Paolo Ascenzi
email: ascenzi@uniroma3.it

Roberto Santucci

Massimo Coletta

Fabio Polticelli

Cytochromes: Reactivity of the "dark side" of the heme

Keywords: Heme-proteins, Alcaligenes xylosoxidans cytochrome c´, Horse heart cytochrome c, Nitrogen monoxide, Carbon monoxide, Ligand binding, Peroxidase activity, Heme-ligand geometry, Allostery

Ligand binding to the heme distal side is a paradigm of heme-protein biochemistry, the proximal axial ligand being in most cases a His residue. NO binds to ferrous heme-Fe-atom giving rise to hexa-coordinated adducts (as in myoglobin and hemoglobin) with His and NO as proximal and distal axial ligands, respectively, or to penta-coordinated adducts (as in soluble guanylate cyclase) with NO as the axial distal ligand. Over the last decade, the ferrous derivative of Alcaligenes xylosoxidans cytochrome c´ (Axcyt c') and of cardiolipin-bound horse heart cytochrome c (CL-hhcyt c) have been reported to bind NO to the "dark side" of the heme (i.e., as the proximal axial ligand) replacing the endogenous ligand His. Conversely, CL-free hhcyt c behaves as ferrous myoglobin by binding NO to the heme distal side, keeping His as the proximal axial ligand. Moreover, the ferrous derivative of CL-hhcyt c binds CO at the heme distal side, the proximal axial ligand being His. Furthermore, CL-hhcyt c shows peroxidase activity. In contrast, CL-free hhcyt c does not bind CO and does not show peroxidase activity. This suggests that heme-proteins may utilize both sides of the heme for ligand discrimination, which appears to be modulated allosterically. Here, structural and functional aspects of NO binding to ferrous Axcyt c' and (CL-)hhcyt c are reviewed.

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The free radical NO is pivotal in many different biological functions [START_REF] Moncada | The discovery of nitric oxide as the endogenous nitrovasodilator[END_REF][START_REF] Furchgott | Endothelium-derived relaxing and contracting factors[END_REF][START_REF] Moncada | Nitric oxide: physiology, pathophysiology, and pharmacology[END_REF][START_REF] Sciorati | Nitric oxide effects on cell growth: GMP-dependent stimulation of the AP-1 transcription complex and cGMPindependent slowing of cell cycling[END_REF][START_REF] Koppenol | The basic chemistry of nitrogen monoxide and peroxinitrite[END_REF][START_REF] Ignarro | Nitric oxide as a unique signaling molecule in the vascular system: a historical overview[END_REF][START_REF] Moncada | Does nitric oxide modulate mitochondrial energy generation and apoptosis?[END_REF][START_REF] Bian | Nitric oxide (NO): biogeneration, regulation, and relevance to human diseases[END_REF][START_REF] Bruckdorfer | The basics about nitric oxide[END_REF][START_REF] Clementi | Nitric oxide and mitochondrial biogenesis: a key to longterm regulation of cellular metabolism[END_REF][START_REF] Brunelli | Nitric oxide: emerging concepts about its use in cell-based therapies[END_REF][START_REF] Brunelli | NO release combined with nonsteroidal antinflammatory activity prevents muscular dystrophy pathology and enhances stem cell therapy[END_REF][START_REF] Calabrese | Nitric oxide in the central nervous system: neuroprotection versus neurotoxicity[END_REF][START_REF] Nisoli | Defective mitochondrial biogenesis: a hallmark of the high cardiovascular risk in the metabolic syndrome?[END_REF][START_REF] Perrotta | Nitric oxide and sphingolipids: mechanisms of interaction and role in cellular pathophysiology[END_REF][START_REF] Huang | eNOS, metabolic syndrome and cardiovascular disease[END_REF][START_REF] Napoli | Nitric oxide and pathogenic mechanisms involved in the development of vascular diseases[END_REF], so much so that NO-synthase gene has been proposed in novel gene therapy approaches [START_REF] O'connor | Nitric oxide synthase gene therapy: progress and prospects[END_REF].

Remarkably, heme-proteins are pivotal in the synthesis, trafficking, storage, and metabolism of NO. In fact, the formation of heme-NO complexes occurs in NO generation by NOsynthase [START_REF] Marletta | Nitric oxide synthase: aspects concerning structure and catalysis[END_REF][START_REF] Nathan | Nitric oxide synthases: roles, tolls, and controls[END_REF][START_REF] Roman | Intrinsic and extrinsic modulation of nitric oxide synthase activity[END_REF][START_REF] Li | Structure-function studies on nitric oxide synthases[END_REF], NO sensing by soluble guanylate cyclase [START_REF] Poulos | Soluble guanylate cyclase[END_REF], NO transport and storage by nitrophorins [START_REF] Montfort | Nitrophorins and related antihemostatic lipocalins from Rhodnius prolixus and other blood-sucking arthropods[END_REF][START_REF] Walker | Nitric oxide interaction with insect nitrophorins and thoughts on the electron configuration of the {FeNO} 6 complex[END_REF], and NO metabolism by NO-reductases and -oxidases [START_REF] Kelm | Nitric oxide metabolism and breakdown[END_REF][START_REF] Wu | Microbial globins[END_REF].

As reported for O 2 and CO, NO binds to the distal side of the ferrous heme, the proximal axial ligand of the heme-Fe-atom being in most cases a His residue. Therefore, many NO-binding heme-proteins display hexa-coordinated heme adducts. However, in some heme-proteins, NO binding to the heme distal side induces the dissociation of the proximal His residue, leading to penta-coordinated complexes. Moreover, the temperature-dependent equilibrium mixture of the ferrous nitrosylated hexa-and penta-coordinated heme has been observed in some heme-proteins (see refs 23,25,28,29).

NO binding to the ferrous heme-Fe-atom not only affects the heme-nitrosyl electronic structure and the heme-Fe-atom coordination number, but also triggers conformational changes that could be linked to allosteric regulation in heme-proteins. The relevant role of ferrous nitrosylated heme adducts has stimulated a significant effort to understand the chemistry of penta-and hexa-coordinated species and to identify the driving force for the loss of the proximal His ligand leading to the formation of the penta-coordinated ferrous heme-Fe-NO adduct (see [START_REF] Poulos | Soluble guanylate cyclase[END_REF][START_REF] Pletneva | Probing the cytochrome c' folding landscape[END_REF][START_REF] Perutz | Regulation of oxygen affinity of hemoglobin: influence of structure of the globin on the heme iron[END_REF][START_REF] Usov | ENDOR of NO-ligated cytochrome c[END_REF].

Recently, the possibility that NO could bind to the proximal heme side came to light.

In fact, since the proximal heme ligand (His) can be lost during the penta-coordination of the ferrous heme-Fe-NO complex, it may be possible for the NO ligand to occupy the proximal heme coordination site. Indeed, the crystal structure of a periplasmic class IIa c-type cytochrome c′ from the denitrifying bacterium Alcaligenes xylosoxidans (Axcyt c') showed

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that distal to proximal heme-NO conversion can occur in heme-proteins [START_REF] Lawson | Unprecedented proximal binding of nitric oxide to heme: implications for guanylate cyclase[END_REF][START_REF] Lawson | A twofaced molecule offers NO explanation: the proximal binding of nitric oxide to haem[END_REF][START_REF] Barbieri | Modulation of NO binding to cytochrome c' by distal and proximal haem pocket residues[END_REF]. NO binding to the proximal heme side of Axcyt c' has been postulated to be relevant: (i) in NO-trafficking or defense against nitrosative stress [START_REF] Cross | Cytochrome c' from Rhodobacter capsulatus confers increased resistance to nitric oxide[END_REF][START_REF] Choi | Regulation and function of cytochrome c' in Rhodobacter sphaeroides 2.4.3[END_REF][START_REF] Pletneva | Probing the cytochrome c' folding landscape[END_REF][START_REF] Benini | High resolution crystal structure of Rubrivivax gelatinosus cytochrome c[END_REF], (ii) in discriminating between the gaseous ligands NO, CO, and O 2 [START_REF] Weiss | Cytochromes c': biological models for the S = 3/2, 5/2 spin-state admixture?[END_REF], and (iii) in inducing specific gas-dependent conformational changes [START_REF] Lawson | Unprecedented proximal binding of nitric oxide to heme: implications for guanylate cyclase[END_REF]. NO binding to the cardiolipin-bound horse heart cytochrome c (CL-hhcyt c) is also consistent with the formation of a proximal heme-NO penta-coordinated species [START_REF] Cross | Cytochrome c' from Rhodobacter capsulatus confers increased resistance to nitric oxide[END_REF].

Note that NO acts as a gas modulator in the mitochondria, triggering, among others, apoptosis [START_REF] Pun | Gases in the mitochondria[END_REF]. Lastly, NO binding to soluble guanylate cyclase shows some features that might suggest NO binding to the proximal side of the heme [START_REF] Lawson | Unprecedented proximal binding of nitric oxide to heme: implications for guanylate cyclase[END_REF][START_REF] Zhao | A molecular basis for nitric oxide sensing by soluble guanylate cyclase[END_REF].

Here, the nitrosylation mechanisms of ferrous Axcyt c' and (CL-)hhcyt c are reviewed and functional implications examined.

Diatomic gas discrimination by Alcaligenes xylosoxidans cytochrome c´

Cytochromes c' (cyt c') are a distinct family of class IIa cytochromes found in the periplasm of certain denitrifying, nitrogenfixing, photosynthetic, methanotrophic, and sulfuroxidizing bacteria [START_REF] Lawson | A twofaced molecule offers NO explanation: the proximal binding of nitric oxide to haem[END_REF][START_REF] Weiss | Cytochromes c': biological models for the S = 3/2, 5/2 spin-state admixture?[END_REF][START_REF] Meyer | New perspectives on c-type cytochromes[END_REF][START_REF] Moore | Cytochromes c: Evolutionary, Structural and Physicochemical Aspects[END_REF]. Although it has long been assumed that cyt c' are involved in electron transfer [START_REF] Weiss | Cytochromes c': biological models for the S = 3/2, 5/2 spin-state admixture?[END_REF][START_REF] Bergmann | Primary structure of cytochrome c' of Methylococcus capsulatus Bath: evidence of a phylogenetic link between P460 and c'type cytochromes[END_REF], cyt c' present in the denitrifying bacteria have been suggested to bind NO, helping to alleviate nitrosative stress [START_REF] Cross | Cytochrome c' from Rhodobacter capsulatus confers increased resistance to nitric oxide[END_REF][START_REF] Weiss | Cytochromes c': biological models for the S = 3/2, 5/2 spin-state admixture?[END_REF][START_REF] Cross | Enzymatic removal of nitric oxide catalyzed by cytochrome c' in Rhodobacter capsulatus[END_REF][START_REF] Mayburd | Mechanism and biological role of nitric oxide binding to cytochrome c[END_REF]. Ferrous cyt c' are able to bind also CO, although they do not generate stable O 2 adducts [START_REF] Weiss | Cytochromes c': biological models for the S = 3/2, 5/2 spin-state admixture?[END_REF][START_REF] Taniguchi | On the anomalous interactions of ligands with Rhodospirillum haem protein[END_REF][START_REF] Gibson | Kinetic analysis of the reaction of cytochrome cc' with carbon monoxide[END_REF][START_REF] Cusanovich | Anomalous ligand binding by a class of high spin c-type cytochromes[END_REF][START_REF] Rubinov | Cytochromes c' in their reaction with ethyl isocyanide[END_REF][START_REF] Doyle | Carbon monoxide binding to Rhodospirillum molischianum ferrocytochrome c[END_REF][START_REF] Yoshimura | Spectral properties of nitric oxide complexes of cytochrome c' from Alcaligenes sp[END_REF][START_REF] Patel | Steric and hydrophobic effects in alkyl isocyanide binding to Rhodospirillum molischianum cytochrome c[END_REF][START_REF] Kassner | Ligand binding properties of cytochromes c[END_REF][START_REF] Moir | Cytochrome c' from Paracoccus denitrificans: spectroscopic studies consistent with a role for the protein in nitric oxide metabolism[END_REF].

Cyt c' are usually homodimers, each subunit being composed of an antiparallel fourhelix bundle of approximately 130 residues containing a c-type heme. A conserved Cys-Xxx-Xxx-Cys-His motif provides the proximal His ligand and two Cys-thioether bonds to the heme (residues 116, 119, and 120 in Axcyt c'). The access to the vacant sixth co-ordination site on the distal face of the heme is mediated by the side chain of a hydrophobic residue (Leu16 in Axcyt c'). Indeed, the distal pocket is generally hydrophobic, sterically crowded,

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and less solvent-accessible than the proximal pocket, which is positively polarized, less crowded and readily accessible to the surrounding solvent (see 31).

In the physiological environment, cyt c' seems to function in the reduced oxidation state of the heme-Fe-atom (i.e., in the ferrous form) (see 37). Ferrous Axcyt c' has been reported to bind NO and CO at the proximal and distal side of the heme, respectively, representing the first structurally-characterized example of exogenous ligand binding to both sides of the heme-Fe-atom [START_REF] Lawson | Unprecedented proximal binding of nitric oxide to heme: implications for guanylate cyclase[END_REF][START_REF] Lawson | A twofaced molecule offers NO explanation: the proximal binding of nitric oxide to haem[END_REF][START_REF] Andrew | Six-to five-coordinate heme-nitrosyl conversion in cytochrome c' and its relevance to guanylate cyclase[END_REF][START_REF] George | Stoppedflow infrared spectroscopy reveals a six-coordinate intermediate in the formation of the proximally bound five-coordinate NO adduct of cytochrome c[END_REF].

Under steady-state conditions, ferrous Axcyt c' binds NO to the proximal side of the heme-Fe-atom as the proximal axial ligand, leading to a penta-coordinated NO adduct (Fig. 1). NO is present as two equally populated conformers with an average Fe-N-O angle of 128°.

Overall the structure of the ferrous nitrosylated derivative of Axcyt c' overlaps with that of the ligand-free reduced form, as the positions of the heme, the Fe-atom and Leu16 are highly similar. In both structures, the heme-Fe-atom lies out of the heme plane, which is distinctly puckered, by about 0.3 Å towards the proximal axial ligand (i.e., His120 and NO in the ligand-free and nitrosylated derivatives, respectively). The major differences between the ligand-free and nitrosylated derivatives of ferrous Axcyt c' are the ligation of NO (as the proximal axial ligand of the heme-Fe-atom), the displacement of the proximal His120 and the flipping of Arg124, which stakes against the heme plane and is hydrogen-bonded to conformer 2 of NO [START_REF] Lawson | Unprecedented proximal binding of nitric oxide to heme: implications for guanylate cyclase[END_REF][START_REF] Lawson | A twofaced molecule offers NO explanation: the proximal binding of nitric oxide to haem[END_REF].

As shown in Figure 1, the binding geometry of NO to the heme-Fe-atom of ferrous Axcyt c' is completely different from that observed in ferrous nitrosylated sperm whale myoglobin (Mb) [START_REF] Brucker | Nitric oxide myoglobin: crystal structure and analysis of ligand geometry[END_REF]. In sperm whale Mb, NO is bound at the heme distal side, the proximal axial ligand being HisF8. Specifically, the Fe-N-O angle is 112°, and the Nε atom of the distal HisE7 residue is located 2.8 Å away from the nitrogen atom of the heme-bound NO, implying electrostatic stabilization of the heme-Fe-NO complex.

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Carbonylation of ferrous Axcyt c' leads to a hexa-coordinated adduct. CO binds to the distal face of the heme in a near-linear conformation (the Fe-C-O angle is 167°), thereby displacing the side chain of Leu16. In this latter case, the proximal axial ligand is His120. The heme-Fe-atom is within the plane of the heme, which is essentially flat. In addition, a second CO molecule is hydrogen bonded to the Nδ1 atom of His120 (30,31) (Fig. 1). This mimics the interaction seen in some heme-proteins between the proximal His residue and either a mainchain carbonyl or a side-chain carboxyl group (see refs 58,59). The positions of the proximal His120 residue and of the heme-Fe-atom in the ferrous ligand-free and carbonylated structures of Axcyt c' are essentially identical, and the movement and flattening of the heme restore the Fe-atom into the plane of the heme (30,31) (Fig. 1). This is in contrast with the situation in hemoglobin where the binding of gaseous diatomic ligands pulls the ferrous Featom into the heme plane and the proximal His residue towards the heme, thereby triggering allosteric transition(s) [START_REF] Perutz | Regulation of oxygen affinity of hemoglobin: influence of structure of the globin on the heme iron[END_REF].

The binding geometry of CO to the heme-Fe-atom of ferrous Axcyt c' is reminiscent of that reported for ferrous carbonylated sperm whale Mb (60) (Fig. 1). Indeed, in sperm whale Mb CO is bound at the heme distal side, the proximal axial ligand being HisF8.

Specifically, the Fe-C-O angle is about 130° and the Nε atom of the heme distal HisE7 residue is located 3.3 Å away from the oxygen atom of the heme-bound CO, implying a weaker electrostatic stabilization of the Fe-CO complex with respect to the Fe-NO complex.

Nitrosylation of ferrous Axcyt c' involves at least three species (i.e., A to C in Fig. 2), including the initial and final forms (i.e., A and C, respectively) and one intermediate (i.e., B) [START_REF] Lawson | Unprecedented proximal binding of nitric oxide to heme: implications for guanylate cyclase[END_REF][START_REF] Lawson | A twofaced molecule offers NO explanation: the proximal binding of nitric oxide to haem[END_REF][START_REF] Barbieri | Modulation of NO binding to cytochrome c' by distal and proximal haem pocket residues[END_REF][START_REF] Andrew | Six-to five-coordinate heme-nitrosyl conversion in cytochrome c' and its relevance to guanylate cyclase[END_REF][START_REF] George | Stoppedflow infrared spectroscopy reveals a six-coordinate intermediate in the formation of the proximally bound five-coordinate NO adduct of cytochrome c[END_REF][START_REF] Andrew | Resonance Raman studies of cytochrome c' support the binding of NO and CO to opposite sides of the heme: implications for ligand discrimination in heme-based sensors[END_REF][START_REF] Kruglik | Molecular basis for nitric oxide dynamics and affinity with Alcaligenes xylosoxidans cytochrome c[END_REF][START_REF] Pixton | Activation parameters for heme-NO binding in Alcaligenes xylosoxidans cytochrome c': the putative dinitrosyl intermediate forms via a dissociative mechanism[END_REF]. NO binds to the distal side of the ferrous heme (i.e., A in Fig. 2) to give the hexa-coordinated ferrous His120-Fe-NO adduct (i.e., B in Fig. 2). The value of the second-order rate constant for the A to B reaction (i.e., k 1 in Fig. 2) is 4.6×10 4 M -1 s -1 .

Nitrosylation of ferrous Axcyt c' leads to the weakening and breaking of the His120-Fe bond

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(see Fig. 2). A second NO displaces the proximal His120 residue and binds to the proximal side of the heme, giving a transient hexa-coordinated ferrous NO-Fe-NO species (see Fig. 2), which evolves toward the dissociation of the distally-bound NO, yielding the final ferrous penta-coordinated proximal NO-Fe adduct (i.e., C in Fig. 2). The value of the second order rate constant for the B to C reaction (i.e., k 2 in Fig. 2) is 8.7×10 3 M -1 s -1 . The transient NO-Fe-NO complex was not observed, and thus it could be concluded that the dissociation of the distally-bound NO is not the rate limiting event. Values of k 1 and k 2 for NO binding to ferrous Axcyt c' are smaller by several orders of magnitude with respect to those reported for the nitrosylation of ferrous heme-proteins (e.g., CL-hhcyt c (see below) and Mb) [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF][START_REF] Moore | Cooperativity in the dissociation of nitric oxide from hemoglobin[END_REF].

Denitrosylation of ferrous Axcyt c'-NO involves four species (i.e., C to A in Fig. 2), including the initial and final forms (i.e., C and A, respectively) and two intermediates (i.e., D and E) [START_REF] Kruglik | Molecular basis for nitric oxide dynamics and affinity with Alcaligenes xylosoxidans cytochrome c[END_REF]. Photolysis of NO from the equilibrium state designated C (see Fig. 2) yields a tetra-coordinated species (i.e., D in Fig. 2). NO rebinding to the transient form designated D (see Fig. 2) is fully geminate (≥ 99%) and the value of the rate constant (i.e., k 3 ; see Fig. 2) is 1.4×10 11 s -1 . Therefore, the rate constant for NO escape from the transient tetra-coordinated Axcyt c'•••NO species (i.e., D in Fig. 2; k 5 ) to the bulk solvent is ~ ×10 9 s -1 . This suggests that the proximal binding heme environment of Axcyt c' does not allow for accommodation of both NO and His120 in the proximity of the ferrous heme-Fe-atom, such that His120 can bind the heme only when NO leaves the proximal heme environment [START_REF] Yoo | Rebinding of proximal histidine in the cytochrome c' from Alcaligenes xylosoxidans acts as a molecular trap for nitric oxide[END_REF]. The observed overall rate constant for NO dissociation from C to A (see Fig. 2) is 4.1×10 -4 s -1 ; in view of the high percentage of geminate recombination (i.e., ≥ 99%, see above), this value indicates that the actual rate for NO dissociation (from C to B, see Fig. 2) in the absence of photolysis is ~ 4×10 -2 s -1 . In this framework (see Fig. 2), the binding of His120 to the NO-free tetracoordinated Axcyt c' species is about 100-fold slower (i.e., ~ 10 9 s -1 ) than the rebinding of NO, slowing the formation of species A in Fig. 2.

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The reactivity of ferrous Axcyt c' towards NO has been postulated to be modulated especially by Arg124 [START_REF] Barbieri | Modulation of NO binding to cytochrome c' by distal and proximal haem pocket residues[END_REF]. Indeed, the retention of a basic residue at the 124 position is essential to stabilize by hydrogen-bonding the His120-Fe-NO transient adduct of Axcyt c'.

Surprisingly, this indicates that the proximal Arg124 residue modulates the reactivity towards NO on the heme distal side [START_REF] Barbieri | Modulation of NO binding to cytochrome c' by distal and proximal haem pocket residues[END_REF]. Moreover, the presence of a cationic residue at this proximal position is likely to be relevant for the physiological function of members of the cyt c' family Hhcyt c is involved not only in electron transfer from the bc 1 complex to the terminal acceptor cytochrome c oxidase, but also in apoptosis [START_REF] Pun | Gases in the mitochondria[END_REF][START_REF] Caroppi | Apoptosis and human diseases: mitochondrion damage and lethal role of cytochrome c as proapoptotic protein[END_REF][START_REF] Orrenius | Mitochondrial oxidative stress: implications for cell death[END_REF][START_REF] Ow | Cytochrome c: functions beyond respiration[END_REF]. In this process, hhcyt c interacts with cardiolipin (CL) prior to be released from the mitochondrion [START_REF] Rytömaa | Evidence for two distinct acidic phospholipidsbinding sites in cytochrome c[END_REF][START_REF] Belikova | Peroxidase activity and structural transitions of cytochrome c bound to cardiolipin containing membranes[END_REF][START_REF] Vlasova | Nitric oxide inhibits peroxidase activity of cytochrome c' cardiolipin complex and blocks cardiolipin oxidation[END_REF][START_REF] Basova | Cardiolipin switch in mitochondria: shutting off the reduction of cytochrome c and turning on the peroxidase activity[END_REF][START_REF] Kalanxhi | Cytochrome c impaled: investigation of the extended lipid anchorage of a soluble protein to mitochondrial membrane models[END_REF][START_REF] Sinibaldi | Insights into cytochrome c-cardiolipin interaction: role played by ionic strength[END_REF][START_REF] Sinibaldi | Extended cardiolipin anchorage to cytochrome c: a model for protein-mitochondrial membrane binding[END_REF]. CL, which constitutes about 20% of total lipids of the mitochondrial membrane, is synthesized in the mitochondrion and possesses a unique structure, being composed of four (instead of two, as in most lipids) fatty acid tails. CL binding to hhcyt c plays an important role in determining the function of the heme-protein (and, thus, the cell fate). At the early stage of apoptosis, the CL-hhcyt c complex dissociates favoring hhcyt c release into the cytosol [START_REF] Kagan | Oxidative lipidomics of apoptosis: redox catalytic interactions of cytochrome c with cardiolipin and phosphatidylserine[END_REF]. Cytosolic hhcyt c plays an important pro-apoptotic role activating caspase-9, which is one of the proteases involved in the caspase cascade leading to cell death [START_REF] Garrido | Mechanisms of cytochrome c release from mitochondria[END_REF]. CL peroxidation is indicated as one of the factors responsible for the CL-hhcyt c complex dissociation, although the mechanism governing the event remains still unclear [START_REF] Gogvadze | Multiple pathways of cytochrome c release from mitochondria in apoptosis[END_REF]. This hypothesis is based on the observation that oxidative degradation of CL occurs in the p53-induced apoptosis [START_REF] Polyak | A model for p53induced apoptosis[END_REF], and that CL hydroperoxides have lower binding affinity for hhcyt c with respect to CL [START_REF] Shidoji | Loss of molecular interaction between cytochrome c and cardiolipin due to lipid peroxidation[END_REF]. Thus, CL binding to hhcyt c is impaired by CL peroxidation and phospholipid peroxidation is prevented by BCL-2, the anti-apoptotic protein member of the Bcl-2 family known to inhibit hhcyt c release from mitochondrion [START_REF] Reed | Bcl-2 family proteins[END_REF][START_REF] Bayir | Apoptotic interactions of cytochrome c: redox flirting with anionic phospholipids within and outside of mitochondria[END_REF].

containing
CL binds to hhcyt c at two distinct regions, called the A-site and the C-site, characterized by different affinity for phospholipids [START_REF] Rytömaa | Evidence for two distinct acidic phospholipidsbinding sites in cytochrome c[END_REF][START_REF] Sinibaldi | Insights into cytochrome c-cardiolipin interaction: role played by ionic strength[END_REF]. The binding reaction seems to proceed through the insertion of one acyl chain of CL in the hhcyt c interior [START_REF] Rytömaa | Evidence for two distinct acidic phospholipidsbinding sites in cytochrome c[END_REF][START_REF] Rytömaa | Reversibility of the binding of cytochrome c liposomes: implications for lipid-protein interactions[END_REF]. The socalled "extended lipid anchorage" hypothesis assumes that one acyl chain of CL accommodates into the protein while the other one points to the opposite direction from the head-group; this event seems to keep the protein firmly anchored to the membrane [START_REF] Rytömaa | Reversibility of the binding of cytochrome c liposomes: implications for lipid-protein interactions[END_REF]. The

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region of hhcyt c into which CL binds is uncertain and still matter of controversy [START_REF] Kalanxhi | Cytochrome c impaled: investigation of the extended lipid anchorage of a soluble protein to mitochondrial membrane models[END_REF][START_REF] Sinibaldi | Extended cardiolipin anchorage to cytochrome c: a model for protein-mitochondrial membrane binding[END_REF][START_REF] Rytömaa | Reversibility of the binding of cytochrome c liposomes: implications for lipid-protein interactions[END_REF]; thus, possible models have been proposed. The first model to be formulated [START_REF] Rytömaa | Reversibility of the binding of cytochrome c liposomes: implications for lipid-protein interactions[END_REF] suggests that upon CL binding to hhcyt c one acyl chain of CL protrudes into the protein interior through the hydrophobic channel located close to the invariant Asn52 residue. The CL chain insertion into the protein is assumed to be favored by H-bonding between Asn52 and the protonated phosphate group of CL. Subsequently, an alternative model has been proposed [START_REF] Kalanxhi | Cytochrome c impaled: investigation of the extended lipid anchorage of a soluble protein to mitochondrial membrane models[END_REF]; it asserts that the binding of CL to the protein occurs in the region of the Met80containing loop, and that the acyl chain protrudes into the protein between the hydrophobic strands formed by 67-71 and 82-85 residues. Anchoring of the phospholipid to the protein would occur via electrostatic interactions between the deprotonated phosphate group of the CL liposome and the Lys72 residue of hhcyt c. In addition to these two models, a very recent report considers the possibility that both sites of the protein may participate to CL binding [START_REF] Sinibaldi | Extended cardiolipin anchorage to cytochrome c: a model for protein-mitochondrial membrane binding[END_REF]; accordingly, CL would bind to hhcyt c by two (instead of one) acyl chains, such binding mode having been confirmed to be sterically feasible (Fig. 3).

Upon interaction with CL, hhcyt c has been shown: (i) to change its tertiary structure disrupting the heme-Fe-Met80 distal bond and, in some cases, to vary the spin state of the metal (the proximal axial ligand is His) (74,77,81-83,91,92), (ii) to reduce drastically the midpoint potential out of the range required for its role in the respiratory chain ( 80), (iii) to display peroxidase activity [START_REF] Belikova | Peroxidase activity and structural transitions of cytochrome c bound to cardiolipin containing membranes[END_REF][START_REF] Chen | Protein oxidation of cytochrome c by reactive halogen species enhances its peroxidase activity[END_REF][START_REF] Diederix | Peroxidase activity as a tool for studying the folding of c-type cytochromes[END_REF][START_REF] Kapralov | The hierarchy of structural transitions induced in cytochrome c by anionic phospholipids determines its peroxidase activation and selective peroxidation during apoptosis in cell[END_REF][START_REF] Patriarca | ATP acts as a regulatory effector in modulating structural transitions of cytochrome c: implications for apoptotic activity[END_REF], also through the peroxidation of CL, thus inducing the permeabilization of the mitochondrial membrane [START_REF] Caroppi | Apoptosis and human diseases: mitochondrion damage and lethal role of cytochrome c as proapoptotic protein[END_REF][START_REF] Belikova | Peroxidase activity and structural transitions of cytochrome c bound to cardiolipin containing membranes[END_REF][START_REF] Basova | Cardiolipin switch in mitochondria: shutting off the reduction of cytochrome c and turning on the peroxidase activity[END_REF], and (iv) to bind CO and NO with high affinity [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF][START_REF] Kapetanaki | Interaction of carbon monoxide with the apoptosis-inducing cytochrome c-cardiolipin complex[END_REF], modulating apoptosis (98-100) by inhibiting the CL-hhcyt c peroxidase activity (38,79.97). Interestingly, the CL-hhcyt c complex binds NO and CO at the proximal and distal side of the heme, respectively [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF][START_REF] Kapetanaki | Interaction of carbon monoxide with the apoptosis-inducing cytochrome c-cardiolipin complex[END_REF], as reported for Axcyt c' (see ref. [START_REF] Lawson | A twofaced molecule offers NO explanation: the proximal binding of nitric oxide to haem[END_REF]).

A C C E P T E D M A N U S C R I P T

In the ferrous CL-hhcyt c complex, NO replaces the proximal His residue via an unusually complex set of kinetic steps, suggesting a remarkable mobility of the heme environment induced by CL [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF].

Nitrosylation of ferrous CL-hhcyt c involves five species (i.e., A to E in Fig. 4), including the initial and final forms (i.e., A and E, respectively) and three intermediates (i.e., B, C, and D) [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF]. The first very rapid [NO]-dependent kinetic process (k 1 = 2×10 7 M -1 s -1 )

has been assigned to the initial NO binding to the ferrous heme-Fe-atom. Interestingly, the second order rate constant is several orders of magnitude faster than that for nitrosylation of Axcyt c' [START_REF] Pixton | Activation parameters for heme-NO binding in Alcaligenes xylosoxidans cytochrome c': the putative dinitrosyl intermediate forms via a dissociative mechanism[END_REF] and Met80Ala hhcyt c mutant (101), in which NO binding is rate limited by steric hindrance provided by the tightly packed protein cage. The primary NO binding process (i.e., the A to B reaction; see Fig. 4) leads to a penta-coordinated derivative (i.e., the B species in Fig. 4). The transient NO-bound hexa-coordinated form was not observed, and thus it has been concluded that, once NO is bound to the ferrous heme-Fe-atom, the dissociation of the His18 residue is more rapid than the rapid-mixing stopped-flow dead time (i.e., < 1 ms).

Following the initial binding reaction a slower step occurs (i.e., the B to C reaction; see Fig. 4), whose rate is essentially [NO]-independent (k 2 = 7 s -1 ). This second process has been referred to the binding of an yet unidentified weak field ligand (i.e., X in Fig. 4) trans to the ferrous heme-Fe-NO complex (i.e., the C species in Fig. 4). The nature of the weak field ligand is still unclear, possibly being an intrinsic amino acid residue or a water molecule [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF].

However, at physiological NO concentrations this can be a long lived intermediate that could play a direct role in NO uptake and/or release in mitochondria, with obvious functional implications. Whatever is the nature of the proximal X ligand, it appears to be displaced by a second NO molecule (i.e., C to D reaction; see Fig. 4). Three remarkable, but possible linked, features characterize this process: (i) the reaction is slow, (ii) the time courses are not exponential but zero order (typical for diffusive processes), this being particularly evident at

A C C E P T E D M A N U S C R I P T

high NO concentration, and (iii) the rates are proportional to the NO concentration. Therefore, it has been proposed that in the C to D reaction the NO in solution partitions favourably into the considerable lipid phase provided by CL, and from this phase NO diffuses to the site where it can displace the putative X ligand. The overall rate constant of the C to D reaction (i.e., k 3 ; see Fig. 4) is 1×10 -1 s -1 , clearly indicating that the access pathways of NO to the distal and to the proximal binding sites of ferrous CL-hhcyt c drastically differ. From the spectroscopic viewpoint the species designated D (see Fig. 4) is similar, but not identical, to the final product (i.e., E in Fig. 4) and it has been assigned to a transient penta-coordinated ferrous heme-Fe-NO adduct. The last slow (k 4 = 0.1 s -1 ), [NO]-independent step (D to E reaction; see Fig. 4) takes the protein to the final penta-coordinated ferrous heme-Fe-NO adduct (i.e., E in Fig. 4). The nature of this final transition is unknown; it has been suggested that it reflects some slow structural relaxation of the lipid/protein complex bringing the heme environment to its final equilibrium state (i.e., E in Fig. 4).

Denitrosylation of ferrous CL-hhcyt c-NO involves four species (i.e., E to A in Fig. 4), including the initial and final forms (i.e., E and A, respectively) and two intermediates (i.e., F and G) [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF]. Photolysis of NO from the equilibrium state designated E (see Fig. 4) yields a tetra-coordinated species (i.e., F in Fig. 4). NO rebinding to the transient form designated F (see Fig. 4) is largely geminate (89%) and the value of the rate constant (i.e., k 5 ; see Fig. 4) is 1.3×10 11 s -1 . The remainder of the NO (11%) moves away from the heme pocket reaching the bulk phase or moves to sites removed from the heme but within the CL-hhcyt c complex.

Unlike Axcyt c', in the case of CL-hhcyt c the rate constant for His18 binding (k 6 = 1.6×10 10 M -1 s -1 ; see Fig 4) competes with NO binding, producing an appreciable amount of pentacoordinated species (G in Fig. 4), suggesting that the flexible proximal binding heme environment of CL-hhcyt c allows for accommodation of both NO and His18 in the proximity of the ferrous heme-Fe-atom. The rate at which NO recombines on the proximal side of the

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT 13
heme (reaction G to F in Fig. 4) is dictated by the rate of the His18 dissociation (i.e., k 7 = 4×10 2 s -1 ). The final step (i.e., G to A in Fig. 4) counterbalances the slow structural rearrangement(s) that has been assigned to the transition within the nitrosylated ferrous CLhhcyt c complex from D to E (see Fig. 4). The overall rate constant for NO dissociation from E (see Fig. 4) is 5×10 -4 s -1 .

The CL-induced conformational change(s) is also reflected in CO binding properties of hhcyt c. Thus, while native hhcyt c is unable to bind CO (102), ferrous CL-hhcyt c interacts with CO by a simple second-order process [START_REF] Kapetanaki | Interaction of carbon monoxide with the apoptosis-inducing cytochrome c-cardiolipin complex[END_REF], as reported for the Met80X hhcyt c mutants (101,103), carboxymethylated hhcyt c (102), and Mb (104). The value of the dissociation equilibrium constant for ferrous CL-hhcyt c carbonylation (~ 3×10 -8 M) ( 97) is similar to that reported for carboxymethylated hhcyt c (102) and Mb (104). However, both combination and dissociation rate constants for (de)carbonylation of the ferrous CL-hhcyt c complex (~ 1×10 7 M -1 s -1 and 0.18 s -1 , respectively) are larger than those reported for the Met80X mutants [START_REF] Silkstone | Production and characterisation of Met80X mutants of yeast iso-1-cytochrome c: spectral, photochemical and binding studies on the ferrous derivatives[END_REF], carboxymethylated hhcyt c (102), and Mb (104).

This suggests a very open heme crevice in the CL-hhcyt c complex (accounting for the very fast CO association rate constant) [START_REF] Kapetanaki | Interaction of carbon monoxide with the apoptosis-inducing cytochrome c-cardiolipin complex[END_REF], but also a weaker proximal Fe-His bond (as suggested by the faster CO dissociation rate constant) (105), which seems confirmed also by the very fast dissociation rate for the proximal His residue upon NO binding [START_REF] Silkstone | Nitric oxide binds to the proximal heme coordination site of the ferrocytochrome c/cardiolipin complex: formation mechanism and dynamics[END_REF]. Therefore, CL binding to hhcyt c seems to induce a gross conformational change(s) on both the proximal and distal sides of the heme, bringing about the loss of the electron transfer properties of hhcyt c and its transformation into a peroxidase [START_REF] Belikova | Peroxidase activity and structural transitions of cytochrome c bound to cardiolipin containing membranes[END_REF][START_REF] Basova | Cardiolipin switch in mitochondria: shutting off the reduction of cytochrome c and turning on the peroxidase activity[END_REF].

Conclusion and Perspectives

As a whole, the unusual features of NO and CO binding to the heme-Fe-atom of

  [START_REF] Weiss | Cytochromes c': biological models for the S = 3/2, 5/2 spin-state admixture?[END_REF].Cardiolipin switches allosterically horse heart cytochrome c function from electron transport to gas sensing and peroxidationEukaryotic cytochromes c (cyt c) are water-soluble globular heme-proteins that are located within the compartment delimited by the inner and outer mitochondrial membranes.Horse heart cyt c (hhcyt c) is a single-chain heme-protein composed of 104 amino acids, containing the heme as a prosthetic group lying within a crevice and covalently attached to the polypeptide chain by two thioether bonds formed with residues Cys14 and Cys17; the heme is axially coordinated by His18 and Met80. The protein is relatively rigid[START_REF] Louie | High-resolution refinement of yeast iso-1-cytochrome c and comparisons with other eukaryotic cytochromes c[END_REF][START_REF] Banci | Solution structure of oxidized horse heart cytochrome c[END_REF][START_REF] Flynn | Main chain and side chain dynamics of a heme protein: 15 N and 2 H NMR relaxation studies of R. capsulatus ferrocytochrome c 2[END_REF], in A C C E P T E D M A N U S C R I P T agreement with its electron transfer function. Hhcyt c binds NO with low affinity (70), does not bind CO (71), and does not display peroxidase activity (72-74).

ferrousFigure captions Fig. 1 .

 captions1 Figure captions

Fig. 2 .

 2 Fig. 2. Simplest reaction mechanism for nitrosylation and denitrosylation of ferrous Axcyt

Fig. 3 .

 3 Fig. 3. Molecular model of the putative CL-hhcyt c complex. The CL molecule (shown in

Fig. 4 .

 4 Fig. 4. Simplest reaction mechanism for nitrosylation and denitrosylation of ferrous CL-hhcyt

Figure 1 A

 1 Figure 1

  invariantly either an Arg or a Lys residue (see HOMSTRAD database for a

	multiple	sequence	alignment:	http://tardis.nibio.go.jp/cgi-

bin/homstrad/showpage.cgi?family=cytprime&disp=str).

No kinetic and thermodynamic data concerning Axcyt c' carbonylation are available.

However, it has been reported that the bimolecular rate constant for carbonylation of Rhodopseudomonas palustris cyt c', homologous to Axcyt c', ranges from ~ 1×10 3 M -1 s -1 , in the folded form, to ~ 1×10 6 M -1 s -1 , in the unfolded state. This suggests that electron transfer, occurring only in the folded conformer, triggers the folding-unfolding equilibrium of Rhodopseudomonas palustris cyt c'

[START_REF] Lee | Cytochrome c' folding triggered by electron transfer: fast and slow formation of four-helix bundles[END_REF]

. Moreover, ferrous cyt c' do not generate stable O 2 adducts (see ref.

Acknowledgements

This work was partially supported by grants from the Ministero dell'Istruzione, dell'Università e della Ricerca of Italy (PRIN 2007ECX29E_002 and University Roma Tre, CLAR 2010, to P.A.).


Florent Mathieu
email: florent.mathieu@lmt.ens-cachan.fr

François Hild
email: francois.hild@lmt.ens-cachan.fr

Stéphane Roux
email: stephane.roux@lmt.ens-cachan.fr

Identification of a crack propagation law by digital image correlation

It is proposed to use digital image correlation (DIC) to identify parameters governing crack propagation of commercially pure titanium. To achieve this goal, crack tip location, stress intensity factor, T -stress and plastic zone size are sought. Most of the DIC approaches are based upon local analyses of displacements, and their subsequent projection onto a set of mechanically relevant fields. It is proposed to perform these two sequential steps in a unique (and integrated) way, and to compare the results with a global approach to DIC with subsequent post-processing. A priori performances of two global approaches are compared, and a propagation law is identified form the series of raw images of a fatigue test on commercially pure titanium with the integrated approach that yields better results.

Introduction

. In particular,

for cyclic tests, one key aspect is related to the estimation of crack opening and closure levels when studying the propagation conditions. The latter ones could be determined by analyzing crack opening displacements measured by DIC [START_REF] Riddel | Determining fatigue crack opening loads from near-crack-tip displacement measurements[END_REF][START_REF] Carroll | Investigation of Fatigue Crack Closure using Mulitscale Image Correlation Experiments[END_REF]. These conditions could also be studied thanks to the analysis of SIF histories [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF]. Crack propagation laws were also determined when the crack tip position was optically determined when using a long-distance microscope [START_REF] De Matos | Experimental and numerical investigation of thickness effects in plasticity-induced fatigue crack closure[END_REF], or by using the crack opening displacements [START_REF] Durif | Development of a Digital Image Correlation controlled fatigue crack propagation experiment[END_REF]. In the following, another will be followed to determine the crack tip position. When the propagation path is not straight, edge detection procedures can also be used to determine the shape of the crack [START_REF] Vanlanduit | A digital image correlation method for fatigue test experiments[END_REF], or even the correlation residuals themselves when a global approach to DIC is used [START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF]. Some of the previous aspects can also be studied thanks to computed tomography and in situ tests when resorting to digital volume correlation [START_REF] Limodin | 3D X-ray microtomography volume correlation to study fatigue crack growth[END_REF]. In particular, stress intensity factors along the crack front and crack opening displacement maps allowed the same authors to analyze the closure conditions in a fatigue experiment on nodular graphite cast iron [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3D correlation of laboratory X ray microtomography images[END_REF], and its influence of the propagation conditions [START_REF] Limodin | Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D Volume Correlation[END_REF]. Even though most of the tools developed herein could be extended to deal with 3D situations, the analysis detailed herein is applied to 2D pictures and aims at the direct and automated determination of crack propagation laws based upon the sole use of digital images.

When analyzing initiation conditions, DIC was also used on meso [START_REF] Poncelet | Biaxial High Cycle Fatigue of a type 304L stainless steel: Cyclic strains and crack initiation detection by digital image correlation[END_REF] and microscales [START_REF] Bartali | Fatigue damage analysis in a duplex stainless steel by digital image correlation technique[END_REF][START_REF] Niendorf | Monitoring the fatigue-induced damage evolution in ultrafine-grained interstitialfree steel utilizing digital image correlation[END_REF] for which multiple damage sites occur. Damage mechanisms could be analyzed in a quantitative manner when using the measured displacement fields. When combined with infrared thermography, heat sources and stress levels are identified thanks to inverse thermomechanical analyses [START_REF] Chrysochoos | Local energy analysis of high-cycle fatigue using digital image correlation and infrared thermography[END_REF]. Ad hoc correlation procedure was also developed to count automatically the number of initiated mesocracks [START_REF] Rupil | Fatigue microcrack detection with digital image correlation[END_REF]. This information is then used to tune the parameters of a continuous initiation model [START_REF] Rupil | Probabilistic modeling of mesocrack initiations in 304L stainless steel[END_REF]. In the following, only one long crack will be analyzed.

One key parameter driving crack propagation is the SIF. Most of the approaches to measure an SIF are based on post-processing the measured displacement fields to extract stress intensity factors [START_REF] Sutton | Advances in Two-Dimensional and Three-Dimensional Computer Vision[END_REF][START_REF] Abanto-Bueno | Investigation of crack growth in functionally graded materials using digital image correlation[END_REF][START_REF] Yoneyama | Evaluating mixed-modestress intensity factors from full-field displacement fields obtained by optical methods[END_REF]. Least squares minimization is generally used when the measured data are projected onto reference (i.e. mechanically admissible [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF]) fields. The amplitudes associated with these reference fields give access, for instance, to SIFs. Alternative extracting techniques follow the same path as for numerical simulations (e.g. the interaction integral [START_REF] Réthoré | Estimation of mixed-mode stress intensity factors using digital image correlation and an interaction integral[END_REF]). The latter can be optimized to be the least sensitive to measurement uncertainties [START_REF] Réthoré | Noise-robust Stress Intensity Factor Determination from Kinematic Field Measurements[END_REF]. Recently, it was shown that the interaction integral is only a local minimizer, and that optimal SIF extractors deal with the measured data, namely, the displacement field, and not any of their derivatives since they will be less sensitive to measurement uncertainties [START_REF] Réthoré | Optimal and noise-robust extraction of Fracture Mechanics parameters from kinematic measurements[END_REF].

The first question to address when evaluating an SIF is related to the location of the crack tip. This is clearly an experimental challenge since the experimentalist cannot rely on a manual estimate that neglects sub-pixel crack openings.

Most of the time, the location of the crack tip is part of the minimization procedure, which becomes nonlinear. Another route was proposed by Hamam et al. [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF]. It consists in considering so-called supersingular displacement fields, and more precisely, the first one that is usually discarded in a purely elastic analysis.

The latter one can be used to locate the crack tip by canceling out its amplitude. This type of approach was applied both on numerical simulations [START_REF] Henninger | Enriched kinematic fields of cracked structures[END_REF], and experimental results [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF][START_REF] Réthoré | An extended and integrated digital image correlation technique applied to the analysis fractured samples[END_REF][START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF][START_REF] Réthoré | Optimal and noise-robust extraction of Fracture Mechanics parameters from kinematic measurements[END_REF]. It will be used herein to follow the crack tip in a cyclic test during which propagation occurs.

To evaluate directly (i.e. with no post-processing of the measured displacement field) the SIF and thus the crack tip location and other relevant mechanical parameters, so-called "integrated approaches" of DIC (or I-DIC [START_REF] Hild | Digital image correlation: From measurement to identification of elastic properties -A review[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: Post-processing and integrated approaches[END_REF][START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF][START_REF] Réthoré | An extended and integrated digital image correlation technique applied to the analysis fractured samples[END_REF]) can be followed. In a global approach to DIC, it consists in implementing di-rectly the relevant fields when minimizing the functional describing the gray level conservation. This approach will be used herein to evaluate SIFs, crack tip locations, T -stress components (i.e. the stress level parallel to the crack faces) on a thin cracked sheet made of commercially pure titanium. One of the outputs will be the parameters of a Paris' law [START_REF] Paris | A Critical Analysis of Crack Propagation Laws[END_REF]. Plastic zone size and shape will also be discussed.

In the following section, the experimental configuration is described, and followed by the presentation of two SIF extraction techniques. An a priori analysis allows for the evaluation of the performances of both techniques. Last, experimental results on commercially-pure titanium are discussed.

Experimental configuration

The aim of the present study is to determine the parameters of a crack propagation law for thin sheets of so-called Ti35 (i.e. commercially-pure titanium, max 0.2wt% Fe, 0.18wt% O, 0.08wt% C, 0.03wt% N, 0.015wt% H). A 0.3 mm thick center-cracked sample is subjected to a cyclic tensile load inducing the propagation of two crack tips. Pictures are shot regularly for the analysis proposed herein. The sample is observed by a single camera (definition: 1022 × 1024 pixels, digitization: 12 bits) and a telecentric lens focused on the vicinity of one of the notch tips. The lens allows us to minimize (if not completely cancel out) artifacts related to out-of-plane motions.

To observe crack propagation, a cyclic fatigue test is performed along the longitudinal direction of forming (load ratio: R = 0.1, load frequency: f = 10 Hz). As shown in Figure 1(a), the images are taken every 1 000 cycles up to failure of the sample, for the minimum and maximum load levels. The maximum force F corresponds to approximately 120 MPa of remote stress (i.e. about 50% of the initial yield stress σ y of the tested material). The center-cracked sample is shown in Figure 1(b). The propagation is studied only on one side of the crack. Sample failure occurs around 120 000 cycles, while crack propagation is observable during the last 60 000 cycles.

Parameter extraction

The sample being observed with a telecentric lens (Figure 2) a physical size of 6.1 µm for each pixel is achieved. The image in the deformed configuration g is related to that in the reference configuration f by assuming gray level conservation

f (x) = g(x + u(x)) ( 1
)
where u is the sought displacement field. As such, the measurement problem is ill-posed. A global approach consists in minimizing the (global) residual between the reference image f and the deformed image corrected by the displacement field g(x) = g(x + u(x)) over the whole region of interest Ω

η 2 = Ω [g(x + u(x)) -f (x)] 2 dx (2)
The form of the displacement field u can be chosen by the user. For instance, it may be decomposed over a basis of shape functions N m of general form

u(x) = α,m v αm N m (x)e α (3)
where v αm are the unknown degrees of freedom, and e α is a unit vector of the element frame. In the following, 4-noded elements are considered (i.e. a Q4-DIC approach [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands[END_REF]).

Conversely, a series of mechanically relevant fields ψ n (e.g. Williams' series [START_REF] Williams | On the stress distribution at the base of a stationary crack[END_REF])

u(x) = n w n ψ n (x) (4)
can be chosen, where w n are generalized degrees of freedom (e.g. mode I and II SIFs).

Post-processing approach

The first DIC procedure consists in measuring displacement fields discretized with Q4 elements. The element size was chosen to be equal to 16 pixels. This value is a good compromise between measurement uncertainty and spatial resolution [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands[END_REF]. No enrichment was used [START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF][START_REF] Réthoré | An extended and integrated digital image correlation technique applied to the analysis fractured samples[END_REF] in the present case since the crack tip and path will be masked. A measurement result is shown in Figure 3. The displacement field is subsequently post-processed (via least squares fit) by using reference fields, namely, Williams' series relevant for small scale yielding when the latter region is excluded from the analysis, by resorting to a least squares technique [START_REF] Mcneill | Estimation of stress intensity factor by digital image correlation[END_REF][START_REF] Abanto-Bueno | Investigation of crack growth in functionally graded materials using digital image correlation[END_REF][START_REF] Yoneyama | Evaluating mixed-modestress intensity factors from full-field displacement fields obtained by optical methods[END_REF][START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF]. These fields u = u x + iu y take the following expression in the crack frame (crack tip at the origin, and crack path along the negative x axis) resorting to the complex plane, z = r exp(iθ)

u(z) = n [ω n Ω n (z) + υ n Υ n (z)] (5)
with, for a mode I regime

Ω n (z) = (-1) (1-n)/2 2µ √ 2π r n/2 κ exp inθ 2 - n 2 exp i(4 -n)θ 2 + (-1) n + n 2 exp - inθ 2 (6)
and a mode II regime

Υ n (z) = i(-1) (1-n)/2 2µ √ 2π r n/2 κ exp inθ 2 + n 2 exp i(4 -n)θ 2 + (-1) n - n 2 exp - inθ 2 ( 7
)
where µ is Lamé's modulus, and κ a dimensionless parameter dependent on Poisson's ratio ν (i.e. κ = (3 -ν)/(1 + ν) in plane stress, or 3 -4ν in plane strain). Due to the very small thickness of the considered sample, a plane stress solution is assumed to be a good approximation of the in-plane fields.

Amplitudes ω 1 and υ 1 , associated with fields Ω 1 and Υ 1 , are the mode I and II SIFs, K I and K II , respectively. Amplitudes ω 0 and υ 0 correspond to rigid body translations. Amplitudes ω 2 and υ 2 give access to the T -stress component, and the rigid body rotation. The crack tip is located by canceling out amplitude ω -1 of the first supersingular field. More precisely, a non-zero ω -1 results in a shift d of the crack tip with respect to its exact location [START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF]

d = 2ω -1 ω 1 (8)
The use of supersingular fields allows us to account for local nonlinear effects.

For this purpose, n min = -3 seems relevant, since lower order fields show very small displacements in the analyzed zone (see Section 5.6). On the other hand, subsingular fields account for large-scale effects, such as boundary conditions.

In practice, n max ranging from 5 to 8 is sufficient. In this analysis n max = 5 was chosen to limit the number of fields used, which allows for faster computations.

The extracted fields are compared with the raw measurement as shown in Figure 6. A good agreement is observed. The root mean square (RMS) residual between the two fields is equal to 0.05 pixel. This value is (only) 2 times that of the resolution of the measurement technique for the selected element size, which is of the order of 0.025 pixel. To obtain this value, an artificial image is built by artificially applying to the reference image a 0.5-pixel motion in both directions. The uncertainty is the standard deviation of the measured field [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands[END_REF].

The post-processing approach is very fast, but it relies on measured displacement fields that are not expressed in the final kinematic basis. This intermediate step introduces an additional uncertainty to the computation of SIFs. This is why integrated approaches are proposed to perform in a single step the measurement and identification procedures [START_REF] Roux | Stress intensity factor measurements from digital image correlation: Post-processing and integrated approaches[END_REF][START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF].

Integrated approach

In this approach, the displacement field is directly decomposed over a kinematic basis whose components are those of the mechanically relevant fields [see Equation [START_REF] Hamam | Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue[END_REF]]. The sought mechanical parameters (i.e. ω n and υ n ) are the generalized degrees of freedom of the chosen displacement fields that are measured from the raw images (Figure 6). This approach avoids one step of the former method that may introduce an additional uncertainty to the evaluation of ω n and υ n . The SIFs and the crack tip location are therefore direct outputs of the integrated approach.

The crack tip location is one of the unknowns of the global correlation resid-

Baseline analysis

To evaluate the performances of both techniques, a test case was run in which a reference image was artificially deformed using a pure mode I field with various amplitudes. The displacement amplitudes (i.e. from 3 × 10 -3 to 16 From this analysis, it can be concluded that I-DIC has a better performance than that provided by the post-processing route. For larger levels of SIF, both techniques yield similar results. It is worth noting that special care should be exercised when SIF evaluation is addressed. The value of the latter is directly related to the correct positioning of the crack tip. A slight offset has a direct effect on the SIF estimate. specimen [START_REF] Tada | The stress analysis of cracks handbook[END_REF] by considering the same crack tip position as that determined by the integrated approach and a symmetric propagation

Experimental results

Displacement and correlation residual fields

K = F √ πa 2bW 1 -0.025 a W 2 + 0.06 a W 4 1 cos πa 2W ( 9
)
where F is the applied load, b the thickness of the sample, W its width, and a the length of the center crack. A good agreement is observed except at the end of the experiment for which large scale yielding is likely to occur, or buckling of the sample under the resulting transverse compression. The SIF offset at the beginning of propagation may be explained by the fact that only one crack tip was observed, and therefore a symmetry hypothesis had to be made.

Paris' law determination

The previous results allow us to determine the parameters of the crack propagation law when described by a Paris' model. The latter is expressed with or without SIF threshold as

da dN = A∆K n (10)
da dN = B(∆K -∆K th ) p (11)
where ∆K th denotes the threshold SIF amplitude below which no propagation occurs.

Instead of tuning the parameters directly between the measured SIF range and the numerical crack velocity da/dN , which can be very noisy, the growth law is recast in terms of the crack growth increment ∆a

∆a(N ) = N 0 B(∆K -∆K th ) p dN (12)
This quantity is numerically evaluated when ∆K itself is assumed to vary with ∆a as

∆K = Ce D∆a+E∆a 2 (13)
where C, D and E are parameters to be tuned. This form is chosen because it provides a good compromise between a small sensitivity to measurement uncertainties and has a small number of parameters (i.e. 3), see Figure 9. By comparing the numerically evaluated ∆a [Equation [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3D correlation of laboratory X ray microtomography images[END_REF]] to the measured values, it is possible to determine the best parameters B, K th and p of Paris' law.

Direct fit between the measured SIF range and the crack growth rate da/dN did not yield such good agreement between the measured and the identified crack growth increment. Figure 9 shows the results obtained for the test. Values of the identified parameters are given in Table 1. Even though the fit quality is better for I-DIC results, very similar results in terms of the propagation parameters are obtained. This is presumably fortuitous.

Figure 10 shows plots of Paris' law with and without threshold. The results obtained herein for Paris' law with no threshold [Equation [START_REF] Réthoré | Extended digital image correlation with crack shape optimization[END_REF]] are in close agreement with those reported on a different titanium grade [START_REF] Adib | An exponential equation of fatigue crack growth in titanium[END_REF]. It is not possible to distinguish from the experimental data reported herein the two types of Paris' law. Both give a good fit quality. Consequently, noting that Equation ( 10) is a particular case of Equation ( 11), i.e. p = n, B = A, ∆K th = 0, it means a higher uncertainty on the threshold ∆K th as the latter's effect can be compensated by a change of the other parameters.

T -stress measurement

The T -stress component (i.e. the nonsingular term in Williams' series for cracks) is another contribution of Williams' series that can be measured thanks to Q4-DIC or I-DIC, since it is directly related to ω 2 . Its estimate is compared to that proposed by McClintock [START_REF] Mcclintock | Discussion to C. Laird's paper "The influence of metallurgical microstructure on the mechanisms of fatigue crack propagation[END_REF] for a CCT specimen by considering the same crack tip position as that determined by the integrated approach and a symmetric propagation

T σ ∞ = -0.007 + 0.283 a W -3.268 a W 2 + 6.622 a W 3 -5.995 a W 4 ( 14
)
with

σ ∞ = F bW ( 15
)
where σ ∞ is the remote stress. The result presented in Figure 11 shows that there is a very good agreement with the analytical prediction. The propagation during the experiment occurs essentially with a constant T -stress of the order of -120 MPa. However, at the end of the experiment, large scale yielding induces a change of the T -stress level, or a lack of reliability of the present analysis if buckling takes place.

Plastic zone size

The computation of the stress field corresponding to the measured displacement field [START_REF] Owen | Engineering Fracture Mechanics: Numerical methods and applications[END_REF] allows us to estimate the plastic zone shape, as the zone where the equivalent Von Mises' stress is greater than twice the yield stress. This estimation does not account for stress redistributions around the plastic zone. The contributions of K I , T -stress and all the fields are analyzed in Figure 12(a).

A significant difference is observed. The shape of the plastic zone is mainly due to the n = -3 field. When all the fields are considered except n = -3, the estimated plastic zone is nearly the same as with only the K I and T -stress contributions.

Within the framework of small scale yielding, an estimate of the process zone size depends upon the yield stress of the studied material. In the case of cyclic loading, this estimation reads [START_REF] Rice | Mechanics of crack tip deformation and extension by fatigue[END_REF]

r p = 1 π (1 -R)K I 2σ y 2 ( 16
)
where r p is the radius of the plastic zone. Furthermore, the identified fields also give an indication of the process zone size [START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF][START_REF] Henninger | Enriched kinematic fields of cracked structures[END_REF][START_REF] Réthoré | Optimal and noise-robust extraction of Fracture Mechanics parameters from kinematic measurements[END_REF]. The first mode I supersingular field (i.e. n = -1) gives an indication of the crack tip position along its propagation direction, the second (i.e. n = -3) provides a contribution that can be interpreted in terms of the process zone size, via a term R p homogeneous to a length scale

R p ∝ -8 ω -3 ω 1 ( 17
)
where ω i are amplitudes of order i of the mode I fields [START_REF] De Matos | Experimental and numerical investigation of thickness effects in plasticity-induced fatigue crack closure[END_REF]. These different estimates are plotted all along the propagation in Figure 12(b). This dimensional indication needs to be adjusted to a known set of data. Over the first 40 images, the slope is very similar, and one can extract a proportionality coefficient of 5.1 between these two estimates. This adjustment is plotted in Figure 12(b). Let us finally note that the post-processing route did not provide trustworthy results.

Contribution of different fields

In order to evaluate which of the measured orders dominate over the other ones, the absolute value of the longitudinal displacement fields ( 6) and ( 7) are analyzed in the ligament in front of the crack (i.e. for θ = 0). This plot is presented in Figure 13. The gray background indicates the masked zone, since the analysis region consist of an annulus 50 < r < 400 pixels.

Orders ranging from n = -3 to n = 5 are shown. Since this problem is mostly in a mode I regime (it was checked that the mode II contributions are at least one order of magnitude less than those in mode I), only the mode I fields are shown. Except for order n = 1, one can clearly distinguish subsingular fields, which have a positive slope in a log-log plot, from supersingular fields with a negative slope. The field of order n = 0 is not shown since it corresponds to rigid body translation, which has no effect on crack propagation.

The displacement amplitudes ω n r n/2 (expressed in pixels) are dominant first when n = 1, which corresponds to the usual K I field, and for n = 2, which is representative of the T -stress contribution. This result shows that the DIC analyses were preformed in a K I -dominant zone. Furthermore, it shows that the estimates of SIF and T -stress are extracted from the two dominant fields.

Conversely, one can note that the amplitude of order n = -3, which is used to evaluate the process zone size [see Equation [START_REF] Chrysochoos | Local energy analysis of high-cycle fatigue using digital image correlation and infrared thermography[END_REF]], is only the third most influential field. This may explain why its estimate becomes less accurate. A higher magnification is desirable if this field is sought. The present distribution of amplitudes, only shown for a given picture, is very stable and is observed for all the 60 images analyzed herein.

Conclusions

An integrated digital image correlation (I-DIC) procedure, which allows for the direct identification of various fracture mechanics parameters, was used to tune parameters of a crack propagation law directly from the analysis of a series of 60 pairs of images of a crack propagation test on commercially-pure titanium.

I-DIC was first validated against a more classical post-processing route. The former out-performed the latter in terms of correlation residuals, and resolution for stress intensity factor measurement and crack tip positioning.

This work shows that the identification of a crack propagation law is possible from one single test, provided the crack is observed during all its propagation.

It requires to estimate the crack tip position. This was achieved by resorting to the first supersingular term of Williams' series whose amplitude cancels out for the most appropriate crack tip location. Once the crack tip was determined, the crack growth increment is estimated in addition to other global parameters such as the stress intensity factor range, and T -stress. By analyzing the dominant displacement fields, it can also be concluded that the estimate of the plastic zone size is more difficult and would require analyses with a higher magnification, or even combined kinematic and thermal measurements [START_REF] Maynadier | One-shot measurement of thermal and kinematic fields: Infra-Red Image Correlation (IRIC)[END_REF].

Last, for the end of the propagation analyzed herein, because of the submillimeter thickness of the sample the out-of-plane effects as well as local buckling may induce deviations from a purely in-plane analysis as performed herein.

To check these effects, 3D-DIC [START_REF] Sutton | Experimental study of crack growth in thin sheet 2024-T3 aluminum under tension-torsion loading[END_REF] is a solution of choice to consider.

  ual minimization [Equation[START_REF] Roux | Digital Image Correlation and Fracture: An Advanced Technique for Estimating Stress Intensity Factors of 2D and 3D Cracks[END_REF]]. Consequently, I-DIC requires a longer computation time since the global matrix to be inverted has to be recalculated each time the crack tip is moved to a new position. The computation time takes about 1 hour for the whole post-processing approach (i.e. a series of 60 images, with a Matlab c code), and the integrated approach is performed in about 3 to 5 hours, depending on the value of the convergence criterion in terms of crack tip increment.

  pixels) would correspond approximately to 0.01 to 55 MPa √ m for the tested material in the present configuration. Both extraction techniques are applied to these artificially deformed images with the same mask parameters (inner radius and frame of 10 pixels, external radius of 400 pixels). Results of K I measurements are shown in Figure 4(a). While SIF measurements are generally in good agreement with prescribed values, I-DIC shows a particularly good performance in terms of SIF values. The post-processing route leads to a poorer resolution (i.e. the 'smallest change in a quantity being measured that causes a perceptible change in the corresponding indication' [33]). The error with respect to the prescribed SIF is shown in Figure 4(b). For a prescribed SIF greater than 0.05 MPa √ m, the error is less than 8 × 10 -4 MPa √ m for I-DIC. For the post-processing route, values can be trusted for SIFs greater than 1 MPa √ m so that the error is less than 0.1 MPa √ m. This difference can be justified by the fact that much fewer degrees of freedom are used in the I-DIC approach.

Figure 5

 5 Figure 5 displays the distance between the prescribed and identified crack tip position in pixels. The convergence criterion, which is written in terms of offset d defined in Equation (8), has been set to 0.5 pixel. Both techniques show good agreement for measured SIFs greater than 0.1 MPa √ m for the studied material parameters. I-DIC provides a quasi constant crack tip position forthese values of SIF (mean offset of 0.74 pixel with a standard deviation of 0.17 pixel, which is below the convergence criterion), while the post-processing route shows more scatter (mean offset of 5.5 pixels with a standard deviation of more than 3.5 pixels).

Figure 6

 6 Figure6shows the displacement fields between minimum and maximum loading after 60 000 cycles. For an assumed crack tip location, the correlation residual η 2 is minimized within the region of interest, which is the zone over which the fields are plotted. The external radius of this mask turns out to have little influence on the measurement results provided it is large enough to encompass a significant zone; the internal radius and the box around the crack must be kept to a minimum value (i.e. 50 pixels in the present case) in order not to consider regions affected by plastic strains in the vicinity of the crack tip. The correlation residuals are shown in Figure7(a). Their level remains very low, i.e. of the order of 2 percent of the dynamic range of the reference picture in the considered zone for I-DIC, and less than 4 percent at most for postprocessed Q4-DIC (Figure 7(b)). These low values indicate that the registration was successful, and that the measured displacements are trustworthy. It can be noted that in the immediate vicinity of the crack front the registration would have been less successful, had the corresponding region not been masked. The fact that the mean level is virtually independent of the number of cycles for I-DIC (Figure 7(b)) shows that except for the very last points, the proposed framework is validated.

34 9Figure 1 :

 341 Figure 1: (a) Image capture during the load history. (b) Dimensions and local detail of the CCT sample. Reference results for these samples [34, 36] often use solutions with a centered crack.

Figure 2 :

 2 Figure 2: Test configuration for the cyclic loading. A telecentric lens is used to minimize artifacts induced by out-of-plane motions.

Figure 3 :

 3 Figure 3: Measured vertical displacement field (in pixels) around the crack tip with Q4-DIC (16-pixel elements are used). No mask was used in this preliminary analysis.

Figure 4 :Figure 5 :

 45 Figure 4: (a) Coefficient of order n = 1 (K I ) measured by both techniques with respect to prescribed value K I for the a priori analysis. (b) Error between measured and prescribed K I values for both techniques. The measurement error for I-DIC remains small up to values of the order of 0.05 MPa √ m whereas it is more important up to values of 1 MPa √ m for the post-processing route.

Figure 6 :

 6 Figure 6: Measured displacement field with Q4-DIC (center), extracted field with the postprocessing technique (top), measured field with I-DIC (bottom) and residuals (right). For the post-processing technique, residuals are the raw differences between the measured displacement field with Q4-DIC and the extracted field (for the two directions). For integrated-DIC, residuals are gray level differences |f (x)g(x + u(x))| normalized by the dynamic range of the reference picture.

Figure 7 :

 7 Figure 7: (a) Gray level residuals |f (x)g(x + u(x))| as percentage of the dynamic range of the reference picture for integrated-DIC (left) and the post-processing technique (right).The displacement field u is for each technique obtained by multiplying the reference fields[START_REF] De Matos | Experimental and numerical investigation of thickness effects in plasticity-induced fatigue crack closure[END_REF] and (7) by the measured amplitudes ωn and υn. (b) Mean gray level residuals normalized by the dynamic range of the reference picture for both techniques. The increase of the measured amplitudes makes the correlation residuals increase for the post-processing technique but not for I-DIC.

Figure 8 :

 8 Figure 8: (a) Measured ∆a (in mm) as a function of the number of cycles for both methods.(b) Measured ∆K as a function of the crack length increment ∆a for both methods compared to tabulated values for CCT samples[START_REF] Tada | The stress analysis of cracks handbook[END_REF], and interpolated curves obeying Equation[START_REF] Limodin | Influence of closure on the 3D propagation of fatigue cracks in a nodular cast iron investigated by X-ray tomography and 3D Volume Correlation[END_REF].

Figure 9 :

 9 Figure 9: Measured and fitted crack growth increment without and with a threshold in Paris' law for I-DIC (a) and post-processing (b) routes.

  .

Figure 10 :

 10 Figure 10: Raw measurement (symbols) and interpolation (solid lines) for Paris' law without threshold (a) and with threshold (b). Results without threshold are compared to reported values of another titanium grade [35].

Table 1 :

 1 Identified coefficients for Paris' law without threshold (a) or with threshold (b). Coefficients for the law without threshold are compared to published values for another titanium grade[START_REF] Adib | An exponential equation of fatigue crack growth in titanium[END_REF]. Units: a in mm, ∆K in MPa √ m.

	Parameter	A	m
	Post-processing 4.76 × 10 -8 3.12
	I-DIC	4.65 × 10 -8 3.12
	Adib et al.	7.49 × 10 -8 2.95
		(a)
	Parameter	B	∆K th	p
	Post-processing 2.87 × 10 -6	4.12	1.77
	I-DIC	1.48 × 10 -6	3.46	2.00

Acknowledgements

This work was supported by Region Ile de France (FRESCORT project), and by Agence Nationale de la Recherche through the RUPXCUBE grant (ANR-09-BLAN-0009-01).

List of Tables

Identified coefficients for Paris' law without threshold (a) or with threshold (b). Coefficients for the law without threshold are compared to published values for another titanium grade [START_REF] Adib | An exponential equation of fatigue crack growth in titanium[END_REF]. Units: