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r18 r25 14 14 \usepackage{amssymb,amsmath} 15 15 16 \title{ DRAFT:Modeling planar signalling in AP axis development in \emph{Xenopus laevis}\\16 \title{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 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. 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. 32 34 33 XXX: Citing to the Bio Papers 34 XXX: Small introductions Petri-nets 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. 35 39 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}. 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}. 39 44 40 45 \section{Modeling} 41 To model this process we will take a modular approach using coloured Petri Nets46 To model this process we will take a modular approach using coloured Petri-Nets 42 47 (see Fig~\ref{fig:model}), since the goal of this assignment is to have a 43 48 solution that can be applied to any configuration of cells. We start with a 44 bu lding block that is an abstraction of a cell (figure: circle), which can then49 building block that is an abstraction of a cell (figure: circle), which can then 45 50 be coupled to other cells (figure: arrows). The abstraction contains two 46 51 different types. First the proteins are modelled (figure: red), secondly the 47 52 proteins (figure: blue) are leading in a second process of the creation of 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 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 51 57 get called \texttt{ACTIVATED}. We assume that the proteins to posterisation 52 58 process is taking place at the same time as the proteins distribution. And in a … … 62 68 posterisation present. 63 69 64 The connectors between the cells (the membra ms) has a special properly. One65 can see them as pressure valves others as si ghons (see Fig~\ref{fig:pressure}).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}). 66 72 The moment the 'volume' at complies with the following properly $A / 2 < B$ 67 73 then the pressure closes, else it passes volume from A to B at an certain rate 68 74 (\texttt{flowSpeed}). This rate could depend on the difference, actual value present 69 or something else. 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. 70 77 71 For the case there exists no standard Petri Net 'component', hence this require78 For the case there exists no standard Petri-Net 'component', hence this require 72 79 the creation of a new property (figure: $2:1$), with the following properties: 73 80 74 81 \begin{verbatim} 75 82 flowSpeed = n 76 if A > 2 * B then83 if A > 2 * B and A => flowSpeed then 77 84 A = A - flowSpeed 78 85 B = B + flowSpeed 79 else if B > 2 * A then86 else if B > 2 * A and B => flowSpeed then 80 87 B = B - flowSpeed 81 88 A = A + flowSpeed … … 84 91 85 92 Planar signaling could theoretically start in every cell, by 86 inserting some amount of prot iens. In our model represented as a bunch of87 \texttt{INITIAL} tokens be eing put in a random cell.93 inserting some amount of proteins. In our model represented as a bunch of 94 \texttt{INITIAL} tokens being put in a random cell. 88 95 89 96 \begin{figure}[htp] … … 106 113 107 114 One it the shortcoming of the 'balancing'. It does not allow reading of how 108 many tokens are present in a certain state and base action up pon them. As115 many tokens are present in a certain state and base action upon them. As 109 116 workaround for this (see Fig~\ref{fig:CPNplanar}) we used a 'dump' gradation 110 117 function. In our case it simply take 3 tokens and pushes 1 forward and … … 117 124 at the head or the tail. 118 125 119 In this implementation the prot iens to gradiants process is taking place at120 cell $A$ at the same time that the proteins get transfer ed from cell $A$ to126 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 121 128 $B$. 122 129 123 130 Also it should be noted that it missing a notion of timed firing sequences; 124 131 meaning firing sequences which will occur at an certain time. This could for 125 example used to 'trigger' a tim med activation of the \texttt{INITIAL} to132 example used to 'trigger' a timed activation of the \texttt{INITIAL} to 126 133 \texttt{ACTIVATED} process as modeled in fig~\ref{fig:model}. An initial idea 127 is shown at fig \ref{fig:time-idea} in appendix 1.134 is shown at fig~\ref{fig:time-idea} in appendix~\ref{sec:timer-idea}. 128 135 129 136 … … 138 145 139 146 \section{Conclusion} 140 Using Petri Nets for modeling biology processes is a powerful framework, which141 could be well ex tendable. The Proof Of Concept implementations and147 Using Petri-Nets for modeling biology processes is a powerful framework, which 148 could be well expandable. The Proof Of Concept implementations and 142 149 visualisations how-ever are lacking. \emph{CPNTools} for example does not 143 provide a powerful l enough toolset for the modeling purposes.150 provide a powerful enough tool-set for the modeling purposes. 144 151 152 \bibliographystyle{amsalpha} 145 153 \begin{thebibliography}{10} 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 % 155 154 \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 156 157 \end{thebibliography} 157 \section{*Appendix} 158 \appendix 159 \section{Timer Idea} 160 \label{sec:timer-idea} 158 161 159 162 \begin{figure}[htp]
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