Possible mechanical feedback mechanisms that could improve tissue growth efficiency and homeostasis.

<p>(A) Time evolution of the elimination rate <i>ε</i> with and without mechanical feedback mechanisms. All feedback mechanisms outlined in the text significantly reduced the elimination rate <i>ε</i>. (B) Time evolution of tissue size with and without mechanical feedback mechanisms. Although the density-dependent growth regulation (red line) leads to a drastic decrease in the elimination rate, it takes much more time to attain a certain tissue size (e.g. 20,000 cells) compared to cases without feedback (black line), meaning that it was not necessarily efficient in regard to developmental speed. In contrast, the other feedback mechanisms, including feedback regulation of tissue fluidity given by Eqs <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005651#pcbi.1005651.e007" target="_blank">(6)</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005651#pcbi.1005651.e008" target="_blank">(7)</a> (cyan and blue lines), and that of cell division orientation (green line), could improve tissue growth efficiency by reducing the elimination rate and by maintaining a normal growth speed. (C) (Left) Incompatibility between the reductions in cell size variance and stress magnitude under a feedback model of tissue fluidity (Γ) given by <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1005651#pcbi.1005651.e007" target="_blank">Eq (6)</a>. When the feedback strength <i>c</i> is positive and larger, the cell size/density variance and elimination rate decreased but the variance in stress magnitude increased (see also (D) for an example of the simulation results). In contrast, when <i>c</i> is negative and smaller (i.e., |<i>c</i>| is larger), the opposite is true; although the cell size/density variance and elimination rate increased, the stress state was homogenized. (Right) For comparison, the relationship between the elimination rate and variance in cell size or stress magnitude is shown in the cases without feedback. The variances and elimination rate are lower for higher tissue fluidity, and vice versa. Graph symbols: red circles indicate the variance in cell size; blue diamonds indicate the variance in stress magnitude. Parameter values: (left) <i>c</i> ranges from -0.004 to 0.004. (right) Γ ranges from 0.02 to 0.05. (D) Examples of the spatial distribution of cell size (upper) and stress magnitude (lower) in the presence and absence of mechanical feedback; from the left: without feedback, with density-dependent growth regulation, feedback to tissue fluidity, and feedback to cell division orientation. All feedbacks could reduce the variance in cell size and density, but in the case of the feedback to tissue fluidity, there was incompatibility between the reductions in variances of cell size and stress magnitude. For the other two feedback mechanisms, the both were compatible.