Testing mixing rules for structural and dynamical quantities in multi-component crowded protein solutions

Macromolecular crowding is a ubiquitous and relevant factor in biological fluids such as the cytoplasm and blood serum, expressing the seemingly trivial finding that macromolecules in natural and biotechnological solutions are affected by the presence of other molecules.^1,2 The kinetic and thermodynamic consequences such as varied reaction rates and equilibria³ and lowered solution stability^4,5 govern largely protein diffusion and interaction, as well as the phase behavior and viscosity of protein solutions. For globular proteins, a colloidal picture has proven highly useful for an in-depth and quantitative understanding of concentrated solutions of proteins, which is of high relevance for understanding of function and misfunction in protein solutions, related in particular, to diseases caused by protein condensation.^6,7 Particular examples include the liquid–liquid phase behavior,^8–12 cluster formation^13–17 as well as the slowing down of self- and cage diffusion,^18–23 eventually leading to dynamical arrest.^16,24,25

For pure solutions of one protein, the physical picture and modeling approaches up to rather concentrated solutions are well developed and can be used also to predict properties with qualitative and even reasonably quantitative accuracy. Here, pure solutions also include systems where a protein is dissolved in a complex mixture of different additives such as salts and co-solvents that is then treated as an effective solvent.^12,26 This situation changes once mixtures of different proteins are involved, as the presence of additional components renders the theoretical treatment more complicated.^27–30 In addition, computational approaches are more involved and costly, as interactions between larger particles in a bath of smaller ones require the simulation of a large number of particles.²⁴ This lack of understanding is very unfortunate, as natural fluids, in general, are multi-component systems and show clear effects of the polydisperse crowding on inter alia phase behavior^31,32 and protein diffusion.^33–35 

In this paper, we systematically and comprehensively explore a well-established binary protein system of α and γ_B crystallin protein from the bovine eye lens fluid as a first step toward an approximate understanding of multi-component systems.

The class of crystallin proteins present in the eye lens has been intensively investigated in the past, due to their link to the eye lens function and conditions such as cataract. The motivation for a structural and dynamical characterization of these crystallin solutions is thus threefold: (i) they can serve as a well-defined model system for the general investigation of crowding effects in biological cells; (ii) they are well suited for the study of so-called protein condensation diseases;⁶ and (iii) their dynamic properties and, in particular, the occurrence of an arrest transition have been believed to be the underlying mechanism responsible for the loss of lens flexibility observed in presbyopia.^24,36–39 Briefly, three distinct protein families are found in the vertebrate eye lens, called α, β, and γ crystallin.⁴⁰ In this work, we focus on two crystallin proteins extracted from the bovine lens: α crystallin and γ_B crystallin.

Bovine α crystallin, a multi-subunit protein with a molecular weight Mw $≈$ 800 kDa and an average diameter $d ≈ 15$ nm, has been successfully described with a polydisperse hard sphere model.^18,24,41,42 Combining x-ray photon correlation spectroscopy (XPCS), neutron spin echo spectroscopy (NSE), dynamic (DLS) and static (SLS) light scattering, zero-shear viscosity measurements, and molecular dynamics simulations of highly concentrated α crystallin solutions, the dynamical arrest transition (i.e., glass transition) has been shown to occur at volume fractions around $ϕ ≊ 0.58$⁠,^18,19,24 driven by the so-called caging effect for hard sphere systems.^43,44

A fundamentally different scenario is observed for the monomeric γ_B crystallin fraction (⁠ $M w ≊$ 21 kDa, $d ≊$ 4 nm), a subclass of the bovine γ crystallins. It has been shown to interact via weak, short-range attractions that lead to a liquid–liquid phase separation of γ_B crystallin solutions upon lowering the temperature.^25,45,46 Dynamic properties, such as collective diffusion coefficients measured via DLS, thus show a strong dependence on the temperature close to the liquid–liquid coexistence curve (or binodal) due to critical phenomena.²⁵ The arrest transition for γ_B crystallin occurs at volume fractions significantly lower than hard spheres^43,47 due to short-range attractions and patchy interactions.¹⁹ In fact, interprotein interactions induce the formation of transient networks and/or clusters that slow down the dynamics at relatively low volume fractions, which is further enhanced by the nonspherical monomeric shape.⁴⁸ According to a recent study performed via microrheology measurements, the glass transition of γ_B crystallin occurs at a volume fraction between $ϕ ≈$ 0.27 and 0.31.⁴⁹ Interestingly, the glass transition for the γ_B crystallin shows no temperature dependence in the investigated temperature-range.

A comprehensive characterization of mixtures of crystallin proteins is so far missing. While phase behavior and the underlying protein interactions have been addressed,^29,31,50,51 dynamical and rheological properties remained uncharacterized, despite their high relevance for a complete understanding of eye function and eye conditions such as presbyopia.

This study attempts to explore simple mixing rules based on the detailed knowledge of the individual crystallins and test whether such models are capable of reproducing the concentration and temperature dependence of key dynamic parameters such as the collective diffusion coefficient and the relative viscosity of crystallin mixtures. For binary mixtures of α and γ_B crystallin, we provide a characterization of their (i) phase behavior, (ii) the osmotic compressibility measured by SLS, (iii) structural properties via small angle x-ray scattering (SAXS) (iv) zero-shear viscosities utilizing both DLS- and confocal laser scanning microscopy (CLSM)-based microrheology measurements, and (v) collective diffusion coefficients from DLS and NSE experiments. We then compare the results of the $α γ B$ binary mixtures to the theoretical predictions, calculated by appropriately weighted combinations of the two properties from the individual or pure crystallin solutions. Given the large number of individual parameters discussed below, we provide a glossary of the used variables for pure and binary lens crystallin solutions in Table I.

Before discussing specific results, we present the basic idea of mixing rules for the prediction of relevant structural and dynamic properties of multi-component protein solutions by calculating a weighted average of the corresponding quantities of the different single-component systems. While the general idea of mixing rules is simple, the choice of the specific form invokes an underlying physical picture. Previous theoretical and experimental studies have provided evidence that for complex protein mixtures as for example present in cells with their thousands of different proteins, important features such as the location of phase boundaries or the diffusion of individual proteins are primarily determined by the effective volume fraction of the particular protein species.^52,53 It has been pointed out that overall, specific strong attractions between different protein species in the cell cytosol are largely absent, and they thus interact not unlike an effective hard sphere liquid. As a result, self-diffusion of individual proteins is then for example primarily dependent on the overall protein volume fraction. Phase boundaries for liquid–liquid phase separation (LLPS) or crystallization, on the other hand, depend on the interactions between a particular protein type and thus largely on the effective volume fraction of this protein species.

$S ϕ α , eff ( q )$ and $S ϕ γ B , eff ( q )$ are the structure factors for the pure components taken at the effective volume fraction, as calculated using Eqs. (1) and (2). With this approach, we thus assume that the effect of mutual interactions between α and γ_B crystallins can be represented by an increased excluded volume effect in the interactions between the individual crystallins (i.e., α - α and γ_B - γ_B).

We stress that the simplicity of the mixing rules presented should not be seen as a serious drawback, but rather as a first test of how far simple concepts could be used to describe and predict the behavior of complex mixtures of different proteins under highly crowded conditions as often found in the cytosol of living cells or in blood serum. Section III presents different experimental quantities over a large range of concentrations, which are then compared with the predictions from these mixing rules. Importantly, these experimental quantities also encompass macro-, meso- as well as microscopic properties, and we will discuss the important consequences of these different length scales in the applicability of the mixing rules.

We first test our approach with a macroscopic property and investigate the location of the coexistence curve for liquid–liquid phase separation in $α − γ B$ protein mixtures. LLPS in the eye lens has been investigated intensively for its link to cataract formation and found to be directly related to $γ B − γ B$ protein–protein interactions.^6,8,46 The results from our characterization of the location of the binodal for LLPS in these mixtures as determined from cloud point measurements for three different $α : γ B$ volume ratios (3:1, 1:1, and 1:3) are shown in Fig. 1(a).

When compared to the binodal in pure solutions of γ_B crystallin from an earlier study²⁵ (gray triangles), we see that an increasing amount of α crystallin has a significant effect, shifting the binodal to higher overall protein concentrations. It is interesting to look at this in view of the evolutionary pressure on the eye lens to maintain a cytosol composition that allows it to achieve high transparency at the very high protein concentrations required to generate the necessary index of refraction profile across the lens.^6,59 As we will discuss below, the different crystallins allow indeed for a densely packed cell interior that optimizes index of refraction, transparency as well as viscosity.

If our reasoning in Sec. II is correct, we would expect to find the location of the binodal primarily determined by the relative volume fraction of γ_B corrected by the reduced available volume caused by the α crystallins present, i.e., by $ϕ γ B , eff$ calculated using Eq. (2). This is demonstrated in Fig. 1(b), and we see that indeed all data points collapse around the binodal for the pure γ_B system when using $ϕ γ B , eff$ instead of $ϕ α γ B$⁠.

The use of such a simple mixing rule based on the picture of a protein embedded in an effective solvent, where the additional other protein species present only result in a reduced available sample volume and thus an effectively increased volume fraction, is of course thermodynamically incorrect. We remark that a complete modeling of the phase diagram of $α γ B$ binary mixtures has in fact been performed previously.^29,51 This work was also extended to a complete description of the resulting light scattering intensity based on a model of sticky spheres, allowing for a detailed investigation of the effects of protein–protein interactions (⁠ $α − α , α − γ B$ and $γ B − γ B$⁠) on the intensity I(q) at q = 0.⁶⁰ However, the resulting multidimensional phase diagrams become far too complex to be useful in a simple framework aiming at a qualitative understanding of the stability, the effect of temperature and composition, and the existence and location of an arrest line in crystallin mixtures with compositions similar to those in eye lens cells. Moreover, these calculations are difficult to extend to dynamic quantities such as the relative viscosity or the collective diffusion coefficient. We therefore aim to develop and test simple models to describe phenomenological observations in a semi-quantitative way, amenable for numerical calculations of thermodynamic, structural and dynamic properties that could not be achieved otherwise and would require extensive computer simulations instead.

Next, we extend our approach to a macroscopic dynamic quantity and investigate the relative zero-shear viscosity $η r$ of $α γ B$ binary mixtures. We employ measurements using both DLS-based and MPT-based microrheology, as described in the Materials and Methods section. To ensure that experiments are made on mixtures in the one-phase region of the phase diagram, all measurements are performed at temperatures well above T_cloud.

Figures 2(a) and 2(b) report $η r$ for the pure α crystallin (red triangles) and γ_B crystallin (green squares) solutions as well as for the 1:1 binary mixture as a function of volume fraction $ϕ$ at $T =$ 25 and 35 °C. The results for the $α γ B$ binary mixture are shown as orange (DLS-based microrheology) and yellow (MPT-based microrheology) circles. Representative $g 2 ( q , t ̃ ) − 1$ functions from DLS-based microrheology, as well as mean-squared displacements and Van Hove functions from multiple particle tracking (MPT)-based microrheology are reported in Fig. S1 in supplementary material.

The volume fraction dependence of $η r$ of the mixture is well reproduced by the mixing rule in Eq. (10) [Figs. 2(a) and 2(b)]. (see supplementary material for a detailed description). The combination of the $η r$ of α and γ_B crystallins taken at their effective volume fractions $ϕ γ B , eff$ and $ϕ α , eff$ indeed appears to provide a good prediction of the binary mixture. This is quite remarkable given the very different mechanisms behind dynamical arrest for the two different lens proteins. While α crystallin exhibits a hard sphere-like behavior with an arrest transition at around $ϕ g , α ≈ 0.58$⁠,²⁴  γ_B crystallin has been found to show a behavior described by a model of particles with a weak short range attraction combined with additional attractive patches. This results in a dynamical arrest transition at significantly lower volume fractions of $ϕ g , γ B ≈ 0.3 − 0.35$⁠, presumably caused by the presence of transient clusters.^19,25 This good agreement suggests that on macroscopic length scales probed by the zero shear viscosity, and given strongly differing viscosities, viscoelastic relaxations arising from protein–protein interactions can be decoupled for the two proteins. At the effective volume fractions $ϕ α , eff ≪ ϕ g , α$ measured for our binary mixtures, α crystallin solutions have rather low viscosities and relax quickly, allowing for a treatment of the α crystallin as an effective solvent for γ_B crystallin, which shows significantly higher viscosity and slower relaxation times at these effective volume fractions due to the formation of transient clusters.

Due to the attractive interactions between γ_B crystallins, their solutions undergo LLPS at lower temperatures as demonstrated in Fig. 1. This also has consequences for scattering experiments, as there will be a significant contribution from critical scattering, resulting in a strongly enhanced intensity in SLS and a slowing down of the measured correlation function obtained in DLS as the temperature approaches T_cloud.²⁵ While these additional contributions from critical opalescence and critical slowing down make microrheology measurements using DLS-based tracer microrheology much more difficult and induce additional errors (see supplementary material), this can be overcome by using MPT-based microrheology instead. However, while temperature has a strong effect on the phase behavior and the results from scattering experiments, the relative viscosity shows no measurable temperature dependence. This indicates that there are no or only weak critical contributions to the $η r$ of binary α - γ_B mixtures.

An interesting feature is observed when plotting and comparing relative viscosity data for pure crystallins and the binary mixture as a function of mass concentration c in mg/mL. As shown in Figs. 2(c) and 2(d), $η r$ from the different systems collapse onto a master curve that extends to rather high concentrations. This coincidence suggests that for these specific proteins, $η r$ is determined by the actual weight concentration for low to intermediate protein concentrations, independent of the ratio between the pure components. It is important to point out that this $η r$ master curve holds up to concentrations c of around 300 mg/mL. For higher c, $η r$ diverges at the arrest value c_g of the particular protein species or mixture, which is different for the two proteins. In particular, for α crystallin the arrest transition occurs at $≈$ 341 mg /mL [red dashed lines in Figs. 2(c) and 2(d)], while dynamical arrest for γ_B crystallin occurs at $≈$ 395 mg/mL [green dashed lines in Figs. 2(c) and 2(d)]. However, it is important to point out that we do not expect to find such a behavior for arbitrary proteins and consider this an effect of the very different voluminosities of α and γ_B crystallin that compensate for the different underlying arrest mechanisms. Instead, we expect that a mixing rule such as given in Eq. (10), where effective volume fractions are considered, should have a much wider applicability.

Next, we move on to mesoscopic length scales of many protein diameters and investigate collective gradient diffusion of $α γ B$ binary mixtures using dynamic light scattering. The protein solution is no longer considered to behave as an effective homogeneous liquid, but as a concentration field, in which concentration fluctuations or inhomogeneous density distributions relax with a rate expressed through the gradient diffusion coefficient $D = D ( ϕ , q → 0 )$⁠.

Figure 3 presents the measured diffusion coefficients $D α γ B ( ϕ α γ B )$ for the 1:1 mixture at two temperatures $T = 25$ and $35$ °C for a series of volume fractions $ϕ α γ B$⁠. For both temperatures, we observe a decrease in $D ( ϕ α γ B )$ with increasing volume fraction. Also shown is the concentration dependence of the pure components $D α ( ϕ α )$ and $D γ B ( ϕ γ B )$⁠, respectively.^24,25

While α crystallin solutions exhibit a concentration dependence of $D ( ϕ )$ that is typical for hard spheres with an almost linear increase in $D ( ϕ )$ with increasing volume fraction,²⁴ the $ϕ$-dependence of $D ( ϕ )$ is much more complex for γ_B crystallin. For this protein, we find a non-monotonic $ϕ$-dependence due to the combination of contributions from a strongly T-dependent critical slowing down in the vicinity of the critical point and a regular non-critical contribution leading to dynamical arrest at $ϕ ≈ 0.35$⁠.²⁵ For the 1:1  $α γ B$ mixture, we observe a behavior that is roughly intermediate between the two individual protein solutions.

We now attempt to use Eq. (5) together with the simple mixing rules given in Eqs. (6)–(8) in order to calculate the theoretical predictions for the mixed system. To determine D₀ for the simple mixing rule in Eq. (5), we use $D 0 , α =$ 2.2 × 10⁻¹¹ and $D 0 , γ B =$ 8.74 × 10⁻¹¹ m² s⁻¹ at $T = 20$ °C, which we then adapt for the correct solvent viscosities at the temperatures measured (see supplementary material for details).^24,25 For S(0) in the case of α crystallin, we employ the Carnahan-Starling equation for monodisperse hard spheres $S CS ( 0 ) = [ ( 1 − ϕ ) 4 ] / [ ( 1 + 2 · ϕ ) 2 + ϕ 3 · ( ϕ − 4 ) ]$⁠.²⁴  $S γ B ( 0 )$ for each temperature are taken from Ref. 25 and parameterized with a simple polynomial function.

The agreement between the measured concentration dependence of the gradient diffusion coefficient $D ( ϕ α γ B )$ and the predictions from Eq. (5) together with the simple mixing rules given in Eqs. (6)–(8) is quite remarkable, especially when considering that the individual proteins exhibit such a vastly different behavior (Fig. 3). We believe that the hybrid approach used here is essential for this, where we calculate the thermodynamic driving force for the decay of concentration fluctuations given by the osmotic compressibility S(0) based on the particular $α − α$ and $γ B − γ B$ interactions using the effective volume fractions of both protein components, while we estimate the contributions from hydrodynamic interactions from the overall volume fraction. This allows us to also incorporate contributions from critical fluctuations for proteins that undergo LLPS, while the frictional resistance that the different proteins experience should primarily depend on the total volume fraction of proteins in solution.

After having successfully reproduced macro- and mesoscopic thermodynamic and dynamic properties, we next investigate whether the proposed mixing rules can also describe structural and dynamic solution properties on more local or molecular length scales. We first investigate the solution structure on length scales from a fraction of the protein diameter up to a few protein molecules as obtained from SAXS measurements.

Figure 4 shows the concentration-normalized form factor $I α γ B FF ( q )$ from SAXS for dilute solutions of $α γ B$ binary mixtures at different α:γ_B mixing ratios, i.e., 1:3 (red circles), 1:1 (green squares), and 3:1 (blue triangles). At this dilute volume fraction of $ϕ α γ B = 0.003$⁠, effects from protein–protein interactions can be neglected. This is indeed demonstrated in Fig. 4, where we see that the scattering from the mixed solutions is quantitatively reproduced by a linear combination (color-matched lines) of the individual form factors for the pure α (purple line) and γ_B (orange line) crystallins also shown in Fig. 4. This also clearly indicates the absence of any self-association under these conditions.

We next characterize the concentration and temperature dependence of the solution structure for 1:1 mixtures using SAXS. Figures 5(a) and 5(b) show the experimental SAXS curves and their theoretical predictions from Eq. (4) for $α : γ B = 1 : 1$ binary mixtures at $ϕ α γ B = 0.22$ and two temperatures $T = 25$ (a) and $T = 35$ °C (b), respectively.

The theoretical predictions are based on the scattering intensities measured for the pure α and γ_B crystallin solutions using the mixing rule in Eq. (4) and their effective volume fractions $ϕ α , eff$ and $ϕ γ B , eff$ [Eqs. (1) and (2)], respectively. This requires theoretical expressions for the structure factors of the pure components $S ϕ α ,eff(q)$ and $S ϕ γ B , eff ( q )$⁠.

For the structure factor of α crystallin, we use the Percus–Yevick approximation for polydisperse hard spheres, where the particle size distribution is described with a Schulz distribution with an average particle diameter of 15 nm and a polydispersity of 20%, which has been shown to describe $S α ( q )$ with very good accuracy on the entire accessible volume fraction range up to the arrest transition at $ϕ g , α ≈ 0.58$⁠.²⁴ For pure α crystallin solutions, we moreover expect no measurable temperature dependence of $S ϕ α ( q )$ for the temperatures investigated.

The comparison between the experimental data and the calculated curves shown in Fig. 5 reveals good agreement for higher q-values at $q ≳ 0.5$ nm⁻¹, and also the low-q limit $I ( 0 ) / c α γ B$ obtained from SLS is quite well reproduced at both temperatures. However, at intermediate q-values around 0.07  $nm − 1 ≲ q ≲ 0.5$ nm⁻¹ we observe systematic differences for both temperatures. There appears to be a more prominent local structure on length scales of the α-crystallin diameter than predicted by our mixing rule model. In order to better understand the origins of this discrepancy, it is quite instructive to look at the individual contributions to the scattering intensity arising from the two proteins also shown in Fig. 5. The high-q part of $I α γ B ( q ) / c α γ B$ obtained from SAXS is dominated by the scattering from γ_B-crystallin, while the scattering from α-crystallin dominates for 0.1  $nm − 1 ≲ q ≲ 0.4$ nm⁻¹, i.e., close to the nearest neighbor peak of α-crystallin solutions at comparable protein volume fractions. Finally, at low-q values it depends on temperature whether the scattering for α-crystallin or γ_B-crystallin dominates due to the strong dependence of $S crit ( q , T , ϕ )$ in Eq. (12).

Figure 6 indeed shows that the local structure is well reproduced for $q ≳ 0.5$ nm⁻¹, where the scattered intensity from α crystallins has strongly decreased due to the form factor of the large protein, and where the data thus primarily describe the local structural correlations between γ_B crystallins. Here, $S ϕ γ B ,eff(q)$ is also almost independent of temperature, as these local correlations are dominated by excluded volume effects. At low q-values, $S eff ( q )$ exhibits a clear temperature dependence, reflecting the enhanced contribution form critical scattering from γ_B with decreasing temperature. Moreover, on length scales comparable to the α crystallin diameter and larger, the structural correlations are much more pronounced than predicted. Furthermore, the additional low-q value from SLS indicates the existence of additional weak attractions that result in a significant upturn of $S ϕ γ B ,eff(q)$ at low q. The γ_B contributions from critical fluctuations and the resulting static correlation length at these temperatures are too small to cause this considerable correlation hole and subsequent strong upturn at low q. However, previous results from SANS experiments and computer simulations of $α − γ B$ mixtures have provided compelling evidence for the presence of mutual weak attractions between these two proteins, resulting effectively in a higher stability against phase separation when compared to a situation where the two protein species interact via excluded volume interactions only.^28,50 Computer simulations that allow for the calculation of the different partial structure factors [ $S α α ( q ) , S α γ B ( q )$⁠, and $S γ B γ B ( q )$ in Eq. (3)] indeed show that $S α α ( q )$ exhibits a behavior that is strongly reminiscent of weakly attractive hard spheres, where the γ_B proteins act as temporary bonds between two α crystallins.²⁸ 

It is quite intriguing that the mixing rule approach reproduces the osmotic compressibility of the mixture quite well, despite the fact that local correlations are not well described and strongly underestimated on length scales of the α crystallin diameter. We have characterized S(0) for different mixing ratios, temperatures and total protein volume fraction using SLS, and the results are summarized in Fig. 7. Here, we see that Eq. (8) indeed reproduces the measured compressibility quite well for the higher total volume fractions $ϕ = 0.16$ and $ϕ = 0.22$⁠, but fails to do so at the lowest volume fraction $ϕ = 0.11$⁠. Moreover, the agreement is also less good for the mixing ratio α: $γ B = 1 : 3$ and the lower temperature $T = 25$ °C, where the measured values are much higher than predicted by the mixing rule approach. At high concentrations and higher temperatures where critical fluctuations are not important, the compressibility is primarily determined by strong excluded volume effects between the different proteins, which is reasonably described by our approach. On the other hand, addition of a smaller amount of α crystallin to γ_B crystallins enhances the criticality of γ_B solutions in a non-monotonic way, with a maximum at a mixing ratio of $1 : 3$⁠. Under these conditions, the mixture phase separates at a higher temperature, and critical scattering is thus significantly enhanced already at $T = 25$ °C.²⁸ Moreover, the effects of an additional weak attraction on S(0) become much more visible at lower volume fractions, where S(0) shows also a non-monotonic behavior for attractive hard sphere systems with a maximum that depends on the nature and strength of the attraction.⁶² With γ_B crystallins acting as temporary bonds between α crystallins, this can lead to enhanced scattering even far away from phase separation, and thus qualitatively explain the failure of the mixing rule calculations at lower total protein concentrations.

The analysis of the SAXS data has clearly demonstrated the limits of the simple mixing rules in describing the properties of protein mixtures at high concentrations, i.e., under crowded conditions. While they were able to reproduce the experimental data for the macroscopic and mesoscopic thermodynamic and dynamic properties almost quantitatively, they fail to describe the microscopic structural features for higher concentrations in the 1:1 mixture. This clearly indicates that we would need to explicitly consider the different partial structure factors, which would require us to properly treat also mutual interactions between α and γ_B crystallins. This is also in line with earlier simulation work, which had pointed out that a weak mutual attraction between α and γ_B crystallins resulted in an enhanced stability of the mixture, suppressing LLPS and increasing transparency of the solution. However, it is also interesting to note that these mixing rules are able to reproduce the osmotic compressibility quite well at high concentrations, i.e., under conditions existing in living cells, as long as the solution conditions are not close to a boundary for LLPS for some of its proteins.

As a final test, we explored collective diffusion on the microscopic length scale of individual molecules using neutron spin echo spectroscopy, i.e., the relaxation of density correlations such as cage formation in concentrated protein solutions.

We collected intermediate scattering functions I(q, t) for mixed solutions of α and γ_B crystallin at volume fractions of 0.11 and 0.22. We used several q values covering both the correlation hole below 0.1 nm⁻¹ and the nearest-neighbor scale of γ_B crystallin beyond 2 nm⁻¹. From the rate constant $Γ ( q )$ of the initial decay of the measured correlation functions I(q, t), we obtained q-dependent diffusion coefficients $D ( q ) = Γ / q 2$ (cf. supplementary material for details).

Figure 8 displays the results as a function of the fraction of γ_B crystallin (symbols). We observe clear non-monotonous trends of the diffusion coefficients with increasing γ fraction. In all cases, the 3:1  $α : γ B$ mixtures relax faster than pure α crystallin. From this maximum value, D(q) decays to the value for pure γ_B crystallin. While the signatures at different q values differ mainly by an offset, the observed trend is significantly varied between the two explored volume fractions.

It has previously been reported that for proteins such as γ_B crystallin experiencing patchy attractions, collective diffusion at length scales of the nearest neighbor distance can be dramatically altered when compared to the predictions for simple colloid models based on an isotropic potential due to the formation of transient clusters.¹⁹ While collective diffusion on mesoscopic length scales measured by DLS could be well described by a simple mixing rule given by Eq. (5), we can no longer use such an approach. Instead we calculate a weighted average of the diffusion coefficients of the pure systems $D α ( q )$ and $D γ B ( q )$ at their respective effective volume fractions $ϕ α , eff$ and $ϕ γ B , eff$ as given by Eq. (9). The thus predicted values of D(q) indeed show a qualitatively similar behavior as observed in the NSE experiments.

This particular dependence on composition is a result of the combination of the very different scattering intensities and diffusion coefficients for the two proteins. At both q-values, the scattering intensity from γ_B crystallin is much higher than that of α crystallin, resulting in ratios of $I α ( q ) / I γ B ( q )$ as summarized in Table II. At the same time, the collective diffusion coefficient $D γ B ( q )$ exhibits a much stronger concentration dependence than $D α ( q )$ (see supplementary material), resulting in a non-monotonic dependence of the intensity weighted D(q) as a function of the mixing ratio.

However, while the qualitative trend is correctly reproduced, the actual values calculated are systematically overestimating the measured D(q). This is particularly pronounced for the 3:1 mixture. Moreover, the measured q-dependence of D(q) is much more pronounced than predicted by Eq. (9), with significantly lower values of D(q) at q = 2.2 nm⁻¹ than at q = 0.08 nm⁻¹.

The diffusion coefficients measured by NSE represent short-time collective relaxations on the nearest neighbor distance and are governed by the local neighbor structure as well as by hydrodynamic interactions. The effective static structure factor of the mixture $S eff ( q )$ is quite well reproduced by our mixing rule approach at these q-values (see Fig. 6), indicating that the local structure on these rather short length scales of order the size of γ_B crystallin are not too different from those underlying the mixing rules used, i.e., assuming that they are determined primarily by the γ_B–γ_B and α–α interactions at their corresponding effective volume fractions. On the other hand, in our approach we consider effective volume fractions only, i.e., we treat the other protein species as immobile obstacles that reduce the available free volume but do not contribute to hydrodynamic interactions. While this explains some of the overestimation of D(q) by our model, it can of course not explain the significant q-dependence in the measured data.

For a puristic theoretical understanding, one would need all pair-wise partial structure factors, which are not available for the given experimental systems. While a complete understanding of these signatures would thus require comprehensive simulations or complicated parametrizations of complex colloid theory that also takes into account possible patchy interactions between different proteins, this deviation from the simplistic mixing rule implies that mutual interactions between the two proteins actually strongly affect the dynamics on local length scales, as intuitively expected.

Driven by the need to understand crowding effects in protein mixtures as present for example in the living cell, we have embarked on a systematic test of the application of simple mixing rules in order to reproduce a number of key properties. We focus, in particular, on macroscopic quantities such as the location of phase boundaries for liquid–liquid phase separation, the osmotic compressibility and the zero shear viscosity, properties on mesoscopic length scales such as the gradient diffusion coefficient, and finally structural and dynamic quantities on local length scales such as the static structure factor and the (local) short-time collective diffusion coefficient. There is no applicable theory capable of incorporating different sizes, interaction potentials, and anisotropy in shape and interactions common for crowded protein mixtures. While one could of course use computer simulations, the resulting computational costs still become unaffordable, in particular, when aiming to reproduce dynamic properties on large length and time scales. Therefore, we have defined different mixing rules for these properties that are all based on the same underlying principle: There is no specific interaction between different species other than excluded volume effects, and we estimate the measured quantities from a weighted sum of the properties of the individual proteins in crowded solutions at an effective volume fraction that takes into account the accessible volume due to the presence of other protein species.

Despite their simplicity, we obtain rather good agreement with experimental data on macroscopic and mesoscopic length scales, i.e., phase behavior, osmotic compressibility, zero shear viscosity, and gradient diffusion. It is only at low volume fractions and low temperatures close to the binodal for LLPS that deviations become quite significant. The agreement is less convincing at intermediate length scales of order the protein size of the larger protein α-crystallin, where the lack of a mutual attraction between γ_B and α-crystallins results in significant disagreement between calculated and measured values for both S(q) and D(q). At even shorter length scales of the size of the smaller protein γ_B crystallin, the finding is mixed, with very good agreement for S(q), but significant deviations for D(q). This clearly shows the sensitivity of the local short-time collective diffusion coefficient on patchy attractions that lead to the formation of transient clusters with life times comparable or larger to the decay time probed with NSE.¹⁹ This is quite in contrast to the mesoscopic gradient diffusion coefficient measured with DLS, where characteristic relaxation times are long compared to the cluster lifetime, and where thus the mixing rule relying on weighted individual properties become quite powerful.

While our approach will require further validation with more complex mixtures, the success of such simple rules for the present system of a binary protein solution, where the individual proteins exhibit vastly different solution properties, appears promising for future applications in real life systems such as cells or other highly crowded protein mixtures.

Eye lens proteins were extracted from calf eye lenses and purified according to previous work.^25,51 In brief, α crystallin was extracted from the cortical extract solution via size-exclusion chromatography (SEC, Superdex 200 column), using a 52.4 mM phosphate buffer at pH 7.1 in H₂O. The solvent of the collected fractions was exchanged with a D₂O phosphate buffer with the addition of 20 mM of 1,4-dithiothreitol (DTT, Sigma-Aldrich, SE), 1 mM of ethylenediaminetetraacetic acid (EDTA, Sigma-Aldrich, SE) and 0.02wt. % sodium azide (Sigma-Aldrich, SE), pH 7.1. Finally, the new solution was concentrated via Amicon Ultra centrifugal filters of 10 kDa (Sigma-Aldrich, SE). The concentration determination was achieved via UV absorption spectroscopy measurements at $λ =$ 280 nm and the specific absorption coefficient $E α , 1 cm 0.1 % , 280 nm =$ 0.845  $mL· mg − 1·cm$⁠.⁵¹ We converted the weight concentrations to volume fractions, according to the relation $ϕ = c · ν$⁠, where c is the concentration in mg/mL and ν is the voluminosity of the protein. For α crystallin, $ν =$ 1.7 mL/g.²⁴ 

For γ_B crystallin, the filtered nuclear extract was purified through SEC (same as above), using a 275 mM sodium acetate buffer (Sigma-Aldrich, SE) in H₂O, pH 4.5 as a mobile phase. The collected fractions were then further purified via ion-exchange chromatography (IEX, SP Sepharose Fast Flow, GE Healthcare, USA), using a sodium acetate buffer in H₂O, pH 4.8 with a 0–325 mM sodium chloride (Sigma-Aldrich, SE) gradient. This allowed the collection of the pure γ_B protein according to its isoelectric point (pI).⁶³ The following steps (solvent exchange, sample concentration, protein concentration determination using a specific absorption coefficient $E γ B , 1 cm 0.1 % , 280 nm =$ 2.18 mL mg⁻¹ cm, and conversion to volume fractions) were achieved as described above for α-crystallin. In this case, the voluminosity of γ_B corresponds to $ν =$ 0.71 mL/g.⁴⁶ 

We investigated $α : γ B$ mixing ratios of 3:1, 1:1, and 1:3 per volume for the phase diagram, SLS, SAXS, and NSE, while we focused on the 1:1 mixing ratio for DLS and microrheology.

Static light scattering (SLS) experiments and transmission measurements for the determination of S(0) were performed on a homebuilt multi-angle light scattering (MA-DLS) instrument. This spectrometer not only allows for suppression of multiple scattering using the 3D cross correlation technology, but has the additional advantage of simultaneous measurements at 4 scattering angles θ separated by 30°. The q-range covered by LS is 0.003–0.03 nm⁻¹.⁶⁴ Additional dynamic (DLS) and static (SLS) light scattering measurements were performed with a goniometer light scattering setup (3D LS Spectrometer, LS Instruments, AG), implemented with a 3D cross correlation unit to suppress contributions from multiple scattering.^65,66 Unless stated, all measurements were performed at a θ range from 30° to 130° (steps of 5°), corresponding to a scattering vector range $q = ( 4 π n / λ ) sin ( θ / 2 ) ≈$ 0.006–0.023 nm⁻¹.

Microrheology was performed via DLS⁶¹ and multiple particle tracking (MPT) using confocal laser scanning microscopy (CLSM).⁶⁷ Tracer particles were prepared according to,⁶⁸ using polystyrene particles stabilized with covalently bound 20 kDa poly(ethylene) glycol chains. For DLS-based microrheology, we use a particle diameter d = 300 nm. On the other hand, for CLMS-based MPT we rely on fluorescent-labeled polystyrene particles, d = 1 μm. The colloidal stability of the tracer particles was tested a priori according to previous work.^61,68

For DLS measurements, a volume of 1 μL of the tracer particle solution (concentration $≈$ 2 wt. %, number density $≈$ 10¹² particles $·$ cm⁻³) was added to 100 μl of protein solution. All DLS experiments were carried out at a single $θ =$ 90° at T = 25 and 35 °C. The addition of tracer particles to the protein solutions results in a double-step relaxation process in the $g 2 ( q , t ̃ ) − 1$ function, originating from the scattering contributions from the proteins and the tracer particles.

In our measurements, the slow relaxation mode resulting from the tracer particles (i.e., at large $t ̃$⁠) has an amplitude that is > 95% of the total $g 2 ( q , t ̃ ) − B$ function, making the additional fast relaxation contribution from the proteins negligible.⁶¹