Eadie–Hofstee diagram

Eadie–Hofstee plot of v against v/a for Michaelis–Menten kinetics

In biochemistry, an Eadie–Hofstee plot (or Eadie–Hofstee diagram) is a graphical representation of the Michaelis–Menten equation in enzyme kinetics. It has been known by various different names, including Eadie plot, Hofstee plot and Augustinsson plot. Attribution to Woolf is often omitted, because although Haldane and Stern^[1] credited Woolf with the underlying equation, it was just one of the three linear transformations of the Michaelis–Menten equation that they initially introduced. However, Haldane indicated in 1957 that Woolf had indeed found the three linear forms:^[2]

In 1932, Dr. Kurt Stern published a German translation of my book Enzymes, with numerous additions to the English text. On pp. 119–120, I described some graphical methods, stating that they were due to my friend Dr. Barnett Woolf. [...] Woolf pointed out that linear graphs are obtained when $v$ is plotted against $vx^{-1}$ , $v^{-1}$ against $x^{-1}$ , or $v^{-1}x$ against $x$ , the first plot being most convenient unless inhibition is being studied.

Derivation of the equation for the plot

The simplest equation for the rate $v$ of an enzyme-catalysed reaction as a function of the substrate concentration $a$ is the Michaelis-Menten equation, which can be written as follows:

v={{Va} \over {K_{\mathrm {m} }+a}}

in which $V$ is the rate at substrate saturation (when $a$ approaches infinity, or limiting rate, and $K_{\mathrm {m} }$ is the value of $a$ at half-saturation, i.e. for $v=0.5V$ , known as the Michaelis constant. Eadie^[3] and Hofstee^[4] transformed this into straight-line relationship. Multiplication of both sides by ${{K_{\mathrm {m} }+a} \over {a}}$ gives:

K_{\mathrm {m} }\cdot {v \over a}+v=V

This can be directly rearranged to express a straight-line relationship:

v=V-K_{\mathrm {m} }\cdot {v \over a}

which shows that a plot of $v$ against $v/a$ is a straight line with intercept $V$ on the ordinate, and slope $-K_{\mathrm {m} }$ (Hofstee plot).

In the Eadie plot the axes are reversed:

{v \over a}={V \over K_{\mathrm {m} }}-{1 \over K_{\mathrm {m} }}\cdot v

with intercept $V \over K_{\mathrm {m} }$ on the ordinate, and slope $-{1 \over K_{\mathrm {m} }}$ .

These plots are kinetic versions of the Scatchard plot used in ligand-binding experiments.

Attribution to Augustinsson

The plot is occasionally attributed to Augustinsson^[5] and referred to the Woolf–Augustinsson–Hofstee plot^[6]^[7]^[8] or simply the Augustinsson plot.^[9] However, although Haldane, Woolf or Eadie were not explicitly cited when Augustinsson introduced the $v$ versus $v/a$ equation, both the work of Haldane^[10] and of Eadie^[3] are cited at other places of his work and are listed in his bibliography.^[5]^{: 169 and 171}

Effect of experimental error

Experimental error is usually assumed to affect the rate $v$ and not the substrate concentration $a$ , so $v$ is the dependent variable.^[11] As a result, both ordinate $v$ and abscissa $v/a$ are subject to experimental error, and so the deviations that occur due to error are not parallel with the ordinate axis but towards or away from the origin. As long as the plot is used for illustrating an analysis rather than for estimating the parameters, that matters very little. Regardless of these considerations various authors^[12]^[13]^[14] have compared the suitability of the various plots for displaying and analysing data.

Use for estimating parameters

Like other straight-line forms of the Michaelis–Menten equation, the Eadie–Hofstee plot was used historically for rapid evaluation of the parameters $K_{\mathrm {m} }$ and $V$ , but has been largely superseded by nonlinear regression methods that are significantly more accurate when properly weighted and no longer computationally inaccessible.

Making faults in experimental design visible

Recognizing poor design in Eadie–Hofstee plots, with most $a$ values too large (left) or too small (right)

As the ordinate scale spans the entire range of theoretically possible $v$ vales, from $0$ to $V$ one can see at a glance at an Eadie–Hofstee plot how well the experimental design fills the theoretical design space, and the plot makes it impossible to hide poor design. By contrast, the other well known straight-line plots make it easy to choose scales that suggest that the design is better than it is. Faulty design, as shown in the right-hand diagram, is common with experiments with a substrate that is not soluble enough or too expensive to use concentrations above $K_{\mathrm {m} }$ , and in this case $V/K_{\mathrm {m} }$ cannot be estimated satisfactorily. The opposite case, with $a$ values concentrated above $K_{\mathrm {m} }$ (left-hand diagram) is less common but not unknown, as for example in a study of nitrate reductase.^[15]

Footnotes and references

^ Haldane, John Burdon Sanderson; Stern, Kurt Günter (1932). Allgemeine Chemie der Enzyme. Wissenschaftliche Forschungsberichte, Naturwissenschaftliche Reihe, herausgegeben von Dr. Raphael Eduard Liesegang. Vol. 28. Dresden and Leipzig: Theodor Steinkopff. pp. 119–120. OCLC 964209806.
^ Haldane JB (1957). "Graphical Methods in Enzyme Chemistry". Nature. 179 (4564): 832. Bibcode:1957Natur.179R.832H. doi:10.1038/179832b0. ISSN 1476-4687. S2CID 4162570.
^ ^a ^b Eadie GS (1942). "The Inhibition of Cholinesterase by Physostigmine and Prostigmine". Journal of Biological Chemistry. 146: 85–93. doi:10.1016/S0021-9258(18)72452-6.
^ Hofstee BH (October 1959). "Non-inverted versus inverted plots in enzyme kinetics". Nature. 184 (4695): 1296–1298. Bibcode:1959Natur.184.1296H. doi:10.1038/1841296b0. PMID 14402470. S2CID 4251436.
^ ^a ^b Augustinsson KB (1948). "Cholinesterases: A study in comparative enzymology". Acta Physiologica Scandinavica. 15: Supp. 52.
^ Kobayashi H, Take K, Wada A, Izumi F, Magnoni MS (June 1984). "Angiotensin-converting enzyme activity is reduced in brain microvessels of spontaneously hypertensive rats". Journal of Neurochemistry. 42 (6): 1655–1658. doi:10.1111/j.1471-4159.1984.tb12756.x. PMID 6327909. S2CID 20944420.
^ Barnard JA, Ghishan FK, Wilson FA (March 1985). "Ontogenesis of taurocholate transport by rat ileal brush border membrane vesicles". The Journal of Clinical Investigation. 75 (3): 869–873. doi:10.1172/JCI111785. PMC 423617. PMID 2579978.
^ Quamme GA, Freeman HJ (July 1987). "Evidence for a high-affinity sodium-dependent D-glucose transport system in the kidney". The American Journal of Physiology. 253 (1 Pt 2): F151 – F157. doi:10.1152/ajprenal.1987.253.1.F151. PMID 3605346. S2CID 28199356.
^ Dombi GW (October 1992). "Limitations of Augustinsson plots". Computer Applications in the Biosciences. 8 (5): 475–479. doi:10.1093/bioinformatics/8.5.475. PMID 1422881.
^ Haldane JB (1930). Plimmer RH, Hopkins FG (eds.). Enzymes. London, New York: Longmans, Green, & Company. OCLC 615665842.
^ This is likely to be true, at least approximately, though it is probably optimistic to think that $a$ is known exactly.
^ Dowd JE, Riggs DS (February 1965). "A comparison of estimates of Michaelis-Menten kinetic constants from various linear transformations". The Journal of Biological Chemistry. 240 (2): 863–869. doi:10.1016/S0021-9258(17)45254-9. PMID 14275146.
^ Atkins GL, Nimmo IA (September 1975). "A comparison of seven methods for fitting the Michaelis-Menten equation". The Biochemical Journal. 149 (3): 775–777. doi:10.1042/bj1490775. PMC 1165686. PMID 1201002.
^ Cornish-Bowden A (27 February 2012). Fundamentals of Enzyme Kinetics (4th ed.). Weinheim, Germany: Wiley-Blackwell. pp. 51–53. ISBN 978-3-527-33074-4.
^ Buc, J.; Santini, C. L.; Blasco, F.; Giordani, R.; Cárdenas, M. L.; Chippaux, M.; Cornish-Bowden, A.; Giordano, G. (1995). "Kinetic studies of a soluble αβ complex of nitrate reductase A from Escherichia coli: Use of various αβ mutants with altered β subunits". Eur. J. Biochem. 234 (3): 766–772. doi:10.1111/j.1432-1033.1995.766_a.x.

[Haldane1932-1] Haldane, John Burdon Sanderson; Stern, Kurt Günter (1932). Allgemeine Chemie der Enzyme. Wissenschaftliche Forschungsberichte, Naturwissenschaftliche Reihe, herausgegeben von Dr. Raphael Eduard Liesegang. Vol. 28. Dresden and Leipzig: Theodor Steinkopff. pp. 119–120. OCLC 964209806.

[2] Haldane JB (1957). "Graphical Methods in Enzyme Chemistry". Nature. 179 (4564): 832. Bibcode:1957Natur.179R.832H. doi:10.1038/179832b0. ISSN 1476-4687. S2CID 4162570.

[:0-3] Eadie GS (1942). "The Inhibition of Cholinesterase by Physostigmine and Prostigmine". Journal of Biological Chemistry. 146: 85–93. doi:10.1016/S0021-9258(18)72452-6.

[4] Hofstee BH (October 1959). "Non-inverted versus inverted plots in enzyme kinetics". Nature. 184 (4695): 1296–1298. Bibcode:1959Natur.184.1296H. doi:10.1038/1841296b0. PMID 14402470. S2CID 4251436.

[:1-5] Augustinsson KB (1948). "Cholinesterases: A study in comparative enzymology". Acta Physiologica Scandinavica. 15: Supp. 52.

[6] Kobayashi H, Take K, Wada A, Izumi F, Magnoni MS (June 1984). "Angiotensin-converting enzyme activity is reduced in brain microvessels of spontaneously hypertensive rats". Journal of Neurochemistry. 42 (6): 1655–1658. doi:10.1111/j.1471-4159.1984.tb12756.x. PMID 6327909. S2CID 20944420.

[7] Barnard JA, Ghishan FK, Wilson FA (March 1985). "Ontogenesis of taurocholate transport by rat ileal brush border membrane vesicles". The Journal of Clinical Investigation. 75 (3): 869–873. doi:10.1172/JCI111785. PMC 423617. PMID 2579978.

[8] Quamme GA, Freeman HJ (July 1987). "Evidence for a high-affinity sodium-dependent D-glucose transport system in the kidney". The American Journal of Physiology. 253 (1 Pt 2): F151 – F157. doi:10.1152/ajprenal.1987.253.1.F151. PMID 3605346. S2CID 28199356.

[9] Dombi GW (October 1992). "Limitations of Augustinsson plots". Computer Applications in the Biosciences. 8 (5): 475–479. doi:10.1093/bioinformatics/8.5.475. PMID 1422881.

[10] Haldane JB (1930). Plimmer RH, Hopkins FG (eds.). Enzymes. London, New York: Longmans, Green, & Company. OCLC 615665842.

[11] This is likely to be true, at least approximately, though it is probably optimistic to think that $a$ is known exactly.

[12] Dowd JE, Riggs DS (February 1965). "A comparison of estimates of Michaelis-Menten kinetic constants from various linear transformations". The Journal of Biological Chemistry. 240 (2): 863–869. doi:10.1016/S0021-9258(17)45254-9. PMID 14275146.

[13] Atkins GL, Nimmo IA (September 1975). "A comparison of seven methods for fitting the Michaelis-Menten equation". The Biochemical Journal. 149 (3): 775–777. doi:10.1042/bj1490775. PMC 1165686. PMID 1201002.

[14] Cornish-Bowden A (27 February 2012). Fundamentals of Enzyme Kinetics (4th ed.). Weinheim, Germany: Wiley-Blackwell. pp. 51–53. ISBN 978-3-527-33074-4.

[15] Buc, J.; Santini, C. L.; Blasco, F.; Giordani, R.; Cárdenas, M. L.; Chippaux, M.; Cornish-Bowden, A.; Giordano, G. (1995). "Kinetic studies of a soluble αβ complex of nitrate reductase A from Escherichia coli: Use of various αβ mutants with altered β subunits". Eur. J. Biochem. 234 (3): 766–772. doi:10.1111/j.1432-1033.1995.766_a.x.

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v t e Enzymes
Activity	Active site Binding site Catalytic triad Oxyanion hole Enzyme promiscuity Diffusion-limited enzyme Cofactor Enzyme catalysis
Regulation	Allosteric regulation Cooperativity Enzyme inhibitor Enzyme activator
Classification	EC number Enzyme superfamily Enzyme family List of enzymes
Kinetics	Enzyme kinetics Eadie–Hofstee diagram Hanes–Woolf plot Lineweaver–Burk plot Michaelis–Menten kinetics
Types	EC1 Oxidoreductases (list) EC2 Transferases (list) EC3 Hydrolases (list) EC4 Lyases (list) EC5 Isomerases (list) EC6 Ligases (list) EC7 Translocases (list)