Direct linear plot

The best known plots of the Michaelis–Menten equation, including the double-reciprocal plot of $1/v$ against $1/a$,{{Cite journal|last1=Lineweaver|first1=H.|last2=Burk|first2=D.|date=1934|doi = 10.1021/ja01318a036| title=The Determination of Enzyme Dissociation Constants |journal=J. Amer. Chem. Soc.|language=en|volume=56|issue=3|pages=658–666|bibcode=1934JAChS..56..658L }} the Hanes plot of $a/v$ against $a$,{{cite journal | doi= 10.1042/bj0261406 | journal = Biochem. J. |pages = 1406–1421 | volume = 26 | number = 2 | author = Hanes, C.S. | title = Studies on plant amylases. I. The effect of starch concentration upon the velocity of hydrolysis by the amylase of germinated barley | year = 1932| pmid = 16744959 | pmc = 1261052 }} and the Eadie–Hofstee plot{{cite journal | journal = J. Biol. Chem. | volume = 146 | number = 1 | title = The inhibition of cholinesterase by physostigmine and prostigmine | author = Eadie, G. S. | pages = 85–93 | year =1942| doi = 10.1016/S0021-9258(18)72452-6 | doi-access = free }}{{cite journal | journal=J. Biol. Chem. | title = Specificity of esterases |volume = 199 | number = 1 | year = 1953 | author=Hofstee, B. H. J. | pages = 357–364| doi = 10.1016/S0021-9258(18)44843-0 | doi-access = free }} of $v$ against $v/a$ are all plots in observation space, with each observation represented by a point, and the parameters determined from the slope and intercepts of the lines that result. This is also the case for non-linear plots, such as that of $v$ against $a$, often wrongly called a "Michaelis-Menten plot", and that of $v$ against $\backslash log\; a$ used by Michaelis and Menten.{{Cite journal| last1 = Michaelis|first1 = L.| last2 = Menten| first2 = M. L.| year = 1913| title = Die Kinetik der Invertinwirkung| journal = Biochem. Z.| volume = 49| pages = 333–369}} In contrast to all of these, the direct linear plot is a plot in parameter space, with observations represented by lines rather than as points.

File:Realistic direct linear plot with errors.jpg

The case illustrated above is idealized, because it ignores the effect of experimental error. In practice, with $n$ observations, instead of a unique point of intersection, there is a family of $n(n-1)/2$ intersection points, with each one giving a separate estimate of $K\_\{\backslash mathrm\{m\}\_\{ij\}\}$ and $V\_\{ij\}$ for the lines drawn for the $i\backslash ,\backslash mathrm\{th\}$ and $j\backslash ,\backslash mathrm\{th\}$ observations.{{cite journal | doi= 10.1042/bj1390721 | journal = Biochem. J. | volume = 130 | number = 3 | pages = 721–730 |author1=Cornish-Bowden, A. |author2=Eisenthal, R. | title = Statistical considerations in estimation of enzyme kinetic parameters by the direct linear plot and other methods| year = 1974 | pmid = 4854389 | pmc = 1166336 }} Some of these, when the intersecting lines are almost parallel, will be subject to very large errors, so one must not take the means (weighted or not) as the estimates of $\backslash hat\{K\}\_\backslash mathrm\{m\}$ and $\backslash hat\{V\}$. Instead one can take the medians of each set as estimates $K\_\backslash mathrm\{m\}^*$ and $V^*$.

The great majority of intersection points should occur in the first quadrant (both $K\_\{\backslash mathrm\{m\}\_\{ij\}\}$ and $V\_\{ij\}$ positive).If they do not one should consider the possibility that the Michaelis–Menten equation is not the appropriate equation. Intersection points in the second quadrant ($K\_\{\backslash mathrm\{m\}\_\{ij\}\}$ negative and $V\_\{ij\}$ positive) do not require any special attention. However, intersection points in the third quadrant (both $K\_\{\backslash mathrm\{m\}\_\{ij\}\}$ and $V\_\{ij\}$ negative) should not be taken at face value, because these can occur if both $v$ values are large enough to approach $V$, and indicate that both $K\_\{\backslash mathrm\{m\}\_\{ij\}\}$ and $V\_\{ij\}$ should be taken as infinite and positive: $K\_\{\backslash mathrm\{m\}\_\{ij\}\}\; \backslash rightarrow\; +\backslash infty,\; V\_\{ij\}\; \backslash rightarrow\; +\backslash infty$.Infinite elements are of course disastrous for estimating a mean value, but as long as they are not very numerous they present no problem for estimating a median value.

The illustration is drawn for just four observations, in the interest of clarity, but in most applications there will be much more than that. Determining the location of the medians by inspection becomes increasingly difficult as the number of observations increases, but that is not a problem if the data are processed computationally. In any case, if the experimental errors are reasonably small, as in Fig. 1b of a study of tyrosine aminotransferase with seven observations,{{cite journal | doi= 10.1007/s00425-021-03623-2 | journal = Planta | title = Identification and biochemical characterisation of tyrosine aminotransferase from Anthoceros agrestis unveils the conceivable entry point into rosmarinic acid biosynthesis in hornworts |author1=Busch, T. |author2=Petersen, M. | year = 2021 | volume = 253 | issue = 5 | pages = 98| pmid = 33844079 | pmc = 8041713 | bibcode = 2021Plant.253...98B }} the lines crowd closely enough together around the point $(K\_\backslash mathrm\{m\}^*,\; V^*)$ for this to be located with reasonable precision.

The major merit of the direct linear plot is that median estimates based on it are highly resistant to the presence of outliers. If the underlying distribution of errors in $v$ is not strictly Gaussian, but contains a small proportion of observations with abnormally large errors, this can have a disastrous effect on many regression methods, whether linear or non-linear, but median estimates are very little affected.

In addition, to give satisfactory results regression methods require correct weighting: do the errors $\backslash varepsilon(v)$ follow a normal distribution with uniform standard deviation, or uniform coefficient of variation, or something else? This is very rarely investigated, so the weighting is usually based on preconceptions. Atkins and Nimmo{{cite journal | doi= 10.1042/bj1490775 | journal = Biochem. J. | title = A comparison of seven methods for fitting the Michaelis–Menten equation |author1=Atkins, G. L. |author2=Nimmo, I. A. | volume = 149 | number = 3 | year= 1975 | pages = 775–777| pmid = 1201002 | pmc = 1165686 }} made a comparison of different methods of fitting the Michaelis-Menten equation, and concluded that

We have therefore concluded that, unless the error is definitely known to be normally distributed and of constant magnitude, Eisenthal and Cornish-Bowden's methodThey were referring to median estimation on the basis of the direct linear plot. is the one to use.

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Category:Enzyme kinetics

Category:Catalysis