Amplitude pseudo first order reaction

The conventional data analysis involves the fitting of data to an equation describing the time dependence of the reaction, leading to the best estimates for the constants defining the equations. Analytical solutions to most simple reaction sequences can be obtained (7, 5, 63). Solutions of differential equations describing the series of first-order (or pseudo-first-order) reactions will always be a sum of exponential terms [Eq. (22)]. Thus for a single exponential, the fitting process provides the amplitude (A), the rate of reaction (X), and the end point (C)... 

Kapp, and the on- and off-rate constants of Qg are given in eqs. (1) ano (2) by their definitions under the three appropriate assumptions (i) under our experimental conditions the binding of Qg to the vacant site is a pseudo-first order reaction to the first approximation, (ii) the ratio of the amplitude of the fast decay component over that of the slow decay component in the decay kinetics following an actinic flash, is equal to the ratio of the fraction of the Qg-bound reaction centers over that of the Qg-unbound reaction centers in the dark, and (iii) the dissociation constant of Qg in the dark is equal to that following an actinic flash. 

Figure 8.1 Simulation of protein fluorescence quenching (F) during NADH binding to lactate dehydrogenase. The curve for (1 — a) rqnesents the exponential time course of a pseudo first order reaction calculated from the fluorescence change as described in the text. The displaced trace is an exponential with the amplitude and time constant of the fluorescence change.

The time course of the complex formation was studied under pseudo-first-order conditions with excess ligand. Concentrations after mixing were at least 10 times larger than those of lead(II). Under such conditions the reaction proceeded to completion in a single rate-limiting step with no more initial loss of spectrophotometric amplitude than that expected trom the dead time of the employed instrument. The process was found to be of first order with respect to lead(II). Indeed, the recorded signal could be fitted with excellent statistical confidence to a single-exponential tunction by nonlinear least squares. 

There are two important results from this analysis. First, the rate constants for binding and dissociation can be obtained from the slope and intercept, resp>ec-tively, of a plot of the observed rate versus concentration. In practice this is possible when the rate of dissociation is comparable to ki [S] under conditions that allow measurement of the reaction. At the lower end, resolution of i is limited by the concentration of substrate required to maintain pseudo-first-order kinetics with substrate in excess of enzyme and by the sensitivity of the method, which dictates the concentration of enzyme necessary to observe a signal. Under most circumstances, it may be difficult to resolve a dissociation rate less than 1 sec by extrapolation of the measured rate to zero concentration. Of course, the actual error must be determined by proper regression analysis in fitting the data, and these estimates serve only to illustrate the magnitude of the problem. In the upper extreme, dissociation rates in excess of 200 sec make it difficult to observe any reaction. At a substrate concentration required to observe half of the full amplitude, where [S] = it., the reaction would proceed toward equilibrium at a rate of 400 sec. Thus, depending upon the dead time of the apparatus, much of the reaction may be over before it can be observed at the concentrations required to saturate the enzyme with substrate. 

While the complexity of reactions with larger numbers of steps makes the derivation of amplitudes prohibitively difficult, much can be gained from obtaining numerical or symbolic expressions for eigenvalues. In the case of pseudo first order steps, such as the one characterized by k above, useful information can be obtained from the concentration dependence of time constants. This will be illustrated with examples of experiments in the literature. The time course of the approach to the steady state is illustrated in figure 5.1 for this numerical example. 

---