Asymptotic Analysis and Singular Perturbation Theory

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with the substrate is balanced by the destruction of the complex by the reverse reaction and the decomposition of the complex into the product and the enzyme. Substituting this result into the first equation, we get a first order ODE for u0(τ): du0 dτ The solution of this equation is given by = − λu0 u0 + k . u0(τ) + k log u0(τ) = a − λτ, where a is a constant of integration. This solution is invalid near τ = 0 because no choice of a can satisfy the initial conditions for both u0 and v0. 4.1.2 Inner solution There is a short initial layer, for times t = O(ε), in which u, v adjust from their initial values to values that are compatible with the outer solution found above. We introduce inner variables The inner equations are We look for an innner expansion T = τ , U(T, ε) = u(τ, ε), V (T, ε) = v(τ, ε). ε The leading order inner equations are The solution is dU = ε {−U + (U + k − λ)V } , dT dV = U − (U + k)V, dT U(0, ε) = 1, V (0, ε) = 0. U(T, ε) = U0(T ) + εU1(T ) + O(ε 2 ), V (T, ε) = V0(T ) + εV1(T ) + O(ε 2 ). dU0 = 0, dT dV0 dT = U0 − (U0 + k)V0, U0(0) = 1, V0(0) = 0. U0 = 1, V0 = 1 −(1+k)T 1 − e . 1 + k 52
4.1.3 Matching We assume that the inner and outer expansions are both valid for intermediate times of the order ε ≪ τ ≪ 1. We require that the expansions agree asymptotically in this regime, where T → ∞ and τ → 0 as ε → 0. Hence, the matching condition is This condition implies that lim T →∞ U0(T ) = lim τ→0 + u0(τ), lim v0(τ). + T →∞ V0(T ) = lim τ→0 u0(0) = 1, v0(0) = 1 1 + k , which is satisfied when a = 1 in the outer solution. Hence u0(τ) + k log u0(τ) = 1 − λτ. The slow manifold for the enzyme system is the curve v = u u + k . Trajectories rapidly approach the slow manifold in the initial layer. They then move more slowly along the slow manifold and approach the equilibrium u = v = 0 as τ → ∞. The inner layer corresponds to the small amount of enzyme ‘loading up” on the substrate. The slow manifold corresponds to the enzyme working at full capacity in converting substrate into product. A principle quantity of biological interest is the rate of uptake, It follows from the outer solution that r0 = du0 dτ τ=0 r0 = λ 1 + k . The dimensional form of the rate of uptake is R0 = ds dt = Qs0 . s0 + km where Q = k2e0 is the maximum reaction rate, and km = k0 + k2 k1 is the Michaelis constant. The maximum rate depends only on k2; the rate limiting step is C → P + E. 53
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Asymptotic Analysis and Singular Pe
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4.1.1 Outer solution . . . . . . .
Page 6 and 7: Once we have constructed such an as
Page 8 and 9: For example, setting ε = 0 in (1.2
Page 10 and 11: Solving the first equation, we get
Page 12 and 13: and A ε : R n → R n is a linear
Page 14 and 15: Energy eigenstates are wavefunction
Page 16 and 17: and the constant α, with dimension
Page 18 and 19: The velocity gradient ∇u has dime
Page 20 and 21: Imposing the requirement that R d
Page 23 and 24: Chapter 2 Asymptotic Expansions In
Page 25 and 26: If, as is usually the case, the gau
Page 27 and 28: where denotes the C n -norm. We def
Page 29 and 30: It follows that where Since we have
Page 31 and 32: 2.2.4 Nonuniform asymptotic expansi
Page 33 and 34: Chapter 3 Asymptotic Expansion of I
Page 35 and 36: we find that the optimal truncation
Page 37 and 38: and making the change of variable (
Page 39 and 40: 3.3 The method of stationary phase
Page 41 and 42: Applying this argument n times, we
Page 43 and 44: The standard normalization for the
Page 45 and 46: We assume for simplicity that the i
Page 47 and 48: where w = − t2/3v . 31/3 It follo
Page 49 and 50: 3.5.1 Multiple integrals Propositio
Page 51: When λ > 0, we can deform the inte
Page 54 and 55: plex C, and the complex breaks down
Page 58 and 59: For more information about enzyme k
Page 60 and 61: Thus, the solution involves two ter
Page 62 and 63: where c is an arbitrary constant of
Page 64 and 65: (a) if a(x) > 0, then the boundary
Page 66 and 67: The inner solution near x = −1 is
Page 68 and 69: If the fluid interface is a graph z
Page 70 and 71: where the prime denotes a derivativ
Page 72 and 73: We seek an asymptotic expansion of
Page 74 and 75: It follows that dψ dX = −U0 sec
Page 77 and 78: Chapter 5 Method of Multiple Scales
Page 79 and 80: Using this expansion in the equatio
Page 81 and 82: where the overline denotes the aver
Page 83 and 84: 5.3 The method of averaging Conside
Page 85 and 86: where f : R n ×T → R n is a Lips
Page 87 and 88: 5.5 The WKB method for ODEs Suppose
Page 89: Thus, the amplitude of the oscillat
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