A quantitative view of strategies to engineer cell-selective ligand binding

Abstract

A critical property of many therapies is their selective binding to target populations. Exceptional specificity can arise from high-affinity binding to surface targets expressed exclusively on target cell types. In many cases, however, therapeutic targets are only expressed at subtly different levels relative to off-target cells. More complex binding strategies have been developed to overcome this limitation, including multi-specific and multivalent molecules, creating a combinatorial explosion of design possibilities. Guiding strategies for developing cell-specific binding are critical to employ these tools. Here, we employ a uniquely general multivalent binding model to dissect multi-ligand and multi-receptor interactions. This model allows us to analyze and explore a series of mechanisms to engineer cell selectivity, including mixtures of molecules, affinity adjustments, valency changes, multi-specific molecules and ligand competition. Each of these strategies can optimize selectivity in distinct cases, leading to enhanced selectivity when employed together. The proposed model, therefore, provides a comprehensive toolkit for the model-driven design of selectively binding therapies.

Insight Window

Selective binding to specific target cells is a critical property of many therapies. To enhance selectivity, a series of strategies have been proposed in the drug development literature, including affinity, valency, multi-specificity and other alterations to target cell binding. We employ a simple yet general multivalent ligand–receptor binding model that can help to direct therapeutic engineering. Using this model, we provide generalized and quantitative analyses of the effectiveness and limitations of each strategy. We also demonstrate that combining strategies can offer enhanced selectivity. This work therefore provides guidance for future therapeutic development.

INTRODUCTION

The intricacies of both inter-population expression differences and intrapopulation expression heterogeneity present significant challenges that limit the selectivity of therapies within the body. Many drugs both derive their therapeutic benefit and avoid toxicity through selective binding to specific cells within the body. Often, target cells differ from off-target populations only subtly in surface receptor expression, making selective binding to or activation of target cells difficult to achieve. This can result in reduced effectiveness and increased toxicity. Even with a drug of very specific molecular binding, genetic and non-genetic heterogeneity can create a wide distribution of cell responses. For example, in cancer, resistance to anti-tumor antibodies [1], targeted inhibitors [2], chemotherapies [3] and chimeric antigen receptor T cells [4, 5] all can arise through the selection of poorly targeted cells among heterogeneous cell populations.

Numerous engineering efforts have tried to develop new selective targeting strategies. Engineering therapies to have high-affinity, monomeric binding to antigens uniquely expressed on target cell populations has been used extensively, but only works to the degree that the surface marker is uniquely expressed [6]. More commonly, target and off-target cells express the same collection of receptors and differ only in their magnitude of receptor expression. In such situations, developing selectivity is an area of ongoing research and has inspired a myriad of drug designs [7–9]. There are also extensive efforts to program complex logic into cellular therapies to recognize target cells more specifically, but they suffer from shortcomings in drug access, manufacture and reliability [10–13]. Many cell selectivity engineering strategies have been proposed. A remaining challenge, however, is to consolidate these scattered efforts into a holistic picture. A quantitative, unified cell targeting framework would not only elucidate how each strategy achieves better cell type-selectivity, but also guide the discovery of new strategies and their synergistic combination.

Here, we systematically enumerate a suite of molecular approaches for engineering binding to specific cells and analyze their quantitative characteristics using a multivalent, multi-receptor, multi-ligand binding model. The generality of this model enables a unified approach toward exploration of many different therapeutic formats. We show that strategies including affinity, valency, binding competition, ligand mixtures and hetero-valent complexes provide improvements in cell-specific targeting in distinct situations and through unique mechanisms. We then demonstrate how these strategies can be used in combination to enhance target cell binding specificity. In summary, our work demonstrates that simple, passive binding-based therapies can usually offer selective targeting without the need to engineer complex cellular therapies, and that their design can be guided using a unified computational framework.

RESULTS

A model system to explore the factors contributing to cell selectivity

Virtually any therapy can be thought of as a ligand for cognate receptors expressed on target cells. To investigate ligand binding quantitatively, we employed a general multivalent, multi-ligand equilibrium binding model [14]. This model can accurately account for the interactions between a mixture of multivalent ligands and cells expressing various types of receptors. The amount of binding is predicted given the monomer binding affinities and cell receptor expression (see Materials and Methods). Monovalent binding interactions are simple and governed by both the affinity with which the ligand binds to surface receptor and the abundance of those receptors. Behavior is more complicated when the ligands are multivalent complexes consisting of multiple units, each of which can bind to a receptor (Fig. 1a). To model this behavior, we assume that the complex’s first receptor–ligand interaction proceeds according to the same dynamics that govern monovalent binding during initial association. Subsequent binding events exhibit different behavior, however, due to the increased local concentration of the complex and steric effects. We assume that the effective association constant for the subsequent binding is proportional to that of the free binding, scaled by a crosslinking constant, . The mathematical details and accompanying assumptions of the model are described in previous work [14]. In comparison to previous models, the current formulation is distinguished in its ability to predict the binding of multiple ligand, multiple receptor interactions with higher valency efficiently (Table 1). The assumptions of this model have been successfully applied to a variety of signaling pathways where multivalent interactions play a key role, such as the interaction between T cell receptors and oligomeric major histocompatibility complexes or the interaction between Fcγ receptors and IgG antibodies [15, 16]. In this work, we define cell population selectivity as the ratio of the number of ligands bound to target cell populations divided by the average number of ligands bound to off-target cell populations. We will use the quantitative binding estimation for each cell population to examine each selectivity strategy.

Figure 1.

A model system for exploring the factors contributing to cell selectivity. (a) A simplified schematic of the binding model. In this example, there are two types of receptors and two types of ligand monomers that form a tetravalent complex. The two ligands (represented by red squares and green circles), initially interact with the surface receptors (blue and orange) with various association constants (, , and ). (b) A cartoon for four-cell populations expressing two different receptors at low or high amounts. (c) A sample heat/contour map for the model-predicted log ligand bound given the expression of two types of receptors. (d) Eight arbitrary theoretical cell populations with various receptor expression profiles.

Table 1.

Comparison of selected modeling approaches and their capacity in cell-selective binding analysis

Model	Multiple receptors	Multiple ligands	Valency	Flexibility	Scalability	Dynamics
Context-specific ODE model	Possible	Possible	Possible	No	No	Yes
[49]	No	No	Yes	Yes	Yes	No
Rule-based [50]	Yes	Yes	Yes	Yes	No	Yes
[51]	Yes	Yes	Yes	No	No	Yes
Our model [14]	Yes	Yes	Yes	Yes	Yes	No

As a simplification, we will consider theoretical cell populations that express only two receptors capable of binding ligand, uniformly distributed across the cell surface (Fig. 1b), ranging in abundance from 100 to 1 000 000 per cell. Figure 1c shows the log-scaled predicted amount of binding of a monovalent ligand (, ) given the abundance of two receptors with dissociation constants of and , respectively. Because all axes are log-scaled, the contour lines intuitively indicate the ratio of ligand binding between populations. For instance, in Figure 1c, cell populations at points 1 and 2 are on the same contour line and thus have the same amount of ligand bound; the cell populations at points 1 and 3 are separated by multiple contour lines, indicating that cells at point 3 bind more ligand. (In fact, the ratio can be read as the exponent of the contour line difference. For point 3 to point 1, the ratio is .) Alternatively, moving from one point to another represents a change in a cell population’s expression profile. This situation might correspond to a cue inducing expression of a receptor, such as interferon-induced upregulation of MHC or the interleukin-induced upregulation of IL-2R⍺ in helper T cell populations [17]. If the amount of receptor 1 () increased (moving rightward, e.g. from 1 to 2), the amount of binding would not increase significantly. In contrast, increased expression of (moving upward, e.g. from 1 to 3) would lead to significantly more binding. These trends are governed by the ligand’s high affinity for and relatively low affinity for which leads to binding varying more strongly with changes to expression than .

To analyze more general cases, we arbitrarily defined eight theoretical cell populations according to their expression of two receptor types ( and plotted on x and y axes). As shown in Figure 1d, they either have high (), medium () or low () expression of and . We chose to exclude from our analysis to introduce asymmetry between and ; however, any finding pertaining to can also be generalized to by swapping and . The receptor expression profile within each cell population can also vary widely. To represent cell-to-cell heterogeneity, we arbitrarily defined intrapopulation variability for each population and computationally accounted for this heterogeneity [18]. For instance, the expression profile of has a wider range. We will use this binding model to examine how engineering a ligand using various strategies can improve cell-specific targeting. Although we will only consider two receptor and ligand subunit types, the principles we present can generalize to more complex cases.

Quantitative model grants insights to existing selectivity engineering strategies

To demonstrate that our model provides a unified but flexible platform to examine selectivity strategies, we first applied it to some more commonly explored ligand-engineering techniques, including affinity modification, multivalency and ligand mixtures (Fig. 2a). While these strategies have been investigated experimentally or modeled separately, here we propose that our model provides a unified framework to examine each of these strategies simultaneously. This not only allows us to quantitatively match known trends but also provide novel insight to how they each impart specificity when combined.

Figure 2.

Our model recapitulates known engineering strategies. (a) Schematics of cell selective engineering strategies. (b) The ligand binding ratio between and for ligands of valency ranging from 1 to 8. The shaded areas indicate the variance of binding ratios caused by the intrapopulation heterogeneity. (c, d) Number of ligands bound to each possible number of receptors for cells exposed to octavalent ligand complexes composed of subunits with dissociation constants of , or for receptor 1. Number of octavalent complexes bound at each degree for a cell with receptors (c) or receptors (d).

We first altered the receptor-binding affinity of a monovalent ligand as a cell population targeting strategy (Fig. S1). Affinity modulation is the most intuitive strategy to change the binding profile of a ligand: by enhancing a ligand’s affinity to the receptors target cells express, the ligand will bind more tightly and in higher number to them. Heat/contour maps predicted by the model clearly illustrate this effect (Fig. S1a). To explain how affinity modulation can enhance selectivity, we created four cases of target vs. off-target population pairs (Fig. S1b–e). Unsurprisingly, we found that when a target cell population expresses a receptor not expressed by off-target cell populations, enhancing the affinity to this receptor is a clear and effective strategy to increase selective binding to this population. For example, when only significantly expresses , whereas only significantly expresses , enhancing the affinity to and reducing the affinity to is a strategy to increase selectivity (Fig. S1b). However, the benefit of this strategy is reduced when both on- and off-target cell populations express the same set of receptors and differ only in their magnitude of expression, such as the selective binding to over and over (Fig. S1c and d). The model also showed that affinity tuning fails when both receptors have identical relative abundance in target and off-target populations, such as when comparing to (Fig. S1e). These results thus mirror previous findings and intuition about ligand affinity engineering in a mathematically rigorous fashion.

Next, we sought to recapitulate the effect of multivalency within our model (Fig. S2). Multivalent ligand binds differently than monovalent ligand due to its nonlinear relationship with receptor density, allowing targeting based on receptor abundance [19]. This effect of valency engineering has been well corroborated [20–22]. Our analysis scheme can intuitively illustrate this relationship: in the heat/contour plot, there are roughly the same amount of contour lines between and as there are between and in the monovalent case, whereas in the tetravalent case, there are comparatively more contour lines between and (Fig. S2a). Moreover, the model confirmed that multivalency-derived selectivity to high receptor expression population requires coordinate changes in lower affinity, as many previous studies suggest [20–23]. It shows that the binding ratio between and is maximized by low-affinity ligands but requires greater valency to achieve peak binding selectivity (Fig. 2b). Similar effects were seen in other population pairs (Fig. S2b and c).

We used the model to explore the underlying mechanism of this ‘high-valency, low-affinity’ selectivity effect. We examined the distribution of binding degrees, defined as the number of receptors bound to each complex, achieved by octavalent ligands of differing affinities on cells with differing receptor abundances. Cells expressing receptors displayed similar amounts of binding at each binding degree for ligands with high and low affinities (Fig. 2c). However, cells expressing receptors exhibit extremely low amounts of higher-degree binding with low binding affinity (Fig. 2d). This finding illustrates the ‘Velcro’-like binding behavior of multivalent ligands. Cells with higher receptor abundances can form stable, high-degree binding due to the proximity of receptors upon initial binding. They accumulate multivalent binding as the forward rate of secondary binding events is greater than that of receptor–ligand disassociation (Fig. S2d). This effect becomes particularly apparent when the affinity is low and cells must compensate with higher receptor availability to maintain stable interactions, where cells with lower receptor abundances cannot. Therefore, multivalent low-affinity ligands can selectively target cells with high receptor abundances. The model is thus able to recapitulate the known effects and benefits of valency, its dependence on affinity and elucidate the mechanism by which that relationship is governed.

Finally, we explored the effects of ligand mixture engineering (Fig. S3). Mixtures may enhance selectivity through synergistic combinations of actions [24]; therefore, many co-formulations of monoclonal antibodies or cytokine cocktails are undergoing development and clinical trials for efficacy in the treatment of solid tumors, blood disease and immunodeficiencies, among others [25–27]. Here, we specifically evaluated model-predicted binding while varying the composition between two distinct monovalent ligands, each exhibiting preferential binding to either or (Fig. S3a). Selectivity varies monotonically with composition, such that any mixture combination is no better than simply using the more specific ligand (Fig. S3b). Our model did show that when considering multiple populations and defining selectivity as the amount of ligand bound by target cells divided by the amount of ligand bound by the off-target cell with the greatest amount of binding, there are unique situations where mixtures provide enhanced selectivity, but only modestly (Fig. S3c). While our model confirms that mixtures can rarely enact selectivity through simple binding, ligands can have non-overlapping signaling effects even with identical amounts of binding where the effect of combinations can be distinct from either individual ligand, and therapeutic advantages are conferred by signaling or therapeutic synergy [16, 28].

In total, we showed that our model can accurately model previously studied strategies for selective binding. While previous models have been used to individually identify and characterize these strategies (Table 2), our modeling approach allows us to flexibly explore these trends within a unified framework.

Table 2.

Summary of engineering strategies examined in this work

Strategy	Examples	Characteristics
Affinity	IL-2 Muteins [10, 35, 36] IgG Muteins [43, 44]	Ligands with strong affinities for a particular receptor selectively bind to cells expressing that receptor uniquely when compared with off-target populations.
Valency	Multivalent fibronectin nanorings [22] Anti-Gal small molecules [9] Antimicrobial peptides [45]	Ligands in higher valency formats can preferentially bind populations with higher receptor abundances. Requires coordinated change in affinity to achieve optimum (high-valency, low-affinity effect).
Mixture	MET targeted antibodies [26] IFN-α/IFN-γ co-formulations [27] Bronchodilator co-formulations [46]	Mixtures of monovalent ligands may enhance selectivity slightly when considering two or more off-target cell populations. Most therapeutic benefits come from complementary signaling responses.
Hetero-specificity	Bispecific antibodies [47] Angiogenesis inhibitors [48]	Multispecific complexes can provide unique benefits to selectivity only when simultaneous binding of multiple units is required for therapeutic efficacy.
Monovalent agonist with multivalent antagonist	None	Mixtures of monovalent receptor agonists and multivalent receptor antagonists allow agonists to bind preferentially to cells expressing fewer receptors than off-target populations.

Heterovalent bispecific ligands exhibit unique characteristics when activity is tied to dual receptor engagement

Constructing multispecific drugs has become a promising new strategy for finer target cell specificity with the advancement of expanded protein engineering techniques [29]. The number of possible configurations of multispecific drugs is combinatorially large and impossible to enumerate, however, creating a challenge when optimizing drug design. Therefore, a computational approach is needed to explore the general principles of multispecificity-induced cell specificity and identify the most effective constructions. We use bivalent bispecific ligands as examples to explore the unique benefit of multispecificity distinct from the previous strategies. We compared a bispecific ligand with a 50–50% mixture of two monovalent ligands and a 50–50% mixture of two different homogeneous bivalent ligands (Fig. 3a). These two strategies both have some similarities to bispecific therapeutics. First, they contain two different ligand monomers with equal proportion. By comparing a bispecific with a 50–50% mixture of monovalent ligands we can determine the benefit of tethering these two monomers into one complex. Second, bispecific molecules are also naturally bivalent; by comparing them to homogeneous bivalent drugs, we see how having two different subunits in the same complex modify the behavior of a drug. By examining any unique behavior exhibited by bispecific ligands when compared with these two basic cases, we sought to explore the potential therapeutic benefits conferred by multispecificity.

Figure 3.

Bispecific ligands exhibit unique selectivity when their effect requires both subunits bound. (a) Schematic of bispecific ligands and fully bound ligand. (b–d) Comparing the total bispecific ligand binding (b) with the amount of binding achieved by a 50–50% mixture of monovalent ligands (c) or a 50–50% mixture of bivalent ligands (d). Total bound ligand is dominated by monovalently bound ligands and so the bispecific ligand lacks unique advantages. Here, the ligand concentration () was set to ; binding affinities were 100 nM, 1 , 10 and 10 , respectively. (e–g) The amount of fully bound bispecific ligand depends on the tendency for multimerization, encapsulated by . Crosslinking constants () were set to (e), (f) and (g), respectively. (h, i) Comparing bispecific with mixture selectivity, varying , the crosslinking constant. When the ratios are larger, bispecific ligands bind to target populations more specifically. (h) Bispecific divided by monovalent 50–50% mixture selectivity. (i) Bispecific divided by a 50–50% homogeneous bivalent mixture selectivity.

We first applied the binding model to predict the amount of ligand bound in bispecific drugs (Fig. 3b), a 50–50% mixture of two monomers (Fig. 3c), and a 50–50% mixture of two different homogeneous bivalents (Fig. 3d), with the same set of parameters in ligand concentration and affinities. Surprisingly, the patterns of ligand binding in these three cases are almost identical, and bispecificity appeared to offer no unique properties. However, many bispecifics only impart their therapeutic action when both of their subunits bind to the target population [30]. For example, in the design of bispecific antibodies, it is common to require binding from both subunits for the desired effect [31, 32]. We therefore investigated whether bispecific ligands that must be doubly bound display any special cell population selectivity characteristics.

We used the model to calculate the amount of ligand fully bound (Fig. 3a). With the same set of parameters, the predictions made for bispecific fully bound show a very distinct pattern from general ligand bound (Fig. 3f). The contour plot of fully bound bispecific ligands has more convex contour lines: has about the same level of general ligand bound as (Fig. 3b), but it has significantly more ligands fully bound than (Fig. 3f). This convexity of contour lines indicates that for bispecific complexes, double-positive cells bind more ligands fully.

The specific amount of fully bound ligand is dependent on the ligand’s propensity for crosslinking captured by the constant (Fig. 3a). Less steric hindrance among the subunits of a multivalent drug molecule (e.g., longer/more flexible tether and smaller subunit size) or local receptor clustering on the target cell corresponds to greater secondary binding and a larger [33]. We plotted the pattern of bispecific fully bound with , and (Fig. 3e–g). In general, when was larger, there were more fully bound units. To demonstrate how this characteristic of fully bound bispecific ligand imparts cell population specificity, we compared the selectivity conferred by bispecific ligands via fully bound interactions to the selectivity conferred by other ligand mixtures for two chosen target and off-target cell population pairs drug given a range of (Fig. 3h and i). These plotted numbers are the selectivity imparted by a bispecific drug divided by the selectivity from a drug mixture of either monovalent (Fig. 3h) or homogeneous bivalent (Fig. 3i). When these quotients are larger, it implies that bispecific ligands with both subunits bound have a greater selectivity advantage than its counterpart. Figure 3h compares the selectivity under bispecific ligands with a 50–50% mixture of monovalent ligands. The results show that a fully bound bispecific can grant enhanced binding selectivity when is small enough. This fits with the expectation that when is small and cross-linking is rarer, most ligands will bind monovalently and fully bound bispecifics will especially favor cell populations with higher dual receptor expression. However, when we compared fully bound bispecific to fully bound homogeneous bivalent mixtures (Fig. 3i), the advantage of bispecific drugs does not increase monotonically with smaller except for to selectivity. In cases where the target populations always have equal or more receptor abundance in every type than the off-target ones, such as to , or to , an optimal exists. This indicates that in these situations, the linker optimization may be an important consideration. Together, we show that bispecific ligands only exhibit unique advantages in inducing selective binding when they are only active upon binding of both subunits and highlight the role of crosslinking in their design.

Using binding competition to invert receptor targeting

While the strategies above provided selectivity in many cases, we recognized that they are all limited to a positive relationship between receptor abundance and binding. Therefore, we wondered if binding competition with a signaling/effect deficient multivalent receptor antagonist could invert this relationship and explored its effect with the model.

To investigate the effect of ligand competition with an antagonist, we modeled mixtures of ligands but only quantified the amount of binding for the active ligand (the agonist). We chose to start by only considering a monovalent agonist and tetravalent antagonist for simplicity’s sake (Fig. 4a). We found that combinations of monovalent agonistic ligands and multivalent antagonistic ligands were able to uniquely target populations expressing small or intermediate amounts of receptors, which is demonstrated by comparing ligand binding ratios between to (Fig. 4b). Here, nearly 16-fold more monovalent agonist can be bound to the target population than the off-target population when combined with a tetravalent antagonist (Fig. 4c). This is striking as expresses either equal or lesser abundances of either receptors when compared with . This phenomenon, which could not be achieved without multivalent antagonists (Fig. 4e), occurs due to the preferential binding of multivalent antagonist to populations expressing more (Figs 2b and4d). Thus, in cases where previously discussed ligand engineering strategies and approaches fail to achieve selective binding to cells expressing smaller or similar amounts of receptors to off-target populations, combinations of agonistic and antagonistic ligands may provide unique benefits.

Figure 4.

Mixtures of receptor agonists and antagonists allow for unique population targeting activity. Ligand concentration () was set to . (a) Schematics of the binding activity of monovalent agonist and multivalent antagonist mixtures. (b) Selectivity for against when exposed to a tetravalent antagonist with varying affinities for receptors 1 and 2, and a monovalent therapeutic receptor agonist with affinities optimized for selectivity. Only the amount of agonist bound is considered in determination of the optimal selectivity. (c–d) Heatmap of agonist (c) and antagonist (d) ligand bound for antagonist and agonist ligand combination shown to impart greatest selectivity improvement in (b). (e) Heatmap of agonist bound with the same parameters when no antagonist is present. (f) Optimal selectivity for against achieved when using antagonists of varying valency with an equal concentration of agonist ligand. The affinity () for receptors 1 and 2 for both the agonist and antagonist ligand were allowed to vary between 0.1 nM and 10 mM in addition to . (g) Affinities at which optimal selectivity was achieved for agonist and antagonist ligand for each antagonist valency. (h) Optimal selectivity for against achieved using combining concentration of a tetravalent antagonist ligand with 1 nM monovalent agonist. Agonist and antagonist affinities for receptors 1 and 2 were allowed to vary as above along with . (i) Affinities at which optimal selectivity was achieved for agonist and antagonist ligand for each agonist/antagonist mixture.

We next explored whether the potential selectivity benefits derived from using mixtures of multivalent antagonists with monovalent agonists could be further enhanced by changing the antagonist valency (Fig. 4f). Here, we again optimized the amount of agonist binding to when compared with and allowed the affinity of both agonist and antagonist ligands to vary. We found that antagonists of greater valency could confer even greater selectivity. For example, an optimized octavalent antagonist could allow 25 times more agonist binding to when compared with . These selectivity increases required coordinate changes in affinity of the antagonist: when the valency of the antagonist is higher, its affinity to both receptors should be reduced to achieve optimal selectivity (Fig. 4g). Here, preferential binding of low-affinity, high-valency antagonists to off-target populations with greater abundances of receptors allows for agonists to achieve selective binding to target cells with lower receptor abundances. The optimal agonist affinity does not change with antagonist valency and achieved optimal selectivity via weak interaction with the off-target population, in this case weak binding to and strong binding to (Fig. 4g).

Finally, we wondered whether modulating the amount of agonist and antagonist in a therapeutic cocktail could increase selectivity (Fig. 4h). Here, we kept the concentration of agonist at 1 nM and varied the concentration of antagonist ligand, which effectively changed the mixture profile of the combination. We optimized the affinity of a tetravalent antagonist-monovalent agonist pair in each case. This analysis found that selectivity was only weakly dependent on concentration (Fig. 4h). Unlike when changing valency, optimal agonist affinities to vary with different mixtures (Fig. 4i). However, the optimal agonists maintain weak interaction with the target population with a low affinity, and the antagonist still binds to both receptors with intermediate or high affinities. Together, these results suggest that high valency antagonists may offer unique benefits for increasing ligand selectivity in cases where off-target populations express more receptors than target population. For optimal effect, high valency of the antagonist is critical.

While previous work has explored multivalent antagonists to decrease pathway activation [34], the benefit of combining multivalent antagonists and monovalent agonists to enhance cell type-selective binding has yet to be explored. Our model outlines the potential of this novel approach, which further demonstrates how a unified model can facilitate the discovery of new selectivity strategies.

Combining strategies for superior selectivity

Each strategy described above provided selectivity benefits in distinct situations, suggesting that they might synergistically improve selectivity when combined. With binding affinities, valency, mixture and subunit composition all considered as variables, the search space is enormous, and experimentally examining the selectivity of every possible ligand is unrealistic. Some theoretical guidance on the most promising direction can greatly reduce the workload. Here, we explored mathematical optimization to determine the ligand design that provided optimal selectivity for one of our theoretical cell populations when combining strategies. We started with an ‘unoptimized’ ligand, a monovalent ligand with plausible initial affinities, prior to selectivity engineering. Then, we elected one population as the target while considering all others to be off-target. The optimization algorithm allowed ligand characteristics to vary within biologically plausible bounds (see Materials and Methods). We examined optimizing affinity alone (‘affinity’), mixture along with affinity (‘mixture + affinity’) and valency along with affinity (‘valency + affinity’) and finally combined all three strategies (‘all’) (Fig. 5). We elected to simultaneously vary valency and mixture effects with affinity due to affinity’s critical role in modulating the effects that valency and mixture engineering have in determining binding selectivity. It should be noted that the parameter space is not convex, and initial parameters were selected manually for strategies where valency and affinity were allowed to vary simultaneously to locate the global maximum selectivity. To examine the efficacy and benefits of each approach, we compared the results of these strategies with the selectivity of an ‘unoptimized’ monovalent ligand. The heatmaps and contour lines here are normalized by the amount of ligand bound to the target population to compare target and off-target binding activity more effectively (Fig. 5).

Figure 5.

Combinations of strategies provide superior selectivity. Optimization was performed to target (a–f), (g–l) and (m–r). (a, g, m) Optimal selectivity levels (ligand bound on target population divided by average ligand bound by all other populations) achieved using various ligand engineering techniques. Ligand concentration () was set to . ‘Unoptimized’ ligands are monovalent ligands with affinities of for both receptor 1 and 2. The dissociation constant was allowed to vary between and for both receptors using the ‘affinity’ approach. Valency was allowed to vary from 1 to 16 for the ‘valency’ approach in addition to affinities varying. Mixtures were assumed to be composed of two monovalent ligands, and affinities were allowed to vary in the ‘mixture’ approach. The combined ‘all’ approach allowed all these quantities to vary simultaneously. The cross-linking constant was allowed to vary between and for all approaches. (b–f, h–l, n–f) Heatmap of magnitude of ligand bound for ligand with optimized characteristics according to various ligand engineering strategies. The target population is shown in red.

Optimizing a ligand for selectivity to highlights a situation in which affinity imparts greater specificity, and optimal selectivity is achieved by combining affinity and valency modulation (Fig. 5a–f). Here selectivity is optimized by ligands with selective binding to receptor 2 and higher valency, which allow the ligands to selectively bind to cells with more abundant receptor. One case contradictory to this trend is shown during the optimization for selectivity toward the population (Fig. 5g–l). Affinity engineering is unable to impart selectivity and significant improvement is only achieved when using valency modulation. A more difficult design problem is featured in the optimization of (Fig. 5m–r). Since it lies amid the other populations in receptor expression space, any modulation of affinity, valency or combining it with mixture-based strategies seems ineffective. Engineering the mixture composition was ineffective at imparting selectivity in all cases when the ligand’s design specifications were flexible and is likely only efficacious when using ligands with static properties and considering multiple off-target populations.

Our results highlight that both in singular and combined strategies for therapeutic manipulation, the target and off-target populations dictate the optimal approach. It is also clear that combined approaches do offer context-dependent synergies that can be harnessed.

DISCUSSION

In this work, we analyzed a suite of ligand engineering strategies for population-selective binding with a multivalent, multi-ligand, multi-receptor binding model (Fig. 1). Using a representative set of theoretical cell populations defined by their distinct expression of two receptors, we examined the efficacy of several potential ligand engineering strategies, including changes to affinity, ligand valency, mixtures of species, multi-specificity, antagonist competition and these in combination. Most importantly, this framework provides a unified approach for analyzing all selectivity strategies.

The computational model reveals the general patterns of each strategy’s contribution (Table 2). We found that affinity changes were most effective when the target and off-target populations expressed distinct combinations of receptors (Fig. S1). When target and off-target populations expressed the same pattern of receptors and only differed in receptor abundances, modulations in valency, but not affinity, were effective (Figs 2 and S2). A key determinant of valency’s effectiveness was the secondary binding and unbinding rate, which is dependent on both the kinetics of the receptor–ligand interaction and receptor abundance. Ligand mixtures were mostly ineffective for imparting binding selectivity, and only had modest benefits when considering two or more off-target populations (Fig. S3). Heterovalent bispecific ligands only showed unique advantages over mixtures of monovalent ligands or bivalent ligands if activity required dual receptor binding (Fig. 3). These ligands exhibit preferential binding to target populations that have high expression of both receptors over those with high expression of only a single receptor, with the propensity for secondary binding acting as a key determinant for selectivity. Using this model, we also found that combinations of monovalent therapeutic ligands with multivalent antagonistic ligands uniquely allow for the selection of target populations expressing relatively fewer receptors than off-target populations (Fig. 4). We investigated how the antagonist effect can be maximally harnessed by optimizing valency, affinity and mixture composition. Finally, we found that, while a single ligand engineering strategy dominated in its contributions to cell type selectivity, synergies between these strategies existed in some cases to derive even greater specificity (Fig. 5).

While the multivalent binding model we applied in this work provides both generality and computational efficiency, it relies on several assumptions. For example, it assumes that receptors are uniformly distributed on the surface of the cell and that no pre-association or colocalization of receptors occurs. While the effects of receptor colocalization may be captured to some degree by the parameter in the model, it is not explicitly accounted for by our approach. The model also assumes that an equilibrium ligand concentration is roughly known. These details and possible adjustment for alternative assumptions are discussed in our previous work [14]. However, for any exception, the absolute and relative abundances of available receptors still play a governing role in determining cell type selectivity of ligand binding; thus, we believe our model’s findings remain pertinent in most selectivity engineering efforts.

While our multivalent binding model identified strategies for selective targeting in many cases, it also identified situations for which selective binding is challenging. For example, selectively targeting populations based on their absence of receptor expression remains elusive. While we computationally show the potential of using multivalent antagonists with monovalent agonists to selectively target such populations, implementing this may be challenging in practice. In cases where a target population expresses fewer receptors of any kind than an off-target population, our analysis suggests that targeting other receptors should be considered. However, in cases where target populations express more of any type of receptor than an off-target population, we show that one or more of our formulated ligand engineering strategies can be employed to improve binding selectivity. While we expect the same patterns to apply with greater than two receptors, still other emergent behaviors may exist with trispecific and more complex ligand binding.

A few of the strategies that we explored have been utilized in existing engineered therapies. For example, affinity changes to the cytokine IL-2 have been used to bias its effects toward either effector or regulatory immune populations [35, 36]. Varying the valency of tumor-targeted antibodies leads to selective cell clearance based upon the levels of expressed antigen [23]. Manipulating of the affinities of the fibronectin domains on octavalent nanorings was shown to enhance the selectivity of binding to cancerous cells displaying relatively higher densities of fibronectin receptors compared with native tissue [22]. The tendency of low-affinity, multivalent interactions to target cells expressing high receptor abundances was also described in a study describing the selectivity of multivalent antibody binding to tumor cells bound by a bispecific therapeutic ligand [13]. These examples lend support to the accuracy and translational value of our model. At the same time, recognizing these previously described ligand engineering approaches as separable strategies provides clearer guidance for future engineering.

Some of the optimization strategies described here have not been exploited in part due to the complexity of real biological applications. It may be difficult to achieve the precise affinity indicated by the model. This problem may be exacerbated in cases where specificity is derived through binding to multiple receptors as binding reagents must be designed for each. Potential dynamic changes in the receptor expression profile of a target population also complicate the matter. It is well documented that cancer cells can escape therapeutic targeting by upregulating [37, 38] or downregulating [39] the expression of certain receptors. In this case, both the current and potential abundance of each receptor must be considered. While this work does not address these issues, we propose that using a computational binding model can analyze these strategies quantitatively and collectively from a mechanistic perspective. Even when the absolute mathematically optimal ligand characteristics cannot be achieved biochemically, the model can provide guidance within the bounds of what is attainable and how to approach the optimum, accounting for implementation feasibility and facilitating the implementation of strategy combinations.

In many therapeutic applications where selective engagement of target cell populations is an important performance metric, such as the treatment of cancer, cellular therapies are becoming increasingly popular [40]. Human engineered chimeric antigen receptor (CAR) T cells have enhanced the potential to selectively recognize and attack malignant tissues [41]. These technologies bypass ligand–receptor binding restrictions by allowing recognition in signaling response. However, we have shown here that high selectivity can often be attained with relatively simple therapeutic ligands. This study lays a general framework for how ligand engineering can be directed using computational modeling. It should be noted that the application of this logic is reliant on knowledge of the target and off-target cell population receptor expression levels. Future application of the ligand binding logic described in this study could be guided using high-throughput single-cell profiling techniques, such as RNA-seq or high-parameter flow cytometry. A computational tool that could directly translate such datasets into ligand design criteria, selecting among potential receptor targets, may represent a potential avenue for the translation of our analyses into a more broadly applicable ligand engineering tool.

MATERIALS AND METHODS

Data and software availability

All analysis was implemented in Python v3.9 and can be found at https://github.com/meyer-lab/cell-selective-ligands.

Generalized multi-ligand, multi-receptor multivalent binding model

To model multivalent ligand complexes, we employed a binding model we previously developed to account the multi-ligand case [14]. In this model, we define as the number of distinct monomer ligands and the number of distinct receptors. For the binding between monomer ligand and receptor , their affinity can be described by the association constant, , or its reciprocal, the dissociation constant . During initial association, we assume that the linker portion of the complex does not sterically inhibit binding, and thus, the first subunit on a ligand complex binds according to the same dynamics that govern monovalent binding. For subsequent binding events, we assume that the effective association constant for the subsequent bindings is proportional to that of the free binding, but scaled by a crosslinking constant, , which describes how easily a multivalent ligand bound to a cell monovalently can attain secondary binding. Therefore, multivalent binding interactions after the initial interaction have an association constant of . The concentration of complexes at equilibrium is , and the complexes consist of random ligand monomer assortments according to their relative proportion. For exogenously administered drugs and in vitro experiments, usually the number of ligand complexes in the solution is much greater than that of the receptors, so it is reasonable to assume binding does not deplete the ligand concentration significantly, and we can use the initial concentration as . Otherwise, we need to estimate from the initial concentration (see section 4.3 in our previous work [14] for details). The model implicitly assumes that all receptors are uniformly mixed on the cell surface. Should different receptors be organized in discrete domains, the model would have to be updated to account for different values for crosslinking within and among various domains. The proportion of ligand in all monomers is . By this setup, we know . is the total number of receptors expressed on the cell surface, and the number of unbound receptors on a cell at the equilibrium state during the ligand complex–receptor interaction.

The binding configuration at the equilibrium state between an individual complex and a cell expressing various receptors can be described as a vector of length , where is the number of ligand bound to receptor , and is the number of unbound ligand on that complex in this configuration. The sum of elements in is equal to , the effective avidity. For all in , let when is in , and . The relative number of complexes in the configuration described by at equilibrium is

with being the multinomial coefficient. Then, the total relative amount of bound receptor type at equilibrium is

By conservation of mass, we know that for each receptor type , whereas is a function of . Therefore, each can be solved numerically using . Similarly, the total relative number of complexes bound to at least one receptor on the cell is

Generalized multivalent binding model with defined complexes

When complexes are engineered and ligands are not randomly sorted into multivalent complexes, such as with the Fabs of bispecific antibodies, the proportions of each kind of complex become exogenous variables and are no longer decided by the monomer composition ’s. The monomer composition of a ligand complex can be represented by a vector , where each is the number of monomer ligand type on that complex. Let be the proportion of the complexes in all ligand complexes, and be the set of all possible ’s. By this definition .

The binding between a ligand complex and a cell expressing several types of receptors can still be represented by a series of . The relationship between ’s and is given by . Let the vector , and the corresponding of a binding configuration be . For all in , we define where and . The relative number of complexes bound to a cell with configuration at equilibrium is

Then, we can calculate the relative amount of bound receptor as

By , we can solve numerically for each type of receptor. The total relative amount of ligand binding at equilibrium is

Mathematical optimization

We used the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm as implemented in SciPy to perform selectivity optimization combining several strategies [42]. Unless specified otherwise, the initial values for optimization were for crosslinking coefficient , 1 for valency , 100% ligand 1 for mixture composition and for the affinity dissociation constants, . A previous study of IgG–Fc receptor interactions used a reduced form of this binding model and fit it to in vitro immune complex binding measurements, deriving a similar value for [17]. To optimize ligand characteristics for cell type specificity, we defined different ligand engineering strategies and allowed various ligand characteristics to vary accordingly. For all optimization strategies, was allowed to vary 1000-fold from , thus to . In the ‘affinity’ approach, we allowed the dissociation constant for ligands to vary between and for both receptors. We then allowed the valency of the ligand to vary between 1 and 16 when considering the ‘valency + affinity’ engineering approach. In the ‘affinity + mixture’ approach, the content of a mixture of two ligands was allowed to vary between 100% of each ligand, and the receptor affinities of both ligands were allowed to vary as previously described. Finally, all ligand characteristics were allowed to vary simultaneously to model the ‘all’ approach. Local optimal selectivity was reliant on the initial point as the optimization space was non-convex. Initialization points were manually selected based on their propensity to result in large improvements during optimization. A variety of starting points were tried within the bounds.

Sigma point filter

To consider the intrapopulation variance of a cell population in the optimization, we implemented the sigma point filter [18], a computationally efficient method to approximate the variance propagated through models. It should be noted that while we found that modulating the magnitude or shape of intrapopulation heterogeneity did affect the inter-population variabilities predicted by our model, it only marginally changed the mean selectivity and did not reverse or alter the qualitative trends that any ligand engineering strategy would have on cell type selectivity.

Author contributions statement

A.S.M. conceived of the work. All authors implemented the analysis and wrote the paper.

Funding

National Institutes of Allergy and Infectious Disease at the National Institutes of Health [U01-AI148119 to A.S.M.].

Conflict of interest statement

None declared.