Singular perturbation theory

Singular perturbation theory Marc R. Rouel June 21, 2004 1 Introduction When we apply the teady-tate approximation (SSA) in chemical kinetic, we typically argue that ome of the intermediate are highly reactive, o that they are removed a fat a they are made. We then et the correponding rate of change to zero. What we are aying i not that thee rate are identically zero, of coure, but that they are much maller than the other rate of reaction. The teady-tate approximation i often urpriingly accurate, but the claical argument lack rigor and don t lead to any atifying mathematical generalization which can be ued to improve on the approximation. There are two mathematical theorie which build on the SSA and provide the required generalization: 1. Slow manifold theory: The curve or urface which arie from the SSA can in fact be undertood a approximation to low invariant manifol of the differential equation. We have touched on the relevant idea in the previou lecture. 2. Singular pertubation theory: Perturbation metho generally try to write olution of equation a power erie in a mall parameter. To undertand the SSA, we will find that we are led to problem in which the mall parameter multiplie a derivative. Thee problem have an intereting tructure, which i both practically and theoretically important, and to which we turn our attention in thi lecture. The two approache are not unrelated, a we hall ee in ection 6. 2 Scaling and balancing One of the problem with the claical SSA argument i that they lead u to a quandary regarding the etablihment of quantitative criteria for their validity. Conider the Michaeli-Menten mechanim: k 1 k 2 E S C E P MM k 1 In thi mechanim, E i the enzyme, S the ubtrate, P the product, and C the intermediate (enzymeubtrate/product) complex. 1

The claical argument run omething like thi: 1 Enzyme are highly effective catalyt. We therefore expect the degradation of C to enzyme and product to be a fat proce. Fat compared to what? Compared to formation of the enzyme-ubtrate complex, run the uual anwer. Thi argument i fine, a far a it goe, but what do we mean when we ay that one chemical proce i fater than another? What number are we comparing? It can t be rate, becaue in the teady tate, the rate of formation and of degradation are approximately the ame. It can t be rate contant, becaue k 1 ha different unit from k 1 and k 2. There i no imple way out of thi difficulty until we realize that we re going to get in trouble a long a we are trying to work with value which have unit. If we could get rid of the unit, we might be able to find thing to compare uch that we can make rigorou tatement about what it mean for one proce to be fat compared to another. Let tart by writing down the rate equation for thi mechanim: de k 1 ES k 1 C k 2 C (1a) ds k 1 ES k 1 C (1b) dc k 1ES k 1 C k 2 C (1c) dp k 2C (1d) Ma conervation give u the two equation E 0 E C S 0 S C P Note that thee conervation relation are conequence of the differential equation ince de dc ds dc dp 0. Uing thee algebraic equation therefore doen t change anything about the olution. It only reduce the et of differential equation 1 to the exactly equivalent planar ytem ds k 1 S E 0 C k1 C (2a) dc k 1S E 0 C k1 k 2 C (2b) The inight behind the SSA i that the concentration of highly reactive intermediate like C eventually reach a rough balance between production and detruction, uch that the net rate of change of C i mall. We want to make that balance evident. To do thi, we firt have to chooe new meaurement cale for each of the variable (S, C and t) uch that each i of unit magnitude. In other wor, we want to define new variable, c and τ by (2c) S S (3a) c C C (3b) and τ t t (3c) 1 The claical argument can be put a little better than thi, but the underlying problem remain. 2

where the tilde indicate a meaurement cale whoe unit are the ame a thoe of the correponding variable and uch that the new variable are, inofar a thi i poible, value between 0 and 1. It eay to chooe S: In mot experiment, we will tart with S 0 S 0, o S S 0 (4) hould be a good choice. Chooing C i trickier, but till not all that difficult. If we tart at the initial point SC S 0 0, which again i what we typically do in experiment, then C will initially rie to ome maximum, and then fall toward zero (the equilibrium concentration). From calculu, we know that the function C t will reach a maximum when dc 0, i.e. when C k 1 E 0 S k 1 S k 1 k 2 (5) Note that thi i the claical SSA. If we knew the value of S when C reache it maximum, we could plug it in and ue that a C. However, we have no eay way of etimating thi value of S. That being aid, we don t need to know the maximum value of C exactly. We really only need an order-of-magnitude etimate to be ued a a cale factor. We expect the maximum in C to be reached early in the reaction, before much S ha been conumed. We therefore ubtitute S S 0 into equation 5, and get k 1 E 0 S 0 E 0 S 0 C (6) k 1 S 0 k 1 k 2 S 0 K S where K S k 1 k 2 k 1 i the Michaeli contant, well known from claical enzyme kinetic. It not at all clear what we hould ue for t, o for now we ll jut ubtitute equation 3 into the rate equation 2, uing the caling factor 4 and 6, and ee what we get. S S 0 C c C c E 0S 0 S 0 K S t τ t ds d S 0 d τ t S 0 t dτ k1 S 0 E 0 E 0 S 0 c S 0 K S k E 0 S 0 1c S 0 K S k 2 dτ t k 1 E 0 S 0 1 c S 0 K S k 1c E 0 S 0 K S Similarly, for C: Let dτ tk 1 S 0 K S 1 c α S 0 S 0 K S S 0 S 0 K S c K S S 0 K S 3

Thi i a parameter between 0 and 1. Our two rate equation become dτ t k 1 E 0 1 E 0 αc k 1 c S 0 K S dτ tk 1 S 0 K S 1 αc c 1 α The term in the brace of the econd of thee equation are now clearly balanced: Each i made up of a product of quantitie which are of unit magnitude. There only remain to chooe t. For many enzyme, the back reaction C E S in t particularly ignificant. Under thee condition, dτ hould be dominated by the firt term, k 1 E 0 1 αc t. Thi ugget that we hould take t 1 k 1 E 0 in order to bring out the balance between the derivative and thi term. To put it another way, we expect k 1 E 0 1 to repreent a low time cale controlling the rate at which the ubtrate i ued up. Obviouly, if the above hypothei i fale for ome particular enzyme, we hould make a different choice for t. In any event, our choice lea to Define dτ 1 k 1 αc c 1 k 1 k α 2 S 0 K S 1 dτ E αc c 1 α 0 β and µ k 1 k 1 k 2 E 0 S 0 K S Then we have and dτ 1 αc βc 1 α (7a) µ dτ 1 αc c 1 α (7b) In claical enzymology, it i often aid that the SSA i valid when the enzyme concentration i mall. Thi correpon to a mall value of µ. The above caling provide a jutification for thi tatement, a well a a precie condition for it validity: If µ i very mall, then the differential equation 7b i cloe to the algebraic equation i.e. the SSA. A mall value of µ implie that 0 1 αc c 1 α (8) E 0 S 0 K S 4

i.e. the SSA will be valid under thi condition. Note that thi i a ufficient, but not a neceary condition. If we caled time differently, we might find other condition which lead to the validity of the SSA. Note that there omething funny about what we re doing here. We tarted out with two differential equation, and ended up aying that one of the differential equation degenerate to an algebraic equation. A ytem of two differential equation i quite different from a ingle differential equation with an algebraic contraint: The firt require the value of both and c to pecify the initial condition, while the latter require only a value for, the value of c being computed from equation 8. We have changed the nature of the problem by conidering the limit µ 0, a limit which can never be realized in practice. Problem like thi are called ingular perturbation problem. Becaue the problem we tarted with and the problem we obtain in the limit in which our mall parameter reache zero are different in nature, ingular perturbation problem can be very tricky. However, they how up all over the cience, o it well worth thinking about them a bit more. Depite the oddity of ingular perturbation problem, the ue of the limit µ 0 to approximate the low manifold (which i what we get from equation 8 if we olve for c) i upported by a theorem: Conider a ytem of differential equation of the form In the ingular limit µ dx µ dz 0, we obtain the ytem f xzt (9a) g xzt (9b) dx f xzt z φ xt where the econd of thee equation i the olution of g xzt ytem. Finally, we define the adjoined ytem, which i jut 0. Thi i called the degenerate dz g xzt where now we treat x and t a contant. 2 Theorem 1 (Tikhonov theorem) When µ 0, the olution of the ytem 9 ten to the olution of the degenerate ytem if z φ xt i a table root of the adjoined ytem. (The condition regarding the tability of the adjoined ytem i rarely checked.) The uphot of Tikhonov theorem i that for ufficiently mall µ, there i alway a low manifold whoe formula i approximately given by the SSA. Further detail can be obtained from Klonowki excellent review paper [1]. 2 If you re wondering where the µ went in the adjoined ytem, the time variable can be recaled in uch a way a to make thi coefficient diappear. Since tability doen t depend on how we cale time, there no point carrying thi parameter around. 5

Now imagine that we tart an experiment at the point c 10 for a ytem with a mall value of µ. Initially, becaue c 0, we will have ċ dτ 1 µ. Thi i a large (becaue µ i mall), poitive quantity. The intermediate complex concentration will therefore increae rapidly at firt, with a characteritic rie time of µ. However, a c increae, ċ will become maller. Becaue µ i mall, the right-hand ide of equation 7b hould alo remain mall, i.e. equation 8 will become valid. In other wor, we can eparate the time coure of the reaction into two part: 1. The rapid rie in the intermediate concentration: In chemical kinetic, we call thi the induction period, or ometime the tranient. In ingular perturbation theory, thi i called the inner olution. 3 2. A lower decay toward equilibrium during which the SSA i at leat approximately valid. Thi i the outer olution. 3 3 The outer olution In thi ection, we obtain the outer olution to lowet order in µ. Thi ue elementary technique which you hould have een in your undergraduate coure. Higher order approximation can be obtained. See Heineken, Tuchiya and Ari [2] for detail. Solving equation 8 for c, we get c α 1 α (10) Subtituting thi equation into the differential equation 7a, we get dτ 1 α α 1 α β 1 α α 1 α 1 β α 1 α Thi equation i eaily olved by eparation of variable: τ 1 α α 1 α 0 1 α 1 β dτ τ 0 1 α 1 β τ τ0 α 1 α 0 α 0 1 α ln 0 The contant 0 and t 0 repreent the tate of the ytem at ome time after it ha reached the manifold. Thee are therefore arbitrary quantitie not related (at leat not in any imple way) to 3 The term inner olution and outer olution are uch for hitorical reaon: Singular perturbation problem were firt conidered in fluid dynamic, where the inner olution referred to the behavior near a wall, while the outer olution referred to the behavior away from the wall. 6

the true initial condition of the ytem. We can therefore combine thee contant into a ingle, arbitrary contant A: 1 α 1 β τ A α 1 α ln (11) We will determine the contant A later. 4 The inner olution The inner olution i a bit trickier to obtain. The problem i that the equation 7 are caled wrong. Given our caling of time, the rie time for the intermediate complex i µ, which i a mall quantity. In order to tudy the inner olution, we have to recale the equation uch that the rie time i not mall. The eaiet way to do thi i a follow: τ µθ The new time variable θ i tretched by a factor of 1 µ relative to τ. Our rate equation become Setting µ and dθ µ 1 αc βc 1 α (12a) dθ 1 αc c 1 α (12b) 0, i.e. to the ame level of approximation a our outer olution, we get dθ 0 with equation 12b remaining a i. It follow that we can, to the lowet level of approximation, aume that i approximately contant during the induction period. To ditinguih thi olution from the outer olution, we define in to be the inner olution. Since S S 0, and S 0 i the initial concentration of S, in 1 to lowet order in µ. With in 1, equation 12b reduce to dθ 1 c Thi i a imple linear differential equation which can be integrated from the initial condition c 0: c 0 1 c θ θ 0 dθ ln 1 c or c 1 e θ (13) 5 Matching the inner and outer olution Since they jut repreent two piece of the ame trajectory, the inner and outer olution have to match up. Thi will provide u with condition from which we will determine the parameter A. 7

The outer olution i uppoed to be valid for value of τ which aren t too mall, while the inner olution i valid when τ in t too large. To match the two olution, we conider mall value of τ and large value of θ. At mall value of τ, the outer olution hould repreent early moment after the decay of the tranient, while large value of θ hould correpond to the very late tage of the induction period. If we have done everything right, we hould be able to match up the two olution in thi intermediate regime. If we let τ 0 and in in the outer olution 11, we get A α in 1 α ln in α Thu, atifie 1 α 1 β τ α α 1 α ln (14) in the outer olution region. Matching the value of c yiel no extra information in thi cae: If we put in 1 in equation 10, we get c 1. If we take the limit τ in equation 13, we get c 1. The c component of the olution therefore match up automatically. The overall procedure for doing ingular perturbation work i alway a hown here, at leat in outline: 1. Develop an appropriate caling for the equation. Identify the mall parameter(). 2. Find the outer olution. 3. Find the inner olution. 4. Match up the two olution in an intermediate time range. The only wrinkle are that we may want the olution developed to higher order than we have done here, and that the matching procedure i ometime difficult to carry out properly. 6 Geometric ingular perturbation theory and the outer olution Our work in the previou few ection focued on the temporal evolution of the ytem. Equation 14 for intance relate the time τ to in the outer olution. However, we have hinted that thee conideration are connected to the low manifold, a tructure which i bet undertood in phae pace. We now make thi connection explicit by working out a ingular perturbation expanion of the low manifold. The theoretical framework for doing thi wa firt provided by Fenichel [3] and i known a geometric ingular perturbation theory. Note that equation 10 ugget that the manifold can be written in the form c c. The manifold equation for our ytem would therefore be dτ dτ 8

We have dτ and dτ from equation 7. We know that, for mall µ, the manifold reduce to the form 10. One imple way to ue thi knowledge i to write the manifold a a power erie in µ: c γ i µ i (15) We hould find (and will confirm) that γ 0 i given by equation 10. In order to find the unknown function γ i (written imply a γ i from here on), we ubtitute the power erie 15 into the manifold equation. I will carry out thi procedure in tep: dτ dγ i µi 1 1 µ α i µ γ i 1 α γ i µ i µ α i µ γ i 1 1 α γ i µ i 1 µ α 1 α γ i µ i 1 dτ 1 α i µ γ i β 1 α γ i µ i α β 1 α µ α 1 α i µ γ i 1 dγ i µi γ i µ i α β 1 α γ i µ i (16) We now want to collect term in power of µ. We could jut read thee term directly from the equation. For intance, the leading term are the µ 1 term: Since thi equation mut be true for any µ 1 µ α 1 α γ 0 0 γ 0 0, the term in brace mut be zero, i.e. α 1 α (17) which i exactly what we aid thi term hould be. It not too difficult to do thi provided we re only intereted in mall power. However, if we want to develop a general equation which would allow u to obtain the erie expanion of the manifold to arbitrary order, it i convenient to rewrite equation 16 o that each ide appear a a power erie in µ, rather than a a complicated expreion involving, among other thing, a product of erie. Furthermore, it i convenient to write thee erie o that they are um involving µ i rather than, a on the left-hand ide, µ i 1 or other uch expreion. There are rule for rewriting um which you may have learned in ome of your 9

mathematic coure. If not, the following example will hopefully let you ee the logic behind thee rule, which in t too hard once you ve done a few example. The firt um on the left-hand ide of equation 16 i fairly eay to rewrite in the deired form: γ i µ i 1 i 1 γ i 1µ i If you can t ee thi immediately, try writing out the firt few term of each ide of thi identity. The trickiet bit i the product of um on the right-hand ide of equation 16. Here i the tranformation: j 0 dγ j µj k µ k 0γ k µ i i dγ j j 0 γ i j The renaming of the ummation indicie on the left-hand ide i a convenience introduced for clarity. The key obervation i that multiplying the two um produce term in µ i when j k i. We re now ready to go back to equation 16: µ α 1 α γ i 1µ i i 1 dγ i µi α β 1 α µ i i dγ j j 0 γ i j Reading off term in each power of µ i now eay: µ 1 : γ 0 α 1 α 0 µ i i 0 : γ i 1 α 1 α dγ i α β 1 α i dγ j j 0 γ i j We have een the firt of thee equation, and even written down it olution (equation 17). The econd equation can be rewritten in the form γ i 1 α 1 1 α dγ i α β 1 α i dγ j j 0 γ i j Note that we only need to know the formula of the γ up to i to determine the i erie. For intance, γ 1 i calculated by 1 t term in the 1 γ 1 α 1 α dγ 0 1 α 2 1 β α 1 α 4 α β 1 α dγ 0 γ 0 A you can imagine, thee calculation get tediou pretty quickly. In the old day, we ued to fill up notebook with perturbation erie, carefully implified by hand. It wa very error-prone, and we would often pend more time verifying our anwer than we did doing the initial calculation. Nowaday, we jut ue Maple. It much fater, but it till a good idea to calculate a few term by hand to check that Maple i doing the calculation correctly. 10

Reference [1] W. Klonowki, Biophy. Chem. 18, 73 (1983). [2] F. Heineken, H. Tuchiya, and R. Ari, Math. Bioci. 1, 95 (1967). [3] N. Fenichel, J. Diff. Eq. 31, 53 (1979). 11