Mechanism of polymer molecular weight-dependent suppression and promotion of liquid–liquid phase separation of a protein solution by the addition of polymer Free

It has long been known that polyethylene glycol (PEG) is useful for protein crystallization.¹ Precipitation by PEG has been widely used as a chemical and biomedical method to concentrate proteins at the initial stage of a purification process since PEG is a nontoxic, water-soluble synthetic polymer.² The underlying mechanism of these phenomena has been interpreted in terms of excluded volume effects. One effect is the steric exclusion of the protein by PEG, which reduces the volume available for the translational movement of the protein within the solution.^2,3 Consequently, the effective density of the protein increases, thereby making a liquid–liquid phase separation (LLPS) more likely. This effect is essentially proportional to the size of the protein and the PEG, although deviations from steric exclusion have been observed due to the attractive interactions between protein and PEG.⁴ Another effect is the depletion force, which is an effective attraction between colloidal particles (or proteins) resulting from the translational entropy of the polymer molecule.^5,6 This is due to the exclusion of the polymer from the region between the colloids, which consequently increases the volume available for the translational movement of the polymer molecule in the solution when the proteins are in contact. These effects can be derived by assuming that the polymer–colloid interaction is of an excluded volume interaction.

The precipitation and crystallization of proteins are frequently related to the fact that protein solutions exhibit a metastable liquid–liquid phase separation with an upper critical point.^7–11 The temperatures at which the liquid–liquid phase separation (LLPS) occurs are located in the supercooling region, which is lower than the temperatures at which the solid–liquid phase transition occurs. These temperatures are called the cloud point (T_cloud) due to the fact that transparent protein solutions become turbid at this temperature.^8,9,11 As a protein solution cools down to T_cloud, liquid droplets with a high protein concentration form in a solution with a lower protein concentration.

The effect of PEG addition on T_cloud has been investigated for a variety of proteins. It has been reported that an increase in T_cloud with the addition of PEG accords with the aforementioned effects resulting from excluded volume interactions, including depletion force.^12–16 Nevertheless, in the case of PEG with a low molecular weight, a decrease in T_cloud has been observed in both lysozyme^17,18 and antibody¹⁹ solutions upon the addition of PEG. Therefore, both the suppression and promotion of LLPS by the addition of low and high molecular weight PEG cannot be comprehensively explained solely by the effects of the excluded volume interactions. This is because the excluded volume interactions always result in the promotion of LLPS, i.e., the destabilization of protein solution. Moreover, it has been shown that the impact of PEG addition on protein solubility is contingent upon the protonation/deprotonation of amino acids in response to pH alterations^16,20 and the amino acid substitutions resulting from mutations.^21,22 Bloustine et al. have also indicated that the depletion force alone is not sufficient to explain the behavior of LLPS. They have conducted a theoretical analysis based on a virial expansion demonstrating that an energetic attraction between lysozyme and PEG is necessary to explain the observed decrease in T_cloud for low molecular weight PEG.¹⁸ These preceding studies indicate that, in addition to protein–protein direct interactions, the attractions between protein and PEG play a crucial role in the stability of protein–PEG mixtures.

Given a specific ρ_c,cloud at the cloud point, the corresponding temperature T_cloud can be identified such that the ρ_c value given by Eq. (2) at a specific temperature is equivalent to the ρ_c,cloud. The temperature at which the cloud point is reached, T_cloud, is determined by repeatedly calculating $BT,ρp$ using Eq. (3) at a range of temperatures. First, utilizing Eq. (2) at a lysozyme concentration of 48 mg/ml,^18, ϵ_cc in $VccLJr$ by Eq. (1b) was optimized to quantitatively reproduce T_cloud for the lysozyme solution without PEG. Subsequently, the ϵ_cc value in $VccLJr$ was fixed, and ϵ_pc in $VpcLJr$ by Eq. (1b) was optimized such that the CG model qualitatively reproduced the experimentally observed PEG concentration dependence of T_cloud for both the 1 and 35 kg/mol PEGs.¹⁸ As indicated by Eq. (2), a spinodal decomposition is expected to occur when B is negative, as the right-hand side of Eq. (2) should be positive. In other words, if the effective interaction $Vcceffr=VDLVOr+VccPolymerr$ yields an appropriate attraction between colloids, the spinodal decomposition will occur at T_cloud and ρ_c,cloud, where $VccPolymerr$ is the polymer-induced interaction. The calculations of B were conducted by using integral equations for pair correlation functions between colloids $hccr$⁠, between the CG segment of polymer and colloid $hpcr$⁠, and between the CG segments of the polymer $hppr$⁠. These integral equations were developed based on density functional theory (DFT) for polymer–colloid mixtures^31–33 (see the supplementary material). The integral equations exhibited a monotonic and reliable convergence due to the infinite dilution condition of the colloid. The solution demonstrated sufficient convergence to allow for the determination of T_cloud.

In Fig. 1(a), the experimentally observed T_cloud of a lysozyme–PEG mixture at a lysozyme concentration of 48 mg/ml is shown as a function of PEG concentration C_PEG in the cases of PEG molecular weight 1 and 35 kg/mol.¹⁸ The addition of the high molecular weight PEG promotes the LLPS and increases T_cloud, indicating that the mixture is destabilized. Conversely, the addition of the low molecular weight PEG results in the suppression of LLPS and a decrease in T_cloud, indicating that the mixture is stabilized. It should be noted that, in this experiment, the number densities of the polymer units of the low and high molecular weight PEGs added to the solution are equivalent at each C_PEG. Consequently, the opposing polymer-induced protein–protein interactions should be attributed to the difference in the degree of polymerization of the added PEGs with different molecular weights. In the pair distribution functions between colloids $gccr$ [Fig. 1(b)], which were used for the calculation of B, at the distances smaller than 4.5 nm, $gccr$ with 35 kg/mol PEG almost overlaps $gccr$ without PEG around the first maximum, while $gccr$ with 1 kg/mol PEG is lower than that without PEG. On the other hand, at distances exceeding 4.5 nm, $gccr$ with 35 kg/mol PEG is higher than $gccr$ without PEG, while $gccr$ with 1 kg/mol PEG is nearly superimposed on $gccr$ without PEG but appears slightly higher. The reason for using 4.5 nm here is that, regardless of C_PEG, $gccr$s after adding the 1 kg/mol PEG cross $gccr$ without PEG addition at this distance. To summarize, the high and low molecular weight PEGs mainly influence the polymer-mediated interaction $VccPolymerr$ at large and small distances, respectively.

To quantify the contribution to B made by the polymer-induced interaction $VccPolymerr$⁠, the change in B upon adding PEG, $ΔB≡BCPEG−B0$⁠, was divided into the short-range part ΔB_short (r $≤$ 4.5 nm) and the long-range part ΔB_long (r $>$ 4.5 nm). In light of the abovementioned observation of $gccr$⁠, the dividing length of 4.5 nm was employed to primarily interpret the result of the 1 kg/mol PEG, whereas the choice did not affect the understanding of the result of the 35 kg/mol PEG, as will be discussed below. Upon the addition of 35 kg/mol PEG [Fig. 1(c)], both the long-range part (r $>$ 4.5 nm) and the short-range part (r $≤$ 4.5 nm) of $VccPolymerr$ resulted in a decrease in ΔB_long and ΔB_short, respectively. This indicates that both the long-range and short-range parts of $VccPolymerr$ contribute to the observed increase in T_cloud. Conversely, upon the addition of 1 kg/mol PEG, ΔB_long exhibited a decrease, while ΔB_short exhibited a sufficiently large increase [Fig. 1(d)]. As a result, T_cloud decreases with increasing C_PEG. In conclusion, the difference in the effect of PEG addition in relation to the difference in molecular weights is primarily attributable to the short-range behavior of $VccPolymerr$⁠, which depends on the polymer molecular weight.

To investigate the impact of the polymer–colloid attractive interaction on the polymer-induced interaction $VccPolymerr$⁠, the value of ΔB was calculated for the addition of 30 mg/ml PEG at 279 K (T_cloud without PEG), while varying the polymer–colloid interaction parameter ϵ_pc [Fig. 2(a)]. For comparison, the value of ΔB for excluded volume interaction models was also calculated, where $VpcLJr$ was equal to zero for r ≥ d_pc. This means that all the attractions between polymer and colloid were completely ignored. It should be noted that a positive ΔB indicates repulsive $VccPolymerr$⁠. The excluded volume interaction models consistently yield a negative and nearly constant value of ΔB, regardless of the polymer molecular weight, even if ϵ_pc is varied. Consequently, $VccPolymerr$ is, regardless of the polymer molecular weight, attractive as predicted by the depletion theory^5,6 if we completely ignore the attractive interactions between polymer and colloid. These observations indicate that the polymer-induced repulsive interaction between colloids that is responsible for the decrease in T_cloud by the low molecular weight PEG cannot be reproduced solely by the excluded volume interactions between polymer and colloid. On the other hand, the low and high molecular weight polymer models with attractive interactions between polymer and colloid result in the repulsive and attractive $VccPolymerr$⁠, respectively. In conclusion, the attractive interaction between the polymer and the protein is necessary for a comprehensive explanation of the experimentally observed decrease and increase in T_cloud for the low and high molecular weight PEG, respectively.¹⁸ With regard to the experimental evidence that lends support to the finding, entropic and enthalpic attractive interactions between lysozyme and PEG have been observed for low molecular weight (2 kg/mol) PEG, where the entropic attraction would be due to dehydration from both the lysozyme and the PEG.³⁴ Furthermore, the determined binding constant is in reasonable agreement with that previously reported for higher molecular weight PEGs.³⁵ 

Figure 2(a) illustrates that the low and high molecular weight PEGs exhibit positive and negative values for ΔB, respectively, and that these values decrease slightly and largely, respectively, with increasing ϵ_pc. To characterize the difference in ΔBs between the low and high molecular weight PEGs, ΔB is divided into the short-range part ΔB_short and the long-range part ΔB_long again [Figs. 2(b) and 2(c)]. In the case of 1 kg/mol PEG, ΔB_short is a positive value and almost constant, while ΔB_long takes a small negative value and slightly decreases with increasing ϵ_pc [Fig. 2(b)]. Consequently, the polymer-induced repulsion in $VccPolymerr$ is predominantly attributed to the short-range part, while the weakening in the polymer-induced repulsion along with ϵ_pc [the slight decrease in ΔB as shown in Fig. 2(a)], is due to the strengthening of the long-range attraction in $VccPolymerr$⁠. In the case of 35 kg/mol PEG, ΔB_long is consistently negative, as observed in the case of 1 kg/mol PEG. In contrast, ΔB_short varies from positive to negative with increasing ϵ_pc [Fig. 2(c)]. Consequently, particularly for the small ϵ_pc, the negative value of ΔB, which represents the polymer-induced attraction, is predominantly attributed to the long-range part in $VccPolymerr$⁠. The short-range repulsion is also observed even for the 35 kg/mol PEG, provided that ϵ_pc is sufficiently small [Fig. 2(c)]. While the magnitude of ΔB_short for the 35 kg/mol PEG varies significantly with ϵ_pc, the underlying mechanism of the short-range repulsion would be consistent with that found for 1 kg/mol PEG. This is discussed in greater detail below. The decrease in ΔB, indicative of an increase in polymer-induced attraction, with increasing ϵ_pc [Fig. 2(a)], is due to the concurrent strengthening of both the long-range and short-range parts of $VccPolymerr$ [Fig. 2(c)]. It is notable that the baseline model, which incorporates the attractive interaction between polymer and colloid indicated by the dashed vertical line, yields a polymer-induced attractive interaction between colloids that is comparable with the depletion effect caused by the excluded volume interaction model [Fig. 2(a)]. Furthermore, the polymer-induced attractive interaction is significantly strengthened by the increase in the polymer–colloid attraction [Fig. 2(a)]. This observation indicates that the underlying mechanism of the polymer-induced attraction between colloids should be different from the depletion effect and attributable to the formation of a polymer adsorption layer, namely, the excess free energy due to the density gradient of the polymers.^36,37 This effect will henceforth be simply referred to as interfacial tension. The formation of a polymer depletion region around colloids gives rise to an interfacial tension effect despite the density gradient of the polymers being completely opposite to our systems.^36,37 This results in colloid–colloid attractions, which are known as depletion forces. It is important to note, however, that the effect of the depletion force is not included in the underlying mechanism of the computational results that we have presented here by introducing the polymer–colloid attraction.

To gain insights into the effects of the collision pressure of polymeric segments on colloids and the interfacial tension, the relationship between the intercolloidal distances and the distribution of polymer segments around a colloid was investigated. Here, the polymer cloud surrounding a colloid was defined by the distance r where the pair distribution function between polymer segment and colloid $gpcr$ is larger than 1. Figures 3(a) and 3(b) depict the polymer cloud surrounding a colloid (white circle with a black solid line) as a blue translucent partial circle for the low and high weight PEG, respectively. Another colloid is introduced as a test colloid particle (shaded circle with a blue dashed line) into the polymer cloud surrounding the central colloid at an intercolloidal distance of 4 nm [Figs. 3(a) and 3(b)]. For the sake of simplicity, the polymer cloud surrounding the test colloid is not shown in Figs. 3(a) and 3(b). This intercolloidal distance is slightly smaller than the distance used to divide ΔB, 4.5 nm [Figs. 3(a) and 3(b)]. In the case of low molecular weight PEG [Fig. 3(a)], since the surface of the test colloid is partially covered by the polymer cloud, the pressure exerted by the polymer collision on the test colloid particle from the side near the central colloid is greater than that exerted from the side farther away. The short-range part in $VccPolymerr$ caused by the low molecular weight PEG is attributed to a repulsive interaction resulting from the polymer collision pressure that overcomes the interfacial tension effect due to the polymer adsorption layer mentioned above. On the other hand, in the case of high molecular weight PEG [Fig. 3(b)], the entire surface of the test colloid, including the side farther away from the central colloid, is fully covered by the polymer cloud at this intercolloidal distance. However, $gpcr$ on the side near the central one appears to be higher than $gpcr$ on the side farther away from that, as indicated by the red bidirectional arrows. Consequently, the pressure exerted by polymer collisions on the test colloid particle from the side near the central colloid is stronger than that exerted from the far side. Nevertheless, the interfacial tension effect due to the large polymer adsorption layer formed by the high molecular weight PEG would be more pronounced than the polymer collision effect. As a result, the short-range part of $VccPolymerr$ by the high molecular weight PEG acts as an attractive interaction, depending on ϵ_pc [see Fig. 2(c)].

Next, in order to interpret the long-range attraction in $VccPolymerr$ observed for the low molecular weight PEG [Fig. 2(b)], the polymer cloud surrounding two colloidal particles that are separated by the intercolloidal distances exceeding the dividing distance of 4.5 nm is depicted for the low molecular weight PEG [Fig. 3(c)]. The excess free energy generated in the interface between the polymer cloud and the bulk solution, which is indicated by the red curve in Fig. 3(c), results in the attraction between the colloids up to the distances where the polymer clouds do not overlap. The short- and long-range attractions caused by the high molecular weight PEG [see Fig. 2(c)] result from the interfacial tension similar to that shown in Fig. 3(c). As previously stated, it cannot be ruled out that the possibility that the effect of interfacial tension-induced attraction between the colloids is included in ΔB_short is not only for the high molecular weight PEG but also for the low molecular weight PEG.

In summary, we have elucidated the effects of PEG molecular weight and protein–PEG attraction on PEG-mediated protein interactions using implicit solvent CG models of protein–PEG mixtures. Our findings indicate that the weak effective attractions between protein (colloid) and PEG (polymer) can reproduce both the experimentally observed increase and decrease in T_cloud upon the addition of high and low molecular weight PEG, respectively.¹⁸ This is consistent with a prior theoretical analysis for the low molecular weight PEG based on the virial expansion presented by Bloustine et al.¹⁸ The key point is that the PEG-mediated instability of the protein solution can be explained not only by the depletion of the polymers around the protein but also by the formation of an adsorption layer of polymers to a protein. The effect of the formation of a polymer adsorption layer was necessary to comprehensively reproduce the experimentally observed increase and decrease in T_cloud upon the addition of high and low molecular weight PEGs, respectively.¹⁸ Moreover, the behavior that the stronger the polymer adsorption, the stronger the polymer-mediated attraction between colloids, is essentially different from the depletion force caused by excluded volume interactions between polymer and colloid. These findings indicate that the classical view of the polymer-dependent phase behavior of colloidal dispersions based on the depletion effects solely due to the excluded volume interactions between colloid and polymer should be partially revisited.

It is important to acknowledge the limitations of this study, which include the utilization of an implicit solvent model and a simplified representation of the protein molecule as a colloidal particle in accordance with the DLVO-type model potential. The impact of polymer adsorption on the protein, resulting from the polymer–protein attractive interaction, could be overestimated in our coarse-grained model due to the absence of the explicit solvent molecules, which also solvate the protein. Nevertheless, it is certain that the stabilization of the protein solution by low molecular weight PEG is attributed to the adsorption of the polymer onto the protein. In fact, deviations from a simple exclusion model have been observed, especially for low molecular weight PEG.^38,39 However, the molecular mechanism of a size-dependent effect on the effective interactions between protein and PEG is unclear and has not been considered in our modeling. This may also be the reason why our model predicts the polymer adsorption on the protein even in the case of high molecular weight PEG, contrary to the assumption employed in the depletion theory.

The influence of complex protein structures and the effects of local interactions between sites of the protein surface cannot be taken into consideration directly. It has been shown that isotropic models, such as the DLVO-type potential, yield a liquid–liquid coexistence curve that is too narrow and/or an overestimation of the critical temperature.⁴⁰ In contrast, in the present study, the parameter of the attractive interaction in the DLVO-type model was optimized for a single experimental cloud point, and then the impact of PEG addition on the cloud point temperature was examined. The impact of randomly distributed protein–protein interaction sites on the protein surface has been demonstrated to reproduce the liquid–liquid phase behavior of lysozyme solutions based on Wertheim’s theory of associating fluids.⁴¹ Furthermore, the DLVO-type theory is unable to account for the chemical specificity of the electrolyte ions, as has been observed in the solubility of lysozyme in the presence of various alkaline salts.⁴² It will be necessary in the future to examine the applicability of the findings of this study to antibodies and to examine the capability of the present CG modeling approach to reproduce the experimental observations of PEG-mediated interactions between other proteins.

The supplementary material provides detailed information on the model parameters, an analysis of spinodal decomposition in colloid–polymer mixtures, and a concise explanation of the density functional theory for mixtures of colloid and polymer.

This work was supported in part by JSPS KAKENHI Grant Nos. JP20K05431, JP22H01888, and JP21K06503.

The authors have no conflicts to disclose.

Yoshihiro Osaka: Data curation (lead); Formal analysis (lead); Investigation (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Ryuichi Okamoto: Validation (equal); Writing – review & editing (equal). Tomonari Sumi‡: Conceptualization (lead); Funding acquisition (lead); Methodology (lead); Project administration (lead); Resources (lead); Software (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead). Kenichiro Koga: Supervision (equal); Validation (equal); Writing – review & editing (equal). Hiroshi Imamura: Validation (equal); Writing – review & editing (equal). Tsuyoshi Shirai: Validation (equal); Writing – review & editing (equal). Yasuhiro Isogai: Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material. The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.