Fluid flow beneath a semipermeable membrane during drying processes

Foundations of modeling in cryobiology-II: Heat and mass transport in bulk and at cell membrane and ice-liquid interfaces

Modeling coupled heat and mass transport in biological systems is critical to the understanding of cryobiology. In Part I of this series we derived the transport equation and presented a general thermodynamic derivation of the critical components needed to use the transport equation in cryobiology. Here we refine to more cryobiologically relevant instances of a double free-boundary problem with multiple species. In particular, we present the derivation of appropriate mass and heat transport constitutive equations for a system consisting of a cell or tissue with a free external boundary, surrounded by liquid media with an encroaching free solidification front. This model consists of two parts-namely, transport in the "bulk phases" away from boundaries, and interfacial transport. Here we derive the bulk and interfacial mass, energy, and momentum balance equations and present a simplification of transport within membranes to jump conditions across them. We establish the governing equations for this cell/liquid/solid system whose solution in the case of a ternary mixture is explored in Part III of this series.

Numerical solution of inward solidification of a dilute ternary solution towards a semi-permeable spherical cell

Modeling a cell's response to encroaching ice has informed the development of cryopreservation protocols for four decades. It has been well documented that knowledge of the cellular state as a function of media and cooling rate faciliate informed cryopreservation protocol design and explain mechanisms of damage. However, previous efforts have neglected the interaction between solutes and the encroaching ice front and their effects on the cell state. To address this, here we examine the cryobiologically relevant setting of a spherically-symmetric model of a biological cell separated by a ternary fluid mixture from an encroaching solid-liquid interface. The cell and liquid regions contain cell membrane impermeable intracellular and extracellular salts, respectively, a cell membrane permeable solute commonly used in cryopreservation protocols known as a cryoprotective agent (CPA), and water as a membrane permeable solvent. As cooling and solidification proceed the extracellular chemical environment evolves and leads to mass transport across the cell membrane. Consequently, both the solidification front and the cell membrane are free boundaries whose dynamics are coupled through transport processes in the solid, liquid and cell regions. We describe a numerical procedure to solve this coupled free-boundary problem based on a domain transformation and method of lines approach. We also investigate how the thermal and chemical states inside the cell are influenced by different cooling protocols at the external boundary. Finally, we observe that the previously unaccounted-for partial solute rejection at the advancing solid-liquid interface increases the CPA and salt concentrations in the extracellular liquid as a function of the interface speed and segregation coefficients, suggesting that previous model predictions of the cell state during cryopreservation were inaccurate.

Elastic membranes in confinement

An elastic membrane stretched between two walls takes a shape defined by its length and the volume of fluid it encloses. Many biological structures, such as cells, mitochondria and coiled DNA, have fine internal structure in which a membrane (or elastic member) is geometrically 'confined' by another object. Here, the two-dimensional shape of an elastic membrane in a 'confining' box is studied by introducing a repulsive confinement pressure that prevents the membrane from intersecting the wall. The stage is set by contrasting confined and unconfined solutions. Continuation methods are then used to compute response diagrams, from which we identify the particular membrane mechanics that generate mitochondria-like shapes. Large confinement pressures yield complex response diagrams with secondary bifurcations and multiple turning points where modal identities may change. Regions in parameter space where such behaviour occurs are then mapped.

The effect of glass-forming sugars on vesicle morphology and water distribution during drying

Cryopreservation requires that stored materials be kept at extremely low temperatures and uses cryoprotectants that are toxic to cells at high concentrations. Lyopreservation is a potential alternative where stored materials can remain at room temperatures. That storage process involves desiccating cells filled with special glass-forming sugars. However, current desiccation techniques fail to produce viable cells, and researchers suspect that incomplete vitrification of the cells is to blame. To explore this hypothesis, a cell is modelled as a lipid vesicle to monitor the water content and membrane deformation during desiccation. The vesicle is represented as a moving, bending-resistant, inextensible interface and is tracked by a level set method. The vesicle is placed in a fluid containing a spatially varying sugar concentration field. The glass-forming nature is modelled through a concentration-dependent diffusivity and viscosity. It is found that there are optimal regimes for the values of the osmotic flow parameter and of the concentration dependence of the diffusivity to limit water trapping in the vesicle. Furthermore, it is found that the concentration dependencies of the diffusivity and viscosity can have profound effects on membrane deformations, which may have significant implications for vesicle damage during the desiccation process.

The equilibria of vesicles adhered to substrates by short-ranged potentials

In equilibrium, a vesicle that is adhered to a horizontal substrate by a long-range attractive, short-range repulsive force traps a thin layer of fluid beneath it. In the asymptotic limit that this layer is very thin, there are quasi-two-dimensional boundary-layer structures near the edges of the vesicle, where the membrane's shape is governed by a balance between bending and adhesive stresses. These boundary layers are analysed to obtain corrections to simpler models that instead represent the adhesive interaction by a contact potential, thereby resolving apparent discontinuities that arise when such models are used. Composite expansions of the shapes of two-dimensional vesicles are derived. When, in addition, the adhesive interaction is very strong, there is a nested boundary-layer structure for which the adhesive boundary layers match towards sharp corners where bending stresses remain important but adhesive stresses are negligible. Outside these corners, bending stresses are negligible and the vesicle's shape is given approximately by the arc of a circle. Simple composite expansions of the vesicle's shape are derived that account for the shape of the membrane inside these corners.