Ludovic Pelosi

Laura Morbiato

Arthur Burgardt

Fiorella Tonello

Abigail K Bartlett

Katayoun Rachel M Guerra

Kazemzadeh Ferizhendi

Maria Andrea Desbats

Bérengère Rascalou

Marco Marchi

Katayoun Kazemzadeh Ferizhendi

Luis Vázquez-Fonseca

Caterina Agosto

Giuseppe Zanotti

Morgane Roger-Margueritat

María Alcázar-Fabra

Laura García-Corzo

Ana Sánchez-Cuesta

Plácido Navas

Gloria Brea-Calvo

Eva Trevisson

Volker F Wendisch

David J Pagliarini

Leonardo Salviati
email: leonardo.salviati@unipd.it

Fabien Pierrel
email: fabien.pierrel@univ-grenoble-alpes.fr

COQ4 is required for the oxidative decarboxylation of the C1 carbon of coenzyme Q in eukaryotic cells

Keywords: Coenzyme Q, Coenzyme Q biosynthesis, COQ4, Oxidative decarboxylation, mitochondria, respiratory chain, Coenzyme Q deficiency

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Coenzyme Q (CoQ) is a redox-active lipid that plays crucial roles in the cellular homeostasis of eukaryotic cells [START_REF] Baschiera | The multiple roles of coenzyme Q in cellular homeostasis and their relevance for the pathogenesis of coenzyme Q deficiency[END_REF] . CoQ is composed of a quinone head group and a polyisoprenoid chain of variable length: six isoprene units in the yeast Saccharomyces cerevisiae (CoQ6), ten in humans (CoQ10), and eight in the bacterium Escherichia coli (CoQ8). Despite the crucial functions of CoQ, its biosynthetic pathway in eukaryotes is still incompletely understood. The CoQ biosynthetic machinery includes the products of at least 15 genes (collectively known as COQ genes), most of them with a conserved function [START_REF] Doimo | Genetics of coenzyme q10 deficiency[END_REF][START_REF] Guerra | Coenzyme Q biochemistry and biosynthesis[END_REF] . Primary CoQ deficiency is a clinically and genetically heterogeneous group of conditions characterized by impairment of CoQ biosynthesis and caused by defects in any of the COQ genes [START_REF] Alcázar-Fabra | Primary Coenzyme Q deficiencies: A literature review and online platform of clinical features to uncover genotype-phenotype correlations[END_REF] .

The precursors of CoQ, 4-hydroxybenzoic acid (4-HB) and polyprenyl-pyrophosphate (PPP) are conjugated to form polyprenyl-4-hydroxybenzoic acid (nP-HB). This reaction is catalyzed by COQ2, a prenyltransferase located in the mitochondrial inner membrane [START_REF] Desbats | The COQ2 genotype predicts the severity of coenzyme Q10 deficiency[END_REF] (Fig. 1A). The subsequent biosynthetic steps, which correspond to modifications of the aromatic ring, are carried out by a multiprotein complex (named complex Q), associated with the mitochondrial inner membrane on the matrix side [START_REF] Guerra | Coenzyme Q biochemistry and biosynthesis[END_REF] . The exact sequence of reactions that modify the aromatic ring is still a matter of debate.

Traditional views, based on yeast data, suggested that the first reactions consist of the hydroxylation of position C5 of nP-HB catalysed by COQ6, and the subsequent methylation catalysed by COQ3, followed by the C1 decarboxylation and hydroxylation steps [START_REF] Stefely | Biochemistry of Mitochondrial Coenzyme Q Biosynthesis[END_REF][START_REF] Ozeir | Coenzyme Q biosynthesis: Coq6 Is required for the C5-hydroxylation reaction and substrate analogs rescue Coq6 deficiency[END_REF] (Fig. 1A). However, this sequence of reactions may be flexible since we reported that human and yeast cells with deficient COQ6 accumulate 3-polyprenyl-1,4-benzoquinone (PBQn) indicating that, at least in some situations, the C1 decarboxylation and C1 hydroxylation are the first modifications of the ring to take place [START_REF] Ozeir | Coenzyme Q biosynthesis: Coq6 Is required for the C5-hydroxylation reaction and substrate analogs rescue Coq6 deficiency[END_REF][START_REF] Acosta Lopez | Vanillic Acid Restores Coenzyme Q Biosynthesis and ATP Production in Human Cells Lacking COQ6[END_REF] (Fig 1). Irrespective of the sequence of reactions, the enzymes that catalyze the C1 decarboxylation and C1 hydroxylation in eukaryotes are still uncharacterized [START_REF] Guerra | Coenzyme Q biochemistry and biosynthesis[END_REF][START_REF] Wang | The Complexity of Making Ubiquinone[END_REF] . In bacteria such as E. coli, these reactions are carried out by two distinct enzymes, UbiD (a decarboxylase that uses a prenyl-FMN cofactor produced by UbiX [START_REF] Marshall | The UbiX-UbiD system: The biosynthesis and use of prenylated flavin (prFMN)[END_REF] ) and UbiH (a hydroxylase) [START_REF] Hajj Chehade | A Soluble Metabolon Synthesizes the Isoprenoid Lipid Ubiquinone[END_REF] , which have no eukaryotic counterparts. Proteomic analyses of complex Q in eukaryotic cells failed to identify candidate proteins that could catalyze these reactions [START_REF] Stefely | Biochemistry of Mitochondrial Coenzyme Q Biosynthesis[END_REF][START_REF] Floyd | Mitochondrial Protein Interaction Mapping Identifies Regulators of Respiratory Chain Function[END_REF] .

In addition to unassigned catalytic steps, gaps in our knowledge of eukaryotic CoQ biosynthesis also relate to the poorly characterized function of some of complex Q proteins, such as COQ4. The COQ4 gene is relevant for human disease because its mutations are one of the most common causes of primary coenzyme Q deficiency [START_REF] Hughes | Estimating the occurrence of primary ubiquinone deficiency by analysis of large-scale sequencing data[END_REF] . They are associated with a variety of clinical pictures ranging from fatal neonatal multisystem disorders [START_REF] Brea-Calvo | COQ4 mutations cause a broad spectrum of mitochondrial disorders associated with CoQ10 deficiency[END_REF] , to childhood-onset cerebellar ataxia [START_REF] Mero | New pathogenic variants in COQ4 cause ataxia and neurodevelopmental disorder without detectable CoQ10 deficiency in muscle or skin fibroblasts[END_REF] , or isolated retinitis pigmentosa [START_REF] Jurkute | Biallelic variants in coenzyme Q10 biosynthesis pathway genes cause a retinitis pigmentosa phenotype[END_REF] . Interestingly, COQ4 patients usually display a lower response to oral CoQ10 supplementation compared to patients with other COQ gene defects [START_REF] Salviati | Primary Coenzyme Q10 Deficiency Overview[END_REF] , hence the importance of understanding the precise function of COQ4.

COQ4 homologues exist in virtually all eukaryotes and potentially in some bacterial species, but not in E. coli [START_REF] Marbois | The yeast Coq4 polypeptide organizes a mitochondrial protein complex essential for coenzyme Q biosynthesis[END_REF] . Based on the demonstration that S. cerevisiae coq4 mutants failed to assemble complex Q and did not synthesize CoQ, COQ4 was proposed to fulfill a structural role in the organization of complex Q [START_REF] Marbois | The yeast Coq4 polypeptide organizes a mitochondrial protein complex essential for coenzyme Q biosynthesis[END_REF] . However, the biosynthetic complex is destabilized in most yeast strains deleted for coq genes, and CoQ, or at least late biosynthetic intermediates like demethoxy-CoQ, seem important for complex Q stability [START_REF] Tran | Endogenous synthesis of coenzyme Q in eukaryotes[END_REF] . COQ4 proteins display a conserved motif, HDxxH(x)11E, which has been proposed to be a zinc-binding domain [START_REF] Marbois | The yeast Coq4 polypeptide organizes a mitochondrial protein complex essential for coenzyme Q biosynthesis[END_REF] . Consistent with this hypothesis, the aspartic acid and glutamic acid residues of the HDxxH(x)11E motif coordinate two magnesium ions in the crystal structure of a cyanobacterial homologue of COQ4, Alr8543 from Nostoc sp. PCC 7120 (PDB: 6E12) (Figure S1A), which has been used in the past to model eukaryotic COQ4 [START_REF] Kawamukai | Biosynthesis of coenzyme Q in eukaryotes[END_REF] . Interestingly, mutation of the same glutamic acid residue in the coq4-1 yeast strain impaired the function of COQ4 without affecting its stability [START_REF] Marbois | The yeast Coq4 polypeptide organizes a mitochondrial protein complex essential for coenzyme Q biosynthesis[END_REF] . Adjacent to the metal coordinating pocket, the 6E12 crystal structure shows a hydrophobic tunnel with a bound molecule in which an oleic acid aliphatic chain is bound (Fig. S1B). This tunnel seems suitable to house the polyisoprenoid side chain of CoQ, potentially positioning the aromatic ring in close proximity to the metal-binding site. Based on these elements, we hypothesized that the function of COQ4 might not be limited to a structural role. Here, we provide evidence that COQ4 acts as an oxidative decarboxylase, substituting in a single step the carboxylic acid group with a hydroxyl group on carbon 1 of CoQ precursors.

RESULTS

Cells lacking COQ4 accumulate 3-decaprenyl-4-hydroxybenzoic acid, the carboxylated precursor of CoQ10

We have previously generated a HEK293 cell line in which the COQ4 gene was disrupted by TALEN nucleases (ΔCOQ4 cells). These cells do not produce any CoQ10, but they incorporate some exogenous CoQ10 from the culture medium [START_REF] Giorgio | The idebenone metabolite QS10 restores electron transfer in complex I and coenzyme Q defects[END_REF]

COQ4 binds zinc and metal binding is essential for function

The human COQ4 protein (hCOQ4) lacking the mitochondrial targeting sequence [START_REF] Casarin | Functional characterization of human COQ4, a gene required for Coenzyme Q(10) biosynthesis[END_REF] and containing an N-terminal His Tag was expressed in E. coli. Although hCOQ4 was mainly found in inclusion bodies, we were able to partially solubilize it using 1% sarcosyl in the elution buffer (Fig S2C -D).

Atomic absorption experiments showed that purified recombinant hCOQ4 treated with a zinc chloride solution retained approximately two zinc ions per protein (Fig 2A). We then mutagenized the aspartate in position 164 within the HDxxH(x)11E putative Zinc-binding domain (D164A). The Both WT and D164A COQ4 captured the same interaction profile of proteins involved in CoQ biosynthesis (Fig 2D ), consistent with previous AE-MS experiments describing the nature of complex Q [START_REF] Floyd | Mitochondrial Protein Interaction Mapping Identifies Regulators of Respiratory Chain Function[END_REF] . Taken together, these experiments demonstrate that mutating the zinc binding domain of COQ4 disrupts its catalytic activity while maintaining its ability to participate structurally in complex Q.

COQ4 rescues defects in C1-decarboxylation and C1-hydroxylation steps in the E. coli CoQ8 biosynthetic pathway

To verify if COQ4 might be involved in the decarboxylation step of the eukaryotic CoQ biosynthesis pathway, we set up a heterologous complementation assay with E. coli cells lacking the ubiD gene that encodes the decarboxylase of the bacterial CoQ pathway. E. coli ΔubiD cells are deficient in CoQ8 [START_REF] Zhang | Identification of the ubiD gene on the Escherichia coli chromosome[END_REF] and consequently present a growth defect on respiratory medium but not on fermentation medium. Heterologous expression of both yeast COQ4 (yCOQ4) and hCOQ4 increased the respiratory growth of ΔubiD cells, whereas the functionally inactive D164A mutant did not (Fig 3A).

The respiratory growth reflected a substantial increase in CoQ8 levels, to 14 and 18% of WT levels for hCOQ4 and yCOQ4 (Fig 3B). This increase was found to be specific to COQ4 because expression of other COQ biosynthetic proteins from yeast (COQ3, COQ5, COQ6, and COQ9) induced no significant rescue of CoQ8 levels (Fig 3C). These data establish that COQ4 is sufficient to complement the defective decarboxylation step in the CoQ pathway of E. coli ΔubiD cells.

Since the C1-hydroxylase of the eukaryotic CoQ pathway remains uncharacterized (Fig 1 ), we tested whether this function might also be carried out by COQ4. E. coli cells lacking the C1-hydroxylase UbiH are deficient in CoQ8 and accumulate the biosynthetic intermediate octaprenyl-phenol with a decarboxylated C1 [START_REF] Pelosi | Evolution of Ubiquinone Biosynthesis: Multiple Proteobacterial Enzymes with Various Regioselectivities To Catalyze Three Contiguous Aromatic Hydroxylation Reactions[END_REF] . Neither yCOQ4 nor hCOQ4 complemented the respiratory growth defect or the CoQ8 levels of ΔubiH cells (Fig S3A -B). However, both genes complemented the phenotypes of E. coli cells lacking both the ubiD and the ubiH genes (ΔubiD ΔubiH), and again the nonfunctional hCOQ4-D164A mutant was incapable of doing so (Fig 3D -E).

We verified that CoQ8 synthesis in ΔubiD ΔubiH cells expressing COQ4 occurred from the normal 4-HB precursor. When 13 C7-4HB was added to the growth medium, we observed a comparable labelling of CoQ8 in WT cells and in ΔubiD ΔubiH cells expressing COQ4 (Fig 3F). Thus, we conclude that cells expressing COQ4 use 4-HB as a precursor of CoQ8, which supports that the C1decarboxylation and the C1-hydroxylation are taking place. Finally, we found that the activity of yCOQ4 and hCOQ4 depends on O2 since these proteins did not increase CoQ8 levels in ΔubiD or ΔubiD ΔubiH cells grown under anaerobic conditions (Fig S3C). Overall, our data support a model in which COQ4 is able to perform a coupled decarboxylation and hydroxylation of position C1 in the CoQ biosynthesis pathway. We note that the presence of a carboxyl group on C1 seems necessary to the activity of COQ4 as C1 was not hydroxylated if it had already been decarboxylated by UbiD, as is the case in ΔubiH cells.

COQ4 synthesizes polyprenyl-1,4-benzoquinone in Corynebacterium glutamicum

C. glutamicum, an actinobacterium of biotechnological interest, synthesises a single isoprenoid quinone, MKn(H2), a partially saturated form of menaquinone 26 (Fig 4A). Recently, extensive genetic modification of C. glutamicum coupled to expression of the ubi genes from the E. coli CoQ biosynthesis pathway allowed production of CoQ10 in strain 413 [START_REF] Burgardt | Coenzyme Q(10) Biosynthesis Established in the Non-Ubiquinone Containing Corynebacterium glutamicum by Metabolic Engineering[END_REF] . We reasoned that C. glutamicum would be the organism of choice to test the activity of COQ4, given the absence of endogenous CoQ biosynthesis genes. In comparison to the control strain 401, strain 405 expressing a decaprenyl synthase DdsA (a COQ1 homolog from Paracoccus denitrificans) and E. coli UbiA (the homolog of COQ2) produced several additional compounds that fall into two categories: polyprenyl-4hydroxybenzoic acids (8-11P-HB) eluting between 3 and 7 min, and MK8-11(H2) eluting between 10 and 25 min (Fig S3D) [START_REF] Burgardt | Rational Engineering of Non-Ubiquinone Containing Corynebacterium glutamicum for Enhanced Coenzyme Q10 Production[END_REF] . For both compounds, 9 and 10 were the most abundant isoprenologs (Fig

DISCUSSION

The exact function of COQ4 within CoQ biosynthesis and the nature of the enzymes responsible for the C1 decarboxylase and hydroxylase activities in the eukaryotic CoQ pathway have remained elusive for many years. We now provide evidence that COQ4 carries out these two reactions in single step as an oxidative decarboxylation reaction. In fact, we showed that expression of COQ4 complemented E. coli strains deficient either for the decarboxylase (ΔubiD) or for the decarboxylase and the C1-hydroxylase (ΔubiD ΔubiH). These data suggest that COQ4 converts octaprenyl-4hydroxybenzoic acid (8P-HB, the C1-carboxylated precursor), which accumulates in ΔubiD and ΔubiDΔubiH strains, into a molecule with an oxygen-substituted C1 position like octaprenyl-1,4benzoquinone (PBQ8) or its reduced form (see Fig 1A). Since COQ4 is ineffective in the E. coli ΔubiH strain that accumulates octaprenyl-phenol (8PP), we suggest that COQ4 requires a carboxyl group on C1 and does not function as a bona fide C1 hydroxylase, but rather as an oxidative decarboxylase, similar to the VibMO1 protein from Boreostereum vibrans, a protein which belongs to the family of FAD-dependent monooxygenases [START_REF] Yang | A Monooxygenase from Boreostereum vibrans Catalyzes Oxidative Decarboxylation in a Divergent Vibralactone Biosynthesis Pathway[END_REF] . Experiments in C. glutamicum support this view since cells expressing COQ4 synthesized PBQ9-10 without any accumulation of 9/10PP, contrary to cells expressing UbiD-X in which it was the only product (Fig 4). In addition, a putative requirement of other COQ proteins to assist the oxidative decarboxylation of C1 is ruled out because COQ4 alone was sufficient to form PBQ8-11 from 8-11P-HB in C. glutamicum, an organism that does not produce CoQ naturally and therefore does not contain endogenous COQ proteins.

The fact that ΔCOQ4 cells accumulate mainly 10P-HB (the product of COQ2) and that ΔCOQ6 cells accumulate PBQ10 (the product of COQ4) support the hypothesis that, at least in our HEK293 cell model, the oxidative decarboxylation by COQ4 occurs preferentially before the C5 hydroxylation by COQ6 [START_REF] Acosta Lopez | Vanillic Acid Restores Coenzyme Q Biosynthesis and ATP Production in Human Cells Lacking COQ6[END_REF] . We therefore propose an updated eukaryotic CoQ biosynthesis pathway with this sequence of reactions (path 1, Fig S4). It should be noted that the sensitivity of our HPLC-mass spectrometry analyses is limited, therefore we cannot exclude the formation of small amounts of other biosynthetic intermediates, which may result from alternative sequences of reactions in the knockout cells.

Compounds that accumulate in cells deficient for COQ genes may not necessarily correspond to the substrates of the deficient COQ enzymes as some reactions may occur "out of order" when the CoQ biosynthesis pathway is impaired. This is exemplified by DMQ10, the substrate of COQ7 and penultimate intermediate of the CoQ pathway, which was detected in low amounts in COQ3-COQ6 siRNA cells, but which accumulated at much higher levels in the COQ7 siRNA cells [START_REF] Herebian | Detection of 6-demethoxyubiquinone in CoQ10 deficiency disorders: Insights into enzyme interactions and identification of potential therapeutics[END_REF] . We consider that the "traditional" order of the CoQ pathway (path 2, Fig S4) is also compatible with our findings since previous studies in yeast showed that vanillic acid can replace 4-HB as a substrate of CoQ 7 . As vanillic acid is prenylated by COQ2 before being decarboxylated on C1 [START_REF] Pierrel | Impact of Chemical Analogs of 4-Hydroxybenzoic Acid on Coenzyme Q Biosynthesis: From Inhibition to Bypass of Coenzyme Q Deficiency[END_REF] , yCOQ4 is necessarily able to process a substrate with a methoxyl group on C5 (path 2 in Fig S4). As such, COQ4 promiscuity may allow various sequences of reactions to take place. Interestingly, the order of the reactions in the bacterial CoQ pathway was recently proposed to differ between E. coli and Rhodobacter capsulatus, in which a new flavin-dependent hydroxylase was shown to function as a C1-hydroxylase and also as a C1-decarboxylase when UbiD was inactivated [START_REF] Nagatani | UbiN, a novel Rhodobacter capsulatus decarboxylative hydroxylase involved in aerobic ubiquinone biosynthesis[END_REF] .

A number of open issues still exist around COQ4. First, the exact mechanism of the oxidative decarboxylation reaction is not clear. Molecular oxygen is required as demonstrated by the lack of complementation of E. coli mutants in anaerobic conditions, and by labelling experiments in yeast, which showed that all three oxygen atoms added on CoQ precursors are derived from O2 [START_REF] Ozeir | Coq6 Is Responsible for the C4-deamination Reaction in Coenzyme Q Biosynthesis in Saccharomyces cerevisiae[END_REF] . Furthermore, we still do not know which metal ion is bound by COQ4 under physiological conditions.

Alr8543 was crystallized with magnesium ions, and we showed zinc binding in recombinant human COQ4. In fact, the geometry of the binuclear metallic site of COQ4 is similar to that of binuclear metallohydrolases, which are known to bind different types of metal ions (Fe, Zn, Mn, Co) in their active site [START_REF] Schenk | Binuclear Metallohydrolases: Complex Mechanistic Strategies for a Simple Chemical Reaction[END_REF] . Therefore, the COQ4 active site may accommodate various metal ions, but finding those with physiological relevance will likely necessitate the development of an in vitro activity assay. Our repeated attempts to detect in vitro activity with purified recombinant COQ4 using different combination of metals (zinc, iron, copper), cofactors (NADH, NADPH, FAD, ATP), and substrates (4-HB, vanillic acid, polyprenyl vanillic acid derived from yeast lipid extracts) were unsuccessful.

The presence of a hydrophobic tunnel next to the putative active site of the cyanobacterial homolog of COQ4 suggests that a polyprenyl side chain on the substrate might be necessary for the reaction to take place. However, the hydrophobicity of polyprenylated substrates complicates their use in in vitro assays with aqueous buffers. Moreover, since COQ4 is associated with the inner mitochondrial membrane [START_REF] Marbois | Coq3 and Coq4 define a polypeptide complex in yeast mitochondria for the biosynthesis of coenzyme Q[END_REF] , the presence of lipids could also be critical for enzymatic activity. Several of the issues discussed above might be clarified if structural data were available, which is unfortunately not the case for any eukaryotic COQ4 homologs.

Even though we have obtained only genetic data in this study, our demonstration of COQ4 activity in two different in vivo models provides strong support for our conclusions. In particular, the activity observed in the non-natural CoQ producer C. glutamicum is undoubtedly linked to the heterologously expressed yCOQ4. Our conclusion that COQ4 carries out, in a single step, the oxidative decarboxylation of the C1 carbon of CoQ precursors fills a major gap in the eukaryotic CoQ biosynthesis pathway since specific enzymes have now been assigned to all the steps required for ring modification.

LIMITATIONS OF THE STUDY

One limitation of our study is the lack of structural data on COQ4. Only models are currently available to gain insights in the structuration of COQ4 active site in which the D164 residue seems to play an important role to coordinate the metal center. Despite numerous attempts, we could not obtain structural or functional data on purified COQ4. This is not unexpected as COQ proteins are notorious to behave poorly when purified individually, without their partners from complex Q. Indeed, among COQ proteins belonging to complex Q, only the structure of an heterotetramer composed of COQ7 and COQ9 is available [START_REF] Manicki | Structure and functionality of a multimeric human COQ7:COQ9 complex[END_REF] .

Another limitation of our study pertains to the uncertainty about the position of COQ4 within the sequence of reactions leading to CoQ. As illustrated in Figure S4, COQ4 may act before or after the addition of the methoxy group on C5. We favour the former possibility based on the in vivo data obtained with ΔCOQ4 human cells and with COQ4 expression in C. glutamicum, but obtaining a definitive answer will likely require the development of a functional in vitro assay. B-E). See also Figure S3.

STAR METHODS

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the L.S (leonardo.salviati@unipd.it).

Materials availability

All unique/stable reagents generated in this study are available from the lead contact upon reasonable request.

o The raw proteomic data generated during the current study have been deposited at MassIVE repository and are publicly available under accession code MSV000093201 as of the date of publication. Accession numbers are also listed in the key resources table. This paper does not report original code.

Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Experimental model and subject details

Cell lines

Human Cells and Culture conditions ΔCOQ4, ΔCOQ6 cell lines, and patients fibroblasts have been reported previously [START_REF] Acosta Lopez | Vanillic Acid Restores Coenzyme Q Biosynthesis and ATP Production in Human Cells Lacking COQ6[END_REF][START_REF] Mero | New pathogenic variants in COQ4 cause ataxia and neurodevelopmental disorder without detectable CoQ10 deficiency in muscle or skin fibroblasts[END_REF][START_REF] Cerqua | Vitamin K2 cannot substitute Coenzyme Q10 as electron carrier in the mitochondrial respiratory chain of mammalian cells[END_REF] . Fibroblasts were cultured in DMEM High Glucose (Gibco™ 11965092), supplemented with 10% FBS, and HEK293 cells were cultured in DMEM High Glucose (Gibco™ 11965092), supplemented with 10% FBS, 1% penicillin-streptomycin, 1 mM pyruvate, 2 mM glutamine, and 10 µM uridine.

Microbes

E. coli E. coli strains are listed in the Key resources table. Unless otherwise stated, E. coli cultures were performed in lysogeny broth medium (LB) at 37°C with 180 rpm shaking. Anaerobic cultures were performed in Hungate tubes containing 12 mL LB medium (with 0.5 mL/L antifoam B emulsion, Sigma) deoxygenated by argon (O20.1 ppm) bubbling for 25 min before autoclaving. Hungate tubes were inoculated through the septum with 100 µL of overnight precultures taken with disposable syringes and needles from closed Eppendorf tubes filled to the top. Ampicillin (50 mg/L) or kanamycin (25 mg/L) were added from stock solutions (1,000X, sterilized through 0.22 µm filters and stored at -20°C) to maintain plasmid selection. Media contained 100 M IPTG or 0.05% arabinose for induction of the pTrc-yCOQ4 or pBAD-hCOQ4 plasmids, respectively.

S. cerevisiae

S. cerevisiae lacking COQ4 (W303Δcoq4) was transformed using the PEG-lithium acetate method as previously described [START_REF] Burke | Methods in Yeast Genetics[END_REF] . Transformants were selected and maintained in minimum medium (SM Glu) containing 2% glucose, 0.17% yeast nitrogen base without amino acids, 0.5% ammonium sulfate, leucine 60 mg/L, tryptophan 20 mg/L.

C. glutamicum

C. glutamicum strains 401 and 405 were obtained from strain UBI400 [START_REF] Burgardt | Rational Engineering of Non-Ubiquinone Containing Corynebacterium glutamicum for Enhanced Coenzyme Q10 Production[END_REF] by transforming and empty pRG_Duet2 or pRG_Duet2-ddsAPd-ubiAEc, respectively. Transformation was carried out via electroporation [START_REF] Bott | Handbook of Corynebacterium glutamicum[END_REF] at 2.5 kV, 200 Ω, and 25 µF. Strain 405 was additionally transformed with plasmids pEC-XT99A, pEC-XT99A-yCOQ4 or pEC-XT99A-ubiDX.

Method details

Plasmids construction

Oligonucleotides used in this study are listed in Table 1. Human COQ4 devoid of the mitochondrial targeting sequence (hCOQ4) was amplified from pCRII-TOPO-COQ4 (which contains the entire coding region, the STOP codon and 10 nucleotides of the 3' UTR) [START_REF] Casarin | Functional characterization of human COQ4, a gene required for Coenzyme Q(10) biosynthesis[END_REF] using primers hCOQ4_HisT_NdeI_For (which encodes the His Tag and enterokinase consensus) and hCOQ4_HindIII_R, digested with NdeI and HindIII (NEB) and cloned into the pET21a vector (Snapgene) cut similarly. The D164A mutant was obtained by site directed mutagenesis using the QuikChange lightning site-directed mutagenesis kit (Agilent) which employs a high-fidelity Pfu Ultra DNA polymerase and primers D164A_F and D164A_R. The PCR products were digested with DpnI (Agilent) and then used to transform E. coli Mach1 competent cells. For subsequent cloning into pBAD24, COQ4 was amplified from pET21a-hCOQ4 or pET21a-hCOQ4_D164A using primers hCOQ4_BAC_ NheI_F and hCOQ4_HindIII_ R, digested with NheI and HindIII, and cloned into the pBAD vector similarly digested. The resulting pBAD-hCOQ4 plasmid contained hCOQ4 without the mitochondrial targeting sequence and without any tag. The open reading frame of the yeast COQ4 gene (yCOQ4) was amplified from the genomic DNA of the S. cerevisiae W303 strain using primers yCOQ4_Sac_Fw and yCOQ4_HIII_Rv. The PCR product was digested with SacI and HindIII and cloned into pTrc99a digested similarly to yield pTrc-yCOQ4. pTrc-UbiH was constructed by cloning the PCR-amplified UBIH gene (using primers UbiH5-EcoRI / UbiH3-HindIII and E. coli K-12 MG1655 genomic DNA as template) into pTrc99a after digestion with EcoRI and HindIII. All constructs were verified by Sanger sequencing. For yeast expression studies pCM189_COQ4 [START_REF] Mero | New pathogenic variants in COQ4 cause ataxia and neurodevelopmental disorder without detectable CoQ10 deficiency in muscle or skin fibroblasts[END_REF] was mutagenized using primers D164A_F and D164A_R as described above) to yield the D164A mutant (pCM189_COQ4 D164A ). Plasmids for other COQ biosynthetic proteins were prepared in the pET28b background with an ampicillin resistance cassette. Sequences for NΔ31 yCOQ3, NΔ30 yCOQ5, NΔ16 yCOQ9 were amplified from yeast vectors containing the full S. cerevisiae genomic coding regions for those genes. N-terminal truncations sites were determined using MitoFates [START_REF] Fukasawa | MitoFates: improved prediction of mitochondrial targeting sequences and their cleavage sites[END_REF] . The PCR products for the COQ3/COQ5/COQ9 constructs were integrated into the empty pET28b using a HindIII and XhoI restriction enzyme (NEB) digest. The full-length COQ6 construct was amplified from a gBlock gene synthesis (IDT) containing a codon-optimized gene sequence for yeast COQ6 and was inserted into the pET28b backbone using Gibson cloning via the NEBuilder® HiFi DNA Assembly Cloning Kit (NEB). All four constructs were verified using Sanger sequencing.

C. glutamicum plasmids were constructed via Gibson Assembly 28 after gene amplification with

Phusion High-Fidelity DNA polymerase (New England Biolabs, United Kingdom). The gene ubiX was amplified from genomic DNA of E. coli K-12 MG1655 using the primers ubiX-fw and ubiX-rv.

The gene COQ4 from S. cerevisiae was codon-harmonized [START_REF] Haupka | Codon harmonization[END_REF] , synthesized (Life Technologies GmbH, Darmstadt, Germany), and amplified using the primers yCOQ4-CG-fw and yCOQ4-CG-rv. The plasmid pEC-XT99A-ubiD [START_REF] Burgardt | Coenzyme Q(10) Biosynthesis Established in the Non-Ubiquinone Containing Corynebacterium glutamicum by Metabolic Engineering[END_REF] was cut using XbaI to construct pEC-XT99A-ubiDX and pEC-XT99A [START_REF] Kirchner | Tools for genetic engineering in the amino acid-producing bacterium Corynebacterium glutamicum[END_REF] was cut using BamHI to construct pEC-XT99A-yCOQ4.

Mammalian expression plasmids pcDNA3.1 GFP-FLAG and pcDNA3.1 hCOQ4-FLAG were generated and described previously 12 . The pcDNA3.1 hCOQ4-FLAG construct was mutagenized using the QuikChange lightning site-directed mutagenesis kit (Agilent) and primers hCOQ4_D164A_F and hCOQ4_D164A_R to generate pcDNA3.1 hCOQ4-FLAG D164A.

Construction of E. coli strains

The chloramphenicol cassette was eliminated from the ubiD::cat strain using plasmid pCP20 as described previously [START_REF] Cherepanov | Gene disruption in Escherichia coli: TcR and KmR cassettes with the option of Flp-catalyzed excision of the antibiotic-resistance determinant[END_REF] to yield the ubiD strain. The ubiD ubiH strain was constructed by introducing the ubiH::kan deletion into the ubiD strain with P1 vir transduction [START_REF] Thomason | E. coli genome manipulation by P1 transduction[END_REF] and selection for kanamycin resistance. The proper insertion of the cassette was verified by locus PCR with primers UbiHprom5 and UbiHterm3.

Mammalian cell culture and transfection

HEK293 WT and ΔCOQ4 cells were grown in Dulbecco's modified Eagle's medium (DMEM, LifeTechnologies) supplemented with 10% fetal bovine serum, 1% penicillin-streptomycin, 1 mM pyruvate, 2 mM glutamine, and 10 µM uridine. Cells were subcultured by trypsinization. For affinity enrichment mass spectrometry experiment, 7 x 10 6 HEK293 cells were plated in a 15 cm dish and allowed to grow overnight. On day two, cells were transiently transfected with a mix of 20 µg plasmid (pcDNA3.1 GFP-FLAG, pcDNA3.1 hCOQ4-FLAG WT, or pcDNA3.1 hCOQ4-FLAG D164A), 72 µg linear polyethylenimine (PEI, PolySciences), and 900 µL Opti-MEM (LifeTechnologies). After 48 hours (Day 4), cells were washed with and harvested into phosphate-buffered saline (PBS), were collected at 2,000 rcf, were snap frozen in liquid nitrogen, and were stored at -80°C. For CoQ10 measurements, 2.5 x 10 6 HEK293 WT and ΔCOQ4 were plated in a 10 cm dish and allowed to grow overnight. On day two, cells were transiently transfected with a mix of 10 µg plasmid (pcDNA3.1 GFP-FLAG, pcDNA3.1 hCOQ4-FLAG WT, or pcDNA3.1 hCOQ4-FLAG D164A), 36 µg linear polyethylenimine (PEI, PolySciences), and 500 µL Opti-MEM (LifeTechnologies). After 48 hours (Day 4), cells were washed with and harvested into phosphate-buffered saline (PBS), were collected at 2,000 rcf, were snap frozen in liquid nitrogen, and were stored at -80°C.

Expression and purification of human COQ4

The expression of hCOQ4 and of the D164A allele from the pET21a plasmids was induced by adding 1mM isopropyl-1-thio-β-D-galactopyranoside to the culture of E. coli BL21 (DE3) cells. After incubation for 4 h at 30°C, cells were harvested by centrifugation (for 20 min, at 2400 g, at 4 °C), suspended in the lysis buffer (Na2HPO4 20 mM, NaCl 0.5 M, imidazole 5 mM, DTT 1mM, pH 7.8), and lysed by sonication (Omni Sonic Ruptor 400 Ultrasonic Homogenizer, 30-second cycle: 5 seconds ON, 25 seconds OFF -> 14 cycles total). After centrifugation (for 30 min at 18000 g 4°C) and due to the low solubility of the protein, the inclusion bodies were solubilized O/N at room temperature in the lysis buffer added with 1% sarcosyl. After centrifugation (for 20 min, at 2400g, at 4 °C), the solubilized protein was isolated with a Ni affinity column (HisTrap HP, 5 mL, Cytiva) using an AKTA-FPLC and the eluate was dialyzed against a buffer containing NaCl 150 mM, Hepes 20 mM, DTT 1 mM, sarcosyl 0.1%, and ± ZnCl2 100 µM. The excess of ZnCl2 was eliminated by extensive dialysis against the same buffer without zinc. The recombinant hCOQ4 was further purified with a gel filtration chromatography using a Superdez Yarra SEC-2000 column (Phenomenex). The sample purity and the molecular weight were checked by SDS/PAGE (see Fig S2C). Protein concentration was determined by UV absorption at 280 nm and by Brosted-Lowry method.

Circular dichroism spectroscopy

Measurements of optical ellipticity were made at room temperature on a Jasco J-710 spectropolarimeter in a 0.1 cm path-length quartz cell at a protein concentration of 0.4 mg/mL in 25 mM Tris-HCl, pH 8.0, 100 mM NaCl. Ellipticities were normalized to residue concentration using the relationship [θ]=(θobs/10)•(M/l*c) where θ obs is the observed ellipticity at a given wavelength, M is the mean residue mass, l is the cuvette path length in centimeters, and c is the protein concentration expressed as g/mL. All CD spectra resulted from an averaging of at least three scans and the final spectrum was corrected by subtracting the corresponding buffer spectrum obtained under identical conditions.

Analysis of zinc content

The zinc content of the hCOQ4 and hCOQ4-D164A proteins purified in the presence or absence of zinc was measured using a Perkin-Elmer 4000 atomic absorption flame spectrophotometer. Samples containing 3 µM of protein were measured after standardization in the linear range of 0-0.5 ppm of zinc and the buffer was used as blank. Analyses were performed in triplicate.

Yeast complementation assay

S. cerevisiae W303Δcoq4 harboring the pRS423-yCOQ8 plasmid to overexpress yeast Coq8, was transformed with pCM189_COQ4 or pCM189_COQ4 D164A . For plate dilution assays, each strain was cultured overnight in SM Glu medium. Optical densities (A600 nm) of harvested cells were adjusted to 0.2, and 2 μl of 1:10 serial dilutions were spotted onto agar plates (Bacto-agar 16 g/L) containing SM Glu, YPD (1% yeast extract, 2% peptone, 2% glucose) or YPGly (1% yeast extract, 2% peptone, 3% glycerol) media. The plates were imaged after incubation for 5 days at 30 °C.

in vivo COQ4 activity assays

E. coli serial dilution assay

Overnight cultures in LB medium with antibiotics for plasmid selection were adjusted to 1 OD600 and used for 10 fold serial dilutions. 5 L drops were spotted on agar plates (Bacto-agar 16 g/L) with fermentation medium (LB + 0.2% glucose) or with respiration medium (synthetic medium [START_REF] Pelosi | Ubiquinone Biosynthesis over the Entire O2 Range: Characterization of a Conserved O2-Independent Pathway[END_REF] supplemented with 0.4% succinate adjusted to pH 6). The plates contained either 100 µM IPTG or 0.05% arabinose for induction of pTrc-yCOQ4 or pBAD-hCOQ4, respectively. The plates were imaged after incubation for 20 hours at 37 °C in aerobic conditions (aerobic respiration) or inside a GasPak anaerobic jar fitted with a BD anaerobic sachet (fermentation).

COQ4 expression in C. glutamicum

Pre-cultures were cultivated in brain heart infusion (BHI) medium at 30 °C in baffled shake flasks at 120 rpm. Respective antibiotics were added in concentrations of 25 µg mL -1 kanamycin, 5 µg mL -1 tetracycline or 100 µg mL -1 spectinomycin. After approximately 16 hours, the cells were washed with TN buffer pH 6.3 (10 mM Tris-HCl, 150 mM NaCl), inoculated to an optical density (OD600) of 1 in CGXII minimal medium [START_REF] Bott | Handbook of Corynebacterium glutamicum[END_REF] with 40 g L -1 glucose, 1 mM IPTG, and 0.25 µg mL -1 anhydrotetracycline (ATc), and cultivated for 72 hours at 30 °C and 120 rpm. 100 mg (wet weight) of cells were centrifuged at 20,200 g for 10 min and stored at -20 °C until quantification of CoQ biosynthetic intermediates.

Quantification of coenzyme Q and biosynthetic intermediates

Measurement by HPLC-ECD-MS Non polar lipids were extracted from HEK293 cells, and CoQ10 and 10P-HB were analyzed by HPLCelectrochemical detection-mass spectrometry as previously described [START_REF] Acosta Lopez | Vanillic Acid Restores Coenzyme Q Biosynthesis and ATP Production in Human Cells Lacking COQ6[END_REF] with the following modifications. For analyses in reducing conditions, a potential of -800 mV was used on the precolumn electrode. The cone voltage was set to -80 V for mass spectrometry detection in negative mode. For E. coli cells, extraction and analysis were as described previously 45 .

Targeted Measurement by LC-MS/MS

For determination of the CoQ content in human primary fibroblasts and HEK293 cell rescue experiments, the frozen cell pellets were thawed and resuspended in 100 μL of PBS, then 5 uL of the resuspension was set aside for BCA assay to normalize the protein content. For E. coli pellets, the cell number was determined using optical density and the same number of cells was collected from each culture for centrifugation and freezing. After thawing, 50μL of 150mM KCl was added and used for resuspension. 600 μL of methanol with 0.1 μM CoQ8 (for human cell experiments) or 0.1 μM CoQ6 (for bacterial experiments) internal standard (Avanti Polar Lipids) was added to each sample. The cells were lysed by 5 min of shaking at 4 °C in a Disruptor Genie set to 3000 rpm. To extract lipids, 400 μL of petroleum ether was added to the sample and subjected to shaking as described previously for 3 min. Sample was spun 2 min. x 1000 g at 4°C and the top ether layer was transferred to a new tube. Extraction was repeated twice before the ether layers were combined and dried for 30 min under argon gas at room temperature. Extracted dried lipids were resuspended in mobile phase (78% methanol, 20% isopropanol, 2% 1 M ammonium acetate pH 4.4 in water) and transferred to amber glass vials with inserts. LC-MS analysis was performed using a Thermo Vanquish Horizon UHPLC system coupled to a Thermo Exploris 240 Orbitrap mass spectrometer. For LC separation, a Vanquish binary pump system (Thermo Fisher Scientific) was used with a Waters Acquity CSH C18 column (100 mm × 2.1 mm, 1.7 m particle size) held at 35 °C under 300 μL/min flow rate. Mobile phase A consisted of 5 mM ammonium acetate in acetonitrile:H2O (70:30, v/v) with 125 μL/L acetic acid. Mobile phase B consisted of 5 mM ammonium acetate in isopropanol:acetonitrile (90:10, v/v) with the same additive. For each sample run, mobile phase B was initially held at 2% for 2 min and then increased to 30% over 3 min. Mobile phase B was further increased to 50% over 1 min and 85% over 14 min and then raised to 99% over 1 min and held for 4 min. The column was re-equilibrated for 5 min at 2% B before the next injection. Five microliters of the sample were injected by a Vanquish Split Sampler HT autosampler (Thermo Fisher Scientific), while the autosampler temperature was kept at 4 °C. The samples were ionized by a heated ESI source kept at a vaporizer temperature of 350 °C. Sheath gas was set to 50 units, auxiliary gas to 8 units, sweep gas to 1 unit, and the spray voltage was set to 3500 V for positive mode and 2500 V for negative mode. The inlet ion transfer tube temperature was kept at 325 °C with 70% RF lens. For targeted analysis, the MS was operated in positive parallel reaction monitoring mode with polarity switching acquiring scheduled, targeted scans to CoQ10 (m/z 863.6912), CoQ8 (m/z 727.566), CoQ6 (m/z 591.4408), and CoQ intermediates: DMQ10 (m/z 833.6806), DMeQ10 (m/z 849.6755), DDMQ10 (m/z 836.6915), 10P-HB (m/z 817.6504) and BPQ10 (m/z 789.6555). MS acquisition parameters include resolution of 15,000, stepped HCD collision energy (25%, 30%, 40% for positive mode and 20%, 40%, 60% for negative mode), and 3s dynamic exclusion. Automatic gain control targets were set to standard mode. The resulting CoQ intermediate data were processed using TraceFinder 5.1 (Thermo Fisher Scientific).

C. glutamicum

CoQ biosynthetic intermediates were extracted from C. glutamicum cells (10-25 mg) and analyzed as described previously [START_REF] Burgardt | Rational Engineering of Non-Ubiquinone Containing Corynebacterium glutamicum for Enhanced Coenzyme Q10 Production[END_REF] . Glass beads (100 μL), 50 μL of 150 mM KCl, and a volume of 2 mM MK7 solution (used as an internal standard, Sigma-Aldrich) proportional to the cell weight (2 μL/mg) were added to cell pellet. CoQ intermediates were extracted by adding 0.6 mL of methanol, vortexing for 10 min, then adding 0.4 mL of petroleum ether (boiling range 40-60 °C) and vortexing for 3 min.

The phases were separated by centrifugation at room temperature for 1 min, 4,000 g and the upper petroleum ether layer was transferred to a fresh tube. The extraction was repeated with 0.4 mL petroleum ether, and petroleum ether layers were combined and dried under nitrogen. The dried samples were suspended in 100 μL ethanol and aliquots corresponding to 2 mg of cells wet weight were analyzed by reversed-phase HPLC with a C18 column (Betabasic-18, 5 m, 4.6 × 150 mm ;

Thermo Scientific) at a flow rate of 1 mL/min with a mobile phase composed of 25% isopropyl alcohol, 20% ethanol, 45% methanol and 10% of a mix of 89.9% isopropanol/10% ammonium acetate

CoQ8 labelling in E. coli with 13 C7-4HB

Overnight cultures in LB medium of MG1655 and ubiD ubiH cells containing pBAD-hCOQ4 or pTrc-yCOQ4 were used to inoculate (at OD600= 0.1) 10 mL LB cultures supplemented with 10 µM 13 C7-4HB (Sigma) and the proper inductor. Cultures were stopped on ice when OD600 reached 0.9-1.

Cells were collected and CoQ8 was extracted and analyzed by HPLC-MS [START_REF] Pelosi | Ubiquinone Biosynthesis over the Entire O2 Range: Characterization of a Conserved O2-Independent Pathway[END_REF] with single ion monitoring (6-10 min) for H + adducts at m/z 727-728 (UQ8) and m/z 733-734 ( 13 C6-UQ8).

Affinity Enrichment Mass Spectrometry

Pellets of HEK293 cells expressing WT or D164 versions of COQ4-Flag, or GFP-FLAG as a negative control, were lysed in 200 µL cold lysis buffer (20 mM HEPES, pH 7.4, 100 mM NaCl, 10% glycerol, 3% digitonin (Sigma), 1 mM DTT, and 1X cOmplete protease inhibitor cocktail (Roche). After periodic vortexing on ice, insoluble materials were pelleted (16,000 g, 10 min, 4°C) and the supernatant was retained. The protein concentration was quantified by BCA, and equal masses of cell supernatant were mixed with 30 µL pre-washed anti-FLAG magnetic beads for 2 h at 4°C with endover-end agitation. Following incubation, beads were washed four times in wash buffer (20 mM HEPES, pH 7.4, 100 mM NaCl, 0.05% digitonin, 10% glycerol) and twice in final wash buffer (20 mM HEPES, pH 7.4, 100 mM NaCl) before being subjected to on-bead trypsin digest. The on-bead proteins were denatured with 2M urea in 200 mM Tris pH 8.0, then reduced with 5 mM DTT for 30 min at 56°C and alkylated with 15 mM iodoacetamide for 30 min at RT in the dark. The proteins onbead were digested overnight at RT with 1 ug trypsin (Promega, V5113). The digested supernatant was acidified with 10% TFA to a pH of 2 and desalted with 10 mg StrataX solid phase extraction columns (Phenomenex), then dried under vacuum using a SpeedVac (Thermo Scientific) and stored at -80°C until MS analysis.

Samples were resuspended in 0.2% formic acid and subjected to LC-MS analysis. LC separation was performed using the Thermo Ultimate 3000 RSLCnano system. A 15 cm EASY-Spray™ PepMap™ RSLC C18 column (150 mm × 75 μm, 3 μm) was used at 300 nL/min flow rate with a 90 min gradient using mobile phase A consisting of 0.1% formic acid in H2O, and mobile phase B consisting of 0.1% formic acid in ACN/H2O (80/20, v/v). EASY-Spray source was used and temperature was at 35°C. Each sample run was held at 4.0% B for 5 min and increased to 50% B over 65 min, followed by 8 min at 95% B and back to 4% B for equilibration for 10 min. An Acclaim PepMap C18 HPLC trap column (20 mm × 75 μm, 3 μm) was used for sample loading. MS detection was performed with Thermo Exploris 240 Orbitrap mass spectrometer in positive mode. The source voltage was set to 1.8 kV, ion transfer tube temperature was set to 275°C, RF lens was at 70%. Full MS spectra were acquired from m/z 350 to 1400 at the Orbitrap resolution of 60000, with the normalized AGC target of 300% (3E6). Data-dependent acquisition (DDA) was performed for the top 20 precursor ions with the charge state of 2-6 and an isolated width of 2. Intensity threshold was 5E3. Dynamic exclusion was 30 s with the exclusion of isotopes. Other settings for DDA include Orbitrap resolution of 15000 and HCD collision energy of 30%.

Raw files were analyzed by SequestHT Search Engine incorporated in Proteome Discoverer v.2.5.0.400 software against human databases downloaded from Uniprot. Label-free quantification was enabled in the searches. The resulting data was analyzed by Perseus v 1.6.15.0 software 46 .

Seahorse Analysis

HEK293 WT and ΔCOQ4 cells were split, and 4 x 10 5 cells were plated per well in 6-well plates and allowed to grow overnight. On day two, cells were transiently transfected with a mix of 2 µg plasmid (pcDNA3.1 GFP-FLAG, pcDNA3.1 hCOQ4-FLAG WT, or pcDNA3.1 hCOQ4-FLAG D164A), 7.2 µg linear polyethylenimine (PEI, PolySciences), and 200 µL Opti-MEM (LifeTechnologies). On day 3, 40,000 cells per well were plated to Seahorse eXF24 plates coated with Cell-Tak (Corning) and allowed to adhere to the plate overnight in DMEM supplemented with 10% FBS and 1× penicillinstreptomycin. At 48 h post-transfection, and immediately before the Seahorse run, the medium was aspirated, cells were washed with dPBS, and medium was replaced with Seahorse XF DMEM Medium, pH 7.4 (Agilent no. 103575-100) supplemented with 10 mM glucose, 1 mM pyruvate and 2 mM glutamine. Oxygen-consumption rates (OCR) and extracellular acidification rate was monitored on a Seahorse eXF24 basally and in the presence of a Seahorse XF Cell Mito Stress Test (Agilent no. 103015-100). For the Stress Test, cells were treated with oligomycin (1 μM final concentration), FCCP (1 μM final concentration) and rotenone and antimycin A (0.

  . We analysed lipid extracts from WT and ΔCOQ4 cells by HPLC. Electrochemical detection (ECD) revealed a strong decrease of CoQ10 in ΔCOQ4 cells and the presence of a compound that eluted at 5.6 min, right before the CoQ8 standard (Fig 1B). This compound was not detected in WT or ΔCOQ6 cells and was characterized by HPLC-coupled mass spectrometry. Mass scanning (m/z 660-900) in positive mode revealed a prominent ion at m/z 836.7 at a retention time of 5.6 min in the extracts of ΔCOQ4 cells (Fig S2A). A signal at 5.6 min was indeed specifically detected in ΔCOQ4 cells with single ion monitoring for m/z 836.6 (Fig 1C), a mass (M + NH4 + ) compatible with that of 3-decaprenyl-4-hydroxybenzoic acid (10P-HB). The compound also showed a MS signal at m/z 817.6 in negative mode, consistent with the ionization of the carboxyl moiety by loss of a proton (M-)(Fig S2B). Overall, these data show that ΔCOQ4 cells are impaired for the biosynthesis of CoQ10 and accumulate 10P-HB, the product of the reaction catalysed by COQ2. A compound with a mass compatible with 10P-HB was found also in cultured skin fibroblasts from a patient with primary CoQ deficiency due to COQ4 mutations 15 and was absent in control cells (Fig 1D). These observations can have two explanations. The absence of COQ4 might disrupt the integrity of complex Q, preventing the occurrence of all subsequent ring modifications, leading to an indirect accumulation of the mentioned precursor. Alternatively, we considered the hypothesis that COQ4 could have a direct role in CoQ biosynthesis, catalysing the decarboxylation of the C1 carbon of the aromatic ring.

  D164A mutant was unable to complement respiratory growth in ΔCOQ4 yeast (Fig S2E) and the protein was virtually unable to bind zinc (Fig 2A). Interestingly, apo and holo hCOQ4 displayed far UV circular dichroism spectra comparable to those of the D164A mutant (Fig 2B), suggesting that the purified proteins maintained their fold irrespective of zinc binding. To test the functional importance of the zinc-binding domain, hCOQ4 WT or D164A was expressed in COQ4 HEK293 cells and CoQ10 levels were analysed by targeted mass spectrometry. Expression of hCOQ4 WT was sufficient to fully rescue the CoQ10 defect of COQ4 cells, however hCOQ4 D164A expression only marginally increased CoQ10 levels (Fig 2C). Consistent with previous results, COQ4 cells with empty vector accumulate 10P-HB, whose levels are slightly decreased upon expression of hCOQ4 WT, though this may have been hindered by poor transient transfection efficiency of the constructs (Fig 2C). While a substantial increase in CoQ10 levels was observed upon expressing hCOQ4 WT but not D164A in COQ4 cells, a similar trend in respiratory rate was observed by Seahorse analysis, with the modest rescue again stemming from the limitations of the low transfection efficiency (Fig S2F). To interrogate if the D164A mutation disrupted the previously described scaffolding role of COQ4 in complex Q 23 , we performed affinity enrichment mass spectrometry (AE-MS) of hCOQ4-FLAG WT and hCOQ4-FLAG D164A in WT HEK293 cells.

  S3D). Expression of yCOQ4 in strain 405 yielded several new peaks between 6 and 13 min that corresponded to isoprenologs of polyprenyl-1,4-benzoquinone (PBQ8-11, Fig 4B), as determined by mass spectrometry (Fig 4C and data not shown). This result established that yCOQ4 decarboxylated and hydroxylated position C1 of the 8-11P-HB substrates. Expression of the E. coli decarboxylase UbiD (together with UbiX, which synthesizes the prenyl-FMN cofactor of UbiD 10 ) in strain 405 caused the complete conversion of 9-10P-HB (Fig 4D) into nonaprenyl-and decaprenyl-phenol (9PP and 10PP, the C1-decarboxylated products) (Fig 4E) without formation of any PBQ9-10 (Fig 4C) (the C1-hydroxylated product). Interestingly, 9PP or 10PP were not detected in the strain expressing yCOQ4 (Fig 4E), strongly suggesting that the conversion of 9-10P-HB into PBQ9-10 occurred in a single step, via an oxidative decarboxylation. Together, these results demonstrate that COQ4 decarboxylates nP-HB into a product with an oxygenated substituent at C1, either a hydroxyl or a ketone (PBQn) depending on the redox state of the molecule (Fig 4A).

FFigure 1 .

 1 FIGURE TITLES AND LEGENDS

Figure 2 .

 2 Figure 2. The D164A mutation impairs COQ4 function and metal binding, but does not compromise stability or interaction with complex Q. A) Zn 2+ /protein molar ratio of wild type and

Figure 3 .

 3 Figure 3. COQ4 complements defects in C1-decarboxylation and C1-hydroxylation in E. coli.

Figure 4 .

 4 Figure 4. COQ4 synthesizes polyprenyl-1,4-benzoquinone in Corynebacterium glutamicum. A)

( 1 M

 1 )/0.1% formic acid. A precolumn 5020 guard cell operated in oxidizing mode (E, +650 mV) and compounds were monitored by mass spectrometry with an MSQ Plus spectrometer and by electrochemical detection with an ESA Coulochem III detector equipped with a 5011A analytical cell (E1, -650 mV ; E2, +650 mV). The MSQ Plus was used in positive mode (probe temperature 400 °C, cone voltage 80 V) and single ion monitoring detected the following (NH4 + adducts, scan time 0.2 s) : 9PP, m/z 724.1-725.1, 5-10 min ; 10PP, m/z 792.2-793.2, 6-13 min ; 9P-HB, m/z 768.1-769.1, 2-7 min ; 10P-HB, m/z 836.2-837.2, 3-8 min ; PBQ9, m/z 738.1-739.1, 5-10 min ; PBQ10, m/z 806.2-807.2, 6-13 min.

  [START_REF] Acosta Lopez | Vanillic Acid Restores Coenzyme Q Biosynthesis and ATP Production in Human Cells Lacking COQ6[END_REF] N/A Human primary fibroblasts carrying the COQ4 mutation c.577C>T/p.Pro193Ser in trans with c.718C>T/p.

  MS counts (m/z 806.7) MS counts (m/z 836.7) MS counts (m/z 768.6) MS counts (m/z 724.6) MS counts (m/z 792.7)

  5 μM final 45. Kazemzadeh, K., Pelosi, L., Chenal, C., Chobert, S.-C., Hajj Chehade, M., Jullien, M., Flandrin, L., Schmitt, W., He, Q., Bouvet, E., et al. (2023). Diversification of ubiquinone biosynthesis via gene duplications, transfers, losses, and parallel evolution. Mol Biol Evol, msad219.

	Key Resource Table
	KEY RESOURCES TABLE
	REAGENT or RESOURCE Experimental models: Cell lines 10.1093/molbev/msad219. Antibodies HEK 293 ΔCOQ4	SOURCE Cerqua et al. 35	IDENTIFIER N/A
	46. Tyanova, S., and Cox, J. (2018). Perseus: A Bioinformatics Platform for Integrative Analysis of Not relevant HEK 293 ΔCOQ6 Acosta Lopez et al.
	Proteomics Data in Cancer Research. Methods Mol Biol 1711, 133-148. 10.1007/978-1-4939-
	7493-1_7. Bacterial and virus strains
	E. coli Mach1 competent cells	Thermo Scientific™	Cat#C862003
	E. coli BL21 (DE3) competent cells	Sigma Aldrich (Merck) Cat#69450-M
	E. coli DH5α competent cells	Thermo Scientific™	Cat#18265017
	E. coli MG1655	Lab stock,	N/A
		Pelosi et al. 26
	E. coli ubiD::cat	Pelosi et al. 26	N/A
	E. coli ubiD	This paper	N/A
	E. coli ubiD ubiH	This paper	N/A
	E. coli ubiH	Pelosi et al. 26	N/A
	C. glutamicum UBI400	Burgardt et al. 28	N/A
	C. glutamicum 401	This paper	N/A
	C. glutamicum 405	This paper	N/A
	Biological samples
	Not relevant
	Chemicals, peptides, and recombinant proteins
	isopropyl-1-thio-β-D-galactopyranoside	Euroclone	Cat#EMR066005
	N-Lauroylsarcosine	Sigma Aldrich (Merck) Cat#L9150
	Anhydrotetracycline (hydrochloride)	biomol	Cat#Cay10009542-
			100
	13 C7-4HB	Sigma Aldrich (Merck) Cat#606472
	Digitonin	MilliporeSigma	Cat#D141-500MG
	Coenzyme Q6	Avanti Lipids	Cat#900150O-1mg
	Coenzyme Q8	Avanti Lipids	Cat#900151P-1mg
	Trypsin	Promega	Cat#V5113
	Polyethylenimine	Polysciences	Cat#19850
	CellTak	Corning	Cat#354240
	cOmplete protease inhibitor cocktail	MilliporeSigma	Cat#11697498001
		(Roche)
	Critical commercial assays
	QuikChange lightning site-directed mutagenesis kit	Agilent	Cat#210518
	Seahorse XFe24 FluxPak Mini	Agilent	Cat#102342-100
	Seahorse XF Cell Mito Stress Test	Agilent	Cat#103015-100
	Seahorse XF DMEM Medium	Agilent	Cat#103575-100
	NEBuilder® HiFi DNA Assembly Cloning Kit	New England Biolabs	Cat#E5520S
	Deposited data
	Affinity enrichment mass spectrometry data	MassiVE	MSV000093201

ACKNOWLEDGEMENTS

This work was supported by grants from the ANR to FP (ANR-19-CE44-0014O2-TABOO), from the NIH to DJP (R35GM131795), from Telethon Italy (GGP13222 and 14187c), from Fondazione IRP Città della speranza (to LS), from Fondazione CARIPARO (20/19 FCR) to ET, and from European Regional Development Fund (FEDER)/Andalusian Regional Government (FEDER operational program 2014-2020; P18-RT-4572). Experimental models: Organisms/strains S. cerevisiae W303 Δcoq4 Mero et al. 15 N/A Oligonucleotides Oligonucleotides used in this study are listed in Table S1 Recombinant DNA Plasmid: pRG_Duet2 Burgardt et al. 27 Genbank: MK130721 Plasmid: pRG_Duet2-ddsAPd-ubiAEc Burgardt et al. 27 N/A Plasmid: pEC-XT99A Kirchner and Tauch 40 Genbank: AY219684 Plasmid: pEC-XT99A-ubiD Burgardt et al. 27 N/A Plasmid: pEC-XT99A-ubiDX This paper N/A Plasmid: pEC-XT99A-yCOQ4 This paper N/A Gene synthesis: COQ4 from S. cerevisiae codonharmonized to C. glutamicum ATCC 13032 This paper N/A Plasmid: pET21a Novagen Cat#69740-3 Plasmid: pBAD24 ATTC Cat#87399 Plasmid: pBAD-hCOQ4 This paper N/A Plasmid: pET21a-hCOQ4 This paper N/A Plasmid: pTrc-yCOQ4 This paper N/A Plasmid: pTrc-UbiH This paper N/A Plasmid: pcDNA3.1 GFP-FLAG Floyd et al. 12 N/A Plasmid: pcDNA3.1 hCOQ4-FLAG Floyd et al. 12 N/A Plasmid: pcDNA3.1 hCOQ4-FLAG D164A This paper N/A Plasmid: pET28b NΔ31 yCOQ3 This paper N/A Plasmid: pET28b NΔ30 yCOQ5 This paper N/A Plasmid: pET28b NΔ16 yCOQ9 This paper N/A Plasmid: pET28b yCOQ6 This paper N/A Gene synthesis: COQ6 from S. cerevisiae codonharmonized to E. coli. This paper N/A Plasmid: pRS423-yCOQ8 Ozeir et al. 33 N/A Plasmid: pCM189_COQ4 Mero et al. 15 N/A Plasmid: pCM189_COQ4 D164A This paper N/A Software and algorithms TraceFinder 5.1 ThermoFisher N/A XCalibur 4.4 ThermoFisher N/A Proteome Discoverer v2.5.0.400 ThermoFisher N/A Perseus v1.6.15.0 Tyanova and Cox 45 N/A Prism v9.4.1 GraphPad N/A MitoFates Fukasawa et al. 38 N/A Wave v2.6.0 Agilent N/A Other SnakeSkin Dialysis Tubing, 3,500 MWCO Thermo Scientific™ Cat#68100 Superdez Yarra SEC-2000 column Phenomenex Cat#00H-4512 Ni affinity column HisTrap HP, 5ml Cytiva Cat#17524801 BD GasPak™ EZ Gas Generating Systems Fisher Scientific Cat#260001 Anti-FLAG® M2 Magnetic Beads Sigma Cat#M8823 StrataX solid phase extraction columns Phenomenex Cat# 8B-S100-AAK

Data and code availability

All data were deposited in Cell-Press-recommended, datatype-specific repositories and made publicly accessible.

AUTHORS CONTRIBUTION SECTION

Conceptualization, L.S., F.P.; methodology, L.S., F.P., P.N., G.Z., R.M.G., F.T., M.A.D., C.A., G.B.C., L.V., E.T., D.J.P., W.F.W.; investigation, L.P., L.M., L.V., M.A.D., A.B., A.K.B., R.G., K.K.F., B.R., M.R.M.; formal analysis, F.T., A.K.B., D.J.P., G.B.C., W.F.W., L.S., F.P.; resources, E.T., L.S., F.P., W.F.W.; writingoriginal draft, L.S., F.P.; writingreview & editing, F.P., D.J.P., concentration). To normalize the cell number per well, cells were then fixed with 1% glutaraldehyde and stained with 1.5% crystal violet, and, after release of the stain with 10% acetic acid, each well was read at an absorbance of 590 nm. Data were exported from the Wave software (version 2.6.0) and analyzed using Prism (n = 5, error bars represent standard deviation).

Quantification and statistical analysis

See each individual method and figures' legends for the associated statistical analysis. The majority of p values in this report were calculated using an unpaired, one-sided, Student's t test. In all cases, n represents independent replicates of an experiment and error bars represent standard deviation.

SUPPLEMENTAL INFORMATION TITLES AND LEGENDS


A Skarlatos

A Martínez-De-Guerenu

D J Badiola

I Lelidis

Interpretation of the magnetic susceptibility behaviour of soft carbon steels based on the scaling theory of second order phase transitions for systems with supercritical disorder

Keywords: random field Ising model, Lorentz function, magnetic susceptibility, isothermal annealing, recovery process

The experimental magnetic susceptibility curves of several soft steel grades, like interstitial free (IF) or low carbon (LC) steels, can be interpreted within the framework of universal scaling functions obtained for systems with quenched disorder, such as the one described by the random field Ising model with supercritical disorder. Mean-field theory (both scalar and site-dependent) is used to explore the behaviour at the proximity of the coercive field, and explicit expressions are derived. As a result, the susceptibility values close to the coercive field can be approximated to a good extend by a Lorentzian function. Theoretical results are compared against a number of experimental curves obtained from interstitial free (IF) and extra low carbon (ELC) steels subjected to isothermal annealing.

Introduction

Differential magnetic susceptibility measurements for soft industrial steel grades, like interstitial free (IF) and low or extra-low carbon (LC, ELC) steels, obtained during recovery, reveal a number of common features, among them, a notable resemblance of the susceptibility profile for external magnetic field values close to the coercive field [START_REF] Martínez-De-Guerenu | Recovery during annealing in a cold rolled low carbon steel. Part I: Kinetics and microstructural characterization[END_REF][START_REF] Martínez-De-Guerenu | Recovery during annealing in a cold rolled low carbon steel. Part II: Modelling the kinetics[END_REF][START_REF] Martínez-De-Guerenu | Nondestructive characterization of recovery and recrystallization in cold rolled low carbon steel by magnetic hysteresis loops[END_REF].

The evolution of several magnetic indicators with the annealing temperature and holding time indicated in the above cited articles reveals the underlying link between the steel microstructure and the magnetic properties, which makes them a good tool for the monitoring of the recovery process. It is thus tempting to try to establish the aforementioned relation between structure and magnetic properties on a more solid theoretical background.

The recovery process during the isothermal annealing of steels is characterised by more-or-less constant grain sizes, with the effect of the heating acting primarily in enhancing the mobility of the dislocations yielding an overall softening of the material. One has thus to deal with a reasonably well-controlled microstructure whose main tuning variable is the dislocation density. These are quite favourable conditions, which satisfy to a good extend the main assumptions of the random-field Ising model (RFIM) making the latter a good candidate for the theoretical study of the problem, despite its simplicity [START_REF] Nattermann | Theory of the random field Ising model[END_REF][START_REF] Belanger | The random field Ising model[END_REF][START_REF] Newman | Monte Carlo study of the random-field Ising model[END_REF].

A considerable amount of work has been published on that model, and a good understanding has been estab-lished thanks to the pioneering work of Sethna, Dahmen, Perkovic et al. [START_REF] Sethna | Random-field Ising models of hysteresis[END_REF][START_REF] Sethna | Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations[END_REF][START_REF] Dahmen | Hysteresis avalanches, and disorder-induced critical scaling: A renormalization-group approach[END_REF][START_REF] Kuntz | Noise in disorder systems: The power spectrum and dynamic exponents in avalanche models[END_REF]. In these articles it is shown that the random field (characterised by a disorder parameter R) plays the role of a tuning parameter in a second order phase transition. As a consequence, a critical disorder value R c exists, around which the magnetization curve can be described by a scaling law of the following form

M -M[H c (R c )] ∼ r β M ± h/r βδ (1)
where H c is the coercive field value at the critical disorder, r and h the normalised distance of the disorder and the magnetic field from their respective critical values, and M ± stands for a universal scaling function with β, δ giving the corresponding critical exponents at the transition. The ± sign in the index denotes the side across the critical point that is considered. Following this line of reasoning, the susceptibility profile around H c can be interpreted in terms of the scaling function M ± , whose explicit form we try to derive in this article from the mean-field RFIM theory and confirm via comparison with numerical data obtained by a site-dependent mean-field model. The explicit relation between disorder and the maximum susceptibility value is also established by resorting to both analytical and numerical results using the site-dependent mean-field (SDMF) approach.

Given this link between susceptibility and disorder, one can address at a second level the experimentally established relation between susceptibility and annealing condi-tions in order to indirectly relate the microstructural evolution (characterised by the material disorder) with the annealing conditions. This idea is explored in the second part of the article, where the experimental results of Martinez-de-Guerenu et al. are used to confirm on one hand the Lorentzian susceptibility profile, and to examine common trends between the annealing temperature and holding time with the disorder.

Theory

Mean-field theory

We consider an Ising spin system submerged to a constant random field. Such a system is described by the following Hamiltonian [START_REF] Sethna | Random-field Ising models of hysteresis[END_REF][START_REF] Sethna | Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations[END_REF][START_REF] Dahmen | Hysteresis avalanches, and disorder-induced critical scaling: A renormalization-group approach[END_REF]

H = - 1 2 J {i,j} σ i σ j -gµ B i (H + H i )σ i (2)
where σ i is the spin at the ith site of the Ising lattice, J stands for the exchange constant (which is considered invariant throughout the system lattice), µ B is the Bohr's magneton, g is the gyromagnetic ratio, H is the applied (external) field, and H i stands for the random field at the ith lattice site. Note that the {i, j} index implies summation over the first neighbours.

Following the literature, we shall assume that the random field is described by a Gaussian distribution, acknowledging, however, that other distributions can be applied as well (yet with a minor impact to the results) [START_REF] Sethna | Random-field Ising models of hysteresis[END_REF] ρ

(H i ) = 1 √ 2πR e -H 2 i /2R 2 . ( 3
)
R in (3) stands for the distribution variance, which is also a measure of the field randomness, or equivalently the degree of the system disorder.

Applying the mean-field theory approximation, the exact Hamiltonian (2) reduces to

H = - i [Jzm + gµ B (H + H i )] σ i (4)
with z being the coordination number, which for the 3D case is equal to 6, and m the mean spin (or normalised magnetisation)

m =< σ >= < M > M s . ( 5
)
M s stands for the magnetisation at saturation. In order to simplify the notation, we proceed to the following normalisation for the magnetic field

h i := gµ B H i /J (6)
and the same for H. Upon substitution in the Hamiltonian, we obtain the following simplified form

H/J = - i (zm + h + h i ) σ i (7)
and the random field distribution becomes

ρ(h i ) = 1 √ 2πr e -h 2 i /2r 2 (8)
with the normalised disorder

r := gµ B R/J (9)
In the rest of the article, lower-case letters, such as h, h i and r will imply normalised variables. At zerotemperature, the mean-field approximation for the mean magnetisation m reads

m(h) = 1 -2 -zm (h)-h -∞ ρ(h i ) dh i . ( 10
)
For disorder values lower than a critical value r c (10) has a non-unique solution at h = 0, and the corresponding m(h) presents a jump there. For r > r c the resulting curve is smooth and single valued at h = 0. This effect is illustrated in Fig. 1a , where the solution of ( 10) is drawn for different values of the disorder r. The magnetisation jump at h = 0 as a function of the normalised distance from the critical disorder 1 -r/r c is also shown in Fig. 1b. This transition from a discontinuous to continuous variation presents all the characteristics of a second-order phase transition with the magnetisation jump δm at zero field playing the role of the order parameter. This transition has been exhaustively studied in the papers of Sethna, Dahmen et al. [7-9, 11, 12].

In the following analysis, we shall be interested in systems with supercritical disorder, i.e. we will consider the case where r > r c . As explained above, for these disorder values the magnetisation curve is smooth around h = 0. Deriving [START_REF] Kuntz | Noise in disorder systems: The power spectrum and dynamic exponents in avalanche models[END_REF] with respect to the external magnetic field h we obtain

dm dh = 2ρ(zm + h) z dm dh + 1 . (11)
Setting dm/dh = χ and isolating the latter, we arrive at the following explicit expression for the susceptibility

χ(h) = 2ρ(zm + h) 1 -2zρ(zm + h) (12)
or equivalently

χ(h) = 1 [2ρ(zm + h)] -1 -z . ( 13
)
Substituting the expression of the random field distribution from (3) yields

χ(H) = 1 π 2 r e (zm+h) 2 /2r 2 -z . (14)
Assuming that sum zm + h admits small values with respect to the system disorder, we can develop the exponential function in Taylor series and keep only the first (linear) term, which will lead us to the approximation

χ(H) ≈ 1 π 2 r -z + π 2 1 2r (zm + h) 2 . (15)
Since (m, h) = (0, 0) is point of the solution described by the implicit expression [START_REF] Kuntz | Noise in disorder systems: The power spectrum and dynamic exponents in avalanche models[END_REF], m + h ≈ 0, implies that both h and m should be small. Let us thus write

m(h) ≈ χ 0 h (16)
with χ 0 being the slope of the m(h) curve at h = 0. Substitution of ( 16) upon ( 15) yields for small field values

χ(h) ≈ 1 π 2 r -z + π 2 1 2r (zχ 0 + 1) 2 h 2 . ( 17
)
With slightly term rearrangement, ( 17) can be brought in the more compact form

χ(h) ≈ χ 0 1 + (h/γ) 2 (18)
which is the Lorentzian distribution with χ 0 being the magnetic susceptibility at h = 0

χ 0 = π 2 r -z -1 (19)
and

γ = 2 π 1/4 2r χ 0 1 (zχ 0 + 1) . ( 20
)
Having a closed form expression for the magnetic susceptibility, the corresponding expression for the magnetisation can be easily obtained by integration over h, which yields

m = χ 0 γ arctan h γ . (21)
In order to test the accuracy of this approximation, the results obtained using the expression ( 18) are compared against the numerical solution of the implicit form [START_REF] Kuntz | Noise in disorder systems: The power spectrum and dynamic exponents in avalanche models[END_REF] in Fig. 2. As one can see from the comparison, (18) yields a very satisfactory description of the susceptibility variation for small fields.

We observe that the derived approximation becomes more accurate at higher field values for increasing r. This conclusion is not surprising regarding the profile of the exact solution in Fig. 1a. In fact, the magnetisation around h = 0 becomes increasingly non-linear for smaller r values, which results in a varying slope in the domain of moderate fields. As a consequence, the approximation ( 16) does not hold for small disorder values just above the critical value except for the immediate neighbourhood of h = 0. The numerical results of (10) (exact solution) are compared in Fig. 2 with the Lorentzian function that is calculated using the asymptotic expression of ( 18) and the one obtained via least-square fit to the exact curve. It turns out that the two approaches yield more or less equivalent results for weak disorders, like the one in Fig. 2a. Note that for more significant r values, the best-fit curve provides a reasonable approximation of the exact numerical solution for the entire h-window, at the expense of a less accurate peak value. This fact signifies that the exact solution is closer to the Lorentzian form for a broader range of h values.

The mean-field approximation, used in the above analysis, does not support hysteresis since the implicit evaluation of ( 10) is a single-valued curve. To introduced hysteresis, (18) can be generalised by shifting the zero of the magnetisation curve to the coercive field (it is the point where m becomes small and hence the main arguments of the analysis hold), obtaining the expression

χ(h) ≈ χ c 1 + [(h -h c )/a] 2 . ( 22
)
where h c stands for the coercive field and χ c the corresponding susceptibility. In principle, ( 19) cannot be applied anymore in order to relate the susceptibility at the coercive field with the random field parameters. Their correlation, can, however, be established numerically by least-square fit, as done in (2). Such generalisation of the mean-field results to hysteretic curves, is a heuristic approach. Nonetheless, some more rigorous justification can be sought by using the results of the site-dependent mean-field solution (SDMF), which natively introduced multi-branch solutions, and thereby hysteresis. This will be the aim of the next section.

Analysis based on the site-dependent mean-field (SDMF)

solutions According to the SDMF approach, the mean-field is (thermally) averaged at each site individually, thus intro-ducing a spatial dependence in the solution. The resulting Hamiltonian reads

H ≈ -J i   j=n.n. < σ j > +h i + h   σ i (23)
which yields for the magnetisation at the ith site

m i = 2θ   j=n.n. m j (h) + h i + h   -1 ( 24
)
where θ(•) stands for the Heaviside step function and

m i =< σ i > ( 25
)
similarly to [START_REF] Belanger | The random field Ising model[END_REF]. It is recalled that the summation is carried out considering only the nearest neighbours (n.n.) of each spin. Notice that (24) is the zero-temperature limit of the SDMF solution. Averaging over the random field values, we obtain the relation

m i (h) = 1 -2 -j mj(h)-h -∞ ρ(h i ) dh i . ( 26
)
which is the generalisation of [START_REF] Kuntz | Noise in disorder systems: The power spectrum and dynamic exponents in avalanche models[END_REF].

Due to the much more complicated mathematical structure of (26), where m i is coupled with all neighbouring spins, we have to proceed to the analysis numerically. Hence, in order to obtain the m i distribution, the system lattice is first truncated along its z/2 spatial dimensions (3 in this case). Let N be the total number of the considered (finite) lattice sites. The random field h i is then evaluated at the lattice sites using a random number generator following the (8) distribution with the disorder r as tuning parameter. Fixing the lower and upper limits of the external field, i.e. setting h ∈ [-h max , h max ], we proceed to a uniform discretisation of the magnetic field, namely

h (k) = -h max + k∆h, k = 0, . . . , N h (27)
with ∆h = 2h max /N h . Varying the field h (k) , the system of N coupled equations ( 24) is solved iteratively using a fixed-point scheme until the relative error drops under a predefined precision threshold , that is

i m (k) i n+1 -m (k) i n 2 i m (k) i n 2 ≤ (28)
n being the iteration number of the fixed-point algorithm. Special care needs to be taken in order to properly account for the hysteresis. Once the system (24) has converged for the kth step of the field discretisation, the obtained solution m (k) i is set as initial value for the solution of the (k + 1)th step. This is important since, if we provide the algorithm with the same initial value at each step (say for example 0 everywhere), the convergence will be biased to a subspace of the original configuration space, thus avoiding the local minima lying at the vicinity of the system state at the kth step. It is in fact these minima which stand for the metastable states responsible for the hysteresis. As far as the initial value for k = 0 is concerned, we set m (0) i = -1, assuming that the system has been prepared to the (negative) saturation.

In order to describe a complete cycle, the field is first increased to its maximum value, according to (27), and then reduced back to -h max by flipping the h (k) sequence. The magnetisation curves obtained by the above described approach for a number of different disorder values are illustrated in Fig. 3a. The magnetisation value depicted is the mean value throughout the lattice, that is

m (k) = 1 N N i=1 m (k) i . ( 29
)
All these solutions demonstrate hysteresis, which is different from the situation we observed with the scalar mean-field theory (cf. Fig. 1a). The explanation should be sought to the nature of the configuration space landscape that the SDMF algorithm explores during the minimisation of (28) and the passage from the possible metastable states, as discussed above. A thorough discussion about the behaviour of SDMF for RFIM solutions out of equilibrium can be found in [START_REF] Grest | Comparative Monte Carlo and mean-field studies of random-field Ising systems[END_REF]. Interestingly, these results could have hardly been obtained by Monte-Carlo simulations, which by construction tend to reach the equilibrium state [START_REF] Newman | Monte Carlo study of the random-field Ising model[END_REF].

The second order transition discussed in the previous paragraph is also observed with the SDMF solution, yet the value of the critical disorder is different. In fact, the SDMF estimate of the critical value is r c ≈ 3.2 in comparison with the higher value of r c ≈ 5.0 obtained via the mean-field theory. The evolution of the magnetisation jump ∆m as a function of the normalised distance from the critical disorder 1 -r/r c is shown in Fig. 3b. Since the SDMF solution fluctuates between successive simulations, a number of independent simulations is carried out and the results are averaged to the curve of Fig. 3b. In order to further test this result, we examine if it exhibits a power-law behaviour in the vicinity of the critical point for r < r c . A first order curve is fitted to the simulation points in a log-log representation

log(∆M ) = β log(1 -r/r c ) + c (30)
and the coefficient β is determined. The resulting curve is compared in Fig. 3b with the average curve thus confirming the power-law behaviour with a critical exponent β = 0.662, close to the 1/2 value obtained by the classical mean-field theory [START_REF] Dahmen | Hysteresis avalanches, and disorder-induced critical scaling: A renormalization-group approach[END_REF]. The magnetic susceptibility along the ascending hysteresis branch obtained by the SDMF solution for three distinct disorder values are depicted in Fig. 4. The sus- ceptibility plots have been obtained by numerical differentiation of the corresponding magnetisation curve, namely

χ (k) ≈ m (k) -m (k-1) ∆h . ( 31
)
The variation of the susceptibility is not smooth as it becomes clear from Fig. 4. This effect is a manifestation of the discontinuous change of the magnetisation value along the hysteresis, in a way that it strongly resembles the Barkhausen jumps. Note that this "noise" reaches its maximum around the coercive field value, which is in accordance with the experiment.

In Fig. 4, the numerical curves are fitted to a Lorentzian and a Gaussian distribution. In a resemblance with the previous results for the mean-field theory, the Lorentzian distribution fits quite nicely with the numerical results for almost the entire window of the field values. The Gaussian distribution gives also satisfactory results for higher disorder values, it deviates slightly with h < h c for r = 3.6, i.e. just after the phase transition. It should be also noticed the better fit of the Gaussian distribution for h > h c . The reciprocal susceptibility values at the coercive field χ c obtained by the fitted Lorentzian (cf. ( 18)) versus the disorder r are plotted in Fig. 5. The plot implies a linear correlation between the two values, which is confirmed by the least-square fit of a first order polynomial. This is in straight analogy with [START_REF] Wilson | High temperature magnetic characterisation of structural steels using Epstein frame[END_REF], obtained analytically for the mean-field theory. It is also interesting to compare the numerical value of the linear term coefficient, determined by the least-square fit as ∼ 0.97 with the π/2 = 1.25 of the analytical expression.

Application to soft steels

The above results have been confronted with the experimental curves obtained for two soft steel grades. The considered specimens stem from cold-rolled laminated sheets subjected thereupon to isothermal annealing at different temperatures and for different holding times [START_REF] Martínez-De-Guerenu | Recovery during annealing in a cold rolled low carbon steel. Part I: Kinetics and microstructural characterization[END_REF][START_REF] Martínez-De-Guerenu | Recovery during annealing in a cold rolled low carbon steel. Part II: Modelling the kinetics[END_REF][START_REF] Martínez-De-Guerenu | Nondestructive characterization of recovery and recrystallization in cold rolled low carbon steel by magnetic hysteresis loops[END_REF].

Two set of samples were considered for this study. One set from industrially produced Ti -Nb stabilized interstitial free (IF) ultra low carbon (ULC) steel, with composition 0.003%C-0.11%Mn-0.02%Nb-0.012%Ti, cold rolled to a final thickness of 0.8 mm through a reduction of 75% [START_REF] Martínez-De Guereñu | Application of coercive field measurements to the evaluation of recovery and recrystallization in cold rolled interstitial free (IF) steel[END_REF] and a second set from extra low carbon steel (ELC), with composition 0.03%C-0.19%Mn-0.13%Al-0.0035%N -0.012%P-0.01% Si, that had been industrially produced and cold rolled to a final thickness of 0.3 mm through a reduction of 84% [START_REF] Martínez-De-Guerenu | Recovery during annealing in a cold rolled low carbon steel. Part I: Kinetics and microstructural characterization[END_REF][START_REF] Martínez-De-Guerenu | Recovery during annealing in a cold rolled low carbon steel. Part II: Modelling the kinetics[END_REF][START_REF] Martínez-De-Guerenu | Use of magnetic techniques for characterisation of the microstructure evolution during the annealing of low carbon steels[END_REF]. The samples had been thermal treated in laboratory. First heated in an argon atmosphere, at a rate of 20 • C/s to reach a temperature within the range of 300-550 • C, followed by holding times at these temperatures in the range of 1 s to 10,000 s (≈ 2.8 h). The annealing treatments were then interrupted and the samples quenched in helium to reach room temperature at a rate close to -60 • C/s.

The annealing time and temperatures were selected so that the thereupon induced microstuctural changes to be dominated by the recovery procedure without triggering recrystallisation. In this way, the major impact of the annealing procedure is restricted to the decrease and rearrangement of the dislocation density with the grain size remaining intact.

Near saturation major magnetic B-H hysteresis loop measurements were made at 1 Hz using a single sheet tester system available at author's laboratory [START_REF] Soto | A completely configurable digital system for simultaneous measurements of hysteresis loops and Barkhausen noise[END_REF], with an encircling coil wound around the samples with maximum magnetic field strengths applied of about 4100 A/m. The effect of the annealing treatment on the experimental BH hysteresis loops can be found for cold rolled IF steel annealed at 350 • C and 550 • C and at 450 • C at [START_REF] Martínez-De Guereñu | Application of coercive field measurements to the evaluation of recovery and recrystallization in cold rolled interstitial free (IF) steel[END_REF] and at [START_REF] Martínez-De-Guerenu | Use of magnetic techniques for characterisation of the microstructure evolution during the annealing of low carbon steels[END_REF] for the cold rolled ELC steel annealed at 300 • C and 450 • C. Assuming that the changes to the specimen microstructure can be captured by a random field with different values of disorder r, and applying the above discussed approach, the measured susceptibility curves are fitted to the Lorentzian approximation described in (22). The results for the two grades and for three different annealing conditions are shown in Fig. 6 and Fig. 7, respectively. The susceptibility curves are plotted as functions of the measured magnetic field H and not the normalised h. It is interesting to trace the evolution of the fitted maximum value of susceptibility χ c as a function of the annealing time. According to the previous analysis, the inverse maximum susceptibility is proportional to the system disorder r. Hence, one could expect that χ -1 c may provide some qualitative image of the recovery process. The corresponding plots of the inverse susceptibility evolution with time for the different annealing temperatures and the two considered steel grades are given in Fig. 8. The nearly monotonous decrease of the inverse susceptibility with the annealing time (with the exception of the short times at high annealing temperature for the IF) seems to be in accordance with the fact that the dislocation density is decreasing during the recovery.

A similar statement can also be made concerning the annealing temperature. In fact, the relative position of the curves, except for some points at low annealing times of the IF steel, follows the decrease in the dislocation density with increasing temperature.

A question that might be arisen concerns the validity of the zero-temperature approach for the studied case. It should be recalled that the Curie point, indicating the temperature region where thermal fluctuations become important, lies around 770 • C (1043 K) for carbon steels. All the magnetic measurements were made at room temperature once the samples had been removed from the annealing simulation device, which means that we are located at 29% of the Curie temperature (ratio calculated after conversion to Kelvin), and we recall by the corresponding phase diagrams that the magnetisation curve descends very steeply just before the Curie point to reach zero [START_REF] Bozorth | Ferromagnetism[END_REF]. Precise measurements at high temperatures for similar steel grades have been communicated in the recent literature which supports this argument [START_REF] Boehm | Measurement of magnetic properties of steel at high temperatures[END_REF][START_REF] Wilson | High temperature magnetic characterisation of structural steels using Epstein frame[END_REF]. We can hence conclude that the random field effect is dominant against thermal fluctuations for these temperatures. Moreover, outside the critical range of the transition (usually a few degrees from the critical temperature at most) the system is expected to be described from the mean-field approach. In our case this condition is fulfilled (T max ∼ 0.29T c ).

One might come to the same conclusion by considering comparisons of the critical exponents relative to Barkhausen avalanche characteristics with the corresponding experimental values [START_REF] Perković | Avalanches, Barkhausen noise, and plain old criticality[END_REF][START_REF] Durin | The Barkhausen effect[END_REF]. The experimental results for these studies have been obtained in room temperature conditions as well.

Conclusions

A very simple random field spin model has been used for the interpretation of the behaviour of the differential susceptibility curves around the coercive field. It appears that the implicit solution of the RFIM for small field values can be adequately approximated by a Lorentzian function, whose peak (corresponding susceptibility at the coercive field) provides a measure for the system disorder. This approximation explains the nearly Lorentzian profile of the experimental susceptibility curves, and is theoretically supported by the scaling laws at the coercive field for supercritical disorder values.

Although the highly simplified conceptual image of the disorder used by the model should not be directly related with the real material randomness translated to a dislocation distribution (in the considered case of steels subjected to recovery), this model "randomness" could still serve as a figure-of-merit for the real randomness, as Fig. 8 implies.

Caution should be also taken concerning the model domain of validity. One must not forget that the Ising model considers spins oriented in one direction, hence it is restricted (at least theoretically) to materials with strong anisotropy, where the spins tend to align with the easy magnetisation axis. A more thorough study should take this limitation into account, and one must be aware that the above analysis may not hold for different families of materials. In such cases the use of more sophisticated models, like the XY or the Heisenberg model may be indispensable.

Figure 1 :

 1 Figure 1: (a) Mean-field solution for the magnetisation curve for different disorder values. (b) Magnetisation jump at zero field as a function of the distance from the critical disorder 1 -r/rc. The critical disorder value can be estimated as rc ≈ 5.

Figure 2 :

 2 Figure 2: Comparison of the exact solution (10) ("Exact") with the expression (18) ("Lorentzian") and a fitted Lorentzian ("Lorentzian fit") for three different disorder values. (a) r = 5.11, (b) r = 5.55 and (c) r = 6.0.

Figure 3 :

 3 Figure 3: (a) SDMF solution for the magnetisation curve for different disorder values. (b) Magnetisation jump at hc as a function of the distance from the critical disorder 1-r/rc. The critical disorder value can be estimated as rc ≈ 3.2. The numerical curve is the average of four successive simulations (pointed by the different coloured dots). The supercritical behaviour can be described by a power law of the type (1 -r/rc) β .

Figure 4 :

 4 Figure 4: Fitted Lorentzian and Gaussian distributions to the susceptibility SDMF solution across the coercive field (ascending branch) for three different disorder values: (a) r = 3.6, (b) r = 5.0 and (c) r = 6.33.

Figure 5 :

 5 Figure 5: Susceptibility values obtained by the fitted Lorentzian distribution at the coercive field as function of the disorder.

Figure 6 :

 6 Figure 6: Fitted Lorentzian to the experimental IF susceptibility curves for three different annealing conditions: (a) Cold-rolled (no treatment), (b) 1000 s at 350 • C (623 K) and (c) 1000 s at 550 • C (823 K).

Figure 7 :

 7 Figure 7: Fitted Lorentzian to the experimental ELC susceptibility curves for three different annealing conditions: (a) 51 s at 300 • C (573 K), (b) 51 s at 400 • C (673 K) and (c) 36 min at 500 • C (773 K).

Figure 8 :

 8 Figure 8: Evolution of the inverse susceptibility with annealing time at different annealing temperatures: (a) IF steel, (b) ELC steel.

Acknowledgements

A. Martínez-de-Guerenu and D. J. Badiola acknowledge funding from the Basque Government under the Elkartek Program, Project Nos. KK-2019/0010 and KK2021/00082.