Binding Efficiency of Protein-Protein Complexes

Abstract

We examine the relationship between binding affinity and interface size for reversible protein-protein interactions (PPI), using cytokines from the tumor necrosis factor (TNF) superfamily and their receptors as a test case. Using surface plasmon resonance, we measured single-site binding affinities for the large receptor TNFR1 binding to its ligands TNFα (K_D = 1.4 ± 0.4 nM) and lymphotoxin-α (K_D = 50 ± 10 nM), and also for the small receptor Fn14 binding to TWEAK (K_D = 70 ± 10 nM). We additionally assembled data for all other TNF/TNFR family complexes for which reliable single site binding affinities have been reported. We used these values to calculate the binding efficiency – defined as binding energy per Ų of surface area buried at the contact interface – for the nine of these complexes for which co-crystal structures are available, and compared the results to those for a set of 144 protein-protein complexes with published affinity values. The results show that the most efficient PPI complexes generate ~20 cal.mol⁻¹/Ų of binding energy. A minimum contact area of ~500 Ų is required for a stable complex, required to generate sufficient interaction energy to pay the entropic cost of co-localizing two proteins from 1 M solution. The most compact and efficient TNF/TNFR complex was BAFF/BR3, which achieved ~80% of the maximum achievable binding efficiency. Other small receptors also gave high binding efficiencies, while the larger receptors generated only 44-49% of this limit despite interacting primarily through just a single small domain. The results provide new insight into how much binding energy can be generated by a PPI interface of a given size, and establish a quantitative method to predict how large a natural or engineered contact interface must be to achieve a given level of binding affinity.

Understanding the molecular origins and the magnitudes of protein-protein binding energies is critical to achieving a quantitative picture of biological function at the molecular level. This knowledge is also important for the development of drugs to modulate such functions, and for designing engineered proteins to achieve desired binding properties. In 1999 Kuntz and coworkers attempted to quantify the maximum binding affinity that a protein can achieve with a small molecule ligand of a given size (1). They showed that relatively small ligands (those containing up to ten or so nonhydrogen atoms) can generate as much as ~1.5 kcal/mol of binding energy per heavy atom of their structure through interactions with their target protein. This seminal publication led to the development of the important concept of ligand efficiency – the assessment of binding strength in terms of binding energy divided by the number of heavy atoms in a ligand’s structure (2) – as a means to quantify how effectively a given small ligand or set of atoms within a ligand interact with a protein to generate binding affinity. The concept of ligand efficiency has become important in drug discovery, where minimizing the size of the ligand (drug) is often critical to achieving good pharmaceutical properties (3), and is now widely used to evaluate the quality of screening hits and to guide drug lead optimization. The aim of the current study is to extend the concept of ligand efficiency to protein-protein interactions (PPI) by asking what is the maximum binding efficiency – i.e. binding affinity per unit contact area – that can be achieved for interaction with a protein ligand of a given size, and how do the binding efficiencies observed for a set of structurally homologous PPI complexes, involving tumor necrosis factor (TNF) family ligands binding to their cognate TNFR family receptors, compare to this maximum value.

For small molecule ligands the number of heavy atoms is a simple and convenient surrogate for molecular size. For protein-protein interactions, on the other hand, the total number of heavy atoms in a participating protein is essentially irrelevant to binding affinity, as the majority of atoms do not directly participate in binding. Nor is the number of heavy atoms actually involved at the binding interface a particularly convenient measure of interface size. Therefore, the ligand efficiency of a PPI complex is best described in terms of binding energy per square Angstrom of interface area. For the purpose of this study we call this quantity “binding efficiency” (units: cal.mol⁻¹/Ų), to distinguish it from the established term “ligand efficiency” with its specific meaning of binding energy per heavy atom (or, sometimes, per unit of molecular weight) of the ligand (2).

Cytokines of the tumor necrosis factor (TNF) superfamily and their receptors provide a good case study for examining variations in the affinities that can be achieved for reversible protein-protein interactions with binding interfaces of different sizes. TNF family cytokines are constitutive trimers that contain three receptor binding sites, one at each subunit interface of the ligand (Figure 1A). TNF family receptors possess a modular structure in their extracellular region, containing one or several cysteine rich domains (CRDs), each CRD itself comprising a pair of modules with conserved disulfide bonding patterns (Figure 1B) (4). The first ligand-receptor co-crystal structure reported for this family, Lymphotoxin-α (LTα, a.k.a. TNFβ or TNFSF1) bound to the extracellular portion of TNF receptor 1 (TNFR1, a.k.a. TNFRp55 or TNFRSF1A) which contains four CRDs, revealed that each trimeric ligand binds three TNFR1 molecules to form a three-fold symmetric complex, with each receptor forming an extensive contact surface primarily involving its second and third CRDs (Figure 1A) (5). Other TNFR superfamily members with multiple CRDs adopt a similar binding geometry with their respective ligands (6-14). In recent years, however, a number of TNFRs have been discovered that are much smaller than the prototypical TNFR1 (15). Some of these contain only a single CRD or, in the case of BR3 (BAFFR, TNFRSF13C), just a fragment of a CRD (16). X-ray and NMR structures of small TNFRs bound to their cognate TNF family ligands show that they conform to the same basic binding topology established for the larger receptors, but occupy only a fraction of the binding footprint defined by TNFR1 with LTα (Figure 1C) (17-19). TNF/TNFR family interactions thus constitute a set of homologous protein-protein complexes that display a range of interface sizes within an overall binding mode that is quite conserved.

Figure 1.

Structure and interaction mode of TNF superfamily ligands with their receptors. (A) Cartoon illustrating how a trimeric TNF family ligand binds to three TNFR family receptor molecules to form a 1:3 complex, with the TNFR molecules binding in the clefts at the subunit interfaces on the ligand. (B) Modular structure of the eight TNFR family receptors considered in this study. The receptors contain up to four cysteine rich domains (CRDs), each comprising a pair of modules designated by a letter (A, B, C or D) to indicate the fold and a number (1 or 2) corresponding to the number of disulfides present. The receptors considered in this study vary in size from TNFR1, CD40 and DcR3, each of which contains four CRDs, down to BR3 which contains only half a CRD. (C) X ray co crystal structures of CD40/CD40L (PDB code 3QD6) and BR3/BAFF (PDB code 1OQE), showing that the overall binding mode is conserved but BR3 occupies a much smaller portion of the binding groove on the ligand.

We recently reported single-site binding affinities for two of the smaller TNFR receptors, BR3 and BCMA, to their cognate ligands BAFF and APRIL (20). By separating affinity from avidity effects we were able to gain new insights into the selectivity of BAFF and APRIL for their preferred receptors versus other receptors that they can also bind. Somewhat to our surprise, we found that these small receptors bind with high affinity, comparable to or in some cases much greater than the single-site affinities we observed for some much larger TNF ligand-receptor pairs such as CD40L/CD40 (21) in which the receptors contain up to 4 CRDs and appear to have more extensive binding interfaces (Figure 1C). Thus, within the TNF/TNFR superfamilies different receptor-ligand pairs appear to manifest a wide range of values for the amount of binding energy they generate per unit of contact surface area.

The relationship between binding energy and interface size for PPI complexes has been addressed in previous studies, using a range of approaches (22-26). Some publications report overall trends between binding affinity and the amount of surface area buried at the interface, typically quantified as the change in solvent-accessible surface area (ΔASA), for particular data sets (22-24, 26-29). But, in general, there is agreement that PPI complexes develop under a variety of evolutionary constraints – such as stability, solubility and selectivity – that in many cases will trump any evolutionary pressure to maximize interaction affinity. Thus, across a diverse set of PPI complexes no strong correlation between ΔG₀ and ΔASA is expected or observed (22, 23, 27). The question of what is the maximum amount of binding energy achievable for a PPI with a given interface size has not been definitively answered, however. Brooijmans et al. have proposed that very small PPIs can generate up to 1.5 kcal/mol of binding energy per nonhydrogen ligand atom at the interface (22), but this conclusion required rather a long, nonlinear extrapolation from a set of PPI complexes all of which had actual measured binding efficiencies ranging from ~0.07 to ~0.31 kcal/mol per nonhydrogen interface atom.

The current study focuses specifically on what available experimental data reveal about the maximum binding efficiencies that can be achieved by proteins when engaging in reversible PPI – or, at least, the highest efficiencies that are actually achieved in real systems. The goal is to gain insight into the physicochemical and structural factors that limit the amount of binding energy that proteins can generate through their interactions with other proteins. To address this question we report a systematic analysis of single-site receptor-ligand binding affinities across a set of TNF/TNFR complexes that includes both larger and smaller binding interfaces, and calculate the amount of binding energy per unit of contact area that is achieved by each. To place the binding efficiency of the TNF/TNFR complexes on a scale relative to the maximum achievable binding efficiency for a reversible protein-protein interaction, we perform a similar analysis for a set of 144 protein-protein complexes using published affinity values, and use these results to propose a maximum binding efficiency (i.e. binding energy per Ų of contact area) that is observed for reversible protein-protein complexes. In addition to shedding light on ligand-receptor recognition in the TNF/TNFR family, the results provide new insight into how much binding energy can be generated by a PPI interface of a given size, and suggest a minimum value for how extensive a natural or engineered PPI contact interface must be to achieve a given level of binding affinity.

MATERIALS AND METHODS

Proteins and Reagents

Recombinant soluble human TWEAK was purified as previously described (30). Briefly, a construct encoding human TWEAK residues A106-H249 was expressed in Pichia pastoris. The fermentation medium was concentrated and diafiltered, and then purified by ion-exchange on a Q Sepharose column, followed by a Zn chelating column, and finally by size exclusion chromotagraphy using a Sephacryl 300 column. LTα was purified from baculovirus infected sf9 (Life Technologies) insect cell culture medium produced as described (31). Cells and cell debris were removed by centrifugation and filtration. The clarified insect cell supernatant containing LTα was concentrated by ultrafiltration, and purified using a TNF-R55-Fc affinity column prepared as described (31). TNFα was expressed and purified as previously described (32). Briefly, the protein was expressed in E. coli and purified using tandem HiTrap S HP columns, followed by tandem HiTrap Q HP columns (Amersham). The protein was dialyzed against 10 mM Tris pH 7.5, 50 mM NaCl, 1 mM β-ME and 1 mM cysteamine at 4 °C overnight to complete native disulfide bond formation. β-ME and cysteamine were removed by dialysis against 10 mM Tris 7.5, 50 mM NaCl overnight at 4 °C. The solution was ultrafiltered before storage at –80 °C. Soluble TNFR1 extracellular domain was expressed and purified as described in (32). Briefly, a construct encoding human TNFR1 extracellular domain, residues 40-201, was expressed in E. coli and purified from inclusion bodies by denaturing in 6 M guanidine-HCl, 100 mM Tris 8.5, 4 mM PMSF, 20 mM DTT. Oxidized glutathione (30 mM) was added, and the solution was slowly diluted 10-fold with 50 mM Tris 10.7, 6 mM cysteine, and 4 mM PMSF, then incubated overnight with gentle stirring at 4 °C. Insoluble material was removed by centrifugation. The supernatant was dialyzed twice against 50 mM Tris 7.5 at 4 °C, and then twice again against 50 mM MES pH 6.2 with 25 mM NaCl at 4 °C. Insoluble material was once again removed by centrifugation and subsequent passage through a 0.2 μm filter. The resulting protein solution was purified over an Uno-S column (Pharmacia), and the fractions containing TNFR1 were sterilized by 0.2 μm filtration before storage. This protein was shown to be monomeric by SDS-PAGE (Supporting Information, Figure S1), and was shown to be fully active by comparison with recombinant soluble TNFR1 from two other, independent sources; all three TNFR1 preparations showed identical affinities for binding to TNFα in a homogeneous time-resolved FRET competition assay measuring their ability to compete against XL665-labeled TNFR for binding to Europium-labeled TNFα (Friedman et al., unpublished). Soluble Myc- plus His-tagged human Fn14 extracellular domain, residues 28-80, was purified and characterized as described previously (33). Briefly, the protein was expressed in Pichia pastoris, and purified by Ni/NTA affinity chromatography followed by size exclusion chromatography. The purified protein was characterized by mass spectrometry, and was shown to be monomeric by SDS-PAGE and by gel filtration (33) (see representative SDS-PAGE data in Supporting Information, Figure S1). Proper folding of the recombinant Fn14 protein was established by showing that all cysteine residues in the protein were fully oxidized, by establishing the disulfide bond connectivity by proteolytic digestion followed by mass spectrometry, and by comparing its binding activity to that of Fn14 protein produced using other expression systems (33).

Biacore Binding Measurements

Biacore chip preparation – All experiments were performed using a Biacore 3000 instrument (GE Healthcare). Soluble TNF family ligands were immobilized on CM5 sensorchips using the Biacore Amine Coupling Kit according to manufacturer’s instructions. Ligands were diluted in 10 mM acetate buffer and injected over Biacore sensorchip flow cells that had been activated with 1:1 N-hydroxysuccinimide (NHS): 1-Ethyl-3(3-dimethylaminopropyl)carbodiimide hydrochloride (EDC). Excess free amine groups were then capped with a 50 μL injection of 1 M ethanolamine. The ligand surfaces were conditioned with 5 × 30 s injections of regeneration solution (see Table 2). A flow rate of 10 μL/min was used for all immobilization procedures. Exact conditions and typical immobilization levels for each ligand are shown in Table 2. For each experiment, underivatized control surfaces were prepared in the same manner as the ligand surface, except that acetate buffer containing no protein was injected in place of ligand. Biacore buffer (10 mM HEPES pH 7, 150 mM NaCl, 3.4 mM EDTA, 0.005% p-20 detergent) was used as the running buffer during TWEAK immobilizations. The same buffer, except containing 0.0005% p-20 (low detergent buffer), was used for TNFα and LTα immobilizations. The activity of the sensorchip surface generated by coupling the TNF family ligand was established by its ability to bind the appropriate receptor saturably, reproducibly and with high affinity. In each case, a number of different running buffers and regeneration conditions were tested, and the best condition was further optimized to find buffers that maintained the integrity of the trimeric TNF family ligands captured on the biosensor surface, while efficiently dissociating any bound receptor at the end of each cycle. Note that the running buffers that were used contained Tween-20 detergent and 0.05% BSA, to minimize the possibility that the soluble monomeric receptor proteins might form any weak, nonspecific aggregates that might affect their binding properties. The stability of each derivatized sensorchip surface, and its robustness to repeated cycles of binding and regeneration, were established in two ways: (i) by monitoring for any progressive decrease in the baseline signal (i.e. the signal level observed after regeneration of the surface and before receptor is passed over the chip) resulting from disruption of the trimeric structure of the coupled TNF family ligand followed by dissociation of the released subunits, and (ii) testing for reproducibility in the receptor-binding capacity of the surface, as measured by the amplitude of the binding signal observed when a given concentration of receptor was flowed over the chip. After careful selection and optimization of the capture conditions, the running buffer, and the regeneration buffer, we were able to find for each TNF family ligand conditions giving a stable surface that showed negligible degradation of its receptor-binding capacity over multiple cycles of binding and regeneration.

Tabe 2.

Conditions used to prepare and regenerate the derivatized Biacore sensorchips used in the binding experiments.

	TWEAK	TNFα	LTα
Ligand concentration (mg/mL)	50	75	50
10 mM Acetate pH	5.0	5.5	6.5
NHS:EDC activation volume (μL)	50	5	5
Ligand injection volume (μL)	50	5	5
Typical Immobilization level (RU)	1000 – 2000	~300	~1000
Regeneration solution	10 mM H3PO4	50 mM glycine pH 4.0 + 500 mM NaCl	50 mM glycine pH 4.0 + 500 mM NaCl

TWEAK/Fn14 kinetic binding assays

Soluble, monomeric human Fn14 was diluted in Biacore assay buffer (Biacore buffer + 0.05% BSA) to the indicated concentrations and injected over the TWEAK derivatized surface, or over an underivatized (i.e. NHS/EDC activated and then ethanolamine-capped) surface as a background control, at a flow rate of 50 μL/min. Following the protein injection, dissociation was monitored for 10 min at the same flow rate. The TWEAK surface was regenerated with 2 × 30 s injections of 10 mM H₃PO₄. In all cases binding to the underivatized chip was negligible.

TNFα/TNFR1 and LTα/TNFR1 kinetic binding assays

Soluble monomeric TNFR1 was diluted in low detergent buffer (10 mM HEPES pH 7, 150 mM NaCl, 3 mM EDTA, 0.05% BSA, 0.0005% p-20) to the indicated concentrations and injected over the TNFα or LTα derivatized surfaces, or over underivatized surfaces as a background control, at a flow rate of 30 μL/min for approximately 8 min (250 μL injection volume). Receptor dissociation was then allowed to proceed for 15 min before surface regeneration by 2 × 30 s injections of 50 mM glycine pH 4.0 + 500 mM NaCl. In all cases binding to the underivatized chip was negligible.

Biacore data analysis

Prior to fitting the data all sensorgrams were “double referenced”; that is, each sensorgram was background-corrected by subtracting the data for the same solution run over the underivatized sensor surface, and then subtracting the similarly treated data for the buffer alone (i.e. [TNFR] = 0) sample run on the ligand-derivatized surface. The binding affinity was determined by performing a global fit of the double-referenced sensorgrams from a given experiment to the kinetic model for simple, single-site binding shown below, using the BIAevaluation software:

$$\begin{array}{cc}\hfill & \mathrm{dAB}∕\mathrm{dt}=\left({\mathrm{ka}}^{\ast }{\mathrm{A}}^{\ast }{\mathrm{B}}^{\ast }\right)-\left({\mathrm{kd}}^{\ast }\mathrm{AB}\right)\hfill \\ \hfill & \mathrm{dB}∕\mathrm{dt}=-\left(\right({{\mathrm{k}}_{\mathrm{a}}}^{\ast }{\mathrm{A}}^{\ast }\mathrm{B})-({\mathrm{kd}}^{\ast }\mathrm{AB}\left)\right)\hfill \\ \hfill & \mathrm{A}=\text{Concentration}\hfill \\ \hfill & \mathrm{B}\left[0\right]={\mathrm{R}}_{\max }\hfill \\ \hfill & \mathrm{AB}\left[0\right]=0\hfill \end{array}$$

For the TWEAK/Fn14 and LTα/TNFR1 interactions the equilibrium binding affinity was also determined by the alternative method of plotting the equilibrium response value seen at long association times (R_Eq) versus receptor concentration [L], and fitting the data to a hyperbolic, single site binding equation: R_Eq = R_Max{[L]/([L] + K_D)}, in which R_max represents the response at saturating receptor concentrations.

Calculation of Buried Surface Areas

The extent of the solvent accessible surface area buried at the interface of each complex was calculated using PyMOL version 1.3, after first modifying the pdb files where necessary to generate the symmetry mates required to recapitulate the full ligand trimer bound to three receptor molecules. Waters and other solvent species were removed prior to the calculation. A probe of radius of 1.4 Å was used, and “dot density” was set to 4. The surface area buried at the interface in each complex was calculated by subtracting the sum of the solvent accessible surface areas for each protein component alone from that for the complex, and dividing the result by three to obtain ΔASA per receptor interface. For complexes in which the unit cell contained multiple distinct complex geometries, or where two independent structures have been reported, the results for the observed ligand-receptor geometries were averaged. PyMol calculates ASA values by the method of Lee and Richards (34), as does the NACCESS program (35) that was used to calculate ΔASA values for the entries in the PPIA database (27). To confirm that these two programs gave comparable results we calculated ΔASA for one of the complexes using both methods, and found the values returned to be in close agreement. Moreover, the complex of APRIL with TACI is also included in the PPIA database, and the ΔASA value reported for this complex agreed with the value we calculated here to within ~1.5%. ΔASA values for individual CRDs were calculated by creating a separate object in PyMol for each CRD using the CRD boundaries for each receptor given by UniProt (36), and then subtracting ASA for the hypothetical ligand/CRD complex from that for the ligand plus CRD alone. In some cases the sum of the ΔASA values for the separate CRDs slightly exceeds the ΔASA value obtained for the full receptor, presumably because regions of the ligand that interact with the receptor close to the junction between adjacent CRDs are partially blocked from solvent by each CRD alone, as well as by the whole receptor, leading to some minor redundancy in the ΔASA calculations for the separate CRDs. Throughout this manuscript, the quantity ΔASA refers to the total solvent accessible surface area buried at the interface, including the surface on both the receptor and ligand sides of the interaction.

RESULTS

Despite the long history of biochemical study of TNF family cytokines and their receptors, until recently the literature contained surprisingly few values for the binding affinity of TNF receptor proteins to individual binding sites on TNF family ligands. Most published binding studies have used receptor proteins attached to a surface (e.g. cell membrane, assay plate or sensor chip), or have used bivalent or multivalent receptor constructs, such that binding to the trimeric ligand is multivalent. Multivalent binding between TNF cytokines and their receptors reflects an important feature of how these proteins interact in vivo, but the apparent K_D values measured in such studies reflect both affinity and avidity effects, and cannot be deconvoluted to determine an accurate number for the interaction energy associated with a single TNF/TNFR binding interface.

For the current study we supplemented the small number of existing single site binding affinities with measurements for three additional TNF/TNFR family interactions for which we could find no published values in the literature. We chose complexes involving receptors at the high and low end of the size range for the TNFR superfamily: TNFR1, which contains four CRDs in its extracellular structure, binding to TNFα and also to its alternate ligand LTα, and Fn14 which contains only one CRD (Figure 1B) binding to TWEAK. Binding affinities were measured by surface plasmon resonance spectroscopy using a Biacore3000 instrument. The homotrimeric ligand (i.e. TWEAK, TNFα or LTα) was immobilized on the surface of a Biacore CM5 sensor chip as described in Materials and Methods, and a soluble monomeric construct of the appropriate receptor extracellular domain was passed over this derivatized sensor surface at various concentrations. Capturing the multivalent binder on the sensor surface and having the monovalent partner in solution ensured that the results reflected single-site receptor binding uninfluenced by avidity effects. We have previously shown for other TNF/TNFR interactions that this experimental configuration results in reliable single-site binding affinities (20). Figure 2A shows the set of binding curves measured for various concentrations of soluble Fn14, ranging from 40-360 nM, binding to TWEAK captured on the sensorchip surface. The black curves show the experimental data, while the overlaid red lines represent a global fit of the entire data set to a single-site binding model. The fit returned a value for the association rate constant of k_a = 1.4 ± 0.3 × 10⁵ M⁻¹s⁻¹ and a dissociation rate constant of k_d = 9.2 ± 1.9 × 10⁻³ s⁻¹ (n = 3). The ratio k_d/k_a gave an equilibrium dissociation constant of K_D = 65 ± 1 nM. The inset plot shows the equilibrium binding signal, R_Eq, as a function of Fn14 concentration, fitted to a hyperbolic, single-site binding equation, giving an alternative means to compute the binding affinity independent of kinetics. The curve fit returned a value of K_D = 76 ± 9 nM (n = 5), in excellent agreement with the value calculated from the kinetic analysis. Figure 2B shows comparable data for 1.5 – 15 nM TNFR1 binding to TNFα immobilized on the sensor chip. Global fitting of the kinetic data returned a value for the association rate constant of k_a = 5.8 ± 1.9 × 10⁵ M⁻¹s⁻¹ and a dissociation rate constant of k_d = 7.5 ± 2.4 × 10⁻⁴ s⁻¹, corresponding to an equilibrium binding affinity of K_D = 1.4 ± 0.4 nM (n = 3). In this experiment the low concentrations of receptor required for binding meant that most of the progress curves did not reach equilibrium during the longest achievable protein injection phase, and therefore a separate analysis of R_Eq versus TNFR1 concentration was not attempted in this case. For TNFR1 (15 – 150 nM) binding to immobilized LTα (Figure 2C), global fitting of the kinetic data returned values of k_a = 1.7 ± 0.2 × 10⁵ M⁻¹s⁻¹ and k_d = 9.2 ± 2.8 × 10⁻³ s⁻¹, corresponding to a binding affinity of K_D = 46 ± 17 nM (n = 3). The global curve fit was not of the same quality for this ligand-receptor combination as for the other two, with the data showing a modest but nonetheless pronounced systematic deviation from the fit, possibly indicating a small amount of slow, nonspecific binding of the receptor to the derivatized sensor surface. However, the plot of R_Eq versus [TNFR1] (Figure 2C, inset) fitted well to a hyperbolic binding equation, giving an affinity of K_D = 57 ± 7 nM (n = 4) that agrees with the results of the kinetic analysis, suggesting that a value of K_D ~ 50 nM is a reasonable estimate for the single-site binding affinity for this ligand-receptor pair. Notably, each of the three K_D values we measure here is 10-100-fold higher literature values for the same three ligand-receptor interactions obtained using bivalent receptor constructs (37-42). This observation, further supports the notion that our K_D values represent single-site binding affinities uncontaminated by avidity effects, as we have shown to be the case for other TNF/TNFR interactions we have characterized using the same method (20). The K_D values for the three ligand-receptor pairs shown in Figure 2, together with the single-site binding affinities for additional TNF/TNFR complexes from our own previous work and all other published values we could find, are collected in Table 1.

Figure 2.

Biacore analysis of soluble monomeric receptors binding to TNF family ligands. Sensorgrams for (A) 40 360 nM soluble monomeric Fn14 binding to TWEAK, (B) 15 150 nM soluble monomeric TNFR1 binding to TNFα and (C) 15 400 nM soluble monomeric TNFR1 binding to LTα, on separate Biacore CM5 sensorchips. The experimental data (black lines) were globally fitted to a 1:1 binding model (red lines) using BIAevaluation software, as described in Materials and Methods, to determine kinetic rate constants. The inset plots in (A) and (C) show the signal observed at equilibrium, R_Eq, plotted as a function of soluble receptor concentration, fitted to a hyperbolic, single site binding equation. Data are representative of at least 3 independent experiments.

Table 1.

Available single-site binding affinities and binding efficiencies of TNF/TNFR family complexes.

Ligand	Receptor (CRDs)	KD(nM)	ΔG0 1(kcal/mol)	ΔASA 2 (Å2)	Binding Efficiency (cal.mol−1/Å2)
					Apparent 3	Intrinsic 4
TNFα	TNFR1 (4)	1.4 ± 0.4 5	−12.1 ± 0.2	2524 ± 93	4.8 ± 0.2	8.8 ± 0.8
LTα		50 ± 10 5	−10.0 ± 0.1	2074	4.8 ± 0.1	9.6 ± 1.0
TWEAK	Fn14 (1)	70 ± 10 5	−9.8 ± 0.1	n.a.	n.a.	n.a.
CD40L	CD40 (4)	13,000 ± 2000 6	−6.7 ± 0.1	1830 ± 53	3.7 ± 0.1	9.1 ± 1.1
BAFF	BR3 (½)	15 ± 5 7	−10.7 ± 0.2	1335 ± 12	8.0 ± 0.7	15.5 ± 1.5
	BCMA (1)	1500 ± 200 7	−8.0 ± 0.1	1278 ± 60	6.2 ± 0.2	14.0 ± 1.6
	TACI (2)	13 8	−10.8	n.a.	n.a.	n.a.
APRIL	BR3 (½)	>3000 7	>−7.5	n.a.	n.a.	n.a.
	BCMA (1)	15 ± 8 7	−10.7 ± 5	1524 ± 27	7.0 ± 0.2	13.6 ± 1.3
	TACI (2)	11 8	−10.9	1675 ± 61	6.5 ± 0.2	12.5 ± 1.2
OX40L	OX40 (3½)	190 9	−9.2	2004 ± 1	4.6 ± 0.02	9.6 ± 1.0
TL1A	DcR3 (4)	56 10	−9.9	2044 ± 9614	4.8 ± 0.12	9.7 ± 1.0
FasL		270 10	−9.0	n.a.	n.a.	n.a.
LIGHT		14 10	−10.7	n.a.	n.a.	n.a.
	LTβR (4)	38 11	−10.1	n.a.	n.a.	n.a.
LTα1β2		48 11	−10.0	n.a.	n.a.	n.a.
TRAIL	OPG (3½)	21 12	−10.5	n.a.	n.a.	n.a.
CD27L	CD27 (2½)	1-10 13	−10.9 to - 12.3	n.a.	n.a.	n.a.

n.a. = not available (i.e. no crystal structure for the complex has been reported).

¹Calculated from ΔG₀ = −RTlnK_D.

²Calculated as described in Materials and Methods. PDB codes are as follows: LTα/TNFR1, 1TNR; CD40L/CD40, 3QD6; BAFF/BR3, 1OQE; BAFF/BCMA, 1OQD; APRIL/BCMA, 1XU2; APRIL/TACI, 1XU1; OX40L/OX40, 2HEV; TL1A/DcR3, 3K51 and 3MI8. The TNFα/TNFR1 co-crystal structure was a personal communication from L. Silvian. Except where otherwise noted, uncertainty limits represent the spread of values where two or more geometries were observed in the unit cell.

³Calculated from ΔG₀/ΔASA, as described in the text.

⁴Calculated from Equation 2, as described in the text. The uncertainty limits derive primarily from the uncertainty in ΔS_config (see text).

⁵This study.

⁶Reference (21).

⁷Reference (20).

⁸Reference (6, 18, 19).

⁹Al-Shamkhani et al. (1997) J Biol Chem. 272:5275-82.

¹⁰Zhan et al. (2011) Structure. 19:162-71.

¹¹Eldredge et al. (2006) Biochem. 45:10117-28.

¹²Lamoureux et al. (2009) Cancer Res. 69:526-36.

¹³Agematsu et al. (1994) J Immunol. 153:1421-9.

¹⁴Uncertainty limits reflect a small difference between two independent structures, PDB codes 3K51 and 3MI8.

To determine how much binding energy each TNF/TNFR binding interface generates per unit of contact area, we calculated the amount of solvent accessible surface area buried per receptor-ligand interface (ΔASA) for each of the complexes in Table 1 for which a high-resolution structure is available. Solvent accessible surface area values were calculated using a spherical probe of radius 1.4 Å, as described in Materials and Methods. The results are collected in Table 1. Figure 3 shows the standard molar free energy of binding, ΔG₀ (= -RTlnK_D) plotted against ΔASA for each of the nine receptor-ligand complexes for which both single-site binding affinities and experimental complex structures are available. To scale the binding efficiencies of the TNF/TNFR complexes from Table 1 relative to the maximum achievable binding efficiency for a reversible protein-protein interaction, we performed the same analysis on the large set of PPI complexes contained in the Cancer Research UK Protein-Protein Interaction Affinity (PPIA) Database (43). This database contains 144 PPI complexes for which both reliable binding affinities and experimental complex structures have been reported. The database also gives ΔASA for complex formation, calculated using an equivalent method to the one we employed for the TNF/TNFR complexes in this study (27). A plot of ΔG₀ against ΔASA for these 144 PPI complexes is also shown in Figure 3, overlaid on the values for the nine TNF/TNFR family complexes from the current study. The results in Figure 3 and Table 1 show that BAFF/BR3, the most compact and efficient TNF/TNFR complex in our test set, displays a binding efficiency that falls close to the highest values observed for complexes in the PPIA database. Other TNF/TNFR family complexes showed a range of lower binding efficiencies. Although no crystal structure is available for the complex of TWEAK/Fn14, based on the small size of this receptor and the relatively high single-site binding affinity that we report here (Table 1), it is expected that this pair also will show a very high binding efficiency.

Figure 3.

Binding energy for the eight TNF/TNFR family complexes (filled circles) and 144 reversible protein protein complexes from the Cancer Research UK Protein Protein Interaction Database (open circles), plotted as a function of the surface area buried at the binding interface. The dotted line represents the estimated maximum observed binding efficiency for these complexes, if this quantity is calculated simply as G₀/ ASA. The dashed line represents the preferred interpretation, in which a threshold level of ~500 Ų of interface area is required to form a measurably stable complex, and each additional square Ångstrom of buried surface area stabilizes the complex by a further ~20 cal/mol.

As expected (22, 23, 27), Figure 3 shows that there is no overall correlation between binding energy and contact area. However, by looking at those complexes that provide the highest binding energy per unit contact area we can establish a value for the maximum binding efficiency that is observed across this large and diverse set of complexes. One approach to analyzing this data set would be simply to divide ΔG₀ by the interface area for each complex, taking the highest values for ΔG₀/ΔASA as the maximum observed binding efficiencies, as illustrated by the dotted line in Figure 3. However, interpreting the data in this way is invalid for the following reason. The K_D values used to calculate ΔG₀ are based on a standard state of 1 M concentration. A value of ΔG₀ = 0 therefore does not correspond to a complete absence of binding, but rather to K_D = 1 M (i.e. no change in free energy upon binding under standard state conditions of 1 M concentration). Thus, the choice of ΔG₀ = 0 as the origin for the dotted line in Figure 3, implying a 1 M K_D at zero contact interface, is arbitrary and likely incorrect. An alternative approach to estimating the maximum binding efficiency of the protein-protein complexes represented in Figure 3 is based on the observation that the upper left boundary of the data can be roughly defined by the dashed line in Figure 3, which has a slope of ~20 cal.mol⁻¹/Ų. This line represents the interpretation that PPI interfaces that bury less than ~500 Ų of surface do not support the formation of a stable complex (defined here as a complex with K_D ≤ 1 M), but above this threshold each additional square Angstrom of contact surface generates a maximum of ~20 cal/mol of binding energy. Interestingly, this notion agrees with previous suggestions that stable PPI complexes have a minimum interface size of ~600 Ų (44, 45).

DISCUSSION

The binding affinity between a protein and its ligand, whether with another protein or with a small molecule, depends not simply on the surface area of contact, but rather on the number and nature of the interatomic interactions that are formed. When binding is multivalent, binding strength can become further increased through avidity effects (46). In the current study, we are interested in obtaining an empirical estimate for the maximum interaction energy that a reversible and monovalent protein-protein binding interface of a given size can generate using the structural elements and interaction modes available to it. The answer to this question not only adds to our fundamental understanding of molecular recognition at protein-protein interfaces, but also has practical applications for the affinity maturation of proteins through protein engineering, and for identifying when a natural protein-protein interface might have evolved to achieve properties beyond simply binding.

The results shown in Figure 3 suggest that a threshold level of contact area, amounting to ~500 Ų, is required for formation of a minimally stable protein-protein complex. The idea of a minimum interface size has been proposed previously by Thornton (44) and by Bogan and Thorn (45), on the basis that their comprehensive surveys of reversible protein-protein complexes for which co-crystal structures were available revealed none with contact areas less than ΔASA ~ 600 Ų, as indeed there are none among the 144 complexes from the PPIA database shown in Figure 3 (43). The existence of a minimum interface size makes intuitive sense when it is considered that, for a reversible complex to exist to any significant degree even at 1 M concentration, some interaction energy is required to compensate for the entropic cost of freezing out the independent rigid-body translational and overall rotational motions of the binding partners so they can come together into a complex. Specifically, for the bound and unbound states to have equal free energy at 1 M concentration (i.e. K_D ~ 1 M), resulting in a standard molar free energy for binding of ΔG₀ = 0 kcal/mol, the interaction between two proteins must generate an amount of binding energy that is equal to TΔS_config, where ΔS_config is the configurational entropy cost of co-localizing two molecules from 1 M solution. Thus, the measured value of ΔG₀ reflects the total binding energy between the interacting proteins, ΔG_Bind, minus TΔS_config. Or, expressed in a different way,

$$\Delta {\mathrm{G}}_{\text{Bind}}=\Delta {\mathrm{G}}_0+\mathrm{T}\Delta {\mathrm{S}}_{\text{config}}$$ (Eq.1)

For small molecule systems ΔS_config has been estimated to have a value of ~35 cal.mol⁻¹K⁻¹ (47), which corresponds to a free energy cost of ~10.5 kcal/mol at 298 K (the temperature of the measurements plotted in Figure 3). Both overall translational entropy and overall rotational entropy are relatively insensitive to molecular size, scaling with the log of the molecular weight (48), suggesting that the corresponding values for protein molecules might not be very much larger than these small-molecule-derived estimates. Moreover, the entropic contribution of new vibrational modes within the complex also scales with molecular size and complexity (47), and will partly compensate for any size-dependent increase in TΔS_config for larger molecules. Indeed, other estimates for TΔS_config at 298 K – including estimates for protein-ligand and protein-protein interactions – fall in the range 6-15 kcal/mol (49-53). Whatever the exact value of ΔS_config, it is clear that ΔG₀ represents only a portion of the overall interaction energy between the two interacting molecules, with the true interaction energy being larger than ΔG₀ by several kcal/mol, as expressed in Equation 1.

The above concept can be applied to the data from Figure 3 as follows: If each Ų of contact area between the two proteins generates up to 20 cal/mol of binding energy, and we take a value for TΔS_config of ~10 kcal/mol, then the minimum amount of contact area required to “pay for” TΔS_config and thus enable a minimally stable complex with K_D = 1 M (i.e. ΔG₀ = 0) will equal 10,000/20 = ~500 Ų. This value is identical to the threshold of ~500 Ų we estimated by extrapolation in Figure 3, and is very close to the minimum PPI interface size of ~600 Ų observed by Thornton (44) and by Bogan and Thorn (45). The agreement with these prior reports becomes closer still if it is considered that 1-10 mM probably better reflects the weakest complexes that can be detected experimentally, which the dashed line in Figure 3 shows to correspond to a minimum interface area of ~600-650 Ų. Our analysis therefore provides a clear, quantitative rationale for these previous observations of a minimum protein-protein interface size. We propose, therefore, that the most efficient reversible protein-protein interactions generate ~20 cal.mol⁻¹ per Ų of surface area buried at the interface, but that the first ~10 kcal/mol of binding energy is not expressed in ΔG₀, but instead goes to pay the entropic cost for complex formation from a 1 M standard state. This interpretation corresponds to the dashed line in Figure 3.

An important consequence of this interpretation is that calculating binding efficiency simply dividing ΔG₀ by ΔASA underestimates the true or intrinsic binding efficiency for these interactions, which is more accurately given by the equation:

$${\mathrm{BE}}_{\text{intr}}=(\Delta {\mathrm{G}}_0+\mathrm{T}\Delta {\mathrm{S}}_{\text{config}})∕\Delta \mathrm{ASA}$$ (Eq.2)

where TΔS_config can conveniently be considered to have a value of approximately 10 kcal/mol at 298 K. Henceforth, we will distinguish binding efficiencies calculated with and without the correction for TΔS_config by terming the value of the simple ratio ΔG₀/ΔASA an “apparent” binding efficiency, while a value calculated from Equation 2 is termed an “intrinsic” binding efficiency, by analogy with the well established distinction between expressed and intrinsic binding energy (54). Intrinsic binding efficiencies reflect the full interaction energy that the proteins generate per Ų of surface area buried at the binding interface, whereas the apparent binding efficiency underestimates the interaction energy density to a degree that depends on the magnitude of ΔG₀.

A maximum intrinsic binding efficiency of ~20 cal.mol⁻¹/Ų is consistent with a variety of previous estimates for the amount of energy generated by burial of different atom types at protein interfaces or protein interiors. For example, Zhou and Zhou (2002) report a range of 12-28 cal.mol⁻¹/Ų for the free energy change accompanying the transfer of different amino acids from aqueous solution to become buried within a protein interior or at a protein-protein interface (55). Similarly, several groups have estimated that the energy arising from the hydrophobic effect for the burial of apolar amino acids at a PPI interface generates ca. 20-30 cal.mol⁻¹/Ų (23, 56-58), though both lower and higher values have been proposed (25, 59, 60). It has been proposed that the hydrophobic effect is generally the dominant force in protein-protein binding (24, 26, 61). We show here that this level of binding energy is indeed characteristic of what is seen experimentally for the most efficient reversible protein-protein interactions.

Importantly, the set of PPI complexes that comprises the PPIA database includes the strongest reversible protein-protein interactions for which affinities and co-crystal structures have been reported to date. These include Colicin E9 nuclease/Im9 (K_D = 24 fM) (62, 63), Ribonuclease A/Ribonuclease Inhibitor (K_D = 59 fM) (64), trypsinogen/BPTI (K_D = 60 fM) (65) and Barnase/Barstar (K_D = 200 fM) (66), as well as many other pM or sub-nM complexes (43). All of these very high affinity complexes are included in Figure 3, and all of them fall within the limit defined by the dashed line in that Figure. Indeed, in searching the literature we are unable to identify any reversible PPI complex that exceeds the upper limit to the single-site binding efficiency defined by the dashed line in Figure 3.

Intrinsic binding efficiencies for the nine TNF/TNFR family complexes considered in this study, calculated using Equation 2, are given in Table 1. The values range from a high of ~16 cal.mol⁻¹/Ų for BAFF/BR3 – corresponding to ca. 80% of the proposed maximum achievable value of 20 cal.mol⁻¹/Ų – to a low of about half of this amount for larger complexes such as TNFα/TNFR1, LTα/TNFR1, CD40L/CD40 and OX40L/OX40. It is noteworthy that, considered in terms of binding efficiency, the different TNF/TNFR complexes appear to group by the number of CRDs present in their receptor extracellular domain. Thus, all of the receptors containing 3½ or 4 CRDs generate from 8.8-9.7 cal.mol⁻¹/Ų of binding energy, while the receptors containing no more than 1 CRD generate 13.6-15.5 cal.mol⁻¹/Ų. TACI, the only one among these nine receptors that contains more than 1 but fewer than 3½ CRDs, shows an intermediate binding efficiency of 12.5 cal.mol⁻¹/Ų in its interaction with APRIL, though Figure 3 suggests it might best be considered to cluster with the small receptors.

It initially seems counter-intuitive that BAFF/BCMA (K_D ~1500 nM) would appear in the high binding efficiency group with BAFF/BR3 (K_D ~15 nM), despite its 100-fold lower binding affinity. Similarly, the complex of the 4-CRD receptor CD40 with CD40L (K_D ~13,000 nM) actually generates more binding energy per Ų than does the much higher affinity complex of TNFα with TNFR1 (K_D ~1.4 nM, also 4 CRDs), both of which complexes fall into the low binding efficiency group. These seeming anomalies highlight the fact that, due to the large TΔS_config term in Equation 2, the intrinsic binding efficiency is more sensitive to differences in ΔASA than it is to differences in ΔG₀. Because ΔG₀ represents only a portion of the observed interaction energy between the proteins, substantial differences in ΔG₀ corresponding to several orders of magnitude differences in binding affinity represent relatively modest variations in the overall interaction energy. Thus, the difference in binding efficiency between the large and small TNFRs manifested in table 1 arises from a significant generic difference in the overall interface area for large versus small receptors, and is not greatly affected by differences of a few kcal/mol in ΔG₀ within each receptor class.

To further explore the relationship between the number of CRDs in a given receptor and the amount of surface buried at the binding interface, we calculated ΔASA for each separate CRD of each receptor. Figure 4 and Table S1 (Supporting Information) show that, for each of these complexes except OX40/OX40L, the buried surface area is dominated by interaction of one CRD, regardless of how many CRDs the receptor contains. For receptors with more than one CRD this dominant domain is invariably CRD2 (where the CRDs are numbered, according to convention, starting from the N-terminal end of the receptor sequence). Moreover, among these eight complexes the dominant domain buries a relatively constant 1600 ± 300 Ų of solvent accessible surface (ligand plus receptor) at the interface, corresponding to between 75 and 100% of the total ΔASA for the entire complex. OX40/OX40L is an exception to this trend, in that CRD1 contributes almost as much to the total contact area as does CRD2. BR3 is also something of a special case; it contains only a fragment of a CRD but, as can be seen in Figure 1C and Figure S2 (Supporting Information), this receptor contains some additional structure that interacts with its ligand adjacent to but somewhat outside the canonical receptor binding groove, towards the “south pole” of the BAFF trimer. Thus, although Figure 4 indicates BR3 as burying ~1300 Ų of surface area at each ligand-receptor interface, not very different from the amount buried by the single CRD receptor BCMA, BR3 achieves this extent of contact partly through additional interactions mediated by residues outside its fragmentary CRD. It is noteworthy that, for TNFR1, DcR3 and CD40L, even were all the binding energy attributed to contact with the dominant CRD alone, those dominant CRDs would still display somewhat a lower binding efficiency than those found for the small receptors. Therefore, the lower binding efficiencies seen for the large receptors do not simply reflect the efficient binding of a single dominant CRD “diluted” by much less efficient contacts through one or more additional CRDs. Instead, the binding efficiency of the dominant CRD is demonstrably lower for the large receptors compared to BR3, BCMA and TACI. Interestingly, the dominant domain in the large receptors (i.e. those with 3½ or 4 CRDs) interacts at a different location within the receptor binding groove on the ligand compared to the small receptors BR3, BCMA and TACI. Specifically, CRD2 of the larger receptors TNFR1, CD40, DcR3 and OX40 binds at the equatorial region of the ligand, while the single CRD receptors and TACI instead bind close to the “southern” end of the ligand furthest away from the N- and C-termini (see Supporting Information, Figure S2). Whether there are generic structural or physicochemical differences between the equatorial and southern sites on TNF family ligands that contribute to the distinct binding efficiencies observed for the TNFR domains that bind at these different locations is unclear.

Figure 4.

Stacked histogram showing the solvent accessible surface area buried in each of the TNF/TNFR family complexes examined in this study, broken down to reveal the contribution of each CRD. Data are stacked in order of amount of surface area buried, as indicated by the key. The bar representing the data for BR3/BAFF is divided into a lower portion indicating the contribution of the fragmentary CRD present in BR3, and an upper portion showing the contribution of the unique non CRD portion of the receptor (see text and Figure S2). Numerical values for ASA buried by each CRD of each receptor are given in Table S1 (Supporting Information). Note that structures are available for four other TNF/TNFR family complexes, not included in this study because no reliable single site receptor binding affinities have been reported. These complexes conform to the pattern shown in the Figure, in that they contain one receptor CRD that buries >1000 Ų of surface area, though in two cases, osteoprotegerin/RANKL (3URF) and DR5/TRAIL (1D4V and 1DU3), domain 3 also buries substantial surface area, similar to OX40/OX40L in the Figure.

These results highlight that the evolution of these ligand-receptor pairs does not appear to place any premium on minimizing the interface area or the number of CRD required in the receptor for strong and specific binding. TNF/TNFR proteins are found in primitive multicellular organisms, including invertebrates such as cnidaria, mollusks and arthropods (67). For example, Drosophila has a single TNF family ligand, eiger, and a single TNFR family receptor, wengen. The extracellular domain of wengen comprises only a single CRD, which most closely resembles the mammalian CRD structure designated A1C2 (68). Interestingly, the small TNFR family receptors in humans, namely BR3, BCMA, Fn14 and TACI, also possess an A1C2 or the closely related A1D2 structure (69), which otherwise is found only in the fourth domain of TNFR1 and not in any of the two dozen other human TNFR family receptors, all of which have multiple CRDs (15, 70). The majority of TNF/TNFR pairs for which single site binding data are available show interaction affinities of 10-100 nM (Table 1). Our results suggest that the footprint made by a single CRD binding to a TNF family ligand, minimally comprising ~1300 Ų, represents more or less the smallest contact region that can achieve a reversible protein-protein interface with this affinity. Notably, even these relatively small interfaces can achieve not only good affinity but also high selectivity. For example, BAFF is 100-fold selective for BR3 versus BCMA, whereas the homologous ligand APRIL shows an even greater opposite selectivity for these two receptors (20). Moreover, TACI, which has two CRDs but uses only one of them for binding, interacts with these same binding sites on BAFF and APRIL with essentially identical affinities of ~10 nM (19). We can therefore conclude that the evolution of receptors with additional CRDs and larger ligand-receptor contact interfaces was not driven by a need for increased affinity or selectivity. The evolutionary pressure behind the development of these larger TNFR family receptors is presently unclear, but perhaps in some cases might relate to additional functional interactions beyond ligand binding, such as the receptor-receptor interactions mediated by the N-terminal PLAD domains present in some family members (71).

It can be argued that by restricting our analysis to reversible protein-protein complexes, while omitting constitutive protein oligomers that might be expected to have higher binding affinities, we might have biased our analysis to exclude the most efficient PPI complexes. However, as has been noted previously (44, 72-74), constitutive and transient protein-protein interactions represent quite different phenomena. This is because proteins that engage in transient PPI complexes must necessarily possess a structure that allows each binding partner to exist as a stable, soluble monomer in the unbound state, which imposes significant constraints on the extent of hydrophobic surface area and other physicochemical properties of the binding site. Constitutive PPI complexes evolved under no such constraint. It is therefore likely that constitutive PPI interactions have a different and possibly higher upper limit to the achievable binding efficiency than we see here for transient, reversible complexes, because constitutive complexes use interaction modes that are not available to the more physicochemically balanced surfaces that necessarily constitute transient PPI binding sites. By restricting our analysis to reversible PPI complexes we have established the maximum binding efficiency associated with PPI binding sites on stable, soluble proteins.

The above estimate for the maximum achievable binding efficiency for a reversible, monovalent protein-protein interaction potentially provides useful guidance for protein engineering efforts to increase binding affinity with another protein or to introduce a binding site for a new protein binding partner. Specifically, it provides an explicit estimate for the minimum size that a monovalent contact interface must possess to achieve the desired level of binding affinity. From the equation for the dashed line in Figure 3 we can say that for a maximally efficient PPI interface the interface area is related to binding affinity by the inequality ΔASA > 500 - 0.05RTlnK_D, where ΔASA is the amount of solvent accessible surface area buried at the interface, in Ų, and R is the molar gas constant expressed in units of cal.K⁻¹.mol⁻¹. At a temperature of 298 K, this expression simplifies to:

$$\Delta \mathrm{ASA}>500-30\mathit{ln}{\mathrm{K}}_{\mathrm{D}}$$ (Eq.3)

Thus, for example, if the goal is to engineer an existing contact interface to increase its binding affinity to a value of 1 nM, our results suggest that such an effort is only likely to succeed for a monovalent interaction if the contact interface has a size of ΔASA > 1100 Ų. To achieve this affinity through modification of a significantly smaller interface would require that the resulting complex exceed the maximum binding efficiency seen in our study. Such an outcome may not be impossible, but at the very least it would presumably require an interaction mechanism that differs from those employed by the large set of proteins included in our study, and in particular it would likely be difficult to achieve such strong binding while maintaining the ability of the engineered protein to exist as a stable, soluble monomer in its unbound form. Our analysis also potentially provides an objective basis for assessing the progress of protein engineering efforts aimed at enhancing binding affinity towards protein targets, for example in the affinity maturation of monovalent binders such as single-domain antibodies and other monovalent antibody constructs (75), fibronectin-based affinity scaffolds (76), engineered ankyrin repeats (77), and stapled peptides (78). The above equation represents a quantitative expectation against which the efficiency of any engineered interface for which a crystal structure is available can be assessed. Complexes with affinities below that given by Equation 3 potentially have room for further improvement whereas, once the proposed limit for binding energy per area has been reached, our analysis implies that further improvement will likely be very difficult. On the other hand, our analysis highlights that relatively modest increases in contact area can result in large increases in binding affinity, if the newly contacted atoms interact with high binding efficiency. For example, to enhance the affinity of a complex by 10-fold, corresponding to ca. 1.4 kcal/mol of binding energy, requires as little as 1400/20 = 70 Ų of additional contact area, provided that this new area contributes the maximal achievable binding efficiency of 20 cal.mol⁻¹/Ų. This is a very small increase in interface area, corresponding to the burial of just an additional two or three non-hydrogen atoms on each partner. Thus, just as adding a small substituent to a small molecule ligand can result in a substantial gain in binding affinity, so increasing the extent of a protein-protein binding interface by a few atoms can do the same for a PPI complex. In cases where Equation 3 suggests that the interface being engineered is at the limit of the affinity achievable for its size, substantial further enhancement in binding might be achieved by making amino acid substitutions at the periphery of the interface to bury a just few tens of additional Ų of protein surface.

Overall, our analysis suggests that binding efficiency represents a useful concept for the quantitative understanding of reversible protein-protein binding, and that its application can be used to derive benchmarks with potential value for protein drug design and protein engineering.

ABBREVIATIONS

ΔASA

amount of solvent-accessible surface area buried at an interface

APRIL

A Proliferation Inducing Ligand, a.k.a. TNFSF13

BAFF

B cell activating factor belonging to the TNF family, a.k.a. BLyS, TALL-1, and TNFSF13B

BAFFR

BAFF receptor, a.k.a. BR3 or TNFRSF13C

BCMA

B Cell Maturation Antigen, a.k.a. TNFRSF17

BR3

BAFF receptor, a.k.a. BAFFR or TNFRSF13C

CD40

Cluster of Differentiation 40, a.k.a. TNFRSF5

CD40L

CD40 Ligand, a.k.a. CD154 or TNFSF5

CRD

cysteine rich domain

DcR3

decoy receptor 3, a.k.a. TNFRSF6B

Fn14

fibronectin growth factor-inducible 14, a.k.a. TWEAKR, CD266 or TNFRSF12A

LTα

lymphotoxin-α, a.k.a. TNFβ or TNFSF1

OX40

MRC OX40 mAb antigen, a.k.a. CD134 or TNFSF4

OX40L

OX40 Ligand, a.k.a. gp34, CD252 or TNFSF4

PPI

protein-protein interaction

PPIA

Database, Protein-Protein Interaction Affinity Database

TACI

transmembrane activator and CAML-interactor, a.k.a. CD267 or TNFRSF13B

TL1A

TNF ligand-related 1A, a.k.a. TNFSF15

TNF

tumor necrosis factor

TNFR

tumor necrosis factor receptor

TWEAK

TNF related weak ligand, a.k.a. APO3L or TNFSF12