A K. phaffii Mut⁺ strain, GS115 (his4), was procured from J. M. Cregg, Keck Graduate Institute, Claremont, CA, United States and was genetically modified to express a recombinant β-galactosidase protein. Details of the strain construction have been presented elsewhere (Singh and Narang, 2020). The resulting strain was called Mut⁺ (pSAOH5-T1) and was used for this study. Stock cultures were stored in 25% glycerol at −80°C.

The minimal medium composition used for shake-flask as well as chemostat cultivations was chosen such as to ensure stoichiometric limitation of the carbon and energy sources, as described in Egli and Fiechter (1981). The defined medium was supplemented with either glycerol (∼3.1 g L⁻¹), a mixture of methanol (∼1.6 g L⁻¹) and glycerol (∼1.5 g L⁻¹) or a mixture of methanol (∼3.2 g L⁻¹) and sorbitol (∼1.5 g L⁻¹) as carbon sources. In addition, the medium contained 100 mM phosphate buffer (pH 5.5), 15.26 g NH₄Cl, 1.18 g MgSO₄⋅7H₂O, 110 mg CaCl₂⋅2H₂O, 45.61 mg FeCl₃, 28 mg MnSO₄⋅H₂O, 44 mg ZnSO₄⋅7H₂O, 8 mg CuSO₄⋅5H₂O, 8.57 mg CoCl₂⋅6H₂O, 6 mg Na₂MoO₄⋅2H₂O, 8 mg H₃BO₃, 1.2 mg KI, 370 mg EDTA disodium salt, 2.4 mg biotin per liter. All components of the defined medium were prepared and sterilised by either filtration or autoclaving as separate stock solutions and then mixed before cultivation.

When required, cells were revived in a 100 ml shake flask containing 10 ml minimal medium supplemented with a suitable carbon source at 30°C and 200 rpm. These primary cultures were sub-cultured once before inoculating the reactor precultures (in the same cultivation medium as prepared for the reactor vessel), which were then used as an inoculum for the bioreactor.

Chemostat cultivations were performed using bench-scale 0.5 L mini bioreactors modified to support chemostat operation and equipped with pH, DO, temperature, level and agitation controls (Applikon Biotechnology, Netherlands) at working volumes of 0.3 L. The cultivation temperature was always maintained at 30°C and pH at 5.5 by the automatic addition of 2 M NaOH. An integrated mass flow controller ensured a constant supply of air to the reactor vessel at 80 ml min⁻¹. Dissolved oxygen levels were monitored by a polarographic probe calibrated with respect to an air-saturated medium. Cultures were agitated to ensure fast mixing as well as aerobic conditions, such that the DO level always remained above 60%. A silicone based anti-foam agent was added to the reactor vessel as and when required to prevent foam formation and wall growth. For chemostat mode operation, the dilution rate was set by fixing the input feed flow rate, while a constant volume was maintained inside the reactor vessel by controlling the output feed flow rate via proportional control based on the on-line monitoring of the change in weight of the reactor vessel. For instance, for a dilution rate of 0.1 h⁻¹, the input feed flow rate was fixed at 30 ml h⁻¹ using a peristaltic pump. When the weight of the reactor vessel increased beyond the set point, the output feed pump was switched on to remove the excess volume. After inoculation, cells were grown in batch phase for some time to allow exhaustion of the initial carbon source (indicated by a rise in DO level), followed by initiating the input and output feed supplies. At any particular dilution rate, steady-state samples were withdrawn after 5-6 liquid residence times. In general, three samples were collected for each dilution rate, separated by an interval of one liquid residence time. For instance, at a dilution rate of 0.04 h⁻¹, the first sample was taken after 150 h (6 liquid residence time), the second after 175 h (7 liquid residence time) and the third after 200 h (8 liquid residence time). Attainment of steady-state was confirmed by analysing the samples for constant dry cell weight and specific enzyme activities.

For determination of residual substrate concentration inside the reactor, samples were withdrawn directly from the vessel. To achieve rapid biomass separation, culture samples were withdrawn using vacuum through a sampling tube attached to a 0.2-micron syringe filter and stored at −20°C until analysis. Samples for determination of biomass and enzyme activities were collected in a sampling bottle kept on ice. Biomass samples were processed immediately, while samples for measuring enzyme activities were pelleted, washed and stored at −20°C until processing.

Glycerol and sorbitol concentrations were estimated by high-performance liquid chromatography (HPLC) analysis (1100 series, Agilent Technologies, Palo Alto, United States) with detection limits of ∼1 mg/L and ∼30 mg/L. An ion-exclusion chromatography column from Phenomenex, California, United States (ROA-Organic acid H⁺ column, 300 × 7.8 mm, 8 µm particle size, 8% cross linkage) with a guard column (Carbo-H cartridges) was used with 5 mM H₂SO₄ in ultrapure water as mobile phase supplied at a constant flow rate of 0.5 ml min⁻¹. The column chamber was maintained at 60°C and a refractive index detector was used for substrate measurement. Methanol concentrations were determined with a gas chromatograph equipped with a flame ionisation detector (GC-FID) (7890A, Agilent Technologies, Palo Alto, United States) using a HP-PLOT/Q column (30 m × 0.32 mm, 20 µm) from Agilent Technologies and nitrogen as the carrier gas. The detection limit for methanol was ∼5 mg/L.

A known volume of the fermentation broth was collected and pelleted in a pre-weighed centrifuge tube. Pellets were washed twice with distilled water and then dried at 80°C to constant weight.

Culture samples were collected on ice and immediately centrifuged at 4°C to collect cells. The cell pellets were washed twice with phosphate buffer (100 mM, pH 7.4) and stored at −20°C until analysis. For cell lysis, pellets were resuspended in 100 µL of chilled breaking buffer (Jungo et al., 2006). Acid-washed glass beads (0.40–0.45 mm diameter) were added to the resulting slurry followed by alternate vortexing (1 min) and resting (on ice for 1 min) steps. This cycle was repeated 4–5 times, after which the cell debris was removed by centrifugation. Cell-free extracts (supernatant) were collected in fresh tubes kept on ice and immediately used for the estimation of enzyme activities. The Bradford assay was used for the estimation of the total protein content of the cell-free extracts for which bovine serum albumin served as standard (Bradford, 1976).

β-galactosidase assays were performed according to the method described by Miller (1972) with modifications. Briefly, cell-free extracts were appropriately diluted and mixed with Z-buffer containing β-mercaptoethanol (Miller, 1972) and incubated at 30°C in a water-bath for 15–20 min. The reaction was started by adding ONPG and stopped by adding Na₂CO₃ when sufficient colour had developed. The specific β-galactosidase activity was calculated with the formula 1000 × O D 420 / R e a c t i o n   t i m e   min ⁡ P r o t e i n   c o n c e n t r a t i o n   i n   e x t r a c t   m g m l × S a m p l e   v o l u m e   m l

and expressed in units mgp⁻¹ where mgp denotes mg of total protein.

Appropriate dilutions of the cell-free extracts were used to measure alcohol oxidase activities based on the method adapted from Jungo et al. (2006). A fresh 2x stock of the assay reaction mixture containing 0.8 mM 4-aminoantipyrine, 50 mM phenolsulfonic acid, freshly prepared 4 U/ml horseradish peroxidase in potassium phosphate buffer (200 mM, pH 7.4) was prepared before setting up the assays. 100 μl of the diluted cell-free extracts were mixed with 25 µl methanol and incubated at 30°C for 10 min. After this, 100 µl of the 2x reaction mixture stock was added to the mix at time t = 0 to start the reaction and the increase in absorbance at 500 nm was monitored every 30 s for 10 min using a microplate reader (SpectraMax M2e, Molecular Devices Corporation, CA, United States). The specific alcohol oxidase activity was calculated with the formula 100,000 × O D 500 / R e a c t i o n   t i m e   s P r o t e i n   c o n c e n t r a t i o n   i n   e x t r a c t   m g m l × S a m p l e   v o l u m e   m l

and reported in units mgp⁻¹.

We are concerned with experiments in which a chemostat is fed with the primary carbon source S 1 (methanol) and a secondary carbon source S 2 which may be repressing (glycerol) or non-repressing (sorbitol). The primary carbon source S 1 induces the synthesis of the enzyme E 1 which represents LacZ or AOX, since the latter is expressed almost entirely from an AOX1 promoter. We are interested in measuring the steady state concentrations of biomass X , primary carbon source S 1 , and secondary carbon source S 2 , as well as the specific activity of enzyme E 1 . These quantities are denoted x , s 1 , s 2 , and e 1 , respectively, and satisfy the mass balances: 0 = d x d t = − D x + μ x (1) 0 = d s 1 d t = D s f , 1 − s 1 − r s , 1 x (2) 0 = d s 2 d t = D s f , 2 − s 2 − r s , 2 x (3) 0 = d e 1 d t = r e , 1 − μ e 1 (4) where s f , 1 , s f , 2 denote the respective feed concentrations of S 1 , S 2 ; and μ , r s , 1 , r s , 2 , r e , 1 denote the respective specific rates of growth, consumption of substrate, and expression of a stable intracellular protein (Pfeffer et al., 2011; Singh and Narang, 2020). It follows from Eqs 1–4 that r s , i = D s f , i − s i x , i = 1,2 (5) r e , 1 = D e 1 (6)

These equations were used to calculate r s , 1 , r s , 2 , and r e , 1 from the measured values of the operating conditions D , s f , i and the steady state concentrations s i , x , and e 1 .

Our goal is to study the kinetics of substrate consumption and P _AOX1 expression during mixed-substrate growth on methanol + glycerol and methanol + sorbitol; however, we also characterized the substrate consumption kinetics during single-substrate growth on glycerol and sorbitol. In batch (shake-flask) cultures grown on glycerol and sorbitol, the biomass yields were quite similar (∼0.6 gdw g⁻¹), but the maximum specific growth rates μ m were dramatically different (Table 1). Due to the exceptionally small μ m of 0.03 h⁻¹ on sorbitol, we could not perform chemostat experiments with pure sorbitol, but we did perform such experiments with glycerol. We found that the biomass and residual glycerol concentrations followed the pattern characteristic of single-substrate growth in continuous cultures (Figure 1A). The specific glycerol consumption rate, calculated from these data using Eq. 5, increased linearly with D with a significant positive y-intercept (Figure 1B). Fitting these data to Pirt’s model (Pirt, 1965) gave a true biomass yield of 0.67 gdw g⁻¹, and specific maintenance rate of 0.07 g gdw⁻¹ h⁻¹. The specific LacZ and AOX activities, which were positively correlated in general, are inversely proportional to D , except for the two data points at the largest D (Figure 1C). This implies that the specific productivity is constant at all but the two largest D (Figure 1D), and the sharp decline at the two largest D may reflect the onset of regulation. Nevertheless, the specific LacZ and AOX productivities, calculated from the data in Figure 1C using Eq. 6, did not exceed ∼1000 and ∼300 units mgp⁻¹ h⁻¹, respectively (Figure 1D).

When the Mut⁺ strain is grown in batch cultures of methanol + glycerol and methanol + sorbitol, there is diauxic growth, but methanol is the unpreferred substrate during growth on methanol + glycerol, and the preferred substrate during growth on methanol + sorbitol (Ramón et al., 2007). Such mixtures, which display diauxic growth in batch cultures, exhibit a characteristic substrate concentration profile in continuous cultures (Egli et al., 1986; Noel and Narang, 2009) (Supplementary Figure S1A). In the dual-limited regime, which extends up to dilution rates approximately equal to the μ m for the unpreferred substrate, both substrates limit growth, because their residual concentrations s i are in the order of their saturation constants K s , i ( s i ∼ K s , i ), and therefore, both substrates are completely consumed ( s i ≪ s f , i ). Beyond the dual-limited regime, only the preferred substrate limits growth because the residual concentration of the unpreferred substrate is well above its saturation constant. At the intermediate D , corresponding to the transition regime, the preferred substrate is still consumed completely, but the unpreferred substrate is only partially consumed. Beyond the transition regime, the unpreferred substrate is not consumed at all.

When methanol + glycerol and methanol + sorbitol were fed to a continuous culture, the variation of the substrate concentrations with D was consistent with the characteristic pattern described above. In the dual-limited regime, both substrates were completely consumed — up to D = 0.08  h − 1 ≈ 0.11 h − 1 = μ m m e t h a n o l (Singh and Narang, 2020) in Figure 2A and D = 0.03 h − 1 = μ m s o r b i t o l in Figure 3A. In the transition regime, the unpreferred substrate was partially consumed up to dilution rates well above its μ m — up to D = 0.2 h − 1 ≈ 2 × μ m m e t h a n o l in Figure 2A, and up to D = 0.08 h − 1 ≈ 3 × μ m s o r b i t o l in Figure 3A.

During single-substrate growth, the specific substrate consumption rate usually increases linearly with D up to washout (Pirt, 1965), but during mixed-substrate growth, the specific substrate consumption rates increase linearly with D only in the dual-limited regime (Egli et al., 1986; Noel and Narang, 2009) (Supplementary Figure S1B). The dashed lines in Figures 2B, 3B show that during growth on methanol + glycerol and methanol + sorbitol, the specific methanol consumption rate is indeed proportional to D up to D = 0.08   h − 1 and D = 0.03   h − 1 , respectively. Beyond the respective dual-limited regimes, the specific methanol consumption rates change non-linearly (Supplementary Figure S1B). In the case of methanol + glycerol, the specific methanol consumption rate decreases non-linearly beyond D = 0.08 h⁻¹ due to repression of methanol consumption by glycerol (Figure 2B); in the case of methanol + sorbitol, the specific methanol consumption rate increases non-linearly beyond D = 0.03 h⁻¹ due to the enhanced methanol consumption that occurs to compensate for repression of sorbitol consumption by methanol (Figure 3B). Using Egli’s model for dual-limited growth (Egli et al., 1993), we chose feed concentrations such that when growth on both the mixtures is dual-limited ( D ≤ 0.03 h⁻¹), the specific methanol consumption rates of the two mixtures are not only proportional to D , but also equal in magnitude. The specific methanol consumption rates of the two mixtures start diverging beyond D = 0.03   h − 1 , but they remain approximately equal up to D = 0.05   h − 1 (compare Figures 2B, 3B).

Although it is widely accepted that glycerol is repressing and sorbitol is non-repressing in batch cultures, we found remarkably similar specific LacZ and AOX activities and productivities in continuous cultures fed with methanol + glycerol and methanol + sorbitol. At low dilution rates ( D ≤ 0.05 h⁻¹), when both mixtures support equal specific methanol consumption rates, the specific LacZ and AOX activities on both mixtures are also equal (Figures 2C, 3C), and hence, their specific LacZ and AOX productivities are also the same (Figures 2D, 3D). At high dilution rates ( D ≥ 0.05 h⁻¹), the specific methanol consumption rates of both mixtures change substantially, but the specific LacZ and AOX productivities are relatively insensitive to this change. Indeed, in the case of methanol + glycerol, the specific methanol consumption rate doubles when D increases from 0.05 h⁻¹ to 0.12 h⁻¹, and decreases 40% when D increases from 0.12 h⁻¹ to 0.20 h⁻¹. But the specific LacZ and AOX activities decrease inversely with D (Figure 2C), and hence, the specific LacZ and AOX productivities calculated from Eq. 6 are expected to be constant. These specific productivities, which are shown in Figure 2D, are constant but show considerable scatter at D ≥ 0.05 h^-1. This is expected since at large D , multiplication of e 1 by D amplifies the errors in the measurement of e 1 . In the case of methanol + sorbitol, the specific methanol consumption rate doubles when D increases from 0.05 h⁻¹ to 0.08 h⁻¹, but the specific LacZ and AOX productivities increase only 25% (Figure 3D). Furthermore, the constant maximum specific LacZ and AOX productivities of 4000–6000 units mgp⁻¹ h⁻¹ and 1200–2000 units mgp⁻¹ h⁻¹, respectively, are close to the corresponding maximum values observed during growth on methanol + glycerol. Taken together, these data suggest that the specific P _AOX1 expression rate is a function of (i.e., completely determined by) the specific methanol consumption rate.

To test this hypothesis, we plotted the specific LacZ and AOX productivities r e , 1 at various D in Figures 2D, 3D against the corresponding specific methanol consumption rate r s , 1 in Figures 2B, 3B. This yielded the graph in Figure 4, which shows that at every specific methanol consumption rate, both mixed-substrate cultures have approximately the same specific P _AOX1 expression rate (measured as either specific LacZ productivity or specific AOX productivity). The specific P _AOX1 expression rate is therefore completely determined by the specific methanol consumption rate regardless of the type (repressing or non-repressing) of the secondary carbon source. More precisely, the specific P _AOX1 expression rate, r e , 1 is proportional to the specific methanol consumption rate, r s , 1 up to the threshold value ∼0.15 g gdw⁻¹ h⁻¹ and remains approximately constant thereafter at the maximum value of ∼5 units gdw⁻¹ h⁻¹. Hence, the specific P _AOX1 expression rates of the mixtures can be approximated by the piecewise linear function r e , 1 = V e , 1 r s , 1 r s , 1 * , r s , 1 ≤ r s , 1 * V e , 1 , r s , 1 > r s , 1 * (7) where V e , 1 denotes the maximum specific P _AOX1 expression rate, and r s , 1 *  denotes the threshold specific methanol consumption rate beyond which the specific P _AOX1 expression rate has its maximum value V e , 1 .

Jungo et al. (Jungo et al., 2007a; Jungo et al., 2007b) performed their mixed-substrate studies by fixing D , s f , 1 + s f , 2 and increasing the fraction of methanol in the feed σ 1 = s f , 1 / s f , 1 + s f , 2 at a slow linear rate aimed at maintaining quasi-steady state. They found that as, σ 1 increased:

a) The residual methanol remained negligibly small, and the biomass concentration decreased linearly.

It follows from a) that the specific methanol consumption rate, which is approximately equal to D s f , 1 + s f , 2 σ 1 / x , increased throughout their experiment. But then b) implies that, as the specific methanol consumption rate increased, the specific avidin expression rate of both mixed-substrate cultures reached essentially the same maximum (cf. Figure 4).

Berrios and co-workers compared the methanol consumption and ROL production rates of the Mut⁺ strain at two different temperatures (22°C and 30°C) during growth on methanol, methanol + glycerol, and methanol + sorbitol (Berrios et al., 2017). These experiments were done in chemostats operated at D = 0.03 h⁻¹, and in the case of mixed-substrate experiments, fed with two feed compositions (40 and 70 C-mole % methanol). They found that “Sorbitol-based cultures led to a higher q p than both glycerol-based and control cultures at most studied conditions.” But closer inspection shows that in all their experiments, the specific expression rates were 0.8–0.9 units gdw⁻¹ h⁻¹, which is close to the maximum specific expression rate of 1-1.1 unit gdw⁻¹ h⁻¹.

Analogous results have also been obtained in studies of lac expression in E. coli. Indeed, batch experiments with mixtures of lactose + glycerol, lactose + glucose, and lactose + glucose-6-phophate show that glycerol is non-repressing, whereas glucose and glucose-6-phosphate are repressing (Magasanik, 1970). However, when chemostat experiments were performed with these three mixtures (Smith and Atkinson, 1980), they yielded the same steady state specific β-galactosidase (LacZ) activity at all D ≲ 0.5 h⁻¹ (Supplementary Figure S2). Furthermore, when the steady state specific LacZ activities at various D were plotted against the corresponding specific lactose consumption rates at the same D , the data for all three mixtures collapsed into a single line (Supplementary Figure S3). This led the authors to conclude that the steady state specific LacZ activity was “an apparently linear function of the rate of lactose utilization independent of the rate of metabolism of substrates other than lactose which are being concurrently utilized.” But then it follows from Eq. 6 that the steady state specific LacZ productivity is also completely determined by the specific lactose consumption rate regardless of the type (repressing or non-repressing) of the secondary carbon source (Supplementary Figure S4).

In conclusion, the specific P _AOX1 expression rate of K. phaffii appears to be completely determined by the specific methanol consumption rate regardless of the type (repressing or non-repressing) of the secondary carbon source. Analysis of the literature shows that the specific expression rate of the lac operon of E. coli is also completely determined by the specific lactose consumption rate regardless of the type of secondary carbon source. It would be interesting to explore if similar results are obtained for other microorganisms and substrate mixtures.