{"id":4297,"date":"2025-05-12T17:01:43","date_gmt":"2025-05-12T08:01:43","guid":{"rendered":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=4297"},"modified":"2026-01-29T09:11:12","modified_gmt":"2026-01-29T00:11:12","slug":"%e7%b5%90%e5%90%88%e9%98%bb%e5%ae%b3%e5%ae%9a%e6%95%b0ki%e3%81%ae%e8%a8%88%e7%ae%97%e5%bc%8f%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6","status":"publish","type":"post","link":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/en\/2025\/05\/12\/%e7%b5%90%e5%90%88%e9%98%bb%e5%ae%b3%e5%ae%9a%e6%95%b0ki%e3%81%ae%e8%a8%88%e7%ae%97%e5%bc%8f%e3%81%ab%e3%81%a4%e3%81%84%e3%81%a6\/","title":{"rendered":"On the Calculation Formula for the Binding Inhibition Constant Ki"},"content":{"rendered":"\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"508\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-1024x508.png\" alt=\"\" class=\"wp-image-6743\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-1024x508.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-300x150.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-768x381.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-1536x762.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-2048x1016.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>In the world of drug discovery, where scientists are the detectives and diseases are the elusive criminals, radioligand binding assays (RLBAs) were once the Sherlock Holmes of the lab.<\/p>\n\n\n\n<p>\u2014 <a href=\"https:\/\/www.oncodesign-services.com\/cpt_ressource\/radioligand-binding-assays-a-lost-art-in-drug-discovery\/\" title=\"\">Ancellin, Nicolas, \u201cRadioligand Binding Assays: A Lost Art in Drug Discovery?\u201d.<\/a><\/p>\n<\/blockquote>\n\n\n\n<p>Memo\uff0eTranslated on May 13, 2025\uff0e<\/p>\n\n\n\n<p>Several methods exist for calculating the inhibition constant (\\(K_{\\mathrm i}\\)). Besides the widely known <strong data-start=\"486\" data-end=\"512\">Cheng\u2013Prusoff equation<\/strong>, modified Cheng\u2013Prusoff equation using [<em>L<\/em>]<sub>50<\/sub> value instead of [<em>L<\/em>]<sub>T<\/sub>, and the exact <strong data-start=\"614\" data-end=\"641\">Munson\u2013Rodbard correction<\/strong>, which allows direct calculation using the total concentration of an unlabeled ligand\u2014namely, the 50% inhibitory concentration \\({\\rm IC}_{50}\\)\u2014without requiring approximations. Additionally, the 50% inhibitory concentration of the <em data-start=\"871\" data-end=\"877\">free<\/em> ligand, denoted \\([I]_{50}\\)<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"mclose\"><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist-s\">\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>, can be estimated either via <strong data-start=\"938\" data-end=\"962\">nonlinear regression<\/strong> or <strong data-start=\"966\" data-end=\"987\">linear regression<\/strong>.<\/p>\n\n\n\n<p>We investigates the deviation between the estimated \\(K_{\\rm i}\\) values obtained by these methods and the true \\(K_{\\rm i}\\) values using simulated data.<\/p>\n\n\n\n<p>To simulate realistic assay conditions, typical experimental values were used for the binding between the protein kinase C \u03b4-C1B domain (as receptor) and the radiolabeled ligand [\u00b3H]phorbol 12,13-dibutyrate ([\u00b3H]PDBu):<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Total receptor concentration: \\([R]_{\\rm T}\\) = 3.45\u2009nM<\/li>\n\n\n\n<li>Total radioligand concentration: \\([L]_{\\rm T}\\) = 17\u2009nM<\/li>\n\n\n\n<li>Radioligand dissociation constant: \\(K_{\\rm d}\\) = 0.53\u2009nM<\/li>\n<\/ul>\n\n\n\n<p>For various true values of \\(K_{\\rm i}\\)\u200b, the ranges of total unlabeled ligand concentration \\(\\log_{10} [I]_{\\rm T}\\) used in the simulations were:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\\(K_{\\rm i}\\) = 0.05\u2009nM: from \u22129.021 to \u22127.457<\/li>\n\n\n\n<li>\\(K_{\\rm i}\\) = 0.1\u2009nM: from \u22129.172 to \u22127.459<\/li>\n\n\n\n<li>\\(K_{\\rm i}\\) = 0.5\u2009nM: from \u22128.908 to \u22126.987<\/li>\n\n\n\n<li>\\(K_{\\rm i}\\) = 1\u2009nM: from \u22128.454 to \u22126.496<\/li>\n\n\n\n<li>\\(K_{\\rm i}\\) = 10\u2009nM: from \u22127.495 to \u22125.500<\/li>\n\n\n\n<li>\\(K_{\\rm i}\\) = 100\u2009nM: from \u22126.500 to \u22124.500<\/li>\n<\/ul>\n\n\n\n<p>(Five data points spaced approximately 0.5 log units apart, centered around the expected 50% inhibition point.)<\/p>\n\n\n\n<p>There was no significant difference between the results of nonlinear regression (fitting to a standard sigmoid function) and linear regression (after logit transformation) in estimating \\({\\rm IC}_{50}\\)\u200b:<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td><strong>True <em>K<\/em><sub>i<\/sub><\/strong><\/td><td><strong>[<em>I<\/em>]<sub>50<\/sub><\/strong> (nM)<\/td><td><strong>True IC<sub>50<\/sub><\/strong> (nM)<\/td><td><strong>IC<sub>50<\/sub><\/strong> ( from nonlinear regression)<\/td><td><strong>IC<sub>50<\/sub><\/strong> (from linear regression)<\/td><\/tr><tr><td>0.05 nM<\/td><td>1.51<\/td><td>3.24<\/td><td>3.18<\/td><td>3.17<\/td><\/tr><tr><td>0.1 nM<\/td><td>3.02<\/td><td>4.75<\/td><td>4.71<\/td><td>4.68<\/td><\/tr><tr><td>0.5 nM<\/td><td>15.1<\/td><td>16.8<\/td><td>16.8<\/td><td>16.8<\/td><\/tr><tr><td>1 nM<\/td><td>30.2<\/td><td>31.9<\/td><td>31.9<\/td><td>31.9<\/td><\/tr><tr><td>10 nM<\/td><td>302<\/td><td>304<\/td><td>303<\/td><td>303<\/td><\/tr><tr><td>100 nM<\/td><td>3019<\/td><td>3020<\/td><td>3015<\/td><td>3014<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<figure class=\"wp-block-flexible-table-block-table is-style-stripes\"><table class=\"has-fixed-layout\"><tbody><tr><td rowspan=\"2\" style=\"border-color:#ffffff\"><strong>True <em>K<\/em><sub>i<\/sub><\/strong><\/td><td colspan=\"4\" style=\"border-color:#ffffff\"><strong><em>K<\/em><sub>i<\/sub> (nM) calculated from True IC<sub>50<\/sub><\/strong><\/td><\/tr><tr><td style=\"border-color:#ffffff\"><strong>Cheng&#8211;Prusoff<\/strong><\/td><td style=\"border-color:#ffffff\"><strong>Cheng&#8211;Prusoff<\/strong><sup><em>a<\/em><\/sup><\/td><td style=\"border-color:#ffffff\"><strong>Munson&#8211;Rodbard (exact) correction<\/strong><\/td><td style=\"border-color:#ffffff\"><strong>IC<sub>50<\/sub> &#8211; [<em>R<\/em>]<sub>T<\/sub>\/2<\/strong> <strong>correction<\/strong><sup><em>b<\/em><\/sup><\/td><\/tr><tr><td style=\"border-color:#ffffff\">0.05 nM<\/td><td style=\"border-color:#ffffff\">0.0980<\/td><td style=\"border-color:#ffffff\">0.108<\/td><td style=\"border-color:#ffffff\">0.500<\/td><td style=\"border-color:#ffffff\">0.502<\/td><\/tr><tr><td style=\"border-color:#ffffff\">0.1 nM<\/td><td style=\"border-color:#ffffff\">0.143<\/td><td style=\"border-color:#ffffff\">0.159<\/td><td style=\"border-color:#ffffff\">0.100<\/td><td style=\"border-color:#ffffff\">0.100<\/td><\/tr><tr><td style=\"border-color:#ffffff\">0.5 nM<\/td><td style=\"border-color:#ffffff\">0.508<\/td><td style=\"border-color:#ffffff\">0.561<\/td><td style=\"border-color:#ffffff\">0.499<\/td><td style=\"border-color:#ffffff\">0.499<\/td><\/tr><tr><td style=\"border-color:#ffffff\">1 nM<\/td><td style=\"border-color:#ffffff\">0.964<\/td><td style=\"border-color:#ffffff\">1.07<\/td><td style=\"border-color:#ffffff\">1.00<\/td><td style=\"border-color:#ffffff\">1.00<\/td><\/tr><tr><td style=\"border-color:#ffffff\">10 nM<\/td><td style=\"border-color:#ffffff\">9.19<\/td><td style=\"border-color:#ffffff\">10.2<\/td><td style=\"border-color:#ffffff\">10.0<\/td><td style=\"border-color:#ffffff\">10.0<\/td><\/tr><tr><td style=\"border-color:#ffffff\">100 nM<\/td><td style=\"border-color:#ffffff\">91.3<\/td><td style=\"border-color:#ffffff\">101<\/td><td style=\"border-color:#ffffff\">100<\/td><td style=\"border-color:#ffffff\">100<\/td><\/tr><\/tbody><tfoot><tr><td colspan=\"5\" style=\"border-color:#ffffff\"><sup><em>a<\/em><\/sup>&nbsp;&nbsp;Using [<em>L<\/em>]<sub>50<\/sub> instead of [<em>L<\/em>]<sub>T<\/sub>&nbsp;\u3002&nbsp;<br><em><sup>b<\/sup><\/em>&nbsp;\\(K_{\\mathrm i} = (\\mathrm{IC}_{50} &#8211; [R]_\\mathrm{T}\/2 )\/(2 [L]_{50}\/[L]_{0} &#8211; 1 + [L]_{50}\/K_\\mathrm{d}) \\).<\/td><\/tr><\/tfoot><\/table><\/figure>\n\n\n\n<p class=\"has-text-align-left has-text-align-center has-normal-font-size\">Conclusions<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><p data-start=\"4154\" data-end=\"4301\" class=\"\"><\/p><p data-start=\"4154\" data-end=\"4301\" class=\"\">The <strong data-start=\"4158\" data-end=\"4193\">original Cheng\u2013Prusoff equation<\/strong> shows unexpectedly large deviations even when the \\(K_{\\mathrm i}\\) value is high.<\/p><\/li>\n\n\n\n<li><p data-start=\"4154\" data-end=\"4301\" class=\"\"><strong>Modified Cheng\u2013Prusoff equation<\/strong> using \\([L]_{50}\\) agrees with the true \\(K_{\\rm i}\\) value to two significant digits when \\(K_{\\rm i}\\) \u2265 10 nM.<\/p><\/li>\n\n\n\n<li><p data-start=\"4602\" data-end=\"4864\" class=\"\">Nowadays, when spreadsheet software is available, there is no reason not to use the exact <strong>Munson\u2013Rodbard correction<\/strong>. At least in cases where \\(K_{\\mathrm i} &lt; 10\\) nM, where the difference between \\([I]_{50}\\) and \\(\\mathrm{IC}_{50}\\) widens, it is necessary to use the Munson\u2013Rodbard equation.<\/p><\/li>\n\n\n\n<li>In practice, the&nbsp;<em>K<\/em><sub>i<\/sub>\u200b value obtained using the <strong>approximation \\([I]_{50} \\simeq \\mathrm{IC}_{50} &#8211; [R]_{\\mathrm T}\/2 \\) <\/strong>is sufficiently accurate.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Binding Assay Calculations<\/h2>\n\n\n\n<p>The binding assay methodology is based on Sharkey &amp; Blumberg (1985), with modifications. Non-specific binding at equilibrium is neglected. Key equations include:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Specific binding<\/strong>: $$[RL]_0 =[\\mbox{pellet(total)}] &#8211; k \\times [\\mbox{supernatant(total)}]\\times \\frac{437}{50} $$<\/li>\n\n\n\n<li><strong>Partition Coefficient <em>k<\/em><\/strong>: $$\\mathbf{\\mathit{k}} =   \\frac{[\\mbox{pellet(nonspecific)}]}{[\\mbox{supernatant(nonspecific)}] \\times \\frac{437}{50}}$$<\/li>\n\n\n\n<li><strong>Total Ligand Concentration:<\/strong> $$\\mathbf{[\\mathit{L}]_{\\mathrm T}} = [\\mbox{pellet(total)}] + [\\mbox{supernatant(total)}] \\times \\frac{437}{50}$$<\/li>\n\n\n\n<li><strong>Corrected Total Ligand Concentration:<\/strong> $$\\mathbf{[\\mathit{L}]^{*}_{\\mathrm T}} =\\frac{[L]_{\\mathrm T}}{1 + k}$$<\/li>\n\n\n\n<li><strong>Free Ligand Concentration:<\/strong> $$\\mathbf{[\\mathit{L}]_0} = [\\mbox{supernatant(total)}] = \\frac{[L]_{\\mathrm T} \u2212 [RL]_0}{1 + k}$$<\/li>\n\n\n\n<li><strong>Ligand Concentration at 50% Binding:<\/strong> $$\\mathbf{[\\mathit{L}]_{50}} = \\frac{[L]_{\\mathrm{T}} \u2212 \\frac{[RL]_0}{2}}{1 + k}$$<\/li>\n\n\n\n<li><strong>Total Receptor Concentration:<\/strong> $$\\mathbf{[\\mathit{R}]_\\mathrm{T}} = \\frac{K_\\mathrm{d} [RL]_0}{[L]_0 + [RL]_0}$$<\/li>\n\n\n\n<li><strong>Conversion Factor <em>N<\/em> from DPM to nM:<\/strong>: $$N = \\frac{1}{60} \\times \\frac{1}{3.7 \\times 10^{10}} \\times \\frac{1}{\\mbox{specific activity (Ci\/mmol)}} \\times \\frac{1}{10^3} \\times \\frac{1}{250 \\times 10^{-6}} \\times 10^9$$<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Non-Specific Binding Considerations<\/h3>\n\n\n\n<p>In membrane preparations, non-specific binding can occur due to low-affinity interactions with membrane proteins, phospholipid partitioning, or filter adsorption. These are typically unsaturable and modeled as the product of the non-specific binding coefficient \\(k\\) and the free radioligand concentration \\([L]\\) (Hulme and Trevethick, 2010).<\/p>\n\n\n\n<p>In [<sup>3<\/sup>H]PDBu binding assays, polyethylene glycol (PEG) is added to precipitate proteins, with \u03b3-globulin included as a co-precipitant, leading to non-specific adsorption (typically \\(k = 3&#8211;5\\%\\)). Calculations show that at \\(k = 0\\), \\([RL]_0 = 3.321\\ \\mbox{nM}\\) and \\([L]_0 = 13.7\\ \\mbox{nM}\\); at \\(k = 0.04\\), \\([RL]_0 = 3.316\\ \\mbox{nM}\\) and \\([L]_0 = 13.2\\ \\mbox{nM}\\). High ligand concentrations minimize the impact on bound \\([RL]\\), allowing correction of free \\([L]\\) using \\(k\\).<\/p>\n\n\n\n<p>For unlabeled ligands, if the partition coefficient is assumed equal to that of [<sup>3<\/sup>H]PDBu, adjusting the added concentration by dividing by \\(1 + k\\) provides a good estimate of \\([I_\\mathrm{T}]\\). For example, with a true \\(K_\\mathrm{i} = 10\\ \\mbox{nM}\\) and \\(k = 0.04\\), the apparent \\(K_\\mathrm{i}\\) calculated using the Munson\u2013Rodbard equation is \\(10.47\\ \\mbox{nM}\\) without correction and \\(10.06\\ \\mbox{nM}\\) with correction.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span style=\"font-weight: bold;\">Cheng\u2013Prusoff equation<\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Parameters Used<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">Total Binding Group:<\/h4>\n\n\n\n<p>$$[R]_0 + [L]_0 \\rightleftharpoons [RL]_]$$ $$K_{\\rm d} = \\frac{[R]_0[L]_0}{[RL]_0}$$ $$[R]_{\\rm T} = [R]_0 + [RL]_0$$ $$K_{\\rm d} = \\frac{([R]_{\\rm T} &#8211; [RL]_0)[L]_0}{[RL]_0}$$ $$[RL]_0 = \\frac{[R]_{\\rm T}[L]_0}{K_{\\rm d} + [L]_0}$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Competitive Inhibition Group:<\/h4>\n\n\n\n<p>\\([I]\\): Concentration of free, non-labeled ligand<\/p>\n\n\n\n<p>$$[RI] \\rightleftharpoons [I] + [R] + [L] \\rightleftharpoons [RL]$$ $$K_{\\rm i} = \\frac{[R][I]}{[RI]}$$ $$[R]_{\\rm T} = [RI] + [R] + [RL]$$ $$[R]_{\\rm T}= \\frac{[R][I]}{K_{\\rm i}} + [R] + [RL]$$ $$[R]_{\\rm T}= \\left(1+\\frac{[I]}{K_{\\rm i}}\\right)[R] + [RL]$$ $$[R]_{\\rm T} = \\left(1+\\frac{[I]}{K_{\\rm i}}\\right)\\frac{K_{\\rm d}[RL]}{[L]}+ [RL]$$ $$[RL] = \\frac{[R]_{\\rm T}[L]}{\\left(1+\\frac{[I]}{K_{\\rm i}}\\right)K_{\\rm d} + [L]}$$<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">At 50% Inhibition of Binding<\/h4>\n\n\n\n<p>Let us consider the situation at 50% inhibition of binding: $$\\frac{1}{2}[RL]_0 = [RL]_{50}$$ $$\\frac{1}{2}\\frac{[R]_{\\rm T}[L]_0}{K_{\\rm d} + [L]_0} = \\frac{[R]_{\\rm T}[L]_{50}}{\\left(1+\\frac{[I]_{50}}{K_{\\rm i}}\\right)K_{\\rm d} + [L]_{50}}$$ Multiply both sides and rearrange: $$2\\frac{K_{\\rm d} }{[L]_0} + 2 = \\left(1+\\frac{[I]_{50}}{K_{\\rm i}}\\right)\\frac{K_{\\rm d} }{[L]_{50}}+ 1$$ $$2\\frac{[L]_{50}}{[L]_0} + \\frac{[L]_{50}}{K_{\\rm d}} = 1+\\frac{[I]_{50}}{K_{\\rm i}}$$ Solving for \\(K_{\\rm i}\\): $$K_{\\rm i} = \\frac{[I]_{50}}{2\\frac{[L]_{50}}{[L]_0} &#8211; 1 + \\frac{[L]_{50}}{K_{\\rm d}}} \\;\\;\\;\\mbox{(Goldstein and Barrett, 1987)}$$ $$= \\frac{[I]_{50}}{1 + \\frac{[L]_{50}}{K_{\\rm d}} + 2\\frac{[L]_{50}-[L]_0}{[L]_0}}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conversion to Cheng\u2013Prusoff Equation<\/h3>\n\n\n\n<p>If the ligand concentration \\([L]\\) is sufficiently large compared to receptor concentration \\([R]\\), then \\([L]_0 \\simeq [L]_{50} \\simeq [L]^{*}_{\\rm T}\\), so $$K_{\\rm i} = \\frac{[I]_{50}}{1 + \\frac{[L]^{*}_{\\rm T}}{K_{\\rm d}}}\\;\\; .$$<\/p>\n\n\n\n<p>If the inhibitor concentration \\([I]\\) is sufficiently large compared to the receptor concentration \\([R]\\), \\([I]_{50}\\) can be approximated as the total ligand concentration at 50% inhibition, <em>i.e.<\/em>, \\(\\rm{IC}_{50}\\):<br>$$K_{\\rm i} = \\frac{\\textrm{IC}_{50}}{1+\\frac{[L]^{*}_{\\rm T}}{K_{\\rm d}}} \\;\\;\\;\\mbox{(Cheng&#8211;Prusoff equation)}\\;\\; .$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Drawing Binding Inhibition Curve<\/h2>\n\n\n\n<p>The value of \\([RL]_0\\)\u200b can be obtained by solving a quadratic equation<\/p>\n\n\n\n<p>$$ [RL]_0 = \\frac{1}{2}\\left[([R]_{\\rm T} + [L]_{\\rm T} + K_{\\rm d}) &#8211; \\sqrt{([R]_{\\rm T} + [L]_{\\rm T} + K_{\\rm d})^2 &#8211; 4[R]_{\\rm T}[L]_{\\rm T}}\\right] \\; .$$<\/p>\n\n\n\n<p>When \\([R]_\\mathrm{T}\u200b=3.45\\), \\([L]_\\mathrm{T}\u200b=17\\), and \\(K_\\mathrm{d}\u200b=0.53\\), \\([RL]_0\u200b=3.321\\).<\/p>\n\n\n\n<p>Similarly, in the presence of an unlabeled ligand, \\([RL]\\) can be expressed as the following function of the free unlabeled ligand concentration \\([I]\\):<\/p>\n\n\n\n<p>$$ [RL] = \\frac{1}{2}\\left\\{\\left[[R]_{\\rm T} + [L]_{\\rm T} + \\left(1+\\frac{[I]}{K_{\\rm i}}\\right)K_{\\rm d}\\right] &#8211; \\sqrt{\\left[[R]_{\\rm T} + [L]_{\\rm T} + \\left(1+\\frac{[I]}{K_{\\rm i}}\\right)K_{\\rm d}\\right]^2 &#8211; 4[R]_{\\rm T}[L]_{\\rm T}}\\right\\} $$<\/p>\n\n\n\n<p>When the fractional radioligand binding<\/p>\n\n\n\n<p> $$\\theta = [RL]\/[RL]_0$$<\/p>\n\n\n\n<p>is plotted against the free unlabeled ligand concentration \\(\\log_{10}\u200b[I]\\), the following graph can be obtained.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"232\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2019\/03\/inhibitioncurvelabel-300x232.png\" alt=\"\" class=\"wp-image-1620\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2019\/03\/inhibitioncurvelabel-300x232.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2019\/03\/inhibitioncurvelabel-768x593.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2019\/03\/inhibitioncurvelabel-1024x791.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2019\/03\/inhibitioncurvelabel-350x270.png 350w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2019\/03\/inhibitioncurvelabel.png 1340w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption class=\"wp-element-caption\"><b>Figure 1.<\/b> Simulation of the inhibition curve. Conditions: \\([R]_{\\textrm T} = 3.45\\; \\textrm{nM}\\), \\([L]_{\\textrm T} = 17\\; \\textrm{nM}\\), \\(K_{\\textrm d} = 0.53\\; \\textrm{nM}\\), \\(K_{\\textrm i} = 11\\; \\textrm{nM}\\). Vertical axis: \\(\\theta = [RL]\/[RL]_0\\). Horizontal axis: logarithmic concentration of the free, unlabeled ligand, \\(\\log_{10} [I]\\).<\/figcaption><\/figure>\n<\/div>\n\n\n<p>Then, the horizontal axis in Figure 1 is changed to \\(\\log_{10} [I]_{\\rm T}\\), the logarithmic total concentration of unlabeled ligand. Note that $$ [I]_{\\mathrm T} = [I] + [RI] $$<\/p>\n\n\n\n<p>and<\/p>\n\n\n\n<p>$$ [RI] = [R]_{\\mathrm{T}} &#8211; \\frac{[L]_0}{[L]_{\\mathrm T} &#8211; [RL]}\\frac{[RL]}{[RL]_0}[R]_{\\mathrm{T}} + \\left( \\frac{[L]_0}{[L]_{\\mathrm T} &#8211; [RL]} &#8211; 1\\right) [RL] \\;\\; .$$<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"237\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/7278bb4d995a5cc1236d2fed58a07f8e-300x237.png\" alt=\"\" class=\"wp-image-3043\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/7278bb4d995a5cc1236d2fed58a07f8e-300x237.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/7278bb4d995a5cc1236d2fed58a07f8e-768x606.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/7278bb4d995a5cc1236d2fed58a07f8e.png 936w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption class=\"wp-element-caption\"><b>Figure 2.<\/b> Simulation of the inhibition curve with horizontal axis as \\(\\log_{10} [I]_{\\textrm T}\\).<\/figcaption><\/figure>\n<\/div>\n\n\n<p>data.csv<\/p>\n\n\n\n<pre class=\"prettyprint\">logIT,theta\n-9.495,0.999\n-8.995,0.997\n-8.495,0.990\n-7.995,0.968\n-7.496,0.906\n-6.996,0.759\n-6.498,0.512\n-5.999,0.259\n-5.500,0.102\n-5.000,0.035\n-4.500,0.011\n-4.000,0.004<\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">Estimating \\({\\rm IC}_{50}\\)by Regression<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Nonlinear Regression<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">1) Fitting to the Complementary Error Function (erfc)<\/h4>\n\n\n\n<p>Fitting to the complementary error function (erfc)<\/p>\n\n\n\n<p>$$\\operatorname{erfc}(x)=\\frac{2}{\\sqrt{\\pi}}\u200b \\int\u200b_{x}^{\\infty} \u200be^{-t^2}dt$$<\/p>\n\n\n\n<p>The probit function is the inverse function of the cumulative distribution function \\(\\Phi\\) of the normal distribution. The complementary error function, erfc, and \\(\\Phi\\) are related by the following expression:<\/p>\n\n\n\n<p>$$\\Phi(x)=\\frac{1}{2}\\operatorname{\u200berfc} \\left(-\\frac{x}{\\sqrt{2}}\\right) \\; .$$<\/p>\n\n\n\n<pre class=\"prettyprint\">&gt; simulation &lt;- read.csv(&#34;data.csv&#34;)\n&gt; erfc &lt;- function(x) 2 * pnorm(x * sqrt(2), lower=FALSE) \n&gt; result2 = nls(theta~1&#47;2*erfc((logIT - logIC50)&#47;a), data = simulation, start = c(logIC50=-6.5, a=1))\n&gt; summary(result2)\n<\/pre>\n<p><b>Formula<\/b>: $$\\theta = \\frac{1}{2}\\cdot\\operatorname{erfc}\\left(\\frac{\\log_{10}[I]_{\\rm T} - \\log_{10}{\\rm IC}_{50}}{a}\\right)$$<\/p>\n\n<p><i>(Note: The complementary error function is multiplied by 1\/2 to constrain the range between 0 and 1)<\/i><\/p>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\nFormula: theta ~ 1&#47;2 * erfc((logIT - logIC50)&#47;a)\n\nParameters:\n         Estimate Std. Error t value Pr(&gt;|t|)    \nlogIC50 -6.476263   0.006851 -945.27  &lt; 2e-16 ***\na        1.082295   0.013710   78.94  2.6e-15 ***\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: 0.005889 on 10 degrees of freedom\n\nNumber of iterations to convergence: 5 \nAchieved convergence tolerance: 6.534e-07\n<\/pre>\n\n\n\n<p><b>Visualizing Fit:<\/b><\/p>\n\n\n\n<pre class=\"prettyprint\">&gt; concpre = seq(-10,-3.5, length=100)\n&gt; plot(theta~logIT,data=simulation,xlim=c(-10,-3.5),ylim=c(0,1))\n&gt; lines(concpre, predict(result2, newdata=list(logIT=concpre)))\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-300x300.png\" alt=\"\" class=\"wp-image-3045\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-300x300.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-1024x1024.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-150x150.png 150w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-768x768.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-1536x1536.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_erfc-2048x2048.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<p>The fit is not very good.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2) Fitting to a decreasing logistic function \\(1\/(1 + \\exp(ax))\\):<\/h4>\n\n\n\n<p>The standard sigmoid function is<\/p>\n\n\n\n<p>$$f(x)=\\frac{1}{1+e^{-ax}}\\; \u200b.$$<\/p>\n\n\n\n<p>The decreasing logistic function is defined as 1 minus the standard sigmoid function:<\/p>\n\n\n\n<p>$$f(x)=1-\\frac{1}{1+e^{-ax}}\u200b=\\frac{1}{1+e^{ax}}\\; \u200b.$$<\/p>\n\n\n\n<pre class=\"prettyprint\">&gt; result3 = nls(theta~1&#47;(1+exp(a*log(10)*(logIT - logIC50))), data = simulation, start = c(logIC50=-6.5, a=1))\n&gt; summary(result3)\n<\/pre>\n<p><b>Formula<\/b>: $$\\theta = \\frac{1}{1 + e^{a \\log_{e} 10 (\\log_{10}[I]_{\\rm T} - \\log_{10}{\\rm IC}_{50})}}$$<\/p>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">Formula: theta ~ 1&#47;(1 + exp(a * log(10) * (logIT - logIC50)))\n\nParameters:\n         Estimate Std. Error t value Pr(&gt;|t|)    \nlogIC50 -6.476238   0.001243 -5209.7   &lt;2e-16 ***\na        0.964276   0.002346   411.1   &lt;2e-16 ***\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: 0.001071 on 10 degrees of freedom\n\nNumber of iterations to convergence: 4 \nAchieved convergence tolerance: 8.128e-08\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"300\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-300x300.png\" alt=\"\" class=\"wp-image-3046\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-300x300.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-1024x1024.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-150x150.png 150w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-768x768.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-1536x1536.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Rplot_sigmoid-2048x2048.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure>\n<\/div>\n\n\n<p>The simulation data appear to fit the decreasing logistic function perfectly. The estimated values were \\(\\log_{10\u200b}\\mathrm{IC}_{50}\u200b= -6.476238\\) and \\(a=0.964276\\). This curve is referred to as an <em>inhibition curve<\/em> in the fields of biology and pharmacology. The parameter \\(a\\) corresponds to the ligand Hill coefficient \\(n_{H}\\)\u200b, which describes the slope of the sigmoidal curve. The transformation is shown below.<\/p>\n\n\n\n<p>First, as shown two sections below, \\(\\theta\\) can be written as<\/p>\n\n\n\n<p>$$\\theta = \\frac{\\frac{[L]}{[L]_0} (K_{\\rm d} + [L]_0)}{\\frac{K_{\\rm d}}{K_{\\rm i}}[I] + K_{\\rm d} + [L]} \\; .$$<\/p>\n\n\n\n<p>If \\([L] \\simeq [L]_0\\) and \\([I] \\simeq [I]_{\\mathrm T}\\),<\/p>\n\n\n\n<p>$$\\theta = \\frac{1}{1 + \\frac{1}{K_{\\rm d} + [L]_0}\\frac{K_{\\rm d}}{K_{\\rm i}} [I]_{\\mathrm T}} \\; .$$<\/p>\n\n\n\n<p>This can be rewritten as<\/p>\n\n\n\n<p>$$\\theta = \\frac{1}{1 + \\exp\\left\\{\\log_{e} 10 \\left[\\log_{10} [I]_{\\mathrm T} - \\log_{10} K_{\\mathrm i}\\left( 1+ \\frac{[L]_0}{K_\\mathrm{d}} \\right) \\right]\\right\\}} \\; .$$<\/p>\n\n\n\n<p>From the Cheng\u2013Prusoff equation,<\/p>\n\n\n\n<p>$$ K_{\\mathrm i}\\left( 1+ \\frac{[L]_0}{K_\\mathrm{d}} \\right) \\simeq \\mathrm{IC}_{50}\\; ,$$<\/p>\n\n\n\n<p>and therefore \\(\\theta\\) can be approximated as<\/p>\n\n\n\n<p>$$\\theta = \\frac{1}{1 + \\exp\\left[\\log_{e} 10 \\left(\\log_{10} [I]_{\\mathrm T} - \\log_{10} \\mathrm{IC}_{50} \\right)\\right]} \\; .$$<\/p>\n\n\n\n<p>In practice, nearly identical IC<sub>50<\/sub> values are obtained even when the complementary error function is used; however, this formulation provides a superior model.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Linear Regression<\/h3>\n\n\n\n<p>Next, to visually evaluate the goodness of fit of the fitted curve, a transformation to a linear graph (<em>e.g.<\/em>, for use in Microsoft Excel) is shown.<\/p>\n\n\n\n<p>From the following expression: $$\\frac{\\theta}{1-\\theta} = \\frac{\\frac{[L]}{[L]_0}(K_{\\rm d} + [L]_0)}{\\frac{K_{\\rm d}}{K_{\\rm i}}[I] + K_{\\rm d} + [L] - \\frac{[L]}{[L]_0}(K_{\\rm d} + [L]_0)} =  \\frac{\\frac{[L]}{[L]_0} K_{\\rm d} + [L]}{\\frac{K_{\\rm d}}{K_{\\rm i}}[I] + \\left(1 - \\frac{[L]}{[L]_0}\\right) K_{\\rm d}} $$<\/p>\n\n\n\n<p>Assuming \\([L]_0 \\simeq [L]\\), it simplifies to: $$\\frac{\\theta}{1-\\theta} = \\frac{K_{\\rm d} + [L]}{\\frac{K_{\\rm d}}{K_{\\rm i}}[I]}$$<\/p>\n\n\n\n<p>Taking the logarithm (base 10): $$\\log \\left(\\frac{\\theta}{1-\\theta}\\right) = -\\log [I] -\\log K_{\\rm d} + \\log K_{\\rm i} + \\log (K_{\\rm d} + [L])$$ This results in a linear relationship with a slope of \\(-1\\) when plotted against \\(\\log[I]\\) (free ligand concentration) (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Logit\">logit function<\/a>) (<b>Figure 3<\/b>).<\/p>\n\n\n<div class=\"wp-block-image wp-block-image aligncenter\">\n<figure class=\"size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"959\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logI_linear-1024x959.png\" alt=\"\" class=\"wp-image-6658\" style=\"aspect-ratio:1.067789540588317;width:343px;height:auto\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logI_linear-1024x959.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logI_linear-300x281.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logI_linear-768x719.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logI_linear-1536x1439.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logI_linear-2048x1918.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><b>Figrue 3.<\/b> Fitting using the logistic function (\\(\\log_{10}(\\frac{p}{1-p})\\). Vertical axis: (\\(\\log_{10}(\\frac{\\theta}{1-\\theta})\\) Horizontal axis: \\([I]\\), the free concentration of unlabeled ligand. The smaller the total receptor concentration \\([R]_{\\rm T}\\), the closer the plot approximates a straight line.<\/figcaption><\/figure>\n<\/div>\n\n\n<p>Although the horizontal axis in the figure above is \\(\\log[I]\\) (free unlabeled ligand), this value is not experimentally accessible. Instead, we use the added total ligand concentration \\(\\log[I]_{\\rm T}\\), and obtain:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"959\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logIT_linear-1024x959.png\" alt=\"\" class=\"wp-image-6659\" style=\"aspect-ratio:1.067789540588317;width:325px;height:auto\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logIT_linear-1024x959.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logIT_linear-300x281.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logIT_linear-768x719.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logIT_linear-1536x1439.png 1536w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/logIT_linear-2048x1918.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>\\(\\log{\\rm IC}_{50} = -6.4765\\) from the linear plot<\/li>\n\n\n\n<li>\\(\\log{\\rm IC}_{50} = -6.4762\\) from nonlinear regression using the standard sigmoid function.<\/li>\n<\/ul>\n\n\n\n<p>These values match up to four significant digits. In nM units:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>333.8 nM (from linear regression),<\/li>\n\n\n\n<li>334.0 nM (from nonlinear regression).<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Simple Conversion from Binding Ratio to \\(K_{\\rm i}\\)<\/h3>\n\n\n\n<p>For weak ligands that do not inhibit binding by more than 50%, \\(K_{\\rm i}\\) can be calculated from a single data point using the following rearranged form: <\/p>\n\n\n\n<p>$$\\frac{\\theta}{1-\\theta} = \\frac{K_{\\rm d} + [L]}{\\frac{K_{\\rm d}}{K_{\\rm i}}[I]} \\rightarrow K_{\\rm i} = \\left(\\frac{\\theta}{1-\\theta}\\right) \\frac{K_{\\rm d}[I]}{K_{\\rm d} + [L]}\\;\\; .$$<\/p>\n\n\n\n<p> If the unlabeled ligand concentration \\([I]\\) is sufficiently large compared to the receptor concentration \\([R]\\) (which is the case for weak ligands), \\([I]\\) can be approximated by the total ligand concentration \\([I]_{\\rm T}\\).<\/p>\n\n\n\n<p>$$K_{\\rm i} = \\left(\\frac{\\theta}{1-\\theta}\\right) \\frac{K_{\\rm d}[I]_{\\rm T}}{K_{\\rm d} + [L]}$$&nbsp;<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Correction for Tight-Binding Unlabeled Ligands<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Correction of IC\u2085\u2080<\/h3>\n\n\n\n<p>When the concentration of the inhibitor \\([I]\\) is not sufficiently high relative to the receptor concentration \\([R]\\) \u2014 for instance, when the binding affinity of the unlabeled ligand is equal to or greater than that of the labeled ligand PDBu \u2014 it is not appropriate to approximate \\([I]\\) with the total ligand concentration \\([I]_{\\rm T}\\). <\/p>\n\n\n\n<p>Therefore,<\/p>\n\n\n\n<p>$$ [I]_{50} = \\mathrm{IC}_{50} - [RI]_{50} $$<\/p>\n\n\n\n<p>and \\([RI]_{50} \\) must be calculated.<\/p>\n\n\n\n<p>Experimentally, \\([RL]_0\\)\u200b and \\([L]_0\\)\u200b can be determined from the measured radioactivity. Using these values and a known \\(K_{\\mathrm d}\\)\u200b, the following quantities can be calculated:<\/p>\n\n\n\n<p>$$[L]_{50} = [L]_{0} + [RL]_{0}\/2\\;\\; ,$$<\/p>\n\n\n\n<p>$$[R]_{\\mathrm{T}} = [RL]_0 (1 + K_{\\mathrm d}\/[L]_0)\\;\\; ,$$<\/p>\n\n\n\n<p>$$[RL]_{50} = [RL]_{0}\/2\\;\\; ,$$<\/p>\n\n\n\n<p>First, consider the following two conservation equations.<\/p>\n\n\n\n<p>In the absence of inhibitor \\(I\\) : $$ [R]_{\\mathrm{T}} = [R]_{0} + [RL]_{0} $$<\/p>\n\n\n\n<p>When the concentration of \\(I\\) is  \\([I]_{50}\\): $$ [R]_{\\mathrm{T}} = [R]_{50} + [RL]_{50} + [RI]_{50} $$<\/p>\n\n\n\n<p>Considering the equilibrium of the labeled ligand \\(L\\),<\/p>\n\n\n\n<p>$$K_{\\mathrm d} = \\frac{[R]_0[L]_0}{[RL]_0} = \\frac{[R]_{50}[L]_{50}}{[RL]_{50}}\\;\\; , $$<\/p>\n\n\n\n<p>Substituting<\/p>\n\n\n\n<p>$$ [R]_{0} = [R]_{\\mathrm{T}} - [RL]_{0}$$<\/p>\n\n\n\n<p>and<\/p>\n\n\n\n<p>$$ [R]_{50} = [R]_{\\mathrm{T}} - [RL]_{50} - [RI]_{50}$$<\/p>\n\n\n\n<p>into this equation and rearranging yields<\/p>\n\n\n\n<p>$$ [RI]_{50} = [R]_{\\mathrm{T}} - \\frac{[L]_0}{[L]_{50}}\\frac{[RL]_{50}}{[RL]_0}R_{\\mathrm{T}} + \\left( \\frac{[L]_0}{[L]_{50}} - 1\\right) [RL]_{50} \\;\\; .$$<\/p>\n\n\n\n<p>Since<\/p>\n\n\n\n<p>$$\\frac{[RL]_{50}}{[RL]_0} = 1\/2\\;\\; ,$$<\/p>\n\n\n\n<p>this simplified to<\/p>\n\n\n\n<p>$$ [RI]_{50} = \\left( 1 - \\frac{1}{2} \\frac{[L]_0}{[L]_{50}} \\right)[R]_{\\mathrm{T}} + \\left( \\frac{[L]_0}{[L]_{50}} - 1\\right) [RL]_{50} \\;\\; .$$<\/p>\n\n\n\n<p>Thus, the correction formula<\/p>\n\n\n\n<p>$$ K_{\\rm i} = \\frac{\\mathrm{IC}_{50} - \\left( 1 - \\frac{1}{2} \\frac{[L]_0}{[L]_{50}} \\right)[R]_{\\mathrm{T}} - \\left( \\frac{[L]_0}{[L]_{50}} - 1\\right) [RL]_{50} }{2\\frac{[L_{50}]}{[L_0]} - 1 + \\frac{[L_{50}]}{K_{\\rm d}}} $$<\/p>\n\n\n\n<p>is obtained. <\/p>\n\n\n\n<p><strong>The <em>K<\/em><sub>i<\/sub>\u200b value obtained from this correction is exact and is identical to the value obtained using the Munson\u2013Rodbard correction described in the following section.<\/strong><\/p>\n\n\n\n<p>If the concentration of the labeled ligand is sufficiently in excess of the receptor concentration such that the approximation<\/p>\n\n\n\n<p>$$ [L]_0 \\approx [L]_{50}$$<\/p>\n\n\n\n<p>can be used, the numerator on the right-hand side becomes simpler, and the equation can be written as<\/p>\n\n\n\n<p>$$ K_{\\rm i} = \\frac{\\mathrm{IC}_{50} - \\frac{[R]_{\\mathrm{T}}}{2}} {2\\frac{[L_{50}]}{[L_0]} - 1 + \\frac{[L_{50}]}{K_{\\rm d}}}  \\;\\; .$$<\/p>\n\n\n\n<p><strong>In practice, the <em>K<\/em><sub>i<\/sub>\u200b values obtained using this simpler correction are sufficiently accurate.<\/strong><\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span style=\"font-weight: bold;\">Munson\u2013Rodbard Equation<\/span><\/h3>\n\n\n\n<p>The Munson\u2013Rodbard correction also allows you to calculate \\(K_{\\rm i}\\) using the actual \\({\\rm IC}_{50}\\) (total concentration), without relying on the approximation \\([I]_{50} \\simeq \\textrm{IC}_{50}\\) (Munson and Rodbard, 1988; Huang, 2003).<\/p>\n\n\n\n<p>If we define \\(y_0 = [RL]_0\/[L]_0\\), then: $$K_{\\rm i} = \\frac{\\textrm{IC}_{50}}{1+ \\frac{[L]_{\\rm T} (y_0 + 2)}{2 K_{\\rm d} (y_0 + 1)} + y_0} - K_{\\rm d} \\left(\\frac{y_0}{y_0 + 2}\\right) \\;\\;\\;\\;\\mbox{(Eq.26; Munson\u2013Rodbard equation) }$$<\/p>\n\n\n\n<p>When \\(y_0\\) is very small, this equation reduces to the <b>Cheng\u2013Prusoff equation<\/b>.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Derivation of the Munson\u2013Rodbard Equation<\/h3>\n\n\n\n<p>Let \\(K_1 = 1\/K_{\\rm d}\\), \\(K_2 = 1\/K_{\\rm i}\\). Then: $$[RL] = K_1 [R] [L]\\;\\;\\;\\;\\mbox{(Eq.27)}$$ $$[RI] = K_2 [R] [I]\\;\\;\\;\\;\\mbox{(Eq.28)}$$ From the conservation laws: $$[L]_{\\rm T} = [RL] + [L]\\;\\;\\;\\;\\mbox{(Eq.29)}$$ $$[I]_{\\rm T} = [RI] + [I]\\;\\;\\;\\;\\mbox{(Eq.30)}$$ $$[R]_{\\rm T} = [RI] + [R] + [RL] = K_2 [R] [I] + [R] + K_1 [R] [L] = [R] (1+ K_1[L] + K_2 [I])\\;\\;\\;\\;\\mbox{(Eq.31)}$$<\/p>\n\n\n\n<p>Now consider the conditions <b>without inhibitor<\/b> and <b>at 50% inhibition<\/b>.<\/p>\n\n\n\n<p>Let the ratio without inhibitor be defined as \\(y_0 = [RL_0]\/[L_0]\\). From Eq.29: $$[L]_{\\rm T} = [L]_0 (1 + y_0)\\;\\;\\;\\;\\mbox{(Eq.32)}$$ $$[L]_0 =\\frac{[L]_{\\rm T}}{1 + y_0}\\;\\;\\;\\;\\mbox{(Eq.33)}$$ $$[RL]_0 = \\frac{ y_0 [L]_{\\rm T}}{1 + y_0}\\;\\;\\;\\;\\mbox{(Eq.34)}$$<\/p>\n\n\n\n<p>At 50% inhibition: $$[RL]_{50} = \\frac{y_0 [L]_{\\rm T}}{2 (1 + y_0)}\\;\\;\\;\\;\\mbox{(Eq.35)}$$ From Eq.29: $$[L]_{50} = [L]_{\\rm T} - [RL_{50}] = \\frac{ [L]_{\\rm T} (2 + y_0) }{2 (1 + y_0)}\\;\\;\\;\\;\\mbox{(Eq.36)}$$<\/p>\n\n\n\n<p>Using Eq.27, the free receptor at 50% inhibition is: $$[R]_{50}= \\frac{[RL]_{50}}{K_1 [L]_{50}} = \\frac{y_0}{K_1 (y_0 + 2)}\\;\\;\\;\\;\\mbox{(Eq.37)}$$<\/p>\n\n\n\n<p>From Eq.28 and Eq.30: $$\\textrm{IC}_{50} = [I]_{{\\rm T} 50} = [RI]_{50} + [I]_{50} = [I]_{50}(1+K_2 [R]_{50})\\;\\;\\;\\;\\mbox{(Eq.38)}$$ Substituting Eq.37 into Eq.38: $$[I]_{50} = \\frac{\\textrm{IC}_{50}}{1+\\left(\\frac{K_2}{K_1}\\right)\\left(\\frac{y_0}{y_0+2}\\right)}\\;\\;\\;\\;\\mbox{(Eq.39)}$$ Solving Eq.31 for \\([R]\\) and substituting into Eq.27: $$[RL] = \\frac{K_1 [R]_{\\rm T} [L] }{1+ K_1 [L] + K_2 [I]}\\;\\;\\;\\;\\mbox{(Eq.40)}$$<\/p>\n\n\n\n<p>Without inhibitor, using Eq.34 and Eq.33: $$[R]_{\\rm T} = y_0 \\left(\\frac{1}{K_1} + \\frac{[L]_{\\rm T}}{1 + y_0}\\right)\\;\\;\\;\\;\\mbox{(Eq.41)}$$<\/p>\n\n\n\n<p>At 50% inhibition: $$[RL]_{50} = \\frac{K_1 [R]_{\\rm T} [L]_{50} }{1+ K_1 [L]_{50} + K_2 [I]_{50}}\\;\\;\\;\\;\\mbox{(Eq.42)}$$<\/p>\n\n\n\n<p>Solving Eq.42 for \\(1\/K_2\\): $$\\frac{1}{K_2} = \\frac{[I]_{50}}{\\frac{K_1[R]_{\\rm T}[L]_{50}}{[RL]_{50}} - K_1[L]_{50} - 1}$$<\/p>\n\n\n\n<p>Substitute Eq.39 into this and simplify: $$\\frac{1}{K_2} + \\frac{1}{K_1}\\left(\\frac{y_0}{y_0+2}\\right) = \\frac{\\mathrm{IC}_{50}}{\\frac{K_1[R]_{\\rm T}[L]_{50}}{[RL]_{50}} - K_1[L]_{50} - 1}$$<\/p>\n\n\n\n<p>Now, substitute:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Eq.36 into \\([L]_{50}\\)<\/li>\n\n\n\n<li>Eq.41 into \\([R]_{\\rm T}\\)<\/li>\n\n\n\n<li>Use \\([L]_{50}[RL]_{50} = \\frac{(y_0 + 2)}{2y_0}\\)<\/li>\n<\/ul>\n\n\n\n<p>Then the denominator on the right becomes: $$K_1 \\times y_0\\left(\\frac{1}{K_1} + \\frac{[L]_\\mathrm{T}}{y_0+1}\\right) \\times \\frac{y_0+2}{y_0} - K_1\\frac{[L]_\\mathrm{T}(y_0+2)}{2(y_0+1)} - 1$$ $$= 2+ y_0 + K_1\\frac{[L]_\\mathrm{T}(y_0+2)}{y_0+1} - K_1\\frac{[L]_\\mathrm{T}(y_0+2)}{2(y_0+1)} - 1 $$ $$= 1+ K_1\\frac{[L]_\\mathrm{T}(y_0+2)}{2(y_0+1)} + y_0 $$<\/p>\n\n\n\n<p>Therefore: $$\\frac{1}{K_2} = \\frac{\\textrm{IC}_{50}}{1+ \\frac{[L]_{\\rm T} (y_0 + 2)}{2 \\left(\\frac{1}{K_1}\\right) (y_0 + 1)} + y_0} - \\left(\\frac{1}{K_1}\\right) \\left(\\frac{y_0}{y_0 + 2}\\right)\\;\\;\\;\\;\\mbox{(Eq.43)}$$<\/p>\n\n\n\n<p>Finally, replacing \\(1\/K_2 = K_{\\rm i}\\), \\(1\/K_1 = K_{\\rm d}\\), we recover Eq.26.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span style=\"font-weight: bold;\">Dandliker's Equation<\/span><\/h2>\n\n\n\n<p>You can calculate \\(K_{\\rm i}\\) from each data point in an inhibition experiment (Dandliker, 1981).<\/p>\n\n\n\n<p>Define: $$\\chi = \\frac{[RL]}{[L]}$$ (Note: \\([L]\\) is computed as \\([L] = [L]_{\\rm T} - [RL]\\))<\/p>\n\n\n\n<p>Then: $$K_i = \\frac{[I]_{\\rm T} K_d \\chi}{[R]_{\\rm T} - \\frac{[L]_{\\rm T}\\chi}{1 + \\chi} - K_d \\chi} - K_d \\chi\\;\\;\\;\\;\\mbox{(Eq.44)}$$<\/p>\n\n\n\n<p>Also, define \\(f_b = [RL]\/[L]_{\\rm T}\\)\/ Since, \\(\\chi = f_b\/(1-f_b)\\), then: $$K_{\\rm i} = \\frac{[I]_{\\rm T}K_{\\rm d} f_b}{[R]_{\\rm T}(1 - f_b) - K_{\\rm d} f_b - [L]_{\\rm T} f_b (1 - f_b)} - \\frac{K_{\\rm d} f_b}{ 1 - f_b}\\;\\;\\;\\;\\mbox{(Eq.45)}$$<\/p>\n\n\n\n<p>Eq.45 is a rearranged form of Equation (11) in Huang, 2003.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Derivation of Dandliker\u2019s Equation<\/h3>\n\n\n\n<p>As before, define \\(K_1 = 1\/K_{\\rm d}\\), \\(K_2 = 1\/K_{\\rm i}\\). $$\\chi = \\frac{[RL]}{[L]} = K_1 ([R]_{\\rm T} - [RL] - [RI])\\;\\;\\;\\;\\mbox{(Eq.46)}$$ $$\\frac{[RI]}{[I]} = K_2 ([R]_{\\rm T} - [RL] - [RI])\\;\\;\\;\\;\\mbox{(Eq.47)}$$ $$[L]_{\\rm T} = [RL] + [L]\\;\\;\\;\\;\\mbox{(Eq.48)}$$ $$[I]_{\\rm T} = [RI] + [I]\\;\\;\\;\\;\\mbox{(Eq.49)}$$ $$[RL] = \\frac{[L]_{\\rm T}\\chi}{1 + \\chi}\\;\\;\\;\\;\\mbox{(Eq.50)}$$ $$[L] = \\frac{[L]_{\\rm T}}{1 + \\chi}\\;\\;\\;\\;\\mbox{(Eq.51)}$$ From Eq.46, express \\([RL]\\) and \\([L]\\) in terms of \\(\\chi\\): $$[RI] = [R]_{\\rm T} - [RL] - \\frac{\\chi}{K_1} = [R]_{\\rm T} - \\frac{[L]_{\\rm T}\\chi}{1 + \\chi} - \\frac{\\chi}{K_1}\\;\\;\\;\\;\\mbox{(Eq.52)}$$<\/p>\n\n\n\n<p>Solving Eq.47 for \\([I]\\): $$[I] = \\frac{[RI]}{K_2} \\times \\frac{1}{[R]_{\\rm T} - [RL] - [RI]}\\;\\;\\;\\;\\mbox{(Eq.53)}$$ Since \\([RI] = [R]_{\\rm T} - [RL] - \\frac{\\chi}{K_1}\\), then: $$[I] = \\frac{[R]_{\\rm T} - [RL] - \\frac{\\chi}{K_1}}{K_2} \\times \\frac{1}{\\frac{\\chi}{K_1}} = \\frac{K_1}{K_2\\chi}\\left([R]_{\\rm T} - \\frac{[L]_{\\rm T}\\chi}{1 + \\chi} - \\frac{\\chi}{K_1}\\right)\\;\\;\\;\\;\\mbox{(Eq.54)}$$ Substitute Eqs. 52 and 54 into Eq.49: $$[I]_{\\rm T} = \\left(1 + \\frac{K_1}{K_2\\chi}\\right) \\left([R]_{\\rm T} - \\frac{[L]_{\\rm T}\\chi}{1 + \\chi} - \\frac{\\chi}{K_1}\\right)\\;\\;\\;\\;\\mbox{(Eq.55)}$$<\/p>\n\n\n\n<p>Solve Eq.55 for \\(1\/K_2\\), giving: $$K_i = \\frac{1}{K_2} = \\frac{[I]_{\\rm T} K_d \\chi}{[R]_{\\rm T} - \\frac{[L]_{\\rm T}\\chi}{1 + \\chi} - K_d \\chi} - K_d \\chi\\;\\;\\;\\;\\mbox{(Eq.56)}$$<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">References<\/h2>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Sharkey, N. A.; Blumberg, P. M. Highly lipophilic phorbol esters as inhibitors of specific [<sup>3<\/sup>H]phorbol 12,13-dibutyrate binding. <span style=\"font-style: italic;\">Cancer Res<\/span>. <span style=\"font-weight: bold;\">1985<\/span>, <span style=\"font-style: italic;\">45<\/span>, 19\u201324. [<a href=\"http:\/\/cancerres.aacrjournals.org\/content\/45\/1\/19.abstract\">URL<\/a>] [PMID]: <a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/3855281\/\">3855281<\/a>.<\/li>\n\n\n\n<li>Dandliker, W. B.; Hsu, M.-L.; Levin, J.; Rao, R. Equilibrium and kinetic inhibition assays based upon fluorescence polarization. <span style=\"font-style: italic;\">Methods Enzymol<\/span>. <span style=\"font-weight: bold;\">1981<\/span>, <span style=\"font-style: italic;\">74<\/span>, 3\u201328. DOI: <a href=\"http:\/\/dx.doi.org\/10.1016\/0076-6879(81)74003-5\">10.1016\/0076-6879(81)74003-5<\/a>. PMID:&nbsp;<a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/7321886\">7321886<\/a>.<\/li>\n\n\n\n<li>Goldstein, A.; Barrett, R. W. Ligand dissociation constants from competition binding assays: errors associated with ligand depletion. <span style=\"font-style: italic;\">Mol. Pharmacol<\/span>. <span style=\"font-weight: bold;\">1987<\/span>, <span style=\"font-style: italic;\">31<\/span>, 603\u2013609. [<a href=\"http:\/\/molpharm.aspetjournals.org\/content\/31\/6\/603.long\">URL<\/a>] PMID:&nbsp;<a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/3600604\">3600604<\/a>.<\/li>\n\n\n\n<li>Hulme, X. C.; Trevethick, M. A. Ligand binding assays at equilibrium: validation and interpretation. <span style=\"font-style: italic;\">Br. J. Pharmacol.<\/span> <span style=\"font-weight: bold;\">2010<\/span>, <span style=\"font-style: italic;\">161<\/span>, 1219\u20131237. DOI:&nbsp;<a href=\"http:\/\/dx.doi.org\/10.1111\/j.1476-5381.2009.00604.x\">10.1111\/j.1476-5381.2009.00604.<\/a>&nbsp;PMID:&nbsp;<a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/20132208\">20132208<\/a>.<\/li>\n\n\n\n<li>Huang, X. Equilibrium competition binding assay: inhibition mechanism from a single dose response. <span style=\"font-style: italic;\">J. Ther. Biol<\/span>. <span style=\"font-weight: bold;\">2003<\/span>, <span style=\"font-style: italic;\">225<\/span>, 369\u2013376. DOI:&nbsp;<a href=\"http:\/\/dx.doi.org\/10.1016\/S0022-5193(03)00265-0\">10.1016\/S0022-5193(03)00265-0<\/a>. PMID:&nbsp;<a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/14604589\">14604589<\/a>.<\/li>\n\n\n\n<li>Munson, P. J.; Rodbard, D. An exact correction to the Cheng-Prusoff correction. <span style=\"font-style: italic;\">J. Receptor Res<\/span>. <span style=\"font-weight: bold;\">1988<\/span>, <span style=\"font-style: italic;\">8<\/span>, 533\u2013546. DOI:&nbsp;<a href=\"http:\/\/dx.doi.org\/10.3109\/10799898809049010\">10.3109\/10799898809049010<\/a>. PMID:&nbsp;<a href=\"https:\/\/www.ncbi.nlm.nih.gov\/pubmed\/3385692\">3385692<\/a>.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\">Appendix: Processing of Actual Experimental Data<\/h2>\n\n\n\n<h3 class=\"wp-block-heading\">Low-affinity ligand<\/h3>\n\n\n\n<p>The following data obtained from a competitive binding assay using a <sup>3<\/sup>H-labeled ligand (specific activity: 18.7 Ci\/mmol) and an unlabeled ligand will be processed. The unit of radioactivity is <strong>dpm<\/strong> (disintegrations per minute; note that Bq represents disintegrations per second).<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td>GROUP<\/td><td>log<sub>10<\/sub> [<em>I<\/em>]<sub>T<\/sub><\/td><td>pellet (dpm)<\/td><td>supernatant (dpm) (50 \u03bcL)<\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>36758<\/td><td>16101<\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>37325<\/td><td>17380<\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>36584<\/td><td>16934<\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>38455<\/td><td>15247<\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>33998<\/td><td>16772<\/td><\/tr><tr><td>Nonspecific binding<\/td><td>\u2014<\/td><td>5592<\/td><td>20072<\/td><\/tr><tr><td>Nonspecific binding<\/td><td>\u2014<\/td><td>4794<\/td><td>18772<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6.5<\/td><td>35734<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6.5<\/td><td>35884<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6.5<\/td><td>32866<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6<\/td><td>33393<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6<\/td><td>33204<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6<\/td><td>34506<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5.5<\/td><td>28644<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5.5<\/td><td>31701<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5.5<\/td><td>30276<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5<\/td><td>22413<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5<\/td><td>22516<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5<\/td><td>21133<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4.5<\/td><td>12318<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4.5<\/td><td>12711<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4.5<\/td><td>11368<\/td><td>\u2014<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Calculation steps<\/strong><\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Convert supernatant values:<\/strong> Multiply the supernatant value by 437\/50 to convert it to the radioactivity of the total supernatant.<\/li>\n\n\n\n<li><strong>Total Radioactivity:<\/strong> Calculate \\(\\mbox{Total Radioactivity} = \\mbox{pellet} + \\mbox{total supernatant}\\) for the Total Binding group.<\/li>\n\n\n\n<li><strong>Partition Coefficient (<em>k<\/em>):<\/strong> Divide the pellet of the Nonspecific Binding group by its total supernatant to calculate the partition coefficient for nonspecific adsorption: <em>k<\/em> = 0.0305\u3002<\/li>\n\n\n\n<li><strong>Specific Binding (Total group):<\/strong> Calculate \\(\\mbox{Specific binding} = \\mbox{pellet} - k \\times \\mbox{total supernatant}\\).<\/li>\n\n\n\n<li><strong>Specific Binding (Unlabeled group):<\/strong> Calculate \\(\\mbox{Specific binding} = \\mbox{pellet} - k \\times (\\mbox{Total Radioactivity} - \\mbox{pellet})\\).<\/li>\n\n\n\n<li><strong>Molar Concentration Conversion:<\/strong> Convert radioactivity (dpm) to molar concentration (nM) within the test tube using the coefficient <em>N<\/em>: \\(N = \\frac{1}{60} \\times \\frac{1}{3.7 \\times 10^{10}} \\times \\frac{1}{\\mbox{Specific activity (Ci\/mmol)}} \\times \\frac{1}{10^3} \\times \\frac{1}{250 \\times 10^{-6}} \\times 10^9\\)<\/li>\n\n\n\n<li><strong>Calculate \\(\\theta\\):<\/strong> For each data point in the Unlabeled ligand group, calculate \\(\\theta = \\mbox{pellet (nM)} \/ \\langle \\mbox{Mean specific binding of Total binding group } ([RL]_0) \\rangle\\).<\/li>\n\n\n\n<li><strong>Logit Transformation:<\/strong> Calculate \\(\\log_{10} \\{\\theta\/(1 - \\theta)\\}\\).<\/li>\n<\/ol>\n\n\n\n<p><strong>Processed Data Table<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td>GROUP<\/td><td>log<sub>10<\/sub> [<em>I<\/em>]<sub>T<\/sub><\/td><td>bound (nM)<\/td><td>free (nM)<\/td><td><em>\u03b8<\/em><\/td><td>log<sub>10<\/sub> {<em>\u03b8<\/em>\/(1 \u2212 <em>\u03b8<\/em>)}<\/td><\/tr><tr><td>Total binding (Mean)<\/td><td>\u2014<\/td><td>3.105 ([<em>RL<\/em>]<sub>0<\/sub>)<\/td><td>13.88 ([<em>L<\/em>]<sub>0<\/sub>)<\/td><td>\u2014<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6.5<\/td><td>3.017<\/td><td>\u2014<\/td><td>0.9715<\/td><td>1.533<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6.5<\/td><td>3.032<\/td><td>\u2014<\/td><td>0.9763<\/td><td>1.615<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6.5<\/td><td>2.732<\/td><td>\u2014<\/td><td>0.8798<\/td><td>0.865<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6<\/td><td>2.785<\/td><td>\u2014<\/td><td>0.8967<\/td><td>0.939<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6<\/td><td>2.766<\/td><td>\u2014<\/td><td>0.8907<\/td><td>0.911<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-6<\/td><td>2.895<\/td><td>\u2014<\/td><td>0.9323<\/td><td>1.139<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5.5<\/td><td>2.313<\/td><td>\u2014<\/td><td>0.7448<\/td><td>0.465<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5.5<\/td><td>2.616<\/td><td>\u2014<\/td><td>0.8426<\/td><td>0.729<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-5.5<\/td><td>2.475<\/td><td>\u2014<\/td><td>0.7970<\/td><td>0.594<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4.5<\/td><td>1.694<\/td><td>\u2014<\/td><td>0.5456<\/td><td>0.079<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4.5<\/td><td>1.704<\/td><td>\u2014<\/td><td>0.5489<\/td><td>0.085<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4.5<\/td><td>1.567<\/td><td>\u2014<\/td><td>0.5047<\/td><td>0.008<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4<\/td><td>0.692<\/td><td>\u2014<\/td><td>0.2228<\/td><td>-0.543<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4<\/td><td>0.731<\/td><td>\u2014<\/td><td>0.2354<\/td><td>-0.512<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-4<\/td><td>0.698<\/td><td>\u2014<\/td><td>0.1924<\/td><td>-0.623<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><em>K<\/em><sub>d<\/sub> = 0.53 nM. Other calculated values (in nM): [<em>L<\/em>]<sub>T<\/sub> = 17.41; [<em>L<\/em>]*<sub>T<\/sub> = 16.90; [<em>R<\/em>]<sub>T<\/sub> = 3.224.<\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">Estimation of IC<sub>50<\/sub><\/h4>\n\n\n\n<pre class=\"prettyprint\">logIT &lt;- c(-6.5,-6.5,-6.5,-6,-6,-6,-5.5,-5.5,-5.5,-5,-5,-5,-4.5,-4.5,-4.5)\ntheta &lt;- c(0.9715, 0.9763, 0.8798, 0.8967, 0.8907, 0.9323, 0.7448, 0.8426, 0.7970, 0.5456, 0.5489, 0.5047, 0.2228, 0.2354, 0.1924)\nresult &lt;- nls(theta~1&#47;(1+exp(a*log(10)*(logIT - logIC50))),start = c(logIC50=-5, a=1))\nsummary(result)\n<\/pre>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\nFormula: theta ~ 1&#47;(1 + exp(a * log(10) * (logIT - logIC50)))\n\nParameters:\n        Estimate Std. Error t value Pr(&gt;|t|)    \nlogIC50 -4.97382    0.02644 -188.15  &lt; 2e-16 ***\na        1.07378    0.06989   15.37 1.03e-09 ***\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: 0.04024 on 13 degrees of freedom\n\nNumber of iterations to convergence: 6 \nAchieved convergence tolerance: 1.317e-06\n<\/pre>\n<pre class=\"prettyprint\">confint(result, level=0.95)\n<\/pre>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\n              2.5%     97.5%\nlogIC50 -5.0301661 -4.916613\na        0.9269206  1.247572\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"966\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Nonlinear_low-1024x966.png\" alt=\"\" class=\"wp-image-6799\" style=\"width:329px;height:auto\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Nonlinear_low-1024x966.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Nonlinear_low-300x283.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Nonlinear_low-768x724.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Nonlinear_low.png 1442w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p><strong>Output Highlights:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The estimated \\(\\log_{10} \\mathrm{IC}_{50}\\) is -4.97382. Converting this to molarity yields <strong>10,621 nM<\/strong> (95%CI: 9,329\u201312,117).<\/li>\n\n\n\n<li>Slope factor (\\(a\\)): 1.07378<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Conversion to <em>K<\/em><sub>i<\/sub><\/h4>\n\n\n\n<p>Assuming that the unlabeled ligand undergoes non-specific binding with the same partition coefficient as [<sup>3<\/sup>H]PDBu, we have:<\/p>\n\n\n\n<p>$$ \\mathrm{IC}_{50} = (1 + k) [I]_{50} + [RI]_{50} $$<\/p>\n\n\n\n<p>$$ [I]_{50} = \\frac{1}{1+ k} (\\mathrm{IC}_{50}  - [RI]_{50}) $$<\/p>\n\n\n\n<p>Therefore, the following exact correction formula for converting \\(\\mathrm{IC}_{50}\\) to \\(K_\\mathrm{i}\\) is obtained:<\/p>\n\n\n\n<p>$$ K_{\\rm i} = \\frac{\\frac{1}{1 + k} \\left[\\mathrm{IC}_{50} - \\left( 1 - \\frac{1}{2} \\frac{[L]_0}{[L]_{50}} \\right)[R]_{\\mathrm{T}} - \\left( \\frac{[L]_0}{[L]_{50}} - 1\\right) \\frac{[RL]_{0}}{2} \\right] }{2\\frac{[L]_{50}}{[L]_0} - 1 + \\frac{[L]_{50}}{K_{\\rm d}}} $$<\/p>\n\n\n\n<p>In this case, since the affinity of the unlabeled ligand is weak and \\(\\mathrm{IC}_{50} \\gg [RI]_{50}\\) holds, the following approximation can be applied:<\/p>\n\n\n\n<p>$$ [I]_{50} \\simeq \\frac{1}{1+ k} \\mathrm{IC}_{50} $$<\/p>\n\n\n\n<p>$$ K_{\\rm i} = \\frac{\\frac{1}{1 + k} \\mathrm{IC}_{50} }{2\\frac{[L]_{50}}{[L]_0} - 1 + \\frac{[L]_{50}}{K_{\\rm d}}} $$<\/p>\n\n\n\n<p>The \\(K_{\\rm i}\\) value obtained using the exact formula was <strong>340.6 nM (95% CI: 299.1\u2013388.6)<\/strong>. The \\(K_{\\rm i}\\) value calculated with the approximation formula was <strong>340.7 nM<\/strong>, demonstrating that there is no significant difference between the two.<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Linear Regression via Logit Transformation (Alternative)<\/h4>\n\n\n\n<p><strong>1) Using all five concentration levels<\/strong><\/p>\n\n\n\n<pre class=\"prettyprint\">logIT &lt;- c(-6.5,-6.5,-6.5,-6,-6,-6,-5.5,-5.5,-5.5,-5,-5,-5,-4.5,-4.5,-4.5)\nlogit &lt;- c(1.533, 1.615, 0.865, 0.939, 0.911, 1.139, 0.465, 0.729, 0.594, 0.079, 0.085, 0.008, -0.543, -0.512, -0.623)\nresult_lm &lt;- lm(logit ~ logIT)\nsummary(result_lm)\n<\/pre>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\nCall:\nlm(formula = logit ~ logIT)\n\nResiduals:\n     Min       1Q   Median       3Q      Max \n-0.56720 -0.04945 -0.00430  0.10460  0.24340 \n\nCoefficients:\n            Estimate Std. Error t value Pr(&gt;|t|)    \n(Intercept) -4.72070    0.40386  -11.69 2.86e-08 ***\nlogIT       -0.94660    0.07283  -13.00 7.98e-09 ***\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: 0.1995 on 13 degrees of freedom\nMultiple R-squared:  0.9285,\tAdjusted R-squared:  0.923 \nF-statistic: 168.9 on 1 and 13 DF,  p-value: 7.975e-09\n<\/pre>\n\n\n\n<p><strong>2) Using the three highest concentration levels<\/strong><\/p>\n\n\n\n<pre class=\"prettyprint\">logIT_a &lt;- c(-5.5,-5.5,-5.5,-5,-5,-5,-4.5,-4.5,-4.5)\nlogit_a &lt;- c(0.465, 0.729, 0.594, 0.079, 0.085, 0.008, -0.543, -0.512, -0.623)\nresult_lm_a &lt;- lm(logit_a ~ logIT_a)\nsummary(result_lm_a)\n<\/pre>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\nCall:\nlm(formula = logit_a ~ logIT_a)\n\nResiduals:\n      Min        1Q    Median        3Q       Max \n-0.144000 -0.023333  0.003333  0.047667  0.120000 \n\nCoefficients:\n            Estimate Std. Error t value Pr(&gt;|t|)    \n(Intercept)  -5.7453     0.3396  -16.92 6.18e-07 ***\nlogIT_a      -1.1553     0.0677  -17.07 5.82e-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: 0.08292 on 7 degrees of freedom\nMultiple R-squared:  0.9765,\tAdjusted R-squared:  0.9732 \nF-statistic: 291.2 on 1 and 7 DF,  p-value: 5.819e-07\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"970\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Linear_low-1024x970.png\" alt=\"\" class=\"wp-image-6800\" style=\"aspect-ratio:1.0556669804756251;width:323px;height:auto\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Linear_low-1024x970.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Linear_low-300x284.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Linear_low-768x728.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/Linear_low.png 1444w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p><strong>Analysis of Results<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Black Line (all five concentration levels)<\/strong>: \\(y = -0.94660x - 4.72070\\). \\(R^2 = 0.929\\). \\(\\log_{10} \\mathrm{IC}_{50} = -4.72070 \/ 0.94660 = -4.987006\\). Converting to molarity gives \\(\\mathrm{IC}_{50} = 10,304\\mbox{ nM}\\).<\/li>\n\n\n\n<li><strong>Red Line (the three highest concentration levels)<\/strong>: \\(y = -1.1553x - 5.7453\\). \\(R^2 = 0.977\\). \\(\\log_{10} \\mathrm{IC}_{50} = -5.7453 \/ 1.1553 = -4.972994\\). Converting to molarity gives \\(\\mathrm{IC}_{50} = 10,642\\mbox{ nM}\\).<\/li>\n<\/ul>\n\n\n\n<p><strong>Interpretation:<\/strong> The \\(\\mathrm{IC}_{50}\\) value estimated from the three-concentration levels dataset\u2014despite having fewer data points\u2014was closer to the results obtained via non-linear regression. This suggests that linear transformation tends to overestimate the weight of data points located at the tails of the sigmoid curve.<\/p>\n\n\n\n<p>Using the same approximation formula from the previous section, the converted \\(K_{\\rm i}\\) values were <strong>330.5 nM<\/strong> (from the five concentration levels) and <strong>341.3 nM<\/strong> (from the three concentration levels).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Tight-binding ligand<\/h3>\n\n\n\n<p>The following data obtained from a competitive binding assay using a <sup>3<\/sup>H-labeled ligand (specific activity: 12.31 Ci\/mmol) and an unlabeled ligand will be processed. The unit of radioactivity is <strong>dpm<\/strong> (disintegrations per minute; note that Bq represents disintegrations per second).<\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td>GROUP<\/td><td>log<sub>10<\/sub> [<em>I<\/em>]<sub>T<\/sub><\/td><td>pellet (dpm)<\/td><td>supernatant (dpm) (50 \u03bcL)<\/td><td><\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>38985<\/td><td>10382<\/td><td><\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>38931<\/td><td>10907<\/td><td><\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>38495<\/td><td>9242<\/td><td><\/td><\/tr><tr><td>Total binding<\/td><td>\u2014<\/td><td>39034<\/td><td>10287<\/td><td><\/td><\/tr><tr><td>Nonspecific binding<\/td><td>\u2014<\/td><td>3629<\/td><td>14208<\/td><td><\/td><\/tr><tr><td>Nonspecific binding<\/td><td>\u2014<\/td><td>3448<\/td><td>13194<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8.5<\/td><td>26327<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8.5<\/td><td>26072<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8.5<\/td><td>25105<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8<\/td><td>13168<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8<\/td><td>11354<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8<\/td><td>11917<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-7.5<\/td><td>6314<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-7.5<\/td><td>6446<\/td><td>\u2014<\/td><td><\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-7.5<\/td><td>5829<\/td><td>\u2014<\/td><td><\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p><strong>Processed Data Table<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-stripes\"><table><tbody><tr><td>GROUP<\/td><td>log<sub>10<\/sub> [<em>I<\/em>]<sub>T<\/sub><\/td><td>bound (nM)<\/td><td>free (nM)<\/td><td><em>\u03b8<\/em><\/td><td>log<sub>10<\/sub> {<em>\u03b8<\/em>\/(1 \u2212 <em>\u03b8<\/em>)}<\/td><\/tr><tr><td>Total binding (Mean)<\/td><td>\u2014<\/td><td>5.302  ([<em>RL<\/em>]<sub>0<\/sub>)<\/td><td>13.05 ([<em>L<\/em>]<sub>0<\/sub>)<\/td><td>\u2014<\/td><td>\u2014<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8.5<\/td><td>3.413<\/td><td>\u2014<\/td><td>0.6438<\/td><td>0.257<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8.5<\/td><td>3.375<\/td><td>\u2014<\/td><td>0.6365<\/td><td>0.243<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8.5<\/td><td>3.229<\/td><td>\u2014<\/td><td>0.6090<\/td><td>0.193<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8<\/td><td>1.430<\/td><td>\u2014<\/td><td>0.2698<\/td><td>-0.432<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8<\/td><td>1.157<\/td><td>\u2014<\/td><td>0.2182<\/td><td>-0.554<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-8<\/td><td>1.242<\/td><td>\u2014<\/td><td>0.2342<\/td><td>-0.515<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-7.5<\/td><td>0.397<\/td><td>\u2014<\/td><td>0.0750<\/td><td>-1.091<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-7.5<\/td><td>0.417<\/td><td>\u2014<\/td><td>0.0787<\/td><td>-1.068<\/td><\/tr><tr><td>Unlabeled ligand (I)<\/td><td>-7.5<\/td><td>0.324<\/td><td>\u2014<\/td><td>0.0612<\/td><td>-1.186<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><em>K<\/em><sub>d<\/sub> = 0.53 nM. Other calculated values (in nM): [<em>L<\/em>]<sub>T<\/sub> = 18.74; [<em>L<\/em>]*<sub>T<\/sub> = 18.20; [<em>R<\/em>]<sub>T<\/sub> = 5.52.<\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">Estimation of IC<sub>50<\/sub><\/h4>\n\n\n\n<pre class=\"prettyprint\">logIT &lt;- c(-8.5,-8.5,-8.5,-8,-8,-8,-7.5,-7.5,-7.5)\ntheta &lt;- c(0.6438,0.6365,0.6090,0.2698,0.2182,0.2342,0.0750,0.0787,0.0612)\nresult = nls(theta~1&#47;(1+exp(a*log(10)*(logIT - logIC50))),start = c(logIC50=-8.5, a=1))\nsummary(result)\n<\/pre>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\nNonlinear regression model\n  model: theta ~ 1&#47;(1 + exp(a * log(10) * (logIT - logIC50)))\n   data: parent.frame()\nlogIC50       a \n -8.341   1.417 \n residual sum-of-squares: 0.002774\n\nNumber of iterations to convergence: 5 \nAchieved convergence tolerance: 8.394e-06\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"991\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_sigmoid-1024x991.png\" alt=\"\" class=\"wp-image-6677\" style=\"aspect-ratio:1.0333096276933131;width:410px;height:auto\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_sigmoid-1024x991.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_sigmoid-300x290.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_sigmoid-768x744.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_sigmoid.png 1444w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p><strong>Output Highlights:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>The estimated \\(\\log_{10} \\mathrm{IC}_{50}\\) is -8.341. Converting this to molarity yields <strong>4.560 nM<\/strong>.<\/li>\n\n\n\n<li>Slope factor (\\(a\\)): 1.417<\/li>\n\n\n\n<li>Using the Munson\u2013Rodbard equation, the converted \\(K_\\mathrm{i}\\) value is <strong>0.0586 nM<\/strong>.<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Linear Regression via Logit Transformation (Alternative)<\/h4>\n\n\n\n<pre class=\"prettyprint\">logIT &lt;- c(-8.5,-8.5,-8.5,-8,-8,-8,-7.5,-7.5,-7.5)\nlogit = c(0.257,0.243,0.193,-0.432,-0.554,-0.515,-1.091,-1.068,-1.186)\nresult_lm = lm(logit ~ logIT)\nsummary(result_lm)\n<\/pre>\n<p><b>Output<\/b>:<\/p>\n<pre class=\"prettyprint\">\nCall:\nlm(formula = logit ~ logIT)\n\nResiduals:\n     Min       1Q   Median       3Q      Max \n-0.09256 -0.05156  0.02944  0.04344  0.06644 \n\nCoefficients:\n             Estimate Std. Error t value Pr(&gt;|t|)    \n(Intercept) -11.22944    0.38820  -28.93 1.52e-08 ***\nlogIT        -1.34600    0.04846  -27.77 2.01e-08 ***\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: 0.05935 on 7 degrees of freedom\nMultiple R-squared:  0.991,\tAdjusted R-squared:  0.9897 \nF-statistic: 771.4 on 1 and 7 DF,  p-value: 2.014e-08\n<\/pre>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"976\" src=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_linear-1024x976.png\" alt=\"\" class=\"wp-image-6678\" style=\"width:409px;height:auto\" srcset=\"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_linear-1024x976.png 1024w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_linear-300x286.png 300w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_linear-768x732.png 768w, https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/ligand_linear.png 1438w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/div>\n\n\n<p><strong>Output Highlights:<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Regression Equation: \\(y = -1.34600x - 11.22944\\)<\/li>\n\n\n\n<li>\\(R^2 = 0.991\\)<\/li>\n\n\n\n<li>\\(\\log_{10} \\mathrm{IC}_{50} = -11.22944 \/ 1.34600 = -8.342823\\). Converting this to molarity yields <strong>4.541 nM<\/strong>.<\/li>\n\n\n\n<li>Using the Munson\u2013Rodbard equation, the converted \\(K_\\mathrm{i}\\) value is <strong>0.0580 nM<\/strong>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>In the world of drug discovery &#8230;<\/p>\n","protected":false},"author":1,"featured_media":6743,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"image","meta":{"om_disable_all_campaigns":false,"_uag_custom_page_level_css":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_uf_show_specific_survey":0,"_uf_disable_surveys":false,"_locale":"en_US","_original_post":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/?p=703","footnotes":""},"categories":[9],"tags":[],"class_list":["post-4297","post","type-post","status-publish","format-image","has-post-thumbnail","hentry","category-9","post_format-post-format-image","en-US"],"aioseo_notices":[],"uagb_featured_image_src":{"full":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-scaled.png",2560,1270,false],"thumbnail":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-150x150.png",150,150,true],"medium":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-300x149.png",300,149,true],"medium_large":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-768x381.png",768,381,true],"large":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-1024x508.png",800,397,true],"1536x1536":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-1536x762.png",1536,762,true],"2048x2048":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-2048x1016.png",2048,1016,true],"onepress-blog-small":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-300x150.png",300,150,true],"onepress-small":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-480x300.png",480,300,true],"onepress-medium":["https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp\/wp-content\/uploads\/2016\/11\/munson_rodbard_2025-640x400.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":"In the world of drug discovery ...","_links":{"self":[{"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/posts\/4297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/comments?post=4297"}],"version-history":[{"count":145,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/posts\/4297\/revisions"}],"predecessor-version":[{"id":6849,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/posts\/4297\/revisions\/6849"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/media\/6743"}],"wp:attachment":[{"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/media?parent=4297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/categories?post=4297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ag.kagawa-u.ac.jp\/charlesy\/wp-json\/wp\/v2\/tags?post=4297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}