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J Biomech. Author manuscript; available in PMC 2010 April 26.

Published in final edited form as:

PMCID: PMC2859701

NIHMSID: NIHMS166626

Departments of Mechanical Engineering and Biomedical Engineering Columbia University

The theory of mixtures is applied to the analysis of the passive response of cells to osmotic loading with neutrally charged solutes. The formulation, which is derived for multiple solute species, incorporates partition coefficients for the solutes in the cytoplasm relative to the external solution, and accounts for cell membrane tension. The mixture formulation provides an explicit dependence of the hydraulic conductivity of the cell membrane on the concentration of permeating solutes. The resulting equations are shown to reduce to the classical equations of Kedem and Katchalsky (Kedem and Katchalsky, 1958, 1961) in the limit when the membrane tension is equal to zero and the solute partition coefficient in the cytoplasm is equal to unity. Numerical simulations demonstrate that the concentration-dependence of the hydraulic conductivity is not negligible; the volume response to osmotic loading is very sensitive to the partition coefficient of the solute in the cytoplasm, which controls the magnitude of cell volume recovery; and the volume response is sensitive to the magnitude of cell membrane tension. Deviations of the Boyle-van't Hoff response from a straight line under hypo-osmotic loading may be indicative of cell membrane tension.

The fundamental physical mechanisms of solute and water transport across the cell membrane have long been studied in the field of cell membrane biophysics, and there exist a number of formalisms aiming to characterize transport through membrane channels and/or lipid bilayers. These formalisms include a one-parameter (solute permeability) model (Mazur et al., 1974), a classic two-parameter (water and solute permeability) model (Jacobs and Stewart, 1932) and a commonly used three-parameter model (water and solute permeability and a solute-solvent interaction term) developed by Kedem and Katchalsky (Kedem and Katchalsky, 1958). The parameters of interest (permeabilities) can be extracted from the formulation when the cell volume change is measured in the experiment, assuming that the cell volume change is due purely to the volume of the water (and solute) that enters or exudes from the cell. These formulations have been derived from the general theory of irreversible thermodynamics (Onsager, 1931a, b; Staverman, 1952).

In more recent decades, the field of rational mechanics has addressed problems in the mixture of fluids (solvent and solutes) (Mills, 1966) as well as the mixture of fluids and solids (deformable porous media) (Atkin and Craine, 1976; Bowen, 1980; Mow et al., 1980; Frank and Grodzinsky, 1987; Lai et al., 1991; Huyghe and Janssen, 1997; Gu et al., 1998). In this study we propose to apply the theory of mixtures to the analysis of the passive response of cells to osmotic loading. For ease of comparison with the more classical treatment of Kedem and Katchalsky (Kedem and Katchalsky, 1958, 1961), the analysis is limited to non-electrolytes. However, the formulation is generalized to multiple solute species; it also incorporates partition coefficients for the solutes in the cytoplasm relative to the external solution, and accounts for cell membrane tension.

In this analysis the cell is modeled as a fluid-filled membrane, where the membrane is described by the mixture theory equations presented below. These equations are generally valid for neutral solute transport in uncharged porous media. In the following sections, the general equations of mixture theory are reduced to the case of a thin membrane, specialized to the case of one permeating and one non-permeating solute, and then compared to the classical Kedem-Katchalsky model.

In the presentation of the mixture theory equations we use the notation of Gu et al. (Gu et al., 1998), along with the generalizations to the solvent and solute momentum equations and solute constitutive equation as described by Mauck et al. (Mauck et al., 2003). The mixture consists of a solid matrix (α = *s*), solvent (α = *w*) and solutes (α ≠ *s*, *w*), with each constituent assumed to be intrinsically incompressible and neutrally charged. The momentum equations for the mixture as a whole and for the solvent and solutes are given respectively by

$$-\text{grad}p+\text{div}{\mathbf{\sigma}}^{e}=\mathbf{0}.$$

(1)

$$-{\rho}^{\alpha}\phantom{\rule{thinmathspace}{0ex}}\text{grad}{\tilde{\mu}}^{\alpha}+{\displaystyle \sum _{\beta}{f}_{\alpha \beta}({\mathbf{v}}^{\beta}-{\mathbf{v}}^{\alpha})=\mathbf{0}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s}$$

(2)

where *p* is the fluid (solution) pressure, **σ**^{e} is the effective (or elastic) stress in the solid matrix, ^{α} the chemical potential of the solvent or solute, **v**^{α} is the velocity of constituent α, *f*_{αβ} is the diffusive drag coefficient between constituents α and β (with *f*_{βα} = *f*_{αβ}), and ρ^{α} is the apparent density of constituent α. In this treatment, the solute-solvent mixture is assumed to be dilute and ideal (solute activity coefficients and osmotic coefficients of unity) so that the constitutive relations for the solvent and solute chemical potentials are given by

$${\tilde{\mu}}^{w}={\mu}_{0}^{w}(\theta )+\frac{1}{{\rho}_{T}^{w}}\left(p-R\theta {\displaystyle \sum _{\beta \ne s,w}{c}^{\beta}}\right)\phantom{\rule{thinmathspace}{0ex}},$$

(3)

$${\tilde{\mu}}^{\alpha}={\mu}_{0}^{\alpha}(\theta )+\frac{R\theta}{{M}_{\alpha}}\text{ln}\frac{{c}^{\alpha}}{{\kappa}^{\alpha}{c}_{0}^{\alpha}}\phantom{\rule{thinmathspace}{0ex}},\phantom{\rule{thinmathspace}{0ex}}\alpha \phantom{\rule{thinmathspace}{0ex}}\ne \phantom{\rule{thinmathspace}{0ex}}s,w\phantom{\rule{thinmathspace}{0ex}},$$

(5)

where *c*^{α} is the solute concentration on a solvent volume basis, *M*_{α} is the molecular weight of solute α, *R* is the universal gas constant and θ is the absolute temperature. Since the pore size distribution within the solid matrix may vary, solutes may not have access to all of the solvent available within the matrix due to steric volume exclusion effects. Mauck et al. (Mauck et al., 2003) extended the formulation of the solute chemical potential to account for this effect by incorporating the partition coefficient κ^{α} which, for an ideal solution, is the solubility of solute α inside the mixture relative to free solution. Thus *c*^{α}/κ^{α} is the concentration of solute per volume of accessible solvent; κ^{α} is assumed to be constant in this analysis. ${\rho}_{T}^{w}$ is the true density of the solvent and ${\mu}_{0}^{\alpha}$ is the chemical potential of constituent α in the standard state (when ${c}^{\alpha}/{\kappa}^{\alpha}={c}_{0}^{\alpha}$ in the case of solutes, and when all *c*^{α} = 0 and *p* = 0 in the case of the solvent). The solute standard state is a constant, usually taken to be ${c}_{0}^{\alpha}=1M$ (Tinoco, 2002). Substituting Eqs. (3)–(4) into Eq.(2) and recognizing that ρ^{α} = *M*_{α}^{w}*c*^{α} (α ≠ *s*, *w*), and ${\rho}^{w}={\phi}^{w}{\rho}_{T}^{w}$ where ^{w} is the volume fraction of the solvent in the mixture, yields

$$-{\phi}^{w}\left(\text{grad}p-R\theta {\displaystyle \sum _{\beta \ne s,w}\text{grad}{c}^{\beta}}\right)+{\displaystyle \sum _{\beta}{f}_{w\beta}({\mathbf{v}}^{\beta}-{\mathbf{v}}^{w})=\mathbf{0}},$$

(5)

$$-{\phi}^{w}{c}^{\alpha}R\theta \frac{\text{grad}{c}^{\alpha}}{{c}^{\alpha}}+{\displaystyle \sum _{\beta}{f}_{\alpha \beta}({\mathbf{v}}^{\beta}-{\mathbf{v}}^{\alpha})=\mathbf{0}},\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s,w$$

(6)

This formulation is simplified to account for friction between solutes and the solvent and between solutes and the mixture while neglecting the solute-against-solute diffusive drag. The non-zero diffusive drag coefficients are given by (Lai et al., 1991; Gu et al., 1998; Mauck et al., 2003; Meerveld et al., 2003)

$${f}_{\mathit{\text{ws}}}=\frac{{({\phi}^{w})}^{2}}{k},{f}_{w\alpha}=\frac{{\phi}^{w}R\theta {c}^{\alpha}}{{D}_{0}^{\alpha}},{f}_{w\alpha}+{f}_{s\alpha}=\frac{{\phi}^{w}R\theta {c}^{\alpha}}{{D}^{\alpha}}$$

(7)

where *k* is the hydraulic permeability of the solvent in the solid matrix, *D*^{α} is the solute diffusivity in the mixture and ${D}_{0}^{\alpha}$ is its diffusivity in free solution. In general, due to steric exclusion effects and tortuosity, *D*^{α} is smaller than ${D}_{0}^{\alpha}$. From these equations we can interpret the diffusion coefficient of a solute in free solution to result only from the frictional drag of the solute with the solvent (*f*_{wα}); in the presence of a solid matrix however, the diffusion coefficient results from the frictional drag between solute and solvent and between solute and solid matrix (*f*_{wα} + *f _{ws}*). Substituting Eq.(7) into Eqs.(5)–(6) yields

$$-\text{grad}p+R\theta {\displaystyle \sum _{\beta \ne s,w}\text{grad}{c}^{\beta}+\frac{{\phi}^{w}}{k}({\mathbf{v}}^{s}-{\mathbf{v}}^{w})+{\displaystyle \sum _{\beta \ne s,w}\frac{R\theta {c}^{\beta}}{{D}_{0}^{\beta}}({\mathbf{v}}^{\beta}-{\mathbf{v}}^{w})=\mathbf{0}}},$$

(8)

$$-\text{grad}{c}^{\alpha}+\frac{{c}^{\alpha}}{{D}^{\alpha}}({\mathbf{v}}^{s}-{\mathbf{v}}^{\alpha})+\frac{{c}^{\alpha}}{{D}_{0}^{\alpha}}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})=\mathbf{0},\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s,w$$

(9)

Equation (9) can be solved for **v**^{α} − **v**^{s} in terms of **v**^{w} − **v**^{s}; substituting this result into Eq.(8) and rewriting **v**^{β} − **v**^{w} as (**v**^{β} − **v**^{s})+(**v**^{s}−**v**^{w}), these equations can be solved simultaneously to yield

$${\phi}^{w}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})=-\tilde{k}\phantom{\rule{thinmathspace}{0ex}}\left[\text{grad}\phantom{\rule{thinmathspace}{0ex}}p-R\theta {\displaystyle \sum _{\alpha \ne s,w}\left(1-\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}\right)\phantom{\rule{thinmathspace}{0ex}}\text{grad}{c}^{\alpha}}\right]$$

(10)

$${\phi}^{w}{c}^{\alpha}({\mathbf{v}}^{\alpha}-{\mathbf{v}}^{s})=-{\phi}^{w}{D}^{\alpha}\text{grad}{c}^{\alpha}+\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}{c}^{\alpha}{\phi}^{w}({\mathbf{v}}^{w}-{\mathbf{v}}^{s}),\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s,w$$

(11)

where

$$\tilde{k}=\frac{1}{\frac{1}{k}+\frac{R\theta}{{\phi}^{w}}{\displaystyle \sum _{\alpha \ne s,w}\left(1-\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}\right)\frac{{c}^{\alpha}}{{D}_{0}^{\alpha}}}}.$$

(12)

^{w}(**v**^{w} − **v**^{s}) is the volumetric flux of solvent relative to the solid matrix and ^{w}*c*^{α}(**v**^{α} − **v**^{s}) is the molar flux of solute α relative to the solid matrix. The material properties of the mixture include the hydraulic permeability of the solid matrix to pure solvent, *k*; is the hydraulic permeability of the matrix to the solution (solvent + solutes).

In addition to these momentum equations the equations of conservation of mass for the mixture as a whole and for the individual solutes are given by

$$\text{div}\phantom{\rule{thinmathspace}{0ex}}\left[{\mathbf{v}}^{s}+{\phi}^{w}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})+{\displaystyle \sum _{\alpha \ne s,w}{\phi}^{\alpha}}({\mathbf{v}}^{\alpha}-{\mathbf{v}}^{s})\right]\approx \text{div}\phantom{\rule{thinmathspace}{0ex}}[{\mathbf{v}}^{s}+{\phi}^{w}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})]=0,$$

(13)

$$\frac{\partial ({\phi}^{w}{c}^{\alpha})}{\partial t}-\text{div}\phantom{\rule{thinmathspace}{0ex}}\left[{\phi}^{w}{D}^{\alpha}\text{grad}{c}^{\alpha}-{\phi}^{w}{c}^{\alpha}\left({\mathbf{v}}^{s}+\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})\right)\right]=0,\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s,w$$

(14)

where, for dilute solutions, we have taken into account that the volume fraction of solutes is much smaller than that of the solvent, ^{α} ^{w}.

The equations presented above are generalized relations for diffusion-convection problems in deformable porous media, where the solution is dilute. In these expressions, ^{w} depends on the matrix dilatation according to

$${\phi}^{w}=1-\frac{1-{\phi}_{r}^{w}}{\text{det}\mathbf{F}}$$

(15)

where ${\phi}_{r}^{w}$ is the porosity in the reference configuration of zero deformation and **F** is the deformation gradient of the solid matrix.

The general form of boundary or interface conditions applicable to mixture problem of this type are given by

$$[[{\mathbf{v}}^{s}]]\cdot \mathbf{n}=0,$$

(16)

$$[[{\phi}^{w}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})]]\cdot \mathbf{n}=0,$$

(17)

$$[[{\phi}^{w}{c}^{\alpha}({\mathbf{v}}^{\alpha}-{\mathbf{v}}^{s})]]\cdot \mathbf{n}=0,\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s,w$$

(18)

$$[[-p\mathbf{I}+{\mathbf{\sigma}}^{e}]]\cdot \mathbf{n}=\mathbf{0}$$

(19)

$$\left[\left[p-R\theta {\displaystyle \sum _{\alpha \ne w,s}{c}^{\alpha}}\right]\right]=0\phantom{\rule{thinmathspace}{0ex}}(\text{interface permeable to solvent})$$

(20)

$$\left[\left[\frac{{c}^{\alpha}}{{\kappa}^{\alpha}}\right]\right]=0\phantom{\rule{thinmathspace}{0ex}}(\text{interface permeable to solvent and solute \alpha})$$

(21)

where [**[·]**] denotes the difference, across the interface, of the quantity in the brackets, and **n** is the unit outward normal to the interface. These equations assume that the interface is defined somewhere on the solid matrix, which is the most common case (e.g., the cell membrane surface). Note that if the boundary or interface is not permeable to solute α, then Eq.(18) is equivalent to ^{w}*c*^{α} (**v**^{α} − **v**^{s})·**n** = 0 and Eq.(21) is not applicable. Eqs.(16)–(18) are kinematic conditions that stipulate continuity of solid matrix velocity, relative solvent flux, and relative solute molar flux across the interface, respectively. Eqs.(19)–(21) represent continuity of the total traction, solvent chemical potential, and solute chemical potentials across the interface, respectively.

For transport across a thin membrane of thickness *h*, the gradients in pressure and concentration across the membrane may be represented in scalar form by grad *p* · **n** ≈ −Δ*p/h* and grad *c*^{α} · **n** ≈ −Δ*c*^{α}/*h*, where **n** is the unit normal to the membrane surface. In this notation, Δ*p* = *p*(*r*)− *p*(*r* + *h*), and similarly for Δ*c*^{α}, where *r* is the coordinate direction normal to the membrane. The flux vector components normal to the membrane are given by

$${J}^{w}={\phi}^{w}({\mathbf{v}}^{w}-{\mathbf{v}}^{s})\cdot \mathbf{n},$$

(22)

$${J}^{\alpha}={\phi}^{w}{c}^{\alpha}({\mathbf{v}}^{\alpha}-{\mathbf{v}}^{s})\cdot \mathbf{n},\phantom{\rule{thinmathspace}{0ex}}\alpha \ne s,w.$$

(23)

Using Eqs.(10)–(11), these expressions reduce to

$${J}^{w}=\frac{\tilde{k}}{h}\mathrm{\Delta}p-\frac{R\theta \tilde{k}}{h}{\displaystyle \sum _{\alpha \ne s,w}\left(1-\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}\right)}\phantom{\rule{thinmathspace}{0ex}}\mathrm{\Delta}{c}^{\alpha},$$

(24)

$${J}^{\alpha}=\frac{{\phi}^{w}{D}^{\alpha}}{h}\mathrm{\Delta}{c}^{\alpha}+\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}{\overline{c}}^{\alpha}{J}^{w},\alpha \ne s,w$$

(25)

where ^{α} in the last term now represents the average solute concentration in the membrane.

Evaluating the projection of the vector equation in Eq.(1) along **n**, and reducing the resulting expression to the case of a thin spherical membrane under symmetric conditions yields

$$\mathrm{\Delta}p-\mathrm{\Delta}{\mathrm{\sigma}}_{\mathit{\text{rr}}}^{e}-\frac{2T}{a}=0,$$

(26)

where *T* is the surface tension in the membrane (in units of force per unit length), *a* is the spherical membrane radius, and ${\mathrm{\sigma}}_{\mathit{\text{rr}}}^{e}$ is the normal effective (elastic) stress in the radial direction.

According to the mixture formulation the concentrations *c*^{α} appearing in the above expressions refer to quantities *inside* the membrane. To express these equations using concentrations in the cytoplasm and in the external solution, boundary conditions of the form given in Eq.(21) are needed. For the interface between the membrane and cytoplasm,

$$-p+{\mathrm{\sigma}}_{\mathit{\text{rr}}}^{e}=-{p}_{i},p-R\theta {\displaystyle \sum _{\alpha \ne s,w}{c}^{\alpha}}={p}_{i}-R\theta {\displaystyle \sum _{\beta \ne s,w}{c}_{i}^{\beta}},{c}^{\alpha}=\frac{{\kappa}^{\alpha}}{{\kappa}_{i}^{\alpha}}{c}_{i}^{\alpha},$$

(27)

where *p _{i}* is the fluid pressure in the cytoplasm; ${c}_{i}^{\alpha}$ is the concentration of solute α in the cytoplasm; ${\kappa}_{i}^{\alpha}$ is the partition coefficient of the solute between the cytoplasm and the external solution and κ

$$-p+{\mathrm{\sigma}}_{\mathit{\text{rr}}}^{e}=-{p}_{e},p-R\theta {\displaystyle \sum _{\alpha \ne s,w}{c}^{\alpha}}={p}_{e}-R\theta {\displaystyle \sum _{\beta \ne s,w}{c}_{e}^{\beta}},{c}^{\alpha}={\kappa}^{\alpha}{c}_{e}^{\alpha}$$

(28)

where *p _{e}* is the fluid pressure and ${c}_{e}^{\alpha}$ is the concentrations of solute α in the external solution. Using these relations Δ

$$\mathrm{\Delta}p=\frac{2T}{a}-R\theta \phantom{\rule{thinmathspace}{0ex}}\left[{\displaystyle \sum _{\beta \ne s,w}({c}_{i}^{\beta}-{c}_{e}^{\beta})-}{\displaystyle \sum _{\alpha \ne s,w}\mathrm{\Delta}{c}^{\alpha}}\right],{p}_{i}-{p}_{e}=\frac{2T}{a},$$

(29)

$$\mathrm{\Delta}{c}^{\alpha}={\kappa}^{\alpha}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}-{c}_{e}^{\alpha}\right),{\overline{c}}^{\alpha}=\frac{{\kappa}^{\alpha}}{2}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}+{c}_{e}^{\alpha}\right).$$

(30)

Substituting these expressions, along with Eq.(26), into Eqs.(24)–(25) produces

$${J}^{w}={L}_{p}\phantom{\rule{thinmathspace}{0ex}}\left[\frac{2T}{a}-R\theta {\displaystyle \sum _{\alpha \ne s,w}({c}_{i}^{\alpha}-{c}_{e}^{\alpha})-(1-{\mathrm{\sigma}}^{\alpha})}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}-{c}_{e}^{\alpha}\right)\right]$$

(31)

$${J}^{\alpha}={P}^{\alpha}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}-{c}_{e}^{\alpha}\right)+(1-{\mathrm{\sigma}}^{\alpha})\frac{1}{2}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}+{c}_{e}^{\alpha}\right){J}^{w}$$

(32)

where

$${L}_{p}=\frac{\tilde{k}}{h},{P}^{\alpha}={\kappa}^{\alpha}\frac{{\phi}^{w}{D}^{\alpha}}{h},{\mathrm{\sigma}}^{\alpha}=1-{\kappa}^{\alpha}\frac{{D}^{\alpha}}{{D}_{0}^{\alpha}}$$

(33)

*L _{p}* is the membrane hydraulic conductivity (or conductance),

$${L}_{p}=\frac{1}{\frac{1}{{L}_{p0}}+R\theta {\displaystyle \sum _{\alpha \ne s,w}\frac{({\kappa}^{\alpha}-1+{\mathrm{\sigma}}^{\alpha})(1-{\mathrm{\sigma}}^{\alpha})}{{P}^{\alpha}}}\frac{1}{2}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}+{c}_{e}^{\alpha}\right)},$$

(34)

where *L*_{p0} = *k/h* is the value of *L _{p}* in the absence of solutes.

Integrating the mixture continuity of mass equation in Eq.**(13)** and recognizing that all velocities are equal to zero at the cell center yields *J** ^{w}* =

$$\frac{\mathit{\text{dV}}}{\mathit{\text{dt}}}=-{\mathit{\text{AL}}}_{p}\left[\frac{2T}{a}-R\theta {\displaystyle \sum _{\alpha \ne s,w}({c}_{i}^{\alpha}-{c}_{e}^{\alpha})-(1-{\mathrm{\sigma}}^{\alpha})\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}-{c}_{e}^{\alpha}\right)}\right]\phantom{\rule{thinmathspace}{0ex}},$$

(35)

$$\frac{{\mathit{\text{dn}}}_{i}^{\alpha}}{\mathit{\text{dt}}}=-{\mathit{\text{AP}}}^{\alpha}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}-{c}_{e}^{\alpha}\right)+(1-{\mathrm{\sigma}}^{\alpha})\frac{1}{2}\left(\frac{{c}_{i}^{\alpha}}{{\kappa}_{i}^{\alpha}}+{c}_{e}^{\alpha}\right)\frac{\mathit{\text{dV}}}{\mathit{\text{dt}}}.$$

(36)

In these expressions the internal concentrations of solutes are related to the corresponding number of moles via

$${c}_{i}^{\alpha}=\frac{{n}_{i}^{\alpha}}{{\phi}_{i}^{w}V},$$

(37)

where ${\phi}_{i}^{w}$ is the average water content (porosity) of the cell. Furthermore, from kinematic considerations under finite deformation as given in Eq.(15),

$${\phi}_{i}^{w}=1-(1-{\phi}_{\mathit{\text{ir}}}^{w})\frac{{V}_{r}}{V}$$

(38)

where ${\phi}_{\mathit{\text{ir}}}^{w}$ and *V _{r}* are the cell water content and volume in the reference configuration (in the initial state prior to osmotic loading), respectively. Note that ${c}_{e}^{\alpha}$ for all solute species are prescribed as boundary conditions. For a non-permeating solute we have κ

The analysis can now proceed to consider solutes which can permeate across the membrane, and solutes which cannot. For the case of one permeating (α = *p*) and one non-permeating (α = *n*) solute species, it follows that *J*^{n} = 0, so that the only remaining relations are

$$\frac{\mathit{\text{dV}}}{\mathit{\text{dt}}}=-{\mathit{\text{AL}}}_{p}\left(\frac{2T}{a}-R\theta \phantom{\rule{thinmathspace}{0ex}}\left[{c}_{i}^{p}-{c}_{e}^{p}+{c}_{i}^{n}-{c}_{e}^{n}-(1-{\mathrm{\sigma}}^{P})\left(\frac{{c}_{i}^{P}}{{\kappa}_{i}^{P}}-{c}_{e}^{P}\right)\right]\right)\phantom{\rule{thinmathspace}{0ex}},$$

(39)

$$\frac{{\mathit{\text{dn}}}_{i}^{p}}{\mathit{\text{dt}}}=-{\mathit{\text{AP}}}^{P}\left(\frac{{c}_{i}^{p}}{{\kappa}_{i}^{p}}-{c}_{e}^{p}\right)+(1-{\mathrm{\sigma}}^{p})\frac{1}{2}\left(\frac{{c}_{i}^{p}}{{\kappa}_{i}^{p}}+{c}_{e}^{p}\right)\frac{\mathit{\text{dV}}}{\mathit{\text{dt}}}.$$

(40)

with

$${L}_{p}=\frac{1}{\frac{1}{{L}_{p0}}+R\theta \frac{({\kappa}^{p}-1+{\mathrm{\sigma}}^{p})(1-{\mathrm{\sigma}}^{p})}{2{P}^{p}}\left(\frac{{c}_{i}^{p}}{{\kappa}_{i}^{p}}+{c}_{e}^{p}\right)},$$

(41)

To evaluate the equilibrium response for the system of equations in Eqs.(39)–(40), let *dn ^{p}/dt* = 0 and

$$\begin{array}{c}{c}_{i}^{p}={\kappa}_{i}^{p}{c}_{e}^{p}\hfill \\ {c}_{i}^{n}={c}_{e}^{n}+(1-{\kappa}_{i}^{p}){c}_{e}^{p}+{\frac{2T}{R\theta a}}^{\text{at equilibrium}.}\hfill \end{array}$$

(42)

This equilibrium solution is equally valid in the reference state (assuming that the reference state is at equilibrium), so that

$$\begin{array}{c}{c}_{\mathit{\text{ir}}}^{p}={\kappa}_{i}^{p}{c}_{\mathit{\text{er}}}^{p}\hfill \\ {c}_{\mathit{\text{ir}}}^{n}={c}_{\mathit{\text{er}}}^{n}+(1-{\kappa}_{i}^{p}){c}_{\mathit{\text{er}}}^{p}+\frac{2{T}_{r}}{R\theta {a}_{r}}\hfill \end{array},$$

(43)

where the subscript *r* refers to quantities evaluated in the reference state. Substituting Eqs.(37)–(38) into the above equilibrium solutions and recognizing that the number of moles of non-permeating solute inside the cell remains constant, ${n}_{\mathit{\text{ir}}}^{n}={n}_{i}^{n}$, produces an expression for the equilibrium values of ${n}_{i}^{p}(t)$ and *V* (*t*),

$$\begin{array}{c}{n}_{i}^{p}=\frac{\frac{2{T}_{r}}{R\theta {a}_{r}}+{c}_{\mathit{\text{er}}}^{n}+(1-{\kappa}_{i}^{p}){c}_{\mathit{\text{er}}}^{p}}{\frac{2T}{R\theta a}+{c}_{e}^{n}+(1-{\kappa}_{i}^{p}){c}_{e}^{p}}{\phi}_{\mathit{\text{ir}}}^{w}{V}_{r}{\kappa}_{i}^{p}{c}_{e}^{p}\hfill \\ \frac{V}{{V}_{r}}=1+\left[\frac{\frac{2{T}_{r}}{R\theta {a}_{r}}+{c}_{\mathit{\text{er}}}^{n}+(1-{\kappa}_{i}^{p}){c}_{\mathit{\text{er}}}^{p}}{\frac{2T}{R\theta a}+{c}_{e}^{n}+(1-{\kappa}_{i}^{p}){c}_{e}^{p}}-1\right]{\phi}_{\mathit{\text{ir}}}^{w}\hfill \end{array}\text{at equilibrium}.$$

(44)

Since the cell membrane tension *T* will generally be a function of its areal strain, the expression for *V/V _{r}* in Eq.(44) would need to be solved nonlinearly for the radius

Note that under hyper-osmotic loading with very high concentrations of a non-permeating solute, ${c}_{n}^{e}\to \infty $, the relative volume change can be used to assess the water content ${\phi}_{\mathit{\text{ir}}}^{w}$ (or equivalently, the osmotically inactive solid fraction, $1-{\phi}_{\mathit{\text{ir}}}^{w}$) inside the cell under the reference configuration,

$$\underset{{c}_{n}^{e}\to \infty}{\text{lim}}\frac{V}{{V}_{r}}=1-{\phi}_{\mathit{\text{ir}}}^{w}\phantom{\rule{thinmathspace}{0ex}}\text{at equilibrium}.$$

(45)

It can be noted that the above equations reduce to the classical three-parameter (*L _{P}*,

$${J}^{w}={L}_{p}({p}_{i}-{p}_{e})-R\theta Lp[{\mathrm{\sigma}}^{p}({c}_{i}^{p}-{c}_{e}^{p})+{c}_{i}^{n}-{c}_{e}^{n}]$$

(46)

$${J}^{p}={P}^{p}({c}_{i}^{p}-{c}_{e}^{p})+(1-{\mathrm{\sigma}}^{p})\frac{1}{2}({c}_{i}^{p}+{c}_{e}^{p}){J}^{w}$$

(47)

where we have replaced 2*T/a* with *p _{i}* −

$${L}_{p}=\frac{1}{\frac{1}{{L}_{p0}}+R\theta \frac{({\kappa}^{p}-1+{\mathrm{\sigma}}^{p})(1-{\mathrm{\sigma}}^{p})}{2{P}^{p}}({c}_{i}^{p}+{c}_{e}^{p})}.$$

(48)

This result suggests that the mixture formulation is more general than the classical phenomenological equations.

To analyze the influence of the various material parameters on the response of a cell to osmotic loading, we propose to investigate each parameter independently of others. In contrast to the classical three-parameter K-K model, the general set of equations in (39)–(41) has five parameters, *L*_{p0}, *P ^{p}*, σ

When examining Eqs.(39)–(41), it can be noted that the parameter κ^{p} appears explicitly only in the expression for *L _{p}* in Eq.(41). If we find that the concentration-dependence of

$$0\le {\kappa}^{p}\le 1,{\kappa}_{i}^{p}=1$$

(49)

The response to hyper-osmotic loading with a permeable solute is analyzed, with

$${c}_{\mathit{\text{er}}}^{p}=0\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{p}=1\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{\mathit{\text{er}}}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}}$$

(50)

For this analysis the membrane tension is taken as *T* = 0. The resulting cell volume response, *V/V _{r}*, is presented in Figure 1a. For κ

According to the equations for the equilibrium solution, Eq.(44), the partition coefficient for the permeating solute between the cytoplasm and external solution, ${\kappa}_{i}^{p}$, influences the equilibrium value of *V/V _{r}*. The complete time-dependent response of a cell to hyper-osmotic loading is investigated for various values of ${\kappa}_{i}^{p}$, using the material parameters

$${\kappa}^{p}=1,0\le {\kappa}_{i}^{p}\le 1.$$

(51)

In this analysis, *L _{p}* is concentration-dependent,

To investigate the influence of membrane tension on the volumetric response, we now provide a constitutive relation between *T* and the areal strain. In this formulation it is assumed that the membrane is slack below a threshold value of the areal strain and supports tensile stresses above that threshold (Guilak et al., 2002),

$$T=\{\begin{array}{cc}0& A\le {A}_{0}\\ K{\left(\frac{A}{{A}_{0}}-1\right)}^{\beta}& A\ge {A}_{0}\end{array}$$

(52)

where *A*_{0} is the cell surface area above which tension develops in the membrane. For simplicity in the current analysis, it is assumed that this threshold transition point coincides with the reference configuration of the cell, *A*_{0} = *A _{r}*. Thus tension will exist in the membrane under hypo-osmotic loading (swelling) relative to the reference state, but not under hyper-osmotic loading (shrinking). Furthermore, the power law of Eq.(52) is implemented in its simplest form, with β = 1. This leaves the area expansion modulus

$${\kappa}^{p}=1,{\kappa}_{i}^{p}=1,0\le K\le 1\phantom{\rule{thinmathspace}{0ex}}N/m.$$

(53)

This range of *K* is based on values reported for red blood cells (Evans and Waugh, 1977; Waugh and Evans, 1979) and lipid membranes (Needham and Nunn, 1990). A representative case of hypo-osmotic loading with a permeating solute is obtained with

$${c}_{\mathit{\text{er}}}^{p}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{p}=0\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{\mathit{\text{er}}}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}}$$

(54)

whereas hyper-osmotic loading is obtained with

$${c}_{\mathit{\text{er}}}^{p}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{p}=0.6\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{\mathit{\text{er}}}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}}$$

(55)

The resulting relative volume change of the cell is presented in Figure 3a, showing an asymmetric response between hyper- and hypo-osmotic loading even in the absence of membrane tension, as supported from experiments (Curry et al., 2000). When membrane tension does develop (*K* > 0), the peak volumetric change decreases and pressure rises inside the cell as the membrane expands under hypo-osmotic loading. The pressure then returns to zero as the cell recovers its initial volume (Figure 3b).

Response to hyper-osmotic and hypo-osmotic loading with a permeating solute, for various values of the area expansion modulus *K*. Membrane tension occurs only under hypo-osmotic loading in this example. (a) Relative cell volume *V/V*_{r}, and (b) pressure difference **...**

The response to loading with a non-permeating solute can similarly be simulated with the representative conditions

$${c}_{\mathit{\text{er}}}^{p}=0\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{p}=0\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{er}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{n}=0.2\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}}$$

(56)

for hypo-osmotic loading and

$${c}_{\mathit{\text{er}}}^{p}=0\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{p}=0\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{\mathit{\text{er}}}^{n}=0.3\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}},{c}_{e}^{n}=0.4\phantom{\rule{thinmathspace}{0ex}}\mathit{\text{Osm}}$$

(57)

for hyper-osmotic loading. The asymmetric volume response between hyper- and hypo-osmotic loading is even more apparent with non-permeating solutes (Figure 4a). There is no volume recovery under loading with a non-permeating solute, and for *K* > 0 the internal fluid pressure remains elevated at steady-state, to balance the membrane tension (Figure 4b). The equilibrium volume increase under hypo-osmotic loading is smaller with increasing membrane stiffness.

Response to hyper-osmotic and hypo-osmotic loading with a non-permeating solute, for various values of the area expansion modulus *K*. (a) Relative cell volume *V/V*_{r}, and (b) pressure difference between the inside and outside of the cell, *p*_{i} − *p* **...**

In addition to analyzing the transient response to osmotic loading, we can also plot the equilibrium volume change in response to osmotic loading with a non-permeating solute, as *V/V _{r}* versus ${c}_{\mathit{\text{er}}}^{n}/{c}_{e}^{n}$, for various values of the membrane area expansion modulus

$$\frac{V}{{V}_{r}}=\{\begin{array}{cc}1-{\phi}_{\mathit{\text{ir}}}^{w}+{\phi}_{\mathit{\text{ir}}}^{w}\frac{{c}_{\mathit{\text{er}}}^{n}}{{c}_{e}^{n}}& A\le {A}_{r},{c}_{\mathit{\text{er}}}^{n}\le {c}_{n}^{e}\\ 1+\left[\frac{{c}_{\mathit{\text{er}}}^{n}}{\frac{2}{R\theta a}K\left(\frac{A}{{A}_{r}}-1\right)+{c}_{e}^{n}}-1\right]{\phi}_{\mathit{\text{ir}}}^{w}& A\ge {A}_{r},{c}_{\mathit{\text{er}}}^{n}\ge {c}_{n}^{e}\end{array}$$

(58)

where *V/V _{r}* = (

Mixture theory has been used to formulate the equations for osmotic loading of a fluid-filled spherical membrane representative of a cell. The fundamental principles of mixture theory represent a generalization of the phenomenological equations that are based on irreversible thermodynamics (Onsager, 1931a, b; Staverman, 1952). The mixture theory equations, Eqs. (10)–(12), are not limited to thin membranes but are applicable to any continuum consisting of a mixture of a neutral solid, solvent, and solute constituents. The material parameters appearing in these equations are the familiar properties of transport theory, including the permeability of the solvent through the porous solid matrix, and the diffusivities of the solutes in the mixture and in free solution. By reducing these governing equations to the special case of a spherically-shaped thin membrane a formulation was obtained which is a generalized form of the classical Kedem-Katchalsky equations. Equivalences were established between the material parameters of mixture theory and the classical phenomenological parameters, as presented in Eq.(33). These equivalences are intuitive and consistent with the physical interpretations attributed to the phenomenological parameters (Kedem and Katchalsky, 1961). They are also comparable to the relations presented by Gu et al. (Gu et al., 1993), though it should be noted that these authors included charge effects, which allows them to model the membrane potential, while neglecting the friction between solutes and the solid matrix (*f*_{sα} = 0). One noteworthy generalization over the Kedem-Katchalsky model is the formulation of a concentration-dependent membrane hydraulic conductivity as shown in the general expression of Eq.(34) and in the special case of Eq.(48). The dependence of *L _{p}* on concentration has been alluded to in previous studies (Williams and Comper, 1990), though we believe that the explicit dependence derived from mixture theory in this study has not been previously reported. Another generalization relative to the Kedem-Katchalsky model is the incorporation of surface tension in the reduction to a thin spherical membrane (Evans et al., 1976).

The requisite interface boundary conditions of Eqs.(16)–(21) were applied to the mixture formulation to investigate the cell response to osmotic loading. One particular feature which bears note is the incorporation of a partition coefficient for solutes in the mixture, relative to free solution. The resulting governing equations for osmotic loading of a fluid-filled membrane, Eqs.(31)–(32), are more general than the three-parameter Kedem-Katchalsky model, Eq.(46)–(47), though they can be reduced to that model in the limit when the cytoplasm partition coefficient for permeating solutes is equal to unity and the membrane tension is neglected.

An assumption which is implicit in the current model as well as the K-K model is that the diffusion of solutes within the cytoplasm occur at time scales significantly shorter than the transport of solutes and solvents across the membrane. Therefore homogeneous distributions of solute concentration are assumed inside the cell. The mixture formulation assumes the existence of a solid matrix in the cell (the osmotically inactive volume) since the intracellular (osmotically active) water content (${\phi}_{i}^{w}$) can be less than 100% and the solute partition coefficient (${\kappa}_{i}^{\alpha}$) can be less than unity. However the stiffness of this matrix, and resistance to intracellular flow through the matrix, are neglected in the current formulation, under the assumption that the magnitude of osmotic pressure developing inside the cell (e.g., Figure 3b and Figure 4b) is considerably higher than the stress magnitudes resulting from matrix strains, and resistance to water and solute transport across the cell membrane is significantly greater than for intracellular transport.

The numerical examples investigated in this study suggest that the concentration-dependence of the membrane hydraulic conductivity, *L _{p}*, can be significant, particularly when the membrane partition coefficient for the permeating solute, κ

The theoretical predictions of this model are in very good agreement with experimental results available in the literature. For example, the response to hyperosmotic loading with a permeating solute, as shown in Figure 1 for the case κ^{p} ≠ 0 agrees very well with the experiments of Xu et al. (Xu et al., 2003) where bovine chondrocytes were osmotically loaded with glycerol (Figure 6a). The asymmetric response to hyper- and hypo-osmotic loading with a non-permeating solute predicted in Figure 4 is very similar to the experiments of Curry et al. (Curry et al., 2000) who performed osmotic loading of rabbit spermatozoa (Figure 6b). It is also interesting that varying the partition coefficient of the permeating solute in the cytoplasm can produce partial volume recoveries (Figure 2) which are remarkably similar to experimental measurements by Lucio et al. (Lucio et al., 2003) in kidney cells treated with various doses of vasopressin hormone (Figure 6c). This agreement suggests that the role of the partition coefficient in osmotic loading of cells may be quite significant and should be investigated in greater detail. Quantitative differences in the time scales and magnitude of volume changes between these experimental studies and the theoretical predictions of the current study are simply due to differences in cell sizes (radius *a*) and choices of material coefficients (*L _{p}* and

(a) Hyper-osmotic loading of bovine chondrocytes with a permeating solute (1.4 M glycerol) at 21°C, with solid curve representing prediction from K-K model (Reprinted from Med Eng Phys, Vol. 25, Xu, X., Cui, Z., Urban, J. P., Measurement of the **...**

Finally, the numerical simulations incorporating cell membrane elasticity show that surface tension can have a non-negligible influence on the volumetric response to osmotic loading (Figure 3, Figure 4). Experimental measurements of the equilibrium response to loading with a non-permeating solute at various concentrations may be used to infer the membrane's area expansion modulus if the Boyle-van't Hoff relationship is found to significantly deviate from a straight line under hypo-osmotic loading (Figure 5). For bovine chondrocytes, measurements to date demonstrate a linear relationship (Guilak et al., 2002; Xu et al., 2003) more akin to the case *K* = 0 *N/m* (“perfect osmometer”) in Figure 5, suggesting that membrane tension may be negligible for those cells.

This study was supported with funds from the National Institutes of Health (R21 AR48791, R01 AR46532).

*a*- cell radius
*A*- cell surface area
*c*^{α}- concentration of solute α inside the mixture, on a solvent volume basis
- ${c}_{0}^{\alpha}$
- concentration of solute α in the standard state
^{α}- average concentration of solute α inside the membrane
- ${c}_{e}^{\alpha}$
- concentrations of solute α in external solution
- ${c}_{i}^{\alpha}$
- concentration of solute α in cytoplasm
*D*^{α}- solute diffusivity in mixture of solid and fluid
- ${D}_{0}^{\alpha}$
- solute diffusivity in free solution
- F
- deformation gradient of the solid matrix
*f*_{αβ}- diffusive drag coefficient between constituents α and β
*J*^{w}- volume flux of solvent across membrane
*J*^{α}- molar flux of solute α across membrane
*k*- hydraulic permeability of solid matrix to pure solvent
- hydraulic permeability of solid matrix to solution (solvent + solutes)
*L*_{p}- membrane hydraulic conductivity (or conductance)
*M*_{α}- molecular weight of solute α
- ${n}_{i}^{\alpha}$
- number of moles of solute α inside the cell
*p*- fluid pressure inside the mixture
*p*_{e}- fluid pressure in external solution
*p*_{i}- fluid pressure in cytoplasm
*P*^{α}- membrane permeability of solute α
*R*- universal gas constant
*T*- surface tension in membrane
*V*- cell volume
**v**^{α}- velocity of constituent α
- α =
*n* - refers to non-permeating solute
- α =
*p* - refers to permeating solute
- α =
*s* - refers to solid matrix
- α =
*w* - refers to solvent
^{α}- volume fraction of constituent α in the mixture
- ${\phi}_{i}^{w}$
- volume fraction of osmotically active water in the cell
- κ
^{α} - partition coefficient of solute α inside mixture relative to free solution
- ${\kappa}_{i}^{\alpha}$
- partition coefficient of solute α between cytoplasm and external solution
^{α}- chemical potential of solvent (α =
*w*) or solute - ${\mu}_{0}^{\alpha}$
- chemical potential of constituent α in the standard state
- ρ
^{α} - apparent density of constituent α
- ${\rho}_{T}^{w}$
- true density of solvent
- θ
- absolute temperature
**σ**^{e}- effective stress tensor in solid matrix
- ${\sigma}_{\mathit{\text{rr}}}^{e}$
- normal effective stress in radial direction
- σ
^{α} - Staverman's reflection coefficient for solute α

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