Proliferating Nematic That Collectively Senses an Anisotropic Substrate

TL;DR

The paper investigates how proliferating elongated cells on a nematic substrate develop global nematic order through a collective sensing mechanism. It develops a proliferating nematic hydrodynamic framework, combining Landau–de Gennes free energy with density-dependent carrying and jamming densities, anisotropic friction encoded by the substrate, and a logistic growth law toward $ ho_C(S)$. Analyses of a spatially homogeneous reduced model show seeding-density–dependent final order, while the spatially extended model demonstrates that anisotropic friction together with density gradients drives collective alignment with the substrate, even without flow alignment or active stresses. The results reproduce experimental observations of higher order at lower seeding densities, defect dynamics with density-depleted cores, and jammed states, and predict that halting proliferation should suppress global alignment, with implications for tissue engineering and pattern design on anisotropic substrates.

Abstract

Motivated by recent experiments on growing fibroblasts, we examine the development of nematic order in a colony of elongated cells proliferating on a nematic elastomer substrate. After sparse seeding, the cells divide and grow into locally ordered, but randomly oriented, domains that then interact with each other and the substrate. Global alignment with the substrate is only achieved above a critical density, suggesting a collective mechanism for the sensing of substrate anisotropy. The system jams at high density, where both reorientation and proliferation stop. Using a continuum model of a proliferating nematic liquid crystal, we examine the competition between growth-driven alignment and substrate-driven alignment in controlling the density and structure of the final jammed state. We propose that anisotropic traction forces and the tendency of cells to align perpendicular to the direction of density gradients act in concert to provide a mechanism for collective cell alignment.

Figures (13)

• Figure 1: Analysis of experimental data highlights the interplay of order and density. (a--c) Snapshots of (a) jammed hdF cells growing on a LCE substrate, (b) their orientation fields, and the local density (heat-map). Dots in (a) and (b) are topological defects in the orientation field of the cells with charges $+1/2$ (red dots) and $-1/2$ (cyan dots). (c) Histogram of the angle between the cell orientation ($\mathbf{\hat{n}}$) and the density gradient ($\nabla\rho$) showing preferential cell orientation perpendicular to density gradients in a jammed monolayer grown on a nematic substrate (data collected from $n=5$ distinct experiments). The purple histogram is weighted by $S |\nabla\rho|$, while the gray histogram is unweighted. (d) Cell-substrate order parameter in experiments on hdF monolayers growing on a nematic (red dots) and an isotropic (blue stars) substrate. Each group of points corresponds to a separate experiment recorded at different times after seeding with $\rho_\mathrm{seed} = 50cells/mm^2$, with higher densities corresponding to later times as cells proliferate. The cyan solid lines are numerical solutions to Eqs. (\ref{['eq:S']}--\ref{['eq:rho']}) with parameters: $\gamma_0=0.25$ and $\Pi=0.0015$. (e) Cell-substrate order at jamming as a function of seeding density. Red dots are experimental data from $n=(3, 1, 1, 1, 3, 2)$ experiments respectively with error-bars showing the standard deviation. Cyan crosses represent numerical solutions to Eqs. (\ref{['eq:S']}--\ref{['eq:rho']}) with parameters: $\gamma_0=0.25$ and $\Pi=0.0015$. The error bars show standard deviation obtained from $50$ simulations with random starting angles. (f) Illustration of the alignment mechanism mediated by the interplay of cell proliferation, anisotropic friction and cell-alignment perpendicular to density gradients.
• Figure 2: Illustration of kNN coarse-graining method for $k=4$. At point $\mathbf{x}$, where cells are sparse, the coarse-graining radius, chosen to encompass exactly $k$ cells, is larger than at point $\mathbf{y}$ near which cells are dense. Each neighboring cell to point $\mathbf{x}$ is indexed by $j=1,2,...k$ and is at a distance $r_j$ and makes an angle $\phi_j$ with the $x$-axis. For expressions of coarse-grained density and order parameter fields see Sec. \ref{['sec:experiments']}.
• Figure 3: Cartoons to illustrate the distinction between average local cell-cell order ($\bar{S}$), global cell-cell order ($\bar{S}_\mathrm{g}$), and cell-substrate order ($\bar{S}_\mathrm{cs}$). The direction of nematic order of the substrate is taken as the $x$-axis (double arrows).
• Figure 4: Phase portraits of the ODE model showing the dynamics as light gray trajectories. The black dashed line in each plot is the $S$-nullcline obtained from the solution of $h(\rho,S)=0$. The dotted and dash-dotted lines show the jamming density $\rho_\mathrm{J}(S)$ and carrying capacity $\rho_\mathrm{C}(S)$, respectively. (a) Phase portrait for $r_\mathrm{J}=0.9$ with a single stable fixed point (green dot; $h(\rho_\mathrm{C}, S)=0$). The red crosses show the state at $T=2\tau_g$ for simulations started from $S = 0$ and $\rho$ uniformly spaced between $1.2$ and $2.5$ (red dots) illustrating that the dynamics slows down as the seeding density is increased. (b) Phase portrait for $r_\mathrm{J}=1$ where the jamming and carrying capacity lines coincide, forming a line of fixed points (solid line). Green/orange segments correspond to stable/unstable fixed points. Blue stars indicate exceptional points where the two eigenvalues coalesce. (c) Phase portrait for $r_\mathrm{J}=1.2$ showing the line of fixed points (solid green line) which is always stable. Parameters: $\Pi = 0.02$, $\gamma_0 = 0.67$.
• Figure 5: Emergence of order in numerical simulations of a proliferating nematic. (a) Cell-substrate order as a function of mean density. (b) Local order parameter as a function of mean density. Black and cyan lines show results for the model variant with an external aligning field (average of 10 runs) and anisotropic friction (average of 20 runs), respectively. Red disks and blue stars show experimental data for cells growing on nematic vs isotropic substrates respectively. Simulations for external field and asymmetric friction are for $n=(10, 20)$ runs per $\rho_\mathrm{seed}$ respectively. (c--f) Snapshots of a simulation with asymmetric friction starting at $\rho_\mathrm{seed}= 100mm^{-2}$ showing the local nematic director (dashes) and the local density (color-map). Initially (c) cells are clustered and locally aligned within the cluster, but the clusters are randomly oriented. As cells divide, and the mean densities increases clusters grow, merge, and rotate to align with the substrate (d). As density increases everywhere, significant local order develops in the entire monolayer (e). At the highest density (f) the orientations jam. Comparing (e) and (f) one can see that jamming prevents annihilation of defects pairs. Scale bar: $2mm$. For details about the simulation parameters refer to Sec. \ref{['subsec:numerical_sims']}.