Optimizing cryopreservation protocols of Saccharomyces eubayanus using heat transfer modeling

Abstract

Abstract

Efficient cryopreservation is essential for maintaining the viability of Saccharomyces eubayanus, a cryotolerant wild yeast of industrial importance as the cold-adapted parent of Saccharomyces pastorianus. This study integrated post-thaw viability and vitality assessments of cryopreserved S. eubayanus CRUB 1568ᵀ with experimental measurements of transient heat transfer and numerical simulation of the freezing stage. Two protocols were evaluated: (A) direct freezing of cryovials in cryoboxes at − 80 °C, governed by convection, and (B) freezing inside a CoolCell® device (Corning Inc., Corning, NY, USA), where heat transfer occurs by conduction through the insulated plastic material. A mathematical model was developed to numerically solve the transient heat transfer equation with phase change using the finite element method. Experimental temperature–time data validated the simulations, allowing estimation of overall heat transfer coefficients (UA = 18.04 W m⁻² K⁻¹; UB = 4.76 W m⁻² K⁻¹) and characteristic freezing times (t𝚌 = 10.9 min; 27.6 min, respectively). Calculated Biot numbers confirmed uniform temperature distribution within cryovials. PROTOCOL A achieved optimal cooling rates (5–7 °C min⁻¹) and yielded higher post-thaw viability (71.7 ± 3.5%) compared with PROTOCOL B (51.2 ± 3.6%) after 1 year at − 80 °C. The integration of modeling and experimental data demonstrates that the overall heat transfer coefficient is a key engineering parameter influencing cryopreservation performance. Direct freezing of cryovials in cryoboxes represents a simpler, faster, and lower-cost approach that ensures uniform cooling and higher cell survival, providing a valuable basis for standardizing yeast cryogenic storage in industrial and biotechnological applications.

Key points

• Finite element modeling assessed heat transfer during yeast cryopreservation.

• Direct freezing in cryoboxes achieved higher viability than CoolCell®.

• Overall heat transfer coefficient (U) is key for cryogenic performance analysis.

Supplementary Information

The online version contains supplementary material available at 10.1007/s00253-026-13779-0.

Introduction

Saccharomyces eubayanus is a cryotolerant wild yeast known as the cold-adapted parent of Saccharomyces pastorianus, the hybrid responsible for lager beer production (Libkind et al. 2011; Giorgetti et al. 2025). Due to its industrial relevance, efficient cryopreservation methods of this microorganism are essential, although recovery from cold storage presents challenges and difficulties, which limit the long-term preservation of this strategic genetic resource in culture collections.

The survival of the model yeast Saccharomyces cerevisiae during freezing has been studied in literature and it is strongly dependent on the rate at which it is cooled (Mazur 1970). The relationship between survival and cooling velocity shows an inverted U shape curve, with maximum survival occurring at an optimal rate. If cooling is carried out faster or slower than this optimal rate, it leads to a decrease in survival; therefore, two distinct, rate-dependent factors are responsible for cell death during freezing. This phenomenon is explained by the “two-factor hypothesis,” which establishes that different mechanisms cause cell death linked to the cooling rates. At a slow cooling rate, injury is caused by “solution effects” (Bank and Mazur 1973), since as the external medium freezes, solutes become concentrated, creating a hypertonic environment that draws water out of the cells. While this dehydration prevents the lethal formation of intracellular ice, prolonged exposure to high solute concentrations and the associated cellular dehydration can be damaging. Slower cooling rates translate to longer exposure to the damaging solution effects. If cells cool faster than the optimal rate, injury is primarily caused by the formation of intracellular ice (Mazur 1963). At high rates, there is insufficient time for water to leave the cell and freeze externally. The intracellular water becomes supercooled and eventually freezes inside the cell, causing a lethal event. The optimal cooling rate is therefore a rate which is slow enough to allow sufficient dehydration to prevent intracellular ice formation, but fast enough to minimize the time during which cells are exposed to damaging solution effects. For S.cerevisiae NRRL-Y-2235, extensive data indicates that the optimal cooling rate is approximately 7 °C/min, which yields a maximum survival of about 55%; however, cooling rates below optimal, such as 1.8 °C/min, result in about 43% survival after rapid warming (Seki et al. 2009; Mazur 1970).

While considerable experience is now available on the optimal preservation methods for many groups of microorganisms, these procedures may not be applicable as a general rule to all yeast species and microorganisms; particular care is needed with cryo-tolerant genera and species. A greater range of procedures should be assessed and research carried out to determine optimal protocols in laboratories when managing culture collections.

It is a widespread practice to apply specific devices such as CoolCell® (Corning Inc., Corning, NY, USA) or Mr. Frosty™ (Thermo Fisher Scientific, Waltham, MA, USA) for the freezing of yeast in cryovials for their long-term cold storage. The CoolCell® container is constructed of closed-cell, cross-linked polyethylene foam and a solid thermo-conductive core, which in combination with a −80 °C freezer provides a freezing rate of 1 °C/min. The CoolCell® design uses a combination of insulation foam, radial symmetry, and a heat transfer core to regulate heat loss. The Mr. Frosty™ container consists of the following parts (i) polycarbonate container, (ii) high-density polyethylene closure, and (iii) a tube holder. It uses isopropyl alcohol to achieve the recommended cooling rate of 1 °C/min. The instructions for both devices are similar: the container should be initially at room temperature, the cryovials are placed at the tube holes or vial chambers inside the container and then inserted into the mechanical freezer at −80 °C for a minimum of 4 h before moving to archive storage. In the case of Mr. Frosty™, special care during handling is needed to avoid spilling of the alcohol inside the container during transfer in and out of the ultrafreezer with cryovials. The procedure using both containers requires the transferring of frozen samples to archive storage; therefore, an insulated pan or container with dry ice must be used when removing the Mr. Frosty™ or CoolCell® from the ultrafreezer. A cryogenic environment is necessary while there is manipulation of cryovials such as removal of samples from the vial chambers and insertion/placement into the cryoboxes or other permanent storage devices. The removal of the cryovials from the containers and relocation into the final storage device may result critical since a temperature rise that leads to cell death must be avoided. The yeast suspension in the cryogenic vial can rise from −80 °C to over −50 °C in less than 1 min if exposed to room temperature (CoolCell® manual). Additionally, it represents a time-consuming activity for the personnel in charge of applying these protocols for culture collection with a limited amount of vials that can be processed in a batch. In these cases (Mr. Frosty™ and CoolCell®), where a container is used, the cryovial is cooled, and the heat transfer is governed by a conduction process since the wall of the cryovial is in intimate contact with a plastic material of the container.

Another method that could be used as a simpler and more efficient alternative is the direct freezing of cryovials from ambient temperature to −80 °C in cryoboxes for long-term storage, without the removal from the mechanical freezer or the need to use dry ice in the process. There is no information regarding the performance of this procedure for S. eubayanus in terms of the cooling rate velocity that the system achieves or the survival outcome. It is, however, a better alternative in practical and economic terms since it is less expensive (no container is used) and the cryovials are directly frozen for long-term storage in the cryobox; in addition, no dry ice is used since no extraction or relocation is necessary. In this procedure or protocol, the cryovial is cooled by the surrounding cold air at −80 °C inside the plastic cryobox. The base of the cryobox has holes that allow air to circulate; therefore, heat transfer is governed by a convection process.

All the methods mentioned involve different freezing stages governed by transient heat transfer. Key parameters include overall surface heat transfer coefficients (U), which reflect the heat exchange between the cryovial and its surrounded environment. The performance of cryopreservation procedures often relies on the use of cooling rates, which are calculated using the slope of the thermal history curve at a given position in the sample. This calculation when applied to a freezing curve shows three different cooling rates, corresponding to different zones; V1 corresponds to the cooling of the sample until the initial freezing temperature is reached; V2 to the phase change zone or plateau region, and V3 to the cooling of the fully frozen sample. Santos et al. (2024) applied the concept of characteristic freezing time (tc) in an attempt to find a single variable for a better visualization of the performance of cryopreservation protocols; this variable can be calculated from the thermal history of the system (temperature vs. time curve) and can be defined as the time elapsed between the initial freezing temperature of the fluid and a reference temperature of −40 °C (Tf), since at this temperature, more than 80% of the total water content is in a frozen state (Santos et al. 2013, 2024).

In the present work, in order to analyze the freezing rates of the cryovials during cryopreservation of yeasts, this concept of tc was applied.

There is scarce information related to the overall heat transfer coefficients that govern each protocol implemented for the cryopreservation of yeasts relevant for the brewing industry. In addition, the characteristic freezing times (that are representative of the reciprocal of the freezing rates) and the comparison of protocols for the cryopreservation of yeasts using cryovials filled with cells suspended in glycerol water mixture have yet not been reported.

This information could be useful since the cryopreservation of yeast strains is time-consuming and several steps using devices increase processing times greatly. Additionally some procedures that use specific containers such as Mr.Frosty™ and CoolCell® could be expensive and hard to implement for some laboratories that maintain large microbial culture collections.

Numerical simulations of the transient heat transfer partial differential equation using the finite element method are efficient when trying to predict temperature vs. time curves in different points of the systems that include several domains such as the liquid suspension of cells (LS) contained in the plastic cryoval (PC) and the headspace air (A); however, it is a challenging task since a phase change transition due to the transformation of water into ice is involved. The use of a numerical technique to describe heat transfer phenomena during the cryopreservation of yeast has yet not been reported, and it might help visualize the bottleneck of the freezing process by determining relevant heat transfer parameters and develop standardized procedures for yeast cryopreservation.

This work aimed to (1) compare the cooling rates of two protocols: (A) direct placement of cryovials inside a cryobox in the Ultrafreezer and (B) cryovials placed in a device that retards the cooling process such as CoolCell® in an Ultrafreezer (UF), (2) to evaluate the characteristic freezing times (tc) and the cooling rate velocities during the cryopreservation of S. eubayanus CRUB 1568 ^T, (3) to determine by numerical simulations using the finite element method the overall heat transfer coefficients of the two protocols, and (4) to correlate these parameters with the post-thaw viability and vitality of S. eubayanus CRUB 1568 ^T.

Materials and methods

Yeast culture and cryopreservation protocols

S. eubayanus CRUB 1568^T (CBS 12357) was obtained from the culture collection Colección de Recursos Microbianos de la Patagonia Andina (CReMPa), Río Negro, Argentina. This strain was inoculated into 100-mL conical flask containing 45 mL of yeast malt (YM) media (0.3 g yeast extract, 0.3 g malt, 0.5 g meat peptone, and 1 g glucose in 100 mL of distilled water), which was shaken at 20 °C and 180 rpm. The liquid culture was incubated overnight and afterwards, 1 mL of yeast suspension was transferred to a 250-mL conical flask containing 99 mL of YM media, obtaining a final volume of 100 mL. This culture was grown under the same conditions (20 °C and 180 rpm) until cells entered the early stationary phase. At this point (20 h of incubation), cells were harvested by centrifugation (10 min, 5000 rpm) in 15-mL falcons. The pellet was suspended in a 10 mL glycerol (99.5% v/v purity, Sigma-Aldrich, St. Louis, MO, USA) dissolved in distilled water at a concentration of 10 w/w. Afterwards, 100 µL of the cell suspension was transferred to cryovials (Greiner Bio-One, Frickenhausen, Germany) containing 900 µL of glycerol (10% w/w) having a final cell concentration of 10⁶ cells/mL. Cryovials were sealed and mixed as described by Shah et al. (2022), before freeze-thaw protocols.

The performance of both protocols was assessed; in PROTOCOL A, cryovials are in contact with air at −80 °C in UF, and in PROTOCOL B, cryovials are in contact with the plastic material of the CoolCell® placed in inside the UF (Fig. 1). It must be remarked that in PROTOCOL B, the cryovial is not in contact with air but with a plastic material as can be observed in Fig. 1.

Fig. 1.

Schematic representation of the cryopreservation protocols evaluated. Photographs of the cryovials with the thermocouples connected to the data-loggers

Experimental determination of the characteristic freezing time and cooling and warming rates

The temperature vs. time curve for each protocol was measured using a type T-thermocouple (copper/copper-nickel) connected to a data-logger TESTO (Testo SE & Co. KGaA, Lenzkirch, Germany), which recorded temperature every 10 s at the center and at the interface plastic/liquid of the cryovial. A second thermocouple was used to record the temperature in the external air inside the ultrafreezer. Figure 1 shows the photographs of the experimental setup using the cryovials, the thermocouples, and dataloggers for each protocol. The cooling rates for each protocol were calculated based on the initial straight-line slope (Fennema et al. 1973) of each stage during the freezing process. The characteristic freezing time (tc) was calculated for each protocol.

An independent set of experiments using distilled water in the cryovials (that has known thermo-physical properties) submitted to the same protocols as the biological samples (described in Fig. 1) was conducted in order to determine the overall surface heat transfer coefficients. The thermal histories were recorded and used for the comparison of the numerical simulations with the experimental data.

After 1 year of storage in an independent rack without risk of potential warming, cryovials were thawed in a controlled temperature water bath at 25 °C with mild stirring. Warming and cooling rates were obtained as described in Santos et al. (2024).

Viability and vitality

After 1-year frozen storage in the ultrafreezer at −80 °C, cells were thawed using a controlled-temperature water bath at 25 °C with agitation. Viability was obtained by plating cells in YM media supplemented with 15 g L⁻¹ agar after serial dilutions in physiological solution. Colony-forming units (CFU) were counted after 48–72 h. Vitality was assessed by measuring the colony diameter after 48 h using ImagePro Plus software. Percentage vitality was calculated by dividing the average colony diameter obtained in each protocol by the average colony diameter of the control treatment. In both cases, the data obtained were compared with the control treatment (cells from the same culture suspended in distilled water without any cryopreservation protocol) to determine viability and vitality.

Statistical analysis (analysis of variance, determination of standard deviation (SD) and Tukey test) was performed by SigmaPlot 12.0 software. Each protocol was done in triplicate, and for each replica, 2 separate plates were analyzed.

Mathematical modeling of the freezing process during the cryopreservation protocols

The cryovials used for simulation purposes were purchased from Greiner Bio-One (Greiner Bio-One, Frickenhausen, Germany) and have a 1-ml volume capacity. The geometrical dimensions of the cryovial were measured using a caliper. For the mathematical modeling and numerical simulation, the cryovial can be considered a solid of revolution having a symmetry plane at r = 0; therefore, cylindrical coordinates were implemented. The geometry and domains were rendered within the engineering simulation environment of Comsol Multiphysics (COMSOL AB, Stockholm, Sweden, license number 2097640), and the domains can be seen in Fig. 2.

Fig. 2.

a Photograph of the cross-sectional area of the cryovial used for the numerical simulations, b cylindrical coordinate system used and relevant dimensions of the cryovial, c material domains considered in the model (solution refers to distilled water or a 10% w/w glycerol suspension with the yeast culture, PP refers to the plastic material of the cryovial (polypropylene)

The cryopreservation protocols involve a freezing process; there is a phase change transition of water into ice; therefore, the thermo-physical properties change abruptly in the temperature range where the transition occurs. This constitutes a highly non-linear mathematical problem since thermal conductivity, density, and specific heat are strongly temperature-dependent. The heat conduction equation in cylindrical coordinates with phase change transition can be written as follows for the fluid inside the cryovial (Carslaw and Jaeger 1959; Pérez-Calderón et al. 2019):

$${\rho }_{\mathrm{f}}\left(\mathrm{T}\right){\mathrm{C}}_{\mathrm{pf}}\left(\mathrm{T}\right)\frac{\partial \mathrm{T}}{\partial \mathrm{t}}\mathrm{r}=\frac{\partial }{\partial \mathrm{r}}\left({\mathrm{k}}_{\mathrm{f}},\left(\mathrm{T}\right),,\mathrm{r},\frac{\partial \mathrm{T}}{\partial \mathrm{r}}\right)+\frac{\partial }{\partial \mathrm{z}}\left({\mathrm{k}}_{\mathrm{f}},\left(\mathrm{T}\right),,\mathrm{r},\frac{\partial \mathrm{T}}{\partial \mathrm{z}}\right)in{\mathrm{\Omega }}_f$$ 1

This equation is valid in the liquid domain Ω_f (solution: either distilled water or glycerol-water mixture) being T the temperature, t the time of the process, r and z the geometric coordinates, k_f(T) is the thermal conductivity, C_pf(T) the apparent specific heat, and ρ_f(T) the density, and subscript “f” corresponds to the properties of glycerol-water or distilled water according to the experiment simulated.

The equation for the air space domain (Ω_a) at the upper part of the cryovial was:

$${\rho }_{\mathrm{a}}{\mathrm{C}}_{\mathrm{pa}}(\mathrm{T})\mathrm{p}\frac{\partial \mathrm{T}}{\partial \mathrm{t}}\mathrm{r}=\frac{\partial }{\partial \mathrm{r}}\left({\mathrm{k}}_{\mathrm{a}},(\mathrm{T}),\mathrm{r},\frac{\partial \mathrm{T}}{\partial \mathrm{r}}\right)+\frac{\partial }{\partial \mathrm{z}}\left({\mathrm{k}}_{\mathrm{a}},(\mathrm{T}),\mathrm{r},\frac{\partial \mathrm{T}}{\partial \mathrm{z}}\right)in{\mathrm{\Omega }}_a$$ 2

where subscript “a” corresponds to the air thermal properties which are temperature-dependent. The headspace was considered a conductive layer of air, with average properties in the applied temperature range (Lespinard et al. 2019; Jiang et al. 2016).

The plastic material domain (Ω_p) of the cryovial can be described with the following equation:

$${\rho }_p{C}_{\mathit{pp}}\frac{\partial T}{\partial t}r=\frac{\partial }{\partial r}(k_{p}r\frac{\partial T}{\partial r})+\frac{\partial }{\partial z}(k_{p}r\frac{\partial T}{\partial z})in{\mathrm{\Omega }}_p$$ 3

where subscript “p” corresponds to the polypropylene thermal properties, which were considered constant with temperature in the studied temperature range.

The boundary and initial conditions are:

$$\left(\frac{\partial T}{\partial z},.,n_z,+,\frac{\partial T}{\partial r},.,n_r\right)k=Uj\left(T_{\mathit{ext}},-,T\right)t>0in\partial {\mathrm{\Omega }}_1$$ 4

$$\left(\frac{\partial T}{\partial z},.,n_z,+,\frac{\partial T}{\partial r},.,n_r\right)k\left(T\right)=0t>0in\partial {\mathrm{\Omega }}_2$$ 5

$$T=T_{0}t=0in{\mathrm{\Omega }}_a,{\mathrm{\Omega }}_p,and{\mathrm{\Omega }}_f$$ 6

where Uj is the overall surface heat transfer coefficient of the system analyzed (j = A, B corresponding to PROTOCOL A and PROTOCOL B, respectively); $\partial {\mathrm{\Omega }}_1$ corresponds to the external plastic surface interface of the cryovial exposed to the cold air in the ultrafreezer or PROTOCOL A, or to the plastic material of the CoolCell® for PROTOCOL B; $\partial {\mathrm{\Omega }}_2$ to the symmetry axis at r = 0; n_r and n_z are normal outward vector components at the radial and axial axis, respectively; n is the outward-pointing normal unit vector at the surface; T₀ is the initial temperature; T_ext is the external cold air temperature.

The numerical model was solved using the COMSOL Multiphysics (COMSOL AB, Stockholm, Sweden, lic. #2097640) software incorporating the thermo-physical properties of each domain, overall surface heat transfer coefficients, and external air temperature.

Numerical simulations and experimental measurements were carried out in a two-step process; first, the tested substance (water) was modeled in the cryovial since its temperature-dependent properties are fully documented in literature and were used as known input information in the model. The experiments with cryovials containing water that were submitted to the PROTOCOLS A and B allowed the rigorous determination of the overall surface heat transfer coefficients applying independent experiments. The advantage relies on the fact that water-ice thermo-physical properties during the temperature range of the freezing process are well known; therefore, the numerical simulations allowed the determination of the overall heat transfer coefficient by contrasting experimental and numerical temperature vs. time curves and finding the UA and UB that minimize the residual sum of squares (RSS). The second step consisted in introducing these calculated UA and UB values in the numerical model using as input information the glycerol + yeast suspension in order to simulate the freezing process and compare with the laboratory experiments. The agreement between experimental and numerical results allows the validation of the heat transfer model, confirming the accuracy of the UA and UB values.

Thermo-physical properties

Plastic material

The plastic (PP) properties of the cryovial were considered constant throughout the freezing experiments: the thermal conductivity, density, and apparent specific heat at 25 °C of polypropylene were 0.22 W/m K, 900 kg/m³, and 1680 J/Kg K, respectively (Sansinena et al. 2010; Jiang et al. 2016).

Air headspace in the cryovial

The properties of air in the cryovial were obtained from previous reports on literature; they can be observed in supplementary material (Supplemental Table S1).

Thermo-physical properties of water

The thermal conductivity and density of water vs. temperature were obtained from published literature values (Choi and Bischof 2010; Petrenko and Whitworth 1999; Santos et al. 2018). Supplemental Figs. S1 and S2 in supplementary material show k(T) and ρ(T) for pure water.

The apparent specific heat was calculated using the following equations considering the amount of ice formed (Miles et al. 1983):

$${\mathrm{x}}_{\mathrm{i}}\left(\mathrm{T}\right)={\mathrm{x}}_{\mathrm{w}}(1-\frac{{\mathrm{T}}_{\mathrm{f}}}{\mathrm{T}})$$ 7

where T_f is the initial melting temperature of the sample, x_i is the mass fraction of ice present at temperature T, and x_w is the water fraction of the sample.

The apparent specific heat was estimated by the equation proposed by Mellor (1976):

$$\mathrm{Cp}\left(\mathrm{T}\right)={\mathrm{x}}_{\mathrm{i}}\left(\mathrm{T}\right){\mathrm{Cp}}_{\mathrm{i}}\left(\mathrm{T}\right)-{\mathrm{x}}_{\mathrm{w}}\frac{{\mathrm{T}}_{\mathrm{f}}}{\mathrm{T}}{\mathrm{Cp}}_{\mathrm{w}}\left(\mathrm{T}\right)-{\mathrm{\Delta }\mathrm{H}}_{\mathrm{w}}(\mathrm{T}){\mathrm{x}}_{\mathrm{w}}\frac{{\mathrm{T}}_{\mathrm{f}}}{{\mathrm{T}}^2}$$ 8

where ${\mathrm{\Delta }H}_{\mathrm{w}}$ is the latent heat of melting of pure water taking into account its temperature dependence (Nesvadba 2008).

Supplemental Fig. S3 shows the apparent specific heat of water vs. temperature in supplementary material.

When Eq. 8 is used as an input in the software, the solution diverges due to the abrupt functionality of the specific heat around the phase change temperature range of water. Therefore, a reformulation using a combined Heaviside and Gaussian function was used in order to have a smoother mathematical function with second-order derivatives that are continuous, which avoids the numerical instabilities of the finite element method, associated with the freezing simulation in the phase change zone as recommended by the software Comsol Multiphysics (COMSOL AB, Stockholm, Sweden, vs. 3.4 Manual). This technique was successfully applied for freezing simulations in biological systems (Santos et al. 2013, 2018, 2024).

The reformulation combining Heaviside and Gaussian functions is:

$$\mathrm{Cp}\left(\mathrm{T}\right)={\mathrm{Cp}}_{\mathrm{cc}}+\frac{\mathrm{\Delta }{\mathrm{H}}_{\mathrm{m}}}{{\mathrm{T}}_{\mathrm{m}}}\mathrm{f}\left({\mathrm{H}}_{\mathrm{ea}}\right)+\mathrm{D}\ast \mathrm{\Delta }{\mathrm{H}}_{\mathrm{m}}$$ 9

where Cp_cc corresponds to the specific heat of the fully frozen state, T_m is the temperature of the freezing peak (equivalent to the median μ for the Gaussian curve), ∆H_m is the latent heat of melting, and D is the Gaussian function defined as:

$$\mathrm{D}\left(\mathrm{T}\right)=\frac{{\mathrm{e}}^{(\frac{{\left(\mathrm{T},-,{\mathrm{T}}_{\mathrm{m}}\right)}^2}{{\mathrm{dT}}^2})}}{\sqrt{\mathrm{\pi }{\mathrm{dT}}^2}}$$ 10

where dT is the half-width of transition, that is 2dT, the temperature interval where 84% of the latent heat occurs; f(H_ea) is the Heaviside function which is a built-in function in COMSOL MATLAB environment that has continuous second-order derivatives. This function enables the numerical finite element software to successfully deal with the abrupt change in the apparent specific heat.

The mathematical model for each thermo-physical function for distilled water was coded in the MATLAB environment using own-generated codes; afterwards, these functions k_s(T), ρ_s(T), and Cp_s(T) were inserted in the software COMSOL Multiphysics, which solves the heat conduction equation in transient state with convective boundary condition using the finite element method.

Thermo-physical properties of the glycerol-water mixture

The thermal conductivity and density function of the binary solution (10% w/w glycerol suspension) was determined using the models proposed by Choi and Okos (1986) considering the composition of the fluid.

The thermal conductivity as a function of temperature is:

$${\mathrm{k}}_{\mathrm{s}}\left(\mathrm{T}\right)={\mathrm{x}}_{\mathrm{i}}^{\mathrm{v}}{\mathrm{k}}_{\mathrm{i}}\left(\mathrm{T}\right)+{\mathrm{x}}_{\mathrm{w}}^{\mathrm{v}}{\mathrm{k}}_{\mathrm{w}}\left(\mathrm{T}\right)+{\mathrm{x}}_{\mathrm{g}}^{\mathrm{v}}{\mathrm{k}}_{\mathrm{g}}\left(\mathrm{T}\right)$$ 11

where k_s is the global thermal conductivity and k_i, k_w, and k_g are the thermal conductivities of ice, water, and glycerol, respectively. The fraction “$x^v$” corresponds to the volumetric fraction of each component, subscripts i, w, and g correspond to ice, water and glycerol, respectively.

The amount of ice formed was estimated using Eq. 7.

The initial freezing temperature Tf was determined by the experimental freezing curves using the tangent method described by Fennema et al. (1973).

The density of the biological fluid was obtained using Choi and Okos (1986):

$$\mathrm{\rho }\left(\mathrm{T}\right)=\frac{1}{\frac{{\mathrm{x}}_{\mathrm{i}}}{{\mathrm{\rho }}_{\mathrm{i}}}+\frac{{\mathrm{x}}_{\mathrm{w}}}{{\mathrm{\rho }}_{\mathrm{w}}}+\frac{{\mathrm{x}}_{\mathrm{g}}}{{\mathrm{\rho }}_{\mathrm{g}}}}$$ 12

where ρ(T) is the global density and ρ_i, ρ_w, and ρ_g are the densities of ice, water, and glycerol, respectively (the component ice is computed if the temperature is below the melting temperature). The fraction “x” corresponds to the mass fraction of each component; subscripts i, w, and g correspond to ice, water, and glycerol, respectively.

The apparent specific heat can be estimated by Eq. 4 adding the term $x_{\mathrm{g}}{\mathrm{Cp}}_{\mathrm{g}}\left(T\right)$ on the right side of the equation which considers the contribution of glycerol in the Cp(T) of the solution; $x_{\mathrm{g}}$ is the mass fraction of glycerol and ${\mathrm{Cp}}_{\mathrm{g}}\left(T\right)$ the specific heat of pure glycerol vs. temperature. The properties of pure glycerol were obtained using literature data shown in Supplementary Table S2 (Sandberg et al. 1977; Blazhnov et al. 2004).

Experimental design to determine the overall heat transfer coefficients

Experimental time–temperature data obtained with cryovials containing distilled water were processed using numerical simulations in order to obtain accurate average values of U for each protocol.

A minimum of three experiments for each protocol were carried out and this procedure allowed the calculation of the standard deviations for the overall heat transfer coefficients. Different U values were introduced in the numerical code to simulate the temperature–time curve for cryovials containing water; experimental and predicted temperatures for each proposed U were compared. The overall heat transfer coefficient that minimized the residual sum of squares given by Eq. (13) was selected for each protocol:

$$\mathrm{RSS}=\sum {\left({\mathrm{T}}_{\mathrm{ext}},-,{\mathrm{T}}_{\mathrm{pred}}\right)}^2$$ 13

Afterwards, the U values calculated from the experiments with pure water were introduced as input data in the numerical model corresponding to the glycerol-water mixture having different thermophysical properties. The predicted temperatures were compared with experimental data, further validating the numerical model with experimental temperature vs. time measurements.

Results

Experimental freezing temperature versus time curves

Experimental time–temperature curves at the center of the cryovial submitted to PROTOCOLS A and B can be observed in Fig. 3a for distilled water and Fig. 3b for yeast culture and water-glycerol mixture (10% w/w). Figure 3c shows schematic typical freezing curves for the cryovials cooled using an ultrafreezer at −80 °C. In Fig. 3c, V1, V2, and V3 are the rates calculated using the slope in three different regions of the same time–temperature curve; tc is the characteristic freezing time.

Fig. 3.

Experimental time–temperature curves at the center of the cryovial submitted to PROTOCOL A and B, a for distilled water and b yeast culture and water-glycerol mixture (10% w/w), c schematic typical freezing curves for the cryovials cooled using an ultrafreezer at −80 °C, V1, V2, and V3 are the rates calculated using the slope in three different regions of the same time–temperature curve; tc is the characteristic freezing time

As can be seen from Fig. 3 a and b, PROTOCOL A reaches the final temperature of −80 °C much faster than PROTOCOL B for both fluids (distilled water and yeast culture and glycerol + water mixture). The plateau region for distilled water is clearly visible occurring at 0 °C (initial freezing temperature, Tf), while for the glycerol + water mixture, the experimental value calculated using Fennema et al. (1973) is Tf = −2.38 °C, which is in agreement with reported values in Lane (1925).

Mathematical modeling of the freezing process during cryopreservation

Thermophysical properties

Figure 4 a, b, and c show the thermal conductivity, density, and apparent specific heat of glycerol-water mixture (10% w/w) vs. temperature, respectively, which were used as input information in the Comsol Multiphysics (COMSOL AB, Stockholm, Sweden). In order to circumvent convergence difficulties of the numerical model, the reformulation of the apparent specific heat of the glycerol-water mixture using Heaviside and Gaussian functions was applied in the Comsol Software (as was previously explained for water under the Thermo-physical properties of water section.

Fig. 4.

Thermo-physical properties of glycerol-water (10% w/w) vs. temperature: a thermal conductivity; b density; and c apparent specific heat

Spatial and time discretization

A triangular mesh using Lagrange elements of order 2 was applied to discretize the three domains that constitute the cryovial system: plastic, the solution (distilled water or glycerol-water mixture 10% w/w), and the headspace air (see Fig. 5). The total number of elements that constituted the mesh for the three domains was 334 (polypropylene 119, solution 168, headspace air 47). The use of a finer mesh showed no significant effect on the accuracy of the solution. A fixed mesh was used because the phase change was modeled using a continuum approach, where the latent heat of melting was incorporated directly into the material properties via highly non-linear temperature-dependent functions for specific heat, thermal conductivity, and density; the phase transition that occurs over a discrete temperature interval was modeled using Heaviside and Gaussian functions. In this way, the model treated the interface as a property gradient rather than a geometric discontinuity, and the fluid domain remains geometrically fixed during the transition, eliminating the need for dynamic meshing. The time discretization scheme used was a backward Euler differentiation (minimum order 1 and maximum order 5) with a tuning step having a maximum of 0.1 s and a minimum initial starting value of 0.001 s. The absolute and relative tolerances for each integration step were 0.001 and 0.01, respectively. The numerical runs were tested for their computational speed using a PC Intel(R) Core(TM) i3 6300 with a processor speed of 3.80 GHz and a random access memory (RAM) of 4 GB having a maximum central processing unit (CPU) time of 4 min.

Fig. 5.

Triangular mesh used in the numerical simulations (left) and temperature spatial distribution predicted from the numerical simulation for the cryovial after 2000 s of being submitted in the ultrafreezer inside the CoolCell®, Ti = 19.5 °C, Text = −78 °C, UB = 4 W/m² K

Cooling rates of each protocol

The cooling rates V1, V2, and V3 for each protocol were calculated based on the straight line slope of the freezing curve as shown in Fig. 3c). Tables 1 and 2 show the cooling rates for distilled water and yeast culture in glycerol water mixture 10% w/w measured at the center and interface fluid-plastic wall of the cryovial for PROTOCOLS A and B, respectively.

Table 1.

Cooling rates for distilled water and yeast culture in glycerol water mixture 10% w/w measured at the center and interface fluid-plastic wall of the cryovial for PROTOCOL A

	Position	N (replicates)	V1 (°C/min)	V2 (°C/min)	V3 (°C/min)
Distilled water	Center	5	5.68 (0.33)	0.06 (0.03)	11.71 (0.98)
	Wall	5	6.28 (0.11)	0.53 (0.22)	10.96 (0.64)
Yeast culture in glycerol water mixture 10% w/w	Center	3	6.86 (1.37)	0.11 (0.06)	6.99 (0.38)
	Wall	3	6.73 (1.23)	0.45 (0.15)	6.73 (0.36)

Table 2.

Cooling rates for distilled water and yeast culture in glycerol water mixture 10% w/w measured at the center and interface fluid-plastic wall of the cryovial for PROTOCOL B

			PROTOCOL B
	Position	N (replicates)	V1 (°C/min)	V2 (°C/min)	V3 (°C/min)
Distilled water	Center	6	2.37 (0.16)	0.02 (0.01)	4.38 (1.69)
	Wall	6	2.67 (0.40)	0.09 (0.03)	3.89 (1.90)
Yeast culture in glycerol water mixture 10% w/w	Center	3	2.27 (0.28)	0.78 (0.47)	2.91 (0.53)
	Wall	3	2.28 (0.33)	0.70 (0.37)	2.62 (0.56)

As can be seen from the results in Table 3, the cooling rate V1 is higher than 1 °C/min, and it corresponds to the initial cooling velocity. The cooling rates V3 show larger values and can be attributed to the higher thermal conductivity of ice as temperature decreases in the fully frozen temperature region.

Table 3.

Characteristic freezing time tc (min), overall heat transfer coefficient U (W/m² K), and percentage viability of S. eubayanus CRUB 1568 ^T (1 year frozen storage). Standard deviations are between parentheses. * shows significant differences between PROTOCOLS (p < 0.05)

	PROTOCOL A			PROTOCOL B
	Distilled water		Yeast + glycerol	Distilled water		Yeast + glycerol
tc (min)	10.86 (0.35)		10.93 (0.59)	26.85 (5.79)		28.29 (2.44)
U (W/m2 K)		UA = 18.04 (1.16)			UB = 4.76 (0.84)
%viability			71.67* (3.46)			51.21 (3.61)
%vitality			72.90 (2.19)			74.76 (1.00)

It is noteworthy that for each protocol, the cooling rates at the center and wall of the cryovials were very similar. This experimental result will be explained in depth in the section Validation of the mathematical model and numerical simulation.

PROTOCOL A shows for both fluid types (water or yeast culture) higher cooling rates (V1, V2, V3) compared to PROTOCOL B.

Warming rates

The warming rates of cryovials (initial straight line slope) containing water and a water + glycerol mixture 10% w/w were 105.33 °C/min (standard deviation, s.d. = 17.63) and 104.46 °C/min (s.d. 14.92), respectively. The values found for both fluids were similar and correspond to high warming rates. This is important since a high warming rate helps to avoid the growth and recrystallization of small ice into larger, more damaging crystals which can occur during slow warming. Rapid warming melts the small crystals before they have time to grow. It has been reported that, for slowly cooled cells (where injury is due to solution effects), slow warming is often more damaging than rapid warming (Mazur 1970).

Determination of the overall surface heat transfer coefficients in water-filled cryovial for PROTOCOLS A and B

Figure 6 shows as an example two of the experiments carried out and the numerical predictions considering phase change of water into ice during the cryopreservation protocol. An excellent agreement can be observed; the overall heat transfer coefficients and their standard deviations (between parentheses) that fitted with minimum RSS were in average for all experiments: UA = 18.7 W/m² K (0.5) and UB = 4.7 W/m² K (1.2), for PROTOCOLS A and B, respectively.

Fig. 6.

Experimental results of temperature vs. time (●) during freezing of cryovials containing water and predicted temperatures obtained by the numerical simulations (➖) for a PROTOCOL A (Ti = 22 °C, UA = 18 W/m² K) and b PROTOCOL B (Ti = 21.6 °C, UB = 3.8 W/m² K)

Validation of the mathematical model and numerical simulation

The UA and UB values, obtained for cryovials containing water in the previous section, were used as input information in the numerical program in order to simulate the cryopreservation process of yeast cultures considering the variable thermo-physical properties of glycerol + water mixture 10% (w/w). The numerical predictions and experimental results were contrasted, and an excellent agreement was observed; therefore, this implies that UA and UB were accurately determined and represent the heat transfer process correctly.

Figure 7 shows as an example the experiments carried out and the temperature vs. time prediction using the numerical model. See also Fig. 5, which shows an example the temperature distribution predicted from the numerical simulation for the cryovial with yeast culture after 2000 s of being submitted in the ultrafreezer inside the CoolCell®, Ti = 19.5 °C, Text = −78 °C, UB = 4 W/m² K.

Fig. 7.

Experimental results of temperature vs. time (●) during freezing of cryovials containing yeast culture and glycerol + water solution (10% w/w), and predicted temperatures (Tpred) obtained by the numerical simulations (➖) for a PROTOCOL A (Ti = 21 °C, UA = 20 W/m² K) and b PROTOCOL B (Ti = 23 °C, UB = 4.6 W/m² K)

Table 3 shows the characteristic freezing times tc (min) when cryovials were filled with water and yeast culture in a water/glycerol solution and the overall heat transfer coefficients U (W/m² K) for PROTOCOL A and B. Also, the %viability of S. eubayanus CRUB 1568 ^T after 1 year frozen storage using both protocols is presented.

In PROTOCOL A, which corresponds to the use of cryovials directly into cryoboxes, tc reaches values that are on average 10.89 min in comparison to 27.57 min for PROTOCOL B. This is in agreement with the obtained values of UA > UB. The overall heat transfer coefficient U defines the rate at which heat flows in the system, and for PROTOCOL A, the heat transfer was faster than for PROTOCOL B.

The value of UA corresponding to the air surrounding the cryovals inside the cryobox was compared with literature reports using the formula from Earle (1988). This equation considers air as the circulating fluid with a velocity < 5 m/s; this represents the case of air inside an ultrafreezer, which is a natural convection condition (Santos et al. 2010). The values obtained for 2.5 m/s were in the range 15.5–20 W/m² K; therefore, it can be concluded that the UA value estimated through the numerical program using the finite element method (18.04 W/m² K) is within the range of the heat transfer coefficients estimated by literature correlations.

The UA value was used to calculate the Biot number:

$$\mathrm{Biot}=\frac{\mathrm{UA}\mathrm{R}}{\mathrm{kf}}$$ 14

where R is the radius of the cryoval (m), kf is the thermal conductivity of the fluid yeast suspension inside the cryovial. Due to the fact that the yeast system was submitted to a freezing process, thermal conductivity of the fluid changed from 0.5 W/mK for the unfrozen suspension at 0 °C to 2.5 W/m K for the frozen yeast suspension at −30 °C. Therefore, both Biot numbers were calculated obtaining 0.238 for the unfrozen state and 0.047 for the frozen fluid.

The same Eq. 10 was applied for PROTOCOL B, and the obtained Biot numbers 0.063 and 0.013, for the unfrozen and frozen conditions respectively.

Biot numbers (Bi) < 1/6 ensure a uniform temperature profile inside the cryovial; this means that the temperature at all points between the plastic wall and the center is the same at each time. However, when Bi < 1 the internal heat transfer resistance of the yeast fluid suspension is very low, resulting in a nearly uniform temperature distribution within the cryovial. This result was also observed from the numerical simulations (see Fig. 1c).

It is important to remark that uniform temperatures inside the cryovials imply the same freezing velocities at all points (see Tables 1 and 2).

Supplemental Fig. S4 shows experimental curves of temperatures vs. time for PROTOCOLS A and B where temperatures of the glycerol + water mixture were measured at the center and at the internal side of the plastic wall in the cryovial (only one of the replicates for each Protocol is shown). Both curves are very similar due to the uniform temperature distribution during freezing inside the cryovial; this result was attributed to the fact that heat transfer is governed by the external resistance for both PROTOCOLS.

Discussion

The results of this study demonstrate that the direct freezing protocol (PROTOCOL A), characterized by a significantly faster cooling rate (tc = 10.9 min; UA = 18.04 W/m² K), is more suitable for the cryopreservation of S.eubayanus CRUB 1568ᵀ than the standard CoolCell® method (PROTOCOL B; tc = 27.57 min; UB = 4.76 W/m² K). PROTOCOL A yielded a significantly higher post-thaw viability of 71.7% compared with 51.2% for PROTOCOL B. These findings challenge the generalized assumption that a controlled cooling rate of 1 °C/min is universally optimal for all eukaryotic cells. This specific rate is widely regarded as the “gold standard” for the cryopreservation of human mammalian cells, particularly for peripheral blood lymphocytes (Farrant et al. 1972), mesenchymal stem cells (Linkova et al. 2022), and hematopoietic stem cells (Shu et al. 2010), where it is essential to maintain a delicate balance between avoiding lethal intracellular ice and minimizing osmotic injury (Hunt 2011). However, yeast cells possess distinct biophysical properties that shift this optimal window. Specifically, the higher hydraulic permeability and significantly higher surface-area-to-volume ratio of yeast (Dumont et al. 2003, 2004) allow sufficient water efflux to reach osmotic equilibrium even at cooling rates higher than 1 °C/min. The experimental data and calculations confirm that the optimal cooling range for S.eubayanus is higher. Additionally, literature reports show that optimal cooling velocities for yeast are in the range of 3–10 °C/min (Morris et al. 1988; Mazur and Schmidt 1968), and Seki et al. (2009) showed an optimal of 7 °C/min for S.cerevisiae; this value is in agreement with the results in this work which resulted in an average V1 = 6.75 °C/min (initial cooling period of the yeast + glycerol solution) and V3 = 6.86 °C/min (freezing period of the yeast + glycerol solution). Therefore, the lower viability observed in PROTOCOL B suggests that for S.eubayanus, the 1 °C/min rate is unnecessarily slow. At such slow rates, cells are likely subjected to prolonged “solution effects,” where excessive dehydration and exposure to concentrated solutes cause irreversible damage (Bank and Mazur 1973). The direct freezing protocol (PROTOCOL A) avoids lethal intracellular ice formation while minimizing the toxic solute exposure typical of slow-rate devices. From a practical and economic standpoint, this method offers a cost-effective and operationally simple alternative, eliminating the need for expensive cooling devices while ensuring higher cell survival and reduced handling time.

While CFU-based viability is a practical and widely used endpoint (Mazur 1961; Meyer et al. 1975; Morris et al. 1988; Dumont et al. 2003; Gujjari et al. 2010; de Arruda Moura Pietrowski et al. 2015), it mainly reports the ability to regrow (often interpreted as preserved membrane integrity) and can miss sublethal effects or injuries that compromise function after thawing (e.g., osmotic/oxidative stress, membrane and organelle perturbations, and the energetic burden of repair). In this study, vitality, which was determined by colony diameter after 48 h, was reduced by approximately 30% in all treatments, regardless of the viability differences observed between protocols. This indicates that a fraction of the viable cells likely sustained sublethal injury that slows recovery and metabolism, so growth performance provides a more sensitive readout of post-thaw health. With that objective, our research group is currently standardizing yeast cryopreservation by combining viability with functional metrics, including lag phase duration, growth kinetics, and fermentation outputs (CO₂ production and ethanol yield). This approach aligns with recent evidence suggesting that high post-thaw recovery coincides with substantial proteome remodeling (Ramasamy et al. 2025), which emphasizes that optimal viability does not equate to minimal metabolic disturbance but rather depends on metabolic readiness and active injury-response pathways.

An important aspect considered in the present work was the potential warming of the cryovials during their frozen storage; it is important to maintain them undisturbed once placed in the ultrafreezer for a 1-year period. In this study, they were not removed from the freezer or submitted to potential warming cycles since they were maintained in an independent rack. Several authors have reported risks associated with thawing yeast cultures (repeated freezing and thawing cycles). Cabrera et al. (2020) recommended taking precautions when sampling from a frozen stock to avoid thawing the entire batch, as cell viability decreases with each freeze–thaw cycle. Bond (2007) advised minimizing the time the straws remain out of liquid nitrogen during sample retrieval, and Meyer et al. (1975) performed three freeze–thaw cycles with Candida utilis and Lipomyces stokesii stored at − 60 °C and observed a reduction in both cell viability and fermentative capacity after each cycle.

A mathematical model based on transient heat transfer partial differential equations with phase change transition that occurs during the cryopreservation of S. eubayanus CRUB 1568 ^T was developed and validated. The equations for the different domains (yeast suspension, plastic wall of the cryovial and surrounding air or insulated material according to the tested protocol) were numerically solved by the finite element method. The model was applied to determine overall surface heat transfer coefficients when two distinct cryopreservation protocols are applied: (A) direct freezing of cryovials placed in cryoboxes at − 80 °C, where heat transfer occurs mainly by air convection, and (B) freezing within a CoolCell® device, where heat is transferred by conduction through the plastic material.

Numerical simulations reproduced the experimental temperature–time curves with high precision, enabling accurate estimation of the overall surface heat transfer coefficients. The values obtained were UA = 18.04 W/m² K for PROTOCOL A and UB = 4.76 W/m² K for PROTOCOL B. These values explain the observed differences in the characteristic freezing times (t𝚌), which was 10.9 min for PROTOCOL A in comparison to 27.6 min for PROTOCOL B, indicating that direct exposure to cold air enables more efficient thermal exchange than the contact with plastic insulation. The UA is consistent with air natural convection conditions around cryovials inside an ultrafreezer. The calculated Biot numbers indicated a nearly uniform temperature distribution within the cryovials for both protocols, confirming that the internal resistance to heat transfer is negligible compared to external resistance.

The observed cooling rates for PROTOCOL A were V1 = 6.75 °C/min (initial cooling period of the yeast + glycerol solution) and V3 = 6.86 °C/min (freezing period of the yeast + glycerol solution); this PROTOCOL showed the best performance and aligned with the optimal range reported for yeasts, ensuring a balance between minimizing intracellular ice formation and avoiding excessive solute concentration effects. Consequently, post-thaw viability after 1 year of storage reached 71.7 ± 3.5% for PROTOCOL A, compared to 51.2 ± 3.6% for PROTOCOL B.

From a mechanistic standpoint, the study demonstrates that the overall heat transfer coefficient (U) is a critical variable determining cryopreservation efficiency. By integrating experimental and numerical analyses, a relationship was established between heat-transfer performance, cooling rates, and biological survival. The results highlight that direct freezing of cryovials in cryoboxes not only enhances cell viability but also simplifies handling and reduces operational costs by eliminating intermediary devices which are expensive.

In conclusion, direct freezing in cryoboxes represents a simpler, faster, and cost-effective method that ensures uniform temperature fields, higher survival, and reduced handling time compared with container-based systems.

The results emphasize the relevance of mechanistic heat transfer modeling in defining reproducible, scalable cryogenic practices for microbial culture collections and industrial biotechnology.

Supplementary Information

Below is the link to the electronic supplementary material.

Author Contribution

MVS and DL conceived and designed the research. AC conducted experiments. DL and MVS contributed devices, reagents, and containers for the experiments. MVS and NZ analyzed data. MVS and AC wrote the initial manuscript. MVS, NZ, and DL revised and proofread the text. All authors read and approved the manuscript.

Funding

We would like to thank the financial support for this research which was provided by the following institutions: Consejo Nacional de Investigaciones Científicas y Técnicas (PIP 11220210100223CO and PIP 11220200102948CO), the project “Adecuación de las Condiciones de Preservación De Cultivos Microbianos Estratégicos de la Patagonia Andina y Consolidación de la Base de Datos Tecnológicos y Moleculares de la Colección,” PIP 28520210100030CO, and Agencia Nacional de Promoción Científica y Tecnológica (PICT-2020-SERIEA-00226) of Argentina. The funding sources had no involvement in the development and analysis of the results presented in the present work.

Data availability

Data sets generated during the current study are available from the corresponding author on reasonable request.

Declarations

Ethics approval

Not applicable.

Competing interests

The authors declare no competing interests.