Magdeburg Centre for Systems Biology, Otto-von-Guericke-Universität, Magdeburg, Germany

LIMOS (Laboratoire d'Informatique, Modélisation et Optimisation des Systèmes), University Blaise Pascal (Faculty of Sciences and Technologies) and CNRS, Clermont-Ferrand, France

Abstract

Background

Network inference methods reconstruct mathematical models of molecular or genetic networks directly from experimental data sets. We have previously reported a mathematical method which is exclusively data-driven, does not involve any heuristic decisions within the reconstruction process, and deliveres all possible alternative minimal networks in terms of simple place/transition Petri nets that are consistent with a given discrete time series data set.

Results

We fundamentally extended the previously published algorithm to consider catalysis and inhibition of the reactions that occur in the underlying network. The results of the reconstruction algorithm are encoded in the form of an extended Petri net involving control arcs. This allows the consideration of processes involving mass flow and/or regulatory interactions. As a non-trivial test case, the phosphate regulatory network of enterobacteria was reconstructed using

Conclusions

The new exact algorithm reconstructs extended Petri nets from time series data sets by finding all alternative minimal networks that are consistent with the data. It suggested alternative molecular mechanisms for certain reactions in the network. The algorithm is useful to combine data from wild-type and mutant cells and may potentially integrate physiological, biochemical, pharmacological, and genetic data in the form of a single model.

Background

Network reconstruction methods infere mathematical models of real world networks directly from experimental data (

The output of the algorithm can be encoded as simple place/transition Petri net (Figure

Petri net elements and the representation of a chemical reaction in the form of a Petri net

**Petri net elements and the representation of a chemical reaction in the form of a Petri net**. a) Petri nets are weighted, directed, bipartite graphs consisting of nodes and arcs. The nodes of a Petri net, places and transitions, are interconnected by arcs. An arc always connects a place with a transition or

The principle of automatic network reconstruction explained with the help of a trivial example

**The principle of automatic network reconstruction explained with the help of a trivial example**. a) The input for the reconstruction algorithm is a time series data set that describes the time-course of the components of interest (A,B,C) with discrete values as a causal sequence of events. At time _{2 }the system reached its terminal state, i.e. the values of all components have reached their final level. In the simplest form, the entries are boolean (0,1). b) Shows the reaction vector of the transition in e). A reaction vector corresponds to the incidence matrix of an individual transition or to a column in the incidence matrix of a Petri net. c,d) The presence of the components at given time points is represented by tokens in places assigned to the components. The algorithm evaluates those places the marking of which has changed between two successive time points and e) connects these places with transitions that cause the observed flow of tokens in the reconstructed Petri net.

For the trivial example shown in Figure

According to the sampling theorem, the number of time points taken in a series needs to be sufficiently high to correctly capture the time-dependent change of the measured components in the form of a time-discrete characteristics (Figure

Constraints for the quality of data to be suitable for automatic network reconstruction

**Constraints for the quality of data to be suitable for automatic network reconstruction**. The algorithm requires that the time course of the considered components is known, no matter whether the data are boolean or continuous (and are subsequently discretized) as only those changes are considered that in fact are reflected by the data set. a) The number of time points measured is so low that the time course of the signal is not reflected with a sufficient resolution in detail and interpolation of the data points may mislead the algorithm as phases of formation and decay of the component are missed. b) The number of time points is sufficiently high in order to correctly reflect the time course of the component.

Alternative ways to represent a catalytic (enzymatic) reaction

**Alternative ways to represent a catalytic (enzymatic) reaction**. a) An enzymatic reaction converting substrate S into product P is represented in the form of a place/transition Petri net by modeling the formation and decay of the enzyme-substrate complex ES. Upon firing of

Although the concept of providing a complete list of solutions remains and the basic principle of reconstructing the network in part is the same as in reference

(1) While in

(2) Terminal states of the network are now applied in a new way which again reduces the number of combinations to be analysed.

(3) Reaction vectors and catalytic events in the form of control arcs are now treated completely independent from each other by introducing control functions. This again greatly reduces the number of combinations to be analysed.

(4) The new algorithm allows that reactions may depend on arbitrary logic combinations of the presence or absence of components allowing to represent complex regulatory dependencies.

A detailed description of the mathematical model extended by the new concept of control functions and the mathematical proof of the completeness of solutions provided by the algorithm will be publised in a compagnion paper

A standard experimental approach in molecular biology is to introduce structural changes into a network and to study the resulting change in its static or dynamic behaviour. Such structural changes can be introduced by genetic or pharmacological intervention, for example. Many reactions in a biochemical network depend on the presence or the absence of a certain component while there may be no obvious, measurable time-dependent change in concentration or abundance of these respective components. Enzymes catalyze biochemical reactions, like e.g. the phosphorylation or dephosphorylation of proteins in a regulatory network. Formally, and based on the original definition of catalysis by Berzelius, a gene may be seen to catalyse an even far downstream process. Deleting an enzyme-encoding gene for example may abolish a certain biochemical reaction. Although transcription and translation are in between, the gene indirectly acts as a catalyst of the biochemical reaction in the sense that the gene is necessary for the biochemical reaction to occur, but it is not consumed by the reaction.

Reactions in biochemical or genetic networks may be subject to the control by inhibitors as well. Deletion of a gene encoding an inhibitory subunit of a specific protein may render this protein constitutively active. A biochemical reaction or the expression of a gene may be controled by different factors and/or might occur through alternative mechanisms. The described regulatory dependencies of reactions can be modeled with the help of read arcs (bidirected arcs) and inhibitory arcs in the extended Petri net formalism. These control arcs determine whether a given transition is able to fire depening on the marking of its pre-places (Figure

Implicit representation of extended Petri nets by controled reactions

**Implicit representation of extended Petri nets by controled reactions**. A controled reaction ** R**is composed of the reaction vector

Results

Neglecting catalysis and inhibition, each transition **r**
^{
T
}and the collection of all such vectors yields the incidence matrix of the studied network. Considering catalysis and inhibition we deal with

- controled reactions

Essential function of the algorithm

We describe the essential function of the algorithm by taking the small Petri net of Figure **x _{4 }
**in Figure

Essential steps in the reconstruction of an extended Petri net

**Essential steps in the reconstruction of an extended Petri net**. The simple extended Petri net with places A to F as shown in panel a) was used to generate a time series data set which in turn was taken to explain essential the steps of network reconstruction as shown in panels b) to g).

From the state matrix, the difference vector matrix is computed with difference vectors indicating how the marking of the net (i.e. the number of tokens in all places) changes from one state to the next. The two states **x _{0 }
**and

Each individual transition might potentially be under the control of test- and/or of inhibitory arcs that might emerge from

- ANY

To summarize, when a reaction **r **is applied to a certain state **x**, it has to be enabled by its control function at that state **( f
_{r}(x) = 1)**. If the reaction is applicable at some terminal state

(I) A reaction that is not applicable to any terminal state can be enabled at all times **f**
_{
r
}
**≡ 1**) which means that by minimality no control arcs are pointing to the respective transition in the reconstructed Petri nets.

(II) A reaction which is applicable to an intermediate state of the system which has the same state vector **x **as a terminal state has, must be deleted (ruled out), because the control function **f**
_{
r
}cannot be 1 (on) and 0 (off) at identical marking of the Petri net. Hence no appropriate control arcs exist in this case.

(III) In all other cases the control of the reaction by control arcs is possible (Figure

This allows to identify places that potentially do control the reaction through a control arc. Mathematical details are given in

Exhaustive decomposition of all difference vectors into reaction vectors with all possible permutations in the order (sequence) of the reactions and testing them against the state vectors finally yields a complete list of controled reactions **( R**as potential elements of a reconstructed extended Petri net. Reverse engineered Petri nets are then composed simply by combining for each difference vector one arbitrarily chosen set of controled reactions that results from the decomposition of each of the subsequent difference vectors. Figure

Composition of Petri nets from controled reactions

**Composition of Petri nets from controled reactions**. a) The algorithm described with the help of Figure 6 provides the complete set of possible controled reactions ** R**for each difference vector

The phosphate regulatory network as a test case

The phosphate regulatory network is a network of interacting phosphate-sensing and signal transducing proteins regulating the expression of a battery of genes which are arranged in the

Biology of phosphate regulation

Growth of micro-organisms requires the presence of inorganic phosphate (P_{i}), an essential component for the synthesis of nucleic acids (DNA and RNA). Inorganic phosphate is taken up by the PstSCAB complex, which transports inorganic phosphate into the cytoplasm against its concentration gradient (see legend of Figure

Implicit representation of the phosphate regulatory network reconstructed and graphically displayed in extended Petri net format

**Implicit representation of the phosphate regulatory network reconstructed and graphically displayed in extended Petri net format**. a) This Petri net model of the phosphate regulatory network in enteric bacteria

**Supplementary material to Figure 8**. The pdf contains a detailed explanation of the alternative network motifs of Figure 8 b-e.

Click here for file

Note that this paper does not make any scientific contribution to the biology of phosphate regulation. The Petri net model of the phosphate regulatory network is only used as a test case for the network reconstruction algorithm.

Petri net model of the phosphate regulatory network

The biochemical interaction of the proteins of the phosphate regulatory network and established feed-back mechanisms

Reconstructing the phosphate regulatory network

For reconstructing the phosphate regulatory network, we started with protein components. The task was to find the wiring diagram based on the simulated time-series data sets. The time series data indicated the phosphorylation status of PstSCAB, PhoR, and PhoB in response to external inorganic phosphate. We did not use any kinetic information on the interconversion of the PhoU protein between its inactive and its active state supposing that the corresponding conformational change cannot be directly measured, but the algorithm was told that these two forms, active and inactive, exist. We then generated

**Experiment**

**Genetic background**

**Experimental perturbation**

**Petri net implementation**

Exp #1

Wild-type

Addition of organic and inorganic phosphate

token in pi_pp and po_pp

Exp #2

Wild-type

Absence of organic and inorganic phosphate

Petri net as shown in Figure 8

Exp #3

Wild-type

Inhibition of transcription/translation

no token in

Exp #4

Δ

Absence of organic and inorganic phosphate

no token in Pst-P

Exp #5

Δ

Addition of organic phosphate

no token in Pst-P, token in po_pp

Exp #6

Δ

Addition of organic and inorganic phosphate

no token in Pst-P, token in pi_pp and po_pp

Exp #7

Δ

Absence of organic and inorganic phosphate

no token in PhoU-I

Exp #8

Δ

Absence of organic and inorganic phosphate

no token in PhoR

Exp #9

Δ

Absence of organic and inorganic phosphate

no token in PhoB

Exp #10

Absence of organic and inorganic phosphate

no token in PhoR-S

Exp #11

Absence of organic and inorganic phosphate

no token in PhoB-S

For each experiment, the type of experimental perturbation and the implementation of this perturbation in the form of the intitial marking of the Petri net of Figure 8a is indicated. Genetic background column: Δ

Modeled molecular events that, starting from the initial marking, lead to successive states of the Petri net

**Process**

**leads to State**

Initial marking, inorganic phosphate present

x0

Uptake of Pi, depleting Pi from the periplasm

x1

Dephosphorylation of PstSCAB-P

x2

Activation of PhoU

x3

Phosphorylation of PhoR

x4

Phosphorylation of PhoB

x5

Biosynthesis of PhoA

x6

Transport of PhoA into the periplasm

x7

Degradation of Po_PP (organic phosphate)

x8

Phosphorylation of PstSCAB

x9

Deactivation of PhoU

x10

Dephosphorylation of PhoB-P

x11

The table is useful to assign the state vectors listed in Tables 3 and 4 to the subsequently occurring molecular events.

Starting with the initial marking as shown in Figure

State vectors used for reconstructing the phosphate regulatory network

**Exp. #**

**1**

**2**

**3**

**4**

**5**

**6**

**7**

**8**

**State vectors (compiled from time-series)**

**Vector #**

**0**

**1**

**2**

**3**

**4**

**5**

**6**

**7**

**8**

**9**

**10**

**11**

**12**

**13**

**14**

**15**

**16**

**17**

**18**

**19**

**20**

**21**

**22**

**23**

**24**

**25**

**26**

**27**

**28**

**29**

**30**

**31**

**32**

**33**

**34**

**pi-pp**

1

0

0

0

0

0

0

0

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

**pi-cp**

0

1

1

1

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

**po-pp**

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1

1

1

0

1

1

0

0

0

0

0

**Pst-P**

1

1

0

0

0

0

0

0

0

1

1

1

1

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

1

0

0

**Pst**

0

0

1

1

1

1

1

1

1

0

0

0

0

1

1

1

1

1

1

1

1

0

1

1

1

1

1

0

0

0

0

1

0

1

1

**PhoU-I**

1

1

1

0

0

0

0

0

0

0

1

1

1

1

0

0

0

0

0

0

0

1

1

0

0

0

0

1

1

1

0

0

1

1

0

**PhoU-A**

0

0

0

1

1

1

1

1

1

1

0

0

0

0

1

1

1

1

1

1

1

0

0

1

1

1

1

0

0

0

0

0

0

0

1

**PhoR**

1

1

1

1

0

1

1

1

1

1

1

1

1

1

1

0

1

1

1

1

0

1

1

1

0

1

0

1

1

1

1

1

0

0

0

**PhoR-P**

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

1

0

0

0

1

0

1

0

0

0

0

0

0

0

0

**PhoR-S**

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

**PhoB**

1

1

1

1

1

0

0

0

0

0

0

1

1

1

1

1

0

0

0

0

0

1

1

1

1

0

0

1

1

1

1

1

1

1

1

**PhoB-P**

0

0

0

0

0

1

1

1

1

1

1

0

0

0

0

0

1

1

1

1

1

0

0

0

0

1

1

0

0

0

0

0

0

0

0

**PhoB-S**

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

**PhoA-T**

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

0

0

0

0

0

0

1

1

1

1

1

1

1

1

**PhoA**

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

**PhoA-pp**

0

0

0

0

0

0

0

1

1

1

1

1

0

0

0

0

0

0

1

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

**Term. state**

T

T

T

T

T

T

T

States are assigned to the experiments during which they occurred. The state vectors of experiments 1 to 8 are shown in this table, the state vectors of experiments 9 to 11 are shown in Table 4. States of Pst, PhoU, PhoR, and PhoB were considered as P-invariants. Phosphate, PhoR, PhoRP and the cytoplasmic PhoA protein were excluded as catalytic factors based on biological knowledge.

State vectors used for reconstructing the phosphate regulatory network, difference vectors and additional decomposed reactions as obtained in the course of network reconstruction

**Exp. #**

**9**

**10**

**11**

**State vectors (compiled from time-series)**

**Difference vectors**

**Additional decomposed reactions**

**Vector #**

**35**

**36**

**37**

**38**

**39**

**40**

**41**

**42**

**43**

**44**

**45**

**1**

**2**

**3**

**4**

**5**

**6**

**7**

**8**

**9**

**10**

**11**

**12**

**13**

**14**

**15**

**16**

**17**

**18**

**19**

**pi-pp**

0

0

0

0

0

0

0

0

0

0

0

-1

+1

-1

+1

**pi-cp**

0

0

0

0

0

0

0

0

0

0

0

+1

+1

**po-pp**

0

0

0

0

0

0

0

0

0

0

0

-1

-1

**Pst-P**

1

0

0

0

1

0

0

1

0

0

0

-1

+1

**Pst**

0

1

1

1

0

1

1

0

1

1

1

+1

-1

**PhoU-I**

1

1

0

0

1

1

0

1

1

0

0

-1

+1

**PhoU-A**

0

0

1

1

0

0

1

0

0

1

1

+1

-1

**PhoR**

1

1

1

0

1

1

1

1

1

1

0

-1

+1

+1

**PhoR-P**

0

0

0

1

0

0

0

0

0

0

1

+1

-1

-1

**PhoR-S**

1

1

1

1

0

0

0

1

1

1

1

**PhoB**

0

0

0

0

1

1

1

1

1

1

1

-1

+1

-1

**PhoB-P**

0

0

0

0

0

0

0

0

0

0

0

+1

-1

+1

**PhoB-S**

1

1

1

1

1

1

1

0

0

0

0

**PhoA-T**

1

1

1

1

1

1

1

1

1

1

1

**PhoA**

0

0

0

0

0

0

0

0

0

0

0

+1

-1

-1

**PhoA-pp**

0

0

0

0

0

0

0

0

0

0

0

+1

+1

**Term. state**

T

T

T

States are assigned to the experiments during which they occurred. The state vectors of experiments 1 to 8 are shown in Table 3. Difference vectors and additional decomposed reactions that appeared in more than one experiment are listed only once. States of Pst, PhoU, PhoR, and PhoB were considered as P-invariants. Phosphate, PhoR, PhoRP and the cytoplasmic PhoA protein were excluded as catalytic factors based on biological knowledge.

Each single reaction vector was then analysed whether it is applied to carry the system from one state x_{i }to a subsequent state x_{i+1}. If a reaction was applicable to a state x_{i }which is also a terminal state, the reaction must be disabled in the terminal state, as defined by the appropriate control function (see Figure

The original phosphate regulatory network and the result of the reconstruction procedure are shown in Figure

The algorithm suggested that inorganic phosphate (P_{i}) is transported by the PstSCAB complex into the cell (Figure _{i }disappears in the periplasm and appears in the cytoplasm mediated by independent, non-coupled reactions both catalyzed by the PstSCAB complex would imply the existence of additional pools of inorganic phosphate different from cytoplasm and periplasm that would serve as source and sink, respectively. This alternative is ruled out as unlikely mechanism.

Alternative mechanisms of PhoU inactivation that occur by conformational coupling with the PstSCAB complex are more difficult to distinguish (Figure _{i}. This alternative is discarded since the two molecules reside in different spatial compartments.

A reaction where the molecular mechanism is unclear based on the results of the reconstruction algorithm concerns the dephosphorylation of PhoBP (Figure _{i}. While direct interaction with periplasmic P_{i }again is discarded, other alternatives cannot be ruled out by argumentation. PhoBP dephosphorylation may be inhibited by active PhoU or by the dephosphorylated form of the PstSCAB complex.

A final ambiguity in mechanisms can simply be resolved (Figure

In summary, the algorithm found a core network with defined wiring (Figure

Discussion

We have described a new algorithm for the reconstruction of extended Petri nets from time series data sets. The algorithm delivers a complete list of solutions expressed in the form of Petri nets all of which are compatible with the input data. As for the previously published method

Using a model of the phosphate regulatory network of enterobacteria as a test case, the algorithm correctly reconstructed the Petri net which was used to generate

Attempts to reconstruct a network from a time series data set where multiple components were measured in response to a specific perturbation (stimulation) typically gives a large number of alternative networks. The implicit representation of these networks e.g. as a column of controled reactions listed for each difference vector (Figure

Comparative analysis of time series measured with wild-type and mutants in different genes is an efficient way to reduce the number of alternative network structures delivered by the reconstruction algorithm. When sufficient experiments with different mutants are evaluated, the algorithm may give only one or few solutions as in the phosphate regulatory network that served as a test case in the present study.

Feeding the algorithm with even more data indeed might lead to a situation where not even a single solution is found without that additional components (in the form of additional places) are introduced. The current algorithm does not support the introduction of additional components as an earlier implementation without read arcs did. This feature is to be implemented in a future version of the algorithm.

According to the original definition, Petri nets represent concurrent processes. Transitions that have the licence to fire do not necessarily fire immediately which makes the behaviour of the network nondeterministic. Reaction rates corresponding to rate constants in chemical kinetics can be introduced by assigning a probablilistic hazard function to each transition, yielding a stochastic Petri net

Conclusions

The algorithm described in this work reconstructs extended Petri nets from time series data sets by finding all alternative minimal networks that are consistent with the data. It suggested reasonable alternative molecular mechanisms for certain reactions in the network that can be tested experimentally. The algorithm integrated data obtained for wild-type and for mutant cells and may, in the way shown, be useful to integrate physiological, biochemical, pharmacological, and genetic data into a consistent Petri net model. The algorithm works with discretized data or with data that are

Methods

The Petri net model of the phosphate regulatory network was drawn using the graphical editor of the Petri net tool Snoopy

Starting with the state vector matrix, the difference vector matrix was computed by substracting each state vector from its successor. Each difference vector was then split into a corresponding set consisting of all possible reaction vectors having entries which are either equal to or partially contribute to the entries of the difference vector. The complete set of reaction vectors is called reaction vector matrix.

For each reaction vector, the set of terminal states was identified at which that reaction vector is applicable. This was done by testing for each terminal state and any reaction vector, whether the entries of the sum of the two vectors is within the bounds to give a valid state vector.

For each difference vector, the list of all combinatorial possibilities to write the difference vector as a sum of vectors from its reaction vector matrix was computed.

For each combination of reaction vectors as a decomposition of a difference vector, all permutations in the sequences of the reaction vectors were generated.

For each permutation, the sequence of intermediate states was computed by starting from the first state of the difference vector and successively adding the reaction vectors in the order as they appear in that sequence. For each reaction vector in such a sequence, the information about its control function was evaluated by comparing the corresponding intermediate state vector (at which the function is 1) with the previously computed set of terminal states (at which the function is 0). If for any reaction the intermediate state was equal to a terminal state, then no control function exists and therefore the sequence was deleted. Otherwise, all logical representations with a minimal number of terms that give the required values for the control function were computed using the Quine-McCluskey Algorithm

The described procedures delivered, for each difference vector, the complete list of sequences of controled reactions which are able to generate that difference vector. For each of those sequences of controled reactions, the corresponding Petri net representation was drawn in Snoopy as an alternative sub network as part of a possible solution.

All described computational steps were executed manually.

Authors' contributions

MD, AW and WM co-developed the reported method for reconstruction of extended Petri nets. MD and AW worked out the underlying mathematics. MD reconstructed the network presented in the case study supervised by WM. WM designed the case study, wrote the manuscript and designed the figures. MD and AW helped with the preparation of the figures. AW and WM conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.

Acknowledgements

This work was financially supported by the BMBF through the FORSYS program.