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

Matthieu Amor, Juan Wan, Ramon Egli, Julie Carlut, Christophe Gatel, Ingrid Marie Andersen, Etienne Snoeck, Arash Komeili

► To cite this version:

Matthieu Amor, Juan Wan, Ramon Egli, Julie Carlut, Christophe Gatel, et al.. Key signatures of magnetofossils elucidated by mutant magnetotactic bacteria and micromagnetic calculations. Journal of Geophysical Research: Solid Earth, 2022, 127, pp.e2021JB023239. 10.1029/2021JB023239. hal-03943167

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Submitted on 17 Jan 2023

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

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Matthieu Amor¹,²*, Juan Wan¹*, Ramon Egli³,⁴*, Julie Carlut⁴, Christophe Gatel,⁵,⁶ Ingrid Marie Andersen⁵, Etienne Snoeck⁵, Arash Komeili¹,⁷

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¹ Department of Plant and Microbial Biology, University of California, Berkeley, CA 94720-3102

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² Aix-Marseille University, CNRS, CEA, UMR7265 Bioscience and Biotechnology Institute of Aix-Marseille, CEA Cadarache, Saint-Paul-lez-Durance, F-13108, France

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³ Zentralanstalt für Meteorologie und Geodynamik (ZAMG), Hohe Warte 38, A-1190 Vienna, Austria

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⁴ Université de Paris, Institut de Physique du Globe de Paris, CNRS, 75005 Paris, France

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⁵ CEMES CNRS, 29 rue Jeanne Marvig, F-31055 Toulouse, France

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⁶ Université de Toulouse, F-31077 Toulouse, France

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⁷ Department of Molecular and Cell Biology, University of California, Berkeley, CA 94720-3200

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Corresponding author: ramon.egli@zamg.ac.at (RE)
17 **Key points:**

18 • Sediments containing fossilized magnetotactic bacteria bear magnetic signatures
19 apparently associated with different sources of magnetite particles

20 • Intact and fold-collapsed single-stranded and multi-stranded magnetosome chains can
21 explain all observed signatures of fossilized magnetotactic bacteria

22 • We provide criteria for the discrimination of distinct populations of ancient
23 magnetotactic bacteria
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**Abstract**

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Magnetotactic bacteria (MTB) produce single- or multi-stranded chains of magnetic nanoparticles that contribute to the magnetization of sediments and rocks. Their magnetic fingerprint can be detected in ancient geological samples and serve as a unique biosignature of microbial life. However, some fossilized assemblages bear contradictory signatures pointing to magnetic components that have distinct origin(s). Here, using micromagnetic simulations and mutant MTB producing looped magnetosome chains, we demonstrate that the magnetofossil biosignature is produced by a combination of single- and multi-stranded chains, and that diagenetically-induced chain collapse, if occurring, must preserve the strong uniaxial anisotropy of native chains. This anisotropy is the key factor for distinguishing magnetofossils from other populations of natural magnetite particles, including those with similar individual crystal characteristics. Furthermore, the detailed properties of magnetofossil signatures depends on the proportion of equant and elongated magnetosomes, as well as on the relative abundances of single- and multi-stranded chains. This work has important paleoclimatic, paleontological and phylogenetic implications, as it provides a novel tool to differentiate distinct MTB lineages according to their chain and magnetosome morphologies, which will enable the tracking of the evolution of some of the most ancient biomineralizing organisms in a time-resolved manner. It also enables a more accurate discrimination of different sources of magnetite particles, which is pivotal for gaining better environmental and relative paleointensity reconstructions from sedimentary records.
## 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.
## 1. Introduction

Magnetite ($Fe_3O_4$) 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., Evans and Heller, 2003; Tauxe et al., 2010; Egli et al., 2013). In addition to lithogenic magnetite-bearing minerals (e.g., Chang et al., 2016), sediments contain secondary magnetite crystals formed in situ through diagenetic processes (e.g., Oldfield, 2007; Franke et al., 2007), but are also produced by bacteria participating in the sedimentary iron cycle (e.g., Lovely et al., 1987; Fortin and Langley, 2005; Miot et al., 2009).

Magnetotactic bacteria (MTB) are the only known microorganisms that produce intra-cellular 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 (Lin et al., 2017). They are markers of oxic-anoxic transition zones in aquatic and sedimentary environments (Flies et al., 2005; Simmons et al., 2004), making them useful paleoenvironmental indicators (Hesse, 1994; Roberts et al., 2011; Usui et al., 2017; Chang et al., 2018; Wagner et al., 2021). Magnetosomes are assembled in chains, and provide the bacteria with a permanent magnetic dipole for navigation purposes (McCausland and Komeili, 2020). When MTB cells die, their magnetite chains can be trapped into sediments and subsequently fossilized (Petersen et al., 1986; Kopp & Kirschvink, 2008). MTB fossils (hereafter referred to as magnetofossils) can be preserved over geological times and contribute to sedimentary paleomagnetic and environmental records (Roberts et al., 2012; Larrasoaña
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et al., 2014; Ouyang et al., 2014). 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.

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Arrangement of single domain (SD) magnetite crystals in linear chains combines two
robust magnetofossil identification criteria, size and arrangement (Kopp and Kirschvink,
2008), 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., Gibbs-Eggar et al., 1999;
Faivre et al., 2004; Liu et al., 2005). Most importantly, the special conditions required to
form isolated, non-branched strings of abiogenic SD particles (e.g., Ruder et al., 2012)
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., Roberts et al., 2019). An additional complication arises from the
fact that the lack of chain structures (e.g., Weiss et al., 2004) 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.

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First-order reversal curve (FORC) diagrams (Egli, 2021), which rely on the measure-
ment of partial hysteresis curves, can be used to discriminate isolated SD magnetite parti-
cles and magnetosome chains from other structures, such as SD particle clusters and

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larger lithogenic (titano-)magnetite crystals containing magnetic vortices (VO) or multiple magnetic domains (MD) (Roberts et al., 2017; Lascu et al., 2018), through a sharp signature called the central ridge (Egli et al., 2010; Roberts et al., 2012; Heslop et al., 2014a). Principal component analyses (PCA) of FORC diagrams obtained from magnetofossil-rich sediments (Lascu et al., 2015) and selective FORC measurements of secondary ≤300 nm magnetite particles through selective chemical leaching (Ludwig et al., 2013) 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 $M_{rs}/M_s$ and $B_{cr}/B_c$ (Dunlop, 2002) are close to the theoretical limit for non-interacting SD particles and intact magnetosome chains (Table 1). In particular, hysteresis squareness (the ratio $M_{rs}/M_s$ 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 (Ludwig et al., 2013) (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; Egli et al., 2010; 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 (Harrison and Lascu, 2014; Chang et al., 2019). Isolated magnetosome chains collapse if bent beyond their elastic limit, due to the magnetostatic attraction between the opposed magnetic poles at their extremities (Shcherbakov et al., 1997; Kiani et al., 2015). 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 (Galindo-Gonzalez et al., 2009). 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 (Thomas-Kaprta et al., 2000; Golden et al., 2004; Weiss et al., 2004), and the origin of the sudden rise of SD magnetite concentration during the Paleocene-Eocene thermal maximum (Kent et al., 2003; Kopp et al., 2007; Wang et al., 2013; Wagner et al., 2021).
154 Extracted magnetosomes obtained from MTB cultures after cell lysis tend to form
155 large clumps with an excessively low hysteresis squareness (Kobayashi et al., 2006; Li et
156 al., 2012) (Table 1, Figure 1). This is also true for the hysteresis squareness of smaller
157 magnetosome clusters produced by MTB whose genes responsible for chain assembly
158 have been deleted, namely the Δ*mamJ* MSR-1 strain in Katzmann et al. (2013) (see
159 sections below), which might serve as analogue for the full collapse of isolated chains.
160 Finally, the random-walk-based magnetosome assemblage algorithm of Harrison and
161 Lascu (2014) produce a range of magnetic properties (Chang et al., 2019) comprised
162 between the endmembers corresponding to linear chains and fractal-like particle clusters,
163 respectively, depending on the assumed degree of randomness (Figure 1). In these
164 simulations, FORC diagrams containing non-UNISD contributions comparable with
165 those of magnetofossil endmembers are associated with hysteresis squareness values of
166 ~0.4 or lower. Pedogenic magnetite, the only well-characterized natural example of non-
167 biogenic secondary magnetite particles (Dearing et al., 2001), also features mixed UNISD
168 and non-UNISD FORC properties (Table 1); however, in this case, the hysteresis
169 squareness of ~0.2 is compatible with the FORC signature. Thus, current interpretations
170 of magnetofossil signatures appear to miss structures possessing the correct combination
171 of UNISD and non-UNISD magnetic properties.

172 Excluding disordered clusters, which do not possess the correct magnetic signature,
173 and the already investigated chain axis randomization, possible candidates for missing
174 magnetofossil morphologies include native multistranded chains of magnetosomes
175 produced by some MTB (Amor et al., 2020) and mechanically meaningful minimum-
176 energy structures obtained from the collapse of isolated chains, such as rings and double
177 strands resulting from the fold-collapse of single-stranded chains that have been bent
178 beyond the elastic limit (von Dobeneck et al., 1987; Philipse and Maas 2002; Kiani et al.,
179 2018). Single- and multi-stranded chains in MTB and in the derived intact magnetofossils
180 (Figure 2a,b) are in a native SD state that maximizes the total magnetic moment, with the
181 magnetization of all magnetosomes being parallel to the chain axis (Hanzlik et al., 2002).
182 This is the only possible zero-field magnetic configuration for single-stranded chains, but
183 not for multiple strands, in which intermediate states with lower magnetic moments can
184 be nucleated with the application of strong external fields (Hanzlik et al., 2002).
185 Intermediate remanent states are formed when the magnetic moment of one strand is
186 reversed with respect to the others, or when the magnetic moments of individual crystals
187 form complex patterns (Varón et al., 2013). On the other hand, structures arising from
188 chain collapse (Figure 2c,d), are expected to possess native flux-closure (FC)
189 configurations characterized by a small net magnetic moment, as observed for instance in
190 nanoparticle rings (Dunin-Borkowski et al., 2004; Takeno et al., 2014). These
191 configurations minimize the magnetostatic energy by suppressing the external stray field
192 of SD-like magnetic states. The low magnetic moment of collapsed structures makes
193 them a poor sedimentary recorder of the Earth magnetic field, contrary to intact
194 magnetosome chains, so that the fate of fossil magnetosome chains during diagenesis is
195 expected to have important implications for the paleomagnetism of magnetofossil-rich
196 sediments.

197 FORC measurements are sensitive to transitions between high-moment (SD-like) and
198 low-moment (FC-like) magnetic configurations and their in-field stability ranges (Egli,
199 2021). In particular, hysteresis squareness values close to 0.5, typical of randomly
oriented UNISD particles, depend on the room-temperature zero-field stability of high-moment states. This stability, which enables MTB to navigate along the Earth magnetic field, has been demonstrated experimentally for native single- and multistranded chains (Hanzlik et al., 2002; Pan et al., 2005), 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 Kiani et al. (2018) and to some structures obtained after dissolution of cultured cells of the same strain (Philipse and Maas, 2002). Looped chains might form in sediment after cell disruption, but before stabilizing structures such as
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magnetosome membranes and the connecting cytoskeletal filament (Komeili et al., 2006) are dissolved, owing to the larger elastic response of membrane-filled gaps (Shcherbakov et al., 1997). 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 (Shcherbakov et al., 1997).

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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 multi-stranded 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.

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# 2. Material and methods

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## 2.1. Micromagnetic modeling

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

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245 Micromagnetic models of high-resolution FORC measurements have been calculated for
246 selected magnetosome morphologies and chain configurations representative of intact or
247 elastically deformed chains, and of fold-collapsed chains. The magnetic properties of the
248 simulated structures are determined mainly by two factors: the anisotropy of individual
249 magnetosomes, which is controlled by their shape and crystal orientation, and in-chain
250 magnetostatic interactions, which are controlled mainly by the chain geometry (Berndt et
251 al., 2020). Magnetosome shapes can be classified as equant (cuboctahedral, octahedral),
252 elongated (prismatic, elongated octahedral), and highly elongated (tooth- and bullet-sha-
253 ped) (Amor et al., 2020). These categories form well-defined clusters of crystal sizes and
254 shapes within the SD stability range of magnetite (Muxworthy and Williams, 2006;
255 Newell, 2009) (Figure S1).
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256 The geometry of intact single-stranded chains is controlled by the gaps between
257 magnetosomes, the size decrease of immature crystals towards the chain extremities
258 (tapering), and chain bending within the elastic limit. Magnetosomes in native multi-
259 stranded chains are staggered: each magnetosome in one strand faces the gap between
260 two magnetosomes of a nearby strand (Figure 2a,b). This arrangement favors the
261 formation of strand bundles with a consistent magnetic polarity, eliminating repulsive
262 forces (Hanzlik et al., 2002; Ruder et al., 2012). Contrastingly, magnetosomes in fold-
263 collapsed chains are arranged side-to-side, and size tapering occurs only at one end, the
264 other end being the kink point of the native chain (Figure 2d). The side-to-side
265 magnetosome arrangement maximizes lateral attractive forces between strands with
266 opposite magnetic polarities (Varón et al., 2013), as they result from fold-collapse. Fold-
267 collapsed chains can be observed inside freeze-dried cells (e.g., Figure 2b in Lins and 268 Farina, 2004), where collapse is likely induced by cell shrinking.

269 The full variability of natural chain structures can be only partially reproduced 270 because geometric parameters must be extrapolated from few observations. Therefore, we 271 selected combinations of two magnetosome morphologies (cuboctahedral and prismatic), 272 and three chain geometries (single-stranded, native double-stranded, and double-stranded 273 from collapse by folding), as representative examples of magnetic endmembers dictated 274 by the magnetosome aspect ratio (equant vs. elongated), and the existence of flux-closure 275 configurations (single- vs. double-stranded). We generated ~10⁵ synthetic chains with 276 random orientations and realistic geometries for each of the six categories obtained by 277 combining the above-mentioned magnetosome morphologies and chain configurations 278 (see Supporting Information for details).

279 Micromagnetic modeling has been performed in two steps using an energy 280 minimization method. First, randomly oriented synthetic chains have been generated for 281 each of the six configurations. Magnetosome-specific control parameters include size, 282 shape, and crystal axes orientation, based on realistic distributions obtained from the 283 literature (Figure S1). Chain-specific control parameters include magnetosome gaps, 284 number of crystals, size tapering towards the extremities, and chain bending within the 285 elastic limit. The geometry of double-stranded chains is additionally controlled by the lag 286 of one strand with respect to the other, and by twisting of the two strands about a 287 common axis (Figure 2a). The distributions of these parameters (Table S1) have been 288 empirically chosen to match images reported in the literature. Representative examples of 289 synthetic double-stranded chains are shown in Figures S2-S5. Next, FORC measurements
between ±0.3 T have been simulated in 1 mT steps using a micromagnetic model of the chain magnetization. These calculations are based on the local minimization of the total energy determined by the magnetocrystalline and shape anisotropy of individual crystals, the Zeeman energy in the applied field, and the energy of magnetostatic interactions between pairs of crystals (see the Supporting Information for details). Because FORC measurements consists of a series of field sweeps starting at positive saturation, where there is only one local energy minimum (LEM), the evolution of LEM states is uniquely determined.

Like in Harrison and Lascu (2014), 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 (Berndt et al., 2020). 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 Berndt et al. (2020) 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 (Egli,
313 2013), using minimal smoothing parameters ($s = 1$ over the central ridge, and $s = 4$, 314 $\lambda = 0.08$ over the remaining parts of the FORC diagram) to eliminate statistical noise.

315 The role of chain geometry for the stabilization of the remanent magnetization of high- 316 moment states, which is essential for obtaining elevated hysteresis squareness values, has 317 been investigated by calculating the energy barrier that must be overcome for a thermally 318 activated transition to a different state at room temperature. For this purpose, we 319 considered four prototypical arrangements of identical, equidimensional magnetosomes 320 with no intrinsic anisotropy: (1) single-stranded chains, (2) native double-stranded chains, 321 (3) fold-collapsed chains, and (4) circular rings. Optimal transition paths between LEM 322 states have been calculated with the nudge-elastic-band and action minimization 323 technique of Fabian and Shcherbakov (2018), upon identifying the magnetic moments of 324 individual magnetosomes with the magnetic moments of elemental cells used in full 325 micromagnetic simula-tions (see the Supporting Information for details).

326 The choice of an adequate initial path is crucial for obtaining a globally optimized 327 transition between states. Single-stranded chains of >4 crystals are known to reverse their 328 remanent magnetization by nucleation of a reversed domain at one end (Hendriksen et al., 329 1994; Newell, 2009) and subsequent propagation of this domain to the other end. In this 330 case, an adequate initial approximation of the reversing path is generated from an initial 331 two-domain structure obtained by forcing the magnetic moment of the middle magneto- 332 some to a direction perpendicular to the chain axis. This configuration is then relaxed 333 while the strength of dipolar interactions in one of the two domains is slightly weakened 334 to favor its denucleation (Figure S7a). The high-moment configuration of large rings 335 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.

## 2.2. Deletion plasmid construction and generation of the Δ*mamJΔlimJΔMIS* 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) (Murat et al., 2010). The AMB-1 genome also contains a small version of the magnetosome island called the magnetotaxis islet (MIS) (Rioux et al., 2010). The MIS contains several genes, some of which perform redundant functions with their homologs in the MAI (Abreu et al., 2014). 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 (Scheffel et al., 2006). 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
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chain (Draper et al., 2011). We reasoned that redundant functions within the MIS might
account for the dramatic differences between *mamJ* mutants in AMB-1 and MSR-1.

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Thus, a mutant strain (Δ*mamJΔlimJΔMIS*) was produced by deleting the entire MIS and

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three MAI genes (*mamJ* and *limJ*) from AMB-1 genome. Detailed procedures for plasmid

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construction and AMB-1 transformation are provided in the Supporting Information.

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**2.3. Bacterial cultures**

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*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% O₂, 90% N₂) at 30°C after inoculation (1/100). The sole iron source in bacterial growth medium was Fe(III)-citrate added at 150 µM.

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**2.4. Transmission electron microscopy**

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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.

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**2.5. Native magnetic moment measurements**

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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

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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 µL 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.

## 2.6. 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 wild-type 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
404 100 µL of phosphate buffer (PBS). They were then deposited on copper grids coated with
405 an ultra-thin carbon membrane. Off-axis electron holography was carried out using a
406 Hitachi HF3300C microscope operated at 300 kV and equipped with a cold field
407 emission gun and a spherical aberration corrector (CEOS B-Corr). Electron holography
408 experiments were performed in a specific corrected Lorentz mode, allowing a spatial
409 resolution down to 0.5 nm in a magnetic field-free sample environment (Snoeck et al.,
410 2014). 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
411 (Harada et al., 2004). The fringe spacing is equal to 1 nm (7 pixels) allowing for a spatial
412 resolution of 1.5 nm for the treated magnetic phase images. The exposure time was set to
413 1 nm using dynamic automation acquisition for removing instabilities and applying the
414 fringe π-shift method (Gatel et al., 2018; Volkov et al., 2013). Phase and amplitude
415 images were extracted from the holograms by using homemade software based on fast-
416 Fourier transform approach. The magnetic and electrostatic contributions have been
417 separated by acquiring two holograms for which the sample has been switched upside
418 down (flipped 180°): the magnetic contribution was obtained by evaluating the difference
419 of the phase images from the two holograms divided by two.
420

2.7. FORC measurements

421 The magnetic properties of the wild-type and mutant AMB-1 strains were characterized
422 with high-resolution FORC analyses using a VSM. AMB-1 cultures were recovered by
423 centrifugation (8 000 rpm, 10 min). Bacterial pellets were transferred in a 1.5-mL Eppen-
424 dorf tube and dried at room temperature under anoxic conditions in a glove box ([O₂] <
425 1ppm) to prevent magnetite oxidation. Samples were stored under anoxic conditions until
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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 over-
shooting artifacts (Wagner et al., 2021), and a pause of 1 s at reversal. Measurements
have been processed with VARIFORC (Egli, 2013) using a minimum smoothing factor
s = 2 across the central ridge, and s = 4, λ = 0.14 over the remaining parts of the FORC
diagram.

3. Results

3.1 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 Hendriksen et al. (1994). 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 $ΔE = β_0k_BT$, where $k_B$ is the Boltzmann constant, $T$ the absolute temperature, and $β_0$ 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 $τ = τ_0\exp(β_0)$, where $τ$ is the time constant of the spontaneous magnetization decay and $τ_0 ≈ 0.1-1$ ns (Moskowitz et al., 1997), 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, Newell (2009) 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 Harrison and Lascu (2014) and Berndt et al. (2020). Nevertheless, thermal activations are still sufficiently large to produce a measurable vertical offset of the central ridge (Egli, 2013, 2021), due to the intrinsic time asymmetry of the FORC measurement protocol (Berndt et al., 2018).
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The central ridge offset of high-resolution FORC measurements of magnetofossil-rich sediments, which is of the order of 0.3-0.5 mT (Egli et al., 2010; Ludwig et al., 2013; Wagner et al., 2021), can be explained by a Stoner-Wohlfarth model of thermally activated UNISD particles (Lanci and Kent 2018; Berndt et al., 2018) with $β_0 \approx 400-600$.

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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 Lewis et al. (2020), 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). 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 single-stranded 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 $β_0 \approx 27$ for a ring made of 20 magnetosomes with 5 nm gaps. The FC state is much more stable, as seen from the energy required for the opposed transition ($β_0 \approx 1632$). 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., $β_0 \approx 55$ for $N = 16$), but still much less than single-stranded chains with the same number of magnetosomes.

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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 (Thomas et al., 2008) (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 low-moment 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 Simpson et al. (2005), 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 (Hanzlik et al., 2002; Ruder et al., 2012). 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.

## 3.2 Micromagnetic FORC models

Simulated FORC diagrams of single-stranded chains (Figure 5a,d) possess all the expected characteristics UNISD particles (Newell, 2005), also seen in cultured MTB (Li et al., 2012; Jovane et al., 2012; Wang et al., 2013, 2015). These characteristics include
(1) a sharp central ridge along $B_u \approx 0$, 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 (Egli, 2021). FORC measurements define three coercivity distributions (Egli, 2021) 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 high-moment 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 (Chen et al., 2014; 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 (Vali et al., 1987; Housen and Moskowitz, 2006). The maghemite endmember ($\gamma-Fe_2O_3$) 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 (Berndt et al., 2020).

Simulated FORC diagrams of double-stranded chain configurations feature typical flux-closure signatures (Figure 5b,c,e,f) as described in Egli (2021). 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 Egli, 2021, and Figure 57 therein 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 (Figure 5b,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 (Roberts et al., 2012). 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
featuring this type of FORC signature, such as single-vortex (SV) particles (Dumas et al.,
2007; Katzmann et al., 2013; Roberts et al., 2017), the hysteresis squareness of our mo-
delled 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 (Ludwig et al., 2013). Squareness values close to 0.5 for all simulated double-
stranded and fold-collapsed chains means that they possess stable SD states in zero field,
as observed on native chains (Li et al., 2015), and unlike the looped chains of the AMB-1
mutant. Stable high-moment states in zero field require the hysteresis loop to be fully
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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).

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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. Non-
central ridge coercivity distributions of native double-stranded chains are remarkably
similar to the biogenic coercivity components BS and BH (Figure 5b,e).

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**3.3 Mutant AMB-1 strain with looped chains**

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Unlike the Δ*mamJ* MSR-1 mutant, which produces agglomerated clusters of magneto-
somes, 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 single-
stranded chains (Muxworthy and Williams, 2006; Newell, 2009). The magnetic behavior
of bacteria was assessed using the magnetic coefficient (C<sub>mag</sub>), 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<sub>mag</sub> is defined as the ratio of the
maximum and minimum optical density measured and quantifies the capacity of bacteria

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to orientate along an external magnetic field (Schüler et al., 1995). C<sub>mag</sub> 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.

## 3.4 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 (Le Nagard et al., 2019). 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<sub>mag</sub>), 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.

## 3.5 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 (Dunin-Borkowski et al., 1998). 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.

3.6 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 equi-
dimensional crystals, but feature also important differences. The central ridge of wild-
type 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 (Pike et al.,
2001; Lanci and Kent, 2018), and has been observed in the growing phase of MTB
cultures (Carvallo et al., 2009). In case of single-stranded chains of equidimensional
crystals, the lower SD stability limit is comprised between 12 and 17 nm (Newell, 2009)
and is compatible with the smallest crystal sizes observed with TEM (Figure 3). The
hysteresis squareness (M<sub>rs</sub>/M<sub>s</sub> = 0.475) is slightly smaller than expected for single-
stranded 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., Wang et al., 2015), the coercivity distributions are
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left-skewed and include a small, non-zero contribution at $B_c = 0$ caused by thermal
relaxation.

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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 (Egli, 2021), as seen with
VO particles (Dumas et al., 2007; Roberts et al., 2017) and small magnetosome clusters
produced by the Δ*mamJ* mutant of MSR-1 (Katzmann et al., 2013). 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 distribu-
tions, 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,

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explains the existence of a shoulder with similar amplitude in all three coercivity distributions.

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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).

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

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# 4. Discussion

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## 4.1 Distinctive characteristics of the magnetofossil signature

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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; Lascu et al., 2015; Chang et al., 2019). 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 (Ludwig et al., 2013). 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 ($M_{rs}/M_s \ge 0.47$), and $B_{cr}/B_c$ values

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comprised between ~1.2 and ~1.6, close to the limits for narrow and wide SD coercivity distributions, respectively (Dunlop, 2002).

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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 (Kobayashi et al., 2006; Li et al., 2012) follow a trend characterized by a steep $M_{rs}/M_s$ decrease (Figure 1) typical of densely packed synthetic SD magnetite particles (Muxworthy et al., 2003). This decrease is associated with a vertical widening of the FORC function beyond the limits of the memory region (Chen et al., 2007), due to large random internal fields associated with magnetostatic interactions (Muxworthy and Williams, 2005). FORC diagrams of SD particle clusters lack a central ridge (Carvallo et al., 2005); minor central ridge contributions in magnetosome extracts (e.g., Li et al., 2012) are therefore associated with incomplete chain collapse.

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Micromagnetic simulations of magnetofossil collapse by chain axis randomization (Chang et al., 2019) 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 ($D \approx 2$) and of the partially collapsed chains generated by the random walk algorithm of Harrison and Lascu (2014) ($1 < D < 2$), compared to intact or fold-collapsed chains ($D \approx 1$) and dense clusters ($D \approx 3$), respectively. FORC diagrams with non-central ridge contributions and $M_{rs}/M_s \approx 0.5$ are therefore important hallmarks of SD particle systems with a uniaxial anisotropy that is

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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 single-stranded 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 (Ludwig et al., 2013; 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 (Egli, 2021). An interesting counterexample is provided by a mutant MTB producing distant magnetosomes (Wang et al., 2019): in this
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case, the combined action of magnetic viscosity and lack of strong in-chain magnetostatic
interactions shifts the central ridge distribution peak to $B_c = 0$.

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ARM and the central ridge are expected to have different sensitivities towards single-
and double-stranded chains. In case of single-stranded chains, ARM represents the room-
temperature thermodynamic equilibrium between two symmetric SD states in the
decaying AF field, like in UNISD particles (Egli and Lowrie, 2002). The case of systems
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featuring two symmetric SD-like states with denucleation field $B_{n1}$ and two symmetric
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FC states with higher denucleation field $B_{n2}$, as in double-stranded and fold-collapsed
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chains, is more complex: once the AF field has decayed below $H_{n2}$ during ARM
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acquisition, the system becomes trapped into one of the two FC states. In this case, the
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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
$\ge B_{n2}$. The similarity between $B_{n2}$ and the switching field of single-stranded chains made
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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
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mean amplitude of the magnetization jumps associated with FC denucleation, which
amounts to ~40% of the saturation moment ($M_{cr}/M_s$ in Table 1), the ARM response is
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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
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magnetic moment of asymmetric strands, which is significantly smaller than the
saturation moment. FC states are therefore expected to lower the ratio $\chi_a/M_{rs}$ between
the ARM susceptibility and the saturation remanence. A marked decrease of $\chi_a/M_{rs}$ has
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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 (Chen et al., 2014), so that multi-stranded chains are the most likely explanation for $\chi_a/M_{rs}$ variations of magnetofossil components.

Coercivity distributions obtained from other types of magnetization curves, such as the commonly employed isothermal remanent magnetization (IRM) acquisition (Kruiver et al., 2001), 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 $B_c = 0$.

## 4.2 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 $M_{cr}/M_{rs}$, where $M_{cr}$ is the magnetization associated with the central ridge (Egli et al., 2010). Our micromagnetic simulations yield $M_{cr}/M_{rs} \approx 0.704$ and 0.711 for single-stranded chains of equidimensional and prismatic magnetosomes, respectively, compared to 0.544 for the Stoner-Wohlfarth model (Egli et al., 2010), whereby all estimates do not take thermal activations into account. The higher $M_{cr}/M_{rs}$ 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 low-moment 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 $M_{cr}/M_{forc} \approx 0.47-0.57$ and $M_{cr}/M_s \approx 0.38-0.42$ 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
(Ludwig et al., 2013) is characterized by values of $M_{cr}/M_{forc}$ and $M_{cr}/M_s$ that are intermediate between those of simulated single-stranded and double-stranded or fold-collapsed 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 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 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 tooth-shaped 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 (Li et al., 2010).
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.

## 4.3 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 (Egli et al., 2010; 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 (Kopp et al., 2006; Charilaou et al., 2011; Kind et al., 2011). An
additional criterion discussed in this work is the confinement of FORC contributions
within the memory region, which ensures that $M_{cr}/M_s \approx 0.5$ 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
(Berndt et al., 2018), but the associated coercivity distribution and vertical offset are
different from those of magnetofossils. In another example (Till et al., 2017), 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 pri-
mary and secondary origins (e.g., Franke et al., 2007; Just et al., 2012; Li et al., 2020),
magnetofossil signatures need to be isolated using numerical unmixing methods. Numeri-
cal unmixing methods can be divided into parametric ones, which are based on model
properties, such as coercivity distribution functions (Kruiver et al., 2001, Egli, 2003), and
non-parametric ones, which are fundamentally linked to principal component analysis
(PCA), such as non-negative unmixing of remanent magnetization curves (Heslop and
Dillon, 2007) and FORC-PCA (Lascu et al., 2015; Harrison et al., 2018). Parametric
coercivity analyses are very sensitive to noise and to subtle variations of the shape
(skewness) of individual coercivity distributions (Egli, 2003; Zhao et al., 2018).
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 fold-
collapsed chains (Figure 5c), where fcr 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 (Hesse, 1994; Yamazaki and Kawahata, 1998; Wagner et al., 2021), 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 (Channell et al., 2016), 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., Roberts et al., 2011). Endmembers with characteristics similar to those of the above example are commonly found in marine sediments (Lascu et al., 2015; Yamazaki et al., 2020). 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 Lascu et al., 2015).

The ambiguity of numerical unmixing results can be partially overcome by combining numerical methods with particle morphology statistics from TEM observations (Wagner et al., 2021) and with physical or chemical separation techniques, such as the selective dissolution of fine magnetite particles (Ludwig et al., 2013).

## 4.4 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., Gibbs-Eggar et al., 1999; Franke et al., 2007; Odlfield, 2007; Li et al., 2020), 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 (Moskowitz
et al., 1993; Carvallo et al., 2008) and the classical FORC signature of strong
magnetostatic interactions (Miot et al., 2014). Because of size distribution overlapping
with the stable SD range, authigenic magnetite is expected to have a similar preservation
potential in geological materials (Roberts et al., 2012) 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 zero-
coercivity contributions and median fields of 10-25 mT (Moskowitz et al., 1993).
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 χ<sub>a</sub>/M<sub>rs</sub> values typical of non-interacting SD particles
(Egli, 2004a), and, in the case of soils, with a central ridge (Geiss et al, 2008; Egli, 2021).
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., Liu et al., 2005), 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 (Egli, 2021). Isolated SD magnetite particles
with random deviations from equidimensional shapes are also expected to produce low-
coercivity FORC contributions above and below the central ridge (e.g., Fig 51 in Egli,
2021), 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., Miot et al., 2009). For instance, FORC signatures of non-interacting and interacting SD particles not produced by MTB have been identified in organic-rich methanogenic sediments (Ustra et al., 2021). 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 (Yamazaki et al., 2020) 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 (Jiang et al., 2020), while being absent in other crusts (Oda et al., 2018). 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 (Chang et al., 2019) 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
952

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 (Kent et al., 2003; Lippert and Zachos, 2007).

953

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955

956

## 4.5 Paleontological and (paleo)environmental implications

957

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 multi-stranded chains of magnetosomes (Deng et al., 2016). However, they have been now identified in the more recent Alpha-proteobacteria and Gamma-proteobacteria phyla (Taoka et al., 2014; Zhang et al., 2017). 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 (Mao et al., 2014). In MTB, magnetosomes are aligned along cytoskeletal filaments made of actin-like proteins (Müller et al., 2020). 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 (Kolinko et al., 2016). We can thus hypothesize that genetic factors are responsible for multi-stranded chain configurations (Kolinko et al., 2016). They may include gene duplication events (Murat et al., 2010), or horizontal gene transfers (i.e., an exchange of genetic material through direct physical interactions between two organisms)

958

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962

963

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965

966

967

968

969

970

971

972

973

974
975

(Monteil et al., 2020). Once the genetic basis of chain configurations (Du et al., 2019) 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.

976

977

978

979

## 4.6 Paleomagnetic implications

980

Owing to the widespread occurrence of living MTB in the topmost ~10-20 cm of the sedimentary column (Petermann and Bleil, 1993; Flies et al., 2005; Mao et al., 2014) 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 (Larrasoña et al., 2014; Ouyang et al., 2014; Chen et al., 2017). 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.

981

982

983

984

985

986

987

988

989

990

Magnetofossils in non-varved sediments are expected to form close to the depth range inhabited by living MTB (Petermann and Bleil, 1993), 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

991

992

993

994

995

996

997
998 rotational diffusion associated with sediment mixing (Egli and Zhao, 2015; Zhao et al., 999 2016). In this case, the resulting remanent magnetization $M_{\text{nrm}} = M_0S(m_0B/\tau_s)$ is 1000 proportional to the total magnetization $M_0$ of the remanence carriers in the ideal case of 1001 full alignment of their natural magnetic moments, and to a sigmoidal function $S$, with 1002 $S(0) = 0$ and $S(\infty) = 1$, which describes the partial alignment of remanence carriers in 1003 the Earth magnetic field $B$ (Egli and Zhao, 2015). In small fields, this partial alignment is 1004 proportional to the ratio between the magnetic torque $m_0B$ of individual remanence 1005 carriers with net magnetic moment $m_0$, and the typical holding torque $\tau_s$ that resists 1006 particle reorientation. Writing $M_0 = cm_0$, where $c$ is the concentration of remanence 1007 carriers, gives $M_{\text{nrm}} = cm_0^2\kappa_{\text{nrm}}$ for the low-field limit of NRM acquisition, with 1008 $\kappa_{\text{nrm}} \propto \tau_s^{-1}$ being a coefficient that describes the mechanical alignment efficiency of the 1009 remanence carriers in sediments with specific mechanical properties. 1010

RPI records are obtained by normalizing $M_{\text{nrm}}$ with a laboratory magnetization $M_{\text{lab}}$ 1010 assumed to be proportional to $M_0$. In this case, $M_{\text{lab}} = cm_{\text{lab}}$, with $m_0$ being the mean 1011 magnetic moment component along the applied magnetic field, acquired by the 1012 remanence carriers during the acquisition of a laboratory magnetization. The normalized 1013 NRM is finally given by $R = \kappa_{\text{nrm}}m_0^2/m_{\text{lab}}B$. As expected for valid RPI normalizations, 1014 $R$ does not depend on the concentration of remanence carriers. Furthermore, a fixed 1015 magnetic mineralogy yields a constant $m_0^2/m_{\text{lab}}$, so that $R$ is proportional to $B$. 1016

Each magnetic component contributing to the NRM is expected to possess a specific 1017 $m_0^2/m_{\text{lab}}$ value related to its magnetic properties, and a specific $\kappa_{\text{nrm}}$ that reflects its 1018 mechanical properties. Accordingly, sediments containing variable proportions of two or 1019 more magnetic components (e.g., detrital and biogenic), yield RPI records that contain an 1020
unwanted environmental modulation caused by variations of $\kappa_{\text{nrm}}m_0^2/m_{\text{lab}}$ within the limit values of the individual components (Fabian and Leonhardt, 2009). The role of $m_0^2/m_{\text{lab}}$ can be investigated by inspecting selected categories of remanence carriers. For instance, the natural magnetic moment of silicate minerals containing $n \lesssim 50$ SD magnetic inclusions is proportional to the sum of $n$ randomly oriented vectors, which yields $m_0 \approx m_{\text{SD}}\sqrt{8n/3\pi}$, where $m_{\text{SD}}$ is the mean magnetic moment of individual SD particles (Heslop et al., 2014b). For comparison, intact chains of $n$ magnetosomes are characterized by $m_0 = m_{\text{SD}}n$, while fold-collapsed chains have a much smaller native moment $m_0 = m_{\text{SD}}\delta n$, where $\delta n$ is the mean absolute difference between the number of magnetosomes in the two strands (e.g., $\delta n = 0$ 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 ($\kappa_{\text{nrm}}$), intact magnetofossils are expected to possess larger RPI efficiencies relative to
silicate minerals containing SD magnetite inclusions with the same saturation moment, as reported by Ouyang et al. (2014) and Chen et al. (2017), because the magnetic moments of individual crystals add arithmetically to the native net moment, instead of randomly.

## 5. 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 fold-collapsed 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 multi-stranded 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 (Hanzlik et al. 2002; Thomas et al., 2008), 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
1067 $M_{rs}/M_s \approx 0.5$), and (2) signatures typical of SD particle systems with random
1068 magnetostatic interactions (non-central-ridge contributions in the FORC diagram).

1069 Full chain collapse leading to a complete loss of the original structure is likely
1070 prevented by the electrostatic adhesion of magnetosomes to sediment particles.
1071 Therefore, the strong uniaxial anisotropy of intact and fold-collapsed magnetosome
1072 chains appears to be a key feature for the preservation of a unique magnetic signature of
1073 magnetofossils, which, besides the elevated hysteresis squareness and a central ridge,
1074 includes two coercivity components (BS and BH in Egli, 2004a) with inferiorly limited,
1075 Gaussian-like switching field distributions distinct from those of isolated or clustered SD
1076 magnetite particles. Our micromagnetic models suggest that the low-coercivity portion of
1077 the central ridge of natural sediments (Figure 8) likely represents the contribution of
1078 isolated SD particles of authigenic origin. ARM is another key parameter of
1079 magnetofossil fingerprints: its coercivity distribution is similar to that of the central ridge,
1080 but, unlike to the central ridge, it is more selective towards single-stranded chains.

1081 Preservation of a distinctive magnetofossil fingerprint, as demonstrated in our work,
1082 is pivotal for the interpretation of the origin of secondary SD magnetite particles in sedi-
1083 ments, since this fingerprint is the only practical mean for verifying, in a quantitative
1084 manner, the existence of isolated chain-like structures of SD magnetite particle as the
1085 most essential magnetofossil identification criterion. The detailed characteristics of
1086 magnetofossil fingerprints depend on the relative abundances of equidimensional and
1087 elongated magnetosomes, which affect the coercivity distribution, and on the relative
1088 abundances single- and multi-stranded chains, which affect the fraction of magnetic
1089 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.

**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 M²OZART. All data generated in this work are available on https://figshare.com/collections/Magnetic_Signatures_of_Magnetofossils/5579547.
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**Table 1.** Summary of measured and modelled magnetic properties of MTB, magnetofossils, and other SD particle assemblages. $M_{rs}/M_s$ is the ratio between the central ridge magnetization and the saturation remanent magnetization, $\chi_a/M_{rs}$ is the so-called ARM ratio, and $M_{cr}/M_s$ the ratio between the total magnetization of the central ridge and the saturation magnetization.

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

<footnote n="1">1 Citrate-Bicarbonate-Dithionite (CBD) extractable component calculated from the difference between pre- and post-CBD measurements.</footnote>
<table><caption>**Table 2.** Boltzmann factors $\beta_0$ 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 $g$.</caption><thead><tr><th>Chain geometry</th><th>Gap g</th><th>N</th><th>β<sub>0</sub> (HMS)</th><th>β<sub>0</sub> (LMS)</th></tr></thead><tbody><tr><td rowspan="7">Single-stranded</td><td>0.1</td><td>2</td><td>439.9</td><td>—</td></tr><tr><td>0.1</td><td>7</td><td>763.9</td><td>—</td></tr><tr><td>0.1</td><td>8</td><td>784.2</td><td>—</td></tr><tr><td>0.1</td><td>9</td><td>795.0</td><td>—</td></tr><tr><td>0.1</td><td>15</td><td>835.9</td><td>—</td></tr><tr><td>0.1</td><td>16</td><td>839.7</td><td>—</td></tr><tr><td>0.1</td><td>17</td><td>842.9</td><td>—</td></tr><tr><td>0.05</td><td>17</td><td>969.2</td><td>—</td></tr><tr><td rowspan="5">Double-stranded (native)</td><td>0.1</td><td>10+9</td><td>667.8</td><td>800.4</td></tr><tr><td>0.1</td><td>10+10</td><td>688.5</td><td>803.1</td></tr><tr><td>0.1</td><td>11+10</td><td>705.2</td><td>801.6</td></tr><tr><td>0.1</td><td>11+11</td><td>751.5</td><td>829.2</td></tr><tr><td>0.05</td><td>11+11</td><td>864.0</td><td>953.4</td></tr><tr><td rowspan="5">Double-stranded (fold-collapsed)</td><td>0.1</td><td>10+9</td><td>235.0</td><td>1119</td></tr><tr><td>0.1</td><td>10+10</td><td>268.4</td><td>1345</td></tr><tr><td>0.1</td><td>11+10</td><td>277.7</td><td>1210</td></tr><tr><td>0.1</td><td>11+11</td><td>242.2</td><td>1367</td></tr><tr><td>0.05</td><td>11+11</td><td>278.5</td><td>1572</td></tr><tr><td rowspan="4">Ring</td><td>0.1</td><td>12</td><td>65.2</td><td>1518</td></tr><tr><td>0.1</td><td>16</td><td>55.5</td><td>1606</td></tr><tr><td>0.1</td><td>20</td><td>27.0</td><td>1633</td></tr><tr><td>0.05</td><td>20</td><td>31.0</td><td>1877</td></tr></tbody></table>
**Table 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, (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 (Ludwig et al., 2013).

<table><thead><tr><th>SD system</th><th><i>M</i><sub>cr</sub>/<i>M</i><sub>s</sub></th><th><i>M</i><sub>forc</sub>/<i>M</i><sub>s</sub></th><th><i>M</i><sub>cr</sub>/<i>M</i><sub>rs</sub></th><th><i>M</i><sub>cr</sub>/<i>M</i><sub>forc</sub></th></tr></thead><tbody><tr><td>Stoner-Wohlfarth</td><td>0.272</td><td>0.272</td><td>0.544</td><td>1.0</td></tr><tr><td>Single-stranded equidimensional</td><td>0.347</td><td>0.379</td><td>0.704</td><td>0.914</td></tr><tr><td>Single-stranded prismatic</td><td>0.355</td><td>0.374</td><td>0.711</td><td>0.947</td></tr><tr><td>Double-stranded equidimensional</td><td>0.194</td><td>0.396</td><td>0.407</td><td>0.489</td></tr><tr><td>Double-stranded prismatic</td><td>0.209</td><td>0.367</td><td>0.424</td><td>0.570</td></tr><tr><td>Fold-collapsed equidimensional</td><td>0.171</td><td>0.397</td><td>0.366</td><td>0.431</td></tr><tr><td>Fold-collapsed prismatic</td><td>0.187</td><td>0.354</td><td>0.381</td><td>0.528</td></tr><tr><td>Composite chains (intact)</td><td>0.274</td><td>0.382</td><td>0.561</td><td>0.718</td></tr><tr><td>Composite chains (all)</td><td>0.242</td><td>0.382</td><td>0.500</td><td>0.632</td></tr><tr><td>Pelagic carbonate (CBD-extractable)</td><td>—</td><td>—</td><td>0.653</td><td>0.666</td></tr></tbody></table>
<table><caption>**Table 4.** RPI parameters for different types of SD remanence carriers made of *n* SD crystals with mean magnetic moment *m*<sub>SD</sub>. Other symbols: *m*<sub>0</sub> — native magnetic moment, *m*<sub>IRM</sub> — magnetic moment after IRM acquisition, *m*<sub>ARM</sub> — magnetic moment after ARM acquisition, *r*<sub>a</sub> — ARM/IRM of isolated SD crystals.</caption><thead><tr><th>SD configuration</th><th>NRM<br>m<sub>0</sub>/m<sub>SD</sub></th><th>IRM<br>m<sub>IRM</sub>/m<sub>SD</sub></th><th>ARM<br>m<sub>ARM</sub>/r<sub>a</sub>m<sub>SD</sub></th><th>R (IRM)<br>m<sub>0</sub><sup>2</sup>/m<sub>IRM</sub></th><th>R (ARM)<br>r<sub>a</sub>m<sub>0</sub><sup>2</sup>/m<sub>ARM</sub></th></tr></thead><tbody><tr><td>Silicate inclusions</td><td>~0.9√n</td><td>~0.5n</td><td>~0.9√n</td><td>~1.7</td><td>~0.9√n</td></tr><tr><td>Single-stranded chains</td><td>n</td><td>0.5n</td><td>n</td><td>2n</td><td>n</td></tr><tr><td>Double-stranded chains</td><td>n</td><td>0.5n</td><td>δn</td><td>2n</td><td>n<sup>2</sup>/δn</td></tr><tr><td>Fold-collapsed chains</td><td>δn</td><td>0.5n</td><td>δn</td><td>2δn<sup>2</sup>/n</td><td>δn</td></tr></tbody></table>
1655 **Figure captions**

1656 **Fig. 1.** So-called Day diagram (Dunlop, 2002) showing the hysteresis properties *M*<sub>rs</sub>*/*M*<sub>s</sub> and *B*<sub>cr</sub>*/*B*<sub>c</sub> of intact cultured cells (Li et al., 2012; Moskowitz et al., 1993; Pan et al., 1657 2005), SD particles in a magnetofossil-rich sediment (Ludwig et al., 2013), the AMB-1 1658 mutant producing looped magnetosome arrangements (this work), and a Δ*mamJ* mutant 1659 producing magnetosome clusters (Katzmann et al., 2013). The properties of extracted 1660 magnetosomes (Li et al., 2012), micromagnetically simulated chains with increasing 1661 degree of axis randomization (Chang et al., 2019), and our micromagnetic simulations of 1662 intact and fold-collapsed chains are shown for comparison. Departures from *M*<sub>rs</sub>*/*M*<sub>s</sub> = 1663 0.5 for ideal, non-interacting, uniaxial SD particles (cultured cells), follow trends with 1664 increasing slopes, from intact and fold-collapsed configurations with strong uniaxial 1665 anisotropy to complete randomization (extracted magnetosomes), through the 1666 intermediate trend of looped and randomized chains. 1667

1668 **Fig. 2.** Representative examples of native and collapsed double-stranded magnetosome 1669 chains obtained from TEM micrographs. (a) Native, twisted double-stranded chain of 1670 equant magnetosomes produced by *M. gryphiswaldense*. Notice the magnetosome size 1671 tapering at both ends. (b) Native double-stranded chain of prismatic magnetosomes 1672 (Lefèvre et al., 2011). Notice the staggered magnetosome arrangement. (c) Top: kinked 1673 single-stranded chain of prismatic magnetosomes (Shcherbakov et al., 1997). Bottom: 1674 complete folding of a single-stranded chain around a kink point. Notice the side-by-side 1675 arrangement of magnetosomes, and one-sided tapering of magnetosome size. (d) Two 1676 examples of rings obtained from magnetophoresis of magnetosome chains extracted from 1677 cultures of *M. magnetotacticum* (Philipse and Maas, 2002).
1678 **Fig. 3.** Electron microscopy images of wild-type and mutant AMB-1 cells.

1679 **Fig. 4.** Numerically simulated zero-field magnetic states in idealized chains of identical
1680 magnetosomes with no magnetic anisotropy and dipolar interactions. The magnetic mo-
1681 ments of individual crystals are indicated by arrows with the color coding corresponding
1682 to the axial component. (a) High-moment (SD-like) state of a single-stranded chain with
1683 15 magnetosomes. (b) Two-domain state of the same chain as in (a), during a thermally
1684 activated magnetization reversal. (c) High-moment state of a ring with 20 magnetosomes,
1685 consisting of two symmetric domains with opposed vorticity. (d) Low-moment (FC) state
1686 of the same ring as in (c). (e) High-moment (SD-like) state of a double-stranded chain
1687 with 11+10 magnetosomes and staggered arrangement. (f) Low-moment (FC) state of the
1688 same double-stranded chain as in (e). See Supporting Movies S1-S4 for the
1689 corresponding transitions between states.
**Fig. 5.** Simulated FORC diagrams of single-stranded (left), native double stranded (middle) and fold-collapsed (right) chains of equidimensional (top) and prismatic (bottom) magnetite magnetosomes. Each FORC diagram corresponds to ~$10^5$ randomly oriented chains with a realistic distribution of geometric parameters as explained in the text. Coercivity distributions obtained from the simulated FORC data ($f_{dc}$ — DC demagnetization of saturation remanence, $f_{hys}$ — irreversible component of the ascending hysteresis branch, $f_{cr}$ — central ridge) are shown below, together with coercivity distributions of the biogenic component BS (gray lines in a-c) and BH (gray lines in d-f), obtained from AF demagnetization curves of ARM (from Egli, 2004a). All FORC diagrams share the same color scale (blue — negative values, green to violet — positive values) with $q = 5, 10, 20, 30, 40, 50, 60, 70, 80$, and 90% quantile contours. The FORC region enclosed by the *q*-quantile contour contributes to a fraction $1-q$ of the total FORC integral (Egli, 2021). Dashed lines indicate the left boundary of the so-called memory region.

**Fig. 6.** Native magnetic states of wild-type AMB-1 and the AMB-1 mutant. Magnetization of a suspension of (a) wild-type and (d) mutant AMB-1 cells in small fields varying between ±2 mT, before (black) and after (gray) saturation in a 200 mT field. Electron microscopy and corresponding magnetic phase contours determined by off-axis electron holography images of (b-c) wild-type and (e-f) mutant magnetite chains.
**Fig. 7.** Magnetic properties of wild-type AMB-1 and the AMB-1 mutant. (a-b) FORC diagrams with 2, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90% quantile contours. Insets are schematic representations of the magnetization of an individual cell as a function of the applied field. Numbers highlight the following features: 1 – central ridge created by the magnetization jump through which the lowest curve merges with curve I; 2 – positive and negative contributions from reversible magnetic moment rotation (difference between the slopes of consecutive curves); 3 – signatures of magnetically viscous particles (vertical ridge and central ridge extending to the origin); 4 – nucleation of a FC state from a negative SD state (contribution of the first magnetization jump in curve III to the difference between curves II and III); 5 – annihilation of a FC state (contribution of the magnetization jump in curve II to the difference between curves I and II); 6 – same as 5, but for the difference between curve II and III; 7 – annihilation of a FC state (contribution of the second magnetization jump in curve III to the difference between curves II and III). (c-d) Coercivity distributions of wild-type AMB-1 and the AMB-1 mutant, obtained from subsets of FORC data: $f_{\text{hys}}$ — irreversible component of the ascending hysteresis branch, $f_{\text{dcd}}$ — DC demagnetization of $M_{\text{rs}}$, $f_{\text{cr}}$ — central ridge. The shaded area represents the $f_{\text{hys}}$ contribution of SD magnetic states that cannot exist in a null field.
**Fig. 8.** Comparison between the FORC signature of SD particles in a magnetofossil-rich sediment and a simulated magnetofossil composite. (a) FORC diagram of in-situ <1µm magnetite particles (mostly SD) in a pelagic sediment (data from Ludwig et al., 2013). (b) FORC diagram of a simulated composite obtained from the six micromagnetically model-led chain structures, with the following relative contributions to the saturation magnetization, chosen for a visual match with the FORC diagram in (a): 1/3 single-stranded chains of equant magnetosomes, 1/6 single-stranded chains of prismatic magnetosomes, 1/3 native double-stranded chains of equant magnetosomes, 1/6 native double-stranded chains of prismatic magnetosomes, 1/3 fold-collapsed chains of equant magnetosomes, 1/6 fold-collapsed chains of prismatic magnetosomes. The same field range and contour levels with respect to the maximum amplitude are used in both plots to ease the comparison. Numbers highlight the following features: 1 — nucleation/annihilation of FC states, 2 — high-field limit of FC state nucleation/annihilation (asymmetric about the central ridge), 3 — residual negative amplitude from the annihilation of positive FC states, uncompensated by positive amplitudes from intermediate coercivity contributions.

1743
<caption>Figure 1.</caption>
<figure><img src="image_2.png" /></figure>
<caption>Figure 2.</caption>
(a)

<figure><img src="image_3.png" /></figure>

(b)

<figure><img src="image_4.png" /></figure>

(c)

<figure><img src="image_5.png" /></figure>

(d)

<figure><img src="image_6.png" /></figure>
<caption>Figure 3.</caption>
<figure><img src="image_7.png" /><figcaption>Wild-type</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Mutant</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Mutant</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Mutant</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Mutant</figcaption></figure>

<figure><img src="image_12.png" /></figure>
<caption>Figure 4.</caption>
<figure><img src="image_13.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(e)</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(f)</figcaption></figure>

<figure><img src="image_18.png" /></figure>
<caption>Figure 5.</caption>
<figure><img src="image_19.png" /><figcaption>(a) single-stranded</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(b) double-stranded</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(c) fold-collapsed</figcaption></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>(e)</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>(f)</figcaption></figure>

<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

equant

prismatic
<caption>Figure 6.</caption>
<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /></figure>

<figure><img src="image_32.png" /></figure>

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>
<caption>Figure 7.</caption>
<figure><img src="image_35.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_38.png" /></figure>
<caption>Figure 8.</caption>
<figure><img src="image_39.png" /><figcaption>(b) Micromagnetic composite</figcaption></figure>

<figure><img src="image_40.png" /></figure>