Structural Characterization and Statistical-Mechanical Model of Epidermal Patterns

Abstract

In proliferating epithelia of mammalian skin, cells of irregular polygon-like shapes pack into complex, nearly flat two-dimensional structures that are pliable to deformations. In this work, we employ various sensitive correlation functions to quantitatively characterize structural features of evolving packings of epithelial cells across length scales in mouse skin. We find that the pair statistics in direct space (correlation function) and Fourier space (structure factor) of the cell centroids in the early stages of embryonic development show structural directional dependence (statistical anisotropy), which is a reflection of the fact that cells are stretched, which promotes uniaxial growth along the epithelial plane. In the late stages, the patterns tend toward statistically isotropic states, as cells attain global polarization and epidermal growth shifts to produce the skin's outer stratified layers. We construct a minimalist four-component statistical-mechanical model involving effective isotropic pair interactions consisting of hard-core repulsion and extra short-range soft-core repulsion beyond the hard core, whose length scale is roughly the same as the hard core. The model parameters are optimized to match the sample pair statistics in both direct and Fourier spaces. By doing this, the parameters are biologically constrained. In contrast with many vertex-based models, our statistical-mechanical model does not explicitly incorporate information about the cell shapes and interfacial energy between cells; nonetheless, our model predicts essentially the same polygonal shape distribution and size disparity of cells found in experiments, as measured by Voronoi statistics. Moreover, our simulated equilibrium liquid-like configurations are able to match other nontrivial unconstrained statistics, which is a testament to the power and novelty of the model. The array of structural descriptors that we deploy enable us to distinguish between normal, mechanically deformed, and pathological skin tissues. Our statistical-mechanical model enables one to generate tissue microstructure at will for further analysis. We also discuss ways in which our model might be extended to better understand morphogenesis (in particular the emergence of planar cell polarity), wound healing, and disease-progression processes in skin, and how it could be applied to the design of synthetic tissues.

Hyperuniform states of matter

“…Figure 54 shows that there is excellent agreement between the structure factors obtained from the aforementioned statisticalmechanical cell model and their experimental counterparts for the total population and the individual cell types. This epithelium system stands in contrast to epithelia in mammalian skin, which has been shown not be hyperuniform [317], since the latter, unlike the former, must be very pliable to deformations. Figure 54: Comparison of the structure factors S (k) of the experimentally obtained and simulated point configurations representing the spatial arrangements of the individual cell types and total population, as presented in Ref.…”

The statistics of noisy growth with mechanical feedback in elastic tissues

“…Qualitatively, our finding of power law correlations says that cells in growing epithelia are more clumped together on large scales than for a totally random spatial distribution, with the effect becoming more pronounced for smaller k/γ0. Correlations in cell density are experimentally accessible (13)(14)(15)(16), and we expect that similar clumping should be observed in spatial distributions of mitotic cells (which, intriguingly, are known cluster in some tissues (1,24)). Estimates of the exponent 2k/γ0 could be used to determine the strength c of mechanical feedback, which has not previously been measured.…”

“…ξ ll − 2ξ [14] ∂t ZT = ξ ⊥ [15] ∂t ZL = ξ ll + 2 k k (d) ξ , [16] where k (d) is defined similarly to k, and ZT = G ⊥ /Ḡ and ZL = [ G ll +2(k/k (d) ) G ]/Ḡ are the amplitudes respectively of transverse and longitudinal soft modes, which do not produce any stress and thus grow diffusively. Before studying the density and clone statistics, we next consider these soft modes more carefully.…”

“…In experiments, noise in growth has most often been probed through the size and spatial distribution of clones of cells (4)(5)(6)(7)(8)(9)(10), especially of neutral clones that are genetically identical to surrounding tissue except for a clone marker. Cell density variation has also been observed directly in culture (11)(12)(13)(14)(15) and in fixed tissues (16), as have correlations in positions of mitotic cells (1), and size asymmetry between contralateral organs can be used as an indirect readout of noise levels (17).…”

Statistics of noisy growth with mechanical feedback in elastic tissues

“…In experiments, noise in growth has most often been probed through the size and spatial distribution of clones of cells (4-10), especially of neutral clones that are genetically identical to surrounding tissue except for a clone marker. Cell density variation has also been observed directly in culture (11)(12)(13)(14)(15) and in fixed tissues (16), as have correlations in positions of mitotic cells (1), and size asymmetry between contralateral organs can be used as an indirect readout of noise levels (17).…”