By P. F. Hsieh, A. W. J. Stoddart
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In addition, there is assumed to be a steady supply rate of S1 , while product S2 is irreversibly removed. 56) k−1 v2 S2 −→. 60) k1 s1 x1 − (k−1 + k2 )x2 . 61) The ﬁfth differential equation is not necessary, because the total available enzyme is k 1 s1 conserved, e + x1 + x2 e0 . 65) e0 k1 k2 2 +k−1 ) −1 where , ν kv2 1e0 , η v2 (k , α k2 +k ( kk−33 )1/γ . 68) αf (σ1 , σ2 ) − ησ2 . 69) The goal of the following analysis is to demonstrate that this system of equations has oscillatory solutions for some range of the supply rate ν.
Indeed, much of our study here is involved in determining appropriate models for production and ﬂux. 1 Fick’s Law The simplest description of the ﬂux of a chemical species is J −D∇u. 4) is called a constitutive relationship, and for chemical species it is called Fick’s law. The scalar D is the diffusion coefﬁcient and is characteristic of the solute and the ﬂuid in which it is dissolved. 4) is called Newton’s law of cooling. Fick’s law is not really a law, but is a reasonable approximation to reality if the concentration of the chemical species is not too high.
104) 30 1: If [E] e0 at time t solution is 0, then x0 Biochemical Reactions 0. 105) and it is valid only for times of order . The exponential term is signiﬁcant only when τ is small of order . Thus, this simple analysis shows that if the reaction is started from arbitrary initial conditions, there is ﬁrst a time span during which the enzyme products rapidly equilibrate, consuming little substrate, and after this initial “layer” the reaction proceeds according to Michaelis–Menten kinetics along the quasi-steady-state curve.
Analytic Theory of Differential Equations by P. F. Hsieh, A. W. J. Stoddart