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\begin{document}
\begin{center}
\section*{\large \bf Supplementary Information}
\end{center}
\medskip
\subsection*{Appendix A}
The volume change $\Delta V_i$ for cell $i$ can be written as:
$$
\Delta V_i \approx \sum_\bee \Delta V_\bee,
$$ where $\Delta V_\bee$ is the volume change associated with edge
$\bee$, and the summation is over all edges of a cell. The volume change
$\Delta V_\bee$ can be written as:
$$
\Delta V_\bee = |\Delta \bv(\bee) \times \bee |,
$$
where
$\Delta \bv(\bee)$ is the displacement vector of vertex $v(\bee)$ and ``$\times$'' represents vector cross product.
We can distribute the volume change
$\Delta V_\bee$ for edge
$\bee$ to its two end-vertices $\{v(\bee)\}$:
$$
\Delta V_\bee
= \frac{1}{2} \sum_{v(\bee)} | \Delta \bv(\bee) \times \bee|.
$$
We have:
\begin{equation}\label{eqn:A}
\begin{split}
\Delta V_i &\approx \sum_\bee \sum_{v(\bee)} \frac{1}{2} |\Delta \bv(\bee) \times \bee| \\
& = \sum_\bee \sum_{v(\bee)} \frac{1}{2} | [\frac{1}{2} \ba_{v(\bee)} (\Delta t)^2]
\times \bee |,
\end{split}
\end{equation}
where
$\ba_{v(\bee)}$ is the
acceleration rate
of vertex $v(\bee)$.
By Newton's law, the acceleration rate is:
$
\ba_{v(\bee)}
= \bF_{v(\bee)} /m.
$
We therefore have:
\begin{equation}\label{eqn:B}
\begin{split}
\Delta V_i &\approx
\sum_\bee \sum_{v(\bee)} \frac{1}{2} \cdot \frac{1}{2} | \frac{\bF_{v(\bee)}}{m} (\Delta{}t)^2
\times \bee |\\
& = \frac{(\Delta t)^2}{2m} \frac{1}{2} \sum_\bee \sum_{v(\bee)}
|\bF_{v(\bee)} \times \bee| \\
& = \frac{1}{2} \sigma \sum_\bee \sum_{v(\bee)} |\bF_{v(\bee)} \times \bee|.
\end{split}
\end{equation}
Here $\sigma = (\Delta t)^2/2m$ in the unit of $s^2/kg$ is the
constant for time integration.
\newpage
\subsection*{Appendix B}
The position $\bv_i$ of vertex $i$ at time $t + 1$ is calculated as:
\begin{equation}\label{eqn:C}
\begin{split}
\bv_i(t+ 1) &= \bv_i(t) + \Delta \bv_i \\
&= \bv(t) + \frac{1}{2} \ba_i (\Delta t)^2\\
&= \bv(t) + \frac{1}{2} \frac{\bF_{v_i}(t)}{m} (\Delta t)^2\\
&= \bv(t) + \sigma \bF_{v_i}(t),
\\
\end{split}
\end{equation}
where
$\bF_{v_i}(t)$ is the net force at vertex $v_i$ at time $t$. We set $\Delta t
= 1$ for one time step.
$\sigma$ controls the
convergence rate of finding the stationary state.
\newpage
\subsection*{Appendix C}
Summary of data structure used in our model.
\begin{lstlisting}[language=C,basicstyle=\footnotesize,backgroundcolor=\color{LightGray}][!t]
class Cell {
Index CI;
Type T;
Color C;
HalfEdge* E;
Radius R;
}
class Vertex {
Index VI;
Coordinate X;
Coordinate Y;
HalfEdge* E;
}
//Each edge is stored as
//two oriented half-edges
class HalfEdge {
Index EI;
Cell* C;
Vertex* V1;
Vertex* V2;
HalfEdge* Reverse;
HalfEdge* Next;
ArcAngle A;
Length L;
}
\end{lstlisting}
Pseudo-code for traversing neighboring cells:
\begin{lstlisting}[language=C,basicstyle=\footnotesize,backgroundcolor=\color{LightGray}][!t]
HalfEdge *E;
Cell *C1,*C2;
E = C1->E;
while(true)
{
C2 = E->Reverse->C;
PRINT( C2->CI is a neighbor of C1->CI);
E = E->Next;
if(E = = C1->E)
break;
}
\end{lstlisting}
\newpage
%\subsection*{Appendix D}
%Movie 3: {\it Inhibition field} model.
%Movie 4: {\it Inhibition field with stripes} model.
%YC: 4 movies added
\clearpage
\subsection*{Appendix D}
Details on the concentrations and spatial patterns of genes {\it ac}, {\it Dl}, and {\it N} for different proposed models were extracted
from literature~\cite{Skeath_GenesDev91,ArtavanisTsakonas_Science95,Oster_MathematicalBiosciences88,Simpson_Development90,Usui_93,Annette_Mech97,Qi_Science99} and given
as an input to the simulation algorithm. Following is the detail of each model:
\paragraph{Pre-destined Model}
As the expression level of genes {\it ac} and {\it sc} is correlated~\cite{Skeath_GenesDev91},
we only use the expression level of {\it ac} gene in our simulations. Initially, the expression level of {\it ac} is set to zero for all cells.
When the tissue size reaches 1000 cells, we randomly choose one cell and assign a value of 0.5-1, representing the expression level of {\it ac} gene to this cell. All its
contacting immediate neighbors are also assigned a value of 0.5-1, representing the expression level of {\it ac} gene. Together, they form a proneural cluster.
We repeat the process of assigning expession level of {\it ac} to cells until we have about
100 cells with the expression of {\it ac} greater than 0.5.
In this models, only a cell with high expression level of {\it ac} can become a bristle.
When a cell from this proneural cluster divides, the expression level of {\it ac} randomly increases or decreases by 10\% in the daughter cells.
After the tissue has grown to 4000 cells, a bristle is formed when the difference in expression of {\it ac} between a cell and at least 4 of its
neighbors is greater than 0.5.
\paragraph{Lateral inhibition}
In this model, after the tissue grows to 4000 cells, each cell is assigned a value between 0-1 randomly, representing the expression level of {\it Dl} gene.
A cell that has a higher expression level of {\it Dl} gene than all of its immediate neighbors becomes a bristle. The bristle cell then reduces the expression level
of {\it Dl} gene to zero in all of its immediate contacting neighbors. This process of identifying a bristle cell is repeated until no new bristles can be formed.
\paragraph{Lateral inhibition with stripes}
In this model, after the tissue grows to 4000 cells, stripes that are 3-5 cells wide are formed as shown in Figure~13a. Green stripes have almost
equal expression of {\it Dl} and {\it N} genes but {\it ac} is not
expressed. Blue stripes have high expression of {\it ac} and {\it
Dl} genes but low expression of {\it N} gene. Red stripes have
high expression of {\it ac} and {\it N} genes but low expression
of {\it Dl} gene.
Bristles only form in the blue stripes. In this model, cells are assigned a value between 0-1 randomly,
representing the expression level of {\it Dl} gene within the blue stripe.
A cell that has a higher expression level of {\it Dl} gene than all of its immediate neighbors becomes a bristle.
The bristle cell then reduces the expression level of {\it Dl} gene to zero in all of its immediate neighbors.
This process of identifying a bristle cell is repeated until no new bristles can be formed.
\paragraph{Inhibition field}
In this model, after the tissue grows to 4000 cells, each cell is assigned a value between 0-1 randomly, representing the expression level of {\it Dl} gene.
A cell that has a higher expression level of {\it Dl} gene than all of its immediate neighbors becomes a bristle. The bristle cell then reduces the expression level
of {\it Dl} gene to zero in all of its layer of neighbor cells, depending on the inhibition radius. This process of identifying a bristle cell is repeated until no
new bristles can be formed.
\paragraph{Inhibition field with stripes}
In this model, after the tissue grows to 4000 cells, stripes that are 3-5 cells wide are formed as shown in Figure~13a. Green stripes have almost
equal expression of {\it Dl} and {\it N} genes but {\it ac} is not
expressed. Blue stripes have high expression of {\it ac} and {\it
Dl} genes but low expression of {\it N} gene. Red stripes have
high expression of {\it ac} and {\it N} genes but low expression
of {\it Dl} gene.
Bristles only form in the blue stripes. Within the blue stripe, cells are assigned a value between 0-1 randomly,
representing the expression level of {\it Dl} gene. A cell that has a higher expression level of {\it Dl} gene than all of its immediate neighbors becomes a bristle.
The bristle cell then reduces the expression level of {\it Dl} gene to zero in all of its layer of neighbor cells depending on the inhibition radius.
This process of identifying a bristle cell is repeated until no new bristles can be formed.
\clearpage
\bibliographystyle{plos2009}
\bibliography{cellmodel}
%\bibliographystyle{omp}
%\begin{thebibliography}{10}
%\bibitem[Sela et~al.(1957)Sela, White, and Anfinsen]{SELA_Science1957}
%Sela,~M.; White,~F.,~Jr.; Anfinsen,~C. \emph{Science} \textbf{1957},
% \emph{125}, 691--692
%\bibitem[Anfinsen(1973)]{Anfinsen_Science1973}
%Anfinsen,~C. \emph{Science} \textbf{1973}, \emph{181}, 223--230
%\end{thebibliography}
\end {document}