In works [1, 2] we have presented
the evidence of development of many spontaneous processes (physical, chemical,
biological, mixed, etc.) according to the certainty law (by Lokhov). For
example, during photosynthesis the photosystem PII absorbs the red quantum of
sunlight (γ) at 680 - 683 nm with the energy not exceeding 1.84 eV. With the
release of this photon from the excited photosystem Р*680 to phaeophytin at a speeds of 10^{-15}-10^{-19}
sec, there occurs disintegration of γ_{II} into an electron-positron
pair in accordance with C-violation effect (1)

During disintegration of 8γ_{II}
the particles exceeding the electron mass not four times, but three times are
formed.

Based on the certainty equation (2) the quantum of time is calculated

within the limits of which the ratio between
the energy release ΔE and the
carry time (Δt), through
the chain of A_{5}-components in photosynthetic space (centre) at a
speeds of K<10^{-10 }s, remains constant (3)

Theoretical calculations of the certainty equation (1) for the launch of mechanisms of the first and second stages of photosynthesis, as well as the new universal mechanism of the genome activation of all living bodies [3] find full experimental confirmation [1, 3 -5].

The modern quantum mechanics was created on the basis of Heisenberg´s uncertainty principles. In this connection, there arises a by no means idle inquiry related to mutual correlation of two discussed principles.

This work is aimed to prove that the certainty principle is the new universal physical-chemical law of nature.

**Formation of homogeneous and isotropic geometric structures of DNA**

In work [6] the direct proof of Euclid´s statement about parallel lines intersection, which was formed as early as in III century BC [7] in the form of a theorem, is provided.

On the examples of reparation (removal) of erroneous sites in double-stranded DNA or RNA chain on the stages of replication (untwining of double-stranded DNA and RNA), mitosis and recombination between sister chromatids it is shown that, in contrast to the heterogeneous and non-isotropic Lobachevsky´s and Riemann´s geometry, or any other, in the geometry proposed by Lokhov the space-and-time becomes homogeneous and isotropic.

There can be created an infinite number of geometries, as logical systems. It is important to just determine in which of an infinite number of indeterminate geometric structures the frequency of errors, such as in

- inclusion of a wrong heterocyclic base into guanine-cytosine (G - C) and adenine-thymin (A-T) pairs in DNA chain;
- untwining of two antiparallel chains of DNA molecules in non-homological and homological sections;
- chromosome disjunction in daughter cells;
- recombination between sister chromatids, etc., which is described by the uncertainty principle

is transformed into the certainty principle in homogeneous and isotropic space-and-time (5)

**Deduction**: the replication mechanism (that of
DNA and RNA double-stranded chains untwining) is universal for аll living bodies - from bacteria to
highest eukaryotes [8-10]. In this complex enzymatic process the DNA-dependent
DNA polymerase provides the replication. For example, in bacteria in а rigorously defined unique region
close to the *ilv** *gene positioned in 74´ in а standard chromosomal mар of the colon bacillus E.Coli the
replication takes place simultaneously in both strictly opposite directions
with the velocity of about 800 nucleotides per second. Two replication forks
(the region, where а
simultaneous untwining of double-chain DNA and RNA and the synthesis of nucleic
acids´ macromolecules оn
every of these chains-matrices happens) are seen nearby the ** trp 25´**
marker in the chromosomal mар (Fig. 1).

The genetic analysis suggests [8-11] that the nucleotides coming out of the replication fork very often undergo spontaneous point mutations. It has bееn found that the errors of а wrong heterocyclic basis insertion in the pair guanine-cytosine (G-C) and adenine-thymine (А-Т) in the DNA chain, from а rigorous sequence, which is а concrete genetic information bееn defined bу, often happen at the stages of replication (l per 100 000 base pairs), mitosis (l per 1000) and, most often, - in the course of genetic recombination between sister chromatides.

The evolution has developed а universal mechanism for the removal (reparation) of wrong points and formation of rigorously equal sequence and length nucleotides in the cells of living bodies [8-10, 13]. Before the complete reconstruction of а damaged region(-s) the DNA polymerases are settled in the homologous region (Fig. 2). In this case the DNA replication comes to а halt as well as the chromosome disjunction to daughter cells. At the stage of mitosis every chromosome remains connected with the spindle fibers.

Let
us assume that two lines АВ and А´В´ intersecting two parallel chains **а** and **b** of аn E.Coli DNA double-stranded
molecules in the region of а base pair А-Т or G-C close to the ilv gene form
together with them concluded angles, the sum of which is equal to the two lines
(Fig. 1). Then during the replication up to the stage of mitosis the ring chromosomes
of the E.Coli will bе
untwined into linear chains from the origin locus* ori *up

*to the homological region*

*trp(c)**(Fig.2). Under these circumstances of the cell´s transition from the metaphase to telophase the parallel lines а and b will intersect each other in the point с as the mechanism of reparation guarantees both equal alteration оf the concluded angles and the equal*

*lengths*

*of the lines*

**А**and

**В**;

**А**

**´**and

**В**

**´**in the triangles

**АВС**and

**А**

**´В**

**´С**(Fig. 3).

In
some prokaryotic cells the replication fork, though moves with the velocity of
about 1000 links per second, the chromosome disjunction to the daughter cells
occurs with unequal speed. However, every chromosome nearby the marker ** trp(c)**
remains constantly connected with the spindle fibers for the period of about 30
minutes normally. This time is quite sufficient for matching of the

**АС**and

**ВС**lengths in the triangle

**АВС**(

**А**

**´С**

**´**and

**В**

**´С**

**in the triangle**

**А**

**´В**

**´С**

**accordingly).**

Although
the reparation mechanism activation decreases the frequency of errors
considerably in the postreplication stage up to оnе per 10^{9} - 10^{10} counted base pairs [8-10, 13] in
the process of molecular recognition some unrepaired mutations remain always.
Otherwise, the bodies of elementary and highest eukaryotes would not have expressing
genes, and the evolution would unlikely bе possible.

Vice
versa, it is as if mutations are аn integral part of homogeneous and isotropic geometric spaces formations
within the interval of replication initiation (** ori**) and cell
division (mitosis). So, with the increase of the DNA mass in the highest
eukaryotes the number of replicons (replication units) increases (yeast - 13,5х10

^{6}base pairs (b.р.) drosophila - 165хl0

^{6}b.р., human - 2,9хl0

^{9}b.р.). In particular, in the haploid genome of mammals there are about 20 000 - 30 000 replicons, in D. melanogaster - 3500 and in S.cerevisiae yeast there are about 500 replicons. With the increase of replicons´ number or, the same, triangles

**АВС**

**(А**

**´В**

**´С**

**)**formed bу the intersection of the parallel lines

**а**and

**b**(Fig.1 and 3) the number of point mutations would have to grow.

However,
by the example of mitosis, we see that every chromosome, for example, in а bacterial cell, remains constantly
connected with the spindle fibers for about 20-30 min and forms 4х10^{9} cells in optimum conditions
less than in 11 hours. Then, under the condition that а normal chromosome disjunction to daughter
cells is broken not more than оnе time per 1000
mitoses [5-7], the number of mutations will make about 10^{7}. This number
is comparable to about 10^{7} mutant cells, which can bе formed out of 10^{13} human
cells at аnу time.

Thus,
from the abovementioned material оnе can conclude
that the key element of the evolution of all living things оn the Earth is the development of
homogeneous and isotropic geometric space in оnе of the planes of the tree-dimensional system **dx-dy**; **dy-dz **and
**dz-dx** coordinates [1] in the form of equally altered right triangle(-s)
(replicon) **АВС**
or **А****´В****´С**. Evidently, it explains that the nucleotides
coming out of the replication fork have discontinuities for а time and serve as the starting
point for the removal (reparation) of the erroneous regions and the development
of rigorously equal sequence nucleotides. In this case аnу chromatid can also become а matrix for the reconstruction of another
onе.

**Genealogy of formation of a homogeneous and isotropic geometric
structure**

Photophosphorylation, as well as oxidative
phosphorylation, is initialized by proton driving force, and consequently, for
its realization a closed space (compartment) is required. For example, light
stage of photosynthesis of O_{2} molecule in a spherical space of
quantum system is associated with C_{3} of plants with the formation of
3ATP and 2NADPH_{2} [13, 14].

In this case, the functional unit of
the light stage of photosynthesis is the square of the time interval (dL)^{2}
(6) in which limits the certainty equation is realized (1)

where Δt is the average time between the excited forms
of photosystems PII and PI and is equal to 20x10^{-9}s; Δt´ is the
average time of photon emission by almost all excited elements (atoms) and is equal
to 1.6x10^{-6}s. For Δt = 20ns the
probability of photons emission belonging to the same excited element is very
high. Index i = 1 and k = 80.

Formulas (1) and (6) allow concluding that in the course of structuring of the spherical geometric space in thylakoid membrane of the quantum system, complex molecular processes of photosynthesis are being formed in parallel as well.

Indeed, let us assume that from the
source of photons, placed in points (**х**^{1}**, х**** ^{2}**) and

**a**of the direct line

^{´}**NK**centre (Fig.4) crossing two parallel direct lines

**A**and

**B**, 680 and 700 nm photons are emitted respectively in the direction to the point (

**х**

^{1}**+dx**). The photons from the sources (

^{1}, x^{2}+dx^{2}**х**

^{1}**, х**

**and**

^{2}**a**) are emitted gradually with the interval of 20ns. For the period of 1÷80(20ns) the wavelengths (or the frequency) of photons

^{´}**λ**

**from the source (**

_{2}**х**

^{1}**, х**

**) and**

^{2}**λ**from

_{1}**a**will be in the following correlation:

^{´}

where **n** and **с** are the average displacement speeds
of photons from (**х**^{1}**, х**** ^{2}**) and

**a**, respectively, and

^{´}**n>>c**;

**∆**

**λ**

**=λ**

_{1}**-λ**

**.**

_{2}
Then for the time of 1.6x10^{-6}s
of the light-stage of photosynthesis O_{2} from H_{2}O, when
based on equation (6) conditions **∆t=∆t ^{1}**, the difference of
angular phases of wavelengths or the oscillation frequency become coherent (8)

In this case, the sensor, located
close to the point (**х**^{1}**+dx ^{1},
x^{2}+dx^{2}**) will register the interference of waves from the sources (

**a**and

^{´}**x**). By summation of similar indexes

^{1}, x^{2}**i**and

**k**(

**i = t; k = 2,3,4,.......80**) from (6) we define the structure of the shortest distance from the point (

**x**) to (

^{1}, x^{2}**х**

^{1}**+dx**) in the form of the matrix tensor

^{1}, x^{2}+dx^{2}**δ**in the new geometry [6]

_{ik}

in which, in contrast to the Lobachevsky´s and
Riemann´s geometry, **δ _{ik}** is the measure of homogeneity and
isotropy of the geometric space. In heterogeneous and non-isotropic environment
it is impossible to draw a direct line through the above mentioned points [2,
6].

From the center a^{´} of the
distance **(dx ^{1})^{2}+(dx^{2})^{2}**, it
is possible to describe a circle with the radius of

**dl[a**, which in crossing with two parallel straight lines

^{´}; (x^{1}+dx^{1}; x^{2}+dx^{2})]**A**and

**B**forms a spherical homogeneous and isotropic geometric space

**b**with them with the equally distorted internal angles, the sum of which is less than two direct lines. The proposed by us model serves as another proof [6] of Euclid´s assertion about concurrence of direct parallel lines as the theorem, which has been formed back in the III century

^{´}c´ d´ e´**BC**.

**A new insight in the mechanisms of energy synthesizing systems
functioning**

Nobel lecture by P.Mitchell delivered in 1979 was a triumph of the difficult creation of the chemosmotic theory of energy synthesizing systems functioning [15]. Currently, this discovery has gained general recognition [16].

However, this concept is extremely schematic as the physical nature itself of photo- and oxidative phosphorylation has not been determined. Consequently, the intrinsic logic of this natural phenomenon manifestation remained unclear. Below, we have presented the following arguments as the evidence of the physical nature of energy synthesizing systems functioning.

*1. Emergence of electrochemical and membrane potentials difference*

Photosynthetic electron transfer and
photophosphorylation in chloroplasts is similar to the electrons transfer and
oxidative phosphorylation mitochondria [13]. According to P. Mitchell, the
electrons transfer and ATP synthesis is ensured by the protonic gradient.
Electron transfer along the respiratory chain leads to the emission of protons
from the matrix to the cytoplasmic side of the inner mitochondrial membrane. As
a result of the occurring growth in the concentration of ions Н^{+} there occurs the generation of the membrane
potential with a positive charge on the cytoplasmic side of the membrane. This
is the proton driving force that initiates the synthesis of ATP by ATP-ase
complex.

In its turn, the flow of electrons through the electron-transport chain from photosystem PII to photosystem PI leads to the occurrence of a proton gradient leading to the synthesis of ATP.

Thus, we can conclude from the chemosmotic
theory that the mechanism of energy synthesizing systems functioning in
mitochondria, chloroplasts and bacterial cells in general is the same, and the
general electrochemical potential ∆р is formed of the membrane potential (∆ψ) and the gradient of ions
concentration Н^{+}(∆рН)

where R is a gas constant, T is the absolute temperature and F is Faraday number. In (10) ∆р=0.224V corresponds to free energy 21.76 kJ/mole per 1 mole of protons [13, 16].

However, on the basis of (10) it is not possible to find out the reason of distribution of electric charges (∆ψ и ∆рН) on both sides of the membrane.

At the same time, the equation of certainty
(1) just allows to disclose the physical nature of emergence of both the
membrane potential (∆ψ) and the gradient of ions concentration of Н^{+ }(∆рН).

Let us assume the charge separation
as the criterion of ATP synthesis in chloroplasts, photosynthetic bacteria and
mitochondria. Then, taking into account that for ∆t = 8(20ns) out of eight
photons only 6γ_{II} will split into three electron-positron pairs, and
2γ_{II} - into an electron, the equation of certainty (1) can be
written in general terms as follows (11)

where the difference of charges ∆рН = 1,348 units, which coincides with the experimental data [13, 16].

In the works [3, 5] we have shown
that at the concentrations of ATP, ADP and Pi equal to 40, 0.93 and 8.05 mM respectively and рН values of 7.0 and the temperature
of 25^{0}С, the
true free energy of the substrate phosphorylation in cells (∆G) is equal to

where ∆G^{01} is standard free energy.

In standard thermodynamic conditions out of 51.9 kJ/mole of energy 30.5 kJ/mole are required for the synthesis of one molecule of ATP from ADP and Pi [13]. Then the difference of 21.4 kJ/mole is the equivalent of the proton driving force equal to 0.224 V [13, 17], as in (10).

On the other hand, the difference of
21.4 kJ is free energy of electron transfer (charge *z*) through the
membrane of energy synthesizing systems. If for the transfer of one electron
(charge) through the membrane with transmembrane potential ∆ψ=10 mV it is required *z*F∆ψ=0.965 kJ/mole [17], then the energy
of membrane potential in (11) is equal to ∆ψ = 0.222V, which conforms well the literary
data [13, 15, 16].

Summarizing the equations (10) and (11), and taking into account that 4е are formed [13] during the decay of 8γ

where ∆pH =[(0.826е^{-} + 2.174е^{+})+е]=4, which indicates that in conditions when in the process of synthesis
and decomposition of ATP the concentration of ATP will be equal to the
concentration of ADP, the potential on the membrane will be ∆ψ = 0. In this case, the
synthesis of ATP can be done at the expense of the difference in protons concentration
on both sides of the membrane equal to ∆рН=4[17]. In order to transfer 2Н^{+} through the membrane the following potential
is required

which also conforms well the literary data [13, 16].

In particular, purple photosynthetic
bacterium does not produce oxygen, and instead of chlorophyll a and b in chloroplasts
it contains bacteriochlorophyll. While absorbing a quantum of energy,
bacteriochlorophyll transfers into the excited state and further rapidly
transmits an electron through acceptor chains А_{1}, А_{2}, А_{3} ......А_{i} at a speed of about 10^{-11}. This
electron, while moving in the secondary acceptor, initiates the cascade of
events of "apochlorotic photosynthesis", for the full completion of
which several seconds are required. Reduced speed of electron transfer to A_{i}
leads to annihilation of the electron and the positron. The released energy
leads to the emergence of the concentration gradient of ions Н^{+} on both sides of the membrane that underlies
the functioning of bacteriorhodopsin as the light dependent proton pump.

Vice-versa, in thermodynamic physiologically significant [17] equilibrium conditions (the ratio of ATP/ADP<1) the gradient of protons on both sides of the membrane is absent. For example, in chloroplasts (∆pH = 0) the value of membrane potential reaches the value of (15)

At the same time, the functioning of
mitochondrial organelle is formed of the membrane potential (∆ψ) and the
gradient of ions concentration Н^{+}(∆рН). Provided that the synthesis of
ATP can be done at the expense of the difference of the protons concentration
on both sides of the membrane of energy synthesizing systems, the units formed
by the difference of charges of the electron and positron-ion 1.348 in (13), can be
written as in (16)

where from ∆ψ=0.1411V, which corresponds well to the chemosmotic theory of P. Mitchell [13, 15]. It was known that pH in the outer side is 1.4 units is lower than from the inner side, and the membrane potential is equal to 0.14 V.

*2. Convergence of 4e and 4Н*^{+}* during O _{2 }restoration and H_{2}O decomposition into
4e and 4Н*

^{+}
In literature, the mechanism of
"amazing" convergence in the reaction co-ordinate of 4e and 4Н^{+} for the restoration of oxygen during
associated oxidative phosphorylation (17).

as well as during water decomposition at photosynthesis (18)

has not been established [13, 15, 16].

A.A. Logunov in work [18] showed that in Lobachevskiy´s, Riemann´s geometry, or in any other, the space-and-time is not homogeneous and not isotropic. In such space-and-time, the collisional frequency of interacting particles (like the frequency of errors at simultaneous measuring of the position and the speed of particles or the energy of the system at a given time) will be described by Heisenberg´s principle of uncertainty:

However, in the geometry proposed by R. Lohov [6], with the transition from a relatively large and, possibly distorted, three-dimensional space into the infinitely small volume of a geometric structure the space-and-time becomes homogeneous and isotropic [2, 6]. Accordingly, in such a space the principle of uncertainty is transformed into the principle of certainty (20)

that underlies the evolutionary development of living and plant organisms.

Equation (20) suggests that the
squared distance between the neighbour points (x^{1}, x^{2})
and (x^{1} + dx^{1}, x^{2} + dx^{2}) shall be
written in the form of a straight line, in contrast to Lobachevsky´s and
Riemann´s geometry (Chapter III):

where **δ _{ik} **is the matrix
tensor, measure of homogeneity and isotropy.

Thus, the key element in the evolution of all living bodies on the Earth is the formation in vivo homogeneous and isotropic geometric space on one of the planes of three-dimensional coordinates.

Apparently, the formation of mutations in the interval of replication initiating and cell division (mitosis) in accordance with the principle of uncertainty serves as the starting point for the initiation of the mechanisms of reparation (removal) of erroneous heterogeneous and non-isotropic sections of DNA (RNA) molecules and the formation of nucleic acids strictly in accordance with the certainty principle in restructuring and functional organization.

**References**

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6. Lokhov, R.Ye. Deduction of Euclid´s statement about parallel lines intersection within the interval of replication initiation and cytokinesis. International Journal of Experimental Education. №3, 81-85 (2008).

7. Euclid. Beginnings, III Century B.C., Book IX, Theorem 20.

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11.
Watson, J.D., Crick, F.A.C. Molecular Structure of
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12.
Watson, J.D.,
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Stryer, L. Biochemistry (4^{th}
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16. Michal, G. Biochemical Pathways. An Atlas of Biochemistry and Molecular Biology. (John Wiley & sons, New-York, NY, USA, 1999).

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