μ MAX of Saccharomyces Cerevisiae: So Often Used, So Seldom Put into Perspective

: The maximum specific growth rate of a microbe in a given growth condition is of primary relevance for biological research and bioprocess development. In the case of the unicellular yeast Saccharomyces cerevisiae , this physiological parameter is routinely calculated in (almost) every laboratory, but this procedure conceals several challenges that are often neglected in scientific works, which might lead to misinterpretation of the reported data and of phenomena. We present here several pitfalls involved in µ MAX calculation and interpretation, which was achieved through comparative analyses of: 1) the use of different methodologies for determining cell concentration, 2) different calibration procedures to correlate indirect (e.g. absorbance) to direct (e.g. dry cell mass) cell concentration measurements, 3) different statistical methods for determining the significance of µ MAX differences, 4) the influence of culture media composition, and 5) the influence of the cultivation system (e.g. microplate, shake-flask or bioreactor). It becomes clear that each of these factors has a great influence on µ MAX calculation and interpretation. We also present a case study involving three yeast strains and three different carbon sources, illustrating that opposite conclusions can be drawn in a screening procedure, if proper caution is not taken during data generation and analysis. Last but not least, we conclude this work with a series of recommendations that we believe could make experimental planning, data generation, µ MAX calculation and interpretation more meaningful and scientifically sound, contributing to the improvement of yeast research and of microbiology in general.

correlate indirect (e.g. absorbance) to direct (e.g. dry cell mass) cell concentration measurements, 3) different statistical methods for determining the significance of µMAX differences, 4) the influence of culture media composition, and 5) the influence of the cultivation system (e.g. microplate, shake-flask or bioreactor). It becomes clear that each of these factors has a great influence on µMAX calculation and interpretation. We also present a case study involving three yeast strains and three different carbon sources, illustrating that opposite conclusions can be drawn in a screening procedure, if proper caution is not taken during data generation and analysis. Last but not least, we conclude this work with a series of recommendations that we believe could make experimental planning, data generation, µMAX

Conflicts of interest
The authors have no conflicts of interest to declare that are relevant to the content of this article.
Introduction 1 Growth of a microbial population is not the increase in size of the individual cells, but 2 rather the increase in total cell number, total cell mass or even total cell volume in the 3 population (Wheals and Lord 1992). Determining the rate at which a microbial population 4 grows is one of the main interests of the fundamental microbiologist, as well as presumably 5 the most important piece of information in an industrial bioprocess. This aspect is captured in 6 a parameter referred to as the specific growth rate, most commonly represented by the Greek 7 letter μ (Doran 2012; Clarke 2013; Liu 2016; Stanbury et al. 2017). Cell growth is an 8 autocatalytic reaction, meaning that the catalyst itself is a product of the reaction (Doran 9 2012). Hence, the cell (or biomass) specific growth rate, rather than a simple growth rate, is 10 the most appropriate parameter to describe microbial growth. Mathematically, 11 = 1 * 1 where X = cell concentration (e.g. in cells/volume or dry cell mass/volume) and t is the 12 reaction time (e.g. in hours). 13 From equation 1, it can be observed that μ is similar to the kinetic constant of a 1 st -14 order chemical reaction and has dimensions of time -1 . Other formulations for rates, such as 15 total or volumetric rates, are scale-dependent and do not directly reflect catalyst performance. 16 The exponential growth phase (EGP) occurs very often both in research and in 17 applied cases, and is typically the longest phase of a conventional batch cultivation. During 18 the EGP, cells encounter neither any nutrient limitation nor any inhibition. The population 19 then grows at the maximum possible rate (the maximum specific growth rate, μMAX) under the 20 applied conditions, until one nutrient becomes growth-limiting or some compound achieves 21 inhibitory concentrations. The term "balanced growth" is often used to describe the 22 physiology of cells during the EGP, since the cell composition typically does not change, 23 although the composition of the nutrient medium is constantly changing (Campbell 1957).
Instead of μ, some professionals prefer to use the doubling (or generation) time (tG) to 25 quantify the rate of microbial growth. tG is the time required for the microbial population to 26 double its size (e.g. in terms of cell number or dry cell mass). The two parameters are 27 intrinsically related by the following equation: 28 = ln 2 2 29 We will here only use μ for all our analyses and discussions. 30 μ cannot be directly measured. Nevertheless, measurements of cell concentration at a 31 minimum of two time points allow for the estimation/calculation of this parameter. 32 There are several methods to determine cell concentration, including direct cell count, 33 dry cell mass, particle count and colony forming units, among other direct off-line methods 34 (Sonnleitner et al. 1992). Moreover, cell concentration is usually assessed by light-scattering 35 measurements, such as those performed with the use of a spectrophotometer, a ubiquitous 36 laboratory piece of equipment. Other terms used to designate this type of measurement are 37 optical density (OD), turbidity, and absorbance. However, the results of such an indirect 38 analysis need to be calibrated against a direct method, and this requires some caution. 39 Calibration should be performed under a particular condition and applied to this circumstance 40 only. Otherwise, the correlation could be compromised. Even analyses performed with cells 41 from a single cultivation but collected at different growth phases represent a source of error 42 due to inadequate calibration. The possibly different cell morphologies in each growth phase 43 affect deviation of light and compromise the translation of the indirectly assessed cell 44 concentrations into real cell concentrations. 45 Further options for indirect determination of cell concentration rely on the 46 measurement of a cell component, for instance protein or DNA. In this case, calibration is 47 also necessary and, as discussed above, care should be taken in the sense that cell composition 48 during growth might differ from the one employed during the calibration procedure. 49 Furthermore, although online methods centered on turbidity, permittivity (Harris et al. 1987), 50 or fluorescence can as well be used to assess cell concentration, as yet they have not 51 substituted the above mentioned off-line methods, which require sampling. 52 Another relevant aspect for the analysis of microbial growth is the cell cycle, which 53 for yeast comprises the phases G1, S, G2, and M (Juanes 2017). Individual cells in a 54 population are in different phases of the cell cycle, meaning they are not synchronised 55 (Cooper 2019). Thus, for a sample withdrawn from a cultivation, be it a microtiter plate or a 56 million-liter-scale bioreactor, the measured cell concentration involves billions of cells at 57 different stages of the cell cycle. 58 Fermentation Technology and/or Bioprocess Engineering textbooks usually do not 59 provide a discussion on how cell concentration measurements affect the calculation of the 60 specific growth rate. In one case, it is even stated that "During balanced growth, the net 61 specific growth rate determined from either cell number or cell mass would be the same" 62 without any connection to cell concentration determination methods. In one exception, Clarke 64 (Clarke 2013) points out that "μMAX can vary significantly depending on the method used to 65 measure the cell concentration". This author also mentions that, in the case of the budding 66 yeast S. cerevisiae, there might be a difference in μMAX calculated from cell mass and cell 67 number. This is because, in the beginning of the EGP, yeast cells tend to present many buds 68 (small cells) of lower mass than fully grown cells, which are counted by direct cell counting 69 methods. The opposite is observed towards the end of the EGP, when the budding rate 70

decreases. 71
There are basically two different approaches to calculate μMAX from cell concentration 72 measurements. One of them is based on a first adjustment of a growth model to data from an 73 entire batch cultivation, including all growth phases (lag, log, de-acceleration and stationary). 74 Frequently used models include the logistic model, the Gompertz and the Richards models, 75 among others (Pylvänäinen 2005). The second method consists of the integration of equation 1 under the assumption that in the EGP μ is constant and equal to μMAX. While early 77 researchers used a log2 or log10 transformation to linearize this equation (Clarke 2013), 78 nowadays, the use of the natural logarithm is common practice: 79 where X0 = cell concentration at the beginning of the EGP, corresponding to t = 0. 80 This transformation allows us to calculate μMAX by plotting ln(X) values along time 81 and taking the slope of the linear region as μMAX. This procedure also results in the 82 identification of the duration of the EGP. Due to the use of the natural logarithm, μMAX 83 represents the number of "e-fold" generations in a given time point t, or the exponential 84 increase of biomass by a factor of e (Manhart and Shakhnovich 2018). We will restrict our 85 analysis and discussion here to this approach, because it is by far the most frequently 86 For the sake of completeness, it should be mentioned that there are methods to 94 calculate μMAX using continuous cultivation data (Jannasch and Egli 1993) and methods that 95 take substrate and product concentrations into account (Oner et al. 1986). We will not discuss 96 them here. 97 Finally, it is important to mention that not only the analytical method used to 98 determine cell concentration influences μMAX calculations, but also other factors such as the cultivation system. Potvin et al (Potvin et al. 1997)

compared μMAX values obtained for 100
Lactobacillus plantarum cells grown in an automated plate reader, in shake-flasks and in a 101 bioreactor, otherwise under similar conditions. Bioreactor cultivations led to higher μMAX 102 values as compared to shake-flask cultivations, which the authors attributed to external pH 103 control in bioreactors. These authors also showed that the μMAX calculated from direct 104 absorbance measurements in an automated plate reader, without sample dilution, differed 105 from the values obtained with samples from shake-flask cultivations that were diluted prior to 106 the absorbance measurements. Although these observations seem obvious, this matter has 107 only been given proper attention in few published works. 108 In the only report we identified involving yeast, Stevenson  Saccharomyces cerevisiae cultures with respect to particle size and shape, refractive index, 111 cultivation volume, spectrophotometer model, cell growth phase, among others. The authors 112 concluded that the cell size effect on the calibration between OD and cell counts was stronger 113 in bacteria than in yeast. This is because the size of the bacterial cells is closer to the 114 wavelength of light (600 nm) used in the OD measurements. In this sense, the bigger size of 115 yeast cells makes them more suitable than bacteria for the application of light scattering 116 techniques at 600 nm or similar wavelengths. Moreover, they demonstrated that the difference 117 between the refractive index of the medium and that of the cells influences the calibration 118 curve. This has implications for yeast research, since sugars commonly used in yeast media, 119 such as sucrose, change the refractive index of the medium significantly. 120 This context motivated us to investigate how different cell concentration 121 determination methods, statistical analyses, cultivation systems, and also culture media 122 influence μMAX calculations during yeast cultivations performed with different strains, 123 including wild isolates, laboratory and industrial ones. 124

Cultivation media 133
Yeast cultivations were carried out using either a defined medium (Verduyn et al. 134 1992), the composition of which altered depending on the cultivation system ( Table 2), or a 135 complex medium (YPD). Microplate cultivations were performed using both media, whilst 136 shake-flask and bioreactor cultivations were restricted to the defined medium. When needed, 137 urea was used as the sole nitrogen source in replacement for ammonium sulphate, to avoid 138 drastic changes in the broth's pH caused by proton release during ammonium consumption. suspension with absorbance at 600 nm equal to 1, was then collected. The aliquot was 155 centrifuged at 974 g for 5 min (MIKRO200 centrifuge, Hettich, Tuttlingen, Germany), the 156 supernatant discarded and the pellet washed with 1 ml of fresh culture medium. This washing 157 procedure was performed twice. Next, 10 µl of the cell suspension was transferred to one well 158 of a microplate that had already been filled with 90 µl of the same culture medium used for 159 inoculum growth. Once all the desired wells were filled with both medium and cell 160 suspension, the microplate was sealed with PCR sealing film (AMPLlSeal TM -Greiner bio-161 one, Kremsmunster, Austria). The cultivation was carried out in quintuplicate (5 wells on the 162 same plate) at 30 °C with an orbital agitation amplitude of 3.5 mm and frequency of 198.4 163 rpm. Absorbance at 600 nm wavelength and 9 nm bandwidth was measured every 15 min 164 during 24 h. 165 Table 1 166 Table 2 167 Using Microsoft Excel, data from independent replicates were analyzed separately, 212 each one yielding a µMAX value of its own fitted by the least-squares regression method. The 213 average and the standard deviation of these µMAX values were then calculated (Fig. 1, Method  214 A). Significant changes in µMAX were evaluated using t-tests with 95% and 99% confidence 215 levels. On the other hand, using GraphPad Prism, data from independent replicates of each 216 experiment were analyzed together, generating one single µMAX value from one regression 217 line also fitted by the least-squares method. This procedure also generated the standard error 218 of the slope (Fig. 1, Method B). Significant changes in µMAX were evaluated using F-tests 219 with 95% and 99% confidence levels. 220 from Abs and true cell concentration measurements will be the same. 239

Shake-Flask cultivations
In our experience at least, b is usually different from zero. We demonstrate this here 240 with μMAX calculations from data obtained during bioreactor cultivations of three different 241 yeast strains on glucose, namely CEN.PK113-7D, UFMG-CM-Y259, JP1 (Table 3). Samples 242 taken throughout the cultivation had their absorbances measured after proper dilution and 243 their cell concentration determined by a direct method (dry cell mass). μMAX was calculated 244 using four different approaches: 1) directly from Abs data; 2) directly from dry cell mass data; 245 3) from calculated dry cell mass values obtained using a calibration equation established 246 between the Abs and the dry cell mass data, including all data points in the cultivation; 4) 247 from calculated dry cell mass values obtained using a calibration equation established 248 between the Abs and the dry cell mass data, including only data points in the EGP (as 249 identified from the dry cell mass data used for calibration). 250 Table 3  251 Remarkably, μMAX values calculated based on approach 1 were in the range of 25 to 252 50% higher than those calculated from dry cell mass data (approach 2). Because the latter 253 approach is based on a direct assessment of cell concentration, widely considered as an critical, since it is the basis for statistical comparisons. One approach to determine the 277 absolute error that affects µMAX was proposed by Borzani (Borzani 1980(Borzani , 1994 independent. This is due to the assumption that "Whether one point is above or below the line 287 is a matter of chance, and does not influence whether another point is above or below the 288 line" (Motulsky 2020). Hence, we also proceeded this way in this work. 289 Using Abs values from exponential growth of strain CAT-1 in microplates, two 290 methods for statistical comparison of µMAX on defined and complex media were evaluated 291 ( Another analysis we carried out was the removal of outliers, since this is a common 295 procedure adopted by scientists in research. After visual inspection, some data appeared much 296 more distant to the regression lines than the others, with no apparent reason. The removal of 297 outliers based on an informal, visual approach is not recommended; thus the ROUT (Robust 298 regression followed by Outlier identification) method was used. This is an automatic routine, 299 based only on the distance of the point from the robust best-fit curve (Motulsky and Brown 300 2006). We evaluated all data points again in GraphPad Prism software using the ROUT 301 method, set up to eliminate outliers with a coefficient Q = 1% (Motulsky and Brown 2006).
We then calculated µMAX with the remaining data points by Method B (Table 4, 303 Supplementary Material). As expected, different µMAX values were calculated and their 304 standard errors were lower than the ones obtained by Method B without removal of outliers. 305 Table 4  306 Next, we performed statistical comparisons of the data from Table 4 to check if the 307 methods would yield the same results. Method A required a t-test to compare the averages 308 from different treatments (in this case, the two cultivation media) and define whether their 309 difference was statistically significant or not. A two-tailed, pooled t-test was chosen because 310 we assumed that both populations were independent and normally distributed, their variances 311 were unknown but equal, and the sample sizes were small (n = 5 for each data set) 312 Supplementary Material). For both methods, the null hypothesis was H0: µMAX,1 = µMAX,2, 317 and the alternative hypothesis was H1: µMAX,1 ≠ µMAX,2. If the calculated p-value was less 318 than the significance level α (0.05 or 0.01), we would reject the null hypothesis and the µmax 319 from the two cultivation media could be considered different at the significance level used 320 (Table 5). At α = 0.05 both methods resulted in a significant difference between defined and complex 328 media, whereas that was not the case at α = 0.01. Other strains were also tested, but the same conclusions were achieved from both methods and significance levels (Supplementary 330

Material). After the removal of outliers, Method B resulted in completely different 331
conclusions at both α for strain CAT-1, when compared to the same method using all data 332 points. 333 Even though Method A is widely used due to its simplicity and straightforwardness, it 334 may not be the best way to calculate the error associated to µMAX values. Each replicate µMAX, 335 once calculated independently, already has its own error associated to the fitness of the 336 regression line itself. But these errors are not taken into account by Method A as they are 337 simply not calculated, differently from Method B. Additionally, we showed that the removal 338 of outliers was decisive for the results. One can easily see that the comparison between µ MAX 339 values calculated using distinct methods is extremely discouraged. First, because results from 340 statistical comparisons are always to be taken with caution. Second, poorly described statistics 341 in microbial physiology papers makes it difficult to understand how data were obtained and 342 even more difficult to know whether interlaboratory comparisons can be performed. 343

Influence of the type of medium on μMAX calculations 344
Researchers often report μMAX values of a yeast strain on a given carbon and energy 345 source, such as glucose. However, whether this carbon source is provided in a synthetic 346 defined medium or in a complex undefined medium will influence the growth rate of a 347 microbial population. In principle, μMAX values should be higher in the latter environment, 348 because cells benefit from compounds that can be taken up directly from the medium, instead 349 of having to synthesize them from metabolic intermediates at the expense of energy. To verify 350 to which extent μMAX values vary between these two types of media, we evaluated this 351 physiological parameter for eight different S. cerevisiae strains cultivated in microplates (Fig.  352   2). 353

Fig. 2 354
Overall, the μMAX values were higher for a given strain in YPD medium than in 355 defined Verduyn medium, as expected. Nevertheless, the level to which this occurs varies 356 among strains (Table 6), and, for a few cases, the difference between the pair of μMAX values 357 was not significant at 95% or higher confidence level. The complex/defined μMAX ratio ranged 358 from 1.12 to 2.33, which is quite remarkable, considering that all strains belong to the same 359 species and that both media employed here are commonly used in experimental research. We 360 were not able to identify any trend in these data, e.g. whether the haploid CEN.PK113-7D 361 strain would present a different behavior than the diploid ones, or whether industrial strains 362 (CAT-1, JP-1, PE-2) would behave differently than the laboratory, the baker's or the wild 363 isolates. This indicates that these results are probably related to cell morphology, which 364 strongly influence Abs measurements (Stevenson et al. 2016 Table 6 377

Influence of the cultivation system on μMAX calculations 378
We assessed how the cultivation system affects the calculation of μMAX by comparing 379 the calculated values obtained from microplate, shake flask, and bioreactor cultivations of values of distinct samples from the EGP as described in the Material and Methods section 382 ( Fig. 1, Method B). For any particular strain, the three systems led to different μMAX values, 383 with the lowest values always being achieved using microplate cultivations. This is consistent 384 with our expectations, and has been observed before with bacteria (Potvin et al. 1997). Cells 385 are exposed to different growth conditions in the three systems, leading to varying oxygen 386 availabilities and pH values. This per se should lead to different physiologies. 387 However, the measuring peculiarities of each system also contribute to the observed 388 differences in μMAX. While in microplates the absorbance is usually measured without prior 389 dilution of the cell broth, in the other two setups, dilution is performed to assure the measured The results obtained in microplates do not necessarily corroborate those obtained in 421 shake-flask cultivations (Fig. 4). For instance, the UFMG-CM-Y259 strain displayed faster 422 growth on sucrose in the microscale system, compared to its growth on either of the hexoses. 423 In shake-flask cultivations, however it grew with a smaller µmax on sucrose, again compared to 424 growth on glucose or fructose. The CEN.PK113-7D strain also displayed a higher μMAX on 425 sucrose in microplate cultivations, but no significant difference was observed in the μMAX 426 values on the three substrates during shake-flask cultivations. 427 When considering growth on fructose, in comparison to glucose only, the UFMG-428 CM-Y257 strain showed higher μMAX on glucose for cultivations using microplates, whereas 429 equivalent growth rates on both substrates were observed during shake-flask cultivations. The 430 opposite was observed for the JP1 strain. Resolving the mechanisms underlying such different 431 behaviors is beyond the scope of this work. Here, the importance relies on the fact that one 432 could easily miss the cultivation system-dependency of μMAX in S. cerevisiae, if a careful study, more than one cultivation system is seldom employed. In spite of this, comparisons 435 with literature data are often reported, without properly highlighting the differences in the 436 experimental setup between the evaluated studies, which frequently leads to misinterpretation.  Table  474 S1, for S. cerevisiae strains CEN.PK113-7D, JP1, and UFMG-CM-Y259 cultivated in aerobic 475 batch bioreactors with glucose as sole carbon and energy source. 476 on either defined or complex medium supplemented with glucose as sole carbon and energy 484 source, using microplate as cultivation system. Experiments were carried out in five replicates, and for each replicate one µMAX was calculated from Abs600 data within the 486 exponential growth phase (EGP). 487 Table S4 Comparative statistical analysis, based on method A a , of the maximum specific 488 growth rates showed in Table S3. 489 Table S5 Comparative statistical analysis, based on method B 1 , of the maximum specific 490 growth rates (µMAX) of different S. cerevisiae strains grown on either defined or complex 491 medium supplemented with glucose as sole carbon and energy source, using microplate as 492 cultivation system. Experiments were performed in five replicates. One single µMAX was 493 calculated from Abs600 data from all replicates. 494

Table S8
Raw Abs600 data from the exponential phase of growth of S. cerevisiae CAT-1 495 cultivated on either defined or complex medium supplemented with glucose as sole carbon 496 and energy source, using microplate as cultivation system. 497 Table S7 Calculations for testing for significant differences among slopes for the S. 498 cerevisiae strain CAT-1. 499 Table S8 Summary of the statistical outcome of the F-test for the S. cerevisiae strain CAT-1. 500 Table S9 Raw Abs600 data from the exponential phase of growth of S. cerevisiae CAT-1 501 cultivated on either defined or complex medium supplemented with glucose as sole carbon 502 and energy source, using microplate as cultivation system. The crossed out data represent the 503 outliers identified using ROUT option on GraphPad Prism software with Q = 1%. 504 Table S10 Calculations for testing for significant differences among slopes for the S. 505 cerevisiae strain CAT-1 after removal of outliers. 506 Table S11 Summary of the statistical outcome of the F-test for the S. cerevisiae strain CAT-1 507 after removal of outliers. 508 Table S12 Comparative statistical analysis, based on method B 1 , of the maximum specific 509 growth rates displayed by S. cerevisiae CEN.PK113-7D, JP1,and UFMG-CM-Y259 during 510 growth on synthetic medium supplemented with glucose as sole carbon and energy source, 511 using either microplate, shake-flask, or bioreactor as cultivation system.       Methods used for calculating and comparing the slope of regression lines (µmax). Method A yields an average µmax and a standard deviation while Method B yields a unique µmax and a standard error.

Figure 2
Maximum speci c growth rates (µMAX) of strains grown in microplates in two cultivation media, calculated using two different regression methods (A and B). Depending on the statistical method and the signi cance level used, distinct conclusions can be drawn. * represent the p-value at which a signi cant difference between the treatments were observed; ns (p > 0.05); * (p ≤ 0.05); ** (p ≤ 0.01); *** (p ≤ 0.001); **** (p ≤ 0.0001) Figure 3 Maximum speci c growth rates (µmax) for three S. cerevisiae strains grown in a de ned medium in three different cultivation systems. Data from different systems were used to calculate and statistically compare μMAX values using Method B and GraphPad Prism software. This yielded a p-value ≤ 0.0001 (****) for all strains Figure 4 Maximum speci c growth rates (μMAX) of S. cerevisiae strains grown in microplates or in shake-asks in a de ned medium supplemented with sucrose, glucose or fructose as sole carbon and energy source, calculated by Method B

Supplementary Files
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