Automatic Network Reconstruction

Transcription

1 Automatic Network Reconstruction Annegret K. WAGLER Laboratoire d Informatique, de Modélisation et d Optimisation des Systèmes (LIMOS) UMR CNRS 658 Université Blaise Pascal, Clermont-Ferrand, France BioPPN 4, Tunis Tutorial: Petri nets for multiscale Systems Biology A. Wagler June 4, 4 / 46

2 Modeling and Model Validation in Biosciences!"%$C%*5)G$-<"59H)!"#$%&'()*'#)!"#$%)+*%&#*,"') &')-.$)/&"&$'$) 34$5&6$'-*%)7*-*)) "8)*%%)9&'#) :'-$545$-*,"') ;/5*&')<"59=) :'8"56*%) E.$6$) E.$6$)F*%&#*,"') 34$5&6$'-*%)7$&(') I.$)%*&*%)%*J)*445"*.H) B$4$*-)C',%)-.$)6"#$%)"55$-%A)45$#&-)*%%)$34$5&6$'-*%)5$C%-D) Modeling and Model Validation in Biosciences Biological models: often cartoon-like informal schemes Classical lab approaches: heuristic and hypothesis-driven methods, yield potentially incomplete solution sets A. Wagler June 4, 4 / 46

3 Mathematical Models and Methods for Biological Systems exp. design validation Biological System Biological Process exp. data hypothesis Structural properties Predicted dynamics Math. Algorithms Math. proof techniques analysis simulation Mathematical Model Mathematical Problem reconstruction interpretation Mathematics for Biosciences Mathematical models: formal schemes representing structure and function of the studied system Mathematical approaches: data-driven methods, can guarantee completeness by mathematical proofs A. Wagler June 4, 4 3 / 46

4 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 4 / 46

5 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 5 / 46

6 Petri nets: The networks Standard networks P = (P, T, A, w) P set of involved components ( places ), (molecules, receptors, genes) T set of involved reactions ( transitions ), (reactions, activations,...) A (P T ) (T P) set of directed links ( arcs ), arc weights w as stoichiometric coefficients. standard network Extended networks P = (P, T, (A A R A I ), w) A R P T read-arcs A I P T inhibitor-arcs A. Wagler June 4, 4 6 / 46

7 Petri nets: The networks Standard networks P = (P, T, A, w) P set of involved components ( places ), (molecules, receptors, genes) T set of involved reactions ( transitions ), (reactions, activations,...) A (P T ) (T P) set of directed links ( arcs ), arc weights w as stoichiometric coefficients. standard network extended network Extended networks P = (P, T, (A A R A I ), w) A R P T read-arcs A I P T inhibitor-arcs A. Wagler June 4, 4 6 / 46

8 Petri nets: States and state changes States Each place p P can be marked with an integral number x p of tokens. A state can be represented as a vector x N P with entries x p for all p P. a b c d e f x = Switching enabled transitions In a standard network, t T is enabled at a state x if x p w(p, t) for all p with (p, t) A; switching t yiels a new state x with x p = x p + w(p, t) (p, t), (t, p) A. In an extended network, t T is enabled at a state x if in addition x p w(p, t) for all p with (p, t) A R and x p < w(p, t) for all p with (p, t) A I. A. Wagler June 4, 4 7 / 46

9 Petri nets: States and state changes States Each place p P can be marked with an integral number x p of tokens. A state can be represented as a vector x N P with entries x p for all p P. a b c d e f x = Switching enabled transitions In a standard network, t T is enabled at a state x if x p w(p, t) for all p with (p, t) A; switching t yiels a new state x with x p = x p + w(p, t) (p, t), (t, p) A. In an extended network, t T is enabled at a state x if in addition x p w(p, t) for all p with (p, t) A R and x p < w(p, t) for all p with (p, t) A I. A. Wagler June 4, 4 7 / 46

10 Dynamic processes and reachability Dynamic processes Dynamic processes correspond to sequences of system states, obtained by sequences of transition switches. Starting from an initial state x, explore the dynamic behavior of the system by consecutively switching enabled transitions, analyzing the reachability of certain system states Can we control transition switches and, thus, the dynamic behavior? A. Wagler June 4, 4 8 / 46

11 Prediction of the dynamic behavior Priority relations for transitions (Marwan, W. & Weismantel 8) Let G be a network and O a priority relation on its transitions. If there are two or more transitions enabled at a state, this transition with highest priority will be switched. This allows to predict the dynamic behavior of the system (instead of listing all potentially possible switching sequences). Example. Consider the network G with O = {t > t 3 } A. Wagler June 4, 4 9 / 46

12 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 / 46

13 When does a network fit given experimental data? Experimental data can be seen as sequences X = (x, x,..., x k ) of measured system states: x x x = B C B C B A = x A network fits the given data if it can reproduce the observations: Conformal network A network G = (P, T, (A A R A I ), w) with priorities O is conformal with X if it contains and correctly selects transitions to reach, from every observed state x j, its observed successor x j+. A. Wagler June 4, 4 / 46

14 When does a network fit given experimental data? Experimental data can be seen as sequences X = (x, x,..., x k ) of measured system states: x = B B C C C A x B B C C C A x B B C C C A = x A network fits the given data if it can reproduce the observations: 3 Conformal network A network G = (P, T, (A A R A I ), w) with priorities O is conformal with X if it contains and correctly selects transitions to reach, from every observed state x j, its observed successor x j+. A. Wagler June 4, 4 / 46

15 The network Reconstruction Problem Network Reconstruction Problem Input: Output: a set P of components (considered to be crucial for the studied phenomenon), experimental time series data X = (x, x,..., x m ) obtained by stimulating a biological system and observing how its states change over time. all extended networks G = (P, T, (A A R A I ), w) with priorities O being conformal with the given data X, i.e., being able to reproduce the experimental observations. Remark: Finding all networks fitting the data means to create all minimal ones (that do not reproduce the data anymore after removing any element). All reconstructed networks have the same set P of places but differ in their transitions, (control-)arcs, and priorities. A. Wagler June 4, 4 / 46

16 The network Reconstruction Problem Network Reconstruction Problem Input: Output: a set P of components (considered to be crucial for the studied phenomenon), experimental time series data X = (x, x,..., x m ) obtained by stimulating a biological system and observing how its states change over time. all extended networks G = (P, T, (A A R A I ), w) with priorities O being conformal with the given data X, i.e., being able to reproduce the experimental observations. Remark: Finding all networks fitting the data means to create all minimal ones (that do not reproduce the data anymore after removing any element). All reconstructed networks have the same set P of places but differ in their transitions, (control-)arcs, and priorities. A. Wagler June 4, 4 / 46

17 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 3 / 46

18 Overview Reconstruction Approaches Based on previous works on reconstructing standard networks (Marwan, Wagler & Weismantel 8; Durzinsky, Wagler & Weismantel ) standard networks with priorities (Marwan, Wagler & Weismantel, 8; Torres & Wagler ) extended Petri nets (Durzinsky, Marwan & Wagler and 3) and on techniques for preprocessing the experimental data (Durzinsky, Wagler & Weismantel 8 and, Wagler & Wegener 3) postprocessing the solution set (Wagler & Wegener, 3) we developed an integrative approach for reconstructing extended Petri nets with priorities (Favre & Wagler 3) A. Wagler June 4, 4 4 / 46

19 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 5 / 46

20 Time series data In an experiment, the system is triggered in some state x to generate an initial state x, then it is observed how the system responds x,..., x k. A time series X = (x ; x,..., x k ) contains discrete (or discretized) values x j p = x p (t j ) for each component p P at several time points t,..., t k : P x(t )... x(t j )... x(t k ) a x a(t )... x a(t j )... x a(t k ).... p x p(t )... x p(t j )... x p(t k ) Thereby: choose enough time points t j to capture the true time course of the values x j p for each single component p P: as we require monotone data for the reconstruction process. A. Wagler June 4, 4 6 / 46

21 Time series data In an experiment, the system is triggered in some state x to generate an initial state x, then it is observed how the system responds x,..., x k. A time series X = (x ; x,..., x k ) contains discrete (or discretized) values x j p = x p (t j ) for each component p P at several time points t,..., t k : P x(t )... x(t j )... x(t k ) a x a(t )... x a(t j )... x a(t k ).... p x p(t )... x p(t j )... x p(t k ) Thereby: choose enough time points t j to capture the true time course of the values x j p for each single component p P: x i x x i x x x x x xx x x x x x x x as we require monotone data for the reconstruction process. t x x x x x x t A. Wagler June 4, 4 6 / 46

22 The Experimental Data As several experiments, starting from different initial states, may be necessary to describe the whole phenomenon, we combine several time series to obtain the experimental data X. x = x = x = x 3 = x 4 = FR R Thereby, we assume the data X to be reproducible: stimulating the system in the same initial state must also result in the same sequence; in particular, each observed state x j must have a unique successor x j+ in X. A. Wagler June 4, 4 7 / 46

23 The Experimental Data As several experiments, starting from different initial states, may be necessary to describe the whole phenomenon, we combine several time series to obtain the experimental data X. x = x = x = x 3 = x 4 = FR R Thereby, we assume the data X to be reproducible: stimulating the system in the same initial state must also result in the same sequence; in particular, each observed state x j must have a unique successor x j+ in X. A. Wagler June 4, 4 7 / 46

24 Example: Light-induced Sporulation of P. polycephalum The states are vectors of the form x = (x FR, x R, x Pfr, x Pr, x Sp ) T having /-entries only. x = x = x = x 3 = x 4 = FR R The scheme represents experimental data X consisting of two time series (x ; x, x, x 3 ) and (x ; x 4, x ). sporulation SP triggered by far-red light FR (turns P fr into P r ) sporulation suppressed by red light R (turns P r back to P fr ) A. Wagler June 4, 4 8 / 46

25 Example: Light-induced Sporulation of P. polycephalum The states are vectors of the form x = (x FR, x R, x Pfr, x Pr, x Sp ) T having /-entries only. x = x = x = x 3 = x 4 = FR R The scheme represents experimental data X consisting of two time series (x ; x, x, x 3 ) and (x ; x 4, x ). sporulation SP triggered by far-red light FR (turns P fr into P r ) sporulation suppressed by red light R (turns P r back to P fr ) A. Wagler June 4, 4 8 / 46

26 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 9 / 46

27 The principle idea for the representation Recall that a conformal (standard) network G = (P, T, A, w) can reproduce 3 the experimentally observed sequences of system states: x t B C x = C A Representation of observed responses C A x t 3 B Use the update vectors r t Z P of the transitions with { w(p, t) if (p, t) A rp t = w(t, p) if (t, p) A else C A = x to realize the changes from one observed state x j to its observed successor x j+. A. Wagler June 4, 4 / 46

28 The principle idea for the representation Recall that a conformal (standard) network G = (P, T, A, w) can reproduce 3 the experimentally observed sequences of system states: x t B C x = C A Representation of observed responses C A x t 3 B Use the update vectors r t Z P of the transitions with { w(p, t) if (p, t) A rp t = w(t, p) if (t, p) A else C A = x to realize the changes from one observed state x j to its observed successor x j+. A. Wagler June 4, 4 / 46

29 The Difficulty: Non-observed Intermediate States Observation The given time series X = (x ; x..., x k ) may not consist of consecutive system states, as between one observed state x j and its observed successor state x j+, there may exist non-observed intermediate states.?????? To reach x j+ from x j, we may need a sequence of transitions t i,..., t k whose update vectors satisfy x j+ x j = r t i r t k : x j = C A r B C A B C A r 3 B C A C A = xj+ A. Wagler June 4, 4 / 46

30 The Difficulty: Non-observed Intermediate States Observation The given time series X = (x ; x..., x k ) may not consist of consecutive system states, as between one observed state x j and its observed successor state x j+, there may exist non-observed intermediate states.?????? To reach x j+ from x j, we may need a sequence of transitions t i,..., t k whose update vectors satisfy x j+ x j = r t i r t k : x j = C A r B C A B C A r 3 B C A C A = xj+ A. Wagler June 4, 4 / 46

31 How to find suitable update vectors? For monotone experimental data X, the intermediate states of the sequence x j = y, y,..., y k, y k+ = x j+ between any two consecutively measured states x j, x j+ X satisfy yp yp... yp k yp k+ for all p P with xp j xp j+ and yp yp... yp k yp k+ for all p P with xp j xp j+. Theorem (Durzinsky, W., Weismantel 8) To represent the difference d = x j+ x j, use only sign-compatibel vectors from r p d p if d p > Box(d) = r Z P : d p r p if d p < \ {}. r p = if d p = Remark. Other constraints may restrict this set further: if P P is a P-invariant of the system, then p P r p = follows for all suitable update vectors r. A. Wagler June 4, 4 / 46

32 How to find suitable update vectors? For monotone experimental data X, the intermediate states of the sequence x j = y, y,..., y k, y k+ = x j+ between any two consecutively measured states x j, x j+ X satisfy yp yp... yp k yp k+ for all p P with xp j xp j+ and yp yp... yp k yp k+ for all p P with xp j xp j+. Theorem (Durzinsky, W., Weismantel 8) To represent the difference d = x j+ x j, use only sign-compatibel vectors from r p d p if d p > Box(d) = r Z P : d p r p if d p < \ {}. r p = if d p = Remark. Other constraints may restrict this set further: if P P is a P-invariant of the system, then p P r p = follows for all suitable update vectors r. A. Wagler June 4, 4 / 46

33 How to find suitable update vectors? For monotone experimental data X, the intermediate states of the sequence x j = y, y,..., y k, y k+ = x j+ between any two consecutively measured states x j, x j+ X satisfy yp yp... yp k yp k+ for all p P with xp j xp j+ and yp yp... yp k yp k+ for all p P with xp j xp j+. Theorem (Durzinsky, W., Weismantel 8) To represent the difference d = x j+ x j, use only sign-compatibel vectors from r p d p if d p > Box(d) = r Z P : d p r p if d p < \ {}. r p = if d p = Remark. Other constraints may restrict this set further: if P P is a P-invariant of the system, then p P r p = follows for all suitable update vectors r. A. Wagler June 4, 4 / 46

34 Example: Light-induced Sporulation (cont.) Consider the time series (x ; x 4, x ) and d = x x 4 = (,,,, ) T. x = FR x = = x 4 R x We obtain the following possible update vectors: = r FR = r R Box(d) = r Z 5 : = r Pfr = r Pr = r Sp = = x 3 = The p-invariant P = {P fr, P r } implies further that r Pfr + r Pr = holds. A. Wagler June 4, 4 3 / 46

35 Example: Light-induced Sporulation (cont.) Consider the time series (x ; x 4, x ) and d = x x 4 = (,,,, ) T. x = FR x = = x 4 R x We obtain the following possible update vectors: = r FR = r R Box(d) = r Z 5 : = r Pfr = r Pr = r Sp = = x 3 = The p-invariant P = {P fr, P r } implies further that r Pfr + r Pr = holds. A. Wagler June 4, 4 3 / 46

36 Example: Light-induced Sporulation (cont.) Consider the time series (x ; x 4, x ) and d = x x 4 = (,,,, ) T. x = FR x = = x 4 R x We obtain the following possible update vectors: = r FR = r R Box(d) = r Z 5 : = r Pfr = r Pr = r Sp = = x 3 = The p-invariant P = {P fr, P r } implies further that r Pfr + r Pr = holds. A. Wagler June 4, 4 3 / 46

37 Example: Light-induced Sporulation (cont.) Consider the time series (x ; x 4, x ) and d = x x 4 = (,,,, ) T. x = FR x = = x 4 R x We obtain the following possible update vectors: = r FR = r R Box(d) = r Z 5 : = r Pfr = r Pr = r Sp = = x 3 = The p-invariant P = {P fr, P r } implies further that r Pfr + r Pr = holds. A. Wagler June 4, 4 4 / 46

38 Representing observed state changes Theorem (Durzinsky, W., Weismantel 8) For each pair x j, x j+ X, we can represent d = x j+ x j by d = λ t r t, λ t Z + r t Box(d) and let R(d, λ) = {r t Box(d) : λ t } be the set of used transitions. Example (cont.) For d 4 = x x 4, the previous considerations showed Box(d 4 ) = = {r 3, r 4, d 4 } and we obtain two representations: λ = (,, ) T with R(d 4, λ)={r 3, r 4 } and λ = (,, ) T with R(d 4, λ )={d 4 } A. Wagler June 4, 4 5 / 46

39 Representing observed state changes Theorem (Durzinsky, W., Weismantel 8) For each pair x j, x j+ X, we can represent d = x j+ x j by d = λ t r t, λ t Z + r t Box(d) and let R(d, λ) = {r t Box(d) : λ t } be the set of used transitions. Example (cont.) For d 4 = x x 4, the previous considerations showed Box(d 4 ) = = {r 3, r 4, d 4 } and we obtain two representations: λ = (,, ) T with R(d 4, λ)={r 3, r 4 } and λ = (,, ) T with R(d 4, λ )={d 4 } A. Wagler June 4, 4 5 / 46

40 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 6 / 46

41 Sequences Sequences Every permutation π = (r t,..., r tm ) of the elements of a set R(d j, λ) yields a sequence of intermediate states x j = y, y,..., y m, y m+ = x j+ with σ π,λ (x j, d j ) = ( (y, r t ), (y, r t ),..., (y m, r tm ) ). Every sequence σ respects monotonicity and induces a priority relation O σ. Example: R(d, λ ) = {r, r } yields σ π,λ (x, d ) = ((x, r ), (x,r )) with r > r and r > σ π,λ (x, d ) = ((x, r ), (x 5, r )) with r > r and x 5 = (,,,, ) T A. Wagler June 4, 4 7 / 46

42 Sequences Sequences Every permutation π = (r t,..., r tm ) of the elements of a set R(d j, λ) yields a sequence of intermediate states x j = y, y,..., y m, y m+ = x j+ with σ π,λ (x j, d j ) = ( (y, r t ), (y, r t ),..., (y m, r tm ) ). Every sequence σ respects monotonicity and induces a priority relation O σ. Example: R(d, λ ) = {r, r } yields σ π,λ (x, d ) = ((x, r ), (x,r )) with r > r and r > σ π,λ x x r r d x σ π,λ (x, d ) = ((x, r ), (x 5, r )) with r > r and x 5 = (,,,, ) T σ π,λ r r x 5 A. Wagler June 4, 4 7 / 46

43 Running example: all obtained sequences For the running example we obtain three difference vectors x xf R xr xpf R xpr xspo x FR d 4 x x 4 d d R x x3 which can be resolved in the following sequences r 4 x FR x r 3 r r x d 4 x 6 x 5 d x r 3 r 4 r r with x 5 = (,,,, ) T and x 6 = (,,,, ) T. x 4 R d x x 3 A. Wagler June 4, 4 8 / 46

44 Priority conflicts Sequences and their conflicts Sequences σ and σ are in priority conflict if there are update vectors r t r t and intermediate states y, y such that both t, t are enabled at y, y but (y, r t ) σ and (y, r t ) σ (since this implies t > t in O σ but t > t in O σ ). weak priority conflict (WPC) if y y, the priority conflict can be resolved by adding appropriate control-arcs strong priority conflict (SPC) if y = y, the priority conflict cannot be resolved by adding appropriate control-arcs Note: we have a SPC between the trivial sequence σ(x k, ) for any terminal state x k and any sequence σ containing x k as intermediate state. Example: σ π,λ (x, d ) = ((x, r ), (x,r )) and σ(x, ) are in SPC. A. Wagler June 4, 4 9 / 46

45 Priority conflicts Sequences and their conflicts Sequences σ and σ are in priority conflict if there are update vectors r t r t and intermediate states y, y such that both t, t are enabled at y, y but (y, r t ) σ and (y, r t ) σ (since this implies t > t in O σ but t > t in O σ ). weak priority conflict (WPC) if y y, the priority conflict can be resolved by adding appropriate control-arcs strong priority conflict (SPC) if y = y, the priority conflict cannot be resolved by adding appropriate control-arcs Note: we have a SPC between the trivial sequence σ(x k, ) for any terminal state x k and any sequence σ containing x k as intermediate state. Example: σ π,λ (x, d ) = ((x, r ), (x,r )) and σ(x, ) are in SPC. A. Wagler June 4, 4 9 / 46

46 Construction of the priority conflict graph Priority conflict graph G = (V D V term, E D E W E S ) the nodes correspond to sequences: V D : the sequence σ π,λ (x j, d j ) for all permutations π of R(d j, λ) V term : the trivial sequence σ(x k, ) for all terminal states x k the edges correspond to their priority conflicts: E D : all SPCs between two sequences σ, σ from the same difference vector E S : all SPCs between sequences σ, σ from distinct difference vectors E W : all WPCs between sequences σ, σ from distinct difference vectors Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

47 Construction of the priority conflict graph Priority conflict graph G = (V D V term, E D E W E S ) the nodes correspond to sequences: V D : the sequence σ π,λ (x j, d j ) for all permutations π of R(d j, λ) V term : the trivial sequence σ(x k, ) for all terminal states x k the edges correspond to their priority conflicts: E D : all SPCs between two sequences σ, σ from the same difference vector E S : all SPCs between sequences σ, σ from distinct difference vectors E W : all WPCs between sequences σ, σ from distinct difference vectors Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

48 Construction of the priority conflict graph Priority conflict graph G = (V D V term, E D E W E S ) the nodes correspond to sequences: V D : the sequence σ π,λ (x j, d j ) for all permutations π of R(d j, λ) V term : the trivial sequence σ(x k, ) for all terminal states x k the edges correspond to their priority conflicts: E D : all SPCs between two sequences σ, σ from the same difference vector E S : all SPCs between sequences σ, σ from distinct difference vectors E W : all WPCs between sequences σ, σ from distinct difference vectors Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

49 Construction of the priority conflict graph Priority conflict graph G = (V D V term, E D E W E S ) the nodes correspond to sequences: V D : the sequence σ π,λ (x j, d j ) for all permutations π of R(d j, λ) V term : the trivial sequence σ(x k, ) for all terminal states x k the edges correspond to their priority conflicts: E D : all SPCs between two sequences σ, σ from the same difference vector E S : all SPCs between sequences σ, σ from distinct difference vectors E W : all WPCs between sequences σ, σ from distinct difference vectors Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

50 Construction of the priority conflict graph Priority conflict graph G = (V D V term, E D E W E S ) the nodes correspond to sequences: V D : the sequence σ π,λ (x j, d j ) for all permutations π of R(d j, λ) V term : the trivial sequence σ(x k, ) for all terminal states x k the edges correspond to their priority conflicts: E D : all SPCs between two sequences σ, σ from the same difference vector E S : all SPCs between sequences σ, σ from distinct difference vectors E W : all WPCs between sequences σ, σ from distinct difference vectors Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

51 Construction of the priority conflict graph Priority conflict graph G = (V D V term, E D E W E S ) the nodes correspond to sequences: V D : the sequence σ π,λ (x j, d j ) for all permutations π of R(d j, λ) V term : the trivial sequence σ(x k, ) for all terminal states x k the edges correspond to their priority conflicts: E D : all SPCs between two sequences σ, σ from the same difference vector E S : all SPCs between sequences σ, σ from distinct difference vectors E W : all WPCs between sequences σ, σ from distinct difference vectors Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

52 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 3 / 46

53 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

54 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

55 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

56 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q A. Wagler June 4, 4 3 / 46

57 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q S = {σ (x, d ), σ(x, d ), σ (x 4, d 4 )}, S = {σ 3(x, d ), σ(x, d ), σ (x 4, d 4 )}, S 3 = {σ (x, d ), σ(x, d ), σ 3(x 4, d 4 )}, S 4 = {σ 3(x, d ), σ(x, d ), σ 3(x 4, d 4 )}. A. Wagler June 4, 4 3 / 46

58 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q S = {σ (x, d ), σ(x, d ), σ (x 4, d 4 )}, S = {σ 3(x, d ), σ(x, d ), σ (x 4, d 4 )}, S 3 = {σ (x, d ), σ(x, d ), σ 3(x 4, d 4 )}, S 4 = {σ 3(x, d ), σ(x, d ), σ 3(x 4, d 4 )}. A. Wagler June 4, 4 3 / 46

59 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q S = {σ (x, d ), σ(x, d ), σ (x 4, d 4 )}, S = {σ 3(x, d ), σ(x, d ), σ (x 4, d 4 )}, S 3 = {σ (x, d ), σ(x, d ), σ 3(x 4, d 4 )}, S 4 = {σ 3(x, d ), σ(x, d ), σ 3(x 4, d 4 )}. A. Wagler June 4, 4 3 / 46

60 Selection of suitable sequences Selection of suitable sequences Towards network composition, we have to select in G - a node set S with exactly one sequence σ V D for each difference d j - all nodes from V term such that no SPCs occur in S V term. Example: G contains the following 4 feasible sets S i V term Q Q 3 σ (x, d ) σ(x 3, ) Q 4 σ (x 4, d 4 ) Q 3 σ (x, d ) Q σ(x, d ) σ (x 4, d 4 ) σ 3 (x, d ) σ(x, ) σ 3 (x 4, d 4 ) Q S = {σ (x, d ), σ(x, d ), σ (x 4, d 4 )}, S = {σ 3(x, d ), σ(x, d ), σ (x 4, d 4 )}, S 3 = {σ (x, d ), σ(x, d ), σ 3(x 4, d 4 )}, S 4 = {σ 3(x, d ), σ(x, d ), σ 3(x 4, d 4 )}. A. Wagler June 4, 4 3 / 46

61 Existence of conformal standard networks The nodes in V D from sequences of the same difference vector d form a clique Q d. Lemma A node σ V D can never be selected for any solution S if there is a clique Q d so that σ is in strong conflict with all sequences σ Q d. In particular, no σ V D can be selected for any solution S if the sequence contains a terminal state as intermediate state. We obtain a reduced priority conflict graph G by recursively removing from V D all nodes which are completely joined to a clique Q d. Theorem (Favre, W. 3) For reproducible data X, there is at least one selection S V term in G and in G ; the sets of all possible selections S V term in G and G are equal. Remark: Outgoing from reproducible data, the existence of a solution cannot be ensured in standard networks with priorities, but in extended networks. A. Wagler June 4, 4 33 / 46

62 Existence of conformal standard networks The nodes in V D from sequences of the same difference vector d form a clique Q d. Lemma A node σ V D can never be selected for any solution S if there is a clique Q d so that σ is in strong conflict with all sequences σ Q d. In particular, no σ V D can be selected for any solution S if the sequence contains a terminal state as intermediate state. We obtain a reduced priority conflict graph G by recursively removing from V D all nodes which are completely joined to a clique Q d. Theorem (Favre, W. 3) For reproducible data X, there is at least one selection S V term in G and in G ; the sets of all possible selections S V term in G and G are equal. Remark: Outgoing from reproducible data, the existence of a solution cannot be ensured in standard networks with priorities, but in extended networks. A. Wagler June 4, 4 33 / 46

63 Standard networks with weak priority conflicts Composition of standard networks For every selected set S V term, we obtain a standard network P S by taking as transitions the union of all update vectors in the sets R(d j, λ) of the selected sequences σ S. Example: In solution S selecting σ 3 (x, d ) = {(x, r ), (x 5, r )}, σ(x, d ) and σ (x 4, d 4 ), there are two WPCs: P R between σ(x, d ) and σ(x, ) since d, are enabled at x, x r F R r d 4 R between σ 3 (x, d ) and σ(x, ) since r, are enabled at x, x d P F R Spo We next insert appropriate control-arcs to resolve all WPCs in P S. A. Wagler June 4, 4 34 / 46

64 Standard networks with weak priority conflicts Composition of standard networks For every selected set S V term, we obtain a standard network P S by taking as transitions the union of all update vectors in the sets R(d j, λ) of the selected sequences σ S. Example: In solution S selecting σ 3 (x, d ) = {(x, r ), (x 5, r )}, σ(x, d ) and σ (x 4, d 4 ), there are two WPCs: P R between σ(x, d ) and σ(x, ) since d, are enabled at x, x r F R r d 4 R between σ 3 (x, d ) and σ(x, ) since r, are enabled at x, x d P F R Spo We next insert appropriate control-arcs to resolve all WPCs in P S. A. Wagler June 4, 4 34 / 46

65 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 35 / 46

66 Resolving WPCs by disabling transitions Resolving WPCs by inserting control-arcs A WPC due to (y, r t ) σ and (y, r t ) σ can be resolved by either disabling t at state y (then t, t remain enabled at y and we add priority t > t ) or disabling t at state y (then t, t remain enabled at y and we add priority t < t ) Remark: If one of y, y is a terminal state, say y, one of the alternatives is not possible, then t has to be disabled at y and t > t = holds automatically. Example: For solution S, we can resolve the WPC between σ(x, d ) and σ(x, ) by disabling d at the terminal state x only (by adding an appropriate control-arc) and have automatically d > in x. A. Wagler June 4, 4 36 / 46

67 Resolving WPCs by disabling transitions Resolving WPCs by inserting control-arcs A WPC due to (y, r t ) σ and (y, r t ) σ can be resolved by either disabling t at state y (then t, t remain enabled at y and we add priority t > t ) or disabling t at state y (then t, t remain enabled at y and we add priority t < t ) Remark: If one of y, y is a terminal state, say y, one of the alternatives is not possible, then t has to be disabled at y and t > t = holds automatically. Example: For solution S, we can resolve the WPC between σ(x, d ) and σ(x, ) by disabling d at the terminal state x only (by adding an appropriate control-arc) and have automatically d > in x. A. Wagler June 4, 4 36 / 46

68 Disabling transitions by inserting control-arcs Disabling transitions by inserting control-arcs To resolve a WPC of (y, r t ) σ and (y, r t ) σ by disabling t at y, we insert a read-arc (p, t) with weight w(p, t) > y p for some p with y p > y p or an inhibitor-arc (p, t) with weight w(p, t) < y p for some p with y p < y p Example: To resolve the WPC between σ(x, d ) and σ(x, ), we note that x, x differ in P FR, P R and disable d at x by one of the following control-arcs: P F R P R R d 4 r F R r r F R r d 4 R P R P F R d Spo d Spo read-arc (P R, d ) by x P R > x P R inhibitor-arc (P FR, d ) by x P FR < x P FR A. Wagler June 4, 4 37 / 46

69 Which sets of control-arcs are appropriate? For each yet constructed standard network P S, we shall find control-arcs that resolve all involved WPCs: Let CA(σ, σ ) be the set of all possible control-arcs that can resolve WPC(σ, σ ) and A S be the union of all these control-arcs for WPCs in P S. To resolve the WPCs in P S, we have to find subsets A of A S hitting each CA(σ, σ ). Such hitting sets A exist if and only if none of the sets CA(σ, σ ) is empty. Non-minimal hitting sets yield extended Peri nets with unnecessary control-arcs and, thus, being not minimal. Theorem (Favre, W. 3) All minimal conformal extended Petri nets based on P S can be obtained by computing all minimal hitting sets in A S. A. Wagler June 4, 4 38 / 46

70 Which sets of control-arcs are appropriate? For each yet constructed standard network P S, we shall find control-arcs that resolve all involved WPCs: Let CA(σ, σ ) be the set of all possible control-arcs that can resolve WPC(σ, σ ) and A S be the union of all these control-arcs for WPCs in P S. To resolve the WPCs in P S, we have to find subsets A of A S hitting each CA(σ, σ ). Such hitting sets A exist if and only if none of the sets CA(σ, σ ) is empty. Non-minimal hitting sets yield extended Peri nets with unnecessary control-arcs and, thus, being not minimal. Theorem (Favre, W. 3) All minimal conformal extended Petri nets based on P S can be obtained by computing all minimal hitting sets in A S. A. Wagler June 4, 4 38 / 46

71 Determining priority relations Determining priority relations Deduce priority relations among so-obtained transitions for all selected sequences σ S by setting t > t for (y, r t ) σ and all other t enabled at y. Example: P R P F R r F R r d 4 R R d 4 r F R r P F R P R d Spo O S,P = {(r > r )} O S,P = {(r > r )} d Spo Theorem (Favre, W. 3) The so-obtained priority relations are conflict-free and reproduce the experiments. A. Wagler June 4, 4 39 / 46

72 Determining priority relations Determining priority relations Deduce priority relations among so-obtained transitions for all selected sequences σ S by setting t > t for (y, r t ) σ and all other t enabled at y. Example: P R P F R r F R r d 4 R R d 4 r F R r P F R P R d Spo O S,P = {(r > r )} O S,P = {(r > r )} d Spo Theorem (Favre, W. 3) The so-obtained priority relations are conflict-free and reproduce the experiments. A. Wagler June 4, 4 39 / 46

73 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 4 / 46

74 Solution set Theorem (Favre, W. 3) Applying this procedure to all feasible sets S V term yields the complete list of all extended Petri nets with priorities being conformal with X. Solution set: Running example PR PF R F R d d 4 R R d 4 d F R PF R PR S d Spo O : d Spo O : PR PF R r F R r d 4 R R d 4 r F R r PF R PR S d Spo O : r > r d Spo O : r > r A. Wagler June 4, 4 4 / 46

75 Solution set: Running example PR PF R F R d r 4 r 4 d F R PF R PR S 3 r 3 R R r 3 Spo Spo d d O S3,P = {(r4 > r 3 )} O S3,P = {(r4 > r 3 )} PR PF R F R d r 4 R r 3 r 3 R r 4 d F R PF R PR d Spo d Spo O S3,P = {(r4 > r 3 )} O S3,P = {(r4 > r 3 )} A. Wagler June 4, 4 4 / 46

76 Solution set: Running example PR PF R r F R r r 4 r 4 r F R r PF R PR S 4 r 3 R R r 3 Spo Spo d d O S4,P = {(r > r ), (r 4 > r 3 )} O S4,P = {(r > r ), (r 4 > r 3 )} PR PF R r F R r r 4 R r 3 r 3 R r 4 r F R r PF R PR d Spo d Spo O S4,P = {(r > r ), (r 4 > r 3 )} O S4,P = {(r > r ), (r 4 > r 3 )} A. Wagler June 4, 4 43 / 46

77 Outline Petri Nets and Extensions Network Reconstruction Problem 3 Reconstruction Approach Input Representation of observed responses Sequences and their conflicts Composition of standard networks Deducing control-arcs and priorities Output 4 Concluding Remarks and Perspectives A. Wagler June 4, 4 44 / 46

78 Concluding remarks and perspectives Achieved goals We proposed an integrative reconstruction method to generate all extended Petri nets with priorities being concormal with experimental time-series data by representing the observed difference vectors as in the case of extended networks (Durzinsky, Marwan, W. and 3), distinguishing weak and strong priority conflicts in the construction of the priority conflict graph (in contrast to Marwan, W., Weismantel 8), resolving weak priority conflicts by inserting control-arcs and determining priorities on the so-obtained transitions. Perspectives Integrate (further) biological pre-knowledge in the reconstruction process. Make the new approach accessible by a suitable implementation (ASP). Apply the approach to new biological experimental data. A. Wagler June 4, 4 45 / 46

79 Concluding remarks and perspectives Achieved goals We proposed an integrative reconstruction method to generate all extended Petri nets with priorities being concormal with experimental time-series data by representing the observed difference vectors as in the case of extended networks (Durzinsky, Marwan, W. and 3), distinguishing weak and strong priority conflicts in the construction of the priority conflict graph (in contrast to Marwan, W., Weismantel 8), resolving weak priority conflicts by inserting control-arcs and determining priorities on the so-obtained transitions. Perspectives Integrate (further) biological pre-knowledge in the reconstruction process. Make the new approach accessible by a suitable implementation (ASP). Apply the approach to new biological experimental data. A. Wagler June 4, 4 45 / 46

80 THANK YOU This work was partly founded by the French National Research Agency, the European Commission (Feder funds), the Région Auvergne within LabEx IMobS 3. A. Wagler June 4, 4 46 / 46