Zade-Oppen followed up Lovelock's work in 1961, providing further validation of Lovelock's findings as well as more details on the kinetics of the process. He agreed that post-hypertonic lysis was due to salt loading and attempted to provide a mechanism. Zade-Oppen speculated that there is a structural framework within the cell that prevents the cell from shrinking past some minimum volume. When the cell reaches that volume, water movement can no longer maintain osmotic equilibrium therefore sodium loading occurs.
In 1968, Meryman presented evidence of a minimum volume being reached which seemed to correlate with the onset of damage. He seems to favor the idea that an osmotic pressure gradient develops at the minimum volume causing an elastic failure of the membrane while in the shrunken state.
Farrant was able to show in 1972 that the minimum volume detected by Meryman was actually an artifact of the hematocrit method. Using better techniques, he found that red cells showed no minimum volume, behaving as perfect osmometers to very high concentrations. He also found that the membrane became leaky to cations at about the same concentration (0.8M) that was implicated by Lovelock.
We will assume that the cytoplasm contains a solution of dissociated potassium and chloride ions as well as a significant number of proteins with fixed charges participating in salt bridges with other proteins. The following diagram illustrates such a cell, surrounded by dissociated sodium and chloride ions.
As ice begins to form outside the cell (those little snowflakes around the cell) the external salt becomes concentrated in the unfrozen channels with the cells. In response to the osmotic gradient, the cell loses water and the intracellular salt becomes concentrated as well.
When either the intracellular or extracellular salt concentration reaches a critical level (0.8M at room temperature, as found by Lovelock and later, Farrant), cation channels in the plasma membrane open allowing and exchange of intracellular potassium for extracellular chloride.
The increase in intracellular salt concentration, whether due to sodium or potassium, causes cytoplasmic proteins to be "salted in" to solution. This is a phenomenon by which insoluble proteins are often solubilized. The free ions in solution begin to interact with the fixed charges on the proteins, breaking the salt bridges and bringing the proteins into solution. As the proteins are brought into solution, they have an increasing effect on the osmotic pressure (their osmotic coefficient increases) that is proportional to the number of ions removed from bulk solution. The net effect is to decrease the intracellular ion concentration without changing the osmotic pressure of the cytoplasm.
The extracellular sodium will move down its concentration gradient, through the cation channels, into the cell. Thus we have an increase in the total number of ions within the cell. Since the degree to which cytoplasmic proteins are salted into solution is proportional to the salt concentration, the increase in intracellular ions is similarly dependent on the temperature during freezing (since the concentration of extracellular solute is determined by the temperature) if the cell is allowed to come to equilibrium.
Upon thawing, the ice melts and dilutes the extracellular solution back to isotonic. Water moves in to the cell interior and begins to dilute the salt in the cytoplasm as well. When both the intracellular and extracellular salt concentrations have gone below the critical level, the cation channels in the membrane close once again. As the intracellular salt concentration is lowered, the proteins salt back out of solution and release the ions that were bound to them resulting in a greater number of ions inside the cell that were there initially. This causes water to continue to enter the cell even after it has swelled past its isotonic volume. If the cell swells past its yield volume, then lysis occurs.
The above diagram shows the cell volume changes of red cells equilibrated in 0.5M glycerol and then cooled rapidly (100°C/min) to one of four temperatures (-10, -20, -30, & -40°C) where they were held for one minute before rapid thawing (100°C/min) to room temperature. The final volumes are all above isotonic because of the salt loading that occurs at the subzero temperatures. If we employ some arbitrary linear function to correlate maximum volume with % lysis (in this case, % lysis = 1.036 (Vf) - 93.5) then we can plot the cell death vs. temperature for these conditions:
The damage increases with increasing temperature but then starts to decrease between -30°C and -40°C. The effect of time on the accumulation of damage can be similarly predicted, again using the same conditions except the cells are held at -20°C for various lengths of time:
Again, the final volumes are all greater than isotonic due to the uptake of salt at the low temperatures. The final volume appears to approach a maximum after about 16 minutes. If we use the same relationship between % lysis and maximum volume, we get:
Clearly showing that the damage exponentially approaches an asymptotic value with time.
Qualitatively, then, the model does correspond with the kind of injury that we see during slow cooling (the choice of red cells allows us to ignore the confounding effects of rapid cooling injury since that doesn't begin to show up until cooling rates of about 500°C/min are reached). We now need to see how well this corresponds to real data.
The kinetics are more-or-less on the mark although the magnitude is a little off. This could easily be improved with some optimization but we're just looking for order-of-magnitude results here.
Now here's the real meat and potatoes! The fit is unbelievably good. Sure you can jimmy some equations to give you the right answers when you have free parameters to play with, but that's not where this fit comes from. The parameters were just crudely adjusted by hand (i.e. run a simulation, hmmm...that looks pretty close, lets just boost the equilibrium value a tad...run it again...close enough) and the results all fell out of the sky. The mechanism may be wrong, but there is no question that it can be used to simulate slow cooling injury. This is important since the environment within tissues is complex and hard to examine during freezing. Mathematical modeling may be the only tool available for investigating the results of various freezing and thawing protocols without spending a lifetime testing every permutation.
Not only this, but the proposed mechanism can certainly be tested experimentally. The yield volume, the equilibrium salt loading, and the kinetics of salt loading can all be measured experimentally, reducing the free parameters to zero. As soon as I get some time...