Pierre Millard
email: pierre.millard@insa-toulouse.fr.

Thomas Gosselin-Monplaisir

Sandrine Uttenweiler-Joseph

Brice Enjalbert

From toxic waste to beneficial nutrient: acetate boosts E. coli growth at low glycolytic flux

Keywords: acetate, kinetic model, metabolism, regulation, robustness

Acetate, a major by-product of glycolytic metabolism in Escherichia coli and many other microorganisms, has long been considered a toxic waste compound that inhibits microbial growth. This counterproductive auto-inhibition represents a major problem in biotechnology and has puzzled the scientific community for decades. Recent studies have revealed that acetate is also a co-substrate of glycolytic nutrients and a global regulator of E. coli metabolism and physiology. Here, we used a systems biology strategy to investigate the mutual regulation of glycolytic and acetate metabolism. Computational and experimental results demonstrate that reducing the glycolytic flux enhances co-utilization of acetate with glucose. Acetate metabolism thus compensates for the reduction in glycolytic flux and eventually buffers carbon uptake so that acetate, far from being toxic, actually enhances E. coli growth under these conditions. We validated this mechanism using three orthogonal strategies: chemical inhibition of glucose uptake, glycolytic mutant strains, and alternative substrates with a natively low glycolytic flux. Acetate makes E. coli more robust to glycolytic perturbations and is a valuable nutrient, with a beneficial effect on microbial growth.

Introduction

Metabolism is the fundamental biochemical process that converts nutrients into energy and cellular building blocks. The preferred nutrients of many organisms are glycolytic substrates, such as glucose, which are oxidized into pyruvate by glycolysis. In the presence of oxygen, pyruvate can then be fully oxidized into carbon dioxide and water via respiratory pathways involving the tricarboxylic acid (TCA) cycle and oxidative phosphorylation. When oxygen is limiting, pyruvate is incompletely oxidized through non-respiratory pathways, and fermentation by-products are released into the environment. Despite fermentation being less efficient than respiration, fermentation by-products are still released in the presence of oxygen by most microorganisms, including bacteria [START_REF] Harden | The chemical action of Bacillus coli communis and similar organisms on carbohydrates and allied compounds[END_REF], yeasts [START_REF] Pasteur | Mémoire sur la fermentation alcoolique[END_REF] and mammalian cells [START_REF] Warburg | The metabolism of tumors in the body[END_REF].

Acetate is one of the most common by-products of glycolytic metabolism in Escherichia coli and many other microorganisms [START_REF] Harden | The chemical action of Bacillus coli communis and similar organisms on carbohydrates and allied compounds[END_REF][START_REF] Kutscha | Microbial upgrading of acetate into value-added products-Examining microbial diversity, bioenergetic constraints and metabolic engineering approaches[END_REF][START_REF] Trcek | Adaptation and tolerance of bacteria against acetic acid[END_REF]. The established view is that acetate and other fermentation by-products are toxic and inhibit microbial growth [START_REF] Beck | Environment constrains fitness advantages of division of labor in microbial consortia engineered for metabolite push or pull interactions[END_REF][START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF][START_REF] Hollowell | Basis for the exclusion of Escherichia coli from the rumen ecosystem[END_REF][START_REF] Kutscha | Microbial upgrading of acetate into value-added products-Examining microbial diversity, bioenergetic constraints and metabolic engineering approaches[END_REF][START_REF] Luli | Comparison of growth, acetate production, and acetate inhibition of Escherichia coli strains in batch and fed-batch fermentations[END_REF][START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF][START_REF] Roe | Perturbation of anion balance during inhibition of growth of Escherichia coli by weak acids[END_REF][START_REF] Roe | Inhibition of Escherichia coli growth by acetic acid: a problem with methionine biosynthesis and homocysteine toxicity[END_REF][START_REF] Russel | Another explanation for the toxicity of fermentation acids at low pH: anion accumulation versus uncoupling[END_REF][START_REF] Trcek | Adaptation and tolerance of bacteria against acetic acid[END_REF][START_REF] Warnecke | Organic acid toxicity, tolerance, and production in Escherichia coli biorefining applications[END_REF]. Why E. coli should produce a self-toxic byproduct is an intriguing question, particularly since E. coli's natural environment, the gut, is acetate-rich. Several explanations for the toxic effect of acetate have been proposed, the classical one being the uncoupling effect of organic acids [START_REF] Russel | Another explanation for the toxicity of fermentation acids at low pH: anion accumulation versus uncoupling[END_REF], with acetic acid freely diffusing through the membrane before dissociating intracellularly into acetate and a proton due to its low pKa, and the energy required to expel the excess protons from the cell to maintain pH homeostasis being detrimental to growth. Another hypothesis is that the presence of acetate in the cytoplasm may affect the anion balance of the cell [START_REF] Roe | Perturbation of anion balance during inhibition of growth of Escherichia coli by weak acids[END_REF], reducing the pool of other anions -primarily glutamate -available to maintain osmotic pressure and electroneutrality and thereby impeding cell function. Acetate has also been reported to inhibit a step in the methionine biosynthesis pathway [START_REF] Roe | Inhibition of Escherichia coli growth by acetic acid: a problem with methionine biosynthesis and homocysteine toxicity[END_REF], reducing the intracellular methionine pool and concomitantly increasing the concentration of the toxic intermediate homocysteine. Finally, acetate may modulate the accumulation of acetylphosphate [START_REF] Weinert | Acetyl-phosphate is a critical determinant of lysine acetylation in E. coli[END_REF][START_REF] Wolfe | Evidence that acetyl phosphate functions as a global signal during biofilm development[END_REF], an intermediate of the acetate pathway and a signaling metabolite that can phosphorylate and acetylate enzymes and regulatory proteins with broad physiological consequences. However, none of these studies conclusively explains how acetate inhibits microbial growth [START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF]. The physiological role of fermentation under aerobic conditions also remains unclear, especially since it leads to the accumulation of toxic by-products at the expense of biomass production.

Acetate overflow and its impact on E. coli are also important issues in biotechnology, where acetate is both produced from glycolytic carbon sources and found in many renewable resources [START_REF] Klinke | Inhibition of ethanol-producing yeast and bacteria by degradation products produced during pre-treatment of biomass[END_REF][START_REF] Kutscha | Microbial upgrading of acetate into value-added products-Examining microbial diversity, bioenergetic constraints and metabolic engineering approaches[END_REF][START_REF] Palmqvist | Fermentation of lignocellulosic hydrolysates. I: inhibition and detoxification[END_REF][START_REF] Sun | Complete and efficient conversion of plant cell wall hemicellulose into high-value bioproducts by engineered yeast[END_REF][START_REF] Trcek | Adaptation and tolerance of bacteria against acetic acid[END_REF]. As well as inhibiting growth, acetate production diverts carbon that could otherwise be used to synthesize biomass or valuable compounds. Acetate therefore decreases the productivity of bioprocesses. Since acetate is only produced at high glucose uptake rates [START_REF] Nanchen | Nonlinear dependency of intracellular fluxes on growth rate in miniaturized continuous cultures of Escherichia coli[END_REF][START_REF] Renilla | Acetate scavenging activity in Escherichia coli: interplay of acetyl-CoA synthetase and the PEP-glyoxylate cycle in chemostat cultures[END_REF][START_REF] Valgepea | Systems biology approach reveals that overflow metabolism of acetate in Escherichia coli is triggered by carbon catabolite repression of acetyl-CoA synthetase[END_REF], a common approach to reduce acetate overflow is to limit glucose uptake by process engineering (e.g. glucose-limited chemostat or fed-batch culture) or by metabolic engineering (e.g. deletion of glucose transporters) [START_REF] Bernal | Acetate metabolism regulation in Escherichia coli: carbon overflow, pathogenicity, and beyond[END_REF][START_REF] De Mey | Minimizing acetate formation in E. coli fermentations[END_REF][START_REF] Eiteman | Overcoming acetate in Escherichia coli recombinant protein fermentations[END_REF][START_REF] Forster | Modification of glucose import capacity in Escherichia coli: physiologic consequences and utility for improving DNA vaccine production[END_REF][START_REF] Kutscha | Microbial upgrading of acetate into value-added products-Examining microbial diversity, bioenergetic constraints and metabolic engineering approaches[END_REF][START_REF] Luli | Comparison of growth, acetate production, and acetate inhibition of Escherichia coli strains in batch and fed-batch fermentations[END_REF][START_REF] Peebo | Coordinated activation of PTA-ACS and TCA cycles strongly reduces overflow metabolism of acetate in Escherichia coli[END_REF]. The utilization of acetate in combination with glycolytic substrates could alleviate growth inhibition and allow acetate to act as both a renewable carbon source and a relevant (co-)substrate for bioproducts that are derived from acetyl-CoA [START_REF] Krivoruchko | Microbial acetyl-CoA metabolism and metabolic engineering[END_REF][START_REF] Kutscha | Microbial upgrading of acetate into value-added products-Examining microbial diversity, bioenergetic constraints and metabolic engineering approaches[END_REF][START_REF] Peebo | Coordinated activation of PTA-ACS and TCA cycles strongly reduces overflow metabolism of acetate in Escherichia coli[END_REF][START_REF] Wei | Enhanced biofuel production through coupled acetic acid and xylose consumption by engineered yeast[END_REF]. However, despite decades of attempts to reduce its accumulation and improve its utilization, acetate is still considered a major problem in biotechnology [START_REF] Bernal | Acetate metabolism regulation in Escherichia coli: carbon overflow, pathogenicity, and beyond[END_REF][START_REF] De Mey | Minimizing acetate formation in E. coli fermentations[END_REF][START_REF] Luli | Comparison of growth, acetate production, and acetate inhibition of Escherichia coli strains in batch and fed-batch fermentations[END_REF][START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF].

Recently, we found that acetate can also act as a co-substrate for glycolytic carbon sources such as glucose, fucose and gluconate [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF], and is therefore more than just a waste by-product. The switch between acetate production and consumption was found to be determined thermodynamically by the acetate concentration itself. Acetate is also a global regulator of E. coli metabolism and physiology [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]. These observations suggest an alternative rationale for acetate toxicity, namely that the inhibition of growth and glucose uptake may result from coordinated downregulation of the expression of most central metabolic genes (glucose uptake systems, glycolysis, TCA cycle) and ribosomal genes. To date, the role of acetate as a co-substrate and regulator of glycolytic metabolism has only been investigated under high glycolytic flux, and little is known about the response of E. coli to acetate at low glycolytic fluxes, conditions that are nevertheless commonly experienced by this bacterium in its natural, industrial and laboratory environments [START_REF] Cummings | Fermentation in the human large intestine and the available substrates[END_REF][START_REF] Cummings | Prebiotic digestion and fermentation[END_REF][START_REF] De Graaf | Profiling human gut bacterial metabolism and its kinetics using [U-13 C]glucose and NMR[END_REF][START_REF] Fabich | Comparison of carbon nutrition for pathogenic and commensal Escherichia coli strains in the mouse intestine[END_REF][START_REF] Jones | Respiration of Escherichia coli in the mouse intestine[END_REF][START_REF] Kleman | Acetate metabolism by Escherichia coli in high-cell-density fermentation[END_REF][START_REF] Macfarlane | Comparison of fermentation reactions in different regions of the human colon[END_REF][START_REF] Shen | High-level dietary fibre up-regulates colonic fermentation and relative abundance of saccharolytic bacteria within the human faecal microbiota in vitro[END_REF][START_REF] Wolfe | The acetate switch[END_REF].

In this study, we took a systems biology perspective to clarify the interplay between glycolytic and acetate metabolism in E. coli. We used a kinetic model of E. coli metabolism to generate hypotheses on the relationships between glycolytic, acetate and growth fluxes. We validated model predictions experimentally, explored the underlying biochemical and regulatory mechanisms on glucose, and extended our findings to other glycolytic nutrients.

Results

Model-driven hypotheses on the interplay between glucose and acetate metabolism in

Escherichia coli. We explored the functional relationship between glycolytic and acetate metabolism in Escherichia coli using a coarse-grained kinetic model coupling glucose and acetate metabolism with growth (Figure 1A) [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]. The model included two compartments, six compounds, and six reactions representing (i) glucose uptake and its conversion into acetyl-CoA by glycolysis, (ii) acetyl-CoA utilization in the TCA cycle and anabolic growth pathways, and (iii) the reversible conversion of acetyl-CoA into acetate via phosphotransacetylase (Pta), acetate kinase (AckA), and acetate exchange between the cell and its environment. Acetate utilization via acetylCoA synthetase (Acs), which is not active when glucose is present [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF], was not included. The model also included two inhibitory terms representing on the one hand inhibition of glucose uptake and glycolysis, and on the other inhibition of the TCA cycle and growth, in line with the observed inhibition of catabolic and anabolic processes [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]. This model has been extensively validated and shown to have good predictive capabilities [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]. We used it to simulate the response of glycolytic, acetate and growth fluxes to different degrees of glycolytic limitation by progressively reducing the activity (i.e. Vmax) of the glycolytic pathway -and thereby the glycolytic flux -from 100 % to 20 % of its initial value. Since glycolytic and acetate fluxes are both determined by the concentration of acetate [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF][START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF][START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF], these simulations were carried out over a broad range of acetate concentrations (from 0.1 to 100 mM).

The results obtained (Figure 1B-D) highlight the strong bidirectional interplay between glycolytic and acetate metabolism, which eventually modulate growth. Glucose uptake and acetate flux both respond nonlinearly to perturbations (Figure 1B-C). According to the model, increasing the acetate concentration inhibits glycolysis (Figure 1B) and reverses the acetate flux (Figure 1C) over the full range of glycolytic fluxes, as previously reported at the highest glucose uptake rate [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]. The acetate concentration at which the flux reversal occurs appears to depend on the glycolytic flux (Figure 1C). Reducing the glycolytic flux lowers acetate production when acetate is a by-product of glucose metabolism, but seems to enhance acetate utilization when acetate is a co-substrate of glucose (Figure 1C). Finally, the glycolytic flux and the acetate concentration both have a strong, nonlinear effect on the growth rate (Figure 1D). At maximal glycolytic activity, the predictions of the model are consistent with the established deleterious effect of acetate on microbial growth (Figure 1D). However, despite the dual inhibition of catabolism and anabolism by acetate included in the model, it also predicts that at low glycolytic flux, increasing the acetate concentration should boost E. coli growth (Figure 1D). A global sensitivity analysis showed that the predictions of the model are robust to parameter uncertainty. We present these model-driven hypotheses in detail (including the sensitivity analysis results) in the following sections along with experimental results collected for validation. Note that none of the data obtained in this study were used to build or calibrate the model and therefore represent independent tests of its predictions and inferred hypotheses. A Representation (in Systems Biology Graphical Notation format, http://sbgn.org) of the glucose and acetate metabolism of Escherichia coli, as implemented in the kinetic model used in the study (reproduced from [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]).

B-D Glucose uptake flux (B), acetate flux (C) and growth rate (D) simulated over a broad range of glycolytic activity levels (from 20 to 100 % of the initial Vmax) and acetate concentrations (from 0.1 to 100 mM).

Enhanced acetate utilization at low glycolytic flux buffers carbon uptake. We investigated the glycolytic control of E. coli metabolism by confronting the model predictions with experimental data (Figures 1 and2). The simulations show that, at low acetate concentrations (0 mM in Figure 2A-D), acetate production is predicted to increase with the glycolytic flux, in line with the relationship observed in glucose-limited chemostat experiments [START_REF] Nanchen | Nonlinear dependency of intracellular fluxes on growth rate in miniaturized continuous cultures of Escherichia coli[END_REF][START_REF] Renilla | Acetate scavenging activity in Escherichia coli: interplay of acetyl-CoA synthetase and the PEP-glyoxylate cycle in chemostat cultures[END_REF][START_REF] Valgepea | Systems biology approach reveals that overflow metabolism of acetate in Escherichia coli is triggered by carbon catabolite repression of acetyl-CoA synthetase[END_REF] and in a set of mutants with different levels of glucose uptake [START_REF] Forster | Modification of glucose import capacity in Escherichia coli: physiologic consequences and utility for improving DNA vaccine production[END_REF]. In contrast, when the acetate concentration is high (30 mM in Figure 2A-D), such that it is co-consumed with glucose, acetate utilization is expected to increase when the glycolytic flux decreases. This response implies, as suggested previously [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF] but never verified experimentally, that the control exerted by glycolysis on the acetate flux is non-monotonic.

To test this hypothesis, we measured growth, glucose uptake, and acetate fluxes in E. coli K-12 MG1655 grown on 15 mM glucose and increased the concentration of the glycolytic inhibitor methyl α-D-glucopyranoside (αMG, an analogue of glucose that is taken up and phosphorylated but not metabolized by glycolysis) [START_REF] Bren | Glucose becomes one of the worst carbon sources for E. coli on poor nitrogen sources due to suboptimal levels of cAMP[END_REF][START_REF] Negrete | Glucose uptake regulation in E. coli by the small RNA SgrS: comparative analysis of E. coli K-12 (JM109 and MG1655) and E. coli B (BL21)[END_REF] (Figure 2E-H). Experiments were carried out without acetate or with 30 mM acetate for comparison with the predictions shown in Figure 2A-D. Results confirm that αMG inhibits glucose uptake under both conditions (Figure 2E). In the absence of acetate, the acetate (production) flux decreased when the αMG concentration was increased (Figure 2F), as expected based on the positive control exerted by glycolysis on the acetate flux. In contrast, in the presence of 30 mM acetate, increasing the αMG concentration increased the acetate uptake flux (Figure 2F). These results are consistent with the predictions of the model (Figure 2A-D) and demonstrate that glycolysis controls the acetate flux in a non-monotonic manner. The control exerted by glycolysis on acetate metabolism thus depends on the concentration of acetate itself.

The growth and total carbon uptake rates decreased when the αMG concentration was increased in the absence of acetate (Figure 2G-H), but were more stable in the presence of acetate despite a ~2-fold reduction in glucose uptake (between 5.7 and 3.3 mmol•g DW -1

•h -1 with 30 mM acetate; Figure 2E). Indeed, both the total carbon uptake and the growth rates were less sensitive to glycolytic perturbations in the presence of acetate than in the absence of acetate (Figure 2I-J), suggesting that acetate compensates for the drop in glucose uptake. Consistently, a strong, nonlinear relationship between glucose uptake and acetate flux was observed in all experiments (Figure 2K), which was reflected in the contribution of acetate to the total carbon uptake of E. coli (Figure 2L). At the maximal glucose uptake rate, acetate production represented a 10 % loss in carbon. However, when glucose uptake was below 7 mmol•g DW -1 •h -1 , acetate was co-utilized with glucose and contributed positively to the carbon balance (up to 50 % at a glucose uptake rate of 3.3 mmol•g DW -1

•h -1 ; Figure 2L). These results highlight the coupling between glycolytic and acetate fluxes in E. coli, with acetate buffering carbon uptake at low glycolytic flux, thereby improving the robustness of E. coli to glycolytic perturbations.

Figure 2. Glycolytic control of E. coli metabolism.

A-D Predicted glucose uptake flux (A), acetate flux (B), growth rate (C), and total carbon uptake flux (D) of E. coli on glucose (15 mM) as a function of the glycolytic activity (20-100 % of the maximal activity), at low (0 mM) and high (30 mM) acetate concentrations.

E-H Experimental glucose uptake flux (E), acetate flux (F), and growth rate (G) of E. coli grown on glucose (15 mM) with different concentrations of αMG (0-100 mM), at low (0 mM) and high (30 mM) acetate concentrations. These data were used to quantify the total carbon uptake flux as a function of the αMG concentration (H). Each data point represents an independent biological replicate, and lines represent the best polynomial fits. I, J Total carbon uptake (I) and growth rate (J) as a function of the glucose uptake flux, at low (0 mM) and high (30 mM) acetate concentrations. Each data point represents an independent biological replicate, and lines represent the best linear fits. K, L Relationships between the acetate flux and glucose uptake (K), and between the contribution of acetate to carbon uptake and glucose uptake (L). Each data point represents an independent biological replicate, and lines represent the best polynomial fits.

The glycolytic flux determines the acetate concentration threshold at which the acetate flux reverses. As already reported at maximal glucose uptake rates [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF][START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF], the acetate concentration threshold at which the acetate flux switches between production and consumption is about 10 mM. However, this threshold was predicted to depend on the glycolytic flux (Figures 1C and3A-B). According to the model, the switch should occur at an acetate concentration of 3.8 mM when the glycolytic activity is 60 % of its maximum value, and at a concentration of just 0.6 mM when the glycolytic activity is reduced to 20 % of the maximum. This suggests that the glycolytic flux determines the role of acetate as a co-substrate or by-product of glycolytic metabolism.

We tested this hypothesis by measuring acetate fluxes in E. coli grown on glucose (15 mM) plus acetate (from 0.1 to 100 mM) at different concentrations of αMG (0, 20, 40 or 100 mM, Figure 3C). As previously observed at full glycolytic activity (i.e., 0 mM αMG) [START_REF] Beck | Environment constrains fitness advantages of division of labor in microbial consortia engineered for metabolite push or pull interactions[END_REF][START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF][START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF][START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF], acetate production decreased when the acetate concentration was increased (Figure 3C), and the acetate flux reversed at a concentration of ~6 mM (Figure 3C-D), resulting in co-consumption of glucose and acetate. This threshold concentration gradually decreased when the glycolytic flux was reduced (from 2 mM acetate at 20 mM αMG to below 0.1 mM acetate at 40 or 100 mM αMG).

Here again, the experimental profiles are qualitatively similar to the simulated profiles, albeit with quantitative differences. These results demonstrate that the acetate concentration threshold at which the acetate flux reverses depends on the glycolytic flux. It therefore follows that when the glycolytic flux is limited, acetate is utilized as a co-substrate at lower acetate concentrations. These results also confirm that inhibiting glycolysis enhances acetate utilization over the full range of acetate concentrations tested here, further validating the unusual control patterns identified above, and generalizing their functional relevance for acetate concentrations spanning three orders of magnitude. C Experimental acetate flux of E. coli K-12 MG1655 grown on glucose (15 mM) at different acetate concentrations in the presence of αMG (0, 20, 40 or 100 mM). The data measured in the absence of αMG are reproduced from [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF]. Each data point represents an independent biological replicate, and the lines represent best fits using a logistic function. D Experimental acetate concentration threshold at which the acetate flux reverses as a function of the αMG concentration, determined from the best fits shown in panel (C).

Acetate enhances E. coli growth at low glycolytic flux. Acetate is considered toxic for microbial cells, but the model predicts this might not always be the case. At full glycolytic activity, the model predicts that maximal growth should occur at low acetate concentrations, with the growth rate remaining above 95 % of the maximum only at acetate concentrations below 12 mM (Figures 1D and4A). In agreement with the reported toxicity of acetate, the growth rate then monotonously decreases for acetate concentrations above 12 mM. However, when the glycolytic flux is reduced, maximal growth is predicted to occur at high acetate concentrations (Figures 1D and4A). For instance, when the glycolytic activity is reduced to 20 %, the growth rate is expected to be maximal at an acetate concentration of 28 mM, and should remain above 95 % of the maximum over a broad range of acetate concentrations (between 12 and 86 mM). According to the model therefore, acetate should enhance E. coli growth at low glycolytic flux.

To test this prediction, we measured the growth rate of E. coli on glucose (15 mM) with acetate (0.1 to 100 mM) in the presence of 100 mM αMG to inhibit glycolysis (Figure 4B). The data measured in the absence of αMG are shown for comparison. At maximal glycolytic flux, the growth rate decreased monotonically with the acetate concentration, but this relationship became non-monotonic when glycolysis was inhibited, which is qualitatively consistent with model predictions. In the presence of αMG, the growth rate was lower in the absence of acetate than at high acetate concentrations (0.28 ± 0.02 h -1 with acetate < 0.5 mM versus 0.34 ± 0.01 h -1 with 60 mM acetate, p-value = 8.10 -3 ). As predicted by the model therefore, acetate appears to be beneficial rather than toxic to E. coli when the glycolytic flux is reduced. The growth rate was slightly lower at 100 mM acetate (0.30 ± 0.03 h -1 ) than at 60 mM acetate, but the similar growth rates observed across the concentration range demonstrate that acetate is not toxic when the glycolytic flux is low, even at high concentrations.

As an orthogonal, alternative strategy to the use of a glycolytic inhibitor, we tested the impact of acetate on the growth of mutant strains with low glycolytic flux, namely the E. coli K-12 BW25113 Δpgi and ΔpfkA strains, where the glycolytic flux is 24 and 29 % that of the wildtype strain, respectively [START_REF] Long | Metabolic flux responses to deletion of 20 core enzymes reveal flexibility and limits of E. coli metabolism[END_REF]. Their growth rates were measured on 15 mM glucose, with or without 10 mM acetate and the effect of acetate was quantified as the relative change in growth rate. The wild-type isogenic strain was grown in the absence and presence of αMG (100 mM) as a control. As expected, acetate reduced the growth rate of the wild-type strain in the absence of αMG (-6 ± 1 %, p-value = 4.10 -5 ) but increased its growth in the presence of 100 mM αMG (+9 ± 2 %, p-value = 3.10 -5 ) (Figure 4C). The beneficial effect of acetate on growth was also observed in the Δpgi and ΔpfkA strains (Figure 4D), whose growth rate strongly increased despite the absence of αMG (Δpgi : +60 ± 19 %, p-value = 3.10 -2 ; ΔpfkA : +117 ± 21 %, p-value = 1.10 -2 ). These results confirm that the beneficial effect of acetate is not due to the presence of αMG but is a consequence of the decrease in glycolytic flux.

Overall, all results are in line with the predictions of the model and demonstrate that acetate enhances E. coli growth when the glycolytic flux is reduced by either chemical or genetic perturbations. [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]) that the apparent "toxicity" of acetate is due to its inhibition of most metabolic genes -including those for the glucose phosphotransferase system (PTS), glycolysis and TCA cycle. Acetate also increases the expression of genes related to stress responses and downregulates genes related to transport, energy biosynthesis and mobility for instance. To determine if acetate has a similar regulatory effect at low glycolytic flux, we compared the transcriptomes of E. coli grown on glucose (15 mM) plus acetate (0, 10, 50 or 100 mM) in the presence of αMG (100 mM), with those measured in the absence of αMG at the same acetate concentrations [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF].

We observed that contrary to what occurs at maximal glycolytic flux (0 mM αMG) acetate did not trigger a stress response at low glycolytic flux (100 mM αMG) (Figure 5A). Conversely, acetate increased gene expression for the respiratory chain and some biosynthetic processes (ribonucleotide monophosphate biosynthesis, amino acid production). Other hallmarks of the effect of acetate at maximal glycolytic flux such as the inhibition of mobility-related gene expression were also not observed at low glycolytic flux. The activation of glutamate catabolism, the repression of transport related genes, and the repression of carbohydrate transport and catabolism were among the few similarities between the responses to acetate at high and low glycolytic flux.

At the metabolic level, acetate-driven inhibition of the expression of central metabolic genes (PTS, TCA and glycolytic enzymes) and growth machinery (ribosomal proteins) was not observed at low glycolytic flux (Figure 5B). In fact, expression of most of these genes (47/65) was slightly higher in the presence of 50 mM acetate and 100 mM αMG, whereas at maximal glycolytic flux it was lower at these acetate concentrations for a similar proportion (46/65). When the glycolytic flux was inhibited, ptsG expression, which controls glucose uptake [START_REF] Rohwer | Understanding glucose transport by the bacterial phosphoenolpyruvate:glycose phosphotransferase system on the basis of kinetic measurements in vitro[END_REF], was 70 % higher at 10 and 50 mM acetate. The expression of most glycolytic genes was 15 to 110 % higher at 50 mM acetate, while the expression of TCA cycle genes remained stable. Regarding acetate metabolism, the expression of pta and ackA was slightly higher at 10 and 50 mM acetate, while alternative acetate utilization via Acs was inhibited by acetate (5-fold reduction in expression at 100 mM). Finally, expression of genes encoding ribosomal proteins (rpl, rpm and rps operons) was increased by 57 % on average at 50 mM acetate and 48 % on average at 100 mM acetate (Dataset S1).

These results indicate that the stress response and lower metabolic activity triggered by acetate at high glycolytic flux do not occur at low glycolytic flux. In this situation, acetate enhances the expression of ribosomal and central metabolic genes, in line with its positive effect on growth. This is the opposite of what is observed at high glycolytic flux, demonstrating that the regulatory role of acetate is largely determined by the glycolytic flux. Acetate-enhanced growth is supported by the Pta-AckA pathway. According to the model (Figure 1A), the reversible Pta-AckA pathway (acetate consumption or production) explains by itself how acetate promotes E. coli growth, without the need for Acs (acetate consumption only). To confirm this hypothesis, we measured the growth rates of E. coli K-12 BW25113 wild-type and of Δpta and Δacs isogenic mutant strains cultivated on 15 mM glucose plus 100 mM αMG, with or without 10 mM acetate. The effect of acetate was quantified as the relative change in growth rate induced. Acetate had a similar, beneficial effect on the growth of the wild-type strain (+9 ± 2 %, p-value = 3.10 -5 ) and of the Δacs strain (+8 ± 1 %, p-value = 1.10 - 3 ), but not for the Δpta strain, whose growth rate decreased slightly but significantly (-6 ± 1 %, p-value = 7.10 -3

). These results confirm that the beneficial effect of acetate on E. coli growth at low glycolytic flux is mediated by the Pta-AckA pathway, with no involvement of Acs in this phenomenon. Acetate boosts growth of E. coli on glycerol and galactose. Assuming that the beneficial impact of acetate on E. coli growth stems from the tight interplay of glycolytic and acetate fluxes, this should be independent of the nature of the carbon source, implying that acetate should boost E. coli growth on glycolytic substrates with low glycolytic flux, such as glycerol and galactose [START_REF] Gerosa | Pseudo-transition analysis identifies the key regulators of dynamic metabolic adaptations from steady-state data[END_REF]. And indeed, wild-type E. coli growth was enhanced on both glycerol (+7 ± 1 %, p-value = 1.10 -2

) and galactose (+52 ± 1 %, p-value = 2.10 -4 ) in the presence of 10 mM acetate (Figure 7). Similar levels of enhancement were observed for the Δacs strain (+8 ± 1 % on glycerol, p-value = 3.10 -4 ; +53 ± 5 % on galactose, p-value = 3.10 -3 ), but not for the Δpta strain (-8 ± 2 % on glycerol, p-value = 1.10 -2 ; +2 ± 12 % on galactose, pvalue = 0.70). These results demonstrate that acetate promotes E. coli growth on galactose and glycerol. As observed for glucose, the growth enhancement is mediated exclusively by the Pta-AckA pathway, with no involvement of Acs. These results extend and generalize our findings to glycolytic carbon sources other than glucose and support our findings in more physiological conditions, i.e. without an inhibitor or deletion of metabolic genes.

Discussion

Following a systems biology approach, we used a kinetic model of E. coli metabolism to generate a comprehensive map of the functional relationship between glycolytic and acetate metabolisms and predict their combined impact on E. coli growth. In agreement with the predictions of the model, experimental results revealed a strong interplay between acetate and glycolytic fluxes in E. coli, with wide-ranging physiological consequences for the cells. Despite its established toxicity for microbial growth, we found that acetate improves the robustness of E. coli to glycolytic perturbations, and even boosts growth when the glycolytic flux is low.

Previous studies of the glycolytic control of acetate metabolism were carried out in the absence of acetate. In this situation, as observed here, glycolysis as a positive effect on acetate production. However, our results also reveal that in the presence of acetate, glycolysis negatively affects acetate utilization. Glycolytic control of the acetate flux is thus determined by the concentration of acetate itself. Because of this non-monotonous control pattern, reduced glycolytic flux increases acetate uptake and lowers the acetate concentration threshold at which acetate is co-consumed with glucose, with the overall effect of increasing acetate utilization by E. coli. This demonstrates that the metabolic status of acetate as a byproduct or co-substrate of glycolytic carbon sources is determined both by the concentration of acetate [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF][START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF] and the glycolytic flux, and thus by the availability of glycolytic nutrients. This mechanism is exclusively mediated by the Pta-AckA pathway and its role and activity are determined by the nutritional conditions experienced by E. coli. The Pta-AckA pathway can therefore be seen as a metabolic valve that integrates information on the nutritional environment of the cells to rapidly adjust the acetate flux depending on the availability of glycolytic substrates and acetate. Our results indicate that these properties, which emerge simply from the distribution of control throughout the network, buffers the total carbon uptake flux and makes E. coli more robust to changes in the nutritional environment.

When the glycolytic flux is low, enhanced acetate utilization promotes E. coli growth, demonstrating that acetate can be beneficial to E. coli, even at high concentrations. Whereas 60 mM acetate has been shown to inhibit growth by ~35 % at maximal glucose uptake rates [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF][START_REF] Luli | Comparison of growth, acetate production, and acetate inhibition of Escherichia coli strains in batch and fed-batch fermentations[END_REF][START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF], the same concentration of acetate had a positive effect on growth at low glucose uptake rates (+21 % in presence of 100 mM αMG). We confirmed the beneficial effect of acetate on E. coli growth using three orthogonal strategies to reduce the glycolytic flux: i) chemical inhibition of glucose uptake with αMG, ii) mutant strains deleted for glycolytic genes, and ii) glycolytic substrates with a naturally low glycolytic flux (glycerol and galactose). Since the impact of acetate on growth is partly determined by the glycolytic flux, this metabolic by-product is not toxic per se. Importantly, these results also weaken several hypotheses that have been proposed to explain the apparent "toxicity" of acetate for microbial growth. Indeed, perturbations of the proton gradient, of the anion composition of the cell, or of methionine biosynthesis, would have a monotonic impact on growth. The slight but global activating effect of acetate on the transcriptome at low glycolytic flux, which correlates with the growth response, also contrasts with its inhibitory effect at high glycolytic flux. The results of this study indicate that mechanisms that only explain growth inhibition at high glycolytic flux are inadequate. Likewise, the recent suggestion that gluconeogenic substrates such as acetate enhance E. coli growth on glycolytic substrates (as observed for mixtures of oxaloacetate with glycerol, xylose or fucose) [START_REF] Okano | Regulation underlying hierarchical and simultaneous utilization of carbon substrates by flux sensors in Escherichia coli[END_REF]) also fails to capture the non-monotonic growth response of E. coli to acetate. The regulatory effect of acetate on E. coli metabolism, which depends both on the acetate concentration and on the glycolytic flux, is therefore more complex than previously suggested [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF] and should be investigated further.

Previous reports have shown that the "toxicity" of acetate for E. coli growth depends on the pH [START_REF] Lawford | Effects of pH and acetic acid on glucose and xylose metabolism by a genetically engineered ethanologenic Escherichia coli[END_REF][START_REF] Orr | Extracellular acidic pH inhibits acetate consumption by decreasing gene transcription of the tricarboxylic acid cycle and the glyoxylate shunt[END_REF][START_REF] Pinhal | Acetate metabolism and the inhibition of bacterial growth by acetate[END_REF]. In our experiments, the pH was initially set to 7.0 but was not adjusted during growth. Nevertheless, the high buffer capacity of the M9 cultivation medium ensured that the pH remained stable in the exponential growth phase where growth rates and extracellular fluxes were measured, with changes of less than 0.3 pH unit in all experiments (Figure S1). Therefore, the metabolic and physiological responses of E. coli to acetate highlighted in this study are not due to changes in pH. Another potential issue is that the effect of acetate might also depend on the concentration of salts [START_REF] Bae | Effect of salt addition on acid resistance response of Escherichia coli O157:H7 against acetic acid[END_REF][START_REF] Lee | Antagonistic effect of acetic acid and salt for inactivating Escherichia coli O157:H7 in cucumber puree[END_REF]. The effect of acetate added as sodium salt may differ from its effect when added as acetic acid. However, repeat experiments performed with acetic acid instead of sodium acetate showed that the beneficial effect of acetate on E. coli growth on glucose, galactose and glycerol was maintained (Figure S2).

The measured responses of E. coli to a broad range of perturbations of the glycolytic and acetate metabolisms were qualitatively consistent with model predictions, though quantitative differences were observed. These differences may stem from the data used to construct and calibrate the model, which were obtained exclusively at maximal glycolytic flux. Therefore, although it was not initially developed to address the questions of this study, the kinetic model used here successfully predicts the observed interplay between glycolytic and acetate metabolisms, as well as the beneficial effect of acetate on E. coli growth at low glycolytic flux. Since the model does not account for the activation by acetate of the expression of glucose uptake, glycolytic, and ribosomal genes, this suggests that transcriptional activation by acetate may not be crucial for the flux response of E. coli. The metabolic network may indeed directly sense and integrate the availability of glucose and acetate to coordinate glucose and acetate fluxes, as already reported for the coordination other metabolic processes [START_REF] Doucette | alpha-Ketoglutarate coordinates carbon and nitrogen utilization via enzyme I inhibition[END_REF][START_REF] Millard | Metabolic regulation is sufficient for global and robust coordination of glucose uptake, catabolism, energy production and growth in Escherichia coli[END_REF][START_REF] Xiao | Glucose feeds the tricarboxylic acid cycle via excreted ethanol in fermenting yeast[END_REF]. In keeping with this hypothesis, our results support the concept of acetate flux being thermodynamically controlled by acetate itself, as detailed in [START_REF] Enjalbert | Acetate fluxes in Escherichia coli are determined by the thermodynamic control of the Pta-AckA pathway[END_REF], and we show that the thermodynamically-driven flux reversal is also controlled by the glycolytic flux. The reduction in glucose-derived acetyl-CoA is counterbalanced by an increased contribution of acetate, which sustains growth. Metabolic regulation is thus (at least partly) responsible for the interplay between acetate and glycolytic flux. The precise regulatory program driven by acetate remains to be identified.

Acetate concentrations in the gut range from 30 to 100 mM and glycolytic nutrients, such as glycerol and galactose [START_REF] Fabich | Comparison of carbon nutrition for pathogenic and commensal Escherichia coli strains in the mouse intestine[END_REF][START_REF] Matamouros | Adaptation of commensal proliferating Escherichia coli to the intestinal tract of young children with cystic fibrosis[END_REF][START_REF] Snider | E. coli O157:H7 catabolism of intestinal mucin-derived carbohydrates and colonization[END_REF], are generally scarce. Given the established inhibition of microbial growth by acetate, these high acetate levels were considered detrimental to E. coli colonization. Our results indicate that acetate, far from being toxic, may in fact be beneficial to E. coli in this environment. Acetate enhances E. coli growth when glycolytic substrates are limiting and may therefore facilitate its colonization and survival in the gut. Moreover, acetate stabilizes growth when the glycolytic flux is perturbed by temporary nutrient shortages, which are frequent in the gut. We suggest that the underlying regulatory mechanism may therefore be optimal for E. coli in its natural environment. Our results also call for the role of acetate in the gut to be reconsidered, since it represents a valuable and abundant resource that may benefit E. coli and other microorganisms.

In biotechnology, developing efficient strategies to overcome the negative effects of acetate requires understanding what controls the acetate flux and how in turn acetate controls microbial metabolism. By addressing these two needs, our findings should immediately help design bioprocesses with reduced acetate overflow and accelerate the development of mixed feed processes in which acetate is utilized alongside glucose or other glycolytic substrates. Current strategies are mainly based on increasing Acs activity to (re)use acetate present in the feedstock, provided as a co-substrate or produced by the Pta-AckA pathway. Given the Pta-AckA pathway produces one ATP and Acs consumes two ATPs, strategies based on activating the Pta-AckA-Acs cycle are highly energy-consuming [START_REF] Valgepea | Systems biology approach reveals that overflow metabolism of acetate in Escherichia coli is triggered by carbon catabolite repression of acetyl-CoA synthetase[END_REF]. Strategies involving the Pta-AckA pathway alone are inherently less costly for the cell. Our results show that the Pta-AckA pathway enables efficient co-utilization of acetate and glycolytic substrates (glucose, glycerol, galactose) in E. coli by itself, with a positive effect on growth. The comprehensive understanding of the relationship between glycolytic and acetate metabolisms provided by this study may also represents a valuable guide to increase the productivity of valuable products synthesized from acetate-derived AcCoA [START_REF] Becker | Advanced biotechnology: metabolically engineered cells for the bio-based production of chemicals and fuels, materials, and health-care products[END_REF][START_REF] Zhang | Engineering Escherichia coli coculture systems for the production of biochemical products[END_REF]. E. coli is not the only microorganism that co-consumes glucose with fermentation byproducts. Saccharomyces cerevisiae can co-consume ethanol and glucose [START_REF] Raamsdonk | Co-consumption of sugars or ethanol and glucose in a Saccharomyces cerevisiae strain deleted in the HXK2 gene[END_REF][START_REF] Xiao | Glucose feeds the tricarboxylic acid cycle via excreted ethanol in fermenting yeast[END_REF], and mammalian cells co-consume lactate with glucose [START_REF] Faubert | Lactate metabolism in human lung tumors[END_REF][START_REF] Hui | Quantitative fluxomics of circulating metabolites[END_REF][START_REF] Hui | Glucose feeds the TCA cycle via circulating lactate[END_REF][START_REF] Rabinowitz | Lactate: the ugly duckling of energy metabolism[END_REF]. Our findings may thus be generalizable to other microorganisms. Glycolytic control of fermentative pathways has indeed been observed, albeit poorly understood, in virtually all microorganisms. Our results explain why co-consumption of ethanol with glucose in yeast is observed at low glucose uptake [START_REF] Raamsdonk | Co-consumption of sugars or ethanol and glucose in a Saccharomyces cerevisiae strain deleted in the HXK2 gene[END_REF]. A similar relationship between glycolytic and lactate fluxes has also been reported in vitro in mammalian cells [START_REF] Mulukutla | On metabolic shift to lactate consumption in fedbatch culture of mammalian cells[END_REF], with the switch between lactate production and utilization being determined by the glycolytic flux. It has been suggested that lactate rather than glucose is the primary circulating energy source in the human body [START_REF] Faubert | Lactate metabolism in human lung tumors[END_REF][START_REF] Rabinowitz | Lactate: the ugly duckling of energy metabolism[END_REF]. Our study demonstrates that similarly to lactate, acetate is not a deleterious waste product but a valuable nutrient that can be shuttled between cells. The reversible Pta-AckA pathway enables uncoupling of glycolysis from the TCA cycle, indicating that this evolutionarily conserved design principle of eukaryotic metabolism [START_REF] Xiao | Glucose feeds the tricarboxylic acid cycle via excreted ethanol in fermenting yeast[END_REF] is also present in bacteria. As a nutrient and global regulator of microbial metabolism, acetate plays a major role in cellular, microbial community and host-level carbon and energy homeostasis.

Materials and Methods

Strains. Escherichia coli K-12 MG1655 was chosen as the model wild-type strain. For experiments involving mutant strains, E. coli K-12 BW25113 was chosen as the reference wild-type strain, and its isogenic Δpta, Δpgi and ΔpfkA mutants were constructed from the KEIO collection [START_REF] Baba | Construction of Escherichia coli K-12 in-frame, single-gene knockout mutants: the Keio collection[END_REF] by removing the kanamycin cassette [START_REF] Datsenko | One-step inactivation of chromosomal genes in Escherichia coli K-12 using PCR products[END_REF]. The Δacs strain was constructed by CRISPR-Cas9 genome editing using the method and plasmids described in Wei et al. (Jiang et al, 2015). pta deletion was confirmed by genome sequencing and acs, pgi and pfkA deletions by PCR (with primers listed in Table S1).

Growth conditions. E. coli was grown in M9 minimal media [START_REF] Millard | Sampling of intracellular metabolites for stationary and non-stationary 13 C metabolic flux analysis in Escherichia coli[END_REF] supplemented with 15 mM glucose, 15 mM galactose, or 30 mM glycerol. Sodium acetate (prepared as a 1.2 M solution at pH 7.0 by addition of potassium hydroxide), acetic acid (prepared as a 1.2 M solution at pH 7.0 by addition of ammonium hydroxide) and methyl α-Dglucopyranoside (αMG, prepared as a 1.2 M solution) were added up to the required concentrations. The cells were grown in shake flasks at 37 °C and 200 rpm, in 50 mL of medium. Growth was monitored by measuring the optical density (OD) at 600 nm using a Genesys 6 spectrophotometer (Thermo, USA). Biomass concentrations were determined using a conversion factor of 0.37 g DW /L/OD unit [START_REF] Revelles | The carbon storage regulator (Csr) system exerts a nutrient-specific control over central metabolism in Escherichia coli strain Nissle 1917[END_REF].

Transcriptomics experiments. Cells were grown in M9 medium with 15 mM glucose, 100 mM αMG, and 0, 10, 50, or 100 mM sodium acetate. Sample preparations and transcriptomics analyses were carried out as described previously [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]. Three independent replicates were analyzed for each condition. The transcriptomics data can be downloaded from the ArrayExpress database (www.ebi.ac.uk/arrayexpress) under accession number E-MTAB-11717. Transcriptomics data in the absence of glycolytic inhibition (ArrayExpress database accession number E-MTAB-9086 [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]) were measured under the exact same conditions but without αMG.

Metabolomics experiments. Extracellular concentrations of glucose and acetate were quantified in 180 µL of filtered broth (0.2 μm syringe filter, Sartorius, Germany) by 1D 1 H-NMR on a Bruker Avance 500 MHz spectrometer equipped with a 5-mm z-gradient BBI probe (Bruker, Germany), as described previously [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF].

Flux calculation. Glucose uptake, acetate and growth rates were calculated from glucose, acetate and biomass concentration-time profiles using PhysioFit (v1.0.2, https://github.com/MetaSys-LISBP/PhysioFit) [START_REF] Peiro | Chemical and metabolic controls on dihydroxyacetone metabolism lead to suboptimal growth of Escherichia coli[END_REF].

Kinetic modeling.

We used a kinetic model of E. coli [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF] available from the Biomodels database (https://www.ebi.ac.uk/biomodels) [START_REF] Chelliah | BioModels: ten-year anniversary[END_REF] under identifier MODEL2005050001. Simulations were carried out as described in the main text, using COPASI (Hoops et al, 2006) (v4.34) with the CoRC package [START_REF] Forster | Modification of glucose import capacity in Escherichia coli: physiologic consequences and utility for improving DNA vaccine production[END_REF] (COPASI R Connector v0.11, https://github.com/jpahle/CoRC) in R (v4.1.3, https://www.r-project.org). The R notebook used to perform the simulations and to generate the figures is provided at https://github.com/MetaSys-LISBP/glucose_acetate_interplay. Sensitivity analyses. We used a Monte-Carlo approach [START_REF] Saa | Formulation, construction and analysis of kinetic models of metabolism: A review of modelling frameworks[END_REF] to estimate the robustness of model predictions. Following the procedure detailed in [START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF], 500 simulated sets of calibration data were generated with noise added according to experimental standard deviations. For each of these artificially noisy data sets, we estimated model parameters and ran all simulations from the optimal parameters values. We calculated the mean value and standard deviation of each flux from the distribution of values obtained for the 500 sets of parameters.

Statistical analyses.

Comparisons between strains or conditions were performed using Student's t-tests with two-tailed distributions. The R notebook used to perform the calculations is provided at https://github.com/MetaSys-LISBP/glucose_acetate_interplay.

Figure 1 .

 1 Figure 1. Predicted response of E. coli to glycolytic and acetate perturbations.

Figure 3 .

 3 Figure 3. Acetate flux reversal is controlled by the glycolytic flux. A Predicted acetate flux in E. coli on glucose (15 mM) as a function of the acetate concentration at different values of glycolytic activity (from 20 to 100 % of the maximal value). Shaded areas correspond to ± one standard deviation on model predictions. B Predicted acetate concentration threshold at which the acetate flux reverses as a function of the glycolytic activity, determined from the data shown in panel (A). Shaded areas correspond to ± one standard deviation on model predictions.

Figure 4 .

 4 Figure 4. Effect of acetate on E. coli growth on glucose. A Predicted growth rate of E. coli on glucose (15 mM) as a function of the acetate concentration at 100 and 20 % of the maximal glycolytic activity. Shaded areas correspond to ± one standard deviation. B Experimental growth rate of E. coli K-12 MG1655 grown on glucose (15 mM) as a function of the acetate concentration at high (0 mM αMG, light blue, reproduced from (Enjalbert et al., 2017)) or low (100 mM αMG, dark blue) glycolytic flux. Each data point represents an independent biological replicate and the lines represent best polynomial fits. C Relative change in the growth rate of E. coli K-12 BW25113 wild-type in the presence of 10 mM acetate at high (0 mM αMG) and low (100 mM αMG) glycolytic flux. Mean values ± standard deviations (error bars) were estimated from n independent biological replicates, as indicated on the figure. D Relative change in the growth rates of E. coli K-12 BW25113 Δpgi and ΔpfkA strains in the presence of 10 mM acetate. Mean values ± standard deviations (error bars) were estimated from three independent biological replicates.At low glycolytic flux, acetate activates E. coli metabolism and growth at the transcriptional level. At high glycolytic flux, it has been proposed[START_REF] Millard | Control and regulation of acetate overflow in Escherichia coli[END_REF]) that the apparent "toxicity" of acetate is due to its inhibition of most metabolic genes -including those for the glucose phosphotransferase system (PTS), glycolysis and TCA cycle. Acetate also increases the expression of genes related to stress responses and downregulates genes related to transport, energy biosynthesis and mobility for instance. To determine if acetate has a similar regulatory effect at low glycolytic flux, we compared the transcriptomes of E. coli grown on glucose (15 mM) plus acetate (0, 10, 50 or

Figure 5 .

 5 Figure 5. Regulation of E. coli transcriptome by acetate at high and low glycolytic flux. A, B Transcriptomic response of E. coli K-12 MG1655 to acetate (10, 50 or 100 mM versus 0 mM) at high (0 mM αMG) and low (100 mM αMG) glycolytic flux. The data obtained in the absence of αMG are reproduced from (Millard et al., 2021). The biological functions modulated by the presence of acetate (based on Gene Ontology analysis) are shown in part (A), with the corresponding p-values. The expression levels of central metabolic genes are shown in part (B). Gene expression is normalized to the value measured at the same αMG concentration but without acetate. Green and red indicate decreased and increased expression, respectively.

Figure 6 .

 6 Figure 6. Response of wild-type, Δpta, and Δacs strains of E. coli to acetate during growth on glucose. Mean relative change in the growth rates of wild-type, Δpta, and Δacs E. coli K-12 BW25113 strains in the presence of 10 mM acetate at low glycolytic flux (100 mM αMG). Mean values and standard deviations (error bars) were estimated from n independent biological replicates, as indicated on the figure.

Figure 7 .

 7 Figure 7. Response of E. coli to acetate during growth on glycerol or galactose. A,B Impact of the presence of 10 mM acetate on the growth rate of wild-type, Δpta, and Δacs E. coli K-12 BW25113 strains grown on (A) 30 mM glycerol and (B) 15 mM galactose. Mean values and standard deviations (error bars) were estimated from three independent biological replicates.

Acknowledgments

This work was supported by a grant to Pierre Millard and Thomas Gosselin-Monplaisir from the Département MICA of INRAE and the Région Occitanie (grant COCA-COLI) and from the Toulouse Biotechnology Institute (grant Master-2022). The authors thank MetaboHub-

Data Availability

The data used to make the figures can be found in the Source Data files. The data, model and computer code produced in this study are available in the following databases: • Transcriptomics data : ArrayExpress E-MTAB-11717 (https://www.ebi.ac.uk/biostudies/arrayexpress/studies/E-MTAB-11717) • Model: BioModels MODEL2005050001 (https://www.ebi.ac.uk/biomodels/MODEL2005050001) • Code: GitHub (https://github.com/MetaSys-LISBP/glucose_acetate_interplay)

MetaToul (Metabolomics and Fluxomics facilities, Toulouse, France, http://www.metatoul.fr), part of the French National Infrastructure for Metabolomics and Fluxomics (http://www.metabohub.fr) funded by the ANR (MetaboHUB-ANR-11-INBS-0010), for access to NMR facilities. The authors are grateful to Edith Vidal and Abdulrahman Khabbaz for help with the growth experiments, and the following INSA Toulouse students for help with the growth and transcriptomics experiments: Lola Blayac, Arno Bruel, Alexis Charbinat, Romane Ducloux, Justine Dumas-Perdriau, Solène Frapard, Sophie Germain, Benjamin Jung, Oliver Larrousse, Nicolas Papadopoulos, Marine Rodeghiero, Leyre Sarrias and Auriane Thomas. The authors also thank the Département Génie Biologique of INSA Toulouse for ongoing support, and Muriel Cocaign-Bousquet and Guy Lippens for insightful discussions on the manuscript.

Author contributions

Pierre Millard: Conceptualization; data curation; formal analysis; funding acquisition; investigation; methodology; project administration; resources; software; supervision; validation; visualization; writing -original draft; writing -review and editing. Thomas Gosselin-Monplaisir: Data curation; formal analysis; investigation; methodology; resources; validation; writing -review and editing. Sandrine Uttenweiler-Joseph: Methodology; resources; writing -review and editing. Brice Enjalbert: Data curation; formal analysis; investigation; methodology; resources; supervision; validation; visualization; writing -review and editing.

Disclosure and competing interests statement

The authors declare that they have no conflict of interest.


Siddharatha Thakur

Philippe Tamarat

Antonine Rochet

Joschka Birk

Ivan S Veshchunov

Malo Bezard

Alexander I Buzdin

Jean-Baptiste Trebbia

Brahim Lounis
email: brahim.lounis@u-bordeaux.fr

High-resolution optical imaging of single magnetic flux quanta with a solid immersion lens

Magneto-optical imaging of quantized magnetic flux tubes in superconductors -Abrikosov vortices -is based on Faraday rotation of light polarization within a magneto-optical indicator placed on top of the superconductor. Due to severe aberrations induced by the thick indicator substrate, the spatial resolution of vortices is usually well beyond the optical diffraction limit. Using a high refractive index solid immersion lens placed onto the indicator garnet substrate, we demonstrate wide field optical imaging of single flux quanta in a Niobium film with a resolution better than 600 nm and sub-second acquisition periods, paving the way to high-precision and fast vortex manipulation. Vectorial field simulations are also performed to reproduce and optimize the experimental features of vortex images.

Introduction

In type II superconductors, an external magnetic field can partially penetrate as filament like flux lines named Abrikosov vortices carrying a single quantum of flux [START_REF] Abrikosov | On the Magnetic Properties of Superconductors of the Second Group[END_REF]. With a nanometric core surrounded by circulating supercurrents over a distance given by the London penetration depth 𝜆 ! (typically 𝜆 ! ~100 nm in Niobium), these vortices represent the most compact magnetic objects in superconductors. It has been shown that they have the potential of widespread use as quantized information bits [START_REF] Golod | Single Abrikosov vortices as quantized information bits[END_REF] to design high-density digital cryoelectronics (tens of Josephson junctions per µm 2 [START_REF] Tolpygo | A 150-nm Node of an Eight-Nb-layer Fully Planarized Process for Superconductor Electronics[END_REF][START_REF] Soloviev | Miniaturization of Josephson Junctions for Digital Superconducting Circuits[END_REF]). The local supercurrents produced by a single vortex placed close to a Josephson junction can also induce Josephson phase shifts and thus affect the Josephson transport properties [START_REF] Golod | Detection of the Phase Shift from a Single Abrikosov Vortex[END_REF][START_REF] Mironov | Anomalous Josephson effect controlled by an Abrikosov vortex[END_REF][START_REF] Golod | Two mechanisms of Josephson phase shift generation by an Abrikosov vortex[END_REF][START_REF] Golod | Reconfigurable Josephson Phase Shifter[END_REF]. For the development of various schemes in superconducting electronics [START_REF] Mironov | Anomalous Josephson effect controlled by an Abrikosov vortex[END_REF], it is crucial to implement simple and precise vortex imaging and manipulation methods.

Over the past decades, various vortex imaging techniques have been introduced, including pioneering Bitter decoration techniques [START_REF] Essmann | The direct observation of individual flux lines in type II superconductors[END_REF], Lorentz microscopy [START_REF] Harada | Real-time observation of vortex lattices in a superconductor by electron microscopy[END_REF], scanning Hall probe microscopy [START_REF] Chang | Scanning Hall-Probe Microscopy of a Vortex and Field Fluctuations in La 1.85 Sr 0.15 CuO 4 Films[END_REF], magnetic force microscopy [START_REF] Moser | Low temperature magnetic force microscopy on YBa[END_REF], scanning electron microscopy [START_REF] Koelle | Vortex imaging by low-temperature scanning electron microscopy and correlation with low-frequency noise in YBCO DC SQUIDs[END_REF], scanning superconducting quantum interference device (SQUID) microscopy [START_REF] Gardner | Manipulation of single vortices in YBa 2 Cu 3 O 6.354 with a locally applied magnetic field[END_REF]. While scanning tunneling microscopy is capable of resolving a vortex core (~10 nm) [START_REF] Roditchev | Direct observation of Josephson vortex cores[END_REF] and SQUID microscopy provides sensitivity down to the single electron spin level [START_REF] Vasyukov | A scanning superconducting quantum interference device with single electron spin sensitivity[END_REF], these local probe techniques are heavy to implement in a cryogenic environment. On the other hand, magnetooptical (MO) imaging [START_REF] Goa | Magneto-optical imaging setup for single vortex observation[END_REF][START_REF] Tokunaga | Improvement of vortex imaging in magneto-optical technique and Bitter decoration[END_REF][START_REF] Tsuchiya | Development of surface magneto-optical imaging method[END_REF] provides far-field optical images with a wide field of view and can be combined with other optical tools recently developed for single vortex manipulation [START_REF] Veshchunov | Optical manipulation of single flux quanta[END_REF] and generation [START_REF] Rochet | On-demand Optical Generation of Single Flux Quanta[END_REF]. MO imaging of individual vortices is based on the Faraday rotation of light polarization in a MO indicator placed onto the superconductor, using a beam path configuration with nearly crossed polarizers [START_REF] Goa | Real-time magneto-optical imaging of vortices in superconducting NbSe 2[END_REF]. The most commonly used MO indicators are ferrite garnets such as Bismuth substituted lutetium iron garnets (Bi:LuIG), that are grown with a thickness of a few micrometers on a thick (~500 µm thickness) gadolinium-gallium-garnet (Gd3Ga5O12, GGG) transparent substrate. However, this MO imaging method suffers from poor spatial resolution of the vortices (apparent vortex diameters ≳ 1.5 µm [START_REF] Goa | Magneto-optical imaging setup for single vortex observation[END_REF]). Indeed, the thickness of the GGG substrate introduces severe spherical aberrations that broaden the pointspread function (PSF), i.e. the spatial distribution of a point-source image, well beyond the optical diffraction limit. Another source of degradation of the vortex images is the residual gap between the superconductor layer and the MO indicator pressed on top of it. This gap places the indicator away from the region where the magnetic field stemming from vortices is the most concentrated [START_REF] Tokunaga | Improvement of vortex imaging in magneto-optical technique and Bitter decoration[END_REF]. To eliminate both problems, it has been proposed to grow a nanometric layer of MO indicator directly on the superconductor. Using a 40 nm-thick EuSe indicator film, apparent vortex diameters as small as 800 nm have been reported [START_REF] Golubchik | A high resolution magneto-optical system for imaging of individual magnetic flux quanta[END_REF]. Yet, this technique has limited versatility since EuSe layers are difficult to fabricate and cause strong light absorption. Moreover, they undergo a magnetic ordering phase transition at 4.6 K that restrict their range of use to liquid helium temperatures and thus superconductors with low critical temperatures [START_REF] Schoenes | Magnetooptic Spectroscopy of EuS, EuSe and EuTe[END_REF].

The implementation of optical microscopy with a high numerical aperture (NA) remains a crucial step to achieve high-resolution vortex images associated with strong MO signals. It is challenging since the common high-NA microscope objectives have many lenses subjected to mechanical strains and phase changes at cryogenic temperatures. Moreover, such objectives would not compensate for the spherical aberrations introduced by the thick garnet substrate. In this context, implementing a solid immersion lens (SIL) combined with a single, low-NA aspheric lens is a powerful approach. SIL-microscopy allows immersion with a high refractive index adapted to the sample and was originally developed to achieve higher optical resolutions [START_REF] Mansfield | Solid immersion microscope[END_REF]. It has been successfully used to improve the light extraction efficiency of solid-state single photon sources such as individual quantum dots [START_REF] Wu | Imaging Spectroscopy of Two-Dimensional Excitons in a Narrow GaAs/AlGaAs Quantum Well[END_REF][START_REF] Zwiller | Single quantum dots emit single photons at a time: Antibunching experiments[END_REF][START_REF] Barnes | Solidstate single photon sources: light collection strategies[END_REF] or molecules [START_REF] Trebbia | Efficient generation of near infra-red single photons from the zero-phonon line of a single molecule[END_REF][START_REF] Maser | Few-photon coherent nonlinear optics with a single molecule[END_REF]. In MO imaging, a SIL-based objective should also eliminate the strong spherical aberrations introduced by the thick garnet substrate. Here, we address the fundamental optical limitations of vortex MO imaging by coupling the indicator on its substrate to a SIL with a high refractive index n ~2 matching that of the GGG substrate. Using the Weierstrass (i.e. super-hemispherical) SIL geometry [START_REF] Zwiller | Improved light extraction from emitters in high refractive index materials using solid immersion lenses[END_REF] to focus and collect light from a subwavelength region in close contact with a superconducting Niobium layer, we achieve MO imaging with single-vortex apparent diameters less than 600 nm. We also demonstrate a significant increase in the vortex MO signals and image acquisition rates. The features of the vortex optical images are well reproduced with vectorial electromagnetic-field simulations, which help optimizing the performance of vortex imaging.

Experimental setup

The experimental setup used for single vortex MO imaging is depicted in Fig. 1. A 100 nmthick Niobium film deposited on a sapphire substrate is cooled below its critical temperature 𝑇 " = 8.6 K under a weak external magnetic field that penetrates the superconducting film in the form of quantized flux tubes. The MO indicator pressed on top of the Niobium film is a 2.5 µm-thick garnet of composition BixLu3-xFe5-zGazO12 with x ~ 0.9 and z ~ 1.0 [START_REF] Veshchunov | Direct Evidence of Flexomagnetoelectric Effect Revealed by Single-Molecule Spectroscopy[END_REF]. It is grown on a GGG substrate with a thickness ℎ ~500 µm. This sample is illuminated with a white light LED source filtered at the optical wavelength 𝜆 = 550 nm with a bandwidth of 15 nm. This wavelength corresponds to a local maximum in the MO indicator's rotational power (Verdet constant 𝑉 ~ 0.16° mT -1 µm -1 ) together with a garnet transmission factor of 0.43. The SIL used here is a half-ball lens made of S-LAH79 glass with refractive index 𝑛 = 2.011 at 𝜆 = 550 nm, which nearly matches the GGG index 𝑛 ### = 1.97 at this wavelength. We operate this SIL combined with the GGG garnet in the Weierstrass geometry, which exploits rigorous stigmatism in the conjugation of two specific points through a spherical refractive surface [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light[END_REF]. The radius 𝑟 of the SIL is indeed chosen so that the nearest Weierstrass point coincides at best with the MO indicator. The requirement 𝑟 = 𝑛 × ℎ for such geometry is nearly fulfilled with the choice of 𝑟 = 1 mm. The residual mismatch between 𝑛 and 𝑛 ### produces negligible spherical aberrations: The Strehl ratio, which corresponds to the ratio of maximal intensities in the PSF and Airy spots, is indeed greater than 0.989. The overall NA of a SIL-based objective operating in the Weierstrass configuration can be as high as 𝑛, provided that the aspheric lens has a NA greater than 1 𝑛 ⁄ . We use here an aspheric lens with focal length 10 mm and NA = 0.49. However, a holder used to press the SIL onto the GGG garnet (See Fig. 1b) reduces the overall NA of this objective to 0.68. stemming from a vortex in the half space above the Niobium surface (𝑧 =0), 𝑟 and 𝑧 being cylindrical coordinates, 𝑧 " = -1.27𝜆 ' and 𝜆 ' =100 nm [START_REF] Carneiro | Vortex lines in films: Fields and interactions[END_REF].

Results and discussion

Vortex images with a solid immersion lens

Figure 2a is a MO image of a spontaneous vortex distribution in a Niobium film at 4 K after field cooling under an external magnetic field of 0.3 Oe, using the aspheric lens alone as in conventional MO imaging setups. A background subtraction scheme is used: An image is acquired above 𝑇 " (at 10 K), then subtracted from the image taken at 4 K. The vortices are located at pinning sites that are randomly distributed in the Niobium film. While the real vortex size given by 𝜆 ! is ~100 nm in Niobium, the single vortex spots have a full width half maximum (FWHM) diameter of 1.7 µm (Fig. 2a). The addition of the SIL drastically reduces the size of the vortex images by a factor of three (Fig. 2b). As demonstrated in Fig. 2c, a FWHM diameter of 583 nm is achieved. The threefold increase in resolution with the SIL design is evidenced in the case of a higher concentration of vortices: In Fig. 2 d-e, the magnetic field applied during field cooling of the Niobium film is raised to 7.1 Oe, which sets the average inter-vortex distance in the micrometer range. While individual vortices can not be distinguished with a single aspheric lens, they are distinctly resolved with the SIL-based objective.

In order to model the vortex images without and with the SIL, we have computed the vectorial field propagation all along the optical path (z-axis) from the source to the sample and back to the camera with a homemade ray-tracing program [START_REF] Trebbia | 3D optical nanoscopy with excited state saturation at liquid helium temperatures[END_REF], taking into account the optical aberrations and changes in polarization at each interface with their Fresnel coefficients. A vortex is considered as a punctual object diffracting light isotropically and producing Faraday rotation of light polarization. Initializing the incident polarization along the x-axis, the PSFs associated to the intensity components 𝐼 $ and 𝐼 % along the x-and y-axes are calculated at the output of the aspheric lens, then projected onto the analyzer axis oriented as in the experiment. The corresponding simulated PSFs are shown in Fig. 2f for the configurations without and with the SIL, keeping the same aspheric lens in both imaging systems. In the case of the aspheric lens used alone (blue curve), the pronounced lobes are due to severe spherical aberrations introduced by the propagation through the GGG substrate. We indeed estimate the peak-tovalley wave-front distortion of ~0.6 𝜆 in this configuration. For a better match with the experimental images, we have added a slight contribution of wavefront distortion 0.03l caused by deformation of the GGG substrate under non-uniform pressure exerted on its edges to minimize the gap between the Nb film and the MO indicator. This deformation is not taken into account in the SIL configuration, since the flat surface of the SIL exerts a uniform pressure over the central area of the GGG substrate. With the SIL-based objective (red curve), the PSF is nearly diffraction-limited due to the inherent elimination of spherical aberrations. The evolution of the FWHM vortex spot size, averaged over ~30 vortices, is shown in Fig. 3a as a function of the NA of the SIL-based objective. The NA is varied by controlling the aperture of a diaphragm placed on the detection path in the Fourier plane (See Fig. 1). As expected, the increase in NA leads to sharpening of the vortex spots, whose average size follows the NA-dependence of the diffraction-limited spot diameter 0.61𝜆 NA ⁄ arising from a point source (solid curve in Fig. 3a). The deviations between the experimental spot sizes and these lower bounds are attributed to residual Nb-BiLuIG gap and to the spreading of the vortex field lines penetrating the MO indicator (see Fig. 1c). At the highest NAs, stronger deviations are observed. They may be due to enhanced optical aberrations, e.g. induced by a slight misalignment of the aspheric lens and SIL optical axes.

Signal-to-noise ratio

The quality of magneto-optical imaging is also characterized by the signal-to-noise ratio (SNR) of the vortex images. In Fig. 3b, the SNR averaged over ~20 vortices is plotted for both cases, without and with the SIL, as a function of the analyzer offset angle 𝜙 with respect to the crossed polarizers configuration. Assuming that the noise essentially arises from the photon shot-noise, and that the Faraday rotation angle 𝜃 & of light polarization after the double pass through the indicator is small (𝜃 & ~ 5 mrad), the SNR can be modeled as

SNR ∝ |( ! )*+(-.)| 0)*+ " (.)12 "4)(-.) (1).
For both configurations, the experimental data are well reproduced with this expression, using the extinction ratios 𝜀 = 5 × 10 56 and 𝜀 = 4.4 × 10 5-measured without and with the SIL, respectively. On one hand, the higher NA of the SIL-based objective introduces undesired components of light polarization that degrade the extinction ratio. On the other hand, it offers higher photon collection efficiency due to its enhanced NA, which ultimately results in a SNR three times stronger under the same illumination beam, as shown in Fig. 3b. An important experimental parameter in vortex MO imaging is the aperture of the light source. While increasing the aperture enhances the illumination intensity in the sample plane, it also promotes depolarization effects that degrade 𝜀 and impact the SNR. To model the aperture dependence of 𝜀, the fibered source with a diameter of 3 mm is modeled by a sum of incoherent, non-polarized and uniformly distributed point sources. Launching 10,000 rays, the three field components are propagated along the optical z-axis towards the sample and reflected towards the analyzer and camera. These fields are then summed incoherently to obtain the intensity components 𝐼 $ , 𝐼 % and 𝐼 7 as a function of the diameter of the source aperture. The spatial distributions of these three components in the camera plane are displayed as insets in Fig. 4a, in the case of a single source point. As expected, the 𝐼 % and 𝐼 7 components are mainly created when propagating the beam through the aspheric lens and the SIL, and take their greatest values at the largest incidence angles. From these intensity profiles, the extinction ratio 𝜖 = 𝐼 % 𝐼 $ ⁄ is computed and shown in Fig. 4a as a function of the source diameter. A degradation of 𝜖 by two orders of magnitude is obtained at the largest diameters. The experimental extinction ratio is displayed in Fig. 4b as a function of the diameter of the source iris. As the source size is reduced below 10 mm down to 2 mm, there is a gain in the quality of extinction by an order of magnitude and a threefold gain in the SNR (Fig. 4c). On the other hand, using large source apertures allows high illumination intensities of the sample and photon flux collection with the SIL, and thus fast vortex imaging. This is illustrated in the series of singleshot images displayed in Figs 4 d-f, recorded for various acquisition times and a fixed aperture of 10 mm. Sub-second imaging with a SNR above unity and a sub-micron resolution is evidenced, which offers great prospects for fast and precise vortex imaging.

Conclusion

In conclusion, we have developed a MO imaging setup based on a SIL operating in the Weierstrass configuration and leading to sub-600 nm optical resolution of single Abrikosov vortices. This represents a three-fold improvement in the resolution with respect to single lens imaging systems. Such performance is combined with a higher efficiency in light collection, which is essential for fast acquisition of vortex images. Among the avenues for improvement of the vortex image quality, higher SNRs could be achieved using a quasi-punctual source for aberration-free illumination of the sample. Greater vortex resolutions and faster vortex imaging could also be reached by increasing the effective NA of the SIL, for instance using a thinner indicator substrate. High-NA MO imaging of vortices with a SIL is thus promising for precise optical localization and manipulation of single vortices applied to the optical operation of Josephson transport [START_REF] Mironov | Anomalous Josephson effect controlled by an Abrikosov vortex[END_REF]. Achieving optical resolutions of the order of the London penetration depth will also enable real-time studies of the forces involved in dense vortex lattices.

FIG. 1 .

 1 FIG. 1. Optical design for single vortex magneto-optical imaging. (a) The general setup operates on Faraday rotation of light polarization in a Bi:LuIG MO indicator placed on the superconductor in a nearly-crossed-polarizer beam path configuration. (b) The superconductor placed in the cryostat is a Niobium film deposited on a sapphire substrate. The magnetic field stemming from vortices penetrates the indicator and is imaged with a SIL-based objective comprising the SIL and an aspheric lens. The SIL placed on top of the GGG substrate is a halfball lens operated in the Weierstrass configuration. The white lamp has a fibered output with diameter 3 mm. A source iris controls the aperture of the illumination beam, while a NA iris is placed on the detection path in the Fourier plane of a telescope. A 15 nm bandpass filter is used to collect enough light without introducing significant chromatic aberrations. (c) Simulation of the vertical component of magnetic field 𝐵 ! (𝑟, 𝑧) = (𝜙 " 2𝜋 ⁄ )(𝑧 -𝑧 " )[𝑟 # + (𝑧 -𝑧 " ) # ] $% # ⁄

FIG. 2 .

 2 FIG. 2. Optical resolution of vortex images. All scale bars are 2 µm. Single vortices in a Nb film with thickness 100 nm are imaged at 4 K with the aspheric lens alone (a) and with the SIL-based objective (b), under magnetic fields of 0.3 Oe and 2.2 Oe, respectively. The vortices are imaged by a CCD (charge-coupled device) camera (MicroMAX EEVx512FT CCD57, Princeton Instruments) having 512 × 512 pixels with an area of 13 µm × 13 µm. One pixel maps an area of 500 nm × 500 nm in (a). When adding the SIL (b), the magnification of the imaging system is increased by a factor 𝑛 # , so that one pixel maps an area 4 × 4 times less. (c) Profiles of the smallest vortex spots imaged without (blue circles) and with (red circles) the SIL, and fitted with Lorentzian curves (blue and red solid curves, respectively). The FWHM diameters are respectively 1.7 µm and 583 nm. Vortices in greater concentration are imaged after field cooling at 7.1 Oe (d) without and (e) with the SIL. (f) Vectorial field simulations of the PSF with the aspheric lens alone (left inset) and with the SIL-based objective (right inset). The blue and red solid curves are the intensity profiles of the respective PSFs calculated from the Debye integral. The black circles represent the cross-section of the left-inset image, taking pixilation into account (pixel size 500 x 500 nm). The dashed blue curve is a Lorentzian profile fitting the circles. Its FWHM of 1.7 µm is in accordance with the experimental vortex spot diameters measured with the aspheric lens used alone.

FIG. 3 .

 3 FIG. 3. (a) Average FWHM of vortex images as a function of the effective NA controlled with a diaphragm. The diffraction-limited FWHM is given by the solid curve. (b) The SNR (red squares) of the vortex images is derived as the average ratio of the peak vortex signals to the noise in the background. It is plotted as a function of the analyzer offset angle for the case without (blue) and with the SIL (red) under similar illumination powers. The error bars are indicative of the signal variations amongst vortex spots in the recorded image. The solid curves are plots of Eq. (1), taking 𝜀 = 4.4 10 $# (red curve) and 𝜀 = 5 10 $% (blue curve), using the SNR proportionality factor as the only fitting parameter.

FIG. 4 .

 4 FIG. 4. Influence of the light source aperture. (a) The extinction ratio is computed in the Weierstrass configuration for various diameters of the source beam, using 100 point sources. The insets show the spatial distribution of the intensity components along the x, y and z axes, calculated with a single point source. Experimental evolutions of the extinction ratio (b) and the SNR (c) as a function of the source beam diameter. d-f: Normalized images taken with an integration time of 200 ms (d), 400 ms (e) and 1 s (f), for a source diameter of 10 mm. The SNRs are respectively 0.73, 1.23, 2.45. The scale bars are 2 µm.

Acknowledgments. We acknowledge the financial support from the French National Agency for Research, Région Aquitaine, Idex Bordeaux (Research Program GPR Light), and the EUR Light S&T. B.L. acknowledges the Institut universitaire de France.

Data availability.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Disclosures.

The authors declare no conflicts of interest.