Modeling Microbial Electrosynthesis

  • Benjamin KorthEmail author
  • Falk Harnisch
Part of the Advances in Biochemical Engineering/Biotechnology book series (ABE, volume 167)


Mathematical modeling is an overarching approach for assessing the complexity of microbial electrosynthesis (MES) and for complementing the relevant experimental research. By describing and linking compartments, components, and processes with appropriate mathematical equations, MES and the corresponding bioelectrodes and complete bioelectrochemical systems can be analyzed and predicted across several temporal and local scales. Thereby, insights into fundamental phenomena and mechanisms, in addition to process engineering and design can be obtained. However, a substantial lack of knowledge about extracellular electron transfer mechanisms and electrotrophic microorganisms presumably prevented the development of adequate models of MES, especially of biocathodes, so far. To propel efforts regarding this demanding task, this chapter provides a comprehensive overview of the relevant compartments, components and processes, appropriate model strategies, and a discussion on potential modeling pitfalls. By adapting an established approach to assessing the energetics of microorganism, an instruction for calculating stoichiometry, thermodynamics, and kinetics, with the example of electro-autotrophic growth at cathodes, is presented. Models of bioanodes and fundamental electrochemical equations are described to provided strategies for calculating cathodic electron-uptake reactions and connecting them to the microbial metabolism. Finally, differential equations are detailed for coupling the distinct compartments of a bioelectrochemical system. Although MES comprises anodic and cathodic reactions, the present chapter focuses on biocathodes representing a functional connection between cathode and electron-accepting microorganisms.

Graphical Abstract


Autotrophy Biocathode Cathodic extracellular electron transfer Microbial electrochemical technologies Microbial fuel cells 

List of Symbols


Cathode area, m2


Gas–liquid–interface area, m


Charge transfer coefficient

\( {C}_{F,i}^0 \)

Initial concentration of the ith component in the inflow of a chemostat, mol m−3


Concentration of the ith component, mol m−3

CA, i

Concentration of the ith component in the anodic reactor volume, mol m−3

CB, i

Concentration of the ith component within biofilm, mol m−3

CC, i

Concentration of the ith component in cathodic bulk volume, mol m−3

CG, i

Concentration of the ith component in the gas phase, mol m−3


Concentration of the ith component in the diffusion boundary layer, mol m−3

CFeed, i

Concentration of the ith component in added liquid during a fed-batch process, mol m−3


Biomass concentration, C-mol m−3

CX, i

Concentration of the ith biomass fraction, C-mol m−3

DB, i

Diffusion coefficient of the ith component within the biofilm, m2 s−1


Diffusion coefficient of the ith component within the diffusion boundary layer, m2 s−1


Effective electron diffusion coefficient, m2 s−1

DM, i

Diffusion coefficient of the ith component through the membrane, m2 s−1


Standard potential for biochemical standard conditions (1 mol L−1 of respective reactants, 298.15 K, 101.325 kPa, pH = 7), V


Potential of the conductive biofilm matrix, V


Cathode potential, V

\( {E}_i^f \)

Formal potential of the ith component, V


Potential for the half-maximum rate, V


Potential of the liquid phase within the biofilm, V

\( {E}_{\mathrm{MED}}^{0\prime } \)

Standard potential of redox mediators for biochemical standard conditions (1 mol L−1 of respective reactants, 298.15 K, 101.325 kPa, pH = 7), V


Dielectric constant of the vacuum, 8.85 × 10−12 F m−1


Dielectric constant of the membrane, F m−1


Faraday constant, 96,485.34 C mol−1


Multiplication factor for catabolic reaction


Flow rate, m3 s−1


Gibbs free energy, kJ mol−1


Standard Gibbs free energy of formation for biochemical standard conditions (1 mol L−1 of respective reactants, 298.15 K, 101.325 kPa, pH = 7), kJ mol−1


Standard Gibbs free energy of reaction for biochemical standard conditions (1 mol L−1 of respective reactants, 298.15 K, 101.325 kPa, pH = 7), kJ mol−1


Gibbs free energy of the anabolic reaction, kJ mol−1


Gibbs free energy of the catabolic reaction, kJ mol−1


Gibbs free energy dissipation, kJ mol−1


Enthalpy, kJ mol−1


Standard enthalpy of formation for biochemical standard conditions (1 mol L−1 of respective reactants, 298.15 K, 101.325 kPa, pH = 7), kJ mol−1


Standard enthalpy of reaction for biochemical standard conditions (1 mol L−1 of respective reactants, 298.15 K, 101.325 kPa, pH = 7), kJ mol−1


Henry’s law proportionality coefficient defined via concentration and partial pressure, mol m−3 Pa−1


Current density, A m−2


Exchange current density, A m−2

JB, i

Flux of the ith component at interface biofilm/cathodic reactor volume, mol m−2 s−1

JG, i

Flux of the ith component at gas–liquid interface, mol s−1

JM, i

Flux of the ith component through membrane, mol m−2 s−1

JMed, x

Flux of redox mediators at layer x, mol m−2 s−1

\( {k}_{\mathrm{Het}}^0 \)

Standard (heterogeneous) electron transfer rate constant at biofilm–electrode interface, s−1

kf, kr

Forward and reverse reaction rate constants for chemical equilibrium reactions, s−1


Liquid-phase mass transfer coefficient, mol m−2 s−1

kox, kred

Electron transfer rate constants for the oxidation and reduction rate in the Butler–Volmer equation, cm s−1


Equilibrium constant for the acid-base pair AH/A


Monod affinity constant of the ith component, mol m−3


Parameters in the Butler–Volmer–Monod equation


Biofilm matrix conductivity, S m−1


Biofilm thickness, m


Thickness of the cell layer directly in contact with the electrode, m


Membrane thickness, m


Added ith component in a continuous flow or fed-batch mode, mol m−3 s−1


Substrate specific maintenance rate, mol C-mol−1 s−1


Maximum growth rate, s−1


Growth rate, s−1


Number of carbon atoms in the carbon source


Amount of the ith component, mol


Overpotential, V


Partial pressure of the ith component, Pa

\( {q}_i^{\mathrm{max}} \)

Maximum biomass-specific uptake rate of the ith component, mol C-mol−1 s−1


Logarithmic acid dissociation constant


Biomass-specific uptake rate of the ith component, mol C-mol−1 s−1


Universal gas constant, 8.31 J mol−1 K−1

req, i

Chemical equilibrium rate of the ith component, mol m−3 s−1


Total conversion rate of the ith component, mol m−3 s−1

rB, i

Net rate of the ith component within the biofilm summarizing all rates, mol m−3 s−1


Biomass decay rate, C-mol m−3 s−1


Total redox mediator oxidation rate, mol m−3 s−1


Total redox mediator reduction rate at the cathode, mol s−1


Reduction rate of cytochromes within the biofilm, mol m−3 s−1


Biomass production rate, C-mol m−3 s−1


Entropy, kJ mol−1


Time, s


Temperature, K


Standard temperature, 298.15 K


Velocity of the biofilm thickness increase, m s−1


Velocity of the biofilm surface increase, m s−1


Cathodic reactor volume, m3


Volume of added liquid, m3


Stoichiometric factor of the ith component


Stoichiometric factor of the ith component in the overall growth reaction, mol C-mol−1


Degree of reduction of the carbon source


Distance of a layer within biofilm or the diffusion boundary layer from the cathode, x = 0 is designated as the cathode surface, m


Mole fraction of the ith compound


Number of transferred electrons


Charge of the ith component



Boundary condition

Allocation of a defined value (Dirichlet boundary condition, e.g., concentration) or of a derivative of a solution (Neumann boundary condition, e.g., flux) to the border between a compartment and the external world or to a border between two compartments to connect these compartments.


Defined one-, two- or three-dimensional section of a model (e.g., biofilm and reactor volume) with specifically assigned parameters and variables.


A charged or uncharged chemical species (e.g., acetate, bicarbonate, redox mediator, other ions) or sum of chemical species (e.g., biomass).


Starting component of a chemical reaction.


Transport phenomenon describing the rate of movement of a component per area (e.g., flux of a component through a membrane).


A parameter/variable valid for all compartments of a model.


Entirety of spatial fragments constituting a compartment and defining the spatial resolution of the compartment.


A parameter/variable valid only for a certain compartment of the model.


Specific position (x, y, z) within a compartment.

Mass balance

Equation that considers all transport and transformation reactions of a component and thus fulfills the conservation of mass. Needs to be established for every component in every compartment.


A given value for biological/physical/chemical processes or properties.


End component of a chemical reaction.


Starting or end component of a chemical reaction.

Reactor volume

Denotes the cathodic compartment of a bioelectrochemical system in this chapter.


Process changing the chemical nature of a component (e.g., chemical equilibrium reaction, oxidation, biomass synthesis).


Process changing the position of a component (e.g., diffusion and migration).


A value for biological/physical/chemical processes or properties calculated according to local parameters and variables (e.g., local concentrations, temperature).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Environmental MicrobiologyHelmholtz-Centre for Environmental Research – UFZLeipzigGermany

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