Constructive criticism

What follows is solely stimulating and constructive criticism for educational purposes. Being prone to mistakes, I made many of them in my career, but I corrected them appropriately, listening at constructive criticism as that presented here. 

Ignoring recommendations

Ignoring (but maybe also overlooking, disregarding, forgetting, avoiding) recommendations released by authorities, chiefly the International Union of Biochemistry and Molecular Biology (IUBMB, formerly International Union of Biochemistry, IUB) does not support the exchange of scientific information. After acknowledging that the Recommendations for symbolism and terminology in enzyme kinetics have not been updated since 1981, there is no reason for disregarding them. There is nothing wrong in the recommendations. They just need updates to include progress made in the elapsed time.

While recommendations are not compulsory, their purpose is to spread a common language that can be understood by scientist all around the world. In this context, the term noncompetitive inhibition has been “discouraged for all purposes” by the recommendations because of its application to several unrelated mechanisms. Evident conflicts with the IUB-recommendations of 1981 are obviously tolerated by Editorial Boards and Reviewers of scientific journals.

Noncompetitive inhibition has been used for describing the following mechanisms and possibly others (mouse-hover for full names):

LMx(Sp>Ca)I, LMx(Sp=Ca)I, LMx(Sp<Ca)I HMx(Sp<Ca)I, HMx(Sp=Ca)I, HMxD(A/I)

Each one of these mechanisms owns distinct kinetic properties and disparate physiological regulatory functions, precisely targeted by evolution! Readers of scientific reports may be confused in finding such a variety of mechanisms grouped under the same, non-recommended name.

Improper use of the double-reciprocal plot

The most popular method to analyze steady-state data of enzyme-catalyzed reactions, to graph results and calculate kinetic constants is still the double-reciprocal plot (1, 2). The statistical problems of this method have been discussed by the authors themselves (1), even before describing its applications in  more detail (2).

Lineweaver, Burk and Deming recommended:

The estimated limits of experimental error … , based on a weighting according to the reciprocals of the squares of the assigned deviations, … [(1), p. 227]

Other authors analyzed in detail the problems of the double-reciprocal plot, starting with Dowd and Riggs, who came to the conclusion:

…  the Lineweaver-Burk transformation tends to give a deceptively “good” fit, even with unreliable points. The marked inferiority of the Lineweaver-Burk plot strongly suggests that it should be abandoned as a method for estimating Km and V from unweighted points …  [(3), p. 869]

The way out of the problem is straightforward: raw data analysis by nonlinear regression without prior linearization. This works immediately for calculating V and Km for substrates (S) in the absence of modifiers (X). In the presence of modifiers, sets of initial velocities at variable [S] and [X] are first subjected to nonlinear regression analysis to calculate Vapp and Kmapp. Further analysis for determining the basic modifier mechanism and calculating the kinetic parameters is straightforward using the methods outlined in Data analysis and Data validation.

Don’t throw the baby out with the bathwater

Since 1934, and increasingly in recent years (≥ 2000), publications in which the double-reciprocal plot has been used, ignored the statistical limitations of the method by using either single or unweighted observations. Studies in which statistics has been respected do exist and the results therein are reliable, but unfortunately, the number of publications that violate the rules is larger. However, the double-reciprocal plot can freely be used as additional test to confirm the identification of a mechanism by an independent method.

In the analysis of enzyme-modifier interactions with this method, a further major problem arises when 1 (one!) or two fixed modifier concentrations and variable [S] are considered pretending to demonstrate the mechanism and to calculate graphically the kinetic parameters. Also, re-plotting the slopes of the primary plot against modifier concentration to ascertain any non-linear dependence is unpopular.

As discussed in this website, the modifiers involved in allosteric regulation, which have a hyperbolic character, are at risk of escaping attention because of insufficient experimental information. Why? With the control rate measured in the absence of modifier, and one modifier concentration, two experimental points are obtained, i.e. the only relationship that can be deduced between apparent parameters and modifier concentration is that represented by a straight line, meaning that any hyperbolic behavior is excluded a priory from data analysis.

With two fixed modifier concentrations, i.e. three experimental points, one can be lucky in seeing a curvature resembling a hyperbola that either increases or decreases with modifier concentration. However, in the presence of errors all luck can vanish. Hence, in a project that takes months of work, is there any reason to avoid one-two more hours in the laboratory to gather initial rates at a few additional modifier concentrations?

Instead of classifying incorrectly a new compound as a boring “competitive” inhibitor, gratification may arise from the discovery of a new, efficient allosteric modifier, possibly of pharmacological interest. And this with just one-two additional hours around a photometer or similar device.

Overseeing hyperbolic inhibition

The double-reciprocal plot, even if drawn with a sufficient number of replicated measurements and corrects statistics, may fail to reveal hyperbolic mechanisms if the slopes of the plots are not re-plotted against the corresponding modifier concentrations. The secondary plots are linear or hyperbolic for linear and hyperbolic modifiers, respectively.

Experiments wit a single modifier concentration

Biochemical education

In enzyme kinetics, hyperbolic inhibition and nonessential activation constitute the realm of allosteric interactions. A published statement that partial (=hyperbolic) inhibition is fortunately quite rare is altogether wrong and unacceptable (4). Since this statement has been formulated by leading authorities in enzyme kinetics, I have no doubts that this it is an involuntary lapse that will be amended in any future communication. Hyperbolic inhibition and nonessential activation (always hyperbolic) are characteristic properties of allosteric modifiers, whether or not accompanied by cooperativity, remembering that allostery and cooperativity are no synonyms.

Here just two citations:

The distinguishing feature of an allosteric inhibition mechanism, therefore, is its partial nature. (5, p. 395)

Allosteric activation of enzymes is a widespread phenomenon … (6, p. 269)

Allosteric interactions play an important role in physiology, in modern pharmacological strategies, as well as in industrial and biotechnological applications. See a minute assortment of examples in this website.

In 2013, the golden jubilee of allosteric interactions, the milestone paper by Monod, Changeux and Jacob (7) has been celebrated with meetings and dedicated publications. In (7), allosteric interactions as enzyme-regulatory mechanisms have been thoroughly discussed, though in a qualitative way and without kinetics. Although not accompanied by kinetic equations, the gap has been filled two years later with the description of the K-system and V-system (8). Despite two groups of mechanisms with three members each cannot be discriminated using the K- and V-systems, this is still a good systematic classification of regulatory kinetic mechanisms in enzymology: brief summary in this website, and more details in (9,  chapter 3).

Whence cometh the allosterome?

This is the title of a short, significant perspective article written by Janet Lindsley and Jared Rutter (12).

The authors stress that “Allostery is the most direct, rapid, and efficient regulatory mechanism to sense changes in the concentration of small molecules and alter cellular responses to maintain homeostasis.” Also, they argument that “… allosteric regulation is underappreciated in the systems biology world …”.

A number of articles that dealt with the kinetics of allosteric interactions may have been overlooked because the word allostery was not manifestly presented in the papers. In some excellent studies cited in this website, the words allostery or allosteric do not even appear in the text, title or abstract even if the study is centered on allosteric interactions with correct analysis.

Historical correctness and the concept of ‘too difficult’

Besides the 50th anniversary of allostery, in 2013 also the 100th anniversary of the Michaelis and Menten publication (10) was celebrated. A third milestone that let science move forward was overseen, the paper of Botts and Morales, whose core is the general modifier mechanism, in its 60th anniversary (11):

on the effects of modifiers upon the steady state catalyzed reaction rate

I have already called attention (9, p. 66) to the fundamental role of this paper, which anticipated by 10 years, without pretense and drumbeat, the theory of allostery of Monod et al. (7). I quote here a passage from Botts and Morales (11, p. 697) that includes also the section title for better understanding:

THE INTERACTION OF SUBSTRATE AND MODIFIER UPON THE ENZYME, AND ITS EFFECT ON THE STEADY STATE REACTION RATE. — For the discussion of this effect one may visualize an enzymatic “patch” upon the enzyme molecule, consisting of an enzymatic site which binds substrate molecules (S) and an adjacent site which binds modifier molecules (Y).

And below the text of Monod et al. from (7, p. 307):

These proteins are assumed to possess two, or at least two, stereospecifically different, non-overlapping receptor sites. One of these, the active site, binds the substrate and is responsible for the biological activity of the protein. The other, or allosteric site, is complementary to the structure of another metabolite, the allosteric effector, which it binds specifically and reversibly.

Boldface in the quotations above is mine and wants to emphasize that the “adjacent site” of Botts and Morales is a less spectacular definition of “allosteric site” by Monod et al.

If someone cannot accept this view, can simply go in depth into the paper of Botts and Morales (11) to discover that it contains exactly the description of the K- and V-system, with rigorous mathematics not provided by Monod et al. (8). From the treatment of Botts and Morales one can derive the properties of seventeen basic modifier mechanisms removing the black box around the six entities that cannot be differentiated with the K- and V-systems. Two of the most sophisticated, fine-tuning kinetic strategies to control enzyme activity optimized by evolution, the dual mechanisms, HMxD(A/I) and HMxD(I/A), are located in this group.

Monod and coworkers did not mention the contribution by Botts and Morales. Also, some books and original papers, which benefit from the general modifier mechanisms, do not put its ‘copyright’ on record.

Is hyperbolic enzyme modification too difficult to be taught and used in practice?

Despite receiving appreciation, the paper by Botts and Morales (11) was cited only 198 times. In comparison, the citations of Monod et al. (7) were 1,720 (Web of Science, January 2019).
A possible reason for the few citations of the Botts-Morales paper is the way the theory is presented.

Jean Botts and her husband Manuel Francisco Morales devoted their careers to studying the thermodynamics of muscle contraction. As customary in thermodynamics, they represented equilibria as association constants, as in their 1953 paper, where inhibition constants and the Michaelis constant are shown in this form.

At first impact, the non-appealing rate equation may disorient readers used to dissociation constants, as routine in enzyme kinetics. In 1972, when I read the paper, recommended by my friend Giorgio Semenza, I understood nothing. So, I first translated the symbols, re-wrote the reaction scheme and the rate equation before analyzing its details discovering amazing properties (9, chapter 2).

Download the Botts-Morales equation of the general modifier mechanism


  1. Lineweaver H, Burk D, Deming WE (1934) The dissociation constant of nitrogen-nitrogenase in Azotobacter. J Am Chem Soc 56: 225-230.
  2. Lineweaver H, Burk D (1934) The determination of enzyme dissociation constants. J Am Chem Soc 56: 658-666.
  3. Dowd JE, Riggs DS (1965) A comparison of estimates of Michaelis-Menten kinetic constants from various linear transformations. J Biol Chem 240: 863-869.
  4. Tipton KF, Armstrong RN, Bakker BM, Bairoch A, Cornish-Bowden A, Halling PJ, Hofmeyr J-H, Leyh TS, Kettner C, Raushel FM, Rohwer J, Schomburg D et al. (2014) Standards for reporting enzyme data: The STRENDA Consortium: What it aims to do and why it should be helpful. Perspect Sci 1: 131-137.
  5. Reinhart GD (1983) The determination of thermodynamic allosteric parameters of an enzyme undergoing steady-state turnover. Arch Biochem Biophys 224: 389-401.
  6. Di Cera E (2009) Kinetics of allosteric activation. Meth Enzymol 466: 259-271.
  7. Monod J, Changeux JP, Jacob F (1963) Allosteric proteins and cellular control systems. J Mol Biol 6: 306-329.
  8. Monod J, Wyman J, Changeux JP (1965) On the nature of allosteric transitions: a plausible model. J Mol Biol 12: 88-118.
  9. Baici A (2015) Kinetics of Enzyme-Modifier Interactions – Selected Topics in the Theory and Diagnosis of Inhibition and Activation Mechanisms. Springer, Vienna.
  10. Michaelis L, Menten ML (1913) Die Kinetik der Invertinwirkung. Biochem Z 49: 333-369. [A link to the original is difficult, here the English translation thanks to Roger S. Goody and Kenneth A. Johnson. See also the comments to of this translation. ]
  11. Botts J, Morales M (1953) Analytical description of the effects of modifiers and of enzyme multivalency upon the steady state catalyzed reaction rate. Trans Faraday Soc 49: 696-707.
  12. Lindsley JE, Rutter J (2006) Whence cometh the allosterome? Proc Natl Acad Sci USA 103: 10533-10535. doi:10.1073/pnas.0604452103.