**2. Cell membrane transport models and mathematical formulatins**

To date, a number of formalisms exist for describing the cell membrane transport process. These include a one-parameter model, a two-parameter model, and a three-parameter model, considering solute-solvent interactions.

i. one-parameter model (Mazur et al, 1974, 1976),

The one-parameter model utilizes the hydraulic permeability (*Lp*) of cell membrane as the only parameter to describe the water transport across cell membrane. The model can be formulized as follows.

$$\frac{d\mathbf{V}\_{\rm w}^{i}}{d\mathbf{t}} = -L\_{p}A\_{c}\left(\Pi\_{e} - \Pi\_{i}\right) \tag{1}$$

where, *<sup>i</sup> Vw* is the volume of intracellular water, *Ac* is the area of cell membrane surface, Π*<sup>e</sup>* and Π*i* are the extracellular and intracellular osmotic pressures.

Prevention of Lethal Osmotic Injury to Cells

addition and removal of CPAs.

glycerol in human sperm

**2.1 Materials and methods** 

**Preparation of sperm suspension** 

**Assessment of human sperm membrane integrity** 

During Addition and Removal of Cryoprotective Agents: Theory and Technology 105

In the following context, two examples are demonstrated to show how to use cell membrane transport models and mathematical formulations to develop optimal conditions and technology/instrument for the addition and/or removal of the permeating CPAs in cells. **An important hypothesis** is that the degree of cell volume excursion can be used as an independent indicator to evaluate and predict the possible osmotic injury of the cells during

Example 1: Development of optimal "multi-step methods" for addition and dilution of

the addition and removal of high concentrations of glycerol in human spermatozoa.

Human semen samples were obtained by masturbation from healthy donors after at least 2 days of sexual abstinence. Samples were allowed to liquefy in an incubator (5% CO2, 95% air, 37C, and high humidity) for ~1h. A total of 5 ul of the liquefied semen were used for a computer-assisted semen analysis (CASA) using CellSoft (Version 3.2/C, CRYO Resources, Ltd, Montgomery, NY, USA) (Jequier and Crich, 1986; Crister et al., 1988b). A swim-up procedure was performed to separate motile form immotile cells [layering 500ul of modified Tyrode's medium (TALP: Bavister et al., 1983) over 250 ul of semen, incubating for ~1 h in the incubator and carefully aspirating 400 ul of the supernatant in which >95% of spermatozoa were motile]. The motile cell suspensions were centrifuged at 400g for 7min and resuspended in the TALP medium (286~290 Osmol) supplemented with pyruvate (0.01 mg/ml) and bovine serum albumin (4 mg/ml), at a cell concentration of 1×109 cell/ml.

A methodology for the assessment of sperm membrane integrity, using dual florescent staining and flow cytometric analysis, has been developed by Garner et al. (1986) and previously validated in our laboratory (Gao et al., 1992, 1993; Noilles et al., 1993). Propidium iodide (catalogue no. P4170; Sigma Chemical Co., St Louis MO, USA) is a bright red, nucleic acid-specific fluorophore which permeates poorly into spermatozoa with intact plasma

Glycerol is the most commonly used CPA in the cryopreservation of spermatozoa (Polge et al, 1949; Watson, 1979; Critser etl al., 1988a). Glycerol permeability characteristics for human spermatozoa have been very well studied and reported (Du et al, 1994; Gao et al., 1992). The hypothesis above was tested first using the following procedures: (i) to determine sperm osmotic injury as a function of its volume excursion limits (swelling/shrinking) in anisosmotic solutions containing only non-permeating solutes without glycerol; (ii) to simulate, by computer, the kinetics of water-glycerol transport through the sperm plasma membrane and to calculate the sperm volume excursion during different glycerol addition and removal processes using membrane transport equations and previously determined sperm membrane permeability coefficients for glycerol and water; (ii) combining information obtained from procedures (i) and (ii), to predict sperm osmotic injury caused by different procedures of glycerol addition and removal; and (iv) to perform experiments to test the predictions. If the hypothesis is confirmed, the above procedures also provide a methodology for predicting optimal protocols to reduce the osmotic injury associated with

#### ii. Two-parameter model

The two-parameter model was firstly presented by Jacob (1932-1933), and further developed by Kleinhans (1998), Katkov (2000) recently. The model utilizes the parameters *Lp* and *Ps* (CPA solute permeability) to characterize membrane permeability when water, a permeable solute and a nonpermeable solute are present:

$$\frac{dV\_w^i}{dt} = -L\_p A\_c RT \left(M^e - M^i\right) \tag{2}$$

$$\frac{dN\_s^i}{dt} = P\_s A\_c \left(M\_s^e - M\_s^i\right) \tag{3}$$

where *Ns* is the number of osmoles of solute inside cell, *R* is the universal gas constant, *T* is the absolute temperature, *Mi* and *Me* are the intracellular and extracellular osmolality, respectively. The subscript 's' refers to permeable solute, and remaining symbols are as previously defined.

#### iii. Three-parameter model

The classical formulation of coupled, passive membrane transport was developed by Kedem and Katchalshy (1958) using the theory of linear irreversible thermodynamics. The formulation includes two coupled first-order non-linear ordinary equations which describe the total transmembrane volume flux and the transmembrane permeable solute flux respectively.

In the model (so called Kedem-Katchalssky transport formalism or KK formalism), a reflection coefficient (σ) was introduced with Lp and Ps to describe water and solute (CPA) transport across the plasma membrane:

$$\frac{dV\_c}{dt} = -L\_p A\_c RT \left[ \left( M\_n^e - M\_n^i \right) + \sigma \left( M\_s^e - M\_s^i \right) \right] \tag{4}$$

$$\frac{dN\_s^i}{dt} = \left(1 - \sigma\right)\overline{M}\_s \frac{dV\_c}{dt} + P\_s A\_c \left(M\_s^e - M\_s^i\right) \tag{5}$$

Where *Vc* is cell volume, *Ms* is the average osmolality of intracellular and extracellular solution, and the subscript '*n*' refers to nonpermeable solute, respectively.

The KK formalism used to be the most general of the three mentioned. However, more recent literature suggests that aquaporins in cell membrane are highly selective, with nonionic solute transport occurring mainly through the lipid bilayer or through other channels that are distinct from the aquaporins (Gilmore et al, 1995; Preston et al, 1992). In this case, the estimation of σ as independent parameter may be inappropriate and may not be relevant from a biological stand point (Kleinhans, 1998). By assuming that there is no interaction between water and solute during their transport through the membrane, the value of σ can be determined as 1 *PV RTL s s <sup>p</sup>* , where *Vs* = partial molar volume of permeating solute. In this manner, the KK formalism can still get correct result as two parameter model.

In the following context, two examples are demonstrated to show how to use cell membrane transport models and mathematical formulations to develop optimal conditions and technology/instrument for the addition and/or removal of the permeating CPAs in cells. **An important hypothesis** is that the degree of cell volume excursion can be used as an independent indicator to evaluate and predict the possible osmotic injury of the cells during addition and removal of CPAs.

Example 1: Development of optimal "multi-step methods" for addition and dilution of glycerol in human sperm

Glycerol is the most commonly used CPA in the cryopreservation of spermatozoa (Polge et al, 1949; Watson, 1979; Critser etl al., 1988a). Glycerol permeability characteristics for human spermatozoa have been very well studied and reported (Du et al, 1994; Gao et al., 1992). The hypothesis above was tested first using the following procedures: (i) to determine sperm osmotic injury as a function of its volume excursion limits (swelling/shrinking) in anisosmotic solutions containing only non-permeating solutes without glycerol; (ii) to simulate, by computer, the kinetics of water-glycerol transport through the sperm plasma membrane and to calculate the sperm volume excursion during different glycerol addition and removal processes using membrane transport equations and previously determined sperm membrane permeability coefficients for glycerol and water; (ii) combining information obtained from procedures (i) and (ii), to predict sperm osmotic injury caused by different procedures of glycerol addition and removal; and (iv) to perform experiments to test the predictions. If the hypothesis is confirmed, the above procedures also provide a methodology for predicting optimal protocols to reduce the osmotic injury associated with the addition and removal of high concentrations of glycerol in human spermatozoa.
