Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Models and Languages for ComputationalSystems Biology – Lecture 16

Quantitative Analysis of Petri Nets

Jane Hillston.LFCS and CSBE, University of Edinburgh

10th March 2011

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Outline

Introduction

Quantified Analysis – Discrete case

Quantified Analysis — Continuous case

References

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Formal definition of a stochastic Petri net

A Petri net is a quintuple N = (P,T , f , v ,m0), where

• P and T are finite, non empty and disjoint sets.P is the set of places and T is the set of transitions.

• f , sometimes called the flow relation, is a functionf : ((P × T) ∪ (T × P)) −→ N0 which defines the set ofdirected arcs, weighted by nonegative integer values.

• v : T −→ H, a function which assigns a stochastic hazardfunction ht to each transition t , where H is the set of all

stochastic hazard functions: H :=⋃

t∈T

{ht | ht : N|

•t |0 −→ R+

}and v(t) = ht for all transitions t ∈ T .

• m0 : P −→ N0 is the initial marking.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Marking dependent firing rates

What the stochastic hazard function ht does is define amarking-dependent rate λt(m) for firing the transition t .

The dependency for firing transition t is restricted to those placeswhich are input places for the transition, i.e. the set •t .

H :=⋃t∈T

{ht | ht : N|

•t |0 −→ R+

}

In general, in stochastic Petri nets more general forms of markingdependency are sometimes allowed, but for systems biology thisrestriction makes sense as it says the rate of a reaction can onlydepend on its reactants.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Timed transition firing

The marking dependent rate is generally assumed to be theparameter of an exponential distribution, meaning that the delayassociated with the firing of a transition is exponentially distributed.

This delay has probability density function:

fXt (τ) = λt(m) · e(−λt (m)·τ) τ ≥ 0

As mentioned earlier, firing remains instantaneous — after thetransition is enabled and before any tokens are consumed orproduced.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Timed transition firing

When a timed transition becomes enabled, we imagine that thesystem now has potential to carry out the corresponding reaction.

This potential will remain until either the reaction occurs or thepotential is removed or interrupted because another reactionoccurs (when transitions are in conflict).

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Timed transition firing

We assume that each transition has a timer or local clockassociated with it: when the transition is enabled the timer is set toa value.

Since we are using a random variable to represent activitydurations, this value is sampled from the exponential distributionwith the appropriate marking dependent rate.

While the transition is enabled we assume that the timerdecrements at a constant speed.

When the timer reaches zero the transition fires.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Timed transition firing

In order to make sense of this interpretation we need to think of thesame reaction, but with a different number of molecules availableto join the reaction, as a distinct instance of the reaction.

This corresponds in the Petri net to the transition remainingenabled after it has fired.

However, in general, the transition’s enabling degree will havechanged, and so its marking dependent firing rate will also havechanged, and we think of it as a fresh instance of the transition.

A new sample will be drawn from the distribution, now governed bythe new marking dependent rate.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Mapping to a continuous time Markov chain

The relationship between the SPN and an underlying CTMC isessentially the same as for an SPA model.

The firing semantics give rise to a labelled transition system, whichin this case is called the reachability graph, which can then beinterpreted as a CTMC.

Given a Petri net model (complete with initial marking):

• we associate a state in the CTMC with every marking in thereachability graph of the Petri net;

• we associate an event, or transition, in the CTMC with eachfiring of a transition in the Petri net which causes thecorresponding change of marking.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Molecular stochastic mass-action hazard function

Just as with the functional rates in Bio-PEPA the exact form of thehazard function in a SPN will depend on the interpretation andanalysis which is be done on the SPN.

For example, if token are interpreted as molecules and we plan tosolve the state space explicitly or use a Gillespie stochasticsimulation algorithm, then we will wish to have reactions governedby mass action kinetics.

ht := ct ·∏p∈•t

(f(p, t)m(p)

)

where ct is the reaction/transition specific stochastic rate constant,and m(p) is the current number of tokens on the preplace p oftransition t .

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Stochastic level hazard function

In an alternative interpretation, tokens can also be regarded asdiscretised levels of concentration (cf. Bio-PEPA).

We assume that species levels range over 0, . . . ,N, each onerepresenting an equivalence class of (infinitely many) continuousstates.

Again based on mass action kinetics we get:

ht := kt · N ·∏p∈•t

(m(p)N

)where kt is the reaction/transition specific deterministic rateconstant, and N is the maximum level of concentration.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Qualitative analysis of quantitative models

In the qualitative analysis we made various conclusions about thepossible behaviours of the system based on the structure of thePetri net.

This ignored any dynamics and so the question arises of whetherthe conclusions drawn from qualitative analysis are still valid whendelays are associated with the firing of transitions.

Fortunately the answer to this question is positive as long as thedistribution governing the firing of transitions has infinite support.

The support of a distribution is those values over which thedistribution has a positive value.

For an exponential distribution the support of the distribution is(0,∞), i.e. infinite.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Qualitative analysis of quantitative models

Some structural properties, such as reversibility tell us usefulproperties of the CTMC.

For example:

• boundedness ensures that the CTMC is finite;

• reversibility ensures that the CTMC is ergodic — that it willhave a steady state behaviour.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Quantitative analysis

Just as with Bio-PEPA, the CTMC underlying an SPN model maybe

• solved numerically,

• simulated using Gillespie’s stochastic simulation algorithm,

• subjected to CSL model checking, either based on explicitstate representations or via statistical model checking.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Continuous Petri nets

In a continuous Petri net the marking function of places is nolonger integer firings are no longer discrete.

Instead the marking of a place is a positive real number, termedthe token value.

In the systems biology context this continuous value is interpretedas the concentration of the molecular species associated with thatplace.

Moreover, the instantaneous firing of a transition is now acontinuous flow.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Formal definition of a continuous Petri net

A continuous Petri net is a quintuple N = (P,T , f , v ,m0), where

• P and T are finite, non empty and disjoint sets.P is the set of continuous places and T is the set ofcontinuous transitions.

• f , the flow relation, is a function f : ((P × T) ∪ (T × P)) −→ N0

which defines the set of directed arcs, weighted by nonegativereal values.

• v : T −→ H, a function which assigns a firing rate function ht

to each transition t , where H is the set of all firing rate

functions: H :=⋃

t∈T

{ht | ht : R|

•t |0 −→ R

}and v(t) = ht for all

transitions t ∈ T .

• m0 : P −→ R+0 is the initial marking.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Firing rate function

The firing rate function ht defines the marking dependentcontinuous transition rate for transition t .

As in the stochastic case the domain of ht is restricted to the set ofpre-places of t .

Technically any mathematical function which satisfies thisconstraint can be used for ht , but in practice in the systems biologycontext the functions used are typically those reflecting massaction, Michaelis-Menten or hill kinetics.

Reversible reactions may be modelled by a function whichsometimes assumes a negative rate.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Firing semantics

A continuous marking is a place vector with real values, and m(p)is the token value for place p in marking m.

A continuous transition is enabled in m, if ∀p ∈ •t , m(p) > f(p, t).

Unlike the stochastic case, there is no delay associated with thefiring of a continuous transition, and the transition starts to fire assoon as it is enabled.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Firing semantics

The semantics of a continuous Petri net is given by a set ofordinary differential equations (ODEs).

In the ODEs one variable is associated with the token value in onecontinuous place, and the ODEs record the impact of eachcontinuous transition on that token value.

Each place p has an associated rate equation:

d m(p)dt

=∑t∈•p

f(t , p)v(t) −∑t∈•p

f(p, t)v(t)

Each equation corresponds to a line in the incidence matrix.

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

Quantitative evaluation

Once the set of ODEs corresponding to a continuous Petri nethave been derived they can be solved to explore the dynamicbehaviour of the system.

Since this is a deterministic representation of the system there isno variability and the model approximates the expected behaviourof the stochastic representation.

If there are transitions with more than one pre-place (reactions withmore than one reactant) the system of ODEs will be non-linear,meaning that in almost all cases solution is via numericalsimulation (just as with Bio-PEPA).

Introduction Quantified Analysis – Discrete case Quantified Analysis — Continuous case References

References

M. Heiner, D. Gilbert and R. DonaldsonPetri Nets for Systems and Synthetic Biologyin Formal Methods for Computational Systems Biology, LNCSVolume 5016, pp. 215–264, June 2008.

http://genome.ib.sci.yamaguchi-u.ac.jp/˜pnp