J. Chem. Thermodynamics 53 (2012) 125–130

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J. Chem. Thermodynamics

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Thermodynamic analysis of thermal hysteresis: Mechanistic insightsinto biological antifreezes

Sen Wang a,b, Natapol Amornwittawat a,1, Xin Wen a,⇑a Department of Chemistry and Biochemistry, California State University Los Angeles, Los Angeles, CA 90032, United Statesb Molecular Imaging Program, Stanford University, Stanford, CA 94305, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 November 2011Received in revised form 23 April 2012Accepted 27 April 2012Available online 7 May 2012

Keywords:AdsorptionCrystal growth inhibitionIceFreezing point depressionAntifreeze proteins

0021-9614/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.jct.2012.04.028

⇑ Corresponding author. Tel.: +1 323 343 2310; faxE-mail address: xwen3@calstatela.edu (X. Wen).

1 Current address: Chulabhorn Research Institute, Ba

Antifreeze proteins (AFPs) bind to ice crystal surfaces and thus inhibit the ice growth. The mechanism forhow AFPs suppress freezing is commonly modeled as an adsorption–inhibition process by the Gibbs–Thomson effect. Here we develop an improved adsorption–inhibition model for AFP action based onthe thermodynamics of impurity adsorption on the crystal surfaces. We demonstrate the derivation ofa realistic relationship between surface protein coverage and the protein concentration. We show thatthe improved model provides a quantitatively better fit to the experimental antifreeze activities of AFPsfrom distinct structural classes, including fish and insect AFPs, in a wide range of concentrations. Our the-oretical results yielded the adsorption coefficients of the AFPs on ice, suggesting that, despite the distinctdifference in their antifreeze activities and structures, the affinities of the AFPs to ice are very close andthe mechanism of AFP action is a kinetically controlled, reversible process. The applications of the modelto more complex systems along with its potential limitations are also discussed.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Antifreeze proteins (AFPs), perhaps the most extensively stud-ied biological antifreezes, are a structurally diverse group of pro-teins that allow certain organisms, such as fish, insects, andplants, to survive subzero temperatures [1–4]. In contrast to com-mon antifreeze compounds, such as sodium chloride and glycerol,AFPs can inhibit ice growth by binding (adsorption) to the surfacesof ice crystals and depress the freezing point through a non-colligative effect. This inhibition process has been generallyconsidered as one of the many cases of crystal-growth inhibitionby impurity adsorptions. The resulting difference between themelting and freezing points termed thermal hysteresis (TH) isusually a measure of antifreeze activity by AFPs. Adsorption–inhibition mechanism was first proposed by Raymond and DeVriesto explain the freezing resistance in the case of antifreeze glyco-proteins (AFGPs) from polar fishes [5]. In this model, the adsorp-tion of AFPs was assumed to be irreversibly onto the surfaces inthe paths of growing steps of ice crystals, which obstructs the stepgrowth at these sites and increases the curvature of the growthsteps. According to the Gibbs–Thomson or Kelvin effect, the growthfront will experience a freezing temperature depression as a conse-quence of the effects of surface tension [5,6]. In their original

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ngkok 10210, Thailand.

analysis, surface antifreeze coverage was assumed to be propor-tional to the concentration of the antifreeze and the freezing pointdepression can be described as a function of antifreeze concentra-tion by the given relationship [5]:

DT ¼ 2cMT0

Lqi

2caMT0

1000MW

� �1=2

; ð1Þ

where c is the surface tension, M is the molar mass of water, T0 isthe normal freezing point, L is the molar latent heat, qi is the densityof ice, a is the distribution coefficient, r is the radius of the protein, Cis the concentration of the protein in mg/mL, 2raC is the surfaceantifreeze coverage, N is Avogadro’s number and MW is the molarmass of the protein.

Equation (1) provides a reasonable fit to the experimentalobservations in the low antifreeze concentration range, but isbiased toward fitting the data in the high antifreeze concentrationrange, especially in the cases that the TH activities of the anti-freezes appear to approach maximum values at high concentra-tions [5]. The assumption that surface antifreeze coverage isproportional to the antifreeze concentration overestimates theadsorption coverage [7–10] and has been shown to be unrealisticby modern experiments with (solid + fluid) systems [11]. In orderto overcome the large deviation of the fitting, especially at highantifreeze concentrations, kinetic descriptions of TH activity wereproposed [7,12,13,14]. These kinetic models are able to describethe experimental observations at high antifreeze concentrations,but inadequate to account for the observed TH activities at low

FIGURE 1. Illustration of even adsorption of AFPs on an ice crystal surface describedby the Cabreca–Vermiliyea (C–V) model. The ice binding sites of AFPs shown inblack dots are considered as impurity particles on the ice crystal surface and everytwo adsorbed AFPs are separated by a distance, L, along the y axis. The black curvesare the arrested growth segments. h is the terrace height and rc is the critical radiusof the curved step. The growths of step segments between AFPs are inhibited when2rc is equal to L. Here, h does not necessarily to be as big as a whole AFP moleculesince only a few amino acids on the so-called ice-binding motif are generallyinvolved in the ice affinity of an AFP [34]. According to X-ray crystallographicstudies, the distance between two adjacent oxygen atoms in ice can be x Å [67] andh can be n � x Å (n = 1,2,3, . . .).

126 S. Wang et al. / J. Chem. Thermodynamics 53 (2012) 125–130

antifreeze concentrations. Langmuir isotherm has been used in thekinetic models by Burcham et al. and Can and Holland to relate thefractional ice surface coverage to the protein concentration, how-ever, a questionable assumption that the fractional ice surface cov-erage is proportional to the observed TH activity of specific AFPwas used in the derivation of the relationship between TH activityand the protein concentration [7,12]. It has been recently sug-gested that different AFPs may have different structures and THactivities, but have very similar ice surface coverage [13].

In this study, we provide an improved model for the adsorptioninhibition mechanism of action of antifreeze proteins. We make athermodynamic argument for the depressed freezing point bythese biological antifreezes (i.e., the TH activity of AFPs) by com-bining the Langmuir isotherm that is to describe the adsorptionof AFPs onto the ice crystal surface, and the extreme limit in thestep-pinning model of Cabrera and Vermilyea (C–V model) thatthe growth of step is inhibited when step length are shorter thanthe critical length. A more realistic relationship of the fractionalice surface coverage of AFPs and the AFP concentration is derivedin this work. The theoretical TH values obtained by using the im-proved model fit the experimental results of different types of AFPsvery well. The apparent maximum freezing point depression andthe free energy change of the adsorption of an AFP can also be esti-mated using the theoretical model, which are comparable to thepreviously reported values. New insights into the mechanism ofaction of biological antifreezes are provided.

2. Theoretical methods

According to the adsorption–inhibition mechanism, the adsorp-tion of AFPs onto the ice surfaces will inhibit the ice furthergrowth. Assuming that the adsorption of antifreezes on the icecrystal surface obeys the Langmuir isotherm (no interaction be-tween the proteins adsorbed onto the ice crystal surface), the frac-tional ice crystal surface coverage of the adsorbates (i.e., AFPs) canbe described as [15]

h ¼ Kc1þ Kc

; ð2Þ

K is the Langmuir adsorption coefficient and is defined by theexpression K = exp (�DGa/RT), where DGa is the Gibbs free energychange of adsorption, R is the gas constant, and T is the temperaturein Kelvin; and c is the concentration of the adsorbate (or theantifreeze).

The inhibitory effect of impurity particles on crystal growth hasbeen successfully described using the step-pinning model of Cab-rera and Vermilyea (C–V model) [16,17]. As the molar amount ofAFPs is much less than that of ice, we can consider AFPs as impurityparticles in the C–V model. We assume that AFPs adsorb evenlyonto the surfaces of ice crystals and every two adsorbed AFPs areseparated by a distance, L, along the y axis (figure 1). Here we ap-peal to thermodynamic reasoning by considering an extreme con-dition in the C–V model that the ice growths of step segmentsbetween AFPs are inhibited when twice the critical radius of thecurved step, rc, is equal to L (i.e., L = 2rc) (figure 1). Half-cylindricalice with the radius rc and the terrace height h is thus resulted bythe adsorption of AFPs (figure 1). With the consideration that thenumber of sites available for adsorption per unit area of the growthfront is n and the number of sites adsorbed by an AFP is m, the icesurface coverage by AFPs can be defined as:

h ¼ m

ndL2 ; ð3Þ

where d = h/L. Thus,

cc ¼12

L ¼ Ahmax

h

� �1=2

; ð4Þ

where A ¼ 12

mndhmax

� �1=2, a constant related to the specific AFP and

hmax is the maximum surface coverage of the ice by the AFP, aconstant.

When the radius of half-cylindrical ice resulted by the adsorp-tion of AFPs is very close to rc, the ice continues to grow until2rc = L. The Gibbs free energy change of the AFP solution is:

DG ¼ liDN � lWDN; ð5Þ

where li and lw are the chemical potentials of the growing tiny iceafter AFPs adsorb onto the bulk ice and the tiny amount of waterthat will become ice, respectively, and DN is the change of numberof water molecules from liquid phase to ice (the bulk ice remains nochange). If the contact portions of the bulk ice and the newlyformed ice are in the same phase (i.e., the single-crystal or theice–water interface), there is no surface tension. The surface tensioncan be neglected even though the contact portions of the bulk iceand the newly formed ice are in different phases (e.g., the tiny iceis still in the ice–water interface) since the portion in the ice–waterinterface will become ice crystal very quickly and the total amountof the ice (both the bulk and the newly formed tiny ice) in the solu-tion is very small. li and lw can be expressed as functions of tem-perature and pressure,

lwðT; pÞ ¼ lwðTm;p0Þ � SwDT ð6Þ

and

liðT;pÞ ¼ lwðTm;p0Þ � SiDT þ v iDp; ð7Þ

where Tm is the melting temperature of pure water, p0 is the equi-librium water vapor pressure at the melting temperature of purewater, Si and Sw are the entropies of water and ice, vi is the specificmolecular volume of ice, DT is the freezing point depression by AFPs(i.e., the TH activity of AFPs), and Dp (=pi� pw) is the vapor pressuredifference between the ice–water interface and the ice phase.

At equilibrium, DG = 0, thus

liðT;pÞ ¼ lwðT;pÞ: ð8Þ

Combining equations (6) to (8), gives

ðSw � SiÞDT þ v iDp ¼ 0: ð9Þ

FIGURE 2. Thermal hysteresis (TH) activities of a Type I fish AFP at variousconcentrations. The experimental data are adopted from reference [22] shown assolid dots and standard deviations were not available. The theoretical valuesobtained by equation (12) are shown in a solid line.

FIGURE 3. Thermal hysteresis (TH) activities of a Ca2+-dependent C-type lectin-likeType II fish AFP at various concentrations. The experimental data are adopted fromreference [23] shown as solid dots and standard deviations were not available. Thetheoretical values obtained by equation (12) are shown in a solid line.

S. Wang et al. / J. Chem. Thermodynamics 53 (2012) 125–130 127

Substituting (Sw � Si) = DHf/Tm, where DHf is the latent heat offusion of ice, and pw � pi = c/rc, where c is surface tension intoequation (9), and rearranging the equation, we get

DT ¼ micTm

ccDHf: ð10Þ

Then, substituting equation (4) into equation (10), we get

DTDTmax

¼ ðhÞ1=2

ðhmaxÞ1=2 ; ð11Þ

where DTmax ¼ micTmADHf

, a constant representing the maximum freezing

point depression (i.e., the maximum TH activity) by AFPs. It is im-plied from equation (11) that the maximum freezing point depres-sion is reached until the surface coverage of the ice by the specificAFPs reaches its maximum value.

After substitute equation (2) into equation (11) and rearrange,we obtain

DT ¼ DT 0maxKc

1þ Kc

� �1=2

; ð12Þ

where DT 0max ¼ DTmax

ðhmaxÞ1=2, a coefficient related to the apparent maxi-

mum TH activity of a specific AFP (i.e., the TH activity at the maxi-mum surface coverage of the ice by the specific AFPs).

3. Experimental methods

3.1. Materials

Type III AFP from ocean pout (6500 Da) was purchased from A/FProtein Inc. (Waltham, MA) and used as received. Fresh Milli-Qwater from a Millipore Synergy Ultrapure water system (Billerica,MA) with a minimum resistivity of 18 MX � cm was used in allthe experiments. All the weight measurements were carried outwith an Ohaus Voyager Pro analytical and precision balance (Par-sippany, NJ). The stock type III AFP solution was prepared byweighing the solute and dissolving the solute in a known volumeof water. Dilutions were made, when necessary, to achieve desir-able concentrations of sample solutions.

3.2. Thermal hysteresis measurements

The measurements of thermal hysteresis (TH) activities of typeIII AFP solutions were performed using a DSC 823e (Mettler Toledo,OH) with an HSS7 high sensitivity sensor and a Julabo FT900intracooler chiller (Julabo Company, PA). The method was as de-scribed previously [18–20]. The concentrations of the protein solu-tions were varied from (0.0 to 1.5) mM. Each sample solution wasprepared at least twice and measured three times. The averages ofthe TH values are plotted in figure 5, and the standard deviationswere within 2% of the mean values.

4. Results and discussion

As shown is equation (12), the theoretical TH activity is a func-tion of the AFP concentration. If the concentration of the AFP isvery low, DT increases proportionally to the square root of the con-centration of the AFP (i.e., c ? 0, DT ? 0), which is the case inequation (1); if the concentration of the AFP reaches a high criticalpoint, then h ? hmax and DT ? DTmax. The theoretical saturationcoverage of AFPs agrees well with the one estimated by ellipsom-etry [21]. Furthermore, the TH activity is dependent on the contactarea of the AFP on the ice. Therefore, equation (12) can generally beused to fit the experimental data on various AFPs regardless of thestructures of the specific AFPs.

Here we demonstrated the fitting of the experimental data offive AFPs from distinct structural classes including a rod likea-helical type I fish AFP from shorthorn sculpin (Type I AFP)(figure 2) [22], a Ca2+-dependent C-type lectin-like type II AFPsfrom herring (Type II AFP-Ca2+) (figure 3) [23], a Ca2+-independentC-type lectin-like type II AFP from longsnout poacher (Type II AFP)(figure 4) [24], a globular type III AFP from ocean pout (Type IIIAFP) (figure 5) [25], and a b-helical AFP from the yellow mealwormbeetle (TmAFP) (figure 6) [26]. In order to test the derived relation-ship between the freezing point depression by AFP and the concen-tration of AFP in a wide concentration range, we measured the THactivity of type III AFP using differential scanning calorimetry(DSC). The experimental TH values of type III AFP obtained in thisstudy were in good agreement with the known data [25,27–30].

The theoretical TH activities of the five representative AFPs cal-culated by equation (12) shown as solid lines were in quantitativeagreement with the experimental data shown as solid circles (fig-ures 2 to 6). As shown in figures 2 to 6, the theoretical results are inexcellent agreement with the empirical values. And the estimated

FIGURE 5. Thermal hysteresis (TH) activities of a globular Type III fish AFP atvarious concentrations. The experimental data reported here were measured in thislaboratory and standard deviations were within 2% of the mean values. Thetheoretical values obtained by equation (12) (as shown in a solid line) demonstratea better fit for the high concentration data than those obtained through the bestpossible fit using equation (1) [5] (as shown in a dashed line).

FIGURE 4. Thermal hysteresis (TH) activities of a Ca2+-independent C-type lectin-like Type II fish AFP at various concentrations. The experimental data are adoptedfrom reference 24 shown as solid dots and standard deviations were within 6% ofthe mean values. The theoretical values obtained by equation (12) (as shown in asolid line) demonstrate a better fit for these low concentration data than thoseobtained through the best possible fit using equation (6) in reference [7] (as shownin a dotted line).

FIGURE 6. Thermal hysteresis (TH) activities of an insect AFP, TmAFP, at variousconcentrations. The experimental data are adopted from reference [26] shown assolid dots and standard deviations were within 8% of the mean values. Thetheoretical values obtained by equation (12) are shown in a solid line.

TABLE 1The estimated apparent maximum freezing point depression, adsorption coefficientsand free energy change of adsorption for the AFPs.a

Antifreezeprotein (AFP)

DT 0max (K) K (10�3 M�1) DG at 273 K/(kJ �mol�1)

Type I AFP 0.37 (0.01) 0.968 (0.08) �15.60 (0.08)Type II AFP-Ca2+ 0.58 (0.03) 0.981 (0.05) �15.64 (0.05)Type II AFP 1.44 (0.01) 0.990 (0.05) �15.66 (0.05)Type III AFP 1.10 (0.01) 0.991 (0.03) �15.66 (0.03)TmAFP 10.9 (0.3) 0.980 (0.06) �15.63 (0.06)

a The Langmuir adsorption coefficient, K, was estimated by fitting the experimentaldata using equation (12). The free energy change of adsorption is estimated fromDGa = �RT lnK, where R = 8.314 J �mol�1 � K�1 and T = 273 K. Standard errors are inthe parentheses.

128 S. Wang et al. / J. Chem. Thermodynamics 53 (2012) 125–130

Langmuir adsorption coefficients of the AFPs are listed in table 1.Despite the differences in antifreeze activities and structures, theassociation constants for the AFPs bound to ice are very similarto each other (table 1), suggesting that the AFPs with higherantifreeze activities do not bind to ice more readily, nor do theybind more tightly.

Due to the inapplicability or the poor sensitivity of usual spec-troscopic techniques in ice, a highly scattering and anisotropicmedium, it is nearly impossible to directly measure the affinityof an AFP to ice [12,13,31]. The indirect experimental results ofthe adsorption affinities of fish and insect AFPs to ice demonstratedthat insect AFPs do not bind to ice with higher affinity than fishAFPs regardless of their distinct differences in antifreeze activitiesand structures [13]. Our results agree very well with their experi-mental results and provide a theoretical support for this earlier re-port [13]. By combining the experimental and theoretical results,

we can also estimate the free energy change for the adsorptionof the AFP onto the ice crystal surface by the equation,DGa = �RT lnK (table 1). The estimated values of the free energychange of the adsorption are reasonably comparable to earlier pre-dictions [32,33].

The Langmuir model of impurity adsorption dynamics has beenaccepted [11,17,34]. In this study, a simple Langmuir isotherm isused to describe the surface coverage of AFPs, which has an impli-cation of a possible reversible adsorption of AFPs. It has been de-bated for decades whether the adsorption of antifreeze(glyco)proteins, AF(G)Ps, on ice surfaces is reversible or irreversible[35–38]. The important concerns regarding the irreversibilityadsorption are summarized below.

First, adsorption dynamics of some AF(G)Ps on ice surfaces hasbeen observed by the methods including surface second harmonicgeneration (SSHG) [39,40], temperature gradient thermometry[41], NMR [42,43], FTIR [43] spectroscopy, and Langmuir adsorp-tion process for the binding of the AF(G)Ps to ice was suggested.Recently, the adsorption kinetics of the fluorescein isothiocyanate(FITC) labeled AFGPs on ice surfaces was studied using confocalfluorescence microscopy [44]. The small fluorescence label, FTIC,has advantage over macromolecular fluorescence label (e.g., greenfluorescent proteins) [45] since the small molecule label can limitthe effect of slow diffusion. Similar cases have been reported forinhibition of mineralization [46]. Moreover, irreversible adsorptionis not generally accepted for adsorbed polymers [47]. Polydisperis-ty effects of polymers and the slow diffusion of polymers maymask their desorption [48].

S. Wang et al. / J. Chem. Thermodynamics 53 (2012) 125–130 129

Second, hydrogen bonding [49–52], van der Waals, hydrophobicinteraction [53–58], and/or some combination of the three are ac-count for the binding of AF(G)Ps to ice. But irreversible adsorptionis attributed to stronger bond formation between the adsorbateand the adsorbent [29,59,60]. No such strong bonds are detectedbetween antifreeze molecules and ice to account for irreversibleadsorption.

Third, the heterogeneity and highly fluidic nature of the ice sur-face make it less possible for irreversible adsorption [44]. The qua-si-liquid layer (QLL) is involved in antifreeze function as suggestedby computer simulations [61]. Experimental and theoretical resultshave suggested that an ice–water interface should be considered inthe adsorption to ice and the accumulation of the antifreeze mole-cules at such ice/water interface is thus implied as a reversible pro-cess [61–66]. Here, the reported thermodynamic model applies tothe ice–water interface. For example, when n = 2, h = 2 � 4.48 Å (fig-ure 1) [67], close to the previously determined 10 Å interface thick-ness [66]. It should also be noted that the value of h varies underdifferent experimental conditions, such as the cooling rate andpressure.

Finally, there is no evidence supporting that the complete inhi-bition of ice growth must need the irreversibility of the adsorptionof antifreeze molecules to the ice [37]. Under the reversibleadsorption conditions, the growth of ice can still be completelyinhibited provided the kinetics of adsorption of AF(G)Ps overcomesthat of water on ice surfaces.

The improved model is simple and can fit the experimental dataof various AFPs in a wide concentration range very well. However,several potential limitations should be noted. In the derivation ofthis model, we assume that AFPs are evenly distributed on theice surface. Experimentally, it is true only when we allow the tem-perature to cool down very slowly, so that distribution equilibriumis reached. In addition, the Langmuir isotherm ignores the possibil-ity that the initial overlayer may act as a substrate for furtheradsorption. In this case, more complex models, such as BET iso-therm [68], need to be considered. Furthermore, as more and moreAFPs adsorb on the ice surface, the further adsorption of the AFPscannot be considered as homogeneous. Statistical model may beused for the non-homogeneous cases. Finally, we cannot rule outthe existence of more complex mechanisms.

5. Conclusions

The detailed derivation of an improved adsorption–inhibitionmodel for simple AFPs is presented in this study. This model quan-titatively describes the TH phenomenon in spite of the diversestructures of antifreeze molecules. The apparent maximum freez-ing point depression and the free energy change of the adsorptionof AFPs onto ice surfaces have been first estimated using equation(12). The results are reliable and meaningful, which have muchbroader utility for a better understanding the mechanisms of bio-logical antifreezes. This work demonstrates the power of thermo-dynamics as a tool to derive useful information for a complexsystem and suggests that the mechanism of AFP action is a kineti-cally controlled, reversible process.

Acknowledgments

This work was supported by National Institutes of Health GrantGM086249 and Research Corporation Cottrell College ScienceAward CC10492 to X.W.

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