Overview

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 (n) of contiguous residues in a manner that prevents binding by additional ligands.  (A)  Symmetric ligands (n = 1) binding to an isotropic lattice.  (B)  Asymmetric ligands (n = 2) wth distinguishable "head" and "tail" ends binding to an isotropic lattice.  (C)  Asymmetric ligands (n = 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, n = 2 when the head of the bound ligand is to the left, and n = 1 when the head of the bound ligand is to the right.

Symmetric Ligands of One Type Binding to a Lattice

Schematic depiction of the binding of symmetric ligands of a single type to a linear lattice.  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 ω 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 ω 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.

Derivation of Equations for Symmetric Ligands of One Type

The following treatment is adapted from that given by Wolfe & Meehan (1992).  The derivations of the equations for the other cases are quite similar to that of this simplest case.  


(a) Non-Interacting Binding Sites

        Let (Subscript[L, T]), (Subscript[L, F]), and (Subscript[L, B]) 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 𝒟, without any competition from other types of ligands.  If binding sites are uniform and noninteracting, the mass-action expression for the association constant is:  

	K  =  ((Subscript[L, B])/(( (Subscript[L, T]) –  (Subscript[L, B]))(𝒟  –  (Subscript[L, B]))))  =    ((ν/ (Subscript[L, F]))/(1– ν)), 	 	(n, ω = 1)†

†  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).  

where ν ≡  (Subscript[L, B])/𝒟.  This can be rearranged into the Scatchard (1949) form as:  

		ν/(Subscript[L, F]) =  (Subscript[𝒮, F]) K ,	  		(1a)	 (ω = 1)
where:  
		(Subscript[𝒮, F])  = 1 – ν 	  		(1b)	(n = 1)

is the fraction of binding sites that are free.  The plot of ν/(Subscript[L, F]) versus ν suggested by the Scatchard equation is the one most commonly used to display binding data.  


(b) The Effect of Binding Site Overlap

        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 n consecutive residues (where n is a positive integer), and that a given lattice residue can only be bound by one ligand at a time.  (Non-integral values of n are algebraically permissible, and arise frequently when n 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 n residues are free; bound sites are those in which all n 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 (𝒟) 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., ν and (Subscript[𝒮, F]) 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  ƒ, is:  

		ƒ  =   1 – n ν .  			(2)

When n is greater than one, only a fraction of the binding sites can be bound at one time, and (Subscript[𝒮, F]) will not vary linearly with ν.  
        Conditional probabilities describe the distribution of free residues and bound ligands within a lattice and provide a simple means of obtaining an expression for (Subscript[𝒮, F]).  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 αβ is the density of α-β contacts of the specified orientation divided by the density of  α .  When α and β differ, only contacts with the same orientation (with β to a given side of  α) 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 α might be flanked by β on both sides.  (The orientation problem does not arise when α and β 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  ff .  
        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   bf , and the probability that it is adjacent on a given side to another bound ligand will be denoted  bb .  The probability that a free residue is bordered on a given side by a bound ligand is  fb .  Because there are only two types of neighbors, we may write:  

		ff  +  fb  =  1			(3a)
and		bf  +  bb  =  1 .  			(3b)

According to the definition of a conditional probability given above, the ratio of conditional probabilities αβ/γδ is the density ratio of γ  to α multiplied by the ratio of the density of α-β contacts to the density of γ -δ contacts.  In a ratio of mirror-image conditional probabilities of the form αβ/βα, 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  ƒ and ν, giving:  

		fb / bf  =  ν/ƒ.			(4)

The quantity ν/ƒ  will be denoted x.  Substituting from equation (4) into equation (3a) gives:  

		ff  +  x  bf  =  1 .  			(5)

This 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),  bf  must equal  ff ; equation (5) then gives:  

		ff  =   (1/(1  +   x)) . 			(6)	(ω = 1)	 
	
The probability that a binding site is free, (Subscript[𝒮, F]), equals the probability that any given residue within it is free, ƒ , times the probability that the  n–1  adjacent residues that make up the rest of the site are also free.  Thus:  

		(Subscript[𝒮, F])  =   ƒ  (ff^( n – 1))			(7)

(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 n = 3 to circular lattices of five residues.  At (Subscript[L, F]) = ∞, each lattice would have one bound ligand, giving ƒ  = 0.4 and ff  = 0.5.  However, (Subscript[𝒮, F]) would be zero, not 0.1 as indicated by equation (7).  
        On the basis of equations (1a), (2), (6), and (7), the binding equation for the noncooperative case is:  
        
		(ν/(Subscript[L, F])) =  (ν/x)(K/(((1  +   x))^( n – 1)))		(8)	(ω = 1)
	 	 
where  x  ≡ ν/(1 – n ν).  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 ν does not need to appear in it at all, except through x  (this is also true of binding equations for more complicated situations).  When n > 1, the binding isotherms are concave upward and hit the ν-axis intercept at 1/n  (representing complete coverage of the lattice) at a shallow angle.  When n = 1, equation (8) reduces to equation (1) and the plot is linear.  


(c) Effects of Cooperativity on Binding  

        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.  
        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 ν.  This introduces additional curvature into the binding isotherm, so equation (1a) (using equation ( 7)) must be rewritten:  

	ν/(Subscript[L, F]) =  (Subscript[𝒮, F]) (Subscript[\(K\&_\), F]) =   ƒ  (ff^( n – 1)) (Subscript[\(K\&_\), F]), 		(9)

where  (Subscript[\(K\&_\), F]) 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, (Subscript[\(K\&_\), F]) 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, (Subscript[\(K\&_\), F])  must equal the reciprocal of the average dissociation constant of bound sites.  
        The cooperativity parameter ω is a unitless binding constant for the interaction between bound ligands, i.e. the equilibrium constant for moving a ligand from a site with g  adjacent bound ligands to a site with g+1 adjacent ligands (where g = 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  h neighboring bound ligands is thus  K (ω^( h)) (where h =  0, 1, or 2).  Binding is cooperative if ω > 1, noncooperative if ω = 1, and anticooperative if ω < 1.  If ω = 0, a bound ligand prohibits binding to adjacent sites (a phenomenon known as neighbor exclusion).  
        Since the probability of binding is proportional to binding affinity, ω  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 ω.  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  bf fb bf  and  bb bf ff, respectively; thus: 
        
		ω  =  ((bb  ff)/(fb  bf))	  		(10)
	  			
(McGhee & von Hippel, 1974, equation (12)).  Movement of the ligand for a greater distance does not alter the result.  
        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.  
        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 ℰ , which satisfies the relation:  

		(Subscript[\(K\&_\), F])   =  K (ℰ^( 2)). 			(11a)  

ℰ  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).  ℰ  is thus a statistical analogue of the cooperativity parameter ω.  
        Conditional probabilities may be used to derive an expression for ℰ  as a function of ν.  The probability that the residue neighboring a given end of a free site is also free is  ff , and the probability that it is bound is  1 – ff .  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 (ff^( 2)), 2 ff (1 – ff ), and (((1 – ff ))^2), respectively.  The expression for (Subscript[\(K\&_\), F]) is thus:  

	(Subscript[\(K\&_\), F])   =  K ( (ff^( 2))+ 2 ff (1 – ff ) ω  +  (((1 – ff ))^2)(ω^( 2)))		(11b)  
        
Comparison of equations (11a) and (11b) yields:  

		ℰ  =  ff  + ω (1 – ff ) , 		(12)  

or:	 	ff  =  ((ω  – ℰ)/(ω  –  1)). 	 		(13)	(ω ≠ 1)

As a lattice is saturated,  ff  and  bf  go from one to zero,  fb  and  bb  go from zero to one, and ℰ  goes from one to ω  (the change as a function of ν is monotonic in each case).  When ν = 1/n ,  ff  and  fb  are, strictly speaking, undefined.  
        Substituting from equation (10) into equation (12) and simplifying with equations (3a) and (3b) gives:  

		ℰ  =  ff / bf .  			(14)

This relationship can be rationalized by noting that ff / bf  is a measure of the extent to which free residues are clustered, which increases or decreases (depending on ω) as the lattice is saturated.  Using equation (14) to eliminate  bf  from equation (5) gives:  

	 	ff  =  (( 1)/( 1+ (x/ℰ))). 			(15a)

On the basis of equations (14), (4), and (10), the expressions for the other conditional probabilities in terms of ff  are:  

	bf  =  (ff/ℰ) , 	fb  =  ((x  ff)/ℰ) , 	and	bb  =  ((x  ω  ff)/(((ℰ )^2))) . 		(15b,c,d)
	
Combining equations (13) and (15a) gives an expression for ℰ  without conditional probabilities:  

	  ℰ  =  (( 1–  x  +  (Sqrt[(((1 –  x))^( 2)) +  4 x ω]))/2)		(16)

(only one root of the quadratic gives physically relevant positive values for ℰ ).  When 
ν = 1/(n + 1), x  = 1 and ℰ = (ω^( 1/2)).  
        On the basis of equations (2), (9), (11a), and (15a), the binding equation for symmetric cooperative ligands is:  

		(ν/(Subscript[L, F])) =  (ν/x)((K (ℰ^( 2)))/(((1 +   x/ ℰ))^( n – 1))), 		(17)

where  x  ≡ ν/(1 – n ν).  This is equivalent to the solutions of Zasedatelev et al. (1971, equation (16)) and McGhee & von Hippel (1974, equation (15); see erratum).  Values of ω  above one tend to produce a humped, convex upward plot that hits the ν-axis intercept at 1/ n  at a steep angle (a tendency which is opposed by the effect of a value of n > 1).  It is obvious from equation (12) or (16) that ℰ = 1 when ω = 1; in that case equation (17) reduces to equation (8).  Values of ω less than one lead to a concave upward plot.  When ω = 0, the limiting value of ν is 1/(n + 1), and ℰ = ff  = 1 – x, from which one can show that  (Subscript[𝒮, F]) (Subscript[\(K\&_\), F])  for ω = 0, n = m is equal to (Subscript[𝒮, F]) (Subscript[\(K\&_\), F])  for ω = 1, n = m+1.  When n = 1, none of the ν/(Subscript[L, F]) plots have inflection points.  When both ω and n are > 1, an inflection point (where the second derivative is increasing) may be present at high ν, although in most cases it probably would not be experimentally discernible.  

References

McGhee, 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.  J. Mol. Biol. 86, 469-489.  Erratum (1976) 103, 679.  

Scatchard, G. (1949).  The Attractions of Proteins for Small Molecules and Ions.  Ann. N. Y. Acad. Sci. 51, 660-672.  

Wolfe, 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.  J. Mol. Biol. 223, 1063-1087.  

Zasedatelev, A. S., Gurskii, G. V. & Vol'kenshtein, M. V. (1971).  Theory of One-Dimensional Adsorption.  I.  Adsorption of Small Molecules on a Homopolymer.  Mol. Biol. 5, 245-251.

Asymmetric Ligands  of One Type Binding to an Isotropic Lattice

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 (Subscript[ω, 1]), that for tail-tail interactions is (Subscript[ω,  2]), and that for head-tail or tail-head interactions is (Subscript[ω,  3]).  (C). The association constant for ligand-lattice interactions.  (D). Binding affinity of ligands to selected sites.  (E and F).  The cooperativity parameter (Subscript[ω, 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 (Subscript[b, 1]) and (Subscript[b, 2]), respectively, while  f  denotes a free lattice residue.

Asymmetric Ligands of One Type Binding to an Anisotropic Lattice

Schematic depiction of the binding of asymmetric ligands to an anisotropic lattice.  (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 (Subscript[K, 1]) or (Subscript[K, 2]), respectively.  (D). Binding affinity of ligands to selected sites.  (E and F).  The cooperativity parameter (Subscript[ω, 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 (Subscript[b, 1 L]), and a ligand tail at the right end, (Subscript[b, 2R]), while  f  denotes a free lattice residue.

Symmetric Ligands of Two Types Binding to an Isotropic Lattice

Schematic depiction of the binding of symmetric ligands of two types (n = 1 and n = 2) to an isotropic lattice.  (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 (Subscript[K, L 1]) or (Subscript[K, L 2]), respectively.  (D). Binding affinity of ligands to selected sites.  (E and F).  The cooperativity parameter (Subscript[ω,  L 12]) 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 bound ligand of type 1 is designated (Subscript[b, L 1]), one of type 2 is designated (Subscript[b, L 2]), while  f  denotes a free lattice residue.