Multivalency in Drug Delivery â when is it too much of a good thing

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Review

Multivalency in Drug Delivery – when is it too much of a good thing? Kristel C. Tjandra, and Pall Thordarson Bioconjugate Chem., Just Accepted Manuscript • DOI: 10.1021/acs.bioconjchem.8b00804 • Publication Date (Web): 11 Feb 2019 Downloaded from http://pubs.acs.org on February 11, 2019

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Bioconjugate Chemistry

Multivalency in Drug Delivery – when is it too much of a good thing?

Kristel C. Tjandra and Pall Thordarson* School of Chemistry, the Australian Centre for Nanomedicine and the ARC Centre of Excellence in Convergent Bio-Nano Science and Technology, the University of New South Wales, Sydney, NSW 2052, Australia

E-mail: [email protected]

Abstract Multivalency plays a large role in many biological and synthetic systems. The past 20 years of research have seen an explosion in the study of multivalent drug delivery systems based on scaffolds such as dendrimers, polymers and other nanoparticles. The results from these studies suggest that when it comes to number of ligands, sometimes, to quote Shakespeare: “too much of a good thing”, is an apt description. Recent theoretical studies on multivalency indicate that the field may have had a misplaced emphasis on maximizing binding strength where in fact it is the selectivity of multivalent drug delivery systems that is the key to success. This review will summarize these theoretical developments. We will then illustrate how these developments can be used to rationalize the immunoresponses and drug uptake mechanisms for multivalent systems and show the path forward towards the design of better multivalent drug delivery systems.

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1. INTRODUCTION A multivalent interaction can be defined as an interaction between two entities (a ligand and a receptor) involving multiple (i) instances of weak non-covalent binding occurring simultaneously.1,2 Often likened to Velcro,3 strong and reversible binding can only be achieved in the presence of multiple simultaneous recognition events. Due to its robustness in biological systems, multivalency has received increased attention in the area of targeted drug delivery, where ligand-receptor interactions are heavily relied upon. More is, however, not always better. In a study using a dendrimer-based nanoparticle with k = 1-14 folic acid ligands (k: number of ligands), a plateau of k = 5 is observed for both the overall association constant or avidity constant and the cell adhesion as measured by flow cytometry (Figure 1A).4 A similar phenomenon was observed by Alewood and co-workers, where a hundredfold increase in inhibition was achieved in a k = 2 (divalent) -conotoxin ImI dendrimer compared to the monovalent azido- -conotoxin ImI peptide counterpart. However, when the valency was increased to k = 4 (tetrameric), no further inhibition was attained (Figure 1B).5 Unexpected trends relating to multivalent system designs have also emerged in studies on the immunoresponse and uptake mechanism of multivalent drug-delivery system. In somewhat simplistic terms the message is clear: multivalency in drug delivery does not always work the way researchers expected to do and, more is not always better.


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Figure 1. (A) Folic acid binding (right y-axis in blue) as measured by Surface Plasmon Resonance (SPR) and in vitro cellular binding (left y-axis in red) as measured by flow cytometry, initially increases exponentially with the number of ligands before it plateaus at k ≈ 5.4 (B) Voltage clamp measurements show that the activity of 7-nicotinic receptor blocker azido--conotoxin ImI peptide increases 100-fold upon dimerization (k =2) of the ligand, however, the increase in inhibitory activity was more modest as the number of the ligand was increased to four (k = 4) copies. Adapted with permission from (A) Hong et al.4 and (B) Wan et al.5 The root-cause for the challenges in designing multivalent drug-delivery systems relate to knowledge gaps in our understanding of multivalency. This is despite the fact that researchers in the field have been highlighting these problems for over 20 years.1 We argue that attempts to transpose, without any modifications, design principles from small-molecule research to multivalent systems are largely to blame. For instance, there is sometimes a naïve tendency to take lessons from welldefined small-molecule supramolecular multivalent systems (with small i’s and k’s) and impose those unchanged to larger more complex biological systems. In biology, however, unlike in supramolecular chemistry, the receptor is usually in excess. Likewise, multivalency does play a significant role in the electrostatic interactions between charged (bio)macromolecules including DNA-DNA or coiled-coiled protein structure interactions. In these cases, more is usually better, Page 3 of 33 ACS Paragon Plus Environment


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however, these sort of multivalent electrostatic interactions are of very little relevance in most targeted drug delivery systems, and hence there is not much to be learned from these in understanding the multivalency effect in drug delivery. Classical ideas from medicinal chemistry relating to rigidity and “tight” fit are another example of ideas that may not translate well to research in multivalent drug-delivery and targeting involving peptide or protein-based ligands. In this topical review we will highlight recent significant advances in our theoretical understanding of multivalency, based largely on taking a statistical thermodynamics approach to the problem. In addition to describing these physical chemistry developments, we will discuss recent developments in understanding how the immunoresponse, the update of multivalent drug delivery systems and other contemporary issues in the field may also relate to the theory. This review will end by giving our perspective on how these latest developments could be applied in the design of more effective multivalent drug delivery systems.

2. THE NATURE OF MULTIVALENCY The role of multivalency in the binding of proteins to antibodies and other targets has been wellknown for decades with the first papers on the implications of avidity6 and specificity7 (see definitions below) in the binding of multivalent systems appearing in the 1970s. Whitesides and coworkers expounded on this robust interaction and how it could be applied in biomedical research in a ground-breaking review in 1998.1 This seminal paper introduced the ‘multivalency effect’ to a larger audience, which since then has continued to be a subject of interest in various biological and synthetic systems including (but not limited to) vaccines,8–11 antibodies,12,13 carbohydrate and glycoproteins,14,15 dendrimers,16 polymers,17 peptides,18 oligonucleotides,19 and supramolecular systems.20 An example of a frequently studied model for multivalency is the interaction between multiple copies of hemagglutinin proteins on the surface of an influenza virus with sialic acids on the host cells.21 Page 4 of 33 ACS Paragon Plus Environment

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Basic definitions In defining multivalency (Figure 2), the exact terminology and the symbols used vary somewhat between papers. Here, we will use terminology that is similar to Whitesides’ original review1 in combination with that used by Frenkel.22 Further, we note that this overview here is not comprehensive and we therefore urge interested readers to read the papers cited for a more detailed discussion. We will use here the terms receptors and ligands as they are conventionally in biochemistry with receptors generally being target protein / protein clusters (nR – see Figure 2C for a full list of abbreviations used in this section) on a surface of a cell and the ligand being a small molecule, peptide or a protein that inhibits or alters the function of the receptors. In supramolecular chemistry, the receptor would usually be designated as the host and the ligand as the guest.23 Multivalent drug delivery systems would generally consist of k-number of ligands linked to a “core”, e.g,, a dendrimer, polymer or a nanoparticle. The nR receptor and k-ligands can form up to inumber of bonds with i defined by whichever is smaller nR or k (this assumes that only one (multi)valent ligand can be bound to each receptor site at any given time, either for steric reasons or because the receptor is in excess).



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Figure 2. Key concepts in multivalency. N.b. statistical or degeneracy pre-factors (i) are not included in the schemes shown. While the receptors (red) as shown as a regular array for clarity it is assumed they can move randomly around on the cell surface. The (multivalent) ligands (blue) are assumed to be fully flexible. (A) The binding of a monovalent (k = 1) ligand to a multivalent receptor (nR = 3). Assuming there is no cooperativity (and after correcting for degeneracy), the individual (microscopic) binding constants23 are all the same (Kmono is a constant). (B) The binding of a multivalent ligand with k = 3 to the same receptor as in (A) with nR = 3. The first step is intermolecular = KA and differs from the subsequent intramolecular Kintra steps. In the absence of cooperativity Kintra is a constant. (C) Key abbreviations used in the discussion on multivalency in this review. In the example we start with, we will compare the equilibrium constant Kmono for a simple 1:1 receptor (host) – ligand (guest) interaction (Figure 2A) with the overall avidity constant 𝐾𝑎𝑣 𝐴 of a multivalent systems with say 3 receptors and 3 ligands or 3:3 binding (Figure 2B). The avidity constant may be thought as the accumulated strength of the multiple affinities. In the method for 22 calculating 𝐾𝑎𝑣 𝐴 outlined by Frenkel and co-workers, the avidity constant is obtained by combining

the first intermolecular receptor-ligand interaction (KA), followed by the i-1 subsequent ones intramolecular (Kintra) interactions, i.e., in our example, the two subsequent Kintra binding constants. Furthermore, one has to take into account the degeneracy pre-factor (i)24 of bound states or what is also referred to as the statistical pre-factor ().25 From Frenkel and co-workers the avidity for the Page 6 of 33 ACS Paragon Plus Environment

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2 3:3 multivalent binding would be then be calculated as: 𝐾𝑎𝑣 𝐴 = 1𝐾𝐴 + 2𝐾𝐴𝐾𝑖𝑛𝑡𝑟𝑎 + 3𝐾𝐴𝐾𝑖𝑛𝑡𝑟𝑎 .

For i number of interactions we can write a generic version of this model for 𝐾𝑎𝑣 𝐴 : 2 𝑖―1 𝐾𝑎𝑣 𝐴 = 1𝐾𝐴 + 2𝐾𝐴𝐾𝑖𝑛𝑡𝑟𝑎 + 3𝐾𝐴𝐾𝑖𝑛𝑡𝑟𝑎 … + 𝑖𝐾𝐴𝐾𝑖𝑛𝑡𝑟𝑎

(1)

It should be noted that eq 1 ignores potential cooperative (allosteric) effect2 between the binding sites. In many multivalent drug delivery systems this is a fair assumption, as the nR receptors are typically too far away from each other on the cell surface to influence each other. The ratio of the intra- and inter-molecular constants is frequently used to define the empirical concept of effective molarity EM = Kintra/KA.22 The effective molarity is at the center of attempts and debates2 on how the chelate cooperativity  should be defined.2 However, for practical applications in drug delivery, the definition and value of  may not be very useful. Rather surprisingly, it turns out that the degeneracy pre-factor (i) is much more important and the key to understanding multivalency in drug delivery. Kitov and Bundle illustrated this clearly by considering the free energy (G) for the formation of the i-th complex between nR number of receptors on a target (e.g., cell surface) and i-number of ligands from a multivalent drug delivery system with k-ligands where k ≤ nR and k ≥ i:24 ∆𝐺°𝑖 = ∆𝐺°𝐴 + (𝑖 ― 1)∆𝐺°𝑖𝑛𝑡𝑟𝑎 ― 𝑅𝑇 ln 𝑖

(2)

The value of the degeneracy pre-factor  depends not only on nR, k and i, but whether the receptor and ligands are rigid or flexible and how they are organized. Kitov and Bundle described the difference between a linear topology where the receptor and ligand must bind in a linear (sequential) fashion (eq 3) akin to complementary nucleotide binding versus the flexible situation where all the k ligands can bind to any of the nR receptors (eq 4)24 and Figure 2B: 𝑖(linear) = (𝑛𝑅 ― 𝑖 + 1 )(𝑘 ― 𝑖 + 1 )

(3)

𝑛𝑅!𝑘! 𝑖(flexible) = ( 𝑛𝑅 ― 𝑖 )!(𝑘 ― 𝑖! )𝑖!

(4)



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It should be clear from eqs 1, 2 and 4 that degeneracy pre-factor  for systems with flexible ligands will make a significant contribution towards the overall binding strength. When compared to mono-valent systems, the magnitude of the degeneracy pre-factor  rises very steeply as k and i increase. In the example shown in Figure 3, as the number of receptors nR doubles,  only doubles (3→6) in the monovalent case but goes up by a factor of 20.22 The difference between linear array of ligands (eq 3) and the flexible (eq 4) is equally drastic. For example, with 4 bound ligands (k and i = 4) and nR = 4, (linear) = 1 and (flexible) = 24. With excess receptors the difference is even greater; with k = i = 4 and nR = 8 we obtained (linear) = 5 and (flexible) = 1680. For a system with 8 receptors and ligands fully bound; k = i = nR = 8 we obtain (linear) = 1 and (flexible) = 40320. In energy terms (-RT ln ) using eq 2, this equates to a difference of -26 kJ mol-1 at 298 K!

Figure 3. The importance of degeneracy in the binding of monovalent versus multivalent ligands. (A) Increasing receptor number from 3 to 6 contributed to only 2-fold increase in binding probability in monovalent ligands. (B) In the case of multivalent ligands, degeneracy i increases by a factor of 20 (see eq 4). Adapted with permission from Curk et al.22 At this point it is worth noting that eqs 1 and 2 are not the only ones that could be derived to ° show how the overall avidity constant 𝐾𝑎𝑣 𝐴 or free energy change ∆𝐺𝑖 could be derived. Elaborating

on their derivation in their break-through paper on multivalency,1 Whitesides and co-workers expand their definition of the avidity constant26 to include amongst other, the degeneracy pre-factor  from Kitov and Bundle.24 Additionally, this derivations attempts to separate the various other contributions to multivalent interactions as shown in eq 5:26 Page 8 of 33 ACS Paragon Plus Environment

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∆𝐺°𝑖 = 𝑖∆𝐻°𝑎𝑓𝑓𝑖𝑛𝑖𝑡𝑦 ― 𝑖𝑇∆𝑆°𝑎𝑓𝑓𝑖𝑛𝑖𝑡𝑦 + (𝑖 ― 1)𝑇∆𝑆°𝑡𝑟𝑎𝑛𝑠 + 𝑟𝑜𝑡 + (𝑖 ― 1)∆𝐻°𝑙𝑖𝑛𝑘𝑒𝑟 ― (𝑖 ― 1)𝑇∆𝑆°𝑐𝑜𝑛𝑓 + (𝑖 ― 1)∆𝐺°𝑐𝑜𝑜𝑝 ― 𝑅𝑇 ln(𝑖/0)

(5)

Noting that with  = 1, the last right-hand term in eq 5 is the same as the last right-handed term in eq 2 and accounts for the degeneracy in the binding. The two right-hand terms in eq 5 with 𝑖∆𝐻°𝑎𝑓𝑓𝑖𝑛𝑖𝑡𝑦 and 𝑖𝑇∆𝑆°𝑎𝑓𝑓𝑖𝑛𝑖𝑡𝑦 represent the enthalpic and entropic components, respectively, of the inumber of monovalent ligand-receptor interactions in the system. The remaining five right-hand terms in eq 5 then measure the various factors that contribute to the observed multivalent effect including: the (𝑖 ― 1)𝑇∆𝑆°𝑡𝑟𝑎𝑛𝑠 + 𝑟𝑜𝑡 – the unfavorable translational and rotational entropies associated with binding which the authors argue is the same for monovalent and multivalent systems and hence they “add” (i-1) of these to the overall energy – this they argue is the classic “chelate effect”;26 (𝑖 ― 1)∆𝐻°𝑙𝑖𝑛𝑘𝑒𝑟, which represents any (un)favorable enthalpic interactions between the linkers used and the multivalent receptor; (𝑖 ― 1)𝑇∆𝑆°𝑐𝑜𝑛𝑓 which corrects for the loss of conformational entropy (if any) between the ligands and the receptors; (𝑖 ― 1)∆𝐺°𝑐𝑜𝑜𝑝, which accounts for any cooperativity in the system;2 and the final term as already mentioned, accounts then for the degeneracy pre-factor  Whitesides and co-worker then discuss how each of these terms could be optimized to enhance the overall avidity,26 however, as we will now discuss in more detail, when it comes to drug delivery with multivalent systems, overall avidity or the “brute strength” of the multivalency effect may not be as important as the selectivity of these systems.

Bound fraction and selectivity Currently, there is a strong push to develop strategies that target specific/overexpressed receptors on the surface of the target cells or bacteria. The conventional wisdom is that the best strategy is to develop systems and drug carriers that bind strongly to the target receptors. But is that really the best strategy? In the real “messy” world of diseases such as cancer, the differences in expression levels between healthy and cancerous cells can be fairly modest. The ideal scenario would therefore Page 9 of 33 ACS Paragon Plus Environment


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be a highly selective multivalent system where, just on the bases of fairly small differences in receptor (nR) concentrations, all the drug particles in the solution (blood) bind to the cancer cell receptors before they start to interact with the same receptor on the healthy cell. The pioneering experimental work by the Kiessling group on how low affinity multivalent ligands can be used to selectively target tumor cells27 provided the motivation for Frenkel and co-workers to investigate where selectivity in multivalent systems originates from.28 To understand how to maximize the selectivity of multivalent drug delivery system we first need understand the factors that control the fraction bound () of the multivalent ligands. Following Frenkel we start off by calculating the fraction of bound multivalent particles using the familiar Langmuir isotherm and the avidity constant 𝐾𝑎𝑣 𝐴 to obtain the adsorption profile for the multivalent ligand and thence the fraction bound:22,28 θ=

𝜌𝐾𝑎𝑣 𝐴 1 + 𝜌𝐾𝑎𝑣 𝐴

(6)

Here  = concentration of the multivalent ligand. Using an approach rooted in statistical thermodynamics, Frenkel has then shown that if either individual bonds are weak (Kintra > k),22,28 the degeneracy i from eq 4 can be used to simplify eq 1 and hence eq 7 to obtain:22 𝜌 θ=

𝐾𝐴

(7)

[(1 + 𝑛R𝐾𝑖𝑛𝑡𝑟𝑎)𝐾 ― 1]

𝐾𝑖𝑛𝑡𝑟𝑎

1+𝜌

𝐾𝐴

[(1 + 𝑛R𝐾𝑖𝑛𝑡𝑟𝑎)𝐾 ― 1]


The term outside the square brackets in the numerator is called by Frenkel the dimensionless activity z of the multivalent ligand.22,28 Noting the empirical nature of the effective molarity mentioned above, z and EM appear to closely linked: 𝑧=𝜌

𝐾𝐴

≅


[𝑚𝑢𝑙𝑡𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑙𝑖𝑔𝑎𝑛𝑑] 𝐸𝑀

(8)


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Furthermore, the selectivity  of a multivalent system can also be quantified by considering the change or the gradient of the adsorption profile. The selectivity  is reminiscent of the Hill coefficient (Hill-slope) in the so-called Hill curves (c.f., oxygen binding to hemoglobin).29 The selectively  is therefore obtained through eq 7:22,28 𝛼=

𝑑 ln θ 𝑑 ln 𝑛R

(9)

Frenkel has then used eqs 7 and 9 (and related equations) to investigate how the selectivity of multivalent systems depends on factors such as binding strength (Kintra), flexibility (and hence i), the number of ligands (k) and number of receptors available (nR).28 Some of the most interesting results from this work are shown in Figure 4. For instance, when compared to monovalent systems, the “switch” like behavior of multivalent systems is readily evident. Even if the binding of the “strong” monovalent system is ≈ 8000 stronger than the multivalent (k = 10) one, the slope of the adsorption curve is much steeper and hence the selectivity  is much higher (Figure 4B).28



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Figure 4. Selectivity in multivalent systems. (A) For clarity, the key equations from the text used in the legends and to calculate the plots, are shown here again. (B) Monovalent versus multivalent adsorption profiles (eq 7).28 In all three curves z = 0.003 is used with “strong” monovalent: KA = 1096, “weak” monovalent: KA = 2.7 and multivalent (k = 10): Kintra = 0.13. (C) The adsorption profile (eq 7) for monovalent and three multivalent systems are compared by adjusting the KA/Kintra 22 values so that overall avidity 𝐾𝑎𝑣 𝐴 at  = 0.5 (50% bound) is kept constant. (D) Same data as in (C) plotted on a log-log scale.22 (E) The selectivity  (eq 9) for the data from (C) and (D).22 (F) The effect of varying the activity z (eq 8) on  (eq 9) in a multivalent ligand system with k = 10.28 The Kintra and z values used are shown in the legend. (G) The effect of valency k (number of ligands) shown in the legend on the selectivity  (eq 9) with Kintra and z at fixed values as indicted in the legend.28 Adapted (Figure legends have been modified for consistency) with permission for (B), (F) & (G) from Martinez-Veracoechea et al.28 and (C), (D) & (E) from Curk et al.22


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If we compare multivalent systems with varying degree of ligands (k = 1-10) and adjust the Kintra such that the overall avidity 𝐾𝑎𝑣 𝐴 at  = 0.5 (50% bound) is kept constant, we see clearly how the slope and hence  increases with increasing number of ligands k (Figure 4C).22 This difference is even clearer if we look at a log-log plot of this data where a slope of 1 (see eq 9) would correspond to  = 1 (Figure 4D).22 Plotting the slope or selectivity of this data demonstrates the super-selective22 nature of multivalent ligand system unambiguously (Figure 4E).22 While the monovalent system does not show any regions with  > 1 and the divalent does not quite get to an  of 2, the system with 10 ligands (k = 1) shows super-selectivity with  > 3 but only in a fairly narrow region of the receptor concentration (here with nR ≈ 3-10). The activity z (eq 8) has an unexpected relation to selectivity  with lower z yielding higher  (Figure 4F).28 This means that a lower concentration () of the multivalent ligand generally increases selectivity. Further, even at fairly high concentration, z can be low, and the selectivity  hence high, if the KA/Kintra ≈ 1/EM ratio is low. In other words, weak initial binding intermolecular KA could be highly beneficial in terms of selectivity. Most importantly, the relationship between number of ligands k (valency) and selectivity  is highly non-linear (Figure 4G).28 Given the same activity z and intramolecular binding constant Kintra,  increases first with k until it peaks at a certain kmax before  goes down again. In the example shown in Figure 4G the highest selectivity  is found around k = 5. What happens, however, if the target surface (cell) continues not only the target receptors but also some other “off-target” receptors that nevertheless may still interact weakly with the multivalent drug delivery system? Given the above, weak but multiple binding interactions may, in fact, enhance the binding selectivity. But for “off-target” receptors that is not what we actually want! Angioletti-Uberti has proposed an ingenious solution to this problem, backed by statistical mechanical modelling studies.30 In simple terms, the solution is to add “protective” receptors to the



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surface of the multivalent delivery system itself that bind more strongly to the targeting ligand on multivalent system than the “off-target” receptors (but not as strong as “on-target” receptors).

Receptor clusters, receptor-ligand matching and the angular restriction factor In several important cases, the target receptors are clustered as discrete multimers (dimers, trimers, tetramers…) with the trimeric hemagglutinin on the influenza virus surface being a well-known example.1 In fact, a significant portion of the early efforts in this field1,26 concerned systems where the receptors were (assumed to be) forming discrete clusters. The discrete receptor clustering scenario is in contrast to the assumption used in the modelling by Frenkel and co-workers discussed in the previous section. So what role does flexibility have in systems that display discrete receptor clustering and which other design factors do researchers that want to target such clusters need to take into account? Very recently, Liese and Netz have revisited this problem using statistical mechanics modelling methods,31 showing that the angular restriction factor = 𝜔𝑘LU (Figure 5), which describes the interaction between the ligands and the linkers, plays a pivotal role in these systems.

Figure 5. The angular restriction factor 𝜔𝑘LU.31 (A) Schematic picture (drawn to scale) of the ligand unit that was used in an experimental study to bind to the anthrax receptor. This ligand unit consisted a helical peptide connected to a PEG15 linker.32 (B) Visualization of the angular restriction of the ligand relative to the linker chain (left) compared to the free ligand (right). For the latter, the 82 is the angular space available to a rigid body and it can rotate freely around all three axis in space. Adapted with permission from Liese and Netz.31 Using the interaction between the heptameric anthrax receptor cluster and a heptavalent ligand (nR = k = i = 7) as an example, the calculations by Liese and Netz show that the highest gain in binding affinity is generally obtained when the size of the ligand core and size of the receptor cluster (radius ≈ 3.5 nm) are matched. This matches well with experimental data from Joshi and coworkers32 who used a heptameric cyclodextrin scaffold with a radius of ≈ 1.5 nm to link the Page 14 of 33 ACS Paragon Plus Environment

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anthrax-binding peptide ligands via PEG-based linker. The experimental data and the calculations by Liese and Netz both show that the optimal PEG-linker length is ≈ 2.0 nm, yielding an effective ligand core radius of ≈ 3.5 nm – which matches the anthrax receptor cluster. Surprisingly, these calculations also show that the more restricted the ligand orientation is with regards to the PEG spacer, i.e., the lower the angular restriction factor 𝜔𝑘LU, the stronger the multivalent ligand – receptor binding is! This, Liese and Netz explain, is due to that fact that in the limiting case of receptor clusters and ligands where nR = k, the overall binding constant 𝐾𝑎𝑣 𝐴

(𝑛R = 𝑘 = 𝑖) can be expressed according to eq 10 (note that in the original paper this equation is expressed in the form of a dissociation constant, and hence it has been slightly modified for consistency with other equations in the current discussion):31 𝐾𝑎𝑣 𝐴 (𝑛R = 𝑘 = 𝑖) =

𝑚𝑘𝐶𝑛 𝐾𝑘𝐴𝜔𝑘LU

(10)

In eq 10 all the interactions that do not depend on the number of ligands (k), i.e., the terms that factorize and enter eq 10 with the power of k, are collated in Cn – which the authors describe as a cooperativity factor. The remaining terms all factorize and include the monomeric interaction (KA) and m, which here denotes the number of binding pockets per receptor (nR) subunits. For hemagglutinin and cholera toxin m = 1 whereas for the anthrax receptor m = 2. Finally, eq 10 now shows clearly why a low angular restriction factor 𝜔𝑘LU enhances the multivalency effect. This may at first seem to contradict the model from Kitov and Bundel24 and the subsequent work by Frenkel and co-workers22 discussed above. However, as Liese and Netz point out,31 their model assumes matching valences between the receptor and the multivalent ligands whereas the former models assume the opposite. Arguably, these two approaches could be viewed as limiting cases of a yet to be derived generic model that describes both extremes equally well; the scenario with randomly distributed excess of monomeric receptors on a cell surface (Frenkel and co-



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workers) vs. a perfect match in valency between the receptor cluster and the multivalent ligand (Liese and Netz). An example of the rigid vs. flexible spacer conundrum is illustrated in a recent study by Seitz and co-workers (including Liese and Netz) on the binding of divalent ligand to a receptor pair that was separated by more than 50 Å (5 nm).33 Using a combination of experimental and theoretical data, the authors showed that if the length of a divalent ligand can be tailored to precisely match the receptor pair distance, rigid linkers based on peptide-nucleic acid (PNA)-DNA conjugates bound stronger than a flexible PEG-linked divalent ligand, presumably due to smaller loss of entropy with the rigid spacer. However, if one needs a divalent ligand that is more robust towards separation mismatches, the flexible ligand is more preferable. In other words, for a receptor pair or a cluster, a rigid linker between the ligands on the multivalent drug delivery system is only likely to be beneficial if the inter-receptor distances are fixed and well-known – otherwise the flexible ligand design is likely to work better.

Take home messages on selectivity in multivalency The above discussions can now be summarized in terms of how to maximize the potential of multivalent systems in drug delivery. To reiterate – the conditions relevant to drug delivery differ quite considerably from those of simple “beaker-based” supramolecular chemistry systems. In drug delivery, extremely low concentrations (nM scale) are desired whereas supramolecular studies are typically conducted on the mM (NMR studies) or sub-mM (isothermal titration calorimetry and UV-Vis spectroscopy) scale.2 Furthermore, in drug delivery, the receptors are often in excess – the aim is to saturate the binding of the drug, not the receptor as often is the case in supramolecular studies. There are special situations where these receptors form discrete clusters but it seems now that the early literature was perhaps too fixated on such “ideal” system. Finally, in drug delivery, selectivity is often important, e.g., cancerous versus healthy cell, whereas analogous competition


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studies in supramolecular chemistry, if of interested at all, are usually focused on ligandcompetition. With this in mind we summarize the take-home messages from above:

1. Selectivity and hence the drug action can be achieved by maximizing the selectivity constant  – which set the concentration range whereby the drug activity increases most rapidly. In the context of drug delivery, the therapeutic window between desired and (toxic) side-effects can be targeted by adjusting .

2. Flexible-linker design is the best in terms of maximizing selectivity (with the notable exception when the valency of the receptor cluster and the multivalent ligand are matched31). The “multivalent effect” is all about degeneracy  (see eqs 1, 3 and 4) which is maximized for flexible ligand systems.

3. Low activity z (eq 8) can enhance selectivity. This can achieved through using low ligand concentrations but also by lowering the initial intermolecular binding affinity KA. This could be achieved by adding non-specific repulsion to the multivalent system (e.g., between a nanoparticle and a cell) but in such a way that once a ligand has manage to bind, this repulsion has been conquered and the remaining k-1 ligands can then bind more readily (Kintra is still reasonably high).22

4. Historically, the “ideal” scenario in the study of multivalent system has been the one where the receptors form discrete clusters. For such receptor clusters, the valency of the receptor cluster and the multivalent ligand should be matched (nR = k = i) for optimal binding. Furthermore, two factors need to be considered. Firstly, and somewhat obviously, the overall sizes of the receptor cluster and the multivalent system should be well-matched. Secondly, a small angular restriction factor = 𝜔𝑘LU, which describes the interaction between the ligand and Page 17 of 33 ACS Paragon Plus Environment


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the linker, will enhance the binding strength.31 While one would expect this ligand-linker interaction to be strong when the spacer is rigid,33 the angular restriction factor = 𝜔𝑘LU plays also a major role in systems based on flexible spacers such as PEG.31-32

5. And finally, and perhaps most counter-intuitively, while multivalent systems are clearly more selective than monovalent systems, selectivity may actually go down once a certain number of ligands k has been added. This may explain why there can be too much of a good thing in multivalent systems.

These are the key considerations that come from better insight into the physical chemistry of multivalency. For practical applications, the immunoresponse and the update of multivalent drug delivery systems are also very important. Before we discuss these, a brief discussion on the link between the biology and physical chemistry of multivalent systems is required.

Biology and multivalency – the different modes of multivalent ligand-receptor binding The above discussion highlighted the role of degeneracy/flexibility in multivalent systems, purely from a physical chemistry perspective. The importance of flexibility and degeneracy also becomes evident when the rich variety of possible binding mechanisms that have been observed in biology is considered. Kiessling and co-workers suggested five possible mechanisms by which multivalent ligands could bind to oligomeric receptors on the cell surface (Figure 6).9,34 The first mode of binding, called here the chelation effect (Figure 6A), involves multivalent ligand binding to oligomeric receptors on the cell surface. This is simply 𝐾𝑎𝑣 𝐴 in action. Multivalent ligands can also bind to multiple non-oligomeric receptors which are facilitated through a two-dimensional diffusion of the receptors in the fluid bilayer (Figure 5B). This is more often referred to as a receptor clustering


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mechanism. Successful clustering of these receptors often leads to activation of receptor signalling pathways.

Figure 6. Five mechanisms by which a multivalent ligand can interact with the cell-surface receptor. Adapted (color has been added and legends modified for consistency) with permission for (A), (B), (D) & (E) from Kiessling et al.9 and (C) from Gestwicki et al.34 In an inhibitory event, the binding of multivalent ligands results in steric stabilization whereby the physical size of the ligands prevents the binding of other competitive ligands (Figure 6C). In other cases, a multivalent ligand can bind to a primary and secondary (subsite) binding site of the same protein (Figure 6D). This type of interaction could either occur in an oligomeric or monomeric multi-ligand context. Lastly, a statistical increase in the local concentration of ligands through a multivalent arrangement often leads to an enhancement in the apparent binding affinity of these ligands (Figure 6E).9 This is one of the manifestations of the degeneracy pre-factor  in action.

3. IMMUNORESPONSE Multivalent interactions have been understood to have modulating effects on both innate and adaptive immunities.35 Drug-delivery vehicles possessing this feature can, therefore, interact with the immune system in either safe or detrimental ways. The pioneering works by Dintzis and coPage 19 of 33 ACS Paragon Plus Environment


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workers alluded to the effect of size and valency (k) on the type of immune responses induced by studying dinitrophenyl-polyacrylamide molecules of different molecular weight (0.5, 0.8, 1.4, 1.8, 1.3, and 3.3 x 105 Da) and valences (k = 14, 25, 48, 66, 8, and 17 of dinitrophenyl groups respectively).36 They observed that immunogenicity was not dictated solely by the size of the molecule but also the ligand valency k. Molecules with lower molecular weight were shown to be non-immunogenic, while those above 1.4 x 105 Da were. Interestingly, unlike the 1.4 x 105 Da with k = 48, the 1.3 x 105 Da molecule bearing only 8 dinitrophenyl group (k = 8) was not immunogenic, suggesting that there was a signalling threshold that needed to be met before any immunological responses could occur. This signalling threshold relied not only on the sufficient ligand valency but also on the appropriate spacing and distance between the ligands as they interacted with the cell receptors.11,36 Clearly with respect to the discussion above, it appears that the root cause of these observations relate to selectivity  in these systems. Kiessling and co-workers have since expanded the knowledge on the correlations between multivalency and immune responses by exploring the receptor clustering effect induced by particles before immune responses occur. They observed that the ligand density is a key factor in determining the ability of the multivalent ligand to cluster receptors which are required for triggering subsequent signalling events.37 This echo’s again the requirement for receptors (nR) to be in excess for optimal selectivity . Kiessling also showed that careful tuning of a polymer with high epitope valency k could yield multivalent antigens that were capable of activating B and T cells through receptor clustering and antigen internalisation.38,39 This need for receptor clustering-mediated immune response was also observed in other instances by Stone and Stern where T cell activation was triggered only in the presence of a multivalent (k =2 or 4) major histocompatibility complex (MHC)-peptide, where a T cell receptor clustering could occur. The monovalent (free; k = 1) peptides were unable to activate the same T cells.40 Page 20 of 33 ACS Paragon Plus Environment

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4. UPTAKE OF MULTIVALENT DRUG SYSTEMS Nanoparticle-based drug delivery vehicles are emerging as carriers with the capacity to display ligands in a multivalent fashion. With increasing varieties of nanoparticles intended for biological applications,41 extensive efforts have been put into understanding the fate of these nanoparticles in vitro and in vivo. The mechanisms underlying the uptake of these nanoparticles into cells are of particular interest for understanding their mechanism of action. Many have utilized different chemical probes to study the intracellular trafficking of these particles which is crucial in determining specific receptor target and hence giving valuable information for future designs.42 Dalal and co-workers explored the role of multivalency on the subcellular trafficking of 35−50 nm nanoparticles with valency ranging from k = 10-40 for the development of nanoparticlebased bioprobes. The study revealed that particles with higher multivalency (k ≈ 40 ligands per particle) were internalized through clathrin-mediated endocytosis. Lysosomal trafficking of these nanoprobes restricted their subcellular targeting, whereas the particles with lower multivalency (k = 10-20 ligands) entered the cell via caveolae-mediated endocytosis and were localized to the perinuclear region without trafficking at the lysosome. The transition between clathrin- and caveolae-mediated transports affected the degradation of the material, with the former directing the material to acidic endosomal/lysosomal compartments, and the latter directing the particles to the nucleus, endoplasmic reticulum and the Golgi bodies. The caveolae-mediated transport, in this case, led to a more efficient mode of perinuclear trafficking (i.e., localization at one side of the perinuclear region). Finally, exocytosis was initiated as the multivalency increased further (k > 40 ligands per particle).43,44 In a separate study, Stolnik and co-workers investigated the effect of altering folate ligand density on the surface of ovalbumin-decorated polystyrene nanoparticles on the nanoparticle Page 21 of 33 ACS Paragon Plus Environment


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internalisation.45 They concluded from this study that the uptake of the particle increased with increasing ligand density, up to a saturation level (Figure 7). Upon treating with two different endosome inhibitors (chlorpromazine for clathrin-mediated endocytosis and filipin for caveolaemediated endocytosis), it became apparent that the particles with higher folate density were more affected by filipin, while the particles with lower folate levels were more affected by chlorpromazine. This indicated that there was a shift from clathrin to caveolae-dependence uptake as the ligand density increased (Figure 7B).45

Figure 7. Effect of ligand density on cellular internalisation of folate-ovalbumin (FA-OVA) conjugates. FA-OVA was produced at increasing FA to OVA molar substitution ratios (k = 1.5– 13.2). ‘OVA’ represents nanoparticles with unconjugated OVA (control). (A) Uptake expressed as % of dose applied apically (containing 1.69 x 1013 nanoparticles) per one cell layer (average 1 x 106 cells). Increasing FA to OVA molar substitution ratios resulted in an increased nanoparticle cellular uptake. (B) Internalisation of folate-modified nanoparticles in the presence of endocytosis inhibitors for clathrin (chlorpromazine) or caveolae (filipin)-mediated pathways. Error bar represents mean ± standard deviation (n = 3). *, ** and *** denote p < 0.05, p < 0.01 and p < 0.001, respectively. Adapted with permission from Stolnik et al.45


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Both these examples appear to show k-dependent transitions between different transport mechanisms. Clearly this seems to relate to earlier discussions on how the selectivity  may be linked to some optimal value of k, depending on the strength of interaction and the receptor density. The Hammond group has given another good example of how cellular update of multivalent nanoparticles can be modulated.46 In this work, PEG-terminated dendrons self-assemble to form a micelle (nanoparticles). By mixing non-functionalized and folate-functionalized PEG-terminated dendrons, the Hammond group was able to create nanoparticles with varying folate ligand concentration on the surface and where those ligands were organized in clusters – hence the resulting particles were “patchy”. In both in vitro studies on KB tumor cells and in vivo studies on mice bearing KB tumors, micellar nanoparticles with ca 20% folate surface coverage showed the most efficient uptake,46 i.e., beyond 20% coverage, too much of a good thing seems to be an apt description for this system. The examples elaborated so far showed that multivalency effect has a far-reaching impact on the pharmacokinetics of the ligands. Parameters such as (1) the local receptor concentration (nR), (2) the type of linker (flexible or not), (3) the distance between ligands and the receptor (which amongst other, may relate to the magnitude of Kintra), and (4) the valency (k) of the ligand all contribute in various ways towards a multivalent interaction.47 With such complexity, designing an effective multivalent scaffold is often difficult.

5. CONTEMPORARY CHALLENGES IN MULTIVALENT SYSTEM DESIGN Scaffolds To optimize the multivalent effect, the fundamental physical chemistry and the dynamic nature of biological interactions requires a careful balance between flexibility, binding strength and other relevant parameters. For these reasons, deciding on a scaffold to best present the ligands for a particular system remains challenging. Various scaffolds with different geometries have been Page 23 of 33 ACS Paragon Plus Environment


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explored in the past, ranging from dendrimers,48 metal-based nanoparticles,49 polymers,17 peptidebased scaffolds,18 glycoprotein,10,50 cyclodextrin/calixarene,51 and oligonucleotides.52–54 The difficulty in synthesizing these molecules varies with some being commercially available (like polyamidoamine (PAMAM), polypropyleneimine (PPI) dendrimers, and some calixarene-based scaffolds) while others are based on well-studied polymerization approaches, such as ring-opening metathesis polymerization (ROMP)55 and reversible addition−fragmentation chain transfer (RAFT)56 developed to minimize molecular weight polydispersity. Others still, would require either intricate genetic engineering (like elastin-like polypeptides57 and virus-like particles58,59), or extensive organic synthesis.60 In addition to the scaffold, linker choices (in terms of length, flexibility, spatial spacing, and chemical functionality) and valency (k) optimization (which often depends on the receptor concentration nR) are not straightforward,28,61 as we already noted in this review. However, it does seem that as a general rule, researchers may want to aim for scaffolds that are flexible or at least readily allow the attachment of flexible linkers. The work of Liese and Netz31 underlines also that the interactions between the scaffold and the targeting ligand – the angular restriction factor – can make very significant contributions towards the multivalency.

Quantifying multivalency The better physical chemistry insight we get into the multivalency effect, the more important being able to measure these interactions accurately becomes. Effective design strategies rooted in our latest theory understanding requires good data. Quantification of multivalent interactions have been done using various techniques such as isothermal titration calorimetry, cell-based assays, various microscopy (such as atomic force microscopy, fluorescence microscopy, and transmission electron microscopy) and spectroscopic (such as NMR, total internal reflection fluorescence, UV/Vis, surface plasmon resonance) techniques47 Although these are all highly familiar techniques in the field, ensuring physiological conditions that best represent the binding events at the time of measurement can still be a challenge, especially with regards to measurements in the nM range – Page 24 of 33 ACS Paragon Plus Environment

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which is both the biologically more relevant region as well as the one with the best selectivity as discussed above. This review has highlighted the importance of detailed theoretical analysis of multivalent systems. Much is still left to do on this front. Molecular dynamics studies are one avenue that has been used to verify and expand on theoretical studies,28 as summarized in a recent review by Angioletti-Uberti.62 Clearly, more effort could be put into linking molecular modelling and dynamics studies to theory, measurements and the design of better synthetic multivalent systems for drug delivery. As Angioletti-Uberti points out,62 deeper integration of theory and computational modelling in the design of multivalent drug delivery systems could lead to more selective targeting and advancement of biomedical nanotechnology.

Kinetics We have so far focused on the thermodynamic perspective of multivalency. However, each of the microscopic receptor–ligand interaction could also be viewed from a kinetic perspective, i.e., as the ratio of on (kon) and off-rates (koff). The normal way we investigate a multivalent system with the eye on the ones that give the best thermodynamic outcome. But on the receptor-ligand energy landscape, other pathways are possible and potentially kinetically accessible, even if they don’t yield lowest energy configuration. If the local energy minima are deep enough, the system could get kinetically trapped. This applies in particular to the “trapping” of the last un-binding event – the koff rate relating to the intermolecular binding constant KA. Furthermore, non-equilibrium energy input, e.g., ionic gradients, which conceivably could be quite important under biological conditions, may close existing or open up new kinetic pathways. Given the size and complexity of biological receptors and synthetic multivalent systems alike – it is not difficult to envision several populations of different conformations that are not too dissimilar in energy but each with quite different kinetic profiles.



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Li and co-workers have investigated the binding behaviour of a multivalent system with high ligand numbers (k), where the results suggests that the system cannot reach a true equilibrium. A series of single-stranded DNA (ssDNA)-functionalized dendrimers were doped against complementary oligonucleotides on a functionalized surface plasmon resonance surface to measure binding. They observed that the Surface Plasmon Resonance association binding curve never reached equilibrium even after an extended association time. In this case, an increase in the ligand binding to the surface continued to occur. From this result, they suggested that two subpopulations existed in the multivalent ligand interaction: a major population exerting weaker binding with rapid equilibrium and faster dissociation rate, and a second minor population with a strong association and minimal dissociation (Figure 8), possibly representing kinetically trapped states. Hence, the association rate of the multivalent ligands continued to increase with no complete dissociation (Figure 8B), whereas the monovalent ligand showed equilibrium association and rapid total dissociation afterwards (Figure 8A).19


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Figure 8. Surface plasmon resonance association and dissociation binding curve showing the different binding behaviour of monovalent vs. multivalent single-stranded DNA (ssDNA) oligonucleotides (A) Binding of free-oligonucleotide (monovalent) ssDNA to a surface decorated with complementary ssDNA. (B) Binding of multivalent ssDNA oligo-decorated dendrimer (k = 6) to the same surface as in (A). (C) Schematic showing the association and dissociation of the particles. Adapted with permission from Li et al.19 As far as we know, there are no examples yet of multivalent drug delivery systems where the selectivity / targeting of the system is based on kinetically trapping certain conformers of the ligand system on the target receptors. In a similar vein, one could envision using non-equilibrium chemistry approaches, e.g., via light, mechanical forces or chemical gradients, to “push” the equilibrium between various conformers of the multivalent drug delivery system from one which was dominated by low-affinity confirmers towards highly selective stronger binding confirmers. By focusing on and ultimately controlling the kinetics of multivalent ligand-receptor interactions, Page 27 of 33 ACS Paragon Plus Environment


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researchers could create even more selective and “smarter” forms of drug delivery systems based on the multivalency effect.

6. PERSPECTIVE As the drug delivery field moves towards a site-specific, target-oriented approach, multivalent interactions have become an invaluable tool in the design of targeted drug delivery systems. This is particularly so when it comes to developing therapeutic strategies that target cellular receptors. This review has highlighted the need for a proper theoretical understanding of the multivalent effect. In particular how selectivity () relates to the design of the multivalent systems. There are strong indications from the experimental literature that it is indeed selectivity that is the most important design factor in terms of optimizing drug activity, the immunoresponse and the uptake mechanism of multivalent drug systems. Theory and experiment alike indicate that flexibility is more important than the individual binding strength. In the special limiting case of systems where receptors form discrete clusters and the valency of the ligand and clusters are matched, ligand-linker (angular restriction factor) interactions are at least as important as the size-match between the receptor cluster and the multivalent ligand system.31 More generally, adding more ligands is not always the best strategy as selectivity appears to correlate very non-linearly to the number of ligands present. Although the key framework is now in place thanks to the work of Frenkel and co-workers,22,28 more theoretical and experimental work to allow for more reliable predictions of selectively in multivalent systems. This would require amongst other better methods for modelling and measurements in multivalent systems. Small-molecule-based supramolecular chemists could play a significant role but the current approach in small-molecule-based work may need to be modified to more accurately reflect what sort of environment multivalent drug delivery systems need to be designed. Selectivity could as Frenkel points out, be improved by reducing KA, i.e., the initial binding event, by specifically engineering systems with a repulsive factor.22 Related to this is AngiolettiPage 28 of 33 ACS Paragon Plus Environment

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Uberti’s suggestion to add “protective” receptors to the multivalent ligand system in order to suppress non-specific binding to “off-target” receptors.30 Neither strategy has been widely used in the design of multivalent drug delivery systems. The other strategy that could be employed is to utilize kinetic trapping or non-equilibrium effects to fixate the delivery system on the receptor once it is bound. Biological systems are de facto non-equilibrium and hence it is not inconceivable that directional or concentration chemical gradients in the body (e.g., tissue-dependent changes in blood chemistry) could be used to enhance the binding of multivalent systems at their target site. This relates also to understanding better what happens to the receptor-ligand complex once it is formed. The field itself will therefore benefit more from studies that look at effects beyond the initial ligandreceptor interaction, as it will reveal more impacts that multivalent interaction has on the pharmacokinetics of the ligands. Echoing similar words from a recent review by Angioletti-Uberti,62 perhaps the most important message of this review is that the recent advances in both theory and computational methods mean that those that are focused on designing and synthesizing multivalent systems for drug delivery have now everything to gain (and nothing to lose!) by joining forces with theory and computational experts to design better, more effective and more selective multivalent systems for drug delivery.

ACKNOWLEDGEMENTS This work was supported by the Australian Research Council (ARC) through a Centre of Excellence Grant (CE140100036) to P.T. which also provided a PhD scholarship to K. T. We thank Susanne Liese and Roland R. Netz for valuable discussions during the revision of this manuscript.

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Figure 1. (A) Folic acid binding (right y-axis in blue) as measured by Surface Plasmon Resonance (SPR) and in vitro cellular binding (left y-axis in red) as measured by flow cytometry, initially increases exponentially with the number of ligands before it plateaus at k ≈ 5.4 (B) Voltage clamp measurements show that the activity of α7-nicotinic receptor blocker azido-α-conotoxin ImI peptide increases 100-fold upon dimerization (k = 2) of the ligand, however, the increase in inhibitory activity was more modest as the number of the ligand was increased to four (k = 4) copies. Adapted with permission from (A) Hong et al.4 and (B) Wan et al.5 81x112mm (300 x 300 DPI)

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Figure 2. Key concepts in multivalency. N.b. statistical or degeneracy pre-factors (Ωi) are not included in the schemes shown. While the receptors (red) as shown as a regular array for clarity it is assumed they can move randomly around on the cell surface. The (multivalent) ligands (blue) are assumed to be fully flexible. (A) The binding of a monovalent (k = 1) ligand to a multivalent receptor (nR = 3). Assuming there is no cooperativity (and after correcting for degeneracy), the individual (microscopic) binding constants23 are all the same (Kmono is a constant). (B) The binding of a multivalent ligand with k = 3 to the same receptor as in (A) with nR = 3. The first step is intermolecular = KA and differs from the subsequent intramolecular Kintra steps. In the absence of cooperativity Kintra is a constant. (C) Key abbreviations used in the discussion on multivalency in this review. 81x110mm (300 x 300 DPI)



Figure 3. The importance of degeneracy in the binding of monovalent versus multivalent ligands. (A) Increasing receptor number from 3 to 6 contributed to only 2-fold increase in binding probability in monovalent ligands. (B) In the case of multivalent ligands, degeneracy Ωi increases by a factor of 20 (see eq 4). Adapted with permission from Curk et al.22 77x64mm (300 x 300 DPI)


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Figure 4. Selectivity in multivalent systems. (A) For clarity, the key equations from the text used in the legends and to calculate the plots, are shown here again. (B) Monovalent versus multivalent adsorption profiles (eq 7).28 In all three curves z = 0.003 is used with “strong” monovalent: KA = 1096, “weak” monovalent: KA = 2.7 and multivalent (k

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