Application of Differential Algebra to the Quasi-Steady State - PowerPoint PPT Presentation

Application of Differential Algebra to the Quasi-Steady State Approximation in Biology and Physics François Lemaire Université de Lille I (France) Symbolic Computation Team (Boulier, Oussous, Petitot, Sedoglavic) Differential Algebra and Related Topics 2010, Beijing Supported by the ANR LÉDA (Logistique des Équations Différentielles Algébriques) Lemaire (Lille I) Differential Algebra and QSSA DART 2010 1 / 49

Plan Background 1 Differential elimination Slow/fast dynamics Tikhonov theorem Slow/fast chemical reaction systems reduction 2 Michaelis Menten example General method Application to physics 3 Communicating vessels Diffusion Pendulum Other examples ... MABSys 4 Conclusion 5 Lemaire (Lille I) Differential Algebra and QSSA DART 2010 2 / 49

Summary Background 1 Differential elimination Slow/fast dynamics Tikhonov theorem Slow/fast chemical reaction systems reduction 2 Michaelis Menten example General method Application to physics 3 Communicating vessels Diffusion Pendulum Other examples ... MABSys 4 Conclusion 5 Lemaire (Lille I) Differential Algebra and QSSA DART 2010 3 / 49

Differential Algebra founders of differential algebra: Ritt (50), Kolchin (73) many people in this room have contributed to differential algebra key notions: ranking → an ordering on the derivatives characteristic sets (regular differential chains, characterizable sets) → membership test to an ideal package for computing the characteristic sets: diffalg package (Boulier, Hubert, Lemaire) evolution of diffalg: → Differential Algebra (Boulier, Maple 14), based on BLAD (Boulier, C library, GPL) Lemaire (Lille I) Differential Algebra and QSSA DART 2010 4 / 49

Differential elimination System (diff notation) dC ( t ) = F 1 ( t ) − k 2 C ( t ) dt dS ( t ) = − F 1 ( t ) dt dE ( t ) = − F 1 ( t ) + k 2 C ( t ) dt dP ( t ) = k 2 C ( t ) dt k 1 E ( t ) S ( t ) = k − 1 C ( t ) Symbols differential indeterminates: C ( t ) , S ( t ) , E ( t ) , P ( t ) , F 1 ( t ) parameters (=constants): k 1 , k − 1 , k 2 Lemaire (Lille I) Differential Algebra and QSSA DART 2010 5 / 49

Differential elimination System (dotted notation) ˙ C = F 1 − k 2 C ˙ S = − F 1 ˙ E = − F 1 + k 2 C ˙ P = k 2 C k 1 E S = k − 1 C Symbols differential indeterminates: C , S , E , P , F 1 derivatives: C , ˙ C , S , ˙ S , E , ˙ E , P , ˙ P , F 1 ranking [ F 1 ] ≫ [ C , E , P , S ] : any derivative of F 1 is greater than any derivative of C , S , E , P . → elimination ranking (which eliminates F 1 ) parameters (=constants): k 1 , k − 1 , k 2 In this talk, all equations are ordinary (i.e. no partial derivatives) Lemaire (Lille I) Differential Algebra and QSSA DART 2010 6 / 49

Rosenfeld-Gröbner algorithm compute a list of r.d.c. C 1 , . . . , C s from a input system Σ [Σ] = ∩ s p i = 1 Sat ( C i ) with Sat ( C i ) = [ C i ] : H C i ∞ each r.d.c. yields a rewritting system such that p ∈ Sat ( C i ) ⇐ 0 ∗ ⇒ p − → C i E S k 1 C → k − 1 − k 2 k 1 ES ( k 1 S + k − 1 ) ˙ S → k − 1 ( k − 1 + k 1 S + k 1 E ) ˙ P . . . → ˙ E . . . → k 2 k 1 ES ( k 1 S + k − 1 ) F 1 → k − 1 ( k − 1 + k 1 S + k 1 E ) F 1 > F 1 > · · · > ˙ P > ˙ ranking: · · · > ˙ C > ˙ E > ˙ S > C > E > P > S Lemaire (Lille I) Differential Algebra and QSSA DART 2010 7 / 49

Normal forms one has a normal form for polynomials (BL Issac01): p = q mod Sat ( C i ) ⇐ ⇒ NF ( p , C i ) = NF ( q , C i ) the normal form of a polynomial is a rational fractions f / g recent paper : A Normal Form Algorithm For Regular Differential Chains , → Boulier Lemaire, AADIOS09MCS one can also consider normal forms of a rational fraction (provided its denominator is not a zero divisor). Lemaire (Lille I) Differential Algebra and QSSA DART 2010 8 / 49

Timescales Different timescales in physics, biology, . . . : phenomena can have very different timescales exemple: communicating vessels with an input u Two phenomena: input: u ( t ) ( m 3 / s ) exchange between 1 and 2 Variables: x2 V2,S2 V1,S1 volumes: V 1 ( t ) , V 2 ( t ) (in m 3 ) x1 sections: S 1 , S 2 (in m 2 ) water heights: x 1 ( t ) , x 2 ( t ) (in m ) Objective Get rid of the fast timescale Lemaire (Lille I) Differential Algebra and QSSA DART 2010 9 / 49

Removing the fast timescale (1/2) First case: assume the input is much slower than the exchange the two vessels balance instantly one can assume x 1 ( t ) = x 2 ( t ) (after some transient time) the input is split between the two vessels w.r.t. their surfaces u If one assumes it starts balanced: 1 ˙ x 1 = u S 1 + S 2 1 x2 V2,S2 ˙ x 2 = u V1,S1 x1 S 1 + S 2 x 1 ( 0 ) x 2 ( 0 ) = Lemaire (Lille I) Differential Algebra and QSSA DART 2010 10 / 49

Removing the fast timescale (2/2) Second case: assume the input is much faster than the exchange the two vessels have no time for balancing the input only goes in vessel 1 u the vessel 1 will be full in a very short time removing the fast timescale here makes no sense indeed, the fast phenomena does not x2 V2,S2 V1,S1 reach a equilibrium x1 one expects the fast phenomena to balance Remark One can easily remove the slow timescale: ˙ x 2 = 0 x 1 = u / S 1 , ˙ Lemaire (Lille I) Differential Algebra and QSSA DART 2010 11 / 49

Intuition of the Tikhonov theorem  ˙ x = f ( x , y ) Σ 1 ˙ y = ε g ( x , y ) ε is small x is the slow variable y is the fast variable (because of 1 /ε ) a solution starting from ( x 0 , y 0 ) behaves in two steps: fast transient x ( t ) does not change y ( t ) quickly reaches the curve g ( x 0 , y ) = 0 slow the solution slowly drifts on g ( x , y ) = 0 following ˙ x = f ( x , y ) . The variety g ( x , y ) = 0 is called the slow variety Lemaire (Lille I) Differential Algebra and QSSA DART 2010 12 / 49

Example of the Tikhonov theorem ˙ x = x + y 1 ε ( 1 − xy ) ˙ y = 3 2.5 2 y 1.5 1 0.5 2 4 6 8 10 x for ε = 0 . 01 and three different initial conditions ( x 0 , y 0 ) = ( 2 , 3 ) , ( 3 , 1 ) and ( 1 , 0 . 5 ) . The transient is (almost) vertical, and solutions slides along y = 1 / x . Lemaire (Lille I) Differential Algebra and QSSA DART 2010 13 / 49

Tikhonov theorem Idea of the theorem it is a limit theorem: the limits of the solutions (when ε → 0 ) are the solutions when ε = 0 in the system. The theorem (rough version)  ˙ x = f ( x , y ) Σ 1 ˙ = ε g ( x , y ) y y = 1 If for any ( x 0 , y 0 ) , the solution of ˙ ε g ( x 0 , y ) , y ( 0 ) = y 0 converges to ¯ y with y ) = 0, then g ( x 0 , ¯ there exist α, T > 0 such that the solution of Σ uniformally tends (for t ∈ [ α, T ] ) towards the solution of the reduced system  ˙ x = f ( x , y ) ¯ Σ 0 = g ( x , y ) Σ is the quasi steady-state approximation of Σ . ¯ Lemaire (Lille I) Differential Algebra and QSSA DART 2010 14 / 49

Tikhonov for the vessels u Input slower than balancing u / S 1 − 1 ˙ x 1 = ε F ( x 1 , x 2 ) / S 1 x2 V2,S2 1 ˙ x 2 = ε F ( x 1 , x 2 ) / S 2 V1,S1 x1 x 1 ( 0 ) x 2 ( 0 ) = the exchange satisfies: F ( x 1 , x 2 ) = 0 ⇐ ⇒ x 1 = x 2 if you assume that x 1 = x 2 , you get: ˙ x 1 = u / S 1 and ˙ x 2 = 0 pb: 1 /ε times something small is not zero exact algorithms do not like ≈ 0 Lemaire (Lille I) Differential Algebra and QSSA DART 2010 15 / 49

Tikhonov for the vessels u Input slower than balancing u / S 1 − 1 ˙ x 1 = ε F ( x 1 , x 2 ) / S 1 x2 V2,S2 1 ˙ x 2 = ε F ( x 1 , x 2 ) / S 2 V1,S1 x1 x 1 ( 0 ) x 2 ( 0 ) = The theorem does not apply (not in Tikhonov form). x 1 and x 2 are both "fast". introducing y = S 1 x 1 + S 2 x 2 , the system is under Tikhonov form ( y is slow . . . ) indeed, y is the total volume of liquid, so it is not affected by the exchange phenomenon ˙ y = u 1 ˙ x 2 = ε F (( y − S 2 x 2 ) / S 1 , x 2 ) / S 2 since x 1 = ( y − S 2 x 2 ) / S 1 Applying the Tikhonov theorem . . . Lemaire (Lille I) Differential Algebra and QSSA DART 2010 16 / 49

Tikhonov for the vessels u Input slower than balancing u / S 1 − 1 ˙ x 1 = ε F ( x 1 , x 2 ) / S 1 x2 V2,S2 1 ˙ x 2 = ε F ( x 1 , x 2 ) / S 2 V1,S1 x1 x 1 ( 0 ) x 2 ( 0 ) = The theorem does not apply (not in Tikhonov form). x 1 and x 2 are both "fast". introducing y = S 1 x 1 + S 2 x 2 , the system is under Tikhonov form ( y is slow . . . ) indeed, y is the total volume of liquid, so it is not affected by the exchange phenomenon ˙ y = u 0 = F (( y − S 2 x 2 ) / S 1 , x 2 ) / S 2 Since F ( x 1 , x 2 ) = 0 ⇐ ⇒ x 1 = x 2 , and replacing y by its value Lemaire (Lille I) Differential Algebra and QSSA DART 2010 16 / 49

Tikhonov for the vessels u Input slower than balancing u / S 1 − 1 ˙ x 1 = ε F ( x 1 , x 2 ) / S 1 x2 V2,S2 1 ˙ x 2 = ε F ( x 1 , x 2 ) / S 2 V1,S1 x1 x 1 ( 0 ) x 2 ( 0 ) = The theorem does not apply (not in Tikhonov form). x 1 and x 2 are both "fast". introducing y = S 1 x 1 + S 2 x 2 , the system is under Tikhonov form ( y is slow . . . ) indeed, y is the total volume of liquid, so it is not affected by the exchange phenomenon S 1 ˙ x 1 + S 2 ˙ x 2 = u x 1 = x 2 and finally. . . Lemaire (Lille I) Differential Algebra and QSSA DART 2010 16 / 49