Semantics of Biological Regulatory Networks

Transcription

1 Electronic Notes in Theoretical Computer Science 180 (2007) Semantics of Biological Regulator Networks Gilles Bernot b Franck Casse a Jean-Paul Comet b Franck Delaplace b Céline Müller b Olivier Rou a a IRCCN, UMR CNRS 6597, BP rue de la Noë, Nantes Cede 3, France b LaMI-genopole, UMR CNRS 8042, UEVE, 523 Place des Terrasses, Evr Cede, France Abstract The aim of the paper is to revisit the model of Biological Regulator Networks (BRN) which was proposed b René Thomas to model the interactions between a set of genes. We give a formal semantics for BRN in terms of transition sstems which formalies the evolution rules given b René Thomas. Then we show how to use this model to find interesting properties of a BRN like the set of stable states, ccles etc using tools for analing transition sstems. Kewords: Biological regulator network, BRN, biological sstem 1 Introduction Modeling of Biological Sstems. The arrival of massive amount of epression data puts the emphasis on computational methods to overcome the difficulties of interpretation of eperimental data. Instead of providing a clear eplanation of biological sstems, data reveals the difficult for analing them. The variet of components and their interacting capabilities lead to cope with their compleit. This opens a field of modeling to investigate computational biological sstems. Computational sstems biolog [10] tries to establish methods and techniques that enable us to understand the structure of the sstem, such as gene/metabolic/signal transduction networks. The modeling of the dnamics of such sstems is a first step towards the control, the design and the modification of the sstems in order to ensure some desired properties[4]. Formal Methods. Formal methods have been used for a decade or more in the area of verification of safet critical sstems. The techniques and tools that have emerged from this /$ see front matter 2007 Elsevier B.V. All rights reserved. doi: /j.entcs

2 4 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) 3 14 field to anale the behaviors of such sstems, makes it possible to model and verif comple concurrent sstems (huge number of states) even with continuous information (dense time) or parameters. It is then natural to tr and use such techniques to model and anale biological sstems especiall when one wants to find properties about their behaviors. Biological Regulator Networks. Biological Regulator Networks (in the sequel BRN) modelie interactions between biological entities (RNA or Proteins). Their regulations involve a lot of comple processes, but it is common to simplif the compleit of the regulations b taking into account onl two actions: activation and inhibition. BRNs are staticall represented b graphs: vertices abstract genes and edges represent their interactions (activation or inhibition). Moreover at a given time, a numerical value is associated to each verte to describe the concentration level of the corresponding entit. The René Thomas boolean approach has been justified as a discretiation of the continuous differential equation sstem[5], it has been confronted to the more classical analsis in terms of differential equations[3]. Then Thomas and Snoussi showed that all stead states can be found via the discrete approach[6]. More recentl Thomas and Kaufman have shown that the discrete description provides a qualitative fit of the differential equations with a small number of possible combinations of values for the parameters[9]. Works of René Thomas and co-workers provide the basis to develop a formal computational framework for gene regulation and its analsis [1]. Our Contribution. In this paper we propose a semantics for an etended gene regulator model of R. Thomas theor. In our etended model a gene can be activator at a certain level and inhibitor at another. This is to our knowledge the first time a formal semantics is proposed for BRN. This enables us to derive automaticall a behavioral model of a BRN and use eisting tools for analing finite state models (e.g. model-checking tools). Outline of the Paper. The paper is organied as follows: section 2 gives the basics of BRN. The core of the paper is in section 3 where we give a formal semantics for BRN. In section 4 we show on a small eample how to use the tool HTech [2] toanaleabrn. 2 Biological Regulator Networks Notations Given a finite set E, E denotes the cardinalit of E. Wedenote2 E the set of subsets of E. If φ is a formula of propositional logic over a set X, [φ] denotes the set of values of the variables satisfing φ. B convention, if U =, φ() are propositional formulas, U φ() =true.

3 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) Biological eamples often rel on intervals: an integer interval [a, b] stands for the set of values { N,a b}, and we denote [] the empt interval. The original model of Biological Regulator Networks [3] makes the assumption that the actual concentration of the products of the genes can be approimated b integer levels: the continuous concentration function is approimated b a piecewise constant function. Those constant levels give the epression levels of the genes. In our formal description of a Biological Regulator Network, a set V of variables stands for the genes of the network. An oriented edge from a variable to indicates that is a regulator (activator, inhibitor) of. Definition 2.1 [Biological Regulator Networks.] Network (BRN) is a 3-uple R =(V,E,π) where: V is a finite set of vertices, E V V is a finite set of edges, A Biological Regulator π =(π +,π )withπ : E 2 N 2 N are respectivel the activation and inhibition functions associated to an edge e E. Moreover, we assume: e E, π + (e) π (e) []: an edge corresponds to a regulation. π + (e) π (e) = []: for a given level, a gene cannot be both activator and inhibitor. Remark 2.2 For v V we will use v as well to denote the epression level of the gene v. π + (, ) (resp. π (, )) gives the interval inside which activates (resp. inhibits). Notethatπ + (, ) =[](resp. π (, ) = []), means that never activates (resp. inhibits). Note also that Def. 2.1 rules out edges (, ) for which π + (, ) =π (, ) = [] which would have no observable effect in the network. ([1,1], []) ([1,1], [2,2]) ([], [1,1]) ([1,1], []) ([], [1,1]) Eemple 1 Eemple 2 Fig. 1. Eamples of BRNs Eample 2.3 Figure 1 gives two eamples of BRN. In the first eample, V = {,, } and π(, ) =([1, 1], []): activates when has the level 1; π(, ) =

4 6 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) 3 14 ([], [1, 1]): inhibits when has the level 1; π(,) =([1, 1], []): activates when has the level 1. The meaning of the second eample is defined accordingl. Definition 2.4 [Activators and Inhibitors.] Let R =(V,E,π) be a Biological Regulator Network, we define the following sets: (i) V,R + () ={ V,π + (,) []} is the set of activators of, (ii) V,R () ={ V,π (, ) []} is the set of inhibitors of. Eample 2.5 For eample 1 of Figure 1, R + () =, R () ={}, R + () ={,}, R () = and R + () =, R () =. Ineample2R + () =, R () ={} and R + () ={}, R () ={}. 3 Formal Semantics of Biological Regulator Networks In René Thomas theor, the evolution of the epression levels of the genes is described b an original notion of attractor. Informall, it represents an upper or a lower bound which is attained if no change occurs in the rest of the BRN. Hence, the computation of the evolution of concentrations is based on the attractors. The are defined b a set of parameters. The evolution of the BRN highl depends on the choice of those parameters. In this section, we formalie the evolution of the states of a BRN b a transition sstem. This semantics also involves some evolution parameters as defined in [3]. 3.1 State Space of a BRN Definition 3.1 [State Space of a BRN.] Let R =(V,E,π) beabrn.thestate space S of a variable V is defined b S =[0, ma V π + (, ) π (, )]. The state space of R is defined b S(R) = V S. A state of the network R is a mapping ν : V N such that V,ν() S. The previous definition of the set S requires that 0 belong to the state space. Eample 3.2 For the eample 2 of Figure 1, S =[0, 2],S =[0, 1]. 3.2 Parameters of a BRN As alread mentioned in the beginning of this section, the behavior of a BRN depends on some parameters. Those parameters pla the role of attractors and give the epression levels towards which a gene is attracted, depending on which genes activate or inhibit it. Definition 3.3 [Parameters of a BRN.] Let R =(V,E,π) beabrn.theset Para(R) ofparameters of R is defined b Para(R) ={K,A,B A R + (),B R ()}

5 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) A valuation of the parameters Para(R) is a mapping κ : Para(R) N such that V,A R + (),B R (),κ(k,a,b ) S. In the sequel we use K,U,V instead of κ(k,u,v ) when the meaning is clear from the contet. Definition 3.4 [Activit Assumption.] Let R =(V,E,π) beabrnandκ a valuation for Para(R). κ satisfies the activit assumption iff V : R + (), X + R + (), X R (),κ(k,x+ {},X ) >κ(k,x+,x ) R (), X + R + (), X R (),κ(k,x+,x {}) <κ(k,x+,x ) Definition 3.5 [Monotonicit Assumption.] Let R =(V,E,π) beabrnand κ be a valuation for Para(R). κ satisfies the monotonicit assumption iff: X + R + (), X R (), X + R +(), X R (), X + X +,X X κ(k,x+,x ) κ(k,x +,X ). The activit assumption stands for the observabilit of the action of a gene on another. Without this assumption it is possible that an combination of activators of a gene does not have an observable effect on the target gene because its level of epression would remain the same. It seems then quite obvious that an valuation of the parameters should satisf this propert. The monotonicit propert is is a biological eperimental fact, alread pointed out b René Thomas. Anwa our framework does not rel on these assumptions motivated b biolog. Eample 3.6 In the eample 2 of Figure 1, a possible valuation of the parameters is: K,, =0,K,, =1andK,, =0,K,, =0,K,, =0,K,, =1and K,, = 0. Notice that this valuation does not satisf the activit propert as K,, K,,.IfwetakeK,, =1,K,, = 0 this propert is satisfied. 3.3 Transition Sstem of a BRN Let us consider a BRN R =(V,E,π). Following [8,7] the evolution of the state of the network depends (i) on the epression level of the genes (ii) on a set of parameters (see Def. 3.3). The epression level of a gene ma either decrease or increase according to which other genes of the network activate or inhibit it. If X + and X are respectivel the set of genes that currentl activate and the set that currentl inhibit, then the value of evolves towards the value defined b the parameter K,X+,X. Which genes are currentl activating or inhibiting is defined according to the levels given in the network (e.g. for eample 2 of Figure 1, activates when its epression level is 1, inhibits when its epression level is 2, and has no effect on when it is 0.) We formall define the different configurations of a network according to the activators and inhibitors of a gene in Def Definition 3.7 [State constraints of a BRN.] For V, X + R + (), X R (), we define A,X+, I,X and C,X+,X b: ( ) ( ) A,X+ = X ( π + +(, )) R + ()\X (/ π + +(, )) ( ) ( ) I,X = X ( π (, )) R ()\X (/ π (, ))

6 8 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) 3 14 C,X+,X = A,X+ I,X C,X+,X is true iff the values of the genes in X + are in the intervals in which the activate and the values of the genes in X are in the intervals in which the inhibit. Eample 3.8 For eample 2 of Fig. 1 the activation and inhibition functions are: π + (, ) =[1, 1],π (, ) =[2, 2] and π + (, ) =,π (, ) =[1, 1]. The activators and inhibitors sets are given b R + () =, R () = and R + () =,R () =. The set of constraints are given b: C,, = / [1, 1], C,, = [1, 1] / [2, 2] and C,, = / [1, 1] [2, 2]. Another feature of the evolution of the state of a network is that the epression level of a gene evolves step-b-step i.e. it cannot go from 1 to 3 in a single step, it must evolve b one unit from 1 to 2 and if some conditions are met 1 will go from 2 to 3. This is captured in the definition of an evolution operator: Definition 3.9 [Evolution Operator.] Let N and k N. The evolution operator is defined b: 1iff>k k = +1 iff<k otherwise Notice that in the case = k the net value of will remain equal to k. We can now define a transition sstem giving the semantics of a BRN. Definition 3.10 [Transition Sstem of a BRN.] Let R =(V,E,π) beabrn and κ a valuation of the parameters in Para(R). The semantics of R with valuation κ is the labeled transition sstem S R (κ) =(S(R),V, ) with S(R) V S(R) such that: A R + (), B R (),ν [C,A,B ] ν ν ν() K,A,B ν () = K,A,B, ν () =ν() Remark 3.11 Note that according to Def. 3.7, there is a unique ν such that ν ν. The transition sstem S R (κ) is (partiall) deterministic in the sense it is deterministic for each -transition. Nevertheless, there ma be another -transition from the state ν and thus S R (κ) is not deterministic. The nondeterminism models the fact that the epression levels of the genes evolve asnchronousl. Note also that there is an -transition onl when has not reached the value it tends to get closer to (i.e. K,A,B for the right 1 it could be that from level 2 it is impossible to reach level 3.

7 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) A and B). This will enable us to define the stable states of a network as those states that have no outgoing transitions (the deadlock states). Definition 3.12 [Stable State of a BRN.] Let R =(V,E,π) beabrnandκ a valuation of the parameters in Para(R) ands R (κ) =(S(R),V, ) its semantics. A state ν S(R) isnon stable iff ν S(R), V such that ν ν.astateν is a stable state if it is not a non stable state (i.e. a stable state is a deadlock state). 4 Simple Case-Stud We consider in this section the eample 1 of Fig. 1. We use the verification tool Htech [2] to automaticall compute the results. Of course we could have chosen an model-checker to anale our models but Htech enables us to compute some constraints on the parameters such that certain properties are satisfied (we will not cover this in this paper.) The Htech input files and results are given in appendi A. The set of activators and inhibitors are given in Eample 2.5, page 4. Thestate space is S =[0, 1], S =[0, 1] and S =[0, 1]. The parameters are K,,, K,, (for ); K,,, K,,, K,,, K,{,}, (for ) andk,, (for ). 4.1 Eample with Regular Stabiliation Let us fi the following values for the parameters: K,, =1 K,, =0 K,, =0 K,, =0 K,, =0 K,{,}, =1 K,, =0 The monotonicit and activit assumptions are satisfied b these parameters. For this eample we obtain the transition sstem given in Fig. A.1 in the appendi A. The Htech input file is given in appendi A. We can easil compute the set of stable states and non stable states as given in the output file Figure A.3, appendi A, 12. Note that the ccle reveals indeed an equilibrium state which is not stable. 4.2 Eample without Regular Stabiliation Let us now fi the parameters to: K,, =1 K,, =0 K,, =0 K,, =1 K,, =1 K,, =1 K,, =0

8 10 G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) 3 14 The transition sstem obtained in this case is given in Fig. A.1 on the right, appendi A. Again the results (Figure A.3 right hand side, appendi A, page 12) obtained with Htech show that there is no regular stable state in this case. Note that the ccle is indeed a stable state, which is called singular in the R.Thomas approach. 5 Conclusion and Future Work In this paper we have given a formal semantics for Biological Regulator Network. The main advantages of this work are (i) the formal semantics enables us to build automaticall a (behavioral) model of a network (ii) this model can then be analed b verification tools eactl as safet critical programs can be (e.g. the formal semantics characteries the stable and non stable states). Our future work will consist in adding timing constraints in the network to build a more accurate model. Our semantics is read to be etended with timing constraints: in this case we will derive a timed or hbrid automata model and use tools for analing this tpes of models to prove properties of the network. Acknowledgement The authors thank genopole-research in Evr (H. Pollard and P. Tambourin) for constant support. Comments from the anonmous referees have also been ver constructive. References [1] O. Cinquin and J. Demongeot. Roles of positive and negative feedback in biological sstem. C. R. Biol, 325(11): , [2] Thomas A. Heninger, Pei-Hsin Ho, and Howard Wong-Toi. HYTECH: A model checker for hbrid sstems. International Journal on Software Tools for Technolog Transfer, 1(1-2): , [3] M. Kaufman and R. Thomas. Model analsis of the bases of multistationarit in the humoral immune response. J. Theor. Biol., 129(2):141 62, [4] H. Kitano. Looking beond the details: a rise in sstem-oriented approaches in genetics and molecular biolog. Curr. Genet., 41(1):1 10, [5] E.H Snoussi. Qualitative dnamics of a piecewise-linear differential equations : a discrete mapping approach. Dnamics and stabilit of Sstems, 4: , [6] E.H. Snoussi and R. Thomas. Logical identification of all stead states : the concept of feedback loop caracteristic states. Bull. Math. Biol., 55(5): , [7] R. Thomas. Logical analsis of sstems comprising feedback loops. J. Theor. Biol., 73(4):631 56, [8] R. Thomas, A.M. Gathoe, and L. Lambert. A comple control circuit. regulation of immunit in temperate bacteriophages. Eur. J. Biochem., 71(1):211 27, [9] R. Thomas and M. Kaufman. Multistationarit, the basis of cell differentiation and memor. I. & II. Chaos, 11: , [10] O. Wolkenhauer. Sstems biolog: the reincarnation of sstems theor applied in biolog? Brief Bioinform., 2(3):258 70, 2001.

9 A G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) Appendi In the appendi, we show the use of Htech in the contet of the analsis of the Biological Regulator Network. For the eample of section 4, we use the input file of Fig. A.2 to model our network. The transition sstems is given in Figure A.1. The result is given in Figure A.3.

10 12 (0,0,0) (0,0,1) (1,0,0) (0,1,0) (1,0,1) (1,1,0) (0,0,0) (1,1,1) (0,1,1) (0,0,1) (1,0,1) (1,0,0) (1,1,1) (0,1,1) (1,1,0) (0,1,0) Fig. A.1 Two transition sstems of the BRN of the Eample 2.3, Fig. 1 with (left) and without (right) regular stabiliation G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) 3 14

11 -- htech input file var k O_O,k O_,k O_O,k O, k O,k 0,k O_O: parameter ; -- parameters,,: discrete ; k: discrete ; -- k changes on ever discrete transition k1,k2,k3: parameter ; -- used for detecting ccles automaton rrb snclabs: ; initiall Start ; loc Start: while >=0 & >=0 & >=0 & <=1 & <=1 & <=1 wait {} -- C_,O,O -> K O_O when < 1 & > k O_O do { =-1,k =1-k} goto Start; when < 1 & < k O_O do { =+1,k =1-k} goto Start; -- C_,O, when >= 1 & < k O_ do { =+1,k =1-k} goto Start; when >= 1 & > k O_ do { =-1,k =1-k} goto Start; -- C_,O,O -> k O_O when <1 & <1 & < k O_O do { =+1,k =1-k} goto Start; when <1 & <1 & > k O_O do { =-1,k =1-k} goto Start; -- C_,,O -> k O_0 when >=1 & <1 & < k O do { =+1,k =1-k} goto Start; when >=1 & <1 & > k O do { =-1,k =1-k} goto Start; -- C_,,O when <1 & >=1 & < k O do { =+1,k =1-k} goto Start; when <1 & >=1 & > k O do { =-1,k =1-k} goto Start; -- C_,,O when >=1 & >=1 & < k 0 do { =+1,k =1-k} goto Start; when >=1 & >=1 & > k 0 do { =-1,k =1-k} goto Start; -- C_,O_O when < k O_O do { =+1,k =1-k} goto Start; when > k O_O do { =-1,k =1-k} goto Start; end var init_reg, f_reachable, stable_states, non_stable_states, _f_reachable, ccle_states : region; init_reg := loc[rrb]=start & >=0 & >=0 & >=0 & <=1 & <=1 & <=1 & k O_O=1 & k O_=0 & k O_O=0 & k O=1 & k O=1 & k 0=1 & k O_O=0; prints "initial values for the K_ parameters and,,:" ; print omit rrb locations hide k,k1,k2,k3 in init_reg endhide ; -- compute the reachable set of states... must be finite -- even if there is a ccle f_reachable := reach forward from init_reg endreach; if empt(f_reachable) then prints "No reachable states..."; else Fig. A.2 Htech Specification of the BRN of Eample 2.3, Figure1. prints "The reachable states are:"; print hide k O_O,k O_,k O_O,k O, k O,k 0,k O_O,k in f_reachable endhide; endif ; -- compute the projection on, of f_reachable _f_reachable := hide k O_O,k O_,k O_O,k O, k O,k 0,k O_O,k in f_reachable endhide; -- compute the set of non stable states i.e. reachable states -- with a successor -- define the strict predecessor operator -- here is a trick to do this with Htech (otherwise Htech computes -- the set of predecessor of a set including the set itself) -- hide k in pre(a & k=0) & k=1 endhide gives the strict predecessor -- of A non_stable_states := f_reachable & hide k in (pre(f_reachable & k=0) & k=1) endhide; -- print the result if empt(non_stable_states) then prints "No non stable states"; else prints "the reachable non stable states are:"; print hide k O_O,k O_,k O_O,k O, k O,k 0,k O_O,k in non_stable_states endhide; endif ; stable_states := f_reachable & ~non_stable_states ; if empt(stable_states) then prints "No stable states...!!!"; else prints "The reachable stable states are:"; print hide k O_O,k O_,k O_O,k O, k O,k 0,k O_O,k in stable_states endhide; endif ; -- now look for ccles... eas in htech with hide -- (eistential quantification) -- first we define the strict sucessor function -- it is a post where k changes followed b a reach ccle_states := =k1 & =k2 & =k3 & f_reachable & reach forward from hide k O_O,k O_, k O_O,k O,k O,k 0,k O_O,k in (post(=k1 & =k2 & =k3 & f_reachable & k=0) & k=1) endhide endreach; -- print the result if empt(ccle_states) then prints "No infinite path in the sstem"; else prints "There is a ccle in the sstem!... from an of these states:"; print hide k O_O,k O_,k O_O,k O, k O,k 0,k O_O,k,k1,k2,k3 in ccle_states endhide; endif ; G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007)

12 14 initial values for the K_ parameters and,,: k O_O = 1 & k O_ = 0 & k O_O = 0 & k O = 0 & k O = 0 & k 0 = 1 & k O_O = 0 & <= 1 & <= 1 & >= 0 & >= 0 & >= 0 & <= 1.Number of iterations required for reachabilit: 1 The reachable states are: <= 1 & >= 0 & <= 1 & >= 0 & <= 1 & >= 0 the reachable non stable states are: = 0 & < 1 & >= 0 & >= 0 & <= 1 = 1 & = 1 & <= 1 & >= 0 = 1 & < 1 & < 1 & >= 0 & >= 0 = 1 & >= 0 & <= 1 & >= 0 & <= 1 The reachable stable states are: < 1 & 0 < & < 1 & >= 0 & <= 1 & >= 0...Number of iterations required for reachabilit: 7 There is a ccle in the sstem!... from an of these states: = 1 & = 0 & = 1 = 1 & = 1 & = 0 = 1 & = 1 & = 1 = 1 & = 0 & = 0 ================================================================= Ma memor used = 0 pages = 0 btes = 0.00 MB Time spent = 57.24u s = sec total ================================================================= initial values for the K_ parameters and,,: k O_O = 1 & k O_ = 0 & k O_O = 0 & k O = 1 & k O = 1 & k 0 = 1 & k O_O = 0 & <= 1 & <= 1 & >= 0 & >= 0 & >= 0 & <= 1.Number of iterations required for reachabilit: 1 The reachable states are: <= 1 & >= 0 & <= 1 & >= 0 & <= 1 & >= 0 the reachable non stable states are: = 0 & < 1 & >= 0 & >= 0 & <= 1 = 1 & = 1 & <= 1 & >= 0 = 1 & < 1 & < 1 & >= 0 & >= 0 = 0 & = 1 & < 1 & >= 0 = 1 & >= 0 & <= 1 & >= 0 & <= 1 The reachable stable states are: 0 < & < 1 & <= 1 & 0 < & >= 0 & < 1 >= 0 & < 1 & 0 < & < 1 & >= 0 & < 1...Number of iterations required for reachabilit: 7 There is a ccle in the sstem!... from an of these states: = 1 & = 0 & >= 0 & < 1 = 0 & = 1 & >= 0 & < 1 = 1 & = 1 & >= 0 & < 1 = 0 & = 0 & >= 0 & < 1 ================================================================= Ma memor used = 0 pages = 0 btes = 0.00 MB Time spent = 73.04u s = sec total ================================================================= Fig. A.3 Htech results for eample of Fig. A.2 (with(left) and without (rigth) regular stabiliation) G. Bernot et al. / Electronic Notes in Theoretical Computer Science 180 (2007) 3 14