Influence of Adsorption Orientation on the Statistical Mechanics Model

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The Influence of Adsorption Orientation on the Statistical Mechanics Model of Type I Antifreeze Protein: The Thermal Hysteresis Temperature Li-fen Li, and Xi-xia Liang J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b06619 • Publication Date (Web): 28 Sep 2017 Downloaded from http://pubs.acs.org on October 3, 2017

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The Journal of Physical Chemistry

The Influence of Adsorption Orientation on the Statistical Mechanics Model of Type I Antifreeze Protein: the Thermal Hysteresis Temperature Li-fen Li1 ,∗, Xi-xia Liang2, 1. Department of Basic Curriculum, North China Institute of Science and Technology, Beijing, 101601, China 2. Department of Physics, Inner Mongolia University, Hohhot, 010021, China

∗

Corresponding author: [email protected] 1

ACS Paragon Plus Environment


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Abstract: The antifreeze activity of type I antifreeze proteins (AFPIs) is studied based on the statistical mechanics theory, by taking the AFP’s adsorption orientation into account. The thermal hysteresis temperatures are calculated by determining the system Gibbs function as well as the AFP molecule coverage rate on the ice-crystal surface. The numerical results for the thermal hysteresis temperatures of AFP9, HPLC-6 and AAAA2kE are obtained for both the cases with and without including the adsorption orientation. The results show that the influence of the adsorption orientation on the thermal hysteresis temperature cannot be neglected. The theoretical results are coincidence preferably with the experimental data.

2


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Introduction Antifreeze proteins (AFPs) are a kind of proteins suppressing ice-crystals growth in organisms and thereby enable their survival in subfreezing habits [1-10]. Despite their different structures, a common characteristic is found that they can be adsorbed onto the surfaces of ice crystals and prevent their growth

[6-8]

. The adsorption lowers the

non-colligative freezing point of water without depressing the melting point. This phenomenon leads to a difference between the freezing and melting temperature, which is referred to as thermal hysteresis

[9-12]

. The thermal hysteresis is used as a

characteristic measure for antifreeze activity of an AFP

[13-16]

. Due to this special

character, antifreeze proteins (AFPs) can highly effectively protect cells and embryos from damages in freezing process [17-20]. As a generally accepted mechanism of action, the adsorption-inhibition mechanism, which is proposed by Raymond and DeVries, suggests that AFPs bind to the surface of a growing ice-crystal, thereby preventing its further growth

[11-12]

. The adsorption of

proteins is thought to prevent macroscopic ice-crystal growth in the hysteresis gap, but microscopic growth occurs at the interface between adsorbed antifreeze molecules in the form of highly curved fronts. This affect causes the decrease of the local freezing temperature because of the Kelvin effect, while leaving the melting temperature relatively unaffected. Based on this mechanism, Li and Luo et al. gave the polymer adsorption model to explain the thermal hysteresis phenomena

[13-16,21]

. Authors and

collaborators have also proposed a statistical mechanics model for type I AFP to calculate the thermal hysteresis activity[22]. Naturally, the coverage rate of AFPs on the ice-crystal surface plays a main role in the antifreeze activity. Moreover, the special adsorption orientation of AFPs on the ice surface will influence the coverage rate

[23]

and then will change the antifreeze activity. Therefore, the effect of adsorption orientation on the thermal hysteresis temperature should be considered for the further understanding of antifreeze mechanisms. AFPI is a class of important antifreeze proteins that has the regular alpha-helical structure, and has been extensively studied theoretically and experimentally

[24-28]

.

AFPI is the alanine-rich antifreeze protein, which consists of several 11-acid repeats. 3



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Authors have analyzed the influence of adsorption orientation on the coverage rate for AFPI in a previous reference

[23]

. In this article, the statistical thermodynamics theory

will be used to discuss the thermal hysteresis temperature of AFPIs, by including the effect of the AFP adsorption orientation on the ice-crystal. As examples, the calculated thermal hysteresis temperatures for AFP9, HPLC-6 and AAAA2kE both with and without including the adsorption orientation will be given and discussed.

Theory Let us consider a two-dimensional lattice-adsorption model: the ice-crystal surface composes of two-dimensional sites and AFPs can adsorb onto the surface. AFPI molecules consist of several amino-acid repeats, such as there are 3 11-amino acid repeats in HPLC-6, 4 such repeats in AFP9 and also 3 repeats in AAAA2kE, respectively. The amino-acid repeats are same both in structure and construction for each AFPI molecule

[3,10]

. Actually, the lattice sites for adsorption composted of ice

molecules and all the ice molecules are indistinguishable from each others. For ease of discussion, we consider that several ice molecules compose a lattice site for AFPI’s adsorption, and then one amino-acid repeat occupies a site. On the ice-crystal surface, the number of lattice sites in the ice-crystal is much more than the sites which can be occupied by AFPs, actually. Let us denote the number of ice molecules as N1 and the site number as Nt on the ice-crystal surface. Then we get the ice molecule number for each "lattice site" as A =N1 /Nt.. For the system without AFP adsorption, only N0 water molecules adsorb on the ice surface at the equilibrium temperature Te (the ice-point of the pure water), the canonical partition function can be written as

Z1 = Ω1e− E1 kTe ,

(1)

where Ω1 is the number of micro-states of N0 water molecules on the ice surface, and Ε1 the energy of the system. As the system composes of N1 ice molecules and N0 water molecules, we have E1 = N1ε1 + N 0 ( −hγ ) . 4


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Where ε1 is the energy of an ice molecule, ‒hγ is the hydrogen bond interact energy of water molecules. Then the free energy of the system is given by F0 = -kTe ln Z1 = − kTe ln Ω1 + N1ε1 + N 0 ( − hγ )

.

(2)

In fact, once the system temperature T lows to the equilibrium temperature Te, some AFPI molecules will adsorb on the ice-crystal surface. Denoting the number of the adsorbed AFPI molecule on the ice-crystal surface as N2 and the water molecule number as N3, the energy of the system becomes E = N 1ε 1 + N 2 ε + N 3 (− hγ ) ,

(3)

where ε is the interaction energy between AFPI and the ice surface. The canonical partition function is then written as Z = Ω2 Ω3e− E kT .

(4)

Here Ω2 andΩ3 are the micro-states numbers of N2 AFPI molecules and N3 water molecules on the ice surface, respectively. The free energy of the system is then F = −kT ln Ω2 − kT ln Ω3 + N1ε1 + N2ε − N3 hγ .

(5)

The micro-states numbers of N0 and N3 water molecules on the ice surface are respectively given by

Ω1 = CNN = 0

1

N1 ! N 0 !( N1 − N 0 ) !

(6)

and

Ω3 = CNN −σ N = 3

1

2

( N1 − σ N 2 )! N 3 !( N1 − σ N 2 − N 3 ) ! ,

(7)

where σ is the ice-molecule number that one AFP molecule combines. It is well known that the coverage rate of the water molecules on the ice surface is 1 [22], so we can get N0 = N1，N3 = N1–σN2 and then Ω1 = 1，Ω3 = 1. As was well-known, the adsorption of AFPI molecules on the ice is along a certain direction, and the AFPI molecule has regular α-helical monomer rigid structure

5


[6,10]

.


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For simplicity, we consider the AFPI molecule adsorption on the ice-crystal surface as one-dimensional rod-like adsorption. And then the micro-states number of N2 AFPI molecules adsorbed on the ice-crystal surface is given by

Ω2 = CNN − yN 2

t

=

2 + N2

( N t − yN 2 + N 2 )! ( N t − yN 2 )! N 2 ! ,

(8)

where y is the number of lattice sites that an AFP molecule occupies. Using Strling’s approximation lnN! = NlnN – N, and the molar number instead of molecular number, the free energy increment of the system due to the AFP adsorption can be obtained as follows ∆F = − RT [(nt − yn 2 + n 2 ) ln(nt − yn 2 + n 2 ) − (nt − yn 2 ) ln(nt − yn 2 ) − n 2 ln n 2 ] + n 2 ε + σn 2 hγ   n − yn 2 + n 2 = − RT (nt − yn 2 ) ln t  nt − yn 2  + n 2 ε + σn 2 hγ

  n − yn 2 + n 2  + n 2 ln t n2  

  

(9)

Using the coverage rate of AFP molecules on the ice surface

θ = yN 2 N t ,

(10)

the increased free energy can be written as   1 − θ + θ y   1−θ +θ y  ∆F = − nt RT  (1 − θ ) ln    + ntθ (σ hγ + ε ) y .  + (θ y ) ln  θ θ 1 − y      

(11)

The change of Gibbs function of the growing ice crystal is then the sum of two parts: the change of ice crystal itself and the increased free energy caused by AFP molecule adsorption. Finally, the change of Gibbs function of the system can be calculated by ∆G = − nL ∆g + ∆F = − ( n1 0.075)λ∆T Te + ∆F = − ( Ant 0.075)λ∆T Te + ∆F ,

(12)

where λ = 1436 cal/mol is the latent heat of phase change of ice [11]. Using the equilibrium condition ∆G = 0, one can get the thermal hysteresis temperature finally

  1 − θ + θ y  1−θ + θ y   + θ (σhγ + ε ) − RTe  y (1 − θ ) ln  + θ ln  1−θ   θ y   ∆T = .   1 − θ + θ y  1−θ + θ y   yAλ (0.075Te ) − R  y (1 − θ ) ln  + θ ln θ 1 − θ y      6


(13)

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The coverage rate θ in Eq. (13) can be determined by solving the following equation [ 23]

− y ln[(1 − θ + θ y ) (1 − θ )] + ln[(1 − θ + θ y ) (θ y )] = − ( µ − ε ) (kT )

(14)

with [29-30] µ = − RT {ln ( nt n − 1) − 1 (1 − n nt ) + x ln ( z − 1) + x ln (1 − xn nt ) − x 2 n ( nt − xn )} + ε 0 . (15)

Results The thermal hysteresis temperature can be calculated using Eq. (13). The ice molecule number for each "lattice site" A in Eq. (13) can be evaluated as follows: the number of ice molecules on the ice-crystal surface covered by the adsorbed AFP molecules can be obtained by the ratio of the AFP molecular weight to the water molecule, that is χ = M/18, where M is the molecular weight of AFP. Meanwhile, an AFPI molecule covers y lattice site, so we have A = χ/y. The coordination number of the solution is taken to be z = 4 in Eq, (15) for the two-dimensional model. As examples, we have numerically calculated the thermal hysteresis temperature for the solutions of AFP9, HPLC-6 and AAAA2kE. As was mentioned above, the AFP9 molecule composes of 4 11-amino acid repeats, so y = 4 HPLC-6

[6]

and AAAA2kE

[31]

; likewise, y = 3 for both

[10,32]

. The other parameters in calculations are given in

Table 1.

Table 1 The used parameters in the calculation of thermal hysteresis temperature. χ[22]

y

σ[22]

x[30]

ε0(kcal/mol)[33]

ε(kcal/mol)[33]

AFP9

244

4

7

20

9.5

－6.1

HPLC-6

183

3

4

15

7.8

－5.1

AAAA2kE

183

3

4

15

8.0

－3.3

Figs. 1‒3 show the thermal hysteresis temperatures as functions of the solution concentration AFP9, HPLC-6 and AAAA2kE, respectively. For comparison, the

7



former results without considering the adsorption orientation as well as the experimental data are also plotted in Figs.

Thermal hysteresis(K)

1.2

1.0

0.8

0.6

0.4

0.2

0.0 0

5

10

15

20

25

30

Concentration(mmol/L)

Fig. 1 Thermal hysteresis temperatures of AFP9 solution as functions of the AFP concentration with (solid line) and without including the adsorption orientation (dashed line). The experimental data [31] are shown as ■.

1.0


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0.8

0.6

0.4

0.2

0.0 0

10

20

30

40


8


50

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Fig. 2 Thermal hysteresis temperatures of HPLC-6 solution as functions of the AFP concentration with (solid line) and without including the adsorption orientation (dashed line). The experimental data [2] are shown as ■.

0.7

0.6


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0.5

0.4

0.3

0.2

0.1

0.0 0

20

40

60

80

100


Fig. 3 Thermal hysteresis temperatures of AAAA2kE solution as functions of the AFP concentration with (solid line) and without including the adsorption orientation (dashed line). The experimental data [2] are shown as ■.

Discussion As well known, AFPI molecules have rigid alpha-helical element structure and can adsorb onto the special ice plane along certain direction. They can be usually bound on the basal ice-crystal planes. It is generally believed based on the ice-etching experiments and geometric considerations that the face of AFPI containing the polar Thr-residues interacts with the{2 0 2 1} ice plane to inhibit ice-crystal growth. Some authors investigated the specific mechanism of AFPI bound to the {2 0 2 1} ice plane and indicated that it has an energetic preference for binding in the {0 1 1 2} direction

[34,35]

. So in our theoretical analysis, the AFPI molecules are considered as

9



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one-dimensional rod-like adsorption on the ice surface, when including the influence of adsorption orientation. It can be easily know from the figures that the thermal hysteresis temperatures including adsorption orientation are obviously different those without including it. Figs 1 and 2 show that for AFP9 and HPLC-6 the thermal hysteresis temperatures involving the adsorption orientation are larger than those without considering adsorption orientation, when the concentration is very dilute. The former becomes smaller than the latter with increasing the concentration. This is due to the fact that the coverage rate involving the adsorption orientation increases faster than that without considering adsorption orientation for the very dilute solutions, but the reversed case appears with increasing the concentration. The thermal hysteresis is caused by the coverage of AFPs on the ice-crystal surface. The influences of adsorption orientation on the thermal hysteresis for AFP9 and HPLC-6 are more visible than that for AAAA2kE. It is also found that, the theoretical prediction are not very good agreement with the experimental data, especially for HPLC-6. This maybe due the fact that, the combined numbers σ for AFPIs are integer in our calculation, but the sizes of AFPI molecules and ice molecules will affect this factor actually, and then it may be not an integer in fact. And the interaction energy between AFPI and ice surface ε may be the influence factor, too.

Conclusion The thermal hysteresis activity of AFPI is studied based on the statistical mechanics theory. The molecular adsorption orientation is considered in the calculations. The thermal hysteresis temperatures for AFP9, HPLC-6 and AAAA2kE solutions are obtained and discussed. The calculated results show that the theoretical result in the mass is in agreement with experimental data and the influence of the adsorption orientation on the thermal hysteresis temperature cannot be neglected. It is pointed out that the statistical mechanics model is established based on the adsorption-inhibition theory of AFPs and has much room for improvement. Here we 10


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focus our attention to the influence of adsorption orientation on the thermal hysteresis activity. The results show that though the adsorption orientation is not the critical factor for the thermal hysteresis, its effect indeed exists and can not be omitted. In the future, the statistical mechanics model can be used to analysis the maximum of the thermal hysteresis temperature of AFPI, and also discuss other AFPs antifreeze activity, such as some insect antifreeze proteins.

Acknowledgment The work was supported by the Special Fund for Basic Scientific Research of Central Colleges, North China Institute of Science and Technology(3142012016), and the Self-raised Fund Project for Technological Research and Development of Langfang(2017011025).

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