Kinetic factors may reshape the dependence of crystal nucleation rate on temperature in protein bulk solution

Abstract

Here we provide an analysis of primary results obtained from a study of apoferritin crystal nucleation in compositionally invariant bulk solution at constant supersaturation ratio of the protein. The temperature dependence of the stationary crystal nucleation rate in the protein bulk solution is obtained with the help of experimentally determined probability for detection of at least one crystal per solution volume until a given time. The stationary crystal nucleation rate demonstrates unusual behavior with temperature. We emphasize that this is caused by kinetic factors that are often disregarded in the frame of the classical nucleation theory but can certainly affect the nucleation kinetics.

Introduction

An important task in the studies of nucleation of protein crystals is to quantify the stationary nucleation rate J (m⁻³ s⁻¹) in the bulk solution of the crystallizing substance [1–4]. Commonly, this can be done in three distinct cases: (i) changing the protein concentration to adjust the supersaturation at constant temperature, pH, and constant salt (precipitant) concentration used to initiate the crystallization; (ii) changing the salt (precipitant) concentration to adjust the supersaturation via modifying the protein solubility (equilibrium concentration) at constant temperature, pH, and constant protein concentration; and (iii) changing the temperature to adjust the protein solubility at constant protein concentration, pH, and constant salt concentration. All of these cases require knowledge about the protein solubility dependence either of salt concentration or temperature for estimation of the supersaturation in buffered solutions. Usually, higher nucleation rate is observed for solutions of higher protein or salt concentration because the supersaturation rises. On the other hand, the majority of proteins exhibit normal dependence of their solubility on temperature [5], which means that the solubility is higher at higher temperatures. This, in turn, leads to higher nucleation rate at lower temperatures because the supersaturation again rises as a result of significant change in the ratio of actual and equilibrium protein concentration (the supersaturation ratio). And this ratio may quite overpower the dependence of the supersaturation itself on temperature, despite the ratio’s logarithmic scale. In the present study, we obtain and analyze the nucleation rate of apoferritin crystals in the bulk solution in a wide temperature range under the constraint of constant supersaturation ratio [6] with a stress on the distortion of the anticipated nucleation rate dependence on temperature, caused by kinetic factors.

Experimental

A detailed description of the experimental setup and the design of the experiments is given in [7]. Here we will provide only a brief description of the most basic features of the experimental approach used.

The composition of the studied crystallization system was 0.5 mg/ml apoferritin, 2.5% (w/v) CdSO₄, and 0.2 M sodium acetate buffer at pH 5. This composition corresponds to an apoferritin supersaturation ratio of 22 (the solubility is taken as 23 μg/ml [8]). Although this supersaturation ratio is relatively high for a protein crystallization system, it still corresponds to a relatively low concentration of apoferritin molecules in the solution.

The crystallization trials were performed in glass crystallization vessel in batch setup, and the solution completely wetted the closed system (Fig. 1). A plastic grid, placed over the glass jacket of the crystallization vessel, was used to fix 25 separate crystallization volume cells (314 nl per cell) (Fig. 1), which can be easily inspected with light microscope. Preliminary test experiments, used to identify appropriate crystallization conditions, demonstrated that the nucleation occurred predominantly in the bulk of the solution. This test experiment was performed with apoferritin concentration 0.375 mg/ml in glass crystallization vessel of 1 mm height at room temperature. We observed crystal population of high crystal number density, and only a negligible portion of the crystals (< 2%) were found on the upper glass of the vessel. In an additional test experiment, using a glass vessel of 0.2 mm height and apoferritin concentration 0.5 mg/ml, no crystals on the upper glass were detected.

Fig. 1.

Scheme of the experimental setup. a Batch crystallization setup in borosilicate glass vessel with internal height 0.1 mm, which determines the height of the protein solution layer. The height of the glass jacket is 15 mm. Plastic grid is placed over the glass jacket to fix inspection crystallization cells of constant volume V = 314 nl. b Cubic crystals of apoferritin, representing the general crystal population for all the temperatures used in the present study. The picture is taken 18 h after a nucleation experiment, performed at 30 °C

Six different crystallization temperatures were used: 10, 20, 30, 40, 50, and 60 °C. These temperatures were kept constant throughout the entire experimental procedure — equilibration of both the protein and CdSO₄ samples (from 10 to 20 min, depending on the temperature), their mixing and mixed sample transfer to the crystallization vessel, which was already equilibrated at the desired temperature. The inspection for emerging detectable crystals was performed every 20 min. This time interval was kept constant for all temperatures for uniformity of the approach, although it could also lead to some flaws in the data collection.

The experiments for determination of the probability P for detection of at least one crystal until time t were considered as Bernoulli trials, i.e., as offering exactly two mutually exclusive outcomes, namely absence of detectable crystals or presence of at least one detectable crystal. For any given t, experimentally, P can be obtained from the equation

$$P\left(t\right)=\frac{N^{+}\left(t\right)}{\left[N^{+}\left(t\right)+N^{-}\left(t\right)\right]}$$ 1

where N⁺ denotes the number of all positive outcomes (presence of at least one detectable crystal) at time t, and N⁻ denotes the number of all negative outcomes (no detectable crystals) at time t.

Up to 225 inspections (the number [N⁺(t) + N⁻(t)] in Eq. (1)) were performed for calculation of a single P(t) value.

The investigated crystallization system is a continuous liquid system. This means that each of the inspected crystallization volumes communicates with the solution volume that is not inspected for crystals. However, such experimental setup should not be a drawback because it would barely influence the measured probability P(t). Statistically, possible outflow of potentially detectable crystals would be equal to the inflow of such crystals into the inspection volumes.

Results and discussion

When experimental data for P(t) are available, the following equation is often used for these data analysis (see Chapter 26 of Ref. [9] and articles cited therein):

$$P\left(t\right)=1-exp\left[-\mathit{JV}\left(t-t_g\right)\right]$$ 2

It provides direct relation between the stationary nucleation rate J (m⁻³ s⁻¹) in a volume V and the probability P(t). The t_g is a correction term that accounts for the time of growth of the crystal nuclei to detectable size. Therefore, it is particularly sensitive to the resolution of the detection technique and the frequency at which the crystallization volumes are inspected. Furthermore, this time could be determined reasonably well only if it clearly dominates the total time for the detection of the crystals (see [9], eq. (33.68), and [10]).

Equation (2) is valid for any type of nucleation. It should be noted, however, that the equation is applicable when the nucleation proceeds at a rate independent of both time and place. This imposes the requirement of time-independent and spatially homogeneous supersaturation and temperature during the nucleation process, which could be met only approximately under experimental conditions. Moreover, the nucleation and the crystal growth proceed simultaneously. This implies that Eq. (2) would generally fit better to data that are obtained for earlier experimental time intervals. Such consideration justifies the fitting procedure that is performed (see below) and helps in minimizing the effects of any eventual time-dependent and solute-consuming process in the solution. Some of the experimental studies, making use of Eq. (2), have been performed elsewhere [11–13].

Figure 2 shows the initial P(t) data obtained for the described apoferritin crystallization system [7]. The curves represent fits of Eq. (2) to the initial (with respect to time) detection probability data. It is worth noting that if, for example, probability plateau regions appear with time (see ref. [7]), which are relatively far below P = 1 (the horizontal asymptote for the curve of the fitting function [11–13]), we would have apparent violation of the conditions for the applicability of Eq. (2). Therefore, either the last cannot be fitted to such data or the resulting fits are very poor.

Fig. 2.

Initial time dependence of the probability P for detection of at least one crystal per inspection volume V. Each point represents a result from 150 (for 10–40 °C) and 225 (for 50 and 60 °C) single inspections. The curves show free fits of Eq. (2) to the experimental data obtained from earliest detections (see also ref. [7]). The squared correlation coefficients r² of the fits from the lowest to the highest temperature are 0.99, 0.99, 0.97, 0.99, 0.95, and 0.95, respectively

The free fitting procedure was performed with two free parameters, namely JV and t_g. The stationary crystal nucleation rate J in the apoferritin bulk solution was determined from the fitting parameter JV. The resulting plot of J (m⁻³ s⁻¹) vs temperature T (K) is shown in Fig. 3.

Fig. 3.

Temperature dependence of the stationary nucleation rate of apoferritin crystals in the bulk solution. The nucleation rate data J(T) are determined from the free parameter JV of Eq. (2) at each of the used temperatures. It is obtained from free fits of Eq. (2) to the experimental data for crystal detection probability at earlier detection times (Fig. 2; see also ref. [7]). Error bars indicate the standard error of free parameter JV. Right eye-guiding line indicates the severe breaking of the J(T) dependence predicted by Eq. (3) (see also Fig. 5)

It can be seen in Fig. 3 that the nucleation rate in the bulk solution passes through a maximum at temperature around 30 °C. According to the classical nucleation theory, the stationary nucleation rate can be generally expressed as [9, 14]:

$$J=Aexp\left(\frac{-{∆G}_D}{k_{B}T}\right)exp\left(\frac{-{∆G}_{n^{\ast }}}{k_{B}T}\right)$$ 3

where A (m⁻³ s⁻¹) is a kinetic prefactor, ΔG_D is the energy barrier for solute transport to and/or incorporation into the crystal, ΔG_n* is the nucleation barrier (work for nucleus formation), k_B is the Boltzmann constant, and T is the absolute temperature. Under the conventional assumptions that ΔG_D does not vary with temperature [14], that the crystal/solution interfacial free energy is temperature-independent [15], and that the dependence of the supersaturation ratio of apoferritin on the temperature is negligible [6], both exponential terms in Eq. (3) should rise with T. Note that, moreover, the work ΔG_n* for nucleus formation is proportional to 1/T² [e.g. Eq. (14) of ref. 16]. This means that the nucleation rate J will certainly rise with temperature, provided that the dependence of A on T is unessential.

As mentioned above, t_g could appear as sensitive parameter, which is dependent on several factors including also the rate of crystal growth and even uncontrollable artifacts of the protein and precipitant mixing during preparation of the crystallization samples [13]. Hence, the relatively low accuracy in the determination of t_g is not generally unexpected, and sometimes even negative values might be obtained (Table 1).

Table 1.

Values of t_g, obtained from the fits of Eq. (2) to the experimental data (Fig. 2)

T (°C)	10	20	30	40	50	60
tg (min)	74 ± 2.6	7.9 ± 2.3	12.7 ± 4.7	−2.2 ± 4.1	5.3 ± 8.2	−58.4 ± 23.5

Because this is physically impossible and t_g strongly influences the value of J (see Eq. (2)), we performed a series of one free parameter fits over the data for the most striking case (Table 1, 60 °C), with physically reasonable values of t_g in the range 2–18 min. The result is shown in Fig. 4.

Fig. 4.

Forced fits of Eq. (2) to the experimental probability data for 60 °C. Only JV is used as free fitting parameter, and nine different values of t_g are used: 2, 4, 6, 8, 10, 12, 14, 16, and 18 min. The inset shows the extensive P(t) dependence for the two extreme values for t_g: 2 min (lower curve) and 18 min (upper curve)

The values of J obtained in this way fall in the range (1.1 − 1.5) × 10⁵ m⁻³ s⁻¹. Although this shift of J for 60 °C corresponds to about a threefold higher value, it is still in the range of J for 40 or 50 °C (Fig. 3). Note that if the bulk nucleation rate J for 40, 50, or the adjusted value for 60 °C (Fig. 4) were even thrice higher, it would be still far from the dependence predicted by Eq. (3). Figure 5 illustrates the change of the stationary nucleation rate in the bulk solution with the temperature, for the case under consideration. For clarity, physically reasonable values ∆G_D = 3 × 10⁻²⁰ joules, ${∆G}_{n^{\ast }}=1.85\times {10}^{-20}$ joules, and A = 2.7 × 10¹⁰ m⁻³ s⁻¹ were used for the illustration. These values were obtained with the help of the data in ref. [7], which were then temperature-averaged.

Fig. 5.

General shape of the dependence J(T) for constant supersaturation ratio (bold curve). The curve is obtained via simulation of Eq. (3) and, for simplicity and comparative purposes, with ∆G_D = 3 × 10⁻²⁰ joules, ${∆G}_{n^{\ast }}=1.85\times {10}^{-20}$ joules, and A = 2.7 × 10¹⁰ m⁻³ s⁻¹. These values were obtained, using the temperature-averaged apoferritin data from ref. [7]. The experimentally measured nucleation rate (m⁻³ s⁻¹) (Fig. 3) is shown in order to highlight the difference between the general shape and the experimental data

This leads to the conclusion that it is the pre-exponential factor A, which controls the temperature dependence of the nucleation rate rather than the intrinsic inaccuracy involved in the determination of t_g.

It has been shown with the protein glucose isomerase that composition-driven formation of nucleation precursors can play important role for the route of protein crystallization, including the selection of polymorphs [17]. In our case, we might have temperature-driven formation of a particular population of prenucleation protein clusters, which should influence the pre-exponential factor A. And this influence would depend primarily on the thermal equilibration time of the protein solution and the temperature of mixing of the protein solution with the precipitant solution. Besides, even though apoferritin possess high thermal stability [18], minor temperature-dependent changes in the protein conformation might possibly affect particular interactions in the crystallizing solution, which would also impact A. The structural dynamics and properties of the water molecules in the immediate vicinity of protein molecules may also play a similar role, as well as potential temperature-dependent dynamics of the cadmium ions involved in the crystalline precipitation of apoferritin molecules. Further analysis of this nucleation-related issue, using an approach that accounts for the presence of nucleation-active sites [19, 20], was performed [7] for exploration and understanding of the problem. It has been found that the stationary nucleation rate J_a (s⁻¹), obtained per nucleation-active site, follows well the temperature dependence that is predicted by the classical nucleation theory. With respect to the case under consideration, it has been found that the number density of the nucleation-active sites (a multiplier in the kinetic prefactor A) is the parameter that reshapes the temperature dependence of nucleation rate of protein crystals in the bulk of the solution.

Conclusions

The performed analysis attempts to specify an uncommon problem which might appear during analysis of experimental data for the nucleation rate J (m⁻³ s⁻¹) of protein crystals in bulk solution. Nevertheless, the observed phenomenon is not limited to the nucleation of protein crystals only and is expected to be independent of the used experimental approach or data analysis as long as the determination of the nucleation rate J (m⁻³ s⁻¹) in the bulk solution is concerned. It was found that the kinetic prefactor A (m⁻³ s⁻¹) in the nucleation rate equation, which is usually considered as a quantity of lesser importance, might significantly modify the nucleation kinetics. The problem under consideration, however, is not merely a one of constant or nearly constant supersaturation ratio as a function of the temperature. It is rather a consequence of the particular importance of the kinetic prefactor A among all the terms in the nucleation rate equation. In this respect, stronger solution conditions (e.g., much higher supersaturation ratios) could probably restore the anticipated general behavior of the nucleation rate J (m⁻³ s⁻¹). Still, unusual results for the nucleation rate might possibly be obtained also for systems with constant temperature, which are influenced by an unexpectedly strong impact of the solute concentration on the kinetic prefactor (e.g., crystallization studies at conditions of macromolecular crowding).

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Conflict of interest

The author declares that he has no conflict of interest.

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