The hydrophobic effect and its role in cold denaturation.

Abstract

The hydrophobic effect is considered the main driving force for protein folding and plays an important role in the stability of those biomolecules. Cold denaturation, where the native state of the protein loses its stability upon cooling, is also attributed to this effect. It is therefore not surprising that a lot of effort has been spent in understanding this phenomenon. Despite these efforts, many unresolved fundamental aspects remain. In this paper we review and summarize the thermodynamics of proteins, the hydrophobic effect and cold denaturation. We start by accounting for these phenomena macroscopically then move to their atomic-level description. We hope this review will help the reader gain insights into the role played by the hydrophobic effect in cold denaturation.

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The hydrophobic effect and its role in cold denaturationq
Cristiano L. Diasa,*, Tapio Ala-Nissilab,c, Jirasak Wong-ekkabuta, Ilpo Vattulainenc,d,e,
Martin Grantf, Mikko Karttunena
aDepartment of Applied Mathematics, The University of Western Ontario, Middlesex College, 1151 Richmond St. N., London, Ont., Canada N6A 5B7
bDepartment of Physics, Brown University, Providence, RI 02912-1843, USA
cCOMP Center of Excellence and Department of Applied Physics, Helsinki University of Technology, P.O. Box 1100, FI-02015 TKK, Espoo, Finland
dInstitute of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland
eMEMPHYS-Center for Biomembrane Physics, University of Southern Denmark, Denmark
fPhysics Department, Rutherford Building, McGill University, 3600 rue University, Montr ´eal, Que., Canada H3A 2T8
a r t i c l e i n f o
Article history:
Received 2 July 2009
Accepted 14 July 2009
Available online 17 July 2009
Keywords:
Hydrophobic effect
Thermodynamics
Clathrate cages
Hydrate cages
Cold denaturation
Proteins
a b s t r a c t
The hydrophobic effect is considered the main driving force for protein folding and plays an important
role in the stability of those biomolecules. Cold denaturation, where the native state of the protein loses
its stability upon cooling, is also attributed to this effect. It is therefore not surprising that a lot of effort
has been spent in understanding this phenomenon. Despite these efforts, many unresolved fundamental
aspects remain. In this paper we review and summarize the thermodynamics of proteins, the hydropho-
bic effect and cold denaturation. We start by accounting for these phenomena macroscopically then move
to their atomic-level description. We hope this review will help the reader gain insights into the role
played by the hydrophobic effect in cold denaturation.
? 2009 Elsevier Inc. All rights reserved.
Introduction
Using cold temperatures in biology and medicine has its origins
in ancient Egypt where they were used for healing already around
2500 BC. The birth of modern cryobiology is often associated with
James Arnott (1797–1883) who applied cold temperatures to de-
stroy cancerous tumors [20]. The temperatures he reached were
not extreme by today’s standards, only to ?24 ?C, but his work
has been the inspiration for using cold temperatures as a cheap
and efficient method for certain surgical operations as well as for
preservation of biological matter. In today’s cryotherapy, liquid
nitrogen temperatures are typically applied.
At a more microscopic level, temperature is one of the most
important parameters in defining proteins’ behavior in living mat-
ter. It is now well established that proteins denature at both high
(typically ?60 ?C) and low (typically ??20 ?C) temperatures
[40,58,57]. The term ‘denaturation’ typically refers to the well
established phenomenon of heat denaturation, whereas the latter
is often called ‘cold denaturation’ and it is the focus of this article.
Denaturation can also be induced by pressure [34,46].
Denaturation, by heat or cooling, refers to the loss of the unique
three-dimensional structure [3] a protein has under physiological
conditions. When a protein experiences this structural instability
it also loses its functionality. Although cold denaturation has been
experimentally established [57,59] and even suggested to be a uni-
versal mechanism present in most proteins [42,52,57], it is much
harder to study experimentally due to the necessary sub-zero tem-
peratures.Whenconsideringcolddenaturationasauniversalmech-
anism, two notable exceptions should be noticed. First, despite
several studies, there is only one report of cold denaturation in
hyperthermophile organisms [16]. Second, the so-called ‘intrinsi-
cally disordered proteins’ are known to be resistant against heat
denaturation and experiments have now shown that to be the case
for their low temperature behavior as well although the kinetic
mechanisms may be different [1]. Historically, cold denaturation
(inthepresenceofurea)wasfirstsuggestedbyHopkinsin1930[33].
Non-covalent bonding (i.e., hydrogen bonds, electrostatic inter-
actions and hydrophobicity) plays a crucial role in the structural
stability of proteins. Hydrogen bonding is important for the forma-
tion of secondary structures, while electrostatic and hydrophobic
interactions are needed for stabilizing the tertiary structure of pro-
teins. The subtle variation in the relative strengths of these interac-
tions lies in the heart of denaturation. Nowadays it is clear that
understanding hydrophobicity [15,24] and the role of entropy vs.
enthalpy [23] are key for a better understanding. However, funda-
mental questions remain unanswered despite several theoretical
0011-2240/$ - see front matter ? 2009 Elsevier Inc. All rights reserved.
doi:10.1016/j.cryobiol.2009.07.005
qThis work was supported by the Natural Sciences and Engineering Research
Council of Canada, and le Fonds Québécois de la recherche sur la nature et les
technologies. T.A.-N. and M.K. wishes to thank support from the Academy of Finland
through its COMP Center of Excellence and TransPoly grants.
* Corresponding author.
E-mail address: diasc@physics.mcgill.ca (C.L. Dias).
Cryobiology 60 (2010) 91–99
Contents lists available at ScienceDirect
Cryobiology
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modeling and computer simulations at lattice and even atomistic
levels [10,12,17,19,23,45,47,54,61,70].
In our earlier work [23], we introduced a microscopic model to
describe heat and cold denaturation within the same framework.
Clathrate cages (particular order structures of water molecules)
around non-polar residues were identified as the crucial structure
leading to both types of denaturation. Their high entropic cost ac-
counts for the folding of the protein at ambient temperature while
their low enthalpy is responsible for the unfolding of the protein at
low temperature. This explains why cold denaturation proceeds
with heat release as opposed to heat absorption seen during heat
denaturation. Notice that a literal interpretation of the word ‘‘cage”
calls for caution since a complete hydrate cage around the solute is
likely to form at only low temperature while at ambient tempera-
ture only incomplete cages survive for a reasonable amount of
time. In this paper we summarize some of the theoretical efforts
to understand microscopically the hydrophobic effect and the role
it plays in cold denaturation. We proceed as follow: in the next sec-
tion we describe the thermodynamics of proteins and identify the
hydrophobic effect as the main mechanism behind cold denatur-
ation. This effect is discussed in ‘‘Hydrophobic effect”. Thermody-
namical and atomic models accounting for cold denaturation are
introduced in ‘‘Cold denaturation”, providing an explanation for
the phenomena. A conclusion is given at the end.
Thermodynamics of proteins
In their natural environment proteins exist in a variety of con-
figurations, which can be mapped into folded and unfolded states
of the protein [21]. In equilibrium, the stability of a configuration
associated with the folded structure is given by the relative free en-
ergy of these two states [40,58,71]: DG ¼ Gu? Gf. The folded state
is stable if DG > 0, and this stability increases with increasing DG.
If DG < 0, then the system unfolds spontaneously, releasing energy
to the environment. Thermodynamics provides a framework for
describing the temperature dependence of DG [6]. This depen-
dence is obtained by expanding the enthalpy and the entropy
around the transition point by means of the heat capacity as de-
scribed below.
At the transition temperature Tc, both the folded and unfolded
configurations have the same energy:
DGðTcÞ ¼ DHðTcÞ ? TcDSðTcÞ ¼ 0;
such that
ð1Þ
DSðTcÞ ¼ DHðTcÞ=Tc;
where DS is the change in entropy and DH is the change in enthal-
py. The enthalpy of transition can be written in terms of the heat
capacity at constant pressure:
ZT
since DCp? ðoDHðTÞ=oTÞP. Whenever DCPðTÞ can be considered to
be constant, this equation reduces to:
ð2Þ
DHðTÞ ¼ DHðTcÞ þ
Tc
dTDCPðTÞ;
ð3Þ
DHðTÞ ¼ DHðTcÞ þ ðT ? TcÞDCP:
For the entropy of transition, we have
ZT
¼DHðTcÞ
Tc
ð4Þ
DSðTÞ ¼ DSðTcÞ þ
Tc
ZT
dT oDSðTÞ=oT
dðlnTÞDCPðTÞ ¼DHðTcÞ
ðÞ
þ
Tc
Tc
þ DCPln
T
Tc
??
;
ð5Þ
where the last equality is obtained by assuming that DCp is
constant.
Now, by considering Eqs. (4) and (5) together, we obtain the
Gibbs energy of unfolding:
DGðTÞ ¼ DHðTÞ ? TDSðTÞ
¼ðTc? TÞ
Tc
DHðTcÞ þ ðT ? TcÞDCP? TDCpln
T
Tc
??
:
ð6Þ
This equation brings about that the temperature dependence of DG
is determined by Tc; DHðTcÞ; and DCP. These quantities can be ob-
tained experimentally from calorimetry experiments [57], which
have shown typical numbers for real proteins to be [63]:
Tc= 60 ?C, DHðTcÞ ¼ 500 kJ mol?1, and DCP¼ 10 kJ mol?1K?1.
Fig. 1 shows the temperature dependence of DG for a typical
protein. This quantity has a convex shape, indicating the presence
of two phase transitions. These transitions take place whenever
DGðTÞ ¼ 0. The transitions correspond to heat denaturation at
T ¼ Tc, and to cold denaturation at T ’ ?30?C. At intermediate
temperatures, between T ’ ?30?C and Tc, the folded configura-
tion is thermodynamically stable, with maximal stability occurring
at about 17 ?C. It is interesting to note that even under most stable
conditions, very little energy is required to unfold the protein:
DGðT ¼ 17?CÞ ’ 32 kJ mol?1. Thus, while nature requires the
folded protein to be stable in order to function, the stability is mar-
ginal, with 32 kJ mol?1being only a minor fraction (about 5%) of
the interaction energy of a single covalent bond between two car-
bon atoms.
In this work, we are mainly interested in the low temperature
transition to the denatured state, i.e., cold denaturation. Thermo-
dynamically this transition results from the convex curvature of
the Gibbs energy. This curvature can be computed from Eq. (6):
o2DGðTÞ
oT2
and it becomes convex, i.e., o2DGðTÞoT2< 0, whenever DCp> 0. For
globular proteins DCpis positive [57] and increases with the length
of the protein [43]. This feature is mostly attributed to the hydra-
tion of non-polar amino acids which have a distinguishable positive
DCp, as opposed to polar amino acids that contribute negatively to
the heat capacity of proteins [36,56].
The temperature Tdat which cold denaturation occurs can be
obtained by solving DGðTdÞ ¼ 0. This task becomes a simple analyt-
ical exercise when the logarithmic term in Eq. (6) is approximated
by its second order Taylor expansion:
?
The temperature at which cold denaturation takes place then reads
[40]:
¼ ?DCP
T
;
ð7Þ
ln
Tc
T
?
’
Tc? T
T
??
?Tc? T
2T2:
ð8Þ
T2
d’
T2
cDCp
2DHðTcÞ þ TcDCp:
This equation shows clearly that cold denaturation becomes acces-
sible at higher temperatures for proteins with larger DCpand Tc.
Notice that DHðTcÞ has the opposite effect on Td.
For a typical protein, the temperature at which cold denatur-
ation occurs is below the freezing point of water, see Fig. 1. This
undesirable feature for experimental studies is usually overcome
by weakening the folded protein through pressure [40,46,59,68]
or chemical denaturants [37,57]. Qualitatively, the weakening
can be seen as shifting of the convex Gibbs energy (shown in
Fig. 1) downwards, thereby increasing Td above freezing, and
decreasing Tc. Pressure has the additional effect of decreasing the
freezing point of water to ?22 ?C (at 200 MPa) such that experi-
ments can be performed within a wider range of temperatures be-
low 0 ?C. The drawback of using pressure or chemistry to weaken
ð9Þ
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the protein is that it becomes difficult to deconvolute the effect of
those denaturants from cold denaturation itself.
In Table 1 we show the experimental value of Tdfor a few se-
lected proteins. Notice that the actual value of Tdis strongly depen-
dent on the experimental technique used to weaken the stability of
a protein.
Summarizing, cold denaturation in globular proteins takes place
because of their large positive heat capacity of unfolding. This is
related to the hydration of non-polar residues and thus to the
hydrophobic effect, which we discuss in the next section. It is
important to realize since the hydrophobic effect is a general prop-
erty of globular proteins [24,35], cold denaturation is also expected
to be a general phenomenon. However, the interpretation of cold
denaturation calls for some caution, since it is technically difficult
to observe and differentiate the related phenomena. For example,
experimental data have recently reported cold denaturation in a
hyperthermophile protein [16]. Meanwhile, due to the lack of
folded structure, intrinsically disordered proteins do not lose solu-
bility either at high or low temperatures [1].
Hydrophobic effect
The hydrophobic effect is one of the main driving forces for the
formation of self-assembled biological structures such as lipid
membranes and proteins [25]. It reflects the tendency of water to
avoid non-polar molecular structures such as hydrocarbons and
hydrophobic amino acids. The hydrophobic effect is largely due
to the special ability of water molecules to form hydrogen bonds
(H-bonds)with themselves, attempting to
arrangements where the network of these H-bonds is perturbed.
Non-polar molecules such as lipids and hydrophobic amino acids
thus tend to aggregate and displace themselves from contact with
water.
The hydrophobic effect can be inferred from the Gibbs energy of
transfer of non-polar molecules from their bulk liquid state into
water: DG ¼ Gwater? Gbulk. If DG is positive, then the solute prefers
to be surrounded by other solutes as opposed to water. The larger
DG becomes, the higher is the tendency of solutes to cluster to-
gether. In contrast, negative DG implies a molecule that is soluble
in water.
Non-polar molecules have a large positive energy of transfer.
For example, the free energy of transfer of methane molecules at
25 ?C is about 26.2 kJ mol?1[60]. Some insight can be gained by
computing the enthalpy DH and entropy DS of transfer. For meth-
ane, the enthalpic contribution to DG is negative (?4.3 kJ mol?1),
while the entropic contribution ?TDS is positive (28.7 kJ mol?1)
and corresponds to 85% of the interaction. Thus, the hydration of
non-polar solutes is characterized by a small favorable enthalpy
and a strong unfavorable entropy. The temperature dependences
of DG; DH; and ? TDS for methane are shown in Fig. 2. As temper-
ature decreases, bulk methane becomes less stable since DG de-
creases. Enthalpy can be held responsible for this destabilization
as DH decreases with decreasing temperature. Entropy has the
opposite behavior, it stabilizes bulk methane since ?TDS increases
with decreasing temperature.
The effects of hydrophobicity can also be measured by the heat
capacity of transfer from liquid to water via DCp¼ Cwater
avoidstructural
p
? Cbulk
p
. For
-40-30 -20-100 10 20 3040
5060
70
Temperature (oC )
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
Free energy ( kJ/mol )
ΔG(T)
ΔH(T)
-TΔS(T)
-40 -200
T ( oC )
20 40 60
0
10
20
30
40
kJ/mol
Fig. 1. Temperature dependence of the Gibbs free energy DGðTÞ, entropic energy ?TDSðTÞ and enthalpy DHðTÞ of folding of a typical protein. Inset shows zoom of the Gibbs
free energy of folding. Tcand Tdare the temperatures at which heat and cold denaturation occur, respectively.
Table 1
Experimental temperature for cold ðTdÞ and heat ðTcÞ denaturation.
Td(?C)
Tc(?C)
Chymotrypsinogen pH 2.07 (0.3 GPa) [32]a
Ribonuclease pH 2 (0.3 GPa) [32]a
Ubiquitin pH 4.0 (0.2 GPa) [37]
Metmyoglobin pH 3.84 [57]b,c
Metmyoglobin pH 3.70 [57]b,c
Apomyoglobin pH 4.70 [57]b,c
Apomyoglobin pH 4.78 [57]b,c
Staphylococcal nuclease pH 6.0 [57]b,c
Staphylococcal nuclease pH 6.5 [57]b,c
Staphylococcal nuclease pH/p2H 5.5 (0.15 GPa) [53]a
B state of ferricytochrome c pH 13 [39]d
?9
?32
?12
49
30
82
55
44
51
54
33
38
48
16
7
10
1
?3
6
4
0
?16
aEstimated from figures by Smeller [68].
bEstimated from figures by us.
cAt ambient pressure.
dIn the presence of NaCl: 0.01 M. Data for other concentrations of NaCl is also
given by the authors.
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simple solutions DCpis small, while for non-polar solutes it is large
and positive [27]. This large heat capacity of transfer has been
shown to be proportional to the surface area around non-polar sol-
utes accessible to water [44], and thus proportional to the number
of solvent molecules around the solute [31].
Thermodynamics
As mentioned above the hydration of non-polar solutes in water
has a large entropic cost and a small favorable enthalpy. This pecu-
liar partition of the free energy imposes constraints on models for
the hydrophobic effect. In this section we discuss a model intro-
duced by Muller [31,41,50] that displays the correct partition of
the free energy and provides insights into the microscopic nature
of the phenomena due to its simplicity.
The focus of Muller’s model is in the H-bond between water
molecules. Those are assumed to exist in two states in mutual
equilibrium:
HbondðintactÞ?HbondðbrokenÞ;
where the equilibrium constant K is given by:
ð10Þ
K ?
f
1 ? f¼ expð?DG?=RTÞ:
In this equation, f is the fraction of broken H-bonds, R is the gas con-
stant and DG?is the difference in Gibbs free energy between broken
and intact states ðDG?¼ Gbroken? GintactÞ. Based on the assumption
that the energy of the system is determined by H-bonds alone,
the enthalpy and entropy are given by:
ð11Þ
H ¼ fHbrokenþ ð1 ? fÞHintact;
S ¼ fSbrokenþ ð1 ? fÞSintact;
and the specific heat is given by:
?
Here, DH?? Hbroken? Hintactand it is assumed to be temperature
independent. The dependence of f on temperature is obtained from
Eq. (11):
ð12Þ
ð13Þ
cp?
oH
oT
?
P
¼
of
oT
??
P
DH?:
ð14Þ
of
oT
?
such that the specific heat [31] is given by:
?
P
¼ DH?fð1 ? fÞ
RT2
;
ð15Þ
cp¼ DH?
Eqs. (12), (13) and (16) account for the enthalpy, entropy and spe-
cific heat of the H-bond network of water. Now, Muller assumes
that hydration energies of non-polar solutes are related to rear-
rangements in this network alone. Thus hydration energies are
computed as the difference in energies between the disturbed and
undisturbed networks. Notice that the H-bond network is only per-
turbed locally by the solute, i.e., in the first hydrated shell, and only
these molecules need to be taken into account for the hydration
energies. This implies that f, DH?and DS?are different for bulk
and first-shell water and we use b and s subscripts to distinguish
them. Therefore the enthalpy and entropy upon hydration are given
by:
?
where DF ? Fs? Fb with Fb? fblnfbþ ð1 ? fbÞlnð1 ? fbÞ and simi-
larly for Fs. These are the ‘‘mixing” entropies characteristic of mix-
ture models. In these equations, n is the number of H-bonds in the
first-shell and it is related to number N of water molecules by
n ¼ 3N=2.1In the same line of thought the heat capacity of hydration
is given by:
h
where cs
and bulk water.
Actual values for the parameters of the model are given in Fig. 3.
The entropy predicted with those values for propane, butane, and
isobutane at 25 ?C are ?88.4 (?75.32), ?101.4 (?93.20) and
?98.9 (?89.14), respectively. Units are in J/K mol?1and experi-
ðÞ2fð1 ? fÞ=RT2:
ð16Þ
DH ¼ n ð1 ? fbÞDH?
DS ¼ n ð1 ? fbÞDS?
b? ð1 ? fsÞDH?
b? ð1 ? fsÞDS?
s
?;
ð17Þ
ð18Þ
s? RDF
??;
DChydration
p
¼ n cs
p? cb
p
i
;
ð19Þ
pand cb
pare the specific heat, given in Eq. (16), of first-shell
0 1020 3040
5060
Temperature (oC )
-20
-10
0
10
20
30
40
50
Energy (kJ /mol)
G
H
-T S
Δ
Δ
Δ
Fig. 2. Experimental data [60] of methane’s free energy (circles), entropy (triangles) and enthalpy (squares).
1To obtain Eq. (17), Muller assumes that the enthalpies of a broken H-bond in the
first-shell and in the bulk are the same [41]. The same assumption is also made for the
entropy of a broken H-bond in Eq. (18).
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mental results are given in parentheses – showing a good agree-
ment. The temperature dependence of the specific heat of hydra-
tion,Eq.(19), ischaracteristic
satisfactorily reproduces the positive DChydrationfor non-polar sol-
ute and the observed decrease in DChydrationwith increasing T [31].
Muller’s model has been refined [41]. However, the original ver-
sion of the model already shows that the energetic states of H-
bonds are predominantly responsible for the thermodynamical
features of the hydrophobic effect. The drawback of the model is
that it requires many parameters and does not explain how the dif-
ferent energetic states of H-bonds correlate with the atomic struc-
ture of water.
oftwo-state models.It
Atomic description
The first ‘‘pictorial representation” of the hydrophobic effect
came from Frank and Evans in 1945 [30]. While studying non-polar
molecules in liquids, they computed anomalous negative entropy
of mixing for aqueous solution. This implied that hydration of
non-polar molecules increased the amount of order in the system.
Also the expected positive heat of mixing related to an increase in
the number of broken H-bonds in shell water was absent. Thus,
they concluded that non-polar solutes perturbed water towards
its crystalline state, locally ordering water and increasing the
amount of H-bonds. Those ordered regions were named ‘‘iceberg”,
although it was noted that the name should not be taken literally.
The structure of water molecules in the iceberg region is still a
question of debate. It is clear that icebergs account for much less
order than ice and thus, the suggested name is misleading. This
can be seen [35] by comparing the entropy released during freez-
ing and during hydration per water molecule. Those are 22.17 J/
K mol?1and 4.18 J/K mol?1, respectively. The latter is much smal-
ler indicating that freezing brings water molecules to a much more
ordered state then iceberg water. On the other hand, the amount of
order in the icebergs is greater than in bulk (by 4.18 J/K mol?1per
water molecule).
Insights into the ‘‘iceberg” structure can be gained by looking at
the distribution of H-bonds around the solute. At room tempera-
ture, water saturates 3–3.5 H-bonds with neighboring molecules.
When a non-polar solute is inserted in water, the number of satu-
rated H-bonds should be much smaller for the molecules in the
first hydrated shell around the solute. This can see by fixing the po-
sition of a shell-water and rotating it in all possible orientations. In
most of those orientations shell-water has at least one H-bond
pointing towards the inert solute, therefore being non-saturated.
On the other hand, a few orientations exist where all the H-bonds
are saturated. We will refer to these two types of situations, as
non-saturated and saturated ones.
Energy wise, the saturated orientations have a lower (more
favorable) enthalpy than the many non-saturated ones. However,
they also have lower (less favorable) entropy since there are not
many of them. This shows a balance between enthalpy and entropy
when the system switches from saturated to non-saturated orien-
tations. This balance does not even out and the free energy of the
system is smaller whenever shell-water are constrained to satu-
rated orientations. In other words, the surface of the solute, which
is tiled with water molecules, has saturated orientations as its til-
ing motif. Different types of tilings are possible. Those are called
clathrate cages [9,29] and the simplest of them is represented by
a dodecahedron with water molecules sitting on each of its vertex
(see Fig. 4). It is likely that a perfect cage only occurs at low tem-
perature and at room temperature only incomplete cages survive
for a reasonable amount of time.
Thus, the formation of clathrate cages around the solute mini-
mizes the free energy of shell water. However, even in those con-
figurations shell-water has a higher free energy than bulk water,
i.e., the hydration free energy is positive. Therefore, when more
than one solute is inserted in water they tend to cluster to reduce
the amount of shell water in the system. This occurs through the
overlap of iceberg regions. The tendency of non-polar solutes to
cluster is knows as the hydrophobic interaction. Since shell water
are more ordered and form more H-bonds than bulk water, the
clustering of solutes increases the entropy and enthalpy of the sys-
tem. Therefore, the hydrophobic interaction is stabilized by entro-
py and destabilized by enthalpy. It is said to be entropically driven.
The behavior of the hydrophobic interaction upon cooling is of
significance to this paper. For most materials, the average strength
of the interaction between atoms increases as thermal energy
decreases. In contrast, the strength of the hydrophobic interaction
decreases with decreasing temperature. This non-intuitive behav-
ior is related to the complex interplay between entropy and enthal-
py – see Fig. 2. Upon cooling, both the stabilizing effect of entropy
and the destabilizing effect of enthalpy increase. This indicates that
the differentiation between shell and bulk water increases with
decreasing temperature, with shell water becoming more ordered
and forming more H-bonds than bulk water. However, the entropic
. 0621
o
b S
Δ
. 63 27
o
s S
mol
J
molK
J
9. 08
o
b
H
696.10
o
s
H
Broken H-bond
Intact H-bond
Δ
Δ
Δ
=
=
=
=
.
Fig. 3. Parameters for Muller’s model. For the number N of water molecules in the
first-shell Muller uses [31]: 25 for propane, 28 for butane and isobutane.
Fig. 4. Schematic representation of a clathrate hydrate caging a non-polar solute.
Water molecules are in the vertices of the dodecahedron.
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and enthalpic terms do not change at the same rate. The enthalpic
penalty increases faster upon cooling, thus accounting for a weak-
ening of the hydrophobic interaction. It should be noted that the
interplay between enthalpy and entropy is not clearly understood
from a microscopic point of view.
Cold denaturation
In the previous section we have discussed the thermodynamics
of the hydrophobic effect and how it arises from the atomic struc-
ture of water. This effect has been shown to be the dominant driv-
ing force for protein folding and is responsible for the stability of
the protein core [24,35,51]. It has been incorporated in simple
models of proteins where the solvent is describe implicitly – an
example is the well known Hydrophobic-Polar model [14]. An im-
plicit description is unlikely to account for both cold and heat
denaturation – unless the parameters in the model are made tem-
perature dependent [17]. When the solvent is described explicitly
both heat and cold denaturation
[4,10,11,13,18,19,23,54,55,61,62,65]. In the proceeding paragraphs
an overview of two approaches used to mimic the hydrophobic ef-
fect in proteins through an explicit solvent are summarized. Explic-
itly, a thermodynamical approach based on Muller’s model for
water and a molecular dynamics approach based on a simple mod-
el that accounts for the relevant structure of shell water.
arerecoverednaturally
Thermodynamical model
One of the first models to account for cold denaturation in the
physics literature was proposed by De Los Rios and Caldarelli
[62]. In this model, the protein corresponds to a self-avoiding ran-
dom walk where each amino acid occupies a site in a lattice. All
other nodes i are occupied by water molecules which can be in
the bulk or form the first shell around the polymer. The bulk state
is considered to be q times degenerate and for simplicity the en-
ergy of this state is set to zero. The first shell can be in an ordered
or a disordered state [50]. The disordered state is considered to be
q ? 1 times degenerate while the ordered state is not degenerate.
This model is therefore described by three parameters: J, K and q
– where J and K are the energies of the ordered and disordered
shell states, respectively. If s describes the state of first shell water
molecules such that s ¼ 0 represents the ordered state and
s ¼ 1;...;q ? 1 corresponds to the disordered states, then the
Hamiltonian is given by:
X
where the sum is over all water molecules that are nearest neigh-
bors of some hydrophobic monomer. Therefore, for each conforma-
tion C of the polymer its energy can be computed. The partition
function of the system can be cast in the form: ZN¼P
ZNðCÞ ¼ qnbðCÞexpðbJÞ þ ðq ? 1Þexpð?bKÞ
where nsand nbare the number of water molecules in the first shell
and bulk, respectively, and b is reciprocal of the thermal energy.
It is possible to classify the polymer according to ns. For self-
avoiding random walks of size N, the fraction of configurations of
perimeter nsis well approximated by a Poisson distribution:
H ¼
j
?Jdsj;0þ Kð1 ? dsj;0Þ
??
;
ð20Þ
CZNðCÞ,
where the partition function of a given conformation C is given by:
½?nsðCÞ;
ð21Þ
PNðnsÞ ? edðN?1ÞdðN ? 1Þ½?2Nþ2?ns
ð2N þ 2 ? nsÞ!
;
ð22Þ
with d ? 0:75. Using this distribution and Eq. (21), the partition
function of the system reads:
ZNðbÞ ¼
X
2Nþ2
nmin
PNðnÞq2Nþ2?nexpðbJÞ þ ðq ? 1Þexpð?bKÞ½?n;
ð23Þ
where the smallest perimeter nmin¼ 2
configuration the system has a circular compact shape. The maxi-
mum number of water sites in contact with the polymer is
2N þ 2. The heat capacity, computed as:
Cv¼ b2o2lnZ=ob2
is shown in Fig. 5. Three peaks appear in the heat capacity. From
zero temperature to the first peak, the polymer is swollen – in
agreement with cold denaturation. As the temperature is raised
above the first peak, the polymer folds and the number of first shell
water is approximately 2
pN
. As temperature increases above the
ffiffiffiffiffiffiffi
pN
p
, assuming that in this
??;
ð24Þ
ffiffiffiffiffiffiffi
p
01234
Temperature
0
100
200
300
400
500
600
700
Cv
N = 100
N = 50
N = 25
Swollen
Folded
Molten globule
Swollen
Fig. 5. Heat capacity, Eq. (24), for different polymer sizes. Here, K/J = 2 and q ¼ 103.
96
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second peak, the polymer occupies globule-molten states. Beyond
the third peak, the polymer reopens.
Consistent with experiments, the free energy of the model has a
convex curvature which is correctly partitioned in entropy and en-
thalpy [62]. The model has been extended successfully to study the
presence of kosmotropes and chaotropes cosolvent [48,49]. How-
ever, as in the case of Muller’s model, it does not provide much in-
sight into the structure of water around the solute. In the next
section we describe an attempt to fill this gap.
Atomic description
Recently, an effort to understand the physical mechanism be-
hind cold denaturation of biopolymers was undertaken by con-
structing a minimal microscopic model for the protein–water
system [23]. For computational efficiency, this study was per-
formed in 2D. The water phase was modeled using the 2D Merce-
des-Benz(MB)model[5,7,25]
represented as 2D disks, with three H-bonding arms resembling
the famous Mercedes-Benz symbol. This model has now been ex-
tended to 3D and the first results look very promising [8,22]. This
simple model (2D) reproduces many important thermodynamic
properties of water, such as the density anomaly, the minimum
in the isothermal compressibility as a function of temperature,
the large heat capacity, and the experimental trends for the ther-
modynamic properties of hydration of non-polar solutes [25,67].
In the study, the parameters of the MB model were chosen to be
those used by Silverstein et al. [66].
To model the protein, the simple bead-spring model of poly-
mers [26] was used: monomers which are adjacent along the back-
bone of the protein are connected to each other by harmonic
springs, and non-adjacent monomers are connected by a shifted
Lennard-Jones potential. The interaction between monomers and
water molecules was also given by a shifted Lennard-Jones poten-
tial with the same binding energy as between the water molecules.
To study the process of cold denaturation at constant pressure,
molecular dynamics simulations were performed on the water–
proteinmodel system inthe
[2,28,38]. The pressure in the system was set such that the MB
model reproduces water-like anomalies seen at ambient pressure
whereH2Omoleculesare
isothermal-isobaricensemble
[25] and hydrates non-polar molecules in a realistic manner [69].
Typically, the simulation box contained 512 molecules comprised
of a 10-monomer long protein and 502 water molecules. To induce
denaturation, simulations were performed at various different
(effective) temperatures.
The main result for the study can be seen in Fig. 6, where the
equilibrium distribution of the size of the protein, as measured
by its radius of gyration RG[26], is shown at three different temper-
atures. In ‘‘hot” water (referring to the largest values of T shown
here), proteins favor more compact configurations with decreasing
temperature. However, a further decrease of temperature results in
reversal of this trend: as the temperature decreases further, the
peak shifts to a larger value indicating that in ‘‘cold” water proteins
become less compact for decreasing temperature. This behavior
can be seen systematically in the inset of Fig. 6, which depicts
the temperature dependence of the protein size. This type of non-
monotonic behavior is characteristic to denaturation of real pro-
teins and in line with previous studies [40,54,61].
Characteristic configurations of the protein at different temper-
atures are shown in Fig. 7. In cold water (upper panels), the mono-
mers are surrounded by an ordered layer of ‘‘shell” water
molecules. Molecules forming this cage are strongly H-bonded to
each other and therefore have a low energy. At T ¼ 0:21, the pro-
tein favors compact configurations. Water molecules close to the
protein have at least one non-saturated H-bond which is pointing
towards the protein. When the temperature is increased to
T ¼ 0:25, most monomers are in contact with the solvent. In Ref.
[23], these observations were further quantified by computing
the average H-bond energy per water molecule for shell and bulk
water.
The molecular dynamics simulations of the MB model provide a
simple microscopic picture for cold denaturation in terms of
changes in hydration: at low temperatures water molecules infil-
trate the folded protein in order to passivate the ‘‘dangling”
water–water H-bonds found in shell water. At the same time,
hydrophobic contacts are destabilized and an ordered layer of
‘‘shell” water molecules forms around the protein monomers such
that they become separated by a layer of solvent in the cold dena-
tured state. Solvent layers around the monomer pairs are highly or-
dered such that their formation decreases the total entropy of the
Fig. 6. Normalized distribution of the radius of gyration of the protein ðRGÞ at three effective temperatures: T ¼ 0:25 (‘‘hot” water), T ¼ 0:21 (intermediate temperature) and
T ¼ 0:17 (‘‘cold” water). Inset shows the detailed temperature dependence of the size of the protein [23].
C.L. Dias et al./Cryobiology 60 (2010) 91–99
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system. The existence of such low entropic states for shell water at
low T explains why cold denaturation proceeds with heat release
as opposed to heat absorption seen during heat denaturation.
Conclusion
In this paper we reviewed and summarized some of the efforts
to model and understand the hydrophobic effect and the role it
plays in the thermodynamics of proteins. This review is not in-
tended to be exhaustive but to focus on the recent advances in
physics and chemistry, and especially on the efforts in computa-
tional modeling and theory. Thermodynamical models for the
hydration of non-polar solutes are successful in reproducing exper-
imental data accurately but they rely on many parameters that
need to be adjusted (see ‘‘Thermodynamics”). They provide a basis
for inferring the molecular structure of water around those solutes
which is responsible for the hydrophobic effect (see ‘‘Atomic
description” in ‘‘Thermodynamics of proteins”). When those ther-
modynamical models are coupled to a coarse-grained structure
of proteins (see ‘‘Thermodynamical model”), non-trivial but realis-
tic phases of these biomolecules are found to coexist. Simulations
of simple models have been performed revealing a potential mech-
anism for cold denaturation of proteins which is consistent with
the thermodynamical models (see ‘‘Atomic description” in ‘‘Cold
denaturation”). Finally, when comparing with experiments, there
are also subtleties which should be considered. For example, recent
results have shown that replacing water with deuterium, as is
common practise, may lead to changes in the physical properties
[64].
Acknowledgments
C.L.D. thank Janet Elliott for suggesting and motivating this
paper.
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