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PN & Systems Biology
May 2005
MSBF, MAY 2005
TO IONS
H NALYSIS
y Cottbus
FROM PETRI NETS DIFFERENTIAL EQUAT
AN INTEGRATIVE APPROAC
FOR BIOCHEMICAL NETWORK A
Monika Heiner
Brandenburg University of Technolog
Dept. of CS
PN & Systems Biology
May 2005
MODEL- BASED SYSTEM ANALYSIS
systemproperties
modelproperties
Petrinetzmodel
Problemsystem
PN & Systems Biology
May 2005
MODEL- BASED SYSTEM ANALYSIS
systemproperties
modelproperties
ementication
Petrinetzmodel
Problemsystem
technicalsystem
requirspecif
verificationCONSTRUCTION
PN & Systems Biology
May 2005
MODEL- BASED SYSTEM ANALYSIS
systemproperties
modelproperties
wn
nown perties
perties
Petrinetzmodel
Problemsystem
biologicalsystem
kno
unkpro
provalidation
behaviour prediction
UNDERSTANDING
PN & Systems Biology
May 2005
BIONETWORKS, SOME PROBLEMS
-> PROBLEM 1
-> PROBLEM 2
ls, . . .
-> PROBLEM 3
NS << - -
❑ various, mostly ambiguous representations
-> verbose descriptions
-> diverse graphical representations
-> contradictory and / or fuzzy statements
❑ knowledge
-> uncertain
-> growing, changing
-> distributed over various data bases, papers, journa
❑ network structures
-> tend to grow fast
-> dense, apparently unstructured
-> hard to read
- - >> models are full of ASSUMPTIO
PN & Systems Biology
May 2005
FRAMEWORK
ding
dation
ve r prediction ODEs
bionetworks knowledge
quantitative modelling
quantitative models
animation /analysis /simulation
understan
model vali
quantitatibehaviou
PN & Systems Biology
May 2005
FRAMEWORK
ding
lidation
prediction
ding
dation
ve r prediction
(invariants)
modelchecking
Petri net theory
ODEs
quantitativeparameters
bionetworks knowledge
qualitative modelling
qualitative models
quantitative modelling
quantitative models
animation /analysis
animation /analysis /simulation
understan
model va
qualitativebehaviour
understan
model vali
quantitatibehaviou
PN & Systems Biology
May 2005
FRAMEWORK
ding
lidation
prediction
ding
dation
ve r prediction
(invariants)
modelchecking
Petri net theory
ODEsLPSLIRG
bionetworks knowledge
qualitative modelling
qualitative models
quantitative modelling
quantitative models
quantitativeparameters
animation /analysis
animation /analysis /simulation
understan
model va
qualitativebehaviour
understan
model vali
quantitatibehaviou
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE STRUCTURE
hemical reactions
output compounds
+ + O2
❑ atomic actions -> Petri net transitions -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE STRUCTURE
hemical reactions
output compounds
+ + O2
H
❑ atomic actions -> Petri net transitions -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
O2
H+
NAD
H2O
NAD+
hyperarcs
2
22
2
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE STRUCTURE
hemical reactions
hemical compounds
output compounds
+ + O2
post-conditions
❑ atomic actions -> Petri net transitions -> c
❑ local conditions -> Petri net places -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
pre-conditions
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE STRUCTURE
hemical reactions
hemical compounds
toichiometric relations
output compounds
+ + O2
❑ atomic actions -> Petri net transitions -> c
❑ local conditions -> Petri net places -> c
❑ multiplicities -> Petri net arc weights -> s
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE STRUCTURE
hemical reactions
hemical compounds
toichiometric relations
vailable amount (e.g. mol)
ompounds distribution
output compounds
+ + O2
❑ atomic actions -> Petri net transitions -> c
❑ local conditions -> Petri net places -> c
❑ multiplicities -> Petri net arc weights -> s
❑ condition’s state -> token(s) in its place -> a
❑ system state -> marking -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE STRUCTURE
hemical reactions
hemical compounds
toichiometric relations
vailable amount (e.g. mol)
ompounds distribution
P -> N0
output compounds
+ + O2
❑ atomic actions -> Petri net transitions -> c
❑ local conditions -> Petri net places -> c
❑ multiplicities -> Petri net arc weights -> s
❑ condition’s state -> token(s) in its place -> a
❑ system state -> marking -> c
❑ PN = (P, T, F, m0), F: (P x T) U (T x P) -> N0, m0:
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE BEHAVIOUR
hemical reactions
output compounds
+ + O2
❑ atomic actions -> Petri net transitions -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE BEHAVIOUR
hemical reactions
output compounds
+ + O2
❑ atomic actions -> Petri net transitions -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
FIRING
PN & Systems Biology
May 2005
PETRI NETS, BASICS - THE BEHAVIOUR
hemical reactions
output compounds
+ + O2
TOKEN GAME
DYNAMIC BEHAVIOUR
(substance flow)
❑ atomic actions -> Petri net transitions -> c
input compounds
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
2 NAD+ + 2 H2O -> 2 NADH + 2 H
2
2
2
2
r1
O2
H+
NADH
H2O
NAD+
FIRING
PN & Systems Biology
May 2005
TYPICAL BASIC STRUCTURES
r3r2
e3e2
r3
r2
r1
❑ metabolic networks
-> substance flows
❑ signal transduction networks
-> signal flows
r1
e1
PN & Systems Biology
May 2005
THE RUNNING EXAMPLE - THE RKIP PATHWAY
[Cho et al.,CMSB 2003]
PN & Systems Biology
May 2005
THE RKIP PATHWAY, PETRI NET
k11
k10k9
RPm10
RKIP-P_RPm11
P
k8
k5k7k6
k4k3
k2k1
RKIP-P
m6
ERK
m5
MEK-PP
m7
Raf-1Star_RKIP_ERK-PP
m4MEK-PP_ERKm8
ERK-PP
m9 Raf-1Star_RKIm3
RKIP
m2
Raf-1Star
m1
PN & Systems Biology
May 2005
THE RKIP PATHWAY, HIERARCHICAL PETRI NET
k11
RPm10
RKIP-P_RPm11
P
k9_k10
k8
k5
RKIP-P
m6
ERKm5
MEK-PP
m7
Raf-1Star_RKIP_ERK-PP
m4MEK-PP_ERKm8
ERK-PP
m9 Raf-1Star_RKIm3
RKIP
m2
Raf-1Star
m1
k6_k7
k3_k4
k1_k2
PN & Systems Biology
May 2005
THE RKIP PATHWAY, HIERARCHICAL PETRI NET
k11
RPm10
RKIP-P_RPm11
P
k9_k10
k8
k5
RKIP-P
m6
ERKm5
MEK-PP
m7
Raf-1Star_RKIP_ERK-PP
m4MEK-PP_ERKm8
ERK-PP
m9 Raf-1Star_RKIm3
RKIP
m2
Raf-1Star
m1
k6_k7
k3_k4
k1_k2
initial marking
PN & Systems Biology
May 2005
BIOCHEMICAL PETRI NETS, SUMMARY
ementary actions) y ]
y (re-) actions
❑ biochemical networks
-> networks of (abstract) chemical reactions
❑ biochemically interpreted Petri net
-> partial order sequences of chemical reactions (= eltransforming input into output compounds / signals[ respecting the given stoichiometric relations, if an
-> set of all pathwaysfrom the input to the output compounds / signals [ respecting the stoichiometric relations, if any ]
❑ pathway
-> self-contained partial order sequence of elementar
❑ typical basic assumption
-> steady state behaviour
PN & Systems Biology
May 2005
ANALYSIS TECHNIQUES, OVERVIEW
nstruction
space construction (RG)
erties,te [with constraints],state [with constraints]
❑ static analyses -> no state space co
-> structural properties (graph theory)
-> P / T - invariants (linear algebra),
❑ dynamic analyses -> total/partial state
-> analysis of general behavioural system properties,
e.g. boundedness, liveness, reversibility, . . .
-> model checking of special behavioural system prope.g. reachability of a given (sub-) system sta
reproducability of a given (sub-) system
expressed in temporal logics (CTL / LTL),very flexible, powerful querry language
PN & Systems Biology
May 2005
INCIDENCE MATRIX C
=> stoichiometric matrix
g by firing of tj
j,pi) - F(pi, tj) = ∆ tj(pi)
enzyme
MB2-catalysed ction
x
cij = 0
j
i
❑ a representation of the net structure
❑ matrix entry cij:token change in place pi by firing of transition tj
❑ matrix column ∆tj:vector describing the change of the whole markin
❑ side-conditions are neglected
PT t1 tj tm
p1
pi
pn
cij cij = (pi, tj) = F(t
. . . . . .
...
C =
∆tj ∆tj = ∆ tj(*)
MB1 enzymerea
x
PN & Systems Biology
May 2005
T-INVARIANTS, BASICS
> multisets of transitions
> Parikh vector
> sets of transitions
n of minimal ones
kx aixii∑=
❑ Lautenbach, 1973
❑ T-invariants -
-> integer solutions x of -
❑ minimal T-invariants
-> there is no T-invariant with a smaller support -
-> gcD of all entries is 1
❑ any T-invariant is a non-negative linear combinatio
-> multiplication with a positive integer
-> addition
-> Division by gcD
❑ Covered by T-Invariants (CTI)
-> each transition belongs to a T-invariant
-> BND & LIVE => CTI (necessary condition)
Cx 0 x 0 x 0≥,≠,=
PN & Systems Biology
May 2005
T-INVARIANTS, INTERPRETATION
RG
ving sequences
invariant, ence
> partial order structure
❑ T-invariants = (multi-) sets of transitions
-> zero effect on marking
-> reproducing a marking / system state
-> steady state substance flows / reaction rates
-> elementary modes [Schuster 1993]
❑ realizable T-invariants correspond to cycles in the
-> RG: concurrent transitions -> all transitions’ interlea
-> if there are concurrent transitions in a realizable T-then there is a RG cycle for each interleaving sequ
-> analogously for conflicts
❑ a T-invariant defines a subnet -
-> the T-invariant’s transitions (the support), + all their pre- and post-places+ the arcs in between
-> pre-sets of supports = post-sets of supports
PN & Systems Biology
May 2005
T-INVARIANTS, THE RKIP PATHWAY
k11
k5
RPm10
RKIP-P
m6
RKIP-P_RPm11
RKIP_ERK-PP
Raf-1Star_RKIP
RKIP
m2
k9
k3
k1
k8
ERKm5
MEK-PP
m7
Raf-1Star_
m4MEK-PP_ERKm8
ERK-PP
m9 m3
Raf-1Star
m1
k6
-> non-trivial T-invariant
+ four trivial ones for reversible reactions
PN & Systems Biology
May 2005
RUN OF THE NON-TRIVIAL T-INVARIANT
k11
k5
RPm10
RKIP-Pm6
RKIP-P_RPm11
Raf-1Star_RKIP_ERK-PPm4
Raf-1Star_RKIPm3
RKIPm2af-1Star
k9
k3
k1
Raf-1Starm1
RK-PP RPm10RKIPm2
RK
❑ partial order structure
❑ T-invariant’s unfoldingto describe its behaviour
❑ labelled condition / event net
-> events - transition occurences
-> conditions - input / output compounds
❑ partial order semantics
-> a net’s all partial order runs
k8
ERKm5
MEK-PPm7
m8
Rm1
k6
ERK_PPm9
MEK-PP m9 Em7
MEK-PP_E
PN & Systems Biology
May 2005
P-INVARIANTS, BASICS
> multisets of places
> sets of places
n of minimal ones
ky aiyii∑=
❑ Lautenbach, 1973
❑ P-invariants -
-> integer solutions y of
❑ minimal P-invariants
-> there is no P-invariant with a smaller support -
-> gcD of all entries is 1
❑ any P-invariant is a non-negative linear combinatio
-> multiplication with a positive integer
-> addition
-> Division by gcD
❑ Covered by P-Invariants (CPI)
-> each transition belongs to a P-invariant
-> CPI => BND (sufficient condition)
yC 0 y 0 y 0≥,≠,=
PN & Systems Biology
May 2005
P-INVARIANTS, INTERPRETATION
weighted sum of tokens on
rom a P-invariant’s placetokens to a P-invariant’s place
m reachable from m0
❑ the firing of any transition has no influence on the the P-invariant’s places
-> for all t: the effect of the arcs, removing tokens fis equal to the effect of the arcs, adding
❑ set of places with
-> a constant weighted sum of tokens for all markingsym = ym0
-> token / compound preservation
❑ a place belonging to a P-invariant is bounded
-> CPI - sufficient condition for BND
❑ a P-invariant defines a subnet
-> the P-invariant’s places (the support), + all their pre- and post-transitions+ the arcs in between
-> pre-sets of supports = post-sets of supports
PN & Systems Biology
May 2005
P-INVARIANTS, THE RKIP PATHWAY
k11
k5
RPm10
RKIP-P
m6
RKIP-P_RPm11
Raf-1Star_RKIP_ERK-PP
m4
Raf-1Star_RKIPm3
RKIP
m2
tar
k9_k10
k3_k4
k1_k2
k11
k5
RPm10
RKIP-P
m6
K
RKIP-P_RPm11
Raf-1Star_RKIP_ERK-PP
m4K
Raf-1Star_RKIPm3
RKIP
m2
Star
k9_k10
k3_k4
k1_k2
k11k8
k5
RPm10
RKIP-P
m6
ERK
m5
MEK-PP
m7
RKIP-P_RPm11
Raf-1Star_RKIP_ERK-PP
m4MEK-PP_ERKm8
ERK-PP
m9 Raf-1Star_RKIPm3
RKIP
m2
Raf-1Star
m1
k9_k10k6_k7
k3_k4
k1_k2
k8
ERK
m5
MEK-PP
m7
MEK-PP_ERKm8
ERK-PP
m9
Raf-1S
m1
k6_k7
k8
ERm5
MEK-PP
m7
MEK-PP_ERm8
ERK-PP
m9
Raf-1
m1
k6_k7
P-INV1: MEK
P-INV2: RAF-1STAR
P-INV3: RP
P-INV4: ERK
P-INV5: RKIP
PN & Systems Biology
May 2005
CONSTRUCTION OF THE INITIAL MARKING
❑ each P-invariant gets at least one token
-> P-invariants are structural deadlocks and traps
❑ all (non-trivial) T-invariants get realizable
-> to make the net live
❑ minimal marking
-> minimization of the state space
❑ assumption: top-to-bottom reading of the figure
-> but, all reachable markings are equivalent (= produce same state space)
-> UNIQUE INITIAL MARKING
PN & Systems Biology
May 2005
STATIC ANALYSIS, SUMMARY
tion subnet (cycle)
lic behaviour
pF0 MG SM FC EFC ESN N N N N YV L&SY Y
❑ structural properties
❑ CPI
-> structural bounded (SB)
-> each P-invariant represents a substance conserva
❑ CTI
-> Live & BND -> CTI
-> 4 trivial T-invariants for reversible reactions
-> 1 non-trivial T-invariant describing the essential cyc
❑ DTP & ES -> Live
INAORD HOM NBM PUR CSV SCF CON SC Ft0 tF0 Fp0 Y Y Y Y N N Y Y N N N DTP CPI CTI B SB REV DSt BSt DTr DCF L L Y Y Y Y Y Y N ? N N Y
PN & Systems Biology
May 2005
DYNAMIC ANALYSIS - REACHABILITY GRAPH
-> single step firing rule
-> interleaving semantics
-> model checking
❑ simple construction algorithm
-> nodes - system states
-> arcs - the (single) firing transition
❑ unbounded Petri net -> infinite RGbounded Petri net -> finite RG
❑ concurrency
-> enumeration of all interleaving sequences
❑ branching arcs in the RG
-> conflict OR concurrency
❑ RG tend to be very large
-> automatic evaluation necessary
❑ worst case: over-exponential growth
-> alternative analyses techniques ?
PN & Systems Biology
May 2005
MODEL CHECKING, EXAMPLES
?
nor using MEK-PP ?
nce of RKIP ?
) );
❑ property 1
Is a given (sub-) marking (system state) reachable
EF ( ERK * RP );
❑ property 2
Liveness of transition k8 ?
AG EF ( MEK-PP_ERK );
❑ property 3
Is it possible to produce ERK-PP neither creating
E ( ! MEK-PP U ERK-PP );
❑ property 4
Is there cyclic behaviour w.r.t. the presence / abse
EG ( ( RKIP -> EF ( ! RKIP ) ) * ( ! RKIP -> EF ( RKIP )
PN & Systems Biology
May 2005
QUALITATIVE ANALYSIS RESULTS, SUMMARY
-> static analysis
l interpretation
the essential behaviour
-> dynamic analysis
ND
IVE
EV
ofessional understanding
❑ structural decisions of behavioural properties
-> CPI -> BND
-> ES & DTP -> LIVE
❑ CPI & CTI
-> all minimal T-invariant / P-invariants enjoy biologica
-> non-trivial T-invariant -> partial order description of
❑ reachability graph
-> finite -> B
-> the only SCC contains all transitions -> L
-> one Strongly Connected Component (SCC) -> R
❑ model checking -> requires pr
-> all expected properties are valid
PN & Systems Biology
May 2005
QUALITATIVE ANALYSIS RESULTS, SUMMARY
-> static analysis
l interpretation
the essential behaviour
-> dynamic analysis
ND
IVE
EV
rofessional understanding
VE MODEL
❑ structural decisions of behavioural properties
-> CPI -> BND
-> ES & DTP -> LIVE
❑ CPI & CTI
-> all minimal T-invariant / P-invariants enjoy biologica
-> non-trivial T-invariant -> partial order description of
❑ reachability graph
-> finite -> B
-> the only SCC contains all transitions -> L
-> one Strongly Connected Component (SCC) -> R
❑ model checking -> requires p
-> all expected properties are valid
-> VALIDATED QUALITATI
PN & Systems Biology
May 2005
BIONETWORKS, VALIDATION
ion
ng T-invariant
tion (?)
❑ validation criterion 0
-> all expected structural properties hold
-> all expected general behavioural properties hold
❑ validation criterion 1
-> CTI
-> no minimal T-invariant without biological interpretat
-> no known biological behaviour without correspondi
❑ validation criterion 2
-> CPI
-> no minimal P-invariant without biological interpreta
❑ validation criterion 3
-> all expected special behavioural properties hold
-> temporal-logic properties -> TRUE
PN & Systems Biology
May 2005
DY ED SES !
NOW WE ARE REA
FOR SOPHISTICAT
QUANTITATIVE ANALY
PN & Systems Biology
May 2005
QUANTITATIVE ANALYSIS
ive parameters
ndent
uous nodes !
❑ quantitative model = qualitative model + quantitat
-> BUT: quantitative parameters often unknown
❑ typical quantitative parameters of bionetworks
-> compound concentrations -> real numbers
-> reaction rates / fluxes -> concentration-depe
❑ continuous Petri nets
p1Cont
p2Cont
p3Contt1Contm1
m2
m3
v1 = k1*m1*m2
d [p1Cont] / dt = d [p2Cont] / dt = - v1d [p3Cont] / dt = v1 - v2
contin
ODEs
v2 = k2*m3
t2Cont
PN & Systems Biology
May 2005
EXAMPLE - MICHAELIS-MENTEN REACTION
50 100 150 200
t
03333
06667
1
50 100 150 200
t
03333
06667
1
0
0,
0,
0,m1
0
0,
0,
0,m2
-> Visual Object Nets-> GON / cell illustrator (?)
s p tCont
m1
v = 0.005 * m1 / (0.1375 + m1)
m2
Vmax = 0.005 (maximal reaction rate)Km = 0.1375 (Michaelis constant)
d[s]/dt = d[p]/dt = Vmax*[s]/(Km+[s])dm1/dt = dm2/dt = Vmax*m1/(Km+m1)
PN & Systems Biology
May 2005
ODEL
RIPTION MODEL !
THE QUALITATIVE MBECOMES
THE STRUCTURAL DESC
OF THE QUANTITATIVE
PN & Systems Biology
May 2005
CHALLENGES
l order semantics
der development)
Es
❑ extensions
-> read arcs -> interleaving / partia
-> inhibitor arcs !? -> Turing power !
❑ efficient computation of minimal invariants
-> exponential complexity
-> compositional / step-wise refinement approach (un
❑ analysis of unbounded nets
-> besides T-invariant analysis ?
❑ model checking
-> relevant properties ?
❑ comparision: continuous / hybrid Petri nets <-> OD
-> Petri net simulation versus classical ODEs solver
-> is there a winner (for certain structures) ?
PN & Systems Biology
May 2005
SUMMARY
sound analysis techniques
> to experience the model
> to increase confidence
> new insights
DEs
es
❑ representation of bionetworks by Petri nets
-> partial order representation
-> formal semantics -> various
-> unifying view
❑ purposes
-> animation -
-> model validation against consistency criteria -
-> qualitative / quantitative behaviour prediction -
❑ two-step model development
-> qualitative model -> discrete Petri nets
-> quantitative model -> continuous Petri nets = O
❑ many challenging questions for analysis techniqu
-> qualitative as well as quantitative ones