Solid nuclei and liquid droplets: A parallel treatment for 3 phase systems

Abstract

For solid phase self assembly into crystals or large diameter polymers, the presence of a liquid‐liquid demixing transition has been known to have an accelerating effect on the nucleation process. We present a novel approach to the description of accelerated nucleation in which the formation of solid phase aggregates and liquid‐like aggregates compete as parallel pathways to formation of dense phases. The central idea is that the small aggregates that would ultimately form the liquid phase are sufficiently labile to sample the configurations that would form the solid, so that the growing cluster begins as a liquid, and switches into growth as a solid when the aggregates have equal free energies. This can accelerate the reaction even when the liquid‐demixed state is thermodynamically unfavorable. The rate‐limiting barrier is therefore the energy at which there is a transition between liquid and solid, and the effective nucleus size is then concentration independent, even though for both nucleated demixing and nucleated crystallization, the nucleus size does depend on concentration. These ideas can be expressed in a chemical potential formalism that has been successfully used in nucleation of sickle hemoglobin, but not to our knowledge previously employed in describing LLD processes. The method is illustrated by considering existing data on Lysozyme.

Introduction

This article is concerned with nucleation‐controlled reactions, specifically how formation of a liquid phase can accelerate nucleation. This issue is not new, and has been addressed both numerically and analytically. In the analytic approach, attempts to merge nucleation and liquid‐formation have often been colored by the questions that dominate the discussions of phase transitions, such as the location of phase boundaries, and scaling relations (cf. e.g., Ref. 1). In contrast, nucleus formation in solids tends to be formulated in terms of microscopic, mechanistic constructs. This article presents a single unified framework for both solid nucleus formation and liquid phase formation, which then allows their mutual interactions to emerge in a natural way. The description has an overall, qualitative conceptual framework as well as a specific analytic formulation, which will be used to discuss nucleation of lysozyme crystals as an example of the method.

The formation of solid phases from molecules in solution is of wide interest in basic science and technology. From issues of efficient crystallization to the problems of pathological aggregates that underlie diseases, the rate of solid phase formation plays an important role. Solid formation from isolated solutes may simply commence once the system is prepared (“quenched”) in a state where the assembled phase is more stable than the dispersed one.

However, the aggregates that first form en route to the dense phase may be less stable than isolated monomers from which they are composed even though larger aggregates are more stable. Thus at some critical number of molecules the relative stability of the aggregate finally begins to improve with each successive step. That turning point is designated as a nucleus. Even beyond the nucleus, though, the aggregates are still unstable until sufficient numbers have associated. Nucleation thus represents a bottleneck in the kinetic scheme, and with reasonable accuracy one may construe the assembly process of many kinetic steps as the equivalent to the likelihood of forming a nucleus—a thermodynamic rather than kinetic construct—combined with the likelihood of adding to that nucleus and transitioning into a downhill regime. (This is based on the classic approach of Ref. 2.)

The foregoing description can also apply to the formation of a liquid aggregate. Formation of a liquid phase from a collection of molecules is conceptually straightforward in the case where the initial configuration is a gas. When the dispersed molecules are already in solution, the process is referred to as liquid‐liquid demixing (LLD), since the dispersed, solvated molecules must abandon their solvent to form a different liquid, made of their own molecular type. In the nomenclature of LLD, the equivalent of solid phase nucleation is nucleated demixing. It is also possible to press the system beyond nucleation so that every monomer added attains sufficient stability to move the aggregate to larger and larger sizes. Substantial demixing will begin when the barrier has dropped to kT, the scale of the thermal fluctuations.

In a seminal work, ten Wolde and Frenkel3 showed by simulation that the presence of the liquid phase could have a dramatic effect on the rate of formation of the solid phase, providing theoretical support to an observed phenomenon. Subsequently, others have worked on the development of simple ways to conceptualize this, often by the construction of a pathway along a free energy trajectory that was viewed to possess multiple sequential barriers.4

In the approach adopted here, parallel pathways are considered that represent these two extremes of solid and liquid. As Frenkel observed, more complicated trajectories are possible with mixed regions of liquid and solid in the nucleating cluster.3 This complication is analogous to the problem in nucleation theory of molecules that are not in their lowest energy configuration. The spirit of this work is to follow that “best” configuration to see what this approach offers, hopeful that what it may lose in rigor it gains in insight. We then make the critical “ansatz” that the liquid can make a transition into the solid phase simply and rapidly. This is a one‐way process, since by construction the liquid is quickly sampling all its condensed arrangements, while the solid is kinetically prohibited from doing so on any relevant time‐scales. Thus, the simple notion is that a liquid cluster, in sampling its various configurations, stumbles upon the geometry corresponding to the solid. Because the geometry of the solid phase is the more stable going forward, the cluster enlarges as a solid thereafter. It then remains to formulate expressions that are able to describe these small aggregates with some fidelity.

To formalize these concepts, we adopt a theory that has successfully described the nucleation process of sickle hemoglobin, a natural mutant of adult Hb in which a single charged, hydrophilic glu on the two β‐subunits has been mutated to a hydrophobic val. The result of this mutation is that the HbS molecule is functionally equivalent to HbA except for a propensity to aggregate in the deoxygenated form when Hb concentrations exceed some solubility. While the aggregates are polymers, and might therefore seem quite distinct from crystals, the polymer cross section is substantial, numbering 14 molecules. On a polymer of this size, spherical aggregates of ∼65 molecules can be accommodated before one must explicitly consider the edges. Since nuclei in sickle hemoglobin are much smaller than this, the description is apt for crystal nucleation as well as forming these polymers, and a variant of HbS, known as HbCHarlem,5 which does form crystals, has also been described by this theory.

While this theory is much in the spirit of classical nucleation theory, the approaches actually have substantial differences. Classical nucleation theory is built on the distinction between a bulk free energy, and a surface free energy, or surface tension, an approach that is difficult to justify for clusters in the 2–10 ranges. The approach employed here uses instead a combination of energy that comes from intermolecular contacts and a free energy that is the result of vibrations of the aggregate molecules about the centers. The latter term is entropic in nature and scales primarily with the number of particles in the aggregate (because it is the number of particles that sets the number of vibrational modes.)

The Conceptual Framework

Before considering the effects of competing pathways, it is useful to consider the behavior of a single nucleation barrier, as shown in Figure 1. The figure shows the free energy in units of kT of an aggregate of various sizes, with curves showing aggregates of increasing internal stability. The central concept in all nucleation theories is that the initial stabilization of the aggregates increases with the number of molecules in the aggregate, so that a reaction that is initially unfavorable (at small sizes) eventually becomes favorable at large size.

Figure 1.

Energetic barrier to association, as a function of aggregate size. The energy is the chemical potential of an aggregate of the given size, relative to the monomers. The net cost in energy for small aggregates creates an effective barrier. The aggregate size at which the barrier peaks is the nucleus. The energy turning point is achieved when the cost of removing an additional monomer from solution and adding it to the aggregate is less than the energetic gain from the joining the aggregate. The curves can become lower either because the cost is less (high concentration of monomer) or because the gain is greater (strong intermolecular contacts for example). As the height decreases, the size of the nucleus also gets smaller until ultimately the barrier vanishes and the monomers are immediately unstable relative to any size aggregate. A dashed line drawn at ΔG = kT. A barrier of that height becomes insignificant against thermal fluctuations, and is the cloud point for LLD.

A key construct is a graph of the free energy of the aggregate as a function of aggregate size: when such a graph has a local maximum the system has a barrier to increasing size, and this is the nucleus. We thus consider in Figure 1a family of curves that differ solely in the incremental stability conferred by each added monomer. At one extreme as exemplified by the top curve, the parameters of the system do not favor assembly at any size even though the stability is gradually improving (as seen by the decreasing slope). Therefore, every aggregate is more costly in energy than its predecessor and the system remains trapped as monomers. At the next curve, incremental aggregate stability is greater so that the asymptotic slope is flat as n approaches infinity (which is not visible in this graph), and the system at very large sizes is just about to become stable, that is, it is poised at its solubility. Subsequent curves have more and more stability, and finite size aggregates eventually are more stable than their constituent isolated monomers, as seen by their ΔG becoming negative. Where the system turns over (called the nucleus) is a thermodynamic tipping point, rather than some special structure, and the assumption in such calculations is that the aggregates are simply excised pieces of the final, macroscopic structures. The energy at the nucleus represents the free energy barrier to growth from monomers to large solid entities. The rate of formation of this solid phase is governed by the likelihood of having a nucleus, so the rate goes as exp(–ΔG*/kT). ΔG* designated the maximum ΔG, achieved at the nucleus, which has a size designated as n*. Eventually the barrier ΔG* shrinks to kT, for which a dashed horizontal line is drawn. For a barrier lower than this, thermal fluctuations readily populate larger aggregates. As the barrier drops, the nucleus size also gets smaller and smaller. Finally, the nucleus drops to 1 for the lowest curve, and even the tiny barrier completely vanishes.

The same set of curves describes LLD, where the aggregates are liquid droplets, which can have a tiny size. In the language of LLD, the asymptotically infinite turning point, which was the solubility above, is known as the binodal point. Greater stability (e.g., concentration greater than the binodal) thus leads to a process known as nucleated demixing. As for solids, the rate at which the system demixes is controlled by the barrier height, and even though one has crossed the binodal, the rate for having a liquid phase emerge can be prohibitively slow. One approach to locating the binodal is to sweep temperature (which affects stability) until a flurry of droplets emerge at what is known as the cloud point. The abrupt emergence of a multiplicity of droplets would correspond to the barrier hitting the kT line. This is sometimes labeled as the binodal, which is clearly not the case here. A second singular point is the spinodal line, which is where the system no longer nucleates to demix. Once prepared (e.g., by a sudden temperature change) the system promptly separates. It is apparent that such a point must be when the nucleus has vanished, which is the case where the nucleus is formally equal to unity. Once concentration exceeds the spinodal, the system is said to undergo spinodal decomposition (cf. e.g., Ref. 6).

We now turn to the case with both solid and liquid phases. Certainly one could imagine an aggregate of which some part is solid, covered in a liquid coating, and indeed this emerged in simulations.3 We adopt a simpler view and assume there are only the two extremes, pure solid aggregates, or pure liquids. The central premise is that the liquid aggregates, being less strongly bound together, will be able to interconvert among different arrangements, while the solid aggregates cannot do so. For simplicity of language, we will talk about forming a crystal, but the same principles apply to thick polymers.

Figure 2 shows the principle involved. The solid black curve shows a crystal nucleation barrier, with a nucleus size of about five molecules and a barrier height of 11.4 kT. The dashed curves show the energies of liquid‐like clusters held by increasingly strong internal energy, which would be reflected in the binodal. Three cases are shown. For curve a, the highest liquid case, aggregates of the solid, even when unstable, are still more stable than formation of liquid clusters. While at the start, liquid or solid are quite comparable in energetic cost, the rate of solid formation is governed exclusively by the height of the solid phase barrier.

Figure 2.

Three possible types of interaction between solid nucleation (solid curve) and a coexisting liquid aggregates (dashed lines). The liquid curve may become isoenergetic with the solid curve before the nucleus (a), and then the transformation of liquid droplets into solid nuclei produces no increment in rate. When the liquid curve crosses the solid nucleus curve at higher aggregates than the nucleus as in (b), a growing droplet that transforms into a solid will not need to surmount as high a barrier as the solid does. This can happen even when the liquid phase is still unstable. The point at which the curves cross then functions as the nucleus. A third possibility (c) is that the liquid barrier is enclosed in the solid curve, meaning that liquid droplet nucleation controls the reaction. In the case shown, the droplets would transform to solids at a point at which they had become more stable than monomers. In all three cases, however, the final structures are equivalent.

For curve b, there is no possibility of forming liquid phases: curve b does not show demixing. However, the intersection of b with the nucleation free energy curve for crystal formation provides a new option. Clusters can grow as a liquid (the most likely path) up to around 7 monomers, since they have lower free energy than solids. Beyond 7 monomers, the crystal aggregates are the more stable entity. Thus, if the reorientation of the molecules in the liquid cluster lets them “discover” the crystal geometry, the aggregates then will enlarge as crystals. The crossover happens some 1.4 kT lower than the crystal nucleation barrier, and would thus allow nucleation to proceed some 4× faster than if they were controlled solely by solid nucleus formation. This happens even though the system would never form a liquid phase of its own, that is, even though there is no LLD.

For curve c, there is a bona fide LLD. The dashed curve c, in the absence of a solid phase option, represents nucleated demixing, as can be seen in the downward slope of the free energy curve at large sizes. Since this demixing has a barrier > kT, it will therefore show the characteristics of nucleation, with a liquid nucleus that has a size of about 4 monomers. At about 11 monomers, the liquid droplets/aggregates have become more stable than monomers, and one might well then see aggregates with liquid‐type lability between 11 and 14, but by size 15, the crystal structure would take over, and the crystal (or polymer, if that is the case) forms. The solid aggregates that then emerge would show no evidence of the fact that their pathway began with the disordered liquid nuclei.

Thus, the three cases described show the conceptual framework for nucleation of crystal or thick polymer formation as controlled in three different ways: by a solid crystal nucleus, a crossover point between liquid and solid clusters, and a liquid nucleus. All three approaches ultimately generate the same solid phase in this framework.

Theoretical Framework: Thermodynamics

We now introduce a thermodynamic framework to describe in detail the processes, which have been described above. The framework has been previously described,7, 8 but it is useful to recapitulate it briefly. The framework represents a specific model for aggregation, but it must be stressed that the ideas of the previous section do not depend on the accuracy of this model to retain validity as means for understanding such assisted solid‐phase nucleation.

There are three fundamental chemical potentials in play. μ_TR is the translational and rotational term for molecules in solution (coming from the entropy their motion entails), and is the equivalent to molecules free in a gas. Surrendering this entropy is the fundamental price any aggregating system must pay. Compensation for this lost entropy comes first in terms of the chemical potential due to the contacts made in the crystal aggregate, designated μ_XC. The calculation involved will strictly count neighbors to scale the number of bonds, so the assumption is also that each intermolecular contact represents a fixed amount of energy, independent of its neighbors. This is particularly apt for hydrophobic bonds, which are typically calculated based on buried surface area. μ_XC can have both enthalpic and entropic contribution, in contrast to μ_RT, which is solely of entropic origin. The most rudimentary estimate would easily reveal that far too much free energy (in the form of entropy) is lost compared to that which might be recovered by the types of buried surface energies available, and this is because a third chemical potential has been omitted thus far. When molecules associate, they do not become immobile, but rather undertake motion about their center of mass within the condensed phase. In a crystal, this is the very motion from which one derives the specific heat of that solid. Therefore, one must include μ_XV, the chemical potential due to center of mass vibrations of the molecule in the aggregate, another entropic term. It does not include internal vibration of the molecule, which are typically assumed unchanged. Thus, there is a transformation of the entropy of molecules that are free to wander in solution into an entropy of molecules that jiggle about their equilibrium positions. Nothing requires that these be equal, and indeed they are not typically so.

In describing the liquid phase, the same three chemical potentials are used, appropriately modified. No modification is needed for μ_TR, representing solution phase motion of monomers. μ_XC will become μ_LC representing the contact of the molecules in the liquid phase with their neighbors, in some averaged (mean field) sense. Most complex is μ_LV, the chemical potential that accounts for center of mass motion of the molecules within the liquid phase. This issue is a central conundrum of the theory of liquids, and it will not be resolved here. For its use in this calculation, the critical feature is the way in which it scales with the number of particles in a liquid‐like aggregate. For the solid, the approximation that has been used is that the chemical potential of an aggregate due to the entropy of the internal vibrations of its constituent particles, μ_nV, is, to first order, scaled by the number of modes, which is the number of particles (minus one to account for the aggregate itself moving).7 In doing this, it is assumed that there is no distinction between vibrations of a surface molecule and vibrations of a bulk molecule. This clearly is not correct, but the introduction of surface corrections creates a model with too many free parameters to be readily tractable. The assumption here is essentially that the distinction is small relative to the increases due to number of particles (and thus number of modes). The goal is thus to see what insights emerge from this simple model. We will use the same idea regarding vibrational entropy for the liquid.

The solid will aggregate when the activity of monomers exceeds the activity at solubility. The liquids are formed when the activity of monomers exceeds the activity at the binodal. Thus the system may be supersaturated with respect to the solid phase (via solubility), and labeled S, or liquid phase (via binodal), and labeled S′.

When the liquid is supersaturated, the metastable region of the phase diagram adjoins an unstable region at a concentration known as the spinodal concentration, which we label c _λ. When metastable, droplets are formed by nucleation, while when they are unstable, the quench of the system produces substantial numbers of droplets (i.e., demixing) at once. In both regions, the system is demixing. A large barrier makes the rate small, and a small barrier produces a rapid demixing. The barrier height diminishes as S′ increases, which also results in smaller and smaller nuclei. Eventually, the barrier disappears at which point the nucleus has diminished to 1. However, even before the barrier disappears, the rate of demixing can be large if the barrier is small relative to kT. The cloud point c _π is the concentration at which droplets rapidly form, and thus we take it as the point where the barrier is insignificant, i.e., ΔG*/kT = 1. Once cloud point and spinodal concentrations are identified, one can deduce the binodal concentration, and then the underlying chemical potentials μ_LC and μ_LV are determined.

Theoretical Framework: Kinetics

The explicit interest here is how the possibility of LLD can assist formation of the solid phase. Note that the conditions need not favor LLD. We postulate that the liquid will spontaneously discover the solid‐phase arrangement at some size n_x, and thus there will be a transition from the liquid‐forming to solid‐forming trajectory through the aggregate sizes.

Therefore, it is essential to identify the crossover point, n_x where liquid and solid aggregates have the same energy. When the crossover occurs after the liquid clusters reach their nucleus size, n′*, the barrier is not ΔG* but ΔG _x, the energy at the crossing point. The consequent enhancement in nucleation rate will be a factor of exp[(ΔG* – ΔG _x)/kT] where ΔG _x is the energy at crossover. When the crossover occurs after the liquid cluster nucleation, that is, n′* < n_x, then the barrier is still not what one expected, but instead is the barrier to nucleated LLD demixing in the metastable region. Now the enhancement is by a factor of exp[(ΔG* – ΔG′*)/kT].

Even though both solid and liquid nucleation processes have barriers with a size that depend on concentration, rather remarkably the crossover point n _x, does not depend on concentration. A simple way to understand this is that the cross over represents an isomerization: the conversion of n molecules in liquid arrangement into n solid molecules. Isomerizations are simply not concentration dependent. In terms of the chemical potentials, as shown in the Supporting Information material, the crossover point n _x is given by

$$\frac{n_x-1}{\ln \hspace{0.17em}{n}_x}=\frac{{\delta }_1({\mu }_{\mathrm{LC}}-{\mu }_{\mathrm{XC}})}{({\mu }_{\mathrm{XC}}-{\mu }_{\mathrm{LC}})+({\mu }_{\mathrm{XV}}-{\mu }_{\mathrm{LV}})}$$ (1)

δ ₁ is a parameter that describes the scaling of contact number with aggregate size as also discussed in the Supporting Information Material. Knowledge of the cross‐over point is particularly essential when the cross‐over acts as the nucleus size.

As the concentration (and thus the supersaturation) rises, for solid or liquid, the nucleation rate rises as well, thus capturing the well‐known effect seen as one approaches a spinodal. However, at some point, very close to the spinodal, significant numbers of molecules form droplets, leading to depletion of the monomer concentration. This means the supersaturation will then fall, effectively locking out nucleation by raising its barrier.

Let the rate of nucleation be designated as f _o where the subscript designates that this is the initial nucleation rate, since as monomers are used in the formation of aggregates, the rate will fall. This rate is taken as the product of the activity of the monomers and the activity of the nuclei, that is,

$$f_{\mathrm{o}}=k_{+}\hspace{0.17em}\cdot \gamma c\cdot {\gamma }_{n\ast }{c}_{n\ast }/{\gamma }^{‡}$$ (2)

γ ^‡ is the activity of the activated complex, that is, the nucleus plus one monomer. For the lysozyme case we are about to consider, the excluded‐volume effects on γ ^‡ are <5% away from unity, but this term can be highly significant if the concentration of the monomers begins to occupy a significant fraction of space.9 When the nucleation rate is controlled by the crossover from liquid to solid‐like aggregates, the rate is controlled by n_x rather than n*.

Lysozyme

The monomeric protein, lysozyme, of MW 14,320 forms crystals as well as demixing into distinct droplets. Both processes have been extensively studied, allowing the application of the above concepts. Because we are seeking to compare nucleation rates as well as LLD, it is essential to view data under comparable conditions. Unfortunately, despite the considerable literature on this system, obtaining a fully consistent set of data at comparable solution conditions was problematic. Nucleation data has been collected by Bhamidi et al.10 at a variety of temperatures and conditions, but none where cloud point and spinodal lines are known. Fortunately, Bhamidi also show that the nucleation rates are the same for a given concentration so long as the solubility is the same, irrespective of the conditions by which it is achieved. For example, at solubility of 2.0 mg/mL, the nucleation rates are indistinguishable for 16°C data with 4.0% NaCl and 9.5°C data with 3.0% NaCl. This observation is thus an illustration of a supersaturation relationship, that is, the nucleation rate depends on the ratio of the monomer concentration to the solubility.11, 12 For crowded solutions, this becomes a ratio of activities rather than concentrations, where activity is defined as the product of a concentration and an activity coefficient. Since solubilities have been determined for a range of conditions, including temperatures,13, 14 we chose to interpolate to nucleation rates at 13°C, 4% NaCl and pH 4.5, where cloud point and spinodal lines are known.15 (Although there is a published data set that does contain nucleation as well as demixing data, it has been the subject of significant methodological controversy,16, 17, 18, 19 and singularly does not agree with other published spinodal lines15, 20 or nucleation rates.2) Thus, to achieve equivalent supersaturation, a change in solubility must be matched by a change in protein concentration, but once this is done, the data remains valid.

To relate nucleation to LLD, it is necessary to determine the binodal of the liquid phase. The binodal is customarily determined by observing the cloud point, although these are not identical, strictly speaking. We here use the equation for the liquid demixing barrier, and find the point at which the barrier drops to kT, since it is that point at which the solution will abruptly become opaque. Knowing the spinodal allows one more parameter to be eliminated (see Supporting Information Material). This produces a single equation (Supporting Information S15) that can be numerically solved for the binodal, given the cloud point and the spinodal. At the higher temperatures of its observation, the binodal is indeed close to the cloud point, while at lower temperatures they differ. For example, at 13°C, the cloud point is measured at 6.30 mM, and the binodal is deduced as being at 6.21 mM. In contrast, at 7.4°C the cloud point is 1.90 mM whereas the binodal is 1.43 mM.

At 13°C, where the nucleation rates are being considered, we find the liquid contact energy μ_LC = 1.9 kcal/mol. Vibrational motion contributes −28.1 kcal/mol at this temperature. In contrast, the loss in rotational and translational entropy accounts for −31.4 kcal/mol.

Having a reasonable estimate of the parameters of the liquid state at 13°C, we turn to the crystal nucleation measurements.10 The data is shown in Figure 3, on a log‐log plot. It was collected for concentrations ranging from 26 to 43 mg/mL at 16°C, allowing us to determine its supersaturation since its solubility is known to be 2.0 mg/mL there. This in turn allows us to apply the same data to 13°C for comparison with the liquid analysis just presented.

Figure 3.

Log (base 10) of nucleation rate (number mL⁻¹ min⁻¹) of lysozyme crystals as a function of log (base 10) of supersaturation, defined as the activity of the monomers divided by the activity at solubility. Data is from Bhamidi et al.10 for 4% NaCl, pH 4.5, at 16°C, but the supersaturation relation allows us to treat the case of 13°C by using the solubility at that point. The solid line is the best linear fit to the upper region. The dashed line extrapolates the fit. In the theory for simple nucleation processes, linear behavior in this plot is not expected, in contrast to the straight line behavior expected when a crossing process acts as a nucleus. At lower supersaturation, as shown in Figure 4(a), the crossover pathway and the straightforward nucleation pathway have similar rates, suggesting that both pathways might be active, and that the net rate would exceed the extrapolated value from the high concentration, similar to what is observed.

Were the crystal nucleation process to be analyzed by the conventional methodology we have previously used, without regard for the influence of a liquid phase, the process would entail a nucleus of 6.25, and a nucleation barrier (as seen in Fig. 1) of just under 18 kcal/mol. At the same conditions, the binodal concentration is not reached until 87 mg/mL, so the nucleation experiments were performed well below this phase separation threshold and conventional wisdom holds that the two processes would be well separated. From this analysis of the liquid phase, the droplets of size 6.25 are unstable against the monomers by about 13 kcal/mol. Thus, while nuclei and droplets at equivalent conditions are both unstable entities, the droplets are the more favored of the two, and thus could provide a pathway to crystal formation. If that is indeed the case, however, the analysis that produced the above nucleus size and barrier height for crystal formation is faulty, since the crystals have not formed via this classic type of nucleation but by a hybrid path.

We therefore turn to the approach presented in this work, in which the nucleation phenomenon is controlled by the liquid‐solid crossover. Taking the crossover as the nucleus allows us to employ Eq. (1). The two unknown chemical potentials for the liquid are both determined, and for the solid, sum of the two unknown chemical potentials is known via the measurement of solubility. This provides two equations (crossover and solubility) with two unknowns (μ_xC and μ_xV), which can be solved. Doing so, we get μ_xC = −8.2 kcal/mol and μ_xV = −25.4 kcal/mol.

Figure 4 shows the barriers. At the high concentration end (panel b) we see that the transition from liquid to solid has lowered the barrier by 1.7 kT, and thus represents a more advantageous pathway to forming the crystal even though no LLD is present in those conditions. At lower concentrations, the nucleus size is the same, as the crossover point does not migrate along the size axis. The predicted constancy of the nucleus size agrees well with the data of Figure 3 in the high concentration regime. As the dashed line in Figure 3 illustrates, extrapolating that high concentration region to lower supersaturation shows the lower concentration nucleation is faster than one might have expected. When we plot the free energy curve, the reason becomes clear, as seen in panel a of Figure 4. The advantage of the crossover has diminished, and there is little advantage in growth via a switch instead of pure nucleation of the solid, since now both pathways become plausible. The small difference in free energy shown in panel a of Figure 4 predicts that with both pathways operative, the rate is 1.8 times that of the cross‐over step alone. The dashed line shows what one expects from cross‐over nucleation alone, and indeed the experimental value at the lowest point is 1.6 times the crossover expectation.

Figure 4.

Free energy barriers deduced for lysozyme at 13°C. The liquid clusters grow almost linearly, while the solid clusters display a clear maximum. Panel (a) is for a supersaturation of 14, which corresponds to c = 1.17 mM at 13°C. Panel (b) is for supersaturation of 29 which corresponds to c = 2.55 mM at 13°C. In panel (b), the intersection of the free energies effectively lowers the barrier for crystal formation by 1.7 kT (or 0.57 kcal/mol). This would speed up the rate by 5.5 times. In panel (a), the intersection is almost at the peak of the crystal nucleation barrier, and the acceleration of crystal formation is only around 22%, making both nucleation and growth via liquid intermediate almost equally likely.

Discussion

The model we have presented provides insight into the nucleation of lysozyme. The molecular parameters derived in this analysis are logical and self‐consistent, even though nothing requires this a priori. The solid's contact energy is −8.2 kcal/mol, while the liquid's mean contact energy is −2.0 kcal/mol. This liquid value is weaker, just as expected. The vibrational contributions (at 13°C) are −25.4 kcal/mol for the crystal, and −28.1 kcal/mol for the liquid. The liquid, as expected, has greater motional freedom in its condensed phase than the solid.

Sickle hemoglobin is currently the only other assembly system that has been analyzed in this framework. While the contact energies would be expected to be substantially different than lysozyme due to the nature of the molecules' mutual interactions, the free energy from vibrations might be expected to have similar values. HbS polymers have a μ_PV = −25.5 kcal/mol at comparable temperatures, a value almost identical to that of lysozyme as deduced here.21 In a sickle hemoglobin mutant which forms crystals instead of polymers, HbCHarlem, the best estimate is that μ_PV = −18.7 kcal/mol, though the exact temperature dependence for that molecule has not been determined and so this represents extrapolation based on results of HbS. The consistency of these vibrational chemical potentials is all the more remarkable given that they represent liquids, crystals, and polymers.

The interplay of liquid phase intermediates in the generation of a crystal structure creates interesting complications for attempts to study intermediate structures. The intermediates will necessarily look rather disordered, if somehow they have been immobilized either in time (by capturing images) or by some technique such as rapid freezing.

Perhaps the most interesting finding of this model, however, is the ability of a covert liquid‐liquid transition to influence a solid phase nucleation process. A characteristic that should help to distinguish such processes is the degree of linearity versus curvature of log‐log plots of the rates of nucleation as a function of concentration, since the cross‐over from liquid to solid is predicted to be completely linear. It is interesting to observe that, were the system of lysozyme not so fully investigated, the presence of the liquid phase might be unknown, since it lies a good two‐fold above the range used for nucleation rate measurement. This would have left its contribution undiscovered, and the nucleation process incorrectly analyzed via straightforward solid cluster models. This suggests that investigating nucleating systems for such labile disordered phases could provide unexpected insights.

Supporting information

Supporting Information 1