A two-dimensional adsorption kinetic model for thermal hysteresis activity in antifreeze proteins.

Abstract

Antifreeze proteins (AFPs) and antifreeze glycoproteins (AFGPs), collectively abbreviated as AF(G)Ps, are synthesized by various organisms to enable their cells to survive in subzero environments. Although the AF(G)Ps are markedly diverse in structure, they all function by adsorbing to the surface of embryonic ice crystals to inhibit their growth. This adsorption results in a freezing temperature depression without an appreciable change in the melting temperature. The difference between the melting and freezing temperatures, termed thermal hysteresis (TH), is used to detect and quantify the antifreeze activity. Insights from crystallographic structures of a number of AFPs have led to a good understanding of the ice-protein interaction features. Computational studies have focused either on verifying a specific model of AFP-ice interaction or on understanding the protein-induced changes in the ice crystal morphology. In order to explain the origin of TH, we propose a novel two-dimensional adsorption kinetic model between AFPs and ice crystal surfaces. The validity of the model has been demonstrated by reproducing the TH curve on two different beta-helical AFPs upon increasing the protein concentration. In particular, this model is able to accommodate the change in the TH behavior observed experimentally when the size of the AFPs is increased systematically. Our results suggest that in addition to the specificity of the AFPs for the ice, the coverage of the AFPs on the ice surface is an equally necessary condition for their TH activity.

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Title:
A two-dimensional adsorption kinetic model for thermal hysteresis activity in antifreeze proteins
Author:
Li, Q Z
Yeh, Y
Liu, J J
Feeney, R E
Krishnan, V V
Publication Date:
05-01-2006
Publication Info:
Postprints, Multi-Campus
Permalink:
http://escholarship.org/uc/item/8kj1p8tt
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The following article appeared in Journal of Chemical Physics. Copyright 2006 American Institute
of Physics. This article may be downloaded for personal use only. Any other use requires prior
permission of the author and the American Institute of Physics.
Abstract:
Antifreeze proteins (AFPs) and antifreeze glycoproteins (AFGPs), collectively abbreviated as
AF(G)Ps, are synthesized by various organisms to enable their cells to survive in subzero
environments. Although the AF(G)Ps are markedly diverse in structure, they all function by
adsorbing to the surface of embryonic ice crystals to inhibit their growth. This adsorption results
in a freezing temperature depression without an appreciable change in the melting temperature.
The difference between the melting and freezing temperatures, termed thermal hysteresis (TH),
is used to detect and quantify the antifreeze activity. Insights from crystallographic structures
of a number of AFPs have led to a good understanding of the ice-protein interaction features.
Computational studies have focused either on verifying a specific model of AFP-ice interaction or
on understanding the protein-induced changes in the ice crystal morphology. In order to explain
the origin of TH, we propose a novel two-dimensional adsorption kinetic model between AFPs and
ice crystal surfaces. The validity of the model has been demonstrated by reproducing the TH curve
on two different beta-helical AFPs upon increasing the protein concentration. In particular, this
model is able to accommodate the change in the TH behavior observed experimentally when the
size of the AFPs is increased systematically. Our results suggest that in addition to the specificity
of the AFPs for the ice, the coverage of the AFPs on the ice surface is an equally necessary
condition for their TH activity. (c) 2006 American Institute of Physics.

Page 2

A two-dimensional adsorption kinetic model for thermal hysteresis activity
in antifreeze proteins
Q. Z. Li
The Theoretical Physics and Biology Laboratory, Department of Physics, College of Sciences
and Technology, Inner Mongolia University, Hohhot 010021, China, and Department of Applied Science,
University of California, Davis, California 95616
Y. Yeh
Department of Applied Science, University of California, Davis, California 95616
J. J. Liu
The Theoretical Physics and Biology Laboratory, Department of Physics, College of Sciences
and Technology, Inner Mongolia University, Hohhot 010021, China
R. E. Feeney
Department of Food Science and Technology, University of California, Davis, California 95616
V. V. Krishnana?
Department of Molecular, Cell, and Developmental Biology, University of California, Santa Cruz,
Santa Cruz, California 95064, and Department of Applied Science and Center for Comparative Medicine,
University of California, Davis, California 95616
?Received 12 January 2006; accepted 16 February 2006; published online 23 May 2006?
Antifreeze proteins ?AFPs? and antifreeze glycoproteins ?AFGPs?, collectively abbreviated as
AF?G?Ps, are synthesized by various organisms to enable their cells to survive in subzero
environments. Although the AF?G?Ps are markedly diverse in structure, they all function by
adsorbing to the surface of embryonic ice crystals to inhibit their growth. This adsorption results in
a freezing temperature depression without an appreciable change in the melting temperature. The
difference between the melting and freezing temperatures, termed thermal hysteresis ?TH?, is used
to detect and quantify the antifreeze activity. Insights from crystallographic structures of a number
of AFPs have led to a good understanding of the ice-protein interaction features. Computational
studies have focused either on verifying a specific model ofAFP-ice interaction or on understanding
the protein-induced changes in the ice crystal morphology. In order to explain the origin of TH, we
propose a novel two-dimensional adsorption kinetic model between AFPs and ice crystal surfaces.
The validity of the model has been demonstrated by reproducing the TH curve on two different
?-helical AFPs upon increasing the protein concentration. In particular, this model is able to
accommodate the change in the TH behavior observed experimentally when the size of the AFPs is
increased systematically. Our results suggest that in addition to the specificity of the AFPs for the
ice, the coverage of the AFPs on the ice surface is an equally necessary condition for their TH
activity. © 2006 American Institute of Physics. ?DOI: 10.1063/1.2186309?
I. INTRODUCTION
Adaptation schemes for survival in extreme environ-
ments have resulted in the evolution of unique protein super-
families that permit organisms to exist and thrive over a
broad geographic distribution of the earth. In addition to the
set of proteins that interact with biological solids in the
biomineralization process, a class of proteins known as anti-
freeze proteins ?AFPs? exists in low-temperature environ-
ments that inhibit lethal intra- or extracellular ice formation.
AFPs and their glycosylated forms antifreeze glycoproteins
?AFGPs?, also known as thermal hysteresis proteins, play an
important biochemical role in many animals,1,2insects,3–5
and plants6that is a product of their cold adaptation pro-
cesses to defend their tissues from freezing injury. The solu-
tion of AFP exhibits a noncolligative freezing temperature
depression that is from 300 to 500-fold effective compared
with the equilibrium colligative depression observed for or-
dinary solutes ?on a molar basis?.7Extensive research in this
area suggests that the mechanism for the freezing tempera-
ture depression of AFP is a result of its ice binding ability,
which inhibits the growth of embryonic ice crystals that
naturally emerge in supercooling water. This unique inhibi-
tory function of AFP provides great potential for cryoindus-
trial usages, such as cryopreservation of tissues and cells and
maintaining the texture of frozen materials.8–10
In vitro, the function of AFP is quantified using two ob-
servable effects due to presumed interaction of these proteins
with ice: modification of the crystal growth morphology and
thermal hysteresis ?TH?. Recently, Wathen et al.11have de-
veloped a model that combines three-dimensional ?3D? mo-
a?Author to whom correspondence should be addressed. Electronic mail:
vvkrishnan@ucdavis.edu
THE JOURNAL OF CHEMICAL PHYSICS 124, 204702 ?2006?
0021-9606/2006/124?20?/204702/8/$23.00© 2006 American Institute of Physics
124, 204702-1
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Page 3

lecular representation and detailed energetic calculations of
molecular mechanics techniques with a statistical probabilis-
tic approach to study the inhibitory effects of AFPs on ice
crystal formation. This modeling has produced results con-
sistent with experimental observations, including the replica-
tion of ice-etching patterns, ice growth inhibition, and spe-
cific AFP-induced ice morphologies.11,12Although ice-
surface recognition is undoubtedly crucial for AFP activity, it
is not completely clear just how AFP-ice interaction is re-
sponsible for the phenomenon of TH. However, neither the
computational methods described below nor conventional
molecular dynamics studies of AFP-ice interactions13–16are
able to completely explain the dynamic nature of TH; it is
the hope that analyses such as that presented here will aid in
experimental and modeling techniques that can provide in-
sight into TH activity of the AFPs.
An early theoretical model for computing the thermal
hysteresis of AF?G?P in solution is based on the Gibbs-
Thomson ?Kelvin? model.7,17It elegantly explains the non-
colligative depression of AFP as an “adsorption-inhibition
mechanism,” in which accumulations of AFPs onto the
growing surfaces of ice crystals result in the creation of num-
bers of convex ice surfaces at the limited open spaces of ice
between the bound AFPs.18,19The free energy of the convex
ice surfaces becomes larger with the increase of the surface
curvature. Therefore, further binding of water molecules
onto the convex ice surfaces becomes energetically unfavor-
able, leading to the suppression of the ice crystal growth by
increasing the number of bound AFPs. This irreversible ice
binding model was then modified to a semireversible
model20,21implying that the kinetics of AFP adsorption leads
to ice growth inhibition. Li and co-workers have presented
an analytical kinetic theory to estimate the TH as a function
of the molar concentration of AFP and AFGPs.22–25Recently,
Sander and Tkachenko26have proposed a “stones on a pil-
low” model in which the geometrical shape of the antifreeze
molecule is used to explain the TH activity.
As AFPs tend to adopt different structural scaffolds,27–29
the precise antifreeze mechanism for different kinds of AFPs
adsorbing to ice crystals to inhibit ice growth is also ex-
pected to be different. For example, structure-based models
developed for type I AFP ??-helical structure? may not suf-
ficiently explain the function of a ?-helical AFP. The model
should be able accommodate the fact that all the available
sites on the surface of the ice lattice are expected to be in-
distinguishable and hence the AFP binding is “nonspecific.”
It is important that the model should be general enough to
accommodate the difference in the structural features while
explaining the universal phenomenon of TH in all the AFPs
and AFGPs. To incorporate all these features, we report a
two-dimensional kinetic adsorption model, together with its
initial application to the TH behavior of two AFPs. We find
that this model is useful not only in understanding how
changes in the protein concentration change the TH behavior,
but also the recent experimental observation that alteration of
the total size of AFPs affects their dynamic behavior. In ad-
dition to the binding of AFPs to ice surfaces, this model
clearly suggests that the surface coverage of the proteins on
ice is critical for their function.
II. THE TWO-DIMENSIONAL ADSORPTION KINETIC
MODEL
A. Description of equilibrium kinetics of thermal
hysteresis
Experimental studies performed in the field of AFP
?AFGP? have yielded several characteristics indicating that
these proteins adsorb to ice in a progressive manner. The
kinetics of adsorption of AFPs to ice surfaces are expected to
progress first by their attachment to the ice-water interface.
This is then followed by rearrangement of adsorbed AFPs by
diffusion, reorientation, and/or conformational change. The
last of the steps is the detachment from the interface.20,21,30
As AFPs do not interfere with the nucleation of ice crystals
but somehow prevent their growth,7thermal hysteresis oc-
curs in the presence of ice crystals which have already been
seeded. AFPs will adsorb onto the surface of small ice crys-
tals and interfere with the propagation of their growth, par-
ticularly on certain specific crystalline facets. As described
by Strom et al.,31AFP-induced surface matching of periodic
bond chains in two dimensions enables two-dimensional
regular ice-binding surfaces of the insect AFPs to engage
with the ice surfaces. This observation was originally re-
ported by Wilson and Leader32and elaborated by Du and
co-workers.33–35
We suppose that the interaction between the AFP and the
surface of the ice crystal is a simple reversible adsorption
process that may be represented by
i?A? + I Û Ai:I,
?1?
where an AFP ?A? molecule with “i” binding sites and ICE
?I? represents the appropriate crystal lattice of ICE to form
the AFP bound of ice ?A:I?. This macroscopic equilibrium
between the AFP and ICE is written in terms of several mi-
croscopic binding events given by
A + I0↔
kd
ka
I1;
A + I1↔
kd
ka
I2;
A + I2↔
kd
ka
I3;
¯ A + Ii−1↔
kd
ka
Ii,
?2?
where I0the bare ice lattice and antifreeze proteins ?A? ad-
sorbed onto the ice surface Ii?i=1,2,3,¯?, with the step-
wise microscopic equilibrium constant, and Kiis then de-
fined to be the ratio of on to off rates for each step. In the
absence of cooperativity, each of the stepwise binding events
will be equivalent, leading to K1=K2=K3=¯=Ki=K
=ka/kdas assumed in writing Eq. ?2?.
The concentration of the adsorbed AFP can be expressed
in terms of the products of stepwise equilibrium constant Ki,
the concentration of the unbound AFPs, and the number of
available sites on ice using the principle of mass action as
?I1? = K1?A??I0?;
?I2? = K2?A??I1?;
¯ · · ?Ii? = Kn?A??Ii−1?.
Rewriting Eq. ?3? in terms of ?I0?,
?3?
?Ii? = ?A??
j=1
i
Ki?I0?.
?4?
204702-2Li et al.J. Chem. Phys. 124, 204702 ?2006?
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Page 4

As in the case of AFP-ice system containing multiple
identical binding sites and assuming that binding of an AFP
molecule is independent of the others ?no cooperativity?, the
stepwise binding constants are equal to the intrinsic associa-
tion constant and the statistical factor determined by the
number of microscopic states constituting different stages of
AFP covered ice surface by the AFPs. This binding phenom-
enon is similar to the nonspecific binding of protein ligands
to the periodically spaced minor/major groves of a DNA
molecule.36In general, an analytical isotherm modified from
a Langmurian type of isotherm developed by McGhee and
von Hippel is often used to study macromolecular binding
equilibrium in which the surface of the substrate is modeled
as an infinite one-dimensional lattice.37Because of the finite
size of the ice crystals, the McGhee and von Hippel model
does not readily permit the calculation of the statistical fac-
tors and the populations of the partially ligated states and one
needs to resort to combinatorial methods. The product of Ki
in Eq. ?4? for “k” bound AFPs can be written in general as
?
i=1
k
Ki= CkKk,
?5?
where Ck’s are the statistical weights and these cause sequen-
tial decrease in the macroscopic binding constants with in-
creasing saturation. Equations ?4? and ?5? therefore allow the
estimation of the fraction of ice surface covered by AFPs.
The observed difference between the freezing and melt-
ing temperatures ?thermal hysteresis activity? changes as a
function of the amount of AFPs on the ice surface, which in
turn is a function of the bulk AFP concentration.7Following
the argument of Burcham et al.,38the fraction of sites cov-
ered ?denoted as ?? therefore must be proportional to the
observed thermal hysteresis activity. Consequently, the ex-
perimentally observed plateau must correspond to the lattice
site saturation of the ice surface. If ?T is the difference be-
tween the melting and freezing temperatures of an AFP in
solution and ?Tmaxis the maximum observed thermal hys-
teresis activity ?at saturation?, for a specific AFP molecule
then ? can be formally written as
? =
?T
?Tmax
.
?6?
Therefore, for equilibrium measurement of AFP-ice in-
teraction, with m AFP molecules adsorbing to N ice binding
sites, using Eqs. ?4? and ?5? the coverage factor ? can be
written as
? =m
NC¯m,
?7?
where C¯mis the statistical factor providing the number of
distinct ways for forming an ensemble of microscopic states
for each of the m AFP molecules. Equating Eqs. ?6? and ?7?
the observed thermal hysteresis is then written as
?T = ??Tmax=?m
NC¯m??Tmax.
?8?
By directly estimating the fraction of an ice surface cov-
ered by AFP ???, this model can be used to estimate the
observed difference in freezing and melting temperatures
?the activity?, as a function of the amount of AFP on the
surface, which, in turn, is a function of the bulk AFP con-
centration. The fraction of sites covered ?, therefore must be
proportional to the observed activity, i.e., to thermal hyster-
esis.
B. Analytic assessment of the fractional surface
coverage
1. One-dimensional model
In order to determine ?, we take the combinatorial ap-
proach. Analytical expressions for a one-dimensional lattice
were originally given by Epstein39and have recently been
reviewed by Munro et al.40Though we have extended the
approach to a two-dimensional model that is more appropri-
ate for the AFP binding to the ice surface, following the
above-mentioned references, we briefly review the salient
features of the one-dimensional model. We consider the case
where the protein molecule ?AFP? binds to “n” but covers “
m” units of ice lattice ?ICE?, which is comprised of “N” units
?available binding sites on the ice surface?. The interaction is
governed by the intrinsic association constant K, which is the
ratio of kaand kd, and governed by the equilibrium process
described by Eq. ?4?. Therefore the fraction of the ice surface
covered by AFP with concentration ?A? ?Eq. ?7?? is rewritten
in general terms as
? =mC¯
N
,
?9?
where C¯is the average number of AFP molecules on N ice-
lattice sites and is given by the molar ratio of bound AFP
molecules to total available lattice sites on the ice and is
written as
C¯=
?i=1
??i=0
RiCi?Ii?
RCi?Ii??,
?10?
where R signifies an integer that is less than or equal to N/n,
as the maximum number of AFP molecules which may bind
to the lattice, and where Ciis the number of possible ar-
rangements of the bound AFP molecules and empty available
for distribution on the ice lattice, the statistical factor ?see
Eq. ?5??. Using Eqs. ?9? and ?10? the coverage factor ? can be
written as
? =?i=1
N??i=0
RmiCi?Ii?
RCi?Ii??.
?11?
For a situation of a lattice consisting of N ice binding
sites with sites with each AFP having n sites for binding, the
number of possible places to distribute i AFP molecules will
be N−ni. The total number of items to be arranged in the
lattice can then be given by
Ci=?N − Mi + i?!
?N − Mi?!i!.
?12?
204702-3 Thermal hysteresis activity in antifreeze proteinsJ. Chem. Phys. 124, 204702 ?2006?
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Page 5

For example, with N=12, n=4, and i=2, the number of
configurations available ?Cifrom Eq. ?12?? is 15. Physically
this means that there is a total of 15 microscopic species that
have two ?i? AFP proteins bound on the ice lattice, each
having an intrinsic binding constant of K, while the macro-
scopic binding constant is a result of cumulative effect of the
ensemble of such ?15 here? species.
This combinatorial approach may be extended to include
the possibility of the AFP occupying a total of m units which
only binding to n. For this situation, the statistical weight can
be given by40
Ci=
?N − m?i − 1?n + 1?!
?N − m?i − 1? − n + i?!.
?13?
These expressions can be further extended to include
cooperative binding40If i AFP molecules are bound, then
various possible arrangements contain a maximum of i−1
such adjacencies and the number of possible arrangements is
defined as the number of ways of attaching i ligands with j
adjacencies as
Ci=?
j=0
i−1
Ci?j??j,
?14?
where
Ci?j?=
?N − m?i − 1? − n + 1?!?i − 1?!
?N − m?i − 1? − n − i + j + 1?!?i − j?!j!?i − j − 1?!,
?15?
and ? is the cooperativity factor. Though no cooperativity is
considered in rest of the analysis, Eq. ?15? is provided for the
sake of completeness and it reduces to Eq. ?13? when ?=1
?no cooperativity?.
The generalized one-dimensional model includes the ef-
fect of cooperative binding, in which the intrinsic binding
constants are given by the factor ?.41–44Cooperative binding
considers the number of attachments to adjacent segments ?j?
possible in a particular arrangement of AFPs on a linear lat-
tice. Expressions for cooperative binding as well as those
with end effects for one-dimensional model have been
derived.39,40Figure 1?a? schematically shows one of the sev-
eral possible combinations of binding of a set of AFP mol-
ecules, each with three binding sites, to a finite length of
one-dimensional ice lattice. This model allows the distinction
between the number of sites covered by the AFP molecule to
the number sites it is bound, i.e., some sites covered by AFP
molecule may not have AFP adsorbed to them. For an ice
crystal that has a total of N sites, where each AFP molecule
can cover m sites on the surface of ice crystal but binds only
to n sites ?m?n?, the fraction ?the ratio of the number of
sites covered by AFPs to the total number of sites on all ice
crystal surfaces? of the ice surface covered by AFPs with
concentration can be defined on the basis of Eqs. ?4? and
?11?,
? =?i=1
N??i=0
=?i=1
N??i=0
RmiCi?Ii?
RCi?Ii??=?i=1
RmiCiKi?A?i?I0?
N??i=0
RCiKi?A?i?I0??
RmiCiKi?A?i
RCiKi?A?i?,
?16?
where ?A? is the concentration of AFP molecules ?CAFP?, ?I0?
the concentration of ice crystal unoccupied, ?Ii? the concen-
tration of ice crystal occupied by i AFP molecules, and K is
the microscopic adsorption equilibrium constant.
2. Two-dimensional model
In order to account for the effect of the interaction of
AFP with ice surface, here we present the extension to two
dimensions. Figure 1?b? schematically depicts the situation
where the squares represent the AFP while the lattice shows
the set of possible adsorption sites on the surface of the ice
?no distinction between the different planes is considered?.
Considering that the surface of ice crystals is composed by
M? lines and N identical adsorption sites on each line, an
AFP molecule can cover m? lines. We may calculate the
number of ways of attaching iaAFPs with j adjacencies in
the ath line ?a=1,2¯M? ?M is an integer given by M?/m??
in a manner similar to the one-dimensional model. If iaAFP
molecules are bound, the various possible arrangements con-
tain a maximum of ?ia−1? such adjacencies.
This can be achieved by rewriting the regression rela-
tions after replacing l by iain Eq. ?15?. This leads to the Ci’s
for the two-dimensional model with cooperativity as
Cia?j?= Cia−j
N−m?ia−1?−n+1Cj
ia−1
=
?N − m?ia− 1? − n + 1?!?ia− 1?!
?N − m?ia− 1? − n − ia+ j + 1?!?ia− j?!j!?ia− j − 1?!.
?17?
For noncooperative interactions ??=1?, the Ci’s for the
two-dimensional model become ?using Eq. ?13? instead of
Eqs. ?15? and ?9??
FIG. 1. Schematic representation of the ?a? one- and ?b? two-dimensional
adsorption models. The AFP molecules bind to the ice lattice with an asso-
ciation constant K. The available sites on the surface of ice and on the AFP
are given by large-gray and small-filled circles, respectively. In the two-
dimensional representation of the model ?b?, the importance of AFPs to
cover more sites than they bind to ?here binds to four sites and covers 6? is
highlighted.
204702-4Li et al. J. Chem. Phys. 124, 204702 ?2006?
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