# Quantum-assisted electron transport in microbial protein wires across
macroscopic distances

Jasper R. van der Veen1,2∗, Silvia Hidalgo Martinez3,
Albert Wieland1, Matteo De Pellegrin 1, Rick Verweij1,
Yaroslav M. Blanter1, Herre S.J. van der Zant1 and Filip J.R. Meysman 2,3

1Department of Quantum Nanoscience, Delft University of Technology,
Lorentzweg 1, Delft, 2628CJ, The Netherlands
2Department of Biotechnology, Delft University of Technology,
Van der Maasweg 9, Delft, 2629HZ, The Netherlands
3Excellence center for Microbial Systems Technology, University of Antwerp,
Universiteitsplein 1, Wilrijk, 2610, Belgium
∗To whom correspondence should be addressed; E-mail<EMAIL_ADDRESS>
One Sentence Summary. Electrical conductivity in cable bacteria remains
exceptionally high at cryogenic temperatures, indicative of quantum-assisted
transport.

> Multicellular cable bacteria display an exceptional form of biological
> conduction, channeling electrical currents across centimeter distances
> through a regular network of protein fibers embedded in the cell envelope.
> The fiber conductivity is among the highest recorded for biomaterials,
> providing a promising outlook for new bio-electronic technologies, but the
> underlying mechanism of electron transport remains elusive. Here, we use
> detailed electrical characterization down to cryogenic temperatures, which
> reveals that long-range conduction in these bacterial protein wires is based
> on a unique type of quantum-assisted multistep hopping. The conductance near
> room temperature reveals thermally activated behavior, yet with a low
> activation energy, suggesting that substantial delocalization across charge
> carrier sites contributes to high conductivity. At cryogenic temperatures,
> the conductance becomes virtually independent of temperature, thus
> indicating that quantum vibrations couple to the charge transport. Our
> results demonstrate that quantum effects can manifest themselves in
> biological systems over macroscopic length scales.

## Introduction

Conventionally, the reference point for long-range biological charge transport
are the protein complexes of the electron transport chain in mitochondria or
the photosynthetic reaction centers in chloroplasts (?, ?). These membrane-
bound complexes enable sequential hopping of electrons between closely spaced
cofactors (e.g. hemes or iron-sulphur clusters) over total distances covering
$\leq 10$ nm (?). Recent studies, however, demonstrate that biological
electron transport may greatly surpass this length scale. While metal-reducing
bacteria such as Geobacter and Shewanella can mediate electron transport over
micrometer distances through thin surface appendages (?, ?, ?), the discovery
of cable bacteria extends the range of biological electron transport to the
centimeter scale (?, ?, ?). However, the question as to why biological protein
structures can sustain conduction over these macroscopic distances remains
fundamentally unresolved.

Cable bacteria harbor an internal conductive network, which consists of
centimeter-long protein fibers that run in the cell envelope along the entire
length of the filamentous bacteria (?, ?, ?, ?, ?). Room-temperature
characterisation (?, ?, ?) reveals that these protein fibers possess an
electrical conductivity up to 300 S/cm-1, which surpasses that of biomaterials
by several orders of magnitude, and even exceeds the conductivity of most
organic semiconductors (?). To elucidate the transport mechanism underlying
this extraordinary form of biological long-range conduction, we characterised
the fiber conductance over a wide temperature range from room temperature down
to liquid helium temperature. These measurements reveal that charges are
transported through a unique type of multi-step hopping, involving low
barriers due to delocalized charge carrier wave functions and quantum
vibration effects at temperatures below 75 K.

## Temperature dependence of conductance

Figure 1: Current as a function of temperature and electric field. (A)
Individual fiber sheaths are deposited on Si/SiO2 substrate with prepatterned
gold contacts. (B) A fiber sheath filament is stretched across a series of
gold contacts. (C and D) Representative current ($I$) versus bias voltage
($V$) curves at different temperatures for two different segment lengths, $L$.
The current is normalized to the maximum current of each $I(V)$ trace.
Temperature is indicated by the color scale. The top axis displays the applied
electric field, $E=V/L$. Shorter segments allow to investigate the conductance
at lower temperatures and higher electrical fields. For the longer segment
($L=240$ µm), the current fell below the detection limit below 40 K.

Conductivity probing as a function of temperature is widely applied to
elucidate the charge transport mechanism in electronic materials (?). For
biological materials, investigation of low-temperature conduction ($<10$ K) is
not physiologically relevant, but is instrumental to get insight into the
mechanism of electron transport in proteins, as thermal excitation becomes
suppressed. Up until now, cryogenic characterization of electron transfer has
been primarily applied to photosynthetic reaction centers (?, ?), which are
experimentally accessible through facile light activation and high electron
transfer rates. For other protein systems, however, data on low-temperature
conductance are critically missing, and consequently, it remains uncertain to
what extent insights and parameters from photosynthetic reaction centers can
be extrapolated (?). In particular, low-temperature conduction studies are
lacking for macroscale ($\geq 1~{}\mu$m) protein structures, as the increasing
resistance with length prevents current detection at low temperatures. The
high conductance sustained across millimetres in cable bacteria alleviates
this experimental bottleneck.

To probe the conductance of the periplasmic fibers, long cable bacterium
filaments ($>2$ mm) were individually isolated from enrichment cultures.
Subsequent extraction provides a so-called "fiber sheath" that retains the
conductive fiber network lying on top of a connective carbohydrate sheath. (?)
The number of parallel fibers embedded in one fiber sheath is determined by
microscopy (fig. S1). Combined with the known (?) fiber diameter (26 nm), this
enables the fiber conductivity to be calculated from the measured conductance
(Eq. 2 in Supplementary Text). This procedure has been shown to provide
reliable fiber conductivity data (?, ?), as native filaments and fibers
sheaths provide similar fiber conductivities, indicating that the extraction
procedure does not affect the electrical response (?).

Fiber sheaths of variable length (diameter 4 µm; length up to several mm) were
deposited onto insulating silicon substrates with pre-patterned gold
electrodes (Fig. 1A) for electrical characterization under vacuum. The
multiple electrical contacts per filament allow conduction to be evaluated
over different segments of the fiber sheath (Fig. 1B). Room-temperature, four-
probe measurements on single filaments show a linear dependence of the
resistance $R$ on the probed segment length $L$ over a distance of millimeters
(fig. S2). These measurements demonstrate that the fiber network possesses a
uniform conductivity over mm-scale distances, which hence justifies the
comparison of different segments.

Current-voltage characteristics ($I(V)$ curves) were recorded for 53 segments
of varying length ($L$ = 4 µm to 2 mm), of which 16 were also measured down to
cryogenic temperatures ($4.2-10$ K). Measurements were highly consistent
between filaments, revealing a similar response. Figure 1B and D displays
representative $I(V)$ data for both a long ($L=240$ µm) and short segment
($L=4$ µm) (fig. S7 provides data for all segments). The measured current
shows a distinct response to the temperature, $T$, and to the imposed electric
field, $E=V/L$. At high temperature and low electric field, the $I(V)$ curve
is linear (Fig. 1B). Yet, when temperature decreases and the electric field
strength increases, the $I(V)$ becomes increasingly more non-linear (Fig. 1C
and D).

In a two-probe configuration, the measured resistance accounts for the
intrinsic resistance of the probed filament segment as well as the contact
resistance between the gold electrodes and filaments (?). To verify our
approach, two-probe and four-probe measurements were compared on the same
segment, revealing that contact resistances do not influence the shape of the
$I(V)$ profile, nor the temperature dependence of the conductance (fig. S11).

When replotting the $I(V)$ data in terms of the conductance, $G=I/V$, we
consistently observe the same $G(T,E)$ response across all segments (Fig. 2A
and B; fig. S7). At high temperatures ($T>75~{}$ K), the $G(T)$ data collected
at different $E$-values converge (Fig. 2B), so conductance is not affected by
the electric field. Moreover, the charge transport is thermally activated and
follows a classical exponential Arrhenius-type dependence (?). Below the
cross-over temperature $T_{\rm C}\backsim 75~{}K$, the temperature dependence
becomes noticeably weaker, and for T < 20 K, the conductance remains high and
becomes virtually independent of temperature at moderate electric fields (Fig.
2B). At these low temperatures, the conductance is strongly affected by the
electric field. For example, at 5 K, $G$ increases by 3 orders of magnitude
when $E$ is increased from 1 to 2.5 V/µm (Fig. 2B).

Figure 2: Dependence of conductance on temperature and electrical field. (A
and B) Conductance data $G=I/V$ are plotted as a function of inverse
temperature at a constant electric field strength (i.e. fixed bias voltage)
for two segments of different length. Data are extracted from $I(V,T)$ curves
as shown in Fig. 1. In panel (A), the Arrhenius fit to the high-temperature
conductance data is included for reference (dashed grey line). (C and D) Model
simulation of the conductance data in panels (A) and (B) via a 1D hopping
chain model with one effective vibrational mode (Jortner model). Model
parameters in panel (C): reorganization energy, $\lambda=0.35$ eV and
effective vibrational frequency, $\langle\omega\rangle=$ 122 cm-1 (15 meV).
The number of hopping sites, $N_{\rm S}$, does not change the shape of the
curve. Model parameters in panel (D): $\langle\omega\rangle=$ 58 cm-1 (7.2
meV), $\lambda=0.16$ eV, number of hopping sites, $N_{\rm S}=200$. (E and F)
$I(V,T)$ data replotted as a universal scaling curve (?, ?); $T$ is
temperature, $e$ is the elementary charge, and $k_{\rm B}$ is Boltzmann’s
constant. The green solid line represents a model fit to the data that has two
fitting parameters: the dimensionless exponent $\beta$ and the number of
hopping sites, $N_{\rm S}$. For the longer segment, $N_{\rm S}=1400$ and
$\beta=6.0$; for the shorter segment, $N_{\rm S}=120$ and $\beta=6.5$.

## Quantum vibrations couple to the electron transport

The observation that the conductance remains elevated as the temperature
decreases, and eventually becomes independent of temperature, is remarkable.
The deviation from an exponential temperature dependence is at odds with the
standard semi-classical Marcus theory of electron transfer (?), which assumes
that transitions are thermally activated. The fact that the conductance
remains high at low temperatures thus hints at an energy source other than
thermal energy that assists the charge transfer. In seminal experiments on
photosynthetic reaction centers in purple bacteria, DeVault and Chance
observed a similar temperature dependence that could not be accounted for by
Marcus theory. The charge separation reactions of photosynthesis continued at
cryogenic temperature and the rate of these reactions became independent of
temperature (?, ?).

This finding spurred a host of theoretical developments, which eventually
extended electron transfer theory to a fully quantum-mechanical treatment (?,
?, ?, ?). Quantum theory asserts that at absolute zero, molecular vibrations
retain a finite zero-point energy. At low temperatures, thermal motion is
frozen out, but quantised vibrational modes can assist the electron
tunnelling, in a process referred to as nuclear tunneling (?, ?). Marcus
theory (?) accounts for electron tunneling, but uses a classical harmonic
approximation of the nuclear motion, so it neglects the impact of nuclear
tunneling and therefore underestimates the electron transfer rate at low
temperatures.

To verify whether nuclear tunnelling could explain our observations, we
modelled electron transport subject to quantum vibrational coupling in a one-
dimensional hopping chain that mimics the conduction path in the fiber sheaths
(Supplementary Text). In metalloproteins, vibrational coupling can involve
both high-frequency intramolecular vibrations of the charge carrying cofactor
as well as lower-frequency vibrations of the surrounding protein matrix and
solvent molecules. This vibrational coupling can be described with models of
increasing complexity. As a first step, we applied a model where only one
effective high-frequency mode $\langle\omega\rangle$ stimulates the hopping
process (Jortner model (?), Supplementary Text). These model calculations
adequately capture the essence of our experimental data: the conductance
displays two regimes with a transition at the cross-over temperature $T_{\rm
C}=\hbar\langle\omega\rangle/k_{\rm B}$, where $k_{\rm B}$ is the Boltzmann
constant and $\hbar$ the reduced Planck constant. For $T>T_{\rm C}$, the
vibrational mode remains thermally excited, and the conductance adopts a
classical Arrhenius-type dependence (Fig. 2C). Below $T_{\rm C}$, the
vibrational mode is no longer thermally excited, but due to the quantum
vibrational energy, the conductance remains higher than predicted by the
Arrhenius relation. Upon further cooling below $T_{\rm C}/5\approx$ 25 K, the
conductance becomes virtually independent of temperature (Fig. 2D).

Across all sixteen segments investigated, we find a mean cross-over
temperature $T_{C}$ = 75 $\pm$ 14 K, which corresponds to a characteristic
frequency $\langle\omega\rangle=k_{\rm B}T_{C}/\hbar$ = 52 $\pm$ 10 cm-1 for
the single effective vibrational mode that couples to the electron hopping
(vibrational energy $\hbar\langle\omega\rangle$ = 6.4 $\pm$ 1.2 meV). The
experimental verification of vibronic coupling is highly challenging in
biological systems, and has only been done for photosynthetic reaction
centers, which yields substantially higher characteristic frequencies
$\hbar\langle\omega\rangle\approx 60$ meV (?). Previously, it has been
speculated that the $60$ meV frequency could be a general feature of electron
tunneling reactions in natural proteins (?). Our results invalidate this
hypothesis, and instead, suggest a substantial variability in vibrational
coupling, where the characteristic frequency is likely dependent on the
specific molecular configuration of the cofactor and metalloprotein involved
in the electron transport. To illustrate this, fig. S15 compares the data fit
for a cable bacterium to that of the photosynthetic reaction center in
_Allochromatium vinosum_.

Figure 3: Empirical fits to conductance data. (A) Low-bias conductance, $G$,
as a function of inverse temperature for $T>100$ K. Data for 4 segments. Two-
and four-probe measurements are shown in blue and red markers respectively.
Solid black lines are fits by the non-linear least squares method to the
Marcus equation $G=g_{0}(k_{\rm B}T)^{-3/2}\text{exp}(\frac{\lambda}{4k_{\rm
B}T})$, where $g_{0}$ is a pre-factor and $\lambda$ the reorganization energy,
of which the value is given next to the data. (B) Histogram with
reorganization energies for $n=53$ samples (blue: two-probe measurements; red:
four-probe measurements). (C) Low-bias conductance against temperature in the
region $20-100$ K for 16 segments. Black dashed lines represent power law fits
(determining $\alpha$). (D) Conductance versus applied electric field at the
lowest temperature achieved ($T<10$ K), for the same 16 segments. Yellow
dashed lines represent power law fits (determining $\beta$).

## Conductance follows a universal scaling relation

The single-mode vibrational model predicts that the zero-bias conductance
should become constant at the lowest temperatures. Yet closer data inspection
reveals that the conductance continues to decrease as a power law ($G\propto
T^{\alpha}$; Fig. 3C), which suggests that vibrational modes lower than
$\omega=k_{\rm B}T_{C}/\hbar$ are additionally coupling to the electron
transport. To accommodate for this, we integrated a multi-mode description of
vibrational coupling in our one-dimensional hopping model (Egger model (?),
Supplementary Text). There are several predictions that can be tested. At high
temperatures, all vibrational modes must remain thermally excited, and hence,
one should retrieve the results of Marcus theory. This is indeed the case. For
$T>100$ K, the conductance follows a Marcus-type dependence on temperature,
$G\backsim T^{-3/2}\text{exp}(-\lambda/4k_{\rm B}T)$, where $\lambda$ is the
reorganization energy (Fig. 3A; Supplementary Text). Also, at high
temperatures, the conductance should remain independent of $E$, which provides
linear $I(V)$ curves, as observed (Fig. 1B). The reorganization energy is low
and shows limited variation (Fig. 3B), $\lambda=270\pm 40$ meV ($n=53$
segments). The corresponding activation energy, $U_{\rm A}=\lambda/4=42\pm 8$
meV (fig. S12), aligns well with earlier high-temperature electrical
measurements on both intact cable bacteria and fiber sheaths (?).

The multi-mode model predicts that the conductance should follow a power-law
dependence on temperature at low electric fields, $G\propto T^{\alpha}$, and a
power-law dependence on the electric field at low temperatures, $G\propto
E^{\beta}$. The fiber network conductance indeed shows such a power law
behaviour with exponents $\alpha\approx 3.7-7.7$ and $\beta\approx 4.0-8.3$
(Fig. 3C and D; table S2). Moreover, the multi-mode model asserts that in the
low-temperature regime, the $I(V,T)$ data for a given segment can be suitably
reduced to a single universal scaling curve (?) (see Supplementary Text),
which features the normalized current $I/T^{1+\beta}$ as a function of a
normalized bias $\overline{V}={eV}/{(2\pi k_{\rm B}T)}$. The rescaled I/V data
closely follow this universal scaling curve for all 16 segments investigated
(Fig. 2E and F; fig. S10). This universal scaling behavior has not been seen
before for a biological material, but is characteristic for various synthetic
one-dimensional conductors, such as networks of graphene nanoribbons (?) and
carbonized polymer nanofibers (?), which display similarly high conductivities
as the periplasmic fibers in cable bacteria. Moreover, the power law exponent
$\beta$, which designates the degree of vibronic coupling, is on par with
these synthetic conductors, thus suggesting that cable bacteria may have
evolved a similar electron relay system.

## A unique form of multistep hopping

Our results provide several insights into the mechanism that sustains long-
range conduction in the periplasmic fiber network of cable bacteria. Foremost,
the observed temperature dependence of conduction displays the hallmark of
quantum-mechanical electron tunneling (?). At high temperatures, electron
transport is assisted by thermal fluctuations, thus providing an Arrhenius
dependence, while at low temperatures, quantized molecular vibrations couple
to the electron transfer, thus resulting in higher conductance than
classically expected. Nuclear tunneling has been previously observed in
photosynthetic reactions centers (?), but not in macroscale electron transfer.

Electron tunneling shows a strong exponential dependence of the electron
transfer rate on distance, which limits individual electron transfer events to
distances $\leq$ 1.5 nm (?). Still, it is well known that protein structures
can support electron transport over much longer distances, of which the cm-
scale conduction in cable bacteria represents the most extreme example.
Biology has resolved this problem by arranging cofactors in chains at close
spacings (typically $10-15$ Å). This enables multi-step electron hopping, in
which electrons are moving through the protein matrix by consecutive tunneling
steps, as for example, documented for multiheme cytochromes involved in
extracellular respiration (?) and FeS clusters aligned in the complexes of the
respiratory chain (?). While our data support this multi-step hopping picture,
there are several unique features in which the electron transport in cable
bacteria is distinct from the currently known forms of long-range electron
transport in proteins.

Foremost, the length scale over which hopping takes place in cable bacteria is
extraordinary, as it covers distances of millimetres to centimeters. Raman
spectroscopy has shown that the periplasmic fiber network does not contain FeS
clusters nor cytochromes (?, ?), thus excluding a heme-based conduction
mechanism as found in the surface appendages of metal-reducing bacteria (?, ?,
?). Instead, recent work suggests that the periplasmic fibers in cable
bacteria contain a nickel-sulfur cofactor that mediates the electron transport
(?). The involvement of nickel is remarkable, as all currently known
metalloproteins involved in electron transport rely on either iron- or copper-
containing cofactors, but never nickel (?). It is thus reasonable to suggest a
model in which the quantized vibrational modes of this nickel-sulfur cofactor
are strongly coupled to the electron transport. If such vibronic coupling is
due to a single mode, this mode would have characteristic frequency of
$\omega\approx$ 50 cm-1. The corresponding Huang-Rhys factor, which provides a
dimensionless measure for the coupling strength, is high
($S=\lambda/\hbar\langle\omega\rangle$ = 42), thus indicating strong vibronic
coupling. While the molecular structure of nickel cofactor remains unresolved,
the appearance of distinct Ni-associated Raman peaks in the spectra of fiber
sheaths near this frequency (?) suggests that cofactor modes could be indeed
involved in the reorganization process (?). Most likely, multiple modes are
coupling to the electron transport, which then provides the observed universal
scaling response (Fig. 2E and F) as also seen in abiotic one-dimensional
conductors (?, ?, ?).

A second notable feature is that the reorganization energy at room temperature
(0.27 eV) is markedly lower than observed for other forms of biological
electron transport. Typical values for single electron transfer steps in
enzymes are in the range 0.7 < $\lambda$ < 1.4 eV, while multi-step electron
transport in the surface appendages of the metal-reducing bacteria also
exhibits similarly high reorganization energies (?, ?). Within the thermally
activated regime, and assuming a negligible driving force (as the voltage bias
is distributed over very many transitions in the hopping chain), the rate of
electron transfer scales with $\sim\exp(\lambda/4k_{\rm B}T)$, and so, a low
reorganization energy helps explaining the extraordinary high fiber
conductivity observed in cable bacteria. Moreover, the reorganization energy
is known to inversely scale with the localisation length of the charge
carriers (?). Therefore, high reorganization energies imply that electrons are
strongly localized on the charge carrier site, while low reorganization
energies, as seen here, suggest substantial delocalization of the charge
carrier wave function. Heme-localised electronic states are examples of the
former: they form low energy states (“traps”) that are separated by sizable
activation barriers of 0.2 $-$0.3 eV, which are large compared to thermal
activation barrier ($\approx$ 10 $k_{\rm B}T$ at 300 K) (?). In contrast, the
electronic states on the cable bacterium fibers appear to be far more
delocalized, thus enabling fast transitions. Comparably low reorganization
energies have been reported for highly-mobility organic semiconductors, such
as rubrene and naphtalene (?), in which the charge carrier sites display
sizeable conjugation and delocalization.

## The challenge of high conductivity

Overall, the observed temperature dependence is consistent with multistep
hopping, which forms the cornerstone of the existing theories of long-range
biological electron transfer (?). Still, the high fiber conductivity recorded
(up to $\backsim$ 100 S cm-1; fig. s2) poses a profound challenge to this
hopping model. The key problem is the magnitude of the electron transfer rate
$\Gamma_{R}$, which is related to the fiber conductivity $\sigma$ via the
relation

$\Gamma_{\rm R}=\sigma\frac{k_{\rm B}T}{e^{2}}\frac{\pi N_{\rm S}d_{\rm
F}^{2}}{LN_{\rm C}},$ (1)

In nearly all electron transfer reactions in biology, the center-to-center
spacing between charge carriers falls within the range of $a=0.5-2$ nm.
Consequently, one needs $N_{\rm S}=$ 5-20 x $10^{6}$ sequential charge
carriers to cover the typical $L$ = 1 cm length scale of a cable bacterium
filament. Even if we assume massive parallelization of conduction channels
(each fiber consists of $N_{\rm C}=$ 125 parallel channels of 2 nm diameter;
Supplementary Text), the required transition frequency for nearest neighbor
hopping exceeds 1013 s-1 (see details in Supplementary Text). This < 0.1
picosecond electron transfer rate is orders of magnitude faster than currently
observed for non-light driven electron transfer, which maximally reaches
nanoseconds for inter-heme transfer in cytochrome c oxidase, but is more
typically on the scale of microseconds (?). Effectively, the 1013 s-1 hopping
rate exceeds the speed limit of non-adiabatic electron transfer, where
relaxation times of vibrational modes are slower than the hopping rate itself
(?, ?).

Our data hint at a possible resolution to this problem. A larger center-to-
center distance would allow for the same conductivity with a smaller hopping
rate, thus bringing the electron transport back in the non-adiatabic regime.
The universal scaling curve displays a cross-over point between high and low
temperature regimes, which enables a direct estimate for the number of hopping
sites ($N_{\rm S}=eV/(2\pi k_{\rm B}T)$). We find that $N_{\rm S}$ linearly
scales with the segment length $L$ (table S 2; fig. S14), thus providing a
center-to-center distance between hopping sites in excess of 10 nm. Clearly,
this distance is far too large to enable through-space tunneling of electrons
(?, ?), and largely exceeds the known heme-to-heme distances in cytochromes (<
1 nm) (?, ?). Still, in doped organic semiconducting nanowires, like
polyacetylene, the universal scaling curves provides similar center-to-center
distances ($\sim 10$ nm) (?, ?, ?). A potential model that can accommodate
these large center-to-center distance consists of hopping chains embedding
stacks of tightly spaced cofactors (> 10 nm). Electron tunneling then occurs
across low-potential barriers between the cofactor stacks, while in the
internal part of the stacks, electrons are delocalized and electron transport
occurs largely unobstructed (fig. S13). This speculative model provides a
potential explanation of how the high electrical conductivity in the
periplasmic fibers in cable bacteria can be reconciled with the multistepping
mechanism that emerges from data obtained here, but requires further
experimental validation. Ultimately, such a better understanding of the charge
carrier transport in cable bacteria may enable the design and construction of
new bio-mimetic materials for electronics and energy conversion (?, ?, ?).

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## Acknowledgements

We thank Spiros Skourtis for discussions.

### Funding

J.R.V. and F.J.R.M. were financially supported by the Netherlands Organization
for Scientific Research (VICI grant 016.VICI.170.072) F.J.R.M. and S.H.M. were
additionally supported by Science Foundation Flanders (FWO grant G043119N).
F.J.R.M. and H.S.J.Z. received support from the EIC Pathfinder project
PRINGLE.

### Author contributions

S.H.M. performed cable bacteria culturing and fiber sheath preparation for all
experiments. J.R.V., A.W., M.P. and R.V. carried out conductance experiments.
J.R.V. analysed conductance data and performed modelling, while H.S.J.Z.,
F.J.R.M., and Y.M.B. provided additional input. J.R.V., H.S.J.Z., and F.J.R.M.
wrote the manuscript with additional contributions of all other co-authors.

### Competing interest

The authors declare no competing interests.

### Data and materials availability

All data are made available. The complete data set for each figure is provided
with an accompanying explanation file.

## Supplementary Materials

Materials and Methods
Figs. S1 to S9
Tables S1 and S2
Supplementary Text

## Materials and Methods

### Conductive fiber networks from cable bacteria

Cable bacterium filaments were harvested from enrichment cultures that were
set up using natural sediment collected in the creek bed of a salt marsh
(Rattekaai, The Netherlands). Upon collection, sediment was sieved and
repacked into PVC core liner tubes (diameter 40 mm). The cores were incubated
in aerated artificial seawater at _in situ_ salinity, and the development of
cable bacteria was tracked by microsensor profiling and microscopy (procedure
as in Ref. 11). Under a stereo microscope, individual filaments were gently
pulled out from the top layer of the sediment with custom-made glass hooks.

Through sequential extraction, the conductive fiber network is isolated from
the cell envelope of individual filaments. This extraction removes the
membranes and cytoplasm, but retains the parallel conductive fibers embedded
in a basal sheath (procedure as in Ref. 8). These so-called "fiber sheaths"
form the starting material for all investigations performed here. To produce
these fiber sheaths, freshly isolated cable bacterium filaments were cleaned
by transferring them at least six times between droplets ($\backsim$ 20 µl) of
MilliQ water on a microscope cover slip. Subsequently, filaments were
extracted in a droplet of 1 (w/w) sodium dodecyl sulfate (SDS) for 10 minutes,
followed by six MilliQ washes. Filaments were then incubated for 10 minutes in
a droplet of 1 mM sodium ethylene diamine tetra-acetate (EDTA), pH 8, and
again six times washed in MilliQ.

### Electrical characterization down to cryogenic temperatures

Gold electrode patterns (Fig. 1) were deposited onto $p^{++}$-doped silicon
substrates with a surface layer of silicon dioxide (285 or 500 nm thickness)
via optical lithography. A laser writer illuminates the desired pattern in a
single light-sensitive resist layer (AZ ECI 3007 or 3012). The laser wave
length (365 nm) limits the minimum feature size to approximately 1 µm. The
gold thickness was 100 nm, with 5 nm of titanium underneath to promote
adhesion of the gold layer to the SiO2 surface. Fiber sheaths were positioned
onto patterned substrates immediately after extraction and substrates were
directly transferred to the vacuum chamber of the probe station. In this way,
fiber sheaths retain a stable conductance for up to a period of weeks (?).

The conductance of fiber sheaths was measured down to liquid helium
temperatures in two separate set-ups. The first setup is based around a dewar
of helium, in which a stick containing a vacuum sample chamber can be
inserted. In this setup, the patterned substrates were glued to a chip
carrier, either with silver paint or epoxy glue. The electrode pads of the
substrate were subsequently wire bonded to the electrodes of the chip carrier.
For the lowest temperatures ($T$ < 20 K) the sample chamber is not completely
vacuum, because helium exchange gas is used to reach the desired temperature.
The minimum temperature reached is near the helium condensation point (4.2 K).
For higher temperatures ($T$ > 20 K), this exchange gas was pumped out of the
sample chamber. A resistor was used to heat up the device, while a thermometer
is placed nearby the sample to measure the temperature. The second set-up used
was a cryo-free LakeShore Cryogenic Probe Station (Type CRX 6.5 K). In this
case, the electrode pads did not need to be wire bonded. Of the 16 segments
for which the temperature dependence was measured down to the lowest
temperatures ($T$ < 10 K; fig. S7), segments 1 - 5 were measured in the first
set-up and segments 6 - 16 were measured in the second.

To assess the impact of contact resistances, two-probe and four-probe
measurements were conducted. In a two-probe measurement, a bias voltage,
$V_{\rm B}$, is applied across two electrodes, and the induced current,
$I_{\rm M}$, is measured. This yields the two-probe resistance, $R_{2P}=V_{\rm
B}/I_{\rm M}=R_{\rm i}+R_{\rm C}$, which is composed the intrinsic resistance
of the fiber sheath segment, $R_{\rm i}$, and the two contact resistances
between the fiber sheath and the electrodes, $R_{\rm C}$. In the four-probe
approach, the bias current, $I_{\rm B}$, is injected over the two outer
electrodes, while the voltage, $V_{\rm M}$, is measured over the two inner
electrodes. The four-probe resistance, $R_{\rm 4P}$ = $V_{\rm M}/I_{\rm B}$,
equals the intrinsic resistance of the conductive fiber sheath segment between
the two inner pads. The contact resistance is hence determined as $R_{\rm
C}=R_{\rm 2P}-R_{\rm 4P}$.

## Figs. S1 to S9

### Figure S1

Figure 1: Microscopic images of fiber sheaths from cable bacteria. (A)
Scanning electron micrograph (acceleration voltage 10 kV) of a fiber sheath
deposited silicon-dioxide substrate with gold patterned electrodes. The scale
bar is indicated in the bottom right corner of the image. The dark background
is silicon dioxide, the lighter grey background is the gold electrode. (B)
Atomic force microscopy image of a fiber sheath filament. The colour indicates
the measured height profile (see colour bar). The fiber sheath displays the
conductive fibers as a set of lines running in parallel to the bacterial
filament.

### Figure S2

Figure 2: Linear dependence of electrical resistance on probing length over mm
distances. (A and B) Four-probe measurements of the resistance, $R$, as a
function of the electrode spacing, $L$, at 300 K for three separate fiber
sheath filaments of cable bacteria. In these measurements, the two current-
carrying electrodes and one voltage-sensing electrode were kept at a fixed
position, while the second voltage-sensing electrode was varied along the
filament, for increasing $L$. The resistance was calculated as $R={V}/{I}$,
where $V$ is the voltage measured across the voltage-sensing electrodes and
$I$ is the applied current. The resistance increases linearly with the probing
length $L$, thus demonstrating that the fiber network possesses a uniform
conductivity. The black lines represent linear fits through the data, which
provide an estimate of the fiber conductivity (inset values) derived as
$\sigma_{\rm 0,F}=4/(N_{\rm F}\pi{d_{\rm F}}^{2}b)$, where $b$ is the fitted
slope, $N_{\rm F}$ = 60 is the number of fibers in a filament (as determined
by microscopy), and $d_{\rm F}=$ 26 nm is the cross-sectional diameter of the
conductive core of a fiber (?, ?).

### Figure S3

Figure 7: Current/voltage measurements as a function of temperature. The two
panels in each row display data for a given fiber sheath segment of length
$L$. Data are provided for 16 segments of different length. (A1 to P1) Two-
probe current-voltage ($I(V)$) characteristics recorded at different
temperatures. To illustrate the change in shape of the $I(V)$ curve, the
current is normalized to the maximum current obtained for the given trace. The
top axis shows the applied electric field, $E=V/L$. The colour of the lines
indicates the temperature (linearly changing from red to black in the range
$300-80$ K and from black to blue in the range $80-4$ K). See colour bar next
to panel (P1). (A2 to P2) The same dataset is replotted in terms of the
conductance, $G=I/V$, as a function the inverse temperature, $100/T$, for
different bias voltages, $V$. The colour scale linearly ranges from black
(zero-bias voltage) to blue (maximum bias voltage). Note that this voltage
range differs per segment. The segment number and segment length are indicated
in the right panels (see also table S1).

### Figure S4

Figure 10: Universal scaling behaviour of the conductance at low temperatures.
(A to P) Scaling curves are displayed for the 16 segments shown in Fig. 7. The
plots display the scaled current $I/T^{\beta+1}$ on the $y$-axis versus the
scaled bias $eV/2\pi k_{\rm B}T$ on the $x$-axis. The $I(V)$ data in the
temperature range below 100 K are used. The temperature at which individual
data points were collected is indicated by the colour scale next to panel (O).
The green solid line provides a fit of the universal scaling relation (Eq. 22)
to the data. The power law exponent $\beta$ is visually calibrated to obtain
the best fit (precision $\pm\ 0.5$). The transition from the low-bias, high-
temperature regime to the high-bias, low-temperature regime occurs at a bias
$eV\approx 2\pi N_{\rm S}k_{\rm B}T$. This transition point hence determines
the parameter $N_{\rm S}$, which signifies the apparent number of electron
transfer steps in the hopping chain. In panel (O) this transition point is
indicated by the dotted line. The resulting values for the power law exponent,
$\beta$ and the apparent number of hopping sites, $N_{\rm S}$, are listed in
table S2. The parameter $N_{\rm S}$ scales with the segment length (see fig.
S14).

### Figure S5

Figure 11: High-temperature conductivity analysis. (A) Temperature dependence
of conductance, $G$, as measured by the two-probe and four-probe approach for
three different segments. As expected, the four-probe conductance is
systematically higher as it does not include the contact resistance. The two-
probe and four-probe data show a very similar temperature dependence,
demonstrating that both methods capture the same intrinsic temperature
response of the conductance. The activation energy, $U_{\rm A}$, is determined
as the slope from an Arrhenius plot, i.e., a linear fit of $\text{ln}(G)$ vs.
$1/(k_{\rm B}T)$ for $T>100$ K (see also fig. S12). We find similar activation
energies for the two-probe and four-probe data: first segment (squares):
$38.1\pm 0.2$ meV vs. $38.2\pm 2$ meV; second segment (stars): $41.2\pm 6.7$
meV vs. $35.7\pm 2.3$ meV; third segment (circles): $48.5\pm 2.3$ meV vs.
$50.6\pm 3.6$ meV. (B) Histogram of the room-temperature conductivity for all
fiber sheath segments investigated ($n=53$). Conductivity was calculated via
Eq. 2 as detailed in the Supplementary Text. Both four-probe (red bars) and
two-probe (blue bars) estimates are shown. Note, that four-probe values are
typically higher than two-probe values, due to the absence of the contact
resistance. (C) Activation energy $U_{\rm A}$ of all segments investigated
($n$ = 53) plotted against the room-temperature fiber conductivity,
$\sigma_{\rm 0,F}$. Two-probe data (blue markers) and four-probe data (red
markers) are provided. The activation energy, $U_{\rm A}$, and the room
temperature conductivity, $\sigma_{0,F}$, are uncorrelated (Pearson
correlation coefficient = 0.29). While the variance in $\sigma_{0,F}$ is
rather large, the variation in $U_{\rm A}$ is small.

### Figure S6

Figure 12: Arrhenius temperature dependence of the conductance. (A) Low-bias
conductance as a function of temperature plotted above 50 K for two- and four-
probe measurements, shown in blue and red respectively. Measurements taken on
four different segments. Arrhenius fits (performed for data where $T>100$ K)
and corresponding activation energies are indicated. (B) Histogram with the
activation energies measured in $n=53$ segments (blue: two-probe measurements;
red: four-probe measurements). The mean activation energy is $42\pm 8$ meV
($n=$ 53) and the mean goodness of fit across all samples as expressed by the
coefficient of determination is $\langle R^{2}\rangle\approx 0.982$.

### Figure S7

Figure 13: Schematic of a one-dimensional hopping chain. The vertical axis
denotes the energy, $U$, and the horizontal axis the position, $x$, along the
chain. The yellow rectangles represent the electrodes that have an applied
voltage bias $V$ ($U=0$ at the right electrode and $U=eV$ at the left
electrode). The horizontal black lines indicate the charge carrier sites,
numbered from $i=1$ to $i=N_{\rm S}$, where $N_{\rm S}$ is the number of
sites. The parameter $a$ is the center-to-center distance between adjacent
sites. There is an equal energy drop between neighboring sites,
$\Delta=eV/N_{\rm S}$. There are forward hopping rates, $\Gamma_{\rm R}$, and
backwards hopping rates, $\Gamma_{\rm L}$, between any pairs of sites, but
only hopping rates between nearest neighbour sites are taken into account. The
reorganization energy, $\lambda$, determines the activation barrier for
hopping (see Supplementary Text).

### Figure S8

Figure 14: Linear scaling of the number of hopping sites with the segment
length. The number of hopping steps, $N_{\rm S}$, is determined in two ways.
In a first approach (red markers), $N_{\rm S}$ is derived from the transition
point in the universal scaling curves (see fig. S10). In a second approach
(green markers), $N_{\rm S}$ is calculated via Eq. 27 in the Supplementary
Text. In both cases, $N_{\rm S}$ linearly scales with the segment length $L$
(solid lines provide linear fits, excluding offset). The center-to-center
distance between hopping sites, $a$, is given by the slope $a=L/N_{\rm S}$.
Both methods imply $a$ values that are in excess of 10 nm. (A) Data from three
independent fiber sheaths. (B) Data obtained from different segments on a
single fiber sheath.

### Figure S9

Figure 15: Comparison of electron transfer rates in the purple bacterium
_Allochromatium vinosum_ versus the cable bacterium _Electrothrix gigas_.
Experimental data are respectively obtained from Ref. 18 for _A. vinosum_
(green dots) and segment 6 (L = 40 $\mu$m) from the dataset provided here
(green dots). The hopping model with a single effective vibronic mode (Jortner
model; Suppl. Text) is fitted to both datasets (solid lines). Fitting
parameters for _A. vinosum_ : $\hbar\langle\omega\rangle=60$ meV,
$\lambda=1.2$ eV and $V_{\rm DA}=1$ meV. For _E. gigas_ , the measured current
was first converted into an electron transfer rate via Eq. 8 with parameters:
voltage bias $V=0.5$ V, $N_{\rm S}=850$, $N_{\rm F}=60$ and $N_{\rm C}=125$.
The parameters used for the Jortner model fit are:
$\hbar\langle\omega\rangle=9$ meV, $\lambda=0.24$ eV and $V_{\rm DA}=6$ meV.

## Tables S1 and S2

### Table S1

Table 1: Parameters estimated from high-temperature conductance data. The
first column displays the sample identification number (16 segments have been
analysed). $L$ denotes the segment length probed between two gold electrode
pads. $G_{0}$ is the conductance measured at room temperature, $T=300$ K.
$\sigma_{\rm 0,F}$ is the fiber conductivity at $T=300$ K calculated via Eq.
2. The activation energy, $U_{\rm A}$, is determined by fitting the Arrhenius
relation (Eq. 16) to the high-temperature conductance data ($T>$ 100 K; see
fig. S11). The cross-over temperature, $T_{\rm C}$, is the first temperature
where the conductance as predicted by the Arrhenius fit is 20% lower than the
actually measured conductance.

### Table S2

Table 2: Parameters estimated from fitting the proposed hopping model. The
first column displays the sample identification number (same 16 segments as in
table S1). $L$ is the segment length between two electrode pads. The
reorganization energy, $\lambda$, is obtained by fitting the Marcus expression
(Eq. 13) to the high-temperature conductance data ($T$ > 100 K; see Fig. 3A).
The electronic coupling, $V_{\rm DA}$, is obtained via Eq. 28. The transition
rate, $\Gamma_{0}$, at 300 K is calculated from the Marcus rate expression
(Eq. 12) using the values for $\lambda$ and $V_{\rm DA}$ listed. The power law
exponent, $\alpha$, was obtained by fitting the temperature-based power law
$G\propto T^{\alpha}$ to the conductance data in the region 20 - 100 K (Fig.
3C). The power law exponent, $\beta$, was obtained by fitting the electric
field-based power law (Eq. 23 and Eq. 9) to the conductance data for $T<10$ K
(Fig. 3D). An alternative value for this exponent, $\beta_{\rm USC}$, is
obtained by fitting the universal scaling curve (fig. S10). The characteristic
vibration energy, $\hbar\omega_{\rm c}$, was determined via Eq. 20. The number
of sites $N_{\rm S,NUM}$ was obtained by Eq. 27 and the corresponding center-
to-center distance is $a_{\rm NUM}=L/N_{\rm S,NUM}$. Alternatively, the number
of sites $N_{\rm S,USC}$ was also independently obtained from the universal
scaling curves (fig. S10) and the corresponding center-to-center distance
between sites, $a_{\rm USC}=L/N_{\rm S,USC}$ is also given.

## Supplementary Text

### Calculation of the fiber conductivity

The fiber conductivity, $\sigma_{0,F}$, at room temperature ($T=$ 300 K) was
calculated from the experimental I/V data as:

$\sigma_{\rm 0,F}=G_{0}\frac{L}{N_{\rm F}A_{F}}=G_{0}\frac{4L}{N_{\rm F}\pi
d_{\rm F}^{2}}.$ (2)

In this, $G_{0}$ represents the low-bias conductance of a single fiber sheath
as determined at $T=300$ K from the slope of the I/V curve, $L$ is the length
of the fiber sheath segment investigated as determined by microscopy, $A_{F}$
is the cross-sectional area of one fiber, and $N_{\rm F}$ is the total number
of fibers embedded in the fiber sheath. Experimentally determined values for
$G_{0}$ and $L$ are given for different segments in table S1. The number of
parallel fibers $N_{\rm F}$ is determined by scanning electron microscopy or
atomic force microscopy on individual filaments (?) and is here $N_{\rm F}$ =
60 (see fig. S1). The cross-sectional area is taken the same value for all
segments: $d_{\rm F}=$ 26 nm is the diameter of the conductive core of an
individual fiber, which is invariant across different filaments (?).

### Hopping model of electron transport in cable bacteria

#### Geometry of the conductive network

Cable bacteria possess a set of $N_{\rm F}$ fiber structures in their cell
envelope, which are arranged in parallel and run continuously along the whole
length of cm-long filaments (?) ($N_{\rm F}$ = 60; see above). The fibers
themselves consist of a conductive core (diameter $d_{\rm F}$ = 26 nm),
surrounded by an electrically insulating shell (?). The large ratio of length
(1 cm) to diameter (26 nm) suggests that the electron transport in the fibers
is one-dimensional. Still, there could be multiple conduction channels acting
in parallel in a single fiber, as $d_{\rm F}$ far exceeds the typical spacing
between charge carrier sites in metalloproteins ($1-2$ nm).

At present, it is not known how many conduction channels are actually present.
Here, we assume that a conductive fiber core consists of multiple parallel
molecular conduction paths, each $d_{\rm P}=2$ nm in diameter. Assuming a
hexagonal packing of cylinders, the number of parallel channels in one
periplasmic fiber can then be estimated as $N_{\rm C}=125$. The total number
of parallel paths in a given cable bacterium filament thus amounts to $N_{\rm
P}=N_{\rm F}\cdot N_{\rm C}=$ 7500\. This large number of conduction channels
should be regarded as an upper limit of what’s physically possible, thus
providing a conservative estimate for the electron transfer rate. If less
channels are present, then the conductivity of individual channels must be
higher, and this in turn also necessitates higher electron transfer rates (see
discussion in main text).

#### Current through a single conduction channel

Each conduction channel is modelled as a one-dimensional chain of $N_{\rm S}$
equidistant hopping sites, which serve as temporary localization centers of an
electron during the hopping process (fig. S13). The site at position, $i$,
along the chain is characterised by the site energy, $U_{\rm i}$. Transitions
between sites represent the quantum-mechanical tunneling of electrons, and are
characterised by a forward rate, $\Gamma_{\rm i,j}$, and a backward rate,
$\Gamma_{\rm j,i}$. The ratio between forward and backward rates is governed
by the detailed balance relation (?, ?):

$\Gamma_{\rm i,j}/\Gamma_{\rm j,i}=\text{exp}(\Delta/k_{\rm B}T).$ (3)

Here, $k_{\rm B}$ is the Boltzmann constant, $T$ is temperature, and
$\Delta=U_{\rm i}-U_{\rm j}$ is the driving force, i.e., the difference
between the site energy of the initial position of the electron, $U_{\rm i}$,
and that of its final position, $U_{\rm j}$. In our model, only nearest
neighbour hopping is considered ($j=i\pm 1$). Moreover, the hopping sites are
assumed to be identical and periodic in structure, with the same center-to-
center distance $a$ between them (fig. S13). As sites are identical, the
transition rate is the same for every pair of sites ($\Gamma_{\rm
i,i+1}=\Gamma_{\rm R}$; $\Gamma_{\rm i+1,i}=\Gamma_{\rm L}$). Moreover, if we
connect the hopping chain to an electrode on either side, and impose a voltage
bias $V$ between the electrodes, the energy difference between two sites is
constant and amounts to $\Delta=eV/N_{\rm S}$ (assuming there is no voltage
drop over the electrode interface). The current between two consecutive
hopping sites along a single conduction path is given by:

$I_{\rm P}=e\left[\Gamma_{\rm R}p_{\rm i}\left(1-p_{\rm
i+1}\right)-\Gamma_{\rm L}p_{\rm i+1}\left(1-p_{\rm i}\right)\right],$ (4)

where $p_{\rm i}$ and $p_{\rm{i+1}}$ are the occupation probabilities at the
consecutive hopping sites, and $e$ is the elementary charge (i.e., the
magnitude of the charge carried by a single electron). This expression embeds
the Pauli exclusion principle: an electron can only jump to a charge carrier
site when this site is vacant.

Because the current is not dependent on the pair of sites where it is
calculated, only certain combinations of $p_{i}$ are possible. We make the
simplification that our hopping chain is optimally filled for conductance
($p_{\rm i}=1/2$), using the following justification. Because in the electrode
many electron states can contribute to the effective hopping rates at the
interface, these interfacial rates can be expected to be far greater than the
internal hopping rates, $\Gamma_{\rm R,L}$. The occupation probability of the
edge hopping sites, $p_{\rm 1}$ and $p_{\rm N}$, are then expected to follow
the Fermi-Dirac distribution in the metal electrodes:

$p_{\rm FD}=\frac{1}{1+\text{exp}\left(\frac{\epsilon}{k_{\rm B}T}\right)}.$
(5)

Here, $p_{\rm FD}$ is the occupation probability of an electron state inside
the electrode and $\epsilon$ is the energy difference of this electron state
with respect to the energy of the electrode. If $\epsilon=0$ for $i={1,N}$
(see fig. S13), then $p_{1}=p_{N}=1/2$ and no occupancy difference is expected
to build up in the hopping chain, because all internal hopping rates
$\Gamma_{\rm R,L}$ are the same. Therefore, all sites in the chain can be
expected to have the occupancy $p_{i}=1/2$.

Accounting for $N_{\rm P}$ parallel conduction channels, and implementing the
detailed balance relation, the total current along a single filament, can be
expressed as:

$I=N_{\rm P}I_{\rm P}=e\frac{1}{4}N_{\rm P}(\Gamma_{\rm R}-\Gamma_{\rm L}).$
(6)

Implementing the detailed balance relation, we obtain:

$I=e\frac{1}{4}N_{\rm P}\Gamma_{\rm R}\left(1-\text{exp}(-\Delta/k_{\rm
B}T)\right).$ (7)

Two separate regimes can be discerned. In the low-bias, high-temperature
regime ($k_{\rm B}T>eV/N_{\rm S})$, the current in Eq. 7 simplifies to:

$I\approx\frac{e}{4}N_{\rm P}\Gamma_{\rm R}\frac{eV}{N_{\rm S}k_{\rm B}T}.$
(8)

Alternatively, in the low-temperature, high-bias regime ($eV/N_{\rm S}>k_{\rm
B}T$), the exponential term in Eq. 7 vanishes, and the current reduces to:

$I\approx e\frac{1}{4}N_{\rm P}\Gamma_{\rm R}.$ (9)

The objective is now to find a suitable expression for the transition rate,
$\Gamma_{\rm R}$.

#### Transition rate estimates

The fiber conductivity $\sigma$ can be obtained from the current expression in
the low-temperature, high-bias regime, Eq. 8, as:

$\sigma=\frac{I}{V}\frac{L}{N_{\rm F}A_{F}}=\Gamma_{\rm R}\frac{e^{2}}{k_{\rm
B}T}\frac{LN_{\rm C}}{N_{\rm S}\pi d_{\rm F}^{2}},$ (10)

As described above, the fiber sheath has $N_{\rm F}=60$ fibers with a diameter
of $d_{\rm F}=26$ nm, and a cross-sectional area $A_{F}=\pi d_{\rm F}^{2}/4$.
Moreover, we estimate that a maximum of $N_{\rm C}=125$ parallel conduction
channels can be integrated in a single fiber (see ’Geometry of the conductive
network’). The above expression can be rearranged to provide the transition
rate $\Gamma_{R}$ as a function of the measured conductivity

$\Gamma_{\rm R}=\sigma\frac{k_{\rm B}T}{e^{2}}\frac{\pi d_{\rm F}^{2}}{aN_{\rm
C}},$ (11)

In this, $a=L/N_{\rm S}$ represents the mean center-to-center distance between
two consecutive sites.

Conductivities on the order of $\sigma=100$ S/cm have been estimated for the
cable bacterium fibers (see also fig. S2). Moreover, typical center-to-center
distances, $a$, between the cofactors in metalloproteins are in the range of
0.5 < $a$ < 2 nm. For a typical cable bacterium filament ($L$ = 1 cm) this
hence translates in between 5 and 20 million sites along the hopping chain.
Using these numbers, Eq. 11 predicts that the transition rate must range
between $\Gamma_{R}$ = 13 $-$ 54 x 1012 s-1. This hopping rate exceeds the
speed limit of 1013 s-1 for non-adiabiatic electron transfer (?), thus
illustrating the theoretical challenge of applying multistep hopping models in
cable bacteria (see discussion in main text).

### Transition rate models

At low temperatures, the energy stored in vibrational modes can stimulate the
hopping process. To account for this effect, several authors have formulated
an extended, quantum version of the semi-classical Marcus electron transfer
theory. These models explicitly account for vibrational modes that suitably
couple to the electron transfer. Two prominent examples are the single
effective mode formulation presented by Jortner (?) and the multiple mode
formulation developed by Egger (?). Below, we will integrate these two
vibrational model formulations in the multistep hopping chain model introduced
above. But first, we introduce the classical Marcus rate formalism as a
reference for these more elaborate quantum-based models.

#### High-temperature regime: the Marcus model formulation

At a sufficiently high temperatures, the thermal energy, $k_{\rm B}T$,
sufficiently exceeds the quantum energy, $\hbar\omega$, of the intramolecular
vibrations, and all vibrations are then thermally excited. Under these
conditions, the transition rate between neighbouring sites is given by the
classical Marcus rate expression (?),

$\Gamma_{\rm R}=\frac{2\pi}{\hbar}\frac{V^{2}_{\rm DA}}{\sqrt{4\pi\lambda
k_{\rm B}T}}\text{exp}\left[\frac{-(\lambda-\Delta)^{2}}{4\lambda k_{\rm
B}T}\right].$ (12)

Here, $\lambda$ is the reorganization energy, $V_{\rm DA}$ is the electronic
coupling, and $\Delta=eV/N_{\rm S}$ is the energy difference between adjacent
sites introduced above. Upon substitution of this expression into the high-
temperature regime current (Eq. 8), we obtain the conductance:

$G=\frac{e^{2}N_{\rm P}}{4N_{\rm S}k_{\rm
B}T}\frac{2\pi}{\hbar}\frac{V^{2}_{DA}}{\sqrt{4\pi\lambda k_{\rm
B}T}}\text{exp}\left[{\frac{-(\lambda-\frac{eV}{N_{\rm S}})^{2}}{4\lambda
k_{\rm B}T}}\right].$ (13)

Because the bacterial filaments investigated are very long ($N_{\rm
S}>10^{2}$), the driving force $\Delta=eV/N_{\rm S}$ in the low-bias regime
($V<0.5$ V) will be < 5 meV, which is substantially smaller than the
reorganization energy, $\lambda\approx 270$ meV (Fig. 3). Consequently, the
exponent in Eq. 13 reduces to $e^{-\lambda/4k_{\rm B}T}$ and the conductance
relation simplifies to:

$G=g_{0}(k_{\rm B}T)^{-3/2}\text{exp}(-\frac{\lambda}{4k_{\rm B}T}).$ (14)

The pre-factor $g_{0}$ is given by:

$g_{0}=\frac{e^{2}N_{\rm P}}{4N_{\rm S}}\frac{2\pi}{\hbar}\frac{V^{2}_{\rm
DA}}{\sqrt{4\pi\lambda}}.$ (15)

Eq. 14 was fitted to the $G(T)$ data to provide values for $\lambda$ and
$g_{0}$ (non-linear least squares fit of $G$ versus $1/T$ as displayed in Fig.
3A; values for $\lambda$ and $V_{\rm DA}$ tabulated in table S2).

This fitting procedure assumes that $g_{0}$ itself is temperature independent.
However, it is known that the electronic coupling $V_{\rm DA}$ is weakly
temperature dependent (?). Consequently, the exact temperature dependence of
the pre-exponential factor in Eq. 14 is unknown. As a simplification, one can
leave the $T^{-3/2}$ dependence out of the equation, to arrive at the
classical Arrhenius relation:

$G=g_{1}\text{exp}\left(-\frac{U_{\rm A}}{k_{\rm B}T}\right).$ (16)

Here, the pre-factor $g_{1}=g_{0}(k_{\rm B}T)^{-3/2}$ is now considered to be
a temperature-independent parameter. This basic Arrhenius relation was also
fitted to the high temperature conductance data to arrive at values for the
activation energy $U_{\rm A}$ (fig. S11; table S1).

#### Low-temperature regime, one effective vibrational mode: the Jortner model
formulation

When the quantum energy of vibrations $\hbar\omega$ becomes smaller than the
thermal energy, $k_{\rm B}T$, the Marcus rate expression (Eq. 12) is no longer
valid. In the Jortner model formulation, vibrational modes are characterised
by a single effective mode $\langle\omega\rangle$, which provides a suitable
average across all molecular vibrations. The corresponding transition rate
becomes (?):

$\Gamma_{\rm R}=\Gamma_{0}\cdot\text{exp}(-S(2\overline{n}_{\rm B}+1))\cdot
I_{\rm q}(2S(\overline{n}_{\rm B}(\overline{n}_{\rm
B}+1))^{1/2})\cdot(\overline{n}_{\rm B}(\overline{n}_{\rm B}+1))^{q/2}.$ (17)

In this expression, $\Gamma_{0}$ is a temperature-independent pre-factor,
$S=\lambda/\hbar\langle\omega\rangle$ is the electron-phonon coupling strength
associated with $\langle\omega\rangle$, $\overline{n}_{\rm
B}=1/(\text{exp}(\hbar\langle\omega\rangle/k_{\rm B}T)-1)$ is the Bose-
Einstein distribution for the single effective mode $\langle\omega\rangle$,
the exponent $q=\frac{\Delta}{\hbar\langle\omega\rangle}$ is the ratio between
the site energy drop between hopping sites and the mean quantum vibrational
energy, and $I_{\rm q}$ is the modified Bessel function (?). One can show that
at high temperatures (when $\hbar\langle\omega\rangle\ll k_{\rm B}T$), the
Marcus rate expression is recovered.

The Jortner expression for the transition rate, Eq. 17, can be combined with
the expression for the current along a single filament, Eq. 7, in order to
estimate the conductance $G$ as a function of temperature and the electric
field strength. The results are given in Fig. 2C and D. At high temperatures,
the conductance follows an Arrhenius type dependence (Fig. 2C), while at low
temperatures ($k_{\rm B}T\ll\hbar\langle\omega\rangle$), the conductance
becomes temperature-independent (Fig. 2D). This calculated $G(T,E)$ response
shows a good agreement with the data.

#### Low-temperature regime, multiple vibrational modes: the Egger model
formulation

In the Egger model formulation (?), it is assumed that many vibrational modes
are coupled to the electron hopping process. The phonon spectral density
function $J(\omega)$ describes how much a certain vibrational mode contributes
to the reorganization energy (?):

$\lambda=\hbar\int_{0}^{\infty}\omega{J(\omega)}{\rm d}\omega.$ (18)

The Egger model formulation assumes that the phonon spectral density function
has an Ohmic form and can be expressed as:

$J(\omega)=(\beta+2)\frac{e^{-\omega/\omega_{\rm c}}}{\omega}.$ (19)

Here, $\omega_{\rm c}$ is the characteristic frequency of the spectrum. For
the Ohmic spectral density, the reorganization energy is directly proportional
to the characteristic frequency, $\omega_{\rm c}$, as stated in the original
derivation of the Egger model (?). :

$\lambda=(\beta+2)\hbar\omega_{\rm c},$ (20)

Since the spectrum starts at zero frequency, some modes remain thermally
activated at the lowest temperatures. This explains an important difference
between the Jortner and Egger model formulations. Whereas in the Jortner
model, the low-bias conductance remains constant at cryogenic temperatures,
this is not the case in the Egger model: the low-bias conductance shows a
power law behaviour, in which the conductance further decreases as the
temperature decreases. Effectively, the modes that remain thermally activated
at low temperatures force the low-bias conductance to die out.

When the temperature is much lower than the characteristic energy of vibration
($k_{\rm B}T\ll\hbar\omega_{\rm c}$), the Egger model provides following
expression for the transition rate (?):

$\Gamma_{\rm R}=\frac{1}{\gamma(\beta+2)\hbar}\frac{V_{\rm
DA}^{2}}{\hbar\omega_{\rm c}}\left(\frac{2\pi k_{\rm B}T}{\hbar\omega_{\rm
c}}\right)^{\beta+1}\left\lvert\gamma\left(1+\beta/2+i\frac{eV}{2\pi N_{\rm
S}k_{\rm B}T}\right)\right\rvert^{2}\text{exp}\left(\frac{eV}{2N_{\rm S}k_{\rm
B}T}\right).$ (21)

Here, $\gamma$ is the complex gamma function and the exponent $\beta$ provides
a measure for the electron-phonon coupling strength. If we combine the
expression for the current along a single filament, Eq. 7, with the above
expression for the transition rate, Eq. 21 , we obtain:

$I=A_{0}T^{\beta+1}\text{sinh}\left(\frac{eV}{2\pi N_{\rm S}k_{\rm
B}T}\right)\left\lvert\gamma\left(1+\beta/2+i\frac{eV}{2\pi N_{\rm S}k_{\rm
B}T}\right)\right\rvert^{2}.$ (22)

The pre-factor $A_{0}$ combines all temperature-independent parameters. Note,
that the Egger model centrally depends on the assumption that the phonon
spectral density function has an Ohmic form. However, at present, the details
of electron-phonon coupling in the conductive fibers of cable bacteria are
unknown, and so, alternative shapes phonon spectral density function could be
relevant (?).

#### Universal scaling curve

Closer inspection of Eq. 22 reveals that the $I(V)$ data can be replotted as a
single universal scaling curve, provided one normalizes the bias voltage as
$eV/(2\pi k_{\rm B}T)$ (the horizontal axis) and rescales the current as
$I/T^{\beta+1}$ (vertical axis). If the Egger model applies, then the $I(V)$
data collected at different temperatures should fall onto a single curve, only
depending on $A_{0}$, $\beta$, and the number of hopping sites $N_{\rm S}$.
These universal scaling plots are shown in fig. S10. In the low-bias, high-
temperature regime ($eV/N_{\rm S}<k_{\rm B}T$), the approximation
$\text{sinh}(x)\approx x$ can be made and then $G=I/V\propto T^{\beta}$.
Accordingly, one expects the conductance to follow a power law dependence on
temperature, which is also seen in the data (Fig. 3C). Oppositely, in the
high-voltage, low-temperature regime ($eV/N_{\rm S}>k_{\rm B}T$), the
transition rate is given by (?):

$\Gamma_{\rm
R}=\frac{2\pi}{\hbar}\frac{V^{2}_{DA}}{\gamma(\beta+2)\hbar\omega_{\rm
c}}\left(\frac{eV}{N_{\rm S}\hbar\omega_{\rm
c}}\right)^{\beta+1}\text{exp}\left(-\frac{eV}{N_{\rm S}\hbar\omega_{\rm
c}}\right).$ (23)

If we combine this expression with that for the current in the high-bias, low-
temperature regime (Eq. 9), the conductance becomes:

$G=\frac{e^{2}}{4}\frac{N_{\rm P}}{N_{\rm
S}}\frac{2\pi}{\hbar}\frac{V^{2}_{DA}}{\gamma(\beta+2)(\hbar\omega_{\rm
c})^{2}}\left(\frac{eV}{N_{\rm S}\hbar\omega_{\rm c}}\right)^{\beta}.$ (24)

In this equation, we have dropped the exponential decay factor in Eq. 23, as
one can prove that for small $x=eV/(N_{\rm S}\hbar\omega_{\rm c})$, this term
is very close to one. The overall result is that the conductance depends on
the bias voltage via a power law: $G\propto V^{\beta}$. This relation is
indeed seen in the data (Fig. 3D).

#### The distance between hopping sites

Each conduction channel is modelled as a one-dimensional chain of $N_{\rm S}$
equidistant hopping sites with the same center-to-center distance $a=L/N_{\rm
S}$ between them (fig. S13). We can estimate the number of hopping sites
$N_{\rm S}$ from our data in two independent ways. In a first approach, we can
extract $N_{\rm S}$ directly from the universal scaling curve (fig. S10). The
scaling curves show a pronounced inflection point at the transition from the
low-bias, high-temperature regime towards the high-bias, low-temperature
regime. This transition occurs at a bias

$eV\approx 2\pi N_{\rm S}k_{\rm B}T$ (25)

As a result, the value of $eV/2\pi k_{\rm B}T$ at the inflection point of the
scaling curve provides a direct estimate of the apparent number of hopping
steps $N_{\rm S}$. In fig. S10, panel (O), this transition point is indicated
by the dotted line. In table S2, the resulting values of $N_{\rm S,USC}$ are
listed for each of the 16 segments investigated. The corresponding estimate
for the center-to-center distance $a_{\rm USC}=L/N_{\rm S,USC}$ ranges between
33 and 102 nm.

In the second approach, a value for $N_{\rm S}$ is numerically calculated from
other parameters available. To this end, we first note that for low
temperatures and high voltages ($eV/N_{\rm S}\gg k_{\rm B}T$), the conductance
scales as $G=B_{0}V^{\beta}$, where $B_{0}$ is the pre-factor in the power law
fit. Using Eq. 24, this pre-factor can be explicitly written as:

$B_{0}=\frac{e^{2}}{4}\frac{N_{\rm P}}{N_{\rm
S}}\frac{2\pi}{\hbar}\frac{V^{2}_{DA}}{\gamma(\beta+2)(\hbar\omega_{\rm
c})^{2}}\left(\frac{e}{N_{\rm S}\hbar\omega_{\rm c}}\right)^{\beta}.$ (26)

Rearrangement provides an expression for the number of hopping sites:

$N_{\rm S}=\frac{e}{\hbar\omega_{\rm c}}\left[\frac{eN_{\rm
P}}{4}\frac{2\pi}{\hbar}\frac{V^{2}_{\rm DA}}{\hbar\omega_{\rm
c}\gamma(\beta+2)}\frac{1}{B_{0}}\right]^{1/(\beta+1)}.$ (27)

This expression requires an estimate for the electronic coupling $V_{\rm DA}$.
To arrive at this, we note that the pre-factor $g_{0}$ (as derived from the
fitting of the Marcus rate expression, Eq. 15) can be transformed into an
expression for the electronic coupling:

$V_{\rm DA}^{2}=\left[g_{0}\frac{\hbar}{2\pi}\frac{4N_{\rm S}}{e^{2}N_{\rm
P}}\sqrt{4\pi\lambda}\right].$ (28)

Substitution of Eq. 28 into Eq. 27 finally gives:

$N_{\rm
S}=\frac{e}{\hbar\omega_{c}}\left[\frac{\sqrt{4\pi\lambda}}{\gamma(\beta+2)(\hbar\omega_{\rm
c})^{2}}\frac{g_{0}}{B_{0}}\right]^{1/\beta}.$ (29)

From this, $N_{\rm S}$ can be calculated provided that we supply parameters
that are fitted in the low-temperature regime (the pre-factor $B_{0}$, the
coupling exponent $\beta$, the characteristic frequency $\hbar\omega_{\rm c}$)
as well as parameters that are determined in the high-temperature regime (pre-
factor $g_{0}$, reorganization energy $\lambda$).

The above derivation essentially assumes that the electronic coupling is
independent of temperature (the high-temperature value is substituted into the
low-temperature expression). It is, however, known that $V_{\rm DA}$ is
enhanced by thermal fluctuations. So in reality, the value of $V_{\rm DA}$
could be smaller at lower temperatures, and as a consequence, the value of
$N_{\rm S}$ will be overestimated by Eq. 29. This could explain the
differences obtained for $N_{\rm S}$ by the two separate estimation procedures
(see fig. S14). However, in high-mobility organic semiconductors, this
temperature effect changes the $V_{\rm DA}$ value by less than one order of
magnitude (?). Because of the power law dependence $1/\beta$ in Eq. 29, such a
tenfold difference in electronic coupling only causes a two-fold difference in
the $N_{\rm S}$ estimate.

The resulting values of $N_{\rm S,NUM}$ are listed in table S2 for all 16
segments investigated. The associated center-to-center distance $a_{\rm
NUM}=L/N_{\rm S,NUM}$ ranges between 10 and 36 nm, which is a factor of 3
lower than the corresponding estimate based on the universal scaling curve.
fig. S14 shows that the number of hopping sites linearly scales with the
segment length. This trend is seen for both approaches.