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source: liacs/pnbm/project/report.tex@ 38

Last change on this file since 38 was 25, checked in by Rick van der Zwet, 15 years ago
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[11]	1	%
	2	% $Id: report.tex 571 2008-04-20 17:31:04Z rick $
	3	%
	4
	5	\documentclass[12pt,a4paper]{article}
	6
	7	\frenchspacing
[12]	8	\usepackage[english]{babel}
	9	\selectlanguage{english}
[11]	10	\usepackage{graphicx}
	11	\usepackage{url}
	12	\usepackage{multicol}
	13	\usepackage{fancybox}
	14	\usepackage{amssymb,amsmath}
	15
[25]	16	\title{Modeling planar signalling in AP axis development in \emph{Xenopus laevis}\\
[11]	17	\large{using Petri Nets in Higher Level Developmental Biology}}
	18	\author{Rick van der Zwet, Tiago Borges Coelho \\
	19	\texttt{<hvdzwet@liacs.nl>,<borges.coelho@gmail.com>}\\
	20	LIACS - Leiden University, The Netherlands}
	21	\date{\today}
	22
	23	\begin{document}
	24	\maketitle
	25	\section{Abstract}
[25]	26	Planar signaling is the process within the development of the AP axis
	27	development of the \emph{Xenopus laevis} \cite{Bertens09} in which cells
	28	accumulate proteins based on the saturation of nearby cells. If one cell
	29	produces n amount of proteins, it will initiate a transferring cascade to cells
	30	in the vicinity. This dissemination of proteins will eventually cease,
	31	considering that n is a finite variable. There is a gradation in the amount of
	32	proteins transferred, meaning that neighbouring cells get n/2 the amount of
	33	proteins of the most saturated cell.
[11]	34
[25]	35	We are going to model this into Petri-Nets beeing a mathematical modeling
	36	language, which suit well for this purpose as we could nicely model the process
	37	in graphical interactive representation and could also be used for automated
	38	model tracking and analyze.
[11]	39
[25]	40	\section{Approach}
	41	First a Petri-Net model will be defined textually and using graphs next the
	42	modeling will be taking into practice using the modeling tool\emph{CPNTools}
	43	\footnote{http://wiki.daimi.au.dk/cpntools/cpntools.wiki}.
[11]	44
	45	\section{Modeling}
[25]	46	To model this process we will take a modular approach using coloured Petri-Nets
[12]	47	(see Fig~\ref{fig:model}), since the goal of this assignment is to have a
	48	solution that can be applied to any configuration of cells. We start with a
[25]	49	building block that is an abstraction of a cell (figure: circle), which can then
[12]	50	be coupled to other cells (figure: arrows). The abstraction contains two
	51	different types. First the proteins are modelled (figure: red), secondly the
	52	proteins (figure: blue) are leading in a second process of the creation of
[25]	53	posterisation which also needs modeling. We assume a 1:1 mapping between the amount
	54	of proteins and the posterisation -this taken into consideration- ones an
	55	\texttt{INITIAL} protein is 'used' (e.g. has on posterisation counterpart) in
	56	this process it
[18]	57	get called \texttt{ACTIVATED}. We assume that the proteins to posterisation
	58	process is taking place at the same time as the proteins distribution. And in a
	59	special format (figure: object B). It tries to matches the posterisation to the
	60	same level as the proteins present. But the moment the protein level lowers,
	61	the posterisation will remain the same. In pseudo-code:
	62	\begin{verbatim}
	63	if numPos < numProteins then
	64	numPos = numPos + 1
	65	endif
	66	\end{verbatim}
	67	\texttt{numProteins} is the proteins available and \texttt{numPos} is the
	68	posterisation present.
[11]	69
[25]	70	The connectors between the cells (the membranes) has a special properly. One
	71	can see them as pressure valves others as siphons (see Fig~\ref{fig:pressure}).
[11]	72	The moment the 'volume' at complies with the following properly $A / 2 < B$
[18]	73	then the pressure closes, else it passes volume from A to B at an certain rate
	74	(\texttt{flowSpeed}). This rate could depend on the difference, actual value present
[25]	75	or something else. Please do mind that negative values could ever appear hence
	76	the checking whether the source is bigger or equal then the flowSpeed.
[18]	77
[25]	78	For the case there exists no standard Petri-Net 'component', hence this require
[12]	79	the creation of a new property (figure: $2:1$), with the following properties:
[11]	80
[12]	81	\begin{verbatim}
[18]	82	flowSpeed = n
[25]	83	if A > 2 * B and A => flowSpeed then
[12]	84	A = A - flowSpeed
	85	B = B + flowSpeed
[25]	86	else if B > 2 * A and B => flowSpeed then
[12]	87	B = B - flowSpeed
	88	A = A + flowSpeed
	89	endif
	90	\end{verbatim}
	91
[11]	92	Planar signaling could theoretically start in every cell, by
[25]	93	inserting some amount of proteins. In our model represented as a bunch of
	94	\texttt{INITIAL} tokens being put in a random cell.
[11]	95
[12]	96	\begin{figure}[htp]
	97	\centering
	98	\caption{Planar signaling model}
[18]	99	\includegraphics[width=100mm]{planar-signaling-model.eps}
[12]	100	\label{fig:model}
	101	\end{figure}
[11]	102
	103	\begin{figure}[htp]
	104	\centering
	105	\caption{Pressure valve example}
	106	\includegraphics[height=60mm]{pressure-valve.eps}
	107	\label{fig:pressure}
	108	\end{figure}
	109
	110	\section{CPNTools 'implementation'}
	111	CPNTools has quite some shortcomings when it comes to modeling (higher level
[12]	112	developmental) biology.
[11]	113
[12]	114	One it the shortcoming of the 'balancing'. It does not allow reading of how
[25]	115	many tokens are present in a certain state and base action upon them. As
[12]	116	workaround for this (see Fig~\ref{fig:CPNplanar}) we used a 'dump' gradation
	117	function. In our case it simply take 3 tokens and pushes 1 forward and
[18]	118	converting 2 directly to \texttt{ACTIVATED}. This does not take in
	119	consideration if the amount get changed in 'further-up', by some external
	120	source.
[12]	121
[11]	122	Secondly it is missing a possibility to for easy random initialisation for
	123	modeling purposes. As a dirty quirk we 'hacked' it to choose between starting
	124	at the head or the tail.
	125
[25]	126	In this implementation the proteins to gradients process is taking place at
	127	cell $A$ at the same time that the proteins get transferred from cell $A$ to
[12]	128	$B$.
[11]	129
[12]	130	Also it should be noted that it missing a notion of timed firing sequences;
	131	meaning firing sequences which will occur at an certain time. This could for
[25]	132	example used to 'trigger' a timed activation of the \texttt{INITIAL} to
[18]	133	\texttt{ACTIVATED} process as modeled in fig~\ref{fig:model}. An initial idea
[25]	134	is shown at fig~\ref{fig:time-idea} in appendix~\ref{sec:timer-idea}.
[12]	135
[18]	136
[11]	137	\begin{figure}[htp]
	138	\centering
	139	\caption{CPNTools implementation}
	140	\advance\leftskip-2cm
	141	\advance\rightskip+2cm
	142	\includegraphics[width=1.3\textwidth]{planer-signaling.eps}
	143	\label{fig:CPNplanar}
	144	\end{figure}
	145
	146	\section{Conclusion}
[25]	147	Using Petri-Nets for modeling biology processes is a powerful framework, which
	148	could be well expandable. The Proof Of Concept implementations and
[11]	149	visualisations how-ever are lacking. \emph{CPNTools} for example does not
[25]	150	provide a powerful enough tool-set for the modeling purposes.
[11]	151
[25]	152	\bibliographystyle{amsalpha}
[11]	153	\begin{thebibliography}{10}
[25]	154	\bibitem[Bertens09]{Bertens09}Laura M.F. Bertens et al., Using Petri Nets in Higher
	155	Level Developmental Biology: A case study on the AP axis development in Xenopus
	156	laevis Extended Abstract, 2009
[11]	157	\end{thebibliography}
[25]	158	\appendix
	159	\section{Timer Idea}
	160	\label{sec:timer-idea}
[18]	161
	162	\begin{figure}[htp]
	163	\centering
	164	\caption{Timed transition idea}
	165	\includegraphics[width=0.5\textwidth]{timer-proposal.eps}
	166	\label{fig:time-idea}
	167	\end{figure}
[11]	168	\end{document}

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