Rebuttal to the Response of Lee and Kong

Abstract

Although I appreciate the response from Drs. Lee and Kong I am disappointed that the main question I posed remains unanswered, which is how a probabilistic/statistic approach can lead to a quantitative/deterministic equation based on mass-action law. Apparently, it is not feasible. As noted in my original Letter, synergism is a physicochemical mass-action law issue, not a statistical issue, so synergism is determined with combination index (CI) values, not with P values (1–5). Although all biological measurements are subject to variability analysis, the chief problem is how to quantify synergism in drug combination when accurate experimental data are obtained. The confidence interval asserted by Lee and Kong can be determined by general statistical method; analysis has also been considered for the r value and the confidence interval for the CI method (1, 3). Here I persist in asking how an explicit derivation can be made for each equation, rather than taking an existing empirical formula, replacing it with different symbols, and labeling it a “model” without any physicochemical bearing [e.g., the “y” in Eqs. 3 and 4, and “β” in Eqs. 7 and 9 indicated by Drs. Lee and Kong (6, 7)]. In this rebuttal, I comment in a point by point manner on Eqs. 1 to 9 provided by Drs. Lee and Kong in their response. As the codeveloper of the accepted and widely cited “combination index” (CI) method (1, 8, 9), I feel compelled to address contradictions that exist, favoring the CI method which has a sound theoretical basis, simple experimental design, and explicit algorithm for automated computerized analysis.Equations 1 and 2 are referred to by Lee and Kong (7) as the Loewe additivity (1928, 1953) based on Fraser's work in the 1870s. However, this reference is inappropriate because a rigorous viable method was, in fact, never developed nor were methods ever applied for synergy quantification in the literature. Additivity, synergism, and antagonism were explicitly derived and mathematically defined for the first time by Chou and Talalay in the multiple drug-effect equation, which is exactly the Eqs. 1 and 2 above along with its experimental design, algorithm of analysis, and computerized simulation (9, 10). The term “combination index” (CI) was coined by Chou and Talalay, where CI <1, = 1, and >1 indicated synergism, additive effect, and antagonism, respectively, along with the Fa–CI plot for all effect levels (8, 9). It is later referred to as the combination index equation or the CI theorem (1–3, 10). These works also include the first explicit derivation of the isobologram equation, which is used for computerized construction or simulation and which is the special case of the CI equation (8–10). Most importantly, the CI theorem developed by Chou and Talalay is based on the median-effect equation (MEE) of the mass-action law which was mathematically derived through several hundred equations (8–15). In contrast, Eqs. 1 and 2 referred to by Lee and Kong do not constitute a “derivation” nor do they introduce anything new.Equation 3 raises problems for the remaining equations. This equation is a modified form of the MEE published in 1976 by Chou for the first time (15), except with an altered notation where “fa” is replaced by “y” and Emax simply inserted. Equation 4 is the correct MEE of Chou except that “y” has replaced fa. Equation 3 cannot be derived, as argued, using the statistical approach. Equation 3 was copied from the MEE with incorrect denotation by arbitrarily adding the coefficient “Emax,” which is not justified. Equation 3 is not the Hill equation (16) as referred to by Drs. Lee and Kong (7). The differences between MEE and the Hill equation are indicated in ref. 1, Table 2, where MEE is for the inhibitor (the reference ligand) and the Hill equation is for substrate (the primary ligand). The term (d/Dm)m appears in Eqs. 3 to 9 and unnumbered Eq. [10], which is well known as the hallmark of the MEE of Chou which is for the inhibitory ligand (1–4). The term Dm used in Eqs. 3–9 is exactly the median-effect dose of the MEE. Thus, Eq. 3 is identical to MEE. In a related manner, Eqs. 4, 5, 8, 9, and [10] (the unnumbered equation similar to the CI equation) are in essence identical to MEE with different symbols (9, 11, 15). In the unnumbered equation termed here Eq. [10], an irrelevant coefficient “β” has been added that has no physicochemical bearing to the analogous combination index equation of Chou–Talalay. Overall, these equations correspond to Eqs. 9, 8, 9, 10, and 16, respectively, in one of my recent review articles (1), which reference earlier work beginning in the 1970s conducted over a decade (8–15). Given that the MEE can be applied to yield the mathematical form of the Michaelis–Menten, Hill, Henderson–Hasselbalch, and Scatchard equations, which are all well known in biological science, the MEE has proven to provide a unifying equation in biochemistry and biophysics (1, 2, 10, 17). The logarithmic transformation of MEE results in the median-effect plot (Chou plot) which allows all dose-effect curves, whether hyperbolic (m = 1), sigmoidal (m > 1), or flat sigmoidal (m < 1), to convert into a linear form (3, 15) that allows computerized simulation of the dose-effect curve, Fa–CI plot (the combination index plot of Chou–Talalay), Fa–DRI plot (Chou–Martin plot), isobologram, normalized isobologram (Chou–Chou plot), and the automated construction of polygonogram (Chou–Chou graphics) (1–3). Thus, the MEE is the unified general equation, whereas the Hill equation is the specified general equation. It is inaccurate to equalize them.The MEE and the CI theorem developed from it are original, without precedent in the biomedical literature. They used the unique approach of merging the mass-action law with mathematical induction and deduction to generate a general theory that is applicable for n drugs combination, different dynamic orders, different mechanisms of the target (e.g., ordered, ping-pong, sequential, or random) and different mechanisms of the drugs that are combined (e.g., competitive, noncompetitve, and uncompetitive inhibitions). Many hundreds of mass-action law–based equations have been derived and published before the general theory is obtained (1, 8–15). The theory development is completely independent and, in contradiction to the assertion of Lee Kong, did not rely on any previously published equations or formulas for drug combination synergy quantification (1, 8–15). As alluded to above, the MEE has been referred to as the Unified Theory where the median is the reference point for the first order to the higher order dynamics, and is the common link for the dynamic action of single entity and multiple entities (1, 2, 17). Thus, the MEE and its plot play the central role for the development of the CI theorem and its algorithms for experimental applications, as well as providing the essential concept for the automated computer simulation. In essence, the MEE has been used by Lee and Kong for the data analysis and software “S-PLUS/R” (7), making automation feasible, but it should neither be taken for granted nor erroneously attributed with findings and features of the Hill equation, which is derived from the MEE (rather than the other way round).I remain unconvinced that life is equal to death in cell biology, as asserted by Lee and Kong. They argue that if fractional kill is 0.3 then fractional survival is 0.7. This is obvious. But, as 0.3 ≠ 0.7, cell survival (S) and cell killing (K) cannot have the same equation as they assert for Eqs. 6 and 7 (for S) and for Eqs. 8 and 9 (for K). Therefore, Eqs. 6 and 7 for survival, as they put forth, have either never been derived or have been derived incorrectly. I argue that their equations were not based on actual derivations because y = S = K is incorrect. In Eq. 6, the median-effect dose normalized term [d/Dm]m should be inverted, based on Chou's MEE of the mass-action law. This is a clear contradiction which illustrates the unconvincing nature of the argument they advance.In conclusion, I assert that the interaction method of Lee and Kong and colleagues (7) for drug combination synergy quantification has merely been added with statistics but without actual derivation of the synergy quantification. Given the primacy, importance, and clarity of CI theory, my concern is that the assertions of Drs. Lee and Kong will serve only to obfuscate the field and confuse biomedical researchers, many of whom have conducted or will conduct all the drug combination studies of the future.See the Response, p. 2794No potential conflicts of interest were disclosed.