Petri Nets, Hybrid Functional Petri Nets

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Petri Nets, Hybrid Functional Petri Nets

• CS 6280 Lecture 2
• P.S. Thiagarajan

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Petri nets

• A well-established system model in CS.
• distributed discrete event systems.
• Has an appealing graphical representation (for
  small models!).
• Many variants
• Has been studied for a long time
• Applied in many settings
• http//www.informatik.uni-hamburg.de/TGI/PetriNets
  /

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System Biology Applications.

• Many !
• Ordinary Petri nets
• Metabolic pathway analysis
• Hybrid Functional Petri nets.
• signaling pathways
• Supported by the tool Cell Illustrator
• Stochastic Petri nets, Colored Petri nets,

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Petri Nets

• Finite State Machines (FSMs)
• (Global) states
• (Global) transitions
• Also called (finite) transition systems.
• Petri nets
• Local states
• Local transitions.

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FSM
Turn-on
Off
On
Turn-off
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State
a/a
Action Input/output
b/b
c
d
Transition
Initial state
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Key_on / Strt_Timer
wait
end_timer / alarm_on
Key_off OR belt_on / reset_timer
alarm
off
Alarm_10 OR belt_on OR key_off / alarm_off
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Petri Nets Local States and Transitions
a
s
s
a
s
s
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Local States and Transitions
s
e
The (local) state s currently holds. The event e
(labeled a) can occur when s holds. When e occurs
, s ceases to hold and s begins to hold. This
can be generalized in many ways!
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Transitions in Petri nets
b3
b1
b4
e
b2
b5

• e can occur when b1 and b2 hold.
• When e occurs b1 and b2 cease to hold b3, b4
  and b5
• begin to hold.

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Transitions in Petri nets
b3
b1
b4
e
b2
b5

• e can occur when b1 and b2 hold.
• When e occurs b1 and b2 cease to hold b3, b4
  and b5
• begin to hold.

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Transitions in a Petri net.
b3
b1
b4
e
b2
b5

• e can occur when b1 and b2 hold.
• When e occurs b1 and b2 cease to hold b3, b4
  and b5
• begin to hold.

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Conflict
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• But only one of them can occur (due to the
  shared b2)
• Conflict!

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Conflict
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• But only one of them can occur (due to the
  shared b2)
• Conflict!

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Conflict
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• But only one of them can occur (due to the
  shared b2)
• Conflict!

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Conflict
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• But only one of them can occur (due to the
  shared b2)
• Conflict!

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Concurrency

• Both e and e are enabled.
• They can occur independently (in any order
  with no
• order,..)
• Concurrency!

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Markings
b3
b1
b4
e
b2
b5

• e can occur.

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Markings
b3
b1
b4
e
b2
b5

• e can occur.

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Markings
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• Both of them can occur one after another or
  simultaneously with no order.

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Markings
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• Both of them can occur one after another or
  simultaneously with no order.

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Markings
b3
b1
b4
e
b2
b5
e
b6
b7

• Both e and e are enabled.
• Both of them can occur one after another or
  simultaneously with no order.

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Self Loops
Producer
consumer
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Self Loops
Producer
consumer
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Self Loops
Producer
consumer
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Self Loops
Producer
consumer
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Arc Multiplicities

• t can occur whenever p1 contains at least 2
  tokens and p2 contains at least one token
• When t occurs (fires) 2 tokens are removed from
  p1, 1 token from p2 3 tokens are added to p3, 1
  tokens to p4, and 2 tokens to p5

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Arc Multiplicities
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Arc Multiplicities
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Petri nets

• PN (P, T, F, Min)
• P, Places ------
• T, Transitions ---
• F, Flow Relations -
• Min P ? N 0,1,2,... initial marking (state)

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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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An example
V2
V3
V1
S1
2S2
V4
S3
V3
S3
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• A Calcium (Ca2) channel is an ion channel which
  displays selective permeabiltiy to calcium ions.
• Inositol triphosphate (IP3) can open a Ca2
  channel on ER and help pumping Ca2 from outside
  to the cytosol. 

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Upside

• Concurrency modeling is important for producing
  succinct descriptions..
• Petri nets are simple, expressive and capture the
  basic concepts of causality, conflict and
  concurrency elegantly.
• They have an attractive visual representation
• A picture is worth a thousand words.
• But..

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Downside

• Petri net do not handle data abstractions well.
• Colored Petri nets.
• They do not support hierarchical structures well.
• Crucial for scaling up!
• These features can be added but
• Must preserve quantitative analysis methods!

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Additional Features

• Inhibitory arcs
• Read-only arcs
• Priorities
• Places/transitions/tokens with time.

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Inhibitory Arcs
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Inhibitory arcs increase the modeling
power! Useful for pathway modeling Gene regulation
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Read (only) Arcs
2
2
Inhibitory arcs increase the modeling
power! Useful for pathway modeling (Enzymes!)
2
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Hybrid Functional Petri Nets

• HFPNs considerably extend the features of Petri
  nets
• Increased modeling power/decreased analysis
  prospects.
• Front end of Cell Illustrator
• Easy to translate structural (cartoon) diagrams
  of bio-pathways to an HFPN model.

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Hybrid Functional Petri Nets
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Hybrid Functional Petri Nets

• Input Arcs (place to transition)
• normal
• From a discrete place to a discrete transition
• Continuous place to a discrete transition

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Hybrid Functional Petri Nets

• Input Arcs (place to transition)
• Continuous
• Continuous place to a continuous transition

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Hybrid Functional Petri Nets

• Input Arcs (place to transition)
• Test arc / Inhibition arc
• From any type of place to any type of transitions
• Test arc (with weight w) allows firing iff
  content of input place is equal or more than w
• Inhibitory arc (with weight w) allows firing iff
  content of input place is less than w

Test arcs
Inhibitory arcs
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Hybrid Functional Petri Nets

• Output Arcs (transition to place)
• Discrete
• From a discrete transition to any kind of place
• Continuous
• Continuous transition to a continuous place

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Firing a discrete transition

• A Discrete transition T of HFPN
• Normal/Test input arcs a1,..,ap from places
  P1,,Pp
• Normal output arcs b1,,bq to places Q1,,Qq
• m1(t),,mp(t) contents of input places and
• n1(t),,nq(t) contents of output places
• Arc weight
• fi(m1(t),,mp(t)) gt 0 number of tokens to be
  removed from Pi through arc ai
• gj(m1(t),,mp(t)) gt 0 number of tokens to be
  added to Qj through arc bj

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Firing a discrete transition
2
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Firing a discrete transition
2
1
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Firing a discrete transition

• A Discrete transition T of HFPN
• Discrete/Test input arcs a1,..,ap from places
  P1,,Pp
• Discrete output arcs b1,,bq to places Q1,,Qq
• m1(t),,mp(t) contents of input places
• n1(t),,nq(t) contents of output places
• Delay function
• d(m1(t),,mp(t)) a non-negative integer value
  function
• Arc weight
• fi(m1(t),,mp(t)) gt 0 number of tokens to be
  removed from Pi through arc ai
• gj(m1(t),,mp(t)) gt 0 number of tokens to be
  added to Qj through arc bj

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Firing a discrete transition
2
1
3
Delay function
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Firing a discrete transition
After 3 seconds
2
1
3
Delay function
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Firing a continuous transition

• A Continuous transition T of HFPN
• Continuous/Test input arcs a1,..,ap from places
  P1,,Pp
• Continuous output arcs b1,,bq to places Q1,,Qq
• m1(t),,mp(t) contents of input places and
• n1(t),,nq(t) contents of output places
• Firing condition given by predicate
• c(m1(t),,mp(t))
• As long as predicate is true, transition fires
  continuously
• Firing speed
• fi(m1(t),,mp(t)) gt 0 speed of consumption from
  Pi through ai
• gj(m1(t),,mp(t)) gt 0 speed of production to Qj
  through bj

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Firing a continuous transition
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Firing Transitions
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Michaelis-Menton Kinetics
k1
k2
E S
ES
E P
k -1
mass action law model
dS/dt -k1 ES k-1 (ES)
dES/dt k1 E.S (k-1 k2) ES
dE/dt -k1ES (k-1 k2) ES
dP/dt k2 ES
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The HPFN model
E
S
P
k1.SE
ES
k2ES
k-1ES
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The HPFN model of the Michaelis-Menten
Approximation
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Hybrid Functional Petri Nets

• Enhances modeling power
• Protein concentration changes
• Discrete transcriptional switches
• Reduced analytic capabilities
• Reachability analysis, S-invariants,
  T-invariants etc. become irrelevant.
• Exposes the structural relationsships, locality
  and flow of information in bio-pathways.

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Other variants

• Stochastic Petri nets
• Transitions occur probabilistically.
• Will consider them when we deal with stochastic
  simulations (Gillespies algorithm)
• Colored Petri nets
• Process algebras.

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