Reagents
The base medium used was isosmotic Tyrode's lactate HEPES (TL HEPES)-buffered solution (285 ± 5 mOsm/kg) (Bavister et al., 1983). Osmolality was determined using a freezing-point depression osmometer (Model 3D2; Advanced Instruments, Needham Heights, MA, USA). The cryoprotectant treatment solutions were prepared by mixing glycerol, DMSO, PG or EG with isosmotic TL HEPES to yield final CPA concentrations of 1 mol/l (glycerol, DMSO and PG) and 2 mol/l (EG). Those concentrations were chosen to ensure accurate permeability estimates from the electronic particle counter (EPC). All media components used in the experiments were obtained from Sigma Chemical Company (St Louis, MO, USA).
Human spermatozoa preparation
Semen samples were obtained from healthy donors by masturbation, after informed consent. The samples were obtained after a minimum of 48 h of sexual abstinence. A minimum concentration of 2x10^{7} spermatozoa per ml, with at least 40% motility, was required for the samples to be included in the study. The ejaculates were allowed to liquefy in an incubator (5% CO_{2}/95% air, 37°C, and high humidity) for 30 min. Samples were layered on a discontinuous (90 and 47%) Percoll gradient to select for motile cells and then washed with TL HEPES supplemented with pyruvate (0.01 mg/ml) and resuspended to ~2.5 ml final sample volume. A fraction of the sample (5 µl) was analysed using computer-assisted semen analysis (CASA) (Cell Soft, Version 3.2/C; CRYOResources, Ltd, Montgomery, NY, USA). Subsequent CASA analysis was done at the completion of each experiment.
Electronic particle counter
An EPC (ZM model; Coulter Electronics, Inc., Hialeah, FL, USA) with a standard 50 µm aperture tube was used for all measurements. Volumes (V) were calibrated via spherical styrene beads (Duke Scientific Corporation, Palo Alto, CA, USA) with a diameter of 3.98 ± 0.03 µm (V = 33.1 µm^{3}) at 22, 11 and 0°C. The relationship between conductivity and styrene bead volume was assumed to be the same as the relationship between conductivity and sperm volume.
Temperature control
Experiments were conducted using a cooled methanol circulating bath. Experimental temperatures were measured using a thermocouple, directly before and after each experimental run and kept within ±2°C (room temperature at 22°C, and a cooling bath at 11 and 0°C). At each experimental temperature, the experimental media, the cell sample, the aperture tube, the calibration beads and the pipette tips were pre-equilibrated to the experimental temperature.
Data analysis
Statistical analysis
Data were analysed using standard analysis of variance and multiple range test approaches with the SAS^{®} General Linear Models program (SAS Institute Inc., Cary, NC, USA).
Determination of membrane permeability coefficients
A pair of coupled non-linear equations (Kedem and Katchalsky, 1958) was used as the theoretical model for analysis of cell membrane permeability in a ternary solution consisting of a permeable solute (cryoprotectant, subscript `s') and an impermeable solute (NaCl, subscript `n') and water. The cell volume and amount of intracellular solute concentration as functions of time are presented as:
and
where V is cell volume, V
_{b} is osmotically inactive cell volume and m
^{i}_{s} is the molal concentration of CPA inside the cell, and
The superscripts `i' and `e' refer to the intra- and extracellular compartments respectively. The terms L
_{p}^{CPA}, P
_{CPA} and
are parameters for the hydraulic conductivity of water in the presence of CPA, the permeability coefficient of the CPA and the reflection coefficient, respectively. The temperature and universal gas constant are given by T and R, respectively. For the impermeable solute (NaCl), the intracellular osmolality is given by:
where
V_{s} =
N^{i}_{s}^{·} is the CPA volume (
is the partial molar volume of the CPA). The superscript `(o)' represents the initial values at t = 0. A fixed value of V
_{b} (Gilmore
et al., 1995
) was used in the analysis.
Evaluation of fitting calculation
The curve-fitting procedure was used to find parameter values that made the theoretical curve most resemble the plot of experimental data using the least-sum-of-squares method. The sum-of-squares (S) can be considered a function of the parameters S(P_{1}, P_{2}....P_{m}) in an (m + 1)-dimensional space. For the transport parameters L_{p}^{CPA} and P_{CPA} the sum-of-squares is given by:
where
X^{th}_{i} represents theoretical data calculated from the modified KK equations and
X ^{exp}_{i} represents experimental data. The values of S(L
_{p}^{CPA}, P
_{CPA}) serve as the criteria for estimating the best fit parameters. In order to evaluate the fitting calculation, the numerical values of sum-of-squares at different values of L
_{p}^{CPA} and P
_{CPA}, varying from –50% to +50% of the best fitted L
_{p}^{CPA} and P
_{CPA} estimates were calculated. These values were then plotted in three-dimensional space. The fitting calculation was very sensitive to the P
_{CPA} value; even very slight changes in P
_{CPA} estimates (5%) resulted in significant changes in values of the sum-of-squares. The sensitivity of the calculation to the value of L
_{p}^{CPA} was much less; however, it was sufficient to determine a minimum sum-of-squares and therefore a robust estimate of L
_{p}^{CPA}.
Activation energies for parameters L_{p}^{CPA} and P_{CPA} The Arrhenius relationship was used to determine the E_{a} of the parameters L_{p}^{CPA} and P_{CPA} (Levin et al., 1976). The permeability value (L_{p}^{CPA} or P_{CPA}) at any temperature T can be plotted as ln[P_{a}(T)] versus 1/T (Arrhenius plot):
where E
_{a} is the activation energy for the process, expressed in kcal/mol, which can be determined by linear regression calculation of the slope of the Arrhenius plot as:
Simulation of intracellular water volume during freezing and thawing.
The changes of intracellular water volume and the mol number of intracellular CPA during temperature change of rate B can be calculated using the following coupled equations (Liu et al., 1997):
and
(W is the weight of a substance). The terms V
_{w} and A are cell water volume and surface area of a cell, and n
^{i}_{s} is the number of mol of CPA inside the cell (see Table I
for a complete listing of symbols and abbreviations). The temperature and universal gas constant are given by T and R respectively. C is the total solute concentration in g/100 g, MW is the molecular weight, and v
_{10} is the molar volume of water. The value of parameters L
_{p} (T) and P
_{CPA} (T) can be calculated with the Arrhenius relationships, using E
_{a} for L
_{p} and P
_{CPA}, respectively.
The extracellular solute concentration, C = C(T, R
_{t}), at temperature T can be determined by solving the following equation of the melting point for CPA/NaCl/water ternary solution:
where coefficients A
_{1}, B
_{1}, and C
_{1} are functions of R
_{t}. The coefficients for DMSO and glycerol have been determined (Pegg, 1983
, 1986
). The coefficients for EG have been determined recently (Woods
et al., 1999
). Definitions of all major symbols presented in the equations, along with corresponding units and values are given in Table I
.
Data acquisition.
The EPC was interfaced to a microcomputer using a CSA-1 interface (The Great Canadian Computer Company, Edmonton, Alberta, Canada). Aliquots of 100 µl of 2 mol/l glycerol, 2 mol/l DMSO, 2 mol/l PG, or 4 mol/l EG were added dropwise over 60 s to 100 µl of cell suspension, yielding final CPA concentrations of 1 and 2 mol/l, respectively. Cells were allowed to equilibrate for ~3 min, at which time they return to near normal volumes, before all 200 µl were returned to isosmotic media. The cells abruptly swell upon return to isosmotic conditions due to the influx of water (determined predominantly by L_{p}^{CPA}) and then return at a lower rate to normal volumes as CPA diffuses out (determined predominantly by P_{CPA}). The resulting changes in cell volume were measured over time. A total of three donors were used and the experiments were performed at 22, 11 and 0°C. Data were analysed using a two parameter fitting method as previously described (Gilmore et al., 1998).