Langmuir 1999, 15, 1905-1912
1905
Articles Phenomenology and Mechanism of Antifreeze Peptide Activity Denver G. Hall North East Wales Institute, Plas Coch. Mold Road, Wrexham LL11 2AW, U.K.
Alexander Lips*,† Unilever Research Colworth, Colworth House, Sharnbrook, Bedford MK44 1LQ, U.K. Received June 4, 1998. In Final Form: December 18, 1998 So far it has not been explained convincingly how antifreeze peptides (AFP) prevent the growth of ice crystals. Step-pinning models as proposed by DeVries, Knight, and others imply a degree of permanent adsorption which it is difficult to reconcile with the observed concentration dependence of the hysteresis gap width. We propose a novel mechanism whereby the role of the antifreeze peptide is to increase the step energy (line tension) associated with the formation of 2D nuclei thereby reducing the nucleation temperature. This is achieved by fairly strong positive adsorption on the crystal faces with reduced adsorption close to a step leading to a negative line excess. The model predicts, in excellent agreement with experiment, a linear dependence of δT on loge c where δT is the freezing point lowering and c the peptide concentration. The magnitudes of the slopes lead to reasonable values of the negative line excess and to estimates of the peptide concentration at which the amount adsorbed should be significantly below the saturation level. They also lead to estimates of adsorption free energies for isolated molecules of order 10kT. These findings provide good circumstantial evidence that the proposed mechanism is at least partially correct. The concept of positive adsorption on crystal faces with negative adsorption close to steps may have important implications not just for AFP but for the general area of additive effects on crystal growth kinetics from melts.
Introduction Antifreeze peptides and glycopeptides are molecules produced by certain species of Arctic and Antarctic fish, notably the Winter Flounder. Their solutions prevent the growth of ice crystals placed in contact therewith over a temperature range of order 1 K below the equilibrium melting point of the solution concerned. In contrast, their influence on the temperature at which ice crystals melt is attributable entirely to their colligative effect and is trivial. The undercooling is clearly a hysteresis phenomenon and the temperature range over which it operates is known as the hysteresis gap. Despite a considerable amount of experimentation and speculation, the mechanism of this effect remains unclear (for recent reviews, see ref 1 and 2). The aims of this paper are to survey some of the salient experimental findings, to offer a critique of existing explanations which exposes their weaknesses, and then to forward an alternative mechanism which is free from these weaknesses. Salient Experimental Facts DeVries and Lin3 and others4 claim that no detectable change in crystal shape or size occurs when an ice crystal † Current address: Unilever Research U.S., 45 River Road, Edgewater, NJ 07020.
(1) Avanove, A. Y. Mol. Biol. 1990, 24, 473. (2) Hew,C. L.; Yang, D. S. C. Eur. J. Biochem. 1992, 203, 33. (3) DeVries,A. L.; Lin, Y. In Adaptations Within Antarctic Ecosystems; Llano, G. A., Ed.; Gulf Publishing Co.: Houston, TX, 1977; p 439. (4) Harrison, K.; Hallet, J.; Burcham, T. S.; Feeney, R. E.; Kerr, W. L.; Yeh, Y. Nature (London), 1987, 328, 241.
is contacted with an AFP solution at temperatures within the hysteresis gap. More recently, large single crystals have been studied by Raymond et al.5 who claim that the AFPs prevent growth on the prism faces but allow some growth on the basal plane along the c axis. This growth is accompanied by the formation of pits and ceases either when the basal plane becomes fully pitted or when the crystal assumes the form of a hexagonal pyramid. Significant changes in crystal habit have been observed at concentrations well below those which prevent growth6 so that higher energy, fast growing faces, which do not appear in the absence of antifreeze, are expressed in its presence. The F/P (flounder, plaice) (202 h 1) and the sculpin adsorption face (2110) are depicted in Figure 3 of ref 6. For some of these faces, there is a reasonable match between repeat distances thereon and the helical structure of the peptide.6 Molecular modeling studies7 lend some support to this viewpoint. They show that adsorption can occur on more than one face but that binding to one face is likely to be more specific than for the others. They also suggest the possibility of cooperative binding. The width of the hysteresis gap depends on the peptide concentration as is shown in Figures 1 and 2.8-11 Some (5) Raymond, J. A.; Wilson, P.; DeVries, A. L. Proc. Natl. Acad. Sci. U.S.A. 1989, 86, 881. (6) Knight, C. A.; Cheng, C. C.; DeVries, A. L. Biophys. J. 1991, 59, 409. (7) Lal, M.; Clark, A. H.; Lips, A.; Ruddock, J. N. White, D. N. J. Faraday Discussion, 1993, 95 299. (8) Knight, C. A.; DeVries, A. L.; Oolman, L. D. Nature (London) 1984, 308, 295.
10.1021/la980657m CCC: $18.00 © 1999 American Chemical Society Published on Web 02/19/1999
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Hall and Lips
Figure 3. Proposed mechanisms of growth inhibition based on the Kelvin effect: (A) step pinning model (ref 16); (B) mattress model (ref 6).
Figure 1. Dependence of hysteresis δT on concentration c for AFGPs (data from ref 8; model fits to eq 19).
Very recently, estimates of amounts adsorbed on the prism faces and basal plane obtained from ellipsometry measurements have been reported.13 The conclusions that emerge therefrom are as follows. For both low and high molecular weight AFGPs, adsorption on the prism faces does not appear to depend on the peptide concentration over the concentration range studied. Adsorption on the basal plane seems to be somewhat lower than on the prism faces. Adsorption of the high molecular weight peptide again shows no obvious dependence on concentration, whereas that of the lower molecular weight peptides appears to approach saturation at a concentration of about 2 mg‚mL-1. However, the experiments involved are clearly fraught with difficulties, and the experimental accuracy is not good. Consequently, these conclusions are somewhat tentative. Other evidence for substantial adsorption has been obtained from grain boundary analysis.14,15 Proposed Mechanisms of Growth Inhibition
Figure 2. Dependence of hysteresis δT on concentration c for AFPs (data from references as indicated; model fits to eq 19).
measurements of the peptide partitioning between ice and water when freezing occurs can be found in Table 1 of ref 12. The quantity, R, is defined as
R)
concentration of additive in ice concentration of additive inwater
(1)
It is notable that the values of R are all less than 1. If R exceeds 1, then once an ice crystal starts to grow, the concentration of peptide in solution is depleted and growth can be expected to accelerate. This is clearly against the fish’s best interest. In contrast, if R < 1, then growth tends to be self-inhibiting. Unfortunately, no information is given in ref 12 about the dependence of R on peptide concentration. (9) Scott, G. K.; Davies, P. L.; Shears, M. A.; Fletcher, G. L. Eur. J. Biochem. 1987, 168, 629. (10) Slaughter, D.; Fletcher, G. L.; Anantharayanan, V. S. Hew, C. L. J. Biochem. 1981, 256, 2022. (11) Sonichsen, F. D.; Sykes, B. D.; Chao, H.; Davies, P. L. Science 1993, 259, 115. (12) Raymond, J. A.; DeVries, A. L. Proc. Natl. Acad. Sci. U.S.A., 1977, 74 25894.
It is generally accepted that the mechanism of inhibition probably involves a 2D or 3D Kelvin effect12,16 rather than the blocking (i.e., poisoning) of sites for molecular addition. Two such mechanisms are depicted in Figure 3. In mechanism A, the adsorbed molecules block the growth of a step, whereas in mechanism B they prevent growth perpendicular to the surface.16,17 Three features common to both mechanisms are first that adsorption is essentially permanent and second that growth requires the formation of a region akin to a nucleus. Third, the undercooling is governed by the amount adsorbed. In addition, it is assumed in model A that the step energy per unit length is of the same order as the product of the surface energy per unit area and the step height. Criticism of Proposed Mechanisms Mechanisms A and B. If the adsorption of peptide is effectively permanent, and the free energy of adsorption is large, one expects fairly large levels of adsorption at low concentrations, and no significant increase with concentration as saturation adsorption is approached. (13) Wilson, P. W.; Beaglehole, D.; DeVries, A. L. Biophys. J. 1993, 64, 1878. (14) Kerr, W. L.; Feeny, R. G.; Osuga, D. T.; Reid, D. S. Cryoletters 1985, 6, 371. (15) Reid, D. S.; Kerr, W. I.; Zhao, J.; Wada, Y. Food Macromolecules and Colloids; Special Publications of the Royal Society of Chemists; RSC: London, 1995; Vol. 156, p 141. (16) Wilson, P. W.; Cryoletters 1993, 14, 31. (17) Knight,C. A.; DeVries, A. L. Science 1989, 245, 505.
Mechanism of Antifreeze Peptide Activity
Langmuir, Vol. 15, No. 6, 1999 1907 Table 1 xδT vs loge c
δT vs xc
type
dxT/d loge c, K1/2
intercept c(0) at δT ) 0 (eq 19), mg‚ml-1
r2
dδT/dxc, K‚mL1/2‚mg-1/2
intercept c(0) at δT ) 0, mg‚mL-1
r2
AFP 1 (flounder) AFP 2 (yellow tail) AFP 2 (sea raven) AFP 3 (ocean pout) AFGP 1-5 AFGP 7
0.186 0.174 0.119 0.133 0.186 0.148
0.12 0.15 0.02 0.01 0.13 0.18
0.995 0.997 0.985 0.970 0.997 0.992
0.212 0.170 0.195 0.341 0.174 0.108
-0.010 -0.014 0.044 0.055 0.082 -0.002
0.994 0.992 0.989 0.992 0.975 0.994
Alternatively, if the free energy of adsorption is small, adsorption can be effectively permanent only if the rate of adsorption is very slow. If this is the case, significant growth should occur for a substantial time after the exposure of an ice crystal to a solution of AFP. Raymond and DeVries argue12 that any permanently adsorbed material is likely to be trapped in the ice when freezing takes place and that because of this, the values of R in Table 1 should indicate amounts permanently adsorbed. These amounts are small. They are much lower than estimates of adsorption densities based on ellipsometry studies13 and are about 2 orders of magnitude below estimates based on molecular close packing. Also, the concentrations in solution are quite large and, as Figures 1 and 2 show, the hysteresis gap δT depends quite strongly on peptide concentration. To reconcile the values of R in Table 1 with the ellipsometry studies and the adsorption levels expected for saturation coverage, it can be argued that most, but not all, of the adsorbed material can desorb and is therefore ineffective. It can also be argued that fairly large concentrations in solution are required to ensure that R < 1 so that growth is self-inhibiting. However, if the amount permanently adsorbed is independent of concentration and δT is governed by this amount, then it is clear that neither mechanism can explain the observed dependence of δT on concentration. If instead the peptides act by blocking sites for molecular addition18 but are not permanently adsorbed, it can be argued that the growth rate per site should be proportional to the extent of undercooling and the fraction of time that the growth sites are unblocked. If this is so and the surface density of binding sites is fixed then, for a given growth rate, one might expect δT to vary linearly with concentration. However this is not the observed dependence. If, on the other hand, growth occurs by a screw dislocation mechanism, the number of binding sites is governed by the pitch of the growth spiral and increases linearly with δT. This results in the prediction that for a given growth rate, δT should vary linearly with xc, which, as Figures 7 and 8 show, fits the experimental data quite well. However in both cases, site blocking should lead to gradual rather than the observed sudden changes in growth rate with concentration and temperature. A sudden change in growth rate could occur if the adsorption isotherm of the peptide exhibits a 2D phase transition. In this case, contrary to experiment, a linear dependence of δT on c is predicted. These points and the reasoning behind them are discussed more fully in the Appendix. Mechanism A Only. For mechanism A to apply, a substantial energy of step formation is required. In the absence of screw dislocations this should lead to a nucleation barrier inhibiting step formation and a significant undercooling in the absence of additive. If screw dislocations are present, then some growth at all levels of supersaturation can be expected but the growth rate (18) Burcham, T. S.; Osuga, D. T.; Yeh, Y.; Feeney, R. E. J. Biol. Chem. 1986, 261, 6390.
Figure 4. Negative line excess of close-packed disks.
Figure 5. xδT vs loge c for data in Figure 1 (same symbols).
should vary as the square of the supersaturation. Neither effect is observed, and it is noteworthy that snowflakes are normally dislocation free.19 It has been reported that growth normal to the basal plane occurs at -0.03 °C and that the growth kinetics are consistent with 2D homogeneous nucleation.20 Also, the prism faces in contact with air exhibit a roughening transition at -2 °C.21 Both observations indicate that the energies associated with step formation are very small. An interesting implication of this is that if adsorption along a step reduces step free energies, these could become negative, in which case, steps should self-propagate spontaneously. These considerations lead to the conclusion that mechanisms based on the quasi permanent adsorption of the (19) Frank, C.; Foreword in Handbook of Crystal Growth; Hurle, G. T. J., Ed.; Elsevier Science: Amsterdam, The Netherlands, 1993. (20) Hillig, W. Growth and Perfection of Crystals; Doremus, R. H., Roberts, W. B., Turnbull, D., Eds.; Wiley: New York, 1958. (21) Elbaum, M. Phys. Rev. Lett. 1991, 67, 2982.
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Figure 6. xδT vs loge c for data in Figure 2 (same symbols).
Figure 8. δT vs xc for data in Figure 2 (same symbols).
or disks) per unit area away from the line. For low coverages, in the limit that the area fraction occupied by the spheres f 0, it is clear that the centers of the spheres cannot get nearer to the step than their radius, r. In this case, in the limit that Γs f 0, the line excess is given by
Γl ) -2rΓs
(3)
For rodlike molecules, similar principles apply, but the calculations are more complex. In this case, as Γs f 0, Γl is given by
Γl ) -ΓsL
Figure 7. δT vs xc for data in Figure 1 (same symbols).
peptide are seriously flawed and probably incorrect. It is appropriate, therefore, to seek an alternative mechanism which is not highly specific for particular faces and which allows for dynamic equilibrium between adsorbed and bulk additives. We now turn to this issue. Proposed Alternative Mechanism The proposed mechanism is that a negative line excess of the additive at a step leads to an increase in the line tension (step energy) of the step, thus providing a barrier to 2D nucleation and related mechanisms. Such a negative line excess will arise if the peptide molecules adsorb strongly on smooth crystal surfaces but avoid adsorbing onto steps. The effect is depicted schematically in Figure 4, which shows that, for close-packed spheres, there are fewer spheres in a patch, which includes part of the step than in a patch which does not. For this case, the line excess Γl is given by
Γl ) -(2 - x3)rΓs
(π2 - 1)
(4)
where L is the length of the rod. It should be noted that exclusion from both sides of the line is incorporated in eqs 2-4. The above mechanism is exactly analogous to the effect of adsorption on the surface tension of an aqueous solution. In this case molecules such as surfactants accumulate at the interface, are said to be positively adsorbed, and reduce the surface tension. In contrast, electrolytes such as KCl, which have a lower concentration at the surface than in bulk solution, are said to be negatively adsorbed and increase the surface tension. The effect of a negative line excess on the line tension may be described by the 2D analogue of the Gibbs adsorption equation22 which at constant T may be written as
dλ ) -Γl dµ
(5)
where λ and µ, respectively denote line tension and the peptide chemical potential. Bearing in mind that undercooling in the absence of peptide is very small and that the step free energy at a roughening transition is zero,23 it is probably reasonable to assume that any significant increase in λ due to the presence of the peptide may be equated to λ itself; however, this assumption is not necessary in the treatment which follows.
(2)
where Γs is the number of molecules (in this case spheres
(22) Lane, J. E.; Trans Faraday Soc. 1968, 64, 221. (23) Liu, X. Y.; van Hoof, P.; Bennema, P. Phys. Rev. Lett. 1993, 71, 97.
Mechanism of Antifreeze Peptide Activity
Langmuir, Vol. 15, No. 6, 1999 1909
At any temperature T lower than the equilibrium freezing point, there is a critical 2D nucleus in unstable equilibrium with the solution. Nuclei smaller than the critical size tend to shrink, whereas nuclei which are larger tend to grow. The critical size is governed by the expression
∆H λa ) µw - µice ) δT r T
( )
(6)
where µw and µice respectively refer to the chemical potentials of bulk water and bulk ice, δT is the undercooling, ∆H is the molar heat of fusion, and a is the area of the 2D nucleus divided by the number of water molecules it contains. The relationship between the line tension λ and the dimensionless step free energy γ may be written as24
γ)λ
a xWRT
(7)
where R denotes the gas constant and W is the work of forming a critical nucleus given by
W ) πrλ
(8)
Once W falls below a critical value, the nucleation rate increases dramatically and rapid growth is obtained. Eliminating r from eqs 6 and 8, we obtain
δT )
πaTλ2 W∆H
(9)
Putting W ) 40kT, similar to the value for homogeneous nucleation,25 T ) 273 K, ∆H ) 6.02 × 103 J mol-1, and a ) 5.8 × 104 m2 mol-1 for a 2D nucleus assumed to be one molecule high, we find that if δT ) 0.64 K, then λ ∼ 3.4 × 10-12 N. To quantify the notion of a positive adsorption onto a face with negative adsorption at a step it is useful to introduce the concept of an exclusion length, as is done in some treatments of negative adsorption at an interface. Here we use two lengths, l and l′ defined by
λ ) l∆σ
(10a)
Γl) -l′Γs
(10b)
where ∆σ is the interfacial tension due to adsorption onto the crystal surface. ∆σ is sometimes termed a surface pressure and is the 2D analogue of a bulk pressure or an osmotic pressure. Γs is the amount adsorbed per unit area on the crystal surface remote from a step. Given the above value of λ, we find on this basis that if ∆σ ) 20 mN m-1, l ) 1.7 × 10-10 m, whereas if ∆σ ) 5 m Nm-1, l ) 6.8 × 10-10 m. Since the estimated ice/water interfacial tension26 is of order 20 m Nm-1, ∆σ can hardly be expected to exceed this figure. It is concluded therefore that l can be expected to take a value between about 10-10 m and the molecular dimensions of the additive. The above values of ∆σ, 5 m Nm-1 < ∆σ < 20 mN m-1 are quite large and indicate strong lateral repulsive interactions between adsorbed molecules. This in turn suggests that amounts adsorbed may be fairly close to saturation. Under these conditions, it is reasonable to expect the dependence of Γl and hence l′ on concentration (24) Elwenspoek, M.; van der Eerden, J. P. J. Phys. A. 1987, 20, 669. (25) Defay, R.; Prigogine, I.; Bellemans, A.; A. Everett, A. Surface Tension and Adsorption; Longmans Green & Co.: London, 1966. (26) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarenden Press: Oxford, U.K., 1984.
to be small. If, in addition, the peptide behaves ideally in solution so that dµ ) RT d loge c, it follows from eqs 5 and 9 that, for small values of δT, we have approximately
dx(δT) πaT ) RTΓl d loge c W∆H
(
1/2
)
(11)
where, for simplicity, πaT/W∆H has been taken as constant and the dependence of λ on T at constant c has been ignored. Equation 11 shows that, when c is large, δT1/2 should scale linearly with the logarithm of c. As Figures 5 and 6 illustrate, this is the case. The slopes lie in the range 0.11-0.19 K1/2. The main features of the data are summarized in Table 1. Experimental values of δT and c were read off from the published graphs. This inevitably leads to errors in loge c and c1/2, which are most pronounced when c is small. Also shown in Table 1 and in Figures 7 and 8 are the results of plotting δT vs c1/2 as suggested by Raymond and DeVries.12 Although, for reasons given above, we believe that their theory underpinning this latter plot is fundamentally flawed, it is clear from the values of the regression parameter r2 there is little to choose between the two types of plot for quality of fit to experimental data. Consequently, it appears that more evidence is required to validate the model based on negative adsorption. This is provided by demonstrating that the magnitudes of the slopes in Figures 5 and 6 are not unreasonable. Analysis of Slopes of xδT vs loge c According to the Gibbs adsorption isotherm
d∆σ ) Γs dµ
(12)
Also, from the definitions of l and l′ as given by eqs 10a,b, it is apparent that
dλ ) l d∆σ + ∆σ dl
(13a)
dλ ) -Γl dµ ) l′ Γs dµ
(13b)
Equating the RHS’s of this pair of equations and rearranging gives
l′ - l )
()
( )( )
∆σ ∂l ∆σ ∂l ∂Γs ) Γs ∂µ Γs ∂Γs ∂µ
(14)
As µ is increased and Γs approaches its saturation value, Γ∞, it is to be expected that (∂Γs/∂µ) will decrease to zero asymptotically and that changes in ∆σ/Γ∞ are approximately equal to the corresponding changes in µ. It follows that the product, (∆σ/Γs)(∂Γs/∂µ), should approach zero with increasing µ so that unless (∂l/∂Γs) f 0 as Γs f ∞
xδT ∆σ ) RTΓ∞ dxδT/d log c e
(15)
This equation has the advantage that W and a, whose values are uncertain, have been eliminated. The data for AFGP 78 gives δT ∼ 0.64 K when c ∼ 30 mg‚mL-1. For this system, the denominator of eq 15 is 0.15 K1/2. Hence, at this concentration, ∆σ/RTΓ∞ ) 5.33. A reasonable value of Γ∞ for the low molecular weight glycopeptide is about 3 × 1017 molecules‚m-2. This gives ∆σ ) 6.4 mN m-1 and a value of l of ∼5.4 × 10-10 m. For close-packed disks of radius 10-9 m which can touch but cannot overlap a straight line, Γ∞ ) 2.8 × 1017 molecules‚m-2 and l ) 2.7 × 10-10 m. This suggests that the above estimate of l is
1910 Langmuir, Vol. 15, No. 6, 1999
Hall and Lips
not unreasonable. It also suggests that small molecules are unlikely to be effective because the change in σ required to obtain a sufficiently large value of λ is too large. Factors other than packing which may influence l and l′ include the possibility that steps are not completely straight but include ripples akin to the capillary waves found on normal surfaces.25 Thus, if λ ) 4 × 10-7 mJ‚m-1, a line of length 10-7 cm has a step energy of order kT, and one might expect significant fluctuations in step length over distances of this order. Since the scale of these fluctuations is of similar order or smaller than the peptide molecules, they may well make a positive contribution to the exclusion lengths l and l′. As c f 0, it is clear that the linear dependence of xδT on loge c depicted in Figures 5 and 6 cannot continue indefinitely as c is reduced. The precise form of this dependence is governed by the dependence of Γl on c, which in turn may be understood in terms of the dependence of Γs on c and that of l′ on Γs. In the absence of this detailed information, it is sensible to proceed as follows. When Γl approaches a limiting value, Γl∞, as c is increased, we may define a concentration c* by writing
∫0c*Γld loge c ) ∫c*∞(Γl∞ - Γl) d loge c
∫c*Γl∞ d loge c ∞ RT ∫c (Γl∞ - Γl) d loge c
c c*
( )
∆λ ≈ -RTΓl∞ loge
c
1/2
(18)
where S is the high c value of (d xδT/d loge c) and δT0 is the value of δT when c ) 0. In general, we expect δT0 > 0 and therefore c* > c(0). When c < c(0), both Γl/Γl∞ and Γs/Γ∞ can be expected to be significantly less than 1. Hence, if it is found experimentally that Γs/Γ∞ ≈ 1 at concentrations much below c(0), then the above treatment cannot be correct. The experimental data in ref 13 does not appear to be sufficiently accurate for any firm conclusion concerning this point to be drawn. The above analysis shows that when (δT)1/2 vs loge c is linear, we have
[
δT ) S2 loge
c c(0)
]
2
( )
(19)
This is the function which has been used to obtain the model fits in Figures 1 and 2, where S and c(0) were obtained from a linear regression of (δT)1/2 vs loge c giving equal weight to all reported data points. It should be noted that, when δT0 ) 0, the adsorption isotherm, which corresponds to the case where eq 19 holds
(20)
In general, w includes a contribution from the interaction between adsorbed molecules, but as c and Γ both approach zero, w is the work of transferring an isolated molecule in bulk to a bare surface. In the case that adsorption conforms to the Langmuir-Syzskowski equation, we have
Γ ) Kc Γ∞ - Γ
(21)
∆σ ) Γ∞kT loge(1 + Kc)
(22)
and
where Γ∞ denotes the saturation adsorption and K is a constant. As c f 0, it follows from eq 21 that Γ/c f Γ∞K. Hence, under these conditions
exp
Let c(0) be the value of c at which the linear portion of (δT)1/2 vs loge c extrapolates to give δT ) 0. When ∆λ(c) ) 0, c ) c*. It is a necessary consequence of the definitions of the quantities c* and c(0) that
( )
w Γ ) -loge kT cx
(17a)
Γl/Γl∞ f 1 for large c (17b)
(δT0) c* ) loge S c(0)
Free Energy of Adsorption If it is reasonable to argue that the adsorbed molecules are uniformly distributed over a region of thickness x, it follows from elementary statistical mechanics that the work required to transfer a molecule from bulk solution to the adsorbed state, w, is given by
(16)
Equation 16 in effect represents an equal area argument, and it should be noted that a low value of c* indicates strong adsorption of the peptide to the ice surface. It follows from eqs 5 and 16 that for c > c*
∆λ ) λ(c) - λ(c ) 0) ) -RT
at all c, is the step function Γs ) 0 for c < c*, 0 e Γs e Γ∞ for c ) c*, Γs ) Γ∞ for c > c*. This isotherm represents a 2D phase transition and describes the limiting situation where adsorption is totally cooperative.
) (-w kT )
Γ∞K x
(23)
Eliminating K from eqs 22 and 23, we obtain for the free energy of adsorption in the limit that c f 0.
( )
( ( ) )
Γ∞ -w ∆σ ) loge + loge exp -1 kT cx Γ∞
(24)
For AFGP 7, an undercooling of 0.64 K results when c ) 30 mg‚mL-1. For a molecular weight of 3000, this corresponds to a number density in solution of approximately 6 × 1024 molecules m-3. Taking this value for c together with ∆σ/RTΓ∞ ) 5.33, Γ∞ ) 3 × 1017 molecule m-2, and x ) 10-7 cm, we obtain w/kT ) -9.24. This is of similar order to the adsorption free energy of typical surfactants at the oil/water interface27 and suggests that adsorption equilibrium may be fairly labile. This conclusion is in good qualitative accord with the values of R in Table 1 and the ellipsometric data.13 It is noticeable that the adsorption free energy of the higher molecular weight glycopeptide AFGP 1-5, calculated in the same way, is not a lot greater than that of AFGP 7. The above numbers for AFGP 7 also indicate that, at a concentration of 0.15 mg‚mL-1, the level of adsorption calculated from the Langmuir-Syzskowski isotherm is about half the saturation value. Conclusions The main points which emerge from the above analysis are as follows. First, the proposed mechanism predicts the linear dependence of xδT on loge c observed for a variety of systems. Second, it is consistent with the observation that the onset of growth as temperature is reduced is sudden. Moreover, the mechanism does not
Mechanism of Antifreeze Peptide Activity
depend on particular details of the adsorption isotherms, which can be expected to differ for different systems. Hence, it accounts for the fact that most of the systems studied exhibit remarkably similar behavior. Third, the values of the slopes are physically sensible. Fourth, the estimated values of adsorption free energies are consistent with fairly strong but labile adsorption. This accounts for the observation that estimates of amounts adsorbed from ellipsometry appear to be much greater than the amounts calculated from the values of R in Table 1. Fourth, the model leads to the conclusion, supported to some extent experimentally,13 that saturation coverage should occur at low peptide concentration. Taken overall, these points provide reasonable evidence that the proposed mechanism which appears to be fairly robust, despite the simplifications involved, is at least partially correct. No other mechanism proposed to date agrees as well with the key observations cited above. The mechanism also indicates that strong adsorption onto faces is a necessary condition for the attainment of freezing point hysteresis but may not be sufficient. It can be argued that if the peptide adsorption lowers the ice/water interfacial tension by the amounts cited above, then there should be some effect of the peptide on the temperature at which homogeneous nucleation of ice occurs. No such effect has been observed. The most likely explanation for this is that the nuclei are too small for peptide adsorption to occur thereon. Estimated dimensions of ice nuclei are of order 10-7 cm, and this is smaller than even the smallest antifreeze peptide molecules. If these molecules avoid steps and ledges on a crystal surface, they are unlikely to attach to ice nuclei. In practice, if adsorption occurs on all faces, each face can be expected to have its own hysteresis gap. The measured gap is presumably that for the faces which resist growth most. These in turn can be expected to have the highest free energy of adsorption. The ellipsometry work13 gives some indication that the level of adsorption and the adsorption free energy are lower on the basal plane than on the prism faces. This ties in well with the findings of Raymond and co-workers.5 The adsorption free energies which correspond to the adsorption levels found from the values of R in Table 1 are very close to zero and, in some cases, may even be negative. This further detracts from explanations of the hysteresis gap based on irreversible adsorption. In two recent papers28,29 structural studies of a low MW AFP 1 (HPLC 6) and an AFP 3 have been reported. In both cases it was noticed that the “ice binding surface” of the AFP is remarkably flat. For the AFP 1 it was also found that the structural match between the H bonding sites on the AFP and the ice surface it binds to appears to be less perfect than was thought previously. This latter finding is consistent with the notion that AFP adsorption is not permanent. A flat “ice binding surface” is clearly conducive to adsorption onto a flat ice surface and can also be expected to avoid adsorption onto steps. Appendix: Growth Inhibition by NonPermanent Adsorption at Growth Sites (a) Growth Rate per Site. We suppose that the crystal grows by the attachment of molecules to particular growth (27) For example, w/kT ∼ -11.2 for a surfactant which lowers the oil/water interfacial tension by 45 mN m-1 at a concentration of 10-2 M and which has a Γ∞ corresponding to an area per molecule of 50 × 10-20 m-2. (28) Sicheri, F.; Yang, D. S. C. Nature 1995, 375, 427. (29) Jia, Z.; DeLuca, C. I.; Chao, H.; Davies, P. L. Nature 1996, 384, 285.
Langmuir, Vol. 15, No. 6, 1999 1911
sites on the crystal surface and that the net growth rate may be regarded as the difference between forward and backward rates. Let ra and rp respectively denote net growth rates per site in the absence and presence of inhibitor, and let A be the affinity for growth given by
A ) µl - µs
(A1)
where l and s respectively denote liquid and solid and µ denotes chemical potential. When A , RT, it is reasonable to write
ra ) rae
A RT
(A2)
where rae is the equilibrium forward or backward rate per site for addition or removal of a molecule. It is now assumed that in the presence of inhibitor
rp ) raf
(A3)
where f denotes the fraction of time that a growth site is blocked due to adsorption of the inhibitor. If the number of blocked growth sites is proportional to the amount of adsorbed inhibitor, it is reasonable to write
f)
Γ∞ - Γ Γ∞
(A4)
where Γ∞ is the number of adsorption sites per unit area and Γ/Γ∞ is the fraction that are occupied. If the adsorption isotherm is a Langmuir isotherm
Γ ) Kc Γ∞ - Γ
(A5a)
or a Frumkin isotherm
Γ exp(-BΓ) ) Kc Γ∞ - Γ
(A5b)
where K is a constant, c denotes concentration, and B is a parameter which allows for interactions between adsorbed molecules on adjacent sites, then, in both cases, as Γ f Γ∞
Γ∞ - Γ f const × c Γ∞
(A6)
Let δT ) (Tf - T) where Tf denotes the equilibrium freezing temperature. It is easily shown that
A)
∆H δT RT2
(A7)
where ∆H is the heat of melting. It now follows from eqs A2-A4, A6, and A7 that
rp ) R
∆H δT RT2 c
(A8)
where R is constant. Equation A8 shows that for a given value of rp δT is a linear function of c. As an explanation for the activity of AFPs eq A8 suffers from two drawbacks. It gives the wrong functional dependence of δT on c and it does not account for the observed sudden onset of growth as δT is increased at constant c.
1912 Langmuir, Vol. 15, No. 6, 1999
Hall and Lips
(b) 2D Phase Transition. If the adsorption isotherm of the inhibitor exhibits a 2D phase transition when c is varied at constant T, this may be accompanied by a large change in f and hence in rp. Let ′ and ′′ denote the two surface phases. When the additive behaves ideally, the Gibbs adsorption equations for ′ and ′′ may be written as
d
H′ σ′ ) - 2 dT - Γ′R d ln c T T
(A9a)
H′′ σ′′ ) - 2 dT - Γ′′R d ln c T T
(A9b)
d
rp ) const × A2/c
where σ denotes surface tension and where H′ and H′′ are the heats of forming unit area of surfaces ′ and ′′ respectively.30 Noting that at equilibrium σ′ ) σ′′ we obtain from eq A9
RT2
H′′ - H′ d ln c )dT Γ′′ - Γ′
the 2D transition occurs should be unaltered. In other words, the nonadsorbing solute should reduce the width of the hysteresis gap. (c) Growth via a Screw Dislocation Mechanism. According to this mechanism when A , RT the growth rate is proportional to A2. As before, the growth rate per site should be proportional to A and to 1/c. However the number of growth sites per unit area is also proportional to A because increasing A by a given factor decreases the pitch of the spiral by the same factor which in turn increases its length. Consequently rp is given by
(A10)
which predicts that the conditions where rp changes dramatically are such that δT vs ln c is linear. Equation A10 also suffers from two drawbacks. The functional form does not agree with experiment. Also, when a nonadsorbing solute is present which lowers the freezing point through its colligative effect, the temperature at which (30) Guggenheim, E. A. Trans. Faraday Soc. 1940, 36, 397.
(A11)
Substituting for A as given by eq A7 we find that for a fixed rp
δT ) const × xc
(A12)
The functional form of eq A12 fits the observed dependence of δT on c very well. However the approach does not account for the sudden onset of growth as δT is increased at constant c. Also to obtain eq A12 it is necessary to assume that the line tension λ is not affected significantly by the adsorption of inhibitor. This will be the case when λ is fairly large, but the available evidence indicates strongly that this is not the case. Moreover, it follows from Frank’s comment19 that screw dislocations are unlikely to be present in ice crystals grown at low supersaturations. LA980657M
