Stochastic Mechanism of Formation of Mutational Protein Nanoparticles and Mesoobjects

cluster sizes: Abstract We consider briefly a possible stochastic mechanism of formation of mutational protein nanoparticles and mesoobjects (cells and organelles). Results based on the uncertainty relation “coordinate – momentum” in the space of object (molecular cluster) sizes are indicative of the fact that in the system of amino acid molecules accidental formation of quasi-crystalline nanoparticles and mesoobjects corresponding in their sizes to essential proteins and cells is possible without of the preliminary formation of polypeptide nanochains. The developed kinetic approach and calculated results can be useful for laboratory investigation of a behavior of biological systems at


Introduction
One of topical trends of nanoscience and nanotechnology consists in the creation and the study of biological materials, in particular, the study of physical mechanisms of protein biosynthesis [1,2]. In accordance with the results of the investigations the commonly accepted scheme of proteins construction is presented in such a manner that a volumetric-packed nanoparticle that represents an aperiodic crystal, is formed from nanochains with the C-N peptide bonds (primary structures) as a result of the twisting and the mutual arrangement of various polypeptides in the presence of nucleic acid molecules. At the same time, in works [3,4], there is another scheme of arrangement of biological nanoparticles, in which as a result of rotational-vibrational interaction of amino acid molecules, their direct volumetric polycondensation can occur. The seeding centers of polycondensation can be the molecules of nucleic acids, around which amino acid molecules are grouped. They are grouped in a certain order given by preferable formation of C-N bonds as the most short and strong bonds as compared to C-C and C-O bonds. In the proposed way of considering the corresponding mechanism of protein nanoparticle synthesis, the principle of uncertainty assumes the possibility of mutation of biological objects at a molecular level. In this work we specify and extend model ideas about stochastic kinetics cluster mechanism of formation of mutational protein nanoparticles as well as mesoobjects -cells and organelles.

Methodology
We will describe the process of the irreversible aggregation of objects using the concept of distribution density wave ( ) , a t φ in the space of cluster sizes a . The wave propagates with the time t toward an increase in the cluster size. Such one-dimensional approach allows one not to take into account deviation of a geometric shape of the object from the ideal one. On the assumption of the above the universal relation 1 4 a k π ∆ ⋅ ∆ ≥ [5] ( k is wave number) for a wave packet half-width and a spectral line half-width for coherent processes one can write down the following uncertainty relation for a coordinate and a momentum in the space of cluster sizes: Here, is cluster mass, α is geometric factor, ρ is the density of cluster matter, 0 0 , m a are the mass and size of germ,  is reduced Planck constant. Momentum uncertainty is equal to the momentum itself in the order of magnitude, i.e., interaction of objects either occurs or does not occur. Physical meaning of relation (1) is in the fact that during the time interval t ∆ of elementary (single) act of interaction between objects the exact cluster size cannot be determined until this interaction is finished either through capture of one object by another or their partial or complete disruption, or elastic scattering. It is connected with the fact that unless the elementary act is finished, it is impossible to determine to which object each of the interacting surface elements is assigned. In ours case such elements are amino acid moleculesthe germs of biological objects. This means that uncertainty principle admits mutations of biological objects at a molecular level.
An evolution of distribution density function ( )  ( 2) (1 ) β λ for germ. Formula (3) indicates on the existence of discrete set of most probable cluster sizes: General growth law of mean size a has the following form: Here, , Κ Ζ are constants determined by the mode of cluster aggregation. At the first step we consider two limiting growth modes: 1) small flux of germs, when each germ has time to occupy its energy advantageous place on cluster surface before interaction with the following seed begins; 2) large flux of germs, when they affect the cluster almost simultaneously which is equivalent to cluster collision. In the first case i t parameter does not depend on a cluster size and the growth law can be written as is velocity of perturbation spread in cluster substance (sonic velocity). Then the law of growth can be written as Here cl un t is the time unit in the germs large flux mode (cluster collision).
In number of problems connected with investigations of high-intensive processes with high energy (for example, at high temperatures T of crystal objects) it is reasonable to determine the value of t ∆ time interval of elementary interaction act based on the "energy-time" uncertainty relation (Heisenberg rule): Here E ∆ is energy level width of an isolated excited state of a quantum mechanical system. Substituting expression (8) in relation (1) rewritten for the mean size, and taking into account that the mean size is changed with the time by uninterrupted manner, one can write the following differential relation: Here value is determined by nature of objects and a process mode and can be depends from the mean size a of the excited field. Having solved differential relation (9) in quadratures one can find approximate laws of mean size a growth with time in high-intensive processes of object aggregation. On the basis of the relation (1) and the condition of mass conservation ( ) In the case of hollow nanofiber (nanotube) the diameter is determined by analogy with single-wall carbon nanotube [6] with the help of following formula: Here e ω is a wave number of characteristic oscillations which for the given bonds is equal approximately to 2·10 5 m -1 [7]. From here we have i v t t ≡ ≈ 3.3·10 -14 s. Figure 1 presents the example of relative distribution density (3) and demonstrates the existence of discrete set of sizes of nanoparticles formed from germs. Set of calculations performed with the help of formulae (3), (4) for all mentioned germs at above time i r t t ≡ , corresponding to basic rotational level, allows one to single out the following typical neigh-borhood and "magic" sizes:  Let us consider the expressions for mean sizes of protein nanoparticles and mesoobjects. At Debye approximation one can rewrite the formulae (6), (7) by following manner: i.
for small flux of germs ii. for large flux of germs It has been supposed that Debye parameter D θ corresponds approximately to average characteristic temperature v θ ≈ 1500 K [9] of extension vibration of bonds C-C, C-O, C-N. Using expression (9) for high intensive processes one can write the growth laws by following manner: iii.
for small flux of germs s of conformational motion [2] envelop characteristic sizes [1,2] of protein nanoparticles from ≈2nm to ≈60nm. Tangent interaction between amino acid molecules at shear processes resulting from mechanical action to system leads to significant magnification of sizes of biological objects under consideration to 800nm at t ≈10 -7 s); and simultaneous excitation of the system at absorption of energy of emanation having different nature leads to formation of low density lipoproteins (≈20nm at s) and leucocytes (≈20µm at t ≈10 -7 s).
With the typical parameter [9] corresponding to rotational molecule isomerism (b is a number of crystal-forming bonds, R θ is the typical rotational bond temperature (≈2.6 K), T is ambient temperature), we derive from the formula (11) that for the germs like glycine and tryptophan at T ≅ 310 K, the diameters d of nanofibres are equal to 1.27nm and 2.35nm.
These values approximately correspond to the thicknesses of collagen protein (~1 nm [1]) and myosin protein (2.5nm [2]). If we assume that to form the nanofibre thickness, it is sufficient to have a single oscillation of a "last bond" of the longest molecule of tryptophan and i t ≈ 3·10 -14 s , then from the formula (11) we get d ≅ 12 m . This approximately corresponds to the thickness of a continuous neurofibril of a human, neurofilament, approximately equal to 10nm [10]. If we consider alanine molecule as the germ, then in case of vibrational interaction we get from the formula (12) d ≈ 20 nm. This approximately corresponds to the diameter of human hollow neurofibril, approximately equal to 20-25 nm [10]. Calculation shows that to form a neurofilament 300nm long, one requires approximately 100 μs. The calculated dependences (10) of maximal sizes of biological nanoparticles and mesoobjects from germ sizes at basic state and at first excited of rotatory and oscillating levels are given in Figure 2. When molecules move from the first rotary level (j = 1) to the second level (j = 2) or from the "zero" oscillating level (v = 0) to the first level (v = 1), the maximum sizes will increase a factor of 3, since the energy of rotations is quantized ( [7]). Formally this means that with rotary motions when germs are glycine molecules (a 0 = 0.42nm) a size of about 40nm corresponding to a ribosome of eukaryotes is reached. If germs are tryptophane molecules (a 0 = 0.67nm), a size of about 450 nm is reached which corresponds to nanoarcheot [11]. At oscillating motions in case of alanine as the germ (a 0 = 0.47nm) a size of 3µm is reached which corresponds to a thrombocyte [1]; in case of tryptophan seed a size of 20µm is reached. The latter means that in the system there can be formed leucocytes whose sizes are within the range from 8 to 15µm [1]. At the second level of oscillations (v = 2) sizes of leucocytes can be reached if germs are valine molecules (a 0 = 0.54nm). This fact does not contradict to the known medical facts of formation of mutations and tumors or development of atherosclerosis and leukemia under the effect of ray penetration into an organism.
Thereby, the results from the uncertainty relationship "coordinate-momentum" in the space of object sizes are indicative of the fact that in the system of amino acid molecules, accidental formation of quasi-crystalline nanoparticles and mesoobjects corresponding in their sizes to essential proteins and cells is possible. These "incorrect" (mutational) objects can grow on these or those crystallization centers without formation of polypeptide bonds, i.e., without formation of "correct" biological code. At the same time formation and growth of such nanoparticles and mesoobjects is possible on fragments of damaged proteins and cells as on the centers of crystallization. All this is in compliance with the generally known concepts concerning mutations of biological structures at a molecular level.
It is necessary to note that the above formulae deduced from the uncertainty relations in the space of object sizes correspond to condition of mass conservation during elementary act of interaction between a germ and an object. These formulae are not tied together with the condition of mass conservation in the whole system. A system can be unclosed in accordance with a general nature of biological processes [1,2,8]. Organic-silicon mesoobjects have been considered also in [12]. The basis of these meso objects is siloxane sceleton -the chain of alternate Si and O atoms. Elementary portion of such polymer chain consists of two adjacent atoms Si and attached atoms C, H, O. Replacement of some H atoms by N, P, S, and Fe atoms results in analogy with biological polypeptide nanochains [1,2]. Such analogy expands a range of questions related to the study of life as a nanoscale phenomenon [8]. Based on the well-known data [7] on atomic radii and the lengths of interatomic bonds, the characteristic sizes of globular organic-silicon nanocrystals have been evaluated in [12]. The obtained sizes 60nm and 4μm correspond to the sizes of biological equivalents (ribosomes) and to biological mesoobjects [1]. The issue of existence of such hypothetical biological structures, con-taining silicon, remains open. It looks reasonable to search for the mentioned structures in places, where rubber-bearing plants grow. The developed kinetic approach and calculated results can be useful for laboratory investigation of a behavior of biological systems at different environments.

Conclusion
The proposed approach for studying kinetics of formation of protein nanoparticles and mesoobjects from amino acid molecules reflects it stochastic nature. This circumstance is caused by the uncertainty principle which assumes the possibility of mutation of biological objects at a molecular level. The results from the uncertainty relationship "coordinate-momentum" in the space of object sizes are indicative of the fact that in the system of amino acid molecules, accidental formation of quasi-crystalline nanoparticles and mesoobjects corresponding in their sizes to essential proteins and cells is possible.