_{1}

^{*}

The experimental data represented in the bibliography are theoretically analyzed to construct an adequate model for dynamics of an open nonequilibrium living system. It is shown that the viable microorganisms are capable of forming the fractal structure, whose dimensionality is certainly non-integral. In addition, we have attempted to provide a generalized description of the properties of living and nonliving matter (in the addition to that described in work [1]). Relevant published data were used to demonstrate a fractal structure of the space in the vicinity of centrally gravitating bodies with satellites revolving around them along closed trajectories and serving as a kind of testers of the neighboring space. A local violation of its discontinuity is likely to be a necessary (yet not sufficient!) dynamic characteristic of the spatiotemporal continuum for self-organization of molecules into a living, i.e., self-replicating, system.

The experimental data represented in the bibliography are theoretically analyzed to construct an adequate model for dynamics of an open nonequilibrium living system. It is shown that the viable microorganisms are capable of forming the fractal structure, whose dimensionality is certainly non-integral. In addition, we have attempted to provide a generalized description of the properties of living and nonliving matter (in the addition to that described in work [

Fractal Structure; Anthropic Principle; Origin of Life

Over 50 years ago, two Moscow microbiologists, V.A. Elin and V.O. Kalinenko, independently from one another discovered a paradoxical phenomenon, namely, the ability of organotrophic microbial E. coli cells after a certain pretreatment to reproduce in the saline solutions completely deprived of any organic substances. Independently of the composition of this saline solution (be it physiological solution or phosphate buffer) and the initial concentration of viable cells (10^{3} - 10^{5} cm^{–3}), all populations over 1 - 2 days of incubation at 37˚C reached the same limit concentration of about 10^{6} cm^{–3} and retained a long-term viability in this state without any access to organics. Results of these experiments were published in the journal Mikrobiologiya (Microbiology) [

Thirty years ago, our research team by a lucky confluence of circumstances discovered an analogous phenomenon of reproduction of an organotrophic microorganism, E. coli, under even more stringent conditions, in tetradistilled water. These experiments are described in [

It is easy to calculate, that the quantity of organic substratum, necessary for the amplification of cells 10^{3} - 10^{4} times cannot be contained as the spare substances in the polysaccharide capsule or the walls of initial cells. Furthermore, as showed authors [

cellular components for the reproduction of daughterly cells. Here one should again note that independent of the degree of the isolation of cells from the possible sources of organic substrata, in all works indicated the authors observed one general regularity. With the arbitrary initial concentration of cells from 10^{2} to 10^{6} cm^{−3} final concentration in 1 - 2 days of incubation at a temperature of 22˚C - 37˚C composed (2 - 5) × 10^{6} cm^{−3} (

In our works [

Let us begin the analysis of nonlinear effects in the population of the living systems of microorganisms from the completely nontrivial phenomenon—the disturbance “elementary statistical regularities”. In

The first, what does cause surprise and does raise the question—why relative dispersion does vary, at that time when on the conditions for experiment it must be constant? Moreover, according to laws the statisticians Poisson (namely to this statistics must be subordinated the distribution of microorganisms on the separate tests in accordance with the procedure of setting experiment!)

Hence follows the obvious conclusion: the population of living i.e., nonequilibrium systems is permanently reconstructed in the space in such a way that tendency toward the regular order, when

It is extremely important to note that this phenomenon is not connected directly with the displacement in the space of molecular structures, i.e., the cells of microorganisms. We specially conducted the examination of our own mobility of microorganisms in the given specific conditions and showed that the speed of their three-dimensional displacement does not exceed 50 mkm/sec [

Comparing these data and maximum mobility of the cells of microorganisms, it is possible to draw the single-valued conclusion that variations in the relative dispersion on such temporary scales cannot be provide ford only due to the three-dimensional displacement of the molecular structures of cells as whole.

To explain such sufficiently regular variations in the relative dispersion on the different temporary scales possible, if one assumes that on molecular many potentially “viable” objects is formed the self-similar fractal structure, whose nature, apparently, is not molecular, but is sooner anything field-electromagnetic or gravitaRelative dispersion

Relative dispersion

tional there is no as yet single-valued answer. One is clear—these structures cannot be stationary and equilibrium. This is the first difficultly refutable conclusion, which follows from represented experimental data.

Let us be distracted for some time from variations in the relative dispersion, and let us examine nonlinear effects in the population of the microorganisms.

The initially linear dependence for small sampled volumes deviates from a linear pattern in a statistically significant manner with increase in this volume. The value and pattern of such deviations are not strictly fixed for an individual sample in population yet change with time (see

Consider one more nonlinear effect associated with a change in concentration of microbial suspension with its dilution with water (_{0} cells per unit volume will change the cell as:

where denominator is the theoretically expected value of microbial concentration. However, the experimental data transform this dependence to:

where denominator is the experimentally observed microbial concentration.

The nonlinearity coefficient α is not a strictly fixed constant. Its value is mainly determined by the time of relaxation to stationary state after mechanical disturbance of population (see

^{8} cm^{–3}) to the concentration K ≈ 10^{3} cm^{–3}. Within the error, the parameters of relaxation curve are independent of the degree of population dilution.

Note that immediately after dilution of cell suspension, NVC drops strictly twofold independently of the dilution factor and then again relaxes to the initial concentration. Evidently, there is no question of any irreversible mechanical destruction of half cell population. Presumably, after a mechanical disturbance of population, a rapid destruction of autonomously stable spatial population structure takes place with its subsequent slow relaxation to

the state that provides all cells with the amount of free environmental energy sufficient for maintaining the viability of the entire population.

The experimental data suggest that:

• Nonlinear phenomena are observed when sampling the population with a constant cell concentration, namely, the NVC in samples increases not directly proportionally to their volume;

• The change in NVC is also not directly proportional to the dilution factor after changing cell concentration (diluting cell suspension via addition of water);

• The spatial structure of population is rather sensitive to external mechanical disturbance but is able to relax to the initial state.

To determine the physical meaning of nonlinearity coefficients, we use fractal geometry methods. As is known, the objects of fractal geometry are the structures with a characteristic hierarchy and a scale invariance.

See

method was selected to level the accuracy error inevitably introduced when counting NVC.

These data suggest that a cell concentration of 8 × 10^{2} cm^{–3} is the critical point below which the variance^{2} cm^{–3} in the volume to heterogeneous distribution, when a hierarchy of clusters with different concentrations of viable cells is formed. In addition, a complex evolution of these clusters is observed. This conclusion was made based on the variations of relative variance around the mean value with time; this mean value is determined by the concentration of viable cells (see

Without getting into the physical mechanism of this phenomenon, consider its formal aspect. In the case of a hierarchical self-similar distribution of elementary units forming a fractal object in three-dimensional Euclidean space, their density in a volume is not a constant but rather depends on the sample volume as follows:

where _{o}, average density over the entire volume occupied by fractal object; V_{o}, the volume of one elementary unit (in this case, the volume per one cell, a uniform distribution of the overall population provided); and D_{R}, fractal dimensionality of the object. It follows from Equation (4) that at m-fold dilution of the population displaying fractal properties, the corresponding densities

As it follows from Equations (2) and (5), the fractal dimensionality and nonlinearity coefficient are related as:

At a cell concentration exceeding 3 × 10^{3} cm^{–3}, the curve in figure commences significantly deviating from asymptotic value of 3, characteristic of small concentrations.

The essence of experiment was as follows: suspension of microbial cells was intensively stirred and then, as soon as possible (as a rule, over 15 - 20 s), 5 ml of this mechanically disturbed suspension was pumped using a peristaltic pump into a separate glass container (tube). Then samples were taken from this tube and multiply diluted with water (as a rule with a step of two). The diluted samples were plated onto Petri dishes containing agar nutrient medium to determine the concentration of viable cells. The dishes were incubated overnight at 37˚C to count the colonies on the next day and determine the fractal dimensionality using Equation (6).

As is evident from

To construct a self-consistent spatiotemporal model for living systems, we then need the experimental data on the time drift and dynamics of microbial populations. As has been demonstrated [

In both the everyday live and scientific research, it is a priori considered that the space where we live has integer dimensionalities: line is one-dimensional, plane is two-dimensional, and space has only three dimensions. It seems that it is impossible to imagine the world and the objects in it with fractional dimensionalities. However, this is a mere consequence of our conservatism and habits. A coastline, clouds, porous structures, electrical discharges in atmosphere, and many other natural phenomena are much easier describable with the help of the language of mathematical equations by introducing the concept of fractality, i.e., a fractional dimensionality. The branched network of blood vessels of warm-blooded animals as well as their neuronal network, branching structures of trees, and so on can be described in an analogous manner. It appeared a surprising fact for us that the distribution of viable aquatic microbial cells in a volume can also be described using the language of fractals.

The fractal dimensionality of time is determined when analyzing long-term time variations in NVC in microbial population via the Hurst constant [

where K_{max} and K_{min} are the maximal and minimal NVC in the array of samples taken during the time T with an interval t and s, the variance for this array. The number of samples in the array,

where D_{T} is the fractal dimensionality of time for the data array.

It is easy to calculate the fractal dimensionality of population, D_{T} = 2 – H_{st}(K) = 2 – 0.309 = 1.691 ± 0.004.

A formal description of objects and processes, accepted in physics, is performed by specifying the corresponding functions strong-minded in the space of three variables, the coordinates, and the variable independent of them, time. In relativistic physics, these variables are regarded as 3 + 1, i.e., four-dimensional, spatiotemporal continuum. In our case, this sum is represented as D + D_{T} = (2.35 ± 0.07) + (1.691 ± 0.004) = 4.04 ± 0.07. The correctness for such summing of the dimensionalities of fractal sets is based in the Mandelbrot empirical rule [

Similar to all the previous arithmetic and algebraic cases with living systems [

Modern physics, which is the foundation for all natural sciences, is unable to explain both the phenomenon of life itself and the molecular mechanism of the origin of life. i.e., the conditions required for self-organization of atoms into a self-replicating system, the elementary unit of which is a microbial cell. In other words, based on the physical postulates—the conservation laws, which are the corollaries of the abstract concepts of uniformity

and isotropy of space and uniformity of time—does not provide for construction of a consistent model of the Universe, during the evolution of which self-replication molecular systems could emerge. The overall experimental and theoretical pool of knowledge tested so far since the 1920s has given neither positive nor negative answer to the question on the natural origin of the terrestrial life formulated by Oparin [

It is possible that such current of thought of the fathers of modern science was initially determined by the laws underlying the evolution of the Universe, which thereby tabooed unraveling of the secret and meaning of all living on the Earth. In an allegoric form, we can follow the idea of Strugatskie [

The only thing left to scientists is to debate on whether the origin of the life on Earth was natural or imported as “seeds” somewhere from the Universe. We can state that a periodic domination of either of the competing hypotheses—panspermia and natural chemical evolution—is the result of a technological breakthrough in either biochemistry or astrophysics. However, neither hypothesis has any constructive origin, because the carriers of these ideas are the scientists whose worldview has been formed within the frame of the current science, unable to explain the phenomenon of life.

Philip Morrison, an American physicist, noted that the discovery of life on one other planet can “transform the origin of life from a miracle to a statistic”, that is, to a certain experimentally reproducible thing, which forms the background for the methodology of scientific cognition of the world. However, neither physics nor biology can answer the question on what is the essence of the experiment that could simulate the conditions of the origin of living systems^{1}.

Appealing to the opinion of the founders of quantum mechanics, we should admit that a constructive solution for this problem (i.e., the problem of the origin of life) requires formulation of a “crazy” hypothesis. This work is an attempt in this direction.

Based on an anthropic principle [

Gorelik in his monograph [_{3} from the geometries and physics in R_{n}?” Thus formulated, these questions are possibly senseless. Undoubtedly, they are subject to criticism, because this entails the questions on whether the space “exists” at all, whether it is three-dimensional, and the very questions “why?” and what “physics” means in the case of R_{3} or R_{7}?

Gorelik did not try to find a better wording for these questions. Presumably, others will succeed in specifying more singular features of R_{3}, and it will then become clear what correct questions match the answers provided by our considerations [

Ehrenfest pioneered in formulating the problem of three-dimensionality of the physical space of the observable Universe at both micro and macro levels as long ago as 1917 [

Thus, the following statement is made: living objects could have appeared only in such Universe as it shows up to the observer. This statement is actually one of the simple definitions of the anthropic principle, introduced into scientific use by Dicke [

Further analysis of the properties of living matter will be based on the following postulates:

• Space does not exist by itself in the absence of elementary structural units of nature and

• The geometry of space, its dimensionality, as well as topological and metrical characteristics are determined by the pattern of mutual arrangements of structural units and the laws of their physical interactions.

These definitions contain nothing principally new and, therefore, contradictory to modern physics. On the other hand, the definition of space as a derivative of physical objects and the laws of their interaction allows its dimensionality to be also considered as a derivative rather than and axiomatic notion. In this case, the problem of space dimensionality and its integer value D = 3 transits from axioms to an interpretable category [

Commence constructing the hypothesis with Kepler’s third law:

where M is the mass of central gravitating body; T, the period of satellite revolution around it; R, radius of the major semiaxis of the orbit; and G, gravitational constant. It is convenient to represent Equation (9) in a dimensionless form:

where R_{o} is the radius of the central gravitating body and T_{o}, the period of satellite revolution at this distance. Such nondimensionalization provides for analyzing both planets and their satellites in a unified system of coordinates.

Analysis of the astronomic data for planets of the Solar system and their satellites [^{2} (in astronomic units, where R_{E} is the distance between the Earth and Sun), Equation (10) starts to considerably deviate from unity; moreover, the variations around unity increase with a decrease in the distance of satellites from planets. Formally, it should be interpreted so that the laws of celestial mechanics formulated for a three-dimensional space fail to precisely describe the movement of bodies at the scales

The only object falling out of the entire data array for satellites is the Moon (log r = –2.59). As for the planetary scale from the Mercury to Neptune, the parameter D = 3 falls within the measurement error.

The author does not assume the responsibility for interpretation of the physical reasons underlying this phenomenon. However, since the goal of this work is to formulate a hypothesis verifiable for the origin of life to the extent possible that would fundamentally differ from all the previously proposed hypotheses, we well use the results obtained by analyzing variations in the parameter D to formulate the following hypothesis:

In the vicinity of gravitating masses in the space on the average uniform and isotropic, spherical layers of a fractal space structure are formed, the dimensionality of which is subject to small variations in the neighborhood of the integer value D = 3.

Consider the possible consequences of such variation in space dimensionality in the context of the stated problem.

Assuming that the observed variation is not a consequence of an accuracy error (which is actually impossible, because otherwise it would be similarly impossible to observe the variations in D_{R} but only a monotonic drift), we can postulate that the physical vacuum additionally influences the pattern of satellite motion. Vacuum is among the most mysterious material substances, whose properties are being discussed among theoretical physicists for already several decades. We will take advantage of the current concepts on physical vacuum by various authors [

During the last decade, the terms dark energy and antigravitation [

From the standpoint of the author, in addition to the standard physiologically necessary conditions for sustainable existence of life, namely, “normal” pressure, temperature, and humidity of environment and the absence of pathogenic physical and chemical factors, polarization of the physical vacuum is a basically important factor for self-organization of molecules into a self-replicating system [

The consequence of this is a fractal, i.e., noninteger, space-time dimensionality, which entails nonlinearity of all molecular processes eventually giving rise to a spatiotemporal ordering of the molecular processes that had led to formation of self-replicating informational structures. Note that the noninteger, i.e., fractional, dimensionalities of both space and time separately also gave in sum an integer, an invariant of four!

The author is grateful to Prof. A.D. Gruzdev, Prof. S.A. Shnoll, Prof. R.G. Khlebopros for fruitful discussions.