Challenge from the simple: some caveats in linearization of the Boyle-van't Hoff and Arrhenius plots

Mathematical Modeling and Optimization of Cryopreservation in Single Cells

Cryobiology is a multiscale and interdisciplinary field. The scope and scale of interactions limit the gains that can be made by one theory or experiment alone. Because of this, modeling has played a critical role in both explaining cryobiological phenomena and predicting improved protocols. Modeling facilitates understanding of the biophysical and some of the biochemical mechanisms of damage during all phases of cryopreservation including CPA equilibration and cooling and warming. Moreover, as a tool for optimization of cryopreservation protocols, modeling has yielded many successes. Modern cryobiological modeling includes very detailed descriptions of the physical phenomena that occur during freezing, including ice growth kinetics and spatial gradients that define heat and mass transport models. Here we reduce the complexity and approach only a small but classic subset of these problems. Namely, here we describe the process of building and using a mathematical model of a cell in suspension where spatial homogeneity is assumed for all quantities. We define the models that describe the critical cell quantities used to describe optimal and suboptimal protocols and then give an overview of classical methods of how to determine optimal protocols using these models. We include practical considerations of modeling in cryobiology, including fitting transport models to cell volume data, performing optimization with cell volume constraints, and a look at expanding cost functions to cooling regimes.

Osmotic behaviour of human mesenchymal stem cells: Implications for cryopreservation

Aimed at providing a contribution to the optimization of cryopreservation processes, the present work focuses on the osmotic behavior of human mesenchymal stem cells (hMSCs). Once isolated from the umbilical cord blood (UCB) of three different donors, hMSCs were characterized in terms of size distribution and their osmotic properties suitably evaluated through the exposure to hypertonic and isotonic aqueous solutions at three different temperatures. More specifically, inactive cell volume and cell permeability to water and di-methyl sulfoxide (DMSO) were measured, being cell size determined using impedance measurements under both equilibrium and dynamic conditions. Experimental findings indicate that positive cell volume excursions are limited by the apparent increase of inactive volume, which occurs during both the shrink-swell process following DMSO addition and the subsequent restoration of isotonic conditions in the presence of hypertonic solutions of impermeant or permeant solutes. Based on this evidence, hMSCs must be regarded as imperfect osmometers, and their osmotic behavior described within a scenario no longer compatible with the simple two-parameter model usually utilized in the literature. In this respect, the activation of mechano-sensitive ion-channels seemingly represents a reasonable hypothesis for rationalizing the observed osmotic behavior of hMSCs from UCB.

Determination of the membrane hydraulic permeability of MSCs

A successful cryopreservation is based on knowledge of the optimal cooling rate. So far, this is often determined by way of complex parameter studies. Alternatively, the identification of cell specific characteristics, such as osmotic behaviour, membrane hydraulic permeability and activation energy could be used to calculate the optimal cooling rate. These parameters should be determined for supra-zero and sub-zero temperatures. In this study cryomicroscopy was used. Mesenchymal stromal cells (MSCs) from bone marrow were analysed. The determined membrane hydraulic permeability for sub-zero temperatures is significantly lower than that for supra-zero temperatures. On the contrary the activation energy is significantly higher in the presence of ice. The addition of a cryoprotective agent (CPA) such as dimethyl sulfoxid (DMSO) shows an additional influence on the characteristics of the membrane of the cell. The optimal cooling rate was determined with these parameters. For cryopreservation without DMSO the optimal cooling rate was found to be 12.82 K/min. If the MSCs were frozen with 5% (v/v) DMSO the optimal cooling rate is 16.25 K/min.

Modeling and optimization of cryopreservation

Modeling plays a critical role in understanding the biophysical processes behind cryopreservation. It facilitates understanding of the biophysical and some of the biochemical mechanisms of damage during all phases of cryopreservation including CPA equilibration, cooling, and warming. Modeling also provides a tool for optimization of cryopreservation protocols and has yielded a number of successes in this regard. While modern cryobiological modeling includes very detailed descriptions of the physical phenomena that occur during freezing, including ice growth kinetics and spatial gradients that define heat and mass transport models, here we reduce the complexity and approach only a small but classic subset of these problems. Namely, here we describe the process of building and using a mathematical model of a cell in suspension where spatial homogeneity is assumed for all quantities. We define the models that describe the critical cell quantities used to describe optimal and suboptimal protocols and then give an overview of classical methods of how to determine optimal protocols using these models.

Rationally optimized cryopreservation of multiple mouse embryonic stem cell lines: I--Comparative fundamental cryobiology of multiple mouse embryonic stem cell lines and the implications for embryonic stem cell cryopreservation protocols

The post-thaw recovery of mouse embryonic stem cells (mESCs) is often assumed to be adequate with current methods. However as this publication will show, this recovery of viable cells actually varies significantly by genetic background. Therefore there is a need to improve the efficiency and reduce the variability of current mESC cryopreservation methods. To address this need, we employed the principles of fundamental cryobiology to improve the cryopreservation protocol of four mESC lines from different genetic backgrounds (BALB/c, CBA, FVB, and 129R1 mESCs) through a comparative study characterizing the membrane permeability characteristics and membrane integrity osmotic tolerance limits of each cell line. In the companion paper, these values were used to predict optimal cryoprotectants, cooling rates, warming rates, and plunge temperatures, and then these predicted optimal protocols were validated against standard freezing protocols.

Some comments on recent discussion of the Boyle van't Hoff relationship

The estimation of several cellular biophysical parameters must be made in order to mathematically predict optimal cryopreservation protocols. These parameters include total cell volume, osmotically inactive volume, cell surface area, and relative water and solute permeabilities. Recent attention has been paid to the determination of the osmotically inactive volume and, specifically, an argument was made suggesting that this volume was incorrectly determined in the literature [4]. Here we show that this assertion is false.

Amicus Plato, sed magis amica veritas: plots must obey the laws they refer to and models shall describe biophysical reality!

In the companion paper, we discussed in details proper linearization, calculation of the inactive osmotic volume, and analysis of the results on the Boyle-vant' Hoff plots. In this Letter, we briefly address some common errors and misconceptions in osmotic modeling and propose some approaches, namely: (1) inapplicability of the Kedem-Katchalsky formalism model in regards to the cryobiophysical reality, (2) calculation of the membrane hydraulic conductivity L(p) in the presence of permeable solutes, (3) proper linearization of the Arrhenius plots for the solute membrane permeability, (4) erroneous use of the term "toxicity" for the cryoprotective agents, and (5) advantages of the relativistic permeability approach (RP) developed by us vs. traditional ("classic") 2-parameter model.

On proper linearization, construction and analysis of the Boyle-van't Hoff plots and correct calculation of the osmotically inactive volume

The Boyle-van't Hoff (BVH) law of physics has been widely used in cryobiology for calculation of the key osmotic parameters of cells and optimization of cryo-protocols. The proper use of linearization of the Boyle-vant'Hoff relationship for the osmotically inactive volume (v(b)) has been discussed in a rigorous way in (Katkov, Cryobiology, 2008, 57:142-149). Nevertheless, scientists in the field have been continuing to use inappropriate methods of linearization (and curve fitting) of the BVH data, plotting the BVH line and calculation of v(b). Here, we discuss the sources of incorrect linearization of the BVH relationship using concrete examples of recent publications, analyze the properties of the correct BVH line (which is unique for a given v(b)), provide appropriate statistical formulas for calculation of v(b) from the experimental data, and propose simplistic instructions (standard operation procedure, SOP) for proper normalization of the data, appropriate linearization and construction of the BVH plots, and correct calculation of v(b). The possible sources of non-linear behavior or poor fit of the data to the proper BVH line such as active water and/or solute transports, which can result in large discrepancy between the hyperosmotic and hypoosmotic parts of the BVH plot, are also discussed.