{"id":4442,"date":"2025-05-16T17:14:06","date_gmt":"2025-05-16T08:14:06","guid":{"rendered":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=4442"},"modified":"2026-01-06T09:42:44","modified_gmt":"2026-01-06T00:42:44","slug":"%e8%a7%a3%e9%9b%a2%e5%ae%9a%e6%95%b0%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e3%81%ae%e3%81%abscatchard-plot%e3%82%92%e4%bd%bf%e3%81%86%e3%81%b9%e3%81%8d%e3%81%a7%e3%81%af%e3%81%aa%e3%81%84%e3%81%a8","status":"publish","type":"post","link":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/en\/2025\/05\/16\/%e8%a7%a3%e9%9b%a2%e5%ae%9a%e6%95%b0%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e3%81%ae%e3%81%abscatchard-plot%e3%82%92%e4%bd%bf%e3%81%86%e3%81%b9%e3%81%8d%e3%81%a7%e3%81%af%e3%81%aa%e3%81%84%e3%81%a8\/","title":{"rendered":"Why You Shouldn\u2019t Use Scatchard Plots to Determine Dissociation Constants"},"content":{"rendered":"\n\n\n\n<p class=\"wp-block-paragraph\">It\u2019s long been known that applying linear regression to equations transformed from nonlinear forms is generally not advisable. However, I had never actually examined how much the resulting values might differ in practice\u2014so I decided to test it myself.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">There are several methods that transform nonlinear equations into linear ones to estimate parameters. These include:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Scatchard plot (used in ligand binding assays)<\/li>\n\n\n\n<li>Lineweaver\u2013Burk plot (used in enzyme kinetics)<\/li>\n\n\n\n<li>Eadie-Hofstee plot<\/li>\n\n\n\n<li>Hanes\u2013Woolf plot<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Nowadays, these methods have been replaced by nonlinear regression, which is both far more accurate and easily accessible, so they should not be used except for data visualization purpose.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<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"wp-block-paragraph\"><strong>Avoid Scatchard, Lineweaver-Burke and similar transforms<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">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 These methods are outdated, and should not be used to analyze data.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<p class=\"wp-block-paragraph\">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&#8217;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<p class=\"wp-block-paragraph\">\u2014 <a href=\"https:\/\/cdn.graphpad.com\/faq\/1283\/file\/AnalyzingDataPrism3.pdf\" title=\"\">Motulsky, Harvey. Linear Regression. In: Analyzing Data with GraphPad Prism. 1999. GraphPad Software, Inc.<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">Scatchard Equation and Plot<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Derivation<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">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 <b>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 <b>Scatchard equation<\/b>, which corresponds to the <b>Eadie\u2013Hofstee plot<\/b>, a linear representation of the Michaelis\u2013Menten equation.<\nFrom this equation, if you plot \\([RL]\\) on the <i>x<\/i>-axis and \\([RL]\/[L]\\) on the <i>y<\/i>-axis, and perform linear regression, you will get a straight line where:\n<ul>\n<li>The <b>slope<\/b> is \\(-1\/K_{\\rm d}\\)<\/li>\n<li>The <b><i>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 <b>concave curve<\/b> (bowing downward), rather than a straight line.\nIf the purpose is to estimate parameters, a <b>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<h3 class=\"wp-block-heading\">Problems<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">The main problems with the Scatchard plot include:\n<ul>\n<li>The <b><i>x<\/i>-axis (\\([RL]\\))<\/b> and <b><i>y<\/i>-axis (\\([RL]\/[L\\)])<\/b> are <b>not independent variables<\/b>, since both contain \\([RL]\\).<\/li>\n<li><b>Least squares regression<\/b> is fundamentally <b>not applicable<\/b> here, because it assumes:\n<ul>\n<li>The <i>x<\/i>-axis is the explanatory variable (with no experimental error)<\/li>\n<li>The <i>y<\/i>-axis is the dependent variable (which contains experimental error)<\/li><\/ul>\n<li>You cannot apply the <b>correlation coefficient<\/b> as a goodness-of-fit indicator.<\/li>\n<li>The method is <b>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 (<i>e.g.<\/i>, Braun, 2005), rather than used for actual parameter estimation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Practice Problem<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">I tried solving a practice problem from the lecture material archived at the following URL:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><a href=\"https:\/\/web.archive.org\/web\/20091007171045\/http:\/\/www.biochem.oulu.fi\/Biocomputing\/juffer\/Teaching\/PhysicalBiochemistry\/PhysBiochem-protein-ligand.pdf\">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<p class=\"wp-block-paragraph\"><blockquote>Mg<sup>2+<\/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<figure class=\"wp-block-table\"><table><tbody><tr><td>Total Mg<sup>2+<\/sup> (\u00b5M)<\/td><td>Mg<sup>2+<\/sup> bound to ADP (\u00b5M)<\/td><\/tr><tr><td>20<\/td><td>11.6<\/td><\/tr><tr><td>50<\/td><td>26.0<\/td><\/tr><tr><td>100<\/td><td>42.7<\/td><\/tr><tr><td>150<\/td><td>52.8<\/td><\/tr><tr><td>200<\/td><td>59.0<\/td><\/tr><tr><td>400<\/td><td>69.5<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">First, from the equation \\([RL] = \\frac{[R_{\\rm T}][L]}{(K_{\\rm d} + [L]}\\), we know that when \\([L] = K_{\\rm d}\\), \\([RL] = \\frac{[R_{\\rm T}]}{2} = 40\\ \\mbox{\u00b5M}\\). So we can estimate that the \\(K_{\\rm d}\\) is roughly around this value.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">We analyze the data using R:\u3002<\/p>\n\n\n\n<pre class=\"prettyprint\">#  R Software\n&gt; TotalMg = c(20, 50, 100, 150, 200, 400) # Total Mg2+\n&gt; B = c(11.6, 26.0, 42.7, 52.8, 59.0, 69.5) # Bound Mg&lt;sup&gt;2+&lt;&#47;sup&gt;, i.e., &#091;RL&#093;\n&gt; F = TotalMg - B # Free Mg\u00b2\u207a, i.e., &#091;L&#093;\n# Nonlinear regression\n&gt; result_nonlinear = nls(B ~ Rt*F&#47;(Kd + F), start=c(Rt=10, Kd=10))\n&gt; summary(result_nonlinear)\n<\/pre>\n<b>Output:<\/b>\n<pre class=\"prettyprint\">\nFormula: B ~ Rt * F&#47;(Kd + F)\n\nParameters:\n   Estimate Std. Error t value Pr(&gt;|t|)    \nRt 79.92642    0.08296   963.5 6.96e-12 ***\nKd 49.86861    0.15975   312.2 6.32e-10 ***\n---\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\n\nResidual standard error: 0.05816 on 4 degrees of freedom\n\nNumber of iterations to convergence: 5 \nAchieved convergence tolerance: 1.579e-06\n\n&gt; confint(result_nonlinear)\nWaiting for profiling to be done...\n       2.5%    97.5%\nRt 79.69637 80.15752\nKd 49.42630 50.31459\n<\/pre>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" class=\"wp-image-3376\" style=\"width: 300px;\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation.png\" alt=\"\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation.png 4200w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation-300x300.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation-1024x1024.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation-150x150.png 150w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation-768x768.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation-1536x1536.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/nonlinearsaturation-2048x2048.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From the nonlinear regression, we obtained:\n<ul>\n<li>Dissociation constant \\(K_{\\rm d}\\) = 49.87 \u00b5M<\/li>\n<li>\\([R_{\\rm T}]\\) = 79.93 \u00b5M<\/li>\n<\/ul>\nWhen we fix \\([R_{\\rm T}]\\) to 80 and rerun the regression, we get \\(K_{\\rm d}\\) = 49.99 \u00b5M.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Now, using the Scatchard plot to determine \\(K_{\\rm d}\\) and \\([R_{\\rm T}]\\):<\/p>\n\n\n\n<pre class=\"wp-block-preformatted prettyprint\"># R Software\n&gt; result_linear = lm(B\/F ~ B)\n&gt; summary(result_linear)\n<\/pre>\n<b>Output:<\/b>\n<pre class=\"prettyprint\">Call:\nlm(formula = B&#47;F ~ B)\nResiduals:\n         1          2          3          4          5          6 \n 0.0038911 -0.0026571 -0.0032285 -0.0010654 -0.0005134  0.0035734 \nCoefficients:\n              Estimate Std. Error t value Pr(&gt;|t|)    \n(Intercept)  1.6115350  0.0033919   475.1 1.18e-10 ***\nB           -0.0202132  0.0000709  -285.1 9.08e-10 ***\n---\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\nResidual standard error: 0.00342 on 4 degrees of freedom\nMultiple R-squared:      1,\tAdjusted R-squared:  0.9999 \nF-statistic: 8.128e+04 on 1 and 4 DF,  p-value: 9.081e-10\n<\/pre>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" class=\"wp-image-3377\" style=\"width: 300px;\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard.png\" alt=\"\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard.png 4200w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-300x300.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-1024x1024.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-150x150.png 150w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-768x768.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-1536x1536.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-2048x2048.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From the regression line:\n<ul>\n<li>Slope = \\(-1\/K_{\\rm d}\\) = -0.0202132<\/li>\n<li><i>x<\/i>-intercept (<i>i.e.<\/i>, \\([R_{\\rm T}]\\)) = &#8211;<i>y<\/i>-intercept \/ slope = 1.6115350 \/ 0.0202132<\/li>\n<\/ul>\nThis gives:\n<ul>\n<li>\\(K_{\\rm d}\\) = 49.47 \u00b5M<\/li>\n<li>\\([R_{\\rm T}]\\) = 79.72 \u00b5M<\/li>\n<\/ul>\n<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This exercise used \\(K_{\\rm d}\\) = 50 \u00b5M to generate the data points. However, rounding of the bound Mg<sup>2+<\/sup> values introduces small errors, which result in slight differences in the \\(K_{\\rm d}\\) and \\([R_{\\rm T}]\\) values obtained via nonlinear regression and the Scatchard method.\nWhen you calculate using values rounded to the second decimal place, you get results very close to the theoretical ones:\n<ul>\n<li>Nonlinear regression: \\(K_{\\rm d}\\) = 49.99 \u00b5M<\/li>\n<li>Scatchard method: \\(K_{\\rm d}\\) = 49.97 \u00b5M<\/li>\n<\/ul>\n<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Actual Data<\/h2>\n\n\n\n<p class=\"wp-block-paragraph\">In the previous practice problem using idealized values, there was almost no difference between the results obtained via nonlinear regression and the Scatchard plot. Now, I tried applying both methods to <b>real experimental data<\/b> to see how they compare.<\/p>\n\n\n\n<pre class=\"prettyprint\">&gt; B = c(0.729,0.805,0.748,1.328,1.317,1.314,1.905,1.936,1.851,2.546,2.414,2.355) # Units: nM\n&gt; F = c(6.102,5.532,5.490,10.448,11.710,11.092,22.863,24.570,24.401,48.750,46.621,46.682) # Units: nM\n&gt; result = nls(B~Rt*F&#47;(Kd+F),start=c(Rt=3,Kd=0.2))\n&gt; summary(result)\n<\/pre>\n<b>Output:<\/b>\n<pre class=\"prettyprint\">Formula: B ~ Rt * F&#47;(Kd + F)\n\nParameters:\n   Estimate Std. Error t value Pr(&gt;|t|)    \nRt   3.3825     0.1127   30.01 3.95e-11 ***\nKd  18.3799     1.4360   12.80 1.59e-07 ***\n---\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\n\nResidual standard error: 0.06848 on 10 degrees of freedom\n\nNumber of iterations to convergence: 6 \nAchieved convergence tolerance: 1.785e-07\n\n&gt; confint(result) # 95% confidence interval\nWaiting for profiling to be done...\n        2.5%     97.5%\nRt  3.148358  3.652665\nKd 15.470323 21.906326<\/pre>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" class=\"wp-image-3406\" style=\"width: 300px;\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve.png\" alt=\"\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve.png 2100w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve-300x300.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve-1024x1024.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve-150x150.png 150w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve-768x768.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve-1536x1536.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_curve-2048x2048.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">From the nonlinear regression, we obtained:\n<ul>\n<li>Dissociation constant \\(K_{\\rm d}\\) = 18.4 nM<\/li>\n<li>\\([R_{\\rm T}]\\)  = 3.38 nM<\/li>\n<\/ul><\/p>\n\n\n\nNext, using the Scatchard plot:\n<pre class=\"prettyprint\">&gt; result_lm = lm(B&#47;F ~ B)\n&gt; summary(result_lm)\n<\/pre>\n<b>Output:<\/b>\n<pre class=\"prettyprint\">Call:\nlm(formula = B&#47;F ~ B)\n\nResiduals:\n       Min         1Q     Median         3Q        Max \n-0.0207551 -0.0043123  0.0007946  0.0048720  0.0171733 \n\nCoefficients:\n             Estimate Std. Error t value Pr(&gt;|t|)    \n(Intercept)  0.177090   0.008028   22.06 8.21e-10 ***\nB           -0.050571   0.004659  -10.85 7.46e-07 ***\n---\nSignif. codes:  0 \u2018***\u2019 0.001 \u2018**\u2019 0.01 \u2018*\u2019 0.05 \u2018.\u2019 0.1 \u2018 \u2019 1\n\nResidual standard error: 0.01015 on 10 degrees of freedom\nMultiple R-squared:  0.9218,\tAdjusted R-squared:  0.9139 \nF-statistic: 117.8 on 1 and 10 DF,  p-value: 7.464e-07\n&gt; confint(result_lm)\n                  2.5 %     97.5 %\n(Intercept)  0.15920357  0.1949766\nB           -0.06095189 -0.0401892\n&gt; -1&#47;result_lm$coefficients&#091;2&#093;\n       B \n19.77436 \n&gt; -result_lm$coefficients&#091;1&#093;&#47;result_lm$coefficients&#091;2&#093;\n(Intercept) \n   3.501842 \n<\/pre>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" class=\"wp-image-3407\" style=\"width: 300px;\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard.png\" alt=\"\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard.png 2100w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard-300x300.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard-1024x1024.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard-150x150.png 150w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard-768x768.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard-1536x1536.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/expdata_scatchard-2048x2048.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The two lowest-concentration <b>B\/F values<\/b> overlap in range. (Because the inverse of F is used, the error becomes significantly amplified.)\nFrom the Scatchard plot, we obtained:\n<ul>\n<li>Dissociation constant \\(K_{\\rm d}\\) = 19.8 nM<\/li>\n<li>\\([R_{\\rm T}]\\) = 3.50 nM<\/li>\n<\/ul>\nCompared to the values from nonlinear regression, the Scatchard-derived \\(K_{\\rm d}\\) is approximately <b>8% higher<\/b>, and \\([R_{\\rm T}]\\) is approximately <b>4% higher<\/b>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Comparing the <b>95% confidence intervals<\/b> for \\(K_{\\rm d}\\):\n<ul>\n<li><b>Nonlinear regression<\/b>: \\(K_{\\rm d}\\) = 18.4 nM (95% CI: 15.5\u201321.9)<\/li>\n<li><b>Scatchard plot<\/b>: \\(K_{\\rm d}\\) = 19.8 nM (95% CI: 16.4\u201324.9)<\/li>\n<\/ul>\n<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">References<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Braun, Derek C.; Garfield, Susan H.; Blumberg, Peter M. &#8220;Analysis by Fluorescence Resonance Energy Transfer of the Interaction between Ligands and Protein Kinase C\u03b4 in the Intact Cell&#8221;. <em style=\"font-size: revert;\">J. Biol. Chem<\/em><span style=\"font-size: revert;\">. <\/span><strong style=\"font-size: revert;\">2005<\/strong><span style=\"font-size: revert;\">, <\/span><em style=\"font-size: revert;\">280<\/em><span style=\"font-size: revert;\">(9), 8164-8171. doi: <\/span><a style=\"font-size: revert;\" href=\"https:\/\/doi.org\/10.1074\/jbc.M413896200\">10.1074\/jbc.M413896200<\/a> <\/li>\n\n\n\n<li>Scatchard, George &#8220;The Attraction of Proteins for Small Molecules and Ions&#8221;. <i>Annals of the New York Academy of Sciences <\/i><strong>1949<\/strong>, <em>51<\/em>(4), 660\u2013672. doi: <a class=\"external text\" href=\"https:\/\/doi.org\/10.1111%2Fj.1749-6632.1949.tb27297.x\" rel=\"nofollow\">10.1111\/j.1749-6632.1949.tb27297.x<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>It\u2019s long been known that appl 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Constants","link":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/en\/2025\/05\/16\/%e8%a7%a3%e9%9b%a2%e5%ae%9a%e6%95%b0%e3%82%92%e6%b1%82%e3%82%81%e3%82%8b%e3%81%ae%e3%81%abscatchard-plot%e3%82%92%e4%bd%bf%e3%81%86%e3%81%b9%e3%81%8d%e3%81%a7%e3%81%af%e3%81%aa%e3%81%84%e3%81%a8\/"}],"jetpack_featured_media_url":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard.png","uagb_featured_image_src":{"full":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard.png",4200,4200,false],"thumbnail":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-150x150.png",150,150,true],"medium":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-300x300.png",300,300,true],"medium_large":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-768x768.png",768,768,true],"large":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-1024x1024.png",800,800,true],"1536x1536":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-1536x1536.png",1536,1536,true],"2048x2048":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-2048x2048.png",2048,2048,true],"onepress-blog-small":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-300x150.png",300,150,true],"onepress-small":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-480x300.png",480,300,true],"onepress-medium":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2021\/10\/scatchard-640x400.png",640,400,true]},"uagb_author_info":{"display_name":"RCY","author_link":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/author\/charlesy\/"},"uagb_comment_info":0,"uagb_excerpt":"It\u2019s long been known that appl 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