Temperature Dependence and Anisotropic E ﬀ ects in the Thermal Properties of Hen Egg-White Lysozyme Crystals

: We measured the thermal conductivity ( λ ) and thermal di ﬀ usivity ( α ) of tetragonal hen egg-white lysozyme (HEWL) crystals by the transient short-hot wire method. The crystals were grown by two di ﬀ erent methods: Magnetically levitated crystals were realized with a superconducting magnet, the c -axis of which was perpendicularly orientated in the direction of the wire, and naturally grown crystals realized by the two-liquid method, grown randomly. We conﬁrmed the temperature dependence in both the λ and α properties by measuring the variations in temperature and by statistical analysis. These properties di ﬀ ered slightly depending on the presence or absence of a magnetic ﬁeld applied during the crystal growth. We hypothesize that the di ﬀ erence originated from the orientation of the crystals caused by the magnetic ﬁeld. The statistical analyses demonstrated the possibility that asymmetric thermal conduction in the protein crystals provides anisotropic e ﬀ ects of the thermal properties.


Introduction
Biophysical researchers know from experience that the appropriate temperature control is one of the most dominant factors in realizing high-quality protein crystals. Thermophysical researchers are confident that the thermal control should be an approach in considering the crystallization process as a heat and mass transfer phenomena. We believe that the thermal properties of protein crystals would advance the understanding of crystal growth as a heat and mass transfer phenomena, and could provide guidance for engineered protein crystal applications. Unfortunately, most of the thermal properties of protein crystals have long remained unknown. This is attributed to various practical difficulties in the measurement of the crystals, which are characterized by smallness, fragility, and delicacy.
We recently succeeded in measuring the thermal conductivity λ (W/(mK)) and thermal diffusivity α (m 2 /s) of hen egg-white lysozyme (HEWL) crystals [1,2]. A key to this success is that we jointly utilized the transient short-hot wire method [3][4][5][6] and the technique of magneto-Archimedes levitation [7,8]. The thermal properties of λ and α can quantitatively clarify which of the crystals and the solution play a major role in the process of heat transfer at the solid-fluid interface of the crystals. The measurement of λ and α has the potential to open up new technologies for quality protein crystallization. In actuality, however, the magneto-Archimedes levitation requires the large apparatus of a superconducting magnet. In this paper, HEWL was crystallized not only by the trustworthy method of magnetic levitation, but also by a simpler two-liquid method. Here, we present the temperature dependencies of λ and α of HEWL crystals, and statistically analyze both results. We will refer to the effect of magnetic field on the thermal properties.

Research Strategy
Protein crystallization has been conventionally carried out, in many cases, by the hanging-drop method or the sitting drop method, where the solution volume is only a few µL. Only small-sized crystals can be grown by these methods, and they are unsuitable for measurement by the transient short-hot wire method. Therefore, we adopted the batch method to produce large tetragonal HEWL crystals.
In order to grow the tetragonal HEWL crystals in the vicinity of the wire, we used two different methods: The magneto-Archimedes effect and the two-liquid method. The magneto-Archimedes effect [8] is a trustworthy method, and realizes a containerless condition [9][10][11][12][13][14]. Owing to the repulsive force between the HEWL crystals (diamagnetic subject) and the paramagnetic protein solution, the crystals levitate to the air-liquid interface of the solution, and grow with the crystal's c-axis coinciding with the direction of magnetic field. These crystals shall be labeled as magnetically levitated crystals (MLC) in this paper. By precisely positioning the wire to the liquid-air or liquid-liquid interface, the MLC includes the wire into the crystals. This method is superior in the reproducibility of measurement. More detailed information about this method has been presented elsewhere [8,15].
In the other method, naturally grown crystals (NGC) were realized by the two-liquid method [16,17], which does not use a superconducting magnet. The principal of this method is very simple, only requiring the mounting of the lighter density of a protein solution on a higher density solution. The crystals grow randomly between the interface of the solutions. All that is needed is how to fix the wire in the vicinity of the interface. The experimental conditions of the MLC and NGC in the present study were the same, except for the application of the magnetic field.

Transient Short-Hot Wire Probe
In previous studies [1,2], the probe was made of tungsten (W), but the W probe was so fragile that long-time use in measuring was almost impossible. In this study, the probe was made of platinum (Pt). Figure 1A is a photo of the new transient short-hot wire probe made of Pt. Figure 1B shows a schematic illustration of this probe. The diameter and the length of the Pt wire (a) were 41.4 µm and 7.1 mm, respectively. The diameter of the lead terminals was 1.0 mm. Both the short-hot wire and the lead terminals were made of the same Pt material; as a consequence, this prevents thermoelectric noise signals at the connection between the wire and the terminals from heating the short-hot wire during the measurement. The wire was horizontally fixed to the terminals (b), and the terminals were fastened with acrylic jigs (c). The responses of the current (d) and the voltage (e) were detected with high sensitivity. Figure 2A is a photo of the crystallization vessel that was newly developed for the two-liquid method. Figure 2B shows the schematic illustration of the vessel. The inert solvent of fluorinert (FC-3283, 3M Japan, Co. Ltd., Tokyo, Japan.), with a density of 1830 kg/m 3 at 298 K, was placed under the protein solution. The remarkable aspect of this vessel is that the level of the interface is adjustable by compressing a Symmetry 2020, 12, 1279 3 of 12 piston syringe connected to the vessel bottom. Owing to this system, the wire was completely buried into the crystals. The diameter and the length of the wire were 50 µm and 7.5 mm, respectively. application of the magnetic field.

Transient Short-Hot Wire Probe
In previous studies [1,2], the probe was made of tungsten (W), but the W probe was so fragile that long-time use in measuring was almost impossible. In this study, the probe was made of platinum (Pt). Figure  1A is a photo of the new transient short-hot wire probe made of Pt. Figure 1B shows a schematic illustration of this probe. The diameter and the length of the Pt wire (a) were 41.4 µm and 7.1 mm, respectively. The diameter of the lead terminals was 1.0 mm. Both the short-hot wire and the lead terminals were made of the same Pt material; as a consequence, this prevents thermoelectric noise signals at the connection between the wire and the terminals from heating the short-hot wire during the measurement. The wire was horizontally fixed to the terminals (b), and the terminals were fastened with acrylic jigs (c). The responses of the current (d) and the voltage (e) were detected with high sensitivity.   Figure 2A is a photo of the crystallization vessel that was newly developed for the two-liquid method. Figure 2B shows the schematic illustration of the vessel. The inert solvent of fluorinert (FC-3283, 3M Japan, Co. Ltd., Tokyo, Japan.), with a density of 1830 kg/m 3 at 298 K, was placed under the protein solution. The remarkable aspect of this vessel is that the level of the interface is adjustable by compressing a piston syringe connected to the vessel bottom. Owing to this system, the wire was completely buried into the crystals. The diameter and the length of the wire were 50 µm and 7.5 mm, respectively.  Figure 3 shows a schematic illustration of the electric circuit system. The use of a new AD converter (USB-2408, Measurement Computing Co. Ltd., USA, (Japanese domestic agency is NT techno Commerce Co., Ltd., Yokohama, Japan)) made it simpler to control the operation of the electric heating wire than in the previous system. We also designed an automatic measurement system with the equipment, e.g., a DC power supply (6240B, ADCMT Co. Ltd., Saitama, Japan), and standard resistance (10 Ω, Yokogawa Test & Measurement Co. Ltd., Tokyo, Japan). These improvements enabled an excellent performance for long-term use.  Figure 3 shows a schematic illustration of the electric circuit system. The use of a new AD converter (USB-2408, Measurement Computing Co. Ltd., USA, (Japanese domestic agency is NT techno Commerce Co., Ltd., Yokohama, Japan)) made it simpler to control the operation of the electric heating wire than in the previous system. We also designed an automatic measurement system with the equipment, e.g., a DC power supply (6240B, ADCMT Co. Ltd., Saitama, Japan), and standard resistance (10 Ω, Yokogawa Test & Measurement Co. Ltd., Tokyo, Japan). These improvements enabled an excellent performance for long-term use.

Crystallization Conditions
Gadolinium chloride hexahydrate (GdCl3 6H2O, Wako Co. Ltd., Osaka, Japan) was used as a precipitant agent to make the protein solution paramagnetic. The concentrations of the GdCl3 and HEWL (MP Biomedicals, Inc., USA, (Japanese domestic agency is Yashima Pure Chemicals Co., Ltd., Osaka, Japan))) were 0.362 mol/kg and 6.53 wt%, respectively. The pH of the solution was adjusted to 3.30. These conditions were optimized so that both the magneto-Archimedes levitation and tetragonal HEWL crystallization could be simultaneously realized at 293 K. The binding site of the gadolinium atom in the lysozyme crystals has been clarified [18].

Magnetic Conditions
In the case of the MLC, the magnetic flux density (B) was set to 4.00 T at the center of the coil. The vertically upward magnitude of (B⋅∇)B became −148.1 T 2 m −1 at the top surface of the protein solution, where the vertically maximum value of (B⋅∇)B was realized in a superconductive magnet (JMTD-6T100EF3, JASTEC SUPERCONDUCTOR Co., Ltd., Kobe, Japan). These conditions were the same as in previous studies [1,2]. When supplying the current in the measurement of the MLC, we briefly stopped applying the magnetic force to avoid the induction of the Lorentz force.

Visualization System
Prior to the measurements, we checked whether the short-hot wire was completely included into the crystals. Figure 4a shows the equipment used in monitoring the crystallization process in the magnet. In this photo, the vessel and other equipment were taken out from the superconducting magnet bore. In the normal setting, the crystallization can be observed from the side of the vessel by using a Charge-Coupled Device (CCD) camera (OH414, Olympus Co., Ltd., Tokyo, Japan) and a rigid scope (Industrial Rigid Scope Type 5, Olympus Co., Ltd., Tokyo, Japan). The crystal growth was automatically photographed every 15 min, using a dedicated software (Image Pro Express, Media Cybernetics Inc., USA).

Crystallization Conditions
Gadolinium chloride hexahydrate (GdCl 3 6H 2 O, Wako Co. Ltd., Osaka, Japan) was used as a precipitant agent to make the protein solution paramagnetic. The concentrations of the GdCl 3 and HEWL (MP Biomedicals, Inc., USA, (Japanese domestic agency is Yashima Pure Chemicals Co., Ltd., Osaka, Japan))) were 0.362 mol/kg and 6.53 wt%, respectively. The pH of the solution was adjusted to 3.30. These conditions were optimized so that both the magneto-Archimedes levitation and tetragonal HEWL crystallization could be simultaneously realized at 293 K. The binding site of the gadolinium atom in the lysozyme crystals has been clarified [18].

Magnetic Conditions
In the case of the MLC, the magnetic flux density (B) was set to 4.00 T at the center of the coil. The vertically upward magnitude of (B·∇)B became −148.1 T 2 m −1 at the top surface of the protein solution, where the vertically maximum value of (B·∇)B was realized in a superconductive magnet (JMTD-6T100EF3, JASTEC SUPERCONDUCTOR Co., Ltd., Kobe, Japan). These conditions were the same as in previous studies [1,2]. When supplying the current in the measurement of the MLC, we briefly stopped applying the magnetic force to avoid the induction of the Lorentz force.

Visualization System
Prior to the measurements, we checked whether the short-hot wire was completely included into the crystals. Figure 4a shows the equipment used in monitoring the crystallization process in the magnet. In this photo, the vessel and other equipment were taken out from the superconducting magnet bore. In the normal setting, the crystallization can be observed from the side of the vessel by using a Charge-Coupled Device (CCD) camera (OH414, Olympus Co., Ltd., Tokyo, Japan) and a rigid scope (Industrial Rigid Scope Type 5, Olympus Co., Ltd., Tokyo, Japan). The crystal growth was automatically photographed every 15 min, using a dedicated software (Image Pro Express, Media Cybernetics Inc., USA). Figure 4b-d is a series of photographs taken from the initial state of the solution. As can be seen in Figure 4b, the short-hot wire was located at a slightly lower position from the interface. Figure 4c is a photo taken at 15 h after the start. Many fine HEWL crystals appeared at the interface, and the wire was partially covered with crystals. At 48 h from the start (Figure 4d), the wire was completely included into the crystals, and measurement became possible. Figure 5a shows the measurement system for the two-liquid method. Figure 5b is a photo at the initial state, and Figure 5c is a photo taken at three days after the start, respectively. As seen from these photos, many fine crystals appeared on the vessel sidewall.  Figure 4b-d is a series of photographs taken from the initial state of the solution. As can be seen in Figure 4b, the short-hot wire was located at a slightly lower position from the interface. Figure 4c is a photo taken at 15 h after the start. Many fine HEWL crystals appeared at the interface, and the wire was partially covered with crystals. At 48 h from the start (Figure 4d), the wire was completely included into the crystals, and measurement became possible. Figure 5a shows the measurement system for the two-liquid method. Figure 5b is a photo at the initial state, and Figure 5c is a photo taken at three days after the start, respectively. As seen from these photos, many fine crystals appeared on the vessel sidewall.

Measurement
As the first step of the measurement, a low current (14 mA) was applied to the circuit to detect the temperature of the wire. Next, a large current was instantly supplied, and the probe was transiently heated. The magnitude of the current was optimized in advance so that only the wire was locally heated and the increase in temperature of the lead terminals became negligibly small. In this study, the electric current was set to 200 mA, and was maintained for 1.5 s to heat the wire. The wire temperature Tv against the current-flow time t was linearly approximated as follows: where T is the initial temperature of the wire. The coefficient a and the intercept b are determined by the

Measurement
As the first step of the measurement, a low current (14 mA) was applied to the circuit to detect the temperature of the wire. Next, a large current was instantly supplied, and the probe was transiently heated. The magnitude of the current was optimized in advance so that only the wire was locally heated and the increase in temperature of the lead terminals became negligibly small. In this study, the electric current was set to 200 mA, and was maintained for 1.5 s to heat the wire. The wire temperature T v against the current-flow time t was linearly approximated as follows: where T 0 is the initial temperature of the wire. The coefficient a and the intercept b are determined by the least-squares (LSQ) fitting. In this study, the a and the b were decided by the data in the range of 0.1 to 0.6 s. Figure 6 shows an example of fitting the data to the above equation.  The heat transfer volume and the response were numerically estimated, and the following relation was arrived at [3].
where θ v and Fo are the non-dimensional averaged temperature of the short-hot wire and the Fourier number, respectively. The θ v and Fo are given by the following equation: where q v , r i , and t are the volumetric heat generation, the short-hot wire radius, and the current-flow time, respectively. The λ and α are the thermal conductivity and thermal diffusivity of the object surrounding the hot wire, respectively. Finally, the λ and α of the protein crystals are obtained by solving Equations (1)- (3).
After the experiments, we removed the wire from the vessel and verified that the wire was thoroughly included into the HEWL crystals (data not shown).

Results
Measurements were carried out by varying the temperature of the MLC to 282.1, 287.2, 290.4, and 294.9 K, and varying the temperature of the NGC to 284.3 ± 0.1, 286.5 ± 0.6, 289.1, 291.5 ± 0.3, 293.2, and 295.2 K. All the measurements were repeated five times at regular intervals of 3-5 min.
The results of λ and α are summarized in Figure 7a,b, respectively. The data series of the MLC by the Pt wire are shown by the circles (O). The previous data of the MLC by the W wire [1,2] are shown by the squares ( ). The data series of the NGC are shown by the triangles (∆). The thermal conductivities of the MLC and the NGC were expressed as λ MLC and λ NGC , respectively. Similarly, the thermal diffusivities of the MLC and the NGC were expressed as α MLC and α NGC , respectively. Since these results reflect the thermal properties of the crystals in the vicinity of the wire, the effects of the solution or the air conditions become small. The thermal properties of the protein solution and the fluorinert are exhibited by the crosses (×) and the diamonds (◊), respectively. Both measurements were carried out with the Pt wire, and repeated five times at regular intervals of 3 min. The dash-dotted line indicates the reference data of water [19]. The bold broken line represents the reference data of fluorinert [20].

Thermal Conductivity
As shown in Figure 7a, the λ MLC by the Pt wire was in accord with the previous results by the W wire within 4.8-12.0%, regardless of using a different material for the probe. This agreement represents the measurement reproducibility. The fluorinert shows almost equal values to the reference, demonstrating the advantages of our methods. Another finding of great interest is that λ MLC and λ NGC increase linearly with the temperature. This characterizes the temperature dependence of the thermal conductivity of the HEWL crystals. These results will be analyzed in the section 5.
Both λ MLC and λ NGC were smaller than those of water and the protein solution. The property of the protein solution was much closer to that of water than that of the crystals. This means that the heat on the crystal can be more easily dissipated to the solution than to the crystals. It is interesting that the thermal five times at regular intervals of 3 min. The dash-dotted line indicates the reference data of water [19]. The bold broken line represents the reference data of fluorinert [20].

Thermal Conductivity
As shown in Figure 7a, the λ MLC by the Pt wire was in accord with the previous results by the W wire within 4.8-12.0%, regardless of using a different material for the probe. This agreement represents the measurement reproducibility. The fluorinert shows almost equal values to the reference, demonstrating the advantages of our methods.
Another finding of great interest is that λ MLC and λ NGC increase linearly with the temperature. This characterizes the temperature dependence of the thermal conductivity of the HEWL crystals. These results will be analyzed in the Section 5.
Both λ MLC and λ NGC were smaller than those of water and the protein solution. The property of the protein solution was much closer to that of water than that of the crystals. This means that the heat on the crystal can be more easily dissipated to the solution than to the crystals. It is interesting that the thermal conductivity of the crystals was intermediate between the values of the fluorinert and the protein solution (or water).
The data points of λ MLC and λ NGC were approximated with the LSQ fitting in the range from 282.1 to 294.9 K. The λ MLC and its R square value (R 2 λ_MLC ) were presented as follows: The LSQ fitting result of λ MLC is displayed by a solid line in Figure 7a.
The LSQ fitting result of λ NGC is represented by the small dotted line in Figure 7a. We are aware of the small difference between the data of λ MLC and λ NGC . This will be discussed later.

Thermal Diffusivity
There was a relative fluctuation in the data points of α MLC and α NGC because a small experimental error is likely to be magnified by the power term of the exponential function in Equation (4). Regardless of the fluctuation of the data, there were several findings. For example, the α MLC gradually increased by the temperature, whereas the α NGC slightly decreased. Both results indicate, at least, that the thermal diffusivity of the HEWL crystals changes by temperature.
The present results with a Pt wire were not the same as the previous results with a W wire, especially in the thermal diffusivity. We first suspected that small air bubbles were included into or between the crystals during the magneto-Archimedes levitation. However, since the results with the two-liquid method were close to those with the magneto-Archimedes levitation, the concern about the air bubbles was gone. In the present stage, there is little reliable data as to the thermophysical properties of protein crystals, e.g., HEWL. It was impossible for us to evaluate the accuracy of our results. Further research data is necessary to solve these problems.
The data series of α MLC and α NGC were approximated by the LSQ fitting method in the range of 282.1 to 294.9 K. The α MLC and its R square value (R 2 α_MLC ) were presented as follows: The LSQ fitting result of α MLC is displayed by the solid line in Figure 7b. In the temperature range of 284.2 to 295.2 K, the α NGC and its R square value (R 2 α_NGC ) were presented as follows: α NGC = −2.448 × 10 −9 T + 9.432 × 10 −7 , R 2 fΩ_NGC = 0.748 The LSQ fitting result of α NGC is shown by the small dotted line in Figure 7b. The trends of α MLC and α NGC are statistically analyzed in the next section.

Temperature Dependence
As for each data series of λ MLC , λ NGC , α MLC , and α NGC , the temperature dependence was statistically evaluated by the Kruskal-Wallis test [21]. All the analyses were performed by means of the SPSS ver.15.0, statistical software (SPSS Statics, IBM Co. Ltd., USA) According to the analyses, the p-values of λ MLC , λ NGC , α MLC , and α NGC resulted in 8.84 × 10 −4 , 6.63 × 10 −5 , 8.73 × 10 −4 , and 1.40 × 10 −4 , respectively. These small values suggest that we should reject the null hypothesis that a temperature dependence does not exist. Hence, we decide to adopt the alternative hypothesis that tetragonal HEWL crystals are dependent on temperature in both properties of thermal conductivity and thermal diffusivity.
On the other hand, we do not know why λ increased with temperature and only the data of α NGC decreased. Except for the data of α MLC at 282 K, the trend of α MLC decreases with temperature in common with that of α NGC . With regards to the temperature trend of α, further measurements are necessary.

Magnetic Effect in the Thermal Properties
As shown in Figure 7, the LSQ fitting lines of λ MLC and λ NGC were slightly different from each other. Similarly, the LSQ fitting lines of α MLC and α NGC were not in accord. We suspect that the difference in these may be derived from the application of the magnetic field in the crystal growth. In order to clarify the magnetic effect in the properties, the difference between the data of λ MLC and λ NGC were statistically investigated by using the analysis of covariance [22]. The difference between the data of α MLC and α NGC was also examined in a similar way. In practice, the thermal properties (λ or α) were defined as dependent variables, and the temperatures were used as independent variables (covariates). The magnetic field was the factor to characterize the properties of the MLC and NGC.
According to the analyses, the p-value between the data of λ MLC and λ NGC was 8.46 × 10 −12 . Such a small value suggests that we should reject the null hypothesis that the temperature change of the λ MLC is regarded as similar to that of λ NGC . Similarly, the p-value between the data of α MLC and α NGC was 1.54 × 10 −5 , also making it valid to reject the null hypothesis. These results support the possibility that the thermal properties of HEWL crystals were affected by the application of the magnetic field.

Is There Anisotropy in the Thermal Properties?
We will discuss why the thermal properties of the HEWL crystals were changed simply by the application of the magnetic field. Figure 8 shows levitating tetragonal HEWL crystals viewed from the top, grown in the same crystallization conditions as the magnetic induction of 4.00 T. As can be seen, the crystals are magnetically orientated along the c-axes of the crystals, being vertical to the paper surface, at the air-liquid interface of the solution [23,24]. The radiating arrows in Figure 9 schematically illustrate the heat transfer process from the wire to the crystals when the wire is heated instantly. The c-axes of the MLC are perpendicular to the direction of the wire (see Figure 9a); in contrast, the NGC grow randomly (see Figure 9b). Considering that the three-dimensional geometries of protein molecules are asymmetric, it is no wonder that the thermal conduction process of MLC was different from that of NGC. If the differences in the thermal properties originated in the macroscopic phenomena of the crystal orientation by the magnetic field, an anisotropic effect in the thermal properties may be considered. Further study is in progress to ensure more comprehensive evidence about the anisotropy of protein crystals.

Conclusions
The thermal conductivity and thermal diffusivity of tetragonal HEWL crystals were measured by means of the transient short-hot wire method. The crystals were grown by two different methods. The magnetically levitated crystals were realized by a superconducting magnet, the c-axis of which was perpendicularly orientated to the direction of the wire. In contrast, naturally grown crystals were realized by the two-liquid method, grown randomly. The temperature dependence in the thermal properties was confirmed by varying the temperature for the measurements. The statistical analyses also demonstrated the possibility that protein crystals may provide anisotropic effects in the thermal properties. In a future study, the anisotropy will be proven by a measurement where the c-axis of the HEWL crystals is orientated parallel to the direction of the wire. We will present an account of these related matters in our next report.