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 
, 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 
. 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 
. For E. coli
, the growth rate is maximized in conditions corresponding to external osmotic pressures of around
. Furthermore, the turgor pressure has been estimated to be around
in an AFM study of the magnetotactic Gram negative bacteria Magnetospirillium gryphiswaldense
. Consequently, the internal osmotic pressure is thought to be around
atm, which corresponds to an effective concentration of
mM of solute. Recent measurements have revealed that some metabolite pools can become very large, such as fructose-1,6-bisphosphate (
mM) and glutamate (
. These large metabolite pools could contribute significantly to the overall internal osmotic pressure of the cells. In general, pools that are large even when growth-limiting will potentially be very large when non-growth-limiting and may cause osmotic imbalance. Such pools in particular may require ultrasensitive feedback mechanisms to restrict their sizes. Experimental manipulation of feedback sensitivities (e.g. by enzyme mutation, knockout of enzymes involved in covalent modification cascades, etc.) should help shed light on the role of ultrasensitive feedback mechanisms.
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 
(). 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
seconds in the nutrient-switching experiment described above. The fast dynamics of metabolite-pool sizes suggests the need for fast feedback mechanisms. Fast regulation can be realized through various post-translational mechanisms – allosteric regulation of protein aggregates, e.g. ATP molecules bind cooperatively to a homodimer of pantothenate kinase 
, competition, e.g. Wee1 regulation of Cdk1 
, or covalent modifications, e.g. reversible phosphorylation of isocitrate dehydrogenase 
and the bicyclic cascade of covalent modifications in glutamine regulation 
(). Thus, the need for fast ultrasensitive feedback mechanisms may be a key driver of the observed complexity in metabolic regulation.
Examples of transcriptional and post-transcriptional regulation schemes for ultrasensitive feedback.
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 
. 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.