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Hybrid Petri net representation of Gene Regulatory Network Introduction Some models have been used to represent gene regulatory networks such as electrical circuit models, boolean networks and differential equations McAdams and Shapiro proposed a hybrid model that integrates conventional biochemical kinetic modeling within the framework of an electrical circuit simulation. In this paper, S Miyano attempts to use a hybrid form of Petri net to model gene regulatory pathways Why Hybrid Petri nets (HPN)? Petri nets can capture the basic aspects of concurrent systems conceptually and mathematically Hybrid Petri nets allow us to express explicitly the relationship between continuous values and discrete values while keeping the characteristics of ordinary Petri nets soundly (This aspect of continuity is not present in ordinary Petri nets) Other features, such as stochastic factors can also be included in the modeling So what is a Petri net? Petri net is a modeling tool that consists of places, transitions and arcs connecting them Places (circles) represents passive entities of the real world such as conditions, resources, waiting pools, channels, states etc Transitions (rectangle) represents active elements such as events and actions Arcs connects places to transitions or vice versa (note – place to place or transition to transition is a violation) and it represents the action or event a place element will participate and what will happen to it after the event Example Classic Example of the Producer/ Consumer problem Producer Consumer Hybrid Petri nets In Hybrid Petri nets, the concept of continuous variables are added in Now the places and transitions have 2 types each – discrete and continuous Definition 1 – We denote a hybrid Petri net as Q = (P, T, h, Pre, Post, M0) where P and T are the set of places and transitions respectively h : P U T → {D, C} indicates for every place and transition, whether or not it is a discrete or continuous one. A non negative integer called the number of tokens is associated with the discrete place and a non-negative real number called the mark is associated with a continuous place Discrete Place Continuous Place Discrete Transition Continuous Transition Hybrid Petri nets Pre(Pi, Tj) and Post(Pi, Tj) are functions that define arcs from place Pi to Tj and from Tj to Pi respectively It has the weight of a non-negative integer if h(Pi) = D and the weight of a non-negative real number if h(Pi) = C. The weights represent a ‘threshold’ value, e.g. The transition T1 will only fire if the mark of P1 is above 3.4134 T1 P1 3.4134 P2 2.7 A variable dTj called the delay time of Tj is assigned to each discrete transition Tj while a variable vTi, called the speed of Ti is assigned to each continuous transition Ti Example of HPN representing Gene Regulatory Network Transcription S1 mRNA S1 Protein S1 Gene S1 Translation S1 Transcription S2 Gene S2 Transcription S2 mRNA S2 Protein S2 λ-Phage switching mechanism λ-Phage is a virus that infects bacteria It is commonly used for applications such as DNA cloning and recombinant as it is completely safe for humans to work with, it is easy to grow, and also it’s genome is small and has already been completely sequenced and functions mapped One of the more commonly studied phenomena is its gene switching mechanism which determines whether or not a phage virus, after infecting a bacteria such as E.Coli, will follow the lytic pathway (where the bacterial cell will lyse and release a large number of newly synthesized virus) or a lysogenic (where the phage DNA is integrated into the bacterial DNA) pathway Diagram showing the Lytic and Lysogenic pathway Which pathway to take? Two regulatory proteins – CI and Cro plays a role in deciding which pathway the λ-Phage will take They are transcribed from genes cI and cRO which are adjacent to each other in the λ-Phage genome In between them is the operator OR which consists of 3 adjacent sites OR1, OR2 and OR3 Map of the λ-Phage DNA PRM turns on cI (Lambda repressor) and genes for integration and lysogeny PRM int xis cIII N cI OR3 OR2 OR1 cro cRII PR O P Q PR turns on cro and the genes of lytic pathway R ….. Role of CI and Cro For the protein CI, when present in certain quantities, it will bind to OR1 and OR2, switching off PR, causing the phage to go into lysogeny and integrate with the bacterial DNA If its concentration is increased, it will bind to OR3 and PRM will also be switched off Similarly for Cro, in certain concentrations, it will bind to OR3, switching off PRM and switching on PR, resulting in cell lysis If its concentration is increased, it will also bind to OR2 and OR1 switching off PR Binding of CI to OR1 and OR2 such that the RNA polymerase can only transcribe at PRM Table showing Proteins and Promoters Concentration UV CI Cro Sites of OR OR3 + ++ @ * OR2 Promoter OR1 PRM PR OFF ON ON OFF OFF OFF OFF ON + ++ OFF OFF * * * * OFF ON + and ++ shows the concentration levels of CI and Cro with ++ being more concentrated. shows that CI is binded to the site and shows that Cro is binded. @ shows that UV is present and * means irregardless of whether CI, Cro is present or any of the sites are binded with proteins, the promoter PR is going to be ON. HPN to show OR Continuous Places showing the concentration of CI and Cro CI A will fire first as it has a lower threshold for both cases BCI 0 ACI ARO CI will bind to OR1 and OR2 first before binding to OR3 OR3 Terminating transitions shows the degradation Shows the presence of UV UV which will inhibit CI Cro 1 Cyclic net shows the dynamics of binding and unbinding with 0 to mean no binding and 1 to mean binding OR3 is not binded, OR2 and OR1 are binded, turning PRM on PRM BRO OR1 OR2 0 1 0 PR Cro will bind to OR3 before binding to OR2 and OR1 1 Discrete Places to denote whether PRM and PR are on or off Hierarchical Feature Each of these HPNs can then be treated as a ‘black box’ The black box can then be inserted into other HPNs Feedback Mechanism of Cro and CI cI can be transcribed by either PRM or PRE activated by CII Anti - Cro If the concentration of CII is high (given by Pre(CII, ACII)), and the promoter PRE is going to be on, then concentration of CI keeps growing during the promoter PRM is on Cro mRNA Transcript initiated at PRE also include an antisense cro sequence which hybridizes with cro mRNA to prevent its translation When CI reaches to high, then PRM will be switched off CI CI is thus self regulated positively and negatively Similarly for Cro which will be produced continuously until it reaches overproduction CROE indicates the termination of transcription gene cro Cro PRE PRM ACII UV CII PR CROE Early Stage Gene Expression So in the same manner, the entire early gene expression of the λ-Phage can be represented using HPNs Results Matsuno, Nagasaki and Miyano has implemented the regulatory network using Visual Object Net++, a Petri-Net CAD/CAE tool Dynamics of the protein concentrations obtained from the simulation corresponds to the biological facts well Figure shows cases where concentrations of CII are different while CIII remains the same If concentration of CII is high, it reaches the threshold level to stimulate promoter PRE If concentration of CII is low, then promoters PRE and PRM is never turned on, instead PR is on, causing the concentration of Cro protein to keep increasing Conclusion Hybrid Petri nets can be a viable model to model biological pathways and simulation Graphical representation is quite similar to those used in biochemistry Can handle probabilistic factors as well Hierarchical Conclusion Compare this to this ……. ;(setq *trace-function-gen* t) ;(setq *trace-nsim* t) ; required by the model ;(defparameter *ODE-RELERR* 1.e-9) ;(defparameter *ODE-ABSERR* 0.0) ;(defparameter *rounding-epsilon* 1.0D-10) ;(defparameter *epsilon* 1.0D-6) change to propagate ; minimum ;(defparameter *absolute-epsilon* 1.e-6) ;(defparameter *ZERO-THRESHOLD* 1.0e-7) ; how close to measure for a 0 axis (defparameter *NsimBlurAbsEpsilon* 1.e-12) ;;; 1.e-7 changed for simulation (defun square (x) (expt x 2)) (defun x10Pow (x y) (* x (expt 10 y))) (defparameter K1 2d-8) (defparameter K2 3d-9) (defun K () (* K1 K2) ) (defun EF (x) (if (> x 0) (/ 1 (+ 1 (/ (K) (square x)))) 0) ) (defun EI (x) (if (> x 0) (sqrt (/ (* (K) x) (- 1 x))) 0) ) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ; this function runs display the readme file ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (defun readme () (format *QSIM-Report* "~2%~a~%~a~2%~a~%" (make-string 80 :initial-element #\*) So what’s next Add probabilistic/ stochastic features Model more complex organisms and extend to other pathways such as metabolic pathways, cell signaling etc. Automatic model construction by referring or reverse engineer from expression levels or gene sequence Consider also positions of genes, movement of cells e.g. using bigraphs etc. Build more robust tools to read and analyse such models (Currently the only software is Cell Illustrator from GNI) Thank you