What is feedback inhibition in metabolic pathways?

PLoS Comput Biol. 2010 Jun; 6(6): e1000802.

Nathan D. Price, Editor

Abstract

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.

Author Summary

Bacteria live in remarkably diverse environments and constantly adapt to changing nutrient conditions. Recent evidence suggests that some bacteria, such as E. coli, are extraordinarily efficient in producing biomass under a variety of different nutrient conditions. This observation raises the question of what physical mechanisms enable such efficiency. Here, we propose that simple product-feedback inhibition by itself is capable of leading to such optimality. Product-feedback inhibition is a metabolic regulatory scheme in which an end product inhibits the first dedicated step of the chain of reactions leading to its own synthesis. Our mathematical analysis of several representative metabolic modules suggests that simple feedback inhibition can indeed allow for optimal and efficient biomass production. 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 pools can be restricted if feedback inhibition is ultrasensitive. We find that the feedback regulation scheme suggested by our mathematical analysis closely aligns with the actual regulation of the nitrogen assimilation network in E. coli and is sufficient to explain much of the dynamical behavior of relevant metabolite pool sizes seen in experiments.

Introduction

Much is known about the metabolic reactions that lead to the production of biomass and energy in cells. However, understanding the logic of metabolic regulation has been challenging due to the network's scale and complexity. Flux-balance analysis (FBA), a constraint-based computational approach, has been used to show that some microorganisms, including E. coli, maximize their growth rates per molecule of carbon source consumed [1]. FBA uses mass conservation to predict optimal growth rates as well as fluxes [2]. In its simplest form, FBA assumes that cells regulate fluxes to produce biomass at the maximum rate possible given a particular limiting input flux. Recently, FBA has been successfully applied to additional microorganisms [3]–[5], and to objective functions other then maximizing biomass [6], e.g. maximization of ATP production [7] or minimization of metabolic adjustment in response to perturbations in metabolic network [8]. Attempts to include regulatory [9], [10], thermodynamic [11], [12], and environment-specific constraints have resulted in insights into the structure of metabolic networks, e.g. the organization of redundant pathways [13], [14]. (For a comprehensive list of FBA achievements see reviews by Kauffman et al, 2003 and Lee et al, 2006). Despite their predictive strength and wide applicability, FBA-based methods are limited; FBA assumes that fluxes are optimal (thereby assuming perfect regulation) but does not reveal how these optimal fluxes are achieved. This leaves open the question: how can cells achieve nearly optimal fluxes for efficient growth?

Previously, some complex bio-molecular networks have been successfully analyzed and understood in terms of simple modules [15], e.g. the eukaryotic cell cycle [16], [17]. In the same spirit, we address the question of how to achieve optimal growth using several representative modules drawn from real metabolism. In particular we consider four modules, each of which captures an essential feature of the real metabolic network - i) a linear pathway, ii) a bidirectional pathway, iii) a metabolic cycle, and iv) integration of two different nutrient inputs. Linear pathways, in addition to being common, suggest simple rules for achieving optimal growth. In the second module, representing a bidirectional pathway, metabolites are interconverted, albeit at a cost, with the consequent risk of running a futile cycle (e.g., interconversion of fructose-6-phosphate and fructose-1,6-bisphosphate (FBP)). In the third module we analyze a metabolic assimilation cycle. A metabolic cycle can be visualized as a linear pathway where the end product is essential for the first step of the pathway. Two important examples of metabolic cycles are the TCA cycle and the glutamine-glutamate nitrogen-assimilation cycle. Finally, the fourth module addresses the problem of balancing two different inputs, carbon and nitrogen. This module takes into account the ability of microbes to assimilate nitrogen in the form of ammonium via an ATP-independent pathway or a higher affinity ATP-dependent one. When nitrogen is scarce, the ATP-dependent pathway is utilized, whereas when carbon is scarce, it is avoided.

For regulation of these modules we invoke only product-feedback inhibition. Since its discovery in the late 1950's, product-feedback inhibition has become recognized as one of the cornerstones of metabolic regulation [18], [19]. This form of regulation was first hypothesized by Novick and Szilard [20] for the tryptophane biosynthetic pathway from chemostat experiments, and has since been found in almost every biosynthetic pathway [21]. Product-feedback inhibition is a regulatory scheme in which the product of metabolism inhibits its own synthetic pathway. Remarkably, in all four of the modules studied, we find that simple product-feedback inhibition is sufficient to control fluxes so as to enable nearly maximally efficient growth.

To test our understanding of the physiological role of product-feedback inhibition, we compared our simple models to actual regulation of the glutamine-glutamate nitrogen assimilation cycle, including its integration with carbon metabolism. We find important similarities between the product-feedback inhibition scheme that we propose based on general principles and the actual regulatory mechanisms present in E. coli.

If, as we will argue, simple product-feedback inhibition is enough to achieve nearly optimal growth, why is real metabolic regulation so complex? Metabolic feedback regulation exists at various levels, such as, control of enzyme mRNA transcription [22], reversible enzyme phosphorylation [23], non-competitive allosteric regulation [24], and competition for enzyme active sites [25]. There are many cases where multiple feedback mechanisms work together, e.g. glutamine synthetase is regulated by a bicyclic cascade of covalent modifications and transcriptionally by the NtrC two-component system [26]. Our mathematical analysis suggests that simple feedback regulation, while adequate for flux control, could lead to large metabolite pools, and that accumulation of these pools may be prevented by multiple regulatory mechanisms working in concert to produce ultrasensitive feedback.

Results

Models

Linear pathway: minimal model

To elucidate the main findings of our mathematical analysis, we first consider a minimal metabolic circuit (Fig. 1A) in which an input flux of magnitude

Analysis of metabolic modules: (A) minimal linear pathway and (D) bidirectional pathway, two different regulation schemes are considered – Min-FI scheme: feedbacks only on the input nutrient fluxes (dashed lines), and Full-FI scheme: feedbacks on all the fluxes.

(B,C) Results for linear pathway from Eq. 3: (B)

To go beyond FBA and explicitly consider the regulation of fluxes, we assume product-feedback inhibition acts on the input flux such that

(1)

where

In our simple linear pathway model, the growth rate

(2)

which satisfies the above constraints. This function was obtained as the growth rate of a heteropolymer made from equal stoichiometries of monomers with pool sizes

Combining Eqs. 1 and 2 (with

(3)

The steady-state metabolite-pool size is obtained by setting the above time derivative to zero, and the growth rate is then calculated using Eq. 2. Intuitively, as long as input flux is limiting for growth (

However, there is a trade-off between the growth rate and the metabolite-pool size (Fig. 1C). For non-cooperative feedback (

(4)

In the asymptotic limit of small input flux,

Cooperative or ultrasensitive feedback (

Simple feedback regulation without ultrasensitivity has two important features: (1) simple product-feedback inhibition is enough to approach the optimal flux-balance growth rate, and (2) metabolite-pool sizes are small when growth limiting but become large when not growth limiting. These large non-growth-limiting metabolite pools can be restricted by more complex ultrasensitive feedback regulation. We test the generality of these features for various metabolic modules drawn from real metabolism.

Bidirectional pathway

Bidirectional pathways, such as glycolysis/gluconeogenesis, are used for switching between different nutrient sources, e.g. glucose (a 6-carbon unit) and lactate (a 3-carbon unit). At the heart of these bidirectional pathways are metabolites that are linked by two different enzymatic reactions (or pathways) of differing energetics due to different cofactor requirements, e.g. fructose-6-phosphate and fructose-1,6-bisphosphate, linked by phosphofructokinase in glycolysis and fructose-bisphosphatase in gluconeogenesis. Since these interconversions may allow cycling, limiting futile cycles between these metabolites is essential for achieving optimal growth.

Here we consider a simple module of two interconverting metabolites shown in Fig. 1D. The module has two input nutrient fluxes,

Limited availability of interconversion enzymes is modeled by the constraint on the interconversion fluxes

We compare two different regulatory schemes for this module. The simpler of the two schemes, minimal product-feedback inhibition (Min-FI) assumes feedbacks only on the input nutrient fluxes (the minimum number of feedbacks required to have a stable-steady state solution), while full product-feedback inhibition (Full-FI) assumes feedbacks on all the fluxes. Full-FI yields the following kinetic equations for the metabolite pools

(5)

where

To achieve optimal growth, the feedback-inhibition constants are chosen according to the logic of flux-balance analysis, i.e. to avoid futile cycling while allowing adequate flux from non-growth-limiting metabolite pool to growth-limiting metabolite pool. To avoid futile cycling, the interconversion flux should preferentially flow from the non-growth limiting pool to the growth-limiting pool. This is achieved by choosing the Michaelis-Menten constant for each outgoing interconversion flux to be much larger than the growth-saturating substrate pool size, e.g.

Numerical solutions for the steady-state growth rate and metabolite-pool sizes for the two alternative regulatory schemes are shown in Fig. 1E,F. For simplicity, we have chosen parameters to make the network symmetric with respect to the two metabolites. The growth-rate deficit and the metabolite pools for the Min-FI scheme follow the same trends seen in the linear pathway: the growth-rate deficit

Metabolic cycle

Organisms metabolize some nutrients using metabolic cycles, e.g. the TCA cycle in carbon metabolism. A metabolic cycle is a wrapped linear pathway where the end product is essential for the first step of the pathway. Consequently, the import of nutrients is slowed or stopped if there is not enough end product available. Therefore, an adequate pool of the end product must always be maintained in order to achieve optimal growth. Here we analyze a module based on the two-intermediate glutamine-glutamate nitrogen-assimilation cycle. In this cycle, ammonium (NH

The cyclic module considered here is shown in Fig. 2A. The input nitrogen flux

Analysis of metabolic modules: (A) metabolic cycle and (D) module for integrating carbon and nitrogen inputs.

(B,C) Results for metabolic cycle module from Eq. 6: (B)

The optimal flux-balance growth rate

As in the previous case, we compare different regulatory schemes for this module. The Min-FI schemes have only one feedback on the input flux from either glutamate or glutamine. In the Full-FI scheme, there is product-feedback inhibition of both the input flux and the conversion flux of glutamine (Q) to glutamate (E). The kinetic equations for the the Full-FI scheme are

(6)

where

Interestingly, we find that neither of the two Min-FI schemes yield steady-state solutions that are stable in all of the three regimes:

For the two-feedback Full-FI scheme, to maximize the growth rate in the

The numerical solution of the kinetic equations for the Full-FI scheme (Fig. 2A) shows that the growth-rate deficit

Integrating carbon and nitrogen inputs: partitioning of carbon into biomass and energy

Microorganisms integrate various nutrients to produce biomass. Since carbon sources (e.g. glucose, glycerol) are used for both biomass and energy, optimal partitioning of the carbon flux is essential for optimal growth. Here, we consider a simple module that integerates carbon and nitrogen fluxes. In E. coli, nitrogen in the form of ammonium (NH

The metabolic module shown in Fig. 2D integrates two elemental nutrients, carbon (C) and nitrogen (N). The module has one input carbon flux

Depending on the constraints on the input fluxes:

Like previous modules, we assume product-feedback inhibition of all the input fluxes (Fig. 2D). This yields the following kinetic equations for the metabolite-pool sizes

(7)

where

To achieve optimal growth, the feedback-inhibition constants are chosen according to the logic of flux-balance analysis, i.e. the carbon-dependent nitrogen flux is turned on only after the carbon-independent nitrogen flux reaches its maximum. This is accomplished by choosing

The kinetic equations (7) are readily solved numerically for the steady-state growth rate and metabolite-pool sizes (Fig. 2E,F). As for the linear pathway, the growth-rate deficit

In experiments, it has been shown that the ATP-independent GDH pathway is preferred under glucose-limited growth [32], [33], which is also consistent with the optimal FBA behavior that we find in our nutrient-integration module. Furthermore, when both carbon and nitrogen are available in excess, the ATP-independent GDH pathway is largely inactive, corresponding to

The results show that simple product-feedback inhibition is sufficient to achieve the optimal flux-balance growth rate in all regimes. As for the other modules considered, larger feedback-inhibition constants improve growth rate but result in large pools of non-growth-limiting metabolites. Increasing the Hill coefficients of the feedbacks restricts pool sizes and simultaneously reduces the growth-rate deficits.

Nitrogen assimilation in E. coli

Regulation of nitrogen assimilation in E. coli has been studied in great detail, perhaps more carefully than any other metabolic sub-network [25], [35], [36] (see also cites in [25]). As nitrogen assimilation involves both a metabolic cycle and nutrient integration, it offers a chance to examine the extent to which actual metabolic networks, beyond the much studied linear or branched biosynthetic pathways, are regulated by feedback inhibition circuits of the sort that we hypothesize above.

Our mathematical analysis of metabolic cycle and nutrient integration suggest a simple regulation scheme that allows near optimal steady-state growth. For the nitrogen assimilation GS/GOGAT cycle the analysis suggests feedback inhibition by glutamine and glutamate on GS and GOGAT, respectively. Feedback inhibition of GS by glutamine is well known. It does not involve standard allostery, but instead a bicyclic cascade of covalent modifications [37]. Interestingly, consistent with our suggestion that ultrasensitive feedback might be necessary for adequate control of metabolite pool sizes, it has been proposed that the purpose of this bicyclic cascade is to yield ultra-sensitive feedback [38]. Feedback inhibition of GOGAT by glutamate, in contrast, had not been explicitly considered until recent efforts at data-driven modeling of the network [25]. These efforts revealed that such feedback inhibition is essential to obtain models that match experimental data. Furthermore, examination of older literature reveals biochemical evidence for such feedback inhibition: glutamate and aspartate both inhibit GOGAT activity [39]. The effect of glutamate is an example of standard product inhibition of an enzyme, and was considered initially insignificant due to the high inhibition constant (i.e., the feedback is weak). However, given the large cellular pool size of glutamate (

For the ATP-independent nitrogen flux via GDH the analysis suggests feedback inhibition of GDH by the key nitrogen intermediates, glutamine and glutamate, which is again consistent with biochemical studies of purified GDH enzyme and with the existence of product inhibition of all enzymatic reactions [40], [41].

A prediction from our analysis is that large changes in metabolite pools will occur upon the onset of nutrient limitation. This also agrees well with experimental observations. For example, consider the dynamics of

Fig. 3A shows the experimental metabolite pool size dynamics following nitrogen limitation and subsequent upshift for wild-type E. coli, as well as E. coli lacking the covalent modification enzyme responsible for feedback inhibition of glutamine synthetase (GS) by glutamine (

Within our model, the WT strain is described by the module with all three feedbacks present (Fig. 2D), while the feedback-defective strain is described by the same basic module but without the feedback on carbon-dependent nitrogen input flux

Some of the system's dynamics, in particular the overshoot of glutamine in the wild-type strain, are not captured by our simple feedback model. Generally, time-delay in the feedback may result in an overshoot in a feedback-inhibited system. This is consistent with the specific implementation of feedback by glutamine on GS: a cascade of covalent modification reactions which occur on the

We also compared the growth rate response of the wild-type and feedback-defective strains to relief of nitrogen limitation. Consistent with experimental results, the simulations predicted a bigger increase in the growth rate in the WT strain than in the feedback-defective strain following nitrogen upshift (Fig. 3C,D). In the simulation, the reason for the slower growth in the feedback-defective strain post nitrogen up-shift is excessive drainage of the carbon metabolite pool (e.g.,

Discussion

Understanding metabolism and its regulation have long been central goals of biochemistry. Recently, flux-balance analysis (FBA), a constraint-based computational approach, has been used to predict the optimal metabolic fluxes and growth rates of microorganisms in different environments. In several cases, in particular involving E. coli, the FBA-predicted optima agree remarkably well with experiments [1], [34], raising the question “for cells to realize optimal growth how complex must metabolic regulation be?” We have addressed this question using a set of representative metabolic modules. We find that, in all the cases studied, simple product-feedback inhibition is enough to achieve nearly optimal growth. Furthermore, the divergence from optimality becomes arbitrarily small as the feedback-inhibition constants are increased.

An important trade-off is that larger inhibition constants result in larger pool sizes of non-growth-limiting metabolites, which can be detrimental to growth. However, ultrasensitive feedback mechanisms (i.e. those with high Hill coefficients) can substantially restrict these pool sizes; the higher the Hill coefficient of the feedbacks, the smaller the increase in pool size required to achieve the same degree of inhibition. This suggests that the need for ultrasensitive mechanisms to control metabolite pool sizes may account for some of the complexity found in metabolic regulation in real cells at both the transcriptional and post-transcriptional levels.

Can we hope to gain insight into real metabolism using the very simple models we studied? To address this question we examined the nitrogen assimilation network in E. coli, which involves both nutrient integration and a metabolic cycle. First, we found the feedback regulation scheme proposed by our mathematical analysis of representative modules aligns closely with the known regulation of the network. Second, we found reasonable agreement between simulations based on our simple feedback models and actual experimental results, for both wild type and feedback-defective E. coli. Comparing strains with different regulatory schemes allowed us to directly ask the question “is product-feedback inhibition essential for achieving optimal growth?” At least in the case of nitrogen up-shift, both our simulations and experimental data argue that it is: the feedback-defective strain grew substantially slower than wild type after the up-shift.

One of the central predictions of our feedback framework is that pool sizes will be large for non-growth-limiting metabolites. Since few metabolites are growth-limiting under any nutrient condition, the cells are likely to have large pools of multiple metabolites under a wide range of conditions. Therefore, we need to consider the possible impact of large pool sizes on cell physiology. Can large sizes of metabolite pools be detrimental to the well-being of cells? In fact, many metabolic intermediates, such as glyoxylate and formaldehyde, are toxic at high concentrations. Even the biosynthetic end-products required for growth (e.g. amino acids, nucleotides, etc.) can be detrimental to a cell's growth at high enough concentrations. Metabolites at high concentration can interact nonspecifically with various enzymes and disrupt metabolic reactions [43]. Furthermore, metabolite pools contribute to intracellular osmolarity and consequently to the osmotic pressure inside cells. Dedicated mechanisms to respond to osmotic stress have evolved in microorganisms, reflecting the harmful effects of osmotic imbalance [44]–[46]. For E. coli, the growth rate is maximized in conditions corresponding to external osmotic pressures of around

Ultrasensitivity is a common feature of feedback inhibition. At the transcriptional level, multiple promoter binding sites along with other cooperative mechanisms like DNA looping yield ultrasensitive responses [49] (Fig. 4A). The response time for transcriptional feedback is limited by protein degradation (and dilution), which in microorganisms is typically of the order of tens of minutes to hours. Metabolite-pool sizes, on the other hand, may change in just few seconds, e.g. the glutamine pool increased by over 10-fold in

Our study of simple representative metabolic modules is an attempt to identify the design principles underlying the regulatory mechanisms that optimize metabolic function, such as biomass production [53]. In addition to highlighting general lessons in metabolic regulation, our analysis raises new fundamental questions. How many feedbacks are required in a metabolic network, in particular the metabolic network of a real cell? What principles, in addition to optimal growth and stability, guide the evolutionary selection of feedbacks and feedback mechanisms? Has the complexity and dynamics of the cellular environment led to additional constraints on feedback strategies? And finally, given the apparent sufficiency of feedback inhibition, why are other regulatory motifs, such as allosteric enzyme activation, also found in metabolism? Further experiments in which metabolic feedbacks are eliminated, modified, and/or rewired, in concert with additional theoretical analyses, should facilitate answering these questions.

Materials and Methods

The analyses were carried out using kinetic equations (Eqs. 3, 5, 6, 7). The equations account for the concentration of each component in the metabolic modules and the steady-state solutions were numerically obtained using MATLAB. For details on the flux-balance analysis (FBA) see Text S1.

Supporting Information

Text S1

Additional information.

(0.25 MB PDF)

Acknowledgments

We thank B. Bennett and M. Reaves for helpful discussions and for sharing their measured metabolite concentrations in E. coli, and K. Huang, Pan-Jun Kim, P. Mehta, A. Motter, and D. Segre for critically reviewing the manuscript.

Footnotes

The authors have declared that no competing interests exist.

This work was partially supported by the Defense Advanced Research Projects Agency (DARPA) under grant HR0011-05-1-0057, and the Burroughs Wellcome Fund Graduate Training Program. JDR thanks NSF Career award NCB-0643859. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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