On the Use of Least Squares To Fit Data in Linear Form Delano P. Chong 2036 Main Mall, University of British Columbia, Vancouver, BC, Canada V6T 121
Frequently, physical scientists wish to extract parameters from observed data by the method of least squares. When ex~erimentaldata can be cast in linear forms. linear plots are-often both more revealing and more appealing to the eye. However, althouch other scientists are well aware of the danger of applying linear regression indiscriminately (131, chemists seem less well informed (4-12). Although we choose Michaelis-Menten kinetics (13,141 as a concrete example in this study, the Idndemam-Hinshelwood pseudo-first-order rate wnstants for unimolecular reactions behave in a similar manner. Other common chemical problems, such as first-order kinetics, second-order kinetics, and Arrhenius activation energy, are often converted to linear forms.
uncertainty is assumed, as shown in Table 2, each linearized form gives results that differ from those for the corresponding reverse version. The linearized forms of Michaelis-Menten kinetics are expected to have nonuniform errors in both the x and y values. Weighting removes the assumption of uniform uncertainties, although few programs in chemistry treat errors in both axes. When these errors are properly accounted for Table 1. Raw Dataa
Alternate Forms In this work, the Michaelis-Menten equation is used to study enzyme kinetics of the raw data in Table 1. u=- " m S
KM+S
(1)
where u is the initial rate; V,, is the maximum rate; S is the substrate concentration; and KM is the Michaelis wnstant. It is well-known that eq 1can be cast in alternative linear forms, as shown below. Wwlf (15)-Lineweauer-Burk
1.460
0.044
0.493
0.025
"Data of Atkinson et a1 quoted by Wilkinson (ZO), augmented by estimates of the uncertainties. The substrate concentrations were assigned uncertainties d approximately 8 to 3%. and the initial rates were given standad erron of about lOt05%. ?he mncentration of nicotinamide mononuoleotide per millimole. %molesof nicotinamide-adenine dinucleotidefarmed in 1 minlmg of enzyme protein.
Table 2. Results of Least-Squares its*,^
(161
uniform nonzerob nonzerob nonzerob 0 uniform 0 nonzerob nonzerob nonzerob 0 uniform 0 nonzerob nonzerob nonzerob 0 uniform 0 nonzerob nonzerob nonzerob 0 uniform 0 nonzerob nonzerob nonzerob 0 uniform 0 nonzerob nonzerob nonzerob 0 uniform 0 nonzerob , , nonzerob nonzerob 0.5860(708) 0 0
Hanes (18)-Hofstee (19) S =S +- KM u
v,,
v,
(4)
Of course, one can interchange the dependent and independent variables and get the following. Reverse Wwlf-Lineweauer-Burk
Reverse Woolf-Eadie
Reverse Hanes-Hofstee I Removing the Assumption of Uniform Uncertainties There are a few drawbacks in applying linear regression to analyze Michaelis-Menten kinetics: Many black box linear regression programs assume uniform uncertainty in they values and no error in the x values. When uniform
0.6854(429)
'Values in parentheses represent one standard deviation in the final digits of the quoted values. b ~ a ~ on e dthe data in Table 1. 'Michaelis conRant in mM. ' ~ a x i m u minitial a t e in same unitsas vfn Table 1 'These uncertainties, obtained when the x intercepts were also used, are Smaller than those obtained without the use of the xintercepts.
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489
( I ) , each linearized form and its reverse version yield the same extracted parameters. However, the different pairs still yield different results.' This is because the error bars are biased (21, with unequal lengths on the opposite sides of a point. In other words, the random e m r s are not normally distributed on either side of the straight line, and therefore the basis of our use of least-squares method is not valid. The correct solution is to fit thedata to eq 1nonlinearly with realistic uncertainties, as indicated by the
''
last of entries in The necessary a'gorithm (3) is straightforward to program, although subroutines for the fitted function (e.g., eq 1) and its analytical derivatives must be coded for each specific case.2 ~ ] t h the ~ ~ special ~ hcase of Michae]is-Menten kinetics does not represent all cases of extracting parameters from experimental data in general, it is clear that caution 'When the experimental errors are small,the problem is less seriand the different pairs of linearized forms are expected to yield vely similar results. 2~ copy of the program for data analysis of Michaelis-Menten kinetics on microcomputers will be sent to the interested reader upon receipt of a floppy disk and reusable mailer. OUS,
490
Journal of Chemical Education
should be exercised in any data analysis. It is therefore my hope that fellow chemists would choose their programs carefully and would teach their students to use correct Procedures. Acknowledgment
I am grateful to the referee for valuable suggestions. L
R cited ~
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I. D PA^. J. P ~ F . IWI, 59,472 and refeenee. therein. 2. Thompson,W.J.;Madonald, J . R h . J Phys IWl,59,854. 3. Msdonald,J.R.;Thompson,WJ.Am. J.Phys. 1992,60,66andreferencestherein. 4. sands, D. E. J cfiem. ~ d1974.51.473. ~ ~ . 5. Chrktian, S. D.;Lsne, E. H.;Garisnd, E J. Chem.Edu. 1ST4,51,475. JI, E, E.J Chem, Educ. 19,T,54, 6, 7. I-", J. A.; Q U ~ C ~ I.~ J.~c h~ r m ~ ~, dTu c.196%6o,111. . 8. Ju", ?I C. Computw Sofluam Appiicofims in Chomisfv; Wilw: New Yak. 1986; Chapter 3. 9. K & , ~ , A . H. J. ckm. E ~ Z C1981,64,28. . 10. T a H . S.; Jon'" W.E. J. Chm. Ed"'. 1 9 8 9 , 6 6 5 0 . 11. Zdravkov8ki.Z. IT Ckm. Educ. 1992,69,A242. 12. Ogren. P J.;Norton,J. R. J. Chsm.Educ. 1992,69,A130. 13. Michaelir, L.; Menten,M. L.Biahem. Z.1913,49,333. 14. B1iggs.G. E.;Haldsne, J. B. SBiahem. J. 1925,19,338. 15. wooif, B. quoted in &ldane, J. B s.;stem, K.G. ~ ~chrmipdor~niyme; i ~ steinkwffvdsg: ~ r e s d e nand ~ e i p d g1932; , p 119. 16. Linewesuer, H.; Burk, D. J. A m Chsm Soc. 1934.56.658,
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20. wllkinson, G. ~ . ~ i ~ e h .J.m1961,80,324. .
