Thermodynamics of Multivalent Interactions: Influence of the Linker

Abstract

This paper describes a thermodynamic analysis of multivalent interactions, with the goal of clarifying the influence of the linker on the enhancement in avidity due to multivalency. The use of multivalency represents a promising approach to inhibit undesired biological interactions, promote desired cellular responses, and control recognition events at surfaces. Several groups have synthesized multivalent ligands that are orders of magnitude more potent than the corresponding monovalent ligands. A better understanding of the theoretical basis for the large enhancements in avidity would help guide the design of more potent synthetic multivalent ligands. In particular, there has been significant controversy regarding the extent to which the loss of conformational entropy of the linker influences the enhancement in avidity due to multivalency. To help clarify this issue, we present the thermodynamic analysis of a heterodivalent ligand-receptor interaction. Our analysis helps reconcile seemingly competing theoretical analyses of multivalent binding. Our results indicate that the dependence of the free energy of multivalent binding on linker length can be weak even if there is a signficant decrease in the conformational entropy of the linker on binding. Our results are also consistent with studies demonstrating that the use of flexible linkers represents an effective strategy to design potent multivalent ligands.

INTRODUCTION

Nature makes extensive use of multivalent interactions, which involve the simultaneous binding of multiple ligands on one biological entity to multiple receptors on another¹. Examples of “natural” multivalent interactions include the attachment of viruses to target cells and the binding of antibodies to pathogens. A major advantage of multivalent interactions is that they can be collectively much stronger than the corresponding monovalent interactions¹. There has also been a growing interest in using this advantage to design potent multivalent molecules that influence biological interactions^1–4. For instance, synthetic multivalent inhibitors have been used to inhibit viruses^{1, 5–10} and bacterial toxins^11–19 and to promote desired cellular responses^20–22. Multivalency can also be used to control recognition events at surfaces^{4, 23}.

There are several reports of the design of synthetic multivalent inhibitors that are orders of magnitude more potent than the corresponding monovalent ligands^{1, 24}. For instance, Whitesides and co-workers have synthesized polymers presenting multiple copies of sialic acid that are approximately 10⁹ fold more potent than monovalent sialic acid at blocking the adhesion of influenza virus particles to erythrocytes²⁵. Kitov et al.¹⁵ have designed oligovalent inhibitors of Shiga-like toxin 1 that are 1–10 million-fold more active than the corresponding monovalent ligands. Understanding the theoretical basis for these enhancements in avidity due to multivalency might guide the design of other potent synthetic multivalent ligands; several groups have made important contributions in this context^{1, 4, 14, 24, 26–34}. One major unresolved issue, however, involves the influence of the linker – the structural element that connects the binding moieties in a multivalent ligand³⁵ – on the enhancement in avidity due to multivalency. While some models predict a severe loss in conformational entropy for flexible linkers (as high as RTln3 per freely rotating single bond)^{24, 36}, other models suggest a much weaker influence of the linker length ^{4, 14, 24, 28, 31}.

In this manuscript, we present a simple thermodynamic analysis to clarify the influence of the linker on the enhancement in avidity due to multivalency. We derive the result for the special case of a heterodivalent interaction between a ligand and receptor; however, the key results derived in this manuscript can be readily generalized to homodivalent systems, as well as to systems with higher valencies. Our results are consistent with previous studies demonstrating that the use of flexible linkers represents an effective strategy to design potent multivalent ligands.

RESULTS

Previous Theoretical Treatments of Multivalent Interactions

Krishnamurthy et al.²⁴ recently provided an expression for the free energy of binding (ΔG⁰ _N(i)) for a multivalent molecule with N ligands as a function of the number of ligands (i) that are bound to a multivalent receptor, where 0≤ i ≤ N:

$$\begin{array}{lll}\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathrm{N}}(\mathrm{i})\hfill & =\hfill & \mathrm{i}\mathrm{\Delta }{{\mathrm{H}}^0}_{\text{affinity}}-\text{iT}\mathrm{\Delta }{{\mathrm{S}}^0}_{\text{affinity}}+(\mathrm{i}-1)\mathrm{T}\mathrm{\Delta }{{\mathrm{S}}^0}_{\text{trans}+\text{rot}}+(\mathrm{i}-1)\mathrm{\Delta }{{\mathrm{H}}^0}_{\text{linker}}-(\mathrm{i}-1)\mathrm{T}\mathrm{\Delta }{{\mathrm{S}}^0}_{\text{conf}}+\hfill \\ \hfill & \hfill & (\mathrm{i}-1)\mathrm{\Delta }{{\mathrm{G}}^0}_{\text{coop}}-\text{RT}\text{ln}({\mathrm{\Omega }}_{\mathrm{i}}/{\mathrm{\Omega }}_0)\hfill \end{array}$$ (1)

In this expression, ΔH⁰ _affinity is the monovalent enthalpy of binding, ΔS⁰ _affinity is the monovalent entropy of binding, and ΔS⁰ _trans+rot represents the monovalent translational and rotational entropy of binding. The term [(i-1) TΔS ⁰>_trans+rot] is based on the assumption that the translational and rotational entropy of binding is approximately the same for a multivalent interaction as for a monovalent one. The term [(i-1) ΔH⁰>_linker] represents the change in enthalpy due to interactions between the linker and the receptor. The term [−(i-1) TΔS⁰ _conf] represents the contribution from the loss of conformational entropy of the linkers following binding to the receptor. The term [(i-1) ΔG⁰ _coop] represents contributions from cooperativity – the influence of one binding event on subsequent events. The term [RT ln(Ω_i/Ω₀)] is a statistical factor described by Kitov et al.³⁰, where Ω_i represents the degeneracy of a complex in which i ligands are bound to the receptor.

The extent to which the loss of conformational entropy of the linker influences the enhancement in avidity due to multivalency has been a source of controversy in the field. Several researchers have suggested that linkers for multivalent ligands should be rigid in order to minimize the loss in conformational entropy that occurs upon binding the multivalent receptor^{24, 37–39}. Whitesides and co-workers previously argued that flexible oligomers should not function effectively as linkers because of the severe loss in conformational entropy of the linker (~ RT ln3 per freely rotating single bond of the linker when it is bound at both ends)^{1, 24, 36}.

Flexible linkers have, however, been used to design potent multivalent ligands^{1, 24, 40}. Theoretical models based on the idea of effective concentration (C_eff, units of concentration) also predict a significantly weaker influence of linker length on multivalent binding than those that assume that bonds are free rotors which become completely restricted following multivalent binding^{4, 14,24, 28, 31}. The effective concentration is a parameter that was originally introduced in the context of intramolecular cyclization reactions^{35, 41, 42}. Lees and co-workers¹⁴ provided an expression for C_eff(r), which is related to the probability that the two ends of a polymer chain (i.e., the two ends of a linker) are a distance r apart.

For chains that obey random walk statistics,

$$C_{\mathit{\text{eff}}}(r)=\frac{1}{N_A{(<r^2>)}^{3/2}}{\left(\frac{3}{2\pi }\right)}^{3/2}\text{exp}\left(\frac{-3r^2}{2<r^2>}\right)$$ (2)

where <r²>^1/2 is the root mean square end-to-end distance for the chain in decimeters^{14, 35}. If necessary, excluded volume effects can be taken into account by multiplying the right hand side of the above expression (eq. 2) by a parameter p^{14, 35}. Lees and co-workers proposed a value of two for this parameter p, when the chain has only a hemisphere of free access when constrained by the receptor as opposed to a sphere in the absence of the receptor^{14, 35}. Lees and co-workers used their expression for C_eff(r) to calculate the enhancement in avidity due to multivalency and found a relatively weak dependence on linker length¹⁴. In a recent study of an intramolecular protein-ligand system, Whitesides and co-workers also observed a weak dependence of dissociation constants and “effective molarities” on linker length³⁵.

Thermodynamic Analysis of a Heterodivalent Interaction

To resolve the controversy regarding the influence of the linker on the enhancement in avidity due to multivalency, we analyzed the thermodynamics for a heterodivalent interaction between a ligand and receptor (Fig. 1A). Heterodivalency has been used to design potent ligands and is particularly promising because it can be used to target monomeric proteins, which do not contain multiple identical binding sites^{24, 43–48}. Moreover, key results derived for a heterodivalent system can be readily generalized to other multivalent systems.

Figure 1.

Thermodynamic analysis of a heterodivalent interaction. A) The heterdivalent interaction between a receptor and molecule composed of ligands A and B connected by a linker. B) Conceptual steps involved in the formation of the heterodivalent complex: Step 1. Binding of ligand A to the receptor. Step 2. Confinement of ligand B to a volume Δv in the vicinity of the binding site on the receptor. In this conceptual step, binding interactions between ligand B and the receptor are not yet in effect. Step 3. Breaking of the bond between ligand B and the linker. Step 4. Delocalization of ligand B, which is no longer confined to a volume Δv. Step 5. Binding of ligand B to the receptor. Step 6. Formation of the bond between ligand B and the linker.

Consider a system consisting of 1 mole of a heterodivalent ligand and 1 mole of a target heterodivalent receptor in a volume (V_T) of 1 liter. We assume that there is no linker-receptor interaction and no cooperativity in binding. We will now develop an expression for the free energy of heterodivalent binding assuming that both binding sites are occupied in the complex. We stress that free energy is a state function and that the value of the free energy of heterodivalent binding is path-independent. To derive an expression for the free energy of heterodivalent binding, we assume that the formation of the complex involves six sequential steps (Fig. 1B) as described below:

Step 1. The binding of ligand A. The free energy change for this step (ΔG₁) is given by

$${\mathrm{\Delta }\mathrm{G}}_1=\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},A}$$ (3-i)

where ΔG⁰ _{mono, A} is the free energy of binding for the monovalent interaction between ligand A and its receptor.

Step 2. The confinement of ligand B to a volume Δv in the vicinity of the binding site – the volume within which the center of mass of the ligand would be confined in the bound state. We stress that the binding interactions between ligand B and the receptor are not yet in effect in this conceptual step. We assume that the confinement of ligand B also results in the confinement of the chain end (i.e., the end of the linker) to a volume Δv. The free energy change for this step (ΔG₂) is therefore directly related to the loss of conformational entropy of the linker, and is given by

$${\mathrm{\Delta }\mathrm{G}}_2=-\text{RT}\text{ln}{\displaystyle {\int }_{\mathrm{\Delta }\nu }P(\overrightarrow{r})dv}$$ (3-iia)

where P(r⃗) is the probability density and P(r⃗) dv represents the positional probability – the probability that the chain end (i.e., the end of the linker) is confined within a volume element dv located at a chain displacement vector r⃗. In equation 3-iia, the integral ${\displaystyle {\int }_{\mathrm{\Delta }v}P(\overrightarrow{r})dv}$ therefore represents the probability that the chain end is confined to a volume Δv in the vicinity of the binding site.

For the special case of a linker consisting of n three-fold rotors (i.e., n freely rotating bonds, each of which assumes one of three degenerate conformational states), where one conformation is populated exclusively upon ligand-receptor binding, this probability would be ~ 1/3ⁿ and the value of ΔG₂ would be ~ nRT ln3 or ~ RT ln3 per bond, as predicted by Whitesides and co-workers³⁵.

Furthermore, if the value of Δv is small, equation 3-iia can be approximated by

$${\mathrm{\Delta }\mathrm{G}}_2=-\text{RT}\text{ln}[\mathrm{P}(\overrightarrow{r})\mathrm{\Delta }\mathrm{v}]$$ (3-iib)

Step 3. The breaking of the “bond” between ligand B and the linker. The free energy change for this step (ΔG₃) is given by

$${\mathrm{\Delta }\mathrm{G}}_3=\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{break}}}$$ (3-iii)

Step 4. The “delocalization” of ligand B. Ligand B is now no longer confined to a volume Δv near its binding site, but may be located anywhere within the total volume of the system (V_T). Since the system contains one mole of receptor and thus one mole of binding sites for ligand B with a “confinement volume” Δv near each site, the free energy change for this step (ΔG₄) can be approximated by

$${\mathrm{\Delta }\mathrm{G}}_4=-\text{RT}\text{ln}[\frac{{\mathrm{V}}_{\mathrm{T}}}{{\mathrm{N}}_{\mathrm{A}}\mathrm{\Delta }\mathrm{v}}]$$ (3-iv)

where N_A is Avogadro’s number, and we have neglected contributions from changes in rotational entropy.

Step 5. The monovalent binding of ligand B to the receptor. The free energy change for this step (ΔG₅) is given by

$${\mathrm{\Delta }\mathrm{G}}_5=\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},B}$$ (3-v)

where ΔG⁰ _{mono, B} is the free energy of binding for the monovalent interaction between ligand B and the receptor.

Step 6. Formation of the “bond” between ligand B and the linker. The free energy change for this step (ΔG₆) is given by

$${\mathrm{\Delta }\mathrm{G}}_6=-\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{break}}}$$ (3-vi)

By combining equation 3-i, equation 3-iib, equation 3-iii, equation 3-iv, equation 3-v, and equation 3-vi, we get an expression for the free energy for the heterodivalent interaction,

$$\mathrm{\Delta }{{\mathrm{G}}^0}_{\text{div}}=\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},A}+\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},B}-\text{RT}\text{ln}[\frac{P(\overrightarrow{r})V_T}{N_A}]$$ (4)

Values of P(r⃗) for specific values of P(r⃗) can be calculated by using molecular dynamics simulations³¹. For the special case of a chain obeying random walk statistics, P(r⃗) is given by the expression

$$P(\overrightarrow{r})=\frac{1}{{(<r^2>)}^{3/2}}{\left(\frac{3}{2\pi }\right)}^{3/2}\text{exp}\left(\frac{-3r^2}{2<r^2>}\right)$$ (5)

where <r²>^1/2 is the root mean squared end-to-end distance for the chain.

From equation 2, equation 4, and equation 5, noting that the total system volume (V_T) is 1 liter (i.e., 1 dm³), we get

$$\mathrm{\Delta }{{\mathrm{G}}^0}_{\text{div}}=\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},A}+\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},B}-\text{RT}\text{ln}[C_{\mathit{\text{eff}}}(r)]$$ (6)

We note that we derived eq. 6 based on the assumption that there is no linker-receptor interaction and no cooperativity in binding. If necessary, the expression above can be readily modified to accommodate these or other³⁰ terms (e.g., by adding ΔH⁰ _linker or ΔG⁰ _coop on the right hand side of eq. 6).

Based on equation 6, the overall binding constant for the heterodivalent interaction (K_div) is given by

$${\mathrm{K}}_{\text{div}}={\mathrm{K}}_{\mathrm{A}}{\mathrm{K}}_{\mathrm{B}}{\mathrm{C}}_{\text{eff}}(r)$$ (7)

where K_A and K_B are the binding constants for the monovalent interaction between ligand A and the receptor, and ligand B and the receptor, respectively. We note that while equation 3-iia can yield estimates of the loss of conformation entropy of the linker (subject to certain assumptions) that are consistent with the predictions of Whitesides and co-workers³⁵ – approximately RT ln3 per bond – we have also derived a result (eq. 7) that is consistent with the effective concentration model³² and predicts a much weaker influence of linker length on the enhancement in avidity due to multivalency.

DISCUSSION

It is instructive to compare the equation for the free energy of heterodivalent binding that we have derived (eq. 6) with the expression provided by Krishnamurthy et al.²⁴ (eq. 1). In particular, for a heterodivalent interaction, with no cooperativity, no linker-receptor interaction, and with both binding sites occupied in the complex, the expression provided by Krishnamurthy et al.²⁴ (eq. 1) can be simplified to yield:

$$\mathrm{\Delta }{{\mathrm{G}}^0}_{\text{div}}=\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},A}+\mathrm{\Delta }{{\mathrm{G}}^0}_{\mathit{\text{mono}},B}+\mathrm{T}\mathrm{\Delta }{{\mathrm{S}}^0}_{\text{trans}+\text{rot}}-\mathrm{T}\mathrm{\Delta }{{\mathrm{S}}^0}_{\text{conf}}$$ (8)

Comparing equation 6 and equation 8, we see that the term in equation 6 involving effective concentration (-RT ln [C_eff(r)]), corresponds to the sum of the terms in equation 8 involving the translational and rotational entropy of binding and the conformational entropy of the linker. Moreover, this correspondence is consistent with our derivation. Collectively, equation 3-iv, equation 3-iib, and equation 6 indicate that (-RT ln [C_eff(r)]) represents the sum of two terms, one favorable term (eq. 3-iv) related to the delocalization of ligand B, which results in an increase in its translational entropy (we neglected contributions from rotational entropy in our derivation) and a second unfavorable term (eq. 3-iib) related to the loss of conformational entropy of the linker.

This comparison of equation 6 and equation 8 can also help explain the observed weak dependence of the free energy of multivalent binding on linker length. Even in a situation where the linker is severely restricted upon association of the ligand with the binding site on the receptor (i.e., in a situation where the chain end is confined to a very small volume “Δv”) and where the loss of conformational entropy of the linker ( ΔG₂ in eq. 3-iib) is therefore significantly unfavorable, the dependence of the free energy of binding on linker length can still be weak (see eq. 4–eq. 6). This weak dependence is observed because the unfavorable loss of conformational entropy (eq. 3-iib) is partially offset by a favorable entropic term involving the delocalization of the ligand (eq. 3-iv); the net contribution to ΔG⁰ _div from eq. 3-iib and eq. 3-iv, $(-\text{RT}\text{ln}[\frac{P(\overrightarrow{r})V_T}{N_A}])$, therefore does not depend on the value of Δv.

In other words, models that assume that first the linker pays an entropic penalty by confining the chain end (and hence ligand B) to a volume Δv near the receptor, and that second the ligand B also pays an entropic penalty by being confined to the same volume Δv on the way to binding (this latter assumption is implicit in using the free energy of monovalent binding of B to receptor for the multivalent calculation), are wrong in that they effectively double count entropic penalties. Once the linker has paid the entropic penalty, ligand B is already where it needs to be to bind the receptor and its free energy of binding to receptor should no longer contain the same contribution from loss in translational and rotational entropy of binding. As a result, the overall free energy for heterodivalent binding is more favorable, and there is effectively less of an effect of the loss in conformational entropy of the linker. Our results therefore indicate that the two ostensibly competing theoretical models – eq. 1 and the effective concentration model – are both consistent with a weak dependence of multivalent binding on linker length for a long and flexible linker. We note, of course, that the avidity of multivalent binding may be strongly influenced by linker length if the extended length of the linker is smaller than the separation between binding sites on the receptor; in this case, the value of P(r⃗) would approach zero and the free energy of multivalent binding would be extremely unfavorable. Conversely, our results indicate that the optimum linker length is one that maximizes the value of P(r⃗), i.e., one for which the maximum value of P(r⃗) is obtained for a value of |r⃗| that matches the separation between the binding sites on the receptor⁴⁰.

While the above discussion indicates that the dependence of the free energy of multivalent binding on linker length may be weak even if the linker becomes severely restricted upon binding, it is still worth exploring whether the “severe restriction” assumption is realistic for a long and flexible linker. Consider a flexible “alkane-like” linker composed of 30 bonds (n =30). Assuming a length of 0.153 nm for each bond²⁴ and that the chain can be treated as an ideal random-flight polymer, the value of <r²>^1/2 would be given by

$$<r^2{>}^{1/2}={(30)}^{1/2}*0.153=0.84\text{nm}$$ (9)

if we assume that the separation between binding sites on the receptor (r) is also 0.84 nm, we can estimate the value of Δv that satisfies equation 10:

$$P(\overrightarrow{r})\mathrm{\Delta }\mathrm{v}=\frac{1}{3^n}$$ (10)

The expression above equates the probability that the chain end is confined to a volume Δv (for a chain that obeys random walk statistics) to the probability that the bonds in the linker become completely restricted following multivalent binding (assuming three degenerate conformational states per bond)³⁵. For the parameter values indicated above, the value of Δv that satisfies equation 10, (3.9×10⁻¹⁴ nm³ or (3.4×10⁻⁵ nm)³) is significantly smaller than the size of an atom, which represents an unrealistically small volume of confinement for the chain end in the bound state. While we have only carried out this calculation for a specific set of values of n, r, and P(r⃗), the results clearly suggest that long and flexible linkers may not always be completely restricted upon multivalent binding.

CONCLUSIONS

The thermodynamic analysis presented in this manuscript attempts to resolve the controversy regarding the influence of the linker on the enhancement in avidity due to multivalency. Our results indicate that the two ostensibly competing theoretical models of multivalent interactions are consistent with each other. Our results also indicate that even if the loss in conformational entropy of the linker on binding is extremely unfavorable, the dependence of the free energy of multivalent binding on linker length may still be weak. While we presented a derivation for the special case of a heterodivalent interaction between a ligand and receptor, our key results can be readily extended to other multivalent interactions. Moreover, our results are not restricted solely to linkers that obey random walk statistics, but also apply to linkers that can be described by other probability distribution functions. Finally, the predicted weak dependence of the free energy of multivalent binding on linker length is consistent with the emerging consensus that flexible linkers can be used to design potent multivalent ligands.