Elsevier

Biochemical Engineering Journal

Volume 63, 15 April 2012, Pages 116-123
Biochemical Engineering Journal

Review
On the Lambert W function and its utility in biochemical kinetics

https://doi.org/10.1016/j.bej.2012.01.010Get rights and content

Abstract

This article presents closed-form analytic solutions to three illustrative problems in biochemical kinetics that have usually been considered solvable only by various numerical methods. The problems solved concern two enzyme-catalyzed reaction systems that obey diversely modified Michaelis–Menten rate equations, and biomolecule surface binding that is limited by mass transport. These problems involve the solutions of transcendental equations that include products of variables and their logarithms. Such equations are solvable by the use of the Lambert W(x) function. Thus, these standard kinetics examples are solved in terms of W(x) to show the applicability of this commonly unknown function to the biochemical community. Hence, this review first of all describes the mathematical definition and properties of the W(x) function and its numerical evaluations, together with analytical approximations, and then it describes the use of the W(x) function in biochemical kinetics. Other applications of the function in various engineering sciences are also cited, although not described.

Highlights

► This review describes the Lambert W function, and its use in biochemical kinetics. ► Numerical evaluations of W, together with analytical approximations, are presented. ► Three illustrative problems that are solvable by the use of W are demonstrated. ► The utility of W in various engineering sciences are also cited but not described.

Introduction

A function that is convenient for solving transcendental equations of the type y + ln(y) = x was derived and used independently by several researchers [1], [2], [3] before Corless et al. settled on a common notation in the mid-1990s [4], [5]. They called it the Lambert W function (the letter W was chosen following the use of early Maple software [4]), on the consideration that this function can be traced back to Johann Lambert in around 1758, and it was considered later by Leonhard Euler [for references, see 5]. These two mathematicians developed a series solution for the trinomial equation, although they left it unnamed. However, in the years during the Lambert/Euler era and through the modern history of Maple [4], this function did not disappear entirely, even if the literature remained widely scattered and obscure until the function acquired the name of the Lambert W function. Meanwhile, Wright [1] analyzed the solution of the equation z exp(z) = a, in 1959, and later, in 1973, Fritsch et al. [2] presented an efficient algorithm for the root computation of such nonlinear transcendental equations. However, even a decade later, in the early 1980s, Beal still evaluated the solutions to the Michaelis–Menten equation using either a table for the so-called function ‘F’ [3] or a relatively slow Newton's root-finding method [6].

However, across numerous scientific disciplines, the modern knowledge of the Lambert W function as a mathematical tool has allowed the derivation of closed-form solutions for models for which explicit or exact solutions were not known, and therefore where alternative iterative methods or approximate solutions had been used. Thus the Lambert W function can arise in a wide scope of practical problems in mathematics, computer science [5], mathematical physics [5], [7] and engineering (see [8] and references therein), although it received little attention in biochemistry and biotechnology. In these latter disciplines, the solutions to relatively simple model equations are frequently given by a standard family of transcendental equations of the following type:A·y+ln(B+C·y)=ln(D)where A, B, C and D do not depend on y. For many life scientists that are faced with Eq. (1) in this implicit form, the usual next step is to search for an available and efficient root-finding computer algorithm that can numerically evaluate the dependent variable y. Thus, they are not aware that Eq. (1) has the solution as given in its closed form:y=1A·WA·DC·expA·BCBCwhere W stands for the Lambert W function. Therefore, I believe that it is expedient to illustrate here this function and how to use it in practice. This is the subject of the present article, in which examples of specific biochemical kinetics are given that can be solved algebraically and expressed in the closed form only in terms of the Lambert W function. Other applications of the use of the Lambert W function in various engineering disciplines are also cited at the end of this review, although these are not described in detail.

Section snippets

Mathematical properties of the Lambert W function

The Lambert W function is defined as the solution to the following transcendental equation:y·exp(y)=xalthough using a logarithmic transformation, Eq. (3) can also be rewritten as:y+ln(y)=ln(x)However, the solution to Eqs. (3), (4) is the Lambert W function, and it is written as:y=W(x)

Actually, when x is a general complex number, there are an infinite number of solution branches that can be labeled with an integer subscript, as Wk(x) for k = 0, ±1, ±2, …. However, if x is a real number, the only

Computing the Lambert W function

A specificity of the Lambert W function is that it is defined as an inverse function, and it cannot be expressed in terms of elementary mathematical functions. As a consequence, arbitrary precision evaluations can be obtained through iterative root-finding techniques. Numerous numerical methods are available for this purpose. The choice has to find the trade-off between complexity of implementation, conditions, and number of iterations to convergence at a given precision. These properties are

Analytic approximations for the Lambert W function

Although the Lambert W function is useful in many scientific disciplines, it is not yet available in the standard mathematical software libraries that are also widely used in the life sciences. Hence, it is very interesting to express it approximately with other elementary mathematical functions that are readily available with any computer program. Thus, in this section, I will present such approximations for W−1(x) and W0(x) from [8], although some other approximations for W−1(x) have also

Use of the Lambert W function in biochemical kinetics

The Lambert W function is nowadays mostly applied in computational sciences and mathematical physics [5], where it has been discussed in great detail. However, over the last decade, the use of the Lambert W function has broadened widely in chemical kinetics [13], [14], [15], [16] and engineering [17], [18], [19]. In contrast, biochemistry and biotechnology are still seldom used to demonstrate the use of the Lambert W function, although biochemical kinetics provide the opportunity to apply it

Other uses of the Lambert W function in engineering and life sciences

Although this review is mainly focused on problems considering the Lambert W function that appear in the biochemical and biotechnological disciplines [22], [23], [24], [25], [26], [27], [28], [29], [34], [35], [36], [37], [38], [39], and in chemical engineering [13], [14], [15], [16], [17], [18], [19], other applications of W of practical import have been discovered in various engineering and life sciences, and I cite here only recently published literature. Thus, W appears in solutions to a

Conclusion

In this review, I have presented some of the calculus properties of the Lambert W function, along with its numerical evaluation and the approximation functions to W(x). I have combined these with some biochemical kinetics examples of its applications and the available references on this function. Furthermore, I believe that awareness of the Lambert W function in the fields of biochemistry and biotechnology will also increase in the years to come [15], [52], and many more applications will be

Acknowledgment

This work was supported by the Slovenian Research Agency (grant P1-170).

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