Computational Insights into Avidity of Polymeric Multivalent Binders

Abstract

Multivalent binding interactions are commonly found throughout biology to enhance weak monovalent binding such as between glycoligands and protein receptors. Designing multivalent polymers to bind to viruses and toxic proteins is a promising avenue for inhibiting their attachment and subsequent infection of cells. Several studies have focused on oligomeric multivalent inhibitors and how changing parameters such as ligand shape, size, linker length, and flexibility affect binding. However, experimental studies of how larger structural parameters of multivalent polymers, such as degree of polymerization, affect binding avidity to targets have mixed results, with some finding an improvement with longer polymers and some finding no effect. Here, we use Brownian dynamics simulations to provide a theoretical understanding of how the degree of polymerization affects the binding avidity of multivalent polymers. We show that longer polymers increase binding avidity to multivalent targets but reach a limit in binding avidity at high degrees of polymerization. We also show that when interacting with multiple targets simultaneously, longer polymers are able to use intertarget interactions to promote clustering and improve binding efficiency. We expect our results to narrow the design space for optimizing the structure and effectiveness of multivalent inhibitors as well as be useful to understand biological design strategies for multivalent binding.

Significance

Multivalent polymers show promise as inhibitors of toxins and microbial infection. Experimental studies have demonstrated that only certain traits of such polymers are useful for binding proteins. Here, we demonstrate that the length of the polymer is an important parameter to consider as well as the weak protein-protein interactions of the toxins themselves. Our results provide a guide for the design of a new generation of polymeric binders that will have enhanced avidity.

Introduction

Biology uses multivalent interactions for a variety of reasons including enhancing weak monovalent interactions, creating conformal interfaces such as those between cells or those inducing endocytosis, or increasing specificity and affinity of binding using a limited number of receptor and ligand types (1). Multivalent binding occurs when multiple ligands on one species bind to multiple receptors on another species simultaneously. Although each individual binding site and ligand interaction might have weak binding affinity, when multiple sites bind simultaneously, they can produce a much stronger binding avidity than the sum of the corresponding monovalent interactions (1). Here, we use the term “avidity” as the overall binding affinity of multivalent interactions and “affinity” to refer to the binding affinity of a single binding site interaction (2).

Because multivalent binding can be used to enhance low-affinity binding interactions such as those commonly found between glycoligands and sugar-binding proteins called lectins, designing synthetic multivalent polymers that target specific lectins is of great interest (3). Binding strongly to lectins is a promising avenue for treating common diseases from diarrhea and colitis to influenza by inhibiting protein targets such as AB5 toxins including Shiga or Cholera toxin (4, 5, 6, 7, 8, 9) or the hemagglutinin receptor on the influenza virus (10, 11).

To narrow the design space for multivalent inhibitors, several theoretical and experimental models have looked at how spacing of binding sites and flexibilities of linkers affect binding avidity (12, 13, 14, 15, 16). Previous studies have explored optimizing parameters on the size scale of individual binding sites. For example, Liese et al. (17) modeled how changing the linker length and flexibility between two ligands changes their binding avidity, whereas Papp et al. (11) explored matching the size between ligands exactly to the target. In contrast, the field has had relatively few studies on how large polyvalent materials and design decisions at size scales much larger than individual binding sites control polyvalent interactions. Some experimental studies have attempted to understand the effect of degree of polymerization on a polymer’s multivalent enhancement. Several of these studies found that longer polymers are more effective binders for influenza virus (10, 18, 19), but other groups have found that there is a limit to this binding enhancement from increased length when interacting with proteins (20, 21). A general theoretical understanding of how polymer length contributes to multivalent binding has yet to be developed.

In this article, we show that although polymers with higher degrees of polymerization bind more tightly to multivalent targets, the enhancement in binding energy tapers off with polymer length. We find that the entropic penalty from forming long loops is a likely explanation for this effect. We also demonstrate that favorable interactions between targets, such as hydrophobic attraction between proteins, can enhance the binding of the inhibiting polymer to the target but only for higher degrees of polymerization. We use a coarse-grain Brownian dynamics simulation to establish rules for how the degree of polymerization can influence the strength of multivalent binding interactions between a polymer and a globular target such as a lectin.

Methods

We model our target as a single bead with M binding sites and n inhibitors as N_P freely jointed beads connected by harmonic springs as shown in Fig. 1. Each inhibitor bead has a single ligand. We model the chain using Brownian dynamics in which the position of each polymer bead and target is governed as follows:

$$r_i\left(t+\mathit{\Delta }t\right)=r_i\left(t\right)+\left(-\frac{\mathit{\nabla }U}{\zeta }\right)\mathit{\Delta }t+R\sqrt{2D\mathit{\Delta }t},$$ (1)

where r_i is the position of the bead at time t in the direction i = x, y, or z, R is a random number drawn from a normal distribution with a mean of 0 and a standard deviation of 1, ζ is the drag coefficient, and D = k_BT/ζ is the diffusion coefficient. The forces each bead experiences due to interactions with the surrounding polymer or target are captured in $\mathit{\nabla }U$ where U is a potential energy that combines contributions from connectivity, excluded volume, and binding. These are added together as U = U_sp + U_LJ + U_bind.

Figure 1.

Inhibitors are represented by spherical beads (light blue) connected by Gaussian springs. Each inhibitor bead has a single ligand. Targets can have multiple binding sites and are represented by a single spherical bead (red). Inhibitor ligands and target binding sites interact when they are within a reaction radius that is dependent on the timestep. Within this reaction radius, they have a probability of binding P_B that depends on the depicted free-energy landscape. Once bound, the target and inhibitor beads are connected by a Gaussian spring, and with some probability, P_UB, can return to being unbound and interacting solely through a Lennard-Jones potential. The rendering is from the Protein Data Bank (37, 38). To see this figure in color, go online.

The connectivity potential between adjacent polymer beads U_sp is modeled like a harmonic spring as follows:

$$U_{\text{sp}}=\frac{\kappa }{2}{\text{k}}_{\text{B}}\text{T}\sum _{i=1}^{N_{\text{P}}-1}{\left(r_{i+1,i}-2a\right)}^2,$$ (2)

where r_ij is the distance between polymer beads, a is the radius of a simulation bead, and κ was chosen to be $50/a^2$, a value sufficiently large enough to prevent the polymer from stretching apart under normal Brownian forces.

A Lennard-Jones potential U_LJ was applied between bead pairs as follows:

$$U_{\text{LJ}}=\mathit{ϵ}{\text{k}}_{\text{B}}\text{T}\sum _{ij}\left({\left(\frac{2a}{r_{ij}}\right)}^{12}-2{\left(\frac{2a}{r_{ij}}\right)}^6\right),$$ (3)

where the value of ϵ can be adjusted to control the solvent quality (22). Across the simulations, we used ${\mathit{ϵ}}_{\text{PP}}=\left(5/12\right)$ to mimic polymer configurations in a θ solvent and ${\mathit{ϵ}}_{\text{PP}}=\left(1/12\right)$ to mimic a good solvent as shown in Fig. S1. We chose to run both θ solvent and good solvent because these bound the solvent conditions for soluble polymers, putting a bound on any characteristics that depend on solvent quality. For reference, the excluded volume parameters for each of our simulation scenarios are listed in Table 1.

Table 1.

ϵ values for Polymer-Polymer, Polymer-Target, and Target-Target Bead Lennard-Jones Interactions

Case No.	${\mathit{ϵ}}_{PP}$	${\mathit{ϵ}}_{PT}$	${\mathit{ϵ}}_{TT}$	Case Description
1	5/12	1/12	N/A	θ solvent, dilute targets
2	1/12	1/12	N/A	good solvent, dilute targets
3	5/12	1/12	1/12	θ solvent, many low attraction targets
4	5/12	1/12	18/12	θ solvent, many high attraction targets

N/A, not applicable; PP, polymer-polymer; PT, polymer-target; TT, target-target.

To simulate reactive binding, we apply a harmonic potential when two beads are bound and turn it on and off using a prefactor $\mathit{\Omega }\left(i,j\right)$ as follows:

$$U_{\text{bind}}=\frac{\kappa }{2}{\text{k}}_{\text{B}}\text{T}\sum _{i=1}^M\sum _{j=1}^{n{N}_{\text{P}}}\mathit{\Omega }\left(i,j\right){\left(r_{ij}-2a\right)}^2.$$ (4)

$\mathit{\Omega }\left(i,j\right)=1$ when the ith binding site on the target is bound to the jth bead of the inhibitor, and $\mathit{\Omega }\left(i,j\right)=0$ when the target binding site or inhibitor bead is unbound. To control the probability of binding and unbinding, we use a piecewise function based on the energy barriers for the binding reaction from Sing and Alexander-Katz (23).

$$\begin{array}{l}\mathit{\Omega }\left(i,j,t\right)=\{\begin{array}{ll}\{\begin{array}{ll}1\hfill & \mathit{\Xi }<e^{-\mathit{\Delta }{E}_{\text{B}}}\hfill \\ 0\hfill & \mathit{\Xi }>e^{-\mathit{\Delta }{E}_{\text{B}}}\hfill \end{array}\hfill & \text{if}\mathit{\Omega }(i,j,t-{\tau }_0)=0\cap r_{ij}<r_{\text{rxn}}\hfill \\ \{\begin{array}{ll}0\hfill & \mathit{\Xi }<e^{-\mathit{\Delta }{E}_{\text{UB}}}\hfill \\ 1\hfill & \mathit{\Xi }>e^{-\mathit{\Delta }{E}_{\text{UB}}}\hfill \end{array}\hfill & \text{if}\mathit{\Omega }(i,j,t-{\tau }_0)=1\hfill \end{array}\hfill \end{array}.$$ (5)

Here, $\mathit{\Xi }$ is a random number between 0 and 1, $\mathit{\Delta }{E}_{\text{B}}$ is the energy barrier to bind normalized by k_BT, and ΔE_UB is the energy barrier to unbind normalized by k_BT as shown in Fig. 1. Without loss of generality, these energies are considered to be always positive, and the kinetics of binding are held constant by keeping ΔE_B at $\left(1/2\right)$k_BT so that binding is an accessibly frequent event. Increasing or decreasing the energy barrier will respectively slow or accelerate the kinetics of binding and unbinding equally, but not change the system’s thermodynamics. The thermodynamic drive of binding is controlled by varying ΔE₀ = ΔE_B − ΔE_UB. Binding becomes more favorable as ΔE₀ is made more and more negative. Binding reactions are evaluated every time interval τ₀ = 100Δt, where Δt is the length of one timestep and t is the current time. The reaction radius r_rxn = 1.1 is the distance apart two beads would be if their surfaces were touching plus 0.1. Choosing $0.1<{\left(6D{\tau }_0\right)}^{1/2}$ allows time for a target that unbinds to diffuse out of the polymer radius of influence in τ₀ and makes binding events independent (23). We have applied the constraint that at any time, an inhibitor bead can only bind to one target binding site $\sum _j\mathit{\Omega }\left(i,j,t\right)\le 1$, and a target site can only be bound to one inhibitor bead $\sum _i\mathit{\Omega }\left(i,j,t\right)\le 1$. Competing reactions are sampled randomly. Note that we do not include the effect of forces in the breaking of the bonds; this is because of the fact that for forces on the order of k_BT/a, this effect is negligible if the characteristic bond length is less than 1 nm. For reference, a discussion of the subject is given in (24).

The potentials are applied over the timestep $\mathit{\Delta }t=\left(6\pi \eta a^3/{\text{k}}_{\text{B}}\text{T}\right)\mathit{\Delta }\tilde{t}$, where $\left(6\pi \eta a^3/{\text{k}}_{\text{B}}\text{T}\right)$ is the characteristic monomer diffusion time or the time that it takes a bead to diffuse its radius a, and the dimensionless timestep is $\mathit{\Delta }\tilde{t}={10}^{-4}$. These equations can all be made dimensionless by scaling energies by thermal energy k_BT, lengths by bead radius a, and times by the characteristic diffusion time $\left(6\pi \eta a^3/{\text{k}}_{\text{B}}\text{T}\right)$.

The simulation code will be made available by request.

Results and Discussion

To better inform the design of multivalent polymeric binders, we seek to determine how the degree of polymerization changes the inhibitor’s binding avidity. We examine two cases. In the first case, we observe how the binding avidity between a single monovalent or multivalent target and a polymer changes with the degree of polymerization of the polymer. In the second case, we probe how this result changes when the polymer is in the presence of multiple targets that have some favorable intertarget interactions.

Biologically relevant binding affinities

To establish a baseline binding affinity and ensure we are at biologically relevant binding affinities for individual binding sites, we first characterize the monovalent binding interactions of monovalent targets and monovalent free inhibitor beads. We place a single monovalent target in a box with a constant concentration of free inhibitor beads and measure the fraction of time the target is bound and unbound from an inhibitor bead. We run these simulations for 10⁸ timesteps and run either 50 or 100 simulations in parallel to ensure that we capture sufficiently long timescales that are much longer than the typical time for a single binding and unbinding to have accurate averaging. This is repeated at several different inhibitor bead concentrations. The resulting fraction of time bound plotted for different inhibitor concentrations is shown in Fig. S2. As expected, these lines fit the Langmuir adsorption curve $\left(\theta =\left[I\right]/\left(\left[I\right]+K_{\text{D}}\right)\right)$, where θ is the fraction of time the target spends bound, $\left[I\right]$ is the inhibitor concentration, and $K_{\text{D}}$ is the dissociation constant. By looking at the K_D, we ensure that we choose a ΔE₀ of binding (Fig. 1) corresponding to an appropriate binding affinity for biologically relevant multivalent interactions. Assuming a target diameter of 5 nm, we find that using a ΔE₀ = −5 k_BT, −4 k_BT, and −2 k_BT corresponds to a K_D on the order of $4\times {10}^{-5}$, $1\times {10}^{-4}$, and $8\times {10}^{-4}$ M, respectively. This binding affinity is similar to the monovalent binding interactions between lectins and their corresponding sugars, which typically have a K_D in the mM to μM range (25, 26).

Effect of length on binding avidity to an individual target

Using ΔE₀ = −4 k_BT as an experimentally relevant individual binding site affinity, we then explored how the degree of polymerization of our inhibiting polymer changed its binding avidity to a monovalent or multivalent target. We compared binding and unbinding kinetics of a single target with one or two binding sites to a constant concentration of inhibitor beads with varying connectivity. This scenario is depicted in Fig. 2 in which a single target is placed with 64 inhibitor beads with increasing degrees of connectivity, such as 64 free inhibitor beads, 32 dimers, or a single 64mer.

Figure 2.

Schematic of the scenarios tested when comparing binding avidity’s dependence on the degree of polymerization. The volume and number of inhibitor ligands are held constant to maintain a constant concentration of ligands at 64 ligands per box. The connectivity of the inhibitor ligands was varied from monomers to 64mers in multiples of 2 so that degrees of polymerization, 1, 2, 4, 8, 16, 32, and 64, were all investigated. This ensured that all polymers in each simulation were monodisperse. To see this figure in color, go online.

To compare the binding avidity of the polymeric inhibitors to the target, we counted the time that a target stayed bound to an inhibitor bead, for which a target was considered bound whenever at least one of the target’s binding sites was occupied. Unsurprisingly, the average time interval spent bound for monovalent targets does not depend on length (Fig. 3). The target’s single binding site can only interact with one inhibitor ligand at a time, so interactions with neighboring ligands have no impact on the duration the target spends bound. Therefore, the polymeric structure and valency of the inhibitor do not affect the average time bound for a single or dilute monovalent target.

Figure 3.

Average time interval a monovalent or divalent target are bound to polymeric inhibitors of various lengths, normalized by the average time bound for a monovalent target, ${\tau }_{\text{B}}^0$. The time interval bound for a monovalent target does not depend on the inhibitor length (dashed blue line and dashed red line overlap). For the divalent target (solid), oligomeric inhibitors spend significantly more time bound than monomeric inhibitors, exhibiting the enhancement of multivalent binding avidities over monovalent binding. At high degrees of polymerization of the inhibiting polymer, as the length of the inhibitors is increased further, there is only a small gain in the average time interval the target is bound. The time spent bound approaches some maximal value with increasing inhibitor length. Inhibitors in θ solvent (red, case 1 in Table 1) spend more time bound than good solvent (blue, case 2 in Table 1). Error bars are smaller than the symbol size. To see this figure in color, go online.

In contrast, for the divalent targets, switching from monomeric inhibitors to polymeric inhibitors shows a significant increase in time the target spends bound (Fig. 3), demonstrating the enhancement in binding avidity created by multivalent binding. This follows the multivalent binding theories of increased local concentration (27) and decreased loss of entropy over free ligands (1, 2, 28). Both the constant duration of time spent bound for our monovalent target and the increase in duration of time spent bound with the lengthening of our inhibiting polymer is consistent with these previous theories.

However, these theories have not previously captured how degrees of polymerization much larger than the size scale of the target can change binding avidity. Revisiting Fig. 3, we can see that the time spent bound approaches a limit at high degrees of polymerization for both θ solvent (case 1 in Table 1) and good solvent qualities (case 2 in Table 1). To explain this phenomenon, we considered the proportion of time a target is bound in a system with a given degree of polymerization. This proportion can be transformed into a free energy of binding, which we term ΔG_B:

$$\mathit{\Delta }{G}_{\text{B}}\approx -{\text{k}}_{\text{B}}\text{T}\ln \left(\frac{{\tau }_{\text{B}}}{{\tau }_{\text{UB}}}\right),$$ (6)

where τ_B and τ_UB are the average time spent bound and unbound, respectively. Relative to the K_D of monovalent binding, we can find the dissociation constant of multivalent binding by using the difference in free energies of monovalent and multivalent binding. For example, assuming a bead diameter of 5 nm, we can estimate the 64mer-divalent target dissociation constant in θ solvent to be approximately 6 × 10⁻⁶ M. Whereas τ_B as examined in Fig. 3 is difficult to treat theoretically, we found this ΔG_B more theoretically tractable. Note that in Eq. 6 we have not included second-order corrections for finite size effects, which will reduce the binding affinity measured in small simulations (29), but we expect this will not change qualitative results. We developed a model predicting ΔG_B as a function of the degree of polymerization and valency of the target as well as other factors, described in detail in Supporting Materials and Methods. Briefly, the model is loosely inspired by the Poland-Scheraga model of DNA denaturation in that a polymer bound multivalently to a target can be represented as a sequence of loops alternating with sites bound to the target (30). In the general case, the partition function of this model can only be evaluated numerically, but in the limit of high N_P, where N_P is the degree of polymerization, there is an analytical result for ΔG_B. The full function, given in Eq. S22, is complex, but the dependence on N_P, the number of polymers n, and volume V_box, is simple and is as follows:

$$\mathit{\Delta }{G}_{\text{B}}=C-{\text{k}}_{\text{B}}\text{T}\ln \left(\frac{n{N}_{\text{P}}}{V_{\text{box}}}\right),$$ (7)

where C is a value not dependent on N_P related to the persistence length, solvent quality, ligand density, and valency of the target. Note that in our simulations, both nN_P and V_box are held constant; specifically, nN_P = 64. Thus, Eq. 7 predicts that at high N_P, polyvalency no longer increases avidity. So, for example, if N_P = 32 is high enough to approach the limit (a question we address shortly), ΔG_B should be the same for two 32-mers and one 64mer. Our theoretical treatment successfully reproduces the qualitative behavior of ΔG_B: as predicted, we observe that ΔG_B initially decreases sharply, representing the benefits of polyvalency, before reaching a limit at higher degrees of polymerization, shown in Fig. 4.

Figure 4.

A plot of the free energy of binding for the degree of polymerization of the inhibitor. The free energy of binding is calculated using the average time interval the target spends bound to the polymer τ_B (meaning one or more binding sites is bound) divided by the average time interval the target spends completely unbound τ_UB. Longer loops are entropically unfavorable, so although they are possible in longer polymers, they are unlikely to form. We can see that this leads to a limiting minimal binding energy as the degree of polymerization of our inhibitor increases. This is true in both good (blue) and θ (red) solvents. To see this figure in color, go online.

The leveling off of ΔG_B in our theoretical model is due to loop entropy: when two faraway monomers each bind the target, the polymer is forced into a large loop, which restricts the conformational degrees of freedom of the polymer chain. This free-energy penalty increases with the size of the loop, and for large enough loops, the free-energy penalty becomes larger than the free energy of binding. Beyond the length where binding loops are no longer thermodynamically favored (which corresponds to where the high N_P limit begins to be reached), increasing the degree of polymerization will provide no benefit. Thus, we predicted that the flattening of the ΔG_B curve should coincide with the length at which loops stop forming. Indeed, the frequency of loops drops precipitously with loop lengths, and loops larger than a length of 9 for good solvent and 13 for θ solvent are vanishingly rare as shown in Fig. 5. This is in agreement with the fact that ΔG_B flattens beyond N_P = 8 for good solvent and N_P = 16 for θ solvent (Fig. 4). Note that if entropic costs were turned off, multivalency would continue to yield increases in avidity for longer polymers. As Kitov and Bundle (31) show, when all possible binding sites on a multivalent ligand can bind equally to the receptor, ΔG will not plateau. Our theory describes the plateau in binding behavior well for an ideal chain, and the good solvent scenario also seems to follow. Thus, loop entropy is the likely culprit for the diminishing returns of increasing N_P.

Figure 5.

Log-log plot of the percent of time a divalent target forms various loop sizes for different length polymers. For reference, 1% frequency is shown with the dashed black line. In θ solvent (A) and good solvent (B), loops larger than 13 and 9 beads, respectively, are formed less than 1% of the time. m_ref = −3v is a reference slope where ν is the Flory exponent. To see this figure in color, go online.

Ultimately, our simulation and model results match excitingly well with the experimental results that increasing polymer length leads to only a limited increase in polymer avidity to lectins (20, 21).

Effect of length on binding avidity in the presence of multiple targets

In vivo, environments can be crowded, and multiple targets might interact with a single inhibiting polymer. If the target is a protein, hydrophobicity and charge can create target-target interactions leading to a wide range of solubility maximums from 1 mg/mL for wheat germ agglutinin to more than 50 mg/mL for serum albumin (32, 33). In this section, we examine binding between multiple targets and the inhibiting polymer and consider how target-target interactions influence binding avidity. To investigate the affect that target-target interactions have on the binding avidity of the inhibitors, we added a Lennard-Jones potential between targets and explored how changing the attraction between the targets modified their binding with the inhibiting polymer.

To examine the effect of multiple targets interacting with the inhibitor simultaneously, we placed 64 divalent targets in a box with inhibiting polymers. To compare the effect of polymer length, we again varied the connectivity of the inhibiting beads while maintaining the same total concentration of polymer binding sites, as depicted in Fig. 6. We modified our target-target attraction by changing ϵ in Eq. 3 and compared two target-target attraction scenarios: a relatively neutral condition in which ${\mathit{ϵ}}_{\text{TT}}=\left(1/12\right)$ (case 3 in Table 1) and a weakly attractive condition in which ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$ (case 4 in Table 1). These ${\mathit{ϵ}}_{\text{TT}}$ values correspond to scenarios in which the target-target interaction has a positive and negative second virial coefficient, respectively. To ensure we were at biologically relevant target-target interactions, we calculated the concentration of our targets by making the following assumptions. Assuming a target diameter of 5 nm and molecular weight of 70 kDa, 64 targets corresponds to a concentration of 7 mg/mL. By running 64 targets in a box without an inhibiting polymer present, we confirmed that at both ${\mathit{ϵ}}_{\text{TT}}=\left(1/12\right)$ and ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$, the targets do not aggregate and phase separate. This shows that both levels of target-target Lennard-Jones interactions are within the range of relevant protein solubilities.

Figure 6.

Schematic of the scenarios tested when comparing binding avidity’s dependence on the degree of polymerization with multiple targets present. The volume and number of inhibitor ligands are held constant to maintain a constant concentration of ligands at 64 ligands per box. The connectivity of the inhibitor ligands was varied from monomers to 64mers in multiples of 2 so that degrees of polymerization, 1, 2, 4, 8, 16, 32, and 64, were all investigated. This ensured that all polymers in each simulation were monodisperse. The concentration of targets was held constant in all simulations. To see this figure in color, go online.

Increased competition

Normally, one does not have isolated targets but a finite concentration of them. Thus, it is interesting to ask the following question: if one had multiple targets with a given degree of solubility binding to the same inhibitor, would that have a marked effect on the kinetics? To answer this, we examined the binding kinetics of 64 targets to our inhibiting polymers to compare to our single or dilute target case. Similarly to when interacting with single targets, the binding avidity of polymers initially increases with increasing degree of polymerization before tapering off at high polymerization as shown in Fig. 7. More interestingly, in the presence of multiple targets, increased attraction between targets decreased the maximal τ_B. To investigate this phenomenon, we compared the rate of unbinding in Fig. 8. Here, we see two timescales at which targets unbind, a fast and a slow timescale. The fast timescale represents targets that only become singly bound before unbinding, whereas the slow timescale represents targets that transition from being doubly bound to unbound. By comparing the slope of the linear best fit line in both regions, we find that the rate of unbinding for single bonds is unchanged when there is intertarget attraction, but the rate of unbinding for doubly bound targets increases with intertarget attraction. The increased rate of unbinding for doubly bound targets leads to the decrease in average τ_B seen in Fig. 7.

Figure 7.

Plot of average time bound (τ_B) for targets when multiple targets are present. The y axis is normalized by the average time bound for monomeric inhibitors, ${\tau }_{\text{B}}^0$. The data presented are for polymer-target binding energy of ΔE₀ = −4 k_BT in θ solvent. Similar to with a single target, τ_B has a limited increase with degree of polymerization of the inhibiting polymer. More attractive intertarget potentials (orange, ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$) decrease the maximal τ_B. Error bars are smaller than the symbol size. To see this figure in color, go online.

Figure 8.

Distribution of times spent bound for single-target scenarios (yellow) and scenarios with multiple targets (orange and blue). The rate of unbinding corresponds to the slope of the line in these two regions, shown in black. There is a fast and a slow timescale on which targets unbind. The former corresponds to singly bound targets unbinding and does not change with intertarget potentials. The second, longer timescale corresponds to doubly bound targets that unbind. When there are favorable target-target interactions (orange), the decay in doubly bound times is faster than if there is not an attraction between targets (blue) or if there is no competition from other targets (yellow). To see this figure in color, go online.

The higher probability that doubly bound targets unbind can be explained by increased competition. If a lone or very dilute target becomes doubly bound and then unbinds with one binding site, this unbound site could easily rebind. In contrast, in a crowded environment with many targets, a site that unbinds has to compete with neighboring targets to rebind. This increase in competition comes from both neighboring bound targets (Fig. 9 A) as well as unbound targets that are aggregated by a high density of bound targets (Fig. 9 B). We will show that the increase in intertarget attraction from ${\mathit{ϵ}}_{\text{TT}}=\left(1/12\right)$ to ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$ leads to a drastically higher local concentration of targets in the polymer’s radius of influence, exacerbating this competition and shortening the maximal τ_B.

Figure 9.

There are two types of competition bound targets experience that lead to shorter times bound for divalently bound targets. (A) shows competition from neighboring bound targets and (B) shows competition from nearby unbound targets, which is increased for more favorable intertarget potentials. To see this figure in color, go online.

Polymer-induced phase separation

Because the kinetic changes were correlated to changes in local concentration of the target, we next considered the thermodynamics of the system for which we found an increased concentration of targets bound to the inhibitor. In Fig. 10 A, we show that in θ solvent for ∼0.1 mM binding affinity (ΔE₀ = −4 k_BT), the average number of targets bound to the polymer increased for higher target-target attraction, for both mono and divalent targets. Therefore, although individual targets unbind more quickly, intertarget attraction leads to higher inhibiting polymer avidity overall. Attraction between targets causes a significant increase in the number of targets bound because it induces a collapse transition in which bound targets collapse the polymer and themselves into a globule or liquid phase. When the polymer/bound target system collapses to form a globule, the enthalpic benefit of an additional target joining the globule becomes greater than the loss of entropy of binding, leading to a significant increase in the number of targets bound to the target. This leads to a target-rich liquid-like phase attached to the polymer and a low concentration gas-like target phase in the supernatant. Similar data for good solvent can be seen in Fig. S3.

Figure 10.

Percent of inhibitor beads bound in θ solvent when the target-polymer binding affinity is ΔE₀ = −4 k_BT (A) and −2 k_BT (B). (A) As inhibitor length increases, a transition occurs that allows the polymer to bind a significantly higher percentage of targets when there is some intertarget attraction. This transition happens at a degree of polymerization of approximately 10 for monovalent targets and a degree of polymerization of around 3 for divalent targets. (B) At very low polymer-target binding affinities, such as −2 k_BT, a critical percentage of targets never bind to the inhibiting polymer, so even at high degrees of polymerization, a transition in binding does not occur. Error bars are smaller than the symbol size. To see this figure in color, go online.

We confirmed that the marked increase in avidity was caused by a polymer collapse transition by examining the end-to-end distance. Fig. 11 shows the decrease in the average end-to-end distance for a 64mer polymer in θ solvent interacting with divalent (Fig. 11 A) and monovalent (Fig. 11 B) targets with ${\mathit{ϵ}}_{\text{TT}}=\left(1/12\right)$ and ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$ attractions between targets. End-to-end distances for polymers in good solvent interacting with multiple targets can be seen in Fig. S4. As expected, the θ polymer is at its normal random walk size of eight with no targets present, but when divalent targets are added, the polymer collapses to a globule for both levels of intertarget attraction. For 64mers interacting with monovalent targets, we only see a collapse in the end-to-end distance when the intertarget attraction is ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$, meaning that the collapse transition does not occur when ${\mathit{ϵ}}_{\text{TT}}=\left(1/12\right)$. Instead, for monovalent targets with positive second virial coefficient $\left({\mathit{ϵ}}_{\text{TT}}=1/12\right)$, bound targets have high enough excluded volume to extend the polymer chain, causing the swelling seen in Fig. 11 B.

Figure 11.

End-to-end distance for 64mer polymers in θ solvent in the presence of divalent targets (A) and monovalent targets (B). (A) Increasing binding affinity between the targets and polymers induces a transition in which the polymer collapses in size for both intertarget attractions. (B) Only high intertarget attraction leads to a collapse transition (orange). Low intertarget attraction (blue) does not provide enough enthalpic gain to overcome the entropic loss of phase separation. Error bars are smaller than the symbol size. To see this figure in color, go online.

This collapse transition that leads to globular polymers and higher target binding is caused by a competition between entropy and enthalpy and can be induced by increasing the polymer length. This can be seen in the large jump in targets bound for monovalent targets with ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$ in Fig. 10 A as the degree of polymerization is increased from 8 to 32 beads. Examining this case more closely, we see that as degree of polymerization is increased, the percent of inhibitor beads bound initially drops before a sudden increase in binding after a polymerization of ∼10 beads.

When the targets have a positive second virial coefficient, a bound target reduces the volume that a target on a neighboring bead has to bind. Because of this, targets prefer to bind to monomeric inhibitors or polymer ends that have more available volume around them, or targets prefer to be unbound. When the targets have high attraction, a neighboring target creates favorable interactions and smaller excluded volume so targets prefer to bind places that have more neighbors such as the center of the polymer. This can be seen in Fig. 12 in which low interattraction monovalent targets prefer to bind polymer ends and monomeric inhibitors. Conversely, high interattraction targets prefer to bind the center of the polymeric inhibitors.

Figure 12.

The amount of time each inhibitor bead spends bound when interacting with monovalent, −4 k_BT binding targets. Plots compare binding times for monomeric inhibitor beads (red) and beads that are part of a 64mer polymer (blue). (A) Time bound when interacting with targets that have low ($\mathit{ϵ}=\left(1/12\right)$k_BT) target-target attraction is shown. Monomers are each bound for a uniform amount of time, but the polymer ends are bound much more frequently than the polymer beads in the center of the chain. (B) When interacting with targets that have higher target-target attraction ($\mathit{ϵ}=\left(18/12\right)$k_BT), the polymer collapses, making the center beads bound more frequently than the chain ends. Monomeric inhibitor beads continue to experience uniform binding preference. To see this figure in color, go online.

Targets can also overcome the unfavorable excluded volume created by their neighbors when the polymer-target binding affinity is high enough, such as in the case in which the target is divalent and ΔE₀ = −4 k_BT. Divalent targets benefit from two factors: they get the energy benefit of binding twice to the polymer and the benefit of monopolizing two polymer beads worth of space, reducing interactions with neighboring bound targets.

In addition to increased binding of targets, the polymeric inhibitor promotes aggregation and increased local concentration of unbound targets. By measuring the minimal distance between all unbound targets and the polymer and normalizing by the volume of the shell, we compared the concentration of targets at each distance R away from the polymer as shown in Fig. 13 for θ solvent and Fig. S5 for good solvent. From these plots, it is clear that at small intertarget potentials such as ${\mathit{ϵ}}_{\text{TT}}=\left(1/12\right)$, there is a negligible increase in the concentration of unbound targets near the polymer. In contrast, with an attractive intertarget potential of ${\mathit{ϵ}}_{\text{TT}}=\left(18/12\right)$, there is a significant increase in the concentration of unbound targets near the polymer: almost five times the bulk concentration for θ solvent with ΔE₀ = −4 k_BT. Overall, this means that intertarget attraction leads to significant increases in both bound targets and unbound target clustering, encouraging the collapse transition that makes the polymeric inhibitors more effective binders.

Figure 13.

Plot of the minimal distance away from the polymer that unbound targets are found, normalized by the volume of a sphere with radius R, where R is the distance the center of the target is from the center of the nearest polymer bead. Data are shown for 64mer polymers in θ solvent with polymer-target binding affinities of (A) −4 k_BT and (B) −2 k_BT. (A) The concentration of unbound targets is approximately the same as the bulk when there is low intertarget attraction, but the concentration of unbound target near the polymer is higher than the bulk concentration when the intertarget potential is increased. The rendering in the inset shows unbound targets (yellow) clustered inside the polymer (blue) by the bound targets (orange). (B) Fewer targets have bound to the polymer, so the polymer has not collapsed. This makes the local concentration of unbound targets near the polymer approximately the same as the bulk concentration for both high and low intertarget attractions. To see this figure in color, go online.

At a target-polymer binding affinity of −4 k_BT, this effect is not specific to the divalent targets, and increased aggregation can also be seen for monovalent targets, although less extreme. But at lower binding affinities such as −2 k_BT shown in Figs. 10 B and 13 B, targets are unaffected by target-target attraction because they do not bind strongly enough to the polymer to create the critical concentration needed on the polymer to attract more targets. Therefore, the average number of targets bound to the polymer barely increases at higher intertarget attraction.

Above a critical length or a critical binding affinity, polymers are able to take advantage of weakly attractive intertarget interactions, increasing inhibitor binding avidity. Although competition for binding sites lowers the τ_B for individual targets, intertarget attraction allows the polymer to induce a collapse transition that clusters unbound targets, significantly increasing the binding of the polymer overall. With a sharper collapse transition and a diminished entropy of collapse and reswelling, longer polymers should show an amplified effect.

Conclusions

This work has shown that increasing the degree of polymerization of a multivalent inhibitor increases the overall avidity of binding, but there is a limited increase in avidity at high degrees of polymerization. To explore the effect of multivalent polymer structure, we used a Brownian dynamics bead-spring model coupled with a reactive polymer-target binding model to investigate how the degree of polymerization influences a polymeric inhibitor’s avidity. First, we examined how the length of our inhibiting polymer modulates binding interactions with a single mono- and divalent target model. We found that consistent with previously reported experimental results for polymer binding to lectins, increasing the inhibitor length did increase binding avidity for multivalent targets, but interestingly, this effect was limited. We provide evidence that this limit can be explained by the entropic penalty of forming large loops; long polymers theoretically provide more possible loops when bound to a target in two places, but the entropic cost of forming long loops makes them unachievable in practice. Therefore, if the target is a globular protein, polymers longer than the maximal achievable loop length will demonstrate the maximal binding avidity. From an inhibitor design perspective, this means that the easier it is for loops to form, the greater the benefits from multivalent binding and lengthening a polymer. For example, our simulations show that increasing solvent quality discourages loop formation and causes ΔG to plateau more quickly and at a less favorable value. Likewise, our theory predicts that factors that discourage loops such as increased polymer stiffness and high amounts of swelling will reduce avidity. However, we do not address precise ligand engineering in this work; if ligands are spaced exactly to fit the receptor’s binding sites, making a polymer stiffer may be an effective method to increase avidity (17).

Because of its estimation of the targets as point particles, our model works well for systems in which the binding sites are clustered in areas smaller than the distance between polymer binding sites, such as lectins or possibly clustered receptors on a surface. Our model does not address the experimental results that increasing polymer length continues to increase avidity for larger many-valent surfaces such as viruses. In this case, longer polymers may continue to show increased avidity because they are able to reach more binding sites along the surface and benefit from increased combinatorial entropy (34). Consequently, for targeting viruses, researchers may want to continue creating polymers with higher degrees of polymerization.

In the presence of multiple targets, we found that longer polymers are able to use intertarget interactions to increase their avidity further. We show that despite decreased time bound for individual targets, longer polymers are able to bind to more targets simultaneously in the presence of favorable intertarget interactions. When intertarget attraction is present, longer polymers are able to induce a collapse transition in which targets precipitate into a globule with the polymer, helping the polymer draw in a significant number of unbound targets. Increasing the concentration of unbound targets near the polymer makes the polymer better at clustering and binding targets. This could be a desirable effect in both the inhibition of targets and in other scenarios such as controlling biological signaling (35).

Our results suggest design rules for creating multivalent polymeric binders. With the understanding that increasing the degree of polymerization has a limited effect on avidity in low target concentration environments and that inhibitor length can be used to induce phase separation in high concentration environments, future designers can focus on other variables when creating multivalent polymeric binders for proteins.

Author Contributions

Designed research, E.Z. and A.A.-K.; Performed research, E.Z., J.W., and A.A.-K.; Analyzed data, E.Z., J.W., and A.A.-K.; Wrote the manuscript, E.Z., J.W., and A.A.-K.