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Origin of Anomalous Mesoscopic Phases in Protein Solutions Weichun Pan,† Peter G. Vekilov,†,‡ and Vassiliy Lubchenko*,‡ Department of Chemical and Biomolecular Engineering, UniVersity of Houston, Houston, Texas 77204-4004, and Department of Chemistry, UniVersity of Houston, Houston, Texas 77204-5003 ReceiVed: January 21, 2010; ReVised Manuscript ReceiVed: March 15, 2010
Long-living mesoscopic clusters of a dense protein liquid are a necessary kinetic intermediate for the formation of solid aggregates of native and misfolded protein molecules; in turn, these aggregates underlie physiological and pathological processes and laboratory and industrial procedures. We argue that the clusters consist of a nonequilibrium mixture of single protein molecules and long-lived complexes of proteins. The puzzling mesoscopic size of the clusters is determined by the lifetime and diffusivity of these complexes. We predict and observe a crossover of cluster dynamics to critical-like density fluctuations at high protein concentrations. We predict and experimentally confirm that cluster dynamics obey a universal, diffusion-like scaling with time and wave vector, including in the critical-like regime. Nontrivial dependencies of the cluster size and volume fraction on the protein concentration are established. Possible mechanisms of complex formation include domain swapping, hydration forces, dispersive interactions, and other, system-specific, interactions. We highlight the significance of the hydration interaction and domain swapping with regard to the ubiquity of the clusters and their sensitivity to the chemical composition of the solvent. Our findings suggest novel ways to control protein aggregation. Introduction The cooperative behaviors of protein molecules in solution are essential for their function in live cells and for a number of laboratory and technological procedures.1-4 In addition to forming crystals, protein solutions exhibit liquid-liquid separation into protein-rich and protein-poor phases, usually at conditions supersaturated with respect to crystallization.5-8 Surprisingly, mesoscopic clusters of a protein-rich liquid have been recently observed with several proteins well outside the regime of macroscopic liquid-liquid separation and even above the crystal solubility line9-11 (see Figure 1a). The clusters measure as much as 102 protein sizes across, contain 105-106 protein molecules, and exhibit lifetimes of the order of seconds. Even though the dense liquid clusters contain only a small fraction of the total proteins10-6 to 10-4sthey are of interest because they are the predominant sites of nucleation of protein crystals.12 Similarly, protein aggregates involved in several pathologies, such as the hemoglobin polymers in sickle cell anemia13,14 or fibrils of misfolded proteins in several neurological disorders,15,16 likely initiate within these dense phase clusters. To control the formation of the protein crystal and aggregates, it is imperative to establish how the clusters form.6,17-20 Previous studies of clusters in protein solutions have suggested, by analogy with colloidal suspensions,21-24 that the puzzling cluster size results from a competition of short-range dispersion forces and longer-range, screened-Coulomb repulsion. Current estimates23,25 show such clusters would be gel-like objects whose size is determined by the particle charge and the solution’s ionic strength. At the low charge values typical of proteins, such assemblies would not exceed tens of molecules,23 see analysis by Hutchens and Wang.22 This scenario has been applied to explain cooperative behavior in lysozyme solutions * Corresponding author. E-mail:
[email protected]. † Department of Chemical and Biomolecular Engineering. ‡ Department of Chemistry.
with a characteristic length corresponding to a few protein sizes.21,26,27 The mesoscopic clusters of interest in this work are much larger; in addition, they are fluid internally, not gel-like.9-11,28 In further contrast with the electrostatic scenario, the mesoscopic clusters are observed with nearly neutral proteins, such as hemoglobin at near-physiological pH.9-11,29 In addition, the typical protein-protein spacing in the dense phase, 6-10 Å (see Supporting Information), exceeds the short range expected of solvent-mediated dispersive interactions. These contrasting notions suggest the interplay between electrostatic repulsion and short-range attraction is not a key factor in the formation of mesoscopic clusters. An alternative a priori explanation for the puzzling size of clusters is that they are micelles. However, since the protein concentration in the bulk solution continues to increase upon adding more protein even after clusters appear,9-11,29 this possibility can be ruled out. In contrast to those scenarios, here we propose a novel microscopic mechanism, by which the clusters are long-lived, compact droplets of a mixture of protein molecules and transient protein complexes. This mixture exhibits a significantly higher protein concentration than the bulk solution. Because of the high free energy cost of such large concentration fluctuations, the clusters contain only about 1 ppm of the total protein. The lifetime of an individual cluster is on the order of one second and is sufficiently long for the cluster to contribute to light scattering. The protein-rich liquid comprising the clusters is an off-equilibrium, spatially inhomogeneous mixture of protein monomers and transient complexes of proteins. This mixture deviates from equilibrium in a very particular, space-dependent fashion. The cluster size and the decay rate of the complexes are inherently related. The proposed scenario implies that the protein complexes are much more stabilized at protein concentrations typical of the cluster core, i.e., ∼450 mg/mL, than in the bulk solution. Although clusters consist of an off-equilibrium, spatially inhomogeneous mixture, they are in equilibrium
10.1021/jp100617w 2010 American Chemical Society Published on Web 04/27/2010
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Figure 1. Clusters and the phase diagram of the protein solution. (a) The phase diagram of a lysozyme solution determined experimentally at composition specified in the plot. Liquidus, or solubility (from refs 17 and 18), liquid-liquid (L-L) coexistence and respective spinodal (from ref 19), solution-crystal spinodal (from ref 20), and gelation line (from refs 6 and 19). The gray area denotes compositions in which metastable mesoscopic clusters were detected. Solidus line is at ∼800 mg/mL in both a and b. (b) Likely phase diagram of lysozyme solution at composition specified in the plot and used in all experiments reported here. The low temperatures of the liquidus and L-L coexistence lines are a consequence of the absence of additional precipitant and low buffer concentration: lowering the concentration of electrolytes brings all phase lines to lower temperatures.6 The horizontal gray line denotes the conditions of the present study. The gray dot marks the likely composition of dense liquid clusters, FH ) 450 mg/mL.
with the bulk solution. There is no net exchange of protein monomers or protein complexes between clusters and the solution. The proposed microscopic picture yields a number of testable predictions that are not sensitive to the precise nature of the protein complexes. We predict that the cluster radius should not depend on the protein concentration. We argue that relatively well-defined clusters should exist in a window of bulk protein concentration: below a threshold bulk concentration, ∼100 mg/ mL, adding protein to the solution should lead to an increasing volume fraction of clusters, while their size remains fixed. Further increasing the bulk concentration is expected to eventually render the density fluctuations critical-like, whereby the clusters no longer have sharp boundaries. We also predict that at these high concentrations light-scattering autocorrelation functions should exhibit a power law portion scaling as 1/τ (τ is decay time). We further establish that at all protein concentrations, including in the near critical regime, cluster shape fluctuations will exhibit a diffusion-like scaling of the relaxation time with the wave vector: q2τ ) const (q is wave vector). This finding implies that at a given protein concentration the autocorrelation functions should fall on the same master curve upon rescaling the time axis with the square of the scattering wave vector. To test the present microscopic scenario, we employ static and dynamic light scattering (SLS and DLS) to characterize aqueous solutions of hen egg white lysozyme at concentrations up to 334 mg/mL and in the presence of additives that enhance or suppress certain molecular interactions. We obtain good agreement between the observed solution behaviors and the above predictions. Finally, we discuss the mechanism of complex formation and conclude that there are several possible contributions to this mechanism. We specifically test the contributions of partial protein unfolding and hydration interaction and find that they are major determinants of the cluster properties.
Seikagaku, Japan, 6× crystallized), robustness allowing for data collection over many hours and quantitative reproducibility of independently performed experiments, and an extensive available database of physicochemical properties. Lysozyme solutions were prepared without additional purification in 20 mM NaHEPES (N-2-hydroxyethylpiperazine-N′-2-ethanesulfonic) buffer at pH ) 7.8. The ionic strength in this buffer is ∼30 mM, due to dissociation of the amino and sulfonic groups of HEPES. All experiments were carried out at 22 °C. The solutions are prepared from freeze-dried lysozyme at concentrations up to 334 mg/mL, near the solubility limit of the dry powder. Despite the apparent presence of short-range attraction favoring precipitation, the solutions are stable for at least six months, even when refrigerated to 4 °C, a condition favoring the formation of the dense phase (see Figure 1). Dynamic Light Scattering (DLS) from Protein Solutions. Light scattering data were collected with an ALV goniometer equipped with a He-Ne laser (632.8 nm) and ALV-5000/ EPP Multiple tau Digital Correlator (ALV-GmbH, Langen, Germany). Prior to loading in the cuvette, the solutions were filtered through 0.22 µm Millipore filters. Intensity correlation functions were acquired at 60 s at angles ranging between 57 and 140° (for definitions and further details, see ref 9). When the correlation functions indicated the presence of clusters, the mean cluster radius R, volume fraction occupied by all clusters φ2, and number density n2 were determined as discussed in ref 9. We note that lysozyme solutions are completely transparent at 632.8 nm, making DLS a convenient structural probe even at the high protein concentrations employed in this work. Static Light Scattering (SLS) Characterization of the Osmotic Compressibility. The osmotic compressibility was measured with the same ALV device used for DLS. The range of concentrations used was 5-300 mg mL-1. The data were cast as Debye plots of the scattered intensity I using experimentally determined Rθ, F pairs31,32
Materials and Methods Solution Preparation. All experiments reported here were carried out with the protein lysozyme. It offers the advantage of ready availability as a highly purified preparation30 (from
MwKF ) 1 + 2B2MwF + 3B3Mw2F2 + 4B4Mw3F3 Rθ
(1)
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Figure 2. Evaluation of the free-energy density of the protein solution. Debye plot of the MwKF/Rθ ratio as a function of protein mass concentration F. Mw ) 14 300 g/mol is the molecular mass of lysozyme; K is the instrument constant; and Rθ ) Iθ/I0 is the Raleigh ratio of the intensities of light scattered at angle θ and incident light. Points: determination using static light scattering as discussed in ref 32. Dashed line: fit of osmotic virial expansion to data. Solid line: integration of data according to eq 2 to determine the solution free energy ∆G.
Here Rθ ) Iθ/I0 is the Rayleigh ratio of the scattered to the incident light intensity; K is a system constant defined as K ) ˜ 0/λ2)2(dn˜/dF)2 (ref 33), where n˜0 ) 1.331 is the refractive N-1 A (2πn index of the solvent at the wavelength of the laser beam λ from ref 34; NA is Avogadro’s number; and dn˜/dF is the derivative of the refractive index n˜ with respect to protein mass concentration F. Mw is the protein molecular mass. Determinations of dn˜/dF were carried out using a Brookhaven Instruments differential refractometer operating at a wavelength of 620 nm.11,32 Results and Discussion Clusters Consist of a Nonequilibrium Mixture of Protein Monomers and Higher-Order Species. We first demonstrate that the clusters contain transient species other than single protein molecules. Let us first suppose the opposite, i.e., that the clusters represent metastable droplets of an equilibrated protein solution at a higher concentration, similar to liquid droplets in a nearly saturated vapor of that liquid. We will see that this assumption, combined with the measured excess free energy of the “metastable phase”, implies the clusters should be much smaller than is observed. Since clusters are observed outside the stability region of the dense protein liquid,9-11 increasing protein concentration is expected to lead to a free energy excess, ∆g˜, per protein molecule. To evaluate ∆g˜, we use the dependence of the KF/Rθ ratio on the protein mass concentration F, shown in Figure 2. This ratio is measured using static light scattering and is directly related to the inverse osmotic compressibility of the solution: KF/Rθ ) (∂Π/∂F)/ RT (Π is the contribution of the protein to the osmotic pressure, and R is the universal gas constant). The osmotic compressibility can be integrated from 0 to F to compute the osmotic pressure Π of the protein subsystem. The osmotic pressure can be further integrated in density to obtain the free energy ∆G ) -∫FFLHΠdV + ∆(ΠV) needed to increase the concentration of N protein molecules from its value in the dilute solution FL to that in the dense liquid FH. The resulting free energy difference between states with densities FL and FH, per particle, is
∆G(FL, FH) ∆g˜ ≡ ) kBT NkBT
∫FF
H
L
dF F2
( ) [∫ ( ) ]
dF′ + ∆ ∫0F Mw KF′ Rθ 1 F
F
0
Mw
KF′ dF′ Rθ
FH
FL
(2)
where Mw is the molar mass of the protein. We measure the KF/Rθ ratio in Figure 2 for concentrations of up to 300 mg mL-1, which is near the apparent solubility limit of the dry protein powder. The resulting concentration dependence of the KF/Rθ ratio is fitted by a cubic polynomial, eq 1. Possible ambiguities in extrapolating ∆G(FL,FH) to densities above 300 mg/mL, the upper limit of the KF/Rθ(F) data, are partially offset by the decaying 1/F and 1/F2 terms in eq 2. Finally note that since the expression above assumes the solution is equilibrated, it gives a lower bound on the actual free energy excess of the solution comprising the clusters. A standard nucleation theory argument (see Supporting Information) shows that the size of a droplet of a metastable phase does not exceed 6kBT/∆g˜. In all considerations below, we choose FH ) 450 mg/mL, a reasonable estimate for lysozyme solutions.19 The resulting ∆G/NkBT, shown in Figure 2, varies between 10 and 0 for the values of the bulk concentration FL ranging between 100 and 450 mg/mL, respectively. Since ∆g˜ ≡ ∆G(FL,FH)/ N, this estimate implies 6kBT/∆g˜ < 1; i.e., a cluster could contain at most one molecule at the bulk concentration 100 mg/mL. This result is in dramatic disagreement with the observed value of 105-106, indicating that the mesoscopic clusters do not represent droplets of an equilibrated phase consisting of single protein molecules. The same conclusion can be reached using kinetic considerations, based on the above estimate of ∆g˜ . In view of the large size of the protein molecules, their motion is strongly overdamped, allowing one to consider a diffusion scheme in the spirit of Lifshitz and Slyozov.35,36 Within the latter framework, the single-molecule exchange between the solution bulk and a cluster is presented as a sum of two contributions: the first contribution is the regular Fickian diffusion, which stems from chemical potential variations and results in a molecular flux scaling inversely proportionally with the interface curvature, 1/R. Since ∆g˜ > 0, this Fickian contribution favors evaporation. The second contribution, which scales as 1/R2, stems from the surface tension between the phases and always results in an outward flux, even if the dense phase is thermodynamically favorable. We conclude that clusters of monomeric proteins must always evaporate, consistent with the above conclusions from nucleation theory. The thermodynamic and kinetic arguments above are strictly valid when the droplets of an alternative phase are not too small. As a result, these arguments do not strictly apply to the formation of very small transient complexes in the bulk solution, such as those proposed in refs 21, 26, and 27; however, they do suggest such small complexes would be unstable. We thus conclude that for clusters to exist they must contain other protein-containing species. To inquire whether protein complexes that are in equilibrium with single protein molecules could stabilize the clusters, let us suppose, for concreteness, that the complexes are dimers. In equilibrium, the chemical potentials of the monomers and dimers, µ1 and µ2, respectively, must obey the relation 2µ1 ) µ2. As a result, the total Gibbs energy of an equilibrium mixture can be expressed through the total amount of protein in the solution and the chemical potential of the single protein molecules: G ) µ1N1 + µ2N2 ) µ1N. Here, N1 and N2 are the number of monomers and dimers, respectively, and N ) N1 + 2N2 is the total number of protein molecules in the solution. One may similarly show that the expression G ) µ1N remains valid, if other higher-order protein oligomers are present, so long as the solution is in chemical equilibrium. Since µ1 ) G/N, the G(F)/NkBT curve from Figure 2 dictates that the chemical potential of the monomer in an equilibrium solution
Origin of Mesoscopic Phases in Protein Solutions always grows with concentration. Consequently, the chemical potentials of any higher-order species, too, would increase because they are proportional to µ1, analogously to the dimer. Thus the formation of equilibrium complexes does not stabilize the mesoscopic clusters and could not account for their formation. Lysozyme is known to form a covalently bound dimer. The dimer concentration in the Seikagaku preparation used here is about 0.5%.30 This permanent dimer does not accumulate in the dense liquid, and hence its presence cannot affect the G(F) curve in Figure 2. Hence, the covalent lysozyme dimer does not stabilize the clusters. Furthermore, mesoscopic clusters have been observed in solutions of three hemoglobin variants9 and the protein lumazine synthase,11 none of which forms any permanent oligomers. We conclude that covalently bound complexes or other permanent oligomers do not stabilize mesoscopic clusters. Clusters Consisting of a Nonequilibrium Mixture of Monomers and Complexes: Thermodynamics. Since the clusters cannot consist of pure monomers or of an equilibrium mixture of monomers and complexes, we conclude that the clusters contain an off-equilibrium mixture of monomers and transient protein complexes. The presence of transient complexes in lysozyme solutions is consistent with the observed viscoelastic response of high concentration homogeneous solutions in 0.1 M acetate.37 The lack of chemical equilibrium means that the rate of complex formation is not equal to the rate of complex decay. Thus, to obey the mass conservation requirement, there should be monomer excess in some parts of the cluster and monomer depletion in others. Before we explicitly demonstrate these findings, in the next subsection, let us briefly review how particle exchange between an open system and its environment shifts the chemical equilibrium in that open system, while increasing its Gibbs free energy. Consider, again, an equilibrium mixture of monomer and dimer, whereby 2µ1 ) µ2. Now imagine removing the monomer from and adding a dimer to the mixture, at a steady rate, so that the total amount of protein is conserved: N1 + 2N2 ) const. The faster the rate of monomer-removal/dimer-addition, the greater the shift of the chemical equilibrium toward a lower fraction of the monomer, accompanied by a free energy increase ∆G (see Figure 3). The latter increase can be easily expressed through equilibrium osmotic compressibilities of the monomers and the complexes, when the deviation ∆N1 of the monomer number from its equilibrium value is small: Taylor expanding 2 ) 2(∂G/ the free energy around its minimum yields ∆G ≈ Σi)1 ∂Ni)∆Ni + Σ2i,j)1(∂2G/∂Ni∂Nj)∆Ni∆Nj/2 ) (∆N1)2(∂µ1/∂N1 + ∂µ2/ 4∂N2 - ∂µ2/∂N1)/2, where we have used µi ) ∂G/∂Ni and ∆N1 ) -2∆N2. The derivatives are evaluated at equilibrium, ∆N1 ) 0; the first-order term is, of course, equal to zero. To estimate the rate of monomer-withdrawal/dimer-addition needed to create a concentration shift ∆N1 ) -2∆N2, we must make a specific assumption on the kinetics of monomer binding/ dimer decay. Suppose, for simplicity, that these kinetics are simple bimolecular with concentration-independent rate constants, n˙1 ) -k1n12 + k2n2, and that the solution is perfectly mixed. Here, n1 ≡ N1/V and n2 ≡ N2/V are the concentrations of the monomer and dimer, respectively, and V is the solution volume. At a small deviation ∆N1 ) -2∆N2 from equilibrium, the monomer is produced, in the solution, in excess at a rate N˙1 ) -(2k1n1 + k2/2)(∆N1). We thus conclude that if one withdraws the monomer steadily at this rate, while replenishing the dimer at rate N˙2 ) -N˙1/2, the chemical equilibrium will be shifted to a new steady state, in which the dimer is in chemical
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Figure 3. Effect of particle exchange on the shift of chemical equilibrium. The shift of chemical equilibrium upon an externally imposed monomer depletion and dimer excess, in a monomer-dimer mixture. The black dot denotes the equilibrium number Neq 1 of the monomer in the absence of the exchange. The red dot denotes the steady-state value Nst1 of the monomer number in the presence of an exchange. The corresponding free energy excess is indicated with ∆G. The effective free energy of the shifted equilibrium, shown in green, is the sum of the equilibrium free energy and the effective bias, -(µ1 - µ2/2)N1 + const, arising from the externally imposed particle exchange. In the shifted steady state, system fluctuations occur subject to this effective free energy profile. See also the discussion in the text.
excess. Under these conditions, 2µ1 < µ2 and ∆G > 0, as already mentioned. The system fluctuates around the new steady state subject to an effective thermodynamic potential Gext ) G + µextN1, shown by the green line in Figure 3. The coefficient µext, which amounts to an externally imposed thermodynamic force, can be determined from the condition ∂Gext/∂N1 ) 0, whereby we obtain µext ) -(µ1 - µ2/2) (see Figure 3). This construction is entirely analogous to how the volume of a system with Helmholtz free energy F is determined by external pressure, so that in equilibrium, where ∂(F + pextV)/∂V ) 0, pext ) -∂F/∂V. It is essential to recognize that in the shifted steady state there is no free energy dissipation, even though the concentrations deviate from their equilibrium values, because the number of ˙ ) Σi(∂G/∂Ni)N˙i ) 0. Furthermore, since all species is steady: G the number of dimers specifically is steady, the number of monomer-monomer “bonds” is time-independent as well, ˙ ) 0. As a result, there is implying the enthalpy is constant: H ˙ ˙ ˙ )/T ) 0. Only after the no entropy production: S )(H - G external force is removed will the monomer number relax to its equilibrium value, and the entropy will grow. Conversely, to create the shift from the equilibrium values, in the first place, the entropy of the mixture was decreased at the expense of work by the external force. Clusters Consisting of a Nonequilibrium Mixture of Monomers and Complexes: Kinetics. Here we present the simplest possible reaction-diffusion scheme that couples protein complex formation and decay with the transport of all participating species. We assume for concreteness that the protein complexes are dimers; incorporating higher-order species into the analysis would yield analogous results. At the cluster periphery, where both the deviation from equilibrium and concentration gradients are small, one can write the following diffusion/reaction scheme in the steady state
0 ) D1∇2n1 - k1n21 + k2n2
(3)
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0 ) D2∇2n2 + k1n21 - k2n2
Pan et al.
(4)
where n1 and n2 are the number densities of the monomer and dimer, respectively, while D1 and D2 are the corresponding diffusion constants. The concentration-independent rate constants k1 and k2 denote, respectively, the rates of dimer formation and decay. The monomer and dimer concentration far from the cluster are n1out and n2out, respectively. Sufficiently far from the cluster, where there is no net flux of any species, the solution is in local chemical equilibrium: k1(n1out)2 ) k2n2out. Close to equilibrium, the deviations of the concentrations from their equilibrium values, (n1 - n1out) and (n2 - n2out), are small. Equations 3 and 4 can be linearized by keeping only the firstorder terms in these deviations. The corresponding general solution of the linearized equations in spherical geometry is -r/R n1 ) nout /r] 1 [1 + Ak2(R/r) - Be
(5)
out out -r/R n2 ) nout /r] 2 + n1 [A2k1n1 (R/r) + B(D1 /D2)e
(6) The parameter -1/2 R ≡ (2k1nout 1 /D1 + k2 /D2)
(7)
has units of length. The parameters A and B are arbitrary constants determined from the boundary conditions. Direct inspection of eqs 5 and 6 shows that it is possible to make only one, but not both functions n1(r) and n2(r), finite at rf0, implying that the full treatment, which would apply also to strong deviations from equilibrium, must account for the concentration dependence of the rate constants, see also below. Let us turn to the analysis of the solutions in eqs 5 and 6, which do apply sufficiently far from the cluster center, where the deviations from equilibrium are small. The terms containing the coefficient A in eqs 5 and 6 are always of the same sign and account for the outflow of monomer and dimer from the cluster in which the concentration of both of these species is higher than in its environment. Because both monomer and dimer fluxes are directed outward, this type of solution can be steady state only when an external source is present. Since there is no external source, such a cluster would always decay. Alternatively, this could be seen by evaluating the total flux stemming from the terms at the coefficient A. Since r2∂(1/r)/∂r ) const * 0 at any value of r, however large, the presence of the term would contradict the original premise of equilibrium sufficiently far from the cluster. We must thus set A ) 0. The terms at the coefficient B represent a fundamentally different solution, in which the concentrations of the two species change in opposite directions. These terms could describe a situation in which the monomer concentration decreases inward, within a compact region of radius R, causing a net influx of monomer within this compact region (see Figure 5a). At the same time, the dimer concentration increases toward the center, causing a net outflow of the dimer. As a result, a spatially nonuniform steady state, which obeys mass conservation, is possible without an external sink/source. An essential feature of the coefficient B-containing terms is that unlike the scale invariant term 1/r at the coefficient A they exhibit a finite length scale R defined in eq 7. The parameter R represents the
characteristic length of decay of the dimer concentration n2 and increase of the monomer concentration n1 toward their values in the solution bulk; this length thus must be identified as the cluster radius. In view of the low concentration of dimers in the bulk solution, where k1n1out/k2 ) n2out/n1out , 1, eq 7 yields
R ≈ (D2 /k2)1/2
(8)
Equation 8 implies that the cluster radius is determined by the lifetime of the dimer and its diffusion constant in the bulk solution. A similar conclusion would be reached if we included in the treatment reversible formation of higher-order species, such as trimers, etc. In the latter case, the radius of the cluster would be determined by the slowest decaying species, provided its mobility is not too low. The result in eq 8 is not surprising: the cluster radius is limited by the lifetime and diffusivity of complexes because complexes must have a chance to escape from the cluster before they decay. The kinetic scheme from eqs 3 and 4 is consistent with the proposed picture of a steady state cluster also in that the cluster radius is stable against small perturbations. Indeed, let us substitute R ) R0 + δR(t) into eqs 5 and 6 and use the general, time-dependent version of eq 3: n˙1 ) D1∇2n1 - k1n12 + k2n2. This yields, upon linearization, a simple first-order equation for the time-dependent fluctuation δR(t) of the cluster radius from its typical value
δR˙ = -(2D1 /R2)δR
(9)
The latter equation implies the cluster radius fluctuates around a metastable minimum with a relaxation time
τ)2
R2 D1
(10)
The more difficult questions of the nucleation of a new cluster and its eventual decay will not be addressed here. Diffusion-Reaction Scheme near the Cluster Center. To derive a kinetic scheme, such as in eqs 3 and 4, that would apply near the cluster center, we need to know the precise concentration dependence of the rates of complex formation and decay in the crowded solution within the cluster. This task is difficult, in view of the lack of detailed knowledge of complex formation, stabilizations of specific intermediates, and their transport characteristics; this is work in progress. Here, we limit ourselves to an ad hoc phenomenological kinetic scheme, which is still sufficient to explicitly demonstrate that for the present picture to apply, the rate of monomer binding near the center of a cluster should significantly exceed the decay rate of complexes in the bulk solution. First note that one may neglect the concentration dependence of the rate coefficients when concentration variations are small, even if the concentrations themselves are not small. Let us then consider a sufficiently small region near the cluster center so that the latter condition is satisfied. In this small region, we write a general reaction-diffusion scheme for a mixture of protein monomers and protein complexes, with concentrationindependent rate coefficients. Decay of higher-order species will ni ≡ k2inn, where replenish the monomer with a rate Σi)2,...kdecay i ni (i ) 2, ...) denotes the concentrations of those species and n (ni/ ) Σi)2,...ni is their total concentration. The rate kin2 ≡ Σi)2,...kdecay i
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n) is the average decay rate of all higher-order species that results in monomer production. The monomer depletion, due to the creation of higher-order species, is a sum over the respective contributions of all those species and the monomer itself: 1 ni) ≡ kin n1. There is also a contribution to the local n1(Σi)1,...kbind i monomer mass balance that stems from the monomer exchange with adjacent regions: D1in∇2n1. The resulting kinetic equation is in in 2 n˙1 ) Din 1 ∇ n1 - k1 n1 + k2 n
(11)
The mass balance for the other species can be written analogously. In steady state, eq 11 yields in in 2 0 ) Din 1 ∇ n1 - k1 n1 + k2 n
(12)
The above equation can be solved by Taylor expanding the concentration profiles in terms of r2 and determining the higherorder coefficients iteratively from the lower-order coefficients. In the lowest order approximation, the solutions are given by 2 ) and n(r) ) nin, where the functions n1(r) = n1in(1 + r2/2Rin coordinate-independent constants n1in and nin denote the r ) 0 values of the concentrations of the monomer and the higherorder species, respectively. The parameter Rin has units length and corresponds to the characteristic length scale of the n1(r) profile at small r. Substituting the expansions of n1(r) and n(r) in eq 12 directly yields in in in in -1/2 Rin ≡ [(kin 1 - k2 n /n1 )/3D1 ]
(13)
This equation self-consistently demonstrates that a finite-size cluster must consist of a nonequilibrium mixture: k1inn1in * k2innin and more specifically that the monomer is in excess in the cluster core: k1inn1in > k2innin. To make compatible the disparate descriptions at the cluster periphery and center given, respectively, by eqs 3 and 4 and eq 12, we must require Rin < R, where R is the cluster radius from eq 8. This requirement implies, by eqs 8 and 13, that the rate of monomer binding near the center of a cluster should significantly exceed the decay rate of complexes in the bulk solution
kin 1 . k2
(14)
In other words, protein complexes must be stabilized at high protein concentrations. Note that the condition in eq 14 automatically insures that total monomer consumption in the cluster core, due to binding, exceeds the net monomer influx from the periphery. Indeed, the former quantity is about (4π/ 3)R3n1k1in, while the latter is roughly 4πRDn1. In view of eqs 8 and 14, (4π/3)R3n1k1in . 4πRDn1. Internal Consistency of the Theory. In Figure 4b, we sketch the spatial variation of the local chemical potentials that is consistent with (a) the excess of the dimer at the cluster periphery (r > R in Figure 4) and excess of the monomer in the cluster core (r < R) and (b) stationary influx of the monomer and outflow of the dimer, both in the outer and core regions of the cluster. The resulting shape of µ1 is rather peculiars reminiscent of the Schottky barrier38sand implies that to reach the cluster’s center a monomer must surmount a barrier. The exit of complexes from clusters may or may not be subject to
Figure 4. Spatial distribution of concentrations n, chemical potentials µ of monomers and complexes, and the total free energy G. (a) Schematic of concentration profiles in a cluster. nL: total protein concentration in the bulk solution. nH: total protein concentration in the cluster core. r: distance from center of cluster. R: cluster radius. (b) Corresponding profiles of the local chemical potentials of the monomer µ1 and the complex µ2 assumed to be a protein dimer for concreteness. Superscript “out” denotes the chemical potentials of the monomer and dimer in the bulk, where 2µ1 ) µ2. The lines are broken in the interface region to avoid the ambiguity of defining a thermodynamic potential in a mechanically unstable phase. (c) Free-energy profiles in a cluster. The black dashed line denotes the equilibrium free energy G(r) ) G[F(r)], for a fixed number of protein molecules. The red dashed line denotes the free energy of the actual nonequilibrium mixture of monomers and complexes. As in Figure 3, we illustrate the deviation (or lack thereof) from chemical equilibrium near the cluster center, in the cluster periphery, and in solution bulk, in the provided sketches of the local free energy as a function of the monomer fraction.
a small barrier. It is essential that the obtained steady state concentration profiles for the monomer and complexes deviate from the uniform distribution only within a compact region, consistent with the rapid decay of the function e-r/R/r from eqs 5 and 6 for large r. Note that both the regions on the left- and right-hand side of the line r ) R in Figure 4 must be considered part of the cluster. While there is a net exchange of monomers and complexes between the outer region and the core of a cluster, there is no such exchange between clusters and the bulk solution. Indeed, at large distances from the cluster center, r . R, the net particle fluxes decay exponentially rapidly with r, namely, proportionally to e-r/R up to a slower, power-law factor.
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We thus observe that although the mixture inside the clusters is off-equilibrium, the clusters themselves are in equilibrium with the bulk solution. The protein solution as a whole is at equilibrium as well: although clusters spontaneously form and decay, their number is on average steady (see below for the discussion on the cluster lifetime and volume fraction). In view of this interplay of nonequilibrium and equilibrium behaviors exhibited by clustercontaining solutions, it seems instructive to review the cluster solution from eqs 5 and 6 from the perspective of the first and second laws of thermodynamics. The discussion of Figure 3 indicates that in steady state both these laws are indeed obeyed. One may monitor energy conservation from yet another angle, by using Figure 5b: Take, for instance, two monomers in the bulk solution, “pull” them inside the cluster, bind them there, “extract” the resulting dimer from the cluster, and finally break the dimer into the monomers. Likewise, one can outline any thermodynamic cycle of interest, whereby the free energy can be seen explicitly to be a state function. The conformity with the second law in clusters is, perhaps, less obvious because the solution is nonuniform on time scales shorter than the cluster lifetime. Imagine, for instance, an infinitely large cluster, whereby cluster core and cluster periphery would be macroscopic. The detailed balance39,40 is hereby violated, as there are steady net fluxes of monomer and dimer across a macroscopic interface. In contrast, compact finitelifetime clusters do obey the detailed balance since the spatially/ temporally averaged fluxes are zero. In addition, the clusters are mobile. Note that the theory provides an inherent test of its compliance with the second law: According to eq 8, an infinitely large cluster would require that dimers be infinitely long-lived. This would imply lack of equilibrium, in that all the monomers would eventually bind to form permanent dimers. Finally, we note that the free energy penalty due to the protein solution inside clusters being off-equilibrium is part of the free energy cost of the clusters, see below. Cluster Size and Lifetime. The experimentally observed cluster radius is typically of the order of 100 nm (Figure 5e). While the above analysis does not afford an a priori prediction of this radius, it does provide a consistency check and an indication of the response of R to variations in the solution concentration. The viscosity of the dilute solution is about 3 cPs,37 resulting in D1 ≈ 10-7 cm2/s.9 Equation 8 thus implies that the decay rate constant of the complexes k2 is of the order 103 s-1; i.e., the lifetime of the complexes is on the order of a millisecond; these figures are reasonable. Furthermore, since the decay rate k2 and the diffusion constant near the cluster edge do not depend sensitively on the protein concentration in the bulk solution, the present theory predicts the radius of the cluster should not either. In comparing this prediction with our data, we observe that the characteristic diffusion time of the shoulder corresponding to the clusters is consistent between different concentrations in Figure 5b-d and f. This consistency implies that, indeed, the cluster size is only weakly sensitive to changes in the bulk concentration. Data on the radii of clusters in solutions of three hemoglobin variants9 and the protein lumazine synthase11 also show the radius is a weak function of solution concentration. Although the relaxation time τ from eq 10 is not the lifetime of a cluster, it nevertheless gives its lower bound at 1 ms. This bound is consistent with the estimates of 15 ms to several seconds for the lifetimes of clusters obtained previously for several proteins.9,11 Universal Scaling of the Autocorrelation of Scattered Light. While eq 10 was derived assuming spherical geometry of the clusters, it also applies to other geometries, where the
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Figure 5. Characterization of clusters by dynamic light scattering. (a)-(d) Time-correlation functions ψ2 of light scattered from lysozyme solutions at pH ) 7.8 in 20 mM HEPES buffer with concentrations indicated in the plots collected at five angles θ, indicated in (a), for each concentration. The angles were chosen so that the corresponding values of q2 cover evenly the range q2 ) 1.5 × 1010 cm-2 at 57° to 6.3 × 1010 cm-2 at 140° (the scattering vector q ) 4πn sin(θ/2)/λ; n, solution’s refractive index; λ, wavelength). The shoulders at τ sin2(θ/ 2) ≈ 10-2 ms correspond to diffusion of single protein molecules; values of ψ2 lower than 5 × 10-5 are affected by noise (for a detailed analysis, see ref 9). As shown in the same ref 9, all features at intermediate times stem from clusters of protein dense liquid. After rescaling the time with sin2(θ/2), all correlation functions fall on the same master curve, indicating the existence of a universal scaling curve for both diffusion of compact hydrodynamic species, i.e., single molecules and clusters, and cluster size relaxation. In (a), (b), and (c), the diffusion of individual protein molecules and relatively well-defined clusters dominates the signal, while in (d), the long-time tail stems mostly from cluster size relaxation, seen here as the nearly power-law portion of ψ2(τ). (e) Time evolution of the average cluster radius R determined from the characteristic time of diffusion τ extracted from the respective correlation functions ψ2(τ), such as those presented in (a)-(d), following procedures discussed in ref 9. Protein concentration is indicated in the plot. Volume fraction occupied by clusters φ2, determined as in ref 9, is indicated. (f) Data from (a)-(d): ψ21/2 are plotted as functions of τ/τE, where τE ) 6πRη/kBTq2 is the characteristic Einstein-Stokes time for a single protein molecule (η is the solution viscosity). A simulated ψ21/2 for single lysozyme molecules is shown. Comparison with experimental ψ21/2 at 80 mg/mL indicates clusters are present even at the lowest concentration probed. Comparison of data at 150 and 228 mg/mL shows that the contribution of the clusters depends on F only weakly for F > 100 mg/mL.
quantity R would now stand for the inverse wave vector of the spatial variations of the protein concentration. The relation between τ and R in eq 10 implies that cluster-boundary relaxation follows the usual hydrodynamic scaling τ-1 ∝ q2 (where q is the wave vector), despite its distinction from the usual diffusion of compact hydrodynamic species. Hence, even in the presence of these additional relaxations, the DLS profiles
Origin of Mesoscopic Phases in Protein Solutions will be universal functions of tq2. This predicted hydrodynamic scaling agrees with the data in Figure 5a-d. Cluster Volume Fraction. We next use the equilibrium measurement of the excess free energy from Figure 2 to estimate the upper bound on the cluster volume fraction. If the clusters consisted of an equilibrium mixture, we could directly use the concentration dependence of the free energy of the protein subsystem from Figure 2 to estimate the proportion of the protein at the concentration typical of the clusters and thereby the volume fraction of the clusters. A sketch of the equilibrium free energy per monomer, as a function of the coordinate, is shown as the black dashed line in Figure 4c. However, the mixture inside clusters is off equilibrium, implying that the free energy of the mixture is higher than its equilibrium value and, consequently, that the cluster volume fraction is lower than would follow from the free energy cost of compressing an equilibrium assembly of protein molecules. The actual, offequilibrium value of free energy of the mixture is shown by the red dashed line in Figure 4c. Thus, at FL < 100 mg mL-1, the fraction of protein contributing to the clusters should not exceed e-10 ≈ 4 × 10-5 since, according to Figure 2, the free energy difference between the solution at density F ≈ 450 mg/ mL, typical of clusters, and the solution at the bulk concentration F ≈ 100 mg/mL is ∆g˜ ≈ 10kBT per particle. The experimentally determined cluster volume fraction (see Figure 5e) is about 10-6, i.e., indeed well below the equilibrium-based estimate. We note that in estimating the cluster volume fraction we must take into account the standard notion of the additional amplification of light scattering by large objects, as explained in detail in the Supporting Information. Conversely, we may infer from these data that the free-energy penalty due to the deviation from chemical equilibrium inside the clusters is a significant portion of the total free energy cost of the clusters, i.e., three to four kBT per particle, at the bulk concentration F ≈ 100 mg/mL. Despite these complications, one should expect a rapid, exponential dependence of the cluster number on the protein concentration. This notion is consistent with the data in Figure 5: Almost no signal other than that from single protein moleculessrepresented by the solid line in Figure 5fsis seen at 80 mg/mL, whereas already at 150 mg/mL the cluster shoulder is well above the noise level. Somewhat surprisingly, the cluster contribution to the DLS signal does not increase for higher protein concentrations, similarly to observations with other proteins. We see this explicitly by plotting in Figure 5f the square root of the time correlation functions ψ2(t) from Figures 5a-d, to which distinct scatterers contribute linearly. The likely explanation is that even though the cluster number increases, light-scattering off the clusters becomes weaker with bulk concentration because of the progressively lower concentration gradient across the cluster boundary. Informally speaking, there is less “contrast” at the cluster-solution interface, when the densities of the two phases become comparable. This notion is consistent with another feature of the intensity correlation function at very high bulk concentrations: the nearly power law portion in Figure 5d, which we discuss next. Near Critical Fluctuations at High GL Lead to Power Law Autocorrelations. As FL approaches FH, the surface tension between the solution inside and outside the clusters becomes progressively smaller. As a result, density fluctuations become critical-like.41,42 The view of R from eqs 8 and 10 as the typical wavelength of concentration fluctuations leads to another testable prediction that the universal hydrodynamic scaling with tq2 will persist even into this near-critical regime. Figure 5d shows this is indeed the case. Furthermore, in this near-critical regime, we
J. Phys. Chem. B, Vol. 114, No. 22, 2010 7627 no longer expect the DLS signal to be a simple superposition of two exponential decays corresponding to the diffusion of single proteins and well-defined clusters. The slow time component now arises from long-wavelength, nearly barrierfree density fluctuations. Barrier-free density fluctuations often lead to power-law decay in the density-density correlation r1,t) ≡ 〈δF(r b1,0) δF(r b2,t)〉 whose Fourier function χ(r b2 - b transform is the square root of the correlation function ψ2(q,t) (ref 43). In view of the tq2 scaling of the correlation function ψ2(q,t), χ must depend on time in a particular combination with r ∝ F(rt-1/2)d3(r bt-1/2), so that χ(r,t) the spatial coordinate, χ(r,t)d3b ∝ t-3/2F(rt-1/2), where F(x) is a monotonically decaying function of x. Density fluctuations near a critical point give rise to a scale-free, power-law decay of correlations at large distances: F(x) ∝ x-ν. We may thus expect a power-law portion in the DLS signal at high protein concentrations. What is the value of the exponent ν? As the characteristic size R diverges, density equilibration via diffusion of clusters becomes far less efficient than the direct exchange of molecules between the phases, across the now extensive interfaces. The latter exchange corresponds to the cluster-size relaxation from eq 9, whose contribution scales with cluster area and would contribute relatively little to light scattering when clusters are well-defined. Particle exchange across a 2D surface constitutes one-dimensional diffusive motions that exhibit the standard longtime asymptotics χ(t) ∝ t-1/2, resulting in F(x) ∝ x-2 at large x. This leads to [ψ2(q,t)]1/2 ∝ (qt1/2)-1 ∝ t-1/2; i.e., at sufficiently high concentrations, we should see a 1/t tail in the DLSmeasured function ψ2(q,t), which is indeed borne out by the data in Figure 5d. Mechanism of Complex Formation. We have shown that for clusters to exist, the monomer and complex concentration should change in opposite directions, toward the cluster center, as in eqs 5 and 6. As a result, the complex-to-monomer ratio rapidly increases toward the cluster center, implying the complexes are significantly more stabilized at the high densities typical of clusters than at low densities typical of the bulk solution. We note this increase well exceeds its value expected in nearly ideal mixtures with concentration-independent rate constants for interconversion between distinct species. Even in ideal mixtures, compact species are entropically stabilized at higher densities.44 An elementary calculation however shows that in the presence of such entropic-only stabilization the monomer and complex concentration should always change in the same direction. Conversely, dimer formation by itself is not a sufficient condition for cluster formation. In contrast, the complexes must be stabilized by some additional interactions arising specifically at high protein densities. (To avoid confusion, we note such interactions may also have entropic contributions, such as due to steric effects.) The present theory yields a quantitative criterion for this extra stabilization, as given by eq 14: The unimolecular rate of binding of monomer to any other species in the cluster core should greatly exceed the rate of complex decay in the solution bulk. What is the molecular origin of this density-driven stabilization of the complexes? Toward addressing this question, we have carried out two experiments to test for two potential contributions to complex formation in lysozyme solutions. One such contribution may stem from partial unfolding of the protein molecules and the attraction between the solvent-exposed hydrophobic residues, including the special case of “domain swapping”.45,46 A recent study has demonstrated that dimers of partially unfolded proteins exhibit additional stabilization in a crowded environment.47 Since urea can be used to controllably modify the degree of
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Figure 6. Effects of urea and phosphate and acetate. (a) and (b) Time-correlation functions ψ2 of light scattered from lysozyme solutions at pH ) 7.8 in 20 mM HEPES buffer with concentrations indicated in the plots collected at a scattering angle of 90° in the absence and presence of urea, as indicated in the plots. (c) Evaluation of the free energy ∆G density of the solution in the presence of urea. Solid squares: Debye plot of the MWKF/Rθ ratio as a function of protein mass concentration F in the presence of 0.5 M urea determined by static light scattering as in Figure 2. Dashed line: fit of osmotic virial expansion to MWKF/Rθ data. Solid line: Integration of MWKF/Rθ data according to eq 2 to determine the solution free energy ∆G. Dash-dotted line: ∆G for lysozyme solution without urea from Figure 2 is shown for comparison. (d) Time-correlation functions ψ2 for a lysozyme solution at 150 mg/mL in 20 mM phosphate buffer and in 0.1 M acetate buffer are compared to data in HEPES buffer. Phosphate has ionic strength similar to the HEPES buffer.
folding,48 we have added urea to two solutions: at 80 mg/mL, where few clusters are present, and at 150 mg mL-1, where clusters are well-defined and are unambiguously present. The results, shown in Figure 6, are consistent with partial unfolding of lysozyme contributing to cluster formation: At 80 mg/mL (Figure 6a), well-defined clusters appear upon adding a sufficient amount of urea. On the other hand, at 150 mg/mL (Figure 6b), the cluster volume fraction first declines mildly and then returns to its original value. The initial decline may come from partial disruption of water layering in the interprotein space. Upon further addition of urea, this disruption is compensated by enhanced protein unfolding and binding. The decreased cluster volume fraction at 0.5 M urea implies an increase in the excess free energy ∆g˜ of the dense phase. After employing the same procedure as in Figure 2, we indeed find that upon the addition of urea ∆g˜ increases from ∼10kBT to ∼11kBT. Water structuring66-69 is expected to contribute to cluster formation because at the protein density ∼450 mg/mL, expected of the dense phase, protein molecules are apart by a distance approximately equal to the thickness of two water layers (see Supporting Information). Secondary minima of significant depth exist in the potential of mean force between nanoscopic solutes in aqueous solutions.49-54 These minima are caused by the free energy cost of removal of the second and third hydration layers and are located at 7 and 10 Å from the surface of the protein molecule. Regardless of the detailed electrostatic characteristics of the protein surface, the minima are separated by barriers of up to 10kBT in height. Similar conclusions about the role of water structuring in interactions between protein molecules transpire from analyses of databases of protein-protein complexes,55 protein-DNA interactions,56 and the thermodynamics of protein phase behavior.19,57-59 Note that the depths of these
secondary minima are comparable to those needed to condense a vapor, thus strongly suggesting the importance of hydration interactions. To test the significance of water layering in the interprotein space, we compare clustering behavior in several chemically distinct buffers, i.e., HEPES, phosphate, and acetate. The results, shown in Figure 6d, are consistent with the presence of hydration effects: The clusters are absent in the phosphate buffer, even though its ionic strength is similar to that of the HEPES buffer employed in the majority of the experiments discussed here. Yet the phosphate is a significantly smaller ion and, as such, is more efficient both in binding to the protein surface and in getting within the protein-protein interface regions, resulting in more efficient disruption of water layering in those regions. (Small ions are known to destabilize the secondary minima in the potential of mean force between nanoscopic solutes.54) Similar amounts of phosphate also have been found to significantly enhance repulsion between molecules of another studied protein, lumazine synthase.11 We note that the contribution of water structuring to cluster formation is not independent of partial protein unfolding, in view of the “slaving” of protein conformational dynamics to solvent motions.60,61 While possibly present in lysozyme,62 partial unfolding appears less likely, for example, in tetrameric hemoglobin, suggesting that this specific mechanism is not universal. Our experiments in Figure 6 lend strong support to the significance of partial unfolding to stabilization of clusters in lysozyme solutions. The very long lifetimes of protein complexes, well exceeding several microseconds, are consistent with the possibility that protein conformational motions accompany complex formation. This is the subject of future studies.
Origin of Mesoscopic Phases in Protein Solutions It is also conceivable that the acid-base equilibrium of the residues on the protein surface is shifted in the crowded environment of the cluster, owing to water restructuring or other mechanisms. If this shift causes a decrease in the charge of the protein molecule, it will also result in a decreased Coulomb repulsion between monomers, which would be particularly important in the specific case of lysozyme, a highly charged molecule. Other system-specific effects likely contribute to cluster formation: For instance, hemoglobin solutions exhibit liquid-liquid separation when considerable quantities of PEG are added.63 It is likely that depletion interaction contributes to the effective attraction between hemoglobin molecules.64,65 Clusters and Protein Aggregation. The clusters are crucial precursors for the nucleation of several types of protein aggregates, including:12-16,70 crystals, amyloid fibrils, sickle cell polymer fibers, etc. Uncovering the mechanism of proteincomplex formation and, hence, the emergence of the clusters may help one to control those types of aggregations. For instance, in view of the stabilizing contribution of water structuring to cluster formation, the latter process can be partially inhibited by amphiphilic molecules, such as urea, glycerol, or polyethylene glycol. These bind to or adsorb on the protein surface and thus partially disrupt the hydration layers:71 Conversely, cluster formation may be enhanced by compounds that facilitate protein-protein binding. For instance, free heme, as released from hemoglobin after autoxidation,72,73 is known to augment the attraction between sickle cell hemoglobin molecules. Recent results indicate that indeed, in the presence of free heme (at 10-100 µM), the volume fraction of the clusters increases by 2 orders of magnitude, while the rate of fiber nucleation is enhanced by the same factor.74 The present conclusions likely apply also to protein concentrations significantly below 80 mg/mL, the lowest concentration tested here. Such relatively low concentrations are employed, for instance, in protein crystallization trials. We point out that long-lived mesoscopic clusters have been detected in solutions of the protein lumazine synthase at concentrations as low as 1.3 mg/mL.10,11 Their properties, size, number, volume fraction, and response to temperature and concentration variations, suggest that the lumazine synthase clusters at low concentrations are analogous to the lysozyme clusters in the present study. Conclusions We have shown that the recently discovered mesoscopic clusters of protein-rich liquid in protein solutions emerge as a result of the formation of transient protein complexes stabilized at high protein concentrations. We have put forth a coupled diffusion-reaction scheme, which demonstrates that the cluster size is directly related to the lifetime of the complexes. Several predictions of this scenario of cluster formation have been successfully tested using dynamic and static light scattering. These predictions concern the dependence of the autocorrelation functions of scattered light on time, the wave vector, and protein concentration; the concentration dependence of the cluster size and volume fraction; and the emergence of a power-law portion in the autocorrelation function at sufficiently high protein concentration. The detailed mechanism of complex formation remains an outstanding question; it likely has several contributions, some system-specific as discussed above. We have proposed that one universal contribution to the stabilization of the dense phase stems from water structuring at the protein molecular surface. Other potential contributions include partial protein unfolding and acid-base equilibrium at the protein-protein interface.
J. Phys. Chem. B, Vol. 114, No. 22, 2010 7629 Finally, since the separations between protein molecules in the cytosol of live cells are comparable to those in the dense liquid, the described cooperative effects in protein assemblies add an extra layer of complexity but may serve as handles for controlling cytosol processes. Acknowledgment. We are indebted to Peter G. Wolynes for many inspiring comments and critiques. We thank B. M. Pettitt for discussions of water structuring. The Cover illustration is based on the photograph made by Oleg Galkin and Peter G. Vekilov. We thank Maria Volodina for help with graphics. The present work was partially supported by NSF through Grants CBET 0609387 and MCB 0843726 and the Welch Foundation through Grant number E-1641. V. L. gratefully acknowledges the support from the GEAR and Small Grant Programs at the University of Houston, and the Arnold and Mabel Beckman Foundation Beckman Young Investigator Award. Supporting Information Available: The Supporting Information file contains two sections. The first section explains how to infer the cluster size distribution from the DLS signal. The second section contains an estimate of the dependence of the interprotein distance on the protein concentration. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Eaton, W. A. Biophys. Chem. 2003, 100, 109. (2) Koo, E. H.; Lansbury, P. T., Jr.; Kelly, J. W. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 9989. (3) McPherson, A. Crystallization of biological macromolecules; Cold Spring Harbor Laboratory Press: Cold Spring Harbor New York, 1999. (4) Dodson, G.; Steiner, D. Curr. Opin. Struct. Biol.o 1998, 8, 189. (5) Thomson, J. A.; Schurtenberger, P.; Thurston, G. M.; Benedek, G. B. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 7079. (6) Muschol, M.; Rosenberger, F. J. Chem. Phys. 1997, 107, 1953. (7) Shah, M.; Galkin, O.; Vekilov, P. G. J. Chem. Phys. 2004, 121, 7505. (8) Casselyn, M.; Perez, J.; Tardieu, A.; Vachette, P.; Witz, J.; Delacroix, H. Acta Crystallogr. D Biol. Crystallogr. 2001, 57, 1799. (9) Pan, W.; Galkin, O.; Filobelo, L.; Nagel, R. L.; Vekilov, P. G Biophys. J. 2007, 92, 267. (10) Gliko, O.; Neumaier, N.; Pan, W.; Haase, I.; Fischer, M.; Bacher, A.; Weinkauf, S.; Vekilov, P. G. J. Am. Chem. Soc. 2005, 127, 3433. (11) Gliko, O.; Pan, W.; Katsonis, P.; Neumaier, N.; Galkin, O.; Weinkauf, S.; Vekilov, P. G. J. Phys. Chem. B 2007, 111, 3106. (12) Vekilov, P. G. Cryst. Growth Des. 2004, 4, 671. (13) Galkin, O.; Pan, W.; Filobelo, L.; Hirsch, R. E.; Nagel, R. L.; Vekilov, P. G. Biophys. J. 2007, 92, 902. (14) Vekilov, P. Br. J. Hamaetol. 2007, 139, 173. (15) Lomakin, A.; Chung, D. S.; Benedek, G. B.; Kirschner, D. A.; Teplow, D. B. Proc. Natl. Acad. Sci. U.S.A. 1996, 93, 1125. (16) Krishnan, R.; Lindquist, S. L. Nature 2005, 435, 765. (17) Cacioppo, E.; Pusey, M. L. J. Cryst. Growth 1991, 114, 286. (18) Howard, S. B.; Twigg, P. J.; Baird, J. K.; Meehan, E. J. J. Cryst. Growth 1988, 90, 94. (19) Petsev, D. N.; Wu, X.; Galkin, O.; Vekilov, P. G. J. Phys. Chem. B 2003, 107, 3921. (20) Filobelo, L. F.; Galkin, O.; Vekilov, P. G. J. Chem. Phys. 2005, 123, 014904. (21) Stradner, A.; Sedgwick, H.; Cardinaux, F.; Poon, W. C. K.; Egelhaaf, S. U.; Schurtenberger, P. Nature 2004, 432, 492. (22) Hutchens, S. B.; Wang, Z.-G. J. Chem. Phys. 2007, 127, 084912. (23) Sciortino, F.; Mossa, S.; Zaccarelli, E.; Tartaglia, P. Phys. ReV. Lett. 2004, 93, 055701. (24) Groenewold, J.; Kegel, W. K. J. Phys. Chem. B 2001, 105, 11702. (25) Mossa, S.; Sciortino, F.; Tartaglia, P.; Zaccarelli, E. Langmuir 2004, 20, 10756. (26) Shukla, A.; Mylonas, E.; Di Cola, E.; Finet, S.; Timmins, P.; Narayanan, T.; Svergun, D. I. Proc. Natl. Acad. Sci. 2008, 105, 5075. (27) Porcar, L.; Falus, P.; Chen, W.-R.; Faraone, A.; Fratini, E.; Hong, K.; Baglioni, P.; Liu, Y. J. Phys. Chem. Lett. 2009, 1, 126. (28) Gliko, O.; Neumaier, N.; Pan, W.; Haase, I.; Fischer, M.; Bacher, A.; Weinkauf, S.; Vekilov, P. G. J. Cryst. Growth 2005, 275, e1409. (29) Vekilov, P. G.; Pan, W.; Gliko, O.; Katsonis, P.; Galkin, O. Metastable mesoscopic phases in concentrated protein solutions. In Lecture
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