Difference between revisions of "20.109(S08):Data analysis (Day7)"
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==Introduction== | ==Introduction== | ||
− | This is it, folks! Moment of truth. Time to find out how the proteins that you worked so hard to make, purify, and test really behave. Although you should be able to produce reasonable | + | This is it, folks! Moment of truth. Time to find out how the proteins that you worked so hard to make, purify, and test really behave. Although you should be able to produce reasonable titration curves by following the example of Nagai, the introduction/review of binding constants below may help contextualize your analysis. |
− | + | [[Image:20.109_Binding.png|thumb|200px|right|'''Simple Binding Curve''' The binding fraction ''y'' at first increases linearly as the starting ligand concentration is increased, then asymptotically approaches full saturation (''y''=1). The dissociation constant <math>K_D</math> is equal to the ligand concentration ''[L]'' for which ''y'' = 1/2.]] | |
− | + | [[Image:20.109_Affinity.png|thumb|200px|right|'''Semilog Binding Curves''' By converting ligand concentrations to logspace, the dissociation constants are readily determined from the sigmoidal curves' inflection points. The three curves each represent different ligand species. The middle curve has a <math>K_D</math> close to 10 nM, while the right-hand curve has a higher <math>K_D</math> and therefore lower affinity between ligand and receptor (vice-versa for the left-hand curve).]] | |
− | + | Let s start by considering the simple case of a receptor-ligand pair that are exclusive to each other, and in which the receptor is monovalent. The ligand (L) and receptor (R) form a complex (C), which reaction can be written | |
<center> | <center> | ||
− | R + L | + | <math> R + L \rightleftharpoons\ ^{k_f}_{k_r} C </math> |
</center> | </center> | ||
− | At equilibrium, the rates of the forward and reverse | + | At equilibrium, the rates of the forward reaction (rate constant = <math>k_f</math>) and reverse reaction (rate constant = <math>k_r</math>) must be equivalent. Solving this equivalence yields an equilibrium dissociation constant <math>K_D</math>, which may be defined either as <math>k_r/k_f</math>, or as <math>[R][L]/[C]</math>, where brackets indicate the molar concentration of a species. Meanwhile, the fraction of receptors that are bound to ligand at equilibrium, often called ''y'', is <math>C/R_{TOT}</math>, where <math>R_{TOT}</math> indicates total (both bound and unbound) receptors. Note that the position of the equilibrium (i.e., ''y'') depends on the starting concentrations of the reactants; however, <math>K_D</math> is always the same value. The total number of receptors <math>R_{TOT}</math>= ''[C]'' (ligand-bound receptors) + ''[R]'' (unbound receptors). Thus, |
− | + | <center> | |
+ | <math>\qquad y = {[C] \over R_{TOT}} \qquad = \qquad {[C] \over [C] + [R]} \qquad = \qquad {[L] \over [L] + [K_D]} \qquad</math> | ||
+ | </center> | ||
− | + | where the right-hand equation was derived by algebraic substitution. If the ligand concentration is in excess of that of the receptor, ''[L]'' may be approximated as a constant, ''L'', for any given equilibrium. Let s explore the implications of this result: | |
− | What happens when | + | *What happens when ''L'' << <math>K_D</math>? |
+ | ::→Then ''y'' ~ <math>L/K_D</math>, and the binding fraction increases in a first-order fashion, directly proportional to ''L''. | ||
− | What happens when | + | *What happens when ''L'' >> <math>K_D</math>? |
+ | ::→In this case ''y'' ~1, so the binding fraction becomes approximately constant, and the receptors are saturated. | ||
− | + | *What happens when ''L'' = <math>K_D</math>? | |
+ | ::→Then ''y'' = 0.5, and the fraction of receptors that are bound to ligand is 50%. This is why you can read <math>K_D</math> directly off of the plots in Nagai s paper (compare Figure 3 and Table 1). When y = 0.5, the concentration of free calcium (our ''[L]'') is equal to <math>K_D</math>. '''This is a great rule of thumb to know.''' | ||
− | < | + | The figures at right demonstrate how to read <math>K_D</math> from binding curves. You will find semilog plots (bottom) particularly useful today, but the linear plot (top) can be a helpful visualization as well. Keep in mind that every ''L'' value is associated with a particular equilbrium value of ''y'', while the curve as a whole gives information on the global equilibrium constant <math>K_D</math>. |
+ | |||
+ | Of course, inverse pericam has multiple binding sites, and thus IPC-calcium binding is actually more complicated than in the example above. The <math>K_D</math> reported by Nagai is called an apparent <math>K_D</math> |