Nowadays, these methods have been replaced by nonlinear regression, which is both far more accurate and easily accessible, so they should not be used except for data visualization purpose.<\/p>\n\n\n\n
\nAvoid Scatchard, Lineweaver-Burke and similar transforms<\/strong><\/p>\n\n\n\n
Before nonlinear regression was readily available, the best way to analyze nonlinear data was to transform the data to create a linear graph, and then analyze the transformed data with linear regression. Examples include Lineweaver-Burke plots of enzyme kinetic data, Scatchard plots of binding data, and logarithmic plots of kinetic data. These methods are outdated, and should not be used to analyze data.<\/p>\n\n\n\n
The problem with these methods is that the transformation distorts the experimental error. Linear regression assumes that the scatter of points around the line follows a Gaussian distribution and that the standard deviation is the same at every value of X. These assumptions are rarely true after transforming data. Furthermore, some transformations alter the relationship between X and Y. For example, in a Scatchard plot the value of X (bound) is used to calculate Y (bound\/free), and this violates the assumption of linear regression that all uncertainty is in Y while X is known precisely. It doesn’t make sense to minimize the sum of squares of the vertical distances of points from the line, if the same experimental error appears in both X and Y directions.<\/p>\n\n\n\n
\u2014 Motulsky, Harvey. Linear Regression. In: Analyzing Data with GraphPad Prism. 1999. GraphPad Software, Inc.<\/a><\/p>\n<\/blockquote>\n\n\n\n
Scatchard Equation and Plot<\/h2>\n\n\n\n
Derivation<\/h3>\n\n\n\n
Let\u2019s assume an equilibrium between a receptor and a ligand as follows:\n$$ [R] + [L] \\rightleftharpoons [RL] $$\nThe dissociation constant \\(K_{\\rm d}\\) is defined as:\n$$K_{\\mathrm d} = \\frac{[R][L]}{[RL]} $$\nAlso,\n$$[R_{\\mathrm T}] = [R] + [RL]$$\nSo, \n$$[RL] = \\frac{[R_{\\mathrm T}][L]}{K_{\\mathrm d} + [L]} $$\nThis equation is a nonlinear expression equivalent to the Michaelis\u2013Menten equation<\/b> in enzyme kinetics.\nOn the other hand, if we rearrange the same equation, we get:\n$$ \\frac{K_{\\mathrm d}[RL]}{[L]} = -[RL] + [R_{\\mathrm T}] $$\nDividing both sides by \\(K_{\\rm d}\\):\n$$\\frac{[RL]}{[L]} = -\\frac{1}{K_{\\mathrm d}}[RL] + \\frac{[R_{\\mathrm T}]}{K_{\\mathrm d}} $$\nThis is the Scatchard equation<\/b>, which corresponds to the Eadie\u2013Hofstee plot<\/b>, a linear representation of the Michaelis\u2013Menten equation.<\nFrom this equation, if you plot \\([RL]\\) on the x<\/i>-axis and \\([RL]\/[L]\\) on the y<\/i>-axis, and perform linear regression, you will get a straight line where:\n
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- The slope<\/b> is \\(-1\/K_{\\rm d}\\)<\/li>\n
- The x<\/i>-intercept<\/b> is \\([R_{\\rm T}]\\)\n<\/ul>\nAn important feature of this plot: if there are multiple binding sites with different affinities for the ligand, the plot becomes a concave curve<\/b> (bowing downward), rather than a straight line.\nIf the purpose is to estimate parameters, a Hanes\u2013Woolf type plot<\/b> like the one below is more accurate than the Scatchard plot:\n$$\\frac{[L]}{[RL]} = \\frac{1}{[R_{\\mathrm T}]}[L] + \\frac{K_{\\mathrm d}}{[R_{\\mathrm T}]} $$\nHowever, such a plot makes it difficult to discern the presence of multiple binding sites with different affinities.<\/p>\n\n\n\n
Problems<\/h3>\n\n\n\n
The main problems with the Scatchard plot include:\n
\n
- The x<\/i>-axis (\\([RL]\\))<\/b> and y<\/i>-axis (\\([RL]\/[L\\)])<\/b> are not independent variables<\/b>, since both contain \\([RL]\\).<\/li>\n
- Least squares regression<\/b> is fundamentally not applicable<\/b> here, because it assumes:\n
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- The x<\/i>-axis is the explanatory variable (with no experimental error)<\/li>\n
- The y<\/i>-axis is the dependent variable (which contains experimental error)<\/li><\/ul>\n
- You cannot apply the correlation coefficient<\/b> as a goodness-of-fit indicator.<\/li>\n
- The method is indirect and extrapolative<\/b>, rather than directly estimating parameters.<\/li>\n<\/ul>\nFor these reasons, in most research papers, the Scatchard plot is typically included only as a small figure next to the binding saturation plot (e.g.<\/i>, Braun, 2005), rather than used for actual parameter estimation.<\/p>\n\n\n\n
Practice Problem<\/h2>\n\n\n\n
I tried solving a practice problem from the lecture material archived at the following URL:<\/p>\n\n\n\n
https:\/\/web.archive.org\/web\/20091007171045\/http:\/\/www.biochem.oulu.fi\/Biocomputing\/juffer\/Teaching\/PhysicalBiochemistry\/PhysBiochem-protein-ligand.pdf<\/a><\/p>\n\n\n\n
Mg2+<\/sup> and ADP form a 1:1 complex. In the binding experiment, the total concentration of ADP is kept constant at 80 \u00b5M. From the data in the following table, determine the dissociation constant \\(K_{\\rm d}\\).<\/blockquote><\/p>\n\n\n\n
![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation.png\")
<\/p>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard.png\")
<\/p>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve.png\")
<\/p>\n\n\n\n![\"\"](\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard.png\")
<\/p>\n\n\n\n