Real-time monitoring of hydrophobic aggregation reveals a critical role of cooperativity in hydrophobic effect

Abstract

The hydrophobic interaction drives nonpolar solutes to aggregate in aqueous solution, and hence plays a critical role in many fundamental processes in nature. An important property intrinsic to hydrophobic interaction is its cooperative nature, which is originated from the collective motions of water hydrogen bond networks surrounding hydrophobic solutes. This property is widely believed to enhance the formation of hydrophobic core in proteins. However, cooperativity in hydrophobic interactions has not been successfully characterized by experiments. Here, we quantify cooperativity in hydrophobic interactions by real-time monitoring the aggregation of hydrophobic solute (hexaphenylsilole, HPS) in a microfluidic mixer. We show that association of a HPS molecule to its aggregate in water occurs at sub-microsecond, and the free energy change is −5.8 to −13.6 kcal mol⁻¹. Most strikingly, we discover that cooperativity constitutes up to 40% of this free energy. Our results provide quantitative evidence for the critical role of cooperativity in hydrophobic interactions.

Hydrophobic interactions occur between nonpolar molecules in water and their experimental quantification can help the understanding of biological self-assembly. Here Jiang et al. examine the kinetics and thermodynamics of hydrophobic aggregation in a bulk environment and characterize its cooperativity.

Hydrophobic interactions drive nonpolar solutes to aggregate in aqueous solution¹,², and hence play an important role in many fundamental processes in nature. Molecular theories³,⁴,⁵,⁶,⁷,⁸ and computer simulations⁹,¹⁰ of hydrophobicity suggest that water molecules need to break some of their hydrogen bonds to accommodate large nonpolar solutes (radius larger than 1 nm). This loss of water–water hydrogen bonds will induce fluctuations and depletion of water density near large hydrophobic solutes¹⁰,¹¹,¹²,¹³, and further lead them to collapse¹⁴,¹⁵. Therefore, hydrophobic interactions are originated from the collective motions of water hydrogen bond networks surrounding hydrophobic solutes¹⁶,¹⁷,¹⁸,¹⁹; hence, they are naturally cooperative²⁰,²¹,²². This is in contrast to fundamental intermolecular interactions that are often treated as pairwise additive, such as ionic interactions, dipolar interactions and dispersion forces.

Cooperativity is a phenomenon whereby the overall interaction for a system containing multiple molecules is stronger than the summation of individual pairwise interactions. Such cooperative feature of hydrophobic interactions is widely believed to accelerate, stabilize and enhance the formation of hydrophobic core in proteins, the aggregation of misfolded proteins and the formation of lipid vesicles and micelles. However, the extent of contributions from cooperativity to these processes still remains unclear. Addressing this issue requires, first and foremost, quantitative measurements of hydrophobic interactions, which are very challenging as the size, chemistry and topography of solutes can affect the strength of hydrophobic interactions³,²³,²⁴,²⁵,²⁶,²⁷.

Microfluidic mixing techniques have been widely applied to investigate many important chemical and biological processes, including protein and RNA folding²⁸,²⁹,³⁰,³¹,³², enzyme activities³³,³⁴ and vesicle formations³⁵,³⁶. In this study, we quantified the thermodynamics and kinetics of hydrophobic interactions in bulk solution by real-time monitoring of fluorescence induced by the aggregation of hexaphenylsilole³⁷ (HPS, C₄₀H₃₀Si) in a state-of-the-art microfluidic mixer at microsecond timescale. Using this technique, we show that in the attachment of a HPS molecule to its aggregate, the free energy change is −5.8 to −13.6 kcal mol⁻¹, the timescale is sub-microsecond, and the cooperativity constitutes up to 40% of the free energy change.

Results

Microfluidic experiment and model fitting

We employed HPS molecules (Fig. 1a left) to study hydrophobic interactions in bulk solutions using a microfluidic mixer as depicted in Fig. 1b (Supplementary Note 1). The aggregation of HPS molecules is mainly driven by solvent-induced hydrophobic interactions, as six aromatic rings render its hydrophobic property and direct π−π stacking interactions between HPS molecules are negligible³⁸. Importantly, HPS molecules emit strong fluorescence upon aggregation³⁷. This unique feature allows us to track the progress of HPS aggregation in the microfluidic mixer (Fig. 2a), where the molecular aggregation occurs in a sample stream that was hydrodynamically sheathed to tens of nanometres in width within a few microseconds upon rapid solvent exchange. We can determine the time course of HPS aggregation via dividing the travelling distance of mixture solution along the exit microchannel by its flow velocity. The measured fluorescence intensities (Fig. 2b,c, symbol points) were linearly correlated with the total amount of aggregated HPS. This linear correlation was validated by quantum mechanics/molecular mechanics calculations³⁹, fluorescence AFM experiments and spectrophotometer experiments (Supplementary Figs 1–4, 13, 19 and 20; Supplementary Notes 2 and 6).

Figure 1. Quantifying cooperativity in hydrophobic interactions by monitoring HPS aggregation.

(a, left) The chemical structure of a HPS molecule. (right) A scheme of HPS aggregation thermodynamics and kinetics. Cooperativity (Q_n) is defined as the difference between the associated free energy of attaching a HPS monomer to the aggregate (ΔΔG_n) and the summarization of the two-body potential of mean force upon the attachment of the (n+1)^th monomer to the aggregate. (b) The principle of real-time monitoring of HPS aggregation in the microfluidic mixer. HPS (red dots) dissolved in DMSO (blue dots) is continuously pumped into the centre microchannel, and then squeezed by two side water (cyan dots) streams to form an extremely narrow stream with tenths of nanometres in width. Thus, rapid solvents exchange occurs in a pure diffusion manner, and HPS molecules aggregate in downstream with strong fluorescence emission under confocal optical microscopy. Blue arrows indicate the directions of continuous fluid flow. Black arrows indicate specific locations. (c,d) Thermodynamics (ΔΔG) (c), cooperativity (Q_n) (c) and kinetics (t) (d) of attaching a HPS monomer to aggregates in pure water.

Figure 2. Time evolution of fluorescence intensity measured by experiment and fitted by theory.

(a) A representative fluorescence image of HPS aggregation in the microfluidic mixer (with subtraction of background fluorescence). White dashed lines indicate the outline of microfluidic mixer. (b) Kinetic profiles of HPS aggregation at various initial HPS concentrations (the solvent condition: DMSO mole fraction of 0.16). The solute–solvent surface tension (γ_sl) through theoretical fitting (solid lines) of experimental data (symbol points) are 20.9 (±0.6), 20.7 (±0.6), 20.5 (±0.6), and 20.3 (±0.6) cal mol⁻¹ Å⁻² at initial HPS concentration of 6, 4, 3 and 2 mM, respectively. (c) Kinetic profiles of HPS aggregation in various solvent conditions (initial HPS concentration of 6 mM). For clear illustrations, the relative fluorescence curves in part (b) corresponding to HPS concentrations of 3, 4 and 6 mM are shifted along y axis by 0.2, 0.4 and 0.6, respectively. Similarly, relative fluorescence curves in part (c) corresponding to DMSO mole fractions of 0.26 and 0.21 are shifted by 0.2 and 0.4, respectively. The solute–solvent surface tension (γ_sl) through theoretical fitting (solid lines) of experimental data (symbol points) are 19.5 (±0.6), 18.5 (±0.6) and 17.0 (±0.6) cal mol⁻¹ Å⁻² in the solvent condition with a DMSO mole fraction of 0.21, 0.26 and 0.32, respectively. The Pearson correlation coefficients for all fitted curves are larger than 0.98 (Supplementary Fig. 11).

Next, we fitted the measured fluorescence data to the classical nucleation-growth model⁴⁰ (Fig. 3a; Supplementary Note 3). At each aggregation time point in this model, new nuclei are being formed and simultaneously existing aggregates are growing. Once new nuclei are formed, they continue to grow as long as the solution is supersaturated. Accordingly, the total amount of aggregated HPS per unit volume at time t is the integration of the product of nuclei generated at a previous time s and its size growing during the remaining time τ

Figure 3. Thermodynamics and kinetics as well as cooperativity of HPS aggregation.

(a) A schematic illustration of the nucleation-growth model for HPS aggregation. At each aggregation time point in this model, new nuclei are being formed at a rate of J determined by the classical nucleation theory, and then are growing in size in a barrier-free and diffusion-controlled manner. (b) The free energy change and cooperativity associated with attaching a HPS monomer to aggregates in various DMSO/water solvent mixtures. (c) The time taken for aggregates to grow by one HPS molecule in various solvent mixtures at initial HPS concentration of 6 mM.

where J(s)is the number of nucleus generated in unit volume per time, n*(s) is the critical nucleus size, and is the growth rate of the nucleus that formed at time s. By fitting the measured fluorescence to the model (Fig. 2b,c, solid lines), we obtained the free energy change associated with attaching a HPS monomer to its aggregate and resolved the kinetics of HPS aggregation in various solvent mixtures of dimethyl sulfoxide (DMSO) and water. The results were then extrapolated to the pure water condition (Supplementary Fig. 5).

Free energy change associated with hydrophobic aggregation

We determined that the strength of hydrophobic interaction associated with attaching a HPS monomer to an existing aggregate (for example, 5.8–13.6 kcal mol⁻¹ in pure water, Fig. 1c) is comparable to that of several water–water hydrogen bonds⁴¹. As shown in Figs 1c and 3b, hydrophobic interaction increases rapidly with aggregate size and decreases with the addition of DMSO solvent. For instance, the free energy of hydrophobic interaction by attaching a HPS monomer to an aggregate containing 10 molecules (∼29 Å diameter) is −8.2 kcal mol⁻¹ in pure water, while it is only −3.8 kcal mol⁻¹ in the solvent mixture with a DMSO mole fraction of 0.32. As the aggregate grows in size, the hydrophobic interactions become stronger; when the aggregate is infinitely large, they reach the asymptotic values of −13.6 and −6.7 kcal mol⁻¹ in pure water and in the solvent mixture, respectively. In addition, the hydrophobic free energy per solvent accessible surface area (SASA) of a HPS molecule (SASA: 750 Ų) in pure water was calculated to be 18.1 cal mol⁻¹ Å⁻², a value close to the estimated one (∼20 cal mol⁻¹ Å⁻²) for aromatic hydrocarbons from previous experiment⁴² and the predicted one (∼16 cal mol⁻¹ Å⁻²) for benzene from MD simulations⁴³. Interestingly, Chandler and co-workers predicted that there should exist a crossover in the length-scale for the hydrophobic effect at around 1 nm in solute radius¹,³. Our experiment demonstrates the existence of this crossover by showing a kink during the transition from volume-based hydration free energy for monomer (smaller than 1 nm in radius) to area-based hydration free energy of aggregates (larger than 1 nm in radius, see Supplementary Note 7 and Supplementary Fig. 21 for details).

Cooperativity of hydrophobic interactions

We found that cooperativity constitutes up to 40% of the free energy for the hydrophobic association in aggregate formation. To compute the magnitude of cooperativity, we followed Wang et al.²¹ to define it as the excess multibody free energy upon attaching a monomer to the aggregate (Fig. 3b). Figures 1c and 3b show the dependence of cooperativity on the size of hydrophobic aggregate in different solvent conditions. The strength of cooperativity in attaching a HPS monomer to aggregates increases from 2.6 kcal mol⁻¹ for aggregate containing five molecules (∼23 Å diameter) to 4.2 kcal mol⁻¹for aggregate containing 50 molecules (∼45 Å diameter) in pure water, whereas the free energy change associated with these processes are –6.9 and –10.4 kcal mol⁻¹, respectively. To evaluate the total amount of cooperativity in the formation of HPS aggregate in pure water, we integrated the contribution of cooperativity in Fig. 1c and compared this accumulated cooperativity with the aggregate formation free energy (that is the integrated hydrophobic interactions in Fig. 1c). As shown in Supplementary Fig. 6, both the aggregate formation free energy and the accumulated cooperativity increase linearly with aggregate size. Interestingly, our reported hydrophobic free energy at large n limit becomes equivalent to the free energy of transferring a HPS molecule from water to the HPS phase. Importantly, cooperativity constitutes up to 40% of the formation free energy in the aggregation process. This finding demonstrates a critical role of cooperativity in hydrophobic aggregation.

Kinetics of hydrophobic aggregation

Through our kinetics analysis, the attachment of a HPS molecule to an aggregate was estimated to occur at tens to hundreds of nanoseconds. We investigated the kinetics of HPS aggregation by tracking the growth of the first nucleus formed in the theoretical model (Supplementary Fig. 7). At the initial stage of aggregation when aggregate size is smaller than 10 molecules, the time required for attaching a monomer to aggregates ranges from 30 to 200 ns (Figs 1d and 3c). When the aggregate grows in size, the kinetics of hydrophobic aggregation first accelerate with increased surface area available for attaching, and then gradually slow down due to the depletion of HPS monomers. Also, the time required for HPS molecules in pure water to form an aggregate containing 50 molecules (∼45 Å diameter) is 1.9 μs (Supplementary Fig. 7). In addition, the initial HPS concentration also influences the aggregation kinetics (Supplementary Fig. 8). For instance, at an initial HPS concentration of 2 mM in the solvent with 0.16 DMSO mole fraction, the time required for monomers to form an aggregate containing 50 molecules is 11.8 μs (Supplementary Fig. 7). Our observation of microsecond timescale of HPS aggregation is comparable to the timescale of early stage protein folding (see the Methods section for details), during which nonspecific hydrophobic interactions among hydrophobic residues induce the initial collapse of polypeptide chain at microsecond timescale⁴⁴.

Discussions

Organic molecules, such as proteins and lipid, bury their hydrophobic components to form stable cores. Hydrophobic interaction plays a crucial role in facilitating the collapse of protein chains into a globular shape²⁵,⁴⁵,⁴⁶,⁴⁷. The faster kinetics of hydrophobic aggregations (at microsecond), in contrast to protein folding (at millisecond or longer), suggest that the formation of protein cores by the aggregation of hydrophobic side-chains occurs at the early stage in the process of globular protein folding. Most importantly, we show that hydrophobic interaction, which is an interaction induced by collective behaviours of many water molecules, is strongly cooperative, and thus substantially enhance its strength during the aggregation of dispersed hydrophobic molecules in solution. We anticipate that our findings have profound implications in protein folding, as the protein core formation involves the collapse of hydrophobic side-chains. We acknowledge that these two processes are different in many aspects. For instance, flexible protein chains contain numerous conformations, while HPS molecules are relatively rigid. In addition, multiple sources may contribute to cooperativity in protein folding, such as the cooperative helix melting process due to hydrogen bonding⁴⁸. Furthermore, even the extent of contributions by hydrophobic interactions to protein folding remains elusive⁴⁹. In spite of these differences, our findings highlight the important role of hydrophobic cooperativity (as large as 40%) in the initial collapse of protein chain to form into a globular shape. We expect that our experimental platform will have promising applications in studying hydrophobic interactions of a wide range of organic molecules with aggregation-induced emission⁵⁰, and in investigating the impact of important factors such as temperature on hydrophobic effect.

Methods

Microfluidic experiments

A solution of HPS molecules dissolved in DMSO was continuously pumped into the centre microchannel. This central stream was hydrodynamically squeezed by two side water streams to form an extremely narrow stream with tenths of nanometres in width. Hence, rapid solvents exchange occurred in a pure diffusion manner, and the immediate environment for HPS aggregation was reached within a few microseconds. In the mixer, the time course of HPS aggregation was determined by dividing the travelling distance of mixture solution along the exit microchannel by its flow velocity. Thus, the progress of HPS aggregation in downstream was monitored by the integrated confocal system at sub-microsecond temporal resolution. Fluorescence images were captured with spatial resolutions of 1 and 0.5 μm in vertical (depth) and horizontal (width) directions, respectively. A diode laser at 405 nm was employed as the excitation source. The excitation beam was focused into the centre layer of microchannels by an oil immersion objective lens (× 63/1.4 NA). Then, fluorescence was collected by the same objective and detected by a photomultiplier tube detector at wavelength of 420–600 nm. Pleases refer to Supplementary Note 1 for more details of the microfluidic mixing experimental setup.

The classical nucleation-growth theory

In this paper, we employed the classical nucleation-growth theory⁴⁰ to model the time evolution of the total amount of aggregated HPS in solution. At each aggregation time point in this model, new nuclei are being formed and the rate of nuclei formation is determined by the classical nucleation theory (CNT). At the same time, existing aggregates are growing in size in a barrier-free and diffusion-controlled manner. Once new nuclei are formed, they continue to grow as long as the solution is supersaturated.

The nucleation work W(n) to form an aggregate containing n molecules, the critical nucleus size n* and the nucleation rate J(t), which is the number rate of new nuclei generated per volume per time, are obtained from the CNT⁴⁰ with the assumption that HPS monomer and HPS aggregates are spherical as suggested by previous studies⁵¹:

where γ_sl is the surface tension of HPS aggregates in the solvent mixture, is the surface area of an aggregate containing n molecules, k is the Boltzmann constant, T is the temperature, is the supersaturation ratio of HPS monomer in the solvent mixture changing with time t, v is the molecular volume of HPS monomer, p is the packing density of aggregates, which is the fraction of the molecular volume over the average volume that is occupied by a molecule in the aggregate, D is the diffusion coefficient of HPS molecule, C_e is the solubility of HPS in the solvent mixture and θ=γ_sl(36π)^1/3(v/p)^2/3/kT is related to the surface energy of aggregates. We adopted the same solute–solvent surface tension γ_sl for aggregates in the same solvent mixture.

The HPS molecular volume, the packing density and the diffusion constant of HPS molecule were obtained as follows: by assuming that HPS molecules are spherical, the volume of HPS molecule can be calculated as , where R₁ is the molecular radius. For spherical aggregate, the packing density was chosen to be p=0.7 as suggested by previous studies⁴³. The diffusion constant of HPS monomer D was calculated from the Einstein–Stokes relation, , where η is the viscosity of solvent mixtures obtained from the reported data⁵², and is the diffusive radius of HPS monomer. The molecular radius R₁ and the diffusive radius of HPS molecule were computed from the surface area of the HPS molecule with and . Here R_W=1.4 Å is the radius of water molecule, A_HPS is the surface area of HPS molecule, and is the SASA of HPS molecule. With DFT calculations using the B3LYP functional⁵³,⁵⁴,⁵⁵ and 6–31 g basis set⁵⁶, we optimized the structure of HPS molecule (Fig. 1a, left), and obtained A_HPS=437 Ų and =750 Ų. These values give the radius of the HPS molecule R₁=5.9 Å and the diffusive radius =6.3 Å, respectively. The calculated diffusion constants of HPS monomer in different solvent mixtures are listed in Supplementary Table 1.

For the growth of HPS aggregates, we adopted a diffusion-controlled kinetic model in the absence of activation free energy barrier for HPS molecules attaching to an existing aggregate, and this model has been adopted by previous studies⁵⁷. In our model, the monomer attachment frequency f_n is the product of two terms: the flux of incoming molecules to encounter the surface of the existing aggregate containing n molecules, and the surface area of the aggregate:

where D is the monomer diffusion constant, C₁(t) is the monomer concentration at time t, R_n is the radius of the aggregate containing n molecules and n(t) is the size of the aggregate at time t. The molecular detachment frequency is independent of monomer concentration⁵⁸, and is approximated to the attachment frequency at equilibrium . Hence, the growth rate of an aggregate containing n molecules can be written as

Fitting fluorescence data to the classical nucleation-growth model

We fitted the experimental measured fluorescence data to a theoretical model, which was built upon the classical nucleation-growth theory (equations (2)–(5), , , ). In our model, new nuclei are being formed at each aggregation time point, and simultaneously existing aggregates are growing. Once new nuclei are formed, they continue to grow as long as the solution is supersaturated. Accordingly, the total amount of aggregated HPS (V(t)) per unit volume at time t is the integration of the product of nuclei generated at a previous time and its size growing during the remaining time τ. Meanwhile, the total amount of aggregated HPS (V(t)) can be independently calculated from supersaturation ratio (S(t), see Supplementary Note 3; Supplementary equation (8) for details). On the basis of these relations at a given initial concentration (C₀), the solution of V(t) can be obtained numerically. As we showed that the total volume of aggregated HPS is proportional to the total fluorescence intensity, the measured dynamics of normalized fluorescence I(t)/I_T should directly correlate to the HPS aggregation kinetics V(t)/V_T. Therefore, we used the normalized total aggregates volume to directly fit the measured normalized fluorescence intensity. In our model, γ_sl, the surface tension of HPS aggregates in the solvent mixture, is the only fitting parameter. As our quantitative analysis heavily relies on the CNT, we have examined an alternative theory: a non-CNT⁵⁹,⁶⁰ involving stable prenucleation clusters (Supplementary Note 4). In non-CNT, the existence of stable prenucleation clusters will introduce a second free energy barrier, ɛ (Supplementary Note 4), in comparison with CNT containing only a single barrier. Applying non-CNT to our system, we show that ɛ has to be within 1.2kT (comparable with thermal fluctuations) in order to obtain reasonable fitting to experimental fluorescence (Supplementary Figs 14 and 15). Not surprisingly, with these small values of ɛ, the non-CNT theory provides consistent results with our CNT theory in both ΔΔG and cooperativity (Supplementary Fig. 16). These results suggest that the formation of stable prenulceation clusters is not substantial in the aggregation of HPS molecules, and thus CNT is well applicable to our system. In our CNT model, we treat the DMSO concentration as a constant, while the DMSO concentration profiles display noticeable drift along the microfluidic tube (see Supplementary Fig. 9d). Our further analysis shows that considering the DMSO concentration gradient does not have significant impact on the results (Supplementary Figs 22 and 23 and Supplementary Note 8).

Free energy change of attaching a HPS monomer to an existing aggregate

The free energy change associated with attaching a HPS monomer to an existing spherical aggregate containing n molecules is

where the first term γ_slΔA corresponds to the free energy cost associated with creating additional solute–solvent interface when the size of aggregate increases from n to (n+1). θ=γ_sl(36π)^1/3(v/p)^2/3/kT. The second term, h, refers to the free energy of transferring a HPS molecule from the infinite-sized aggregate phase to the solution, where no additional interface between the aggregate and solution needs to be created. We followed ref. 61 to compute h:

where C_e is the solubility of HPS monomer in the solvent mixture and C_agg=p/(vN_a)=1.352 M is the HPS molar concentration in aggregates. Here, v is the molecular volume of a HPS monomer, p is the packing density and N_a is the Avogadro's constant. As the solute–solvent surface tension γ_sl have already been obtained previously, we can then calculate the values of θ in various solvent mixtures (see Supplementary Table 2). As the solubility C_e was measured in our experiments (see Supplementary Fig. 10 for details), we can also obtain the values of h in various solvent mixtures (see Supplementary Table 2). Thus, free energy changes associated with attaching a HPS monomer to an existing aggregate in various DMSO/water solvent mixtures can be computed based on equation (7) and the results were shown in Fig. 3b.

We further determined the free energy associated with attaching a HPS monomer to aggregates in pure water (Fig. 1c). To achieve this, we obtained h and solute–solvent surface tension γ_sl in pure water as follows: first, by plotting the transferring free energy h against the surface tension of solvent γ_s at 20 °C (ref. ⁶²; Supplementary Fig. 5a and Supplementary Table 1), we identified the relationship between h and γ_s to be linear (R² of 0.98). We then extrapolated this fitted linear curve (from solvent mixture condition) to obtain h to be 13.6 kcal mol⁻¹ in pure water, yielding a HPS solubility of 9.6 × 10⁻⁵ μM. Next, we plotted the solute–solvent surface tension of HPS aggregates γ_sl against the surface tension of the solvent (γ_s) (Supplementary Fig. 5b), and a linear relationship was obtained (R² of 0.99). By extrapolating the fitted cure to the pure water condition, we obtained a solute–solvent surface tension of 31.8 cal mol⁻¹ Å⁻² for HPS aggregates in pure water. These two linear relationships (h versus γ_s, and γ_sl versus γ_s) were also shown in previous studies⁶³,⁶⁴.

Cooperativity in hydrophobic aggregation

We followed Wang et al.²¹ to define the cooperativity in hydrophobic interactions as the excess multibody free energy upon attaching a HPS monomer to an aggregate. For example, the cooperativity in a trimer formation is defined as the difference between the associated free energy to form a trimer and the summarized two-body potential of mean forces²⁰,²². For systems containing more than three molecules, the free energy of an aggregate containing n molecules, W_n, can be decomposed into the single-body term , the pairwise term and the multibody term :

Then, the free energy of the aggregate containing (n+1) molecules, W_n+1, can be decomposed into the single-body term , the pairwise term and the multibody term , where Q_n is the excess multibody term, which is the quantity of the cooperativity in our definition. Therefore, we have

After simple rearrangement, we obtained,

where is the free energy change associated with attaching a HPS monomer to an aggregate containing n molecules, and is the summarization of the two-body potential of mean force between the (n+1)^th molecule and the i^th molecule in the aggregate. The cooperativity Q_n defined by equation (11) can be zero (additive), negative (cooperative) or positive (anti-cooperative).

To compute the hydrophobic cooperativity as defined in equation (11), we need to obtain both and . The first term has been obtained from our experiment (Figs 1c and 3b). To compute the second term, we need to consider the potential of mean force to bring together the (n+1)^th molecule and the i^th molecule in the aggregate from infinite distance away to a particular distance determined by the location of i^th molecule in the aggregate. If the (n+1)^th molecule is in contact with the i^th molecule in the newly formed aggregate containing (n+1) molecules, the magnitude of δF_i,n+1 is significant due to the hydrophobic interactions associated with the desolvation between these two molecules. However, if the (n+1)^th molecule is separated from the i^th molecule by other HPS molecules, their centre of mass distance is at least two nanometres (∼2 diameters of HPS). In this case, no desolvation is needed and δF_i,n+1 is thus negligible. Therefore, we only consider the contributions of these contact pairs to , and then need to estimate the number of contact pairs in the aggregate. Bezdek and Reid⁶⁵ proved that the number of contact pairs in an arbitrary lattice packing of n unit balls in three-dimensional Euclidian space is always greater than (6n−7.862n^2/3), but less than (6n−3.665n^2/3). To make the best estimation of contact pairs in a spherical aggregate, we proved that the number of contact pairs in the aggregate containing n molecules is (see Supplementary Note 9 and Supplementary Fig. 24 for details):

The estimated contact pairs are shown in Supplementary Fig. 12. Next, we need to obtain the two-body potential of mean force (δF) of a contact pair. δF can be obtained from directly computing the free energy difference by bringing a pair of monomers from infinite distance to be in contact as suggested by Wang et al.²¹. In our study, it is challenging to directly measure δF in this way, because the smallest stable aggregate (that is the critical size plus one, according to the CNT) in experiment contains more than two molecules (see Supplementary Table 2 for details). Therefore, we estimated δF by approximating it to the pair strength in the smallest stable aggregate in a given solvent:

In particular, the smallest stable aggregate (n*+1) contains 3–7 molecules in the solvent with a DMSO mole fraction of 0, 0.16, 0.21, 0.26 and 0.32, respectively. ΔΔG can be obtained from equation (7). Thus, the summarization of the two-body potential of mean force between the (n+1)^th molecule and the i^th molecule in the aggregate is obtained as follows:

Substituting equations (7) and (14) into equation (11), we determined the hydrophobic cooperativity (Figs 1c and 3b). We note that the data reported in Figs 1c and 3b,c start from the smallest stable aggregate in the corresponding experimental condition. As our reported hydrophobic free energies at large n limit (ΔΔG_n→∞) become equivalent to the free energy of transferring a HPS molecule from water to the HPS phase, this makes it possible to directly estimate the cooperativity even without performing microfluidics experiments, provided that one could also obtain the pair potential of mean force (for example, from MD simulations discussed in Supplementary Note 5 and Supplementary Figs 17 and 18).

Kinetics of the hydrophobic aggregation

To resolve the kinetics of HPS aggregation, we have tracked the growth of the first nucleus formed. Based on the theoretical model obtained previously, we derived the equation for nucleus growth as follows:

where n(t−Δt) is the size of the aggregate at time t−Δt. On the basis of equation (15), we obtained the nucleus growth curve as shown in Supplementary Fig. 7. Then, we defined the time for attaching a HPS monomer to the aggregate as the time required for the aggregate to grow by one molecule in size, and the results are shown in Figs 1d and 3c. The nucleation-growth curve (Supplementary Fig. 7) suggests the microsecond timescale of HPS aggregation, which is comparable to the timescale of early stage protein folding. This comparison is made between similar concentration of HPS in our experiment and local concentration of residues on a protein chain. We estimated local concentrations of protein residues by computing number of residues within a distance of the protein chain length (for example, a protein domain containing 89–100 residues has a chain length of 28–30 nm). The resulting local concentration of protein residues (1.3–1.8 mM) is comparable with our initial HPS concentration in the microfluidics tube: 2.0–6.0 mM.

Data availability

The data that support the findings of this study as well as the computer code for the CNT and non-CNT theories are available from the corresponding authors upon reasonable request.

Additional information

How to cite this article: Jiang, L. et al. Real-time monitoring of hydrophobic aggregation reveals a critical role of cooperativity in hydrophobic effect. Nat. Commun. 8, 15639 doi: 10.1038/ncomms15639 (2017).

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Supplementary Material

Supplementary Information

Supplementary Figures, Supplementary Tables, Supplementary Notes and Supplementary References