Integrating prior knowledge in Automatic Network Reconstruction

Transcription

1 Integrating prior knowledge in Automatic Network Reconstruction M. Favre W. Marwan A. Wagler Laboratoire d Informatique, de Modélisation et d Optimisation des Systèmes (LIMOS) UMR CNRS 658 Université Blaise Pascal,Clermont-Ferrand, France Magdeburg Center for Systems Biology (MaCS), Otto-von-Guericke-Universität, Magdeburg, Germany BioPPN 4 TUNIS June 3, 4 A. Wagler June 3, 4 / 9

2 Outline Petri Nets and Extensions Reconstruction Approach (Sketch) 3 Minimality Aspect 4 Integrating Prior Knowledge 5 Summary and Conclusions A. Wagler June 3, 4 / 9

3 Outline Petri Nets and Extensions Reconstruction Approach (Sketch) 3 Minimality Aspect 4 Integrating Prior Knowledge 5 Summary and Conclusions A. Wagler June 3, 4 3 / 9

4 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 3, 4 4 / 9

5 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 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, and 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 3, 4 5 / 9

6 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 3, 4 6 / 9

7 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 3, 4 6 / 9

8 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 3, 4 7 / 9

9 Outline Petri Nets and Extensions Reconstruction Approach (Sketch) 3 Minimality Aspect 4 Integrating Prior Knowledge 5 Summary and Conclusions A. Wagler June 3, 4 8 / 9

10 The Input The input (P, I P, X ) consists of a set P of studied components the set I P of all known P-invariants experimental time-series data X = {X (x, x k ) : x X ini, xk X initial states X ini X, terminal states X term X time series X (x, x k ) = (x ; x,..., x j,..., x k ) Required properties of X term} with reproducibility: for each x j X there is a unique observed successor state succ X (x j ) = x j+ X (can be ensured by pre-processing (W., Wegener 3)) monotonicity: for each pair (x j, x j+ ) X, the values of the elements do not oscillate in the possible intermediate states between x j and x j+ (depends on the chosen time-points (Durzinsky, W., Weismantel 8)) A. Wagler June 3, 4 9 / 9

11 The Input The input (P, I P, X ) consists of a set P of studied components the set I P of all known P-invariants experimental time-series data X = {X (x, x k ) : x X ini, xk X initial states X ini X, terminal states X term X time series X (x, x k ) = (x ; x,..., x j,..., x k ) Required properties of X term} with reproducibility: for each x j X there is a unique observed successor state succ X (x j ) = x j+ X (can be ensured by pre-processing (W., Wegener 3)) monotonicity: for each pair (x j, x j+ ) X, the values of the elements do not oscillate in the possible intermediate states between x j and x j+ (depends on the chosen time-points (Durzinsky, W., Weismantel 8)) A. Wagler June 3, 4 9 / 9

12 Running example: Input data Example: Light-induced sporulation of Physarum polycephalum Experimental biological data from Starostzik and Marwan (995) P = {FR, R, P FR, P R, G, Spo}, X (x, x 4 ) = (x ; x, x, x 3, x 4 ), X ini = {x, x 5, x 6 }, I P = {P FR, P R }, X (x 5, x ) = (x ; x 5, x ), X term = {x 4, x, x 8 } X (x 6, x 8 ) = (x 3 ; x 6, x 7, x 8 ), serve as input for the algorithm, we present all observed states schematically: x xf R xr xpf R xpr xg xspo x x x x3 x4 x5 x6 x7 x8 FR R R d d d 3 d 4 d 5 d 6 A. Wagler June 3, 4 / 9

13 Output: X -deterministic extended Petri nets with priorities Time series (x, x,..., x k ) are sequences of measured system states. A network fits the given data X if it can reproduce the observations: 3 Output For given X, a network P = (P, T, (A A R A I ), w) with priorities O is X -deterministic if it contains and correctly selects transitions to reach, from every observed state x j X, its observed successor x j+ X. all X -deterministic extended Petri nets with priorities Remark: Finding all networks fitting the data means to create all minimal ones (that do not reproduce the data anymore after removing any element). A. Wagler June 3, 4 / 9

14 Output: X -deterministic extended Petri nets with priorities Time series (x, x,..., x k ) are sequences of measured system states. A network fits the given data X if it can reproduce the observations: 3 Output For given X, a network P = (P, T, (A A R A I ), w) with priorities O is X -deterministic if it contains and correctly selects transitions to reach, from every observed state x j X, its observed successor x j+ X. all X -deterministic extended Petri nets with priorities Remark: Finding all networks fitting the data means to create all minimal ones (that do not reproduce the data anymore after removing any element). A. Wagler June 3, 4 / 9

15 Step : Representation of observed responses Time series (x ; x..., x k ) may not consist of consecutive system states, as between (x j, x j+ ) X, there can be non-observed intermediate states. To reach x j+ from x j, we then need a sequence of transitions t i,..., t k whose update vectors satisfy x j+ x j = r t i r t k. Theorem (Durzinsky, W., Weismantel 8) To represent the difference d = x j+ x j, use only sign-compatibel vectors from Box(d) = r Z P : r p d p if d p > d p r p if d p < r p = if d p = p P r p = P I P For each d = x j+ x j, determine all solutions λ Λ(d) with 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 update vectors. A. Wagler June 3, 4 / 9

16 Step : Representation of observed responses Time series (x ; x..., x k ) may not consist of consecutive system states, as between (x j, x j+ ) X, there can be non-observed intermediate states. To reach x j+ from x j, we then need a sequence of transitions t i,..., t k whose update vectors satisfy x j+ x j = r t i r t k. Theorem (Durzinsky, W., Weismantel 8) To represent the difference d = x j+ x j, use only sign-compatibel vectors from Box(d) = r Z P : r p d p if d p > d p r p if d p < r p = if d p = p P r p = P I P For each d = x j+ x j, determine all solutions λ Λ(d) with 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 update vectors. A. Wagler June 3, 4 / 9

17 Step : Representation of observed responses Time series (x ; x..., x k ) may not consist of consecutive system states, as between (x j, x j+ ) X, there can be non-observed intermediate states. To reach x j+ from x j, we then need a sequence of transitions t i,..., t k whose update vectors satisfy x j+ x j = r t i r t k. Theorem (Durzinsky, W., Weismantel 8) To represent the difference d = x j+ x j, use only sign-compatibel vectors from Box(d) = r Z P : r p d p if d p > d p r p if d p < r p = if d p = p P r p = P I P For each d = x j+ x j, determine all solutions λ Λ(d) with 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 update vectors. A. Wagler June 3, 4 / 9

18 Step : Representation of observed responses (cont.) 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 For the running example we obtain as sequences x 7 d 3 = d 6 x 8 r 4. r 4. x FR x x x 3 r 4. r 4. r. r. d4 = d 5 x d 4 x x 9 d x r 4. x 6 r 4. r 4. r 4. r. r. R x 5 x d x 3 d 3 x 4 R with x 9 = (,,,,, ) T, x = (,,,,, ) T and x = (,,,,, ) T. A. Wagler June 3, 4 3 / 9

19 Step : Detecting priority conflicts between sequences 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. Task: select one sequence per d j s.t. no SPCs occur between them. A. Wagler June 3, 4 4 / 9

20 Step : Detecting priority conflicts between sequences 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. Task: select one sequence per d j s.t. no SPCs occur between them. A. Wagler June 3, 4 4 / 9

21 Step : Detecting priority conflicts between sequences 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 σ(x, ) Q3 σ(x, d ) σ(x 7, d 3 ) σ(x 3, d 3 ) σ(x 6, d 4 ) σ(x 5, d 4 ) σ(x, d ) σ3(x, d ) σ3(x 6, d 4 ) σ3(x 5, d 4 ) σ(x 8, ) σ(x 4, ) Q3 In G, there are 8 possible selections S avoiding SPCs. A. Wagler June 3, 4 5 / 9

22 Step : Detecting priority conflicts between sequences 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 σ(x, ) Q3 σ(x, d ) σ(x 7, d 3 ) σ(x 3, d 3 ) σ(x 6, d 4 ) σ(x 5, d 4 ) σ(x, d ) σ3(x, d ) σ3(x 6, d 4 ) σ3(x 5, d 4 ) σ(x 8, ) σ(x 4, ) Q3 In G, there are 8 possible selections S avoiding SPCs. A. Wagler June 3, 4 5 / 9

23 Step 3: Constructing X -deterministic Petri nets Composition of standard networks For every selected set S, we obtain a standard network P S by deriving transitions from the update vectors in the sets R(d j, λ) of the selected sequences σ S. Example In solution S selecting all canonical sequences σ (x j, d j ), there are three WPCs: P F R between σ (x, d ) and σ(x, ) since d, are enabled at x, x R d 4 d F R between σ (x 3, d 3 ) and σ(x, ) since d 3, are enabled at x 3, x P R between σ (x 7, d 3 ) and σ(x, ) since d 3, are enabled at x 7, x d d 3 G Spo A. Wagler June 3, 4 6 / 9

24 Step 3: Constructing X -deterministic Petri nets (cont.) A WPC(σ, σ ) due to (y, r t ) σ and (y, r t ) σ can be resolved by either disabling t at state y (and adding priority t > t ), or disabling t at state y (and adding priority t < t ). This can be done by inserting control-arcs from CA(σ, σ ) which partitions into a subset CA t>t (σ, σ ) containing a read-arc (p, t) with weight w(p, t) > y p for all p with y p > y p, an inhibitor-arc (p, t) with w(p, t) < y p for all p with y p < y p, and a subset CA t<t (σ, σ ) defined analogously. 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. Inserting one control-arc from each CA(σ, σ ) resolves all WPCs. The obtained extended Petri nets can be made X -deterministic by adding the requested priorities t > t or t < t. A. Wagler June 3, 4 7 / 9

25 Step 3: Constructing X -deterministic Petri nets (cont.) A WPC(σ, σ ) due to (y, r t ) σ and (y, r t ) σ can be resolved by either disabling t at state y (and adding priority t > t ), or disabling t at state y (and adding priority t < t ). This can be done by inserting control-arcs from CA(σ, σ ) which partitions into a subset CA t>t (σ, σ ) containing a read-arc (p, t) with weight w(p, t) > y p for all p with y p > y p, an inhibitor-arc (p, t) with w(p, t) < y p for all p with y p < y p, and a subset CA t<t (σ, σ ) defined analogously. 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. Inserting one control-arc from each CA(σ, σ ) resolves all WPCs. The obtained extended Petri nets can be made X -deterministic by adding the requested priorities t > t or t < t. A. Wagler June 3, 4 7 / 9

26 Step 3: Constructing X -deterministic Petri nets (cont.) A WPC(σ, σ ) due to (y, r t ) σ and (y, r t ) σ can be resolved by either disabling t at state y (and adding priority t > t ), or disabling t at state y (and adding priority t < t ). This can be done by inserting control-arcs from CA(σ, σ ) which partitions into a subset CA t>t (σ, σ ) containing a read-arc (p, t) with weight w(p, t) > y p for all p with y p > y p, an inhibitor-arc (p, t) with w(p, t) < y p for all p with y p < y p, and a subset CA t<t (σ, σ ) defined analogously. 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. Inserting one control-arc from each CA(σ, σ ) resolves all WPCs. The obtained extended Petri nets can be made X -deterministic by adding the requested priorities t > t or t < t. A. Wagler June 3, 4 7 / 9

27 Outline Petri Nets and Extensions Reconstruction Approach (Sketch) 3 Minimality Aspect 4 Integrating Prior Knowledge 5 Summary and Conclusions A. Wagler June 3, 4 8 / 9

28 Inserting minimal sets of control-arcs In a standard network P S, we have to add control-arcs that resolve all WPCs: Hereby, inserting one control-arc from CA(σ, σ ) resolves WPC(σ, σ ), but possibly also another WPC in P S. Let A S be the union of all the control-arcs from the sets CA(σ, σ ) for all WPC(σ, σ ) in P S. To resolve the WPCs in P S, we have to find subsets A of A S hitting each CA(σ, σ ). Non-minimal hitting sets yield extended Peri nets with unnecessary control-arcs and, thus, being not minimal. Theorem All minimal extended Petri nets based on P S can be obtained by computing all minimal hitting sets in A S. A. Wagler June 3, 4 9 / 9

29 Inserting minimal sets of control-arcs In a standard network P S, we have to add control-arcs that resolve all WPCs: Hereby, inserting one control-arc from CA(σ, σ ) resolves WPC(σ, σ ), but possibly also another WPC in P S. Let A S be the union of all the control-arcs from the sets CA(σ, σ ) for all WPC(σ, σ ) in P S. To resolve the WPCs in P S, we have to find subsets A of A S hitting each CA(σ, σ ). Non-minimal hitting sets yield extended Peri nets with unnecessary control-arcs and, thus, being not minimal. Theorem All minimal extended Petri nets based on P S can be obtained by computing all minimal hitting sets in A S. A. Wagler June 3, 4 9 / 9

30 Inserting minimal sets of control-arcs: Example Example For the standard network P S with T = {d, d, d 3 = d 6, d 4 = d 5 }, we have: (P FR,d ) A I (P R,d ) A R (P FR,d 3 ) A I (P R,d 3 ) A R (G,d 3 ) A R WPC x x WPC x x x WPC3 x Two minimal hitting sets result in two minimal extended Petri nets based on P S : PF R PR R d 4 d F R R d 4 d F R PR PF R d G d G d 3 Spo d 3 Spo The total number of minimal X -deterministic extended Petri nets is 66. A. Wagler June 3, 4 / 9

31 Outline Petri Nets and Extensions Reconstruction Approach (Sketch) 3 Minimality Aspect 4 Integrating Prior Knowledge 5 Summary and Conclusions A. Wagler June 3, 4 / 9

32 Indecomposability of difference vectors In Step Representing the observed responses, the set Box(d j ) of update vectors to refine the sequence between (x j, x j+ ) X can be restricted if d j exactly corresponds to a well-known biochemical reaction (including the correct stoichiometry) or to a well-known mechanism (that a certain trigger is detected by a suitable receptor), experiments have shown that subsets of the input components of d j do not lead to the observed response, d j is treated as black box-like reaction where only input and output matter, but not the intermediate mechanism due to the chosen level of detail. Indecomposability of difference vectors Exclude the corresponding response d j D from decomposition, just define: Box(d j ) := {d j } A. Wagler June 3, 4 / 9

33 Indecomposability of difference vectors In Step Representing the observed responses, the set Box(d j ) of update vectors to refine the sequence between (x j, x j+ ) X can be restricted if d j exactly corresponds to a well-known biochemical reaction (including the correct stoichiometry) or to a well-known mechanism (that a certain trigger is detected by a suitable receptor), experiments have shown that subsets of the input components of d j do not lead to the observed response, d j is treated as black box-like reaction where only input and output matter, but not the intermediate mechanism due to the chosen level of detail. Indecomposability of difference vectors Exclude the corresponding response d j D from decomposition, just define: Box(d j ) := {d j } A. Wagler June 3, 4 / 9

34 Indecomposability of difference vectors: Example Example Since the light-dependent reactions of the photoreceptor are so much faster than the subsequent processes, the difference vectors d, d 4 and d 5 describing the photoconversions will not be decomposed due to the chosen level of detail. x 7 d 3 = d 6 x 8 r 4. r 4. x FR x x x 3 r 4. r 4. r. r. d4 = d 5 x d 4 x x 9 d x r 4. x 6 r 4. r 4. r 4. r. r. R x 5 x d x 3 d 3 x 4 R Thus, only solutions based on P S with T = {d, d, d 3 = d 6, d 4 = d 5 } remain. Accordingly, the total number of solutions reduces from 66 to. A. Wagler June 3, 4 3 / 9

35 Treating equal difference vectors in the same way If two difference vectors d i, d j D are equal, we have Box(d i ) = Box(d j ) and Λ(d i ) = Λ(d j ). In Step Detecting priority conflicts between sequences, it is natural to require that d i = d j are decomposed in the same way: using the same λ Λ(d i ) = Λ(d j ) means that the same subset of molecules involved in a reaction will not interact according to different mechanisms, permutation π of the elements in R(d i, λ) = R(d j, λ) means that the order in which the reactions are applied reflects the relative rates of the reactions in R(d i, λ) = R(d j, λ), which leads to the same priorities within the resulting sequences. We call σ π,λ (x i, d i ) and σ π,λ (xj, d j ) twin sequences if d i = d j and the same λ, π has been used. Treating equal difference vectors in the same way Force that twin sequences are always selected together by adding strong priority conflicts in G between all other sequences stemming from a pair d i = d j D. A. Wagler June 3, 4 4 / 9

36 Treating equal difference vectors in the same way If two difference vectors d i, d j D are equal, we have Box(d i ) = Box(d j ) and Λ(d i ) = Λ(d j ). In Step Detecting priority conflicts between sequences, it is natural to require that d i = d j are decomposed in the same way: using the same λ Λ(d i ) = Λ(d j ) means that the same subset of molecules involved in a reaction will not interact according to different mechanisms, permutation π of the elements in R(d i, λ) = R(d j, λ) means that the order in which the reactions are applied reflects the relative rates of the reactions in R(d i, λ) = R(d j, λ), which leads to the same priorities within the resulting sequences. We call σ π,λ (x i, d i ) and σ π,λ (xj, d j ) twin sequences if d i = d j and the same λ, π has been used. Treating equal difference vectors in the same way Force that twin sequences are always selected together by adding strong priority conflicts in G between all other sequences stemming from a pair d i = d j D. A. Wagler June 3, 4 4 / 9

37 Treating equal difference vectors in the same way: Example Example In our running example, we have d 3 = d 6 and d 4 = d 5. The latter vectors can be decomposed in different ways, among the resulting sequences we have σ (x 5, d 4 ), σ (x 6, d 5 ) and σ 3 (x 5, d 4 ), σ 3 (x 6, d 5 ) as pairs of twin sequences. Q σ(x, ) Q3 Q Q6 Q3 Q5 Q4 Q σ(x, d ) σ(x 7, d 6 ) σ(x 3, d 3 ) σ(x 6, d 5 ) σ(x 5, d 4 ) σ(x, d ) σ3(x, d ) σ3(x 6, d 5 ) σ3(x 5, d 4 ) Q8 Q7 σ(x 8, ) σ(x 4, ) Q3 Forcing to select these pairs together rules out 4 of the previously possible 8 selections. Accordingly, the total number of solutions reduces from 66 to 8. A. Wagler June 3, 4 5 / 9

38 Integrating knowledge on relative reaction rates In Step 3 Constructing X -deterministic Petri nets, to resolve a WPC(σ, σ ) between σ, σ involving update vectors r t r t and intermediate states y y, either transition t has to be disabled at y or transition t at y, while the decision between t and t on the other state can be handled by a priority. Prior knowledge about the relative reaction rates of t and t can help to decide whether t > t or t < t better reflects the reality: Integrating knowledge on relative reaction rates If y is a terminal state and σ = σ(y, ) its trivial sequence, then t > holds automatically at y, but t has to be disabled at y using control-arcs from CA t> (σ, σ ) whereas CA t< (σ, σ ) is empty. For a WPC(σ, σ ) where the time-scale of the corresponding experimental observations clearly differs, deduce the correct priority t > t or t < t and reduce CA(σ, σ ) accordingly either to CA t>t (σ, σ ) or to CA t >t(σ, σ ). A. Wagler June 3, 4 6 / 9

39 Integrating knowledge on relative reaction rates In Step 3 Constructing X -deterministic Petri nets, to resolve a WPC(σ, σ ) between σ, σ involving update vectors r t r t and intermediate states y y, either transition t has to be disabled at y or transition t at y, while the decision between t and t on the other state can be handled by a priority. Prior knowledge about the relative reaction rates of t and t can help to decide whether t > t or t < t better reflects the reality: Integrating knowledge on relative reaction rates If y is a terminal state and σ = σ(y, ) its trivial sequence, then t > holds automatically at y, but t has to be disabled at y using control-arcs from CA t> (σ, σ ) whereas CA t< (σ, σ ) is empty. For a WPC(σ, σ ) where the time-scale of the corresponding experimental observations clearly differs, deduce the correct priority t > t or t < t and reduce CA(σ, σ ) accordingly either to CA t>t (σ, σ ) or to CA t >t(σ, σ ). A. Wagler June 3, 4 6 / 9

40 Integrating knowledge on relative reaction rates: Example Example In our running example, we have different time-scales for the experimental observations as d and d 4 = d 5 need only milliseconds to occur, d needs about hour to occur, and d 3 = d 6 need at least hours which implies d, d 4, d 5 > d > d 3, d 6. Consequently, we can reduce the sets CA(σ, σ ) of four WPCs as follows: for WPC4 between σ 3 (x, d ), σ(x 7, d 3 ) to CA r. >d 3(WPC4); for WPC7 between σ 3 (x 5, d 4 ), σ(x, d ) to CA r 4. >d (WPC7); for WPC8 between σ 3 (x 5, d 4 ), σ(x 3, d 3 ) to CA r 4. >d 3(WPC8); for WPC9 between σ 3 (x 6, d 4 ), σ(x 3, d 3 ) to CA r 4. >d 3(WPC9). Accordingly, the number of solutions decreases from 66 to 36. A. Wagler June 3, 4 7 / 9

41 Outline Petri Nets and Extensions Reconstruction Approach (Sketch) 3 Minimality Aspect 4 Integrating Prior Knowledge 5 Summary and Conclusions A. Wagler June 3, 4 8 / 9

42 Summary and Conclusions ANR aims at reconstructing all X -deterministic extended Petri nets that fit given experimental data X which typically results in large solution sets. To keep this solution set reasonably small while still guaranteeing its completeness, we firstly generate only minimal solutions. To integrate prior knowledge in the reconstruction procedure to make the right decisions in some intermediate steps, we extend the input from (P, I P, X ) to (P, I P, X, D in, D eq, O D ) where D in contains all indecomposable difference vectors; D eq contains all pairs of equal difference vectors to be treated in the same way; O D contains relative reaction rates between difference vectors; which results in substantial reductions of solution alternatives. Impact of ANR Models computed by an exact reconstruction approach have predictive ability due to the completeness of the solution set guaranteed by mathematical proofs. A. Wagler June 3, 4 9 / 9

43 Summary and Conclusions ANR aims at reconstructing all X -deterministic extended Petri nets that fit given experimental data X which typically results in large solution sets. To keep this solution set reasonably small while still guaranteeing its completeness, we firstly generate only minimal solutions. To integrate prior knowledge in the reconstruction procedure to make the right decisions in some intermediate steps, we extend the input from (P, I P, X ) to (P, I P, X, D in, D eq, O D ) where D in contains all indecomposable difference vectors; D eq contains all pairs of equal difference vectors to be treated in the same way; O D contains relative reaction rates between difference vectors; which results in substantial reductions of solution alternatives. Impact of ANR Models computed by an exact reconstruction approach have predictive ability due to the completeness of the solution set guaranteed by mathematical proofs. A. Wagler June 3, 4 9 / 9