Humboldt- Universität zu Berlin Edda Klipp Systembiologie 8 – Petri Nets Sommersemester 2010...
-
Upload
marybeth-lucas -
Category
Documents
-
view
213 -
download
0
Transcript of Humboldt- Universität zu Berlin Edda Klipp Systembiologie 8 – Petri Nets Sommersemester 2010...
Humboldt-Universität
zu Berlin
Edda Klipp
Systembiologie 8 – Petri Nets
Sommersemester 2010
Humboldt-Universität zu BerlinInstitut für BiologieTheoretische Biophysik
Humboldt-Universität
zu Berlin
Petri Nets: Literature
Petri
Humboldt-Universität
zu Berlin
Petri Nets Invention
Humboldt-Universität
zu Berlin
Reminder: Stoichiometry
Stoichiometric matrix
Vector of metabolite concentrations
Vector of reaction rates
Parameter vector
ijnN ni ...1 rj ...1
T1,..., nSSS
T1 rvv ,...,v T
1,..., mppp
Systems equations in matrix formd
d
Sv
tN
00 NKNvIn steady state:
K represents the basis vector for all possible steady state fluxes.
Humboldt-Universität
zu Berlin
Petri Nets – General Remarks
PetriNets
Structural PropertiesInvariants
Dynamical PropertiesSimulation
ModelsHypotheses Representation and Generation
Interaction PatternsKnowledge/Query Representation
Humboldt-Universität
zu Berlin
Petri Nets – Semantics
Humboldt-Universität
zu Berlin
Petri Nets – Definitions
Petri nets are bipartite directed multi-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.
e.g. metabolites and reactions
node
transition
Humboldt-Universität
zu Berlin
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
Humboldt-Universität
zu Berlin
Petri Nets – Definitions
The arcs in the net describe the causal relation between activeand passive elements. They are illustrated as arrows. They can be labeled with their weight (if appropriate).
Humboldt-Universität
zu Berlin
Petri Nets – Definitions
A Petri net is a 5-tuple, PN = (P,T,F,W,M0) where:
TPTP
,...,,PM
,...,,FW
PTTPF
,...,,T
,...,,P
and
210:
321:
0
21
21
n
m
ttt
ppp 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).
Humboldt-Universität
zu Berlin
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.
Humboldt-Universität
zu Berlin
Tokens as Dynamic Elements
In short: transitions fire, when enough tokens are present
1
2
3
2
1
2
3
2
Tokens are removed and add as indicated by arc weights.
Humboldt-Universität
zu Berlin
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. 1
2
3
2
1
2
3
2
e.g. enzyme or activator necessary to convert substrates into their products
Humboldt-Universität
zu Berlin
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 systemstate before any transition has fired.
1
2
3
2
Humboldt-Universität
zu Berlin
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 cij gives 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.
1
2
3
2
2
3
2
1
C
Humboldt-Universität
zu Berlin
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
Humboldt-Universität
zu Berlin
Types of Petri Nets
Humboldt-Universität
zu Berlin
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 thatall transitions of the t-invariant are able to fire in a suitable partial order.
Compare: steady state rates, NK=0
Humboldt-Universität
zu Berlin
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
Humboldt-Universität
zu Berlin
Self-loops
Humboldt-Universität
zu Berlin
Autocatalysis
Humboldt-Universität
zu Berlin
Reversible Reactions
A+B C+D
AB C
D
Humboldt-Universität
zu Berlin
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.
Humboldt-Universität
zu Berlin
Model of Yeast Pheromone Pathway
Ste2
G
Fus3 Sst2
Ste12Bar1
MAPKscaffold
Far1Cdc28
Plasma membrane
Gene expression Complex formation
Signalingcascade
G proteincycle
Receptor activation
Pheromone
Humboldt-Universität
zu Berlin
Model of Yeast Pheromone Pathway
Sackmann et al., 2006
Humboldt-Universität
zu Berlin
Model of Yeast Pheromone Pathway
Sackmann et al., 2006
Humboldt-Universität
zu Berlin
Marwan W, et al., 2005, JTB
Reconstructing the regulatory network controlling commitment and sporulation in Physarium polycephalum based on hierachical Petri Net modeling and simulation.
Humboldt-Universität
zu Berlin
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.
Humboldt-Universität
zu Berlin
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
Humboldt-Universität
zu Berlin
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.