Singular perturbations, algorithmic algebra and scalings Sebastian - PowerPoint PPT Presentation

Singular perturbations, algorithmic algebra and scalings Sebastian Walcher, RWTH Aachen Symbiont Meeting, Bonn, July 2018

Based on work by and work with : Lena N¨ othen Alexandra Goeke Christian Lax Eva Zerz

Overview Objective : Dimension reduction for polynomial and rational ODEs. Motivated by chemical reaction networks, with slow and fast reactions, slow and fast variables (QSS). Mathematics : Singular perturbations. Contents : ◮ Method of choice (and restriction): Classical reduction theorems by Tikhonov and Fenichel; coordinate-free. (Reliable convergence results, verifiable conditions.) ◮ Tikhonov-Fenichel and QSS. ◮ Tikhonov-Fenichel and scaling approaches.

Singular perturbation reduction

Tikhonov and Fenichel (reminder) System with small parameter ε in standard form = ε f ( y 1 , y 2 ) + ε 2 ( . . . ) , y 1 ∈ D ⊆ R s , y 1 ˙ y 2 ∈ G ⊆ R r y 2 ˙ = g ( y 1 , y 2 ) + ε ( . . . ) , y ′ ε y ′ Slow time τ = ε t : 1 = f ( y 1 , y 2 )+ · · · , 2 = g ( y 1 , y 2 )+ ε · · · . Assumptions: (i) Critical manifold � � ( y 1 , y 2 ) T ∈ D × G ; g ( y 1 , y 2 ) = 0 � � = ∅ ; Z := (ii) there exists ν > 0 such that every eigenvalue of D 2 g ( y 1 , y 2 ), ( y 1 , y 2 ) ∈ � Z has real part ≤ − ν . Theorem . There exist T > 0 and a neighborhood of � Z in which all solutions converge uniformly to solutions of y ′ 1 = f ( y 1 , y 2 ) , g ( y 1 , y 2 ) = 0 on [ t 0 , T ] ; t 0 > 0 arbitrary .

Ordinary differential equations for CRN We consider: Parameter dependent ordinary differential equations x ∈ R n , π ∈ R m x = h ( x , π ) , ˙ with polynomial (or rational) right hand side. Motivation : Chemical reaction networks with mass action kinetics and constant thermodynamical parameters; spatially homogeneous. Parameters : Rate constants, initial concentrations. Question : Where (and how) do singular perturbations enter the picture?

Tikhonov-Fenichel, coordinate-free: First step We have x = h ( x , π ) ˙ versus ε f ( y 1 , y 2 ) + ε 2 ( . . . ) y 1 ˙ = y 2 ˙ = g ( y 1 , y 2 ) + ε ( . . . ) Fact and problems: No a priori separation in slow and fast variables. No small parameter ε . Preliminary step : For suitable (?!) � π consider system h (1) ( x ) + ε 2 · · · . π + ερ + · · · ) =: � h (0) ( x ) + ε � x = h ( x , � ˙ Suitability of � π : Scenario must be singular ; i.e. the vanishing set V ( h (0) ) contains a submanifold Z with dimension s > 0.

Tikhonov-Fenichel, coordinate-free: Identification Proposition . Assume dim Z = s > 0. Then h (1) ( x ) + ε 2 . . . x = � h (0) ( x ) + ε � ˙ admits a coordinate transformation into standard form and subsequent Tikhonov-Fenichel reduction near every point of Z if and only if (i) rank D � h (0) ( x ) = r := n − s for all x ∈ Z ; (ii) for each x ∈ Z there exists a direct sum decomposition R n = Ker D � h (0) ( x ) ⊕ Im D � h (0) ( x ); (iii) for each x ∈ Z the nonzero eigenvalues of D � h (0) ( x ) have real parts ≤ − ν < 0. Remaining problem : Direct application (via explicitly computable coordinate transformation) may be impossible.

Tikhonov-Fenichel, coordinate-free: Reduction x ′ = ε − 1 � h (0) ( x ) + � h (1) ( x ) + . . . with a ∈ Z ⊆ V ( � h (0) ), satisfying conditions (i), (ii) und (iii). Decomposition: There is a Zariski-open neighborhood U a of a such that h (0) ( x ) = P ( x ) µ ( x ) , � with µ ( x ) ∈ R ( x ) r × 1 , P ( x ) ∈ R ( x ) n × r , rank P ( a ) = r , rank D µ ( a ) = r , and (wlog) V ( � h (0) ) ∩ U a = V ( µ ) ∩ U a = Z . Reduction: The system � � x ′ = I n − P ( x ) A ( x ) − 1 D µ ( x ) h (1) ( x ) , � with A ( x ) := D µ ( x ) P ( x ) is defined on U a and admits Z as invariant set. The restriction to Z yields the reduction from Tikhonov’s theorem as ε → 0.

Tikhonov-Fenichel, coordinate-free: Parameters Definition : We call � π a Tikhonov-Fenichel parameter value (TFPV) for dimension s (1 ≤ s ≤ n − 1) of ˙ x = h ( x , π ) if the following hold: (i) The vanishing set V ( h ( · , � π )) of x �→ h ( x , � π ) contains a component � Y of dimension s . (ii) There is x 0 ∈ � Y and neighborhood Z of x 0 in � Y such that rank D 1 h ( x , � π ) = n − s and R n = Ker D 1 h ( x , � π ) ⊕ Im D 1 h ( x , � π ) , for all x ∈ Z . (iii) The nonzero eigenvalues of D 1 h ( x 0 , � π ) have real parts < 0. Note : Conditions simply were copied from above. Therefore reduction works for small perturbations � π + ερ + · · · .

Determining TFPV (algorithmic starting point) Proposition. If � π is a TFPV for dimension s (1 ≤ s ≤ n − 1) of x = h ( x , π ) then there exists x 0 ∈ R n such that: ˙ (i) h ( x 0 , � π ) = 0; (ii) for each k > r := n − s all k × k minors of the Jacobian D 1 h ( x 0 , � π ) vanish. Observation : More equations for x than variables. Enter algorithmic algebra (elimination theory). More theory: Theorem . The TFPV of a polynomial (or rational) system x = h ( x , π ) with nonnegative parameters form a semi-algebraic ˙ subset of R m .

Standard example: Michaelis-Menten (irreversible) Reaction network k 1 k − 1 C k 2 E + S ⇀ E + P ⇋ Differential equation system for concentrations: s ˙ = − k 1 es + k − 1 c , c ˙ = k 1 es − ( k − 1 + k 2 ) c , e ˙ = − k 1 es + ( k − 1 + k 2 ) c ; typical initial values s (0) = s 0 , c (0) = 0, e (0) = e 0 . With stoichiometry (linear first integral e + c ): s ˙ = − k 1 e 0 s + ( k 1 s + k − 1 ) c , c ˙ = k 1 e 0 s − ( k 1 s + k − 1 + k 2 ) c .

Michaelis-Menten: Determine TFPV System s ˙ = − k 1 e 0 s + ( k 1 s + k − 1 ) c , c ˙ = k 1 e 0 s − ( k 1 s + k − 1 + k 2 ) c with Jacobian determinant d = k 1 k 2 ( e 0 − c ). Eliminate s and c from the three equations (manageable by hand). e 0 , � k 1 , � k − 1 , � Result : A TFPV ( � k 2 ) satisfies e 0 � k 2 � k 1 = 0 . � Small perturbations yield (all) relevant cases:         ε e ∗ e 0 e 0 � � e 0 � 0         � � � k 1 ε k ∗ k 1 k 1         1  or  or  or         � � �  k − 1  k − 1   ε k ∗  k − 1 − 1 � � ε k ∗ k 2 ε k ∗ k 2 2 2

Michaelis-Menten; some reductions ◮ Michaelis-Menten with little enzyme e 0 = ε e ∗ 0 : Familiar result. ◮ Michaelis-Menten with slow product formation: s ˙ = − k 1 e 0 s + ( k 1 s + k − 1 ) c − − ε k ∗ c ˙ = k 1 e 0 s ( k 1 s + k − 1 ) c 2 c . h (0) = P · µ with Decomposition � � � 1 P = , µ = k 1 e 0 s − ( k 1 s + k − 1 ) c . − 1 Reduced equation (on Z = V ( µ )): � � � � � � 1 s ′ ∗ k 1 s + k − 1 0 · = . c ′ ∗ k 1 ( e 0 − c ) − k ∗ 2 c k 1 ( e 0 − c ) + k 1 s + k − 1

Further example: TFPV for competitive inhibition Michaelis-Menten network with inhibitor (binding reversibly to enzyme) leads to three dimensional system = k − 1 c 1 − k 1 s ( e 0 − c 1 − c 2 ) , s ˙ c 1 ˙ = k 1 s ( e 0 − c 1 − c 2 ) − ( k − 1 + k 2 ) c 1 , c 2 ˙ = k 3 ( e 0 − c 1 − c 2 )( i 0 − c 2 ) − k − 3 c 2 with Jacobian determinant d ( x , π ) = − k 1 k 2 ( e 0 − c 1 + c 2 )( k − 3 + k 3 ( i 0 + e 0 ) − k 3 (2 c 2 − c 1 )) . Elimination ideal has radical 3 ( e 0 − i 0 ) 2 + k − 3 ( k − 3 + 2 k 3 ( e 0 + i 0 )) � I = � e 0 k 1 k 2 k − 3 ( k 2 with single generator.

Tihkonov-Fenichel approach: Review ◮ The approach is (in principle) universal. (Restriction to two time scales due to theoretical foundation.) ◮ Defining equations and inequalities algorithmically accessible (in weak sense: termination guaranteed but no criterion). ◮ Methods in current use not quite adequate (algebra rather than real algebra). ◮ Obvious practical problem: Feasibility. More restrictive searches for slow manifolds (e.g. by scaling) remain relevant, from scientific (chemistry) and from pragmatic perspective.

Quasi-steady reduction

“Classical” QSS reduction Quasi-steady state (QSS) reduction of a parameter dependent system ˙ x = h ( x , π ). ◮ Loose description : Designate some variables as being in “quasi-equilibrium” (zero rate of change) after short initial phase. This yields a system of algebraic equations which are used to eliminate these variables. (Heuristical procedure.) ◮ First explicit statement: Briggs and Haldane (1925) for Michaelis-Menten: QSS for c . ◮ Obvious : Consistency and validity of this procedure depend on conditions for parameters (Briggs and Haldane require small enzyme concentration e 0 ).

Tikhonov-Fenichel vs.“classical” QSS reduction ◮ Reductions agree for Michaelis-Menten with small enzyme concentration. (Heineken, Tsuchiya and Aris 1967: Possibly the first mathematically rigorous application of Tikhonov to a reaction network.) ◮ Concepts are frequently conflated, but should not be. ◮ QSS reduction may yield incorrect results. ◮ Singular perturbation reductions do not only target slow and fast species: For instance, slow and fast reactions are equally relevant.