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Cell as elastic rod behavior model is proposed to describe its contractile activity. The model takes into account the result of the transduction of external influences, which is resulting in the formation of internal deformation, and evaluates the mobility and/or the tension in the muscle cells under the external influence.

Mechanical properties of both living and nonliving objects appear in its reaction against external action, mainly mechanical. Living cell like every system in external mechanical field stays in tense. External physical signals transformation results in corresponding cell response. Consequently, changes in external field lead to mechanical tension change in cell and deformations arise.

Interaction between cell and external mechanical field still remains one of the modern cell biophysics unsolved problems, due to the fact that determination of cell mechanosensor is an extremely difficult task. At the same time the need to solve this problem is necessary for different tissue cells protection methods development under changes of the external mechanical conditions, for example, during spaceflight, especially long-term spaceflight.

Special interest in mechanical properties study represent cells can change their own mechanical parameters in response to external influence by contractile-relaxation cycle initiation specific to muscle cells-myocytes. We supposed earlier the mechanosensor, being the most general for different cell types, to be connected to submembrane cytoskeleton [

On the other hand, muscle cells have specific structure, advanced cytoskeleton, which takes most of cell volume and forms a contractile apparatus. Taking into consideration these muscle cells features and contractile apparatus important role in mechanical tension generation, muscle cell mechanosensor may be connected with its contractile apparatus, for example with M-line [

At present time the most powerful tool for different organisms functioning theoretical research is mathematical modeling in terms of continuum mechanics. With the help of equations and relations from this theory one can formulate a closed system of equations. Their solution allows us to study deformable media behavior and to obtain information about its state and motion.

Let’s consider a cell like a structure with continuously distributed mass. To describe processes inside we use a system of equation for Cosserat’s continuum [

where

Cauchy boundary conditions are given by:

Let’s transform the system above according to biomechanic medium specific features.

The system given depicts both continuous medium static and dynamic behavior, as it takes into consideration inertia parameters: mass, eccentricity, inertia tensor. This values describes mass distribution in the system. Eccentricity vector

An external bulk moment

One should note that direct physical influence on biological structures are rather light. Mostly their action is mediated by internal signaling systems. For this reason, to simplify calculations we can take

In this case, the balance equation form is significantly simplified:

Despite a small magnitude of external influence on cell biomechanical parameters, mechanical signal transduction results in significant intracellular changes and forming an internal deformation (intracellular compartments structural changes). These deformations in turn generate motion, contractile activity. For this influence to be stated completely, we need knowledge about every mechanical parameter regulation pathways inside a cell. Data collected at present time are insufficient for this task. Nevertheless there are models depicted a mobility generation kinetic rather profound. Their essence is a chemical reaction kinetic of subcellular structures interaction analysis.

In this paper, during cell biomechanics consideration we only postulate a final result of all chemical interactions aimed at motion generation. Obviously, this result is some internal deformation causes internal tension arise and subcellular structures mutual displacement change. Parameters of “internal deformation” (for its ma-

thematical designation we will use tensor

In this case the free energy becomes a function of not only known deformation tensor and temperature, but

also of tensor

Complexity of explicit determination internal deformations tensor

Without loss of generality we can take a reference configuration strainless, i.e.

So, the cell biomechanics closed system of equations with Cauchy boundary conditions is given by:

where

specifying bending in 3 different planes, since there are no external moments inducing torsion),

Taking into account muscle cell structural features (see Introduction), one can consider a myocyte as a thin long cylindrical body. This assumption allows us to apply special methods of rods mechanic to describe the cell.

Cell biomechanics system of equation may be significantly simplified, in the same way we’ve done it with such system for rods, basing on continuum mechanics. We will consider motion in regard to stationary Cartesian coordinate system (OXYZ). For angle orientation being established we associate orthogonal trihedron

For example, radius-vector may be given by:

where:

At the same time vector

In mechanics conventionally transversal shifts is concerned in cases, when we consider a short thick rod. Considerable cell structures are thin long bodies, for this reason we can neglect transversal shifts. Moreover, muscle filaments motions are aligned, in the case of transversal shift a myofibrilla and sole myocyte structure would be broken and they won’t function.

All mentioned above leads us to some modification of geometrical relations. We will consider a Kirchhoff model, where stretching and compression are allowed, but transversal shift is forbidden. Vector

Consequently:

where

According with

forms from

Crosstalk tensor

Problem setting up for muscle cell derived from such formulation in cell biomechanics is given by:

where

Initial and boundary conditions are:

Or, if we fix one end:

Thus we have a muscle cell spatial motion description problem formulation as a result of contractile activity.

For the problem stated to be solved we’ll use a variational procedure, formulating the problem statement as variational principle in a following way [

where

Let’s take into consideration forces and moments expressions, variation formulas, initial and boundary conditions obtained above. The problem formulation will be given by:

where

Rotation tensor

quently,

are connected by constraint equation

or

where n―degrees of freedom number;

We have:

where

in reference configuration,

The

We will calculate displacement as:

For muscle cells we have correspondingly:

where

Taking into account preconceived idea about motion, we will take

In the case when stiffness coefficient depends on both coordinate and time, equation solving seems to be difficult.

Let’s assume the stiffness coefficient b does not depend on time. Then we can express

where g(t):

Solution

Then:

where

So, for muscle cell we obtain:

where

Mechanosensitivity problem still remains to be one of the least studied. Most complicated case is one that deals with gravity change, since its magnitude, proportional to cell mass, is extremely small. Even more complex situation is possible when gravity magnitude is constant, but its direction varies. However for cells inside tissue such gravity vector change induces a number of nerve activity level change processes (for soleus muscle) or hydrostatic pressure redistribution (for cardiomyocytes). For this reason cell mechanosensor determination is extremely difficult task.

Extracellular matrix, membrane proteins, ion-channels components, cytoskeletal structures, intracellular structures could be considered as mechanosensors. It was shown that the strengthened force applying to neuron or smooth muscle cell culture via extracellular matrix results in microtubes polymerization increase [

One of the most useful ways for on-cell influence significance estimation is mathematical modeling. In this paper we propose a living cell as biomechanical medium mathematical model, built on Cosserat’s system of equations. Such model allows us to evaluate cell response to both direct and transducted external mechanical conditions change.

Since muscle cells have specific structure we suggest developing such approach for all filamentic objects. Most of earlier crated models pay attention primarily on muscle cell internal processes kinetic, inducing contractile mechanism. In our method, we focused on cell level mechanics. At the same time internal deformations may be given by kinetic models. Obtained solution allows us to find given internal deformation

The financial support of the Russian Fond of the Basic Research (RFBR grant 13-04-00755-a) and Program of Presidium of Russian Academy of Sciences “Molecular and Cell Biology” is greatly acknowledged.