In the first question in Section 3.2.4 we sought the value of three unitary rate constants. However, there were only two steady-state parameters so it was necessary to assume the value of one of either k1 or k-1 ; k1 was chosen because there is a rational choice that can be made for the value of such a second-order rate constant. Could we have chosen a much larger value than the 1 x 107mol-1 L s 1 that was used? The answer is "yes, sort of!" as there is an upper bound on the value; this is called the 'diffusion limit' for a second-order rate constant.
The upper limit of second-order rate constants can be determined experimentally by using rapid-reaction methods. At ~37°C the value has been found to be ~1 x 108 mol-1 L s-1 . This value applies to solutions such as those found in the cytoplasm of a cell, or in the plasma of blood. A simple theoretical analysis shows how this value comes about.
Consider two reacting molecular species, A and B, in solution. If the rate of the reaction is proportional to the rate at which a molecule of A collides with one of B, then the effective mean intermolecular (inter-collision) distance and the diffusion coefficient of each reactant will dictate the rate. Since the molecules are in motion we can, without loss of generality, conveniently focus attention on just one molecule of A and place its centre at the origin of a spherical polar coordinate system. The relative motion of the two molecules can then be characterized by the sum of their diffusion coefficients (DA + Db). If the inward flux of molecules of B through a sphere of radius r around the single molecule of A is denoted by J (flux; units, mol m-2 s-1), then the magnitude of the flux is given by Fick's first law of bulk diffusion, d\nB]
dr where the derivative term expresses the gradient in number-concentration (number of molecules per unit volume) of B. Thus, the number of molecules of B diffusing across the surface of a sphere of radius r (surface area = 4 p r2) per second is given by I = J 4 p r2. Equation [3.12] can be integrated with respect to the radius and [nB ] in the bulk medium,
where (\nB])r=M is defined as the concentration in the bulk solution, namely, \nB]bulk . Thus,
4 p RM 4 p Rab and at r = RM the flux is assumed to be zero. Eqn [3.14] is rearranged to give (dropping the "bulk" subscript) the expression that has units of (number of molecules)-1 m3 s-1,
I (number of molecules s-1)
\nB] (number of molecules m-3)
[nB] is now expressed as a molar concentration by dividing Eqn [3.15] by Avogadro's number, N, and multiplying by 10-3 to convert the units of m-3 to L-1. Hence, multiplying the Eqn [3.15] by N x 103 gives the expression for the second-order rate constant, k, with the appropriate units (mol-1 L s-1), k = k' N 103 = 4 p Rab (Da + Db) N 103 . [3.16]
Q: Does the analysis in the previous section (Eqn [3.16]) yield a realistic value for the upper limit of a second-order rate constant that characterizes the reaction between a metabolite and enzyme in a cellular environment?
A: The diffusion coefficient of bulk water at 37°C is ~2 x 10-9 m2 s-1. Suppose that the metabolite B has a diffusion coefficient that is an order of magnitude less than water because of the higher viscosity in the cell, and the enzyme's diffusion coefficient is smaller again by a factor of 10 because of its much larger molecular mass. Also, the radius of many metabolites is ~ 0.2 to 1 nm while that of a typical enzyme is ~3 nM. By using Eqn [3.16] the value of k is calculated to be k = 4 n * 4 x 10-9 m (2.2 x 10-10 m2 s-1) 6.022 1023 mol-1 103 L m-3
This value is about two orders of magnitude greater that we expected if the claim of 1 x 108 mol-1 L 5 1is correct. The explanation for the discrepancy lies in the very simplistic model used in our analysis. We did not take into account the fact that molecules need to be correctly aligned in order to react, so not all collisions lead to products. Furthermore, the value of the diffusion coefficients under cellular conditions are likely to be significantly less than that used here. So, 1 x 108 mol-1 L s~l appears to be a useful, albeit conservatively low, value to use in simulating enzymic systems. This value is useful if we are faced with having to choose an arbitrary one for a second-order rate constant in simulations of metabolism, and it is used in some of the following sections.
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