Divalent ligand-monovalent molecule binding

Simultaneous binding of divalent ligands to two identical molecules is a widespread phenomenon in biology and chemistry. Here, we describe this binding event as a divalent ligand AA that can bind to two identical monovalent molecules B to form the complex AA · B2. Cases where the total concentration [AA]T is either much larger or much smaller than the total concentration [B]T have been studied earlier, but a description of intermediate concentrations is missing. In this paper, we describe the general case of any ratio of ξ ≡ [B]T /[AA]T. We show that the concentration of the intermediate complex AA · B is governed by a cubic equation and discuss several scenarios in which this cubic equation simplifies. Our numerical results, which cover the entire range of 0 < ξ < ∞, are relevant to processes wherein the concentrations of free ligands and proteins both decrease upon binding. Such ligand and protein depletion is expected to be important in cellular contexts, e.g., in antigen detection and in coincidence detection of proteins or lipids.


I. INTRODUCTION
Chemical binding is at the heart of many processes in biology, including oxygen binding to hemoglobin, self assembly, antibodies binding to antigens, and growth factors binding to their transmembrane receptors [1][2][3][4][5][6]. In many cases, binding interactions should be specific and strong, yet reversible [7][8][9][10]. One way to accomplish such a "molecular velcro" [7] is through ligands containing many ligating units per molecule: Multivalent ligands are known to bind transmembrane receptors more readily than their monovalent counterparts (with one binding site per ligand). This makes multivalent ligands interesting in clinical applications, for example, where less therapeutic cargo is needed for the same response. The intuitive explanation why multivalent ligands bind more readily to, for instance, receptors on a plasma membrane or a viral envelope, is that, after the binding of a first ligating unit with association constant K 1 , the other ligating units are close to other membrane-bound receptors as well. Around a first bound unit, a second ligating unit is thought to sweep out a semi circle with a radius set by the (fixed) distance between ligating units [11][12][13][14]. This is typically a nanometers length, meaning that the effective concentration of ligating units belonging to a partly-bound multivalent ligand is much higher than the concentration of unbound ligands nearby. More generally, for flexible rather than stiff linkers between ligating units [15,16], increased effective concentrations can be determined rigorously by statistical mechanics [17,18].
In turn, high effective concentrations are reflected in a high association constant K 2 for binding a second ligating unit of a multivalent ligand, and the same for further binding steps. Systems for which K 2 /K 1 > 1 are * mathijsj@uio.no † acarlson@math.uio.no called cooperative [19][20][21][22]. In the above example of large effective concentrations, one speaks of apparent cooperativity. This is to distinguish it from true cooperativity, which refers to binding pockets whose binding affinity changes when nearby pockets are occupied, as happens for the binding of oxygen to hemoglobin [23]. In either way, the hallmark of cooperativity is the switching from mostly-unbound to mostly-bound ligands over a narrow protein-concentration range [19]. Ligand-protein binding models often have governing equations that simplify when one molecular species is assumed to be present in excess compared to other species. While this assumption may be appropriate to certain systems and experiments, it is not always the case. One example is when two types of ligands compete for the binding of one type of receptor. In this case, the relative concentrations of the ligands must be important-unless the receptor is in excess to both types of ligand, in which case there would be no competition for it. When no molecular species is in excess to the other present in the system, binding can significantly reduce the concentration of unbound species. Such depletion is difficult to capture in theoretical models, even for the steady state, as governing algebraic equations are typically nonlinear and with a high polynomial order. Two notable exceptions where the concentrations of all species can be expressed analytically are monovalent ligand-monovalent receptor binding [1] and the competitive binding of two different monovalent ligands to one type of monovalent protein [24].
Several recent review articles [19,20,22] discuss the reversible binding of a divalent ligand AA to two identical monovalent proteins B [ Fig. 1(a)], as it is the simplest example of a binding reaction with nontrivial effects of multivalency and cooperativity. Yet, Eq.
From the law of mass action follow the reaction-rate equations associated with Eq. (1). In turn, the steady state of these equations yields two association constants K 1 and K 2 as [see Appendix A] where factors of 1/2 and 2 account for the degeneracy of the intermediate complex AA · B. An assumption underlying the derivation of Eq. (2) in terms of concentrations, is that all species are well mixed. This assumption may be violated when receptors cluster at the plasma membrane [31,32].
The reaction in Eq. (1) does not affect the total concentration [AA] T and [B] T of ligands AA and proteins B-both bound and unbound-and, hence, needs to be satisfied.
We rewrite Eq. (5b) to Inserting Eq. (6) into Eq. (5a) yields While Eq. (7) for x 3 can be solved analytically with Cardano's formula, unfortunately, its solution, presented in Appendix B, is too cumbersome to be helpful. In Appendices C-E, we analyse Eq. (7) for limiting values of the ligand-to-receptor ratio, ξ 1 and ξ 1, and for the case where the cooperativity parameter α = K 2 /K 1 takes the value α = 1. The analytical results obtained there help us interpret the numerical solutions of Eq. (7) that we present below.

III. RESULTS
After Eq. (4), we reduced the four parameters [AA] T , [B] T , K 1 , and K 2 of our original problem [Eqs. (2) and (3)] to three dimensionless combinations κ 1 , κ 2 , and ξ thereof. We choose these particular combinations to tidy up the calculations of Section II and Appendices B-F. But for the description of particular systems or experiments, other dimensionless combinations of the four dimensional parameters may be more appropriate. Accordingly, to recover the results of Ref. [29], we first vary   7) and (6) as a function of Here, panel (b) generalizes the "cross linking curves" of Fig. 3 of Ref. [29] to ξ values away from the limit ξ → 0. For ξ = 0.2, we show Eq. (D5) (purple dashed line) as obtained by Ref. [29]. Small difference are visible in Fig. 2(b) between the purple dashed line and the purple diamonds, which means that, for ξ = 0.2, Eq. (D5) approximates the numerical solution to Eqs. (6) and (7) well, but not perfectly. This finding is in line with Appendix D, where we find that Eq. (D5) contains errors of O(ξ 3 ). Reference [29] showed that max(  Fig. 2(b) up to ξ = 1. Moreover, the bell shape of Eq. (D5) was shown to be symmetric around its maximal value [29]. With increasing ξ, however, we see that this symmetry is broken. For ξ > 2, [AA · B 2 ]/[AA] T and [AA · B 2 ]/[B] T are sigmoidal instead. Next, we show Eq. (C2) (thick grey line), which corresponds to the limit ξ → ∞ [19]. Small differences between this expression and the numerical solution to Eqs. (6) and (7) for ξ = 100 are visible in both panels of Fig. 2 T . Different from before, we fix the dimensionless cooperativity parameter α ≡ K 2 /K 1 , as it is often set solely by (fixed) molecular properties [17,18].  Fig. 3(a). Notably, the maximal plateau height occurs at ξ = 1, as also follows from Eq. (E4). Next, we compare our numerical results for ξ = 0.2 (purple diamonds) to the expressions derived in Ref. [29] [cf. Eqs. (D4) and (D5)] (purple dashed lines). These panels reinforce our analytical insights of Appendix D, namely, that the expressions derived in Ref. [29]  For the same parameters as in Fig. 3, Fig. 4 shows the receptor occupancy θ ≡ Fig. 4(a)] and α = 100 [ Fig. 4(b)]. We see that increasing cooperativity shifts θ curves to smaller K 1 [B] T values and that θ switches from θ ≈ 0 to θ ≈ 1 over a narrower range of K 1 [B] T . To characterise the slope of θ, we numerically determined the Hill coefficient where κ * 1 is such that θ(κ * 1 ) = 1/2; hence,    Fig. 6 of Ref. [19], which showed n H for ξ 1 [Eq. (C4)], indicated here with a thick grey line. We see that, for ξ = 100, the numerically determined n H is close to predictions from Eq. (C4). Conversely, we see that n H → 0 for ξ → 1. For ξ < 1, we see in Fig. 4 (a) and (b) that θ < 1/2, leaving n H undefined. The dots in Fig. 4(c) for α = 1 represent the analytical expression Eq. (E7), which gives a perfect match when compared with the numerical prediction.
At last, we mimic a titration experiment by varying [B] T at fixed K 1 , K 2 and [AA] T , i.e., varying κ 1 at fixed α and K 1 [AA] T (or κ 2 /ξ, in terms of our original dimensionless parameters).   Second, to model antibody binding to surface-bound antigens, Refs. [11,12] expressed concentrations of antigens and (partly) bound complexes in numbers per unit area [12,26]. We note, however, that the governing equations of Refs. [12,26] could also be cast into the form of Eqs. (2) and (3), that is, with volumetric concentrations only, and the effect of reduced positional freedom of surface-bound molecules absorbed into the constants K 1 and K 2 . Conversely, though volumetric concentrations appear in our Eqs. (2) and (3), this set of equations can just as well describe a binding process wherein either AA or B is confined to a thin (membrane) surface (see also page 13 of Ref. [1]). Next, Refs. [11,12] postulated specific relations between K 2 and K 1 . Here, we studied Eqs. (2) and (3) for general K 1 and K 2 instead. Hence, in order to apply our mathematical framework to specific binding reactions, one should first determine α = K 2 /K 1 , for example, with the methods of Refs. [17,18].
Third, the constraint of particle conservation in homobivalent ligand-monovalent receptor binding-described in this article-can be especially relevant in cellular contexts, where few molecules of either species may be present. However, for tiny systems with small numbers of particles, the reaction rate equation-type modelling that underlies our results breaks down. One should then account for stochasticity [33], possibly using our continuum results as a benchmark.

V. CONCLUSION
We have laid out a unified description of the reversible binding of a bivalent ligand to two identical monovalent proteins. The same process has been studied previously, but only in concentration limits of either much more ligands than proteins or vice versa. We have described the binding process for any concentration of ligands and proteins. Comparable concentrations of species can occur both in in vivo and in synthetic biological systems. Our theoretical work is built on classical reaction-rate equations. At steady state these reduce to four coupled equations for the concentrations [ Our work can be a stepping stone to study the effect of nontrivial protein-to-ligand ratios on hetero bivalent interactions [14,[34][35][36]. Future work could include how comparable molecular concentrations affect the competition between monovalent and divalent receptors for divalent ligands [28,37].
which need to be supplement with initial concentrations of the four species, which we choose as Time-dependent concentrations were studied in [13].