(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 531110, 7388]*) (*NotebookOutlinePosition[ 532025, 7418]*) (* CellTagsIndexPosition[ 531981, 7414]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Overview", "Subsection", Editable->False], Cell[TextData[{ "The scheme below depicts a number of different ligand-lattice binding \ possiblities. The lattice is assumed to be an infinite linear array of \ regularly spaced identical lattice residues. Ligands are assumed to \ reversibly bind to an integer number (", StyleBox["n", FontSlant->"Italic"], ") of contiguous residues in a manner that prevents binding by additional \ ligands. (A) Symmetric ligands (", StyleBox["n", FontSlant->"Italic"], " = 1) binding to an isotropic lattice. (B) Asymmetric ligands (", StyleBox["n", FontSlant->"Italic"], " = 2) wth distinguishable \"head\" and \"tail\" ends binding to an \ isotropic lattice. (C) Asymmetric ligands (", StyleBox["n", FontSlant->"Italic"], " = 3) binding to an anisotropic lattice in which left and right are \ distinguishable. (D) Two types of symmetric ligands binding competitively \ to an isotropic lattice. (E) Two types of symmetric ligands binding \ competitively to an anisotropic lattice. (F) Flexible asymmetric ligands \ binding to an anisotropic lattice in two different binding modes, depending \ on their orientation. In this case, ", StyleBox["n", FontSlant->"Italic"], " = 2 when the head of the bound ligand is to the left, and ", StyleBox["n", FontSlant->"Italic"], " = 1 when the head of the bound ligand is to the right. 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Whether the lattice is isotropic or anisotropic actually \ makes no difference in this case. (A). The association constants for \ interactions between adjacent bound ligands and for ligand-lattice binding \ are \[Omega] and K, respectively. Nonadjacent bound ligands are assumed not \ to interact. (B). The binding affinities of ligands to sites with zero, one \ or two neighbors (adjacent bound ligands). (C and D). The use of conditional \ probabilities to calculate the relative likelihood of two lattice \ configurations (see derivation). The cooperativity parameter \[Omega] is the \ ratio of the probability of configuration D to that of configuration C. \ These probabilities are proportional to the products of the conditional \ probabilities shown. A free lattice residue is denoted f, and a bound \ ligand, b. 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that given by Wolfe & Meehan \ (1992). The derivations of the equations for the other cases are quite \ similar to that of this simplest case. \n\n\n(a) Non-Interacting Binding \ Sites\n\n Let ", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["T", FontSlant->"Plain"]], TraditionalForm]]], ", ", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["B", FontSlant->"Plain"]], TraditionalForm]]], " be the total, free, and bound concentrations of a ligand (generally a \ small molecule) that binds to a macromolecule whose concentration (in terms \ of binding sites) is \[ScriptCapitalD], without any competition from other \ types of ligands. If binding sites are uniform and noninteracting, the \ mass-action expression for the association constant is: \n\n\tK = ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ FractionBox[ StyleBox[ SubscriptBox["L", StyleBox["B", FontSlant->"Plain"]], FontSize->12], RowBox[{ RowBox[{"(", " ", RowBox[{ SubscriptBox["L", StyleBox["T", FontSlant->"Plain"]], " ", "\[Dash]", " ", SubscriptBox["L", StyleBox["B", FontSlant->"Plain"]]}], ")"}], RowBox[{"(", RowBox[{"\[ScriptCapitalD]", " ", "\[Dash]", " ", SubscriptBox["L", StyleBox["B", FontSlant->"Plain"]]}], ")"}]}]], FontSize->17], " ", "=", " ", StyleBox[ FractionBox[ RowBox[{ StyleBox["\[Nu]", FontSize->12], StyleBox["/", FontSize->12], StyleBox[" ", FontSize->12], SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]]}], RowBox[{ StyleBox["1", FontSize->12], StyleBox["\[Dash]", FontSize->12], StyleBox[" ", FontSize->12], "\[Nu]"}]], FontSize->17]}], TraditionalForm]]], ", \t \t(", StyleBox["n", FontSlant->"Italic"], ", \[Omega] = 1)\[Dagger]\n\n\[Dagger] ", StyleBox[ "Restrictions on the applicability of the equations in this derivation are \ given in parentheses to the right of the equation (those noted here are \ explained in the next two sections). ", FontSize->10], "\n\nwhere \[Nu] ", StyleBox["\[Congruent]", FontFamily->"Symbol"], " ", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["B", FontSlant->"Plain"]], TraditionalForm]]], "/\[ScriptCapitalD]. This can be rearranged into the Scatchard (1949) form \ as: \n\n\t\t\[Nu]/", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " K ,\t \t\t(1a)\t (\[Omega] = 1)\nwhere: \n\t\t", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " = 1", StyleBox[" ", FontSize->9], "\[Dash] \[Nu] \t \t\t(1b)\t(", StyleBox["n", FontSlant->"Italic"], " = 1)\n\nis the fraction of binding sites that are free. The plot of \ \[Nu]/", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]], TraditionalForm]]], " versus \[Nu] suggested by the Scatchard equation is the one most commonly \ used to display binding data. \n\n\n(b) The Effect of Binding Site Overlap\n\ \n In order to analyze the interactions of ligands with a long linear \ macromolecule such as a polynucleotide, it is convenient to model the \ macromolecule as an infinite one-dimensional lattice of identical residues. \ It will be assumed that a bound ligand occupies a string of ", StyleBox["n", FontSlant->"Italic"], " consecutive residues (where ", StyleBox["n", FontSlant->"Italic"], " is a positive integer), and that a given lattice residue can only be \ bound by one ligand at a time. (Non-integral values of ", StyleBox["n", FontSlant->"Italic"], " are algebraically permissible, and arise frequently when ", StyleBox["n", FontSlant->"Italic"], " is used as a fitting parameter in data analysis, but the physical \ interpretation of such values may be ambiguous.) Free binding sites are \ those in which all ", StyleBox["n", FontSlant->"Italic"], " residues are free; bound sites are those in which all ", StyleBox["n", FontSlant->"Italic"], " residues are bound by the same ligand. Sites in which some residues are \ free and others bound, or in which residues are bound by two different \ ligands, will be termed occluded; they do not participate in binding. A \ binding site can begin at any lattice residue, so the total concentration of \ potential binding sites (\[ScriptCapitalD]) equals the total concentration of \ residues. Since it is not generally necessary to be concerned with lattice \ concentration, it is convenient to convert the concentration of a \ lattice-associated item into the unitless density of the item on the lattice \ by dividing the item concentration by the lattice residue concentration \ (e.g., \[Nu] and ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " are the densities of bound ligands and free binding sites). The density \ of free residues, i.e. the fraction of residues that are free, denoted \ \[Florin], is: \n\n\t\t\[Florin] = 1", StyleBox[" ", FontSize->9], "\[Dash] ", StyleBox["n", FontSlant->"Italic"], " \[Nu] . \t\t\t(2)\n\nWhen ", StyleBox["n", FontSlant->"Italic"], " is greater than one, only a fraction of the binding sites can be bound at \ one time, and ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " will not vary linearly with \[Nu]. \n Conditional probabilities \ describe the distribution of free residues and bound ligands within a lattice \ and provide a simple means of obtaining an expression for ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], ". In general, a conditional probability is the probability that a \ randomly chosen example of a given type of lattice-associated item is \ bordered on a given side by a given type of neighbor. In this case, both \ \"item\" and \"neighbor\" refer to either a free residue or bound ligand. \ Each item of either type has two neighbors. In the notation, the randomly \ chosen item is listed first and the neighbor second. The conditional \ probability ", StyleBox["\[Alpha]\[Beta]", FontFamily->"Symbol"], " is the density of ", StyleBox["\[Alpha]", FontFamily->"Symbol"], "-", StyleBox["\[Beta]", FontFamily->"Symbol"], " contacts of the specified orientation divided by the density of ", StyleBox["\[Alpha]", FontFamily->"Symbol"], StyleBox[" ", FontSize->5], ". When ", StyleBox["\[Alpha]", FontFamily->"Symbol"], " and ", StyleBox["\[Beta]", FontFamily->"Symbol"], " differ, only contacts with the same orientation (with ", StyleBox["\[Beta]", FontFamily->"Symbol"], " to a given side of ", StyleBox["\[Alpha]", FontFamily->"Symbol"], ") can be considered, even if there is no inherent difference in properties \ between contacts of opposite orientations, simply to ensure that no more than \ one contact per item is possible. Otherwise, the value of the probability \ could exceed one, since all examples of ", StyleBox["\[Alpha]", FontFamily->"Symbol"], " might be flanked by ", StyleBox["\[Beta]", FontFamily->"Symbol"], " on both sides. (The orientation problem does not arise when ", StyleBox["\[Alpha]", FontFamily->"Symbol"], " and ", StyleBox["\[Beta]", FontFamily->"Symbol"], " are the same, because in that case the maximum possible density of \ contacts equals the density of items, and each contact can be viewed as \ having either orientation.) The probability that a randomly chosen free \ residue is bordered on a given side by another free residue will be termed", StyleBox[" ", FontSize->10], "ff", StyleBox[" ", FontSize->4], ". \n It will be assumed initially that ligands are symmetric. In \ this case, if there is only one type of ligand, there are three other \ conditional probabilities, and the system is isotropic (even if the lattice \ is not). The probability that a randomly chosen bound ligand is bordered on \ a given side by a free lattice residue will be designated", StyleBox[" ", FontSize->10], "bf", StyleBox[" ", FontSize->4], ", and the probability that it is adjacent on a given side to another bound \ ligand will be denoted", StyleBox[" ", FontSize->10], "bb", StyleBox[" ", FontSize->4], ". The probability that a free residue is bordered on a given side by a \ bound ligand is", StyleBox[" ", FontSize->10], "fb", StyleBox[" ", FontSize->4], ". Because there are only two types of neighbors, we may write: \n\n\t\t\ ff + fb = 1\t\t\t(3a)\nand\t\tbf + bb = 1 . \t\t\t(3b)\n\nAccording \ to the definition of a conditional probability given above, the ratio of \ conditional probabilities ", StyleBox["\[Alpha]\[Beta]", FontFamily->"Symbol"], "/", StyleBox["\[Gamma]\[Delta]", FontFamily->"Symbol"], " is the density ratio of ", StyleBox["\[Gamma]", FontFamily->"Symbol"], " to ", StyleBox["\[Alpha]", FontFamily->"Symbol"], " multiplied by the ratio of the density of ", StyleBox["\[Alpha]", FontFamily->"Symbol"], "-", StyleBox["\[Beta]", FontFamily->"Symbol"], " contacts to the density of ", StyleBox["\[Gamma]", FontFamily->"Symbol"], " -", StyleBox["\[Delta]", FontFamily->"Symbol"], " contacts. In a ratio of mirror-image conditional probabilities of the \ form ", StyleBox["\[Alpha]\[Beta]", FontFamily->"Symbol"], "/", StyleBox["\[Beta]\[Alpha]", FontFamily->"Symbol"], ", the same type of contact is involved in both cases, so the contact \ densities drop out. The densities of free residues and bound ligands are \ \[Florin] and \[Nu], giving: \n\n\t\tfb", StyleBox[" ", FontSize->9], "/", StyleBox[" ", FontSize->9], "bf = \[Nu]/\[Florin].\t\t\t(4)\n\nThe quantity \[Nu]/\[Florin] will be \ denoted ", StyleBox["x", FontSlant->"Italic"], ". Substituting from equation (4) into equation (3a) gives: \n\n\t\tff + \ ", StyleBox["x", FontSlant->"Italic"], " bf = 1 . \t\t\t(5)\n\nThis relationship can also be obtained from \ equation (5) of McGhee & von Hippel (1974). In the absence of cooperativity \ (attractive or repulsive interactions between bound ligands),", StyleBox[" ", FontSize->9], "bf", StyleBox[" ", FontSize->9], "must equal", StyleBox[" ", FontSize->9], "ff ; equation (5) then gives: \n\n\t\tff = ", Cell[BoxData[ FormBox[ StyleBox[\(1\/\(1\ + \ \ x\)\), FontSize->17], TraditionalForm]]], " . \t\t\t(6)\t(\[Omega] = 1)\t \n\t\nThe probability that a binding site \ is free, ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], ", equals the probability that any given residue within it is free, \ \[Florin] , times the probability that the", StyleBox[" ", FontSize->9], StyleBox["n", FontSlant->"Italic"], "\[Dash]1", StyleBox[" ", FontSize->9], "adjacent residues that make up the rest of the site are also free. Thus: \ \n\n\t\t", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " = \[Florin] ", Cell[BoxData[ FormBox[ SuperscriptBox["ff", RowBox[{" ", RowBox[{ StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], "\[Dash]", " ", "1"}]}]], TraditionalForm]]], "\t\t\t(7)\n\n(McGhee & von Hippel, 1974, equation (9)). This expression \ is based on the assumption that the status of one neighbor of a free residue \ is independent of the status of the other. It would not be valid for a \ circular lattice, where an influence from one neighbor of a free residue \ could propagate around the lattice to the other neighbor (an effect which \ would vary inversely with the circumference of the circle). As an extreme \ example, consider the binding of ligands with ", StyleBox["n", FontSlant->"Italic"], " = 3 to circular lattices of five residues. At ", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]], TraditionalForm]]], " = \[Infinity], each lattice would have one bound ligand, giving \[Florin] \ = 0.4 and ff = 0.5. However, ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " would be zero, not 0.1 as indicated by equation (7). \n On the \ basis of equations (1a), (2), (6), and (7), the binding equation for the \ noncooperative case is: \n \n\t\t", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox["\[Nu]", SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]]], " "}], FontSize->17], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ StyleBox[\(\[Nu]\/x\), FontSize->17], TraditionalForm]]], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox["K", FontSlant->"Plain"], SuperscriptBox[ RowBox[{ StyleBox["(", FontSize->17], \(1\ + \ \ x\), StyleBox[")", FontSize->16.0625]}], \(\ n\ \[Dash]\ 1\)]], FontSize->17], TraditionalForm]]], "\t\t(8)\t(\[Omega] = 1)\n\t \t \nwhere ", StyleBox["x", FontSlant->"Italic"], " ", StyleBox["\[Congruent]", FontFamily->"Symbol"], " \[Nu]/(1", StyleBox[" ", FontSize->9], "\[Dash] ", StyleBox["n", FontSlant->"Italic"], " \[Nu]). This is equivalent to the solutions of Zasedatelev et al. (1971, \ equation (13)) and McGhee & von Hippel (1974, equation (10)). The form of \ the equation illustrates that \[Nu] does not need to appear in it at all, \ except through ", StyleBox["x", FontSlant->"Italic"], " (this is also true of binding equations for more complicated \ situations). When ", StyleBox["n", FontSlant->"Italic"], " > 1, the binding isotherms are concave upward and hit the \[Nu]-axis \ intercept at 1/", StyleBox["n", FontSlant->"Italic"], " (representing complete coverage of the lattice) at a shallow angle. \ When ", StyleBox["n", FontSlant->"Italic"], " = 1, equation (8) reduces to equation (1) and the plot is linear. \n\n\n\ (c) Effects of Cooperativity on Binding \n\n If binding is \ cooperative (or anticooperative), a bound ligand increases (or decreases) the \ binding affinity of neighboring sites. For a single type of symmetric \ ligand, only one type of ligand-ligand interaction and one type of \ ligand-lattice interaction are possible (Figure 1), so whether the lattice is \ isotropic or anisotropic is irrelevant. It will be assumed throughout this \ discussion that cooperative effects extend only one lattice residue from the \ end of a bound ligand, i.e. two ligands can interact only when they are in \ adjacent sites. \n In the presence of cooperativity, the average \ affinities of free and bound sites differ from that of isolated sites (K) by \ an amount that (usually) increases with \[Nu]. This introduces additional \ curvature into the binding isotherm, so equation (1a) (using equation ( 7)) \ must be rewritten: \n\n\t\[Nu]/", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]], TraditionalForm]]], " = ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]]}], TraditionalForm]]], " = \[Florin] ", Cell[BoxData[ FormBox[ SuperscriptBox["ff", RowBox[{" ", RowBox[{ StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"], "\[Dash]", " ", "1"}]}]], TraditionalForm]]], " ", Cell[BoxData[ FormBox[ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], ", \t\t(9)\n\nwhere ", Cell[BoxData[ FormBox[ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " is the average association constant of free sites. This quantity is used \ because association is forbidden at bound or occluded sites. In the presence \ of cooperative interactions, ", Cell[BoxData[ FormBox[ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " will always be less than the average affinity of bound sites. However, \ because the law of mass action applies to the lattice as a whole, as well as \ to each individual class of binding sites, ", Cell[BoxData[ FormBox[ FormBox[ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], "TraditionalForm"], TraditionalForm]]], " must equal the reciprocal of the average dissociation constant of bound \ sites. \n The cooperativity parameter \[Omega] is a unitless binding \ constant for the interaction between bound ligands, i.e. the equilibrium \ constant for moving a ligand from a site with ", StyleBox["g", FontSlant->"Italic"], StyleBox[" ", FontSize->10], "adjacent bound ligands to a site with ", StyleBox["g", FontSlant->"Italic"], "+1 adjacent ligands (where ", StyleBox["g", FontSlant->"Italic"], " = 0 or 1). It can be described as the ratio of the binding affinity of a \ site when bordered on a given end by a bound ligand to the affinity of the \ site when bordered on the same end by a free residue (holding the other \ neighbor of the site constant). The binding affinity of a non-occluded site \ with ", StyleBox["h", FontSlant->"Italic"], " neighboring bound ligands is thus K ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^\(\ h\)\)]], " (where ", StyleBox["h", FontSlant->"Italic"], " = 0, 1, or 2). Binding is cooperative if \[Omega] > 1, noncooperative \ if \[Omega] = 1, and anticooperative if \[Omega] < 1. If \[Omega] = 0, a \ bound ligand prohibits binding to adjacent sites (a phenomenon known as \ neighbor exclusion). \n Since the probability of binding is \ proportional to binding affinity, \[Omega]", StyleBox[" ", FontSize->10], "can be expressed in terms of conditional probabilities. This may be \ illustrated by considering two lattice configurations involved when a bound \ ligand moves the distance of a single residue from an isolated site to a site \ neighbored by one ligand (see the figure immediately above this derivation). \ The ratio of the probability of the latter configuration (part D of the \ figure) to that of the former (part C of the figure) is equal to \[Omega]. \ In calculating this ratio, one need only include conditional probabilities \ involving residues where the two configurations differ. Moving left to \ right, the probabilities of the relevant segments of configurations C and D \ are", StyleBox[" ", FontSize->10], "bf fb bf", StyleBox[" ", FontSize->10], "and", StyleBox[" ", FontSize->10], "bb bf ff, respectively; thus: \n \n\t\t\[Omega] = ", Cell[BoxData[ FormBox[ StyleBox[\(\(bb\ \ ff\)\/\(fb\ \ bf\)\), FontSize->17], TraditionalForm]]], "\t \t\t(10)\n\t \t\t\t\n(McGhee & von Hippel, 1974, equation (12)). \ Movement of the ligand for a greater distance does not alter the result. \n \ Since it is assumed that only adjacent bound ligands can interact, the \ two neighbors of a binding site whose status is defined (as being either free \ or bound) do not influence each other. That is, the status of the neighbor \ of one end of the binding site will have no effect on the probability of the \ presence of a given type of neighbor at the other end. The probability of a \ free binding site having a given set of neighbors is the product of the \ independent probabilities of the presence of each of the neighbors. The lack \ of interaction between the two neighbors also implies that the combined \ effect of both neighbors on a given site's binding affinity is the product of \ the characteristic effects of each of the neighbors. Thus, the average net \ effect of the cooperative influences emanating from both neighbors on the \ affinity of a free binding site is the product of the average effects of each \ of the neighbors considered separately. \n If there is only one type \ of ligand-ligand interaction, both neighbors are equivalent, and the average \ effect of the interactions at both ends will be the square of the average \ effect due to one end. In this case, one may define a \"neighbor-effect \ parameter\", denoted \[ScriptCapitalE] , which satisfies the relation: \n\n\t\ \t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], " "}], TraditionalForm]]], " = K ", Cell[BoxData[ \(TraditionalForm\`\[ScriptCapitalE]\^\(\ 2\)\)]], ". \t\t\t(11a) \n\n\[ScriptCapitalE] is the ratio of the average binding \ affinity of a free site if the status of the lattice residue neighboring one \ of its ends is unspecified (left to chance) to the affinity if that residue \ is free (holding the other neighbor of the site constant). \[ScriptCapitalE] \ is thus a statistical analogue of the cooperativity parameter \[Omega]. \n \ Conditional probabilities may be used to derive an expression for \ \[ScriptCapitalE] as a function of \[Nu]. The probability that the residue \ neighboring a given end of a free site is also free is", StyleBox[" ", FontSize->10], "ff", StyleBox[" ", FontSize->4], ", and the probability that it is bound is 1", StyleBox[" ", FontSize->9], "\[Dash] ff", StyleBox[" ", FontSize->4], ". Because of the independence of the two neighbors of a free site, the \ probabilities of the site having zero, one or two neighboring bound ligands \ are ", Cell[BoxData[ \(TraditionalForm\`ff\^\(\ 2\)\)]], ", 2 ff ", StyleBox["(", FontSize->14], "1", StyleBox[" ", FontSize->9], "\[Dash] ff", StyleBox[" ", FontSize->9], StyleBox[")", FontSize->14], ", and ", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{ StyleBox["(", FontSize->14], RowBox[{"1", StyleBox[" ", FontSize->9], "\[Dash]", " ", "ff"}], StyleBox[" ", FontSize->9], StyleBox[")", FontSize->14]}], "2"], TraditionalForm]]], ", respectively. The expression for ", Cell[BoxData[ FormBox[ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], " is thus: \n\n\t", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]], " "}], TraditionalForm]]], " = K ", StyleBox["(", FontSize->19], Cell[BoxData[ \(TraditionalForm\`\(\ ff\^\(\ 2\)\)\)]], "+ 2 ff ", StyleBox["(", FontSize->13], "1", StyleBox[" ", FontSize->9], "\[Dash] ff", StyleBox[" ", FontSize->9], StyleBox[")", FontSize->13], " \[Omega] +", StyleBox[" ", FontSize->9], Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{ StyleBox["(", FontSize->14], RowBox[{"1", StyleBox[" ", FontSize->9], "\[Dash]", " ", "ff"}], StyleBox[" ", FontSize->9], StyleBox[")", FontSize->14]}], "2"], TraditionalForm]]], Cell[BoxData[ \(TraditionalForm\`\[Omega]\^\(\ 2\)\)]], StyleBox[")\t\t", FontSize->18], "(11b) \n \nComparison of equations (11a) and (11b) yields: \n\n\t\ \t\[ScriptCapitalE] = ff + \[Omega] (1", StyleBox[" ", FontSize->9], "\[Dash] ff ) , \t\t(12) \n\nor:\t \tff = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[\(\[Omega]\ \ \[Dash]\ \[ScriptCapitalE]\), RowBox[{"\[Omega]", " ", StyleBox["\[Dash]", FontSize->11.0625], StyleBox[" ", FontSize->11.0625], StyleBox["1", FontSize->12]}]], FontSize->17], TraditionalForm]]], ". \t \t\t(13)\t(\[Omega] \[NotEqual] 1)\n\nAs a lattice is saturated,", StyleBox[" ", FontSize->10], "ff", StyleBox[" ", FontSize->10], "and", StyleBox[" ", FontSize->10], "bf", StyleBox[" ", FontSize->10], "go from one to zero,", StyleBox[" ", FontSize->10], "fb", StyleBox[" ", FontSize->10], "and", StyleBox[" ", FontSize->10], "bb", StyleBox[" ", FontSize->10], "go from zero to one, and \[ScriptCapitalE] goes from one to \[Omega] \ (the change as a function of \[Nu] is monotonic in each case). When \[Nu] = \ 1/", StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSize->4], ",", StyleBox[" ", FontSize->10], "ff", StyleBox[" ", FontSize->10], "and", StyleBox[" ", FontSize->10], "fb", StyleBox[" ", FontSize->10], "are, strictly speaking, undefined. \n Substituting from equation \ (10) into equation (12) and simplifying with equations (3a) and (3b) gives: \ \n\n\t\t\[ScriptCapitalE] = ff", StyleBox[" ", FontSize->9], "/", StyleBox[" ", FontSize->9], "bf . \t\t\t(14)\n\nThis relationship can be rationalized by noting that \ ff", StyleBox[" ", FontSize->9], "/", StyleBox[" ", FontSize->9], "bf is a measure of the extent to which free residues are clustered, which \ increases or decreases (depending on \[Omega]) as the lattice is saturated. \ Using equation (14) to eliminate bf from equation (5) gives: \n\n\t \tff \ = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox[\(\ 1\), FontSize->12], RowBox[{ StyleBox[" ", FontSize->12], RowBox[{ StyleBox["1", FontSize->12], StyleBox["+", FontSize->12], StyleBox[" ", FontSize->12], StyleBox[\(x\/\[ScriptCapitalE]\), FontSize->17]}]}]], FontSize->17], TraditionalForm]]], ". \t\t\t(15a)\n\nOn the basis of equations (14), (4), and (10), the \ expressions for the other conditional probabilities in terms of ff are: \n\n\ \tbf = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ StyleBox["ff", FontSize->12], "\[ScriptCapitalE]"], FontSize->17], TraditionalForm]]], " , \tfb = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{"x", " ", StyleBox["ff", FontSize->12]}], "\[ScriptCapitalE]"], FontSize->17], TraditionalForm]]], " , \tand\tbb = ", Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{"x", " ", "\[Omega]", " ", StyleBox["ff", FontSize->12]}], \(\(\[ScriptCapitalE]\ \)\^2\)], FontSize->17], TraditionalForm]]], " . \t\t(15b,c,d)\n\t\nCombining equations (13) and (15a) gives an \ expression for \[ScriptCapitalE] without conditional probabilities: \n\n\t \ \[ScriptCapitalE] = ", Cell[BoxData[ FormBox[ FractionBox[ RowBox[{ StyleBox[" ", FontSize->12], RowBox[{ RowBox[{ StyleBox["1", FontSize->12], StyleBox["\[Dash]", FontSize->11.0625], StyleBox[" ", FontSize->13], StyleBox["x", FontSize->12]}], StyleBox[ RowBox[{ StyleBox[" ", FontSize->13], " "}]], StyleBox["+", FontSize->12], StyleBox[ RowBox[{ StyleBox[" ", FontSize->12], StyleBox[" ", FontSize->13]}]], StyleBox[ FormBox[ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{ StyleBox["(", FontSize->14], RowBox[{ StyleBox["1", FontSize->12], " ", StyleBox["\[Dash]", FontSize->11.0625], " ", StyleBox["x", FontSize->12, FontSlant->"Italic"]}], StyleBox[")", FontSize->14, FontSlant->"Italic"]}], \(\ 2\)], StyleBox[" ", FontSlant->"Italic"], StyleBox["+", FontSize->12], StyleBox[ RowBox[{" ", StyleBox[" ", FontSize->12]}]], RowBox[{ StyleBox["4", FontSize->12], StyleBox[" ", FontSlant->"Italic"], StyleBox["x", FontSize->12, FontSlant->"Italic"], " ", "\[Omega]"}]}]], "TraditionalForm"], FontSize->13]}]}], StyleBox["2", FontSize->12]], TraditionalForm]]], "\t\t(16)\n\n(only one root of the quadratic gives physically relevant \ positive values for \[ScriptCapitalE] ). When \n\[Nu] = 1/(", StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSize->9], "+", StyleBox[" ", FontSize->9], "1), ", StyleBox["x", FontSlant->"Italic"], " ", StyleBox["=", FontFamily->"Symbol"], " 1 and \[ScriptCapitalE] = ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\^\(\ 1/2\)\)]], ". \n On the basis of equations (2), (9), (11a), and (15a), the \ binding equation for symmetric cooperative ligands is: \n\n\t\t", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ FractionBox["\[Nu]", SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]]], " "}], FontSize->17], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{\(\[Nu]\/x\), FractionBox[ RowBox[{ StyleBox["K", FontSlant->"Plain"], " ", \(\[ScriptCapitalE]\^\(\ 2\)\)}], SuperscriptBox[ RowBox[{ StyleBox["(", FontSize->17], \(1 + \ \ x/\ \[ScriptCapitalE]\), StyleBox[")", FontSize->16.0625]}], \(\ n\ \[Dash]\ 1\)]]}], FontSize->17], TraditionalForm]]], ", \t\t(17)\n\nwhere ", StyleBox["x", FontSlant->"Italic"], " ", StyleBox["\[Congruent]", FontFamily->"Symbol"], " \[Nu]/(1", StyleBox[" ", FontSize->9], "\[Dash] ", StyleBox["n", FontSlant->"Italic"], " \[Nu]). This is equivalent to the solutions of Zasedatelev et al. (1971, \ equation (16)) and McGhee & von Hippel (1974, equation (15); see erratum). \ Values of \[Omega]", StyleBox[" ", FontSize->10], "above one tend to produce a humped, convex upward plot that hits the \ \[Nu]-axis intercept at 1/", StyleBox[" ", FontSize->5], StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSize->10], "at a steep angle (a tendency which is opposed by the effect of a value of \ ", StyleBox["n", FontSlant->"Italic"], " > 1). It is obvious from equation (12) or (16) that \[ScriptCapitalE] = \ 1 when \[Omega] = 1; in that case equation (17) reduces to equation (8). \ Values of \[Omega] less than one lead to a concave upward plot. When \ \[Omega] = 0, the limiting value of \[Nu] is 1/(", StyleBox["n", FontSlant->"Italic"], StyleBox[" ", FontSize->9], "+", StyleBox[" ", FontSize->9], "1), and \[ScriptCapitalE] = ff = 1", StyleBox[" ", FontSize->9], "\[Dash]", StyleBox[" ", FontSize->9], StyleBox["x", FontSlant->"Italic"], ", from which one can show that ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]]}], TraditionalForm]]], " for \[Omega] = 0, ", StyleBox["n", FontSlant->"Italic"], " = m is equal to ", Cell[BoxData[ FormBox[ SubscriptBox["\[ScriptCapitalS]", StyleBox["F", FontSize->9, FontSlant->"Plain"]], TraditionalForm]]], Cell[BoxData[ FormBox[ RowBox[{" ", SubscriptBox[\(K\&_\), StyleBox["F", FontSize->9, FontSlant->"Plain"]]}], TraditionalForm]]], " for \[Omega] = 1, ", StyleBox["n", FontSlant->"Italic"], " = m+1. When ", StyleBox["n", FontSlant->"Italic"], " = 1, none of the \[Nu]/", Cell[BoxData[ FormBox[ SubscriptBox["L", StyleBox["F", FontSlant->"Plain"]], TraditionalForm]]], " plots have inflection points. When both \[Omega] and ", StyleBox["n", FontSlant->"Italic"], " are > 1, an inflection point (where the second derivative is increasing) \ may be present at high \[Nu], although in most cases it probably would not be \ experimentally discernible. \n\nReferences\n\nMcGhee, J. D. & von Hippel, P. \ H. (1974). Theoretical Aspects of DNA-Protein Interactions: Co-operative \ and Non-co-operative Binding of Large Ligands to a One-dimensional Homogenous \ Lattice. ", StyleBox["J. Mol. Biol.", FontSlant->"Italic"], " ", StyleBox["86", FontWeight->"Bold"], ", 469-489. Erratum (1976) ", StyleBox["103", FontWeight->"Bold"], ", 679. \n\nScatchard, G. (1949). The Attractions of Proteins for Small \ Molecules and Ions. ", StyleBox["Ann. N. Y. Acad. Sci.", FontSlant->"Italic"], " ", StyleBox["51", FontWeight->"Bold"], ", 660-672. \n\nWolfe, A. R. & Meehan, T. (1992). The Use of Binding Site \ Neighbor-Effect Parameters to Evaluate the Interactions Between Adjacent \ Ligands on a Linear Lattice. Effects on Ligand-Lattice Association. ", StyleBox["J. Mol. Biol.", FontSlant->"Italic"], " ", StyleBox["223", FontWeight->"Bold"], ", 1063-1087. \n\nZasedatelev, A. S., Gurskii, G. V. & Vol'kenshtein, M. \ V. (1971). Theory of One-Dimensional Adsorption. I. Adsorption of Small \ Molecules on a Homopolymer. ", StyleBox["Mol. Biol.", FontSlant->"Italic"], " ", StyleBox["5", FontWeight->"Bold"], ", 245-251. " }], "Text", Editable->False, FontFamily->"Palatino"] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Asymmetric Ligands of One Type Binding to an Isotropic \ Lattice\ \>", "Subsection", Editable->False], Cell[TextData[{ "Schematic depiction of the binding of asymmetric ligands to an isotropic \ lattice. (A and B). The cooperativity parameter for head-head interactions \ between adjacent bound ligands is ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_1\)]], ", that for tail-tail interactions is ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_\(\ 2\)\)]], ", and that for head-tail or tail-head interactions is ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_\(\ 3\)\)]], ". (C). The association constant for ligand-lattice interactions. (D). \ Binding affinity of ligands to selected sites. (E and F). The cooperativity \ parameter ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_1\)]], " is the ratio of the probability of configuration F to that of \ configuration E. These probabilities are proportional to the products of the \ conditional probabilities shown. The head and tail ends of a bound ligand \ are designated ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["b", FontSlant->"Plain"], "1"], TraditionalForm]]], " and ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["b", FontSlant->"Plain"], "2"], TraditionalForm]]], ", respectively, while f denotes a free lattice residue. 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(A and B). The cooperativity parameters for bound ligand-adjacent \ bound ligand interactions. (C). The binding constant for interactions \ between head-left or head-right ligands and the lattice is ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontSlant->"Plain"], "1"], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontSlant->"Plain"], "2"], TraditionalForm]]], ", respectively. (D). Binding affinity of ligands to selected sites. (E \ and F). The cooperativity parameter ", Cell[BoxData[ \(TraditionalForm\`\[Omega]\_11\)]], " is the ratio of the probability of configuration F to that of \ configuration E. These probabilities are proportional to the products of the \ conditional probabilities shown. A ligand head at the left end of a binding \ site is designated ", Cell[BoxData[ FormBox[ StyleBox[ SubscriptBox[ StyleBox["b", FontSlant->"Plain"], StyleBox[\(1 L\), FontSlant->"Plain"]], FontSlant->"Italic"], TraditionalForm]]], ", and a ligand tail at the right end, ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["b", FontSlant->"Plain"], RowBox[{ StyleBox["2", FontSlant->"Italic"], StyleBox["R", FontSlant->"Plain"]}]], TraditionalForm]]], ", while f denotes a free lattice residue. 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(A and B). The cooperativity parameters \ for interactions between adjacent bound ligands. (C). The binding constant \ for interactions between ligands of type 1 or 2 and the lattice are ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontSlant->"Plain"], RowBox[{ StyleBox["L", FontSlant->"Plain"], " ", "1"}]], TraditionalForm]]], " or ", Cell[BoxData[ FormBox[ SubscriptBox[ StyleBox["K", FontSlant->"Plain"], RowBox[{ StyleBox["L", FontSlant->"Plain"], " ", "2"}]], TraditionalForm]]], ", respectively. (D). Binding affinity of ligands to selected sites. (E \ and F). The cooperativity parameter ", Cell[BoxData[ FormBox[ SubscriptBox["\[Omega]", RowBox[{" ", RowBox[{ StyleBox["L", FontSlant->"Plain"], " ", "12"}]}]], TraditionalForm]]], " is the ratio of the probability of configuration F to that of \ configuration E. These probabilities are proportional to the products of the \ conditional probabilities shown. 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