1. Hybrid Petri net representation of Gene Regulatory Network

2. 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

3. 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

4. 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

5. Example
Classic Example of the Producer/ Consumer problem

6. 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

7. 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
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

8. Example of HPN representing Gene Regulatory Network

9. 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

10. Diagram showing the Lytic and Lysogenic pathway

11. 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

12. Map of the λ-Phage DNA

13. 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

14. Table showing Proteins and Promoters
Concentration
Sites of OR
Promoter UV CI
Cro
OR3
OR2
OR1
PRM PR
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 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.

15. HPN to show OR UV CI OR3 OR1 OR2 1 1 1 PRM PR
Shows the presence of UV which will inhibit CI
Continuous Places showing the concentration of CI and Cro UV CI
Cro
A will fire first as it has a lower threshold for both cases
BCI
ACI
ARO
BRO
CI will bind to OR1 and OR2 first before binding to OR3
Cro will bind to OR3 before binding to OR2 and OR1
OR3
OR1
OR2
Terminating transitions shows the degradation 1 1 1
OR3 is not binded, OR2 and OR1 are binded, turning PRM on
Cyclic net shows the dynamics of binding and unbinding with 0 to mean no binding and 1 to mean binding
Discrete Places to denote whether PRM and PR are on or off
PRM PR

16. Hierarchical Feature
Each of these HPNs can then be treated as a ‘black box’
The black box can then be inserted into other HPNs

17. Feedback Mechanism of Cro and CI
cI can be transcribed by either PRM or PRE activated by CII
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
Transcript initiated at PRE also include an anti-sense cro sequence which hybridizes with cro mRNA to prevent its translation
When CI reaches to high, then PRM will be switched off
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
Anti - Cro
Cro mRNA CI UV
Cro
PRM PR
PRE
CROE
ACII
CII

18. Early Stage Gene Expression
So in the same manner, the entire early gene expression of the λ-Phage can be represented using HPNs

19. 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

20. 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

21. Conclusion Vs 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) ; minimum change to propagate
;(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 #\*) Vs

22. 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)

23. Thank you