A compact modeling approach for deterministic biological ... · IGoss PJE, Peccoud J (1998)Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri

A compact modeling approach fordeterministic biological systems

Luis M. Torres1 Annegret K. Wagler2

1Ecuadorian Research Center on Mathematical Modeling – ModeMatEscuela Politécnica Nacional, Quito

2LIMOS (UMR 6158 CNRS)University Blaise Pascal, Clermont-Ferrand

6th Int. Workshop on Biological Processes & Petri NetsBrussels, June 22nd

BioPPN 2015 1 / 51

Modeling phenomena in cellular networks

Adequate models for the structure and dynamics of biologicalsystems?

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Adequate models for the structure and dynamics of biologicalsystems?

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Systems biology aims at the integrated experimental andtheoretical analysis of cellular networks

I network inferencereconstructing interaction networks of biological entities

I network analysismining the information content of the network

I dynamic modelingconnecting interaction network and dynamic behaviour

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Application areas

I study biological processes “in silico”I response of cells/organs/organisms to environmental

changesI immune responses to virus infectionI effects of gene defectsI internal activities of a cell: proliferation, differentiation,

motility

I identify good conditions for growth

I design intervention strategies (pharmaceutical)

BioPPN 2015 4 / 51

Outline

IntroductionPetri NetsApplications and limitations

Modeling deterministic systemsThe transition conflict graphValid orientations

Model reconstruction from experimental dataPartial orientations and the MVTPThe Optimal Test Set Extension Problem

Summary

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Outline




Summary

Introduction BioPPN 2015 6 / 51

Petri nets

The network is modeled as a weighted bipartite digraphG = (N ∪ T ,A,w) withI N = {1, . . . ,n} places (e.g. metabolites©)I T = {1, . . . , τ} transitions (e.g. reactions �)I arc weights stoichiometric coefficients

The network can be represented through its incidence matrixM ∈ Zn×τ :

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Petri nets: dynamics

I states of the system are modeled through assignments oftokens to places [“markings”]

I some places may have bounded capacities

I a transition is enabled at a certain state if firing it leads toanother valid state

I many transitions may be enabled at the same state

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I some places may have bounded capacitiesI a transition is enabled at a certain state if firing it leads to

another valid state

I many transitions may be enabled at the same state

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I some places may have bounded capacitiesI a transition is enabled at a certain state if firing it leads to

another valid stateI many transitions may be enabled at the same state

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The state digraph

I the potential space state is given by:

X := {x ∈ Zn : 0 ≤ xp ≤ up, ∀p ∈ NB}

I the state digraph G = (X ,A) contains arcs (x , y) ∈ A for ally obtained from x by firing one single transition

I dynamic processes are modeled as sequences oftransition fires starting from an initial state x0:

I movement of tokens in GI paths in reachability (marking) digraph G(x0)


Some central problems

I Reachability:Can the system reach one of a set of target states startingfrom an initial state x0?

I Boundedness:Are there sequences of transition fires that lead tounlimited token accumulation at some place?

I Existence of deadlocks:Can the system reach a state at which no transitions areenabled?

I Liveness:Is it (im)possible to reach a state where some transition ispermanently disabled?


Outline




Summary


Application fields

Petri nets have been widely used to model several kinds ofdynamic systemsI asynchronous hardware circuits

[Yakovlev, Koelmans, Semenov & Kinniment 1996]I production and workflow systems

[Adam, Atluri & Huang 1998]I batch processes

[Gu & Bahri 2002]I distributed algorithms

[Reisig 1998]I biological networks

[Hardy & Robillard 2004; Chaouiya, Remy & Thieffry 2008]


Petri Nets for cellular networks

Some (non-exhaustive) references:I Blätke MA, Heiner M, Marwan W (2015)BioModel Engineering with Petri

Nets. In: Algebraic and Discrete Mathematical Methods for ModernBiology. Elsevier

I Chen M, Hofestaedt R (2003)Quantitative Petri net model of generegulated metabolic networks in the cell. In Silico Biol 3:347-365

I Goss PJE, Peccoud J (1998)Quantitative modeling of stochasticsystems in molecular biology by using stochastic Petri nets. Proc NatlAcad Sci USA 95:6750-6755

I Hardy S, Robillard PN (2005)Phenomenological and molecular-levelPetri net modeling and simulation of long-term potentiation. BioSystems82:26-38

I Hardy S, Robillard PN (2004)Modeling and simulation of molecularbiology systems using Petri nets: modeling goals of variousapproaches. Journal of Bioinformatics and Computational Biology 2(4),619-637


Petri Nets for cellular networks

Some (non-exhaustive) references:I Heiner M, Gilbert D, Donaldson R (2008a)Petri nets for systems and

synthetic biology. In: Bernardo M, Degano P, Zavattoro G (eds) Formalmethods for computational systems biology, LNCS 5016. Springer,Heidelberg, pp 215-264

I Hofestädt R (1994)A Petri net application of metabolic processes. SystAnal Model Simul 16:113-122

I Matsuno H, Tanaka Y, Aoshima H, Doi A, Matsui M, Miyano S(2003)Biopathways representation and simulation on hybrid functionalPetri net. In Silico Biol 3:389-404

I Pinney JW, Westhead RD, McConkey GA (2003)Petri Netrepresentations in systems biology. Biochem Soc Trans 31:1513-1515


Limitations

I Petri Nets are well-suited for studying concurrent systemsI reachability graph G(x0) provides local point of view

I for biological systems a model aiming at a globalunderstanding of the (inferred) network is pursued

I G is not adequate for encoding dynamic behaviors:exponential on the size of the Petri Net

I some biological systems are deterministic:each state x has a unique successor x+, even if severaltransitions are enabled

Challenge:Find a compact adequate way to encode system dynamics!!


Limitations

I Petri Nets are well-suited for studying concurrent systemsI reachability graph G(x0) provides local point of viewI for biological systems a model aiming at a global

understanding of the (inferred) network is pursuedI G is not adequate for encoding dynamic behaviors:

exponential on the size of the Petri NetI some biological systems are deterministic:

each state x has a unique successor x+, even if severaltransitions are enabled



Limitations

I Petri Nets are well-suited for studying concurrent systemsI reachability graph G(x0) provides local point of viewI for biological systems a model aiming at a global

understanding of the (inferred) network is pursuedI G is not adequate for encoding dynamic behaviors:

exponential on the size of the Petri NetI some biological systems are deterministic:

each state x has a unique successor x+, even if severaltransitions are enabled



Example

Light-induced sporulation of Physarum polycephalum

I entering the sporulation pathway is controlled byenvironmental factors like visible light

I a photoreversible photoreceptor occurs in twoconformational states PFR and PR

I PFR + far-red light FR converted into PR, causessporulation

I PR + red light R converted back to PFR, preventssporulation

FR

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Potential system states X :

FR 0 0 0 0 0 0 0 0 1 1 1 1R 0 0 0 0 1 1 1 1 0 0 0 0PFR 1 0 1 0 1 0 1 0 1 0 1 0PR 0 1 0 1 0 1 0 1 0 1 0 1Spo 0 0 1 1 0 0 1 1 0 0 1 1

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

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Outline




Summary

Modeling deterministic systems BioPPN 2015 18 / 51

Modeling determinism via transition priorities

Deterministic systems can be modeled via a successor oracle:

x ∈ X 7→ x+ (successor state)

We consider those systems where the oracle can beimplemented through a transition selection function:

X → Tx 7→ t∗(x) ∈ T (x),

where T (x) is the set of enabled transitions at state x ,so that...I firing t∗(x) puts the system in state x+

I t∗(x) is called the highest-priority transition at x



Deterministic systems can be modeled via a successor oracle:

x ∈ X 7→ x+ (successor state)

We consider those systems where the oracle can beimplemented through a transition selection function:

X → Tx 7→ t∗(x) ∈ T (x),

Idea:Find a compact encoding for t∗.



Can all deterministic systems be modelled in this way?

No!

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j branching state x j T (x j) x j+

0 (1, 1, 0, 0)T {t1, t2} (0, 0, 1, 1)T

1 (0, 1, 1, 0)T {t2, t3} (0, 1, 0, 1)T

2 (1, 0, 0, 1)T {t1, t4} (0, 1, 0, 0)T

3 (1, 1, 1, 0)T {t2, t3} (1, 1, 0, 1)T



Can all deterministic systems be modelled in this way?No!

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j branching state x j T (x j) x j+

0 (1, 1, 0, 0)T {t1, t2} (0, 0, 1, 1)T

1 (0, 1, 1, 0)T {t2, t3} (0, 1, 0, 1)T

2 (1, 0, 0, 1)T {t1, t4} (0, 1, 0, 0)T

3 (1, 1, 1, 0)T {t2, t3} (1, 1, 0, 1)T


The transition conflict graph

I two transitions are in conflict if they are both enabled atsome state

I they are in dynamic conflict if firing one disables the other

Lemma (TW 2011)Transitions t , t ′ ∈ T are in conflict if and only if

wpt ≤ up − wt ′p ∀p ∈ P−(t) ∩ P+(t ′) ∩ B,andwpt ′ ≤ up − wtp ∀p ∈ P−(t ′) ∩ P+(t) ∩ B.



Transition conflicts can be represented by a graph K := (T ,E)

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I for every x ∈ STS, T (x) induces a clique in K

I not all cliques correspond to states




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The structure of K is very difficult to characterize from a graphtheoretical point of view!

Theorem (TW 2011)Let H = (V ,E) be an (arbitrary) undirected graph and Q thefamily containing all cliques in H. Then there exists a networkG = (N ∪ T ,A,w) that satisfiesI H = KI Q := {T (x) : x ∈ X}


Outline




Summary


Valid orientations

IdeaEncode transition priorities by orienting the edges of K

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Definition (Valid orientation)An orientation D of the edges of K is valid ifI ∀x ∈ X , T (x) induces a clique with a unique sink


Valid orientations

IdeaEncode transition priorities by orienting the edges of K

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Definition (Valid orientation)An orientation D of the edges of K is valid ifI ∀x ∈ X , T (x) induces a clique with a unique sink


Valid orientations

Theorem (TW 2009)There is a valid orientation of K iff for any pair of statesx , x ′ ∈ X with t∗(x) ∈ T (x) ∩ T (x ′),I either t∗(x ′) = t∗(x) or t∗(x ′) ∈ T (x ′) \ T (x)

I in this case D encodes the dynamics of the system.I the successor oracle can be implemented as

I select t∗(x) as unique sink in clique in K induced by x

I obtain x+ by swtiching t∗(x)


Successor oracle

Input: (G,u,D), x ∈ XOutput: x+

3: Construct the set T (x) of enabled transitionsif T (x) = ∅ then

return x6: end if

{Compute out-degree of transitions in the subgraph of Dinduced by T (x)}for t ∈ T (x) do

9: δ−(t) := |{tt ′ ∈ A : t ′ ∈ T (x)}|end for{Return successor state}

12: t∗ ← t ∈ T (x) with δ−(t) = 0return x + M·t∗


Predecesor oracle

Input: (G,u,D), x ∈ XOutput: pred(x)

3: {Construct set of transition candidates}

cand(x)←{t ∈ T : xp + wpt ≤ up, ∀p ∈ P−(t) ∩ B;

xp − wtp ≥ 0,∀p ∈ P+(t)}

{Call successor-oracle to construct sets of predecessors}6: pred(x)← ∅

for t ∈ cand(x) doy ← x −M·t

9: if t∗(y) = t thenpred(x)← pred(x) ∪ {y}

end if12: end for

{Return set of possible predecessors}return pred(x)


Example


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Strongly valid orientations

I in general, valid orientations are difficult to characterize

I an orientation is strongly valid, if it is valid for all Petri Netssharing the same conflict graph

I strongly valid orientations have a unique sink for eachclique

Theorem (TW 2009)I D is strongly valid if it does not contain a directed cycle of

length 3I in particular, acyclic orientations are strongly valid


Strongly valid orientations

I in general, valid orientations are difficult to characterize

I an orientation is strongly valid, if it is valid for all Petri Netssharing the same conflict graph

I strongly valid orientations have a unique sink for eachclique

Theorem (TW 2009)I D is strongly valid if it does not contain a directed cycle of

length 3I in particular, acyclic orientations are strongly valid


Outline




Summary

Model reconstruction from experimental data BioPPN 2015 31 / 51

The Minimum Valid Test Set Problem

ProblemFor a deterministic system whose dynamic behavior isprescribed by an unknown valid orientation D,I determine D by observing some pairs (x , x+)

I use as few observations as possible

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DefinitionA set X ′ ⊂ X is a valid test set if knowledge about{

(x , x+) : x ∈ X ′}

is sufficient for determining D.

Minimum Valid Test Set Problem (MVTP)Determine a valid test set of minimum cardinality.



DefinitionA set X ′ ⊂ X is a valid test set if knowledge about{

(x , x+) : x ∈ X ′}

is sufficient for determining D.

Minimum Valid Test Set Problem (MVTP)Determine a valid test set of minimum cardinality.


Partial orientations

I a partial orientation D′ := (T ,A′,E′) is a mixed graphobtained by fixing directions of some edges of K

I D′ is extendible if it is possible to choose directions for theedges in E′ to obtain a valid orientation

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Inferable, dominated, and essential arcs

I an edge tt ′ ∈ E′ is inferable as (t , t ′) if the digraph(T ,A′ ∪ {(t ′, t)},E′ \ {tt ′}) is not extendible

I an edge tt ′ ∈ E′ is dominated if

∀x ∈ X with t , t ′ ∈ T (x), t∗(x) 6∈ {t , t ′}

I D′ is sufficient if all edges in E′ are either inferable ordominated

I an arc (t , t ′) of D is essential ifI tt ′ is not dominatedI the orientation D2 := (T ,A \ {(t , t ′)} ∪ {(t ′, t)}) is valid


Characterizing optimal solutions for MVTP

Theorem (TW 2009)Let A∗ be the set of all essential arcs. Then,I the partial orientation D∗ := (T ,A∗,E∗) is sufficientI for any x ∈ X , knowledge about (x , x+) allows orientation

of at most one essential arcI X ′ ⊂ X is an optimal solution for MVTP iff for every x ∈ X ′

{(t , t∗(x)) : t ∈ T (x), t 6= t∗(x)} ∩ A∗ 6= ∅

We consider in the following the special case when D isacyclic...


Characterizing optimal solutions for MVTP

Theorem (TW 2009)Let A∗ be the set of all essential arcs. Then,I the partial orientation D∗ := (T ,A∗,E∗) is sufficientI for any x ∈ X , knowledge about (x , x+) allows orientation

of at most one essential arcI X ′ ⊂ X is an optimal solution for MVTP iff for every x ∈ X ′

{(t , t∗(x)) : t ∈ T (x), t 6= t∗(x)} ∩ A∗ 6= ∅

We consider in the following the special case when D isacyclic...


Acyclic graphs

Lemma (TW 2015)If D is acyclic, then every clique Q has a unique sink, and adirected hamiltonian path PQ through its nodes.

Lemma (TW 2015)If D is acyclic, the essential arcs of a clique are exactly the arcscorresponding to PQ.


Algorithm

Input: (G, u, t∗) {deterministic system with oracle for computing t∗}Output: D {valid orientation}

3: initialize Q as the set of all inclusion-wise maximal cliques in Kwhile Q 6= ∅ do

retrieve a clique Q from Q6: while |Q| > 1 and ∃x ∈ X with T (x) = Q do

call the oracle and determine t∗ := t∗(x)orient the arcs {(t , t∗) : t ∈ T (x), t 6= t∗}

9: deduce orientations for inferable edgesremove t∗ from Q

end while12: if Q contains more than one node then

compute Q′ := {T (x) ⊂ Q : x ∈ X}remove from Q′ cliques that are not inclusion-wise maximal

15: Q := Q∪Q′end if

end while18: while there are yet unoriented edges do

deduce an orientation for all yet unoriented inferable edgeschoose an arbitrary orientation for a yet unoriented dominated edge

21: end whileModel reconstruction from experimental data BioPPN 2015 38 / 51

What about the “online” problem?

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I states induce either edges or cliques of size 3 (triangles)I for every e ∈ K, there is an acyclic orientation where e is

not essentialI the figures show that the same holds for every triangle

for every x ∈ X , there is at least one valid orientation forwhich (x , x+) does not provide the direction of any essential arc no “winning strategy”


Outline




Summary


Experiment design for model verification

In practice...I network G is not known a priori

I both G and D have to be reconstructed simultaneouslyfrom experimental observations

I it is not always possible to observe pairs (x , x+)

I instead...I system is set to an initial state x0

I sequence sequ(x0) = (x1, . . . , xk ) of state changes inresponse to initial trigger is observed

I there might be intermediate non-observed states



Let X ′ be a collection of states observed in some experimentsI (G,D′) is X ′-deterministic if it fits the experimental dataI P(X ′) is the set of all X ′-deterministic models (G,D′)I P(X ′) can be computed with an exhaustive reconstruction

approach 1

Optimal Test Set Extension Problem

Given X ′ ⊆ X and P(X ′), find an optimal extension X ′ of X ′such that

I each network in P(X ′) can be either completed to anX ′-deterministic network or ruled out as “false positive”,

I all networks in P(X ′) are either equivalent or not furtherdistinguishable

1M. Favre, A. Wagler. Reconstructing X ′-deterministic extended Petri netsfrom experimental time-series data X ′. CEUR Workshop Proceedings988:45–59, 2013. (Special Issue BioPPN 2013)



Let X ′ be a collection of states observed in some experimentsI (G,D′) is X ′-deterministic if it fits the experimental dataI P(X ′) is the set of all X ′-deterministic models (G,D′)I P(X ′) can be computed with an exhaustive reconstruction

approach 1

Optimal Test Set Extension Problem

Given X ′ ⊆ X and P(X ′), find an optimal extension X ′ of X ′such that

I each network in P(X ′) can be either completed to anX ′-deterministic network or ruled out as “false positive”,

I all networks in P(X ′) are either equivalent or not furtherdistinguishable

1M. Favre, A. Wagler. Reconstructing X ′-deterministic extended Petri netsfrom experimental time-series data X ′. CEUR Workshop Proceedings988:45–59, 2013. (Special Issue BioPPN 2013)


Example

Light-induced sporulation of Physarum polycephalum.

Perform two experiments:I sequ(x9) = (x2, x4)

I sequ(x6) = (x1)

I X ′ = {x1, x2, x4, x6, x9}I non-observed intermediate states x10 and x5

P(X ′) contains twelve networks...


Example

Light-induced sporulation of Physarum polycephalum.Perform two experiments:I sequ(x9) = (x2, x4)

I sequ(x6) = (x1)




Example


I sequ(x6) = (x1)




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I sequ(x6) = (x1)




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I sequ(x6) = (x1)



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Example

Potential system states X :

FR 0 0 0 0 0 0 0 0 1 1 1 1R 0 0 0 0 1 1 1 1 0 0 0 0PFR 1 0 1 0 1 0 1 0 1 0 1 0PR 0 1 0 1 0 1 0 1 0 1 0 1Spo 0 0 1 1 0 0 1 1 0 0 1 1

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12

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Example

Possible successors X :

x3+ x5+ x7+ x8+ x10+ x11+ x12+

P1 x3 x5 x7 x3 x12 x4 x12

P2 x3 x5 x7 x3 x2 x12 x4

P3a x3 x1 x3 x7 x12 x4 x12

P3b x3 x1 x3 x4 x12 x4 x12

P4a x3 x1 x3 x7 x2 x12 x4

P4b x3 x1 x3 x4 x2 x12 x4

I a test sequ(x3) = (x3) provides no useful informationI a test sequ(x8) = (x3) rules out P3b and P4b

I a test sequ(x7) = (x3) discards P1 and P2

I a test sequ(x12) = (x4) discards P1,P3a and P3b


Example


x3+ x5+ x7+ x8+ x10+ x11+ x12+

P1 x3 x5 x7 x3 x12 x4 x12

P2 x3 x5 x7 x3 x2 x12 x4

P3a x3 x1 x3 x7 x12 x4 x12

P3b x3 x1 x3 x4 x12 x4 x12

P4a x3 x1 x3 x7 x2 x12 x4

P4b x3 x1 x3 x4 x2 x12 x4

I a test sequ(x3) = (x3) provides no useful information

I a test sequ(x8) = (x3) rules out P3b and P4b




Example


x3+ x5+ x7+ x8+ x10+ x11+ x12+

P1 x3 x5 x7 x3 x12 x4 x12

P2 x3 x5 x7 x3 x2 x12 x4

P3a x3 x1 x3 x7 x12 x4 x12

P3b x3 x1 x3 x4 x12 x4 x12

P4a x3 x1 x3 x7 x2 x12 x4

P4b x3 x1 x3 x4 x2 x12 x4





Example


x3+ x5+ x7+ x8+ x10+ x11+ x12+

P1 x3 x5 x7 x3 x12 x4 x12

P2 x3 x5 x7 x3 x2 x12 x4

P3a x3 x1 x3 x7 x12 x4 x12

P3b x3 x1 x3 x4 x12 x4 x12

P4a x3 x1 x3 x7 x2 x12 x4

P4b x3 x1 x3 x4 x2 x12 x4





Example


x3+ x5+ x7+ x8+ x10+ x11+ x12+

P1 x3 x5 x7 x3 x12 x4 x12

P2 x3 x5 x7 x3 x2 x12 x4

P3a x3 x1 x3 x7 x12 x4 x12

P3b x3 x1 x3 x4 x12 x4 x12

P4a x3 x1 x3 x7 x2 x12 x4

P4b x3 x1 x3 x4 x2 x12 x4





Outline




Summary

Summary BioPPN 2015 46 / 51

Summary

I Petri Nets are a well-established framework for modelingand analyzing several processes in Systems Biology

I interrelations of components in complex systemsI concurrency

I we presented an extension for encoding the dynamics ofcertain deterministic systems in a compact manner

I global point of viewI reconstructing model structure from experimental data

I we assume successor oracle can be represented viapriorities among transitions(not all deterministic systems can be modeled in this way)


Summary







Summary







Summary

I graph-theoretical characterizations of valid orientationsI in general difficult to obtainI acyclic orientations are always (strongly) validI some characterizations in terms of hypergraphs

I we considered the MVTP reconstruction problemI an “optimal offline” strategy in terms of essential edgesI reconstruction algorithmI no (online) “winning strategy”

I experiment design (OTSEP)I reconstruct both network and valid orientationI extend or rule out models that do not fit the dataI assist in the choice of new experiments


Summary





Summary





Future work

I further structural properties of valid orientations

I (algorithmic) implications for MVTP

I algorithms for OTSEP

I integration in available software tools for Petri Nets(Snoopy, CHARLIE, ...)


Future work






Future work






Future work






For Further Reading

Marwan, W., Sujatha, A., Starostzik, C.: Reconstructing the regulatorynetwork controlling commitment and sporulation in physarumpolycephalum based on hierarchical Petri net modeling and simulation.Journal of Theoretical Biology 236, 349–365 (2005)

Marwan, W., Wagler, A., Weismantel, R.: A mathematical approach tosolve the network reconstruction problem.Math. Methods of Operations Research 67, 117–132 (2008)

Torres, L.M., Wagler, A.: Model reconstruction for discrete deterministicsystems.Electronic Notes of Discrete Mathematics 36, 175–182 (2010)

Torres, L.M., Wagler, A.: Encoding the dynamics of deterministicsystems.Mathematical Methods of Operations Research 73(3), 281–300 (2011).

Torres, L.M., Wagler, A.: Analyzing the dynamics of deterministicsystems from a hypergraph theoretical point of view.RAIRO – Operations Research 47(3), 321–330 (2013).


Thank you!!!


Documents

A compact modeling approach for deterministic biological ... · IGoss PJE, Peccoud J (1998)Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri