Effect of antifreeze protein on heterogeneous ice nucleation based on a two-dimensional random-field Ising model

Antifreeze proteins (AFPs) are the key biomolecules that protect many species from suffering the extreme conditions. Their unique properties of antifreezing provide the potential of a wide range of applications. Inspired by the present experimental approaches of creating an antifreeze surface by coating AFPs, here we present a two-dimensional random-field lattice Ising model to study the effect of AFPs on heterogeneous ice nucleation. The model shows that both the size and the free-energy effect of individual AFPs and their surface coverage dominate the antifreeze capacity of an AFP-coated surface. The simulation results are consistent with the recent experiments qualitatively, revealing the origin of the surprisingly low antifreeze capacity of an AFP-coated surface when the coverage is not particularly high as shown in experiment. These results will hopefully deepen our understanding of the antifreeze effects and thus be potentially useful for designing novel antifreeze coating materials based on biomolecules.

Figure 1

The average of $M$ at each temperature in a typical cooling process. $M$ is the mean value of all $S_i$ at each MC step. Here, the freezing temperature is calculated as $-22.7{}^{\circ }\mathrm{C}$.

Figure 2

The freezing temperature of water on surfaces covered with AFPs of various sizes and shapes at different coverage. In the main figure, the size of AFPs range from single site ($1\times 1$) to rectangle ($60\times 40$), and the letter $R$ indicates the AFPs have random orientation. The straight dotted line is from the mean-field theory. Here, $T_{\alpha }=5.6{}^{\circ }\mathrm{C}$. The inset is replotted from the experimental result presented by Liu et al. [28].

Figure 3

Four typical ice nucleation events when substrate surface is covered by AFPs of different sizes and shapes but at the same coverage $40\%$. The surface is represented in pink, and the attached AFPs are indicated by red squares or rectangles. The blue color represents ice. Specifically, the light blue indicates ice on uncovered surface, and the dark blue represents ice on the top of AFPs. Water is considered to be transparent. The four types of AFPs in are: (a) square AFPs of size $50\times 50$ (lattice sites), (b) rectangular AFPs of size $60\times 40$, (c) rectangular AFPs of the same size as in (b) but having random orientation, and (d) pointlike AFPs each covering only single site. The entire panel consists of $750\times 750$ sites in (a), and consists of $720\times 720$ lattice sites in (b), (c), and (d).

Figure 4

The freezing temperature at different cooling rates. Each lattice site takes 1000 MC turning attempts in average at original cooling rate (cyan line), and takes 10000 at the slower cooling rate (red line). The difference between $\mathrm{\Delta }{T}_o$ ($\mathrm{\Delta }T$ at the original cooling rate) and $\mathrm{\Delta }{T}_s$ ($\mathrm{\Delta }T$ at the slower cooling rate) are calculated, as shown by the gray line. The size of AFPs is $20\times 20$ in both simulations, and $T_{\alpha }=5.6{}^{\circ }\mathrm{C}$.

Figure 5

The freezing temperature of water on surfaces covered with square AFPs of different sizes and $T_{\alpha }$. In the legend: number indicates the length of AFP square lattice, and letter indicates the relative magnitude of $T_{\alpha }$ small (S), medium (M), or large (L). The three parameters are $T_{\alpha ,\mathrm{small}}=4.5{}^{\circ }\mathrm{C}, T_{\alpha ,\mathrm{medium}}=5.6{}^{\circ }\mathrm{C}$, and $T_{\alpha ,\mathrm{large}}=9.0{}^{\circ }\mathrm{C}$. Here each freezing temperature is averaged from 500 independent simulations at the same condition but different random realization of AFP positions, without considering the orientation degrees of freedom of AFPs.