Transcript Hybrid Petri Nets

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
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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
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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
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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?
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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
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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
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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
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If its concentration is increased, it will bind to OR3 and PRM will also
be switched off
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Similarly for Cro, in certain concentrations, it will bind to OR3,
switching off PRM and switching on PR, resulting in cell lysis
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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
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Each of these HPNs can then be treated
as a ‘black box’
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The black box can then be inserted into
other HPNs
Feedback Mechanism of Cro and CI

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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
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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
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Similarly for Cro which will be produced
continuously until it reaches overproduction
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CROE indicates the termination of transcription
gene cro
Cro
PRE
PRM
ACII
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UV
CII
PR
CROE
Early Stage Gene Expression
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So in the same manner, the entire
early gene expression of the λ-Phage
can be represented using HPNs
Results
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Matsuno, Nagasaki and Miyano has
implemented the regulatory network using
Visual Object Net++, a Petri-Net CAD/CAE
tool
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Dynamics of the protein concentrations
obtained from the simulation corresponds to
the biological facts well
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Figure shows cases where concentrations of
CII are different while CIII remains the same
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If concentration of CII is high, it reaches the
threshold level to stimulate promoter PRE
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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
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Hybrid Petri nets can be a viable model to model
biological pathways and simulation
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Graphical representation is quite similar to those
used in biochemistry
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Can handle probabilistic factors as well
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Hierarchical
Conclusion
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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
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Add probabilistic/ stochastic features

Model more complex organisms and extend to other pathways such
as metabolic pathways, cell signaling etc.
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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.
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Build more robust tools to read and analyse such models (Currently
the only software is Cell Illustrator from GNI)
Thank you