## Growth on a single nutrient source

The classical system to investigate the efficiency of nutrient utilization in pulsating environments where organisms have sufficient time to fully utilize all available nutrients in a single nutrient pulse are batch cultures. Here we follow growth of *E. coli* until depletion of the initial nutrient source and potential secreted byproducts when stationary phase is reached in M9 minimal medium with glucose, malate, or aspartate as sole carbon sources [16, 17]. These carbon sources were chosen as respiro-fermentative, strictly respiratory, and a degradable biomass component. The produced biomass (\({\Delta }B\)), that is the biomass reached at stationary phase minus the biomass at inoculation, was recorded as the optical density at 600nm, converted to cellular dry weight using a predetermined conversion factor [18], and plotted against the initial nutrient amount (Fig. 1; S1 Fig). The produced biomass shows a good linear fit to the initial amount of the sole carbon source (Fig. 1B) and as such, can be described by [16]:

$${\Delta }B={Y}_{X/D}{N}_{D}$$

(1)

where \({N}_{D}\) is the initial amount of nutrient \(D\) and \({Y}_{X/D}\) the overall biomass yield for organism \(X\) on nutrient \(D\) which describes the efficiency of full utilization of the available nutrient.

To predict the produced biomass, we used a black box formalism [10] that separates the growth reaction of chemotrophic organisms to a two-reaction process (Fig. 2A). The first is a catabolic reaction that releases Gibbs free energy by breakdown of nutrients. The second is the anabolic reaction that uses the released free energy for the synthesis of new biomass. The overall Gibbs energy dissipation \({\Delta }{G}_{X}\) of the growth process is given by ([10], S1A text):

$${\Delta }{G}_{X}=\frac{1}{{Y}_{X/D}}{\Delta }{G}_{cat}+{\Delta }{G}_{an}$$

(2)

where the subscripts *cat*, and *an* refer to the Gibbs energy of dissipation of the catabolic and anabolic reactions, respectively. Given that all secreted byproducts are utilized in the here investigated growth conditions, the free energy of the secreted byproducts can be set to 0, the overall biomass yield may be predicted as [10]:

$${Y}_{X/D}=\frac{{\Delta }{G}_{cat}}{{\Delta }{G}_{X}-{\Delta }{G}_{an}}$$

(3)

Combining equations (1) and (3) predicts a linear correlation of the produced biomass as function of initial nutrient amount with a slope that depends only on the type of nutrient through\({\Delta }{G}_{cat}\). This prediction fits well with all measured nutrients and is consistent with previous results [16, 17] (Fig. 1B, S1 Fig).

## Growth on multiple nutrient sources

Since organisms typically encounter multiple nutrients in natural environments, we next asked whether the availability of one nutrient affects the overall biomass yield of another. To enable a black box model to capture such effects, we added another reaction that depends on the type of second nutrient: A) degradable nutrients that first must be catabolized before they can be used, such as a sugar; B) non-degradable nutrients that can be used only as a biomass precursor, such as the non-degradable amino acid methionine in *E. coli*; and C) nutrients that can be both catabolized or used directly as a biomass precursor, such as the amino acid aspartate in *E. coli*. For the combination of two degradable nutrients the added reaction is catabolic (S1B text, Fig. 2B). In this case, the overall Gibbs energy dissipation gives:

$${\Delta }{G}_{X}=\frac{1}{{Y}_{X/{N}_{1}}}{\Delta }{G}_{cat}^{{N}_{1}}+\frac{1}{{Y}_{X/{N}_{2}}}{\Delta }{G}_{cat}^{{N}_{2}} +{\Delta }{G}_{an}$$

(4)

where \({\Delta }{G}_{cat}^{{N}_{i}}\) is the Gibbs energy of dissipation for the catabolic process of nutrient \(i\). When the second nutrient source is a non-degradable biomass precursor, we split the anabolic reaction into two – a reaction for biosynthesis of the biomass precursor and a reaction for the general anabolic process (S1C text, Fig. 2C). The overall Gibbs energy of dissipation in this case gives:

$${\Delta }{G}_{X}=\frac{1}{{Y}_{X/{N}_{1}}}{\Delta }{G}_{cat}+{\Delta }{G}_{an}^{bsyn}+{\Delta }{G}_{bsyn}(1-{M}_{utl})$$

(5)

where \({\Delta }{G}_{bsyn}\) is the Gibbs energy of dissipation for synthesis of the biomass precursor and \({\Delta }{G}_{an}^{bsyn}\) is the dissipation energy for the general anabolic process minus that of the biomass precursor. The function \({M}_{utl}\) describes the ratio of available biomass precursor to that required to generate the produced biomass during the growth process. It is dependent on biomass precursor availability such that when all the necessary biomass precursor is available in the environment, the function assumes the maximal value of 1 and the cost for this precursor biosynthesis is alleviated.

Combining equations (4) or (5) with Eq. (1) shows that regardless of the type of nutrient supplemented, the produced biomass is predicted to be a linear sum of the biomass gained from the available nutrients and the overall biomass yield of each nutrient is independent of the availability of others (S1B-C text, Fig. 2D):

$${\Delta }B={Y}_{X/N1 }{N}_{1}+{Y}_{X/N2}{N}_{2}$$

(6)

where \({N}_{i}\) is the amount of nutrient \(i\) in the growth medium and \({Y}_{X/Ni }\) is the overall biomass yield of nutrient \(i\).

To test the prediction that the overall biomass yield of a nutrient is independent of the availability of others, we compared the overall biomass yield of *E. coli* for different nutrients, henceforth referred to as the measured nutrient, in the presence or absence of a second nutrient, termed the base nutrient. To do so, the initial amount of the measured nutrient was varied for each batch culture experiment at constant initial amounts of the base nutrient between 0 and 1.2 g/l for glucose, acetate, or aspartate and 0 and 0.06 g/l for methionine. The produced biomass was plotted against the initial amount of the measured nutrient and the overall biomass yield was determined as the slope of a linear fit of that curve (Fig. 3A,B). In combination with glucose, succinate, or acetate as base nutrients, we determined the overall biomass yield of xylose and methionine as measured nutrients, as examples of degradable or non-degradable nutrients, respectively (Fig. 3). The initial amount of base nutrient determines the intercept with the Y-axis and was chosen such that the measured parameters remain within measurable range.

The overall biomass yield was highly dependent on the base nutrient. For xylose, the overall biomass yield was higher on succinate as base nutrient than on glucose or when used alone, and for methionine the overall yield was by far the highest on glucose (Fig. 3C,D). For most combinations, the influence of the second nutrient was monotonous across the tested concentrations, i.e., the overall biomass yield of the measured nutrient can be determined from the slope of a linear fit (Fig. 3A,B). An exception was the non-monotonous behavior of methionine as the measured nutrient in combination with glucose as a base nutrient (Fig. 3B). At low initial amounts of methionine (below 3 \({\mu }\text{g}\)), increasing initial amounts of methionine unexpectedly decreased the produced biomass. In the higher range of initial amounts (above 3 \({\mu }\text{g}\)), increasing methionine initial amounts increased the produced biomass linearly.

Thus, the overall biomass yield of a measured nutrient is dependent on the base nutrient, consequently black box theory cannot capture the produced biomass of multiple nutrient sources. To enable the model to describe such mutual effects, we expand it to include such effects phenomenologically. To do so, we coupled a function that is dependent on the combination of available nutrients to the Gibbs energy dissipation of each reaction in the growth processes. For simplicity, we assumed these functions are linear to the initial nutrient amount.

As such, the overall Gibbs energy dissipation of growth on two degradable nutrient sources is described as (S1D text):

$${\Delta }{G}_{X}=\frac{1}{{Y}_{X/{N}_{1}}}{\Delta }{G}_{cat}^{{N}_{1}}{f}_{cat1}\left({N}_{2}\right)+\frac{1}{{Y}_{X/{N}_{2}}}{\Delta }{G}_{cat}^{{N}_{2}}{f}_{cat2}\left({N}_{1}\right)+{\Delta }{G}_{an}{f}_{an}\left({N}_{1},{N}_{2}\right)$$

(7)

where \({f}_{cat}\left({N}_{i}\right)\), \({f}_{an}\left({N}_{i}\right)\) are linear functions to the initial amounts of nutrient source \(i\), with coefficients \({\text{m}}_{\text{c}\text{a}\text{t}}^{{\text{N}}_{\text{j}}} , {\text{m}}_{\text{a}\text{n}}^{{\text{N}}_{\text{j}}}\)respectively. These functions phenomenologically depict the mutual effect of the nutrient combination on the growth processes. Combining equations (1) and (7) predicts the produced biomass:

$${\Delta }B=\frac{{\Delta }{G}_{cat}^{{N}_{1}}{N}_{1}+{\Delta }{G}_{cat}^{{N}_{2}}{N}_{2}+{\Delta }{\text{m}}_{\text{C}\text{A}\text{T}}^{{\text{N}}_{1}{N}_{2}}{N}_{1}{N}_{2}}{{\Delta }{G}_{X}-{\Delta }{G}_{an}{f}_{an}\left({N}_{1}, {N}_{2}\right)}$$

(8)

where \({\Delta }{\text{m}}_{\text{C}\text{A}\text{T}}^{{\text{N}}_{1}{N}_{2}}={\Delta }{G}_{cat}^{{N}_{1}}{m}_{cat}^{{N}_{1}}+{\Delta }{G}_{cat}^{{N}_{2}}{m}_{cat}^{{N}_{2}}\) and \({f}_{an}\left({N}_{1}, {N}_{2}\right)=1+{m}_{an}^{{N}_{1}}{N}_{1}+{m}_{an}^{{N}_{2}}{N}_{2}\). Given a mutual effect between nutrients, the produced biomass is thus made of three terms, two describing the direct effect of catabolism of the two nutrient sources and a third term describing the mutual catabolic effect depending on availability of both substrates. Although the last term of the model is based on the phenomenological observation, the range of model solutions is limited. Exploring this solution space shows that, depending on the type of mutualism, qualitatively different relationships are predicted between available nutrients and biomass formation (Fig. 4A) – a positive mutual catabolic effect increases the overall biomass yield (Fig. 4A, orange curve) while a negative catabolic effect decreases it (Fig. 4A, purple curve). The expanded model can capture the experimentally observed mutual effect of increased overall biomass yield with a positive mutual catabolic effect (compare increased slope for different base nutrients in Fig. 3A to the orange curve in Fig. 4A).

For growth on a combination of a degradable nutrient and a non-degradable biomass precursor, the overall Gibbs energy dissipation is described as (S1E text):

$${\Delta }{G}_{X}=\frac{1}{{Y}_{X/D }}{\Delta }{G}_{cat}{f}_{cat}\left(M\right)+{\Delta }{G}_{an}{f}_{an}\left(N,M\right)+{\Delta }{G}_{bsyn}^{M} {f}_{bsyn}\left(N\right)(1-{M}_{utl})$$

(9)

where \({f}_{cat}\left({M}_{utl}\right)\), \({f}_{an}\left(N,{M}_{utl}\right)\), \({f}_{bsyn}\left({M}_{utl}\right)\) are linear functions with coefficients \({m}_{cat}\), \({m}_{an }^{N}\), \({m}_{an }^{M}\), \({m}_{sbyn}\) respectively. These functions depict the mutual effect between the nutrient sources on the Gibbs free energy of each growth reaction. Solving equations (1) and (9) for the produced biomass gives a quadratic equation:

$${\Delta }{B}^{2}-\frac{{\Delta }B}{{\Delta }{G}_{A}} \left(N{\Delta }{G}_{cat}+{M}^{{\prime }}\left({m}_{an }^{M}{\Delta }{G}_{an}-\left(1+{m}_{sbyn}N\right){\Delta }{G}_{bsyn }^{M}\right) \right)-N\frac{{\Delta }{G}_{cat}}{{\Delta }{G}_{A}}{m}_{cat}{M}^{{\prime }}=0$$

(10)

$$\text{w}\text{h}\text{e}\text{r}\text{e} {\Delta }{G}_{A}=\left({\Delta }{G}_{X}-{\Delta }{G}_{an}(1-N{m}_{an }^{N})-{\Delta }{G}_{bsyn}^{M}\left(1+{m}_{sbyn}N\right)\right) \text{a}\text{n}\text{d} {M}^{{\prime }}={M}_{utl}{\Delta }B.$$

Unlike the solution for growth on two degradable nutrient sources, solving Eq. (10) for the produced biomass shows that a mutual effect between a biomass precursor and a degradable nutrient can give rise to non-monotonous solutions. Figure 4B explores the solution space of possible mutual effects between a precursor and a degradable nutrient. The case of a negative catabolic effect (Fig. 4B, orange curve) fits qualitatively well with the experimental observation of the biomass precursor methionine on glucose as base nutrient (Fig. 3B, green data points).

The coefficients of the linear functions depicting the mutual effect between the nutrients are a key output of the model since they infer how each combination of nutrients effects the different growth reactions. Fitting these coefficients to the experimental results of methionine growing with glucose as a base nutrient gives a qualitative fit to a negative value for the catabolic parameter (coefficient \({m}_{cat})\), revealing that methionine decreases the catabolic efficiency of glucose. Furthermore, the overall biomass yield of methionine on glucose in the linear region is higher than that on succinate or acetate (Fig. 3D), suggesting a mutual effect on another metabolic process in one of these combinations, potentially the precursor biosynthesis processes (coefficient \({m}_{sbyn})\). For all combinations of two degradable nutrients, the overall biomass yield increased as compared to growth on sole nutrient sources (Fig. 3C), a result that fits a positive mutual effect on the catabolic process (coefficient \({m}_{cat})\).

An unexpected model prediction is noticeable in equations (9) and (10) where the initial amounts of the two available nutrients are coupled in at least one term. Hence, the model predicts that the overall biomass yield of a measured nutrient depends not only on the availability of a base nutrient, but also on the relative initial amounts of the nutrients. For a combination of two degradable nutrient sources with a positive catabolic effect, as observed experimentally for xylose on the two base nutrients (Fig. 3A,C), the overall biomass yield is predicted to increase with increasing initial amounts of the base nutrient (Fig. 4C). For the combination of a degradable nutrient and a biomass precursor, such as methionine on glucose, with a negative catabolic effect and positive effect on precursor biosynthesis, the model predicts a shift of the curves for the non-linear part as well as an increase in the slope of the linear part with increasing initial amounts of base nutrient (Fig. 4D).

To test these predictions, we determined the produced biomass on xylose and methionine as the measured nutrients on different initial amounts of succinate and glucose as the base nutrients, respectively (Fig. 5A,B). The overall biomass yield of xylose (i.e., slope of the curve) increased linearly with the initial amount of the base nutrient succinate (Fig. 5A, C). This observation fits well with the model prediction for a positive catabolic effect between two degradable nutrients (Fig. 4C). The curve of the produced biomass on methionine exhibits a more complex dependency on the initial amount of glucose as the base nutrient. Above \(5 {\mu }\text{g}\) methionine, the slope of all curves increased linearly with the amount of the base nutrient glucose, but below \(5 {\mu }\text{g}\)methionine there was no linear dependency and the amount of base nutrient varied the curve shape (Fig. 5B, D). This observation fits well with the theoretical prediction (Fig. 4D) that this nutrient combination not only has a positive effect on the precursor biosynthesis reaction (i.e., the linear dependency at higher methionine supplementation), but also a negative catabolic effect where at low methionine concentrations, in some cases, more methionine leads to lower biomass gain.

Which mechanism underlies the negative catabolic effect of methionine on glucose? The growth curves followed the classical diauxic shift with exponential growth on glucose and a second phase on previously secreted acetate (S2A Fig). For the example of 160 \({\mu }\text{g}\) glucose as the base nutrient (Fig. 5, pink curve), the first phase lasted 4-4.5 hours and growth on acetate resumed between 7–10 hours (S2A Fig). In both phases, the biomass gain (calculated as the biomass at the end minus the biomass at the beginning) increased linearly with methionine amounts greater than 2 \({\mu }\text{g}\) (S2B, C Fig). The biomass gain was much higher than the trendline in the absence of or at very low methionine concentrations. During exponential growth on glucose in the first phase, methionine decreased the gain in biomass but increased the growth rate (S2D Fig.). Given the diauxic shift from growth on glucose to previously secreted acetate (S2E Fig), the most plausible explanation for the higher biomass gain without or low methionine in the second phase is due to higher acetate secretion in the first phase. To test whether methionine supplementation indeed reduced acetate secretion, we varied acetate secretion rates by altering steady state growth through an inducible promoter for the glucose uptake gene *ptsG* that limits glucose uptake [19]. Comparing acetate secretion in the presence and absence of methionine shows that methionine indeed decreases acetate secretion (S2F Fig). Thus, the negative catabolic effect of methionine on glucose catabolism appears to be a combination of a lower biomass gain during the first growth phase, with a higher growth rate and less acetate secretion, and a lower biomass gain in the second phase because less acetate was secreted.

At the lowest amounts of methionine (0 and 1.43 \({\mu }\text{g}\)) we noted a shorter lag time for growth on acetate (S2A Fig, compare red and black curves to the other curves). Growth with 1.43 \({\mu }\text{g}\) methionine was somewhat special as it followed the biomass trendline in the first growth phase but could not sustain the higher growth rate throughout this growth phase (S2A Fig, red curve between 2-4h), presumably because methionine was used up, which explains why its biomass gain in the second phase was indistinguishable from the no methionine condition (S2C Fig). Consistently, 1.43 \({\mu }\text{g}\) methionine was below the amount necessary to produce the biomass reached at the end of the first growth phase (about 1.7 \({\mu }\text{g}\) of methionine is required to generate 0.8 \(\text{g}\text{D}\text{C}\text{W}\) of biomass [20]).

So far, we focused on degradable nutrients or nutrients that can only be used as biomass precursors. Some nutrients such as degradable amino acids, however, can be directly used both as biomass precursors or energy source. Given the complex curves observed for the combination of biomass precursor and degradable nutrient, we expected that a degradable amino acid in combination with a degradable nutrient would also produce non-monotonous curves. To investigate the effects of such nutrient combinations, we measured the produced biomass on the degradable amino acid aspartate on different initial amounts of glucose and acetate as base nutrients (Fig. 6A,B). The combination of aspartate and acetate led to a complex curve with two linear phases separated by a double shift in slope at intermediate concentrations (between \(100-150 {\mu }\text{g}\), Fig. 6A). The first phase at low initial amounts of aspartate resulted in a linear slope that increases with initial amount of acetate while the slope of the second phase shows only a low dependency on acetate initial amounts (Fig. 6C). Aspartate on glucose also shows a complex curve with two linear phases (Fig. 6B). In this nutrient combination, the slope of the first phase is independent of the initial amount of glucose yet the length of this phase increases with increasing initial glucose amounts (Fig. 6D). The slope of the second linear phase increases with increasing glucose initial amounts. The complex behavior observed in these experiments cannot be captured even with the mutual effect model presented here. We hypothesize that the ratio of how much aspartate is utilized as a biomass precursor to how much is catabolized affects the overall biomass yield. The multiple utilization possibilities add an additional degree of freedom to the system and as such, capturing the behavior of these nutrients in a model requires time-resolved intracellular flux information.