Modeling of the Primary Acts of the Interaction between a Cell and an External Mechanical Field

The mechanism of interaction between a cell and an external mechanical field is still poorly understood, and the accumulated diverse experimental data are often scattered. Therefore, the aim of this work was to systematize the experimental data in a mathematical model of the interaction between a cell and an external mechanical field based on standard kinetic equations and Fick’s diffusion equation. Assuming that the cortical cytoskeleton proteins play a key role in cell mechanosensitivity, we compared the results of mathematical modeling and experimental data concerning the content of cytoskeletal proteins at the early stages of a mechanical field change. In addition, the proposed mathematical model suggests the dynamics of changes of a key transcription factor, which is necessary for the expression of certain genes encoding cytoskeletal proteins.


Introduction
Human exploration of outer space faces a number of unsolved problems, including medical problems. Being in conditions of weightlessness, even during Earth orbit, leads to a number of negative effects, for example, on the musculoskeletal and cardiovascular systems [1] [2] [3]. Existing methods of countermeasures, for the most part, are palliative, which is associated with a lack of un- [10], as well as intracellular structures [11] [12], can also act as a mechanosensor.
The result of mechanotransduction is the formation of an adaptation pattern of proteins and gene expression. Thus, in cultured cells under conditions of altered gravity, there was a change in the cell profile, disorganization of microfilaments and, sometimes, microtubules [13]- [18], and changes in mitochondrial localization [19], which is determined by the state of the intermediate filaments.
In addition, the changes are not limited to the protein content but also occur at the level of the expression of genes encoding cytoskeletal proteins and associated components of signaling cascades [20]- [26].
Our previous studies have suggested the role of the actin-binding proteins of the submembrane cytoskeleton in the primary mechanoreception of cells of various types, in particular skeletal muscle and myocardium. We assume that any change in the external conditions for the cell is reflected in the deformation of its cortical cytoskeleton. However, these strains are fundamentally different with increasing and decreasing loads. The first result is the dissociation of various actin-binding proteins from the cortical cytoskeleton: alpha-actinin-4 with a load decrease and alpha-actinin-1 with an increase [10] [27] [28]. With further development of this process, the deformation leads to the destruction of the structure and, at subsequent early stages of exposure, to an initial decrease in stiffness, which correlates with the content of actin non-muscle isoforms in the membrane fraction, which form the cortical cytoskeleton [29]. Furthermore, in the case of a decrease in external mechanical stress, there is a decrease in the expression of genes encoding cytoskeletal proteins and a further decrease in cortical cytoskeleton stiffness. In contrast, an increase in the external mechanical stress increases the mRNA content of the genes encoding cytoskeletal proteins and proteins directly and increases the stiffness [30] [31]. In general, the process of sensitivity to external stress by cells can be quite universal in the evolutionary series. Thus, Drosophila melanogaster lacks the isoform alpha-actinin-4; however, it is possible that supervillin plays a role in the process of mechanosensitivity [32].
Thus, the multiple components and variability of the mechanisms of cellular mechanosensitivity and mechanotransduction make it difficult to find the "hot

Formulation of the Problem
Based on the experimental results described above, it is possible to suggest the following mechanism for triggering the formation of an adaptive response to changes in the external mechanical stress.
Suppose that a sensitive protein SP associated with microfilaments reacts to any change in mechanical stress. In addition, its antagonist, aSP, is also associated with microfilaments. Protein SP can exist both in connection with microfilaments and in free form. The binding to the microfilament network and dissociation from microfilaments at the initial state occur to maintain the initial level of free protein Suppose also that the efficiency of proteolysis and degradation of mRNA does not depend on the content and type of substrates and the rates of cleavage are constantp v and d v for proteins and mRNA, respectively. In general, we assume that all reactions proceed at a constant rate, bearing in mind that the rate does not depend on the content of the substrate/reaction product. We introduce the following notation: Then, for the proposed mechanism, the standard kinetic dependencies taking into account the diffusion between the compartments, the efficiency of synthesis and degradation for the concentrations of the analyzed proteins and mRNA are: Similar to the previous system, for comparison with the experimental results and to determine the type of dependencies, we write the expressions for those parameters that can be determined (divided into compartments): Solving equations together, we obtain expressions for the content of various proteins. For the modifying protein and transcription factor: For "sensitive" protein: For the antagonist of "sensitive" protein: For microfilaments: For microtubules:

Simulation
In previous studies, we obtained systematic data concerning the contents of various cytoskeletal proteins in the membrane and cytoplasmic fractions of rat soleus muscle fibers [28] [29]. Therefore, for the simulation, we consider this type of cell under changes in the external mechanical field.
We consider the actin-binding proteins alpha-actinin-4 and alpha-actinin-1 as a "sensitive" protein and its antagonist, respectively, and beta-actin as a protein of microfilaments of the submembrane cytoskeleton because its content dominates over the content of gamma-actin in this cell type [28] (Figure 1(a)).
As a result of a change in the external mechanical stress, an adaptive pattern is formed: in the case of an increase, the cytoskeleton becomes more developed, in the case of a decrease, vice versa. Consider the option of decreasing external mechanical stress (Figure 1(b)).
We will follow the "sensitive" protein, its antagonist, and microfilaments and compare the results of the simulation with the experimental data.
Since the experiment evaluated the relative contents of proteins and mRNA as a whole in compartments, then mc z is the "path length" between the cortical cytoskeleton and the cytoplasm, and cn z is the "path length" between the cytoplasm and chromatin ( Figure 1).
We assume that for the fibers of the soleus muscle of rats: Following [33], using the Stokes-Einstein equation, we assume that the diffusion coefficient is: Beta-actin, considered the main protein of microfilaments of the cortical cytoskeleton in rat soleus muscle, has 375 amino acid residues. Then, we will assume that its hydrodynamic radius is 5.28 × 10 -9 m [33]. Assuming that SP and aSP are alpha-actinin-4 and alpha-actinin-1, having 911 and 892 amino acid residues, respectively, we will assume that for them, the hydrodynamic radii are the same and amount to 6.61 × 10 -9 m, based on the extrapolation proposed by [33]. The transcription factor remains unknown in the proposed mechanism, but since many parameters are not determined accurately but are estimated, for simplicity of calculations, we will consider the hydrodynamic radius of the transcription factor as a certain average value, and we will use 6.0 × 10 -9 m. Consequently, all the hydrodynamic radii necessary for calculations have close values, and we can assume that the diffusion coefficient has one order: where , rSPs raSPs k k and rMF k are recruitment coefficients of the transcriptional complex to DNA depending on the content of the activated transcription factor in the nucleus for alpha-actinin-4, alpha-actinin-1 and beta-actin, respectively.
We assume that the half-life of mRNA of genes encoding cytoskeletal proteins, as well as for globin, is approximately 8 hours [36]. We assume that on average, it is approximately 28,800 seconds. Therefore, the reaction rate constant for mRNA degradation in the cytoplasm is: The speed of ribosomes in eukaryotic cells is diverse, but we will assume that on average, including for cytoskeletal proteins, processing proceeds at a speed of 5 amino acid residues per second [37]. Then: for SP (alpha-actinin-4, 911aa)-( ) We assume that proteolysis is carried out using the proteasome. The rate of proteolysis depends on how long the protein has been synthesized but, on average, is 2.5 substrates/minute [38]. Let us assume that on average, for the analyzed proteins, the rate of proteolysis reaction of protein molecules in the cytoplasm is: to the free state when the mechanical stress changes. Without loss of generality, we will assume that the variables are independent and then: Considering gravity as a bulk force, we accept, as before [27], that: where ϕ is the orientation angle, in this case, the soleus muscle in the field of gravity, g is the acceleration of gravity, and ρ is the density of the cortical cy- is 30˚ [39].
We take the initial values of the estimated parameters for 100% and substitute (45) -(58) into (24) - (38). We evaluated the process of perception of a mechanical stimulus and its transduction at several points-6, 12, 18, 24 and 72 hours-to compare the results of mathematical modeling and experimental data obtained by us earlier [28].
The dependences obtained for the content of the "sensitive" proteinalpha-actinin-4 (Figure 2), the antagonist of the "sensitive" proteinalpha-actinin-1 (Figure 3), and microfilaments-beta-actin ( Figure 4) (Figure 4(c)). In addition, at the 12 o'clock point in the experiment, the beta-actin content in the membrane fraction is already reduced and amounts to 51% ± 4% of the control; in a numerical experiment, it does not differ from the control (103%) (Figure 4(b)).
The dynamics of the transcription factor change ( Figure 5) indicates an increase in its content in the nucleus after 6 hours by 35% and a subsequent increase up to 24 hours (235% relative to the control) and then a decrease after 72 hours compared to the maximum accumulation (up to 190%).        chanical stimulus. For mammals, we assume that these proteins can be two calcium-dependent alpha-actinin forms: alpha-actinin-1 and alpha-actinin-4 [10] [27] [28]. However, for example, in Drosophila, there is only one alpha-actinin isoform, but our previous data suggest that another actin-binding protein, supervillin, may be the second participant [32]. Therefore, in this work, we designated this pair of proteins as a sensitive protein SP and an antagonist of the sensitive protein aSP.

Discussion
In system biology, mathematical modeling is often used to estimate the values of unknown parameters. The model of population dynamics of Lotka-Volterra is used especially widely in various kinetic models, for example, when modeling the development of bacterial infection [40]. In this paper, the use of kinetic re-  [28]. In addition, a significant difference between the experimental data and those obtained in a numerical experiment occurred in the membrane fraction of microfilaments after 12 hours, as well as in the cytoplasmic fraction, after 72 hours. However, it should be noted that in the experiment, we evaluated the content of actin isoforms (beta and gamma-) separately, while in a numerical experiment, for simplicity, we followed the total content of proteins forming microfilaments, comparing the data with beta-actin. However, because the content of beta-actin in muscle cells substantially dominates the content of gamma-actin, this approach can be justified. In general, this model, developed for early acts of cellular mechanoreception, gives results that coincide with experimental data up to 72 hours of exposure.
In addition, it is known that one of the candidates for the role of the sensitive protein SP, alpha-actinin-4, can penetrate into the nucleus [41] and bind to the promoter regions of the genes. However, it is still unknown, in the case of changes in external mechanical stress, whether alpha-actinin-4 itself regulates the expression of its potential targets or indirectly regulates them through a transcription factor. Therefore, we introduced a transcription factor modifier into the model, which can be activated by SP, and, in turn, activate the corresponding transcription factor. In a numerical experiment, the dynamics of changes in the content of the transcription factor in the nucleus were estimated, and according to these results, we can assume that the maximum of its accumulation occurs at 24 hours, where its content exceeds the control level by almost 2.5 times. In addition, its content in the nuclei begins to increase after 6 hours of exposure.
Alpha-actinin-4, according to experimental data, has approximately the same initial content in the membrane and cytoplasmic compartments. After 6 hours of exposure in the membrane compartment, its content decreases by 27% and increases in the cytoplasm by 30% [28] [29]. In the numerical experiment, we obtained the same data. However, at the same time, after 6 hours of exposure, according to the simulation data, the content of the activated transcription factor in the nucleus increases by 35%. Comparing these data, we can assume that when the external mechanical stress changes, alpha-actinin-4 does not directly regulate the expression of the genes assessed but indirectly regulates them through the activation of an appropriate transcription factor.
Of course, such an assumption after mathematical modeling needs experimental verification, but it can be assumed that the disclosure of the detailed mechanism of interaction between the cell and the external mechanical field will help in the development of effective preventive measures that are necessary, for example, during deep space exploration.

Conclusions
The results obtained indicate that the model of perception of an external mechanical stimulus by living cells, based on a system of kinetic equations and the second Fick law, adequately describes the process, and the simulation results correlate with the experimental data.
The time dependencies estimated in a numerical experiment suggest that alpha-actinin-4 triggers a signaling cascade, leading to an increase in the content of certain transcription factors whose targets are both alpha-actinin-4 and alpha-actinin-1 and beta-actin. In other words, the answer to the question of whether alpha-actinin-4 regulates gene expression directly when mechanical stress changes (given its ability to penetrate into the nucleus and bind to some promoters) is as follows: according to a numerical experiment, it is more likely that another transcription factor will appear.