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  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryDi�erential Algebra, Regular Chains and ModelingFrançois Boulier, François Lemaire and Mar Moreno Maza(Université de Lille 1, Frane)ACA 200925 June 2009

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryAbstratDi�erential Algebra provides algorithmi tools for dealing withDAE (index redution, omputation of the variety ofonstraints and the missing relations)We will demonstrate an example from modeling inbiohemistryRegular Chains (or equivalently harateristi sets) are thefundametal objets of Di�erential Algebra.Moreover, regular hains provide a bridge from Di�erentialAlgebra to Polynomial Algebra. As a onsequene, anyimprovement on one side bene�ts to the other.The o-authors of this talk have developed various algorithms(RosenfeldGroebner, Triade, Pardi, VCA) and softwareDifferentialAlgebra, MABSys, Modpn, RegularChains foromputing with algebrai and di�erential regular hains.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryAbstratDi�erential Algebra provides algorithmi tools for dealing withDAE (index redution, omputation of the variety ofonstraints and the missing relations)We will demonstrate an example from modeling inbiohemistryRegular Chains (or equivalently harateristi sets) are thefundametal objets of Di�erential Algebra.Moreover, regular hains provide a bridge from Di�erentialAlgebra to Polynomial Algebra. As a onsequene, anyimprovement on one side bene�ts to the other.The o-authors of this talk have developed various algorithms(RosenfeldGroebner, Triade, Pardi, VCA) and softwareDifferentialAlgebra, MABSys, Modpn, RegularChains foromputing with algebrai and di�erential regular hains.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryAbstratDi�erential Algebra provides algorithmi tools for dealing withDAE (index redution, omputation of the variety ofonstraints and the missing relations)We will demonstrate an example from modeling inbiohemistryRegular Chains (or equivalently harateristi sets) are thefundametal objets of Di�erential Algebra.Moreover, regular hains provide a bridge from Di�erentialAlgebra to Polynomial Algebra. As a onsequene, anyimprovement on one side bene�ts to the other.The o-authors of this talk have developed various algorithms(RosenfeldGroebner, Triade, Pardi, VCA) and softwareDifferentialAlgebra, MABSys, Modpn, RegularChains foromputing with algebrai and di�erential regular hains.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryDi�erential algebraA mathematial theory (Ritt, Kolhin) whih permits to proesssystems of di�erential equations symbolially, without integratingthem.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryRegular ChainsFor the following input polynomial system :F :

    x2 + y + z = 1x + y2 + z = 1x + y + z2 = 1The following regular hains desribe its solutions :

    z = 0y = 1x = 0 ⋃ z = 0y = 0x = 1 ⋃ z = 1y = 0x = 0 ⋃ z2 + 2z − 1 = 0y = zx = z

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryDi�erential eliminationPolynomial differential system

    Ranking

    Differential elimination(Rosenfeld-Gröbner)

    "A" differential system"equivalent" to the input system but"simpler" since it involves"hidden" differential equations"consequences" of the system.�The� output system is a list of regular di�erential hains ordi�erential harateristi sets.Rankings indiate the sort of sought di�erential equations.Tehnially, a ranking is an �admissible� total ordering on thederivatives of the dependent variables.In the ase of ODE in two dep. vars. u(t) and v(t) it might be :

    · · · > ü > v̇ > u̇ > v > u.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryBibliographyE. R. Kolhin.Di�erential Algebra and Algebrai Groups.Aademi Press (1973).J. F. Ritt.Di�erential Algebra.Dover Publ. In. (1950).http://www.ams.org/online_bks/oll33.M. KalkbrenerThree ontributions to elimination theoryJohannes Kepler University, Linz (1991)W.-T. Wu.On the foundation of algebrai di�erential geometry.MM, researh preprints (1989).

    http://www.ams.org/online_bks/coll33

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in Biohemistry1 Di�erential elimination and index redution2 Di�erential and Algebrai Regular Chains3 Modeling in Biohemistry

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryExample : a DAE (Hairer, Wanner)The unknowns are three funtions x(t), y(t) and z(t).

    ẋ(t) = 0.7 · y(t) + sin(2.5 · z(t))ẏ(t) = 1.4 · x(t) + os(2.5 · z(t))1 = x2(t) + y2(t).Di�erential elimination helps integrating the DAE by omputingthe underlying ODE ż(t) = somethinga omplete set of onstraints on initial values

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryDemo using Di�erentialAlgebraConvert the DAE into a polynomial DAE

    ẋ(t) = 0.7 · y(t) + s(t)ẏ (t) = 1.4 · x(t) + (t)1 = x2(t) + y2(t) ṡ(t) = 2.5 · ż(t) · (t)̇(t) = −2.5 · ż(t) · s(t)1 = s2(t) + 2(t).Use Di�erentialRing to set the ranking.Use RosenfeldGroebner to perform di�erential elimination.Di�erentialAlgebra is just an interfae MAPLE pakage for theBLAD libraries (open soure, LGPL, C language, 40000 lines)

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryThe set of equations �onsequenes� of the DAEThis set has the struture of a (radial) di�erential ideal.Inferene rules applied by Rosenfeld-GröbnerLet a, b be two di�erential polynomials1 a = 0 and b = 0⇒ a + b = 02 a = 0 and b =?⇒ a b = 03 a = 0⇒ δa = 04 a b = 0⇒ a = 0 or b = 0

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryA word on rankingsLet I be the di�erential ideal assoiated to the DAE.R := Di�erentialRing (derivations = [t℄, bloks = [[x,y,s,,z℄℄)RosenfeldGroebner (DAE, R) ;reghain

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryA word on rankingsLet I be the di�erential ideal assoiated to the DAE.R := Di�erentialRing (derivations = [t℄, bloks = [[x,y,s,,z℄℄)RosenfeldGroebner (DAE, R) ;reghainThe ranking is orderly.The reghain involves the elements of I of lowest order in alldep. vars.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryA word on rankingsLet I be the di�erential ideal assoiated to the DAE.R := Di�erentialRing (derivations = [t℄, bloks = [x,y,s,,z℄)RosenfeldGroebner (DAE, R) ;reghain

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryA word on rankingsLet I be the di�erential ideal assoiated to the DAE.R := Di�erentialRing (derivations = [t℄, bloks = [x,y,s,,z℄)RosenfeldGroebner (DAE, R) ;reghainThe ranking eliminates x w.r.t y , s, , z .It eliminates y w.r.t. s, , z and so on . . .The reghain involves the element of I of lowest order in z ,whih only depends on z .It also involves the element of I of lowest order in , whihonly depends on and z .And so on . . .

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryA word on rankingsLet I be the di�erential ideal assoiated to the DAE.R := Di�erentialRing (derivations = [t℄, bloks = [[x,y℄,[s,,z℄℄)RosenfeldGroebner (DAE, R) ;reghain

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryA word on rankingsLet I be the di�erential ideal assoiated to the DAE.R := Di�erentialRing (derivations = [t℄, bloks = [[x,y℄,[s,,z℄℄)RosenfeldGroebner (DAE, R) ;reghainThe ranking eliminates (x , y) w.r.t. (s, , z).It is a blok elimination ranking.The reghain involves the element of I of lowest orderin s, , z whih only depend on s, , z .It also involves the element of I of lowest order in x , y whihdepend on all the dep. vars.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryImpliitizations, Ranking ConversionsFor R = x > y > z > s > t and R = t > s > z > y > x :onvert(

    x − t3y − s2 − 1z − s t ,R,R) = s t − z(x y + x)s − z3z6 − x2y3 − 3x2y2 − 3x2y − x2For R = · · · > vxx > vxy > · · · > uxy > uyy > vx > vy > ux >uy > v > u andR = · · · ux > uy > u > · · · > vxx > vxy > vyy > vx > vy > v :onvert(

    vxx − ux4 u vy − (ux uy + ux uy u)u2x − 4 uu2y − 2 u R,R) = u − v2yyvxx − 2 vyyvy vxy − v3yy + vyyv4yy − 2 v2yy − 2 v2y + 1

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryBibliographyF. Boulier, D. Lazard, F. Ollivier, M. Petitot.Representations for the radial of a �nitely generated di�erential idealPro. ISSAC. 1995.F. Boulier, F. Lemaire, M. Moreno Maza.PARDI !ISSAC (2001).E. Hairer, G. Wanner.Solving ordinary di�erential equations II. Sti� and Di�erential�AlgebraiProblemsSpringer-Verlag. 1996.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in Biohemistry1 Di�erential elimination and index redution2 Di�erential and Algebrai Regular Chains3 Modeling in Biohemistry

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryHow does Rosenfeld-Gröbner work ?The Rosenfeld-Gröbner algorithm omputes a triangulardeomposition of the input system into di�erential regularhains.This algorithm relies on three key routines : di�erentiation,pseudo-division and GCD modulo algebrai regular hains.The same holds for the Pardi algorithm.Hene, improvements on the algebrai side bene�t to thedi�erential side.Therefore, there is today a great potential of algebraitehnology transfert !

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryModular MethodsF ⊂ Q[x1, . . . , xn]p prime, p > A(F ) −−−−−−−−−−−−−→ Eq.D.Q(F )Triang.(F mod p) LiftingTriang.(F mod p)y

    x

    LiftingTriang.(F mod p) Lifting{T 1, . . . , T s}⊂ Fp[x1, . . . , xn] Split + Merge−−−−−−−−−−−−−→ Eq.D.Fp (F )A(F ) := 2n2d2n+1(3h + 7 log(n + 1) + 5n log d + 10) where hand d upper bound oe�. sizes and total degrees for f ∈ F .Assumes F square and generates a 0-dimensional radial ideal.In pratie we hoose p muh smaller with a probability ofsu

    ess, i.e. > 99% with p ≈ ln(A(F )).

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryModular MethodsF ⊂ Q[x1, . . . , xn]p prime, p > A(F ) −−−−−−−−−−−−−→ Eq.D.Q(F )Triang.(F mod p) LiftingTriang.(F mod p)y

    x

    LiftingTriang.(F mod p) Lifting{T 1, . . . , T s}⊂ Fp[x1, . . . , xn] Split + Merge−−−−−−−−−−−−−→ Eq.D.Fp (F )A(F ) := 2n2d2n+1(3h + 7 log(n + 1) + 5n log d + 10) where hand d upper bound oe�. sizes and total degrees for f ∈ F .Assumes F square and generates a 0-dimensional radial ideal.In pratie we hoose p muh smaller with a probability ofsu

    ess, i.e. > 99% with p ≈ ln(A(F )).

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryFast Computation of Normal FromsNormal form omputations are used for simpli�ation andequality test of algebrai expressions modulo a set of relations.y3x + yx2 ≡ 1− y mod x2 + 1, y3 + xBy extending the fast division trik (Cook 1966) (Sieveking72) (Kung 74) we have obtained �nearly optimal� algorithmswhen the numbers of rules and variables are equalRelaxing this onstraint and adapting to di�erential normalforms is work in progress.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryBibliographyX. Li, M. Moreno Maza and W. PanComputations modulo regular hainsIn pro. of ISSAC 2009.X. Dahan, M. Moreno Maza, É. Shost, W. Wu and Y. XieLifting tehniques for triangular deompositionsIn pro. ISSAC'05M. Moreno Maza and Y. XieBalaned Dense Polynomial Multipliation on Multiores.O. Golubitsky, M. Kondratieva, M. Moreno Maza and A. OvhinnikovA bound for the Rosenfeld-Gröbner AlgorithmJournal of Symboli Computation, 2007

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in Biohemistry1 Di�erential elimination and index redution2 Di�erential and Algebrai Regular Chains3 Modeling in Biohemistry

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryThe Mihaelis-Menten redution revisitedThe basi enzymati reation systemE + S k1−−−→←−−−k2 C k3−−−→ E + PBasi di�erential model :Ė = k3 C − k1 E S + k2 C ,Ṡ = −k1 E S + k2 C ,Ċ = −k3 C + k1 E S − k2 C ,Ṗ = k3 C .The approximation, assuming mainly : k1, k2 ≫ k3Ṡ = −Vmax SK + SVmax and K being parameters.

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryThe Mihaelis-Menten redution revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PRed terms are the ontributions of the fast reation.Ė = k3 C − (k1 E S − k2 C ) ,Ṡ = −(k1 E S − k2 C ) ,Ċ = −k3 C + k1 E S − k2 C ,Ṗ = k3 C .

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryThe Mihaelis-Menten redution revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PEnode the onservation of the �ow by replaing theontribution of the fast reation by a new symbol F1.Ė = k3 C − F1 ,Ṡ = −F1 ,Ċ = −k3 C + F1 ,Ṗ = k3 C .

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryThe Mihaelis-Menten redution revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PEnode the onservation of the �ow by replaing theontribution of the fast reation by a new symbol F1.Enode the speed by adding the equilibrium equation.Ė = k3 C − F1 ,Ṡ = −F1 ,Ċ = −k3 C + F1 ,Ṗ = k3 C ,0 = k1 E S − k2 C .

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryThe Mihaelis-Menten redution revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PEnode the onservation of the �ow by replaing theontribution of the fast reation by a new symbol F1.Enode the speed by adding the equilibrium equation.Ė = k3 C − F1 ,Ṡ = −F1 ,Ċ = −k3 C + F1 ,Ṗ = k3 C ,0 = k1 E S − k2 C .Raw formula by eliminating F1 from Lemaire's DAE.Ṡ = −k21 k3 E S2 + k1 k2 k3 E Sk1 k2 (E + S) + k22 ·

  • Di�erential elimination and index redution Di�erential and Algebrai Regular Chains Modeling in BiohemistryBibliographyV. Van Breusegem, G. Bastin.Redued order dynamial modelling of reation systems : a singularperturbation approah30th IEEE Conf. on Deision and Control. (1991).N. Vora, P. Daoutidis.Nonlinear model redution of hemial reation systems.AIChE Journal vol. 47 (2001).F. Boulier, M. Lefran, F. Lemaire, P.-E. Morant.Model Redution of Chemial Reation Systems using Elimination.MACIS (2007). Submitted to MCS.http://hal.arhives-ouvertes.fr/hal-00184558

    http://hal.archives-ouvertes.fr/hal-00184558

    Differential elimination and index reductionDifferential and Algebraic Regular ChainsModeling in Biochemistry