Double perturbation series in the differential equations of enzyme kinetics Available to Purchase

See, e.g., S. W. Benson, The Foundations of Chemical Kinetics (McGraw-Hill, New York, 1960), p. 53, for references.

G.

Li

,

A. S.

Tomlin

, and

H.

Rabitz

, , ().

A. S.

Tomlin

,

M. J.

Pilling

,

T.

Turányi

,

J. H.

Merkin

, and

J.

Brindley

, , ();

U.

Maas

and

S. B.

Pope

, , ().

T.

Turányi

,

A. S.

Tomlin

, and

M. J.

Pilling

, , ().

See, e.g., C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice-Hall, Englewood Cliffs, NJ, 1971);

J. D. Lambert, Computational Methods in Ordinary Differential Equations (Wiley, New York, 1973).

F. G.

Heineken

,

H. M.

Tsuchiya

, and

R.

Aris

, , ();

S. I.

Rubinow

and

J. L.

Lebowitz

, , ();

C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in the Natural Sciences (Macmillan, New York, 1974), Chaps. 9, 10;

J. D. Murray, Lectures on Nonlinear-Differential-Equation Models in Biology (Clarendon, Oxford, 1977), Chap. 1.

M. R.

Roussel

and

S. J.

Fraser

, , ().

N.

Chafee

, , ().

J. Hale, Ordinary Differential Equations (Wiley, New York, 1969), especially Chap. 7.

A. N.

Yannacopoulios

,

A. S.

Tomlin

,

J.

Brindley

,

J. H.

Merkin

, and

M. J.

Pilling

, , (). This paper describes the construction of inertial manifold by the method of algebraic sets where fast and slow variables are expressed as series in the time variable convergent within a strip containing the real time axis.

J. Carr, Applications of Centre Manifold Theory, Appl. Math. Sciences Vol. 35 (Springer, New York, 1981), Chap. 7.

C.

Foias

,

G. R.

Sell

, and

R.

Temam

, , (), consider nonlinear evolution equations of the form $du/dt+Au+R(u)=0,$ where $u=u(x)$ are, for example, chemical concentrations, A is an unbounded linear operator, typically $−const. ∇2$ for reaction-diffusion equations, and R(u) is the nonlinear reaction term. The dynamics is projected onto a finite-dimensional subspace and its complement in the Hilbert space of the (unbounded) linear operator A that “dominates” the nonlinearity in the evolution equations. (Cf. Refs. 8 and 9.) The successful construction of the inertial manifold for the chosen subspace depends on the existence of a gap in the spectrum of A, since certain inequalities must be satisfied for the analysis to imply the existence of an inertial manifold. Only if the support of u is compact can the spectrum of A have the appropriate gap property. Since A does not appear in our simpler equations [of the form $u̇+R(u)=0$], the gap condition cannot be applied. The method of the construction of a contraction mapping for the inertial manifold in Refs. 8 and 9 depends on Picard’s method and Gronwall’s inequality and is explicitly parameterized by time. The methods discussed in this paper do not use this parameterization.

M. S.

Jolly

, , ().

See, e.g., O. E. Rössler, in Bifurcation Theory and Applications in Scientific Disciplines, edited by O. Gurel and O. E. Rössler (Ann. N.Y. Acad. Sci., NY, 1979), p. 376.

A.

Puhl

and

G.

Nicolis

, , ().

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical, and Bifurcations of Vector Fields, Appl. Math. Sci. Vol. 42 (Springer, New York, 1983).

J. E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Appl. Math. Sciences Vol. 19 (Springer, New York, 1976).

O. H.

Straus

and

A.

Goldstein

, , ();

A.

Goldstein

, , ().

P. A.

Srere

, , ();

A.

Sols

and

R.

Marco

, , ();

M.

Laurent

and

N.

Kellershohn

, , ();

G. F.

Betts

and

D. K.

Srivastava

, , ().

A description of differential topology is given in: D. R. J. Chillingworth, Differential Topology with a View to Applications (Pitman, San Francisco, 1976).

See, e.g., R. O’Malley, Singular Perturbation Methods for Ordinary Differential Equations (Springer, New York, 1991);

E. J. Hinch, Perturbation Methods (Cambridge University Press, New York, 1991).

H. Haken, Synergetics: An Introduction (Springer, New York, 1978);

Advanced Synergetics (Springer, New York, 1983). Haken calls the adiabatic approximation, i.e., the adiabatic elimination method, the “slaving principle.” This is essentially the steady-state method.

A.

Fernández

and

O.

Sinanoǧlu

, , ();

A.

Fernández

and

O.

Sinanoǧlu

, , ();

A.

Fernández

, , ().

J. B.

McLaughlin

and

P. C.

Martin

, , ().

R.

Graham

and

H. J.

Scholz

, , ().

G. L.

Oppo

and

A.

Politi

, , ().

P. E.

Phillipson

, , ().

V.

Henri

, , ();

Lois Générales de l’Action des Diastases (Hermann, Paris, 1903);

L.

Michaelis

and

M. L.

Menten

, , ().

A. H.

Nguyen

and

S. J.

Fraser

, , ().

D. D.

van Slyke

and

G. E.

Cullen

, , ().

The standard singular perturbation scheme uses the initial concentration of substrate, $s(0)=s0,$ to scale concentrations. This scheme has three dimensionless parameters, whereas the method used here has only two. The scaling transformation used by Heineken et al. in Ref. 6 depends on the initial substrate concentration $s0;$ this leads to ODEs with three rather than the minimal two parameters required for the outer solution. This outer solution requires that only the initial concentration of substrate be defined since the enzyme-substrate complex concentration is then determined by entrainment. However, the inner solution requires the initial condition for enzyme-substrate complex also.

B. W. Char, K. O. Geddes, G. H. Gonnet, M. B. Monagan, and S. M. Watt, MAPLE: REFERENCE MANUAL (WATCOM, Waterloo, 1988).

The notation $∂α(•)≡∂(•)∂α,$ the partial derivative with respect to α; the vector operator $∇ξ≡(∂ξ1,∂ξ2,…,∂ξd)$ in $Rd.$

Note, however, that there are important classes of reactions, including, e.g., free radical, chain reactions, or even simple, exactly solvable two-step organic reaction models, where the SSA is not applicable or, at least, cannot be applied without considerable caution. S. J. Fraser (in preparation) solves a kinetic model for xylene-alcohol reaction suggested by R. A. McClelland.

The labeling $γXY(σ)$ means the $X$ approximation in the $Y$ representation, where $X=S$ or $E$ is short for $X=SSA$ or EA. Similarly, the $Y$ representation means $Y$ can take values $S$ or $E.$

For the SSA and EA representations, the SSA, which corresponds to the γ nullcline, leads to a closed equation of motion for σ; however, the EA corresponds to the σ nullcline and for this very reason does not give an equation of motion for σ. The EA in the van Slyke case, which has no reversible steps, is given by $limεD→0γED(σ)=1.$ Geometrically, this corresponds to a branch of a degenerate hyperbola so that the flow in the phase plane $(σ×γ)$ is vertical on this EA nullcline, $γ=1.$

For a discussion of contraction mappings, see e.g.: P. Roman, Some Modern Mathematics for Physicists and Other Outsiders, (Pergamon, New York, 1975), Vol. I, pp. 280–289;

P. Henrici, Elements of Numerical Analysis (Wiley, New York, 1964);

G. F. Simmons, Introduction to Topology and Modern Analysis (McGraw-Hill, London, 1963). The uniqueness of $M=graph{γ(σ)}$ as a solution of the functional equation for ℳ is ensured, under appropriate conditions of smoothness of starting functions for the iteration, by the contraction mapping theorem. A contraction mapping procedure for solving the functional equation for ℳ in various cases has been discussed previously in

S. J.

Fraser

, , (), and in Refs. 29 and 48.

G. E.

Briggs

and

J. B. S.

Haldane

, , ().

Compare the nonunique initialization of the $εS$ expansion in Sec. III A 2.

The operators Δ and $(•̇)$ commute.

K. J.

Laidler

, , ().

This separation procedure is related to boundary layer methods in singular perturbation theory. This point relates to the remarks made in Ref. 31 because, in general, two variables are required to describe the system. See Ref. 21.

A. C. Hindmarsh, ODEPACK: A Systematized Collection of ODE Solvers, in Scientific Computing, edited by R. S. Stepleman et al. (North-Holland, Amsterdam, 1983), pp. 55–64.

The functional coefficient of $εS0$ term has two factors. One factor gives an algebraic equation which partly initializes the recursion defining higher order terms. For this algebraically initiated recursion the functional coefficients of $εSj,$ $j⩾1,$ contain terms in the derivatives $∂σγ̃j$ whose coefficients vanish identically. Thus the hierarchy of equations defining the slow-manifold function is purely algebraic. The other factor of functional coefficient of $εS0$ gives the ODE: $1+ηS∂σγ̃0=0$ with general solution $γ̃0(σ;σ0)=−(σ−σ0)/ηS.$ This $O(εS0)$ coefficient is the leading term in the fast-mode solution for the trajectory equation (9). As $εS→0$ this general term is the exact solution of the trajectory equation. Evidently, fast-mode terms must contain an $ηS−1$ (first-order) pole; this appears immediately in the $εS$ Taylor expansion for $γ̃(σ)$ in Eq. (33) but not in the $ηS$ Taylor expansion in Eq. (22). However, this can be remedied by taking an $O(ηS−1)$ (leading-term) Laurent expansion for γ(σ) in Eq. (22). Thus it appears that appropriately written series capture the fast mode in differential part and the slow mode in algebraic part.

The term $−δ0jσ$ in the denominator in Eq. (38) comes from the term $−σ$ in Eq. (37). This term contributes only to the $O(ηS0)$ coefficient in the denominator in Eq. (38).

For $εS≪1$ the velocity function along the slow manifold is $σ̇=εσ/(1+σ)+O(εS2).$ This ODE produces an $O(1/εS)$ leading term on separation of variables. The corresponding divergence of the time for motion on the slow manifold ℳ corresponds to the fact that ℳ and the SSA nullcline, $γSSA(σ)=σ/(1+σ),$ coincide to form a continuum of equilibria at $εS=0.$

See also the center manifold treatment discussed in Carr’s book, Ref. 11. However, this treatment provides a representation of the slow manifold as a jet (Taylor series) rather than a uniformly convergent (valid for σ→+∞) functional series. Because the nearest pole is at $σ≈−1,$ or $s≈−KS,$ the Taylor series will have finite radius of convergence. The slow manifold’s pole at $σ⩽−1$ arises since it is confined between the hyperbolic EA and SSA nullclines so that explicit expansions in σ, or $s,$ will not converge globally. For the same reason the theory of normal forms see, e.g., A. H. Nayfeh, Method of Normal Forms (Wiley, New York, 1993), Ref. 16 is not applicable since the normal form is a jet. This jet defines a far smaller domain than the region of interest which extends to $s≫KS$ or $σ≫1.$ A related point is the convergence of explicit series in the time variable, τ. In Ref. 27 it is suggested that the difficulty in finding explicit, convergent forms for singular perturbation treatment of the system of ODEs (5) and (6) arises from the use of the SSA and not any intrinsic property of the solution. However, the existence of the pole complicates the issue. The hyperbolic shape is related to the crossover from zeroth to first-order reaction. Similar problems are likely to arise if σ is reexpanded as a series in τ from some point $σ(0)>1$ since such forms are not uniformly convergent (see Ref. 10). Therefore, for geometrical reasons, convergence difficulties are associated with Taylor series solutions in σ or τ. However, such convergence difficulties in configuration space do not appear for functional forms for $γMs(σ)$ discussed in this paper as these functions are implicitly infinite order sums. Functional series divergence may appear as the perturbation parameter becomes sufficiently large.

M. R.

Roussel

and

S. J.

Fraser

, , ().

The Schur product $z=x∧y$ in $Rd$ means (formally) ${zj=xjyj|j=1,2,…,d}.$

For another approach to this problem see Ref. 2; in this reference there is a discussion of a general method for treating the perturbation series using just the set of singular perturbation parameters. What we call η is written as ε in Ref. 2.

A. N.

Tikhonov

, , ();

A. B.

Vasil’eva

, , ().

In the simplest class of phase flows we assume that $M0$ and $Mη,$ which lies nearby, both have global graphs.

Note that $η∧(•)$ has an inverse, $η−1∧(•)$ which is used during these manipulations.

To prove existence theorems for inertial manifolds, the dynamical variables are partitioned into two sets, as in Refs. 8, 9, and 11: a finite set of underlying (independent) variable, and a complementary (possibly infinite) set of (dependent) variables. Provided the velocity function g obeys a uniform Lipschitz condition (see Refs. 11 or 20) in y (and satisfies some additional conditions), there is a class of functions from the finite set onto the complementary set; the corresponding inertial manifold is the graph of the function in this class which is the fixed point of a contraction mapping constructed from the integral equation for the complementary dynamical variables.