, 2015). Igamberdiev noted (2012) that the role of the cytoskeleton in morphogenesis is particularly great. In this case, the cell cytoskeleton may be in a coherent state. Cytoskeleton relaxation can then proceed slowly, while the pure quantum state proceeds for a long time. At the same time, the author notes the importance of the “internal measurements” concept introduced earlier by Matsuno (1996). We also note that this concept can be extended to take into account the directivity of the process. Igamberdiev (2012) proposed to consider the morphogenetic field not in the sense of the concentration field of morphogens, but as a Bohmian field concept – the field of potential possibilities for the collapse of the wave function. In the papers (Montagnier et al., 2017a, 2017b, 2009) the role of water in the interaction between proteins and DNA is discussed. It was shown that London dispersion forces between delocalized electrons of base pairs of DNA are responsible for the formation of dipole modes that can be recognized by Taq polymerase. It was shown, that in water, the DNA was detected by polymerase chain reaction using the Taq polymerase from the heat-tolerant strain T. aquaticus. The results of these experiments generate an important question on, how is it that a bacterial DNA polymerase (Taq) can read water nanostructures imprinted by bacterial DNA sequences? The authors constructed a model of the process based on quantum field theory. According to the authors, water molecules (being dipoles) seem to transmit, through long-range correlations, the electromagnetic image of DNA to neighboring molecules. Note that the role of water in the interaction between biologically important molecules is very important. All without exception, such interactions occur in water, the molecules of which have a large dipole moment. This applies, of course, to the processes of morphogenesis. On the other hand, the results of the experiments (Montagnier et al., 2017a, 2017b, 2009), as well as the theoretical models created by the authors, which lead to the need to take into account the long-range interaction between biologically-important molecules, are very well correlated with the ideas of the long-range interaction, proposed by Melkikh earlier on the basis of other ideas (see discussion of the generalized Levinthal’s paradox 1.2.1.) Convergence of the quantum field model considered in papers (Montagnier et al., 2017a, 2017b) with previously proposed models of authors (Melkikh, 2014b; Melkikh and Meijer, 2018) seems promising. 30

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least from the fact that it is not enough for molecules to meet for a completely definite reaction to occur. It is also necessary that the conformational degrees of freedom (which are the greater, the more complex the reacting molecules are) also come to a definite state. Thus, for biologically important molecules, the equations for chemical kinetics represent only a rough approximation, which does not say anything about the characteristic times for the processes. Bearing the Eqs. in mind (2)–(3) and (4)–(6), we introduce an additional variable ξ, which is responsible for the internal degrees of freedom for the reacting molecules. Then, the kinetics of the biochemical reactions will be determined by the spatial coordinates and coordinates ξ. This separation of variables has been used repeatedly in the modeling of hierarchical systems (for example, see Allahverdyan et al., 2016). Thus, we can write the following master equation with ξ:

In the paper (Melkikh, 2014b), the quantum model for the folding of biologically important molecules and the reactions between them, was proposed with the following equations:

iℏ

∂ψ ˆ + φψ, = Hψ ∂t

∂φ = g (φ , ψ). ∂t

(2) (3)

The first equation is the Schrödinger equation for a particle; in addition to the usual Hamiltonian, this equation also contains a potential related to the collective interaction between particles. The second equation describes the dynamics for many-particle potentials. This particular potential organizes the collective effects so that protein folding and the other processes discussed above occur through a funnel-like landscape. These equations can also be represented in the following form, which is based on explicit control (Melkikh and Khrennikov, 2017b):

⎫ ⎧ ⎛ ∂ γ ⎜ − ieAμ (x ) ⎟⎞ + m ψ (x ) = uψ (x ) ⎬ ⎨ μ ⎝ ∂xμ ⎠ ⎭ ⎩

(4)

⎧ T⎛ ∂ ⎫ γ ⎜ + ieAμ (x ) ⎞⎟ − m ψ (x ) = u*ψ (x ) ⎨ μ ⎝ ∂xμ ⎬ ⎠ ⎭ ⎩

(5)

□Aμ (x ) = −jμ (x ) + vAμ (x )

(6)

∂pn (ξ ) = ∂t

Wmn pm −

m

∑

Wnm pn .

m

(7)

∂u (ξ ) = γf (u, v , ξ ) + Du Δu. ∂t The presence of the variable ξ in this equation means that a welldefined reaction between these substances will substantially depend on their conformations. We denote ξ * as the state of the variable ξ, which corresponds to the native conformation of the macromolecule (or the biologically functional state of the reacting substances, such as proteinprotein, protein-ligand, etc.). Only at this value of ξ can the reaction be considered as being maintained; for all other values, it can be said that the substances are eliminated from the game, as they are unable to perform useful work (the probability that the substances will be useful will be small). Thus, the system of Eq. (1) that takes into account the long-range effects can be transformed to the following form:

ie (ψ (x ) γμ ψ (x ) − ψ c (x ) γμ ψc (x )). 2

The superscript “c” indicates the change in the charge sign. Aμ is the electromagnetic field potential, γμ are the Dirac matrices, e is electron charge, and □ is

?≡

max

∑

This equation can be obtained based on Eqs. (2) and (3). The probabilities Wmn of transitions also depend on the coordinates of other biologically important molecules in the cell (and possibly beyond). Thus, the general form of the reaction-diffusion equations between the substances u and v will be as follows:

where ψ(x) is the wave function (operator) and jμ(x) is the 4-density of the electron current, which is equal to

jμ (x ) =

max

∂2 ∂2 ∂2 ∂2 − − 2 − 2. 2 2 ∂t ∂x ∂y ∂z

Here, u and v are controls; they are components of united control vector:

∂u (ξu) ∂t ∂v (ξv )

⎛⎜u ⎞⎟ v ⎝ ⎠

∂t ∂pn (ξu)

In the absence of the control, the system (4–6) uses the quantum field theory (Akhiezer and Berestetskii, 1965) within the framework of the secondary quantization. The cost function will generally depend on the basic variables and controls:

∂t ∂pn (ξv ) ∂t

= γf (u, v , ξu ) + Du Δu, = γg (u, v , ξ v ) + Dv Δv, mmax

mmax

=

∑

Wmn (ξu ) pm (ξu ) −

=

∑ m

∑

Wnm (ξu ) pn (ξu ),

m

m mmax

mmax

Wmn (ξ v ) pm (ξ v ) −

∑

Wnm (ξ v ) pn (ξ v ).

m

(8)

This system of Eq. (8) explicitly contains long-range interactions; as a result, the formation for each cell depends on the state of the adjacent cells and the many other cells that form the organ.

I = I (ψ, u, v ). A specific type of cost function should be determined based on specifically designed experiments. Considering the above models, we introduce the concept of the internal control. By this term we mean the control for the motion of particles at the quantum level. In many ways, this concept is an extension of the internal measurement term introduced earlier for biosystems modeling (Matsuno, 1996). In the latter case, it is implied that the collapse of the wave function (measurement) leads to definite system states. It can be assumed that not only the collapse occurs in well-defined states (the concept of the collapse of the wave function is relative and represents only the limiting state of the quantum system (for example, see Melkikh, 2015b)), but there are definite mechanisms for controlling the collapse (decoherence) within the system. The presence of these mechanisms is reflected in Eqs. (2), (3), and (4)–(6). Thus, the systems of Eqs. (2)–(3) or (4)–(6) represent the microscopic basis for the biochemical reactions. These reactions themselves underlie morphogenesis. The deficiencies in the Eq. (1) are visible at

2.4. Morphogenesis quasicrystallinity and non-Archimedean dynamics One can note an important similarity between the growth of quasicrystals and morphogenesis. The distinctive property of quasicrystals (for example, see Steinhardt, 2008) is that the growth of such a crystal is determined not only by the atoms that surround a given atom but also by sufficiently distant ones. Similarly, in morphogenesis, the state of a cell depends not only on its nearest neighbors but also on the state of distant cells, and often on the state of the body as a whole. During organ growth, once a command has to be given that stops its growth, this command cannot be given by only neighboring cells because they “do not know” where exactly in the organ they are located. The command should be given by a large set of cells and it must be the result of the nonlinear addition of distant signals, which are not small. If we consider the cells in any organ as cells in space, if the cell state (kind) is the 31

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tree-like structures and hence cognition can be modelled with the aid of ultrametric (non-Archimedean) spaces, see, e.g., Dubischar et al. (1999), Khrennikov (2000, 2004a, 2004b), Murtagh (2012).

motion of a particle in it, then the fundamental principles of Archimedean dynamics, which are characteristic of most physical systems, begins to be violated. One of the main properties of Archimedean dynamics is the well-known triangle inequality. According to (Khrennikov, 2004a), we defined the field F in which the absolute value is a map F → R +, satisfying the following conditions:

Remark 3. We also pay attention of the reader to a series of works on padic modeling genetic code, see (Dragovich and Dragovich, 2007; Khrennikov, 2009; Khrennikov and Kozyrev, 2009). We note that there is a deep similarity between non-Archimedean dynamics and the Bohm-de Broglie quantum theory. Indeed, one of the main properties of Bohm-de Broglie quantum mechanics is its nonlocality; i.e., the dependence of the particle velocity on the wave functions of all other particles. Bohm's theory is a theory with hidden parameters and is based on independence from the observer and determinism. The position and momentum of the particle are considered to be hidden variables; they are defined at any time but are not known to the observer. The initial conditions for the particle are also not known exactly. Thus, from an observer's point of view, there is an uncertainty in the state of the particle that corresponds to the Heisenberg uncertainty principle. The set of particles corresponds to a wave that evolves in accordance with the Schrodinger equation. Each of the particles follows a deterministic trajectory, which is oriented towards the wave function; i.e., the particle density corresponds to the value of the wave function. The wave function is independent of particles and can also exist as an empty wave function. There are also relativistic versions of Bohm's theory (for example, see Tumulka, 2017; Durr et al., 2014). The nonlocality property brings Bohm's theory closer to the quantum models for folding and reactions, discussed above and makes it promising for modeling biological systems, particularly morphogenesis. The connection between biologically important molecules can also be interpreted as obtaining active information about the remote parts of the organ. This information is active in a sense that as a result of it receiving a very definite action takes place. The noted similarity makes it possible to classify the folding and reactions models as Bohmian-like. As noted earlier (Melkikh and Khrennikov, 2017b), in the presence of long-range potentials between biologically important molecules and in the presence of the internal control, the analogue of the action may be a more general cost function. This approach leads to more general dynamics for biologically important molecules and cells. Thus, the equations for motion in a cellular medium in its most general form can be written as follows:

x F = 0 ⇔ x = 0, xy F = x F ⋅ y F , x+yF ≤ xF + yF. The last inequality is known as the triangle inequality. The absolute value is called non-Archimedean if the strengthened triangle inequality is satisfied:

x + y F ≤ max( x F , y F ). Non-Archimedean analysis has found wide application in various fields of mathematics, physics and biology (for example, Khrennikov, 2004b). Known growth algorithms for quasicrystals (see, for example, Steinhardt, 2008) also imply non-Archimedeanness with respect to force (energy), as the superposition principle in our 3D-spce is violated in this case. Indeed, the addition of the next molecules to a growing quasicrystal in such an algorithm is due to the state of the quasicrystal in some bounded neighborhood at a given point. We note that superposition principle can be valid over the Archimedean field. Thus, the fact that the force of interactions between particles will depend on what exactly is occurring in all other parts of the system, suggests that the dynamics of such a system is fundamentally different from the dynamics of most physical systems. One of the most important consequences of non-Archimedeanness is non-locality. Non-locality is a consequence of the fact that the interaction force between particles is nonlinearly dependent on the particles’ environment. Depending on the law of interaction between the particles, the long-range interaction or the power dependence of the interaction force on the distance between them is also possible. For example, if two forces act on the particle, then their vector sum in the non-Archimedean case may turn out to be both smaller and larger than Archimedean one. Moreover, this can be directed in an arbitrary direction (Fig. 3). This clearly manifests itself when modeling the growth of a quasicrystal. In quantum mechanics, the principle of superposition of forces is replaced by the principle of superposition of wave functions. In the nonArchimedean case, this can generally be violated. For example, in (Khrennikov and Yurova, 2017), the protein conformational degrees of freedom are modeled with p-adic numbers, as well as using more general ultrametric spaces that encode these states.

ρ

∂σij ∂ 2u i + Fi [ξ ] = ∂x j ∂t 2

(9)

− where ui is the i-th component of the deformation and [ξ] is the dependence on the set of variables ξ for cells from a certain region (not necessarily adjacent to a given one). For ξ, the (7)-type equations are still satisfied. Thus, the stresses in the extracellular matrix, the functioning of the cytoskeleton, the changes in cell shape and the work of YAP and TAZ proteins can be described based on generalized mechanics with longrange interactions. This mechanism naturally explains the surprisingly precise operation of these complex molecular machines. Long-range interactions do not replace the actions of known intermolecular interactions but modulate them and fine tune the complex molecules. The proposed model can be verified by special experiments. A number of these experiments were proposed earlier by Melkikh and Meijer (2018). In contrast, specific morphogenesis experiments should include the monitoring of this process online. That is, we should be able to trace not only the concentrations of the substances (i.e., inhibitors or activators) and the shape of cells but also have more detailed information about the state of the biologically important molecules involved in morphogenesis, including their spatial conformations. This

Remark 1. We stress that p-adic numbers as well as more general ultrametric (non-Archimedean) spaces have the tree-like structure. Such a structure matches well the hierarchic process of dividing of cells. Therefore non-Archimedean features of interactions between cells are traces of the original tree-like structure of creation of an organism from a single cell. Remark 2. The neuronal networks in the brain also have the hierarchic

Fig. 3. If two forces act on the particle, then their vector sum in the nonArchimedean case may turn out to be both smaller and larger than Archimedean one. 32

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of a new theory of evolution. According to Lyubishchev, the existence of benefit in this or that organ is not sufficient grounds for believing that it was developed by natural selection (Lyubishchev, 1982). For example, he asks how such complex patterns of peacocks or pheasants developed. This required a long series of breeding males. Only mathematical solutions can provide an idea regarding how long this process took. This problem will be discussed here in detail. For now, let us note that these remarks, made more than 60 years ago, remain true and are still valid today. To date, almost no one has seriously tried to answer these questions. Lyubishchev said (Lyubishchev, 1982), “By embarking on this path, we open an extensive perspective of research and study of nature from a new point of view, and the huge amount of facts now lurking in the backyard of biology textbooks, as they do not fit into modern views, will fall into place. It's about the hierarchy of organisms, from the cell to the landscape and, perhaps, to the understanding of the organisms of the whole Earth, as a single organism." Lima de Faria drew attention to the similarity between the symmetry of the biological forms and the symmetry of crystals (Lima-deFaria, 1988, 1997). In the author's opinion, it is the properties of atoms that largely determine the symmetry of the organism. As a consequence, the number of variants of biological systems is substantially limited. In this case, the role of genes and chromosomes is secondary – they only determine which option will be fixed. The author also noted (1988) that there is no mathematical theory of evolution consistent with the levels of theories available in physics and chemistry. According to the author, evolution is firmly established, but its mechanism is unknown. In his book “Evolution without selection”, however, the author does not propose a new theory of evolution, but he proposes a new approach. Meyen recognized the role of selection in the “polishing” of organs but believed that natural selection cannot explain the origin of most organs. At the heart of Meyen's ideas concerning evolution lay the nomothetic (oriented on general properties) approach (Meyen, 1973, 1974, 1984, 1986; Sharov and Igamberdiev, 2014). Meyen studied ancient conifers and noted that many forms (for example, the shape of leaves) are repeated, despite the very distant relationships between organisms. According to Meyen, each organ has its own internal logic. Goodwin was the advocate of the hypothesis that genes cannot fully explain the origins of the complexity of biological systems. He suggested that nonlinear phenomena and fundamental laws of nature are necessary for understanding biology and mechanisms of evolution of biosystems (Goodwin, 2001; Goodwin et al., 1993). His approach to evolutionary biology can be called structuralist. According to Goodwin, many patterns in nature are a by-product of the limitations imposed on the body by its complexity. This is expressed in the limited repertoire of the forms of animals and plants at different levels of the hierarchy of life. Goodwin also considered the role of selection secondary because he considered this force too weak to explain evolution, instead acting as a filter. This statement is in many respects consistent with the views of Berg, according to whom “selection protects the norm." According to Goodwin, modern evolutionary biology fails to explain the origins of biological forms and ignores the role of morphogenesis in evolution. The author proposed the development of a new evolutionary theory in place of Darwinism. Thus, despite the fact that the views of the listed evolutionists are partially obsolete today, they contain a rational grain, which can be used to create a new theory of evolution.

Table 1 The similarity between nonlocal model of folding and Bohmian model. Nonlocal model of folding Basic system of equations

iℏ ∂φ ∂t

Nonlocality. Non-linearity Active information

∂ψ ∂t

2

ℏ = ⎛− Δ + U + φ ⎞ ψ ⎝ 2m ⎠

= g (φ , ψ)

Exists Exists ?

Bohmian model

iℏ

∂ψ ∂t

2

ℏ = ⎛− Δ + U + ⎝ 2m

ψ=

ρ exp

ℏ2 Δ ρ ⎞ψ 2m ρ ⎠

( ) iS ℏ

Exists Exists Exists

task, although difficult, is feasible in the future. It should be noted that there is a deep similarity between the process of morphogenesis and the process of evolution (see, for example, Melkikh and Mahecha, 2017). This similarity is expressed primarily in the fact that in both cases long-range forces between biologically important molecules play a special role (see also Table 1). As will be shown below, this similarity is reflected in the same way in the models of both processes. 3. Evolution of morphogenetic structures of living systems As noted earlier by different authors (for example, Igamberdiev, 2015), the problems of morphogenesis have been relegated to the margins of modern evolutionary theory, which is based primarily on models of the evolution of genes. However, in recent decades, there has been significant progress in the understanding of the relationship between the development of an organism and its evolution (evo-devo). At the same time, this direction has not provided a definitive answer regarding how complex structures evolve. 3.1. Morphogenesis and evolution: alternative theories Evolutionary theories alternative to Darwinism have focused on morphogenesis and genetically independent laws of organ formation. It is important to note the works of Berg, Lyubishchev, Meyen, Lima-deFaria and Goodwin. One of the first theories alternative to Darwinism was created by L. Berg in 1922. At the heart of Berg's “Nomogenesis” (Berg, 1969) is an assertion regarding the regular nature of the variability of organisms and the evolutionary process as a whole. As Berg noted, hereditary variations are limited, and they can only follow certain directions. Lyubishchev (1982) hypothesized that it is necessary to consider the immanent laws of form and system. In his opinion, the diversity and repeatability of forms often do not fit with their adaptive value. The theory of taxonomy of living organisms is a special case of common taxonomy. The claims of a synthetic theory of evolution to universality are unfounded, and there is in fact a directed component in hereditary variability. According to Lyubishchev, many biologists consider only Darwinism and Lamarckism. Commonly, the impossibility of the second is automatically declared proof of the first (Lyubishchev, 1982). To a large extent, this is true now; if Lamarckism is included in an extended synthesis, then competing theories are automatically excluded. Lyubishchev noted that the diversity of many organs is much larger than would be predicted simply by need alone. He suggested that the leading factor in the development of an organism is some factor that binds the whole organism into a single system, including small things, such as papillary patterns on the fingers. Lyubishchev also noted that in physics, the principles of optimization are widespread (e.g., the principle of least action). Why should we completely exclude natural autogenesis in the organic world? Although the principles of optimization with respect to evolution were considered by different researchers (see, for example, Rosen, 1967), this did not lead to the creation on this basis

3.2. Evo-devo direction and its problems The interrelation of evolution and morphogenesis (ontogenesis) has been developing intensively in recent years. These have included such approaches as evo-devo, eco-evo-devo and others (for example, Brakefield, 2011; Laubichler 2009; Muller, 2007). The essence of the results obtained within these approaches is that the formation of 33

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Fig. 4. Homogous hox genes in such different animals as insects and vertebrates control embryonic development and hence the form of adult bodies. Wikipedia. https://upload.wikimedia.org/wikipedia/commons/0/0c/Genes_hox.jpeg.

genes is denied. However, there was another hypothesis (“genetic determinism”) that stated that genes completely determine behavior. At present, these extreme views have not been confirmed. The study of the individual development of an organism shows that the same genotype may be expressed differently in different environments. The most common interpretation is that the phenotype and behavior of an animal depend not only on genes but also on the environment (for example, Agrawal, 2001; Whitman and Agrawal, 2009; Asano et al., 2013, 2015, 2017). In one report (Melkikh and Khrennikov, 2017b), the problem of the evolutionary interpretation of phenotypic plasticity was considered. As is the case with the Baldwin effect (Baldwin proposed, that selected offspring would tend to have an increased capacity for learning new skills rather than being confined to genetically coded, relatively fixed abilities), the mechanisms of plasticity remain unclear. The mechanism of change in the phenotype must be registered somewhere. If this mechanism is innate (i.e., the genes already include various options for phenotypes in response to certain environmental conditions), then this is simply a variant of evolution. If this mechanism is not inherent, how does a phenotype (behavior) form? For example, because complex behaviors require a large amount of information, the problem of storing this information arises. This need for information storage presents a problem for evolution as a whole and for many intracellular processes (for example, Melkikh, 2013). In the paper by Melkikh and Khrennikov (2017b), it was proposed that

organs, tissues, and the body as a whole is determined not only by the genotype itself but also by the way the individual development of the organism (ontogenesis) proceeded, i.e., which genes were active and which were not, as well as the influence of the environment. Within the evo-devo approach, a number of unexpected results were obtained. For example, it was found that the homologous organs of organisms that evolved independently (for example, the eyes of insects, vertebrates and cephalopods) are largely controlled by the same genes, such as pax-6. These genes are ancient and have remained relatively unchanged throughout the evolutionary process. In the process of embryonic development, these genes form the shape and future plan of the body (see, for example, Fig. 4). Another feature is that the species do not differ much in their genes encoding proteins; the most significant differences are the differences between how gene expression is regulated by toolkit genes (for example, Schrader et al., 2015; Linz and Tomoyasu, 2015). These genes form a complex cascade controlling other genes (see, for example, Drapek et al., 2017; Kroos, 2017) in the process of individual development. These control genes are highly conserved because changes in them will significantly affect many body structures simultaneously. New morphological properties, as well as new species, often arise through variations in the expression patterns of the same genes. Another possibility is neo-Lamarckhist mechanisms, according to which epigenetic inheritance occurs, which then begins to be encoded by genes (Jablonka and Lamb, 2014). However, neo-Lamarckism (as part of an extended synthesis) does not by itself solve the problem of the origins of complex biological structures (for example, Melkikh and Khrennikov, 2017b). Homologous hox genes in such diverse animals as insects and vertebrates govern the embryonic development and formation of adult individuals. These genes are also highly conserved. The evolution of morphogenetic structures is closely connected with the notion of “phenotypic plasticity”. Phenotypic plasticity is an important part of modern evolutionary theory. Pigliucci (2007) suggested, “Today we simply can no longer talk about basic concepts like, for instance heritability, without acknowledging its dependence on the sort of genotype–environment interactions that are best summarized by adopting a reaction norm perspective.” There are two opposing hypotheses linking behavior and genotypes. In one theory, conditionally called “behaviorism”, the important role of

– behavioral programs and organism structures exist, which are recorded either in the genes or in other structures; – a part of these programs is stored in a latent form; – with changes in the environment, certain programs or changes of the behavior and the phenotype occur, even though the genetic component of programs may not change; and – when the environmental changes become stable, it is advantageous to change the genetic composition; however, this is done in accordance with a priori information recorded in specific structures. Thus, although within the framework of the evo-devo concept (as well as within the framework of extended synthesis as a whole), the influence of the development of the organism on genes is recognized.

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emergence of longer chains − carriers of genetic information (DNA). Of course, in the early stages of evolution, the DNA chain could be much shorter; as the chain lengthens, however, the number of possible spatial conformations, as well as the number of possible reactions with surrounding proteins, increases exponentially. It remains unclear how selection could cope with the solution of such a problem, since the number of RNA molecules and DNA molecules in any case is limited and substantially lower than the number of possible conformations. It is believed that many organelles (chloroplasts, mitochondria) occurred by capture by prokaryotic cells. As a result, eukaryotes appeared. Can we imagine a relatively simple chain of events of such a capture, with which selection could work? This is difficult for the following reasons. The life of the mitochondria, for example, must be highly coordinated with the behavior of other organelles within the cell. How does the internal immune system understand what is before it – its organelle or another organism's bacteria? For such an “understanding”, there must be receptors capable of recognizing their own proteins. However, the solution of such a problem is difficult by enumeration, since it is again associated with an exponentially large number of variants of such proteins, considering their different conformations (for example, Melkikh and Seleznev, 2012). The appearance of the sexes is a complex process. Even in the simplest version, which is associated with the introduction of genetic material from one type of protocells into protocells of another species, there are insurmountable difficulties for the enumeration of variants. This of course applies to more advanced versions of sexual reproduction. Both mitosis and meiosis are complex and include the work of a large number of molecular machines. The potential mechanisms of transition from one network of machines to another network is not at all obvious, since the overwhelming number of variants of such a network (including protein conformations) are not viable and cannot perform useful work (Melkikh and Khrennikov, 2017a). Multicellular organisms originated from unicellular organisms, but what is the mechanism of such unification? In this process, the cell wall gradually disappeared, and the shape of the cell could vary within a wide range, albeit delimited by the cytoskeleton. However, how did the enumeration of the cytoskeletal variants occur, given that the number of its possible conformations is exponentially large (see 1.2)? Obviously, wherever these variants are coded, their simple enumeration will provide no useful information, since the number of attempts of such a search is too small. The transition from individual organisms to colonies is largely similar to the transition from single-celled to multicellular organisms; in both cases, we face the complexity of communication between members of the colony (cells). What is the complexity of this process? The fact is that immune receptors on the surface of cells, which are actually proteins, must accurately recognize signals, which are quite complex molecules. The more complex the organism and the larger the colony of such organisms is, the greater the number of variants needed to be enumerated. However, the total number of such organisms in the biosphere (that is, the total number of attempts to enumerate) is actually reduced in comparison to bacteria as the organizational complexity of the system increases. The emergence of humans from a common ancestor with monkeys is considered one of the most important events in evolution. The most significant difference between a human and his more primitive ancestors is a more complex brain. During 3.5 million years of human evolution, an enormous increase in the size of the brain has occurred, from a volume of 450 cm3 in Australopithecines to approximately 1350 cm3 in modern Homo sapiens. It is generally accepted that the brain is the most complex organ and biological structure of all. The structure of the brain, one way or another, is connected with the intelligence of animals. For example, Dicke and Roth (2016) have traced the connection between brain structure and intelligence. It was concluded that intellect evolved independently in primates, birds, and other species. In this case, there is a difference in the general properties

However, the evolutionary mechanism nevertheless remains unclear. An adequate explanation for the appearance of complex organs and structures remains unresolved in the framework of the new synthesis. The relationship between complexity and enumeration tasks has been considered in previous works (Melkikh, 2014a, 2015a) in relation to genes. Let us now consider the complexity of certain specific biological structures and discuss possible mechanisms for their origin. 3.3. Critical steps in evolution and morphogenesis A number of studies have analyzed the “critical steps” in evolution (Smith and Szathmary, 1995; Jablonka and Lamb, 2014). These authors discuss the most important steps in the emergence of complex systems, each with a qualitative change in the structure of the system. For example, according to Smith and Szathmary (Smith and Szathmary, 1995; Szathmary, 2015), these critical transitions include changes from replicators to protocells, independent genes to chromosomes, RNA to DNA, prokaryotes to eukaryotes, asexual to sexual reproduction, unicellular to multicellular organisms, individual organisms to colonies and high primates to humans. The authors attributed these transitions to changes in the form of storage and information transmission, and the complexity of living systems continually increased. We will show that, from the point of view of the mechanisms of the emergence of complexity of morphogenetic structures, it is precisely these critical steps that present the greatest difficulty for explanations by the Darwinian theory, or extended synthesis. Indeed, one can consider the transition from replicators to protocells as the transformation from one morphological structure into another. For a replicator to become a protocell, its fundamental transformation is necessary. If we consider only the sequence of monomers as the replicator (although here the question of folding such a molecule arises (Melkikh, 2014c)), it is difficult to imagine effective intermediate links. What did the selection process work on? Over information sequences placed in microspheres (Fox, 1965)? However, a system is needed that controls microsphere division at a certain stage. Such a control system, being built on the basis of proteins, should contain a large number of variants (see discussion of the generalized Levinthal’s paradox). Even the simplest cells – the archaea – represent a very complex system. They contain from 500 to several thousand genes responsible for various ion transport systems, metabolism, cell membrane receptors, and other functions. The enumeration of variants of transport proteins alone would be an impossible task for selection and random mutations. In another study (Melkikh, 2014a), estimates were made for the number of possible protocells that have ever lived in the biosphere. As a result, a value approximately equal to 1045 was obtained. However, this is quite insignificant in comparison with the number of conformations of transport proteins (and possible variants of their reactions) for the transport of only basic ions. The integration of individual genes in chromosomes is also a complex process in terms of enumeration tasks. In this sense, the chromosome is similar in structure to folded proteins. As previously shown (see, for example, Teif and Bohinc, 2011), the DNA density in chromosomes is very large and approaches that of a solid. The combination of genes into chromosomes so that they, despite being in such a tightly folded system, maintain their functional capacity is a difficult task. Even the combination of two genes in such a state is difficult, in the sense that the search for all possible configurations of these two genes cannot be performed in any reasonable time for any population of organisms, both small and large. In other papers (Melkikh and Khrennikov, 2017b; Melkikh and Meijer, 2018), the number of possible DNA conformations was estimated, which clearly indicated that combining genes into chromosomes is a non-trivial task that requires additional assumptions for its solution. How did DNA originate from RNA? If we consider that RNA was the first carrier of genetic information (the hypothesis of the RNA world), then the question arises regarding the mechanisms underlying the 35

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The typical proboscis of Xanthopan morganii consists of several thousand cells. This raises the question of how it evolved. Let us assume that at some stage of evolution, there was a Xanthopan with a proboscis of some intermediate length. Which of these cells (and not only these, but any cells of the body) changed and why? As was shown above, even such a small structure has an exponentially large number of potential opportunities for change. This applies both to a possible change in the shape of the proboscis cells and to a possible change in the number of cells of different types, as well as the connections between them. Why should the cells located at the end of the proboscis be divided to extend it? Why does the proboscis not grow somewhere else? If evolution is not directed (that is, it is not known in advance which of the structures can be useful to Xanthopan), then all other variants should be realized, i.e., the division of other cells located on the periphery. In this case, the shape of the proboscis should resemble an irregular dendrite. Usually, it is answered that the process of selection destroys all these variants. However, such variants are many orders of magnitude greater than there are butterflies in the whole history of life on Earth. Thus, selection could not control this process – it simply would not have had time to accomplish this. Another variant considered in the literature in the framework of extended synthesis is that the organism uses groups of already existing genes for new purposes. That is, it acts “by analogy”. However, such a variant is not compatible with Darwinism. As was shown previously (Melkikh, 2014a, 2015a; Melkikh and Khrennikov, 2017a, 2017b), this requires that the process be a priori directed. Thus, the emergence of such a long proboscis can only be explained within the framework of a partially directed evolution. Let us consider some more examples of the evolution of complex systems.

of the brain. In particular, this is manifested in the organization of the cortex, in which the density of neurons decreases. At the same time, a human has approximately 12 billion cortical neurons. In the human cortex, each neuron has an average of 29,800 synapses, while the entire brain contains approximately 3.6 × 1014 synapses. In whales and elephants, the number of synapses is on the same order but is less well known (Dicke and Roth, 2016). From the point of view of evolution, it is necessary to determine how such a complex structure could arise and evolve in the future. How did a complex brain originate from a simpler one? What did the selection work in this case? In the process of evolution, the brain not only increased in size, but the number of connections in it also increased substantially. However, this is not just an increase in connections, but an increase in the connections between certain neurons. This number of connections is so great (approximately 1014) that the number of their variants is absolutely impossible to enumerate in any population of monkeys or primitive human species. Even assuming these populations to be on the order of a million individuals, and considering that each individual lived for 25 years, we will get absolutely insignificant quantities on the order of 1011 attempts per search. If the connections between 1012 neurons were moving simply chaotically, then there would be an overwhelming probability of an unworkable result for such a number of attempts. On the other hand, as noted earlier (Melkikh, 2014b), the very formation of synaptic connections between neurons is contradictory, since the known physico-chemical mechanisms cannot ensure the accuracy of such a process. Instead of a well-organized neural network, a chaotically intricate structure must arise. The effectiveness of such a structure is not obvious and should be separately justified (see, also Melkikh and Meijer, 2018). Most evolutionary biologists suggest that the emergence of such a complex structure was not through a complete enumeration of variants but occurred by adding to existing structures certain properties that are subsequently subjected to selection, resulting in the formation of a new organ – a more complex and efficiently arranged brain. However, as was shown earlier (Melkikh, 2015a), such logic goes against Darwinism (including extended synthesis). Indeed, the use of what has already been used for something earlier (for example, in the form of a bit string) is possible only if it is known what exactly encodes such a sequence. Otherwise, it is just a meaningless set of bits. Moreover, it should also be known exactly what will encode the changed set of bits (even before the organism is created!). This prediction completely contradicts the non-directivity of Darwinian evolution and can be realized only within the framework of partially directed evolution. Thus, the consideration of critical steps in evolution leads to the conclusion that such steps are inevitably connected with the overcoming of a large barrier of complexity. Even the simplest estimates show that it is not possible to overcome such a barrier by simply enumerating the variants. Thus, it is necessary to consider other mechanisms for the origin of complex structures.

3.4.2. The origin of humans and the complexity of the skeleton The origin of humans is one of the main problems in understanding the evolution of living nature. In particular, one such unsolved problem is the origin of modern people. Until recently, it was generally accepted that modern humans originated in Africa approximately 200,000 years ago. However, recent findings of archaic skulls (Richter et al., 2017) shift the possible age of modern humans to 340 thousand years ago. In this case, the restoration of the shape of the bones of the skull plays an important role in understanding whether a skull represents a modern human. What can be said regarding the evolution and appearance of humans from the point of view of the mechanisms of the appearance of various forms of the skull? It is considered typical that the transitional forms of skulls (and other parts of the body) are mosaic, i.e., they contain signs of both one form (ancient) and another (modern) form. Jaws, superciliary arches, straight or chambered chins and other parts of the skull are critical in this sense. The problem is that these “parts” themselves consist of many thousands of cells connected to each other. Schematically, the structure of the bone can be represented as follows (Fig. 5). Although they may seem simple at first sight, this simplicity must somehow still be obtained in an evolutionary way. This mechanism is

3.4. The complexity of certain evolutionary adaptations and the mechanisms of their origin Let us consider some specific morphological structures and show that their origin requires new mechanisms of evolution. 3.4.1. How did the Morgan's sphinx moth with its anomalously long proboscis (Darwin’s prediction), arise? One of the known predictions of Darwin, which was subsequently confirmed, is that there must be a pollinator with an anomalously long proboscis. Darwin drew attention to a very long spur of a flower approximately 30 cm long with nectar at the very bottom. He suggested that this particular species of orchids may have a special pollinator, most likely a large night butterfly with the corresponding proboscis. Such a Xanthopan was subsequently discovered (for example, Arditti et al., 2012).

Fig. 5. The structure of the bone. 36

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1965; Chaitin, 1966). A long program means that each cell should be placed in a specific location relative to other cells, and this location should be explicitly indicated in this program. Such a program should be of a length on the order of the total number of cells (the work of the cell as such, of course, is a separate problem). A short program means that there are regularities in the arrangement of cells, and an explicit indication of the location of a particular cell is not necessary. The question remains – where to put the next cell? It can continue the existing structure but also not correspond to it. The second variant is much more likely in light of the fact that the shape of the cell can change almost limitlessly. Suppose that there exists a finite automaton controlling the arrangement of cells. The following canonical equations can be used for deterministic automaton (Tsetlin, 1963):

not obvious. For example, it is very difficult to obtain the ball of cells in an evolutionary way if we just sort through the codes of cell positions (this is the way that corresponds to the spirit of Darwinism). If the form of the ball is somewhere encoded in the form of some formula (requiring a shorter record), and the codes are chosen on the basis of such a formula, then such a formula can be rightfully considered a part of the evolution program, and the evolution itself becomes directed. A number of authors attribute such mosaicism of skeletal forms to gene drift. However, why not a drift of nucleotides, because by the laws of nature, the creation of their different combinations is not prohibited. What process is organizing such a drift? According to (Richter et al., 2017), new findings of skulls indicate that the evolutionary spurt was quite long. If we compare this period with the time that was previously considered necessary for such a spurt, then it really can be regarded as a long one. However, 300 thousand years is very small in comparison to the time of enumeration of variants; here, the difference of 200 thousand years is not significant. If modern humans really emerged independently in different places (as suggested in (Richter et al., 2017)), then this fact can be considered a strong argument in favor of the directedness of evolution; if evolution is directed, it is approximately equally directed in different spatial populations.

φ (t + 1) = Φ (φ (t ), s (t + 1)),

f (t ) = F (φ (t )). In this case, we are interested not so much in the details of the automaton itself but in its evolution. Let the program of work of this automaton be written in a bit string: 0111010001110101000. The evolution of such a bit string will essentially depend on the mechanisms for replacing zeroes and ones. Now, we change some bits in our program in a random way. The latter means that we do not have a priori information about what will come out of such a sequence. What do we expect to see as a result? The result, of course, will depend significantly on how much the changed bits are important for our structure (beak). However, if the evolution is not directed a priori (that is, the program does not have an a priori goal of constructing a given surface), then it cannot be known in advance what these randomly altered bits are responsible for! Let the cell coordinates and its orientation be specified with an accuracy of 10%. Then, in order to properly place the cell in space, we need to specify three numbers and two angles. As a result, to build a beak will require approximately 1010–1012 coded coordinates. Thus, wherever the a priori information about the position of each cell in space (or relative to other cells) is contained, such information must exist. This information is not necessarily stored in the genes. Moreover, it is easy to show that such information in genes is certainly not enough for this task. However, this does not change anything fundamentally! If the search for these bits is not directed, then the

3.4.3. Beaks of Darwin’s finches Let us consider an example, which has become a classic – the origin of the beaks of the Darwin finches. From the point of view of Darwinism, this process occurs schematically as follows: In the course of evolution, there are changes in the genome responsible for the shape of the beak; those changes that are successful are fixed in the population, the remaining individuals gradually die out. In the process of evolution, some past versions of genes are often used. For example, Janković (2016) considers evolution to be partially directed and partially undirected. The author considered the origin of the beaks of the Darwin’s finches (Fig. 6) and came to the conclusion that the use of some already existing (or pre-existing) beaks was quite within the Darwinian paradigm. However, remember that the beak (like any other organ) is a complex structure. We estimate approximately the complexity of this structure on the basis of the length of that program, which is necessary for the construction of a beak from individual cells. This approach to complexity was proposed by Kolmogorov and Chaitin (Kolmogorov,

Fig. 6. Beaks of the Darwin’s finches (Darwin, 1845). 37

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evolution is a priori directed. It is in this case that a small change in code that does not affect the basic functions of the limb is possible, merely optimizing them for a specific environment. In the light of the above, the concept of “homology” in relation to organs acquires a new meaning. Organs can be homologous only within partially directed evolution, when they are part of a single process. If evolution is not a priori directed, then it is completely impossible to obtain homologous organs. The above examples, of course, are not unique; on the contrary, they reflect biological structures whose complexity is typical. Examples of more complex structures can be easily found. Returning to the approach to the complexity of Kolmogorov and Chaitin (Kolmogorov, 1965; Chaitin, 1966), it can be noted that such a definition is very useful for determining the complexity of biological systems. Indeed, if there was a simple program for the growth of the beak, it could be considered simple. However, where does such a program come from within the framework of non-directional (blind) evolution? What is the mechanism of its origin, if the blind evolution does not know anything except enumeration of variants (regardless of which ways)? In contrast, the existence of regularities in the construction of parts of the organism (beaks, eyes, wings, etc.) within the framework of partially directed evolution is quite natural. In this case, such programs, as well as behavioral programs, exist a priori even before the onset of evolution and should be launched under certain conditions. We will show that the evolutionary occurrence of complex morphological structures of living systems bears deep resemblance to computation in the broadest sense of the word.

orientation and position of the cells in space will also not be directed. To find the correct position of the cells, it is necessary to perform experiments, the number of which is easy to relate to the required accuracy of the “product”. The same is true for arbitrary technical systems: the more accurate the product, the more a priori information about it we need. As noted above, the number of experiments is directly related to the information that can be derived from them (see Appendix A). How likely is it that the newly constructed structure (beak) will be something better? Obviously, such a probability is exponentially small. Generally, the beak could have an exponentially large number of variants associated with the orientation of each small surface elements, but we do not observe anything of such sort in nature. That is, nature operates within certain given patterns, which correspond to already existing programs. Since such programs exist a priori and do not arise in the process of enumeration, this is confirmation of the fact that evolution is directed. 3.4.4. Origin of limbs It was Darwin who noted that a man's hand, a digging paw of mole, a bat's wing and a dolphin's fin, are constructed according to the same pattern and show our common origin (Fig. 7). There is a system of genes responsible for the formation of this organ, and the entire variety of forelimbs is predetermined by restructuring in this system. Some genes are responsible for the formation of the wrist, others for the fingers (for example, Metscher et al., 2005). A slight genetic shift is sufficient for the fingers to lengthen or partially disappear and the claws to turn into nails. However, this raises the question of the mechanisms of the origin of these limbs. Is it that a case when a slight change in genes leads to significant morphological changes obvious? The answer depends on the mechanisms of evolution of such structures. A typical limb contains approximately 20–50 bones that are connected to each other in a complex system. As noted above, each bone is also a complex system, consisting of thousands and millions of cells of different types. If evolution is not a priori directed, then there is no reason to prefer one nucleotide to another until an organism is built on the basis of this code. However, it is impossible to enumerate all these variants – there will not be enough organisms for this. A different situation arises if the

3.5. Self-organization, modularity and mechanisms of evolution The evolution of many morphological structures shows that they are built on a modular basis, i.e. many new structures are built from parts of the old ones (see, for example, Gordon, 2016). The same applies to genes: it is known that many genes are common to many multicellular organisms. Would it be sufficient to say that the mechanisms for the evolution of such structures are a combination of processes of self-organization and a modular principle? In our opinion, the answer should be negative. This issue has been discussed many times, including in the works of the authors in relation to the model of partially directed evolution (see for example, Melkikh, 2015a; Melkikh and Khrennikov, 2017a; Melkikh and Meijer, 2018). First, note that the term “self-organization” is too vague in relation to evolutionary processes. Self-organization in the most general sense is the emergence of new structures in non-equilibrium processes. However, self-organization is not at all a universal way of solving arbitrary problems. In relation to living systems it is important, that their properties are encoded in some information sequence. In connection with this (Melkikh, 2014a), the following dilemma arises: – Evolution is a priori undirected, but then it is impossible to prove a rational mechanism for the selection of variants of an exponentially large number. This applies to all mechanisms, including sexual reproduction, the selection of alleles in a population, and phenotypic plasticity. – Or, evolution is a priori directed (i.e., it is known a priori that certain blocks encode something good). However, it is then difficult to justify the existence of such mechanisms in the framework of Darwinism. The essence of Darwinism is that a priori evolution will not focus elsewhere, it has no purpose, and species cannot know what they will need in the future. These are the axioms without which Darwinism does not exist.

Fig. 7. The limbs of several tetrapods (schematic).

If for example, we consider the allele variants, not all of the possible 38

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solution is (i.e., how close it is to the target). In this case, there is no need to control the lifetime of the population because it will not be profitable for any structure. However, it is now recognized that the aging process is largely genetically determined. It was previously shown (Melkikh and Khrennikov, 2017a) that the finiteness of the lifetime of organisms is a natural consequence of the directivity of evolution. Since in this case there is a directed movement, the only way to organize such a movement with a limited population size is to limit the lifetime of individuals. In swarm intelligence, an analog of aging is pheromone evaporation (for example, Bonabeau et al., 1999). In this case, populations of solutions are also updated. From an information perspective, the finiteness of life can be interpreted as the need to erase (forget) information within limited populations. When the environment changes rapidly, it is necessary to erase the old information faster. In such circumstances, it is necessary to reduce the life expectancy of the population of solutions. The aim of gene exchange (actually, the exchange of information) is to improve the population. However, such improvement is only possible because the objective function is known in advance. If the objective function were not known in advance, gene exchange would be meaningless. It is important that the finite lifetime of solutions is inextricably linked to the diversity of their population. A finite lifespan is required to maintain this diversity. Variety is beneficial in partially directed evolution in the face of uncertainty. Note that “advantageous” in this case can only be considered for the entire algorithm in general, not for each solution separately. Thus, it can be noted that a variety of information, as well as the associated lifetime of solutions, is a more general concept. This variety is typical for a wide class of computational methods and for the evolution of living systems. In computational methods, these properties have a natural explanation. A similar explanation can be made for the evolution of life. It is possible to examine the evolution of files, as well as any other type of documentation or technology. In this evolution, there is always a random part that is connected with neutral features. However, this does not mean that the evolution of files is necessarily random. For some attributes of the file, evolution can be directed. Note that in the absence of a priori information, all of the computational methods do not work. More precisely, they are no better than a simple enumeration of variants. For example, one can imagine that the objective function will also be obtained using the enumeration of variants. However, the creation of complex algorithms, such as genetic algorithms, would completely lose its meaning because it would be easier to organize a simple exhaustive search of solutions. There is another aspect to the role of prior information in computational processes and search problems. In particular, in the 1990s, several theorems yielded fairly general conclusions concerning search and optimization. The first is the “No Free Lunch Theorem” (Wolpert and Macready, 1997). This theorem demonstrates that there can be no algorithm that is optimal for all tasks. A priori information about the task is crucial. Because evolution is a search process based on certain algorithms, the conclusions of the theorem are fundamental for these processes. The connection between the No Free Lunch Theorem and the mechanisms of biological evolution was first noted in an article by Melkikh (2011). In particular, it was shown that the mechanism of evolution (block-type) could exist only if there is a priori information for a given species in the future. Universal mechanisms of this type cannot exist because they contradict the above-considered No Free Lunch Theorem. If such a mechanism existed, this mechanism could easily solve all NPcomplete (requiring an exponentially large number of steps) problems (such as password cracking).

nucleotide variants, this significantly changes the problem. The number of variants of nucleotides is, of course, much larger. However, this consideration is only a posteriori (that is, comes from what realized), not a priori, which must come from what is permitted by the laws of nature. According to the authors (Melkikh and Khrennikov, 2017a, 2017b), such well-known methods to accelerate evolution compared to the exhaustive search of variants, as cumulative selection and block coding require a priori information for their implementation. Modularity and hierarchicity was considered by Melkikh and Meijer (2018) in relation to the process of protein folding. According to the authors, the very problem of folding or reactions between biologically important molecules could be partially solved in the process of evolution. Indeed, it is known that some proteins are folded hierarchically, i.e. separate parts of the protein chain are folded separately (see, for example, Diaz-Santin et al., 2017). This could, in principle, lead to a reduction in the folding time. However, first, even if such a hierarchical folding is possible, enumeration of variants of the genome for its achievement represents an NP-problem (that is, it requires an exponential number of steps). The solution of this problem, as was shown earlier (Melkikh, 2014a; Melkikh and Khrennikov, 2017a, 2017b), can be obtained in a relatively short time only if there is a priori information about the future states of the system. In this case evolution becomes partially directed. If there are two proteins that, in the course of their undirected evolution, combine into a single complex, the folding of such a complex cannot simply be composed of the folding of individual proteins that existed earlier. In the process of folding, parts of a single molecule interact with each other. As a result, the energy landscape is changing, and the folding problem must be solved for this single molecule separately. Thus, for biological systems it is fundamental that all these structures are somewhere encoded. Therefore, modularity in combination with self-organization, cannot solve the problem of the evolutionary emergence of such structures. To solve the problem, additional assumptions are need, in particular, the assumption that the system has a priori information, according to which it evolves. 3.6. The similarities and differences between the evolutionary process and the computing process Note that between the process of partially directed evolution (evolution of forms) and processes of computation, in the broadest sense, there is a profound similarity. Indeed, processes such as fuzzy logic, genetic algorithms and swarm intelligence can be interpreted from the same perspective as evolution. An especially notable similarity between evolution and the computational process occurs in the case of genetic algorithms (e.g., Koza, 1992). Genetic algorithms have been developed to address certain classes of problems that are very similar to Darwinian evolution. We show, however, that genetic algorithms are not similar to Darwinian (undirected) evolution but are instead similar to partially directed evolution. To achieve this conclusion, we must first make a few comments. First, it is important for genetic algorithms that the objective function, with which the comparison at each step of the algorithm occurs, is known before the process. Specifically, a particular set of data must be approximated by a linear dependence. This requirement is the most important similarity between calculation processes based on genetic algorithms and the process of partially directed evolution. Second, an important feature of genetic algorithms (and other types of computing) is that the lifetime of the population of solutions is finite. Finite lifetimes arise because the objective function is known a priori. Indeed, if this function is known a priori, it is therefore known at each step to what extent obtaining a population of solutions is separated from the desired result. In this case, the generation of a new population becomes necessary to improve the solution. If such information is not present, there is no information regarding how good the resulting 39

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nucleotides but also to an arbitrary complex living system, for example, to the spatial arrangement of cells, irrespective of where exactly the a priori information is encoded. One can consider a situation where there is no a priori information, but an error in the target sequence in K nucleotides is permissible. In this case, the reasoning may remain the same. Hence, it is clear that the presence of some a priori information and the degeneracy of the sequence are closely related. That is, “ignorance” and permissible error in this case play the same role. The information is closely related to the number of measurements, which it can result in the following understanding: the more measurements, the more information can be obtained (see Appendix A). Let us explain this on the basis of the Bayesian formula, which is widely used in decision-making processes in systems of an arbitrary nature. We write the formula for the total probability in the form:

3.7. NP-complete tasks and the evolution of morphogenesis As was shown above, the problem of morphogenesis at the molecular level can be classified as belonging to the NP class in the presence of only short-range interaction potentials between atoms of biologically important molecules. We will show that the problems of the evolution of morphogenetic structures can also be classified as NP in the absence of a priori orientations in evolution. As is known, a priori information regarding an NP-complete problem can play a critical role in its solution. Indeed, if the a priori information on the solution of the problem is largely present (for example, the skeleton of the graph of the route network between cities in the traveling salesman problem is known), the problem can become substantially simpler by going to the P-class. We note that there are approximate methods for solving NP-complete problems based on various heuristics, that is, based on a priori information concerning the solution (for example, Edelcamp and Schrodl, 2012). However, such a question should be solved specifically for each NP-complete problem. Consider in this connection the problem of password cracking, which is the closest to the problem of evolution (see also Melkikh, 2014a). The problem is formulated as follows: let there be a target information sequence, which must be achieved by enumeration variants from some initial sequence. Let such a target sequence be one. How will the a priori information on this sequence affect the characteristic time of its achievement? This question was considered by Melkikh (2014a). Consider, according to (Melkikh, 2014a), a sequence of N nucleotides. Let a priori information be present for K of N nucleotides. Suppose that N1 nucleotides are different for neighboring species of replicators, with N2 representing intraspecies differences. Then, for one of the neighboring empty niches, the following algorithm for partially directed evolution holds:

P (A) P (B / A) . P (B )

P (A/ B ) =

where P (A) is the a priori probability of hypothesis A; P (A/B) is the probability of hypothesis A at the occurrence of event B (a posteriori probability); P (B/A) is the probability of occurrence of event B with the truth of hypothesis A; and P (B) is the probability of event B. Let us consider an example of the application of the Bayesian classifier: the classification of nucleotides in the genome. We will only be interested in two classes of genes: good and bad. Therefore, the question is whether this or that sequence of nucleotides is good? We will assume that genes are selected from several classes of genes that can be represented by a set of nucleotides with (independent) probability such that the i-th nucleotide of a given gene is found in a gene of class A:

P (Bi / A).

– For K nucleotides, to act in accordance with a priori information (i.e., replace them with certain nucleotides), – For the remaining N-K nucleotides, produce nucleotide substitutions randomly.

For this case, suppose that the probability of meeting a nucleotide in a gene is independent of the length of the gene and that all genes have the same length. Then, the probability for a given gene B (which contains a set of nucleotides-hypotheses) and class A can be written in the form:

These operations are repeated as long as the target (terminal) set is reached. In this algorithm, a priori information can be calculated by the number of nucleotides to which it is available. If for each nucleotide, a priori information equals two bits (log24), then for K nucleotides, a priori information will equal 2 K bits. We show that for K ≪ N, this algorithm leads to exponentially larger times required to achieve the terminal set. Indeed, for nucleotides for which a priori information is available, modifications can quickly occur because this information is not associated with the enumeration of large numbers of variants. If these nucleotides are no longer required for change (i.e., they must be protected from further changes) after a small number of generations, these nucleotides do not participate in evolution. Thus, to reach the terminal set, variations in the remaining N-K nucleotides are required. This search will make the largest contribution to the average time to reach the set. As indicated in the paper by Melkikh (2014a), the method of sorting effectively works for only a small number of nucleotides, approximately 103-104. Thus, a method for achieving the terminal set is operational only in the case of N-K ∼ 103-104. Notably, this estimate was also obtained for a large number of replicators in the population. For larger animals with much smaller population sizes, the number of nucleotides for which a priori information is not required will be even smaller (up to 102). Thus, for larger genomes (N ∼ 109), prior information concerning the future states of a vast number of nucleotides must exist before the start of evolution! All this applies, of course, not only to the information sequence of

P (B / A) =

∏

P (Bi / A).

i

The question that we want to answer is, “What is the probability that this gene B belongs to class A?" In other words, what is the value of P (A/ B ) ? For two classes S and ¬S (good and bad), we can write

P (B / S ) =

∏

P (Bi / S )

i

and

P (B / ¬ S ) =

∏

P (Bi / ¬ S ).

i

Using the Bayesian result, we can write:

P (S / B ) =

P (S ) P (B )

P (¬S / B ) =

∏

P (Bi / S ),

i

P (¬S ) P (B )

∏

P (Bi / ¬ S ).

i

Dividing one into another, we will have:

∏ P (Bi/S ) P (S / B ) P (S ) i = . P (¬S / B ) P (¬S ) ∏ P (Bi / ¬ S ) i

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Table 2 Correspondence between morphogenesis and evolution of complex structures. Property

Folding (reactions)

Evolution

Morphogenesis

The number of variants

The number of possible conformations of proteins (possible reactions between biologically important molecules) is exponentially large The number of correct variants is exponentially less than incorrect ones Folding (reactions) is partially directed due to long-range forces

The number of possible variants of the genome is exponentially large

The number of possible variants of connections between cells is exponentially large

The number of correct variants is exponentially less than incorrect ones Evolution is partially directed

Fokker-Planck equation

Fokker-Planck equation

The number of correct variants is exponentially less than incorrect ones The aggregate of the cells of the organ is a whole that exists on the basis of long-range forces Equations of morphogenesis with long-range interactions

The number of correct variants Directivity

Mathematical description

and Khrennikov, 2017b).1 Thus, the forms are formed according to their own laws, not in relation to mutations and natural selection, as previously suggested by Berg, Lyubishev and other scientists (see section 2.1). If we consider, for example, from this point of view the origin of the beaks of the Darwin finches, then we can say that their various variants have been pre-programmed. As a result of feedback to the environment (for example, using the immune system, as well as other sensory organs), the most appropriate ones were selected from these variants. This was expressed in the fact that quite definite cells, on the basis of which the beak was built, began to be divided; as a result, the shape of the beak changed. Selection in this process “preserves the norm” (Berg), that is, rejects the erroneous variants. The regulation of the genome (as well as all other operations on genes and chromosomes) in this sense can be considered part of the morphogenesis at the molecular level. According to a recent report (Melkikh and Khrennikov, 2017b), the equations of partially directed evolution can be written in the following form:

We can re-write this expression in the form:

P (S / B ) P (S ) = P (¬S / B ) P (¬S )

∏ i

P (Bi / S ) . P (Bi / ¬ S )

Taking the logarithm of both sides, we get:

ln

P (S / B ) P (S ) = ln + ln P (¬S / B ) P (¬S )

∏ i

P (Bi / S ) . P (Bi / ¬ S )

Finally, the gene can be classified as follows. This is a bad gene if P (S / B ) ln P (¬ S / B) > 0 , otherwise it is a good gene. In fact, the logarithm value, at which the gene is recognized as bad is rather arbitrary and can be specified on the basis of experience. The quantities on the right-hand side must be known a priori from experience. Thus, the method of recognizing bad genes can be represented in the form of the following sequence of events: from the set of genes, those that the decision-making system refers to as the bad ones are distinguished. These genes contain nucleotides with different frequencies of occurrence. Therefore, it can be concluded that certain nucleotides belong to genes that are classified as bad. After this, when analyzing new genes, one can find in them nucleotides previously found in genes classified as bad. After this, when analyzing new genes, one can find in them nucleotides previously found in genes classified as bad. If the number of such nucleotides is large enough, then the new gene is also classified as bad. Such a sequence of actions could be performed by some controlling genetic system. The problem, however, is that all this must be done when taking into account the state of other genes consisting of nucleotides. That is, to dial the database, we need to go over the number of variants that can be compared with the full set of possible variants of nucleotides, otherwise such a classifier cannot be considered reliable. Thus, the training of such a classifier implies the presence of a priori information about the system. This information can be calculated on the basis of the connection between information and conditional probability (see Appendix A). It is important to note, that in quantum systems, the formula of total probability can be violated (for example, Khrennikov, 2010).

n

dx j

⎞ ⎛ = x j ⎜∑ aji ({ξ }) x i − φ (t ) ⎟, dt = i 1 ⎠ ⎝

where n

φ (t ) =

n

n

∑ ∑ aji xi xj , ∑ xi = 1. j=1 i=1

i=1

The values of xi are the frequencies of the alleles (other codes in the replicator population). In a previous study (Melkikh, 2014c), it was suggested to present the coefficients aji in the form:

aji = ajiD + aji*, where ajiD are elements of the matrix of mutations, a priori not directed at creating new species, and aji* are elements of the mutation matrix, a priori directed at creating them. On the other hand, the process of controlled mutations itself must be connected with the control of certain degrees of freedom. Therefore, in this record, the set {ξ } describes the state of the entire genetic control system. This set includes also epigenetic effects. For the set of variables {ξ } , it is possible to write master equations analogous to (7). Note also that the dynamics of variables of the set {ξ } are non-Archimedean, i.e., in this case, the superposition principle for forces and potentials at DNA regulation and controlled mutations is not fulfilled. Thus, in many respects, the evolution of morphogenesis is similar to morphogenesis itself! This similarity has, in our opinion, a fundamental basis, which is the integrity of life at different levels of its organization (Table 2). We can draw an analogy between the problems of folding

3.8. Partially directed evolution and mechanism of origin of forms In several studies (Melkikh, 2014a; Melkikh and Khrennikov, 2017a, 2017b), a model of partially directed evolution has been proposed. Let us discuss the peculiarities of this model in light of the formation of complex forms in the process of evolution. First, we note that within the framework of directed evolution, a complex structure arises by combining other complex structures. The very rules of such a combination constitute the mechanism of directed evolution. Second, many complex adaptations of animals and plants to the environment arise on the basis of feedback from the environment. The mechanisms of such feedback are considered in a recent study (Melkikh

1 See also the work of Asano et al. (2017) about modeling of such mechanism with the aid or theory of open quantum systems.

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considered from the point of view of complexity. We show that the mechanisms of morphogenesis at the molecular level should be unstable in the case of short-range interaction potentials between biologically important molecules. To solve this problem, a quantum model of morphogenesis is proposed that takes into account long-range interactions between biologically important molecules. The essential point of the model is its non-Archimedean nature, which is expressed in violation of the superposition principle for forces and potentials. The evolutionary aspect of the formation of morphological structures is considered. It is shown that the complexity of the organs does not provide an explanation of their origin in the framework of non-directed evolution. The previously proposed model of partially directed evolution is further refined for the case of the evolution of complex structures. The complexity of several morphological adaptations of animals (e.g., the proboscis of a sailor moth, limbs, the skulls of the ancestors of modern humans, the beaks of the Darwin’s finches) is considered. For all these cases, it is shown that the formation of structures should necessarily include a priori information concerning them.

(reactions) and evolution (see Table 2) (see also, Melkikh, 2014a; Melkikh and Khrennikov, 2017a) on the one hand and morphogenesis on the other. The proposed model of morphogenesis and its evolution requires experimental verification. Several experiments regarding the evolution of complex systems have been proposed previously in papers (Melkikh, 2014a; Melkikh and Khrennikov, 2015, 2017a, 2017b). Promising systems for experiments concerning the evolution of organs and other complex structures could be rapidly evolving systems. As such, one can consider the classical species for evolutionary theory, Drosophila or another insect species, for which generations change rapidly, and it is possible to trace the formation of morphological structures in the process of evolution. We note, however, that experiments of much greater accuracy and greater detail are needed in comparison with those that are currently being performed today (for example, Melkikh and Khrennikov, 2017a, 2017b). 4. Conclusion In this work, the evolutionary mechanisms of morphogenesis are Appendix A Information, measurement and evolution

According to information theory, there is a relationship between information and the number of experiments required to solve a specific problem. Consider the evolution of a sequence of experiments that will result in the achievement of a terminal set or the absence of such an achievement. In information theory, the degree of uncertainty of an experiment β with possible outcomes and their probabilities p (A1), p (A2)… p (Ak) is usually characterized by the Shannon entropy as follows: k

H (β ) = − ∑ pi log 2pi . i=1

Equality of this value to zero means that the outcome of the experiment is known in advance. Any measurement or observation of α, the prior experience of β, may limit the number of possible outcomes of an experiment β and thus reduce the degree of uncertainty. The fact that the implementation experience of α reduces the degree of uncertainty of the experiment β is reflected in the fact that the conditional entropy H (β/α) of the β experience at a condition α is not greater than the initial entropy of the same experience. In this case, if the experience of β does not depend on α, then α exercise does not reduce the entropy of β:

H (β / α ) = H (β ). However, if the result of α completely determines the outcome of β, the entropy of β decreases to zero: Hα (β ) = 0 . Thus, the difference

I (α, β ) = H (β ) − H (β / α ) indicates how the implementation experience of α reduces the uncertainty of the experiment β. This difference is the amount of information regarding the experience β that is contained in the experience α. There is a relationship between the number of experiments and the information obtained as a result of each experiment. For example, if the experiment involves determining whether the terminal set has been reached, each experiment produces one bit of information. If no aprioristic information with which to find the terminal set is available at the beginning of the experiment, the average number of steps required to achieve it will be:

N=

Ω , Ωt

an exponentially large number. Such a number of steps corresponds to Shannon's entropy:

H = log 2N = log 2

Ω . Ωt

If it turns out that the number of experiments can be reduced compared to the enumeration of all variants by using any algorithm, this indicates that a priori information is present. This information should be included in the algorithm even before using it (evolution). We can also make a more general conclusion: speeding up the search in comparison with the enumeration of all variants requires aprioristic information concerning the terminal set. Information theory allows us to provide another interpretation of partially directed evolution. It allows us to answer the question of to what extent the mutation process at a given point of phase space is predetermined by the structure of the fitness landscape. From the viewpoint of information theory, if two random processes X and Y exist, then we can introduce the entropy for each of them:

H (X ) and H (Y ). Then, we say that a random process X contains information about the process of Y that is equal to:

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I (X , Y ) = H (X ) − H (X / Y ) = H (X ) + H (Y ) − H (X , Y ), where H (X / Y ) is the conditional entropy and H (X , Y ) is the entropy of association. For independent random variables the mutual information is zero. In relation to evolution, we can consider two limiting cases. In the first case, a random event, i.e., a “mutation”, does not contain information about the random event “probability of survival”. This case corresponds to Darwinian evolution. In another limiting case, the structure of mutational flow is completely predetermined by the structure of the fitness landscape. Then, we have

I (X , Y ) = H (X ), where H (X ) is the entropy of the fitness landscape. This case corresponds to fully directed evolution in the absence of uncertainty in the environment and negligibly small random mutations. Precisely in this sense, we can discuss how aprioristic information is used in search tasks. The proposed mechanism of evolution also has the advantage that it is more general than Darwinian evolution. Thus, in the simulation of certain situations (the evolution of short sequences of information, with a large degree of uncertainty in the environment and others), evolution can be reduced to a Darwinian mechanism as a special case. The present results were partially obtained in the frame of state task of Ministry of Education and Science of Russia 1.4539.2017/8.9.

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