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Two different simple cases of plane tensegrity cytoskeleton geometries are presented and investigated in terms of stability. The tensegrity frames are used to model adherent cell cytoskeletal behaviour under the application of plane substrate stretching and describe thoroughly the experimentally observed reorientation phenomenon. Both models comprise two elastic bars (microtubules), four elastic strings (actin filaments) and are attached on an elastic substrate. In the absence of external loading shape stability of the cytoskeleton is dominated by its prestress. Upon application of external loading, the cytoskeleton is reorganized in a new direction such that its total potential energy is rendered a global minimum. Considering linear constitutive relations, yet large deformations, it is revealed that the reorientation phenomenon can be successfully treated as a problem of ma- thematical stability. It is found that apart from the magnitude of contractile prestress and the magnitude of extracellular stretching, the reorientation is strongly shape–dependent as well. Numerical applications not only justify laboratory data reported in literature but such experimental evidence as the concurrent appearance of two distinct and symmetric directions of orientation, indicating the cellular coexistence of phases phenomenon, are clearly detected and incorporated in the proposed mathematical treatment.

Active adherent cells alter their orientation direction, defined by their long axis, in response to substrate stretching. In absence of external strain field cells appear with random orientation; yet, the application of extracellular strain results to a concerted reorganization of the components of the cytoskeleton (CSK) in the new direction. The CSK is the intracellular network that consists of different types of biopolymers such as actin and intermediate microfilaments, microtubules, myosin and other filaments, and acts concurrently as a supporting frame and chief regulator of cell deformability. Constant remodelling of the CSK directly affects almost all functions of living cells like growth, differentiation, mitosis, apoptosis, motility, cell locomotion, etc. ([

In the present study the cellular orientation phenomenon is addressed by employing two simple mechanical models belonging to a family of structural systems known as tensegrity. Tensegrities are reticulated structures forming a highly geometric combination of bars and strings in space. In fact, tensegrity is a portmanteau word for “tension-integrity” referring to the integrity of structures as being based in a synergy between balanced continuous tension (elastic strings) and discontinuous compression (elastic bars) components. Pre–existing tensile stress in the string members, termed prestress, is required even before the application of any external loading in order to maintain structural stability. There already exists extensive literature regarding the advanced mathematics and mechanics used for the integral description of these structures [16-20], as well as the successful identification of the principles of tensegrity architecture to cytoskeletal biomechanics [21-24]. In fact, some of the characteristic mechanical properties of the CSK were initially predicted by the cellular tensegrity model and were later verified in laboratory experiments as such [21,24].

Since reorganization of the CSK is observed at high extracellular strains (of the order 10%–110%), Finite Elasticity principles and methods will be followed. Adopting Maxwell’s convention for stability [25,26], two different planar tensegrity CSK geometries are introduced and the stability of their orientation directions under the application of biaxial substrate stretching is studied. As it is the case for their biological counterparts, under the effect of stretching, the planar tensegrity models are considered to deform and reorient in a new direction. In concert to Maxwell’s convention, it is assumed that this new orientation direction, out of all the available ones, renders the total potential energy function of the given tensegrity CSK model a global minimum. Recently, the same analytical methodology presented in this study was used to theoretically investigate the problem of stress fibre reorientation under both static and cyclic substrate stretching [27-29]. The current treatment is an extension of the previous work to the cellular scale, and enhances further the former effort that had not focused on the intracellular microstructure but considered the cell as a generalized Mooney–Rivlin elastic material [

For the theoretical description of the reorientation phenomenon the CSK is modelled by two independent planar tensegrity frames of rectangular and rhombic shaperespectively (

In absence of extracellular strain field the shape stability of the models is controlled by prestrain. Next, biaxial substrate stretching is applied in the directions defined by the angles and (), cf.

The initial configuration of the rectangular tensegrity CSK model (henceforth simply rectangular model) is illustrated in _{1}, i = 5, 6 to the strings (BC) και (DA) with initial length b_{1}, and taking advantage of the model geometry, the initial deformation of every structural member due to prestrain is described by the deformation gradient tensors, in the fashion:

where, , is the initial homogeneous displacement and its gradient (prestrain) along the string members, respectively, while the identity matrix 1 in Eq.1 expresses the fact that at the initial configuration the bars are considered to be undeformed.

The deformation gradient of the biaxial substrate stretching with reference to the axis of the maximum and minimum extracellular normal strain, respectively (

where, , with, are the displacement gradients along the directions and, and:

is the rotation matrix of the coordinate system by the angle. The superscript denotes the transpose matrix. The total deformation gradient of every member of the rectangular model as a result of the superposition of prestrain and biaxial substrate stretching is expressed through the product of the respective deformation gradient tensors as: for.

The model responds to the increase of the substrate strain field by altering its initial configuration. Consequently, the reference placement has changed; in fact, it has been rotated through an angle (

for, where is the stretch, is the right Cauchy–Green tensor [33,34], and:

is the unit vector along the direction of each member in the new (reoriented) configuration. Specifically, defining the angle, see

The initial configuration of the rhombic tensegrity CSK model (henceforth simply rhombic model) is illustrated in

,(13)

with,

where is the initial homogeneous displacement gradient (prestrain) along the strings, is the identity matrix expressing, again, the fact that at the initial configuration the bars are considered to be undeformed, and is the rotation matrix with explicit form:

Defining the angle, see

respectively.

The deformation gradient of the biaxial substrate stretching is provided again from Eq.4. The total deformation gradient of the members of the rhombic model, as a result of prestrain and the strain field of the substrate, is given through the product of the respective deformation gradient tensors as: with. In response to the increased strain the initial configuration of the model is changed to the new direction. In the direction the deformation gradient tensor of each member of the rhombic model due to prestrain is given as:. Hence, the total deformation in the direction due to prestrain and the superimposed biaxial substrate stretching is expressed through the tensors:. Now, the displace-

ment gradient along each member of the model is given from Eqs.6 and 7, after the apparent interchange between indexes and. Specifically, for the unit vector along the direction of each member in the new configuration (

For the reasons of physiological compatibility mentioned in Section 2, it is assumed for both model geometries that the constitutive equations of all members are linear. Thus, the strain energy density per unit length function for each member may be written as:

with,

where is the Young modulus, is the cross–sectional area, and:

is the nonlinear Lagrangian strain. Again for the physiological reasons of Section 2—and recalling that index i corresponds to the rectangular model, whereas index j to the rhombic—it is considered that for to 6, for (bars), and for to 6 (strings). With no harm of the generality, in what follows it is assumed that and. Evidently, the strain energy density, through the Lagrangian strain of Eq.20, may be expressed as a function of the displacement gradient. Then, the total potential energy of each model is written as:

where is the first Piola–Kirchhoff stress along each member, and is its natural (unstressed) length. In the case of the rectangular model, from the geometry of the initial configuration,

,(22) ,

whereas for the rhombic model, from

,(23) ,.

The combination of Eqs.8 to 11, 19 for, and Eq.22 yields the analytical expression of the total potential energy density function of the rectangular model. Similarly, the combination of Eqs.15 to 18, 19 for, and Eq.23 yields the analytical expression of the total potential energy density function of the rhombic model. The explicit form of both functions is not given here due to their large representations; albeit, in compact form they are expressed as:

where, , for the rectangular model, and, , for the rhombic model. Furthermore, it should be stressed that after some elementary algebraic manipulation and factorization, the potential energy functions show a very strong dependence on the ratio rather than on the individual lengths themselves. It is straightforward that the value of this ratio directly controls the shape geometry of the two plane tensegrity models. Assuming that, as already implied in

Finally, the stable equilibrium directions are detected from the minimization of the total potential energy of both models, i.e., when the following two conditions are met simultaneously:

(a) and (b) (25)

It is evident that there exists a number of equilibrium directions for the plane tensegrity models that emerge as solutions to Eq.25(a); among them, the ones that additionally satisfy Eq.25(b) are stable. The investigation of the stability of the equilibrium requires the definition and adoption of the appropriate explicit stability criteria for the given problem. The orientation of active adherent cells, or even of stress fibres within them, is experimentally established to be one of the natural phenomena that exhibit coexistence of phases. Here, coexistence of phases is translated to the emergence of two distinct and concurrent orientation directions in a two–dimensional culture of cells or stress fibres, under the application of the same substrate stretching [3,5]. This kind of behaviour can be met at many different fields of the physical sciences, e.g., the melting of ice, or the coexistence of crystallographic systems in solids [35,36]. In order that our present framework is compatible to this behaviour, we adopt Maxwell’s convention for stability which allows coexistence of phases phenomena and declares that the system state is such that globally minimizes the potential [

For the integral study of the reorientation phenomenon, the mechanical response of both tensegrity frame geometries will be examined for a broad range of the problem parameters. The general outline is sketched through numerical inspection of the stability of the equilibrium solutions of Eqs.25. To this end, a series of graphs is produced that illustrate the variation of the total potential energy density function of each model with respect to the orientation direction. Specifically, for a given set of the parameters (, , ,) the potential energy is plotted for selected, increasing values of the ER until its maximum value equal to unity. In this fashion, the differences in mechanical behaviour between the two shape geometries, and between more or less elongated frames of the same geometry, are both successfully detected. Moreover, comparison between graphs that correspond to the same ER value, yet different substrate stretching or prestrain conditions, allows the interpretation of the individual parameter effect on the stability of the orientation direction and on the control of the phenomenon in general. Finally, in this way, the direct cross–examination between experimental data and theoretical predictions is also possible.

The graphs comprising

The graphs of the first (reference) column, Figures 4(1) to 4(8), have been produced for the set of parameter values:, , , , and outline the general behaviour of the rectangular model under biaxial substrate stretch with respect to the geometry of the frame shape. From this sequence it is deduced that for elongated rectangular frames (low ER values), relatively low prestrain and minimum stretch component values, only one global minimum exists at the direction; that is, the rectangular model is reoriented and aligned with the direction of the maximum substrate stretch, see Figures 4(1) to 4(3). For intermediate ER values, i.e., moderately elongated rectangular frames, the stability character of this direction is maintained (Figures 4(4) and 4(5)). However, as ER tends to its limiting value equal to one, the solution is destabilized and the emergence of two new globally stable solutions is evident (Figures 4(6) to 4(8)). The direction of maximum substrate stretch, , evolves initially to a local, and finally to a global maximum at the square configuration of the frame. Moreover, the two new, globally stable equilibrium directions located at are obviously symmetric with respect to the direction of the maximum substrate stretch.

The second column graphs of

The third column graphs of

The same conclusions hold for the case where prestrain is the parameter increased with respect to the first column value set. For, and all the other parameter values the same as in the reference column, the fourth column graphs are produced. Comparing Figures 4(6), 4(14), 4(22), and 4(30) (that is, the sixth row of

It should be pointed out that the four parameters sets reported above and used to construct the graphs of

(ER = 1), where the rectangular frame evolves to square, there exist two globally stable orientations (), symmetrically arranged with respect to the direction of the maximum substrate stretch, again independently of the magnitude of the stretch or prestrain. The stability character for intermediate ER values is determined primarily by the magnitude of the maximum substrate stretch component. Initially, the direction of the maxi-

mum stretch is globally stable, but as the value of increases, the aforementioned direction destabilizes and the new, symmetric globally stable minima emerge. On the other hand, increase of the minimum stretch or prestrain with respect to, either constant or increasing, maximum stretch, delays the destabilization and the appearance of the symmetric global minima until relatively higher ER values only when the values of and are comparable to the value of. In any case, the increase of, , values does not prevent the destabilization from happening, and their effect is observed to diminish for high values. Hence, as it is concluded by the preceding analysis of the rectangular model, its stability is strongly shape dependent and it is affected by other agents as well, of which the magnitude of the maximum substrate stretch is the primary one.

The graphs comprising

The graphs of the first (reference) column,

Doubling the value of the maximum displacement gradient of the substrate (), while keeping all the other parameter values the same as in the reference column, the second column graphs of

The exact opposite case holds when the parameter values that are increased with respect to the reference column correspond to the minimum displacement gradient of the substrate (third column,), or prestrain (fourth column,). Upon comparing Figures 5(5) with 5(21) and 5(29) it is concluded that increased values of the aforementioned parameters allow the stability transition of directions only for high ER values.

It is pointed out once again that the parameter sets reported above for the construction of the graphs of

maximum substrate stretch.

Summarizing the findings of the stability investigation of both plane tensegrity models in the context of Maxwell’s convention, the following can be said. The orientation direction is defined primarily by the shape geometry of the models (ER value), and secondarily the magnitude of the maximum substrate stretch. The similar contribution that present the minimum stretch and prestrain is rather negligible and is only detected when their values are comparable to the maximum stretch component. The rectangular model, for the greatest part of the ER value range, is aligned parallel to the direction of the maximum substrate stretch, while for the high ER or maximum stretch values it is symmetrically aligned away from the aforementioned direction. The rhombic model is oriented parallel to the direction of the maximum stretch for all ER values; albeit, in the liming configuration where it degenerates to a square (ER = 1), it can also align perpendicular to the direction. In terms of increasing ER values, the orientation of both models evolves from a single, globally stable orientation direction, to the appearance of multiple stable directions,

The discords in the described above orientation evolution of the two models may be attributed both to the different initial shape geometries, and the slightly different definitions of the ER. Although ER expresses essentially the same thing for both frames, it is evident that in the case of the rectangular model it is a ratio of length of elastic strings, whereas for the rhombic model it is a ratio of length of elastic bars,

For the low and intermediate ER values, that correspond to physiologically more compatible cellular configurations, the stability analysis of the two models predicts orientation parallel to the direction of the maximum substrate stretch. This response is consistent to the experimental observations of the behaviour of active adherent cells cultured on an elastic substrate under the effect of static or quasi–static stretching [6-9]. For low ER values (ER→0), it is reasonable to consider that the models now represent stress fibres, rather than the entire CSK [

Apart from this last discrepancy between theoretical predictions and experimental observations there appears to be another one related to the effect of prestrain. For both model geometries, increase of prestrain while keeping all the other parameter values constant seems to promote alignment parallel to the maximum stretch direction. Nevertheless, experimental observations for the case of cyclic substrate stretching report a different response [

A general integral stability analysis of two plane tensegrity frames used for the description of cellular reorientation under biaxial substrate stretching has been presented. Adopting Maxwell’s convention for stability, the analysis was performed in the context of Finite Elasticity Theory. It has been shown that the reorientation is controlled by such parameters as the magnitude of the extracellular stretch components, of which the maximum component is the primary one, and the strength of the intracellular contractile mechanism. This dependence is consistent to numerous well documented laboratory reports (see Refs. in Introduction). It is also in accord with previous theoretical modelling of the reorientation of cellular stress fibres as well [27,28]; yet, the present study focuses on the response of the entire intracellular biopolymer network, which is a substantially more difficult undertaking than studying individual, isolated stress fibres. The new finding, when working with a plane tensegrity CSK, is the governing role of the shape– geometry of the frame. The properties and general behaviour of the two different tensegrity frame geometries have been thoroughly analysed and discussed in the text.

Cellular orientation is a mechanochemical process involving the transduction of a mechanical stretching signal to changes in intracellular biochemistry (and vice versa), while the CSK exhibits dynamic, viscoelastic behaviour. Furthermore, it is certain that cellular architecture is far more complicated than any of its existing mechanical descriptions, and that tensegrity modelling is, rather, a simplification. For these reasons, the good agreement between experimental data and predictions, based on elastic stability considerations of the two, purely mechanical, plane tensegrity models, is quite remarkable. However, the models employed here are simple, physiologically compatible, and the analysis is based on fundamental principles; moreover, no far– fetched assumptions that could bias the results have been introduced. Taking all these into account, it is revealed that mechanical stability is a major determinant of cytoskeletal rearrangement, and that cellular orientation can be successfully treated as a problem of elastic stability. This conclusion does not–in any way–exclude other (chemical) agents that could influence the determination of the orientation direction. Albeit, the consistency of theoretical predictions to the experimental data suggests that the extent of the other agents effect is rather limited, or that their contribution is already integrated to the analysis through their mechanical expressions (e.g., prestrain).