Supporting Information on: Quantitative Prediction of Multivalent Ligand-Receptor Binding Affinities for Influenza, Cholera and Anthrax Inhibition

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1 Supporting Information on: Quantitative Prediction of Multivalent Ligand-Receptor Binding Affinities for Influenza, Cholera and Anthrax Inhibition Susanne Liese, and Roland R. Netz,* Freie Universität Berlin, Department of Physics, 495 Berlin, Germany University of Oslo, Department of Mathematics, 085 Oslo, Norway * rnetz@physik.fu-berlin.de Multivalent Dissociation Constant n In the following we derive the multivalent dissociation constant n Eq. 3 in the main text) of an n-valent ligand-receptor complex from the partition functions of the bound and unbound ligands. The multivalent dissociation constant is defined as n = [L][R] [LR], S) with [L] the concentration of unbound multivalent ligands, [R] the concentration of unbound multivalent receptors and [RL] the concentration of ligand-receptor complexes. There is no unambiguous way to distinguish bound and unbound complexes, which is an issue we will discuss further below. If ligand concentration aneptor concentration are dilute, interactions among the ligands as well as among the receptors are negligible and we can apply an ideal gas approximation to relate the dissociation constant to the partition function of the individual constituents n = q L/V q R /V q LR /V, S) with q L the partition function of an unbound ligand, q R the partition function of an unoccupieeptor and q LR the partition function of a ligand-receptor complex. For an n-valent system, there are n different binding modes by which ligand aneptors can bind to each other, where the i-th binding mode corresponds to i bound ligand units c.f. Fig.a in the main text). We denote the partition function of the i-th binding mode as q i) LR. The partition function q LR follows as a sum over all q i) LR q LR = n According to Eq.S3 we consider a ligand as bound, if at least one ligand unit is bound to a binding pocket. This is a meaningful definition for a large multivalent ligand that competes with a multivalent target for the binding to a multivalent receptor. An influenza virus that targets sialic acid on the cell surface is an example for such a receptor-target system. If the ligand is large enough, it sterically shields the unoccupied binding pockets in addition to the binding pockets that are occupied by ligand units. Hence, in such a system, it is sufficient to bind only one ligand unit to effectively cover the entire receptor. In the following, we first discuss the partition function of an unbound ligand q L and subsequently the partition function q i) LR. The partition function of an unbound ligand q L reads q L = V 8π Ω n LU q i) LR. n S3) dr j P bulk str r j ), S4) with V the system volume, 8π the angular space available to a rigid body that can rotate around all three axes in space, Ω LU the angular space available to each ligand unit when it is in bulk and thus far away from the receptor surface and Pstr bulk r j ) the probability that the j-th linker extends along the end-to-end vector r j schematically shown in Fig.Sa). We impose that Pstr bulk is normalized, such that q L simplifies to q L = V 8π Ω n LU. S5) To derive an expression for the partition function q i) LR of a ligand-receptor complex in the i-th binding mode, we denote as {i} the set of indices of i bound ligand units and as {{i}, n} the set of combinations to bind i ligand units to an n-valent receptor. The S

2 partition function q i) LR factorizes into the partition function of the receptor q R, which includes all internal degrees of freedom of the receptor that remain unchanged during the binding process, and the integrals over the ligand position and orientation q i) LR = q R dr 0 dω 0 {{i},n} j {i} Ω bp V bp e β G P bound str r j )m i j / {i} Ω LU dr j Pstr im r j ), with r 0 the position of the core midpoint relative to the receptor midpoint and ω 0 = ω 0 φ, ψ, θ) the Euler angles by which the core can rotate schematically shown in Fig.Sb). The number of equivalent binding pockets per receptor subunit is denoted as m. As we discuss in the main text, m= for the trivalent hemagglutinin receptor and pentavalent cholera toxin, but m= for the heptavalent anthrax receptor. The angular space of a bound ligand unit is denoted as Ω bp. The position of each bound ligand unit is confined to the volume of the binding pocket V bp. Pstr bound r j ) describes the stretching probability of a bound linker, where r j is a vector that connects the j-th ligand unit and the j-th ligand core corner, and Pstr im r j ) is the stretching probability of an unbound linker. Note that unlike Pstr bulk, which describes the stretching probability of a linker that connects to a ligand that is not bound to a receptor, Pstr bound r j ) and Pstr im r j ) take into account that the linker cannot penetrate the receptor surface. We here assume that the linker is much longer than the extension of the binding pocket r j V /3 bp ) and hence, the stretching probability is approximately independent of the position inside the binding pocket. Furthermore, each bound ligand unit gains the binding free energy G. The factor β denotes the inverse thermal energy. The angular space available to each unbound ligand unit is Ω LU, equivalent to the case of an unbound ligand Eq.S5). Analytic expressions for Pstr bound and Pstr im are derived further below. Based on Eq.S, Eq.S3, Eq.S5 and Eq.S6 the dissociation constant of an n-valent ligand is written as n n = dr 0 dω 0 8π {{i},n} j {i} Ω bp Ω LU V bp e β G P bound str r j )m i j / {i} S6) dr j Pstr im r j ). S7) For monovalent ligands and monovalent receptors n=, m=) the integration over the core position r 0 and orientation ω 0 as well as the integration over the linker end-to-end vector r j and the linker stretching probabilities can be performed. Furthermore, the angular space of the ligand unit Ω LU is given by the angular space of a freely rotating rigid body, Ω LU = 8π. Hence, Eq.S7 simplifies to = 8π e β G. S8) Ω bp V bp Inserting Eq.S8 into Eq.S7 we obtain the dissociation constant n as n n = with C n i) = C i) n m i i ΩLU 8π ) i dr 0 dω 0 = {{i},n} [ n ] C n i) m i i, ωi LU j {i} P bound str r j ) k / {i} dr k Pstr im r k ). For a fully bound ligand, i.e. if we neglect all terms for i < n, the dissociation constant n is given by Eq. 3 from the main text S9) S0) ) n n ΩLU 8π n = C nm n = n ωlu n C nm n, with C n := C n n) = dr 0 dω 0 8π Pstr bound r j ). {{n},n} j {n} S) S) This approximation will be shown to be accurate further below, where we discuss the impact of partially and fully bound ligands. If the angular space Ω LU is smaller than the angular space within the binding pocket, Ω LU < Ω bp, then the angular space of a bound ligand unit is equal to Ω LU. In other words, Ω bp in Eq.S6 should be replaced by Ω LU and consequently, Ω LU in Eq.S should be replaced by Ω bp. So, in writing Eq.S we assume that the angular space in the binding pocket Ω bp is smaller than Ω LU. Cooperativity Factor C n for a Fully Bound Ligand In the following, we derive an analytic expression for the cooperativity factor C n. First, we discuss the steric repulsion between the linker and the receptor that arises because the linker cannot penetrate the receptor surface. We quantify this effect by deriving S

3 the stretching probability for a flexible polymer close to an impenetrable planar wall. On large scales and neglecting self avoidance effects, a flexible polymer can be described as a Gaussian chain and is characterized by the average end-to-end distance. The free energy F r) in dependence of the end-to-end distance r reads, r ) k B T, S3) F r) = 3 with k B T the thermal energy. By construction, the free energy in Eq.S3 leads to an average squared end-to-end distance r = rete in three dimensions. From the free energy, we obtain the stretching probability Pstr bulk, in the absence of the receptor surface, as Pstr bulk exp [ βf r j ) ] ) [ )] 3/ 3 z j, ρ j ) = 4π 0 dr r exp [ βf r )] = πrete exp 3 ρ j + zj rete, S4) with ρ j and z j the radial and the z components of the end-to-end vector r j. a) b), φ ψ θ r r r 3 r 0 drec, r r r 3 d) e) f) z s z e z=0 z s a 0 -z s Figure S: a) Schematic picture of the linker end-to-end vectors for an unbound ligand. b) Schematic picture of the relative position and orientation of the core with respect to the receptor. The vector from the receptor midpoint to the first binding pocket,,, as well as the vector from the core midpoint to the core edge, d core,, are shown. The vector that connects the receptor and core midpoints is denoted as r 0. The rotation of the core around the x, y, z-axis is described by the angles, φ, θ, ψ. Schematic picture of the end-to-end vectors of the linkers for a fully bound ligand. d) Image construction for a flexible polymer at an impenetrable wall. For every path that starts at z s, ends at z e and penetrates the wall, there is an equivalent path that starts at z s and ends at z e. e) Schematic picture of the position and height z s of the first linker segment, which is approximated by a freely rotating rigid rod of length a 0. f) Ligands with long linkers and small cores top) exhibit a large angular space Ω c 8π and all ligand units can reach all binding pockets. The rotation of ligands with short linkers and large cores bottom) are severly restricted Ω c 8π ) and ligand units can reach only the closest binding pocket. Steric Repulsion: At an impenetrable planar wall Fig.Sd), the stretching probability P im str of a linker that starts at height z s and ends at height z e follows from an image construction as 3 Pstr im z e, z s, ρ) = Pstr bulk z e z s, ρ) Pstr bulk z e + z s, ρ), S5) with ρ the linker vector component parallel to the surface of the wall. The second term in Eq.S5 ensures that Pstr im vanishes in the limit z s = 0. To regularize Eq.S5 as z s 0 we treat the first linker segment as a rigid rod of length a 0 c.f Fig.Se). We set a 0 =0.4nm, which is equivalent to the monomer length of PEG as determined from MD simulations. 4 The probability that the first rigid segment reaches a height z s reads P rod z s) = 4 a 0 zs π a, S6) 0 which is normalized as a 0 dz 0 sp rod z s) =. The stretching probability Pstr bound of a bound linker is obtained by integrating Eq.S5 over z s weighted with P rod z s). In the limit that the z-position z s is much smaller the the average end-to-end distance, z s/, S3

4 Pstr bound results as a0 Pstr bound z j, ρ j ) = dz sp rod z s)pstr im z j, z s, ρ j ) 0 ) [ 3/ 3 πrete exp 3 ρ j + z j rete ) [ 3/ 3 πrete exp 3 ρ j + z j rete ] 8a 0 z j ] πr ete 8a 0 z 0 πrete. S7) For a fully bound ligand the heights of the ligand core corners z j are similar to the height of the core midpoint z 0. We therefore approximate z j in the exponential prefactor in Eq.S7 by z 0. Angular Space and Conformational Weight: Inserting Eq.S7 into Eq.S, we see that the cooperativity factor C n depends on the sum over the squared end-to-end distances j ρ j + z j ) = j r j. It is therefore useful to derive an expression for the squared end-to-end distances summed over all linkers. The position of the core corners is obtained by rotating d core,j, the vector from the midpoint of the core to the i-th corner, by φ, θ, ψ around the x, y, z axis. Subsequently, the core midpoint is moved by r 0 relative to the midpoint of the receptor, as schematically depicted in Fig.Sb. The vector from the j-th corner of the ligand core to the k-the binding pocket reads ) ) πkj) cos ) n ) r j k) = πkj) sin n ρ0 cosφ 0 ) d core cos πj n ) ρ 0 sinφ 0 ) + R zφ) R yθ) R xψ) d core sin πj n, z with R x,y,z the rotation matrix around the x, y, z axis and ρ 0, φ 0, z 0 the components of r 0 in cylindrical coordinates. For a divalent ligand the rotation around x is omitted, due to the rotational symmetry of the core. The sum n r j k) cannot be evaluated in closed form for arbitrary combinations of binding pockets and ligand units, i.e. for arbitrary kj). However, for k = j, the sum over r j k) can be performed exactly and yields S8) n r j k = j) = { [drec d core) + ρ 0 + z0 d corecos φ + cos θ cos φ )] n =, n[ d core) + ρ 0 + z0 d coresin φ sin θ sin ψ + cos φ cos ψ + cos θ cos φ )] n >. S9) To discuss the impact of the restriction to ligand unit-binding pocket combinations with k = j, we consider two limiting cases. ) A small, point-like core d core ): If the core is very small, all permutations of ligand unit - binding pocket combinations are equivalent, since the core has the same distance from all binding pockets Fig.Sa). The sum over the number of permutations in Eq.S in this case yields a factor of n! and we obtain C n = dr 0 dω 0 8π n! j Pstr bound rj k = j) ) if d core. S0) ) A ligand core that is similar in size to the receptor d core ): If the ligand core is similar in size to the receptor, the condition k = j in Eq.S9 means that consecutive ligand units bind to consecutive binding pockets Fig.Sb). This condition neglects permutations, for which the linkers are considerably stretched schematically shown in Fig.S. Neglecting these conformations is justified, since extended linkers result in a small stretching probability Pstr bound and therefore do not contribute significantly to the cooperativity factor C n. Further below, we show that this approximation is accurate by comparing the approximate expression for C n Eq.S) with a numerical Monte-Carlo integration of Eq.S0 and S. In the limit d core, there are n permutations of ligand unit - binding pocket combinations that are equivalent to the condition k = j Fig.Sb), hence the cooperativity factor reads C n = dr 0 dω 0 8π n P bound str rj k = j) ) if d core. S) j To combine these two limits into a single expression, we introduce a permutation factor Πn), which describes the number of S4

5 a) b) k= j= j= k= k= j= k= j= j= k= j= k= j= k= j= j= j= k= k= k= Figure S: a) Permutations by which a trivalent ligand with a point-like core can bind to a trivalent receptor. b) Permutations of ligand unit - binding pocket combinations that are equivalent to ki) = i for a liganeptor pair with d core. Schematic conformation that is not equivalent to the condition ki) = i for a receptor pair with d core. permutations by which the n ligand units can bind to the n binding pockets. The cooperativity factor C n then factorizes as C n = Πn) Ωc 8π Qc, π dφ π 0 0 with Ω c = dθ sinθ) [ ] π dψ exp 3 drec cos φ + cos θ cos φ ) 0 rete π dφ π 0 0 dθ sinθ) [ ] π dψ exp 3n d core sin φ sin θ sin ψ + cos φ cos ψ + cos θ cos ψ ) 0 rete ) and Q c = e 3n drec ) n dr 0 Pstr bound r 0 ) V ) = e 3n drec ) 3n/ 3 4π dρ 0 dz πrete exp 3 n ) n rete r j ρ 0, z 0 ) 8a0 z 0 πr ete = ) n/ 3 n [ + n ] π Γ 3 ) 3 ) 7 π 3π n a0 ) n 3) ) 3 4n ) 4n 3 ) e 3n drec. n =, n >, S) S3) S4) S5) The terms Ω c and Q c are the angular and conformational weights, which account for the restriction of the core rotational degrees of freedom as well as the restriction of the core positional degrees of freedom due to the linker stretching. As we discuss in detail further below, the angular weight Ω c exhibits two limits, with Ω c 8π for small, point-like cores and Ω c 0 for large cores. To obtain an expression that interpolates between Πn) = n for large cores and Πn) = n! for point-like cores, we approximate the number of permutations as Πn) = n 8π Ω c 8π + n! Ωc 8π. For all figures shown in the main text, Eq.S6, Eq.S5 and Eq.S3 are inserted into Eq.S, and the angular weight Ω c is evaluated numerically. Analytic Approximation: In the case of a ligand core that is similar in size to the receptor, d core/, as well as for very small, point-like cores, d core/, an analytic approximation for the cooperativity factor C n can be derived, as we show now. If the ligand core is much smaller than the linker, d core/, the restriction of the rotational degrees of freedom of the S6) S5

6 core is negligible, i.e. in the limit 3nd core/r ete) 0, the integration over the Euler angles in Eq.S3 simplifies and we obtain Ω c = 8π. If the linker size is much shorter than the receptor size and the core size, or equivalently 3nd core/r ete), the integration over the Euler angles in Eq.S3 is performed by a saddle-point approximation around φ = θ = ψ = 0. The two limits of Ω c are summarized as Ω c 8π, π 3 ) 3/ 8π, 3/ π3 ) πn ) ) 3 drec 3 d core drec ) ) 4 drec 4 d core drec n = and ) 3/, n = and n > and ), n > and r ete d core > 6 π) /3 r ete d core 6 π) /3 r ete d core > 3n π /4 r ete d core 3n π. /4 S7) According to the discussion leading to Eq.S6, the two limits of the number of permutations Πn) are Πn) { n!, n, d core d core. Inserting Eq.S5, S7 and S8 into Eq.S, we obtain the following expression for the cooperativity factor C n 5 3 ) 3 ) ) 6 ) ) 5 a0 π 3π d 3 rec drec 3 drec e, C n ) ) π 3/ 3/ drec 3 d core 5 a0 ) d 3 rec n! n 3 ) 3 ) 3 ) 7 π a0 ) n d 3) rec n 3/ 8π 3 n π3 ) a0 ) n d 3) rec 3π n 3 ) 3 ) 6 π 3π ) n/ π Γ [ ] +n ) 4n 3 drec e 3n ) n 3 ) 7 4n drec ), 3 ) 3 ) 7 π n = and ) ) 3 drec e, 3π n n = and n > and ) n/ π Γ [ ] +n ) 4n 7 e 3n Maximizing Eq.S9 with respect to, we find the following optimal linker length ete 6 5 drec, n = and ete 3 d rec, 3n 4n 3 drec, 3n 4n 7 drec, drec ), n > and. n = and n > and n > and. S8) S9) S30) Inserting Eq.S30 and Eq.S9 into Eq. 6 in the main text, the monovalent dissociation constant as a function of the enhancement S6

7 factor α reads α m ω 5 LU π e 5 a0 ) d 3 rec, α m ω ) LU 3 π e a 0 d 3 rec α m n ω LU n [ 4 3 n π n n n+5 4n 3) ) 3 ) d core, eπn ) 4n 3 n! Γ π n = and n = and [ + n ] ] a0 ) n d 3 rec, S3a) S3b) S3 α m n ω LU n a0 ) n d 3 rec [ 6 3 n π n+6 n n+7 ) 4n 7 4n 7) eπn ), n > and ) 4n 7 [ + n ] ] π Γ n > and. S3d) In Fig.S3 the monovalent dissociation for α= and the optimal linker length ete based on the numerical evaluation of Eq.S is compared with the analytic approximations Eq.S30 and S3. In the respective limits the analytic approximations agree well with the numerical results. 0 a) n=3 α= b) n=5 α= n=7 α= [M] 0 ete / d core d core / Figure S3: Enhancement diagram and optimal linker length ete based on the numerical evaluation of Eq.S solid lines) and given by the analytic approximations Eq.S3d dotted lines) and Eq.S3e dashed lines). The following input parameters are used: a) n=3, =.6nm, ω LU = 0.03, m= b) n=5, =.5nm, ω LU = 0.03, m= n=7, =3.5nm, ω LU = 0.03, m=. Validation of the permutation factor Πn): The only difference between Eq.S and Eq.S arises from the introduction of the permutation factor Πn), which describes the number of equivalent permutations by which n ligand units can bind to n binding pockets. To validate the approximation for Πn) in Eq.S6, we show in Fig.S4 the dissociation constants 5 and 7 defined in Eq.S for cholera toxin and the anthrax receptor based on Eq.S solid lines) as well as based on a numerical evaluation of Eq.S dashed lines). The numerical integration Eq.S over the core position r 0 and the core angles ω 0 is done using Monte Carlo integration techniques. The experimental data as well as the model parameters,, d core,, ω LU and m are the same as S7

8 in Fig.4 in the main text. The approximate treatment according to Eq.S agrees satisfactorily with the full numerical integration of the multidimensional integral Eq.S. For the pentavalent cholera toxin, the approximation Eq.S agrees slightly better with the experimental results than the numerical evaluation of Eq.S, presumably because the approximation of the permutation factor Πn) in Eq.S6 effectively corrects for volume exclusion effects between neighboring linkers, which is not included in Eq.S. a) b) [M] [M] linker length [nm ] linker length [nm ] Figure S4: Comparison of the experimental dissociation constant with the theoretical model based on the approximation for C n according to Eq.S solid lines) and based on the numerical evaluation of C n in Eq.S dashed lines). a) Heptavalent ligand-receptor pair. b) Pentavalent liganeptor pair. The larger core size d core=0.8nm) is shown in blue, while the smaller core size d core=0.3nm) is shown in red. Enhancement Diagram Eq. 6 in the main text defines the monovalent dissociation constant that is needed to achieve an enhancement in binding efficiency by a factor of α. In Fig.S5a we show the enhancement diagram for tri-, penta- and heptavalent ligand-receptor pairs for four different values of α. a) [M] n=3 n=5 n=7 α α = = 0 3 α= 0 6 α= 0 9 b) [M] ete ete - 0.3nm ete + 0.3nm drec ete / d core d core / d core / d core / Figure S5: a) Enhancement diagram for tri-, penta- and heptavalent ligand-receptor pairs for four different enhancement factors α. The monovalent dissociation constants for hemagglutinin =.5mM), cholera toxin =5mM) and anthrax =4mM) are shown as dashed, black lines. b) Enhancement diagram for tri-, penta- and heptavalent ligand-receptor pairs for enhancement factor α= foimal and non-optimal linker lengths. Optimal rescaled linker length ete / drec. The rescaled linker length that would equal the geometric separation between ligand core corners aneptor units is indicated by a dashed horizontal line. S8

9 In accordance with the receptor structures shown in Fig. a in the main text, we use the following values for m, the number of binding pockets per receptor subunit, m= n=3, n=5), and m= n=7). From a fit of the experimental data to our theoretical model, shown in Fig. 4 a and b in the main text, we determined the angular restriction factor ω LU =0.03 n=5 and n=7). For trivalent ligands we use the same angular restriction factor, ω LU =0.03. The linker length is optimized, such that n becomes minimal. The monovalent dissociation constants for hemagglutinin =.5mM), cholera toxin =5mM) and anthrax =4mM) are shown as dashed horizontal lines. In all three cases a multivalent ligand that has the same binding affinity as the monovalent counterpart, corresponding to an enhancement factor of unity α=, is obtained for a wide range of core sizes. However, to achieve an enhancement by several orders of magnitude, the core size has to match the receptor size quite precisely, in particular for the trivalent case n = 3. If the core size aneptor size are similar, the optimal linker length is slightly larger than the difference between d core and, as shown in Fig.S5c. To study the robustness of the binding enhancement with respect to a non-optimal linker length, Fig.S5b shows for α = for the optimal linker length solid lines), as well as for linkers that are by 0.3nm too short dotted lines) or too long dashed lines). Since the optimal receptor length depends on d core, we have to chose a specific value for or for d core to determine the enhancement diagram for suboptimal linker lengths. Therefore, we set the receptor size to =.6 n=3), =.5 n=5) and =3.5nm n=7), corresponding to hemagglutinin, cholera toxin and anthrax. Similar to the results presented in Fig. 5 in the main text, we find that for ligand cores that are similar to the receptor size, the binding enhancement significantly decreases for non-optimal linker length. In fact, the decrease in binding enhancement is much more pronounced for linkers that are shorter than the optimal length, compared to linkers that are longer. Impact of Partially Bound Ligands To study the impact of partially bound ligands, we show the inverse of the dissociation constant of hepta- and pentavalent ligands in Fig.S6, since the inverse dissociation constant consists of a sum over the individual binding modes. The dissociation constant in Eq.S9 is determined by numerically evaluating the cooperativity factors C n i). In analogy to the results shown in Fig.S4, we determine C n i) by numerically integrating Eq.S0 over the core position r 0, the core angles ω 0 and the linker positions r k using Monte Carlo integration. The model parameters,, d core,, ω LU and m are the same as in Fig.4 in the main text and are identical in all binding modes. Evidently, C n i) changes in each binding mode according to Eq.S0. Fig.S6 shows that the minimum of the dissociation constant, i.e. the maximum of the inverse dissociation constant, is dominated by fully bound ligands. If the linker length is larger or smaller than the optimal value, the contribution of partially bound ligands surpasses the contribution of fully bound ligands. In the three examples studied here, the deviation between the dissociation constant taking all binding modes into account dotted line) and taking only the fully bound binding mode into account dashed lines for i = 7 and i = 5, respectively) differ by less than a factor of two around the maximum. Since the aim of multivalent ligand design is to improve the minimal dissociation constant by several orders of magnitude, we consider these deviations as negligible. 7 [M ] all i=7 i=6 i=5 i=4 i=3 i= min 7 = 0.3nM 0 3 linker length [nm] a) 5 [M ] all i=5 i=4 i=3 i= min 5 = 0.04μM linker length [nm] b) 5 [M ] all i=5 i=4 i=3 i= min 5 = 0.05μM linker length [nm] Figure S6: The inverse of the dissociation constant taking all binding modes into account dotted line) as well as the contributions from all binding modes dashed lines). The minimal dissociation constant 7 min and 5 min, which corresponds to the maximum of the dotted line, is indicated in each subfigure. a) Heptavalent ligand-receptor pair with d core=.5nm and ω LU =0.03. b) Pentavalent liganeptor pair with d core=0.8nm and ω LU =0.03. Pentavalent liganeptor pair with d core=0.3nm and ω LU =0.03. The balance of the contribution of partially and fully bound ligands to the dissociation constant depends on the value of the angular restriction factor ω LU, which we determine from a fit of the fully bound contribution to experimental data in the main text. To check whether our fitting procedure is self-consistent, we in Fig.S7a and b show the inverse dissociation constant of a heptavalent ligand-receptor pair for different angular restriction factors ω LU. For an angular restriction factor ω LU = in Fig.S7a, the fully bound ligand contribution becomes even more dominant compared to the result for ω LU = 0.03 in Fig. S6a, whereas for an angular restriction factor ω LU = 0.3 in Fig.S7b, the contribution of partially bound ligands becomes dominant. In Fig. S7c the experimental dissociation constant of the heptavalent anthrax ligand is compared with the theoretical model including all binding modes for ω LU =0.003, 0.03 and 0.3 the model results correspond to the inverse of the data shown as dotted lines in Fig. S7a, Fig. S6a and Fig. S7b). We also show a curve for the fit value of ω LU = 0.04 including all binding modes continuous line in Fig. S7. The value ω LU = 0.04 is only slightly larger than the value ω LU = 0.03 we obtained by a fit of the model that only considers fully bound ligands. We conclude that our fitting procedure for ω LU that uses the simplified model that only accounts for fully bound ligands is consistent and gives very similar results as the general model that includes all binding modes and which necessarily needs a numerically demanding Monte Carlo evaluation of the configurational integrals. S9

10 7 [M ] all i=7 i=6 i=5 i=4 i=3 i= min 7 = 0.06fM 0 3 linker length [nm] a) 7 [M ] all i=7 i=6 i=5 i=4 i=3 i= min 7 = 0.0mM 0 3 linker length [nm] b) 7 [M] linker length [nm] ωlu = 0.04 ωlu = 0.3 ωlu = 0.03 ωlu = Figure S7: Impact of partially bound ligands and of the angular restriction factor ω LU on the dissociation constant of a heptavalent ligandreceptor pair with d core=.5nm. a) and b) The inverse of the dissociation constants taking all binding modes into account dotted line) as well as the contributions from the i-th binding modes dashed lines) for a) ω LU =0.003 and for b) ω LU =0.3. The minimal dissociation constant 7 min, which correspond to the maximum of the dotted line, is indicated in each subfigure. Comparison of the experimental dissociation constant of the heptavalent anthrax ligand with the theoretical model including all binding modes for different values of ω LU. Variation of the Core Size for Fixed Linker Length In Fig.S8a the dissociation constant 5 of a pentavalent ligand is shown in dependence of the core size d core for three fixed linker lengths, corresponding to PEG linkers with N=0, 0 or 30 monomers. The receptor size =.5nm, indicated by a dotted vertical line) and binding strength =5mM) are representative for cholera toxin. For comparison the equivalent monovalent dissociation constant /5=mM is shown as a black, horizontal line. The results shown in Fig.S8 are based on Eq.S-S6 and are thus obtained on the same level of approximation as all model results shown in the main text. Fig.S8a shows that the lowest dissociation constant is achieved with the shortest linker. The dissociation constant of the ligands with longer linkers is two to three orders of magnitude larger, but exhibits a broad plateau and is hence very robust towards variations of the core size. If the core size is larger than the receptor size, d core >, the dissociation constant 5 continuously increases with increasing core size, i.e. the better the match between core size aneptor size, the stronger becomes the binding. In contrast, if the core size is smaller than the receptor size, d core <, the dissociation constant 5 can increase or decrease with increasing core size depending on the linker length. To explain this behavior, we show in Fig.S8b-d all three terms that contribute to the effective concentration in Eq.S and thereby contribute to the dissociation constant Eq.3 in the main text). The angular space of the core, Ω c, as well as the number of permutations, Π, decrease with increasing core size, while the conformational weight Q c exhibits a maximum if core size aneptor size are equal. Whether Ω c and Π or Q c dominate the effective concentration depends on the linker length. S0

11 C/8 a) 5 [M] b) =.5nm N= 0) =.nm N= 0) =.6nm N= 30) 0 0 d) QC [M 4 ] d core [nm ] Figure S8: a) Dissociation constant 5 of a pentavalent ligand in dependence of the core size d core for three linker lengths, corresponding to PEG linkers with N=0, 0 or 30 monomers. The receptor size =.5nm is indicated as a dotted vertical line. b) Angular space of the ligand core Ω c according to Eq.S3, number of permutations Π according to Eq.S6 and d) conformational weight Q c according to Eq.S5 in dependence of the core size. References [] Diestler, D. J.; napp, E. W. Statistical Mechanics of the Stability of Multivalent Ligand-Receptor Complexes. J. Phys. Chem. C 00, 4, [] rishnamurthy, V. M.; Estroff, L. A.; Whitesides, G. M. In Multivalency in Ligand Design; Jahnke, W., Erlanson, D. A., Eds.; Wiley-VCH Verlag GmbH & Co, 006. [3] Eisenriegler, E. Conformation Properties and Relation to Critical Phenomena Polymers near surfaces; World Scientific Publishing Co. Pte. Ltd., 993. [4] Liese, S.; Gensler, M.; rysiak, S.; Schwarzl, R.; Achazi, A. J.; Paulus, B.; Hugel, T.; Rabe, J. P.; Netz, R. R. Hydration Effects Turn a Highly Stretched Polymer from an Entropic into an Energetic Spring. ACS Nano 07,, S