Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryDi�erential Algebra, Regular Chains and ModelingFrançois Boulier, François Lemaire and Mar Moreno Maza(Université de Lille 1, Fran e)ACA 200925 June 2009

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryAbstra tDi�erential Algebra provides algorithmi tools for dealing withDAE (index redu tion, omputation of the variety of onstraints and the missing relations)We will demonstrate an example from modeling inbio hemistryRegular Chains (or equivalently hara teristi sets) are thefundametal obje ts of Di�erential Algebra.Moreover, regular hains provide a bridge from Di�erentialAlgebra to Polynomial Algebra. As a onsequen e, 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 for omputing with algebrai and di�erential regular hains.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryAbstra tDi�erential Algebra provides algorithmi tools for dealing withDAE (index redu tion, omputation of the variety of onstraints and the missing relations)We will demonstrate an example from modeling inbio hemistryRegular Chains (or equivalently hara teristi sets) are thefundametal obje ts of Di�erential Algebra.Moreover, regular hains provide a bridge from Di�erentialAlgebra to Polynomial Algebra. As a onsequen e, 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 for omputing with algebrai and di�erential regular hains.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryAbstra tDi�erential Algebra provides algorithmi tools for dealing withDAE (index redu tion, omputation of the variety of onstraints and the missing relations)We will demonstrate an example from modeling inbio hemistryRegular Chains (or equivalently hara teristi sets) are thefundametal obje ts of Di�erential Algebra.Moreover, regular hains provide a bridge from Di�erentialAlgebra to Polynomial Algebra. As a onsequen e, 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 for omputing with algebrai and di�erential regular hains.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryDi�erential algebraA mathemati al theory (Ritt, Kol hin) whi h permits to pro esssystems of di�erential equations symboli ally, without integratingthem.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryRegular ChainsFor the following input polynomial system :F :

x2 + y + z = 1x + y2 + z = 1x + y + z2 = 1The following regular hains des ribe 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 redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryDi�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 hara teristi sets.Rankings indi ate the sort of sought di�erential equations.Te hni ally, 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 :

· · · > u > v > u > v > u.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryBibliographyE. R. Kol hin.Di�erential Algebra and Algebrai Groups.A ademi 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, resear h preprints (1989).

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistry1 Di�erential elimination and index redu tion2 Di�erential and Algebrai Regular Chains3 Modeling in Bio hemistry

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryExample : a DAE (Hairer, Wanner)The unknowns are three fun tions x(t), y(t) and z(t).

x(t) = 0.7 · y(t) + sin(2.5 · z(t))y(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 z(t) = somethinga omplete set of onstraints on initial values

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryDemo using Di�erentialAlgebraConvert the DAE into a polynomial DAE

x(t) = 0.7 · y(t) + s(t)y (t) = 1.4 · x(t) + (t)1 = x2(t) + y2(t) s(t) = 2.5 · z(t) · (t) (t) = −2.5 · z(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 interfa e MAPLE pa kage for theBLAD libraries (open sour e, LGPL, C language, 40000 lines)

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryThe set of equations � onsequen es� of the DAEThis set has the stru ture of a (radi al) di�erential ideal.Inferen e 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 redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryA word on rankingsLet I be the di�erential ideal asso iated to the DAE.R := Di�erentialRing (derivations = [t℄, blo ks = [[x,y,s, ,z℄℄)RosenfeldGroebner (DAE, R) ;reg hain

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryA word on rankingsLet I be the di�erential ideal asso iated to the DAE.R := Di�erentialRing (derivations = [t℄, blo ks = [[x,y,s, ,z℄℄)RosenfeldGroebner (DAE, R) ;reg hainThe ranking is orderly.The reg hain involves the elements of I of lowest order in alldep. vars.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryA word on rankingsLet I be the di�erential ideal asso iated to the DAE.R := Di�erentialRing (derivations = [t℄, blo ks = [x,y,s, ,z℄)RosenfeldGroebner (DAE, R) ;reg hain

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryA word on rankingsLet I be the di�erential ideal asso iated to the DAE.R := Di�erentialRing (derivations = [t℄, blo ks = [x,y,s, ,z℄)RosenfeldGroebner (DAE, R) ;reg hainThe ranking eliminates x w.r.t y , s, , z .It eliminates y w.r.t. s, , z and so on . . .The reg hain involves the element of I of lowest order in z ,whi h only depends on z .It also involves the element of I of lowest order in , whi honly depends on and z .And so on . . .

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryA word on rankingsLet I be the di�erential ideal asso iated to the DAE.R := Di�erentialRing (derivations = [t℄, blo ks = [[x,y℄,[s, ,z℄℄)RosenfeldGroebner (DAE, R) ;reg hain

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryA word on rankingsLet I be the di�erential ideal asso iated to the DAE.R := Di�erentialRing (derivations = [t℄, blo ks = [[x,y℄,[s, ,z℄℄)RosenfeldGroebner (DAE, R) ;reg hainThe ranking eliminates (x , y) w.r.t. (s, , z).It is a blo k elimination ranking.The reg hain involves the element of I of lowest orderin s, , z whi h only depend on s, , z .It also involves the element of I of lowest order in x , y whi hdepend on all the dep. vars.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryImpli itizations, 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 and

R = · · · 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 redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryBibliographyF. Boulier, D. Lazard, F. Ollivier, M. Petitot.Representations for the radi al 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�Algebrai ProblemsSpringer-Verlag. 1996.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistry1 Di�erential elimination and index redu tion2 Di�erential and Algebrai Regular Chains3 Modeling in Bio hemistry

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryHow does Rosenfeld-Gröbner work ?The Rosenfeld-Gröbner algorithm omputes a triangularde omposition of the input system into di�erential regular hains.This algorithm relies on three key routines : di�erentiation,pseudo-division and GCD modulo algebrai regular hains.The same holds for the Pardi algorithm.Hen e, improvements on the algebrai side bene�t to thedi�erential side.Therefore, there is today a great potential of algebrai te hnology transfert !

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryModular 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 radi al ideal.In pra ti e we hoose p mu h smaller with a probability ofsu ess, i.e. > 99% with p ≈ ln(A(F )).

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryModular 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 radi al ideal.In pra ti e we hoose p mu h smaller with a probability ofsu ess, i.e. > 99% with p ≈ ln(A(F )).

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryFast 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 tri k (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 redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryBibliographyX. Li, M. Moreno Maza and W. PanComputations modulo regular hainsIn pro . of ISSAC 2009.X. Dahan, M. Moreno Maza, É. S host, W. Wu and Y. XieLifting te hniques for triangular de ompositionsIn pro . ISSAC'05M. Moreno Maza and Y. XieBalan ed Dense Polynomial Multipli ation on Multi ores.O. Golubitsky, M. Kondratieva, M. Moreno Maza and A. Ov hinnikovA bound for the Rosenfeld-Gröbner AlgorithmJournal of Symboli Computation, 2007

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistry1 Di�erential elimination and index redu tion2 Di�erential and Algebrai Regular Chains3 Modeling in Bio hemistry

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryThe Mi haelis-Menten redu tion revisitedThe basi enzymati rea tion systemE + S k1−−−→←−−−k2 C k3−−−→ E + PBasi di�erential model :E = k3 C − k1 E S + k2 C ,S = −k1 E S + k2 C ,C = −k3 C + k1 E S − k2 C ,P = k3 C .The approximation, assuming mainly : k1, k2 ≫ k3S = −

Vmax SK + SVmax and K being parameters.

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryThe Mi haelis-Menten redu tion revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PRed terms are the ontributions of the fast rea tion.E = k3 C − (k1 E S − k2 C ) ,S = −(k1 E S − k2 C ) ,C = −k3 C + k1 E S − k2 C ,P = k3 C .

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryThe Mi haelis-Menten redu tion revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PEn ode the onservation of the �ow by repla ing the ontribution of the fast rea tion by a new symbol F1.E = k3 C − F1 ,S = −F1 ,C = −k3 C + F1 ,P = k3 C .

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryThe Mi haelis-Menten redu tion revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PEn ode the onservation of the �ow by repla ing the ontribution of the fast rea tion by a new symbol F1.En ode the speed by adding the equilibrium equation.E = k3 C − F1 ,S = −F1 ,C = −k3 C + F1 ,P = k3 C ,0 = k1 E S − k2 C .

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryThe Mi haelis-Menten redu tion revisitedE + S k1−−−→←−−−k2 C k3−−−→ E + PEn ode the onservation of the �ow by repla ing the ontribution of the fast rea tion by a new symbol F1.En ode the speed by adding the equilibrium equation.E = k3 C − F1 ,S = −F1 ,C = −k3 C + F1 ,P = k3 C ,0 = k1 E S − k2 C .Raw formula by eliminating F1 from Lemaire's DAE.S = −k21 k3 E S2 + k1 k2 k3 E Sk1 k2 (E + S) + k22 ·

Di�erential elimination and index redu tion Di�erential and Algebrai Regular Chains Modeling in Bio hemistryBibliographyV. Van Breusegem, G. Bastin.Redu ed order dynami al modelling of rea tion systems : a singularperturbation approa h30th IEEE Conf. on De ision and Control. (1991).N. Vora, P. Daoutidis.Nonlinear model redu tion of hemi al rea tion systems.AIChE Journal vol. 47 (2001).F. Boulier, M. Lefran , F. Lemaire, P.-E. Morant.Model Redu tion of Chemi al Rea tion Systems using Elimination.MACIS (2007). Submitted to MCS.http://hal.ar hives-ouvertes.fr/hal-00184558