Physiochemical Modeling of Vesicle Dynamics upon Osmotic Upshift

Abstract

We modeled the relaxation dynamics of (lipid) vesicles upon osmotic upshift, taking into account volume variation, chemical reaction kinetics, and passive transport across the membrane. We focused on the relaxation kinetics upon addition of impermeable osmolytes such as KCl and membrane-permeable solutes such as weak acids. We studied the effect of the most relevant physical parameters on the dynamic behavior of the system, as well as on the relaxation rates. We observe that 1) the dynamic complexity of the relaxation kinetics depends on the number of permeable species; 2) the permeability coefficients (P) and the weak acid strength (pK_a-values) determine the dynamic behavior of the system; 3) the vesicle size does not affect the dynamics, but only the relaxation rates of the system; and 4) heterogeneities in the vesicle size provoke stretching of the relaxation curves. The model was successfully benchmarked for determining permeability coefficients by fitting of our experimental relaxation curves and by comparison of the data with literature values (in this issue of Biophysical Journal). To describe the dynamics of yeast cells upon osmotic upshift, we extended the model to account for turgor pressure and nonosmotic volume.

Figure 1

Simulated curves for an impermeable solute. (a) Variation of the solute concentration gradient across the membrane Δc_s(t) is shown. (b) Vesicle volume variation V(t) is shown. (c) Variation of the internal solute concentrations c_i(t) is shown. (d) Variation of the internal pH is shown. The following parameters were used for calculations: pK₁ = 7.21, M₁ = 18 cm³/mol, pH₀ = ${\text{pH}}_0^{\ast }$ = 7.0, [KPi] = 90 mM, [KPi]^∗ = 100 mM, [calcein] = 10 mM, k₁ = 10⁶ s⁻¹, K_SV = 10² M⁻¹, $c_6^{\ast }$ = 120 mM, r₀ = 100 nm, and P₁ = 0.003 cm/s. For calculation of pH(t), we set [KPi] = 100 mM and [calcein] = 0 M.

Figure 2

Simulated curves for an impermeable solute. (a) Comparison of normalized volume V(t)/V(0) and calcein fluorescence intensity F(t)/F(0) is shown. Variation of F(t)/F(0) is shown as a function of (b) the solute concentration gradient Δc_s, (c) the vesicle radius r₀, and (d) the water permeability coefficient P_wj. The following parameters were used for calculations: pK₁ = 7.21, M₁ = 18 cm³/mol, pH₀ = ${\text{pH}}_0^{\ast }$ = 7.0, [KPi] = 90 mM, [KPi]^∗ = 100 mM, [calcein] = 10 mM, k₁ = 10⁶ s⁻¹, K_SV = 10² M⁻¹, $c_6^{\ast }$ = 100 mM, r₀ = 100 nm, and P₁ = 0.003 cm/s. The last three parameters ($c_6^{\ast }$, r₀, P₁) were modified according to the figure legends.

Figure 3

Simulated curves for permeable weak acids. The permeability coefficient of water is fixed to P₁ = 0.003 cm/s, whereas the weak acid permeability P₆ was modified as indicated in the figure legend. The time evolution of (a) the acid concentration gradient Δ[AH] of AH and (b) the solute concentration gradient Δc_s is shown. The relaxation dynamics of (c) internal pH and (d) normalized volume V(t)/V(0) are shown. The following parameters were used for calculations: pK₁ = 7.21, pK₂ = 4, M₁ = 18 cm³/mol, pH₀ = ${\text{pH}}_0^{\ast }$ = 7.0, [KPi] = 90 mM, [KPi]^∗ = 100 mM, [calcein] = 10 mM, k₁ = 10⁶ s⁻¹, K_SV = 10² M⁻¹, $c_6^{\ast }+c_7^{\ast }$ = 60 mM, r₀ = 100 nm, and P₁ = 0.003 cm/s. For calculation of pH(t), we set [KPi] = 100 mM and [calcein] = 0 M. To see this figure in color, go online.

Figure 4

Simulated curves for a permeable weak acid. The time evolution of internal pH upon variation of the weak acid pK_a is shown. The following parameters were used for calculations: pK₁ = 7.21, pK₂ = 4, M₁ = 18 cm³/mol, pH₀ = ${\text{pH}}_0^{\ast }$ = 7.0, [KPi] = 100 mM, [Kpi]^∗ = 100 mM, [calcein] = 0 M, k₁ = 10⁶ s⁻¹, K_SV = 10² M⁻¹, $c_6^{\ast }+c_7^{\ast }$ = 60 mM, r₀ = 100 nm, and P₁ = 0.003 cm/s. The parameter pK_a was varied as indicated in the legend.

Figure 5

Simulated curves for different vesicle size distributions g(r₀) and a permeable weak acid. The color code is the same in the four panels. (a) Log-normal distributions with mean m = 100 nm are shown. The variance v varies as indicated in the legend. (b) Average volume dynamics $\langle V\left(t\right)\rangle$ calculated according to Eq. 25 are shown. The average pH dynamics $\langle pH\left(t\right)\rangle$ (c) and the normalized average fluorescence intensity $\langle F\left(t\right)\rangle /\langle F\left(0\right)\rangle$ (d) were calculated as described in Appendix B. The following parameters were used for calculations: pK₁ = 7.21, pK₂ = 4, M_w = 18 cm³/mol, pH₀ = ${\text{pH}}_0^{\ast }$ = 7.0, [KPi] = 90 mM, [KPi]^∗ = 100 mM, [calcein] = 10 mM, k₁ = 10⁶ s⁻¹, K_SV = 10² M⁻¹, $c_6^{\ast }+c_7^{\ast }$ = 60 mM, r₀ = 100 nm, P₁ = 0.003, and P₆ = 0.03 cm/s. For calculation of pH(t), we set [KPi] = 100 mM and [calcein] = 0 M. To see this figure in color, go online.