(Note that, more generally, one could deduce the functional form Wint=���Ҧ�ij(1)uij(2)+�¡Ҧ�ii(1)ujj(2) despite for the effective elastic interaction from symmetry considerations. Using this form instead of Eq. 5 would give qualitatively similar results.) Communication via elastic interactions has been experimentally observed between spatially separated, contractile cells (30,31). Below, we discuss how such elastic interactions naturally arise also between contractile striated fibers and guide their relative sliding. Elastic interactions between neighboring fibers can favor smectic ordering We now apply the arguments of the preceding section to the particular case of neighboring striated fibers and their elastic interactions. In a simple, idealized geometry, we consider two infinite fibers, which are parallel and separated by a lateral distance d (see Fig.

2 B). As outlined above (see Eq. 1), we can model the force dipole densities associated with the two fibers as ��(1)ij = ��(1)(x) ��(y) ��ix��jx and ��(2)ij = ��(2)(x) ��(y ? d) ��ix��jx, where ��(1)(x) = ��0 + ��1 cos(2��x/a) and ��(2) (x) = ��0 + ��1 cos(2��(x + ��x))/a, respectively. Note that there is a phase shift ��x between the periodic structures of the two fibers. We insert the specific strain field induced by a single striated fiber, Eq. 2, into the general formula for elastic interactions, Eq. 5, and thus find the elastic interaction energy between the two fibers (per minisarcomere) as a function of the phase shift ��x, Winteraction=��(d/a,��)��12aEmcos(2��x/a). (6) Here W = ��12/(aEm) sets a typical energy of the elastic interactions.

In the Appendix, we estimate the order of magnitude of the interaction energy as W ~ 1aJ �� 250 kBT. The factor ��(d/a, ��) was introduced below Eq. 2 and characterizes the lateral propagation of strain away from the centerline of a fiber. The sign of the propagation factor �� determines whether a configuration with zero phase shift between the two striated fibers is favorable or not: registry of fibers with ��x = 0 is favored for �� < 0. Fig. 3 shows this prefactor as a function of lateral distance d for different values of the Poisson ratio ��. For highly compressible substrates with Poisson ratio �� = 0, we find that the elastic interaction energy is always positive if the two neighboring fibers are in-registry, W(��x = 0) > 0, and negative if they are out-of-registry, W(��x = a/2) < 0.

This implies that elastic interactions would actually disfavor a configuration where striated fibers are in registry if cells were plated on a highly compressible substrate. However, for incompressible substrates with Poisson ratio close to �� = 1/2, such as those used in experiments GSK-3 (13,32), we find that the sign of the prefactor �� of the elastic interaction energy is negative provided the lateral fiber spacing is larger than some threshold d/a > d/a �� 0.247.