A linear algorithm for computing Polynomial Dynamical System
Ines Abdeljaoued-Tej, Alia BenKahla, Ghassen Haddad, Annick Valibouze
AA Linear Algorithm For Computing Polynomial Dynamical Systems
October 10, 2018
Ines Abdeljaoued-Tej , Alia Benkahla , Ghassen Haddad , Annick Valibouze * [email protected] Abstract
Computation biology helps to understand all processes in organisms from interaction of molecules to complex func-tions of whole organs. Therefore, there is a need for mathematical methods and models that deliver logical explanationsin a reasonable time. For the last few years there has been a growing interest in biological theory connected to finite fields:the algebraic modeling tools used up to now are based on Gr¨obner bases or Boolean group. Let n variables representinggene products, changing over the time on p values. A Polynomial dynamical system (PDS) is a function which has severalcomponents; each one is a polynom with n variables and coefficient in the finite field Z / p Z that model the evolution ofgene products. We propose herein a method using algebraic separators, which are special polynomials abundantly studiedin effective Galois theory. This approach avoids heavy calculations and provides a first Polynomial model in linear time. Keywords: Polynomial dynamical system; Finite field; Mathematical model; Gene regulatory network.Code: 37-XX, 37-E-15, 12E20, 12E05, 11Txx, 92C42, 37N25, 12F10, 12Y05, 11Y40
A living cell can be compared to a complex factory animated by DNA and large amounts of biological data are nowavailable. Therefore, the challenge is to extract biological meaning: it consists of developing methodologies for usingthese data to address biological questions [15]. System biology is a study of several networks related to biology data.We choose gene expression data to design regulatory gene network and to build a mathematical model from observationsof the system response to well-constructed perturbations. The data are measurements of concentrations of biochemicals,recorded at time intervals: we propose an adjusted model, defined in a finite field, that fits these data.Several methods have been proposed to infer gene interrelations from expression data. To solve this issue, many re-searchers have proposed various approaches [21]. In fact, mathematical network modelling is an important step towardcovering the dynamic behaviour of biological networks. New areas of mathematics, not traditionally considered applica-ble, are now contributing with powerful tools. Some applied research were focused on investigating dynamical properties.They propose original analysis for regulatory interactions where Boolean models were generated [20, 11]. A linear sys-tem of differential equations, obtained from measured gene expression, can be used to infer gene regulatory network [6].1 a r X i v : . [ q - b i o . M N ] O c t ayesian networks are used to measure gene expression data [12], as well as modeling gene expression data [9].Polynomial dynamical system is an approach used for understanding the behavior of complex systems over time [26].All these approaches deal with biological systems which studies and describes the interactions between micro-biologicaloutputs. It finds its roots in symbolic computation and mathematical modelling. One of the precursors of Polynomialdynamical system is R. Thomas with his Boolean dynamical system [25]. In the last decade, several studies have beenmade, including contributions in Hybrid systems biology [4]. The main objective of the paper is to define an algorithmwhich computes biological systems, over finite fields, in linear time. Our approach is part of a broader framework of amultidisciplinary team working on an algebraic modelling as well as based on EDO or Bayesian networks. With this inmind, we started in [3] by presenting a summary of the main methods adapted to this framework. The algebraic methodpresented in this paper adapts the techniques of Galois theory to issues of bio-systems. This approach can effectivelymodel the important size of the biological data, with relatively simple tools: the calculation of elementary symmetricpolynomials in the case of boolean or the fundamental modules in the polynomial case.The first use of Polynomial dynamical system (PDS) on System biology was published by R. Laubenbacher and B. Stigler[17, 23]. Their model is a deterministic graphical model which depends on the degree of data discretization p ( p = n entries evolve in time, the dynamicalfunction can be represented by n polynomials which describe a table of p n possible state. Our researches were inspiredby a classical method based on Lagrange interpolation [16]. We propose herein two methods using algebraic separators,both are special polynomials abundantly studied in effective Galois theory. Algebraic separators are directly determinedby using symmetric functions, or by linear combinations of fundamental modules. These approaches avoid heavy calcu-lations of Gr¨obner bases and provide a first Polynomial model in linear time, similar to the one produced by the team ofR. Laubenbacher. The computation of a first Polynomial dynamical system, as detailed in this article, can perfectly becomputed in parallel. Tools based on differential equations, Bayesian networks or Boolean networks are commonly usedto model biological networks [5]. Unlike these techniques and despite the important work done within R. Laubenbacher’steam [14], the Polynomial modelling remains underexploited. Very few publications are available in the literature thataddresses the issue of algebraic modelling in biology. Because of the large amount of information present in a PDS,other network forms can easily be derived from it. Further work is needed to discern models that balance these needsoptimally. Issues regarding model personalization and boundary condition tuning are particularly important. Instead, weuse computational algebra to discuss the biochemical networks using the example of gene regulatory networks.We propose a linear algorithm that performs learning in Polynomial dynamical systems. The paper offers an effectivealternative for Gr¨obner basis when modeling biological systems. We introduce a novel method of modelling based onpolynomial over finite fields. The remainder of the paper is organized as follows: Section 2 details an approach allowingseparators computation: we start by introducing useful definitions and notations. Section 2.1 presents a method based onGalois theory tools as the fundamental modules or elementary symmetric functions . Section 2.2 is devoted to introduce amain theorem for computing algebraic separators with optimal degree; it allows us to compute a finite number of otherPolynomial dynamical systems. Section 2.3 summarizes the particular case of Boolean dynamical Systems. Experimentalresults are given in Section 2.4: some basic techniques that enable us to compute affine separators. Section 3 gives analgebraic technique for defining different types of rules between genes. We illustrate our algorithm with a network wherestates are in a finite field. The main feature of the data generated by DNA microarrays, called gene expression data , isthat they have few experiments with regard to the number of genes tested simultaneously. Unfortunately, it is difficult toget all the data’s necessary information to verify if the model is valid. In order to propose a mathematical model basedon the polynomial dynamical systems, we discretize the data by using a double filter for changing genes expressions. Thedata we have chosen to study, are divided in two lines for two different media. Taking into account the expression data ofgenes over time, we discusses an example of computing a Polynomial dynamical system for each cell line computed inlinear time. Finally, Section 4 gives the conclusion with the perspective.2 Polynomial dynamical system
In the following, let p be a prime number and k be a finite field with p elements: k is also noted as Z / p Z . Let E = k n be theset of p n states. A Polynomial dynamical system of dimension n , denoted by PDS, is a polynomial function f = ( f , . . . , f n ) whose components f i are polynomials of the quotient ring k [ x , . . . , x n ] / < x p − x , . . . , x pn − x n > , i.e., polynomials on n variables with coefficients in k and degree smaller than p − n proteins and thechange of these concentrations over m time points. We usually need normalized data but even if normalization is sufficientfor continuous models, it is inappropriate for discrete models, as it is for Boolean or Polynomial Networks. Indeed, in thelatter cases, a discretization is also necessary. It can be done using E. Dimitrova’s method [8]. The latter gives an integer p ,called degree of discretization, and replaces each value by x j with j ∈ [ , n ] , a corresponding value in Z / p Z . At each time t ∈ [ , m ] , the vector s t = ( x , x , . . . , x n ) is called the state of the system at time t . The finite trajectory s (cid:55)→ s (cid:55)→ · · · (cid:55)→ s m is called a discrete trajectory of length m . We consider a PDS, which is a polynomial function f = ( f , . . . , f n ) satisfying,for time i ∈ [ , m − ] : f ( s i ) = s i + . (1)There is an infinite number of such PDS because it is a solution of a multivariable interpolation problem. In fact, let I ( V ) = { h ∈ k [ x , . . . , x n ] | h ( s ) = , ∀ s ∈ V } be the ideal of affine variety V ⊂ E , j ∈ [ , n ] . Let j ∈ [ , n ] . For each i ∈ [ , m − ] , f j ( s i ) = s i + , j . For every g ∈ k [ x , . . . , x n ] there exists h ∈ I ( V ) such as g = f j + h . The polynomial f j − g ∈ I ( V ) verifies also g ( s i ) = s i + , j for i ∈ [ , m − ] .It is possible to separate an element of V with respect to others thanks to a polynomial of k [ x , . . . , x n ] : Definition . Let V ⊂ E and s a state in V . A polynomial r ( x ) ∈ k [ x , . . . , x n ] which equals 1 at x = s and 0 in the otherelements of V is called a (polynomial) separator of s in V .Separators have the same purpose as Lagrange’s polynomials in a univariate interpolation problem. For example, follow-ing the results of [18] and taking into account a Gr¨obner basis’s computation of the ideal I ( V ) , we obtain separators r j of s j in V for j ∈ [ , m − ] : r ( x , x , x ) = − x x − x x + x (2) r ( x , x , x ) = x − x x + x (3) r ( x , x , x ) = − x + x + x (4) r ( x , x , x ) = − x x − x . (5)We compute a PDS f = ( f , . . . , f n ) by interpolation resolution based on separators: starting from the respective con-centrations ( , , , ) , ( , , , ) and ( , , , ) of proteins 1, 2 and 3, we compute a PDS verifying f j ( s i ) = s i + , j for j = , , i = , , , f ( x , x , x ) = r + r + r + r = x + x x + x x − x + x + x + x (6) f ( x , x , x ) = r + r + r + r = x + x x + x x − x + x − x + x (7) f ( x , x , x ) = r + r + r + r = x − x x − x − x + x + x (8) From f = ( f , f , f ) , we can deduce all the trajectories using DVD developed in [32]. This example illustrates a Polyno-mial Networks reproduced by the Buchberger-M¨oller algorithm which is implemented in Computer Algebra Systems likeMacaulay2 [10] or Cocoa [1]. R. Laubenbacher and B. Stigler had discussed the Lagrange interpolation without pushingthe analysis to develop a specific method for computing algebraic separators. However, they adapted Gr¨obner’s basiscomputation of I ( V ) to all PDS, and in particular, to obtain algebraic separators in o ( n m + nm ) . Let s = ( s , . . . , s n ) ∈ E where E = k n is the set of p n states. In k [ x , . . . , x n ] , the maximal ideal M ( s ) of s -relations isgenerated by p = x − s , p = x − s , . . . p n = x n − s n . (9)3igure 1: Space state Graph ( n = p =
5) with respect to (6), (7) and (8)The set p = { p , . . . , p n } is called by N. Tchebotarev the set of fundamental modules of s [24]. The fundamental moduleof s satisfies for all t ∈ E : ∀ j ∈ [ , n ] p j ( t ) = ⇔ t = s . (10)Which means that the variety of M ( s ) is reduced to the single element s . It is a special case of Galois theory in which theGalois group over the field k of the polynomial ( x − s ) · · · ( x − s n ) is the identity group [29]. Let J be the set of indices j for which all elements of V = { s , s , . . . , s m } have the same j -th coordinate: J = { j ∈ [ , n ] | ∀ l ∈ [ , m ] s l , j = s , j } ;On the coordinates indexed by j no separation is possible. Fix S = { , . . . , n }\ J , the minimum subset of { , . . . , n } whichseparates V ’s elements, S is called separator set of V . Note that the set J excludes the gene products (like proteins) whoseconcentration is constant in time. We keep our calculations for genes products that vary over time: these are S ’s elements.In particular, for V = E , the separator set is S = { , . . . , n } . Let us consider this first theorem that lights us on the algebraiccomputation of separators: Theorem 1.
Let be the univariate polynomialg ( x ) = ∏ j ∈ S ∏ l ∈ E ( p j ( x ) − l ) (11) where S is the separator set of V and E all non-zero values taken by the points of V on the ideal’s generators of s -relations(except for those unnecessary to separation):E = { p j ( t ) | j ∈ S ; p j ( t ) (cid:54) = t ∈ V } ⊂ { , . . . , p − } . (12)4 hen the polynomial r ( x ) = g ( x ) g ( s ) (13) is a separator of s in V .Proof. Let g ( s ) (cid:54) = r ( s ) =
1. In fact, for all j ∈ S and l ∈ E , p j ( s ) − l = − l (cid:54) =
0. The product ∏ l ∈ E − l (cid:54) = k because E ⊂ { , . . . , p − } and p is a prime number. Let t ∈ V \{ s } , then it exists j ∈ [ , n ] where p j ( t ) (cid:54) = E , l = p j ( t ) ∈ E ; as the g ’s factor p j ( x ) − l (cid:54) = x = t , we obtain the identity r ( t ) = g ( t ) = Proposition 1.
The complexity of calculation proposed in Theorem 1 equals o ( n ( p − )) .Proof. Let card ( E ) be the cardinal of the set E and card ( S ) the cardinal of the set S . In the worst case, computingthe univariate polynomial g ( x ) = ∏ j ∈ S ∏ l ∈ E ( p j ( x ) − l ) uses card ( S ) = n and card ( E ) = p −
1. On the other hand, p j ( x ) = x j − s j and computing a separator r ( x ) needs 2 n ( p − ) additions, ( n − )( p − ) multiplications and 1 divisionin Z / p Z .To obtain a PDS associated with a discrete path of length m , it suffices to compute m − n , which is very interesting to find a firststudy model (remember that biological data are such that p and m are very small compared to n ). In the previous section, and specially in Theorem 1, we separate repeatedly the same state s from a state t in V . To avoidthis redundant operation, let us reduce the degree of g . For t ∈ V and j ∈ [ , n ] , p t , j ( x ) = x j − t j . Definition . A separate set of two distinct points s and t of V is given by: S ( s , t ) = { j ∈ S | s j (cid:54) = t j } = { j ∈ S | p s , j ( t ) (cid:54) = } where S is the separator set of V . The initial set of separators of s and t is: PS ( s , t ) = { p t , j ( x ) | j ∈ S ( s , t ) } . A minimal initial separator’s set of s in V , denoted by Min ( s , V ) is composed of less than m − g the same applies to each t ∈ V distinct from s , the intersection of Min ( s , V ) with PS ( s , t ) is reduced to a single element.For every distinct point s and t in V and for each g ∈ PS ( s , t ) , we have g ( t ) = p t , j ( t ) = g ( s ) = p t , j ( s ) = − p s , j ( t ) (cid:54) = j ∈ S ( s , t ) . Any product of initial separators of s and t is a separator between these two points. Corollary 1.
Let Min ( s , V ) a minimal initial separator’s set of s in V and let G = ∏ g ∈ Min ( s , V ) g (14) then r ( x ) = G ( x ) G ( s ) (15) is a separator of s in V . We put r s = r or r i = r when s is indexed by i . To illustrate the above definitions, consider the statements s = ( , , ) , s = ( , , ) , s = ( , , ) and s = ( , , ) . The separator set is S = { , , } and the respective separator sets of pair ofelements of V are: S ( s , s ) = { , } , S ( s , s ) = { } , S ( s , s ) = S ( s , s ) = S ( s , s ) = S ( s , s ) = { , , } .We get s ’s initial sets of separators: PS ( s , s ) = { x − , x } , PS ( s , s ) = { x , x , x − } , PS ( s , s ) = { x − , x − , x } .Here are some minimal initial separator’s sets Min ( s , V ) of s in V (there is a finite number): { x − , x } , { x − , x } , { x , x } . We compute a separator r of s = ( , , ) . From the minimum set { x , x } of s in V , we obtain thepolynomial r = x x / = − x x which equals to 1 in s and 0 everywhere in V . This is the same as the separator in [19].5 separator of s obtained by Corollary 1 is r = ( x − )( x − )( x − ) = x x x − x x − x x + x x + x − x − x + V ’s elements are independentfrom each other. So, the computation of a PDS can be done in parallel by applying for each separator the Theorem 1. Notethat the separators obtained by Corollary 1 have degree ≤ n , while those obtained by Theorem 1 have degree ≤ n . card ( E ) .For example, for n = E = { , } , n . card ( E ) = s separator obtained by Theorem 1 (with linear complexity)is r = − x ( x − )( x − ) x x ( x − ) . We suppose, for simplicity, that the separator’s set S of V is { , . . . , n } . In the case p = E is reduced to { } andseparators are in a more compact form. In fact, let k = Z / p Z and let two polynomials in k [ x , . . . , x n ] given by: q = n ∑ r = e r ( p ) and r = q + e ( p ) , . . . , e n ( p ) are elementary symmetric functions in the elements of p = { p , . . . , p n } . To compute, e i ( p ) , weuse tools developed in [28]. For all t ∈ E q ( t ) = ⇔ t = s r ( t ) = ⇔ t (cid:54) = s . In particular, q ( t ) = ∀ t (cid:54) = s and r ( s ) =
0. To see these properties on q and r , we consider the polynomial g in univariate y with coefficients in k [ x , . . . , x n ] whose roots are the fundamental modules, being written as follows: g ( y )( x ) = n ∏ j = ( y − p j ( x )) . The polynomial g ( )( x ) is the same as that of Theorem 1. For t ∈ E , we have the following equivalences: g ( )( t ) = ⇔ ∃ j ∈ [ , n ] : p j ( t ) = ⇔ t (cid:54) = s . So g ( ) is a polynomial separating for s in E . Moreover, according to the following identity on the coefficients ofunivariate polynomials: g ( y ) = y n − e ( p ) · y n − + e ( p ) · y n − · · · + ( − ) n · e n ( p ) , we have r = g ( ) = q + Z / Z .To illustrate this result, let m = p = s = ( , , ) , s = ( , , ) and s = ( , , ) . The fundamental module setsassociated to each state s i is denoted by p i ; we obtain: p = { x , x + , x + } and p = { x + , x + , x } . The elementary symmetric functions of p are: e ( p ) = x + x + x , e ( p ) = x x + x x + x x + x + x + e ( p ) = x x x + x x + x x + x ; so a separator of s is r ( x , x , x ) = x x x + x x . In parallel, we find the separator r of s : r ( x , x , x ) = x x x + x x . For t = ( , ) , t = ( , ) and t = ( , ) , we obtain f = r , f = r + r and f = r . So, f = ( f , f , f ) is defined by: f = x x x + x x , f = x x + x x and f = x x x + x x . (16)6igure 2: Dependency graph of f = ( x , x , x + x ) where n = m = p = f = ( x x x + x x , x x + x x , x x x + x x ) where n = m = p = f is used to draw the dependency graph (shown in Figure 2). Note that usingCocoa, we must consider the states s and s that give r = x and r = x + x . Hence, taking into account s : f =( x , x , x + x ) . This Boolean dynamical system allows us to obtain the dependency graph in Figure 3 (graph whichincludes a smaller number of arcs than the PDS (16)). The dependency graph is a directed graph with vertex set { x , ..., x n } and edge set { ( t , x i ) | t ∈ supp ( f i ) , i = , ..., n } where the support of f i , denoted supp ( f i ) , is the set of variables that appearin f i . We detail in this section a method for determining, in some cases, affine separators. Note CL ( t ) the set of all linearcombinations of generators and M ( t ) the ideal of ( t ) -relations. To find affine separators of s in V , we must determine ( ∩ t (cid:54) = s CL ( t )) \ CL ( s ) . This set is a set of affine separators of s in V . To determine if a relationship does not belong to M ( s ) (i.e., to CL ( s ) ), simplydivide in the fundamental modules of M ( s ) (or what amounts to even evaluate s ). Generally r j ∈ ∩ i (cid:54) = j M ( s i ) and r j / ∈ M ( s j ) , for any separator r j of s j in V . 7et s = ( , , ) , s = ( , , ) , s = ( , , ) , s = ( , , ) ; S = { , , , } . First, we add the generators 2-by-2 ( p i , j + p i , k ),second we subtract ( p i , j − p i , k ) and 2 p i , j + p i , k , etc. Then we pass p i , + p i , + p i , ; for example M ( s ) = < x + , x + , x + > CL ( s ) = { , x + x + x + , x + x + x + , x − x + x + ,..., x − x + , x + x + x + , x + x + ,..., + − ,... } M ( s ) = < x + , x + , x > CL ( s ) = {− x + , − x − x − x + , x + x + x , x − x + x + , − x − x + , x + x + , x + x + x + , x + x + x + , x + x + x + , x + x + x + + − , x + x + x + ,... } M ( s ) = < x + , x + , x + > CL ( s ) = {− x + , x + x + , x + x + x + , x − x + x + ,..., x − x + x + , x + x + } M ( s ) = < x + , x + , x + > CL ( s ) = {− x − x − x + , x − x + x + , − x − x + , x + x + x + , + − , x + x , x + x + x + , − x + ... } We quickly find the separator 2 x + x − s and we can also find a linear identification. r = x + x − ( ∩ i (cid:54) = CL ( s i )) that does not belong to CL ( s ) . In this example, it is not possible to find affine separators of s , s and s .This method can fully be treated dynamically and there is no question to compute full CL ( t ) . In fact, there is an interestingalgorithm for implementation. Several strategies are possible including parallel computing. On the other hand, to avoidrepeating the same relationship in the ideal of ( t ) -relations M ( t ) , we choose a standard format for each polynomial.For example, the polynomial is the same denominator (which we withdrew), it is divided by its content (the gcd of itscoefficients) and multiplying the coefficient of the smallest variable by its sign to make it positive. The authors of [18]explain that the order of x i can be predetermined by information concerning the influence of certain genes over others:e.g. x < x < · · · < x n .Thus, we obtain a first model in the form of a PDS with linear complexity in the order of o ( n . ( p − ) . ( m − )) . Manyother PDS can be determined with our approach optimally. In the other hand, getting all Polynomial dynamical systemsfrom one PDS computed algebraically is also possible: a calculation of the generators of the ideal I ( V ) = { h ∈ k [ x ] | h ( s ) = , ∀ s ∈ V } . The computation of all the PDS through the determination of all separators: a separator r ( x ) of s in V follows from the choice of an element of all { r / ∈ M ( s ) and r ∈ M ( t ) ∀ t ∈ V , t (cid:54) = s } . Let us begin with a brief descriptions and reminders of some of the important concepts, as well as of our example data.Once that has been done; we show how this information can be carried out using PDS. DNA repair pathways maintain theintegrity of the genome. There are at least 150 different proteins that catalyze DNA repair [2]. Conversely, the require-ment for DNA repair and genome maintenance in response to therapy implicates DNA repair proteins. This offers novelapproaches for tumour selective treatment intended to relieve or heal a disorder. To seed new therapies, researches areneeded to explore the detailed consequences of an alteration in each of these repair pathways in general, and in deficientcell lines in particular. Different cell lines provide essential tools to address this need. At first, we selected genes regardingthe homologous recombinations pathway [27].The used information is compared with respect to the evolution of genes in different cell lines, according to the data atdifferent time [31]. Furthermore, we consider the evolution of each gene independently considering the discretization oftheir expression [13]. We assume that given the small changes in gene expressions, Boolean discretization is sufficient for8ur model. Indeed, we calculate the average expression of each gene over time and we discretize in two Boolean stateswith respect to this average. Then, we selected genes that have consistent behaviour compared to the first filter. After thisfiltering, we have kept the genes common to two cell lines, noted C and C . Among these genes, we chose to keep thosewho behave differently on each cell lines: we cnsider for simplicity 5 genes in Table 1, for which we compute a model foreach cell line, which allows us to compare their evolution, over time.Gene cell lines C at 0h C at 24h C at 72h C at 0h C at 24h C at 72h g g g g g C and C in 3 steps of time.We can now assume that the number of variables appearing in each component of the PDS will make the difference. Whatcounts is to have a model that sticks most to the realities of data. We obtain two models: f C describes C cell lines and f C describes C cell lines. Simulation and computation were obtained by using SageMath [22, 7]. f C = ( , , − x x x x x + x x x x , − x x x x x + x x x x + x x x x + x x x x − x x x − x x x − x x x + x x , − x x x x x + x x x x ) f C = ( − x x x x x + x x x x , − x x x x x + x x x x , x x x x + x x x x , , − x x x x x + x x x x ) Here, x is the concentration of g , x the concentration of g , x the concentration of g , x the concentration of g and x the concentration of g . Then we deduce a wiring diagram and a state space graph of the given input data. Describinga gene network in terms of Polynomial dynamical system has advantages. First, it describes gene interactions in anexplicitly numerical form. Second, these are casual relations between genes: a coefficient x i in a function f j determinesthe effect of gene i on gene j . Many biological systems are modeled with discrete models. From the research that has been carried out, it is possibleto conclude that effective alternative for Gr¨obner basis when modeling biological systems is realizable. We use classicalmethod based on Lagrange’s interpolation. This paper details an approach allowing separators’s computation: we presenta method based on Galois theory’s tools as the fundamental modules or elementary symmetric functions.The findings have directly practical relevance. We propose an algorithm that performs learning in Polynomial dynamicalsystems. For this purpose, we introduce a main theorem for computing algebraic separators with optimal degree, it allowsus to compute a finite number of other Polynomial dynamical systems. We also introduce some basic techniques thatenable us to compute affine separators.In this context, we presented an analytical method of easily readable expression and easily interpretable specific data.Thus, we obtain a first model in the form of a Polynomial dynamical systems with linear complexity. Many other Poly-nomial dynamical systems can be determined with our approach optimally. Besides, getting all polynomial dynamicalsystems from one computed algebraically is also possible: a calculation of the generators of the ideal I ( V ) . The calcu-lation of all the PDS through the determination of all separators. The gain is that very quickly we propose models tobio-informatics and molecular biologists in which they can advance and refine their queries.Clearly, further research will be required on experimental data. Continuing research on this field appears fully justifiedbecause of the simplicity of this approach. 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