Oxygen and sulfide in ocean sediments can be consumed biologically over long spatial distances by way of filamentous bacteria in electron-conducting sheaths. To analyse observations, a mathematical model of these filamentous sulfur-oxidising bacteria was developed, including electrical conduction between reactive zones. Mechanisms include Nernst-Planck diffusion and migration of ions coupled with Ohm's law for conduction along filaments, and metabolic activity throughout the filaments. Simulations predict outward biomass growth toward the boundaries of the sediment floor and top surface, resulting in two distinct zones with anode (sulfide consumption) and cathode (oxygen consumption) reactions enabled by electron conduction. Results show inward fluxes of 4.6mmolO2/m2/d and 2.5mmolS/m2/d, with consumption increasing with growth to final fluxes of 8.2mmolO2/m2/d and 4.34mmolS/m2/d. Qualitatively, the effect of varying cell conductivity and substrate affinity is evaluated. Controlling mechanisms are identified to shift from biomass limitation, to substrate limitation, and to conductivity limitations as the lengths of the filaments increase. While most observed data are reflected in the simulation results, a key discrepancy is the lower growth rates, which are largely fixed by thermodynamics, indicating that microbes may utilise secondary substrates or an alternative metabolism.