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\title{DRAFT: Modeling planar signalling in AP axis development in \emph{Xenopus laevis}\\
\large{using Petri Nets in Higher Level Developmental Biology}}
\author{Rick van der Zwet, Tiago Borges Coelho \\
  \texttt{<hvdzwet@liacs.nl>,<borges.coelho@gmail.com>}\\ 
  LIACS - Leiden University, The Netherlands}
\date{\today}

\begin{document}
\maketitle
\section{Abstract}
Planar signaling is the process in which cells accumulate proteins based on the
saturation of nearby cells. If one cell produces n ammount of proteins, it will
initiate a transfering cascade to cells in the vicinity. This dissemination of
proteins will eventually cease, considering that n is a finite variable. There
is a gradation in the ammount of proteins transfered, meaning that neighbouring
cells get n/2 the ammount of proteins of the most saturated cell.

XXX: Citing to the Bio Papers
XXX: Small introductions Petri-nets

\section{Approch}
First a PetriNet model will be defined textually and using graphs next the
modeling will be taking into practice using the modeling tool\emph{CPNTools}. 

\section{Modeling}
To model this process we will take a modular approach, since the goal of this
assignment is to have a solution that can be applied to any configuration of
cells. We start with a bulding block that is an abstraction of a cell,
which can then be coupled to other cells. The abstraction contains two
different types. First the proteins are modelled, secondly the proteins are leading 
in a second process of the creation of gradients which also needs modeling. We
assume a 1:1 mapping between the amount of proteins and gradients -this taken
into consideration- ones an $INITIAL$ protein is used in this process it get
called $ACTIVATED$. We assume that this process is taking place at cell $A$ at
the same time that the proteins get transfered from cell $A$ to $B$. 

The connectors between the cells (the membrams) has a special properly. One
can see them as pressure valves others as sighons (see Fig~\ref{fig:pressure}).
The moment the 'volume' at complies with the following properly $A / 2 < B$
then the pressure closes, else it passes volume from A to B at an certain rate.
For the case there exists no standard PetriNet 'component', hence  this require
the creation of a new property.

Planar signaling could theoretically start in every cell, by
inserting some amount of protiens. In our model represented as a bunch of
$INITIAL$ tokens beeing put in a random cell.


\begin{figure}[htp]
\centering
\caption{Pressure valve example}
\includegraphics[height=60mm]{pressure-valve.eps}
\label{fig:pressure}
\end{figure}

\section{CPNTools 'implementation'}
CPNTools has quite some shortcomings when it comes to modeling (higher level
developmental) biology. One it the shortcoming of the 'balancing'. It does not
allow reading of how many tokens are present in a certain state and base action
uppon them. As workaround for this (see Fig~\ref{fig:CPNplanar}) we used a
'dump' gradation function. In our case it simply take 3 tokens and pushes 1
forward and converting 2 directly to $ACTIVATED$. This does not take in
consideration if the amount get changed in 'further-up', by some external source.

Secondly it is missing a possibility to for easy random initialisation for
modeling purposes. As a dirty quirk we 'hacked' it to choose between starting
at the head or the tail.

Not very important for our case, but also it should be noted that it missing a
notion of timed firing sequences; meaning firing sequences which will occur at
an certain time. This could for example used to 'trigger' a timmed activation
of the $INITIAL$ to $ACTIVATED$ process.

\begin{figure}[htp]
\centering
\caption{CPNTools implementation}
\advance\leftskip-2cm
\advance\rightskip+2cm
\includegraphics[width=1.3\textwidth]{planer-signaling.eps}
\label{fig:CPNplanar}
\end{figure}

\section{Conclusion}
Using PetriNets for modeling biology processes is a powerful framework, which
could be well extendable. The Proof Of Concept implementations and
visualisations how-ever are lacking. \emph{CPNTools} for example does not
provide a powerfull enough toolset for the modeling purposes.

\begin{thebibliography}{10}
%   sing Petri Nets in Higher Level Developmental Biology:
% A case study on the AP axis development in Xenopus laevis
%                    Extended Abstract
% http://www.liacs.nl/~csbpn/COURSE%20DOCUMENTS/extended%20abstract%20Bertens%20Jansen%20Kleijn%20Koutny%20Verbeek.pdf
% Laura M.F. Bertens

% http://www.liacs.nl/~csbpn/
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%

\end{thebibliography}
\end{document}