Kinetic modelling of Mucor circinelloides growth on skim milk agar under dairy-relevant temperature and water activity conditions

Fungal isolate

The fungal isolate M. circinelloides 1L was used for all experiments in the present study. The monitored isolate, which was selected from the collection of the Institute of Food Science and Nutrition (Slovak University of Technology in Bratislava, Slovak Republic), originated from the artisanal Slovak "Bryndza" cheese (Turčianske Teplice, Slovak Republic). Molecular identification was performed based on internal transcribed spacer (ITS) sequencing carried out according to Pangallo et al. (2014). The confirmation was based on the morphological and physiological characteristics following Pitt and Hocking (2022), Botha and Botes (2014) and Samson et al. (2004).

Experimental design

The levels of the factors were chosen to simulate conditions during the manufacturing, ripening and storage of cheeses and to fully cover the growth region of the species to the greatest possible extent. Growth trials with M. circinelloides 1L strain were assessed for fungal growth diameters periodically (10 experimental temperatures × 3 a_w levels × 3 growth replicates = 90 growth assays).

Growth medium

Mycelial growth assessments were performed on Plate Count Skim Milk agar (SMA; Merck, Darmstadt, Germany). Three different a_w scenarios were implemented: 0.992±0.002 (unmodified growth medium), 0.985±0.002 (1 % NaCl, w/v), and 0.975±0.002 (2 % NaCl, w/v). After sterilisation, 50 mL of medium was poured into sterile Petri dishes (diameter 130 mm) and a_w levels were measured with Novasina LabMaster-aw (Novasina, Lachen, Switzerland).

Inoculation and incubation

The strain was stored on an SMA slant under refrigeration (5±0.5 °C) to slow down metabolic activity and prevent excessive growth or sporulation during long-term maintenance. To preserve viability, the culture was sub-cultured monthly. For inoculum preparation, a 5-day culture grown on the surface of a Sabouraud slant (Sabouraud Dextrose agar, Biolife Italiana Srl, Milan, Italy) in the dark at 25±0.5 °C was used to promote active growth and abundant sporulation under optimal conditions, resulting in a heavily sporulating culture. For spore collection, the cultures were flooded with sterile saline solution (8.5 g/L NaCl, 0.1 g peptone) containing Tween 80 as a wetting agent (0.01 % v/v). Immediately after preparation, this spore suspension was diluted down and adjusted to 10³ spores/mL and used for inoculation.

Petri dishes were single-point inoculated (2 μL) with the fresh suspension to form a circular inoculum in the centre of each SMA plate. Petri dishes with the same water activity level were enclosed in sealed polyethene bags to avoid a_w fluctuations, followed by temperature-controlled incubation (Pol-Eko Aparatura, Wodzisław Śląski, Poland.

Surface growth measurements and growth modelling

For all static experiments, time zero was defined as the time the suspension was applied to the surface of the agar plate. The diameters of developing colonies were measured at appropriate time intervals (daily or as needed) to capture the lag and exponential phases. Growth measurements were continued until the colony reached the edge of the Petri dish (i.e., fully covered the plate) or until growth ceased. Colony diameters were recorded using a Vernier calliper (150 mm × 0.02 mm; Sinochem Jiangsu, Nanjing, China) in two directions at right angles to each other, without opening the dishes. The final diameter of the colonies (expressed in micrometres) was calculated as an arithmetic mean. The max

imum storage period was less than 3 months.

Growth modelling

Statistical analyses

The growth parameters (sgr_opt, λ, and d_max) were expressed as mean values ± standard deviation (SD). To evaluate the accuracy of both primary and secondary growth models, the coefficient of determination (R²) and root-mean-square error (RMSE) metrics were calculated using Microsoft Excel (Microsoft 365, Redmond, WA, USA).

Validation

To validate the model, supplementary experiments were conducted using programmed temperature shifts in SMA medium at a_w levels of 0.992 and 0.985 (adjusted with 1 % NaCl). The time/temperature regime 25 °C and 18 °C (each for 2 days), followed by 15 °C and 10 °C (each for 5 days), simulates traditional fermentation and ripening conditions typical of Slovak artisan ewe lump cheese production (Medveďová and Valík, 2012). The growth parameters of two-step modelling performed in this work were used to calculate colony diameters of M. circinelloides. Without a stationary phase, the following general equation (8) was used:

[TeX:] d_{\text{cal}} = d_{0} + {\text{sgr}(CM)}_{\text{cal}} \bullet (t - \lambda)$d_{\mathrm{\text{cal}}}=d_0+{\mathrm{\text{sgr}}(CM)}_{\mathrm{\text{cal}}}•(t-\lambda )$ (10)

where d_cal is a calculated colony diameter, d₀ is the initial colony diameter at each T/t module, which is also the final d_cal of the previous T/t module.

The bias, accuracy and discrepancy factors (B_f, A_f and %D, respectively) representing validation indices were calculated using the following equations (Baranyi et al., 1999):

[TeX:] B_{f} = e^{\frac{\sum_{i = 1}^{n}{{(ln\ y}_{i}^{\text{cal}} - ln\ y_{i}^{\exp}\ )}}{n}}$B_f=e^{\frac{{\sum }_{i=1}^n{(lny}_i^{\mathrm{\text{cal}}}-ln{y}_i^{exp})}{n}}$ (11)

[TeX:] A_{f} = e^{\sqrt{\frac{\sum_{i = 1}^{n}\left( \text{ln\ y}_{i}^{\text{cal}} - ln\ y_{i}^{\exp} \right)^{2}}{n}}}$A_f=e^{\sqrt{\frac{{\sum }_{i=1}^n{({\mathrm{\text{ln y}}}_i^{\mathrm{\text{cal}}}-ln{y}_i^{exp})}^2}{n}}}$ (12)

[TeX:] \% D = {(A}_{f} - 1) \bullet 100\%$\%D={(A}_f-1)•100\%$ (13)

where [TeX:] y_{i}^{\text{cal}}$y_i^{\mathrm{\text{cal}}}$ is the calculated value for the i-th observation and [TeX:] y_{i}^{\exp}$y_i^{exp}$is the experimental value for the i-th observation.

Primary growth of M. circinelloides

The surface growth parameters of M. circinelloides cultivated on SMA are presented in Table 1. As expected, the lag phase shortened with increasing temperature and water activity, while specific growth rate (sgr) values increased until reaching optimal conditions, after which they declined. The average coefficients of determination exceeded 0.978, underscoring the high consistency and smoothness of growth curves observed across triplicate experiments. Root-mean-square error (RMSE) values for the growth curves ranged from 0.73 mm to 5.17 mm, yielding an average of 2.36±1.08 mm. For comparison, the primary model employed by Van Long et al. (2021) similarly described radial growth with fitting quality reflected by coefficients of determination (R²) above 0.80 and RMSE values below 5.32 mm.

Table 1. Average growth parameters of Mucor circinelloides 1L cultivated on SMA, along with statistical indices describing growth curve consistency across triplicate experiments

T - incubation temperature; sgr - surface growth rate; λ - lag phase duration; nl - no lag; ng - no growth; d_max - maximum colony diameter in the stationary phase; N/A - non-applicable.

Secondary modelling

Lag phase

To model the lag phase (λ), reciprocal values were fitted using the Ratkowsky equation, incorporating the environmental factors, temperature (T) and water activity (a_w). Figure 1 displays the model’s application, with estimated parameters as follows: b=0.031±0.005 1/°C/√d, Tₘᵢₙ = -7.77±0.20 °C, and a_w ₘᵢₙ = 0.962±0.002. The model produced a total RMSE of 0.209 1/d and a coefficient of determination (R²) of 0.811, indicating a reasonably good fit. The parameter errors and associated statistical indices were deemed acceptable. For reference, the R² reported in a prior study by Koňuchová et al. (2024) using a cardinal temperature model was similarly robust (R² = 0.716, 0.956, 0.783, and 0.970). In addition, Van Long et al. (2021) explored variability in parameters estimated using the cardinal model for radial growth in 29 Penicillium roqueforti strains. Their model also included T and a_w as environmental variables, yielding R² values ranging from 0.703 to 0.990.

Figure 1. RTK secondary model depicting the effect of temperature and water activity on the reciprocal of lag phase duration (d⁻¹) during radial growth of Mucor circinelloides strain 1L. The solid grid shows values fitted by Eq. 5 to the observed data (open circles). Results represent three independent biological replicates.

Surface growth rate

The effect of temperature and water activity (a_w) on the surface growth rate of M. circinelloides cultivated on SMA medium is depicted in Figure 2. The interaction of these environmental factors was modelled using the cardinal model, with parameter estimates summarised in Table 2. Key parameters, sgr_opt = 33 mm/d, T_opt ≈ 32 °C, and a_w ₘᵢₙ = 0.97, indicate a high growth rate of the strain, along with sensitivity to decreasing a_w adjusted with NaCl. In contrast, the strain studied by Morin-Sardin et al. (2016) exhibited a lower a_{w min} of 0.927 under similar NaCl adjustments. From this perspective, the a_{w min} = 0.97 observed in our strain might appear theoretical; however, in our supplementary experiments, no growth was detected at NaCl concentrations exceeding 3 %. Despite the use of a different medium, this discrepancy likely reflects a strain-specific sensitivity to NaCl. As M. circinelloides belongs to lower fungi, this rapid growth aligns with expected physiological behaviour. Regarding growth rates, Morin-Sardin et al. (2016) reported sgr_opt values ranging from 9.3 to 19.3 mm/d for M. circinelloides grown on Potato Dextrose Agar (PDA). In contrast, Gougouli et al. (2011) observed a higher sgr_opt of 1.53 mm/h - equivalent to approximately 36.7 mm/d on Malt Extract Agar, a value closely aligned with our estimate (Table 2).

Other cardinal model parameters estimated in Table 2 also show close agreement with those reported by Gougouli et al. (2011). For instance, the theoretical Tₘᵢₙ of -3.2 °C aligns well with their reported -3.0 °C, while Tₒₚₜ of 31.7 °C and Tₘₐₓ of 35.9 °C compare favorably with their respective values of 29.0 °C and 37.0 °C. Similarly, Morin-Sardin et al. (2016) documented for M. circinelloides cultivated on PDA medium an almost identical Tₒₚₜ of 29.2 °C, alongside a comparable Tₘᵢₙ of -1.2 °C and a slightly higher Tₘₐₓ of 40.9 °C.

Figure 2. Effect of the temperature on the surface growth rate (sgr) of M. circinelloides strain 1L cultivated on SMA medium. The Cardinal Model (solid grid) was fitted to the observed surface growth rate values (full dots)

Table 2. Estimated parameters of the cardinal model describing the surface growth of Mucor circinelloides 1L cultivated on SMA medium

CM parameters	SE	SE of fit	R 2
sgropt (mm/d)	33.04	1.60	0.393	0.884
Tmin (°C)	-3.18	0.04
Topt (°C)	31.74	0.13
Tmax (°C)	35.94	0.12
aw min	0.970	0.003
aw opt	0.985	0.001
aw max	1.00	fixed

SE - standard error; R² - coefficient of determination

Growth model validation

Validation outcomes shown in Figure 3 express the colony diameters that were predicted modularly across the temperature/time sequence using the previously established models for surface growth rate (sgr) and lag phase.

Figure 3. Comparison of predicted colony diameter (d_cal) and experimentally observed colony diameter (d_exp) of Mucor circinelloides strain 1L cultivated on SMA medium at a_w levels of 0.995 and 0.985. Calculations were performed modularly under a dynamic temperature regime of 25 °C (48 h), 18 °C (48 h), 15 °C (120 h), and 10 °C (120 h)

The statistical indices, RMSE and coefficient of determination (R²), for surface growth modelling of M. circinelloides on SMA medium at both a_w conditions were 9.35 mm and 0.989, respectively (Table 3). Despite increased variability, the progression of predicted colony diameters over time closely mirrored the experimental measurements, with high statistical significance.

Table 3. Statistical and validation indices for growth of Mucor circinelloides in SMA under dynamic temperature regimen of 25 °C (48 h), 18 °C (48 h), 15 °C (120 h), and 10 °C (120 h)

R² - coefficient of determination; RMSE - root mean square error;

A_f - accuracy factor; B_f - bias factor; %D_f - % discrepancy.

Graphical evaluation in Figure 3, along with bias (B_f) and accuracy (A_f) factors, further confirms acceptable model performance in predicting surface growth under dynamic temperature conditions (Burgain et al., 2013). Specifically, the calculated B_f values of 0.977 and 0.926 at a_w levels of 0.992 and 0.985 indicated 2.3 % and 7.4 % underestimation of observed colony diameters, respectively, suggesting a "fail-dangerous" tendency.

In comparison, Baert et al. (2007) modelled the growth of Penicillium expansum on apple purée agar using the Ratkowsky model, and reported B_f values ranging between 0.91 and 1.14, which are of similar magnitude to those observed in our study. Moreover, the A_f values of 1.117 and 1.108 indicate deviations between predicted and observed diameters of up to 11.7 % and 10.8 % (primarily underestimations). These findings are consistent with those of Judet-Correia et al. (2010), who reported A_f values of 1.11 and 1.29 for growth rate predictions of Penicillium expansum and Botrytis cinerea, respectively.

Time required for a visible colony

Cardinal model parameters for lag phase and surface growth rate were applied to predict the time required for a visible colony to form, a valuable metric in dairy practice. Such predictions are especially relevant for fast-growing fungal species like Mucor circinelloides, a member of the lower fungi. In the context of microscopic fungi, a colony diameter of 3 mm (t₃) is generally considered visible. The growth data presented in this study confirm that M. circinelloides is characterised by a short t₃, reflecting its rapid colonisation behaviour. Compared to Geotrichum candidum, commonly known as "machinery mould" (Cai and Snyder, 2019), M. circinelloides demonstrates faster growth. These findings are consistent with Gougouli et al. (2011), who classified M. circinelloides among dairy spoilage fungi exhibiting the shortest t₃ values under high a_w conditions.

Figure 4. Predicted time to visible colony formation (t₃) of Mucor circinelloides strain 1L cultivated on SMA medium, plotted as a function of water activity (a_w) and temperature.

Based on experimental data for M. circinelloides strain 1L cultivated on SMA medium, the predicted t₃ values illustrated in Figure 4 demonstrate a clear dependency on water activity (a_w) and temperature. Notably, combinations of lowered a_w (<0.975) and reduced temperatures (<10 °C) resulted in a pronounced extension of t₃, with visible colony formation delayed beyond 30 days.