Biomechanical Feedback Strengthens Jammed Cellular Packings

Growth in confined spaces can drive cellular populations through a jamming transition from a fluidlike state to a solidlike state. Experiments have found that jammed budding yeast populations can build up extreme compressive pressures (over 1 MPa), which in turn feed back onto cellular physiology by slowing or even stalling cell growth. Using numerical simulations, we investigate how this feedback impacts the mechanical properties of model jammed cell populations. We find that feedback directs growth toward poorly coordinated regions, resulting in an excess number of cell-cell contacts that rigidify cell packings. Cell packings possess anomalously large shear and bulk moduli that depend sensitively on the strength of feedback. These results demonstrate that mechanical feedback on the single-cell level is a simple mechanism by which living systems may tune their population-level mechanical properties.

Figure 1

(a) Schematic of the growth and division process. Each cell grows by bud expansion. Budding culminates via mother-daughter separation when the daughter reaches the size of a mother cell. (b) Schematic of feedback of cell-cell forces onto growth. Daughter buds contact their neighbors as they grow (dashed black lines). The associated contact forces generate pressure on the growing bud, defined here as the ratio of total force magnitude $F_{\mathrm{tot}}$ to perimeter $L_{\mathrm{bud}}$ of the daughter bud (see Supplemental Material [24]): $P_{\mathrm{bud}}=F_{\mathrm{tot}}/L_{\mathrm{bud}}$. This pressure in turn decreases its growth rate as $\gamma \propto e^{-P_{\mathrm{bud}}/P_0}$. (c)–(g) Snapshots from a typical simulation. (c) Each simulation is inoculated with two cells. (d) Cell growth drives population to expand outward. During expansion, cells interact with their neighbors via repulsive elastic forces and completely overdamped dynamics. (e) Population undergoes a “jamming transition” at ${\varphi }_J=0.84$; at which point, a system-spanning network of force-bearing intercellular contacts develops (red lines). At jamming, most mother (gray) and daughter (brown) buds are constrained by their neighbors, but $\approx 25\%$ of buds (yellow) are unconstrained (see Supplemental Material [24]). Above jamming, packings have more unconstrained buds when Fig. 1 cell growth rate is independent of mechanical pressure than when Fig. 1 pressure feeds back onto growth ($P_0/k={10}^{-3}$). Both Figs. 1 and 1 are at $\varphi =0.89$. (h) Pressure that entire population exerts on its surroundings (see Supplemental Material [24]) is zero below jamming ($\varphi <{\varphi }_J$) and increases as the cells grow above jamming ($\varphi >{\varphi }_J$). With no feedback and weak feedback ($P_0/k=5\times {10}^{-3}$), $P$ is almost linear in $\varphi$. For strong feedback ($P_0/k={10}^{-3}$), $P$ increases more slowly with $\varphi$. All pressures are measured in units of the cell-cell modulus $k$ (see Supplemental Material [24]). (i) Number of contacts $Z$ per cell jumps discontinuously from $Z\approx 0$ to $Z=Z_J\approx 5.5$ at jamming at ${\varphi }_J$ and increases more quickly for strong feedback than for weak or no feedback. For periodic boundary conditions, Figs. 1 use box size $L=7\sigma$ and Figs. 1 and 1 use box size $L=15\sigma$, where $\sigma$ is a cell diameter. Figures 1 and 1 show data for one typical population. We find $Z_J$ does not exhibit significant system size effect, with $Z_J\approx 5.5$ for different initial conditions and for large systems {Fig. S2(b) from the Supplemental Material [24]}.

Figure 2

(a) Fraction of unconstrained buds $f_u$ as a function of the population pressure generated by growing budding yeast packings above the jamming point. $f_u^J$ denotes the value of $f_u$ at the jamming threshold ($P\to 0$). Colored lines correspond to different feedback values. Populations without feedback have a finite number of unconstrained buds up to $P\approx P_{\max }$, which corresponds to $\varphi \approx 1$ [Fig. 1]. (b) Distribution of cell growth rates for microbial populations with weak ($P_0/k=5\times {10}^{-3}$) and strong ($P_0/k={10}^{-4}$) feedback. In order to measure growth rates as unconstrained buds make contact, both populations have a value of $f_u$ that is $\approx 50\%$ of that measured at jamming ($f_u\approx 0.13$). The black bar denotes cells under no or very little pressure, thus growing as they would in the absence of feedback. The gray bar denotes cells for which the growth rates are reduced by pressure. The growth rate $\gamma (i)$ of each cell is normalized by ${\gamma }_i^0$: the growth rate that a cell would have without feedback (see Supplemental Material [24]). Simulations have box size $L=15\sigma$. Each data point is averaged over 100 independent inoculations.

Figure 3

(a) Excess number of contacts: $\mathrm{\Delta }Z=Z-Z_{\mathrm{iso}}$. Colored lines correspond to different feedback values, and shaded regions represent one standard deviation. To show where growth has been appreciably slowed by cell-cell forces, dashed colored lines correspond to populations for which the average cell growth rate is reduced by a factor of 10 as compared to growth without feedback (Fig. S5). Solid colored lines correspond to growth rates within a factor of 10 of those without feedback. The dotted black line shows the number of contact resulting from unconstrained bud contacts $\mathrm{\Delta }{Z}_{\mathrm{u}}=2f_u\approx 1$ (see Supplemental Material [24]). (b) Bulk modulus as a function of pressure for populations growing under five different feedback strengths. (c) Shear modulus $G$ for cell packings. Inset: Shear modulus in terms of excess contact number $\mathrm{\Delta }Z$. Line types in Fig. 3 are the same as shown in Fig. 3. Black lines for each panel show known results for disk packings $G\propto \mathrm{\Delta }Z\propto P^{1/2}$ [2]. Simulations have box size $L=15\sigma$, and each data point is averaged over 100 independent inoculations. For this box size, packings have $⟨N⟩\approx 172$ cells at the jamming point, for which system size effects in $\mathrm{\Delta }Z$ and $G$ are expected for smaller pressures ($P/k\ll 1/N^2\approx 3\times {10}^{-5}$ [30]) than presented here. We do not find significant system size effects in $\mathrm{\Delta }Z$ [Fig. S2(b)] or $G$ (Fig. S3).