Basic Limitations of Self-Organization by the Example of High- and Low-Integrated Very Complex Systems (Mammalian Skeleton Elements and Mammalian Fossil Assemblages): From Empirical Evidence to Theory

Abstract

High variety is a characteristic attribute of any material phenomena and processes involving living matter, i.e., very complex systems (VCSs). We have verified the presence of fundamental constraints on the size/shape diversity and self-organization by the example of mammalian skeleton in four orders (41 species). The properties of more than 4700 multidimensional descriptive models of VCSs were studied. A self-organization parameter R (0 ≤ R ≤ 1) was calculated for each model, and its range of variability was mainly limited to the interval from ~0.10 to ~0.31. The concepts of an abstract Ashby regulator and the Shannon–Hartley theorem were used to explain the variation in the empirical data. It has been concluded that there are significant constraints on the quality of morphological diversity regulation and the possible level of self-organization of VCSs for steady states.

INTRODUCTION

In the modern scientific concepts of living matter, the latter is usually not considered as one of the standard forms of matter. At the same time, one cannot deny the empirical fact of evolutionary continuity and continuous coherence between nonliving and living matter (Vernadsky, 1978). As a consequence of this continuity, evolutionary limitations of the “nonliving” form of standard matter are extended to its “living” form.

The deterministic idea of the New Age (the 16th–19th centuries) that the knowledge of the “laws of nature” provides a possibility of making accurate predictions about the evolution of material systems proved to be wrong already at the turn of the 19th and 20th centuries (“the crisis in physics” (Mainzer 2009; Gershunin and Alov, 2019). Biologists and, to a lesser extent, physicists are fully aware of the extreme complexity of the structure and function of living matter. In the second half of the 20th century, physicists (E. Schrödinger) and then specialists in cybernetics (W.R. Ashby, H. Verster, L. von Bertalanffy) proposed “global” solutions to the problem of the evolution of life and human consciousness (Bertalanffy, 1962). However, it is indicative that S. Hawking (Hawking, 2002) did not include the evolution of life in his lecture course “The Theory of Everything: The Origin and Fate of the Universe.” In contrast to the theoretical constructions of the 19th century and the first half of the 20th century, the modern biological theories must be based on extensive empirical data. The movement of scientific thought from models (theory) to real objects should be supplemented by an explicit reverse movement from objects to models. This provision, in our opinion, is of fundamental importance for the cognition of very complex systems (VCSs).

We have proposed (Puzachenko, 2016) a set of axioms that create a basis for consistent inclusion of the evolution of living matter in the context of evolution of standard matter accompanied by a general increase in organizational complexity: from elementary particles to biological and social systems, and the continuity of fundamental limitations, which is provided by “memory” sensu lato.

It is reasonable to combine the phenomena, processes, etc., which involve the living form of matter, in a specific class of material systems: “very complex systems.” Separation of VCSs from the wider class of complex systems is objectively determined by special properties of the living matter (Beer, 1959; Nicolis, 1986; Cilliers, 1998; Heylighen, 1999; Di Marzo Serugendo et al., 2004; Holden, 2005; Green et al., 2008; Ladyman et al., 2013; Bohórquez Arévalo and Espinosa, 2015; Ma’ayan, 2017; Gershenson et al., 2021).

The main subject of the present study is the phenomenon of biodiversity. The objects of research are mammalian skeleton elements, primarily the skull (Puzachenko, 2013, 2016). This study was aimed at describing the quantitative regularities of implementation of the size/shape diversity of skeletal elements invariant with respect to particular forms of variability (ontogenetic, group forms of variability, intrapopulational, geographic, interspecific, and phylogenetic). In the context of this study, it has been attempted to provide theoretical justification for some empirical data.

Our research methodology is based on multivariate analysis (methods for reducing dimensionality of the space of descriptive variables), on the one hand, and on the information theory tools, including the theory of signal transmission, and the ideas formulated in cybernetics, on the other hand (Puzachenko, 2000, 2001, 2011, 2013, 2016, 2020; Ashby, 1956, 1958; Conant and Ashby, 1970; Haken, 2006; Puzachenko and Markova, 2011; Abramov and Puzachenko, 2012; Puzachenko and Korablev, 2014). Information macroparameters, namely, information entropy and its derivatives, are used for quantitative characterization of the structure of a multidimensional descriptive model (Puzachenko, Yu., 1982, 1992, 2009; Shannon, 1948, 1949; Shannon and Weaver, 1949; Wiener, 1961; Brillouin, 1962; Atlan, 1977; Schrödinger, 2012; Schneider, 2000; McCowan et al., 2002; Collier, 2008; Puzachenko, J., 2008; Tkačik and Bialek, 2016). In cybernetics, diversity directly correlates with entropy–information in information theory (Conant and Ashby, 1970). The notion of a “regulator” as a specialized subsystem, generally of unknown nature, which controls and maintains homeostasis (stationary state) of a system, is one of the key abstract concepts of cybernetics (Ashby, 1947, 1956, 1958, 1962). A regulator, according to W.R. Ashby, is an abstract transmitter of information to a system from its environment, which ultimately influences the choice of a particular state by the system (Conant and Ashby, 1970). It is proved that the ability of the Ashby regulator to perform the function of noise suppression is possible only due to its internal diversity (entropy, organizational complexity) and is limited by its throughput capacity (Shannon, 1949), as if it were an information channel (Ashby, 1958).

Interpretation of the coordinates of descriptive models of VCSs as information channels indirectly characterizing the work of an Ashby regulator provides the opportunity to consider the sequence of information flow from the “environment” to VCSs and further to the “observer” (Fig. 1a). Regardless of the specific nature of signal S, it may contain “noise” (D), perturbations distorting its semantics. The regulator decodes the information flow from the environment and forms its information (correcting) channel R, through which it corrects the main signal S. The amount of “noise” that can be removed from S is limited by entropy that can be transmitted through R: H(R) ≥ H(S) (Fig. 1a). The regulator-corrected signal Z is what can be directly or indirectly measured by the researcher as a feature/variable of the system. The observer only has the data on entropy of the descriptive model of VCS (H(M)). Therefore, the indirect measure of the Ashby regulator diversity is H(R) ≥ –H(M) + H(D).

Fig. 1.

(a) The schematic diagram of information transmission between an environment, a VCS, and a hypothetical observer and (b) the graph illustrating the limit of information channel capacity according to the Shannon–Hartley law. (a) (1) The “environment”; (2) the VCS; (3) the measurement system; (4) the observer; E, energy, information, order/disorder; D, environmental perturbations, the “noise” distorting signal S (uncorrected signal) coming to the VCS; C, the basic information channel of the VCS; R, the correcting information channel of the Ashby regulator; Z, the corrected signal; M, the information in the measurement system of the corrected signal; O, the information from the regulator of the observer. (b) C/w₀, the throughput capacity (C/w₀ = $({W \mathord{\left/ {\vphantom {W {{{w}_{0}}}}} \right. \kern-0em} {{{w}_{0}}}}){{\log }_{2}}[1 + ({{{{w}_{0}}} \mathord{\left/ {\vphantom {{{{w}_{0}}} W}} \right. \kern-0em} W})]$) of the information channel; W, the bandwidth; w₀, the signal frequency when its power P is equal to the specific noise power (N₀) per unit of signal frequency measurement (w₀ = P/N₀); R, the measure of VCS organization.

The law of channel capacity is formulated as the Shannon–Hartley theorem (Shannon, 1949). The channel capacity (C) is directly proportional to the bandwidth (W) of signals and the logarithmic ratio of the signal power (P) to noise power (N) C = W log₂(1 + P/N) (Fig. 1b). As the bandwidth expands, the channel capacity rapidly increases until the total noise power becomes close to the power of the signal per se (W/w₀ = 1). Graph C asymptotically tends to the value of w₀log₂e ≈ 1.443w₀, bps.

The concept of a “self-organizing system” was proposed by W.R. Ashby in 1947 (Ashby, 1947) for deterministic (closed) systems and developed in his work in 1962 (Ashby, 1962). However, according to H. Foerster (1960), a self-organizing system must consume energy and order (information) from the environment; i.e., it must be open in the mechanical and thermodynamic sense. The capability of self-organization is not an exclusive attribute of living matter and is widespread in nonliving nature (Haken, 2006). Self-organization is not limited by the increase of internal order in systems. An equal component of self-organization is organizational simplification. The internal order can decrease and increase simultaneously at different hierarchical levels of the system, or cyclically within the same hierarchical level.

In this article we present the results of verification of the hypothesis about the presence of fundamental constraints on the size/shape diversity of morphological structures: mammalian skull and postcranial skeleton elements, which are considered as examples of VCSs. Some elements of the theory explaining empirical observations can be found in traditional information theory (the theory of signal transmission) and cybernetics. The elements of the postcranial skeleton are simpler morphological structures compared to the skull. Hence, it was possible to assess the efficiency of regulation of morphological diversity for skeletal elements with different levels of complexity of the structural and functional organization. We also compare this result with a similar one for complex supraorganismal systems—mammalian communities. The latter are seen as an example of weakly integrated systems compared to skeletal elements.

MATERIALS AND METHODS

The materials of the present study included 7091 skulls of representatives of four orders (Artiodactyla, Carnivora, Perissodactyla, and Rodentia: Equus ferus, E. f. przewalsskii, Coelodonta antiquitatis, Alces alces, A. americanus (A. a. pfizenmayeri), Bison bonasus (B. b. bonasus, B. b. caucasus), B. priscus, Capra sibirica (C. s. sibirica, C. s. alaiana), C. caucasica, C. cylindricornis, C. aegagrus, Ursus spelaeus (U. s. spelaeus), U. kanivetz = ingressus, U. arctos (U. a. arctos, U. a. yesonensis, U. a. collaris, U. a. piscator), Felis margarita (F. m. thinobia, F. m. scheffeli), F. silvestris (F. silvestris), F. lybica, F. catus, Meles meles (M. m. meles, M. m. taxus, M. m. milleri), M. leucurus, M. canescens, Martes martes, Lutra lutra, Mustela lutreola, M. eversmanii, M. erminea, M. sibirica (M. s. sibirica, M. s. manchurica), M. altaica, Vormela peregusna (V. p. peregusna, V. p. koshewnikowi), Vulpes lagopus (V. l. lagopus, V. l. semenovi, V. l. beringianus), Canis lupus (C. l. lupus), Castor fiber, Spermophilus erythrogenys, Sp. fulvus, Sp. major, Sp. pallidicauda, Sp. ralli, Sp. relictus, Spalax microphthalmus, and Cricetulus migratorius.

Metacarpal and metatarsal bones, the talus, and metacarpal phalanges (4080 bones altogether) belonged to cave bears (Ursus deningeri, U. spelaeus, Ursus kanivetz = ingressus, U. kudarensis), caballoid horses (Pleistocene Equus ferus and modern E. f. przewalsskii), and representatives of the order Chiroptera, family Rhinolophidae (metacarpal bones).

The systems and number of measurements varied between different taxa. The generally accepted standard measurements characterizing the overall dimensions of the skull and the postcranial skeleton elements were used in all cases. Previously (Puzachenko, 2013), it was shown that the number of measurements has no effect on the values of the diversity parameters.

The concepts of morphosystem and morphospace. According to G. Simpson (1944), the morphological variability of organisms as a set of various biological processes is related to the measure of difference between individuals at the population level and, therefore, is a statistical characteristic of a sample (within the general population) rather than an individual (Yablokov, 1966). An individual (organism) is taken as the smallest “indivisible unit,” i.e., an element. The properties of an element can be characterized by measuring a usually limited set of attributes (variables).

Let us define the formal object of research, the morphological system, as the intersection of a set (statistical ensemble) of elements, a set of all possible variables, and a set of metrics allowing estimation of differences between elements (Fig. 2). The similarity/difference relationships between all pairs of morphosystem elements are written as a square matrix of metric values (Fig. 2). The variability of element sizes is adequately reproduced by the Euclidean distance. The diversity of proportions (form) is well reproduced by Kendall’s τ_b rank metric (Kendall, 1975; Puzachenko, 2016). The morphological space, or morphospace, is understood to mean a multidimensional descriptive model of diversity of morphosystem elements. Thus, our definition of morphospace differs from its previous interpretations (Foot, 1990, 1997; Raup, 1966; McGhee, 1991, 2007; Pavlinov, 2011). The morphospace was obtained as a result of processing of similarity/difference matrices by nonmetric multidimensional scaling, NMDS (Shepard, 1962; Davison and Jones, 1983).

Fig. 2.

The algorithm for constructing a descriptive model of a VCS (morphospace); (1) the selection of a morphosystem (M) as (A) the intersection of sets of elements, (B) variables, and (C) metrics; (2) the dissimilarity/similarity matrix between all pairs of elements x and y of morphosystem M; (3) the example of 3D descriptive model of a VCS (morphospace) with the X1, X2, and X3 coordinates obtained by one of methods for reducing data dimensionality (dimensionality reduction model); (4) the distribution of the coordinates of morphosystem M elements relative to the coordinates of the descriptive model (the projection of the morphospace on the X1 and X2 coordinates is shown).

The optimal dimensionality (d) of morphospace (=the number of NMDS axes) is estimated using the Kruskal Stress-1 measure of the NMDS model quality (Kruskal, 1964; Kupriyanova et al., 2003; Abramov et al., 2009; Abramov and Puzachenko, 2012; Puzachenko et al., 2017).

Parameters of the morphospace. The morphospace structure is defined by the distribution of the points corresponding to morphosystem elements. The natural parameter of this distribution is entropy, H (Kupriyanova et al., 2003; Puzachenko, 2003, 2011). In general, H and other information parameters depend on the sample volume (n) (Foerster, 1960) (Figs. 3a, 3c). Comparable entropy values can be obtained by their calibration. The dependence of H on log n (using the decimal logarithm) was obtained by finding several H values for descriptive models of morphosystems with different numbers of elements (the minimum volume: ~25 elements) at a fixed d value. The calibrated H value is obtained by subtracting the regression from the original data, followed by restoration of the scale for a fixed n_const: ${{H}_{{{\text{cal}}}}} = [{{H}_{n}} - (a + b\log n)]$ + $(a + b\log {{n}_{{{\text{const}}}}})$, where n_const = 50 (Figs. 3b, 3d). Thus, H₅₀ corresponds to the entropy for the morphospace of a morphosystem consisting of randomly selected 50 elements. Number 50 was chosen because, according to the calculations for a random normally distributed variable, entropy increases only by ~3.5% when n increases from 40 to 100. Number 50 is close to the modal value of n in the collections studied, with more than 45 and 70% of samples falling within the range of 30–40 and 30–80 specimens, respectively (Puzachenko, 2013).

Fig. 3.

The noncalibrated entropy values (H) of the morphospace of (a) cranial size and (c) shape in the Eurasian otter (Lutra lutra) demonstrate the linear dependence of H on the logarithm of the number of elements in a morphosystem described by the following regression equations: 5.6 + 0.67 log n (r = 0.61, p = 0.02) and 3.6 + 0.65 log n (r = 0.71, p = 0.005), respectively (the dashed lines limit the 95% confidence interval for the regression line); (b, d) the calibrated entropies of H₅₀ and ${{\bar {H}}_{{50}}}$ (mean value) after subtracting the linear trend for the same morphospaces of cranial size and shape, respectively.

The most important parameter of the descriptive model space is R, i.e., “redundancy” (Shanon, 1948), or the measure of self-organization/internal order of a system (Foerster, 1960), $R = 1 - {H \mathord{\left/ {\vphantom {H {{{H}_{{\max }}}}}} \right. \kern-0em} {{{H}_{{\max }}}}}$ = $1 - \left[ {{{\left( { - \sum\nolimits_1^d {\sum\nolimits_{i{\kern 1pt} = {\kern 1pt} 1}^k {{{p}_{i}}{{{\log }}_{2}}{{p}_{i}}} } } \right)} \mathord{\left/ {\vphantom {{\left( { - \sum\nolimits_1^d {\sum\nolimits_{i{\kern 1pt} = {\kern 1pt} 1}^k {{{p}_{i}}{{{\log }}_{2}}{{p}_{i}}} } } \right)} {(d{{{\log }}_{2}}n)}}} \right. \kern-0em} {(d{{{\log }}_{2}}n)}}} \right]$, where H_max is the maximum entropy. R varies from 0 (the maximum “disorder”) to 1 (the ideal “order”). When R = 0, the system is in the state of the maximum internal diversity (all of its possible states that preserve the system as such are realized with a certain nonzero and approximately equal probability); when R = 1, the diversity is minimal and only a single state is realized. The R₅₀ values were obtained as described above for entropy. The channel capacity of an Ashby regulator can be written as a linear function of the measure of organization R: C = (1 – R)log₂e. The R value at the inflection point of the graph (W/w₀ = 1, С/w₀ = 1, Fig. 1b) is: R = 1 – H/H_max = 1 – 1/log₂e ≈ 1 – 0.693 = 0.307.

The efficiency of the correction channel (Є_R, %) of the regulator is directly proportional to its channel capacity and inversely proportional to the signal/noise power ratio: Є_R = ${C \mathord{\left/ {\vphantom {C {({P \mathord{\left/ {\vphantom {P N}} \right. \kern-0em} N})}}} \right. \kern-0em} {({P \mathord{\left/ {\vphantom {P N}} \right. \kern-0em} N})}}$ = $\left[ {{{(\ln ({{2}^{{\left( {{{{{H}_{{50}}}} \mathord{\left/ {\vphantom {{{{H}_{{50}}}} d}} \right. \kern-0em} d}} \right)}}} + 1)} \mathord{\left/ {\vphantom {{(\ln ({{2}^{{\left( {{{{{H}_{{50}}}} \mathord{\left/ {\vphantom {{{{H}_{{50}}}} d}} \right. \kern-0em} d}} \right)}}} + 1)} {({{2}^{{\left( {{{{{H}_{{50}}}} \mathord{\left/ {\vphantom {{{{H}_{{50}}}} d}} \right. \kern-0em} d}} \right)}}} - 1)}}} \right. \kern-0em} {({{2}^{{\left( {{{{{H}_{{50}}}} \mathord{\left/ {\vphantom {{{{H}_{{50}}}} d}} \right. \kern-0em} d}} \right)}}} - 1)}}} \right] \times 100$, where Є_R is the “energy efficiency” calculated by analogy with the channel utilization rate with respect to its power (Vasiliev et al., 2008). The higher Є_R, the lower the measured diversity of the morphosystem and the higher the hypothetical diversity of the Ashby regulator.

The following parameters have been measured for the present work: d, H₅₀, H₅₀/d, R₅₀, Є_R for 3512 morphospaces of the size and shape of skull and 1200 morphospaces of the size and shape of postcranial elements in mature individuals. In addition, 290 models for the skulls of immature animals have been studied.

For comparison, we used previously determined R values of mammalian assemblages from the second half of the Late Pleistocene (MIS3–MIS2), the first half of the Holocene (MIS1) of Europe (Puzachenko, 2019) and some mountainous regions of Eurasia in the Late Pleistocene and the Holocene (MIS5–MIS1) (Puzachenko and Markova, 2020).

RESULTS

Diversity parameters of the morphosystem of skull and postcranial skeleton elements. The dimensionality of the descriptive models of morphosystems varies from one to nine for the size and shape of skull and from one to six for the size and shape of postcranial skeleton elements. Most models have 2–4 coordinates for the skulls and two coordinates for the postcranial elements of males and females. In the case when males and females are combined into a single morphosystem, the dimensionality of morphospaces can both decrease and increase with increasing sexual dimorphism. The influence of sexual dimorphism on d of descriptive models is stronger for the skull than for the postcranial bones.

The H₅₀ parameter depends linearly on the morphospace dimensionality. This means that there is a normal increase in the morphospace entropy when dimensionality increases by one, on average. This norm is specified by the H₅₀/d parameter (Table 1). The H₅₀/d median is 2.18–2.19 bits/element for the morphospace of the skull size and 2.38–2.39 bit/element for the morphospace of the skull shape. Hereinafter, we use the median as the estimates of parameters, because in most cases their distributions do not conform to the hypothesis of normality.

Table 1.   The statistics of information parameters of descriptive models (morphospaces) of the skull and the postcranial skeleton elements: H₅₀/d (bit/element), R₅₀, Є_R (%)

For the skull, the medians of parameter R₅₀ were the same for males and females: 0.19 (skull size) and 0.12 (skull shape), respectively. The same parameter in the postcranial skeleton diversity models was slightly higher: 0.22 (0.27) and 0.13 (0.14), respectively. The regulation of the skull size diversity is markedly higher on average compared to that of skull shape diversity. In turn, the postcranial elements show higher regulation of both the size and the shape compared to the skull.

The R₅₀ variability is not the same in representatives of different orders and families of mammals (Fig. 4). At the order level, Rodentia is characterized by the maximum constraints on the skull size diversity, on average, while Artiodactyla is characterized by the minimum constraints. No differences between the orders in the R₅₀ value for skull shape have been found. At the family level, the maximum constraints on skull size diversity have been found in Castoridae (Castor fiber) and Sciurudae (Spermophilus sp.). Relatively high values of the parameter have been shown for Ursidae and Spalacidae (Spalax microphthalmos). Ursidae and Cricetidae (Cricetulus migratorius) are characterized by a relatively high level of constraints on the skull shape diversity.

Fig. 4.

(1) The mean, 95% confidence interval, (2) the median of parameter R₅₀ (the measure of ordering) for the morphospaces of (3) the cranial size and (4) the shape diversity in the orders and families of mammals.

The higher R₅₀, the lower the diversity of the corrected signal Z (Fig. 1) and the higher the potential diversity and channel capacity of the hypothetical Ashby regulator. All R₅₀ medians lie above the critical value C = 1 (R ≈ 0.307) (Fig. 5). Consequently, the Ashby regulator does not provide “tight” control of the skull sizes or shapes and allows for relatively high diversity at the level of morphosystem elements, the skull and the postcranium.

Fig. 5.

The medians of parameter R₅₀ (the measure of ordering) on the graph of the information channel capacity according to the Shannon–Hartley theorem. (1, 2) The morphospaces of (1) cranial size and (2) shape diversity, respectively; (3, 5), the morphospaces of diversity of the sizes of postcranial skeleton elements in (3) males and (5) females; (4, 6) the morphospaces of diversity of the shapes of postcranial skeleton elements in (4) males and (6) females.

Different morphosystems can be located on the graph of the channel capacity both in the zone with the predominantly stochastic variability of their elements with low efficiency of the Ashby regulator (S: R < 0.1, C > 1.3) and in the zone with severe diversity constraints, deterministic dynamics (D: R > 0.304, С < 1), and high efficiency of the Ashby regulator. However, the overwhelming majority of the morphosystems studied are located between these extremes. In other words, these VCSs should be assigned to the category of material systems with probabilistic–deterministic dynamics (Beer, 1959).

The results of studying the dynamics of the R₅₀ parameter in the ontogeny of the skull in the four species representing three mammalian orders (Fig. 6) support the hypothesis that there is no “accumulation” of internal order during self-organization (Ashby, 1962). All species demonstrate multidirectional changes in R₅₀ up until the stage of sexual maturity. The R₅₀ value undergoes fluctuations even in adult animals (Figs. 6b–6d). However, the limits of R₅₀ variability do not go beyond the limits of the parameter variability defined above for mature mammals.

Fig. 6.

The dynamics of parameter R₅₀ (the measure of ordering) in ontogeny of the skull of (a) Vulpes lagopus (I, V. l. lagopus, II, V. l. beringianus), (b) Bison bonasus, (c) Castor fiber, and (d) Spalax microphthalmus: jv, sad, and ad are the age groups of young, semi-mature, and mature animals, respectively. (1) The mean, 95% confidence interval, (2) the median, (3) the morphospaces of cranial size diversity, and (4) the morphospaces of the cranial shape diversity.

The statistics for the parameter characterizing the efficiency of the Ashby regulator are given in Table 1. The Є_R median is ~42.8–43.9% for the skull size and ~39.1–39.5% for the skull shape. Є_R is naturally higher in the postcranial elements (44.4–50.0% and 39.9–40.8%). The efficiency of diversity regulation is usually lower for the shapes of bony structures than for their sizes. The variations in Є_R, as well as in the R parameter, are very limited. Both parameters are highly correlated with each other so that there is linear growth of Є_R along with the increase in R₅₀ values.

DISCUSSION

The empirical data on the parameters of morphological diversity of the size and shape of mammalian skull and postcranial skeleton elements demonstrate significantly limited variations in the parameter of self-organization (R₅₀, Fig. 7) and, accordingly, in the efficiency of diversity control. From 94 to 99% of the parameter values for males and females do not exceed 0.31. The small number of cases above this “threshold” only confirms the general trend.

Fig. 7.

Distributions of the parameter of self-organization (R₅₀) of the (a) sizes and (b) proportions of bone structures of the mammalian skeleton. (1) The median and the interquartile range; (2) the mean, 95% prediction interval; N, the number of observations.

The skull has a more complex organization and is a multifunctional organ with limited possibilities of changes in relationships between its individual parts (Romer and Parsons, 1986; Trainor et al., 2003), while the postcranial elements, i.e., metapodia, talus, and metacarpal phalanges, are individual bones. The parameter of self-organization was slightly higher for cranial size than for cranial shape, which implies that the diversity of shapes (proportions) of the skull and the postcranial bones is less tightly regulated than the diversity of their sizes. The parameter of self-organization was higher for the postcranial skeleton bones than for the skull. Such an parameter ratio most likely indicates the influence of system complexity on the R value and on the efficiency of regulation. It can be assumed that the more complexly organized VCSs encounter stronger constraints on self-regulation.

In our material, R is weakly dependent on or almost independent of sexual dimorphism and relatively weakly depends on the taxonomic affiliation. The effects of taxonomic affiliation, as well as phylogeny, on the efficiency of regulation need to be studied further. The effects of environmental conditions on the R parameter (epigenetic regulation) also need to be studied. Some evidence for the possible influence of environmental conditions (altitude above sea level) has been described for the lower cheek teeth (p4, m1, m2) of fossil cave bears. (Baryshnikov and Puzachenko, 2020). Noteworthy, the R variability for cheek teeth was within the range of 0.1 to 0.3, which falls within the basic range of this parameter for the skull as a whole.

The dynamic variability of the parameter of self-organization for the same VCSs is of particular interest. The data on the ontogenetic variability of R of the skull indicate that its values are not beyond the range estimated for mature animals. There is also no steady increase in the efficiency of regulatory mechanisms in ontogeny. In some cases, a marked decrease in R was observed for the group of sub-adult individuals. In general, the ontogenetic dynamics of the parameter of self-organization demonstrates quasi-cyclic fluctuations for objects with good age resolution.

There are limited data on self-organization at the subcellular level of organization of living matter. We know about the only model proposed by T.D. Schneider (Schneider, 1991, 2010), which describes the relationship between the informational and physical “concepts” of entropy for biological enzymes. In particular, the author determined the efficiency of restriction endonuclease EcoRI, which binds to DNA only at the sites combining two specific triplets of nucleobase, ignoring all other possible combinations. The enzyme, by selecting a certain pattern of the DNA molecule, acts as a molecular machine capable of decision- making. Making the “right decision,” i.e., specific binding to DNA, is accompanied by a decrease in the internal energy of the enzyme due to its dissipation. Nonspecific binding, on the contrary, increases the energy state of the molecule making this state less likely to occur. The information measure characterizing machine operation (channel capacity) is the logarithm of the number of possible choices (for EcoRI, it is 12 bits/per action), each of them corresponding to a particular energy state of the molecule. To describe the work of the molecular machine, T.D. Schneider arrives at an expression equivalent to that for the data throughput capacity of any information communication channel (Schneider, 1991). Direct measurements have shown that the efficiency of self-organization for EcoRI is about 70%. It should be emphasized that this is a very high value, which is close to the theoretical limit of the information channel capacity and exceeds by an order of magnitude the value for human-made electronic signal transmission devices. Similar values of efficiency were obtained for some other proteins (RepA, rhodopsin, photoactive yellow protein). The results of assessing the efficiency of self-organization at the molecular level are fundamental, because eventually it is precisely molecular mechanisms that underlie the values measured, e.g., when studying morphological diversity. The efficiency of information transfer in the long information chains of conjugate reactions from DNA to the differentiating cells and morphological structures being formed should decrease due to accumulation of dissipated energy. Consequently, we can also possibly assume a decrease in the efficiency of regulatory mechanisms in this series.

There is almost no quantitative data on the efficiency of self-organization of modern VCS at the supraorganismal level. At the “ecosystem level,” VCSs are poorly integrated systems with elements (populations) characterized by the high potential diversity of relationships both between themselves and with the environment. In accordance with the hypothesis suggesting a decrease in the efficiency of regulation along with the increase in the hierarchical level and the accompanying increase in the organizational complexity of the living matter, it could be expected that the parameter of self-organization of supraorganismal VCSs is lower, on average, compared to highly integrated organismal VCSs such as the skull.

We have determined the parameter of self-organization for the Late Pleistocene and Holocene fossil mammalian assemblages (Puzachenko, 2019; Puzachenko, Markova, 2020). Here, the parameter of self-organization is associated with the spatial distribution of species; i.e., it is proportional to species distribution by the area of their ranges. When R approaches 0, the relative sizes of the ranges will be aligned with each other within the geographic region under study. If R tends to 1, the distribution of range sizes becomes sharply asymmetric. In this case, there are few species with wide ranges, with a long “tail” of distribution of rare species.

For the second half of the Pleistocene and the first half of the Holocene, the R median was 0.13 and the range of variations in this parameter was 0.05–0.19. For mountainous regions, the R median was 0.11, with a range of variations from 0.03 to 0.17. These results do not contradict the hypothesis of a negative relationship between the structural and functional complexity of systems and their self-organization ability. The dynamics of R over time has a distinct biogeographic interpretation. In the range of isotope–oxygen stages MIS3–MIS1, there was a stable decrease in the R value, which reflected simplification of the spatial pattern of the species ranges: an increase in the dominance of wide-ranging and intrazonal species during the Pleistocene–Holocene transition. The mountain assemblages of mammals showed, along with the decrease in R in the Late Pleistocene, a steady increase in this value within 57 000–29 000 (14 000) calendar years ago, after its global decrease to a minimum within 71 000–57 000 calendar years ago (MIS4). The decrease in the parameter of self-organization in MIS4 correlates with the decline in species richness of fossil mammalian assemblages and was probably due to the intensive development of mountain glaciations at that time (Doughty et al., 2021).

Using some elements of the above theory, it is possible to justify theoretically the empirical threshold value R ~ 0.31. We have shown a consistent relationship between the parameter of self-organization calculated for morphological systems and the potential internal diversity/channel capacity of an abstract Ashby morphoregulator. Other things being equal, this relationship is inverse, i.e., the higher the parameter of self-organization or, in other words, the lower the entropy of a morphosystem, the higher the entropy/channel capacity of its regulator and the efficiency of regulation. The critical value of the parameter of self-organization is probably invariant both for the skull and for the postcranial skeleton of mammals. It marks the upper boundary of the range of parameter values, where realization of the stationary state of morphosystems is most likely. It is essential that this region on a graph is located beyond the boundary of deterministic regulation, which suggests a very high channel capacity of the Ashby regulator. Thus, in the observed range of R values, the hypothetical regulator per se belongs to the class of quasi-deterministic systems.

For the VCSs studied, the realized diversity H(R) probably suggests its correspondence to the optimal ratio between the variety of perturbations coming from the “environment” (H(D)) and the resultant diversity of the system (H(Z)), when the latter remains in the stationary regime.

The fundamental constraints on regulation associated with the channel capacity and additional “costs” such as information code redundancy, duplication of control systems, hierarchical organization of the regulator (Aulin, 1979) and information flows, associated energy costs, etc., are compensated by the potential ability not only to maintain the steady state in a randomly varying environment, but also to but also to support the adaptive and evolutionary potential of VCSs.

The decrease in the efficiency of regulation, as the internal organization of the VCS becomes more complex, should be compensated, at least partially, by a relative (specific) decrease in the dissipated energy. Otherwise, VCSs could not spontaneously (in the physical sense of this concept) evolve towards complexity. At a fixed level of thermal noise within VCS and the rates of matter and energy flows at its boundary with the environment, the evolution of efficiency of the Ashby regulator must be directed not toward the increase in determinism (accuracy) of regulation but toward the decrease in dissipated energy per act of informational “selection” of a particular state of the regulator from the whole set of its possible outcomes (in J/bit). In other words, a necessary condition for the spontaneous evolutionary process is reduction of the energy cost of information synthesis by the regulator between the successive levels of hierarchical complexity of the organization of living matter (in bit/J). Thus, the evolution of VCS complexity satisfies the condition of axiom on its spontaneity (Puzachenko, 2016) and is a natural continuation of the evolution of the nonliving form of standard matter.

The capacity of the Ashby regulator as an information channel is formally limited by the fundamental Shannon–Hartley theorem. Consequently, it could be expected that the revealed constraints on regulation/self-organization of VCSs, by the example of morphological and other VCSs, are fundamental. The available, through still very few, estimates of the parameter of self-organization at the suborganismal and supraorganismal levels of organization of living matter do not contradict this statement.

The problems of regulation and its limitations for VCSs are directly related to the problems of searching for the optimal governance and political organization in social systems. We assume that the possibilities of self-organization (the permissible limits of self- organization in both directions) and evolution of social systems are limited by the fundamental laws that also apply to other VCSs.

From the theory that we are developing, it seems to follow that the quest of social VCSs for “stability” based on hyper-regulation of social/political and economic relationships, which is often encountered in history, leads in the medium-term perspective to an increase of internal “instability” under the conditions of an inevitably changing, both internal and external, political and socio-economic “environment.” Instability develops in spite of the stated goals of regulation, as a natural consequence of constraints on the VCS capacity for evolutionary adaptation and reduction of the efficiency of control in general.

CONCLUSIONS

The observed morphological diversity of elements of the mammalian skeleton is an aspect of manifestation of the more general phenomenon: biological diversity, which in turn is considered as a special case of the phenomenon of diversity of material systems involving living matter and belonging to a specific class of VCSs. This determines the relevance of studying the basic patterns of implementation of diversity for better understanding of the parameters of regulation and self-organization of such systems.

The proposed theoretical and methodological approach goes back to the ideas of V.I. Vernadsky and is based on several axiomatic statements. The main statement postulates the casual unity of the evolution of nonliving and living matter on the Earth. The consequence of this unity is universality of the fundamental laws of (constraints on) self-organization in the evolution of material systems within standard matter.

By the example of highly integrated organismal VCSs (the skull, the postcranial skeleton elements) and poorly integrated supraorganismal systems (the paleocomplexes of mammals), we have tested the hypothesis of existence of constraints on their self-organization and the efficiency of regulation of their diversity. The empirical data presented in this article do not allow us to reject this hypothesis.

The available data on the efficiency of VCS regulation suggest that the degree of determinism in the work of abstract VCS regulators decreases in a series of hierarchical levels of organization of the living matter, from the subcellular level to the entire biosphere.

Possible theoretical explanation for the observed variability of the parameter of VCS self-organization is contained in the Shannon–Hartley theorem/law. If a VCS model is reduced to a model of its abstract Ashby regulator, the theorem defines the limit of increase in the channel capacity (entropy) of the regulator and, accordingly, the limit of regulation efficiency. The result has the signs of scientific fundamentality, because the consequences of the Shannon–Hartley theorem are invariant with respect to the nature of systems and the semantics of information signals.

Despite the fact that the classical cybernetics and information theory are obviously limited by closed systems, the main results of these sciences can and should be used as the simplified models and tests for open VCSs. The theoretical and probabilistic representation of a multidimensional descriptive model and interpretation of its coordinates as independent information channels is applicable to a wide class of VCSs.