3.1 Introduction

Multivalent particles have the ability to form multiple bonds to a substrate. Hence, a multivalent interaction can be strong, even if the individual bonds are weak. However, much more interestingly, multivalency greatly increases the sensitivity of the particle–substrate interaction to external conditions, resulting in an ultra‐sensitive and highly non‐linear dependence of the binding strength on parameters such as temperature, pH or receptor concentration.

In this chapter we focus on super selectivity: the high sensitivity of the strength of multivalent binding to the number of accessible binding sites on the target surface (see the schematic drawing in Figure 3.1). For example, the docking of a multivalent particle on a cell surface can be very sensitive (super selective) to the concentration of the receptors to which the multiple ligands can bind.

We present a theoretical analysis of systems of multivalent particles and describe the mechanism by which multivalency leads to super selectivity. We introduce a simple analytical model that allows us to predict the overall strength of interactions based on physicochemical characteristics of multivalent binders. Finally, we formulate a set of simple design rules for multivalent interactions that yield optimal selectivity.

3.1.1 Background: Ultra‐sensitive Response

Many processes in biology depend ultra‐sensitively on variations in one or more of the parameters that control the process. Such ultra‐sensitivity manifests itself as an almost switch‐like, sigmoidal change in the ‘output’ when the control parameter crosses a threshold value. Understanding such switch‐like behaviour is obviously important to understand many regulatory processes in living systems, but such understanding will also help us design synthetic systems that combine weak supramolecular interactions with high selectivity.

The best known example of ultra‐sensitivity dates back to Hill who, in the beginning of the twentieth century, studied the binding of oxygen to haemoglobin. He found the that the relation between bound oxygen and partial pressure was sigmoidal [1]. Today this phenomenon is explained in terms of allosteric cooperativity whereby the four binding sites on haemoglobin do not act independently but are ‘cooperative’, that is binding of the first oxygen molecule increases the probability that the second oxygen molecule will bind. Hence, haemoglobin is likely to be either fully loaded with oxygen or empty, which makes haemoglobin an efficient transporter of oxygen between lungs and peripheral tissues. Other examples of ultra‐sensitivity include the switch‐like response of bacterial motors [2], or the switch‐like behaviour in gene regulation due to positive feedback loops in nucleosome modification [3]. For more information on this broad topic, the reader is referred to a review by Ferrell [4, 5, 6] and references therein.

Ultra‐sensitive response is usually characterized by a so‐called Hill curve:

(3.1)

where the Hill coefficient n quantifies the degree of cooperativity of the process: the higher the Hill coefficient, the more sensitive the response.¹

Due to cooperativity, blocks that, individually, have limited selectivity can form units that interact selectively. For example, DNA base pairing is highly specific, even though underlying interactions (hydrogen bonding and base‐stacking) are not. Multivalent (or polyvalent) interactions can also lead to an ultra‐sensitive response, for example, the aggregation of multivalent DNA‐coated colloids depends sensitively on temperature [7]. Moreover, ligand–receptor or antibody–antigen interactions, are very sensitive to temperature, but also to ion concentration and pH. Internal protein interactions are also multivalent: protein folding and unfolding depend critically on temperature and other external conditions. The functioning of the biochemical machinery in cells relies (mostly) on multivalent supra‐molecular interactions. These interactions are very sensitive to external conditions which helps explain why the properties of living matter (cells, tissues) are very sensitive to temperature, while those of ‘formerly living’ matter (say, a piece of wood) are not.

In what follows, we focus on the ultra‐sensitivity of multivalent interactions to the density of ‘receptors’ on the substrate surface. In particular, we will derive expressions that show how the binding strength of a multivalent entity (say a ligand‐decorated nanoparticle or a multivalent polymer) to a substrate changes with the concentration of receptors² on the substrate surface (Figure 3.2). It will turn out that multivalent interactions can be designed such that they result in an almost step‐like switch from unbound to bound as the receptor concentration exceeds a well‐defined threshold value. In the remainder of this chapter, we will use the term ‘super selectivity’ to denote this kind of sharp response.

The remainder of this chapter is structured as follows: First, we show how the description of simple chemical equilibria and Langmuir adsorption can be extended to multivalent interactions. We then discuss the conditions under which super selectivity appears and formulate simple design principles to achieve super selectivity. We include an appendix where we discuss how, in simple cases, our approach reduces to the widely used ‘effective molarity’ picture.

3.2 Super Selectivity: An Emergent Property of Multivalency

We first focus on a prototypical system of multivalent particles in solution that can adsorb to a receptor‐decorated surface (Figure 3.2). For simplicity, we assume that the surface is flat and much larger than the multivalent particles. Furthermore, we assume that these particles are larger than the surface receptors such that each particle can attach to many receptor sites simultaneously. Adsorption of particles is governed by the well‐known Langmuir isotherm which states that the fraction of the surface occupied by particles is

(3.4)

with ρ the molar concentration of particles in solution³, is the equilibrium avidity association constant of particles adsorbing to a surface. Note that is different from the affinity equilibrium constant K_A which specifies chemical equilibria of individual ligand–receptor binding. Avidity (functional affinity) is the accumulated strength of multiple affinities [9].

We aim to understand how the overall avidity constant depends on the properties of the system, that is individual bond affinities K_A⁴, the ligand valency k and number of receptors n_R. The avidity constant includes all possible bound states, and is written as a sum over bonds

(3.5)

The first term on the right‐hand side takes into account all states with a single formed bond, the second term represents all doubly bound states, the third term triply bound states and so on K_intra is a constant specifying the internal equilibrium between singly and doubly bonded states. We have assumed that individual bonds form independently and K_intra is a constant, that is we ignore (allosteric) cooperative effects. We do this to clearly distinguish multivalent effects (the subject of this chapter) from cooperative effects [10].⁵

Ω_i is the degeneracy pre‐factor, it measures the number of ways in which i bonds can be formed between two multivalent entities, see Figure 3.3 for representative cartoons. Degeneracy Ω is often labelled as a ‘statistical pre‐factor’ which denotes something that should be included for rigour but is otherwise not essential. However, as we will show, it is precisely this degeneracy that gives rise to super selectivity. The focus of the majority of theoretical papers [9, 11, 12, 13, 14] is on the calculation of the internal equilibrium constant K_intra. Here, instead, we focus on the degeneracy Ω. We will simply assume that K_intra is (or can be) known.

The degeneracy Ω depends on the spatial arrangement of both ligands and receptors. However, it is instructive to consider first the binding of flexible ligands, where all k ligands on a particle can bind to n_R receptors (Figure 3.3b). In this case the degeneracy given by Eq. (3.6) becomes a very steep and non‐linear function of k and n_R. This form was first considered by Kitov and Bundle [15] and has been applied, among others, to super‐selective targeting [8] and modelling the adhesion of influenza virus [16].

A low fraction of bound receptors in the system can arise either because the number of receptors is greater than the number of available ligands: or when individual bonds are weak: ⁶. In this case the avidity constant [Eq. (3.5)], using degeneracy [Eq. (3.6)], can be rewritten⁷ to yield a simple form:

(3.7)

where, as before, K_A is the monomeric single‐bond affinity constant, K_intra the internal association constant, and n_R and k are the number of receptors and ligands respectively. For our purpose it is important to note that for multivalent binding (), is a steep, non‐linear function of n_R (Figure 3.4).

Equation (3.7) could have also been obtained directly by reasoning that for non‐saturated receptors (fraction of bound receptors is low), competition for the same receptor can be ignored. Each of the k ligands can then bind independently to any of the n_R receptors with an equilibrium constant K_intra (weight n_RK_intra). Alternatively, the ligand is unbound (weight 1). Hence, for systems with a low fraction of bound receptors, the factor accounts (approximately) for all possible states. Furthermore, we subtract 1 because we use the convention that at least a single bond needs to be formed for the multivalent particle to be considered bound. The avidity constant has units of inverse molar concentration. To obtain the correct limiting behaviour in the limit , where , we must multiply the expression in square brackets by .

We note that the ratio has the dimension of an effective volume v_eff. The form of Eq. (3.7) suggests that we can view the multivalent particle adsorption as a two‐step process. First, the particle adsorbs from the solution to the surface and comes into a position to start forming bonds, the equilibrium constant of this process is given by the ratio . Once the particle is in this position, all of the k ligands can independently form bonds with surface receptors.

In the monovalent case () the avidity constant reduces to and the standard Langmuir isotherm is obtained. Furthermore, expanding Eq. (3.7) in a binomial series and using a maximum term approximation we can insert the maximum term in Eq. (3.4) and obtain the phenomenological Hill equation [Eq. (3.1)]. In the case of very strong individual bonds () virtually all k bonds are formed and the avidity becomes⁸: .

We have shown how combinatorial entropy (also called ‘avidity entropy’ [9]) gives rise to sharp switching behaviour upon a change in receptor concentration n_R (Figure 3.4). Next, we introduce a measure of the sensitivity of the binding of multivalent particles to the surface concentration of receptors:

(3.10)

α is the slope of the adsorption profile in a log–log plot (Figure 3.5). For mono‐valent binding the selectivity α is never larger than one, while in the multivalent case the selectivity can reach values greater than one, indicating a supra‐linear response. Note that for low surface coverage () the selectivity α is equivalent to the effective Hill coefficient n from Eq. (3.1). However, because we consider all terms (all possible number of bonds) in calculating avidity [Eq. (3.7)], α is not a constant. At very low receptor concentrations the avidity shows a linear dependence on n_R, and ⁹. Selectivity then grows with increasing receptor concentration n_R until reaching a peak just before the saturation of the surface (). We refer to the region with as the ‘super‐selective’ region. In this region, a small change in the receptor density n_R causes a faster‐than‐linear change in adsorption θ.

3.3 Multivalent Polymer Adsorption

To validate the model for super‐selective adsorption described above, we now compare its predictions with experimental data on polymer adsorption. Multivalent glycopolymers have been used as selective probes for protein–carbohydrate interactions in a biochemical setting [20, 21, 22]. More recently, super‐selective targeting was demonstrated in a synthetic system based on host–guest chemistry [17, 18]. We briefly describe multivalency effects in the case of polymers functionalized with many ligands.

We consider a flexible polymer with a contour length much larger than the persistence length. Ligands are randomly attached along the polymer chain (Figure 3.6a). Similar to the nanoparticles case above, a reasonable first assumption is that, due to polymer‐chain flexibility, all k ligands on a polymer can bind to any of the n_R receptors within a domain on the surface with lateral dimensions comparable with those of the polymer. For simplicity, we describe the surface as a square lattice. The cells of the lattice have linear dimensions comparable with the radius of gyration R_g of the polymer. As in the case of soft multivalent particles, any ligand on the polymer can bind to any receptor in one (and only one) lattice cell, see Figure 3.6. The model is expected to offer a faithful description of the real system if the mean distance between ligands is larger than the Kuhn segment length such that even consecutive ligands along the polymer chain can be treated as uncorrelated.

The analytical expression given by Eq. (3.11) captures the essentials of multivalent polymer adsorption¹⁰. Importantly the model allows us to predict adsorption profiles and selectivities depending on the physicochemical properties of multivalent polymers, shown in Figure 3.7. Hence, use of the simple theoretical expression given by Eq. (3.11) allows us to design a multivalent polymer such that it will selectively target a desired receptor density. In other words, Eq. (3.11) offers a tool for the rational design of selective targeting.

3.4 Which Systems are Super Selective?

The discussion thus far has focused on selective adsorption of multivalent particles and polymers. We now generalize our treatment and discuss various practical systems. In particular, we will discuss the key role of disorder that is needed to observe super‐selective behaviour in multivalent interactions. Specifically, what is needed is that a multivalent entity can bind in many different ways to a receptor‐decorated substrate. This kind of disorder is usually not possible for multivalent interactions on the angstrom or nanometre scale, as the interacting units tend to be effectively rigid on that scale. In contrast, larger supramolecular systems (e.g. the binding of a multivalent polymer to a receptor decorated membrane) can sustain the ‘disordered’ interactions.

3.4.1 Rigid Geometry Interactions

A prototypical example of multivalent interactions is the fixed (rigid) geometry multivalency shown in Figure 3.8. Two rigid, multivalent entities bind via multiple bonds: as the geometry is rigid, individual bonds either fit together, or they do not. Examples of this kind of interaction include the base pairing between nucleotides in complementary sequences of single‐stranded DNA.

Another well‐known example of a rigid multivalent interaction is the binding between an enzyme and a substrate. The interaction between a pair of proteins is multivalent, as it involves a number of local interactions of various types (hydrogen bonding, hydrophobic, Van der Waals, electrostatic etc.). To a first approximation the enzyme and substrate can be described as rigid objects. This is a simplification as proteins, even in their native state, are not entirely rigid. In any given relative orientation of the ligand to a substrate we find a two‐dimensional equivalent of the Figure 3.8. We name this class of multivalent interactions ‘rigid geometry multivalency’.

Due to the lack of flexibility of individual bonds, rigid multivalency will generally not show super‐selective behaviour. To understand this, consider a simple one‐dimensional example of a sequence of rigidly positioned ligands that bind to a commensurate sequence of receptors. One cannot increase the binding site density on the substrate without breaking the commensurability of the binding. Hence, increasing the receptor density will normally decrease the binding strength. In other words: commensurate lock‐and‐key interactions are not super selective. Interestingly, it seems that the ability of rigid multivalent particles to detect commensurate structures is exploited in nature, for instance in the activation of certain Toll‐like receptors [19].

3.4.2 Disordered Multivalency

Super‐selective behaviour can be exhibited by multivalent systems that can increase the number of possible bonds as the density of receptors increases. As we saw above, fully ordered multivalent systems only bind optimally to commensurate receptor arrangements. To achieve super selectivity, we typically need some kind of disorder or randomness in the geometry of binding. The ability to increase the number of bonds with increasing receptor density can be due to: (i) long, flexible binders; (ii) mobile receptors; or (iii) random binder positions. Figure 3.9 shows schematic examples of these three cases. Different types of bond disorder will result in different expressions for the bound partition functions (and therefore, for the avidity constants), see Eqs (3.13–3.15). However, they all show similar super‐selective behaviour (Figure 3.10).

At first sight, it would seem that the case of mobile receptors shown in Figure 3.9b should be rather different from the immobile case. However, since the receptors are mobile, each ligand can, in principle, bind to any receptor. In this light the two problems become very similar. Another way of looking at the system with mobile receptors is to consider the receptors as a (two‐dimensional) ‘ideal gas’ of particles that can bind to the ligands with an interaction strength f. Up to a concentration‐independent term , the chemical potential of these receptors is given by . A small change in the receptor concentration n_R leads to a small change in the chemical potential μ_R, which alters the probability of each and every individual ligand binding. For multivalent particles a small change per ligand adds up to a large change per particle.¹¹ Clearly, the binding probability depends on n_R, see Refs [19, 23] for practical examples of super selectivity with mobile receptors. We note that for dilute receptors the chemical potential is dominated by the translational entropy. Hence, the origin of super selectivity is entropic, also for mobile receptors.

Finally, for immobile randomly distributed binders shown in Figure 3.9c the intuitive reasoning for super selectivity follows from our initial discussion in the introduction. Let us consider two ligand/receptor‐decorated multivalent nanoparticles, A and B that can attach through ligand–receptor binding. The binding moieties are randomly distributed on both nanoparticles. From a point of view of a particular ligand on particle A, the probability of it binding, p_1A, is to a first approximation linear in the density n_R of complementary receptors on particle B. The number of possible bonds in the contact area is proportional to the number of ligands k in that area. The net result is that the binding probability depends exponentially on the product of k and n_R, as would follow from Eq. (3.15).

We note that in the cases of fixed short ligands we have only illustrated and discussed the two limiting cases: (i) perfectly complementary rigid interaction (Figure 3.8); and (ii) disordered interaction case (Figure 3.9c). Practical systems will fall between these two extremes. As a rule of thumb, small molecules and macromolecules, such as DNA or proteins, or virus capsids have a rather well defined geometry and we expect their interactions to be closer to the rigid geometry case. On the other hand, the spatial distribution of binders (ligands) on entities larger than a few nanometres is, in general, more disordered; be they man‐made such as DNA coated colloids [24, 25, 26], or natural such as cells.

We have presented simple analytical models that can be used to rationalize and understand super selectivity in various multivalent systems. In the case of polymers, the simple model works very well (3.7). However, certain systems have been studied in a greater detail. For these cases, more sophisticated (and more complex) models have been developed. For example, cell endocytosis of a virus is mediated by a multivalent interaction between membrane proteins (receptors) and virus capsid proteins (ligands). But to model the process, one should account for membrane elasticity and, in some cases, also for active processes [27]. More detailed models of multivalent polymer adsorption have recently been developed [28, 29]. A theory of valence‐limited interactions explicitly taking into account specific positions and different types of tethered binders requires the self‐consistent solution of a system of equations [30, 31], the framework was also extended to mobile ligands [32]. A complementary approach is based on a saddle‐point approximation for the binding free energy [33]. We note that the results presented in these papers support the conclusions about super‐selective behaviour that we have obtained here using much simpler models.

3.5 Design Principles for Super‐Selective Targeting

Clearly, super‐selective targeting has important practical applications (as even viruses seem to ‘know’). It is therefore important to formulate design principles for achieving optimal super selectivity. To formulate design rules, we start once again from the simple model described above: multivalent particle docking to a receptor‐decorated surface (e.g. a cell). The density of receptors on the surface is again measured by n_R, the mean number of receptors in the contact area (i.e. the area accessible to a docked particle). In many cases of practical interest, we aim to target only those surfaces (e.g. a cell surface) that have a receptor concentration above a certain threshold. How should we design the particle to target this surface optimally? Our control parameters are the valency k, the ligand–receptor binding strength f, and the activity of particles in solution z.

The landscape plots of selectivity as a function of the valency k and bond strength f are shown in Figure 3.11. We immediately notice three features: (i) High selectivity appears only in a small region of the parameter space, along the curve predicted by Eq. (3.17). (ii) The selectivity reaches a plateau value at large valencies k and weak individual bonds. (iii) Maximum selectivity is limited by the activity z; lowering the activity (or density) of multivalent particles yields a higher selectivity.

The dimensionless activity depends on the density ρ, but also on the ratio of the equilibrium constants for the formation of the first bond, and for the formation of subsequent ligand–receptor bonds in a particle–substrate complex [see Eq. (3.5)]. Therefore, even at large densities, selectivity can be substantial if the ratio is small. This can be achieved by adding a non‐specific repulsion between the multivalent entities (for instance, by coating the particle with inert polymer that provides steric repulsion [34]). Such a repulsion would present a barrier to particle association but would not prevent additional bonds from forming once the barrier is overcome: the result would be a reduction in K_A due to repulsion, but as K_intra would be less affected, this steric repulsion would decrease the ratio .

Our calculations show that selectivity is suboptimal when using high affinity bonds. However, strong affinity multivalent constructs can still behave super selectively () if their activity (concentration) in the solution is low enough, see Eq. (3.19). This suggests that, although in principle it is possible to design a super‐selective system based on very strong affinity interactions, such as the biotin–streptavidin pair, such a system would only be super selective at extremely low concentrations where the kinetics would be too slow for practical applications.

Multivalency leads to super selectivity, but it also leads to high sensitivity of binding to the variation in other relevant quantities. Therefore, in practical applications, it is important to control (or, at least know) parameters such as temperature, pH, ionic binding strength when using multivalent particles for selective targeting. The parameter range that yields high selectivity is rather small, see Figure 3.11b). A brute‐force ‘random’ search in design‐parameter space is, therefore, unlikely to find the optimal selectivity region. We hope that the theoretical guidelines and design principles set forth in this chapter will enable a more rational design of particles for super‐selective targeting.

We condense the results shown in Figure 3.11 and our theoretical considerations, Eq. (3.18), in a set of simple design rules for multivalent binding that yield maximum selectivity. We use our dimensionless statistical mechanics notation, which can be straightforwardly converted to chemical equilibrium units using and , as discussed in the Notation box.

1. The maximal possible selectivity α is limited by the activity of multivalent particles in solution: so the activity z of multivalent binders should be small.
2. Many weak bonds are better than a few strong ones. The selectivity is also limited by the valency k, until a point of saturation given by . The first two design rules together state that the maximal selectivity is limited by either the valency k or the , whichever is smaller.
3. The relationship between the ligand number k and binding strength f should be obeyed: . Together with the above rule, this one states that to achieve maximal selectivity individual bonds should be very weak . In other words, the fraction of bound receptors/ligands should always remain small.

The main assumptions used to arrive at these design rules are: (i) ligands are identical and bind independently; (ii) all ligands of a (surface bound) multivalent construct can reach all surface attached receptors within a lattice site, but cannot bind to any receptor outside of the site (Figure 3.6); and (iii) receptors, ligands or particles have no interactions except for the steric repulsion and ligand–receptor affinity.

3.6 Summary: It is interesting, but is it useful?

We have shown that weak, multivalent interactions can result in a super‐selective behaviour where the overall interaction strength becomes very sensitive to the concentration of individual binders (receptors). We presented a simple yet powerful analytical model with good predictive power for designing multivalent interactions. We expect that, even in cases where the simple model fails quantitatively, the above design rules will still provide a good starting point for designing super selectivity in practical multivalent systems. Figure 3.12 summarizes advantages of weak multivalent interactions in selective targeting.

We can imagine effective purification devices where nano objects of different valencies are passed through super‐selective sieves. In the field of material self‐assembly, multivalent supramolecular entities could be designed to hierarchically assemble depending on the valency, thus enhancing the precision of self‐assembled constructs [25].

The ability to target diseased cells pathogens based on the surface concentration of certain (over)expressed receptors would be of huge practical importance. At present, the delivery of pharmaceutical compounds to specific cells is usually based on the existence of a specific marker (e.g. a sugar or a peptide fragment) that is unique to the targeted cell type. The current wisdom seems to be to functionalize drugs or drug carriers such that they bind strongly to the specific marker. This strategy is fine if the target cells (e.g. bacteria) are very different from the cells of the host, and carry very different markers.

However, the strong‐binding strategy becomes problematic if one wishes to target, say, cancerous cells, which are usually very similar to our healthy cells. Cancerous cells typically over‐express markers that are also present, be it in smaller quantities, on healthy cell surfaces. Examples are the CD44 (‘don’t eat me’ receptor) or the folic receptor. In such cases, a compound that binds strongly to the over‐expressed marker will also bind to (and kill) healthy cells. The insensitivity of strong binders to the surface concentration of markers is one of the main reasons why antibiotics can be efficient with few side effects (in most patients), while chemotherapy is directly harmful to our body.

As outlined in this chapter, carefully designed multivalent drugs could be targeted super selectively only to cells with cognate receptor concentration above a certain threshold value [8, 40, 41]. Furthermore, in a living cell, receptor interactions and signalling play a major role which can further enhance the non‐linear response of the system [42, 43, 44, 45, 46].

Multivalency extends the sensitivity of interactions into the receptor density domain. Moreover, it enables the design of specific, highly selective interactions based on the concentration of ligands or binders, as well as on their chemical nature, thus opening up the possibility for selective targeting with minimal side effects.

Appendix 3.A: What Is Effective Molarity?

Effective molarity (EM) is an empirical concept that is commonly used to relate the kinetics and equilibria of intramolecular and intermolecular reactions [9, 10, 11]. It is defined as

(3.A.1)

where K_A^intra and K_A^inter are the equilibrium association constants. EM has units of molar concentration and is a useful measure of multivalent interactions efficacy, see Figure 3.A.1. For example, when the concentration ρ of multivalent ligands in solution is high multivalent effects are suppressed and ligands will bind monovalently. On the other hand when multivalent interactions dominate over monovalent binding. Additionally, EM allows us to de‐convolute the intra equilibrium constant into a simple part (K_A) due to bond formation, and a complicated part (EM) related to the change of conformational entropy and free energy upon binding, see Refs [9, 10, 12, 13] for more discussion.

However, it is important not to over‐interpret the meaning of ‘effective’ concentrations. The name suggests that we can calculate the internal chemical equilibria of multivalent interactions simply by using some effective concentrations of ligands. That, however, is not quite the case, as the expressions for association equilibrium between two compounds do not carry over to the situation when the numbers involved are small.

Let us consider a prototypical system: Only two particles (ligands) in a box with volume V. The particles can associate with an equilibrium constant K_A that was predetermined for us, see Figure 3.A.2. We wish to calculate the association probability of these two particles. To obtain the correct result we can calculate the partition functions of the bound/unbound state.

On the other hand, if we naively make use of the expression for chemical equilibrium in a bulk mixture binary chemical equilibrium, we do not reproduce the correct result.

Treating the system as a bulk binary reaction is not valid for only two dimerizing particles. The approach is valid in the thermodynamic limit where the chemical potential of a molecular species can be related to the logarithm of its concentration. What it boils down to is that Stirling’s approximation is valid only for large number of particles , it is clearly wrong when N equals 1 or 2. The same problem occurs when trying to calculate equilibrium constant from molecular dynamics simulations using small system sizes [47].

The above example might seem rather abstract. However, it exposes a potential pitfall of misusing ‘effective’ concentrations. The same pitfall is encountered when calculating binding probabilities of multivalent ligands, because the reactions shown in Figures 3.A.1 and 3.A.2 are very similar. For example, one could naively argue that both the unbound ligand (A) and receptor (B) in Figure 3.A.1 are flexible and can explore some effective volume V and have some effective concentration within this volume. One then applies a ‘Local chemical equilibrium’ (LCE) assumption [24, 48] which, in our simple system is given by Eqs (3.A.5, 3.A.6). But this procedure does not generally give a correct result. It becomes a good approximation only in the limit of weak binding¹² or a very large valency where the Stirling’s approximation becomes applicable.

It should be clear that effective molarity is not really a concentration.¹³ Rather, it is a quantity with the dimensions of concentration, defined by Eq. (3.A.1). We can view the effective molarity as a measure for the probability that an unbound ligand and receptor would overlap in space (and hence come into position to bind). In an idealized system, neglecting the effects of the linker and orientational correlations in the unbound state, this probability is related to an effective concentration of, say, a ligand (B) as experienced by its complementary receptor (A) [9, 12, 14]. This is exactly the ‘cloud of ideal ligands’ approximation we have used as a starting point for our theory of multivalent polymer adsorption, Eq. (3.11). In the case of our simplified system of two dimerizing particles (Figure 3.A.2) the effective concentration c_eff of type‐A, as experienced by type‐B (or vice versa) is

(3.A.7)

where we recall that V is the box volume. We can think of particle A adsorbing to particle B and the ratio of probabilities of being bound to unbound becomes

(3.A.8)

which is consistent with the correct result, Eq. (3.A.4). We could view c_eff p_u as the concentration of unbound A.

Applying this concept to dimer adsorption (Figure 3.A.1) we would find that the empirically calculated effective molarity [Eq. (3.A.1)] is similar to the theoretical effective concentration (in our idealized system they are equal). Therefore, effective concentration, when applied properly, is a useful concept when attempting to theoretically predict equilibria of multivalent binding.

Acknowledgements

We thank Galina V. Dubacheva and Stefano Angioletti‐Uberti for useful suggestions for the manuscript and help with designing figures. This work on multivalency was supported by the ERC Advanced Grant 227758 (COLSTRUCTION), ITN grant 234810 (COMPPLOIDS) and by EPSRC Programme Grant EP/I001352/1. T.C. acknowledges support form the Herchel Smith fund.

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