Long Range Electron Transport Rates Depend on Wire Dimensions in Cytochrome Nanowires

The ability to redirect electron transport to new reactions in living systems opens possibilities to store energy, generate new products, or probe physiological processes. Recent work by Huang et al. showed that 3D crystals of small tetraheme cytochromes (STC) could transport electrons over nanoscopic to mesoscopic distances by an electron hopping mechanism. Such protein-based structures with multiple localized electron carriers are promising materials for nanowires. A potential barrier to protein nanowire adoption for handling long-range electron transport is that fluctuations at room temperature may distort the nanostructure, hindering efficient electron transport. These fluctuations at the nano-and mesoscopic scales are quantified using classical molecular dynamics simulations for a small fragment of a STC nanowire and measure the effective distance distribution between electron carriers. From distance distributions, we develop a graph network representation for electron transport along nanowires with varying dimensions, and through stochastic methods determine the maximum electron flow that can be driven through these STC wires. Longer nanowires were capable of carrying fewer electrons than shorter nanowires with the same diameter, as long electron transfer distances that occasionally arise reduce the efficiency for electron transport. Thicker nanowires permit more alternative transport pathways, increasing electron transport beyond the increase in cross-section. Thus, this model implies that the design of protein-based nanowires that depend on electron hopping between charge carriers must consider control of the inherent protein flexibility, as more flexible protein-protein interfaces impose a limit on the required minimum diameter to carry currents commensurate with conventional electronics.


Introduction
Biology harnesses electrical signals and impulses to regulate and control nanoscale processes.With ongoing miniaturization efforts for conventional electronics and nanoneedles that penetrate cells [1], it will soon be possible to measure these electrical signals with high spatiotemporal accuracy, overcoming a major challenge to in vivo measurements in single cells.Example measurements may include directly measuring protein activity [2], detecting nanomolar concentrations [3,4,5,6], identifying cancer cells [7,8,9,10], and other diseases [11,12,13,14].These applications depend on nanowires, nanopillars, and nanomeshes to conduct electrical signals out of the cell, using materials such as Si, InP, Au, and TiO 2 [15,16].Ranging in thickness starting from 1 nm to 100 nm and with a micrometer to millimeter length [15], conventional nanowires not only nondestructively measure cell properties with minimal impact, but can also supply electrons to catalytic processes within cells, such as synthesizing acetic acid directly from CO 2 [17], or optically modulate action potentials in neurons [18], membrane properties [19] and metabolism [20,21,22].Recently, new promising biological nanowire materials are emerging that are based on DNA [23,24], fibrous proteins [25,26,27,28] or heme-containing proteins [29,30,31,32,33], each aiming to carry current efficiently over long distances with biocompatible molecules.Nanowires synthesized biologically offer the advantage of sustainable sourcing and self-assembly, as well as controlling critical properties such as redox potentials and conductances, without the need for toxic solvents and harsh chemical processes used in inorganic nanowire synthesis [34].While biological nanowires show tremendous promise, the comparatively sparse electron carriers within biological nanowires alter long range electron transport dynamics compared with inorganic atomic nanowires.In this study, we investigate the limits of this long range electron transport through a crystalline network of small tetraheme cytochrome proteins at room temperature in the context of understanding previous electron transport measurements triggered by photoreduction [29].Mechanistically, electron transport within multi-heme proteins in aqueous environments is thought to occur via individually identifiable heme-to-heme electron transfers (hops), which has been confirmed by both experimental and computational methods [35,36,37,31,38,22,39].While coherent tunneling has also been suggested [40], the slow conductivity decay with increasing nanowire or nanocrystal length argues in favor of a heme-toheme hopping mechanism [32,41,29].
The rate at which electrons transfer between heme groups within and between STCs k et depends on the relative orientation of the heme groups, the potential energy differences between the hemes, and local environmental factors [42].The nonadiabatic Marcus theory describes this relationship in terms of a Gibbs free energy ∆G, a reorganization energy λ, and an electron coupling parameter [43].A common approximation to the Marcus theory is the Moser-Dutton ruler [44], in which the electron coupling parameter is replaced by the minimum distance R between donor and acceptor groups, such as the porphyrin rings within a heme group, and a scaling factor.
Crystallography has long been a key tool to estimate the distances between porphyrin rings within single heme-bearing proteins, and indeed early crystal structures for single domain proteins were key to establishing the relationship between structure and electron transfer [44].However, crystallographic data is often collected at cryogenic temperatures, where thermal motion is substantially reduced.At room temperature, where potential nanowires show their greatest application potential, increased motion at the nanoscale may broaden the distribution for inter-porphyrin distances, particularly between heme groups in adjacent protein monomers, and offers a potential mechanism to explain anti-Arrhenius electron transport along nanocrystals [29,45,39], particularly in light of recent work studying physical motion of nanowire components [33].Based on Marcus theory, the broader intermonomer distance distributions would reduce electron transfer rates and limit the electron throughput for biological nanowires.
To simulate the impact of thermal protein fluctuations on the long range electron transport within biological nanowires, we designed a STC nanowire fragment along the crystallographic b-axis solvated in water (Figure 1).After microsecond-scale MD simulations at different temperatures, we measure interporphyrin distance distributions within the nanowire.We translate these distance distributions into graph networks modeling electron transport pathways available within a STC nanowire.From the graph representation for the nanowire, we measure the maximum electron flow for a STC nanowire along the three crystallographic axes (a, b, and c, see Fig. S1, S2).By modulating dimensions for the graph networks, we predict that the maximum electron flow decreases with increasing wire length, as longer wires will be limited by sites where thermalization has increased the interprotein heme-to-heme distance.The simulations suggest that thinner wires should exhibit this limit more strongly than thicker wires, suggesting that bio-derived nanowires may have a minimum effective thickness for use in conventional electronics applications.

Crystal structure relaxation to room temperature
The completely oxidized STC nanowire fragment shown in Figure 1 was simulated at four different temperatures between 273K and 309K, with animations included in the Supplementary Information.Beginning from the initial crystalline structure solved at cryogenic temperatures, the animations highlight rapid crystal expansion upon thermalization, with the structures reaching an expanded equilibrium state at room temperature (Figure S3A).The distance distributions between heme groups within a STC were extracted from the molecular dynamics simulations and quantified for all simulated temperatures in Figures S4 and S5.Qualitatively, we find that the distances between interprotein heme groups increase within our simulations relative to crystal structure expansion due to increased thermal motion (Figure S3B).
Although the interprotein heme-to-heme distance histograms widened slightly with increasing temperature, this was not significant in the measured temperature range, and the mean distances did not shift appreciably (Figure S4).Intraprotein heme-to-heme distances were not affected by temperature (Figure S5).

Individual electron transfer rates
The slightly expanded distances upon thermalization will have an impact on the electron transfer along the nanocrystal.The conversion from distance distributions to electron transfer rates is a two-step process, first fitting the distance distributions determined from simulation to a probability density function to enable random sampling, and subsequently using the Moser-Dutton ruler [44,46] to convert randomly sampled distances to electron transfer rates cost effectively.The inter-heme distance distributions exhibited general skewness, with longer distances more likely than a symmetric gaussian distribution would suggest (Figure S6).Longer distances than the crystal average happen when the individual STCs move apart during thermalization, while exceptionally short distances are sterically prohibited.Thus, the distances between hemes were fit to exponentially modified Gaussian functions f (R) that tailed towards higher distances.
Within the STC nanowire fragment, data from multiple structurally equivalent heme-to-heme distances were combined when fitting Eq. 2 to determine the mean distance µ, standard deviation σ, and rate of exponential component ϕ parameters.Integrating Eq. 2 to create the cumulative distribution function allows for a conventional pseudo-random number generator to sample each individual distance distribution parameterized from our simulations, so that we can rapidly generate statistically meaningful sampling without extensive molecular simulations.The distances R were translated into electron transfer rates k et between individual heme groups using the Moser-Dutton ruler Eq. 1 for endergonic processes [44,46].The activation free energy ∆G † for this process is described by the Gibbs free energy ∆G and the reorganization energy λ.
Intramolecular electron transfers within a single STC are rapid due to short 0.4 nm to 0.6 nm distances between heme groups (Figure 2A and 3), resulting in electron transfer rate constants between 10 8.4 s −1 and 10 7.2 s −1 .The shortest average intermolecular distances between STCs are 0.82 nm, 0.63 nm and 1.73 nm for heme groups along the a-, b-, and c-axis, respectively (Figure 3).Diagonal electron transfers along multiple axes, e.g., the a-and b-axis simultaneously, are rare with distances greater than 2 nm.Therefore, the most efficient electron transport direction in the nanowire fragment is the b-axis, which has the smallest heme distances along the axis, including between individual STC monomers.Using the average distances from the heme-to-heme distance distributions together with Eq. 1 (Figure 2, SI Appendix) yields electron transfer rates constants along the nanowire axes, k a et =10 5.7 s −1 , k b et =10 6.8 s −1 and k c et =10 0.7 s −1 , qualitatively agreeing with experimentally observed electron transfer rate constants for the equivalent system in experiment [29].

Single protein chain maximum electron flow
However, average distances mask significant rate differences in individual electron transfer events because heme-to-heme distances along STC nanowires are not homogeneous.Consider an isolated single linear chain of STC proteins of finite length.In this system, the long range electron transport rate from one end to the other is likely limited by the largest heme-to-heme distance along the chain.For example, the distances between adjacent hemes for the electron transfer from 4 → 5 exceed 8.1 Å, 1.8 Å larger than the average, 1% of the time (Figure S6) based strictly on the population distribution observed from thermalizing the crystal.As a result of the extended distance, the predicted electron transport rate along this chain is k b et =10 5.9 s −1 , 10 times slower than the average transfer rate.Thus, the rare states where the inter-heme distance is longest represents a bottleneck for electron transport.The influence of these bottlenecks on the electron transport rate is highlighted with maximum electron flows in this study.Assuming that all electrons are transported in the same direction from the beginning to the end of the wire, and electrons on neighboring hemes along the transport direction do not obstruct electron transfer, the maximum electron flow I can be understood as a current without a driving electrical potential.
Using Eq 4, the maximum electron flow that would be observed through an STC chain with the bottleneck described above is 1.32 • 10 −13 A (132 fA), which is near the limit of currently detectable currents [47].

Electron transport through a nanowire
STC nanowires feature multiple parallel chains arrayed within the wire, and so while our example above is a useful demonstration of the impact heterogeneity can have in a nanocrystal, it is not the modal behavior.Since electron transfer is possible along any axis, an unusually long heme-to-heme distance along a single axis at a particular location would be expected to be less detrimental to electron transport rates, since a single defect can be circumvented by alternative paths along other wire dimensions.To model these alternative paths, we created mathematical graphs to represent arbitrary STC nanowires with varying dimensions (Figure 4A).Within the graphs, the nodes represent individual heme groups, and edges indicate that the hemes are close enough for electron transfer to take place.Edge weights within the graph are determined based on electron transfer rate constants calculated from randomly chosen values of the heme-to-heme distance distributions.A total of 23100 STC graphs were constructed for each temperature to sample the stochastic electron transport.Conductance was assessed by connecting the ends of the graph, representing the first and last cross-section layer of the nanowire, to a source and sink node, mimicking a metallic plate placed at each end of the nanowire, and measuring the maximal transport rate through the graph from source to sink (Figure S7, S8, and S9).
Since the b-axis features the shortest inter-protein heme distances (Figure S7), and therefore has the fastest electron transport rates, the electron flow along the b-axis is the focus of this study.However, since the geometry along the a-axis features similar heme-to-heme distances to those found in the baxis (Figure S7B,S8A), many of the findings presented below will also apply to the a-axis, albeit at a marginally reduced maximum electron flow.Electron transfer events along the c-axis within the nanowire fragment almost never occur (Figure S10), since the heme-to-heme distances are much larger (Figure S9C).The large interprotein heme distances along the c-axis divide the nanowire into electron transport planes in the a-and b-axes that are largely independent.The planar structure is seen most clearly in Figure 4A and Figure S1, where each transport plane has densely connected fast electron transfer paths within the plane.The planes are interconnected by slow electron transfer paths representing transfer events along the c-axis.
Analogous to what is seen for a single protein chain, in sufficiently long wires, thermal fluctuations may induce regions where the heme-to-heme distances are large in the a-and b-axes simultaneously, forcing substantially slower electron transfer and creating a bottleneck.The probability of such bottlenecks occurring will increase with increasing wire length, slowing the long range electron transport.For a wire with 4x4 STCs in the ac-plane, increasing the length in the b-axis by a factor of 10 results in a reduction of the maximum electron flow along this axis between 34% and 48% (Figure S7B).If the electron transfer rates along the perpendicular axes are fast, i.e., the distances between heme groups are short, bottlenecks can be partially avoided by diverting part of the electron transport to adjacent electron transport pathways.Consequently, blockages induced by lengthening the wire ten-fold along the a-or c-axes and having the electron source and sink in the same direction have smaller effects on the maximum electron flow (Figure S8A,S9C).Defects in the nanowire fragment, where entire proteins are missing, also reduce the maximum electron flow of the nanowire.Missing proteins stop electron transfer by creating an exceptionally long transfer distance, forcing electron transport to be redirected through neighboring electron transport pathways as in the case of long heme-to-heme distances (Figure 4).At 5% defects of missing proteins, the maximum electron flow reduced within one order of magnitude compared to the perfect 1.6 µm (500 proteins) long nanowire.With more than 10% defects, the reduction in maximum electron flow is further deteriorated compared to the perfect nanowire and is within one order of magnitude and 2-3 orders of magnitude for 0.16 µm and 1.6 µm long wires, respectively.Unlike defects and wire elongation, an increase in wire thickness in one dimension leads to an increase in the maximum electron flow linearly by the same factor (Figure S7A,C), since each individual string of proteins within a nanowire contributes proportionally to the total electron flux.

Discussion
Graph theory highlights limitations on the electron transport through nanowires with an underlying electron hopping mechanism.While previous studies have looked at understanding the transport mechanism for STC and multi-heme proteins [31,48,40,49,50], the above work represents a systems-level description of electron transport in larger nanowires.In the further discussion, we will focus on the mechanism of long-range electron transport and, based on the results, potential applications of these nanowires.

Temperature dependence
Huang et al. found that STC electron transfer rates had a strong negative temperature dependence [29] indicating a high activation barrier ∆G † , and thus proposed that the heme-to-heme transfer 4 → 5 along the b-axis must have a high reorganization energy, consistent with prior approaches where anti-Arrhenius behavior has been noted [51,45].One alternative interpretation is that the strong temperature dependence results from structural thermal fluctuations or temperature induced structural changes, modifying distances between heme groups and giving rise to an apparent high ∆G † .In this study, we tested this hypothesis by simulating the nanowire fragment at different temperatures, ignoring the temperature dependence on the factors 3.1 and 0.06 in the Moser-Dutton's ruler [44].We further assume that the reaction and reorganization energies for all electron transfer events are identical, and the electron coupling is only modulated by protein motion.In our graph model, we use the highest ∆G † along the b-axis, namely the interprotein heme-to-heme transfer 4 → 5, as a reference for the free energy values of all electron transfers in the nanowire.
Assuming midpoint potential differences of 0.12 eV and a reorganization energy more typical of biological electron transfer reactions of 1.0 eV [52,29], in the comparatively narrow temperature range studied here, we do not see significantly different heme-to-heme distance distributions (Figure S5, S4), and the simulations show relatively small temperature dependencies in electron transfer rates.These results argue against the hypothesis that thermal fluctuations or structural changes within the nanowire with increasing temperature are responsible for the experimentally observed temperature dependence of the electron transfer rates.However, the longer electron transfer distances for thermalized crystals do neatly retard electron transfer predictions relative to what was previously predicted for neat crystals [29].

Electron transport path
The specific electron transport paths for an electron transport along the b-axis depend on wire dimensions and imperfections of the STC lattice.In perfect nanowires with electrons transported primarily along the b-axis, electrons transfers perpendicular to this axis along the a-axis only represent 1.1 % to 1.5 % of all electron transfers (Figure S10A).Electron transport along the c-axis is very slow and makes up a vanishingly small number of electron transfers within our model (Figure S10C).The probability for electron transfers along the a-axis depends on the wire diameter (ac-cross section plane), but not on the wire length for all the wire lengths tested.Electrons in thicker wires have, on average, more options to escape to neighboring chains in the event of a longer transfer distance along the primary axis, with electron transport chains in the wire center being adjacent to two other chains in the a-axis, while edge and corner chains might only have one neighbor.Therefore, electrons on center transport chains are more likely to find more efficient transport paths along adjacent chains.
For unoccupied protein positions, electron transport for this transport chain is completely interrupted and must be redirected to adjacent transport chains.Nanowires with 10 % missing proteins are twice as likely to have electrons transferring to neighboring a-axis chains, whereas the probability triples for 15 % unoccupied protein positions (Figure S10B).Moreover, a high percentage of unoccupied protein positions requires inefficient electron transport along the c-axis (Figure S10D).Optimizing the protein-protein interface along the c-axis to allow a more densely packed lattice along this axis could increase the robustness of the maximum electron flow to imperfections and thermal fluctuations.This is because shorter heme-to-heme distances along the c-axis would open up more alternative electron transport paths and reduce the likelihood of nanowire bottlenecks.The graph model representing the nanowire is a static environment, in which the heme-to-heme distances do not change with time, while the interheme distances in experimentally observed nanowires change dynamically due to structural vibrations.This raises the question of how the dynamic changes in the heme-to-heme distance might affect the electron transport, especially for the 4 → 5 interprotein transfer that is most frequently rate-limiting due to its longer distance and greater distance variation.The individual distances can vary quickly (Figure 5A), but analyzing the autocorrelation function for these distance measurements (Figure 5B) reveals that there are two timescales encoded in the interheme distance dynamics, modeled with Eq. 5. Two vibrational regimes are identified at 10 7.5 s −1 and 10 11 s −1 , which we assign to STC protein and heme group vibrations, respectively (Figure 5C).Surprisingly, the STC protein vibrations are in the same order of magnitude as the electron transfer rates along the b-axis (k b et =10 6.8 s −1 ), which might partly mediate reductions in electron transfer rates that resulted from long heme-to-heme distances, and potentially even promote faster electron transfer.Other studies in multiheme cytochromes have also indicated that motion between hemes can strongly influence observed electron transfer events [33], and so this hypothesis merits future study.In developing our graph models, the distances between adjacent hemes were assumed to be independent.However, coupling in the interactions between proteins may facilitate concerted lengthening or microfractures within the nanocrystal.The logical consequence for any concerted movement between two adjacent STCs could lengthen the distances in nearby STCs along the a-or c-axis, which are coupled together via protein-protein interactions.This type of cooperativity would drastically reduce the maximum electron flow through the wire, as one long interprotein distance could induce a blockade in a whole crosssection.Our current models argue against this interpretation, as the 4 → 5 heme-to-heme distances between neighboring STC proteins are not correlated or anticorrelated to each other (Figure 5D), although it should be noted that our simulations are based on a computationally tractable subset of a much larger nanocrystal.

Wire thickness dictates electron flow
Depending on growth conditions, protein nanowires can vary substantially in their morphology, particularly in terms of their diameter.Typically, nanowires are up to 100 nm with lengths in the micro-to millimeter range [39,15], but thicker wires are also plausible.Using a full representation for all hemes for millimeter-length nanowires was infeasible computationally.Instead of using graph theory to compute the total flow through the graph along the b-axis, we search for the bottleneck in the ab-plane with the largest influence by summing parallel electron transfer rates along the a-axis and taking the minima as the rate limiting point for electron transport through the wire (see methods).The smallest sum of the electron transfer rates then determines the maximum electron flow through the wire.In this simplified model, electron transport along the cross section axis, the a-axis, happens instantaneously, ensuring the independent measurement of the electron flow between a-axis layers, but potentially overestimating the maximum electron flow through the wire.Electron transport along the c-axis is not considered in this model.Assuming a relatively thin nanowire of 25 nm containing 10 monomers along the diameter, the maximum electron flow decreases by a factor of 10 for a 10 µm long wire compared to a shorter 100 nm wire (Figure 6).By contrast, a wire that is 100 times thicker has nearly as much maximum electron flow over a nanometer distance as it would over a millimeter, highlighting how important alternative pathways are for maximizing electron flows in these nanowires.In scenarios that require long and thin wires with only a few STCs in diameter, such as intracellular sampling [53], the losses in maximum electron flow as a function of distance are substantial, and may pose significant challenges to current detection.As the thickness of the nanowire increases, more pathways for electrons to traverse the crystal are available to avoid bottlenecks.Thus, the decrease in maximum electron flow with increasing length is not nearly as drastic for these thicker nanocrystals, as large distances and lattice defects that prevent electron transport can easily be bypassed.We predict that, if heating and denaturation could be avoided, in principle STC wires with one-millimeter diameter could transport electrons at a flow rate comparable with household currents.

Conclusion
Heme-based biological nanowires are a promising new material as sensors or as replacements for conventional metal wires in special cases.Measuring the distance distributions between heme groups and creating large STC crystals with graph theory have shown that thermal fluctuations lead to significant variability in interprotein heme-to-heme distances.This leads to reduced electron hopping along thin and long STC nanowires due to stochastic bottlenecks.The thicker STC nanowires are, the lower the probability of a bottleneck within a lattice, allowing for greater electron flow beyond what would be expected for introducing more charge carriers.Stochastic bottlenecks are expected to be a feature for all proteinbased nanowires with a heme-to-heme hopping electron transport mechanism.Our results demonstrate that intermolecular stiffness plays a crucial role in the material design of protein-based nanowires.

Nanowire Model
The STC crystal structure from Shewanella oneidensis (pdb: 6ee7 [29]) was copied four, six and four times in the a-, b-and c-axis (Figure 1), respectively, with Chimerax v1.2.5 [54] to form a small nanowire fragment.Subsequently, the structure was solvated in a 22.86 x 16.50 x 14.30 nm 3 cubic box with explicit water in VMD v1.9.4 [55].150 mmol sodium chloride neutralized the system and structural zinc ions as found in the crystal structure were kept in the model.To capture the local protein structural rigidity around the heme groups, HEC and PHEM CHARMM patches were applied to create bonds between local cystine residues, histidine residues, and the heme group.

Molecular Dynamics Simulations
Before the 1 µs long NPT production simulations at 273 K, 281 K, 298 K or 309 K, the nanowire fragment was subjected to minimization and equilibration simulations.First, the nanowire fragment's potential energy was minimized with a conjugated gradient and line search algorithm for 1000 iterations, whereby the protein backbone, ions and heavy atoms in water and porphyrin were position restrained with 1 kcal mol −1 .Afterwards, the position restraints are reduced to 0.01 kcal mol −1 and the system was briefly equilibrated in a NPT ensemble for 4 ns.NAMD 3.0a9[56] integrated the equation of motion every 2 fs by a velocity verlet integrator [57].Proteins, ions and heme groups were described by the CHARMM36 force field [58], while the TIP3P model represented water [59,60].The RATTLE algorithm [61] constraint all bond lengths to their optimal distances.Interaction pair lists are calculated every 20 fs between atoms within 14 Å.Van der Waals and short-range electrostatic interactions are considered up to 12 Å.Above 10 Å, both interaction forces are scaled down to zero with a switching function.Long-range electrostatic interactions are calculated with the particle mesh Ewald summation (PME) [62,63] using a 1.2 Å grid spacing.Temperature is adjusted by a Langevin thermostat [64] to 273 K, 281 K, 298 K or 309 K. Pressure is controlled to 1 atm by a Langevin piston [65] with a 200 fs oscillation period and 100 fs damping time.

Distance Analysis
Intra-and intermolecular heme-to-heme distance distributions are calculated in a python context with VMD 1.9.4a55 [55] from the molecular dynamics trajectories.We defined the heme-to-heme distance as the minimal distance between all pair combinations of nitrogen, carbon and iron atoms of the two porphyrin rings.The resulting distance matrix is efficiently calculated with the cdist function included in the scipy v1.0 module [66].Within one STC all intramolecular heme-to-heme distance combinations were considered.Intermolecular heme-to-heme distances are only calculated for neighboring proteins.All distances between equivalent hemes (Figure 1) are considered within one distribution for the nanowire fragment and trajectory frames.Timescales t for distance fluctuations are calculated from the the heme-to-heme distance autocorrelation function by fitting the sum of two exponential decays: (5) The weighting factors a indicate the relative magnitude of the respective timescale.

Graph Theory and Electron Hopping Rates
Graphs were generated with networkx v2.6.3 [67] in python and visualized in Gephi v0.9.2 [68].Every STC nanowire heme molecule is represented by one node in the graph.Across STC interfaces, all hemes are interconnected with each other, while only intermolecular hemes for neighboring proteins are considered.Distances for connections are randomly picked from the distance distributions for the respective heme pair.Electron transfer rates k et are calculated from distances R with Moser-Dutton's Ruler [44] (Eq.1).
Eq. 1 describes endergonic reactions with the coefficient 3.1 unifying the constants for the quantized nuclear term at room temperature [69] and the coefficient 0.6 representing the packing density [70].The Gibbs free energy ∆G=0.12eV and reorganization energy λ=1 eV were considered isotropic and in accordance with the experimental intermolecular heme-heme transfer in b-axis [29].Electron transfer rates with less then k et =1 s −1 were excluded in the graph.Additionally, all hemes in the first and last ac-axis crosssection plane are connected to a source and sink node, respectively, imitating electrodes.The electron transfers from the source to the first cross section layer of the nanowire and from the last cross section layer of the nanowire to the sink node were set to 10 15 e/s.Edge capacities, representing the amount of electrons that can transfer between two hemes, were set according to the electron transfer rate between the nodes connected by the edge.The maximum electron transport k et,max from the source to the sink node was calculated with the maximum_flow_value function of networkx using the preflow-push method [71].The maximum electron flow is determined by I = e k et,max , analogous to Eq. 4.
For each crystallographic axis of the nanowire, the protein number was set to 1, 2, 5, 10, 15, 20, 50, 100, 150, 200 or 500, while the other two crystallographic axes were kept at 4 proteins each (Figure S7, S8, and S9).Graph generations including maximum electron flow calculations were repeated 100 times for each set of parameters.The reported values are the mean value over all repetitions.

Simplified Model
To better scale our graph-based network model to larger length-scales, we generate a simplified model.In the simplified model, we assume that the electron transport along the wire b-axis is determined by the single slowest cross-section.The electron transfer rate between heme groups is much faster within an STC protein than between adjacent STCs, therefore, in this model only the interprotein transfers 4 → 5 and 8 → 1 are considered.The electron transport in the c-axis is unfavorable compared to the a,bplane and neglected.The overall electron transport rate through the nanowire along the b-axis can be expressed as the a-axis cross section b = 1, .., n with the lowest sum of electron transfer rates: This equation is evaluated in python without by picking random distances from the distance distribution between heme 4 and 5.In the bi-exponential setup, there is one faster vibrational process that happens at 10 11 s −1 , which we attribute to heme group vibrations within the protein, and one slower vibrational process with a peak at around 10

Table of Contents
ToC Entry

Figure 1 :
Figure 1: 4-6-4 STC nanowire fragment in water (blue box) visualized from (A) b-and (B) a-view axis.Unique STC proteins are colored with a gradient (blue-purple-red-orange-yellow) according to their crystal position.The α-helix secondary structure elements are depicted as winding strip.(C) Location of the four heme groups per protein within the STC structure.Every other protein along the b-axis is rotated around the a-axis by 180°.To account for this difference, the hemes are numbered from one to eight.

Figure 2 :
Figure 2: (Top) heme-heme distance distribution and (middle) maximum electron flow along the length, b-axis, of the nanowire.Vertical lines indicate which heme-to-heme electron transfer is calculated.(Middle) The variance is shaded in light gray.(Bottom) Heme groups (orange) along the wire numbered from one to eight.The numbering represents hemes within two adjacent monomers along the b-axis, with hemes 1 → 4 found within one monomer and hemes 5 → 8 found in the crystallographic symmetry partner.

Figure 3 :
Figure 3: (Left) Important inter-and intraprotein minimum heme-to-heme distances over the course of the simulation at 298 K.The individual hemes are numbered based on their position within the nanocrystal, highlighted in Figures 1 and 2. While the black 1 → 2, purple 2 → 3, and blue 3 → 4 lines account for the intramolecular heme-to-heme distances along the b-axis, the green 1 → 2(a), orange 4 → 5(b) and red 3 → 2(c) lines visualize the shortest intermolecular heme-to-heme distances along the a-, b-and c-axis, respectively.(Right) Probability density of the heme-to-heme distances shown in the left plot with the average distance drawn as colored numbers.

Figure 5 :
Figure 5: (A) plots an example time trace for a specific intermonomer electron transfer from 4 → 5 at 298K.(B) Transforms this trace to an autocorrelation function, which is fit to a bi-exponential with two distinct time constants.In (C), we report the fitted timescales of the 4 → 5 heme-to-heme distance fluctuations during the molecular dynamics simulation.In the bi-exponential setup, there is one faster vibrational process that happens at 10 11 s −1 , which we attribute to heme group vibrations within the protein, and one slower vibrational process with a peak at around 10 7.5 s −1 , which may be a slower motion related to protein vibrations within the lattice.(D) reports the correlations between neighboring 4 → 5 heme-to-heme distance along the b-or a-axis.
Figure 5: (A) plots an example time trace for a specific intermonomer electron transfer from 4 → 5 at 298K.(B) Transforms this trace to an autocorrelation function, which is fit to a bi-exponential with two distinct time constants.In (C), we report the fitted timescales of the 4 → 5 heme-to-heme distance fluctuations during the molecular dynamics simulation.In the bi-exponential setup, there is one faster vibrational process that happens at 10 11 s −1 , which we attribute to heme group vibrations within the protein, and one slower vibrational process with a peak at around 10 7.5 s −1 , which may be a slower motion related to protein vibrations within the lattice.(D) reports the correlations between neighboring 4 → 5 heme-to-heme distance along the b-or a-axis.

Figure 6 :
Figure 6: Estimated maximum electron flows at 298K of wires with different lengths along the b-axis.The thickness of the wire is shown as diameter in different colors.