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r25 r18 14 14 \usepackage{amssymb,amsmath} 15 15 16 \title{ Modeling planar signalling in AP axis development in \emph{Xenopus laevis}\\16 \title{DRAFT: Modeling planar signalling in AP axis development in \emph{Xenopus laevis}\\ 17 17 \large{using Petri Nets in Higher Level Developmental Biology}} 18 18 \author{Rick van der Zwet, Tiago Borges Coelho \\ … … 24 24 \maketitle 25 25 \section{Abstract} 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. 26 Planar signaling is the process in which cells accumulate proteins based on the 27 saturation of nearby cells. If one cell produces n ammount of proteins, it will 28 initiate a transfering cascade to cells in the vicinity. This dissemination of 29 proteins will eventually cease, considering that n is a finite variable. There 30 is a gradation in the ammount of proteins transfered, meaning that neighbouring 31 cells get n/2 the ammount of proteins of the most saturated cell. 34 32 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. 33 XXX: Citing to the Bio Papers 34 XXX: Small introductions Petri-nets 39 35 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}. 36 \section{Approch} 37 First a PetriNet model will be defined textually and using graphs next the 38 modeling will be taking into practice using the modeling tool\emph{CPNTools}. 44 39 45 40 \section{Modeling} 46 To model this process we will take a modular approach using coloured Petri -Nets41 To model this process we will take a modular approach using coloured PetriNets 47 42 (see Fig~\ref{fig:model}), since the goal of this assignment is to have a 48 43 solution that can be applied to any configuration of cells. We start with a 49 bu ilding block that is an abstraction of a cell (figure: circle), which can then44 bulding block that is an abstraction of a cell (figure: circle), which can then 50 45 be coupled to other cells (figure: arrows). The abstraction contains two 51 46 different types. First the proteins are modelled (figure: red), secondly the 52 47 proteins (figure: blue) are leading in a second process of the creation of 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 48 gradients which also needs modeling. We assume a 1:1 mapping between the amount 49 of proteins and gradients -this taken into consideration- ones an \texttt{INITIAL} 50 protein is 'used' (e.g. has on posterisation counterpart) in this process it 57 51 get called \texttt{ACTIVATED}. We assume that the proteins to posterisation 58 52 process is taking place at the same time as the proteins distribution. And in a … … 68 62 posterisation present. 69 63 70 The connectors between the cells (the membra nes) has a special properly. One71 can see them as pressure valves others as si phons (see Fig~\ref{fig:pressure}).64 The connectors between the cells (the membrams) has a special properly. One 65 can see them as pressure valves others as sighons (see Fig~\ref{fig:pressure}). 72 66 The moment the 'volume' at complies with the following properly $A / 2 < B$ 73 67 then the pressure closes, else it passes volume from A to B at an certain rate 74 68 (\texttt{flowSpeed}). This rate could depend on the difference, actual value present 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. 69 or something else. 77 70 78 For the case there exists no standard Petri -Net 'component', hence this require71 For the case there exists no standard PetriNet 'component', hence this require 79 72 the creation of a new property (figure: $2:1$), with the following properties: 80 73 81 74 \begin{verbatim} 82 75 flowSpeed = n 83 if A > 2 * B and A => flowSpeedthen76 if A > 2 * B then 84 77 A = A - flowSpeed 85 78 B = B + flowSpeed 86 else if B > 2 * A and B => flowSpeedthen79 else if B > 2 * A then 87 80 B = B - flowSpeed 88 81 A = A + flowSpeed … … 91 84 92 85 Planar signaling could theoretically start in every cell, by 93 inserting some amount of prot eins. In our model represented as a bunch of94 \texttt{INITIAL} tokens be ing put in a random cell.86 inserting some amount of protiens. In our model represented as a bunch of 87 \texttt{INITIAL} tokens beeing put in a random cell. 95 88 96 89 \begin{figure}[htp] … … 113 106 114 107 One it the shortcoming of the 'balancing'. It does not allow reading of how 115 many tokens are present in a certain state and base action up on them. As108 many tokens are present in a certain state and base action uppon them. As 116 109 workaround for this (see Fig~\ref{fig:CPNplanar}) we used a 'dump' gradation 117 110 function. In our case it simply take 3 tokens and pushes 1 forward and … … 124 117 at the head or the tail. 125 118 126 In this implementation the prot eins to gradients process is taking place at127 cell $A$ at the same time that the proteins get transfer red from cell $A$ to119 In this implementation the protiens to gradiants process is taking place at 120 cell $A$ at the same time that the proteins get transfered from cell $A$ to 128 121 $B$. 129 122 130 123 Also it should be noted that it missing a notion of timed firing sequences; 131 124 meaning firing sequences which will occur at an certain time. This could for 132 example used to 'trigger' a tim ed activation of the \texttt{INITIAL} to125 example used to 'trigger' a timmed activation of the \texttt{INITIAL} to 133 126 \texttt{ACTIVATED} process as modeled in fig~\ref{fig:model}. An initial idea 134 is shown at fig ~\ref{fig:time-idea} in appendix~\ref{sec:timer-idea}.127 is shown at fig\ref{fig:time-idea} in appendix 1. 135 128 136 129 … … 145 138 146 139 \section{Conclusion} 147 Using Petri -Nets for modeling biology processes is a powerful framework, which148 could be well ex pandable. The Proof Of Concept implementations and140 Using PetriNets for modeling biology processes is a powerful framework, which 141 could be well extendable. The Proof Of Concept implementations and 149 142 visualisations how-ever are lacking. \emph{CPNTools} for example does not 150 provide a powerful enough tool-set for the modeling purposes.143 provide a powerfull enough toolset for the modeling purposes. 151 144 152 \bibliographystyle{amsalpha}153 145 \begin{thebibliography}{10} 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 146 % sing Petri Nets in Higher Level Developmental Biology: 147 % A case study on the AP axis development in Xenopus laevis 148 % Extended Abstract 149 % http://www.liacs.nl/~csbpn/COURSE%20DOCUMENTS/extended%20abstract%20Bertens%20Jansen%20Kleijn%20Koutny%20Verbeek.pdf 150 % Laura M.F. Bertens 151 152 % http://www.liacs.nl/~csbpn/ 153 % 154 % 155 157 156 \end{thebibliography} 158 \appendix 159 \section{Timer Idea} 160 \label{sec:timer-idea} 157 \section{*Appendix} 161 158 162 159 \begin{figure}[htp]
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