Effect of Poisson Ratio on Cellular Structure Formation

Abstract

Mechanically active cells in soft media act as force dipoles. The resulting elastic interactions are long-ranged and favor the formation of strings. We show analytically that due to screening, the effective interaction between strings decays exponentially, with a decay length determined only by geometry. Both for disordered and ordered arrangements of cells, we predict novel phase transitions from paraelastic to ferroelastic and anti-ferroelastic phases as a function of Poisson ratio.Predicting structure formation and phase behaviour from the microscopic interaction laws is a formidable task in statistical mechanics, especially if the interaction laws are long-ranged or anisotropic. In biological systems, the situation is further complicated because interacting components are active in the sense that informed by internal instructions (e.g. genetic programmes for cells) and fueled by energy reservoirs (e.g. ATP), they react to input signals in a complicated way, which usually does not follow from an energy functional. Therefore these systems are often described by stochastic equations [1,2]. One drawback of this approach is that typically the stochastic equations have to be analyzed by numerical rather than analytical methods. However, for specific systems structure formation of active particles can be predicted from extremum principles. In this case, analytical progress might become feasible again, in particular if analogies exist to classical systems of passive particles. One example of this kind might be hydrodynamic interactions of active particles like swimming bacteria [3]. Here this is demonstrated for another example, namely mechanically active cells interacting through their elastic environment [4,5].Our starting point is the observation that generation and propagation of elastic fields for active particles proceed in a similar way as they do for passive particles like defects in a host crystal, e.g. hydrogen in metal [6,7]. For a local force distribution in the absence of external fields, the overall force (monopole) applied to the elastic medium vanishes due to Newton's third law [8]. Therefore each particle is characterized in leading order of a multipolar expansion by a force dipole tensor P ij . For many situations of interest, including cells in soft media, this force dipole will be anisotropic and can be written as P ij = P n i n j , where n is the unit vector describing particle orientation and P is the force dipolar moment. The perturbation of the surrounding medium resulting from a force dipole P ′ kl positioned at r ′ is described by the strain tensor u ij ( r) = ∂ j ∂ ′ l G ik ( r, r ′ )P ′ kl , where summation over repeated indices is implied and G ij is the Green function for the given geometry, boundary conditions and material properties of the sample. The strain u ij generated by one particle causes a reaction of another particle leading to elastic interactions. The essential difference between active and passive particles is that cells respond to strain in an opposite way as do defects. ...

Collective effects in cellular structure formation mediated by compliant environments: A Monte Carlo study

“…This may reflect the competition between ring-and string-like domains which is modulated by the Poisson ratio. In fact, as we have shown before, at m = 0.5 ring-like structures are only slightly disfavored with respect to the string-like structures [36]. Nevertheless, it may seem surprising that ring-like domains are excited so easily already at very low T w even before coexistence of the equally favorable (1, 0) and (0, 1) domains occurs (see snapshot at T w = 1).…”

“…At low densities cells predominantly optimize locally the interaction between them by forming short strings with no long-range correlation between string-like clusters. We attribute this finding to the strong tendency of dipoles to form strings and the strong elastic screening of string-string interactions [36]. This results in rather robust pattern formation that does not differ qualitatively as the Poisson ratio is varied (see Fig.…”

“…However, it has not been discussed in detail before how this individual behavior translates into collective behavior of cell ensembles in compliant environments. In a recent short report we have shown that elastically interacting cells assemble into strings of cells which then leads to screening of the cellular traction patterns [36]. Although this effect in principle suppresses macroscopic order, we have also found that macroscopic order can exist at high cell density, with an interesting competition between string-and ring-like structures as a function of the Poisson ratio of the surrounding material.…”

Modeling the impact of granular embedding media, and pulling versus pushing cells on growing cell clones

“…Moreover, the fingering pattern was moderately increased with increasing object size of the external medium as well as for increasing adhesion with between B-type objects. The micro-motility of the external objects has no detectable influence on the instability, and the density of B-type objects seem to only influence the growth speed and the saturation size 6 A higher resolution version can be found at http://msysbio.com/videos njp. We compared the relation for the critical wavelength of the instability with that of the Saffman-Taylor instability.…”

“…Biomechanically induced interactions are increasingly discovered to play an important role in the growth control of tissues and tumors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Biomechanical interactions can serve as morphogenetic regulators [3] and permit cells to compare and adjust their growth rate to surrounding cells [5], help cells to choose their orientation according to their environment [6] or to adjust their growth rate to a local geometric property such as tissue curvature; this has led to the conjecture that sometimes functions follow form [7]. Morphogenetic strain rates, cell shape change and intercalation have been demonstrated to be interlinked [18].…”