Universal Features of Metastable State Energies in Cellular Matter

Abstract

Mechanical equilibrium states of cellular matter are overwhelmingly metastable and separated from each other by topology changes. Using theory and simulations, it is shown that for a wide class of energy functionals in 2D, including those describing tissue cell layers, local energy differences between neighboring metastable states as well as global energy differences between initial states and ground states are governed by simple, universal relations. Knowledge of instantaneous length of an edge undergoing a T1 transition is sufficient to predict local energy changes, while the initial edge length distribution yields a successful prediction for the global energy difference. An analytical understanding of the model parameters is provided.

Unconventional rheological properties in systems of deformable particles

“…For the parameter values we have considered, this occurs in the AP model [2][3][4][5][6]. Indeed, figures 2(a) and (b) reveal that the edge-length distribution of the AP model quickly vanishes as l → 0 in the solid phase at small γ, while it saturates to a finite value in the strain weakened regime, implying θ = 0.…”

“…This potential is the work needed to squeeze the particle into its volume, and establishes a close link between the mechanical and the geometrical properties of the system. This link has recently been uncovered in a two-dimensional model, when the energy of a particle depends on its perimeter [2][3][4][5][6]. In this case, an elementary relaxation event involving the disappearance of an edge separating two particles of length l, has an energy cost ∝ l. Hence, the system's mechanical response correlates with the edge-length distribution, P (l), and in particular, the system loses mechanical rigidity when an extensive number of zero-length edges appear, P (l) ∝ l θ with θ = 0 [2][3][4][5][6].…”

“…This link has recently been uncovered in a two-dimensional model, when the energy of a particle depends on its perimeter [2][3][4][5][6]. In this case, an elementary relaxation event involving the disappearance of an edge separating two particles of length l, has an energy cost ∝ l. Hence, the system's mechanical response correlates with the edge-length distribution, P (l), and in particular, the system loses mechanical rigidity when an extensive number of zero-length edges appear, P (l) ∝ l θ with θ = 0 [2][3][4][5][6]. When a particle's energy depends not only on its perimeter but also on other geometrical properties, such as the area, it is not clear how the intimate relationship between geometrical and mechanical properties manifests.…”

“…In the strain weakened state the system rearranges easily and hence possesses an extensive number of soft spots [3][4][5][6][25][26][27], structural rearrangements induced by a small increment s of the local shear stress. Accordingly, the distribution P (s) of the local stress increments needed to trigger a rearrangement scales as s θ with θ > 0 in the solid phase, and θ = 0 in the fluid one.…”

Non-metric interaction rules in models of active matter

“…More generally, one might view this as subset of the possible low-order Taylor expansions that can be written in the various geometric properties of the tessellation. Indeed, in the absence of any area terms precisely this interpretation has been given, where the terms nonlinear in perimeter are what separate out foam-like energy functionals from tissue-like energy functionals [43]. Since we will be simulating this model on a computer, it is convenient to non-dimensionalize the energy by choosing the unit of length to be A , where A is the average area of all of the cells in our simulation, letting k r = K A A /K P , and writing…”