Computation of biological conductance with Liouville quantum master equation

Amdursky, N. et al. Electronic transport via proteins. Adv. Mater. 26, 7142–7161 (2014).Article 
CAS 
PubMed 

Google Scholar 
Bostick, C. D. et al. Protein bioelectronics: A review of what we do and do not know. Rep. Prog. Phys. 81, 026601 (2018).Article 
ADS 
PubMed 

Google Scholar 
Zhang, B. et al. Observation of giant conductance fluctuations in a protein. Nano Futures 1, 035002 (2017).Article 
ADS 
PubMed 
PubMed Central 

Google Scholar 
Zhang, B. & Lindsay, S. Electronic decay length in a protein molecule. Nano Lett. 19, 4017–4022 (2019).Article 
ADS 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Zhang, B. et al. Role of contacts in long-range protein conductance. Proc. Natl. Acad. Sci. 116, 5886–5891 (2019).Article 
ADS 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Kayser, B. et al. Solid-state electron transport via the protein azurin is temperature-independent down to 4 k. J. Phys. Chem. Lett. 11, 144–151 (2019).Article 
PubMed 

Google Scholar 
Sepunaru, L., Pecht, I., Sheves, M. & Cahen, D. Solid-state electron transport across azurin: From a temperature-independent to a temperature-activated mechanism. J. Am. Chem. Soc. 133, 2421–2423 (2011).Article 
CAS 
PubMed 

Google Scholar 
Bera, S. et al. Near-temperature-independent electron transport well beyond expected quantum tunneling range via bacteriorhodopsin multilayers. J. Am. Chem. Soc. 145, 24820–24835 (2023).CAS 
PubMed 
PubMed Central 

Google Scholar 
Lambert, C. Basic concepts of quantum interference and electron transport in single-molecule electronics. Chem. Soc. Rev. 44, 875–888 (2015).Article 
CAS 
PubMed 

Google Scholar 
Marcus, R. A. On the theory of oxidation-reduction reactions involving electron transfer. i. J. Chem. Phys. 24, 966–978 (1956).Article 
ADS 
CAS 

Google Scholar 
Segal, D., Nitzan, A., Davis, W. B., Wasielewski, M. R. & Ratner, M. A. Electron transfer rates in bridged molecular systems 2. A steady-state analysis of coherent tunneling and thermal transitions. J. Phys. Chem. B 104, 3817–3829 (2000).Article 
CAS 

Google Scholar 
Davis, W. B., Wasielewski, M. R., Ratner, M. A., Mujica, V. & Nitzan, A. Electron transfer rates in bridged molecular systems: A phenomenological approach to relaxation. J. Phys. Chem. A 101, 6158–6164 (1997).Article 
CAS 

Google Scholar 
Zahid, F., Paulsson, M. & Datta, S. Electrical conduction through molecules, in Advanced Semiconductor and Organic Nano-Techniques, 1–41 (Elsevier, 2003).Datta, S. Electronic Transport in Mesoscopic Systems (Cambridge University Press, 1997).
Google Scholar 
Gebauer, R. & Car, R. Kinetic theory of quantum transport at the nanoscale. Phys. Rev. B 70, 125324 (2004).Article 
ADS 

Google Scholar 
Fischetti, M. Master-equation approach to the study of electronic transport in small semiconductor devices. Phys. Rev. B 59, 4901 (1999).Article 
ADS 
CAS 

Google Scholar 
Gebauer, R. & Car, R. Current in open quantum systems. Phys. Rev. Lett. 93, 160404 (2004).Article 
ADS 
PubMed 

Google Scholar 
Breuer, H.-P. et al. The Theory of Open Quantum Systems (Oxford University Press on Demand, 2002).
Google Scholar 
Gebauer, R., Burke, K. & Car, R. Kohn-sham master equation approach to transport through single molecules, in Time-Dependent Density Functional Theory, 463–477 (Springer, 2006).Mohseni, M., Rebentrost, P., Lloyd, S. & Aspuru-Guzik, A. Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129, 11B603 (2008).Article 

Google Scholar 
Papp, E., Jelenfi, D. P., Veszeli, M. T. & Vattay, G. A landauer formula for bioelectronic applications. Biomolecules 9, 599 (2019).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Papp, E. et al. Experimental data confirm carrier-cascade model for solid-state conductance across proteins. J. Phys. Chem. B 127, 1728–1734 (2023).Article 
CAS 
PubMed 

Google Scholar 
Filman, D. J. et al. Cryo-em reveals the structural basis of long-range electron transport in a cytochrome-based bacterial nanowire. Commun. Biol. 2, 219 (2019).Article 
PubMed 
PubMed Central 

Google Scholar 
Wang, F. et al. Structure of microbial nanowires reveals stacked hemes that transport electrons over micrometers. Cell 177, 361–369 (2019).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Yalcin, S. E. et al. Electric field stimulates production of highly conductive microbial OmcZ nanowires. Nat. Chem. Biol. 16, 1136–1142 (2020).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Wang, F. et al. Cryo-em structure of an extracellular geobacter omce cytochrome filament reveals tetrahaem packing. Nat. Microbiol. 7, 1291–1300 (2022).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Wang, F. et al. Structure of Geobacter OmcZ filaments suggests extracellular cytochrome polymers evolved independently multiple times. eLife 11, e81551. https://doi.org/10.7554/eLife.81551 (2022).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Baquero, D. P. et al. Extracellular cytochrome nanowires appear to be ubiquitous in prokaryotes. Cell 186(13), 2853–2864 (2023).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Agam, Y., Nandi, R., Kaushansky, A., Peskin, U. & Amdursky, N. The porphyrin ring rather than the metal ion dictates long-range electron transport across proteins suggesting coherence-assisted mechanism. Proc. Natl. Acad. Sci. 117, 32260–32266 (2020).Article 
ADS 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Futera, Z. et al. Coherent electron transport across a 3 nm bioelectronic junction made of multi-heme proteins. J. Phys. Chem. Lett. 11, 9766–9774 (2020).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Malvankar, N. S. & Lovley, D. R. Microbial nanowires for bioenergy applications. Curr. Opin. Biotechnol. 27, 88–95 (2014).Article 
CAS 
PubMed 

Google Scholar 
Liu, X. et al. Power generation from ambient humidity using protein nanowires. Nature 578, 550–554 (2020).Article 
ADS 
CAS 
PubMed 

Google Scholar 
Bond, D. R. & Lovley, D. R. Electricity production by geobacter sulfurreducens attached to electrodes. Appl. Environ. Microbiol. 69, 1548–1555 (2003).Article 
ADS 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Smith, A. F. et al. Bioelectronic protein nanowire sensors for ammonia detection. Nano Res. 13, 1479–1484 (2020).Article 
CAS 

Google Scholar 
Liu, X. et al. Multifunctional protein nanowire humidity sensors for green wearable electronics. Adv. Electron. Mater. 6, 2000721 (2020).Article 
CAS 

Google Scholar 
Futera, Z., Wu, X. & Blumberger, J. Tunneling-to-hopping transition in multiheme cytochrome bioelectronic junctions. J. Phys. Chem. Lett. 14, 445–452 (2023).Article 
CAS 
PubMed 

Google Scholar 
Garg, K. et al. Interface electrostatics dictates the electron transport via bioelectronic junctions. ACS Appl. Mater. Interfaces 10, 41599–41607 (2018).Article 
CAS 
PubMed 

Google Scholar 
Fereiro, J. A. et al. A solid-state protein junction serves as a bias-induced current switch. Angew. Chem. 131, 11978–11985 (2019).Article 
ADS 

Google Scholar 
Gu, Y. et al. Structure of geobacter cytochrome OmcZ identifies mechanism of nanowire assembly and conductivity. Nat. Microbiol. 8, 284–298 (2023).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Lowe, J. P. & Peterson, K. Quantum Chemistry (Elsevier, 2011).
Google Scholar 
Virtanen, P. et al. Fundamental algorithms for scientific computing in python. SciPy 1.0.. Nat. Methods 17, 261–272. https://doi.org/10.1038/s41592-019-0686-2 (2020).Article 
CAS 
PubMed 
PubMed Central 

Google Scholar 
Beckstein, O. et al. Mdanalysis/griddataformats: Release 1.0.1. https://doi.org/10.5281/zenodo.6582343 (2022).Humphrey, W., Dalke, A. & Schulten, K. VMD – Visual molecular dynamics. J. Mol. Graph. 14, 33–38 (1996).Article 
CAS 
PubMed 

Google Scholar 
Stone, J. An Efficient Library for Parallel Ray Tracing and Animation. Master’s thesis (Computer Science Department, University of Missouri-Rolla, 1998).