Determination of kinetic parameters for both reversible and irreversible ﬁrst-order reactions
Serguei A. Bobrovnik
Department of Molecular Immunology, A.V. Palladin Institute of Biochemistry, 9 Leontovicha str., 252030 Kiev, Ukraine Received 9 January 1998; accepted 4 June 1998
Abstract Two methods of calculating the rate constant(s) and thetotal amount of ﬁnal product for both irreversible and reversible ﬁrst-order reactions have been developed. These methods are based on the analytical solution (for a special case) of the system of transcendental equations, which describe the reaction kinetics. The methods are simpler and more convenient to use than the earlier proposed graphic methods of Guggengeim and Kezdy-Swinbourne. © 1998Elsevier Science B.V. All rights reserved. Keywords: First-order reaction; Kinetics; Rate constants
1. Introduction For some biochemical reactions it is rather difﬁcult (and sometimes impossible) to monitor the kinetics of the initial substance concentration, while it is easy to do this for the concentration of the ﬁnal product. In such cases, we are obliged to calculate the rate constant(s) (k foran irreversible or k 11 1 k 21 for a reversible reaction) using the data of the ﬁnal product kinetics. It would not be difﬁcult if we could determine experimentally the concentration of the ﬁnal product [P` ] after completion of the reaction, i.e. the quantity of the reaction product when the time of reaction approaches inﬁnity. However, as a rule, it is very difﬁcult to do this precisely in areal experiment . The two-equation system with two unknowns (k and [P` ] for an irreversible reaction
0165-022X / 98 / $ – see front matter © 1998 Elsevier Science B.V. All rights reserved. PII: S0165-022X( 98 )00019-0
S. A. Bobrovnik / J. Biochem. Biophys. Methods 37 (1998) 53 – 68
or k 11 1 k 21 and [P` ] for a reversible reaction), which describe the dynamics of such reactions, istranscendental and therefore has no general analytical solution. This is the reason why primarily Guggenheim  and later Kezdy et al.  and Swinbourne  proposed the graphical methods for determination of the rate constants without experimental measurement of [P` ]. According to Cornish-Bowden , these two methods are almost equally accurate. Previously we found (unpublished data) thatapplication of these two methods gave slightly different results. In this paper, we calculate the rate constants for a theoretical curve with known k value in order to determine which of these two methods is more accurate. In addition, we develop a method of the analytical solution (for a special case) of the equation system which describes the ﬁrst-order reactions. This enables us to propose moresimple and convenient methods to determine the above-mentioned values. The data presented in this paper show that our methods, like Kezdy-Swinbourne’s method, allow the calculation of the values of k more precisely than Guggenheim’s method. In addition, our methods permit the calculation of [P` ].
2. Theory First let us consider the dynamics of an irreversible ﬁrst-order reaction. If the initialsubstance A transforms irreversibly into a ﬁnal product P A→P (1)
and the rate of the reaction is proportional to the concentration of the initial reactant A, then: d[A] ]] 5 2 k[A] dt (2)
where t is the time of the reaction, [A] the concentration of the initial substance and k the rate constant. The solution of this differential equation gives the known relationship between theconcentration of substance A and time t: [A] 5 [A 0 ]exp(2kt) (3)
were [A 0 ] is the concentration of A at t50. If at the beginning of the reaction (at t50) the concentration of the ﬁnal product [P] 5 0, then, according to the mass conservation equation [P] 5 [A 0 ] 2 [A] (4)
It is obvious that if t→`, then [A] → 0, and [P] → [P` ]. From this it follows that at t5`, [P] 5 [P` ] 5 [A 0 ]. Then, after...