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

ExampleClassic 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

3.4134 2.7

P1 P2T1

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

Gene S1

Gene S2

Transcription S1

Transcription S2

mRNA S1

mRNA S2

Translation S1

Transcription S2

Protein S1

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

cI croOR1OR2OR3 cRII O P Q RNcIIIxisint …..

PRM

PR

PR turns on cro andthe genes of lytic

pathway

PRM turns on cI(Lambda repressor)

and genes forintegration and

lysogeny

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

+ 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.

Concentration Sites of OR Promoter

UV CI Cro OR3 OR2 OR1 PRM PR

OFF ON

+ ON OFF

++ OFF OFF

+ OFF ON

++ OFF OFF

@ * * * * * OFF ON

HPN to show OR

CI Cro

Continuous Places showing the concentration of CI and Cro

PRM PR

Discrete Places to denote whether PRM and PR are on or off

UV

Shows the presence of UV which will inhibit CI

OR1

ACI

OR2OR3

BCI ARO BRO

10 10 10

A will fire first as it has a lower threshold for both cases

CI will bind to OR1 and OR2 first before binding to OR3

Cro will bind to OR3 before binding to OR2 and OR1

Terminating transitions shows the degradation

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

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

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

CIUV

Cro

PRM PR

Anti - Cro

CII

CROE

PRE

ACII

Cro mRNA

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

…….

;(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) )

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; this function runs display the readme file

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(defun readme ()

(format *QSIM-Report* "~2%~a~%~a~2%~a~%"

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to this

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