2.1 The core model

Metabolic shifts are studied using both large genome-scale metabolic models [16,29,35] and small phenomenological models [6,20,36–38]. Although large models contain more biochemical detail, both approaches yield (similar) explanations for the shifts [17]. The reason is that all these models consist of linear relations. The mathematical properties underlying metabolic shifts are thus preserved when condensing a large model to a small one. As long as the same number of (active) constraints and degrees of freedom are taken into account [40], the structure of the solution space of the model is not affected. The reason that a metabolic shift is observed is therefore retained, but may be studied with greater clarity in a smaller model.

The main ingredients for any of these models are (i) the stoichiometric matrix of the metabolic network; (ii) the assumption of steady-state behaviour of metabolism in balanced growth; (iii) the assumption that growth rate is maximised in any given condition; (iv) one or more limitations on cellular processes or resources. We now introduce a small core model featuring one pathway that makes efficient use of the carbon source, and one that is carbon-inefficient. Cells may use these pathways exclusively, or use both at the same time. A metabolic shift occurs if the cell changes between these different regimes.

The core model, which is inspired by the model in [20], is depicted in Fig 1. It consists of four metabolic processes. Assimilation (A) converts the carbon and energy source S into the intermediate X. For glycolytic carbon sources, like glucose, the assimilation pathway consists of uptake and upper glycolysis. So, in these cases, the intermediate X represents DHAP and GA3P. The pathway R (e.g., respiration) converts X into the catabolic product P_R and produces ATP per X. The pathway F (e.g., fermentation) converts X into the catabolic product P_F and produces ATP per X. Finally, biomass is formed from X by consuming ATP and X per unit biomass. Since protein synthesis is the major ATP consuming process during growth, we model biomass formation as protein synthesis. Accordingly, the sum of the cellular protein is denoted by . The processes A, R, F and B represent lumped metabolic pathways.

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Fig 1. The coarse-grained metabolic network described by the core model.

A carbon source S is assimilated (A) and converted into the intermediate X. Two catabolic modes (R, F) generate energy carriers (ATP) while converting X into catabolic waste products (P_R, P_F). Biomass synthesis (B) consumes the remaining carbon and generated ATP.

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The core model depicted in Fig 1 is associated with the following differential equations for the concentrations of X and ATP,

(1)

We denote the stoichiometric matrix by

(2)

and the flux vector by

(3)

At balanced growth, metabolism is in steady state [41], i.e., . The steady-state flux vectors for this core model must have positive entries. The following two steady-state vectors correspond to the modes using exclusively the R and the F pathways, respectively,

(4)

The R pathway is assumed to be most carbon-efficient, in the sense that it generates more ATP per unit of carbon: . Later, we will also introduce the concept of proteome efficiency, which refers to the ATP synthesis rate per unit of protein. Both vectors have been normalised so that the biomass producing flux equals 1, and have the property that each reaction in the corresponding pathway is required to sustain a flux through it. Such flux vectors have been termed Elementary Flux Modes (EFMs) [42].

2.2 Optimisation of the growth rate

Explanations of overflow metabolism often invoke the assumption of growth rate maximisation. Single cells have an evolutionary advantage if their long-term growth rate is high [8]. In constant conditions, this is equal to their steady-state growth rate . The growth rate is bounded by limitations on cell growth: cells have limited biosynthetic resources, the total enzyme concentration in the cell is typically found to be constant across conditions [15,43,44], and cells need to spend some of their enzymes on maintenance and preparation for future conditions such as changes in food availability or possible stresses [1,8,11,45].

To focus on these limitations, we introduce enzyme concentrations into the model, with each metabolic process represented by a reaction catalysed by its unique enzyme. Let be the total enzyme concentration, and assume it is constant across conditions. Then the rate of protein synthesis by ribosomes balances with the rate of dilution by growth [5],

or in other words,

(5)

Each enzymatic reaction j has a corresponding enzyme concentration , catalytic rate constant and saturation constant , with , e.g., . (In Section 9 in S1 Appendix we show how one can estimate mean constants for a set of individual enzymatic reactions that are lumped into one process in a core model.)

In terms of protein concentrations, maximisation of the growth rate is formulated as an optimisation problem by

(6)

We have introduced one new protein pool, , for maintenance and structural processes, and assume it to be fixed across conditions [5]. Moreover, even though is assumed constant, the sum of the protein concentrations is not: this will be clarified in the next section.

The optimisation problem (6) can be rewritten in terms of the growth-associated fluxes as variables

(7)

The set of admissible vectors may be characterised using the special EFM vectors and we defined in (4). Any positive steady-state vector satisfies

(8)

Here, and are convex coefficients that satisfy and .

In more general terms (i.e., for arbitrary stoichiometric matrices ), the set forms a pointed polyhedral cone that is spanned by its extreme rays [46]. These are called Elementary Flux Modes (EFMs) in the context of metabolic networks [42]. Therefore, any steady-state flux vector may be written as a convex combination of these rays. The EFMs are uniquely determined by the stoichiometric matrix . EFMs are used frequently in metabolic pathway analysis, both to obtain fundamental understanding of metabolism [2,47,48] and for biotechnological applications like metabolic engineering [49,50].

de Groot and colleagues [40] showed for this general case that the number of flux-carrying EFMs is bounded by the number of active flux-limiting constraints when a (specific) flux is optimised. We therefore infer (an upper bound on) the number of active EFMs by identifying the active constraints in the model. We thus know the root cause of the onset of overflow in general terms: if one adds extra constraints to (7), the maximiser switches from a single EFM to a mix of two or more. The downside of this formulation in terms of fluxes is that it is hard to interpret these additional constraints biologically; as a result, there are many putative explanations of overflow metabolism in the literature [17].

2.3 The core model for carbon-limited conditions

To provide a biological interpretation of the constraints in terms of actual limitations on cell growth, we adhere to the formulation in terms of protein concentrations (eq. (6)). The extra constraint follows from modeling different experimental conditions, such as carbon limitation. Various ways exist to realise this condition experimentally. During carbon uptake-limited batch cultivation, the carbon source is available in excess, but the experimentalist controls the carbon uptake flux by varying the nutrient quality and/or by titrating the expression of protein involved in nutrient assimilation (e.g., [20,51]). The corresponding optimisation problem is formulated as

(9)

This includes an extra constraint on the carbon assimilation flux ().

Another way to realise carbon-limited growth is through chemostat cultivation. In this setup, the growth rate is set by the experimentalist through the dilution rate. The cells in the culture that survive are the ones that are able to minimise their carbon uptake flux to sustain this growth rate [52]. This also limits the concentration of the carbon source in the chemostat. The optimisation problem is thus formulated as one in which the assimilation rate of carbon is to be minimised per unit of biomass produced,

(10)

Mathematically, the optimisation problems (9) and (10) are equivalent. In both formulations, we are in the end maximising the ratio , which is proportional to the biomass yield per mole carbon source,

(11)

Both modes each have a different biomass yield. Since the fixed vectors of their corresponding EFMs (eq. (4)) are normalised with respect to the biosynthetic flux , their entries are yields [42]. The biomass yield of each mode corresponds to the ratio of the last over the first entry of their EFM vector

(12)

Because the R pathway is assumed to be most carbon-efficient (), the R-mode also has the highest yield: . It therefore requires the lowest assimilation flux to sustain a certain growth rate.

When multiple constraints become active, the -minimising flux vector can be a convex combination (8) of the EFMs [40]. The optimal yield is now a convex combination of the yield of the two strategies,

(13)

Clearly, the optimal yield is then always lower than or equal to the maximal yield .

We now show that if enzyme saturation levels are assumed to be constant (an assumption used in many proteome-constrained models [16,20,29,35]), and cells have a constant total enzyme concentration across different conditions [15,43,44], we can identify a protein pool that is not related to growth but still needs to be made by the cell at low growth rates. As we will see, as the growth rate increases, this pool shrinks until it vanishes, and at this point, the cell needs to start using the carbon-inefficient pathway to grow even faster.

We start from the optimisation problem in eq. (10). Setting implicitly sets a constraint on , since , and and are assumed to be fixed,

(14)

If the total protein constraint is not hit in the optimum, i.e., , then there exists a pool of enzymes not associated with growth,

(15)

So depending on the growth rate, either 1 or 2 protein constraints are hit: certainly (14), and possibly

(16)

From previous work [40], we thus expect a switch from one EFM to a mix of two as the growth rate increases and the second constraint is hit. Other models that exhibit a shift between catabolic modes use a similar decomposition of the proteome in growth-associated and maintenance sectors [6,20,53] but typically without the pool we have identified here. Such a pool has been observed in proteomics data of E. coli [11,45]. It includes proteins that are used to deal with stresses or to prepare for future conditions [1,11,15,29]. Expression of such proteins is at the expense of proteins associated with growth, causing trade-offs between these key cellular functions [8].

2.4 Reducing large genome-scale metabolic models to small core models

The large genome-scale models [16,29,35,53] and small coarse-grained models [6,20,36–38,40] that are used to study metabolic shifts differ in their level of detail. Here, we integrate and unify these two modeling branches by bridging these levels. In Section 9 in S1 Appendix we provide a protocol for deriving a core metabolic network from a large (genome-scale) network, based on the decomposition of the metabolic network in different functional processes as presented in Section 2.1. We derive equations for the (average) metabolic fluxes for each process that are approximated in terms of averages of both the kinetic parameters and protein concentrations per sector. This allows for calibration of the core model with experimental data. Furthermore, this reduction method unifies existing models across different levels of detail. To illustrate this, we apply this protocol to the yeast S. cerevisiae and derive a core model that describes its metabolic shift. Other reduction methods for large models have been derived by [54] and [55].

3.1 The carbon-efficient regime of nutrient-limited growth:

Nutrient-limited growth can be attained in a chemostat [52] and by titration of the expression of the transporter of a nutrient [20,51]. The nutrient we are considering is a source for both carbon and energy. We shall mostly be referring to aerobic growth on a sugar, as this is the best understood condition for the onset of overflow metabolism. The growth rate rises as function of the dilution rate in the chemostat and the titrated level of the transporter from a value close to 0 to the maximal growth rate . When overflow occurs it does so at an intermediate critical growth rate . What all these cases have in common is a shift at from a carbon-efficient mode of metabolism (i.e., a high yield of biomass on the carbon source; respiration) to a less carbon-efficient mode of metabolism (overflow metabolism; fermentation). In all these cases, the carbon-efficient mode of metabolism, active exclusively below , synthesises more ATP per unit carbon source than the carbon-inefficient mode, which gradually overtakes the carbon-efficient mode in the regime .

We investigate the onset of overflow in two experimental conditions. First, we focus on a carbon-limited chemostat. After that, we consider the shift as function of the growth rate on different carbon sources in carbon uptake-limited batch cultures.

One question we need to answer is why the carbon-efficient mode is favoured over the inefficient mode in the regime , that was previously unaddressed by the model proposed in [20], and why the efficient mode cannot be used exclusively for higher growth rates. This becomes clear from the optimisation problem formulation of the chemostat (eq. (10)). The carbon assimilation rate is minimised given a set growth rate (equal to the dilution rate of the chemostat). The set growth rate implies that we set the protein pool of biomass formation equal to . This is the only active protein expression constraint in the regime . Accordingly, a single EFM is active [40]. Since in this case the optimisation problem is equivalent to maximising the yield of biomass on the carbon source , the carbon-efficient mode is the optimal solution because (eq. (12)). This explains why the carbon-inefficient mode of metabolism is not used in the growth rate regime .

In this growth rate regime, the carbon-efficient pathway may therefore be studied in isolation by setting the protein pool of the carbon-inefficient mode to zero (). The steady-state equations (1) and (5) now have a unique solution for the protein pools as function of the (set) growth rate. Since it is customary in the field to use protein fractions , i.e., protein concentrations normalised to the total protein concentration, we define, e.g., .

In terms of protein fractions, the steady-state solution of the carbon-efficient, R-strategy is given by

(17)

as derived in Section 2 in S1 Appendix. Here we introduced the constants to collect corresponding parameter combinations. These constants may be interpreted as (lumped) protein costs for each lumped reaction. These equations indicate that the protein fractions are proportional to the growth rate, which is in agreement with experimental data [29,56]).

Since is fixed, the increase in the growth-associated protein is accompanied by a proportional decrease of the growth-unassociated protein pool. Since is assumed fixed, this decreasing pool has to be , and

(18)

This behaviour is also found in experimental data of protein sectors as function of the growth rate [11,45]. The last equation indicates the decrease with growth rate of the growth-unassociated protein pool and results from substituting the protein fraction relations in the protein fraction conservation equation.

The region in Fig 2A up to the critical growth rate, indicated by the dashed line, presents the protein fractions of the efficient strategy as function of the growth rate. The same region in Fig 2B presents the biomass yield, which is constant and maximal, and thus gives minimal carbon assimilation (rate).

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Fig 2. A shift from the carbon-efficient mode to the inefficient mode occurs when all available protein is invested in growth.

A) Growth-associated protein fractions () increase linearly with the growth rate, at the expense of the NG-sector. Beyond the critical growth rate, the R-sector is replaced by the (smaller) F-sector. The assimilation sector () is omitted for clarity. B) The biomass yield is constant and maximal when only the carbon-efficient mode is active. Beyond the critical growth rate, the yield decreases because the carbon-inefficient mode has a lower yield.

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The cell is able to increase its growth rate using only the carbon-efficient mode until all protein available for growth, i.e., has been invested. The maximum is thus reached when the growth-unassociated protein sector is depleted. The cell has now reached its critical growth rate . Solving eq. (18) at for , we find

(19)

The depletion of the growth-unassociated protein pool constituted the second protein constraint that is hit, in addition to the one that was throughout the regime , i.e., . The growth rate can now only increase further if a part of the protein spent on the carbon-efficient mode is reinvested into another mode that is less carbon-efficient, but is more proteome-efficient, and into anabolism. In terms of the EFM theory [40], a second EFM mixes in with a first when a second protein expression constraint becomes active provided that the second strategy has the required behaviour. In the next section, we will identify under which conditions this can occur.

Finally, we note that equation (18) when solved for , i.e.,

(20)

captures the trade-off between the instantaneous growth rate and the fraction of protein devoted to growth-unassociated processes [8]. These are often alternative nutrient uptake systems and stress tolerance [11,45]. The equation thus indicates that a higher investment in preparation for possible future conditions is concomitant with a decrease in the instantaneous growth rate. Put differently, there is a trade-off between the instantaneous and long-term growth rate [8]. These insights agree with experimental data; E. coli samples taken from an aerobic, glucose-limited chemostat below proved more adaptive and stress-tolerant than samples taken at growth rate close to the maximal growth rate (above ) [9].

3.2 The overflow regime () during nutrient-limited growth

In order for growth to continue at a rate above , another energy-generating strategy is required. The growth rate is directly proportional to the concentration of protein invested. With a fixed total concentration, and no more growth-unassociated protein to spend, the protein spent on the carbon-efficient pathway needs to be reallocated to the other mode. This second, carbon-inefficient strategy should have a higher ATP synthesis rate than the carbon-efficient mode, but require a lower protein fraction. In terms of the model, this means that the protein costs for ATP synthesis are lower for the carbon-inefficient mode than for the carbon-efficient mode. This can be achieved by the inefficient mode having a higher activity per unit protein and/or because the associated metabolism relies on fewer enzymes (e.g., glycolysis + fermentation enzymes < glycolysis + TCA cycle + respiratory chain enzymes). When all the protein that was previously invested in the carbon-efficient mode is completely used and replaced by to catalyse the carbon-inefficient mode exclusively, the maximal growth rate is reached. The cell now only displays the carbon-inefficient strategy.

In the regime, the optimisation problem (10) also admits an analytical solution for the protein fractions. This is derived in Section 4 in S1 Appendix and agrees with [20]. In this growth rate regime, the optimal strategy is a convex combination (8) of both strategies. The corresponding convex coefficients result in linear protein fractions . An increase in the convex coefficient inherently decreases its counterpart as they add up to one, thereby reflecting the exchange of the two strategies. The resulting change in the protein fractions as function of the growth rate is depicted in Fig 2A, after the dashed line.

The carbon-inefficient mode synthesises less ATP per glucose while the needed ATP supply rate increases linearly with growth rate. The expectation is thus that the carbon uptake rate in the chemostat rises more steeply for than below it. This is indeed the case in such studies [2,25] and Fig 5C. The slope of the glucose uptake rate () as function of the growth rate () equals the inverse of the biomass yield on glucose. Since this slope is higher above we conclude that the yield of the carbon-inefficient mode is less than that of the efficient mode, i.e., (as we assumed, cf. eq. (12)).

Since the optimisation still involves minimising the nutrient uptake for a given growth rate, we are still searching for the highest biomass yield on the carbon source, but now in accordance with two mixed strategies (eq. (13)). The resulting yield is therefore always below as the carbon-inefficient mode is mixed in. This is shown in Fig 2B after the dashed line. When the maximal growth rate is reached, the optimal yield equals .

From the above discussion, we can conclude that a necessary condition for the onset of overflow metabolism is for the second metabolic strategy to be carbon-inefficient (so that it is not favoured as the main mode at low growth rates ()), but also that it is able to sustain a growth rate that exceeds the critical growth rate. In other words, .

Analogous to eq. (17)–(19), we study the carbon-inefficient mode in isolation to determine . Setting and solving the steady-state equations (1) and (5) for the remaining growth-associated protein fractions yields

(21)

The maximal growth rate is found in exactly the same way as before, as explained in Section 3 in S1 Appendix. We find

(22)

We immediately observe that when such that for all ’s: the carbon-inefficient mode is more proteome-efficient.

Fig 2 may thus be summarised as follows. For growth rates above the critical growth rate, the (optimal) protein fraction decreases until it is absent. At this point, the employed strategy has completely shifted from the efficient to the inefficient mode and the protein fractions are given by eq. (21). In terms of the convex coefficients, and . For the yield (13), this implies that . Since no alternative strategy exists that can further increase the growth rate, the maximal growth rate of the carbon-inefficient mode equals the absolute maximal growth rate.

3.3 The proteome efficiency of fermentation versus respiration as an overflow condition

In the previous section, we concluded that the carbon-inefficient catabolic mode can take over from the carbon-efficient catabolic mode above when the first has a maximal growth rate that exceeds the maximal growth rate of the second. This higher maximal growth rate is achieved at a lower protein fraction that is allocated to catabolism, since at higher growth rate biomass formation requires more protein. Hence, per unit protein, the carbon-inefficient mode needs to achieve a higher ATP synthesis rate than the carbon-efficient mode. This insight underlies the introduction of the concept of proteome efficiency by others [16,20,29,35]. It is defined as the ATP synthesis rate per unit of protein (although the amount of protein used for normalisation varies between studies [35,57]) and denoted by . In our model, we define it by dividing by the total protein concentration expended to sustain the flux and , respectively,

Using eq. (17), (21) and the steady-state equations (1) and (5), one may simplify these to

showing that these proteome efficiencies are directly related to the respective maximal growth rates. Explaining the shift in terms of the maximal growth rates or the proteome efficiencies of the strategies is therefore equivalent. In Sections 5 and 17 in S1 Appendix, we examine in detail the different definitions for the proteome efficiency. We show that, as long as protein costs are properly accounted for, they all lead to the same result: a shift to the carbon-inefficient mode caused by a constraint on the total protein content implies its higher proteome efficiency.

3.4 The critical and maximal growth rate respond similarly to physiological perturbations that affect both metabolic modes

Both the critical (eq. (19)) and maximal growth rate (eq. (22)) are proportional to . This shows that varying the Q-pool by overexpressing a protein that does not contribute to growth is expected to shift both the critical and maximal growth rate with the same factor. Treatment of cells with antibiotics gives a similar physiological perturbation of both modes by inhibiting translation. This corresponds to the decrease of the catalytic constant of biomass synthesis () in the model, which increases the biosynthetic protein costs (). However, in contrast to the change in , this perturbation decreases the maximal growth rate by a larger factor than the critical growth rate. This indicates a larger effect on the carbon-inefficient mode due to its lower protein costs. Both these effects were indeed observed by [20]. Here, we formulate these in terms of biochemically interpretable parameters.

Finally, we note that the critical and maximal growth rate both depend on the C-coefficients representing the lumped protein costs of the growth-associated processes (each resembling a lumped metabolic pathway). All the C-coefficients (eq. (17) and (21)) are inversely proportional to the enzyme saturation degrees of the lumped pathways. These relations imply that both the critical and the maximal growth rate increase with the enzyme saturation degree. Less enzyme is then needed to reach a certain process flux and therefore higher rates can be reached.

3.5 Physiological parameters that influence the occurrence of overflow metabolism

In the previous section we considered physiological perturbations that change both metabolic modes. Although these affect the onset of overflow metabolism, this will likely not alter its manifestation. This can only be altered when properties of one of the modes change. For instance, the P/O ratio of the respiratory chain may vary between species or varied for one species by perturbation. This parameter is directly related to the parameter in the core model.

To be able to address how physiological parameters influence the occurrence of overflow metabolism, we consider the following dimensionless ratio of the maximal growth rate of the inefficient over the efficient mode,

(23)

As concluded above, a shift occurs if ; hence, when . In general, the critical growth rate increases when the protein costs of the efficient mode decrease. This reduces . Increasing the protein costs of the inefficient strategy causes a similar effect, since this reduces . When a metabolic shift will not occur.

We shall refer to as the break-even condition. Using eq. (23), this condition is written explicitly in terms of protein costs as

(24)

where is the difference in (carbon) assimilation costs for both strategies. This difference is generally positive, as the inefficient mode requires more carbon due to its lower yield.

The break-even condition separates the regime where a metabolic shift will occur (in blue) from the regime without a shift (in brown) in Fig 3A. This figure may therefore be interpreted as a phase diagram, similar to those used in physics. An organism is represented by its protein costs as a coordinate .

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Fig 3. Parameter perturbations affect the occurrence of overflow metabolism.

The break-even analysis in terms of A) catabolic protein costs and B–F) several model parameters. For parameter configurations in the blue region, a metabolic shift occurs. In the brown region a shift will never occur. The arrow in B) and C) indicates the direction in which the break-even condition moves if the protein costs of the carbon-inefficient mode decrease.

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To study this break-even condition for different parameters, it is useful to unpack a little how a large metabolic model may be condensed into a core model. Each reaction in the core model gives an average reaction rate. As detailed in Section 9 in S1 Appendix, expressions for these average rates include the number of individual enzymatic reactions in each lumped reaction. We denote these lengths by for the number of reactions in assimilation, etc.

With these length parameters, the break-even condition can be written explicitly in terms of model parameters as

(25)

This follows from using the definitions of from eq. (17) and (21).

The break-even condition allows us to investigate the dependence of two parameters (assuming all other parameters remain fixed). We generally find a curve that divides the two-parameter plane in regions. To illustrate this, Figs 3B–3F show the (conceptual) phase diagrams for five different parameter pairs that are perturbed. All of these figures have a similar intuitive interpretation: only for values of the parameters that result in , a shift occurs.

For instance, Fig 3B shows that there is a trade-off between the ATP yield and the catalytic constant . The effect of a change in the catalytic constants of both strategies on the metabolic shift is depicted in Fig 3D. Fig 3F shows that for increasing length of the inefficient mode the protein costs increase, so also the ATP yield should decrease for a shift to occur. Additional examples for different parameters are given in Section 6 in S1 Appendix.

This break-even analysis may be used to clarify why some organisms or strains shift between metabolic strategies, but others do not [35]. It may be that . This means that the carbon-efficient mode also has a higher proteome efficiency than the inefficient mode, so the inefficient mode can not reach growth rates exceeding . A different reason for the absence of a shift is that the organism never attains its critical growth rate, and therefore the growth-unassociated protein sector never gets depleted.

3.6 The metabolic shift as function of the growth rate in (carbon uptake-limited) batch conditions with different carbon sources

In nutrient excess conditions, microbes attain their maximal growth rates. The associated growth rate maximisation problem is given in eq. (6). It follows from the previous sections that the growth rate is maximal when the growth-unassociated protein pool is empty (). In nutrient excess conditions, neither the growth rate nor the assimilation rate is fixed. Therefore, there is only one active (proteome) constraint limiting growth in this case, and accordingly only a single EFM is active [40]. This will be the EFM with the lowest protein costs. We therefore predict that the cell uses the carbon-inefficient mode in nutrient excess conditions.

As explained in Section 2.3, during carbon uptake-limited batch cultivation, the cell is limited by the assimilation rate of carbon. Experimentally, this is achieved through titration of the carbon uptake system [20,51]. This sets , and thereby the assimilation rate . The resulting optimisation problem (eq. (9)) is equivalent to the one in eq. (10) that describes chemostat conditions. Their solution in terms of optimal protein fractions is the same, but for eq. (9) it is parameterised in terms of instead of . Therefore also in uptake-limited conditions, the efficient mode is optimal below a critical uptake rate

(26)

For higher carbon uptake rates, the growth-unassociated protein pool is empty and the carbon-inefficient strategy will occur. Due to the equivalence between the optimisations for chemostat and uptake-limited batch conditions, we find that .

Another way to realise uptake limitation experimentally is by varying the carbon source (at a constant titrated or wild type level of the uptake system). For each carbon source, the catalytic rate constant of the assimilation pathway is different and represents the quality of the nutrient [5,58].

Fixing while also fixing the assimilation enzyme concentration again results in a constraint on that limits growth. However, there is a small but important difference in this implementation of uptake limitation with respect to the previous setting. When, instead, the carbon source is fixed (by the experimentalist), uptake limitation is only realised for a constant . Otherwise, the cell may increase this enzyme concentration at low-quality nutrients to alleviate the uptake limitation that constrains the growth rate. Whether it will do so depends on its gene-regulatory strategy. This minor distinction leads to different quantitative results.

The solution to this optimisation problem is now parameterised in terms of the nutrient quality . The shift from the efficient to the inefficient mode now initiates at a critical nutrient quality

(27)

It depends on the constant value of in two ways. This critical nutrient quality corresponds to a critical assimilation rate

(28)

Due to the constant , the assimilation contribution appears as in the numerator, rather than as the costs in the denominator in eq. (26) for . The corresponding critical growth rate

(29)

differs in the same way from the expression

(30)

derived in Section 3.1. The analysis of the optimisation problem with and fixed is presented in more detail in Section 7 in S1 Appendix.

Our analysis of the difference between chemostat settings and varying nutrient quality in batch conditions allows us to understand why the patterns both display are so alike. It is consistent with the experimental data for E. coli obtained by [20]. They studied carbon uptake-limited cultivation of E. coli by varying titrated levels of the carbon uptake system on different glycolytic and non-glycolytic carbon sources. Both for increasing uptake titration and increasing nutrient quality, the authors observed an increase in the acetate production flux with the growth rate after a critical growth rate. This indicates a higher activity of the acetate-formation pathway, which is the more proteome-efficient pathway in E. coli [20].

What is particularly noteworthy is that the data seems to follow the same linear increase in the acetate production flux with the growth rate, both for increasing uptake titration and increasing nutrient quality. This ‘acetate line’, as it is called by the authors, is in the core model represented by the (overflow) flux . In Section 7 in S1 Appendix, we show that this flux, when parameterised by the nutrient quality , is a linear function of the growth rate, with both the slope and intersection with the -axis determined by the critical growth rate (29). Nevertheless, the critical growth rate depends on both the constant assimilation fraction and the protein costs , which can differ per carbon source. Therefore, the linearity of alone does not explain the observation of a single overflow flux line.

Proteomics data [53,59] shows, however, that the assimilation fraction is generally small relative to other protein fractions. Furthermore, for glycolytic carbon sources, the cell uses similar catabolic and anabolic pathways, resulting in comparable protein costs . Together, this results in a value for the critical growth rate that is similar both for (small) values of and for different glycolytic carbon sources. In fact, small is equivalent to low assimilation costs (with respect to other protein costs). In this limit, the different equations (29) and (30) for the critical growth rate reduce to the same expression. Consequently, this results in (approximately) one overflow flux line , because this is completely determined by the critical growth rate. This explains why the E. coli data from [20] for different experimental (carbon) uptake-limiting conditions seem to follow the same acetate line.

We can now also understand why the data for non-glycolytic carbon sources (see Extended Data Fig 1f in [20]) do not follow the acetate line. This is because the employed catabolic (and anabolic) pathways change for these carbon sources, thereby resulting in different associated protein costs, and thus different values for the critical growth rate (29).

3.7 Characteristics of overflow metabolism in yeast are captured by a core model variant that follows from a genome-scale model

As explained in Section 2, large complex models used to analyse metabolic shifts, such as GEMs, may be reduced to a small core model, provided that equivalent constraints and degrees of freedom are taken into account. In this section, we illustrate this by adapting the core model to describe overflow metabolism in S. cerevisiae occurring at high growth rates in aerobic glucose-limited chemostat cultures. The reaction network corresponding to the yeast model is depicted in Fig 4. It is similar to the main core model sketched in Fig 1, but instead of combining carbon transport and phosphorylation into an assimilation pathway A, the yeast model describes this as separated carbon transport T with flux , and glycolysis G with flux and pyruvate as its product. The analysis of this model focuses on the analogue of optimisation problem eq. (10).

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Fig 4. The metabolic network described by the yeast model.

Glucose is imported by a transport reaction (T). Glycolysis (G) converts glucose into pyruvate, yielding 2 ATP per glucose. Pyruvate is further degraded into carbon dioxide by respiration (R) or into ethanol by fermentation (F). Respiration yields an additional 16 ATP per glucose. The generated ATP is used for biomass synthesis (B).

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In Section 9 in S1 Appendix, we show that the rate of lumped reaction j in the core model may be approximated in terms of averaged kinetic properties of all reactions in “sector” j as

(31)

The rate should then be interpreted as the mean rate of the reactions in sector j. Here, is the mean catalytic rate constant of these reactions. The enzyme concentration in (31) signifies the total concentration of all enzymes involved in sector j, distributed among the (metabolically distinct) reactions. Mean saturation levels that change with the growth rate are inferred from proteomics data from [29]. Other specific modifications to the core model for yeast, such as which pathways are included in each sector, are detailed in Sections 8–11 in S1 Appendix.

We calibrate the model parameters to ensure that the critical and maximal growth rates are consistent with those observed during experimental cultivation of yeast. In the chemostat, ethanol formation starts at the critical dilution rate . The maximal growth rate as defined in eq. (22) should correspond to the point where yeast grows fully fermentative. However, this state is never reached in the chemostat. Therefore, we set the maximal growth rate of yeast to its growth rate during batch cultivation with excess glucose, which is for the strain examined by [29]. A Mathematica implementation of the core model adapted to yeast is provided in the source code on Zenodo.

The optimal protein fractions obtained with the yeast model are depicted in Fig 5A as function of the growth rate. They exhibit dynamics qualitatively similar to Fig 2A, and may be interpreted analogously. Once the non-growth protein fraction is depleted at , an extra constraint becomes active and the model shifts from high-yield respiration to low-yield ethanol fermentation. The nonlinear relationship between the protein fractions and growth rate results from the saturations that now also change with the growth rate. Fig 5B shows qualitative agreement with the categorised proteomics data from [29]. The proteomics data show the presence of fermentative protein before the critical growth rate. This may be explained by preparatory expression of these enzymes. For example, the enzyme pyruvate decarboxylase is allosterically regulated and can therefore remain in an inactive state [60,61]. These results are discussed in more detail in Section 14 in S1 Appendix. In Section 15 in S1 Appendix, we show that including changing saturation gives a better quantitative match between the model and the experimental data than for constant saturations.

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Fig 5. The yeast model captures the characteristics of overflow metabolism in yeast.

A) Dynamics of the protein fractions as function of the growth rate. The second figure is enlarged to show the smaller fractions. B) The protein fractions predicted by the yeast model together with proteomics data [29]. C) Specific fluxes and D) Biomass yield predicted by the yeast model together with chemostat data [29]. In all figures, the dashed line indicates the critical growth rate. Data is reused with permission from [29] (Creative Commons Attribution 4.0 International License).

https://doi.org/10.1371/journal.pcbi.1014417.g005

Specific uptake and production fluxes follow directly from normalising the model fluxes by the dry weight density , and accounting for the appropriate stoichiometric coefficient. The glucose uptake flux predicted by the model is given by the orange line in Fig 5C. Its faster increase after the critical growth rate results from the higher carbon requirement of fermentation. The oxygen uptake flux is , as respiration requires six moles of oxygen for the complete oxidation of one mole of glucose to carbon dioxide and water. It declines after the critical growth rate due to the replacement of respiration by fermentation, as the red line in Fig 5C shows. During fermentation, yeast metabolises glucose into two ethanol, carbon dioxide and water. Thus, the ethanol production flux follows from the fermentation flux in the model as . Ethanol excretion, shown by the blue line in Fig 5C, initiates after the critical growth rate. We compute the biomass yield via a unit conversion of the model prediction (eq. (13)), resulting in the green line in Fig 5D. At low growth rates it equals the respiration yield. When the critical growth rate is exceeded, the yield declines to the fermentation yield.

We have had to make a number of adjustments to fit the yield and (some of) the specific fluxes to experimental data, because of assumptions used in the yeast model. The model predictions shown in Fig 5C and 5D are the results after these adjustments. For instance, obtaining the biosynthetic carbon demand would require a detailed dissection of the carbon- and energy fluxes for the catabolic and anabolic part of the metabolic network [62], which is beyond the scope of this work. Since the shift occurs in the catabolic part of yeast’s metabolic network, we instead focus on the catabolic carbon flow. The catabolic carbon uptake predicted by the model is then adjusted to match experimental data for the total carbon uptake. These adjustments are detailed in Sections 12 and 13 in S1 Appendix.

In Fig 6 we give an illustration of the result from Section 3.6 for parameters of the yeast model and constant saturation factors. It shows that, during uptake-limited growth on different carbon sources, there is approximately one ethanol flux line when the transporter fraction () is small. Whether this also applies to yeast remains uncertain and requires experimental results similar to E. coli found by [20].

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Fig 6. The metabolic shift is similar under chemostat and carbon uptake-limited batch conditions.

Ethanol excretion flux as function of the growth rate for different values of constant transporter fraction (). When is small, the corresponding ethanol excretion lines are parallel and close to each other, seemingly resulting in only one line. A Mathematica implementation is provided in the source code on Zenodo.

https://doi.org/10.1371/journal.pcbi.1014417.g006

In Sections 16–18 in S1 Appendix we again analyse the respiration and fermentation strategies independently, compute their proteome efficiencies, and perform a break-even analysis.