@@ -198,7 +198,7 @@ The intercept, shows a fast initial response for smaller $\sigma_x$ and then see

Confidence intervals at the beginning and the end of the range of $\sigma_x$ overlap for the other parameters. The coefficient for $w$ seems to show a downward trend (figure \ref{pspaceiso_lL_a2}), but not enough to become relevant at $\sigma_x =0.025$\,MPa. The coefficient for $E_\text{cells}$ shows both a minimum and a maximum in the response to $\sigma_x$ (figure \ref{pspaceiso_lL_a4}), but includes zero over most of the range. There appears to be a fast initial response with stabilisation for $\nu_\text{film}$ (figure \ref{pspaceiso_lL_a4}) and cell layer Poisson ratio $\nu_\text{cells}$ (figure \ref{pspaceiso_lL_a6}), but the size of the initial response almost drowns in the 95\,\% confidence intervals, and that interval includes zero for almost the entire range of $\sigma_x$ for $\nu_\text{cells}$ (figure \ref{pspaceiso_lL_a6}). \\

\noindent Some coefficients seem to show a minimum or maximum after their (fast) initial response to $\sigma_x$ (e.g.\ $E_\text{film}$ and $t_\text{film}$, figures \ref{pspaceiso_lL_a2} and \ref{pspaceiso_lL_a7}; perhaps $t_\text{cells}$, figure \ref{pspaceiso_lL_a8}). Such minima and maxima were not expected. The values of $\sigma_x$ for where they occur seem similar between the parameters, about 0.007\,MPa. Something seems to happen around this value of $\sigma_x$ that `reverses' these effects. Pending further investigation, we hypothesise that this is related to some geometrical effect in film curling, e.g.\ completing a quarter circle or a full circle.\\

\noindent Some coefficients seem to show a minimum or maximum after their (fast) initial response to $\sigma_x$ (e.g.\ $E_\text{film}$ and $t_\text{film}$, figures \ref{pspaceiso_lL_a2} and \ref{pspaceiso_lL_a7}; perhaps $t_\text{cells}$, figure \ref{pspaceiso_lL_a8}). Such minima and maxima were not expected. The values of $\sigma_x$ for where they occur seem similar between the parameters, about 0.007\,MPa. Something seems to happen around this value of $\sigma_x$ that `reverses' these effects. Pending further investigation, we hypothesise that this is related to some geometrical effect in film curling, e.g.\ completing a quarter circle.\\

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\noindent\textbf{round up} Longer isotropic constructs and isotropic constructs with thicker cell layer curl more, isotropic constructs with stiffer films and isotropic constructs with thicker films curl less.

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@@ -679,14 +679,14 @@ The only two coefficients that seem to matter for $\kappa_\varepsilon$ are $a_0

\subsubsection{Difference between local curvature and $R$}

The 2D radius of curvature as assessed by the projected edge length $l$ lies between the local mean 3D curvature of both edges (figure \ref{anispace_Re_fig}). At $\sigma_x =0.015$\,MPa e.g., we find (table \ref{anispace_Re_tab}):

The absolute value and the spread in the distribution of $R_\varepsilon$ are larger for the right edge (table \ref{anispace_Re_tab}): projected edge length does not assess the mean (3D) curvature of sideways curling films.

The linear models do not fit well ($R_a^2 < 0.5$, SS$_\text{error}$/SS$_\text{tot} > 0.5$), but the effects (signs, sizes) are similar between the left and right edge. Both predominant orientation $\mu$ (SS/SS$_\text{tot} > 0.1$, $p =0.000$) and anisotropic fraction $P$ (SS/SS$_\text{tot} > 0.15$, $p =0.000$) affect $R_\varepsilon$, peak standard variation $\sigma$ has no role in the models (SS/SS$_\text{tot}=0.001$, $p > 0.7$).

\caption{Coefficients of the equations for $R_\varepsilon$ (red) with the left edge on the left (equation \ref{anispace_Rel_eq}) and the right edge on the right (equation \ref{anispace_Rer_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_Re_as}}

\end{figure}

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The evaluation of the coefficients of the model (equation \ref{aregress}) shows that both the intercept ($a_0$, figures \ref{pspaceani_Rel_a0} and \ref{pspaceani_Rer_a0}) and the coefficient for predominant orientation $\mu$ ($a_1$, figures \ref{pspaceani_Rel_a1} and \ref{pspaceani_Rer_a1}) respond to increasing $\sigma_x$. The confidence intervals for peak standard deviation $\sigma$ ($a_2$, figures \ref{pspaceani_Rel_a2} and \ref{pspaceani_Rer_a2}) are wide and include 0, similar to the results at $\sigma_x =0.015$\,MPa (table \ref{anispace_Re_tab}). The effect of anisotropic fraction $P$ ($a_3$, figures \ref{pspaceani_Rel_a3} and \ref{pspaceani_Rer_a3}) seems rather stable over the investigated range of $\sigma_x$ with relatively small coefficients.

\subsection{Minimum for $\frac{l}{L}\left(\sigma_x\right)$}\label{sec_md}

In our preliminary investigations, we found a minimum in the $\frac{l}{L}\left(\sigma_x\right)$ relationship for some constructs (figure \ref{lfmu_sigma}, 110\,\degree, 120\,\degree, and 130\,\degree). This prompted us to investigate the frequency with which such minima may occur in our 100 simulations.

We therefore define $\overline{\min l(\sigma)}$, the normalised difference between $\frac{l}{L}$ at the last converged value of $\sigma_x$ and the minimum value of $\frac{l}{L}$ over the tested range of $\sigma_x$ :

I.e.: there is no minimum for $\overline{\min l(\sigma)}=0$. There \textsl{is} a minimum when $\overline{\min l(\sigma)} > 0$, but we note that the actual value of $\overline{\min l(\sigma)} > 0$ will include effects of simulation convergence (figure \ref{aniconverge}).\\

\noindent More than half of our random models have $\overline{\min l(\sigma)} > 0$ (figure \ref{anispace_md_fig}), and the linear model (equation \ref{aregress}) predicts a positive intercept (table \ref{anispace_md_tab}):

\input{anispace_md_eq.tex}

The confidence interval for this intercept is wide and to all practical purposes seems to include 0 ($95\,\%$\,CI$=\left[7.96\cdot10^{-4}\quad6.02\cdot10^{-2}\right]$, $p =0.044$). Some of the detected minima may still be `numerical quirks'.

Yet, for about 20\,\% of the models $\overline{\min l(\sigma)} > 0.1$ (figure \ref{anispace_md_fig}): in these models there is a range more than 10\,\% for which the calculated values of $\frac{l}{L}$ are not unique in $\sigma_x$.\warning

\begin{table}

\begin{table}[p]

\center\small

\caption{Multiple linear regression model fitted to $\overline{\min l(\sigma)}$ for the 100 random models. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_md_tab}}

\caption{Distribution of (normalised) $\overline{\min l(\sigma)}$ for the 100 anisotropic models. \label{anispace_md_fig}}

\end{figure}

\ No newline at end of file

\end{figure}

\section{Round up}

\subsection{Isotropic constructs}

Isotropic constructs curl well and the parameter space might have evaluated over a larger of $\sigma_x$. However, `full circle curling' was not uncommon in our simulations (figure \ref{isospace_lL_fig}). While we argued that this did not affect the current analysis, it is a limit of the experiments that we wish to analyse. Before increasing the range of $\sigma_x$ for this analysis, it will be worthwhile to investigate this practical limitation with the current range $0\leq\sigma_x\,\text{[MPa]}\leq0.025$.

We are fitting a linear model to parameters that may not respond linearly, if only because of the geometrical effects of film curling. Furthermore, the response of the model as function of $\sigma_x$ shows unexpected non-linear effects around $\sigma_x =0.007$\,MPa for the main read-out $\frac{l}{L}$ (figure \ref{fig:pspaceiso_lL_as}). We hypothesised that some geometrical effect in film curling may predominantly occur around $\sigma_x =0.007$\,MPa, and a more in-depth analysis of the non-linear (geometrical) effects on film curling in this isotropic parameter space is certainly warranted.\\

\noindent The analysis of film curvature from projected edge does not account for asymmetric boundary conditions and curling edge effects (figure \ref{isospace_dk_fig}), but it is a good estimation for the actual mean local curvature over the length of the constructs (figure \ref{isospace_dR_fig}). Furthermore, constructs hardly change length during curling (figure \ref{isospace_dL_fig}), and these small changes in length only affect the analysis for $l > \frac{2L_d}{\piup}$, by a fraction $\abs{\frac{L_d}{L}}\ll1$.

Grossberg \textsl{et al.} remark that \cite{Grosberg2011}: ``For $x$-projections approaching the length of the film, the radius of curvature increases rapidly; however, that does not negatively impact the accuracy of the stress calculations because the stress at large radius of curvature is very small compared to peak systole.'' We already argued that projected edge $\frac{l}{L}$ is a better metric for $\sigma_x$ than curvature $R$, but we note that $l$ too, is more accurately estimated for constructs that `curl more'.

Al in all, we can conclude that the analysis of film curvature $R$ from projected edge $\frac{l}{L}$ is valid assessment of cell traction stresses $\sigma_x$ in isotropic constructs. \\

\noindent Alford \textsl{et al.} performed an analytical parameter space analysis of free curling constructs with $\nu_\text{film}=\nu_\text{cells}=0.5$\cite[figure 5]{Alford2010}. The results of our analysis qualitatively agree with theirs: film thickness and cell layer thickness have an opposite effect, cell layer stiffness has no effect. Alford \textsl{et al.} did not investigate film stiffness $E_\text{film}$. Our analysis suggests that this parameter may be the most important one (based on its coefficient), and that construct length $L$, also not investigated in \cite{Alford2010}, may be as important as cell layer thickness $t_\text{cells}$.

Within the limitations of our analysis we found that in order of decreasing effect, film stiffness $E_\text{film}$, film length $L$, cell layer stiffness $t_\text{cells}$, and film thickness $t_\text{film}$ affect the amount of construct curling (table \ref{isospace_lL_tab}). Of these parameters, construct length $L$ is probably accurately assessed from the image analysis. Proper experimental control or measurement of film stiffness $E_\text{film}$ therefore seems most the important factor to take into account.

Cell layer thickness $t_\text{cells}$ is not under experimental control and is probably also the hardest parameter to assess from these four. Yet, our results suggest that its effect on the accuracy of the analysis should not be neglected.

\subsection{Anisotropic constructs}

Things get more hazy in our anisotropic analysis.

For starters, our analysis is hampered by convergence issues for larger values of $\sigma_x$

Figure \ref{lfmu_sigma} suggests that the occurrence this minimum is most prevalent around a predominant orientation $\mu$ of $90\,\degree\pm30\,\degree$.

effects of $\sigma$ and $P$ best investigated with $\mu=90$\,\degree