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Networks in Metabolism and Signaling Edda Klipp Humboldt University Berlin Lecture / WS 2007/08 Petri Nets PowerPoint Presentation
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Networks in Metabolism and Signaling Edda Klipp Humboldt University Berlin Lecture / WS 2007/08 Petri Nets. Petri Nets: Literature. Petri. Petri Nets Invention. Reminder: Stoichiometry. Stoichiometric matrix. Vector of metabolite concentrations. Vector of reaction rates. Parameter vector.

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slide1
Networks in Metabolism and Signaling Edda Klipp Humboldt University BerlinLecture / WS 2007/08Petri Nets
reminder stoichiometry
Reminder: Stoichiometry

Stoichiometric matrix

Vector of metabolite concentrations

Vector of reaction rates

Parameter vector

Systems equations in matrix form

In steady state:

K represents the basis vector for all possible steady state fluxes.

petri nets general remarks
Petri Nets – General Remarks

Structural Properties

Invariants

Dynamical Properties

Simulation

Petri

Nets

Models

Hypotheses Representation and Generation

Interaction Patterns

Knowledge/Query Representation

petri nets definitions
Petri Nets – Definitions

Petri nets are bipartitedirectedmulti-graphs, i.e.,

they consist of

- two types of nodes, called places and transitions, and

- directed arcs, which are weighted by natural numbers and connect only nodes of different type.

node

transition

e.g. metabolites and reactions

petri nets definitions8
Petri Nets – Definitions

Examples for places:

passive system elements as conditions, states,

or biological species, i.e., chemical compounds as proteins.

Examples for transitions:

active system elements such as events, or chemical reactions such as activation or deactivation

petri nets definitions9
Petri Nets – Definitions

The arcs in the net describe the causal relation between active

and passive elements. They are illustrated as arrows. They can be labeled with their weight (if appropriate).

petri nets definitions10
Petri Nets – Definitions

A Petri net is a 5-tuple, PN = (P,T,F,W,M0) where:

is a finite set of places,

is a finite set of transitions,

is a set of arcs (flow relations),

is a weight function,

is the initial marking

A Petri net structure N = (P,T,F,W) without any specific initial marking

is denoted by N.

A Petri net with the given initial marking is denoted by (N,M0).

tokens as dynamic elements
Tokens as Dynamic Elements

Arcs connect an event with its preconditions, which must be fulfilled to trigger this event, and with its postconditions, which will be fulfilled, when the event takes place.

The fulfillment of a condition is realized via tokens residing in places. Principally, a place in a discrete net may carry any integer number of tokens, indicating different degrees of fulfillment.

If all preplaces of a transition are marked sufficiently (corresponding to the arc weights) with tokens, this transition may fire.

If a transition fires, tokens are removed from all its preplaces and added to all its postplaces, each corresponding to the given arc weights.

tokens as dynamic elements12
Tokens as Dynamic Elements

In short: transitions fire, when enough tokens are present

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1

2

2

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2

2

Tokens are removed and add as indicated by arc weights.

tokens as dynamic elements13
Tokens as Dynamic Elements

If a condition must be fulfilled, but the firing of an adjacent transition does not remove any tokens from it, these nodes are connected via two converse arcs. Such arcs can be represented by bidirectional arrows and

called read arcs.

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1

e.g. enzyme or activator necessary to convert substrates into their products

2

2

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1

2

2

marking of a petri net
Marking of a Petri Net

A current distribution of the tokens over all places, usually given as

M € N0 , describes a certain system state and is called a marking of the net.

Accordingly, the initial marking M0 of a net describes the system

state before any transition has fired.

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1

2

2

marking of a petri net15
Marking of a Petri Net

The incidence matrix C of a given Petri net is an (n×m)-matrix (where n denotes the number of places and m the number of transitions). Every matrix entry cijgives the token change on the place pi by the firing of the transition tj .

The incidence matrix does not reflect read arcs.

Note: similarity to stoichiometric matrix for metabolic networks.

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1

2

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petri nets semantics16
Petri Nets – Semantics
  • Explicit representation of causality relations of events and states
  • Independent events are concurrent (nebenläufig)
  • Spatially and temporally non-sequential distributed systems
  • Hierarchical abstraction levels
  • System properties, system dynamics, proofs
t invariants
T-Invariants

A t-invariant is defined as a non-zero vector x € M0 , fulfilling

C . x = 0.

A t-invariant represents a multiset of transitions, which have altogether a zero effect on the marking, i.e., if all of them have fired the required number of times, a given marking is reproduced. The invariant property holds for an arbitrary initial marking.

A t-invariant is called realizable, if a marking is reachable, such that

all transitions of the t-invariant are able to fire in a suitable partial order.

Compare: steady state rates, NK=0

p invariants
P-Invariants

A p-invariant is defined as a non-zero vector y € M0 , fulfilling

y . C = 0.

A p-invariant characterizes a token conservation rule for a set of places, over which the weighted sum of tokens is constant independently from any firing, i.e., for a p-invariant y and any markings Mi, Mj € N0, which are reachable from M0 by the firing of transitions, it holds

y . Mi= y . Mj

Compare: conservation relations, GN=0

siphons traps deadlocks and liveness
Siphons, Traps, Deadlocks and Liveness

Siphon – place that marked once, remains so.

(Input transition set is included in output set)

Trap – place that once sufficiently marked, never loses all tokens.

(Output transition set included in input set)

Deadlock-free – if for any possible marking there is an enabled transition.

Deadlock – no more transition possible.

Liveness – from the initial marking is a marking reachable, such that a certain transition is enabled.

model of yeast pheromone pathway

a

Model of Yeast Pheromone Pathway

Pheromone

Plasma

membrane

Ste2

Receptor

activation

G protein

cycle

Gbg

MAPK

scaffold

Signaling

cascade

Fus3

Sst2

Far1Cdc28

Bar1

Ste12

Complex formation

Gene expression

slide27

Reconstructing the regulatory network controlling commitment and sporulation in Physarium polycephalum based on hierachical Petri Net modeling and simulation.

Marwan W, et al., 2005, JTB

slide28

Glucose consumption and starvation of a Physarum plasmodium represented by a Petri Net. Whether the plasmodium is fed or starved is indicated by a token in the respective place. When a starved plasmodium (a) is supplied with glucose (b), glucose is used up (c) and the plasmodium is fed (the token moves from the Starved place to the Fed place by switching of transition T1 which functions as logic AND). With time, the metabolic energy provided by the added glucose is used up and the plasmodium starves again (d). Putting more than one token (n>1) into the Glucose place, glucose consumption according to the model would proceed by cyclically running n-times through states (b), (c), (d) removing one token from the Glucose place in each cycle, while the two places indicating the mutually exclusive physiological states of the plasmodium, fed or starved, always are marked by a single token only.

slide29

Sensory control of sporulation in P. polycephalum represented as a Petri Net derived from physiological experiments with wild-type (a) and as a more detailed model (b), refined by including genes involved in sporulation

slide30

Modelling of a time-resolved somatic complementation experiment performed by fusion of two plasmodia carrying mutations at different sites of the sporulation control network: (a) Mutant plasmodia before fusion. The flow of tokens along the sporulation control network depends on the activity of transitions which by themselves are controlled by the cellular concentration of the wild-type gene product. In the α-plasmodium T1α is disabled due to a loss-of-function mutation in Gene 1. In the β-plasmodium T2β is disabled due to a loss-of-function mutation in Gene 2. In the α-plasmodium, a token cannot pass T1α. In the β-plasmodium, a token cannot pass T2β and consequently none of the two plasmodia can sporulate. (b) After fusion of the two plasmodia, their protoplasms and suspended nuclei mix due to the vigorous acto-myosin-powered protoplasmic shuttle streaming. The fluxes of tokens along the signalling pathways superimpose with the fluxes of tokens between the plasmodial halves. Note that the tokens representing the gene products do not enter the sporulation control network, since they are connected to the transitions they regulate via test-arcs. In the example shown, the token which was trapped in P1α before fusion occurred (panel A) has moved in the fused plasmodium (panel B) to P2β via P1β and T1β. It could also have moved through T1α provided this transition became activated by supply of (complementation with) the gene product, which was missing before plasmodial fusion occurred.