where v
frac
b :¼ V b =V
iso is the osmotically inactive fraction of the
isosmotic volume, a commonly reported value in the literature.
2.2 Boyle van ’t Hoff
An intrinsic assumption of membrane mass transport modeling is
that cell water volume, and thus, total cell volume, behaves as an
“ideal osmometer.” In fact, this is a property observed in nearly all
cell types [31–39] over a range of osmolalities from 1/3 Â isosmolal to 10 Â isosmolal. Classically [40], the Boyle van ’t Hoff relationship is defined by πW ¼ π 0 W 0 , where π and π 0 are intracellular
osmolalities, W is the intracellular water volume (as in Eq. 1), and
subscript 0 defines a particular known state. Typically π 0 is isosmolal and W 0 is the corresponding intracellular water volume. Therefore, the cell water volume as a function of osmolality can be
described by the relationship W ¼ π 0 W 0 /π.
Prickett et al. [41] point out that the impermeability of some
solutes allows a variant of the Boyle van ’t Hoff relationship to be
derived from the relationship N ¼ N 0 , where N is the moles of
intracellular non-permeating solute, by showing that this is equivalent to m 0 W 0 ¼ mW where m is the molality of non-permeating
solute. In this case, we have a “molal” version of the Boyle van ’t
Hoff relationship. They show that using this relationship and a
nonideal description of osmolality as a function of molality (see
Subheading 2.3 below) a different and potentially more accurate
estimate for the osmotically inactive volume may be obtained, and
the accuracy of their model over the usual Boyle van ’t Hoff relation
is enhanced in the extremely concentrated solutions of interest to
cryobiology. Nevertheless, in either case, this is how the Boyle
van ’t Hoff relationship is used most often—to allow the replacement of the mole fraction, concentration, or molality of intracellular non-permeating solute in the membrane transport equations
with the inverse of intracellular osmolality or molality. Note that in
the dilute case when π 0 /π > 1, the behavior of the molal and
osmolalal models are nearly identical as a function of molality.
Mathematically, the Boyle van ’t Hoff relationship implies that
the water volume is inversely proportional to the intracellular
osmolality (or molality). If a cell satisfies this relationship over a
range of osmolalities, it is said to behave as a linear osmometer.
However, this relationship is understood in the isothermal case. In
particular, the constant of proportionality is related to several temperature dependent parameters that include the relative density of
intracellular water and possibly also the relative density of water as a
function of the concentration of non-permeating solutes.
This determination has an additional benefit. Using the total
volume Eq. 1, the total equilibrium volume of a cell in anisosmotic
media containing only non-permeating solutes is V ¼ W + V b .
Replacing W with the expression from the Boyle van ’t Hoff relationship gives
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
133
frac
b :¼ V b =V
iso is the osmotically inactive fraction of the
isosmotic volume, a commonly reported value in the literature.
2.2 Boyle van ’t Hoff
An intrinsic assumption of membrane mass transport modeling is
that cell water volume, and thus, total cell volume, behaves as an
“ideal osmometer.” In fact, this is a property observed in nearly all
cell types [31–39] over a range of osmolalities from 1/3 Â isosmolal to 10 Â isosmolal. Classically [40], the Boyle van ’t Hoff relationship is defined by πW ¼ π 0 W 0 , where π and π 0 are intracellular
osmolalities, W is the intracellular water volume (as in Eq. 1), and
subscript 0 defines a particular known state. Typically π 0 is isosmolal and W 0 is the corresponding intracellular water volume. Therefore, the cell water volume as a function of osmolality can be
described by the relationship W ¼ π 0 W 0 /π.
Prickett et al. [41] point out that the impermeability of some
solutes allows a variant of the Boyle van ’t Hoff relationship to be
derived from the relationship N ¼ N 0 , where N is the moles of
intracellular non-permeating solute, by showing that this is equivalent to m 0 W 0 ¼ mW where m is the molality of non-permeating
solute. In this case, we have a “molal” version of the Boyle van ’t
Hoff relationship. They show that using this relationship and a
nonideal description of osmolality as a function of molality (see
Subheading 2.3 below) a different and potentially more accurate
estimate for the osmotically inactive volume may be obtained, and
the accuracy of their model over the usual Boyle van ’t Hoff relation
is enhanced in the extremely concentrated solutions of interest to
cryobiology. Nevertheless, in either case, this is how the Boyle
van ’t Hoff relationship is used most often—to allow the replacement of the mole fraction, concentration, or molality of intracellular non-permeating solute in the membrane transport equations
with the inverse of intracellular osmolality or molality. Note that in
the dilute case when π 0 /π > 1, the behavior of the molal and
osmolalal models are nearly identical as a function of molality.
Mathematically, the Boyle van ’t Hoff relationship implies that
the water volume is inversely proportional to the intracellular
osmolality (or molality). If a cell satisfies this relationship over a
range of osmolalities, it is said to behave as a linear osmometer.
However, this relationship is understood in the isothermal case. In
particular, the constant of proportionality is related to several temperature dependent parameters that include the relative density of
intracellular water and possibly also the relative density of water as a
function of the concentration of non-permeating solutes.
This determination has an additional benefit. Using the total
volume Eq. 1, the total equilibrium volume of a cell in anisosmotic
media containing only non-permeating solutes is V ¼ W + V b .
Replacing W with the expression from the Boyle van ’t Hoff relationship gives
Mathematical Modeling and Optimization of Cryopreservation in Single Cells
133