Achieving Optimal Growth through Product Feedback Inhibition in Metabolism

Recent evidence suggests that the metabolism of some organisms, such as Escherichia coli, is remarkably efficient, producing close to the maximum amount of biomass per unit of nutrient consumed. This observation raises the question of what regulatory mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. We analyze several representative metabolic modules—starting from a linear pathway and advancing to a bidirectional pathway and metabolic cycle, and finally to integration of two different nutrient inputs. In each case, our mathematical analysis shows that product-feedback inhibition is not only homeostatic but also, with appropriate feedback connections, can minimize futile cycling and optimize fluxes. However, the effectiveness of simple product-feedback inhibition comes at the cost of high levels of some metabolite pools, potentially associated with toxicity and osmotic imbalance. These large metabolite pool sizes can be restricted if feedback inhibition is ultrasensitive. Indeed, the multi-layer regulation of metabolism by control of enzyme expression, enzyme covalent modification, and allostery is expected to result in such ultrasensitive feedbacks. To experimentally test whether the qualitative predictions from our analysis of feedback inhibition apply to metabolic modules beyond linear pathways, we examine the case of nitrogen assimilation in E. coli, which involves both nutrient integration and a metabolic cycle. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the network and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes in nutrient-switching experiments.

(1) This along with the linear constraints on input flux and growth rate, yield the following for the FBA growth rate, where g FBA is the maximum allowed growth rate consistent with all the constraints. Including feedback regulation, the kinetic equation for the metabolite-pool size p is dp dt where p is the metabolite-pool size, h is a Hill coefficient and K is an inhibition constant. The steady-state pool size therefore satisfies whereṼ = V max /g max . It is straightforward to calculate the asymptotic behavior of p in the two asymptotic FBA regimes: flux-limited (Ṽ 1) and growth-limited (Ṽ 1). We assume a large feedback-inhibition constant K p , this yields where p is the steady-state metabolite pool; we keep terms to leading order in K.
In the V max -limited regime, i.e. whereṼ 1, the growth rate is where we have used p K. This yields for the growth-rate deficit In the g max -limited regime,Ṽ 1, where p ∼ K, the growth rate is yielding for the growth-rate deficit

B. Bidirectional pathway
The flux-balance condition at steady state for the bidirectional module shown in Fig. 3 of the main text is given by where c 1 , c 2 give the stoichiometry of utilization of the two intermediates. These along with the linear constraints on input fluxes, are used to calculate maximum growth-rate g FBA , first for V max Due to symmetry, the FBA growth rate for V max 1 /c 1 ≥ V max 2 /c 2 can be found by exchanging the 1 ↔ 2 indices. The above calculation assumes growth is limited by nutrients; if the growth is limited by g max , then g FBA = g max .

C. Metabolic cycle
The flux-balance condition at steady state for the metabolic cycle shown in in Fig. 4 of the main text is given by where c Q , c E are the stoichiometry of utilization of glutamine and glutamate, respectively. The growth rate is maximized within the following linear constraints on fluxes, which yields The above calculation assumes growth is limited by nutrients; if the growth is limited by g max , then g FBA = g max .

D. Integration of two different nutrient inputs
The flux-balance condition at steady state for the carbon-nitrogen module shown in Fig. 1 of the main text is given by where we have included facors c C , c N giving the stoichiometry of utilization of the two intermediates. Assuming the following linear constraints on input fluxes, yields the FBA optimal growth rate The above expressions apply if growth in limited by nutrients, otherwise the growth rate is limited by g max , i.e. g FBA = g max . Model for feedback knock-out strain. In the feedback knock-out (KO) strain, the negative feedback from glutamine on its own biosynthetic enzyme glutamine synthetase is absent. The module describing the KO strain is the basic carbon and nitrogen combining module without the feedback on the carbon-dependent nitrogen input flux V N1 . Furthermore, we allow leakage out of the cell from the nitrogen intermediate p N . This yields the following kinetic equations for metabolite pools p C and p N , where h is a Hill coefficient (assumed for simplicity to be the same for all feedbacks), the K i , with i = C, N 1 , N 2 , are feedback-inhibition constants, and the growth rate g is given by Eq. 2 in MT. The auto-regulatory negative feedback on carbon flux and the leakage of p N ensure a steady state, which is guaranteed to be stable by the 1:1 stoichiometry (Goyal and Wingreen, 2007).

A. Nitrogen starvation
One prediction from our analysis is that large changes in metabolite pools will occur upon the onset of nutrient limitation. Such large changes in metabolite-pool sizes have recently been observed in nutrient switching experiments with Escherichia coli (Brauer et al, 2006). When cells growing on filters were moved from a minimal media (no nutrient limitation) agar plate to a no NH + 4 (nitrogen limited) agar plate, the glutamine pool (a nitrogen intermediate) decreased by almost 64 fold, while the pool of α-ketoglutarate (a carbon intermediate) increased by almost 16 fold (Fig. 1). Furthermore, the changes in metabolite pools were monotonic over short times ( 1 hour). However, pools of other metabolites such as ATP and glutamate, which are known to fulfill other functions in cell, did not change much after the nutrient switch (ATP serves as the energy currency of the cell and glutamate is the dominant counter-ion for potassium).
Despite the complexity of the real metabolic network, we find that our simple module for combination of carbon and nitrogen predicts pool size dynamics qualitatively similar to the measurements (see Fig. 1 inset). To make the analogy between our simple module and the real metabolic network, we identify our carbon intermediate p C with α-ketoglutarate and our nitrogen intermediate p N with glutamine. As a simulation of the experiment, we start the carbon-nitrogen module in steady state in the non-nutrient limited regime, and at time t = 0, reduce the input nitrogen availability by simultaneously reducing tenfold the two maximum input nitrogen fluxes V max N1 and V max N2 . The metabolite pools then evolve dynamically as per Eq. 7 in MT. Consistent with the experimental data, the carbon intermediate pool p C increases monotonically while the nitrogen intermediate pool p N decreases monotonically before approaching new steady states. Note that the consistency between simulations and the experimental data degrades at long times ( 1 hour). In order to understand such long-time behavior of the metabolite pools one would have to include the effects of genetic regulation on the various enzyme concentrations.

B. Nitrogen up-shift
Interestingly, the leakage of key nitrogen intermediates, glutamine and glutamate, in the feedback-defective strain following nitrogen upshift depends on the nitrogen source used in the media. Particularly, both glutamate and glutamine leak for ammonia as nitrogen source while only glutamine leaks out for proline (also see Kustu et al., 1984) as nitrogen source (Fig. 3). In cells growing in proline media, synthesis of glutamate from glutamine is suppressed as proline directly substitutes for glutamate. Consequently, the metabolic cycle between glutamine and glutamate is broken in the proline media, i.e. the reaction from glutamate to glutamine is active and the reaction from glutamine to glutamate is inactive. Therefore, in the feedback-defective strain, following nitrogen upshift in proline media, additional glutamate (coming from proline) is readily converted into glutamine leading to large production of glutamine. However, in ammonia media, the full metabolic cycle is active and therefore both glutamine and glutamate are synthesized at high rates following the nitrogen upshift.