Article pubs.acs.org/JPCB
Measuring and Modeling Hemoglobin Aggregation below the Freezing Temperature Mónica Rosa,† Carlos Lopes,‡ Eduardo P. Melo,‡ Satish K. Singh,§ Vitor Geraldes,† and Miguel A. Rodrigues*,† †
Centro de Química Estrutural, Department of Chemical Engineering, Instituto Superior Técnico, Lisboa 1049-001, Portugal Instituto de Biotecnologia e Bioengenharia, Centro de Biomedicina Molecular e Estrutural, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal § Biotherapeutics Pharmaceutical Sciences, Pfizer Inc., Chesterfield, Missouri 63017, United States ‡
ABSTRACT: Freezing of protein solutions is required for many applications such as storage, transport, or lyophilization; however, freezing has inherent risks for protein integrity. It is difficult to study protein stability below the freezing temperature because phase separation constrains solute concentration in solution. In this work, we developed an isochoric method to study protein aggregation in solutions at −5, −10, −15, and −20 °C. Lowering the temperature below the freezing point in a fixed volume prevents the aqueous solution from freezing, as pressure rises until equilibrium (P,T) is reached. Aggregation rates of bovine hemoglobin (BHb) increased at lower temperature (−20 °C) and higher BHb concentration. However, the addition of sucrose substantially decreased the aggregation rate and prevented aggregation when the concentration reached 300 g/L. The unfolding thermodynamics of BHb was studied using fluorescence, and the fraction of unfolded protein as a function of temperature was determined. A mathematical model was applied to describe BHb aggregation below the freezing temperature. This model was able to predict the aggregation curves for various storage temperatures and initial concentrations of BHb. The aggregation mechanism was revealed to be mediated by an unfolded state, followed by a fast growth of aggregates that readily precipitate. The aggregation kinetics increased for lower temperature because of the higher fraction of unfolded BHb closer to the cold denaturation temperature. Overall, the results obtained herein suggest that the isochoric method could provide a relatively simple approach to obtain fundamental thermodynamic information about the protein and the aggregation mechanism, thus providing a new approach to developing accelerated formulation studies below the freezing temperature.
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INTRODUCTION
interfaces, and/or freeze concentration (cryoconcentration) of the protein and solutes as well as crystallization of solutes.5,6 The phenomenon of cold denaturation refers to loss of the compact folded structure of the protein as a result of low temperatures. This is a thermodynamic consequence of the large positive ΔCp of unfolding for proteins. The Gibbs free energy of unfolding (ΔuG) as a function of temperature has a maximum between two roots (ΔuG = 0), the melting temperature (Tm) and the cold denaturation temperature (TCD). Beyond these temperatures (higher than Tm or lower than TCD), the unfolded state is the most stable. The cold denaturation temperature, for a multimeric protein, is dependent on a number of factors that impact protein structural stability, such as the concentration of cosolutes and the pH of the solution.7 However, these factors are difficult to control while studying the impact of freezing on the protein. When water crystallizes as ice, all solutes concentrate in the liquid phase. As a consequence, buffer components may crystallize,
The study of protein aggregation has been important in understanding the pathogenicity of a number of human diseases and has gained increasing importance in the development of biotherapeutic proteins.1 Aggregation is considered to be a critical quality attribute of therapeutic proteins because of the potential for triggering immunogenicity.2,3 There are many causes that can lead to protein aggregation. Formation of aggregates has been observed in therapeutic proteins during purification, storage, and administration. It is therefore an important concern during production and handling of recombinant therapeutic proteins. Freezing is a unit operation that is particularly risk-prone from an aggregation perspective if not carried out properly or if the formulation is not appropriate.4 Frozen storage is convenient for handling protein solutions over the long term and is considered a conservative product storage method. Several mechanisms have been proposed for aggregation of proteins during freezing and during frozen storage. These include protein unfolding due to low temperatures (cold denaturation), interaction with ice−liquid © 2013 American Chemical Society
Received: April 10, 2013 Revised: June 4, 2013 Published: June 28, 2013 8939
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accompanied by significant changes in pH.8,9 Ionic strength also increases in the unfrozen liquid phase. Overall, the composition of the cryoconcentrated liquid phase is much different than that of the initial unfrozen solution. To stabilize proteins in solution, protective solutes such as sugars/polyols are usually added.10,11 Proteins in solution show higher stability in the presence of small molecules, such as disaccharides, because of the preferential exclusion mechanism. Briefly, these compounds are preferentially excluded from the surface of the protein, leading to an increase in the protein chemical potential, especially of the unfolded state. The preferential exclusion of these solutes depends on the amount of exposed surface area of the protein, leading then to greater destabilization of the unfolded state and favoring the more compact native state.11−13 Understanding the influence of protein conformational changes due to temperature variations or solute concentration is important to understanding the mechanisms related to aggregation in the frozen storage, such as the aggregation of cryogloblins at low temperature.14,15 However, it is difficult to quantify the impact of these solutes below the freezing temperature because their concentration changes continuously as water freezes. The study of protein stability and aggregation is therefore much more challenging below the freezing temperature. Freezing is usually carried out at normal pressure, i.e., at constant pressure. Therefore, as water freezes, the solute concentrations in the liquid phase will increase until equilibrium is established. For example, if we suppose that the solution is a water−sugar pseudobinary system, according to Gibbs rule, there is only one degree of freedom in the solid− liquid matrix, i.e., the sugar concentration is determined by the storage temperature.16,17 To better understand the contribution of temperature and solute concentrations, these two variables must be deconvoluted. For this purpose, in this work we developed an isochoric method to study proteins at low temperatures while preventing water from freezing. One degree of freedom is gained by fixing volume instead of pressure, and consequently the influence of the variation in sugar or protein concentration can be studied. This approach enables the solution to remain in the liquid phase below 0 °C, with the obvious inconvenience of raising the system pressure. Pressure can also compromise protein stability. Studies have shown that at ambient temperature, neutral pH, and 200 MPa, hemoglobin should not undergo denaturation.18,19 Balny et al.20 observed that hydrostatic pressure could unfold the cytoplasmic creatine kinase toward one intermediate state if the pressure was as high as 650 MPa. High pressure can therefore contribute to the ΔuG, thus shifting the Tm and TCD, compared to those at normal pressure. In fact, this effect has been used to accelerate protein aggregation. Elevated temperatures are frequently used to accelerate aggregation by destabilizing the native conformation and increasing reaction rates. Both temperature and pressure may accelerate aggregation through different mechanisms. Generally, pressure favors reactions with low activation volumes, whereas elevated temperature favors reactions with low activation energies.21 Seefeldt et al.21 proposed that high pressure can be used as a method analogous to high temperature to conduct accelerated formulation studies of protein aggregation. Also, Meersman et al.22 observed pressure-assisted cold unfolding of metmyoglobin. Pressure variation is an important component to understanding the impact of solute concentrations on protein aggregation below the freezing temperature; however, one must be careful when trying to
establish a correspondence between accelerated studies and normal-pressure reactions. The mechanisms of protein aggregation have been modeled by several authors. These models are presented in the literature for many protein aggregation kinetics, mechanisms, and curvefitting studies intended for different applications.23 Protein aggregation models can be divided into five major types: (i) the sequential monomer addition mechanism; (ii) the reversible association mechanism; (iii) prion aggregation mechanisms; (iv) “Ockham’s razor”/minimalistic model, also known as the Finke−Watzky two-step model; and (v) quantitative structure− activity relationship models.23 We have applied the sequential monomer addition model to the BHb aggregation data in this work. This model is applicable whenever unfolding favors aggregation of proteins.24,25 Roberts26 has developed a sequential monomer addition model that is an extended version of the initial formalism proposed by Lumry−Eyring, applied for the irreversible aggregation of proteins, considering protein unfolding prior to aggregation.26−28 The fundamentals of this model are presented in the Appendix, with particular emphasis on the assumptions and simplifications applicable to our system, i.e., BHb aggregation below the freezing temperature.
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MATERIALS AND METHODS Materials. Bovine hemoglobin (BHb), analytical grade sucrose (Suc), and sodium phosphate dibasic dehydrate were purchased from Sigma Aldrich. Analytical grade potassium dihydrogen phosphate was purchased from Roig Farma (Barcelona, Spain). Distilled water, used to make all the solutions, was treated with Milli-Q Integral 3 Pure (Millipore). The buffer used for BHb freezing studies was phosphate buffer (PB) 1/15 mol L−1 at pH 7.4. Experimental Methods. Solution Preparation. Distilled water was used for all the solutions: 0.5, 1, and 2 g of BHb/L of PB; 1 g of BHb/L of PB with 100 g of Suc/L of PB; and 1 g of BHb/L of PB with 300 g of Suc/L of PB. Isochoric Method. The aggregation experiments were performed in high-pressure reactors (HIP MS-16) with a 6 cm3 volume. Freezing is hindered when temperature is lowered at constant volume because the density of ice is less than that of liquid water in the range of pressure of this study. Pressure will rise as temperature is lowered following the negative slope of water’s solid−liquid equilibrium. However, uncontrolled nucleation temperature would result in abrupt pressure variation. To prevent this, 0.3 g of water was frozen inside the reactor (1 h at −10 °C) to provide ice seeds before loading the BHb solution. BHb solutions were then carefully loaded into the reactors, avoiding formation of air bubbles, and placed in a temperature-controlled ethanol bath cooled by a Haake EK 12 immersion cooler (Karlsruhe, Germany) controlled by a Julabo EC immersion circulator (Seelbach, Germany). The aggregation curves were determined from samples removed at different times. The integrity of the isochoric freezing system was confirmed by using a pressure sensor (PX603 from Omega). We found that pressure increased as temperature was lowered, demonstrating that the solution was not freezing. Maximum final pressure was approximately 200 MPa for −20 °C and 100 MPa for −10 °C. Control samples were frozen normally at −20 °C. Analytical Methods. The concentration of BHb was determined using the cyanomethemoglobin method.29,30The BHb concentration of each sample was measured by UV−vis 8940
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spectroscopy at 540 nm on a Corning 96-well plate using a Versamax tunable spectrophotometer microplate reader (Molecular Devices). A calibration curve was performed for every plate using the absorbance of concentrations 0, 0.25, 0.5, 1, and 2 g of BHb/L, with linear regression coefficients higher than 0.999. Chemically induced unfolding of BHb was measured by fluorescence. Hemoglobin solutions (1.91 g/L) were prepared in phosphate buffer (20 mM, pH 7.4). This BHb stock solution was diluted to a final concentration of 2 μM in guanidinium hydrochloride (GuHCl), solubilized in the same phosphate buffer (20 mM, pH 7.4). Fluorescence studies were carried out in a Fluoromax-4 instrument connected to a heating/refrigeration circulator using an excitation wavelength of 280 nm. Emission spectra were recorded from 295 to 420 nm (5 nm slits). Three replicates were measured for each temperature within the range of 5−35 °C. We assessed the loss of tertiary structure with GuHCl denaturation using intrinsic tryptophan and tyrosine fluorescence. Unfolding curves were generated by plotting the total fluorescence intensity as a function of GuHCl concentration. Circular Dichroism (CD). Circular dichroism experiments were performed using an Applied Photophysics PiStar-180 spectrometer (Leatherhead, U.K.) with a Peltier temperature controller. Spectra were obtained in the range of 260−200 nm at a scanning rate of 50 nm/min in a 1 mm optical path length cuvette. Spectra were measured with 0.19 g/L BHb samples from 5 to 80 °C, and three spectra were recorded and averaged. The corresponding buffer spectrum was subsequently subtracted. BHb Stability Data Analysis. The linear extrapolation method (LEM) was applied to BHb chemical unfolding data measured by fluorescence to determine the free energy of unfolding in water (ΔuG) at different temperatures. These values were plotted against temperature to evaluate the dependence of BHb stability on temperature. The susceptibility of a protein toward unfolding in a given solution as a function of temperature is described by the Gibbs−Helmholtz expression,7 which relates the free energy of unfolding, ΔuG, to the temperature-independent heat capacity change (ΔCp):
fU ≡ fR =
exp( −ΔuG /RT ) 1 + exp( −ΔuG /RT )
(2)
Modeling and Calculations. The model applied to the aggregation curves was Type IB simplification (eq 3) of the extended Lumry−Eyring model described by Roberts25 for the irreversible aggregation of proteins, considering protein unfolding prior to aggregation ⎛ n* ⎞ dM = −2k11(app)⎜ ⎟M2 ⎝2 ⎠ dt
(3)
k11(app)
where is the experimentally observed kinetic constant of the aggregation reaction, which depends on the concentration of the reactive monomers as follows: k11(app) ≡ fR 2 k11
(4)
Equation 3 was used to model and predict the BHb aggregation below the freezing temperature. During experimental runs we made two observations that are in agreement with the Type IB mechanism: presence of visible insoluble precipitates and no detectable soluble aggregates in the BHb solutions using size exclusion chromatography. Therefore, aggregates must either be insoluble (low n*) or, if soluble, grow so rapidly that they are present at levels that are too low to be easily detectable.27 Therefore, we set n* = 2 for all calculations; in other words, this implies that the formation of a dimer is the limiting step to start precipitation. The model was adjusted to the aggregation curves obtained at −20 °C. All fittings consisted of optimizing k11 for minimization of the root-mean-square error (RMSE) between experimental and calculated time using the solver provided with Microsoft Excel software.
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RESULTS AND DISCUSSION Sucrose Protection under Accelerated Aggregation. The results shown in Figures 1 and 2 reveal that the aggre-
⎡ ⎛ T ⎞⎤ Δu G = ΔHTref + ΔCp(T − Tref ) − T ⎢ΔSTref + ΔCp ln⎜ ⎟⎥ ⎢⎣ ⎝ Tref ⎠⎥⎦ (1)
where Tref is any reference temperature of hemoglobin and ΔHTref and ΔSTref are the changes in the enthalpy and entropy at that temperature, respectively. The melting temperature of hemoglobin (Tm) was measured by circular dichroism because light scattering distorts fluorescence spectra particularly because of protein aggregation. The formation of aggregates upon thermal unfolding could have shifted Tm to a lower temperature. However, we expect this shift to be small because aggregation was observed only after the onset of denaturation. The software Origin was used to fit the dependence of BHb stability on temperature with the Gibbs−Helmholtz equation and to determine the enthalpy change and heat capacity change. Once these parameters were determined, the free energy change of unfolding at any temperature is known and the fraction of BHb unfolded is calculated from eq 2:
Figure 1. Consumption of soluble BHb (M0 = 1 g/L) during aggregation at −20 °C: under isochoric conditions in PB buffer pH 7.4 only (blue circles), with 100 g/L sucrose (red squares), with 300 g/L sucrose (green triangles), and under normal pressure conditions (frozen control) in PB buffer pH 7.4 only (open circles).
gation was substantially accelerated in isochoric conditions (in 24 h), if compared to the frozen controls, which show no aggregation under the time scale studied. These figures also reveal that the aggregation rate increases with decreasing hold temperature under isochoric cooling. As BHb aggregates, the 8941
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method conditions is apparently contradictory to the general kinetic behavior of chemical reactions in which the rate is favored by higher temperature. However, temperature has a substantial impact on protein structure and therefore on its reactivity toward aggregation. The higher apparent aggregation rate (k11app) at lower temperature is therefore explained by the extended unfolding of the protein’s native structure as we approach its cold denaturation temperature (TCD). Note that at TCD, as at melting temperature (Tm), half of the total protein molecules are in an unfolded state. Two factors may cause protein unfolding under the isochoric method storage conditions: low temperature (as we approach TCD) or high pressure, which increases with decreasing temperature. The pressure range associated with the isochoric conditions used in this work is 3−5 times lower than the pressure required to aggregate hemoglobin at ambient temperature reported by Suzuki and Kitamura18. Yet pressure could still contribute to accelerating aggregation in the temperature range of this study, which is much lower than that of Suzuki and Kitamuras’s work. To better understand the influence of the unfolded fraction, temperature, and pressure on BHb aggregation kinetics, a thermodynamic study and modeling were carried out. Figure 3
Figure 2. Consumption of soluble BHb (M0 = 1 g/L) during aggregation at −10 °C: under isochoric conditions in PB buffer pH 7.4 only (blue circles), with 100 g/L sucrose (red squares), and under normal pressure conditions (frozen control) in PB buffer pH 7.4 (open circles).
monomer fraction (M) in solution decreases, reaching minima of 0.3 and 0.5 after 24 h for −20 and −10 °C, respectively. The effect of sucrose in preventing aggregation of BHb is also evident in these two figures. The solutions containing 100 g/L sucrose show considerably lower BHb aggregation. When sucrose content was 300 g/L (Figure 1), the aggregation levels were equivalent to the normally frozen samples (control). Several possible mechanisms of cryoprotection with solutes such as sucrose have been proposed. The high viscosity or vitrification mechanism suggests that sucrose causes proteins to have low molecular mobility because of the increased viscosity of its surroundings, which is a consequence of the formation of highly concentrated solutions or a glassy state.31,32 This is not expected to be the case in the present work. The viscosity of the protective 300 g/L sucrose solution (0.022 is the mole fraction of the sugar) can be estimated as 24 mPA s,17 which is 3 orders of magnitude smaller than the corresponding viscosity of the liquid phase if the solution were frozen at −20 °C. Freezing induces phase separation; according to the water−sucrose binary phase diagram, the equilibrium concentration of solutes at −20 °C reaches 0.142 of the mole fraction of the sugar.16,17 Another mechanism that has been proposed is the preferential exclusion of the cryoprotectant from the surface of the protein. Accordingly, the presence of the cryoprotectant increases the chemical potential of both the protein and the cryoprotectant. The protein is thus stabilized against dissociation and denaturation as these would lead to greater thermodynamically unfavorable contact surface area between the protein and the cryoprotectant.11−13,33−35 This is the widely accepted view for the stabilization of protein as observed herein. The protection conferred by the high concentration of solutes also explains why the frozen controls did not aggregate on the time scale of the isochoric freezing experiments. In frozen solutions (stored above Tg), protein aggregation is typically very slow. For example, Miller and co-workers36 have reported approximately 2% aggregation in IgG2 frozen solutions after 230 days of storage at −20 °C (containing 84 g/L trehalose). The same authors also reported that aggregation was slower when the solutions were stored at −10 °C, which is in agreement with the results described herein. Aggregation Mechanism. The observation that aggregation is favored by lower temperature under the isochoric
Figure 3. Fraction of unfolded BHb induced by guanidinium hydrochloride at 6.4 °C (solid green squares), 10.5 °C (blue circles), 15.8 °C (solid red triangles), 21.2 °C (plus signs), 25.4 °C (open green squares), 29.6 °C (open circles), and 32.8 °C (open red triangles). Solid lines were fit according to eq 2.
shows the chemically induced unfolding of BHb at different temperatures. The free-energy change of unfolding (ΔuG) in the absence of denaturant was then calculated according to the LEM method and plotted versus temperature (Figure 4). The melting temperature determined from a thermal scan was 53 °C (also plotted in Figure 4). The Gibbs−Helmholtz equation was then used to fit experimental data and calculate TCD (−28 °C) and ΔuG at any specific temperature.7,37 The f R−T curve (Figure 4B) was used to calculate the intrinsic aggregation kinetic constant (k11) at various temperatures. Figure 5 shows experimental plots of monomer decrease during BHb aggregation at various storage temperatures and the corresponding fitted curves, obtained by optimization of k11 using the Type IB model. The results show that the model describes fairly well the aggregation curves, revealing that the major assumptions inherent in the Type IB model describe the essential characteristics of BHb aggregation. Table 1 presents the fitting 8942
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as expressed by eq 4. This confirms that BHb aggregation is mediated by an unfolded intermediate, i.e., the aggregation is slower at higher temperature because the concentration of reactive BHb is lower, but the intrinsic kinetic constant k11 (independent of f R) increases with temperature as expected. In fact, k11 shows Arrhenius dependence over the temperature range of this study (Figure 6), from which the activation energy
Figure 6. Arrhenius plot of k11 (linear correlation coefficient R = 0.996).
Figure 4. (A) Temperature dependence of BHb stability at pH 7.4. Experimental data were fit (solid line) with the Gibbs−Helmholtz equation. Data points are the mean of three assays. (B) Unfolded fraction (f R) of BHb with temperature given by eq 2.
for BHb aggregation could be determined (61 kJ/mol). The importance of considering the unfolded fraction to assign Arrhenius parameters was highlighted in another study.38 The rate constant k11 has been shown empirically to follow an Arrhenius T dependence over the range from close to 0 °C up to the vicinity of Tm.25 Herein, we show that this temperature dependence also extends to lower temperatures, down to TCD, at least in the case of BHb. The linearity expressed in Figure 6 (R = 0.996) confirms that we are below the threshold of pressure that contributes to BHb aggregation by extending its unfolding. Even though the pressure variation was significant (approximately 50 MPa at −5 °C and 200 MPa at −20 °C), k11 variation is described by temperature dependence only. Note that the unfolded BHb fraction used in k11 calculations was determined at constant pressure (atmospheric). BHb aggregation curves could therefore be predicted with any of the k11 values shown in Table 1, using the Arrhenius dependence described by eq 5:
Figure 5. Consumption of soluble BHb during aggregation in phosphate buffer pH 7.4 at storage temperatures of −5 °C (black circles), −10 °C (red squares), −15 °C (blue diamonds), and −20 °C (green triangles) and corresponding aggregation curves calculated by fitting the Type IB model.
ln k11(T ) = ln k11(Tref ) −
fR
M0 (g/L)
k11 (h−1)
NRMSE
−20 −15 −10 −5
0.113 0.046 0.023 0.013
0.96 1.02 0.96 0.97
9.1 16.1 30.9 44.6
0.09 0.16 0.18 0.13
(5)
Figure 7 shows an example wherein the k11 value obtained for the aggregation curve at −10 °C (M0 = 1 g/L) was used to predict BHb aggregation under different conditions. Overall, the results presented herein suggest that the isochoric method may provide an alternative approach for accelerated formulation studies. Aggregation is usually accelerated by increasing temperature, i.e., approaching Tm. In contrast, herein the aggregation was accelerated by approaching TCD. The major advantage of this approach is the enabling of accelerated studies at typical frozen storage temperatures. In this case, by preventing freezing, the effect of cryoprotective concentrations on protein stability can be studied below the freezing temperature, as exemplified for sucrose (Figures 1 and 2). This method also has potential to provide fundamental
Table 1. Parameters Used for the Fitting of the Type IB Model to the Aggregation Curves, Calculated k11 Values, and Normalized Root-Mean-Square Standard Error (NRMSE) Associated with Each Curve t (°C)
Ea ⎛ 1 1 ⎞ ⎟ ⎜ − R ⎝T Tref ⎠
results, which show that k11 values increase with increasing temperature, contrary to the k11app values. Note that k11app is dependent on the concentration of reactive (unfolded) BHb 8943
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Overall, this work also provided evidence for Arrhenius temperature dependence of the intrinsic rate coefficient for the aggregation, showing that this relation is also applicable below the freezing temperature and approaching the cold denaturation temperature. The results obtained show that the isochoric method provides a relatively simple approach to obtaining fundamental thermodynamic information about the protein and the aggregation mechanism, providing a new approach to develop accelerated formulation studies below the freezing temperature. However, future work is required to clarify whether correspondences can be established with normally frozen solutions undergoing long-term storage.
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APPENDIX Protein aggregation involves many stages. The first stage corresponds to conformational changes involving the transition (A1)
N↔U
Figure 7. Consumption of soluble BHb during aggregation in phosphate buffer pH 7.4 at −20 °C with an initial concentration (M0) of 0.5 g/L (green diamonds), 1 g/L (red squares), and 2 g/L (blue circles) and corresponding aggregation curves predicted by the Type IB model.
where N represents a monomer in its native state and U represents a monomer in its unfolded state, defined by the equilibrium constant Kun. Aggregation is then driven by an initial reversible monomer association, i.e., a “pre-nucleation” event, where a reactive form (R) of the monomer is formed
thermodynamic information regarding the aggregation mechanism of the protein when used together with thermodynamic measures, as shown for BHb. On the basis of the aggregation mechanism observed for BHb (unfolding mediated), one might anticipate that sucrose is stabilizing the folded molecule, lowering the TCD, and consequently hindering aggregation. Nonetheless, to clarify how the partitioning between monomers and higher molecular weight species evolves in the presence of sucrose would require another concerted study relating measurements of unfolding thermodynamics and aggregation kinetics of BHb with sucrose at different temperatures. An obvious disadvantage of this method is the rise of pressure inherent in the solid−liquid equilibrium of water below the freezing temperature in isochoric conditions. In the case of BHb, this range of pressure did not influence aggregation; however, other proteins may show pressure-dependent aggregation, complicating the extrapolation of results for normal pressure conditions. Whenever the proteins are sensitive to the range of pressure of the isochoric method, this approach can be considered equivalent to the work of Seefeldt et al.21 that proposed to accelerate aggregation by increasing pressure. However, in our case the volume expansion of the freezing water provides a “natural piston” pressurizing the system.
iR ↔ R i
(A2)
where Ri is a reversible oligomer (composed of i monomers), defined by an equilibrium constant Ki. Therefore, k11 is defined as the intrinsic rate coefficient for the aggregation of two R monomers (i = 1). Typically, N, U, and R cannot be distinguished experimentally. For this reason, the total (soluble) monomer (M) is considered.26 The monomer is therefore characterized by an unfolded reactive fraction, herein defined as f R. Afterward, the reactive monomers grow by monomer addition, forming aggregates (Aj) composed of j monomers. Aggregates may continue to grow by addition of more monomers (M) or by condensation with other aggregates of the same size or (most likely) of different sizes. Ultimately, the condensation of aggregates leads to the formation of fibrils, precipitates, or gels.26−28 The rate of monomer consumption in the aggregation reaction can therefore be described by eq A3: ⎞ ⎛ n *−1 dM = −2k11 fR 2 M2 − fR ⎜⎜ ∑ k1jA(j)⎟⎟M dt ⎠ ⎝ j=2
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(A3)
where n* represents the size cutoff, i.e., the number of monomers below which aggregates still have appreciable solubility or reactivity with respect to aggregation; when (i, j ≥ n*) aggregates have an extremely low solubility and precipitate.26−28 This model can be simplified into three aggregation types, which are defined by the relative rates of nucleation and growth and by the size at which the aggregates precipitate. In this work, we found that BHb aggregation can be considered a Type IB mechanism, i.e., in which all aggregates that form are insoluble (low n*) or soluble aggregates grow so rapidly to n* that they are present at levels that are too low to be detected. Characteristic features of Type IB kinetics include visible precipitates present at low extents of reaction and essentially undetectable soluble aggregate concentrations.26,27 The dimensionless form of eq A3 is more convenient for numerical processing and analysis. The species are made
CONCLUSION The isochoric method developed in this work provided accelerated aggregation studies for BHb below the freezing temperature. These studies revealed that BHb aggregation rates were higher at lower temperature because of the higher fraction of unfolded BHb closer to the cold denaturation temperature, estimated herein to be −28 °C. The aggregation mechanism, accessed by modeling, revealed that BHb aggregation below the freezing temperature is mediated by the unfolded state. The mechanism is also characterized by a very low solubility of aggregates or very rapid growth of the soluble aggregated species so that they are not easily detectable. The aggregation kinetics was substantially attenuated by the higher thermostability of BHb in the presence of sucrose; aggregation was prevented when the sucrose content was 300 g/L. 8944
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where k1j is the equivalent of kij with the restriction that one of the reactants is an R molecule (i = 1). This is the most relevant case for BHb, characteristic of Type IB kinetics. Inserting eq A6 in eq A4 gives
dimensionless by dividing their respective concentration (M and A(j)) by the characteristic concentration, i.e., the initial monomer concentration (M0). A dimensionless time variable is created by defining the characteristic time (τagg) as the end of the aggregation, i.e., when the monomer concentration approaches zero. The dimensionless form of eq A3 is therefore ⎛ n *−1 ⎞ dm = −2m2 − fR −1 ⎜⎜ ∑ k1j′aj⎟⎟m dτagg ⎝ j=2 ⎠
dm = −n*m2 dτagg
Finally, converting eq A9 from the dimensionless form to the equivalent expression for M(t) gives ⎛ n* ⎞ dM = −2k11(app)⎜ ⎟M2 ⎝2 ⎠ dt
(A4)
where m ≡ M/M0, τagg ≡ k11M0 f R t, kij′ ≡ kij/k11, and aj ≡ A(j)/M0. The aggregated species, i.e., 2 ≤ j ≤ n*, are accounted for by eq A5: 2
daj dτagg 2≤j ca. 1 and/or n* → ∞, and n *−1
∑ j=2
αj ≅ fR
n* − 2 m if kij/k1j <
