Matthieu Amor

Juan Wan

Ramon Egli
email: ramon.egli@zamg.ac.at

Julie Carlut

Christophe Gatel

Ingrid Marie Andersen

Etienne Snoeck

Arash Komeili

Key signatures of magnetofossils elucidated by mutant magnetotactic bacteria and micromagnetic calculations

Sediments containing fossilized magnetotactic bacteria bear magnetic signatures apparently associated with different sources of magnetite particles Intact and fold-collapsed single-stranded and multi-stranded magnetosome chains can explain all observed signatures of fossilized magnetotactic bacteria We provide criteria for the discrimination of distinct populations of ancient magnetotactic bacteria

Plain language summary

Magnetotactic bacteria (MTB) produce chains of magnetic nanoparticles that contribute to the magnetization of sedimentary rocks. Fossilized chains can be detected in ancient geological samples using magnetic measurements. They provide unique evidence for microbial life. However, remains of fossil MTB possess ambiguous magnetic properties that make their detection problematic. Here, we demonstrate, with the help of numerical simulations and a modified MTB producing looped chains, that the magnetic properties of MTB-rich sediments can be explained by a combination of chains made of single and multiple strands, and that chain collapse in the sediment, if occurring, likely preserves key magnetic properties, enabling the identification of fossil MTB. These properties depend also on the proportion of equidimensional and elongated magnetosomes, as well as on the relative abundances of single-and multi-stranded chains. This work has important implications, as it provides a novel tool to differentiate distinct MTB types according to their chain and magnetosome morphologies, enabling the tracking of the evolution of some of the most ancient mineral-producing organisms. It also enables a more accurate discrimination of different sources of magnetite particles, which helps gaining better environmental and relative paleointensity reconstructions from sedimentary records.

Introduction

Magnetite (Fe 3 O 4 ) is a ubiquitous iron oxide found in sediments and sedimentary rocks. It is a major carrier of sediments and rock magnetization used for numerous applications in Earth sciences including paleomagnetic, paleoclimate, paleoenvironmental, paleogeographic and paleontological reconstructions (e.g., [START_REF] Evans | Environmental Magnetism: Principles and Applications of Enviromagnetics[END_REF][START_REF] Tauxe | Essentials of Paleomagnetism[END_REF][START_REF] Egli | Introduction to "Magnetic iron minerals in sediments and their relation to geologic processes, climate, and the geomagnetic field[END_REF]. In addition to lithogenic magnetite-bearing minerals (e.g., [START_REF] Chang | Widespread occurrence of silicate-hosted magnetic mineral inclusions in marine sediments and their contribution to paleomagnetic recording[END_REF], sediments contain secondary magnetite crystals formed in situ through diagenetic processes (e.g., [START_REF] Oldfield | Sources of fine-grained magnetic minerals in sediments: a problem revisited[END_REF][START_REF] Franke | Magnetic petrology of equatorial Atlantic sediments: Electron microscopy results and their implications for environmental magnetic interpretation[END_REF], but are also produced by bacteria participating in the sedimentary iron cycle (e.g., [START_REF] Lovely | Anaerobic production of magnetite by a dissimilatory iron-reducing microorganism[END_REF][START_REF] Fortin | Formation and occurrence of biogenic iron-rich minerals[END_REF][START_REF] Miot | Iron biomineralization by anaerobic neutrophilic iron-oxidizing bacteria[END_REF].

Magnetotactic bacteria (MTB) are the only known microorganisms that produce intracellular nanoparticles of magnetite in organelles called magnetosomes under a genetically controlled pathway (Uebe and Schüler, 2016). They represent some of the most ancient biomineralizing organisms, with a proposed origin of ~3 Ga [START_REF] Lin | Origin of microbial biomineralization and magnetotaxis during the Archean[END_REF]. They are markers of oxic-anoxic transition zones in aquatic and sedimentary environments [START_REF] Flies | Diversity and vertical distribution of magnetotactic bacteria along chemical gradients in freshwater microcosms[END_REF][START_REF] Simmons | Spatiotemporal distribution of marine magnetotactic bacteria in a seasonally stratified coastal salt pond[END_REF], making them useful paleoenvironmental indicators [START_REF] Hesse | Evidence for bacterial palaeoecological origin of mineral magnetic cycles in oxic and sub-oxic Tasman Sea sediments[END_REF][START_REF] Roberts | Magnetotactic bacterial abundance in pelagic marine environments is limited by organic carbon flux and availability of dissolved iron[END_REF][START_REF] Usui | Changing Abundance of Magnetofossil Morphologies in Pelagic Red Clay Around Minamitorishima, Western North Pacific[END_REF][START_REF] Chang | Coupled microbial bloom and oxygenation decline recorded by magnetofossils during the Paleocene-Eocene thermal maximum[END_REF][START_REF] Wagner | Diversification of iron-biomineralizing organisms during the Paleocene-Eocene Thermal Maximum: Evidence from quantitative unmixing of magnetic signatures of conventional and giant magnetofossils[END_REF]. Magnetosomes are assembled in chains, and provide the bacteria with a permanent magnetic dipole for navigation purposes [START_REF] Mccausland | Magnetic genes: Studying the genetics of biomineralization in magnetotactic bacteria[END_REF]. When MTB cells die, their magnetite chains can be trapped into sediments and subsequently fossilized [START_REF] Petersen | Fossil bacterial magnetite in deep-sea sediments from the South Atlantic Ocean[END_REF][START_REF] Kopp | The identification and biogeochemical interpretation of fossil magnetotactic bacteria[END_REF]. MTB fossils (hereafter referred to as magnetofossils) can be preserved over geological times and contribute to sedimentary paleomagnetic and environmental records [START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF][START_REF] Larrasoaña | Paleomagnetic and paleoenvironmental implications of magnetofossil occurrences in late Miocene marine sediments from the Guadalquivir Basin, SW Spain[END_REF][START_REF] Ouyang | Variable remanence acquisition efficiency in sediments containing biogenic and detrital magnetites: Implications for relative paleointensity signal recording[END_REF]. Their unambiguous identification and discrimination from abiotic magnetite would thus provide strong constraints on the evolution of life, biomineralization processes, and environmental conditions over geological times.

Arrangement of single domain (SD) magnetite crystals in linear chains combines two robust magnetofossil identification criteria, size and arrangement [START_REF] Kopp | The identification and biogeochemical interpretation of fossil magnetotactic bacteria[END_REF], which are key parameters for a unique magnetic signature among all sources of natural magnetic minerals. Other types of secondary magnetite particles tend to have a wide size distribution exceeding the SD stability range (e.g., [START_REF] Gibbs-Eggar | Possible evidence for dissimilatory bacterial magnetite dominating the magnetic properties of recent lake sediments[END_REF][START_REF] Faivre | Mineralogical and isotopic properties of inorganic nanocrystalline magnetites[END_REF][START_REF] Liu | Quantifying grain size distribution of pedogenic magnetic particles in Chinese loess and its significance for pedogenesis[END_REF]. Most importantly, the special conditions required to form isolated, non-branched strings of abiogenic SD particles (e.g., [START_REF] Ruder | Biological colloid engineering: Self-assembly of dipolar ferromagnetic chains in a functionalized biogenic ferrofluid[END_REF] are unlikely to occur frequently in natural sediments. Because the magnetic extraction procedure required to prepare sufficiently concentrated samples for statistically significant TEM observations destroys the native arrangement of SD magnetite crystals in the sediment matrix, the in-situ identification of chain structures is only possible through indirect methods based on magnetic measurements of the bulk sediment.

Unfortunately, the identification of magnetofossil signatures with magnetic methods is not always unique (e.g., [START_REF] Roberts | Domain state diagnosis in rock magnetism: Evaluation of potential alternatives to the Day diagram[END_REF]. An additional complication arises from the fact that the lack of chain structures (e.g., [START_REF] Weiss | Magnetic tests for magnetosome chains in Martian meteorite ALH84001[END_REF] does not necessarily exclude a biological origin of SD magnetite particles, since chain collapses that erase the original magnetic fingerprints might occur during early diagenesis.

First-order reversal curve (FORC) diagrams [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF], which rely on the measurement of partial hysteresis curves, can be used to discriminate isolated SD magnetite particles and magnetosome chains from other structures, such as SD particle clusters and larger lithogenic (titano-)magnetite crystals containing magnetic vortices (VO) or multiple magnetic domains (MD) [START_REF] Roberts | Resolving the origin of pseudo-single domain magnetic behavior[END_REF][START_REF] Lascu | The vortex state in geologic materials: A micromagnetic perspective[END_REF], through a sharp signature called the central ridge [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF][START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF]Heslop et al., 2014a). Principal component analyses (PCA) of FORC diagrams obtained from magnetofossil-rich sediments [START_REF] Lascu | Magnetic unmixing of first-order reversal curve diagrams using principal component analysis[END_REF] and selective FORC measurements of secondary 300 nm magnetite particles through selective chemical leaching [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF] contain signatures of VO or strongly interacting SD particle clusters with a large vertical dispersion around the central ridge, which contribute to 35-65% of the total magnetization (Table 1). FORC diagrams containing these features can be explained by mixtures of (i) well-dispersed uniaxial, non-interacting SD particles, or linear chains of such particles, contributing to the central ridge (collectively referred to as UNISD in the following), and (ii) non-SD or clustered SD particles contributing to the remaining parts of the FORC distribution. The associated magnetic hysteresis parameters and [START_REF] Dunlop | Theory and application of the Day plot (M rs /M s versus H cr /H c ) 1, Theoretical curves and tests using titanomagnetite data[END_REF]) are close to the theoretical limit for non-interacting SD particles and intact magnetosome chains (Table 1). In particular, hysteresis squareness (the ratio of remanent and saturation magnetizations) values 0.44 are unrealistically large for samples containing non-SD or strongly interacting SD particles associable with 35-65% non-central ridge FORC contributions [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF] (Figure 1).

The UNISD signature of magnetofossil-rich sediments is usually characterized by two narrow coercivity distributions (called biogenic soft, BS, and biogenic hard, BH), which are particularly evident in SD-selective magnetic responses such as the central ridge in FORC diagrams and the AF demagnetization of an anhysteretic remanent magnetization (ARM) (Egli, 2004a;[START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF]Heslop et al., 2014a). The peculiar narrowness of these coercivity distributions is determined by the uniaxial magnetic anisotropy of the chain structure, which, together with the shape anisotropy of equidimensional (BS) and elongated (BH) magnetosomes, yields median coercivities of ~30-50 mT and ~65-80 mT, respectively, for these two components.

The mixed FORC properties of magnetofossil endmembers have been explained with a form of chain collapse by which the linear magnetosome arrangement becomes increasingly randomized after MTB death, leading to the formation of fractal-like magnetosome clusters [START_REF] Harrison | FORCulator: A micromagnetic tool for simulating first-order reversal curve diagrams[END_REF][START_REF] Chang | Micromagnetic simulation of magnetofossils with realistic size and shape distributions: Linking magnetic proxies with nanoscale observations and implications for magnetofossil identification[END_REF]. Isolated magnetosome chains collapse if bent beyond their elastic limit, due to the magnetostatic attraction between the opposed magnetic poles at their extremities [START_REF] Shcherbakov | Elastic stability of chains of magnetosomes in magnetotactic bacteria[END_REF][START_REF] Kiani | Elastic properties of magnetosome chains[END_REF].

Bending forces might be produced by mechanical interactions with sediment particles during mixing in the bioturbated layer, and during compaction. Chains might also be stabilized by electrostatic adhesion onto larger sediment particles, such as clay platelets [START_REF] Galindo-Gonzalez | Magnetic and microscopic characterization of magnetite nanoparticles adhered to clay surfaces[END_REF]. The lack of direct in situ observation of magnetofossil structures, the mixed character of the magnetic signature of magnetofossil-rich sediments, and the sometimes-poor definition of characteristic coercivity components, leave margins for contrasting interpretations of some geological materials. Most notable examples include the controversial biogenic origin of magnetite nanoparticles in the Martian meteorite ALH84001 [START_REF] Thomas-Keprta | Elongated prismatic magnetite crystals in ALH84001 carbonate globules: Potential Martian magnetofossils[END_REF][START_REF] Golden | Evidence for exclusively inorganic formation of magnetite in Martian meteorite ALH84001[END_REF][START_REF] Weiss | Magnetic tests for magnetosome chains in Martian meteorite ALH84001[END_REF], and the origin of the sudden rise of SD magnetite concentration during the Paleocene-Eocene thermal maximum [START_REF] Kent | A case for a comet impact trigger for the Paleocene/Eocene thermal maximum and carbon isotope excursion[END_REF][START_REF] Kopp | Magnetofossil spike during the Paleocene-Eocene thermal maximum: Ferromagnetic resonance, rock magnetic, and electron microscopy evidence from Ancora[END_REF][START_REF] Wang | Evidence for abundant isolated magnetic nanoparticles at the Paleocene-Eocene boundary[END_REF][START_REF] Wagner | Diversification of iron-biomineralizing organisms during the Paleocene-Eocene Thermal Maximum: Evidence from quantitative unmixing of magnetic signatures of conventional and giant magnetofossils[END_REF].

Extracted magnetosomes obtained from MTB cultures after cell lysis tend to form large clumps with an excessively low hysteresis squareness [START_REF] Kobayashi | Experimental observation of magnetosome chain collapse in magnetotactic bacteria: Sedimentological, paleomagnetic, and evolutionary implications[END_REF][START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF] (Table 1, Figure 1). This is also true for the hysteresis squareness of smaller magnetosome clusters produced by MTB whose genes responsible for chain assembly have been deleted, namely the ∆mamJ MSR-1 strain in [START_REF] Katzmann | Analysis of Magnetosome Chains in Magnetotactic Bacteria by Magnetic Measurements and Automated Image Analysis of Electron Micrographs[END_REF] (see sections below), which might serve as analogue for the full collapse of isolated chains.

Finally, the random-walk-based magnetosome assemblage algorithm of [START_REF] Harrison | FORCulator: A micromagnetic tool for simulating first-order reversal curve diagrams[END_REF] produce a range of magnetic properties [START_REF] Chang | Micromagnetic simulation of magnetofossils with realistic size and shape distributions: Linking magnetic proxies with nanoscale observations and implications for magnetofossil identification[END_REF] comprised between the endmembers corresponding to linear chains and fractal-like particle clusters, respectively, depending on the assumed degree of randomness (Figure 1). In these simulations, FORC diagrams containing non-UNISD contributions comparable with those of magnetofossil endmembers are associated with hysteresis squareness values of ~0.4 or lower. Pedogenic magnetite, the only well-characterized natural example of nonbiogenic secondary magnetite particles [START_REF] Dearing | Magnetic, geochemical and DNA properties of highly magnetic soils in England[END_REF], also features mixed UNISD and non-UNISD FORC properties (Table 1); however, in this case, the hysteresis squareness of ~0.2 is compatible with the FORC signature. Thus, current interpretations of magnetofossil signatures appear to miss structures possessing the correct combination of UNISD and non-UNISD magnetic properties.

Excluding disordered clusters, which do not possess the correct magnetic signature, and the already investigated chain axis randomization, possible candidates for missing magnetofossil morphologies include native multistranded chains of magnetosomes produced by some MTB [START_REF] Amor | Ironbiomineralizing organelle in magnetotactic bacteria: function, synthesis and preservation in ancient rock samples[END_REF] and mechanically meaningful minimumenergy structures obtained from the collapse of isolated chains, such as rings and double strands resulting from the fold-collapse of single-stranded chains that have been bent beyond the elastic limit (von [START_REF] Dobeneck | Bacterial magnetofossils (in German)[END_REF][START_REF] Philipse | Magnetic colloids from magnetotactic bacteria: Chain formation and colloidal stability[END_REF][START_REF] Kiani | Self-organization and stability of magnetosome chains -A simulation study[END_REF]. Single-and multi-stranded chains in MTB and in the derived intact magnetofossils (Figure 2a,b) are in a native SD state that maximizes the total magnetic moment, with the magnetization of all magnetosomes being parallel to the chain axis [START_REF] Hanzlik | Pulsed-field-remanence measurements on individual magnetotactic bacteria[END_REF]. This is the only possible zero-field magnetic configuration for single-stranded chains, but not for multiple strands, in which intermediate states with lower magnetic moments can be nucleated with the application of strong external fields [START_REF] Hanzlik | Pulsed-field-remanence measurements on individual magnetotactic bacteria[END_REF].

Intermediate remanent states are formed when the magnetic moment of one strand is reversed with respect to the others, or when the magnetic moments of individual crystals form complex patterns [START_REF] Varón | Dipolar magnetism in ordered and disordered lowdimensional nanoparticle assemblies[END_REF]. On the other hand, structures arising from chain collapse (Figure 2c,d), are expected to possess native flux-closure (FC) configurations characterized by a small net magnetic moment, as observed for instance in nanoparticle rings [START_REF] Dunin-Borkowski | Off-axis electron holography of magnetic nanowires and chains, rings, and planar arrays of magnetic nanoparticles[END_REF][START_REF] Takeno | Morphology and magnetic flux distribution in superparamagnetic single-crystalline Fe 3 O 4 nanoparticle rings[END_REF]. These configurations minimize the magnetostatic energy by suppressing the external stray field of SD-like magnetic states. The low magnetic moment of collapsed structures makes them a poor sedimentary recorder of the Earth magnetic field, contrary to intact magnetosome chains, so that the fate of fossil magnetosome chains during diagenesis is expected to have important implications for the paleomagnetism of magnetofossil-rich sediments.

FORC measurements are sensitive to transitions between high-moment (SD-like) and low-moment (FC-like) magnetic configurations and their in-field stability ranges [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF]. In particular, hysteresis squareness values close to 0.5, typical of randomly oriented UNISD particles, depend on the room-temperature zero-field stability of highmoment states. This stability, which enables MTB to navigate along the Earth magnetic field, has been demonstrated experimentally for native single-and multistranded chains [START_REF] Hanzlik | Pulsed-field-remanence measurements on individual magnetotactic bacteria[END_REF][START_REF] Pan | Rock magnetic properties of uncultured magnetotactic bacteria[END_REF], but no information is available for collapsed structures.

Here, we demonstrate that the unresolved magnetic signature of magnetofossils (i.e., FORC signatures typical of magnetostatic interactions and FC, which do not lower the hysteresis squareness typical of UNISD particles) can be explained by the predominant contribution of intact or elastically deformed single-or multi-stranded magnetosome chains (non-collapsed). Contributions from fold-collapsed chains are also possible. Nucleation and annihilation of FC magnetic states explains non-UNISD signatures in FORC diagrams of magnetofossil-bearing sediments, while the strong uniaxial anisotropy of intact and fold-collapsed chains ensures that they still possess a UNISD-like hysteresis squareness of ~0.5. Our claim is supported by micromagnetic simulations of native single-and double-stranded chains and fold-collapsed chains. Unfortunately, MTB producing multi-stranded chains are not available for culture in the laboratory; however, the validity of our micromagnetic calculations has been tested by comparing the magnetic signature of intact chains with some structures that may be produced by the spontaneous collapse of isolated chains. For this purpose, we produced a mutant of the magnetotactic bacterium Magnetospirillum magneticum strain AMB-1 synthesizing looped structures similar to the rings described by [START_REF] Kiani | Self-organization and stability of magnetosome chains -A simulation study[END_REF] and to some structures obtained after dissolution of cultured cells of the same strain [START_REF] Philipse | Magnetic colloids from magnetotactic bacteria: Chain formation and colloidal stability[END_REF]. Looped chains might form in sediment after cell disruption, but before stabilizing structures such as magnetosome membranes and the connecting cytoskeletal filament [START_REF] Komeili | Magnetosomes are cell membrane invaginations organized by the actin-like protein MamK[END_REF] are dissolved, owing to the larger elastic response of membrane-filled gaps [START_REF] Shcherbakov | Elastic stability of chains of magnetosomes in magnetotactic bacteria[END_REF]. Fold-collapsed chains, on the other hand, might form after dissolution of all biological structures, as a consequence of the much lower elastic limit associated with contacting magnetosomes [START_REF] Shcherbakov | Elastic stability of chains of magnetosomes in magnetotactic bacteria[END_REF].

From magnetic measurements and off-axis electron holography characterizations, we

show that the looped chains produced by the AMB-1 mutant generate FC magnetic signatures similar to those obtained with our micromagnetic simulations of single-and multistranded chains, while lacking the hysteresis squareness of the simulations and of natural magnetofossil signatures. This demonstrates that the unique magnetic fingerprints of magnetofossils originate from strongly uniaxial magnetosome arrangements, rather than clusters, rings, and other forms of looped chains lacking a uniaxial shape. This work resolves longstanding and ongoing controversies about the interpretation of sedimentary sources of SD magnetite, establishing a theoretical and experimental framework that explains the magnetofossil signature and provides insights into the diversity of ancient MTB populations. A glossary of specialized terms used to describe magnetic measurements and micromagnetic modeling results is provided in the Supporting Information.

Material and methods

Micromagnetic modeling

To determine whether native multi-stranded and fold-collapsed magnetosome chains could explain the observed magnetofossil signatures, we modeled their FORC properties.

Micromagnetic models of high-resolution FORC measurements have been calculated for selected magnetosome morphologies and chain configurations representative of intact or elastically deformed chains, and of fold-collapsed chains. The magnetic properties of the simulated structures are determined mainly by two factors: the anisotropy of individual magnetosomes, which is controlled by their shape and crystal orientation, and in-chain magnetostatic interactions, which are controlled mainly by the chain geometry [START_REF] Berndt | Mind the gap: Towards a biogenic magnetite palaeoenvironmental proxy through an extensive finite-element micromagnetic simulation[END_REF]. Magnetosome shapes can be classified as equant (cuboctahedral, octahedral), elongated (prismatic, elongated octahedral), and highly elongated (tooth-and bullet-shaped) [START_REF] Amor | Ironbiomineralizing organelle in magnetotactic bacteria: function, synthesis and preservation in ancient rock samples[END_REF]. These categories form well-defined clusters of crystal sizes and shapes within the SD stability range of magnetite [START_REF] Muxworthy | Critical single-domain/multidomain grain sizes in noninteracting and interacting elongated magnetite particles: Implications for magnetosomes[END_REF][START_REF] Newell | Transition to superparamagnetism in chains of magnetosome crystals[END_REF] (Figure S1).

The geometry of intact single-stranded chains is controlled by the gaps between magnetosomes, the size decrease of immature crystals towards the chain extremities (tapering), and chain bending within the elastic limit. Magnetosomes in native multistranded chains are staggered: each magnetosome in one strand faces the gap between two magnetosomes of a nearby strand (Figure 2a,b). This arrangement favors the formation of strand bundles with a consistent magnetic polarity, eliminating repulsive forces [START_REF] Hanzlik | Pulsed-field-remanence measurements on individual magnetotactic bacteria[END_REF][START_REF] Ruder | Biological colloid engineering: Self-assembly of dipolar ferromagnetic chains in a functionalized biogenic ferrofluid[END_REF]. Contrastingly, magnetosomes in foldcollapsed chains are arranged side-to-side, and size tapering occurs only at one end, the other end being the kink point of the native chain (Figure 2d). The side-to-side magnetosome arrangement maximizes lateral attractive forces between strands with opposite magnetic polarities [START_REF] Varón | Dipolar magnetism in ordered and disordered lowdimensional nanoparticle assemblies[END_REF], as they result from fold-collapse. Fold-collapsed chains can be observed inside freeze-dried cells (e.g., Figure 2b in [START_REF] Lins | Magnetosome chain arrangement and stability in magnetotactic cocci[END_REF], where collapse is likely induced by cell shrinking.

The full variability of natural chain structures can be only partially reproduced because geometric parameters must be extrapolated from few observations. Therefore, we selected combinations of two magnetosome morphologies (cuboctahedral and prismatic), and three chain geometries (single-stranded, native double-stranded, and double-stranded from collapse by folding), as representative examples of magnetic endmembers dictated by the magnetosome aspect ratio (equant vs. elongated), and the existence of flux-closure configurations (single-vs. double-stranded). We generated 10 5 synthetic chains with random orientations and realistic geometries for each of the six categories obtained by combining the above-mentioned magnetosome morphologies and chain configurations (see Supporting Information for details).

Micromagnetic modeling has been performed in two steps using an energy minimization method. First, randomly oriented synthetic chains have been generated for each of the six configurations. Magnetosome-specific control parameters include size, shape, and crystal axes orientation, based on realistic distributions obtained from the literature (Figure S1). Chain-specific control parameters include magnetosome gaps, number of crystals, size tapering towards the extremities, and chain bending within the elastic limit. The geometry of double-stranded chains is additionally controlled by the lag of one strand with respect to the other, and by twisting of the two strands about a common axis (Figure 2a). The distributions of these parameters (Table S1 Like in [START_REF] Harrison | FORCulator: A micromagnetic tool for simulating first-order reversal curve diagrams[END_REF], we assumed that individual crystals are homogeneously magnetized, but we calculated magnetostatic interactions for the actual magnetosome shape, instead of using a point dipole approximation. The homogeneous magnetization approximation, which is needed to speed up calculations and cover the whole parameter space with a sufficiently large number of high-resolution FORC simulations, holds reasonably well in all applied fields, including those immediately preceding transitions between magnetic states, for magnetosome sizes up to ~50 nm, as seen by comparison with full micromagnetic models [START_REF] Berndt | Mind the gap: Towards a biogenic magnetite palaeoenvironmental proxy through an extensive finite-element micromagnetic simulation[END_REF]. At larger sizes, the validity of this approximation declines progressively, as helical configurations start to nucleate inside the end magnetosomes. Nevertheless, the coercivities of remanence calculated by [START_REF] Berndt | Mind the gap: Towards a biogenic magnetite palaeoenvironmental proxy through an extensive finite-element micromagnetic simulation[END_REF] for single-stranded chains of equidimensional and elongated magnetosomes are compatible with the coercivity distributions obtained from our simulations, so that the uncertainty introduced by our approximation is not expected to exceed the uncertainties associated with a poor knowledge of the real distribution of model parameters. Simulated FORC measurements (Figure S6) have been further processed with VARIFORC [START_REF] Egli | VARIFORC: An optimized protocol for calculating non-regular firstorder reversal curve (FORC) diagrams[END_REF], using minimal smoothing parameters ( over the central ridge, and , over the remaining parts of the FORC diagram) to eliminate statistical noise.

The role of chain geometry for the stabilization of the remanent magnetization of highmoment states, which is essential for obtaining elevated hysteresis squareness values, has been investigated by calculating the energy barrier that must be overcome for a thermally activated transition to a different state at room temperature. For this purpose, we considered four prototypical arrangements of identical, equidimensional magnetosomes with no intrinsic anisotropy: (1) single-stranded chains, (2) native double-stranded chains,

(3) fold-collapsed chains, and (4) circular rings. Optimal transition paths between LEM states have been calculated with the nudge-elastic-band and action minimization technique of [START_REF] Fabian | Energy barriers in three-dimensional micromagnetic models and the physics of thermoviscous magnetization[END_REF], upon identifying the magnetic moments of individual magnetosomes with the magnetic moments of elemental cells used in full micromagnetic simula-tions (see the Supporting Information for details).

The choice of an adequate initial path is crucial for obtaining a globally optimized transition between states. Single-stranded chains of >4 crystals are known to reverse their remanent magnetization by nucleation of a reversed domain at one end [START_REF] Hendriksen | Magnetization reversal in chain-like clusters of interacting particles[END_REF][START_REF] Newell | Transition to superparamagnetism in chains of magnetosome crystals[END_REF] and subsequent propagation of this domain to the other end. In this case, an adequate initial approximation of the reversing path is generated from an initial two-domain structure obtained by forcing the magnetic moment of the middle magnetosome to a direction perpendicular to the chain axis. This configuration is then relaxed while the strength of dipolar interactions in one of the two domains is slightly weakened to favor its denucleation (Figure S7a). The high-moment configuration of large rings consists of two domains separated by perpendicular magnetic moments pointing to the same direction (Figure S7b). The transition to a zero-moment FC state with clockwise or counterclockwise magnetic moment arrangement is simulated in the same manner as for linear chains by denucleating one of the two domains. Finally, the transition path of double-stranded chains from the high-moment state, where the two strands are magnetized in the same direction, to the low-moment state, where the two strands have opposed magnetizations, requires the nucleation of a reversed domain at the extremity of one strand (Figure S7c). The nucleated domain creates an energetically favored FC configuration with the other strand, so that the transition to a complete FC state is obtained by growing the reverse domain until it reaches the other chain extremity.

Deletion plasmid construction and generation of the ∆mamJ∆limJ∆MIS AMB-1 mutant strain

To obtain a term of comparison for possible collapsed chain configurations and validation of our micromagnetic modeling, we generated a mutant AMB-1 strain with looped magnetosome chains (Figure 3). Magnetosome formation in MTB requires genes contained in a specific portion of the genome called the magnetosome island (MAI) [START_REF] Murat | Comprehensive genetic dissection of the magnetosome gene island reveals the step-wise assembly of a prokaryotic organelle[END_REF]. The AMB-1 genome also contains a small version of the magnetosome island called the magnetotaxis islet (MIS) [START_REF] Rioux | A Second Actin-Like MamK Protein in Magnetospirillum magneticum AMB-1 Encoded Outside the Genomic Magnetosome Island[END_REF]. The MIS contains several genes, some of which perform redundant functions with their homologs in the MAI [START_REF] Abreu | Interplay between two bacterial actin homologs, MamK and MamK-like, is required for the alignment of magnetosome organelles in Magnetospirillum magneticum AMB-1[END_REF]. Previous work in Magnetospirillum gryphiswaldense strain MSR-1, a close relative of AMB-1, showed that the loss of the MAI gene mamJ results in collapse and aggregation of magnetosome chains [START_REF] Scheffel | An acidic protein aligns magnetosomes along a filamentous structure in magnetotactic bacteria[END_REF].

Surprisingly, the deletion of mamJ and its other MAI homolog limJ in AMB-1 does not produce collapsed chains and instead results in minor disruptions to the continuity of the chain [START_REF] Draper | MamK, a bacterial actin, forms dynamic filaments in vivo that are regulated by the acidic proteins MamJ and LimJ[END_REF]. We reasoned that redundant functions within the MIS might account for the dramatic differences between mamJ mutants in AMB-1 and MSR-1.

Thus, a mutant strain (∆mamJ∆limJ∆MIS) was produced by deleting the entire MIS and two MAI genes (mamJ and limJ) from AMB-1 genome. Detailed procedures for plasmid construction and AMB-1 transformation are provided in the Supporting Information.

Bacterial cultures

Magnetospirillum magneticum strain AMB-1 (ATCC700264) and the mutant AMB-1 strain described in Section 2.2 were cultivated following ATCC recommendations in 500-mL screw-caped bottles until end of the exponential phase. Bottles were filled with 300 mL of growth medium and placed in a glove box with controlled atmosphere (10% O 2 , 90% N 2 ) at 30°C after inoculation (1/100). The sole iron source in bacterial growth medium was Fe(III)-citrate added at 150 M.

Transmission electron microscopy

Wild type and mutant AMB-1 cells were deposited on copper grids coated with a Formvar and carbon films and characterized with a FEI Tecnai 12 transmission electron microscope operating at 120 kV.

Native magnetic moment measurements

Magnetic characterizations of the mutant AMB-1 strain suggest that its looped magnetosome chain configurations possess a low net magnetic moment (see section 3.3).

We thus investigated the native magnetic state of wild-type and mutant AMB-1 strains at the population level by comparing the magnetization of two different preparations of aqueous cell suspensions in a maximum external field of 2 mT, using a vibrating-sample magnetometer (VSM). The external field is sufficiently large to partially align the cells, but not large enough to alter their native magnetic configuration. In this case, the magnetization of the suspension is expected to be proportional to the mean strength of the magnetic moments of individual cells. The first preparation contains cells directly taken from the culture and is used to assess the native magnetic moments. In the second preparation, the suspension is previously exposed to a 200 mT field. This field is strong enough to reset the native magnetic states, replacing them with a saturation remanent state that corresponds to the maximum magnetic moment that can be maintained in a null field. Cell suspensions for these experiments were prepared by centrifugation of AMB-1 cultures (8000 rpm, 10 min), and subsequent suspension in 10 mL of phosphate buffer (PBS). 100 L of cell suspension were transferred in plastic vials (diameter of 4 mm) and placed in a Lakeshore Micro-Mag 3900 VSM for acquisition of remanent magnetizations at room temperature.

Off-axis electron holography

VSM characterizations of mutant AMB-1 demonstrated a vanishingly small native magnetic moment of the bacteria, which can be explained by the looped chain structures being in a FC state. To further confirm this hypothesis, we mapped the magnetic flux in the looped chains at the single-crystal level with off-axis electron holography. Electron holography is an interferometric method that correlates morphological and local magnetic characterizations of magnetic materials. It allows quantitative mapping of the in-plane flux inside the magnetite chains produced by AMB-1 at the nanometer scale. Both wildtype and mutant AMB-1 strains were cultivated in 10-mL glass tubes following the protocol described above. Cells were centrifuged (8000 rpm, 10 min) and suspended in 100 L of phosphate buffer (PBS). They were then deposited on copper grids coated with an ultra-thin carbon membrane. Off-axis electron holography was carried out using a Hitachi HF3300C microscope operated at 300 kV and equipped with a cold field emission gun and a spherical aberration corrector (CEOS B-Corr). Electron holography experiments were performed in a specific corrected Lorentz mode, allowing a spatial resolution down to 0.5 nm in a magnetic field-free sample environment [START_REF] Snoeck | Off-Axial Aberration Correction using a B-COR for Lorentz and HREM Modes[END_REF]. All holograms were recorded in a 2-biprism configuration to avoid artifacts linked to Fresnel fringes and to set separately the interference area size and the fringe spacing [START_REF] Harada | Doublebiprism electron interferometry[END_REF]. The fringe spacing is equal to 1 nm (7 pixels) allowing for a spatial resolution of 1.5 nm for the treated magnetic phase images. The exposure time was set to 1 nm using dynamic automation acquisition for removing instabilities and applying the fringe π-shift method [START_REF] Gatel | Unlimited acquisition time in electron holography by automated feedback control of transmission electron microscope[END_REF][START_REF] Volkov | Double-resolution electron holography with simple Fourier transform of fringe-shifted holograms[END_REF]. Phase and amplitude images were extracted from the holograms by using homemade software based on fast-Fourier transform approach. The magnetic and electrostatic contributions have been separated by acquiring two holograms for which the sample has been switched upside down (flipped 180°): the magnetic contribution was obtained by evaluating the difference of the phase images from the two holograms divided by two.

FORC measurements

The magnetic properties of the wild-type and mutant AMB-1 strains were characterized with high-resolution FORC analyses using a VSM. AMB-1 cultures were recovered by centrifugation (8 000 rpm, 10 min). Bacterial pellets were transferred in a 1.5-mL Eppendorf tube and dried at room temperature under anoxic conditions in a glove box ([O 2 ] < 1ppm) to prevent magnetite oxidation. Samples were stored under anoxic conditions until

being measured with a Lakeshore 8600 VSM. High-resolution FORC measurements were performed in steps of 0.3 mT with a stepwise approach to the reversal field to avoid overshooting artifacts [START_REF] Wagner | Diversification of iron-biomineralizing organisms during the Paleocene-Eocene Thermal Maximum: Evidence from quantitative unmixing of magnetic signatures of conventional and giant magnetofossils[END_REF], and a pause of 1 s at reversal. Measurements have been processed with VARIFORC (Egli, 2013) using a minimum smoothing factor across the central ridge, and , over the remaining parts of the FORC diagram.

Results

Chain geometry and magnetic states

The existence of one or more than one pair of magnetic states, and the stability of these states against thermal relaxation and applied fields, depends on the interplay between the magnetic anisotropy of individual crystals and the magnetostatic interactions between them. The stabilizing or destabilizing effect of magnetostatic interactions depends in turn on the chain geometry. Because magnetosome elongation is the main factor controlling the magnetic anisotropy of individual crystals, and because elongation, if present, is parallel to the chain axis and has a stabilizing effect, the minimum stability granted by the chain geometry is best investigated assuming equidimensional magnetosomes with no magnetocrystalline anisotropy and purely dipolar interactions, as in [START_REF] Hendriksen | Magnetization reversal in chain-like clusters of interacting particles[END_REF]. In this case, single-stranded chains possess only one pair of remanent states with all magnetic moments parallel or antiparallel to the chain axis (Figure 4a). For chains with >4 magnetosomes, thermally activated transitions between these two states occur by nucleating a domain with reversed magnetization at one end of the chain (Figure 4b). The two domains with opposed axial moments are separated by a transition region of ~3 crystals with magnetic moments that deviate from the chain axis.

Once the two domains are formed, the transition is completed by moving the transition region towards the other end of the chain, until the original domain is denucleated (Movie S1). The energy barrier that needs to be overcome for this transition is , where is the Boltzmann constant, the absolute temperature, and the so-called Boltzmann factor. At room temperature (T = 293 K), β 0 ≈ 836 for N =15 magnetite magnetosomes with a diameter of 50 nm separated by 5 nm gaps. The energy barrier depends only weakly on the number of magnetosomes if N > 6 (Table 2). For comparison, the stability threshold predicted by the Néel-Arrhenius law ,

where τ is the time constant of the spontaneous magnetization decay and τ 0 ≈ 0.1-1 ns [START_REF] Moskowitz | Determination of the preexponential frequency factor for superparamagnetic maghemite particles in magnetoferritin[END_REF], is β 0 ≈ 59 for geologic time scales (τ = 1 Ga), ~32 for the typical lifespan of a cell (τ ≈ 1 day), and ~25 for the typical time required to measure a single FORC (τ ≈ 1 min). This yields a lower stability threshold of ~17 nm for τ ≈ 1 min.

For comparison, [START_REF] Newell | Transition to superparamagnetism in chains of magnetosome crystals[END_REF] obtained a threshold of ~15 nm for the same chain geometry and N = 6, using a purely analytical approach and upon including the magnetocrystalline anisotropy of magnetosomes with 111 axes parallel to the chain axis. The latter adds stability to the magnetic moments, explaining the slightly smaller threshold.

The large energy barrier generated by axial magnetostatic interactions ensures that field-induced transitions between the two stable states of single-stranded chains occur in proximity of the theoretical switching fields predicted by micromagnetic models that neglect thermal activations, such as the one used here and those of [START_REF] Harrison | FORCulator: A micromagnetic tool for simulating first-order reversal curve diagrams[END_REF] and [START_REF] Berndt | Mind the gap: Towards a biogenic magnetite palaeoenvironmental proxy through an extensive finite-element micromagnetic simulation[END_REF]. Nevertheless, thermal activations are still sufficiently large to produce a measurable vertical offset of the central ridge [START_REF] Egli | VARIFORC: An optimized protocol for calculating non-regular firstorder reversal curve (FORC) diagrams[END_REF][START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF], due to the intrinsic time asymmetry of the FORC measurement protocol [START_REF] Berndt | Time-asymmetric FORC diagrams: A new protocol for visualizing thermal fluctuations and distinguishing magnetic mineral mixtures[END_REF].

The central ridge offset of high-resolution FORC measurements of magnetofossil-rich sediments, which is of the order of 0.3-0.5 mT [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF][START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF][START_REF] Wagner | Diversification of iron-biomineralizing organisms during the Paleocene-Eocene Thermal Maximum: Evidence from quantitative unmixing of magnetic signatures of conventional and giant magnetofossils[END_REF], can be explained by a Stoner-Wohlfarth model of thermally activated UNISD particles [START_REF] Lanci | Forward modeling of thermally activated single-domain magnetic particles applied to first-order reversal curves[END_REF][START_REF] Berndt | Time-asymmetric FORC diagrams: A new protocol for visualizing thermal fluctuations and distinguishing magnetic mineral mixtures[END_REF] with β 0 ≈ 400-600.

The high-moment remanent state of magnetosome rings, obtained by decreasing the applied field from saturation to zero, is similar to the "onion state" of toroidal magnetite nanoparticles described by [START_REF] Lewis | Magnetic vortex states in toroidal iron oxide nanoparticles: combining micromagnetics with tomography[END_REF], in which the magnetic flux enters the ring from one side, flows past the central hole, and exits on the other side (Figure 4c). In the limit case of large rings, the "onion state" is equivalent to two magnetic domains with clockwise and counterclockwise fluxes tangential to the circumference, separated by two transition zones similar to that of a single-stranded chain with two domains (Figure 4b). gaps. The FC state is much more stable, as seen from the energy required for the opposed transition (β 0 ≈ 1632). In practice, the saturation remanence of N = 20 rings will decay exponentially to zero (which is the net magnetization of the FC states) in ~9 minutes. The saturation remanence of smaller rings is better stabilized (e.g., β 0 ≈ 55 for N = 16), but still much less than single-stranded chains with the same number of magnetosomes.

The high-moment remanent state of double-stranded chains is similar to that of single stranded chains, with the two strands possessing nearly axial moments with same polarity [START_REF] Thomas | Electron holography for the study of magnetic nanomaterials[END_REF] (Figure 4e). The magnetic moments of the end magnetosomes are not completely aligned with the chain axis, due to the reduced axial magnetostatic coupling and the stray field concentration at the chain extremities. Transitions to the lowmoment remanent state, where the two strands with opposed polarities form a closed flux loop (Figure 4f), occur by nucleating a reverse domain in one of the two strands. An example can be seen in Figure 10 of [START_REF] Simpson | Magnetic induction mapping of magnetite chains in magnetotactic bacteria at room temperature and close to the Verwey transition using electron holography[END_REF], where two antiparallel domains in one strand are stabilized by a kink. The transition is then accomplished by extending the reverse domain until the original domain is fully denucleated (Movies S3-4). Native double-stranded chains with staggered magnetosomes (Figure 2a,b), and double-stranded chains with facing magnetosomes resulting from fold-collapse (Figure 2d), undergo the same type of transitions between high-and low-moment states;

however, the high-moment state of the native double-stranded chains is more stable than that of fold-collapsed chains (e.g., β 0 ≈ 751 vs. 242 for N = 2×11, Table 2), due to the staggered magnetosome arrangement [START_REF] Hanzlik | Pulsed-field-remanence measurements on individual magnetotactic bacteria[END_REF][START_REF] Ruder | Biological colloid engineering: Self-assembly of dipolar ferromagnetic chains in a functionalized biogenic ferrofluid[END_REF]. In both cases, high-moment states are stable over geologic times. The energy barriers that need to be overcome for the reversed transitions, from low-to high-moment states, are slightly larger than those of single-stranded chains.

Micromagnetic FORC models

Simulated FORC diagrams of single-stranded chains (Figure 5a,d) possess all the expected characteristics UNISD particles [START_REF] Newell | A high-precision model of first-order reversal curve (FORC) functions for single-domain ferromagnets with uniaxial anisotropy[END_REF], also seen in cultured MTB [START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF][START_REF] Jovane | Prismatic magnetite magnetosomes from cultivated Magnetovibrio blakemorei strain MV-1: a magnetic fingerprint in marine sediments?[END_REF][START_REF] Wang | Evidence for abundant isolated magnetic nanoparticles at the Paleocene-Eocene boundary[END_REF][START_REF] Wang | Characterizing and optimizing magnetosome production of Magnetospirillum sp. XM-1 isolated from Xi'an City Moat, China[END_REF]. These characteristics include

(1) a sharp central ridge along B u ≈ 0, which is produced by the collective switching of all magnetic moments within individual chains, (2) lack of contributions over the upper quadrant, and (3) a distribution of positive and negative amplitudes in the lower quadrant, nearly antisymmetric with respect to the B c = -B c diagonal, produced by the reversible rotation of the magnetic moments in the applied field [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF]. FORC measurements define three coercivity distributions [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF] associated with (1) irreversible magnetization changes along the lower branch of the hysteresis loop (f hys ), ( 2) the remanent magnetization of a so-called DC or backfield demagnetization curve (f dcd ), and (3) the central ridge (f cr ). In our simulations of single-stranded chains, all three distributions have identical, Gaussian-like shapes dictated by the in-field (f irr , f cr ) and zero-field (f dcd ) magnetic moment changes associated with transitions from negative to positive highmoment states. The coercivity distributions are slightly narrower and biased towards 30% higher fields with respect to the biogenic components BS and BH commonly found in magnetofossil-rich sediments [START_REF] Chen | Magnetic properties of uncultivated magnetotactic bacteria and their contribution to a stratified estuary iron cycle[END_REF]Egli, 2004a;Heslop et al., 2014a).

This difference can be explained in part by the fact that micromagnetic calculations were based on stoichiometric magnetite, while real magnetofossils are often partially or completely maghemitized [START_REF] Vali | Magnetotactic bacteria and their magnetofossils in sediments[END_REF][START_REF] Housen | Depth distribution of magnetofossils in nearsurface sediment from the Blake/Bahama Outer Ridge, western North Atlantic Ocean, determined by low-temperature magnetism[END_REF]. The maghemite endmember (γ-Fe 2 O 3 ) has a ~20% lower spontaneous magnetization, and a similar reduction in coercivity is therefore expected. Furthermore, the magnetosome gap in real magnetofossils might be larger than assumed in our calculations, producing a further reduction of the switching field [START_REF] Berndt | Mind the gap: Towards a biogenic magnetite palaeoenvironmental proxy through an extensive finite-element micromagnetic simulation[END_REF].

Simulated FORC diagrams of double-stranded chain configurations feature typical flux-closure signatures (Figure 5b,c,e,f) as described in [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF]. These features are fully developed only in fold-collapsed chains of equidimensional magnetosomes, resulting in a constricted hysteresis loop (Figure S6e) and clearly identifiable pairs of positive and negative lobes surrounding the central ridge (Figure 5c). These lobes are generated by transitions between high-and low-moment states with different in-field stability ranges (see [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF] featuring this type of FORC signature, such as single-vortex (SV) particles [START_REF] Dumas | Magnetic fingerprints of sub-100 nm Fe dots[END_REF][START_REF] Katzmann | Analysis of Magnetosome Chains in Magnetotactic Bacteria by Magnetic Measurements and Automated Image Analysis of Electron Micrographs[END_REF][START_REF] Roberts | Resolving the origin of pseudo-single domain magnetic behavior[END_REF], the hysteresis squareness of our modelled double-stranded chains is only slightly lowered with respect to the uniaxial SD case (M rs /M s = 0.471, Table 1, Figure 1), and fully compatible with magnetofossil signatures [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF]. Squareness values close to 0.5 for all simulated doublestranded and fold-collapsed chains means that they possess stable SD states in zero field, as observed on native chains [START_REF] Li | Crystal growth of bullet-shaped magnetite in magnetotactic bacteria of the Nitrospirae phylum[END_REF], and unlike the looped chains of the AMB-1 mutant. Stable high-moment states in zero field require the hysteresis loop to be fully reversible until the applied field reverses sign, which means that the FORC function is fully comprised within the so-called memory region (dashed lines in Figure 5).

The central ridge coercivity distributions of double-stranded chains are almost identical to those of single-stranded chains made of the same type of magnetosomes. This is expected from the similar stability of high-moment states in single-stranded chains and of FC states in double-stranded chains (Table 2). Non-central ridge coercivity distributions, on the other hand, contain additional low-field contributions related to the nucleation of FC states. In the case of equant magnetosomes, these contributions produce a second peak (fold-collapsed chains) or a shoulder (native chains) at B c ≈ 15 mT. In the case of prismatic magnetosomes, coercivity distributions remain unimodal but become wider, and the peak position is lowered by 10 mT with respect to the central ridge. Noncentral ridge coercivity distributions of native double-stranded chains are remarkably similar to the biogenic coercivity components BS and BH (Figure 5b,e).

Mutant AMB-1 strain with looped chains

Unlike the ∆mamJ MSR-1 mutant, which produces agglomerated clusters of magnetosomes, the phenotype of our AMB-1 mutant contains looped magnetosome chains with necklace structures located either at one of the poles or at the center of the cell (Figure 3).

Magnetosomes in the mutant strain fall mostly within the stable SD size range of singlestranded chains [START_REF] Muxworthy | Critical single-domain/multidomain grain sizes in noninteracting and interacting elongated magnetite particles: Implications for magnetosomes[END_REF][START_REF] Newell | Transition to superparamagnetism in chains of magnetosome crystals[END_REF]. The magnetic behavior of bacteria was assessed using the magnetic coefficient (C mag ), which relies on the differential measurement of a culture's optical density when a magnet is oriented either vertically or horizontally close to the cell suspension. C mag is defined as the ratio of the maximum and minimum optical density measured and quantifies the capacity of bacteria to orientate along an external magnetic field [START_REF] Schüler | A simple light scattering method to assay magnetism in Magnetospirillum gryphiswaldense[END_REF]. C mag values at the end of bacterial growth were 1.82 ± 0.09 and 1.09 ± 0.01 for wild-type and mutant AMB-1 (three replicates for each strain), respectively, demonstrating a very limited orientation capability for the mutant strain.

Native magnetic states of wild-type and mutant AMB-1 cells

MTB cells containing ideal magnetite chains possess already a saturation moment.

Exposing them to large fields will thus not change the magnetization of a cell suspension.

In practice, a 40% increase is observed for the wild-type AMB-1 after applying a 200 mT field (Figure 6a). This increase can be explained by the growth of aligned chain fragments with opposite native polarities within the same cell [START_REF] Le Nagard | Magnetite magnetosome biomineralization in Magnetospirillum magneticum strain AMB-1: A time course study[END_REF]. In this case, the application of a strong external field imparts the same polarity to all fragments, increasing the cell's total magnetic moment. A drastically different behavior is observed with the AMB-1 mutant, where the application of a 200 mT field produces an 18-fold magnetization increase (Figure 6d). Along with the magnetic behavior assays (C mag ), this result indicates a vanishingly small native magnetic moment of the AMB-1 mutant, which is compatible with a FC configuration of its looped chains. Overall, these results show that virtually all mutant cells contain looped magnetosome chains.

Electron holography

Mapping of the magnetic flux in magnetosome chains produced by wild-type AMB-1 (Figure 6b,c) indicates that the magnetic moments of individual magnetosomes are nearly parallel to the chain axis, as previously observed [START_REF] Dunin-Borkowski | Magnetic microstructure of magnetotactic bacteria by electron holography[END_REF]. In contrast, necklace structures of the mutant strain display a closed magnetic flux (Figure 6e,f).

This configuration has a zero-net magnetic moment, up to small fluctuations due to asymmetries (e.g., larger crystals on one side of the structure), confirming the origin of the vanishing native magnetic moments of the AMB-1 mutant deduced from magnetic measurements.

FORC measurements

The FORC diagrams of the wild-type and the mutant AMB-1 (Figure 7) share many similarities with the numerical simulations of single-and double-stranded chains of equidimensional crystals, but feature also important differences. The central ridge of wildtype cells extends to the origin and is accompanied by a vertical ridge along B c = 0, located mainly in the lower quadrant. These are the typical signatures of thermal relaxation in systems containing particles close to the lower SD stability limit [START_REF] Pike | First-order reversal curve diagrams and thermal relaxation effects in magnetic particles[END_REF][START_REF] Lanci | Forward modeling of thermally activated single-domain magnetic particles applied to first-order reversal curves[END_REF], and has been observed in the growing phase of MTB cultures [START_REF] Carvallo | Formation of magnetite in Magnetospirillum gryphiswaldense studied with FORC diagrams[END_REF]. In case of single-stranded chains of equidimensional crystals, the lower SD stability limit is comprised between 12 and 17 nm [START_REF] Newell | Transition to superparamagnetism in chains of magnetosome crystals[END_REF] and is compatible with the smallest crystal sizes observed with TEM (Figure 3). The hysteresis squareness (M rs /M s = 0.475) is slightly smaller than expected for singlestranded chains, probably because of a minor superparamagnetic contribution associated with immature chains, as confirmed by electron microscopy (Figure 3). The coercivity distributions obtained from FORC measurements of the wild-type cells (Figure 7c) are unimodal with a peak at 28-30 mT and the right tail extending to a maximum of 60-70 mT. Contrary to the micromagnetic simulations of single-stranded chains of equidimensional magnetosomes (Figure 5a) and to some examples of cultured MTB producing single-stranded chains (e.g., [START_REF] Wang | Characterizing and optimizing magnetosome production of Magnetospirillum sp. XM-1 isolated from Xi'an City Moat, China[END_REF], the coercivity distributions are left-skewed and include a small, non-zero contribution at B c = 0 caused by thermal relaxation.

The mutant AMB-1 strain shares the same FORC features (i.e., central ridge, viscous component and rotations of magnetic moments in the applied field), albeit over different field ranges, but also exhibits additional contributions (Figure 7b), consisting of a doublet of positive lobes at (B c , B u ) ≈ (12, ±8) mT, and a doublet of negative lobes at (B c , B u ) ≈ (25, ±3) mT, almost symmetrically arranged above and below the central ridge. These lobes are the typical hallmark of FC nucleation and annihilation [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF], as seen with VO particles [START_REF] Dumas | Magnetic fingerprints of sub-100 nm Fe dots[END_REF][START_REF] Roberts | Resolving the origin of pseudo-single domain magnetic behavior[END_REF] and small magnetosome clusters produced by the ∆mamJ mutant of MSR-1 [START_REF] Katzmann | Analysis of Magnetosome Chains in Magnetotactic Bacteria by Magnetic Measurements and Automated Image Analysis of Electron Micrographs[END_REF]. The hysteresis loop is constricted (Figure S8), and has a much smaller squareness (Table1, Figure 1) caused by the instability of high-moment magnetic states predicted by micromagnetic simulations of magnetosome rings (Section 3.1). This instability is associated with FORC contributions extending beyond the memory region of the FORC diagram (Figure 7b, left of the dashed lines). Constricted hysteresis loops are typically the result of bimodal coercivity distributions, which, in this case, consist of a main peak at 2 mT, and a shoulder at 25 mT (Figure 7d). The main peak of the central ridge, close to the origin, is caused by thermal relaxation of high-moment states, which, as predicted by numerical simulations (Section 3.1), are unstable at room temperature. FC nucleation from the negative high-moment state (feature 4 in Figure 7b) contributes additionally to the main peak of the other two coercivity distributions, explaining their larger amplitude with respect to the central ridge. A single mechanism, namely the annihilation of FC states around +25 mT, explains the existence of a shoulder with similar amplitude in all three coercivity distributions.

About 50% of the high-moment states in looped chains of the AMB-1 mutant possess a stable remanence, as indicated by the portion of the positive lobes above and below the central ridge located within the memory region (i.e., right of the dashed lines in Fig. 7b).

This is explainable by the stability of smaller loops with N ≤ 16 magnetosomes predicted by numerical simulations (Table 2).

Discussion

Distinctive characteristics of the magnetofossil signature

Magnetic assemblages showing coexisting FORC signatures of UNISD particles (central ridge) and non-SD or interacting SD particles (positive contributions above and below the central ridge) were previously interpreted as mixings of several magnetite sources including intact and collapsed magnetofossil assemblages, and non-SD particles of detrital origin (Heslop et al., 2014a;[START_REF] Lascu | Magnetic unmixing of first-order reversal curve diagrams using principal component analysis[END_REF][START_REF] Chang | Micromagnetic simulation of magnetofossils with realistic size and shape distributions: Linking magnetic proxies with nanoscale observations and implications for magnetofossil identification[END_REF]. However, mixings of the above-mentioned magnetic sources cannot provide a satisfactory explanation for the elevated hysteresis squareness typical of secondary magnetite in magnetofossil-rich sediments [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF]. Our work demonstrates that the magnetic signatures of these sediments can be explained by the sole presence of MTB producing single-and multi-stranded magnetosome chains, as well as specific alterations of these native configurations, such as fold-collapse, which maintain the original strong uniaxial anisotropy. Fold-collapsed chains form a uniform trend with the hysteresis properties of intact chains and UNISD particles (Figure 1), which is characterized by a very limited decrease of the hysteresis squareness ( ), and values comprised between ~1.2 and ~1.6, close to the limits for narrow and wide SD coercivity distributions, respectively [START_REF] Dunlop | Theory and application of the Day plot (M rs /M s versus H cr /H c ) 1, Theoretical curves and tests using titanomagnetite data[END_REF].

Other putative chain collapse products, such as disordered magnetosome clusters and looped chains, might possess compatible FORC signatures, but lack the elevated hysteresis squareness of systems with stable SD-like remanent states. The hysteresis properties of large clusters of extracted magnetosomes [START_REF] Kobayashi | Experimental observation of magnetosome chain collapse in magnetotactic bacteria: Sedimentological, paleomagnetic, and evolutionary implications[END_REF][START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF] follow a trend characterized by a steep decrease (Figure 1) typical of densely packed synthetic SD magnetite particles [START_REF] Muxworthy | Effect of magnetostatic interactions on the hysteresis parameters of single-domain and pseudo-single-domain grains[END_REF]. This decrease is associated with a vertical widening of the FORC function beyond the limits of the memory region [START_REF] Chen | First-order reversal curve (FORC) diagrams of natural and cultured biogenic magnetic particles[END_REF], due to large random internal fields associated with magnetostatic interactions [START_REF] Muxworthy | Magnetostatic interaction fields in first-orderreversal-curve diagrams[END_REF]. FORC diagrams of SD particle clusters lack a central ridge [START_REF] Carvallo | Experimental comparison of FORC and remanent Preisach diagrams[END_REF]; minor central ridge contributions in magnetosome extracts (e.g., [START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF] are therefore associated with incomplete chain collapse.

Micromagnetic simulations of magnetofossil collapse by chain axis randomization [START_REF] Chang | Micromagnetic simulation of magnetofossils with realistic size and shape distributions: Linking magnetic proxies with nanoscale observations and implications for magnetofossil identification[END_REF] produce a trend of hysteresis properties that is intermediate between the two cases described above (Figure 1). This trend, which includes the looped chains produced by our mutant AMB-1 as the endmember with lowest squareness, is associated with the intermediate dimensionality of chain loops (D ≈ 2) and of the partially collapsed chains generated by the random walk algorithm of [START_REF] Harrison | FORCulator: A micromagnetic tool for simulating first-order reversal curve diagrams[END_REF] (1 < D < 2), compared to intact or fold-collapsed chains (D ≈ 1) and dense clusters (D ≈ 3), respectively. FORC diagrams with non-central ridge contributions and are therefore important hallmarks of SD particle systems with a uniaxial anisotropy that is sufficiently strong to stabilize SD-like remanent states. The only known natural way for secondary SD magnetite particles to develop such a strong uniaxiality is by inheritance of native magnetosome chain structures, which must therefore survive diagenesis without becoming fully randomized. Fold-collapse is one of the possible fates of single-stranded chains that are not immediately stabilized by electrostatic adhesion to other sediment particles. Multi-stranded chains, on the other hand, are expected to be more resistant to mechanical solicitations, due to the staggered magnetosome arrangement, and might survive diagenesis with little structural modifications.

The bell shape of the central ridge coercivity distribution is the second distinctive element of magnetofossil signatures. It is generated by the denucleation of magnetic states with largest switching field: specifically, the high-moment, SD-like state of singlestranded chains, and the low-moment, FC state of multi-stranded chains. Our simulations show that the switching fields of these states tend to form two relatively narrow distributions with medians of ~40-50 mT in the case of equidimensional magnetosomes and ~70-80 mT in the case of prismatic magnetosomes, prior to maghemitization. These distributions are similar to the biogenic components BS and BH discovered in AF demagnetization curves of ARM (Egli, 2004a) and later identified also in the central ridge of magnetofossil-rich sediments [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF]Heslop et al., 2014a). The existence of a minimum uniaxial anisotropy provided by the chain structure ensures that the coercivity distributions of BS and BH are inferiorly limited by a non-zero switching field of ~10 mT (BS) and ~25 mT (BH). This limit is absent in isolated SD particles and in interacting clusters of such particles [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF]. An interesting counterexample is provided by a mutant MTB producing distant magnetosomes [START_REF] Wang | Novel protein Mg2046 regulates magnetosome synthesis in Magnetospirillum gryphiswaldense MSR-1 by modulating a proper redox status[END_REF]: in this case, the combined action of magnetic viscosity and lack of strong in-chain magnetostatic interactions shifts the central ridge distribution peak to .

ARM and the central ridge are expected to have different sensitivities towards singleand double-stranded chains. In case of single-stranded chains, ARM represents the roomtemperature thermodynamic equilibrium between two symmetric SD states in the decaying AF field, like in UNISD particles [START_REF] Egli | Anhysteretic remanent magnetization of fine magnetic particles[END_REF]. The case of systems featuring two symmetric SD-like states with denucleation field B n1 and two symmetric FC states with higher denucleation field B n2 , as in double-stranded and fold-collapsed chains, is more complex: once the AF field has decayed below H n2 during ARM acquisition, the system becomes trapped into one of the two FC states. In this case, the acquired ARM is a room-temperature thermodynamic equilibrium between two symmetric FC states. This ARM is reset to zero only in AF fields with initial amplitude B n2 . The similarity between B n2 and the switching field of single-stranded chains made of the same type of magnetosomes (Figure 5), yields comparable ARM and central ridge coercivity distributions. However, while the central ridge response is proportional to the mean amplitude of the magnetization jumps associated with FC denucleation, which amounts to ~40% of the saturation moment ( in Table 1), the ARM response is proportional to the remanent moment of FC states, which, in case of identical strands, is nearly zero. Therefore, the ARM of double stranded chains reflects the residual net magnetic moment of asymmetric strands, which is significantly smaller than the saturation moment. FC states are therefore expected to lower the ratio between the ARM susceptibility and the saturation remanence. A marked decrease of has been observed for the BS and BH components of lake sediments during a eutrophication event and was tentatively attributed to the presence of greigite magnetofossils (Egli, 2004a,b). However, greigite-producing MTB appear to have similar magnetic signatures as magnetite-producing MTB [START_REF] Chen | Magnetic properties of uncultivated magnetotactic bacteria and their contribution to a stratified estuary iron cycle[END_REF], so that multi-stranded chains are the most likely explanation for variations of magnetofossil components.

Coercivity distributions obtained from other types of magnetization curves, such as the commonly employed isothermal remanent magnetization (IRM) acquisition [START_REF] Kruiver | Quantification of magnetic coercivity componets by the analysis of acquisition curves of isothermal remanent magnetization[END_REF], contain contributions from all transitions between magnetic states, which, in the case of multi-stranded chains, include the denucleation of SD-like states in B n1 . This additional contribution is easily identifiable as a secondary low-coercivity peak in the case of fold-collapsed chains of equidimensional magnetosomes (Figure 5c), while it merges with the distribution of B n2 in the other case, lowering the median field. As a result, coercivity distributions obtained from IRM acquisition curves respond to a larger number of transitions between magnetic states with respect to the central ridge or ARM, yielding less constrained coercivity components that make the identification of BS and BH more difficult or even impossible. More importantly, these coercivity distributions respond also to unstable remanent magnetic states that are switched in arbitrarily small fields, yielding left-skewed curves with non-zero contributions at B c = 0.

Forward models of magnetofossil FORC signatures

The FORC signature of single-stranded chains is identical to that of UNISD particles:

both systems possess only a pair of antisymmetric SD-like states, and the in-field transition between these states produces the central ridge. The transition is preceded by a reversible rotation of magnetic moments, which generates additional FORC contributions in the lower quadrant. The coercivity distributions f hys and f cr associated with in-field magnetization jumps produced by switching between SD-like states are identical and proportional to the coercivity distribution derived from remanent magnetization curves (e.g., f dcd ). The small differences between f hys and f cr in Figure 5a,d are due to the numerical method used to isolate the central ridge from other FORC contributions. The difference in amplitude between remanent and in-field coercivity distributions is quantified by , where is the magnetization associated with the central ridge [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF]. Our micromagnetic simulations yield 0.704 and 0.711 for single-stranded chains of equidimensional and prismatic magnetosomes, respectively, compared to 0.544 for the Stoner-Wohlfarth model [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF], whereby all estimates do not take thermal activations into account. The higher values of single-stranded chains are explainable by the fact that reversible magnetic moment rotations is limited mostly to the end magnetosomes, while the magnetic moment of central magnetosomes tend to maintain their remanent state. Non-central ridge contributions in the lower quadrant of the FORC diagram are perfectly antisymmetric with respect to B u = -B c and do not contribute to the total magnetization M forc obtained by integrating the FORC function, so that M forc = M cr .

The FORC diagrams of double-stranded and fold-collapsed chains contain additional non-central ridge contributions originating from transitions between high-and lowmoment magnetic states. These contributions create nearly symmetric FORC amplitudes around the central ridge, which contribute to M rs and M forc , but not to M cr . The corresponding ratios 0.47-0.57 and ≈ 0.38-0.42 are therefore significantly smaller than those of single-stranded chains (Table 3). For comparison, the FORC signature of secondary magnetite particles isolated from a magnetofossil-rich sediment [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF] is characterized by values of and that are intermediate between those of simulated single-stranded and double-stranded or foldcollapsed chains. Mixtures of nearly equal parts of single-and double stranded chains (native and/or fold-collapsed) yield magnetization ratios that are compatible with those of magnetofossils (Table 3). Therefore, a composite FORC diagram has been generated using a mixture of equal amounts of all chain configurations simulated in this work, assuming a 2:1 proportion of equidimensional and prismatic magnetosomes (Figure 8b).

This proportion has been chosen to reproduce at best the natural sediment example (Figure 8a). The corresponding FORC diagrams share striking similarities, but also some negative amplitudes above the central ridge are also not completely cancelled. This is understandable considering that simulations were limited to few, well-defined geometries. For instance, single-and multi-stranded chains made of bullet-or toothshaped magnetosomes were not included. Multi-stranded chains of bullet-shaped magnetosomes are characterized by similarly elevated hysteresis squareness values as our double-stranded simulations, with a central ridge peaking at 60 mT [START_REF] Li | Biomineralization, crystallography and magnetic properties of bullet-shaped magnetite magnetosomes in giant rod magnetotactic bacteria[END_REF].

This intermediate coercivity range fills the gaps between simulated chains of equidimensional and prismatic magnetosomes, making the resulting FORC diagram more similar to that of natural magnetofossils.

Magnetofossil fingerprints and numerical unmixing

Our numerical simulations show that magnetofossils possess a unique combination of magnetic properties related to the preservation of native or collapsed chain structures with a strong uniaxial anisotropy. This uniaxial anisotropy is at the basis of all magnetic criteria proposed so far for magnetofossil identification, which include (1) the central ridge [START_REF] Egli | Detection of noninteracting single domain particles using first-order reversal curve diagrams[END_REF]Heslop et al., 2014a), ( 2) relatively narrow coercivity components not extending to B c = 0 (Egli, 2004a), and (3) ferromagnetic resonance (FMR) spectra with positive anisotropy [START_REF] Kopp | Ferromagnetic resonance spectroscopy for assessment of magnetic anisotropy and magnetostatic interactions: A case study of mutant magnetotactic bacteria[END_REF][START_REF] Charilaou | Simulation of ferromagnetic resonance spectra of linear chains of magnetite nanocrystals[END_REF][START_REF] Kind | Combined use of magnetometry and spectroscopy for identifying magnetofossils in sediments[END_REF]). An additional criterion discussed in this work is the confinement of FORC contributions within the memory region, which ensures that ≈ 0.5 also in case of multi-stranded and fold-collapsed chains. Each of these criteria, if taken singularly, is fulfilled by an equivalent assemblage of isolated SD particles with appropriated equivalent anisotropy, but all known natural examples other than magnetofossils do not fulfill all of them. For instance, titanomagnetite needles in the Tiva Canyon Tuff do possess a central ridge [START_REF] Berndt | Time-asymmetric FORC diagrams: A new protocol for visualizing thermal fluctuations and distinguishing magnetic mineral mixtures[END_REF], but the associated coercivity distribution and vertical offset are different from those of magnetofossils. In another example [START_REF] Till | Presumed magnetic biosignatures observed in magnetite derived from abiotic reductive alteration of nanogoethite[END_REF], magnetite derived from the inorganic breakdown of nanocrystalline goethite produces FORC diagrams similar to those of magnetofossil-rich sediments, but with a much lower hysteresis squareness. Finally, SD magnetite can form along dislocation lines of silicate minerals, forming "strings of pearls" as documented in ancient zircons (Tang et al., 2018); however, the large irregular particle spacing resulting from this growth process is not expected to yield the narrow coercivity distribution of single-stranded chains.

Because natural sediments contain mixtures of magnetite particles with different primary and secondary origins (e.g., [START_REF] Franke | Magnetic petrology of equatorial Atlantic sediments: Electron microscopy results and their implications for environmental magnetic interpretation[END_REF][START_REF] Just | Signatures and significance of aeolian, fluvial, bacterial and diagenetic magnetic mineral fractions in Late Quaternary marine sediments off Gambia, NW Africa[END_REF][START_REF] Li | Classification of a complexly mixed magnetic mineral assemblage in Pacific Ocean surface sediment by electron microscopy and supervised magnetic unmixing[END_REF], magnetofossil signatures need to be isolated using numerical unmixing methods. Numerical unmixing methods can be divided into parametric ones, which are based on model properties, such as coercivity distribution functions [START_REF] Kruiver | Quantification of magnetic coercivity componets by the analysis of acquisition curves of isothermal remanent magnetization[END_REF][START_REF] Egli | Analysis of the field dependence of remanent magnetization curves[END_REF], and non-parametric ones, which are fundamentally linked to principal component analysis (PCA), such as non-negative unmixing of remanent magnetization curves [START_REF] Heslop | Unmixing magnetic remanence curves without a priori knowledge[END_REF] and FORC-PCA [START_REF] Lascu | Magnetic unmixing of first-order reversal curve diagrams using principal component analysis[END_REF][START_REF] Harrison | An improved algorithm for unmixing first-order reversal curve diagrams using principal component analysis[END_REF]. Parametric coercivity analyses are very sensitive to noise and to subtle variations of the shape (skewness) of individual coercivity distributions [START_REF] Egli | Analysis of the field dependence of remanent magnetization curves[END_REF][START_REF] Zhao | Applying the Burr Type XII distribution to decompose remanent magnetization curves[END_REF].

Furthermore, individual coercivity components might be associated with the type of irreversible magnetization changes induced by a particular measurement protocol, rather than distinct types of magnetic particles. This is particularly evident in the case of foldcollapsed chains (Figure 5c), where f cr is described by a single, unimodal distribution, while the other coercivity distributions are bimodal and require two coercivity components for a correct fit. In this case, the analysis of magnetization curves that are selective to specific magnetization processes, such as the central ridge and the AF demagnetization of ARM, helps with the identification of magnetofossil-specific coercivity components.

The correct individuation of endmembers in PCA-based unmixing methods is based on the assumption that individual magnetic components possess fixed magnetic properties within the set of analyzed samples. In this case, a single magnetofossil endmember is expected from PCA analyses. More than one endmember is needed to describe magnetofossil signatures that change in time, reflecting, for instance, variable proportions of equidimensional and elongated crystals [START_REF] Hesse | Evidence for bacterial palaeoecological origin of mineral magnetic cycles in oxic and sub-oxic Tasman Sea sediments[END_REF][START_REF] Yamazaki | Organic carbon flux controls the morphology of magnetofossils in marine sediments[END_REF][START_REF] Wagner | Diversification of iron-biomineralizing organisms during the Paleocene-Eocene Thermal Maximum: Evidence from quantitative unmixing of magnetic signatures of conventional and giant magnetofossils[END_REF], or single-and multi-stranded chains.

Our micromagnetic simulations suggest that magnetosome elongation would control mainly the coercivity of magnetofossil endmembers (e.g., the proportion of BS and BH coercivity components), while higher proportions of multi-stranded chains would increase non-central ridge contributions to the FORC diagram. At least two magnetofossil endmembers are required to describe these variations. Splitting between these endmembers is controlled by the PCA covariance matrix, rather than by physical parameters. For instance, variable proportions of single-and multi-stranded chains could be described by the linear combination of a pure central ridge and a pure non-central ridge endmember, instead of the mixed FORC signatures of single-and multi-stranded chains. This might explain FORC-PCA results from the North Atlantic [START_REF] Channell | Magnetic record of deglaciation using FORC-PCA, sortable-silt grain size, and magnetic excursion at 26 ka, from the Rockall Trough (NE Atlantic)[END_REF], where the three identified endmembers include one with the typical FORC signature of detrital magnetite (EM 3 in Fig. 11 of that study), one with a typical UNISD signature consisting of a central ridge, and almost no contributions in the upper quadrant (EM2, ibid), and one that is compatible with non-central ridge contributions from SV magnetite particles or multi-stranded magnetosome chains (EM3, ibid). In this example, the interpretation of the latter endmember is ambiguous; especially when considering that PCA can group the signatures of magnetic components with completely different origins -such as multi-stranded magnetofossils and a fine detrital component SV signature -into a single endmember, if these components co-vary in time. Co-varying concentrations of primary and secondary magnetite particles can be expected if the abundance of magnetofossils is controlled by the flux of nutrients, which, in certain locations, is in turn controlled by primary mineral inputs (e.g., [START_REF] Roberts | Magnetotactic bacterial abundance in pelagic marine environments is limited by organic carbon flux and availability of dissolved iron[END_REF]. Endmembers with characteristics similar to those of the above example are commonly found in marine sediments [START_REF] Lascu | Magnetic unmixing of first-order reversal curve diagrams using principal component analysis[END_REF][START_REF] Yamazaki | Unmixing biogenic and terrigenous magnetic mineral components in red clay of the Pacific Ocean using principal component analyses of first-order reversal curve diagrams and paleoenvironmental implications[END_REF]. FORC-PCA results are also sensitive to the number of principal components and the choice of the endmembers: for example, the FORC signature of a fourth endmember featuring two lobes around the central ridge similar to those of our micromagnetic simulations of double-stranded chains is distributed among the other three endmembers in the case of a three-endmember analysis of the same samples (Fig. 9-10 in [START_REF] Lascu | Magnetic unmixing of first-order reversal curve diagrams using principal component analysis[END_REF].

The ambiguity of numerical unmixing results can be partially overcome by combining numerical methods with particle morphology statistics from TEM observations [START_REF] Wagner | Diversification of iron-biomineralizing organisms during the Paleocene-Eocene Thermal Maximum: Evidence from quantitative unmixing of magnetic signatures of conventional and giant magnetofossils[END_REF] and with physical or chemical separation techniques, such as the selective dissolution of fine magnetite particles [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF].

Authigenic magnetite in magnetofossil signatures?

Fe(III)-reducing and Fe(II)-oxidizing bacteria are known to induce the precipitation of magnetite particles that are expected to provide an authigenic contribution to the magnetic signature of sediments affected by iron diagenesis. Ultrafine magnetite particles with broad volume distribution and irregular shapes have been often observed (e.g., [START_REF] Gibbs-Eggar | Possible evidence for dissimilatory bacterial magnetite dominating the magnetic properties of recent lake sediments[END_REF][START_REF] Franke | Magnetic petrology of equatorial Atlantic sediments: Electron microscopy results and their implications for environmental magnetic interpretation[END_REF]Odlfield, 2007;[START_REF] Li | Classification of a complexly mixed magnetic mineral assemblage in Pacific Ocean surface sediment by electron microscopy and supervised magnetic unmixing[END_REF], but their magnetic signature is uncertain, owing to the uncontrolled grain size distribution and the unknown arrangement in the sediment matrix. Magnetite samples produced by these bacteria in the laboratory (Table 1) are characterized by a small squareness [START_REF] Moskowitz | Rock Magnetic Criteria for the Detection of Biogenic Magnetite[END_REF][START_REF] Carvallo | Biogenic vs. abiogenic magnetite nanoparticles: a XMCD study[END_REF] and the classical FORC signature of strong magnetostatic interactions [START_REF] Miot | Formation of single domain magnetite by green rust oxidation promoted by microbial anaerobic nitrate-dependent iron oxidation[END_REF]. Because of size distribution overlapping with the stable SD range, authigenic magnetite is expected to have a similar preservation potential in geological materials [START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF] and a magnetic signature that might be confused with that of magnetofossils. Irregularly shaped SD magnetite particles lacking a source of minimum uniaxial anisotropy, for instance by particle elongation or chain arrangement, are expected to produce a broad coercivity distribution with zerocoercivity contributions and median fields of 10-25 mT [START_REF] Moskowitz | Rock Magnetic Criteria for the Detection of Biogenic Magnetite[END_REF].

Coercivity components with similar characteristics have been found in freshwater and marine sediments, as well as in soils (EX and P in Egli, 2004a). These components are associated with moderately large values typical of non-interacting SD particles (Egli, 2004a), and, in the case of soils, with a central ridge [START_REF] Geiss | Direct estimates of pedogenic magnetite as a tool to reconstruct past climates from buried soils[END_REF][START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF].

Therefore, authigenic magnetite consists, at least in part, of isolated SD particles dispersed in the sediment matrix. The broad grain size distribution of biologically uncontrolled magnetite precipitation (e.g., [START_REF] Liu | Quantifying grain size distribution of pedogenic magnetic particles in Chinese loess and its significance for pedogenesis[END_REF], which includes SV particles, is expected to yield also FORC signatures typical of non-SD particles, as indeed observed in FORC diagrams of well-developed soils [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF]. Isolated SD magnetite particles with random deviations from equidimensional shapes are also expected to produce lowcoercivity FORC contributions above and below the central ridge (e.g., Fig 51 in [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF], due to the competition between magnetocrystalline and shape anisotropy. Finally, elevated local iron concentrations might lead to the formation of particle clusters with strong magnetostatic interactions that contribute to the vertical dispersion of the FORC function at low coercivities. Indeed, magnetite encrustation of Fe(III)-reducing or Fe(II)oxidizing bacteria has been observed in culture (e.g., [START_REF] Miot | Iron biomineralization by anaerobic neutrophilic iron-oxidizing bacteria[END_REF]. For instance, FORC signatures of non-interacting and interacting SD particles not produced by MTB have been identified in organic-rich methanogenic sediments [START_REF] Ustra | Ultrafine magnetic particles: A DIET-proxy in organic rich sediments?[END_REF]. Both non-SD and clustered SD magnetite contributions are expected to lower the hysteresis squareness, and might therefore be distinguishable from pure magnetofossil contributions.

Overall, the SD fraction of well-dispersed authigenic magnetite particles can produce magnetic signatures that might merge continuously with those of magnetofossils over the low-coercivity range. For instance, the low-coercivity FORC-PCA endmembers EM1 and EM2 in red clays from the Pacific Ocean [START_REF] Yamazaki | Unmixing biogenic and terrigenous magnetic mineral components in red clay of the Pacific Ocean using principal component analyses of first-order reversal curve diagrams and paleoenvironmental implications[END_REF] provide altogether a signature that is similar to that of the pedogenic enhancement in soils. A low-coercivity component peaking at B c = 0 is also particularly evident in the central ridge of magnetofossil-bearing ferromanganese crusts formed mainly through biomineralization [START_REF] Jiang | Characterization and quantification of magnetofossils within abyssal manganese nodules from the western Pacific Ocean and implications for nodule formation[END_REF], while being absent in other crusts [START_REF] Oda | Characterization of marine ferromanganese crust from the Pacific using residues of selective chemical leaching: identification of fossil magnetotactic bacteria with FE-SEM and rock magnetic methods[END_REF]. The interpretation of the low-coercivity range (<20 mT) of the central ridge remains uncertain. Short chain fragments of 1-3 equant magnetosomes possess significantly lower coercivities than fully developed chains [START_REF] Chang | Micromagnetic simulation of magnetofossils with realistic size and shape distributions: Linking magnetic proxies with nanoscale observations and implications for magnetofossil identification[END_REF] and can therefore contribute to the central ridge over this range. It is not known whether such fragments are produced by MTB under natural, slow-growing conditions, or if longer chains are naturally fragmented through adhesion of segments of the same chain to different sediment particles. Overall, the existence of low-coercivity contributions in the central ridge of most sediments means that a biogenic and an inorganic origin of SD magnetite particles, as postulated by different authors for the Paleocene-Eocene Thermal Maximum, are not mutually exclusive [START_REF] Kent | A case for a comet impact trigger for the Paleocene/Eocene thermal maximum and carbon isotope excursion[END_REF][START_REF] Lippert | A biogenic origin for anomalous fine-grained magnetic material at the Paleocene-Eocene boundary at Wilson Lake, New Jersey[END_REF].

Paleontological and (paleo)environmental implications

Discrimination of single-and multi-stranded chains based on the FORC signatures discussed above are of paleontological interest and could provide a temporal resolution on the evolution of magnetosome formation in MTB. Only bacteria belonging to the ancestral Delta-proteobacteria and Nitrospirae phyla were thought to produce multistranded chains of magnetosomes [START_REF] Deng | In vitro assembly of the bacterial actin protein MamK from 'Candidatus Magnetobacterium casensis' in the phylum Nitrospirae[END_REF]. However, they have been now identified in the more recent Alpha-proteobacteria and Gamma-proteobacteria phyla [START_REF] Taoka | Characterization of uncultured giant rod-shaped magnetotactic Gammaproteobacteria from a freshwater pond in Kanazawa, Japan[END_REF][START_REF] Zhang | Magnetotactic Coccus Strain SHHC-1 Affiliated to Alphaproteobacteria Forms Octahedral Magnetite Magnetosomes[END_REF]. Therefore, multi-stranded chain configuration has no specific phylogenetic distribution and might instead correspond to an adaptation to specific environments. For instance, single-stranded chain-producing cocci and the multistranded M. bavaricum display different responses to a sudden change of the oxygen gradient in sediment [START_REF] Mao | Magneto-chemotaxis in sediment: first insights[END_REF]. In MTB, magnetosomes are aligned along cytoskeletal filaments made of actin-like proteins [START_REF] Müller | A compass to boost navigation: Cell biology of bacterial magnetotaxis[END_REF]. Among them, the protein MamK is ubiquitously conserved in all known MTB. Bacteria forming multiple magnetosome chains contain several copies of genes encoding actin-like proteins in addition to MamK [START_REF] Kolinko | Single-cell genomics of uncultivated deep-branching magnetotactic bacteria reveals a conserved set of magnetosome genes[END_REF]. We can thus hypothesize that genetic factors are responsible for multi-stranded chain configurations [START_REF] Kolinko | Single-cell genomics of uncultivated deep-branching magnetotactic bacteria reveals a conserved set of magnetosome genes[END_REF]. They may include gene duplication events [START_REF] Murat | Comprehensive genetic dissection of the magnetosome gene island reveals the step-wise assembly of a prokaryotic organelle[END_REF], or horizontal gene transfers (i.e., an exchange of genetic material through direct physical interactions between two organisms) [START_REF] Monteil | Repeated horizontal gene transfers triggered parallel evolution of magnetotaxis in two evolutionary divergent lineages of magnetotactic bacteria[END_REF]. Once the genetic basis of chain configurations [START_REF] Du | Magnetosome gene duplication as an important river in the evolution of magnetotaxis in the Alphaproteobacteria[END_REF] is fully understood, the methodology we describe in this work can be used to provide a temporal constrain on evolutionary events that occurred in one of the oldest and more diverse group of biomineralizing organisms.

Paleomagnetic implications

Owing to the widespread occurrence of living MTB in the topmost ~10-20 cm of the sedimentary column [START_REF] Petermann | Detection of live magnetotactic bacteria in South Atlantic deep-sea sediments[END_REF][START_REF] Flies | Diversity and vertical distribution of magnetotactic bacteria along chemical gradients in freshwater microcosms[END_REF][START_REF] Mao | Magneto-chemotaxis in sediment: first insights[END_REF] and cell dissolution in the same layer, magnetofossils are expected to contribute to sedimentary records of the Earth magnetic field like other magnetic particles. However, specific differences between the magnetic and mechanical properties of magnetosome chains and other carriers of a natural remanent magnetization (NRM) might affect relative paleointensity records (RPI) of magnetofossil-rich sediments [START_REF] Larrasoaña | Paleomagnetic and paleoenvironmental implications of magnetofossil occurrences in late Miocene marine sediments from the Guadalquivir Basin, SW Spain[END_REF][START_REF] Ouyang | Variable remanence acquisition efficiency in sediments containing biogenic and detrital magnetites: Implications for relative paleointensity signal recording[END_REF][START_REF] Chen | Remanence acquisition efficiency in biogenic and detrital magnetite and recording of geomagnetic paleointensity[END_REF]. Although the exact mechanism of NRM acquisition is unknown, it is possible to make some predictions on the influence of chain geometry on the RPI efficiency of magnetofossils on the basis of our findings.

Magnetofossils in non-varved sediments are expected to form close to the depth range inhabited by living MTB [START_REF] Petermann | Detection of live magnetotactic bacteria in South Atlantic deep-sea sediments[END_REF], inside the surface mixed layer (SML). In this case, magnetofossils would undergo the same continuous reorientation process of other remanence carriers, until they are definitively buried below the SML.

The resulting NRM has been described as the result of a dynamic equilibrium between

(1) the aligning effect of the Earth magnetic field on the net natural magnetic moment of isolated or grouped magnetic particles behaving as rigid mechanical units in the bioturbated sediment (e.g., clusters or inclusions in silicate minerals), and ( 2) the rotational diffusion associated with sediment mixing (Egli and Zhao, 2015;[START_REF] Zhao | Microbially assisted recording of the Earth's magnetic field in sediment[END_REF]. In this case, the resulting remanent magnetization is proportional to the total magnetization of the remanence carriers in the ideal case of full alignment of their natural magnetic moments, and to a sigmoidal function , with and , which describes the partial alignment of remanence carriers in the Earth magnetic field (Egli and Zhao, 2015). In small fields, this partial alignment is proportional to the ratio between the magnetic torque of individual remanence carriers with net magnetic moment , and the typical holding torque that resists particle reorientation. Writing , where is the concentration of remanence carriers, gives for the low-field limit of NRM acquisition, with being a coefficient that describes the mechanical alignment efficiency of the remanence carriers in sediments with specific mechanical properties.

RPI records are obtained by normalizing with a laboratory magnetization assumed to be proportional to . In this case, , with being the mean magnetic moment component along the applied magnetic field, acquired by the remanence carriers during the acquisition of a laboratory magnetization. The normalized NRM is finally given by . As expected for valid RPI normalizations, does not depend on the concentration of remanence carriers. Furthermore, a fixed magnetic mineralogy yields a constant , so that is proportional to .

Each magnetic component contributing to the NRM is expected to possess a specific value related to its magnetic properties, and a specific that reflects its mechanical properties. Accordingly, sediments containing variable proportions of two or more magnetic components (e.g., detrital and biogenic), yield RPI records that contain an unwanted environmental modulation caused by variations of within the limit values of the individual components [START_REF] Fabian | Records of paleomagnetic field variations[END_REF]. The role of can be investigated by inspecting selected categories of remanence carriers. For instance, the natural magnetic moment of silicate minerals containing SD magnetic inclusions is proportional to the sum of randomly oriented vectors, which yields , where is the mean magnetic moment of individual SD particles [START_REF] Heslop | A statistical simulation of magnetic particle alignment in sediments[END_REF]

Conclusions

Rock and sediment magnetism methodologies have been extensively used for the search and identification of magnetofossils. The central ridge observed during FORC analyses was established as a distinctive feature of the strong uniaxial anisotropy of isolated magnetosome chains. Our findings show that intact multi-stranded and foldcollapsed chains can generate additional signatures in FORC diagrams through the nucleation and annihilation of FC magnetic states while maintaining high hysteresis squareness (M rs /M s ). FC magnetic states in multi-stranded chains, which can be described by individual strands possessing opposed magnetic polarities, are similar to FC states observed in small magnetosome clusters and looped magnetosome chains forming after cell dissolution in aqueous solutions, which might serve as term of comparison for diagenetic chain collapse in sediment. However, these artificial magnetosome structures are characterized by a significantly lowered hysteresis squareness that is not observed in magnetofossil-rich sediments. Micromagnetic calculations show that native multistranded chains and fold-collapsed chains feature stable SD states with large magnetic moments and the same hysteresis squareness of intact single-stranded chains. This stability, which has been confirmed experimentally in the case of native chains [START_REF] Hanzlik | Pulsed-field-remanence measurements on individual magnetotactic bacteria[END_REF][START_REF] Thomas | Electron holography for the study of magnetic nanomaterials[END_REF], originates from the strong uniaxial anisotropy of such structures and explains the coexistence of (1) magnetic signatures typical for systems of isolated SD particles and single-stranded magnetosome chains (central ridge and

), and ( 2) signatures typical of SD particle systems with random magnetostatic interactions (non-central-ridge contributions in the FORC diagram).

Full chain collapse leading to a complete loss of the original structure is likely prevented by the electrostatic adhesion of magnetosomes to sediment particles.

Therefore, the strong uniaxial anisotropy of intact and fold-collapsed magnetosome chains appears to be a key feature for the preservation of a unique magnetic signature of magnetofossils, which, besides the elevated hysteresis squareness and a central ridge, includes two coercivity components (BS and BH in Egli, 2004a) with inferiorly limited, Gaussian-like switching field distributions distinct from those of isolated or clustered SD magnetite particles. Our micromagnetic models suggest that the low-coercivity portion of the central ridge of natural sediments (Figure 8) likely represents the contribution of isolated SD particles of authigenic origin. ARM is another key parameter of magnetofossil fingerprints: its coercivity distribution is similar to that of the central ridge, but, unlike to the central ridge, it is more selective towards single-stranded chains.

Preservation of a distinctive magnetofossil fingerprint, as demonstrated in our work, is pivotal for the interpretation of the origin of secondary SD magnetite particles in sediments, since this fingerprint is the only practical mean for verifying, in a quantitative manner, the existence of isolated chain-like structures of SD magnetite particle as the most essential magnetofossil identification criterion. The detailed characteristics of magnetofossil fingerprints depend on the relative abundances of equidimensional and elongated magnetosomes, which affect the coercivity distribution, and on the relative abundances single-and multi-stranded chains, which affect the fraction of magnetic remanence carried by the central ridge. Magnetosome shape and chain structure reflect environmental conditions controlling the growth of different MTB strains. Therefore, detailed magnetofossil records obtained with high-resolution FORC measurements, combined with statistically significant TEM observations, will provide novel methodologies to discriminate distinct populations of ancient MTB and improve our understanding of paleomagnetic and environmental records of magnetofossil-rich sediments.

Tables Table 1. Summary of measured and modelled magnetic properties of MTB, magnetofossils, and other SD particle assemblages. is the ratio between the central ridge magnetization and the saturation remanent magnetization, is the so-called ARM ratio, and the ratio between the total magnetization of the central ridge and the saturation magnetization. (3) a composite of all chain configurations with 2:1 proportion of equant vs. prismatic magnetosomes (Figure 8b), and (4) the CBD-extractable fraction of a magnetofossil-rich pelagic carbonate from the Equatorial Pacific [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF]. [START_REF] Dunlop | Theory and application of the Day plot (M rs /M s versus H cr /H c ) 1, Theoretical curves and tests using titanomagnetite data[END_REF] showing the hysteresis properties and of intact cultured cells [START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF][START_REF] Moskowitz | Rock Magnetic Criteria for the Detection of Biogenic Magnetite[END_REF][START_REF] Pan | Rock magnetic properties of uncultured magnetotactic bacteria[END_REF], SD particles in a magnetofossil-rich sediment [START_REF] Ludwig | Characterization of primary and secondary magnetite in marine sediment by combining chemical and magnetic unmixing techniques[END_REF], the AMB-1 mutant producing looped magnetosome arrangements (this work), and a ∆mamJ mutant producing magnetosome clusters [START_REF] Katzmann | Analysis of Magnetosome Chains in Magnetotactic Bacteria by Magnetic Measurements and Automated Image Analysis of Electron Micrographs[END_REF]. The properties of extracted magnetosomes [START_REF] Li | Magnetic anisotropy, magnetostatic interactions and identification of magnetofossils[END_REF], micromagnetically simulated chains with increasing degree of axis randomization [START_REF] Chang | Micromagnetic simulation of magnetofossils with realistic size and shape distributions: Linking magnetic proxies with nanoscale observations and implications for magnetofossil identification[END_REF], and our micromagnetic simulations of examples of rings obtained from magnetophoresis of magnetosome chains extracted from cultures of M. magnetotacticum [START_REF] Philipse | Magnetic colloids from magnetotactic bacteria: Chain formation and colloidal stability[END_REF].

SD system

  Deviations from the tangential magnetization in the transition zones point to the in-plane component of the field used to impart a saturation remanence. Unlike the case of singlestranded chains, where most of the energy barrier is associated with the nucleation of a new domain, transitions to a zero-moment FC state (Figure 4d) occur with little additional energy by moving one of the two domain boundaries until it merges with the other boundary, leaving a single domain with circular magnetization (Movie S2). The energy barrier of this transition is β 0 ≈ 27 for a ring made of 20 magnetosomes with 5 nm

  Figure57therein for a detailed explanation). The central ridge, on the other hand, is associated exclusively with the denucleation of FC configurations. Magneto-some elongation and staggering in native double-stranded chains add stability to the high-moment state, as seen by the partial suppression of the negative lobes around the central ridge (Figure5b,e,f). The only natural example of FORC signature produced by multi-stranded chains is that of a concentrate of wild-type M. bavaricum cells extracted from a pond sediment[START_REF] Roberts | Searching for single domain magnetite in the "pseudo-single-domain" sedimentary haystack: Implications of biogenic magnetite preservation for sediment magnetism and relative paleointensity determinations[END_REF]. This sample features positive contributions in the upper quadrant of the FORC diagram which might be attributed to the denucleation of FC states.Contrary to the AMB-1 mutant (see below) and other examples of magnetic systems

  important differences. Both feature a central ridge, but the coercivity range of simulations is comprised between 20 and 130 mT, while the natural central ridge covers also the 0-20 mT range (see Section 4.4). Other signatures of our composite FORC diagram are qualitatively similar to those of magnetofossil-bearing sediments, including the slight asymmetry between the upper and the lower quadrant, and the abrupt B c -termination of FC contributions in the upper quadrant. The contour lines of FORC simulation are clearly affected by localized contributions of FC nucleation and annihilation of individual chain endmembers, which are not completely merged into a smooth distribution as in the natural counterpart. Small

Figure captions Fig. 1 .

 captions1 Figure captions

  intact and fold-collapsed chains are shown for comparison. Departures from for ideal, non-interacting, uniaxial SD particles (cultured cells), follow trends with increasing slopes, from intact and fold-collapsed configurations with strong uniaxial anisotropy to complete randomization (extracted magnetosomes), through the intermediate trend of looped and randomized chains.

Fig. 2 .

 2 Fig. 2. Representative examples of native and collapsed double-stranded magnetosome

Fig. 3 .Fig. 4 .

 34 Fig. 3. Electron microscopy images of wild-type and mutant AMB-1 cells.

  magnetofossils are expected to possess larger RPI efficiencies relative to silicate minerals containing SD magnetite inclusions with the same saturation moment, as reported by[START_REF] Ouyang | Variable remanence acquisition efficiency in sediments containing biogenic and detrital magnetites: Implications for relative paleointensity signal recording[END_REF] and[START_REF] Chen | Remanence acquisition efficiency in biogenic and detrital magnetite and recording of geomagnetic paleointensity[END_REF], because the magnetic moments of individual crystals add arithmetically to the native net moment, instead of randomly.

				). For comparison, intact chains of magnetosomes are
	characterized by	, while fold-collapsed chains have a much smaller native
	moment	, where	is the mean absolute difference between the number of
	magnetosomes in the two strands (e.g.,	for a chain that is folded exactly in the
	middle). A similar reasoning can be applied to the estimation of magnetic moments
	produced by the ARM and IRM magnetizations commonly used as RPI normalizers
	(Table 4).
		Magnetofossil RPI values obtained with the above magnetic moment estimates display
	different degrees of sensitivity to the chain structure. If IRM is used as normalizer,
	similar RPI estimates are expected for intact single-and multistranded chains, since they
	carry a saturated magnetic moment in their native state and after application of a
	saturating IRM. Fold-collapsed chains, on the other hand, have a small net native
	moment, making them inefficient NRM carriers. If ARM is used as a normalizer, intact
	double stranded chains are expected to yield larger RPI values than intact single-stranded
	chains, because the ARM sets multistranded chains in a FC state with low magnetic
	moment (See Section 4.1). In case of comparable mechanical alignment efficiencies
	(	), intact

Table 2 .

 2 Boltzmann factors β 0 associated with the denucleation of high-moment states (HMS) and low-moment states (LMS) for various chain geometries with of N equidimensional magnetosomes of 50 nm diameter, separated by a diameter-normalized gap g.

	Structure	Reference			M cr /M rs
				(mm/A)
	Natural sediment
	Lake Baldeggersee	Egli (2004), Wagner et al.,	0.41	2-4	0.37
		(2021)
	Lake Ely	Egli et al. (2010)	0.47	-	0.48
	Pelagic carbonate (CBD) 1 Ludwig et al. (2013)	0.47	2-3	0.65
	Paleosol S1 (CBD) 1	Egli (2004, 2021)	0.20	1.7	0.32
	Single-stranded chains
	Cultured MTB	Li et al. (2012),	0.50	3.1-3.7	0.65
		Moskowitz et al. (1993)
	AMB-1	This study, Li et al. (2012) 0.48-0.50 1.38	0.79
	Simulated	This study	0.49-0.50 -	0.71-0.72
	Simulated	Chang et al. (2019)	0.32-0.50 -	-
	Multistranded chains
	Wild-type MTB	Pan et al. (2005)	0.47-0.51 0.5-0.8	-
	Simulated (2 strands)	This study	0.47-0.49 -	0.40-0.42
	Simulated (fold-collapsed) This study	0.47-0.50 -	0.37-0.38
	Clumps and loops
	AMB-1 extract	Li et al. (2012)	0.22	0.23	-
	MV-1 extract	Wang et al. (2013)	0.31	-	0.092
	ΔmamJ mutant	Katzmann et al. (2013)	0.23	-	0.36
	ΔmamJ ΔlimJ mutant	This study	0.31	-	0.53
	Extracellular magnetite
	Geobacter metallireducens Moskowitz et al. (1993)	0.03	0.25	-
	Shewanella putrefaciens	Carvallo et al. (2008)	0.14	-	-
	Acidovorax on green rust	Miot et al. (2014)	-	-	0

Table 3 .

 3 Central ridge magnetization M cr and total FORC magnetization M forc from micromagnetic simulations of (1) single-and double-stranded chains, (2) a composite of intact chain configurations with a 2:1 proportion of equant vs. prismatic magnetosomes,

Table 4 .

 4 RPI parameters for different types of SD remanence carriers made of n SD crystals with mean magnetic moment m SD . Other symbols: m 0 -native magnetic moment, m IRM -magnetic moment after IRM acquisition, m ARM -magnetic moment after ARM acquisition, r a -ARM/IRM of isolated SD crystals.

	M cr /M s	M forc /M s	M cr /M rs	M cr /M forc

Citrate-Bicarbonate-Dithionite (CBD) extractable component calculated from the difference between pre-and post-CBD measurements.

Acknowledgements

. AK and MA are supported by grants through the National Science Foundation (1504681) and National Institutes of Health (R35GM127114). MA is supported by grant through the Fondation pour la Recherche Médicale (ARF201909009123). The research leading to these results has received funding from the European Union Horizon 2020 research and innovation program under grant agreement No. 823717 -ESTEEM3. The authors acknowledge the French National Research Agency under the "Investissement d'Avenir" program reference No. ANR-10-EQPX-38-01 and the "Conseil Régional Midi-Pyrénées" and the European FEDER for financial support within the CPER program. This work was also supported by the international associated laboratory M 2 OZART. All data generated in this work are available on https://figshare.com/collections/Magnetic_Signatures_of_Magnetofossils/5579547.

 5,10,20,30,40,50,60,70,80, and 90% quantile contours. The FORC region enclosed by the q-quantile contour contributes to a fraction 1 -q of the total FORC integral [START_REF] Egli | Magnetic characterization of geologic materials with first-order reversal curves[END_REF]. Dashed lines indicate the left boundary of the so-called memory region.