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THE STABILITY Of FROZEN AND DEHYDRATED CELLS AND
MEMBRANES IN THE AMORPHOUS CARBOHYDRATE MATRIX:
THE WILLIAMS-LANDEL-FERRY KINETICS
Wendell Q. Sun
School of Biological Sciences, Faculty of Science, National University of Singapore, Kent
Ridge Crescent, Singapore 119260. E-mail: [email protected].
Summary
The kinetic stability of frozen and dehydrated cells and membranes in the amorphous
state was examined. Dynamic processes examined included 1) membrane fusion and solute
leakage of frozen liposomes and dry liposomes in carbohydrate glasses, and 2) ice formation
and hemolysis of frozen human red blood cells during isothermal storage. The rate
parameters of these dynamic processes are Tg-dependent, deviate significantly from the
Arrhenius behaviors, and fit well to the WLF equation. The universal values for C1 (17.44)
and C2 (51.6) in the WLF equation do not apply to these dynamic processes. However, the
derived values for C1 and C2 are in the same order as the universal values. The kinetics for
hemolysis of frozen blood cells is identical to that for fusion and solute leakage of dry
liposomes, but is different from the ice formation kinetics of frozen cells. This is probably
because both hemolysis of blood cells and solute leakage of liposomes are linked to the
membrane stability in the amorphous state. The results are discussed in relation to the
prediction of the stability of biological materials preserved in the amorphous matrix. At
temperatures T > Tg, the stability of frozen and dehydrated cells and membranes decreases
roughly at a rate of 10(T-Tg)/ ( = 15 ~20).
Keywords: Cryopreservation, crystallization, glass transition, hemolysis, membrane
preservation, red blood cell, sugar glass, Williams-Landel-Ferry kinetics
INTRODUCTION
Relaxation processes in an amorphous (glassy) system deviate significantly from the
traditional Arrhenius relationhsip. As the amorphous system cools down towards its glass
transition temperature (Tg), various relaxation processes become increasingly slow, and many
of them are practically arrested at temperature T ≤ Tg because of the extremely high bulk
viscosity (1012-14 Pa s), which retards molecular movement in the system (9, 13-14). This
slow relaxation phenomenon in the glassy state is of great biological significance. In the past
two decades, enormous attention has been paid to the preservation of biological materials in
the glassy state (1-6, 15-19). Many successful protocols used to dry food, pharmaceutical and
biotechnology products exploited this property of the glassy state by the addition of polymers
and excipient stabilizers such as carbohydrates (5). The glassy state has been found to be
essential for the stabilization and preservation of bioactive susbtances, proteins, membranes
and cells during freeze-drying and subsequent dry storage (4, 18). The survival of a number
of life forms in the desiccated state, including seeds, resurrection plants, bacterial and fungal
spores, and certain microscopic animals, is also associated with the presence of intracellular
glasses (12, 20-21).
One of the most important paramenter describing the amorphous system is the Tg. It is
believed that biological materials in the glassy state (i.e., T ≤ Tg) would enjoy the real-time
stability achieved by virtue of the extremely high viscosity (7). If the biological materials are
held in the rubbery state at temperature T > Tg, their stability is estimated to decrease at a rate
proportional to 10T-Tg (1, 10, 11). However, few studies have examined the quantitative
relationship between the Tg and the kinetic stability of preserved biological materials. The
present work is to study the relationship between the Tg and the rate parameters associated to
the kinetic stability of frozen and dehydrated cells and membranes, using the Williams–
Landel–Ferry (WLF) model.
DATA ANALYSIS
For frozen or dehydrated biological systems that are associated with supercooling and
vitrification (glass formation), the dynamics of relaxation processes can be described by the
Williams–Landel–Ferry (WLF) equation, which is written as
log (t/tg) = log (h/hg) = – C1 (T – Tg)/(C2 + T – Tg)
(1)
where t and tg are relaxation times, and h and hg are viscosities at temperature T and Tg,
respectively. C1 and C2 are equation constants. The WLF kinetics applies at temperatures
between Tg and Tg + 100 (°C) for the amorphous systems (22).
After t/tg is substituted with Kg/K, equation (1) can be used to describe the temperature
dependence of the rate parameters for dynamic processes in the amorphous state, such as the
survival of cells and plant seeds (12, 20). K and Kg are rate constants at temperature T and
Tg, respectively. Therefore, equation (1) can be re-written as
log (Kg/K) = – C1 (T – Tg)/(C2 + T – Tg)
(2)
In the present study, equation (2) was used to describe the rate parameters related to the
kinetic stability of frozen and dehydrated cells and membranes in the amorphous state. Kg,
Tg, C1 and C2 were derived with experimental data, through an iterative curve-fitting
procedure that were inserted as a marco in a commercial software, ‘Igor’ (WaveMetrics, Lake
Oswega, OR, USA). C1 is correlated to the inverse of free volume in the amorphous system
at the Tg, while C2 is correlated to the ratio of free volume at Tg to the increase in free volume
with thermal expansion above Tg (13). Free volume is the limiting factor for diffusion-related
processes at temperature T > Tg.
FROZEN AND DRY LIPOSOMAL MEMBRANCES
Carhohydrates, especially sucrose and trehalose, are often used to stabilize liposomal
membranes during freeze-drying (4) and air-drying (18). Sugar concentrations in the sample
solution are typically as high as 5 to 10 % before drying. Therefore, liposomes are stabilized
in the concentrated amorphous solutions during freezing, and preserved in sugar glasses after
freeze-drying. The role of the glassy state in the kinetic stability of liposomes can be easily
seen from Fig. 1, where data of liposomal fusion and solute leakage are plotted against the T –
Tg, the difference between storage temperature and the Tg. Membrane fusion and solute
leakage were extremely slow at temperatures below the Tg. Three sets of data showed an
excellent agreement, although different carbohydrates and phopholipids were used in each
DPPC + trehalose
(Tg = -30°C)
DPPC + dextran
(Tg = -10 °C)
PC + sucrose
(Tg = 48 °C)
600
450
300
150
0
-40
ln (rate constant)
Solute leakage (%)
100
75
50
-20
0
20
T – T g (°C)
40
DPPC + trehalose
(Tg = -30°C)
DPPC + dextran
(Tg = -10 °C)
PC + sucrose
(Tg = 48 °C)
250
0.0
A. Arrhenius plot
-3
0
-6
Fig. 1.
Membrane fusion and solute
leakage of frozen and dry liposomes.
Sizes of liposomes and solute leakage
were determined after 1-h incubation at
various temperatures above and below
Tg. Data of three liposomal preparations
with different Tg were superimposed
together. Liposomes prepared with DPPC
were in freeze-concentrated dextran and
trehalose solutions (Tg = -10 and -30 °C,
respectively). Liposomes prepared with
egg PC were preserved in dry sucrose
glass (Tg = 48 °C). Data were adopted
from ref. 4 and 18, and reinterpreted.
-40
-20
0
T – T g (°C)
20
40
ln (rate constant)
Liposome diameter (nm)
liposomal preparations, and their Tg values were greatly different. The result shows that the
fusion and solute leakage of frozen and dehydrated liposomes is predominantly governed by
the kinetics of the glassy state.
-2.0
-4.0
The rate constants for solute leakage of air-dried phosphatidylcholine liposomes in a
sucrose-9 glass were recently determined at temperatures
both above and below the Tg (18).
-6.0
A. Arrhenius plot
Tg
The Arrhenius plot showed
a break to the linear relationship between 1/T and rate constant
-12
-8.0
near the
Tg. It appeared that the solute leakage conformed
to the Arrhenius relationship at
2.8both 3.0
3.2 below
3.4 the 3.6
temperatures
above and
Tg, with a transitional
deviation
Tg (Fig.
3.5
4.0
4.5at the5.0
5.5 2A).
1000/T (K)
Temperature,
1000/K
A re-analysis of the data, however, indicated that the WLF kineitics was applicable (R2 =
0.99) (Fig. 2B). The rate constants of solute leakage at temperatures T ≥ Tg were used to fit
5.6 (T - Tg )
5.9 (T - Tg )
0
= – constants were calculated
10 0 equation. log(K
the WLF
Theg/K)
WLF
to be 5.6
and
10
log(K
g/K)73.9
= – for C1 and C2,
73.9 + (T-T g)
71.5 + (T - T g)
-3
respectively. The T(Kg,
g of2.97
thex 10liposomal
°C through the WLF
; T g , 48 °C)sample was derived to be(Kg48
, 5.30 x 10-4 ; T g, -96 °C)
-1
10
curve-fitting,
which was the same as that determined by Differential Scanning Calorimetry
10-1
Kg /K
(DSC).
However,
to accommodate the data at temperatures
T < Tg, the equation constants, C1
Kg /K
10-2
and C2, -2had to be changed to 10.6 and 166.0, respectively, while the Kg and Tg remained the
10
same. FROZEN HUMAN RED BLOOD CELLS 10-3
B. WLF
Biophysical
andplot
biological parameters related to the survival
B. WLFof
plotcells in frozen conditions
-3
10
have been extensively studied (8). Numerous investigations
have been carried out to develop
10 -4
0 cryopreservation
15
30
45
60
and optimize
protocols
for human red blood
cells 30
(15-17).
0
15
45 Recently,
60
75 Spieles
90
T - Tg (°C)
et al. (15, 16) studied the stability of human red blood cells frozen inT the
hydroxyethyl stach
- Tg (°C)
(HES) matrix. The cell suspension were rapidly cooled to -196 °C at a rate of 293 °C/min by
Fig. 2. (A) the Arrhenius plot of the rate
Fig. 3. (A)
the Arrhenius
plot for
the
immersion in liquid nitrogen, and then stored isothermally
at various
temperatures.
Excellent
constants for solute leakage of dry egg
hemolysis
of frozen
human
red blood
cell the
experimental
data
have
been
obtained
in
their
study,
including
the
rate
parameters
about
PC liposomes in the sucrose glass. The
preparations
during
isothermal
storage at of
stability
of frozen
red blood
in the amorphous
HES matrix.
However,
the significance
relationship
between
ratecells
constant
and
temperatures
above
Tg.
The
kinetics
of
thetemperature
data, in studying
Tg-dependent
kinetic stability of frozen cells, has not been appreciated.
deviates
from the Arrhenius
hemolysis
deviated
fro
the
Arrhenius
Fig. 3 shows
the rate constants
of hemolysis
of frozen red blood cells during isothermal
bahavior
as temperature
approached
to
behavior. (B) the WLF plot of the same
storage
at Tvarious
temperatures.
The
of hemolysis did not follow the Arrhenius
Tg. The
g was 48°.
(B) the WLF
plotkinetics
of
dataset. The relationship between the
the same and
dataset.
pointsis at
relationship,
significantData
curvature
observed
3A). of
However,
theand
WLF
kineitics
rate(Fig.
constants
hemolysis
storage
temperature T≥Tg were used to fit the
WLF model (solid curve). The Tg value,
estimated from the curve-fitting was also
48 °C. Data were adopted from ref. 18,
and re-analyzed.
temperatures fitted the WLF model
excellently (solid curve).
The rate
constants of hemolysis of blood cells
were adopted from ref. 15 and 16.
was applicable to this kinetic process (Fig. 3B). The data were fitted to the WLF equation,
and the WLF constants, C1 and C2, were derived to be 5.9 and 71.5, respectively. The Tg of
the system was estimated to be -96 °C from the hemolysis data through the WLF curve-fitting
procedure.
It is interesting to find that the values of the WLF constants, C1 and C2, for hemolysis of
frozen red blood cells are so close to those for solute leakage of dry liposomes. In fact, when
these two sets of data are super-imposed, one would see immediately that the two kinetic
processes are identical. This result is not surprising because both hemolysis of blood cells
and solute leakage of liposomes are associated with the membrane stability in the amorphous
state. The WLF curve-fitting with pooled data has derived a common set of equation
constants, C1 = 5.8 and C2 =70.7, for both hemolysis of frozen blood cells and solute leakage
of dry liposomes.
There is evidence showing that cell injury during cryopreservation is correlated with
extracellular or intracellular ice formation. At temperatures below -0.6 °C, biological water in
the cells becomes thermodynamically unstable and will favour the crystalline state (8). Under
many circumstances, however, the amount of ice formed is governed by kinetic rather
thermodynamic factors. When ice forms, the unfrozen fraction of the solution becomes
increasingly concentrated. Consequently, ice nucleation and the subsequent crystal growth
ln (rate constant)
-4.0
-4.5
-5.0
-5.5
-6.0
-6.5
4.2
4.5
4.8
5.1
Temperature, 1000/K
10 0
7
Rate increase at T > Tg (K/Kg )
A. Arrhenius plot
10
Rate increase at T > Tg (K/Kg )
-3.5
100
10
10
10
10
10
10
10
Type I (Universal)
6
Type II
5
Type III
4
3
2
1
A
0
5.4
2.1 (T - T g )
11.2 + (T - T g)
(Kg , 4.81 x 10-4 ; Tg, -87 °C)
log(K g/K) = –
Kg /K
10-1
B. WLF plot
10 -2
ice formation
hemolysis & membrane
leakage
10
B
1
0
0
10
20
30
40
T - Tg (°C)
50
60
Fig. 4. (A) the Arrhenius plot for ice
formation of frozen human red blood cell
preparations during isothermal storage
at temperatures above Tg. The kinetics
deviated significantly from the Arrhenius
behavior. (B) the WLF plot of the same
dataset. Rate constants of ice formation
were plotted in a logarithmic scale againt
storage temperatures.
Solid curves
show predictions from fitted equations.
The rate constants were calculated by
the present author with data from ref. 15
5
10
15
20
T = T - T g (°C)
25
30
²
Fig. 5. (A) Temperature dependence of
the rates of diffusion-limited relaxation
processes in three types of amorphous
systems, as defined by Slade and Levine
(17). (B) Temperature dependence of
the rate parameters that are relevant to
the biological stability of frozen and
dehydrated cells and membranes. he
increase of the rates at T = T – Tg is
calculated by the WLF equation with
constants from Slade and Levine (17) for
(A), and with derived constants in the
present study for (B).
slow down because the solution viscosity is increased. Extracellular ice formation alters the
chemical environment of the cells, dehydrates the cells, mechanically constrains and deforms
the cells, and may also induce ice formation inside the cells.
An attempt was made by Spieles et al. (15) to associate ice formation with the hemolysis
of frozen blood cells during isothermal storage after cooling to -196 °C with liquid nitrogen.
They reconstituted an intracellular solution from dried blood cells and water, and investigated
the devitrification behavior using the DSC method (annealing). Their results of DSC
measurements were used by the present author to calculate the rate constants of ice formation
during storage at different temperatures (Fig. 4). The kinetics of ice formation also deviates
the Arrhenius relationship (Fig. 4A), and follows the WLF kinetics (Fig. 4B). From the ice
formation data, the Tg of the preparation is derived to be -87 °C through the WLF curvefitting procedure, which is in a good agreement with the DSC measurement done by Spieles et
al. (15). [The Tg value of the sample was not reported in the cited paper. The interpretation
(Tg = ~85-90°C) is based on the present author’s own assessment on the published DSC
thermograms. Also note that the devitrification (water crystallization) commonly occurs at
temperature Tg + 10 °C (19).]
The WLF constants for the process of ice formation were calculated to be 2.1 and 11.2
for C1 and C2, respectively. These values are significantly different from those for hemolysis
of frozen blood cells, suggesting that the kinetics for two processes are different. If hemolysis
of frozen blood cells during storage were caused by ice formation during devitrification, the
two sets of the WLF constants would be expected to be similar. A similar conclusion can be
obtained by direct examining the rate constants of hemolysis and ice formation in Figs. 3 and
4. The rate constant of hemolysis remains relatively slow at -60 °C, whereas the rate constant
of ice formation has almost reach the plateau. Therefore, the ice formation of frozen cells
may not always be correlated to hemolysis of blood cells.
DISCUSSION
While numerous studies have examined the kinetics of slow relaxations or diffusioncontrolled reactions in the amorphous state, little systematic research has been devoted to
determining the actual values of the WLF equation constants, C1 and C2, for particular
dynamic processes (28). So-called ‘universal’ constants, 17.44 and 51.6 for C1 and C2
respectively, were originally derived by averaging data for various types of synthetic
polymers (27). Soesanto and Williams (149) reported that the "universal" constants gave a
good description of the temperature dependence of the viscosity of highly concentrated
aqueous solutions of sugar mixtures. Their results experimentally demonstrated that simple
amorphous systems obeyed the WLF quantitatively and as well as synthetic high polymers.
These constants are currently used by many workers "only in the absence of the ability to
obtain specific data (16)". However, the WLF equation constants are system-dependent (10,
17, 18). These "universal" constants are by no means universal (13, 16), and should not be
expected to be valid for a particular dynamic process.
The present study used the general form of WLF equation, and the constants were
allowed to vary so that the best fit of the data could be obtained. ‘Universal’ values for C1
and C2 did not apply to the dynamic processes examined, including solute leakage of frozen
and dry liposomes in carbohydrate glasses, and water crystallization and hemolysis of frozen
human red blood cells during isothermal storage. The values derived from the curve-fitting
methods, however, were generally in the same order of magnitude as the "universal" values.
One interesting observation was that the kinetics of hemolysis was identical to that for
membrane leakage, but was different from the ice formation kinetics of frozen cells. This was
probably because both hemolysis of blood cells and solute leakage of liposomes were linked
to the membrane stability in the amorphous state. It appears that C1 and C2 are not just
system-dependent, but also process-dependent. The general form of the WLF equation
specifies a reference temperature, To. To values derived from the experimental data were
found to be the same as the Tg measured with DSC. The author has just completed a series of
enzyme inactivation experiments with amorphous sucrose, raffinose and trehalose. To values
in all these systems are found to be the same as the system Tg (Sun, unpublished). Therefore,
the To can be substituted by the Tg.
The kinetic properties of biological amorphous systems are poorly understood. In an
attempt to gain some insight in understanding the kinetic stability of biological glasses, the
derived WLF constants in the present study are compared with these for the viscosity change
or relaxation processes in other glass-forming systems. In a comprehensive review, Slade and
Levine (17) defined three classes of amorphous systems. The first class, including
biologically important sugar–water systems, is characterized by a ratio of Tm/Tg ≈ 1.5 (Tm,
crystalline melting temperature). The amorphous systems in this class are said to be ‘wellbehaved’, partly because the ‘universal’ constants apply. The second class of glass-forming
systems has a Tm/Tg >> 1.5. This ‘typical but not well-behaved’ class are readily
crystallizable (e.g., water). For the second class, the WLF constants are about 20 and 155 for
C1 and C2, respectively. The third class of glass-forming systems is described to be ‘atypical
and poorly behaved’ (e.g., native starch and gelatin). The third class has a ratio of T m/Tg
<<1.5 (i.e., Tg near Tm). The WLF constants of C1 = 12.3 and C2 = 22.3 apply to the third
class (17). These WLF constants for three classes are far different from the derived constants
for ice formation, hemolysis of frozen blood cell preparation, and solute leakage of frozen and
dry liposomes in amorphous carbohydrates. Since the WLF constants are also processdependent, it is conceivable that the kinetics of the processes such as hemolysis of frozen
blood cells and solute leakage of dry membrane would be significantly different from that of
the molecular relaxation process of the amorphous system itself. A different set of the WLF
constants is needed to describe a particular dynamic process associated with the amorphous
system.
The prediction of the stability of biological materials preserved in the amorphous matrix
is of particular interest. From the WLF equation, one can easily derive that the stability of
biological materials in the amorphous matrix will decrease at a rate proportional to 10 T-Tg ,
if held in the rubbery state at temperature T > Tg, (1, 14, 15). However, more accurate
estimation is required in practice. A conceptual illustration has been provided by Slade and
Levine (17), who used the WLF equation constants for three classes of amorphous systems as
discussed previously. For the systems of the first class, the stability at ∆T = T-Tg for 0, 3, 7,
11 and 21 °C corresponds to relative rates, 1, 10, 100, 1000, and 100000, respectively. These
rates show the 5 orders-of-magnitude change over a 21 °C interval above the Tg or roughly a
factor of 10 for every 3 °C. For the amorphous systems in the second class and the third class,
their stability may decrease by a factor of 10 for an increase of every 6 and 1 °C, respectively
(17). The author has done a similar calculation with the derived WLF constants in the present
study (Fig. 5). The rate of ice formation in frozen blood cell preparations during
devitrification increases approximately by 10(T-Tg)/6. (i.e., a 10-fold increase over a 6 °C
interval above the Tg). This value corresponds very well to that illustrated by Slade and
Levine (17) for the amorphous systems of the second class (including water). However, the
stability of frozen and dehydrated cells and membranes, as measured by hemolysis and solute
leakage, appears to decrease much slower than expected, roughly at a rate of 10(T-Tg)/15 at
temperatures T > Tg, (i.e., a 10-fold decrease over a 15 °C above the Tg).
Acknowledgements: The orginal idea for this study was stimulated by a comment from an
anonymous referee for my previous paper (CryoLetters 18, 99-106, 1997). The work was
supported by a research grant to WQS. (RP-960366) from National University of Singapore.
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