J. Phys. Chem. B 2009, 113, 2865–2873
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Ice Recrystallization Kinetics in the Presence of Synthetic Antifreeze Glycoprotein Analogues Using the Framework of LSW Theory C. Budke,† C. Heggemann,‡ M. Koch,† N. Sewald,‡ and T. Koop*,† Department of Chemistry, Physical Chemistry, Bielefeld UniVersity, Bielefeld, Germany, and Department of Chemistry, Organic and Bioorganic Chemistry, Bielefeld UniVersity, Bielefeld, Germany ReceiVed: June 30, 2008; ReVised Manuscript ReceiVed: December 5, 2008
The Ostwald ripening of polycrystalline ice in aqueous sucrose solutions was investigated experimentally. The kinetics of this ice recrystallization process was studied at temperatures between -6 and -10 °C and varying ice volume fractions. Using the theory of Lifshitz, Slyozov, and Wagner (LSW), the diffusion-limited rate constant for ice recrystallization was determined. Also, the effects of synthetic analogues of natural antifreeze glycoproteins (AFGP) were studied. These analogues synAFGPmi (i ) 3-5) contained monosaccharide side groups instead of disaccharide side groups that occur in natural AFGP. In order to account for the inhibition effect of the synAFGPmi, we have modified classical LSW theory, allowing for the derivation of inhibition rate constants. It was found that the investigated synAFGPmi inhibit ice recrystallization at concentrations down to ∼3 µg mL-1 or, equivalently, ∼1 µmol L-1 for the largest synAFGPmi investigated: synAFGPm5. Hence, our new method is capable of quantitatively assessing the efficiency of very similar AFGP with a sensitivity that is at least 2 orders of magnitude larger than that typical for quantitative thermal hysteresis measurements. 1. Introduction The aging of polycrystalline materials is important to many disciplines and occurs in various types of crystalline materials.1,2 Such aging is also observed in polycrystalline ice samples with implications for frozen food or storage of frozen biological tissues at low temperature.3-8 The latter process is often termed ice recrystallization, an Ostwald ripening process by which large ice crystals grow at the cost of small ones, leading to an increase of the mean ice crystal size and a decrease of ice crystal number at a constant overall ice volume. In all of these aging processes, the driving force behind Ostwald ripening is a reduction of the total ice crystal/solution interface energy of the system. Ice recrystallization poses a threat to organisms living at subzero temperatures because intracellular ice formation and growth is usually lethal to the affected cells.8-10 A number of plants, fish, and insects have adapted to this situation by following one of two strategies: freeze avoidance or freeze tolerance. Freeze avoiding species inhibit the formation of ice by reducing their ice melting point through an increase in solute concentration in their body fluids (typically sugars and other polyols) and/or by neutralizing or removing ice nucleators, which normally trigger ice nucleation.8 In contrast, freeze tolerant species allow ice formation but inhibit ice crystal growth. This is achieved by producing so-called antifreeze proteins (AFP) or antifreeze glycoproteins (AFGP). These proteins inhibit ice recrystallization via a molecular recognition process by which the proteins adsorb to certain faces of ice.5,7,11 Some AFP and AFGP have been synthesized or expressed11-13 because of their potential to be applied in frozen food or in the cold storage of tissues.3,14 However, both low stability and high cost of AFP and AFGP hamper their use in industrial applications, and hence, cheap * To whom correspondence should be addressed. E-mail: thomas.koop@ uni-bielefeld.de. † Department of Chemistry, Physical Chemistry. ‡ Department of Chemistry, Organic and Bioorganic Chemistry.
synthetic mimetics for AFP and AFGP are worthwhile. Such development is impeded by the fact that, first, the involved synthesis is often challenging and, second, that no methods are available that allow for quantitative comparison of synthesized variants in terms of their ice growth inhibition efficiency when only minute amounts are available. In particular, quantitative comparisons between different studies are challenging if not impossible because the various experimental protocols often only allow a relative comparison. There are several methods employed to determine the activity of AFP or AFGP. For example, the activity of a particular protein can be determined by observing morphology changes of ice crystals in the presence of aqueous solutions containing small amounts of the proteins. There are variants of this type of experiment such as using a very small (∼50 µm) single ice crystal within a small aqueous droplet embedded in oil, often performed inside a nanoliter osmometer,15,16 or using large single ice crystals in an ice dome etching type of experiment.17,18 Both types of experiments allow to determine whether a protein is active or not even at very small protein concentration. However, one disadvantage is that this method does not allow for a distinction between varying degrees of efficiency, i.e., on how strong the affected ice facets are inhibited in their growth due to adsorbed proteins. An alternative method is thermal hysteresis in which a temperature difference is investigated between the equilibrium ice melting temperature and the temperature below which ice crystals begin to grow uninhibited in the presence of antifreeze agents.7 Thermal hysteresis is widely employed as it provides quantitative data on the efficiency of a particular protein. However, it is disadvantageous in that it usually requires milligram samples of the antifreeze agents. For example, differences in efficiency between various types of natural AFGP can be observed only at concentrations larger than about 2-3 mg mL-1.13,19 Another frequently used method is ice recrystallization analysis. Originally, this method was developed by Knight et
10.1021/jp805726e CCC: $40.75 2009 American Chemical Society Published on Web 02/05/2009
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al. as a splat cooling method in which a water droplet containing the protein was dropped at high speed onto a very cold metal surface, thereby producing a polycrystalline ice film (about 50 µm thick) with large ice volume fractions.20 Thereafter, the sample was placed at a warmer temperature under a microscope, and the recrystallization process in which larger ice crystals grow at the expense of smaller ones at a constant total ice volume was observed optically. Several groups have modified this original approach by introducing other sample preparation or cooling protocols,21 adding colligative solutes such as NaCl or sucrose to the droplets for the purpose of increasing sensitivity,6,21 adding fluorescent dyes for better detection of grain boundaries,6 or even using microliter capillaries in which freezing is initiated and observed.22 The ice recrystallization method is fast and, therefore, can be applied as a screening technique in order to quickly check whether a specific polymer or protein reveals antifreeze activity or not.6,22,23 On the other hand, it has the drawback that only rarely quantitative data arise from such measurements, in particular those that can be applied to other conditions (e.g., temperature, ice volume fraction, etc.) or different experimental protocols and apparatus. In this article we use the modified splat cooling method developed by Smallwood et al. and introduce a new analytic procedure that allows for quantitative evaluation of ice recrystallization kinetics, in both the presence and absence of antifreeze agents.21 This evaluation method is based on the theory of Lifshitz, Slyozov, and Wagner (LSW), which was developed for the description of Ostwald ripening processes.24,25 We show exemplarily by using synthetic AFGP that our new method allows to distinguish and quantitatively assess the efficiency of very similar AFGP even at concentrations of j10 µg mL-1, i.e., with a sensitivity that is at least 2 orders of magnitude larger than that typical for thermal hysteresis measurements. 2. Theory Ostwald ripening of polycrystalline ice in a liquid solution occurs by way of two different processes: accretion and diffusional growth. Accretion describes the mechanism by which two ice crystals directly coalesce to form a single bigger ice crystal, a process that occurs preferentially at high ice volume fractions. In contrast, diffusion of water molecules from the smaller to the larger ice crystals through the intermediate liquid typically dominates at smaller ice volume fractions. When diffusion of water molecules through the liquid phase between ice crystals determines the overall kinetics of ice recrystallization, LSW theory suggests that the temporal increase in the mean crystal radius r should follow
r3(t) ) r03 + kdt
(1)
where r0 is the initial mean radius at time t ) 0. kd is the observed rate constant of recrystallization, which depends on several factors such as solution viscosity η, temperature T, and ice volume fraction Q ) Vice/(Vice + Vliq), where Vice and Vliq are respectively the total ice volume and total liquid volume in the sample. Equation 1 for diffusion-controlled coarsening is based on LSW theory,24,25 which has been employed successfully for description of Ostwald ripening processes, including the recrystallization of ice.26,27 In the limit of Q f 0, i.e., at very small ice crystal concentrations, LSW theory predicts that the corresponding rate constant kLSW can be written as
lim kd ≡ kLSW
Qf0
8σΩ2Dwcw,eq ) 9RT
(2)
where σ is the ice/liquid interface energy, e.g., the interface energy between ice and a sucrose solution in our experiments. Ω is the molar volume of ice, Dw is the diffusion coefficient of water molecules in the sucrose solution, and cw,eq is the molar concentration of water in the solution that is in equilibrium with an ice particle of infinite size. R is the gas constant and T absolute temperature. The dependence of kd on Q is often required for real applications with finite ice volume fractions, and several theories have been developed for this purpose in the past.28-30 However, it is difficult to determine which of these theories describes the coarsening process best for a particular application. Therefore, we chose a rather basic nonlinear mean-field theory developed by Voorhees and Glicksman which assumes a constant volume fraction of the crystalline phase and, with a minor modification, is sufficient for our purposes:31
kd(Q) ) kd0
R3 1 - pQ1/3
(3)
Here, kd0 is the apparent rate constant kd(Q) at Q ) 0, and p is a scaling factor that takes into account a variable dependence on Q at different temperatures. Note that p is equal to one in the original derivation by Voorhees and Glicksman.31 The meanfield potential R ) r/rcr is the ratio of the mean crystal radius r and the critical radius rcr, which represents a crystal that neither grows nor melts. A comparison of various theories including those of Voorhees and Glicksman and different forms of the theory by Brailsford and Wynblatt30,32 indicates that R can be approximated by a simple linear relationship R = 1 - mQ, with m = 0.15 at small volume fractions (Q j 0.4). We have made a sensitivity analysis of the choice of R upon the quality of the fit of eq 3 to our data with negligible effects. Therefore, we chose for simplicity reasons a value of R ) 1. Note that the results presented below are not very sensitive to the choice of theory for describing kd(Q). In principle, also a simple polynomial fit to such data is fine as long as only interpolations within the studied range of Q are required. Below, we will use small extrapolations to Q ) 0 using eq 3 and justify this procedure by independent calculations (see section 4.1). 3. Experimental Section 3.1. Antifreeze Glycoproteins. Natural antifreeze glycoproteins AFGP occur in the blood serum of several Antarctic fishes, for example, the Antarctic Nototheniidae.33 AFGP consist of a peptide backbone with a repeating [Ala-Ala-Thr]i motif (i ) 4-50), in which a disaccharide is attached to the hydroxyl oxygen of the Thr residues12 (see Figure 1a). In the present paper, we investigate the effects on ice recrystallization kinetics of synthetic antifreeze glycoproteins synAFGPmi (i ) 3-5), which contain a monosaccharide group instead of a disaccharide (see Figure 1b). These peptides were synthesized by semiautomated, microwave-enhanced solid phase peptide synthesis. A detailed description of each step of the synthesis and the properties of intermediates is given elsewhere;34 here we briefly summarize the main steps. In order to apply Fmoc strategy, Fmoc-protected O-[tetraacetyl-R-D-N-acetylgalactosaminyl]threonine tert-butyl ester was chosen as key precursor. The fully protected amino acid was prepared by R-galactosylation of
Ice Recrystallization Kinetics
Figure 1. Chemical structure of (a) natural AFGP and (b) the synthetic synAFGPmi studied here. The disaccharide and monosaccharide groups are indicated in red and green, respectively.
Fmoc-threonine tert-butyl ester.35 It was transformed into the building block suitable for solid phase peptide synthesis by acidolysis of the tert-butyl ester. The solid phase peptide synthesis was carried out in an automated microwave peptide synthesizer. During all cycles, the system was adjusted to a maximum power of 20 W and a temperature of 40 °C. The synAFGPmi were prepared starting from 2-chlorotrityl resin loaded with Fmoc-alanine. Alanine couplings were accomplished by O-benzotriazol-1-yl-N,N,N′,N′-tetramethyluronium tetrafluoroborate (TBTU) and Hu¨nig’s base, while the bulky glycosylated building block was coupled by the more effective coupling reagent mixture O-(7-azabenzotriazol-1-yl)-N,N,N′,N′tetramethyluronium hexafluorophosphate (HATU) and 1-hydroxy-7-azabenzotriazole (HOAt). The threonine derivative was preactivated with HATU for 2 min before manual addition to the microwave reactor. Cleavage of the glycopeptides from the resin was achieved by treatment with trifluoroacetic acid (2%) in dichloromethane. The O-acetyl groups of the carbohydrates were removed by transesterification at pH 9 with catalytic amounts of sodium methoxide in methanol. Finally, the synAFGPmi were purified by preparative RP-HPLC.34 3.2. Setup and Procedure. All experiments were performed using an Olympus BX51 optical microscope in transmitted light (bright field) mode. The samples were located on a temperaturecontrolled silver block inside a Linkam MDBCS 196 cold stage. Cooling is provided by pumping cold gaseous nitrogen from a liquid nitrogen reservoir through the silver block. Both the interior and the upper window of the cold stage were purged with dry nitrogen to avoid water condensation from ambient air during the experiments. The sample preparation and experimental protocols follow those of the widely applied modified splat cooling approach introduced by Smallwood et al.21 Sample films of about 10 µm in thickness were produced by placing 2 µL droplets of the investigated solutions between two 14 mm circular glass coverslips. Each sample was cooled from room temperature to -50 °C at a cooling rate of 20 °C min-1 in order to induce multiple ice nucleation events within the film. A polycrystalline ice sample was then obtained upon reheating to the final annealing temperature at a heating rate of 10 °C min-1. Solutions containing only varying concentrations of sucrose were annealed at either -6, -8, or -10 °C, whereas solutions
J. Phys. Chem. B, Vol. 113, No. 9, 2009 2867
Figure 2. Optical microphotographs of ice crystals, formed in aqueous 45 wt % sucrose solutions, during annealing at -8 °C for 120 min: (a) control solution without antifreeze glycoproteins; (b) solution containing 60 µg mL-1 synAFGPm4; (c) solution containing 140 µg mL-1. All pictures were contrast-enhanced for better visibility.
containing synAFGPmi in addition to sucrose (45 wt %) were annealed at -8 °C. During the 2 h of annealing time optical microphotographs were taken every minute using a digital video camera (PixeLINK PL-A662). The images were analyzed using UTHSCSA Image Tool Version 3.0, yielding information about the individual and mean ice crystal area, radius, and roundness; see Appendix A.1 for details. The determination of the ice volume fraction and the procedures for the evaluation of the data are explained in detail in Appendices A.2 and A.3. In total, more than 150 experiments with sucrose concentrations of 25-54 wt % have been performed and more than 80 experiments with synAFGPmi concentrations of 2-1400 µg mL-1. 4. Results 4.1. Pure Sucrose Solutions. Figure 2 displays microphotographs of polycrystalline ice samples as a function of annealing time at -8 °C. Panel a shows the results of aqueous solutions containing only 45 wt % sucrose. The initial mean crystal radius of ∼0.6 µm at 0 min changes to ∼2.2 µm at 10 min, ∼3.8 µm at 60 min, and ∼4.7 µm at 120 min, clearly revealing that an Ostwald ripening process is at work. These data were analyzed using eq 1 described above for diffusion-controlled coarsening based on LSW theory. The results of this evaluation are shown in Figure 3a. The data can be fitted to a linear growth function, indicating that the kinetics of the ice recrystallization process is very well described by a bulk diffusion process of water molecules from the smaller to the larger ice crystals, in agreement with LSW theory. Note that although the mean crystal radius increases continuously during the course of the experiments, the total ice volume fraction Q remains constant after an initial period of less than 15 min (see Figure 3c). This clearly reveals that the observed processes are indeed true ice recrystallization experiments, in which the larger ice crystals grow at the expense of smaller ones driven by Ostwald ripening. Similar measurements have been performed at many different sucrose concentrations and at three different temperatures: -6, -8, and -10 °C. Each of these measurements was evaluated with LSW theory as shown in Figure 3a, and the resulting individual rate constants kd as a function of the ice volume fraction Q are shown as symbols in Figure 4. The lines in Figure 4 are derived by fitting eq 3 to the data, with the corresponding
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Figure 4. Dependence of the diffusion-limited rate constant for ice recrystallization kd as a function of ice volume fraction Q in sucrose solutions at different temperatures. Symbols (circles, squares, triangles) denote experimental data, and the lines are fits to the data according to eq 3. The diamonds at Q ) 0 are the values of kLSW derived independently; see text.
TABLE 1: Parameters for the Fits Shown in Figure 4 and the Corresponding Values for kLSW Calculated from Eq 2a
Figure 3. Temporal development of the cube of the mean ice crystal radius, r3, in 45 wt % sucrose solutions at -8 °C. (a) Pure solution without antifreeze glycoproteins. The black circles are the data, and the solid line is a linear fit to the data according to eq 1. (b) r3 in solution containing 60 and 140 µg mL-1 synAFGPm4 (red and blue circles, respectively). The solid lines are fits to the data according to eq 4, and the dotted lines indicate the extension of the individual rate processes to earlier or later time. The inset illustrates the two processes that determine ice recrystallization kinetics. The faster rate is normally due to molecular diffusion of water molecules from the smaller to the larger ice crystals (black), and the slower rate is limited by the kinetics of how fast water molecules pass the liquid/ice interface, i.e., the rate of the liquid-to-ice transfer. The latter process is significantly inhibited by the presence of antifreeze glycoproteins that adsorb to certain faces of the ice crystals. Panel c shows the temporal development of the ice crystal volume fraction Q for the data shown in (a) and (b). After an initial short time period, Q remains constant over the entire experiment, clearly indicating that the observed mean ice crystal growth is a true Ostwald-ripening driven ice recrystallization process.
fit parameters p and kd0 given in Table 1. Clearly, all data sets can be approximated well with eq 3 (coefficient of determination R2 g 0.75). In order to confirm the validity of our approach and the reliability of our ice recrystallization data, we used an independent approach to verify the identity between the apparent rate constants for a zero volume fraction derived from eq 3, kd0, and that derived from LSW theory, kLSW. All parameters required for calculation of kLSW based on eq 2 were taken from the literature from completely independent studies using various theoretical and experimental approaches (see Appendix A.4 for details). The resulting values are shown as diamonds in Figure 4 and are all in good agreement with those derived from the experiments presented here (see also Table 1). This comparison shows that both our experimental data and the theory to describe them are well understood and provide a consistent picture of ice recrystallization in aqueous solutions. Therefore, we propose that it is advisible to provide ice recrystallization rate constants as the kd0 limit because this value is comparable even if different
T [°C]
p
kd0 [µm3 min-1]
-6 -8 -10
1.334 ( 0.106 1.318 ( 0.057 1.263 ( 0.092
1.57 ( 0.46 0.65 ( 0.09 0.44 ( 0.13
kLSW [µm3 min-1] 1.38 ( 0.28 0.66 ( 0.13 0.30 ( 0.06
a The errors for kd0 represent (3σ and those for kLSW an estimated uncertainty of 20%.
types of protocols or different experimental techniques were used to derive the data. The experimental results presented above show that the dependence of kd on Q becomes smaller at lower temperatures (e-8 °C). Moreover, also the rate of ice recrystallization becomes smaller at lower temperatures, making it more difficult to detect changes induced by inhibitors. Therefore, we chose to perform the experiments with synthetic AFGP at -8 °C, thus allowing for the easy detection of the effects induced by AFGP while minimizing the dependence on the ice volume fraction Q for analysis. 4.2. Solutions Containing Synthetic AFGP. In solutions that contained synthetic AFGP in addition to sucrose, the ice recrystallization kinetics was markedly different from those of the pure sucrose solutions described above. As an example, we show ice crystal photomicrographs of such experiments in panels b and c of Figure 2. At an intermediate concentration of 60 µg mL-1 synAFGPm4 (panel b) the ice crystal size initially increases in a way very similar to that of pure sucrose solutions (panels a and b, respectively, at 10 min). Thereafter, ice crystal growth seems to slow down, and differences from pure sucrose solutions become visible at about 60 min. Between 60 and 120 min hardly any changes occur in panel b, implying almost complete inhibition of ice recrystallization. At larger concentrations of 140 µg mL-1 synAFGPm4 (panel c) the ice crystal size remains significantly smaller throughout the entire experiment, indicating an even stronger inhibition effect when compared to the solutions without or with 60 µg mL-1 synAFGPm4 (panels a and b). Note also that the ice crystals in pure sucrose solutions remain circular throughout the experiment while those with synAFGPm4 adopt a hexagonal crystal habitus. Such type of ice structuring effect is well-known from other AFP, AFGP, and PVA.7,11,36 It occurs because active inhibitors only adsorb to certain ice faces. This is due to a molecular
Ice Recrystallization Kinetics
J. Phys. Chem. B, Vol. 113, No. 9, 2009 2869 process is determined by liquid phase diffusion because under the conditions studied here it is usually the rate-limiting process in the absence of inhibitors (see above). In contrast, the slower process is determined by the rate of the liquidto-ice phase transfer of water molecules. The rate of this transfer is reduced when inhibitors such as synAFGPm4 are adsorbed to the ice surfaces. This is supported by the fact that the rate of the slower process becomes smaller as the concentration of synAFGPm4 is increased. In order to account for the effects of inhibitors just described, we modify the original LSW theory by allowing two independent growth processes in the following way: r3(t) ) (1 - β(t))[r03 + kd(Q)t] + β(t)[rh03 + ki(Q)t]
(4)
Here, r0 and kd(Q) are defined as in eqs 1 and 3, rh0 is the hypothetical initial mean radius at time t ) 0 if only the slower inhibition process was at work, and ki(Q) is the observed rate constant for recrystallization inhibition corresponding to the slower process. β(t) is a time-dependent weighting factor that indicates which of the two processes is rate limiting during the course of an experiment:
β(t) ) (1 + exp(ts - t))-1
Figure 5. Overall recrystallization rate constant kl0(c) vs synAFGPmi concentration c: (a) linear and (b) logarithmic plot. Symbols are experimental data for three different synAFGPmi (i ) 3-5), and lines are fits to the data according to eq 6. Each data point is the mean of typically 3-5 individual measurements at the same inhibitor concentration. The black asterisk represents the value of kd0 in pure sucrose solutions without synAFGPmi.
recognition process in which those moieties of the inhibitor capable of forming hydrogen bonds match the pattern of O atoms in a particular ice face. Because of the adsorbed inhibitors, these faces grow at a much slower rate and, hence, are those that are presented by the affected ice crystals yielding a crystal habitus that is very specific for a particular type of AFP or AFGP.7,11 The features observed in Figure 2 can also be quantified by analyzing the mean crystal radius and the corresponding growth rate using LSW theory (see Figure 3b). Clearly, two different growth regimes can be seen in the 60 µg mL-1 data (red circles): one initial regime that shows about the same rate as that in pure sucrose solutions (∼0-30 min) and a second much slower growth regime (∼30-120 min). Although the ice crystal growth processes revealed in Figure 3a,b are quite different, the total ice volume fractions in all experiments are effectively constant. The two different growth regimes observed in Figure 3b can be visualized by two processes that both take place during the growth of ice crystals in a recrystallization process (see the inset to Figure 3b). First, water molecules diffuse through the intermediate liquid from the small to the large ice crystals. In addition, these water molecules are built into the ice crystal lattice; i.e., a liquid-to-ice phase transfer of water molecules occurs. We believe that this model can account for the observed two growth regimes and their kinetics. The faster
(5)
where t is time and ts is the moment where β(t) ) 0.5; i.e., both processes contribute equally to the overall growth kinetics. For β(t) < 0.5 process 1 (diffusional growth) is rate limiting; for β(t) > 0.5 process 2 (liquid-to-ice transfer) is rate limiting. Equation 4 fits the data of the 60 µg mL-1 synAFGPm4 solution nicely (R2 ) 0.99) (see Figure 3b), and similar results were obtained for other synAFGPmi solutions. Note that a time-dependent β(t) was only required for analyzing experiments in which the changeover from diffusion-controlled growth to liquid-to-ice transfer controlled growth occurs during the observation time of 120 min, i.e., for cases when 0 < ts < 120 min. For small synAFGPmi concentrations ts is typically larger than 120 min, and hence, only the diffusion-controlled kd(Q) is obtained because β(t) ) 0 during the entire experiment. In contrast, at large synAFGPmi concentrations ts is very small, yielding only the liquid-to-ice transfer controlled growth rate constant ki(Q), because β(t) ) 1 during the entire experiment. If our interpretation of the two rate constants kd(Q) and ki(Q) is correct, only ki(Q) contains information about the inhibition efficiency of a particular antifreeze protein or antifreeze glycoprotein, while kd(Q) resembles that observed also in pure sucrose solutions. Therefore, we identify that particular rate constant that is limiting the overall ice growth kinetics; i.e., the smaller rate constant in each individual experiment kl(Q) ≡ min(kd(Q), ki(Q)). Note that also for kl(Q) we use the value for Q f 0, kl0, in order to account for the effect of ice volume fraction as discussed above (see also Appendix A.5). The resulting values of kl0(c) are shown in Figure 5 as a function of the inhibitor concentration c. For solutions containing proteins that do not act as inhibitors as well as for solutions with small concentrations of active inhibitors, we expect to observe a rate constant kl0(c) similar to kd0, while for solutions with larger concentrations of active inhibitors we expect a rate constant that is much smaller and representative of the inhibition efficiency of the particular protein (ki0). This behavior is indeed observed in
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Figure 6. Automated object analysis using the UTHSCSA ImageTool. (a) Grayscale image of ice crystal. (b) Determination of the “outer” area and (c) of the “inner” area of an ice crystal; see text. In (b) and (c) the radius is given that was derived from the area.
TABLE 2: Values of the Parameters for the LSW Calculation According to Eq 2 T [°C] ws,eq [wt %] Fsol [kg m-3] cw,eq [mol m-3] -6 -8 -10
46.1 51.9 56.5
1208.4 1239.7 1265.6
Dw [m2 s-1] 1.25 × 10-10 6.42 × 10-11 3.18 × 10-11
36 160 33 128 30 590
Our results also reveal that the monosaccharides synAFGPmi are also potent ice recrystallization inhibitors and that the disaccharide group present in natural AFGP is not required for antifreeze activity, thereby providing clues for the elucidation of the molecular mechanism by which natural AFGP adsorb to and inhibit the growth of ice, a subject much discussed.12,13 We are currently in the process of also synthesizing natural AFGP in order to compare directly the antifreeze inhibition activity of the two classes of peptides. Nevertheless, the synAFGPmi studied here exhibit strong ice growth inhibition activity down to mass concentrations of ci ∼ 3 µg mL-1 or, equivalently, molar concentrations of ∼1 µmol L-1 for synAFGPm5, thus establishing the possibility for potential applications such as increasing the quality and shelf life of frozen food and the cold storage of biological tissues at subfreezing temperatures.38,39 5. Conclusions
Figure 5 for all three synAFGPmi studied here. The lines in Figure 5 are sigmoidal fits to the data of the form
kl0(c) ) kl0(0) -
kl0(0) ci - c 1 + exp s
(
)
(6)
Here, kl0(0) is the value of the growth limiting rate constant kl0(c) for vanishing inhibitor concentration c f 0. Moreover, ci is the inhibitor concentration that represents the turnover concentration from diffusion-limited growth to liquid-to-ice transfer limited growth, i.e., the inflection point of the curve, and s is a parameter that determines the slope of the curve in the turnover region. Note that the above treatment suggests that under ideal conditions at small inhibitor concentrations the observed kl0(c) should tend toward the value of kd0 determined independently above. This is indeed the case as all three curves approach the value of kd0 indicated by the black asterisk in Figure 5. 4.3. Discussion. We suggest that the value of ci determined in the analysis described above is characteristic for the inhibition efficiency of a particular antifreeze agent. Our evaluation reveals that the largest synthetic synAFGPmi studied here, synAFGPm5, shows the strongest inhibition efficiency, as it inhibits ice recrystallization effectively already at a concentration of about ci ) 3.2 ( 0.2 µg mL-1, while synAFGPm4 and synAFGPm3 inhibit ice recrystallization at ci ) 38 ( 7 µg mL-1 and ci ) 978 ( 26 µg mL-1, respectively. This comparison shows that ci decreases by about 1 order of magnitude when going from the trimer to the tetramer and from the tetramer to the pentamer. The most likely reason for this behavior is that larger inhibitors exhibit a larger adsorption constant because they can bind more strongly to the ice faces due to a larger number of moieties that can form hydrogen bonds. This in turn may lead to an irreversible adsorption process, a phenomenon that has been discussed also as a cause for the strong change in thermal hysteresis when going from smaller to larger AFP or AFGP.7 Therefore, it seems reasonable that the strongest changes in ci occur between the smallest effective synAFGPmi, and we expect an asymptotic behavior of ci for very large synAFGPmi because in this picture all of them would adsorb irreversibly. In addition, just because of their geometric size, larger antifreeze molecules are better inhibitors for ice recrystallization. The same sizedependent behavior is also found in natural AFP and AFGP.37 It is interesting to note in this respect that the smallest naturally occurring AFGP is the tetramer.
We have studied experimentally the ice recrystallization kinetics in sucrose solutions in both the absence and presence of synthetic antifreeze glycoprotein analogues synAFGPmi. These data were analyzed with an extension to the classical LSW theory for Ostwald ripening processes. The analysis using this modified LSW theory clearly shows that synAFGPmi exhibit ice growth inhibition activity and that our new method is capable of quantitatively assessing the efficiency of very similar AFGP even at concentrations of j10 µg mL-1, i.e., with a sensitivity that is at least 2 orders of magnitude larger than that typical for quantitative thermal hysteresis measurements. Our measurements also reveal that the disaccharide group present in natural AFGP is not a necessary condition for activity, as the monosaccharide studied here also show ice growth inhibition. Acknowledgment. We are grateful for support by the Deutsche Forschungsgemeinschaft through Sonderforschungsbereich SFB 613. We also thank Bernhard Zobrist for providing the code for the calculation of the water diffusion coefficient and helpful comments on the manuscript. Appendix A.1. Image Analysis. The microphotographs obtained during the annealing time with a digital camera (PixeLINK PL-A662, 1280 × 1024 pixel, field of view at 20× magnification: 383 × 307 µm) were analyzed using the UTHSCSA ImageTool program Version 3.0 (developed at the University of Texas Health Science Center at San Antonio, TX, and available from the Internet at http://ddsdx.uthscsa.edu/dig/itdesc.html). With this software the original microphotographs were first converted into grayscale images, in which black corresponds to a gray level of 0 and white to 255 (see Figure 6a). As the optical depth and, hence, the gray level of the ice crystals are very similar to those of the surrounding liquid, the ice crystal contours were analyzed to derive crystal size. However, both the gray level and the line width of the contour lines varied between different experiments (dependent on crystal size, image focus, image contrast, etc.). Hence, we developed a standard procedure within ImageTool that gave consistent and reproducible results. In this procedure each grayscale image was converted into two different binary images. For this purpose the “find objects” command was used with a typical threshold value of about 210 (depending on the image contrast). First, each pixel with a gray level below the threshold value (i.e., darker pixels) is converted to a red pixel, while pixels with a gray level above the threshold were not altered. The resulting image is shown in Figure 6b. The area of the crystal is now
Ice Recrystallization Kinetics determined as the sum of all red pixels and all pixels within the red circle, i.e., including the full contour line of the crystal. We refer to this value as the “outer” area of the crystal. Second, all pixels in the original grayscale image with a gray level above the threshold (i.e., the brighter pixels) were converted to blue pixels (see Figure 6c). The resulting blue circle was used to derive the corresponding “inner” area of the crystal that excludes the contour line. The realistic area of the crystal is then assumed to be the mean of the two “outer” and “inner” areas values. In addition to the area A, also the number of crystals N, their perimeter P, and their roundness R were determined. Note that R is computed as R ) 4πAP-2 with R values between 0 and 1, where 1 characterizes a perfectly circular object. The crystal radius was calculated as the half of the Feret diameter d, which assumes the object covers the same area as a regular circle, i.e., d ) (4Aπ-1)1/2. These radii are also indicated in Figure 6b,c. In total, two average values of each measured quantity described above were obtained from each image. The mean of these values were used in the analysis presented in the main text. All these procedures were performed for each individual microphotograph containing between 1000 and 50 crystals. In total, the automated analysis of a 120 min experiment on a standard PC took between about 4 min (for a total of ∼15 000 crystals in pure sucrose solutions) and 10 min (for ∼100 000 crystals in solutions containing synAFGPmi). A.2. Ice Volume Fraction. The ice volume fraction Q for every image was determined as the product of the average area times the number of crystals divided by the total investigated image area. Here, we assume a constant thickness of the sample film in the examined region and a cylindrical shape of the crystals between the lower and upper coverslip. Sensitivity studies showed that the errors introduced by ignoring those crystals at the edges of an image that are not fully visible and by using the product of the average area and crystal number instead of the sum of all areas are negligible. The mean ice volume fraction of all images in one experiment was used for further analysis. A.3. Data Evaluation. During the initial phase of this study we encountered several experimental difficulties that retarded a meaningful analysis of the corresponding data. On the basis of this experience, we have established criteria that need to be fulfilled by the experimental data in order to provide an informative and consistent data evaluation. In some of the initial experiments we observed movement of crystals within the film at the start of an experiment. We believe these movements were due to a flow of the liquid sucrose solution induced by evaporation of water at the edges of the coverslips. These effects were minimized by using a thinner film of ∼10 µm. To ensure that evaporation of water did not affect the solute concentration in the film, we monitored the ice volume fraction Q during all experiments. Hence, only those data in which Q varied by less than 1% during the course of an experiment were considered in the analysis. Another potential problem is the fact that accretion of ice crystals contributed significantly to the kinetics of the observed ice recrystallization, thus affecting the rate constants attributed to diffusion-limited growth. We have developed two means of excluding the effects of accretion. First, we studied the roundness R of the investigated ice crystals. A typical round crystal exhibits a value of ∼0.9. However, if two crystals are very close to each other and coalesce, their mutual R value is normally below ∼0.6. As accretion is more important in the initial phase of a particular experiment, data were considered once the roundness in at least 90% of the ice crystals became greater
J. Phys. Chem. B, Vol. 113, No. 9, 2009 2871
Figure 7. Ice recrystallization rate constant as a function of synAFGPm3 concentration. The black triangles show the original data at the measured ice volume fraction Q (mean of several measurements at the same synAFGPm3 concentration). The red circles are the same data scaled to zero ice volume fraction Q ) 0 according to eq 10.
than 0.7 and never fell below 0.7 again. Only time intervals with a minimum duration of 20 min were considered. Second, it has been shown previously that the growth kinetics due to accretion obeys the same r3 vs t dependence as that for diffusionlimited growth described in eq 1,29 with the difference that the accretion rate constant ka is larger than kd. Hence, when accretion dominates initially, but diffusion in the later phase of an experiment, the apparent rate constant will change continuously from ka toward kd. As a result, the r3 vs t dependence becomes nonlinear. Therefore, we inspected the quality of the fits to eqs 1 and 4 by checking the standard deviation of the rate constant derived from the fitting procedure. For measurements performed at -8 and -10 °C, those experiments with an absolute standard deviation in k of larger than 0.03 were not considered, and for measurements at -6 °C (with larger k values) those experiments with an absolute standard deviation in k of larger than 0.12 were not considered. These threshold values were determined empirically by visually inspecting a number of individual data plots and fits. A.4. LSW Rate Constant Calculation. The ice recrystallization rate constant for Q f 0 predicted by LSW theory, kLSW, was calculated according to the equation
kLSW )
8σΩ2Dwcw,eq 9RT
(2a)
In the following, we explain how we derived the values of the individual coefficients on the right-hand side of eq 2a. R ) 8.314 47 J K-1 mol-1 is the gas constant and T absolute temperature in K. σ, the ice/liquid interface energy: We are not aware of any data for σ in aqueous solutions of sucrose (or of any other solute). Therefore, we used in our calculation instead the ice/ liquid interface energy in pure water at 0 °C, σ ) 0.033 J m-2, which is the only available value that was determined experimentally.40 Ice nucleation studies41 indicate that the temperature dependence of this value is minor, with deviations of less than 3% when going from 0 to -8 °C. Ω, the molar volume of ice: We calculated Ω from the published density of hexagonal ice42 at 0 °C, yielding a value of Ω ) 1.96519 × 10-5 m3 mol-1. The temperature dependence of this value in the investigated temperature range is negligible (∼0.2%).42
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Budke et al.
cw,eq, the molar concentration of water in a sucrose solution that is in equilibrium with an ice crystal of infinite size: We calculated cw,eq with
cw,eq )
(
Fsol 1 -
ws,eq 100
)
Mw
(7) kl0 )
where ws,eq is the weight fraction of sucrose (in wt %), Fsol is the solution density, and Mw is the molar mass of water (Mw ) 0.018 0153 kg mol-1). The weight fraction of sucrose in an aqueous solution that is in equilibrium with ice, ws,eq, was derived by fitting the following function to ice melting point data from the literature:43-45
ws,eq ) 90.009 - 29.674 exp(0.47463Tm) 60.215 exp(0.059252Tm)
(8)
with ws,eq given in wt % and the ice melting temperature Tm given in °C. This parametrization is valid over the temperature range from 0 to - 30 °C. The density of sucrose solutions Fsol (in g cm-3) can be calculated according to45
Fsol ) 0.43651 + 0.56152 exp(0.0069027ws)
ice volume fractions, experimental setups, etc.). Therefore, we account for the effect of synAFGPmi on the ice volume fraction Q and, thereby, on the rate kl(Q) in the data analysis by scaling all measured kl(Q) to zero ice volume fraction, similar to pure sucrose solutions according to
(9)
where ws is the sucrose weight fraction in wt % and the fit is valid between 0 and 84 wt % at 20 °C. For the calculation of cw,eq we ignored the temperature dependence of Fsol between 20 and -10 °C. Dw, the diffusion coefficient of water molecules in sucrose solutions: No experimental data of Dw in aqueous sucrose solutions are available for the temperatures and concentrations relevant for this study. Therefore, the values for Dw were derived from the model by He et al.46 This model was developed to predict the water self-diffusion coefficient in aqueous sugar solutions over a wide range of concentrations and temperatures. The model was found to be in good agreement with experimental data for sucrose solutions.47 We also tried to directly extrapolate the available experimental data47-49 to low temperatures using the Vogel-Tammann-Fulcher (VTF) equation Dw ) D0 exp[-CT0/(T - T0)]. However, the fitting parameters (D0, C, T0) at different sucrose concentrations gave inconsistent results, which is why we prefer a physically consistent extrapolation of Dw using the He et al.46 model. Note, however, that Dw values derived from the VTF equation were generally larger by about 60% than those predicted by the model of He et al.46 As kLSW is directly proportional to Dw, a similar deviation in kLSW results, which is within the scatter of the experimental data shown in Figure 4. The various values that resulted form the analysis above at the three different temperatures investigated here are summarized in Table 2. These values were used in the calculation of kLSW as given in Table 1. A.5. Scaling to Zero Ice Volume Fraction. The addition of synAFGPmi or any other solute to a sucrose solution directly affects the ice melting point curve and, hence, the ice volume fraction. However, because of the small amounts of synAFGPmi that were available, it was impractical to determine the corresponding changes in the ice-solution liquidus line of the phase diagram in each case. Moreover, we want to provide rate constants that can be compared also to other conditions and experimental protocols (e.g., other stock solution concentrations,
kd0kl(Q) kd(Q)
(10)
The results of this procedure are shown in Figure 7 for the case of synAFGPm3. The raw data kl(Q) are shown in black, and the red data points are those scaled to Q ) 0, i.e., the values of kl0. It can be seen that the original values are higher than the scaled values, which is consistent with the fact that higher ice volume fractions lead to larger recrystallization rate constants (see eq 3 and Figure 4). In addition, the scaled data for c f 0 are consistent with the results for pure sucrose solutions described above. Furthermore, it is important to note that the derived turnover concentration ci is not affected by the scaling because its value is nearly identical in both data sets (∼995 ( 29 µg mL-1 in the original data vs ∼978 ( 26 µg mL-1 in the scaled data set). References and Notes (1) Talapin, D. V.; Rogach, A. L.; Shevchenko, E. V.; Kornowski, A.; Haase, M.; Weller, H. J. Am. Chem. Soc. 2002, 124, 5782–5790. (2) Shen, Z.; Zhao, Z.; Peng, H.; Nygren, M. Nature (London) 2002, 417, 266–269. (3) Griffith, M.; Ewart, K. V. Biotechnol. AdV. 1995, 13, 375–402. (4) Regand, A.; Goff, H. Food Hydrocolloids 2003, 17, 95–102. (5) Knight, C. A.; DeVries, A. L.; Oolman, L. D. Nature (London) 1984, 308, 295–296. (6) Knight, C. A.; Wen, D.; Laursen, R. A. Cryobiology 1995, 32, 23– 34. (7) Yeh, Y.; Feeney, R. E. Chem. ReV. 1996, 96, 601–617. (8) Zachariassen, K. E.; Kristiansen, E. Cryobiology 2000, 41, 257– 279. (9) Franks, F.; Mathias, S. F.; Hatley, R. H. M. Philos. Trans. R. Soc. London, Ser. B 1990, 326, 517–533. (10) Smallwood, M.; Bowles, D. J. Philos. Trans. R. Soc. London, Ser. B 2002, 357, 831–846. (11) Davies, P. L.; Baardsnes, J.; Kuiper, M. J.; Walker, V. K. Philos. Trans. R. Soc. London, Ser. B 2002, 357, 927–933. (12) Harding, M. M.; Anderberg, P. I.; Haymet, A. D. J. Eur. J. Biochem. 2003, 270, 1381–1392. (13) Tachibana, Y.; Fletcher, G. L.; Fujitani, N.; Tsuda, S.; Monde, K.; Nishimura, S. I. Angew. Chem., Int. Ed. 2004, 43, 856–862. (14) Regand, A.; Goff, H. D. J. Dairy Sci. 2006, 89, 49–57. (15) Yang, D. S. C.; Sax, M.; Chakrabartty, A.; Hew, C. L. Nature (London) 1988, 333, 232–237. (16) Chakrabartty, A.; Yang, D. S. C.; Hew, C. L. J. Biol. Chem. 1989, 264, 11313–11316. (17) Knight, C. A.; Cheng, C. C.; DeVries, A. L. Biophys. J. 1991, 59, 409–418. (18) Knight, C. A.; Wierzbicki, A.; Laursen, R. A.; Zhang, W. Cryst. Growth Des. 2001, 1, 429–438. (19) DeVries, A. L. Science 1971, 172, 1152–1155. (20) Knight, C. A.; Hallett, J.; DeVries, A. L. Cryobiology 1988, 25, 55–60. (21) Smallwood, M.; Worrall, D.; Byass, L.; Elias, L.; Ashford, D.; Doucet, C. J.; Holt, C.; Telford, J.; Lillford, P.; Bowles, D. J. Biochem. J. 1999, 340, 385–391. (22) Tomczak, M. M.; Marshall, C. B.; Gilbert, J. A.; Davies, P. L. Biochem. Biophys. Res. Commun. 2003, 311, 1041–1046. (23) Liu, S. H.; Ben, R. N. Org. Lett. 2005, 7, 2385–2388. (24) Lifshitz, I. M.; Slyozov, V. V. J. Phys. Chem. Solids 1961, 19, 35–50. (25) Wagner, C. Z. Elektrochem. 1961, 65, 581–591. (26) Hagiwara, T.; Hartel, R. W.; Matsukawa, S. Food Biophys. 2006, 1, 74–82. (27) Sutton, R. L.; Lips, A.; Piccirillo, G.; Sztehlo, A. J. Food Sci. 1996, 61, 741–745. (28) C. S. Jayanth, P. N. J. Mater. Sci. 1989, 24, 3041–3052. (29) Baldan, A. J. Mater. Sci. 2002, 37, 2171–2202.
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