How the biomimetic assembly of membrane receptors into multivalent domains is regulated by a small ligand

In living cells, communication requires the action of membrane receptors that are activated following very small environmental changes. A binary all-or-nothing behavior follows, making the organism extremely efficient at responding to specific stimuli. Using a minimal system composed of lipid vesicles, chemical models of a membrane receptor and their ligands, we show that bio-mimetic ON/OFF assembly of high avidity, multivalent domains is triggered by small temperature changes. Moreover, the intensity of the ON signal at the onset of the switch is modulated by the presence of small, weakly binding divalent ligands, reminiscent of the action of primary messengers in biological systems. Based on the analysis of spectroscopic data, we develop a mathematical model that rigorously describes the temperature-dependent switching of the membrane receptor assembly and ligand binding. From this we derive an equation that predicts the intensity of the modulation of the ON signal by the ligand-messenger as a function of the pairwise binding parameters, the number of binding sites that it features and the concentration. The behavior of our system, and the model derived, highlight the usefulness of weakly binding ligands in the regulation of membrane receptors and the pitfalls inherent to their binding promiscuity, such as non-specific binding to the membrane. Our model, and the equations derived from it, offer a valuable tool for the study of membrane receptors in both biological and biomimetic settings. The latter can be exploited to program membrane receptor avidity on sensing vesicles, create hierarchical protocell tissues or develop highly specific drug delivery vehicles.


Model for the in-membrane nucleation-growth assembly of the receptor
For the isodesmic assembly model two forms of the receptor, monomer M and the clustered form C, are considered to be dissolved in their respective solvents, either in lipids (Lip) or the receptor itself (1Ch). The partition process can be written as: for which a clustering constant K c can be written, as a function of the bulk volume concentrations: This equilibrium can also be thought of as a binding event in which a molecule of the monomeric form of the receptor M, initially bound to a lipid molecule Lip, associates with another molecule of the receptor (1Ch). This would yield a molecule of the clustered form of the receptor, C, and release the lipid molecule Lip that was associated to M. That is, a molecular exchange that can be written as: For nucleation growth the cluster will only grow if M binds with 1Ch molecules that are either part of a nucleus (composed of at least two molecules of 1Ch) or as part of the growing cluster, but not as monomer M. We can therefore write the nucleation growth process as: M + (C or Nuc) C + Lip Eq. S7 Where Nuc is the nucleus. Thus, the clustering constant that defines the extent of clustering can be written as a function of the bulk volume concentrations of species: Eq. S8 For simplicity of notation we define the species C 0 , as the sum of all clusters and nucleus, that is: Which substituted in Eq. 6 is: Eq. S10 For a highly cooperative assembly, under conditions where C forms, the concentration of the nucleus is much lower than that of C or M. We can therefore assume that: Eq. S11 [ 0 ] ≈ [ ] which combined with Eq. 4 gives: Eq. S12 Therefore, the maximum in-membrane concentration of the monomer (i.e. the in-membrane solubility) relates to the clustering constant as follows: Eq. S13 , = 1 Calculating the clustering constant K c at specific temperatures during temperature stage 2 (t i < 20 o C) In the growth regime we can assume that there are two different species of 1Ch in the membrane, the monomer M and the clustered form C. The absorbance observed at any given wavelength, A obs , can be written as: Eq. S14 Where  C and  M are the extinction coefficients of the pure C and M forms of the receptor at the wavelength being monitored. The apparent extinction coefficient,  app , can thus be written as: Eq. S15 Which combined with equation S5 yields: Eq. S16 Which can be re-arranged as: Eq. S17 In the growth regime, and substituting in equation 3, equation S15 can be re-written as: Eq. S18 Which can be re-arranged as: Eq. S19 Therefore, the increase in apparent extinction coefficient,  app , can be written as: Eq. S20  is common to the data at all temperatures. We can thus write for any temperature (t i ) below 20 o C: According to the Lambert-Beer law, at any given wavelength of the Soret band the apparent extinction coefficient is the ratio between the observed absorbance, A obs , at this wavelength and the total concentration of receptor: which can be written as a function of the concentration of the chromophore species as: Eq. S26  In stage 2 (t < 20 o C) K app can be written as: The binding affinity for the clustered form of the receptor can be written as a function of the binding affinity for the monomer, K m , and the modulation factor M f , which quantifies the change in binding affinity of the receptor for the ligand upon clustering: Combining equations S27-S29 results in: Which can be written as a function of the in-membrane concentration of the receptor, r 1Ch , and clustering constant, K C , as: After rearranging we show that M f can be written as a function of the different binding constants: Eq. S32

Temperature stage 1 (t > 20 o C)
The experiments with monovalent ligand L R have allowed the determination of the binding constant (K m ) and the modulation factor, M f , for a ligand that bears a chemically identical binding site to those in divalent ligand L. It is therefore assumed that the binding constant of L for the receptor is the same, if it is statistically corrected to account for the presence of two binding sites in L, while each of the imidazole moieties should have the same M f for the binding to the receptor. According to the UV-visible data, in temperature stage 1 (i.e. above 20 o C) the self-assembly of 1Ch is negligible for samples with r 0 = 0.01 (Fig 2b), and the receptor is therefore found in the monomeric form M. In this temperature stage, the binding of divalent L can lead to the formation of a 1 to 1 complex, ML, and by binding to the second binding site of the ligand, the 2 to 1 complex, M 2 L, is formed (Fig 3).
The formation of the complex ML depends on K m according to equation 4: Where 2 is the statistical factor that accounts for the fact that L bears 2 identical binding sites, compared to one binding site for L R . Eq. S34 is applicable for the binding of the ligand found in solution. The ligand can also bind to the membrane interface, according to Eq. 2: From Eq. S38 it can be seen that EM i is the effective molarity of M in the interface for the formation of complex ML, in the hypothetical situation in which the concentration of lipid equals that of monomeric receptor M (i.e. a membrane composed of pure receptor in the form M).
In our system, both the L i and L forms of the ligand are present at the temperatures studied. Eq. 34 can thus be written as Eq. S39 Combining Eq. S39 with Eq. S35 we have that: Eq. S40 And substituting in Eq. 2 we have that: Which can be re-arranged to: Eq. 3 The formation of complex M 2 L from ML and M that are located in the same membrane depends on the in-membrane concentrations of these species. The binding constant for this process, K 2 , can thus be written as: Eq. S42 2 = 2 r M2L , r ML and r M are calculated as the ratio of the concentration, in relation of the total solution volume, of the corresponding species over that of the lipid. Eq. S42 can thus be written as a function of K m and the concentration of the species in relation to the total volume of solvent as (Fig 3): Where the factor 0.5 is the statistical correction factor that accounts for the fact that dissociation of the complex by either of the two L binding sites re-generates the binding partners (M and ML).

Model for temperature stage 1
Equilibrium equations

Mass balances:
Eq. S45 The concentrations of ML and M 2 L are negligible in relation to those of L i and L, and have therefore been disregarded for the mass balance of the ligand (Eq. S46). The apparent extinction coefficient,  The error in the measure is of the order of 20 %.

Temperature stage 2 (t < 20 o C)
In temperature stage 2, below 20 o C, there are two dominant forms of the receptor, the monomer M and the cluster C, whose relative amounts depend on the in-membrane solubility, r M,max , or its reciprocal, the clustering constant K C . The clustering constant can be written as: Eq. S10 In the absence of ligand and for a strongly cooperative lateral self-assembly, we can assume that [C 0 ] is approximately equal to [C]. In the presence of ligand, C 0 is the sum of ligand bound and free forms of the clustered form of the receptor, that is: The fraction of free binding sites within the cluster, xC, is defined as: Eq. S50 The route relating to Eq. 7 is shown in Fig. 3. Alternative routes are shown in Supplementary Fig. S4.
These routes can be written as a combination of the five independent equilibria shown in Fig. 3. Thus, considering the relevant equilibrium shown for the formation of C 2 L: Eq. S51 The equations for the alternative routes leading to the formation of CL and C 2 L as depicted in Supplementary Figure S4 can be written as a function the equilibrium constants K 1 to K 5 as: Eq. S53

Estimating the value of K i in temperature stage 2: K i as a function of the properties of the membrane-water interface.
Changes in the Soret band of the UV-Visible spectrum of 1Ch in stage 1, in the absence of ligand, are attributed to a solvatochromic shift due to changes in the properties of the lipid-water interface. These absorbance changes correlate very well with the increase in the affinity of the ligand for the interface ( Supplementary Fig. S5). It is therefore reasonable to attribute the changes in K i to the same changes in the lipid membrane interface. To account for these changes, we formulate a hypothetical equilibrium between two types of membrane interface, I a and I b , whose change in character is centred at the main lipid transition temperature (that is, at T m [I a ] = [I b ]). The fraction of the high temperature interface population, x Ia , can be written as:

Supplementary
Equation S56 is derived from the van 't Hoff equation for the change of state. 24 H i is the enthalpy associated with the changes in solvation in the membrane interface, T m is the main lipid transition temperature in Kelvin, and t is the temperature in Celsius.
Since no lipid binding to the interface is observed at the highest temperature points (i.e. above 35 o C) we can further assume that the ligand only binds to the low-temperature interface state, I b . The constants calculated at single temperature points, K i , can thus be written as a function of the interface composition: Eq. S58 Where K i,0 is the intrinsic constant for the binding to the low temperature interface state. We fit the values of K i determined for temperature stage 1 to the system composed of equations S45 and S46, and enter T m as a known value, which was obtained from the fitting of the clustering of 1Ch vs temperature ( Supplementary Fig. S6). The value of H i obtained from the fitting allow us to estimate K i at the temperature points above T m (Supplementary Table S4).
Supplementary Fig. S6. Changes in K i with the temperature in temperature stage 1 (blue circles) and fit to the model described by equations S57 and S58 (red line). The dotted grey line is the extrapolation of the model function to temperatures in temperature stage 2 (i.e. above the T m that gives rise to values of K i shown in Supplementary Table S4). Eq. S64

Supplementary
The apparent extinction coefficient,  app , at 4 different wavelengths was fitted to this model (Fig. 4c,   Supplementary Fig. S7). See methods for details on the fitting procedure. See Supplementary Table S5 for values of K da at each temperature point analysed.
In the fitting, an important aspect to consider is the value of the nucleus concentration [Nuc]. For a highly cooperative assembly, the concentration of nucleus is much smaller than that of the assembly.
A mathematically simple way to account for this phenomenon is to define [Nuc] as a small fraction of the total receptor. If it is small enough, the exact value does not have a measurable impact in the speciation, and thus the fitting, and can be assumed to be a constant. 21

Disassembly of clusters upon ligand saturation
The first step of disassembly is the saturation of clustered receptor C with ligand ( Supplementary Fig.  S8a): Eq. S66

In-membrane solubility on a ligand-saturated membrane and correlation with the in-membrane
solubility in the absence of the ligand.
In excess of ligand the dominant form of the cluster is CL. We can therefore assume that: Eq. S68

[ ] = [ ]
Under these conditions, Eq. 8 can be simplified as: [ ] That is, K da is the maximum in-membrane concentration of the complex ML on a ligand saturated membrane.
We can determine the increase in in-membrane solubility of the receptor, in a ligand saturated membrane, by comparing with r M,max , (that is, the inverse of K C ) in the absence of ligand. A graphical representation of K da vs r M,max shows that there is a linear correlation ( Supplementary Fig. S9). From the slope of the trendline we estimate the increase in in-membrane solubility to be 45-fold.
Supplementary Fig. S9. Representation of the values of K da vs r M,max at each of the temperature points in temperature stage 2 (blue circles), fit to a straight line.

Derivation of the van 't Hoff equation for lipid phase change.
The change of lipid phase can be written as an equilibrium between the phases G and L d as: Eq. S70 Eq. S72 =- Where H m is the enthalpy of phase transition, T m is the temperature of phase transition in Kelvin, and t the temperature in degrees Celsius. Combining with Eq. S71, and re-arranging, gives: Eq. 11 Thermodynamic parameters for the lateral assembly of the receptor into clusters C at temperature stage 2 The van 't Hoff equation for the dependence of K C with the temperature is: Eq. 12 which in the classic linearized form can be written as: Plotting lnK C vs 1/T, H C can be calculated from the slope of the liner fit and S C from the intercept ( Supplementary Fig. S10 Eq. S75 Eq. S75 accounts for the spectral changes attributed to the receptor anchored in each form of the interface, with  Ma and  Mb the extinction coefficients for pure I a and I b , respectively. The apparent extinction coefficient,  app , at 445 nm (where the band of the assembled receptor C is dominant) was fitted to this model ( Supplementary Fig. S11). See methods for details on the fitting procedure and Table 1  The apparent extinction coefficient,  app , at 429 nm (i.e. the wavelength that experiences the largest change) was fitted to this model (Fig. 5a). See methods for details on the fitting procedure. See Table   1 for thermodynamic parameters.

Multivalent platform deployment as a function of the ligand concentration
Multivalent platform xMV can be written as a function of the cluster forms as: Eq. S84 Which, in the general case of n binding sites, becomes: Eq. 13 If r 1Ch equals r Mmax , we can simplify to: Eq. 14 At 50% signal deployment xMv = 0.5. In these conditions we have that: Eq. S100 Which can be re-arranged as: Eq. S101 [ ] -1 = 1 And isolating [L] is: