Conceptualizing Biogeochemical Reactions With an Ohm's Law Analogy

In studying problems like plant‐soil‐microbe interactions in environmental biogeochemistry and ecology, one usually has to quantify and model how substrates control the growth of, and interaction among, biological organisms (and abiotic factors, e.g., adsorptive mineral soil surfaces). To address these substrate‐consumer relationships, many substrate kinetics and growth rules have been developed, including the famous Monod kinetics for single‐substrate‐based growth and Liebig's law of the minimum for multiple‐nutrient‐colimited growth. However, the mechanistic basis that leads to these various concepts and mathematical formulations and the implications of their parameters are often quite uncertain. Here, we show that an analogy based on Ohm's law in electric circuit theory is able to unify many of these different concepts and mathematical formulations. In this Ohm's law analogy, a resistor is defined by a combination of consumers’ and substrates’ kinetic traits. In particular, the resistance is equal to the mean first passage time that has been used to derive the Michaelis‐Menten kinetics under substrate replete conditions for a single substrate as well as the predation rate of individual organisms. We further show that this analogy leads to important insights on various biogeochemical problems, such as (a) multiple‐nutrient‐colimited biological growth, (b) denitrification, (c) fermentation under aerobic conditions, (d) metabolic temperature sensitivity, and (e) the legitimacy of Monod kinetics for describing bacterial growth. We expect that our approach will help both modelers and nonmodelers to better understand and formulate hypotheses when studying certain aspects of environmental biogeochemistry and ecology.

2 of 19 et al., 2020). In the past, three approaches have been used to obtain such relationships. The first approach is by fitting empirical response functions to observational data (e.g., Monod, 1949). The second approach is based on an ad hoc heuristic conceptualization of the problem, for example, the logistic equation was derived by adding a quadratic term to dissipate the exponential growth of a population when Pierre-Francois Verhulst was helping his teacher Alphonse Quetelet to model human population dynamics (Cramer, 2002). The third approach is based on systematic applications of some theory, such as the law of mass action (Atkins et al., 2016), statistical mechanics (Ma, 1985), or renewal theory (Doob, 1948). Notably, Michaelis-Menten kinetics (and some of its extensions) can be derived by applying any of these theories (see reviews in Kooijman, 1998;Swenson & Stadie, 2019;Tang & Riley, 2013b, with the renewal theory even being able to show that Michaelis-Menten kinetics is the statistical mean of the stochastic description of a single-enzyme molecule processing the substrate molecules (English et al., 2006;Reuveni et al., 2014).
Compared to the empirically based and ad hoc approaches, which generally provide limited understanding of the processes implied by the parameters, theory-based approaches have the advantage of linking various related, albeit fragmented, knowledge (that is abstracted from a much wider range of observations compared to the limited observational data used by empirically based approaches), thereby enabling a deeper understanding of the processes and systems of interest. For instance, when the law of mass action is employed to derive the Michaelis-Menten kinetics, using related theory of chemical reaction rates (e.g., Smoluchowski's diffusion model of chemical reaction, von Smoluchowski, 1917), Tang and Riley (2019a) were able to upscale the microbially enabled reactions from one permease to a single bacteria cell and then to a representative soil volume (∼O(1 cm 3 )) and used the results to explain why substrate affinity parameters are observed to be highly variable in soil. Additionally, the theory-based approach has been used to derive the temperature response function of microbial activity (Ghosh & Dill, 2010) and to explain why Michaelis-Menten kinetics are more appropriate for microbial uptake of small molecules, while reverse Michaelis-Menten kinetics are more appropriate for enzymatic degradation of organic polymer particles (Tang & Riley, 2019b).
In this study, we first introduce an analogy that uses the Ohm's law from electric circuit theory to interpret substrate-consumer relationships. Similar analogies have been widely used by land models to represent the gradient-driven land-atmosphere exchanges of water, gases, and energy (e.g., Lawrence et al., 2019;Riley et al., 2011;Shuttleworth & Wallace, 1985;Wu et al., 2009; so that in a certain sense, Ohm's law is unifying all three aspects of biogeochemistry into physics). We then exploit this analogy to explain several interesting biogeochemical phenomena that are observed in different contexts. We conclude the paper with recommendations of other potential applications of this analogy.
Although the example problems below are solved with the Ohm's law analogy, we note that they can all be solved using the more accurate equilibrium chemistry approximation (ECA) kinetics (Tang & Riley, 2013b) or the synthesizing unit plus ECA (SUPECA) kinetics . However, the Ohm's law analogy proposed here is more intuitive and can provide an alternative to the ECA and SUPECA kinetics in formulating biogeochemical models.

A Brief Review of Ohm's Law and Circuit Theory
We below briefly review Ohm's law and the theory of series and parallel resistor circuits. More detailed descriptions of circuit theory can be found in Feynman et al. (2011b).
Ohm's law describes the relationship between voltage ( E V ), electric current ( I ), and resistance ( E r ) as To simplify the presentation, we henceforth assume that all variables are properly defined as in the international system of units.
For a series concatenation of resistors j E r , application of Ohm's law yields 5 of 19 assuming that at each step the enzyme and its cofactor together form an integrated enzyme functional unit to process the substrate delivered from a prior step, and the whole chain of enzymatic reactions is in detailed balance (i.e., the whole chain is in steady state without overflow, Cao, 2011, an assumption that is often made in flux balance models, Orth et al., 2010), we can then use the series circuit analogy to calculate the overall enzyme kinetics in a straightforward manner. According to the schema for this configuration (Figure 1b), when the whole enzyme chain is taken as a catalysis unit, the abundance of enzyme at the first step represents the voltage of the battery, and the total resistance is  Several interesting inferences can be drawn from Equations 10-14 that will provide us with a better understanding of the trade-offs in metabolic pathways and their temperature sensitivity, both of which are essential for parameterizing biochemical models, such as microbial respiration (Alster et al., 2020), plant photosynthesis, and respiration (Medlyn et al., 2002;Slot & Kitajima, 2015). First, even though any chain-like metabolic pathway as a whole can be represented similarly with the Michaelis-Menten kinetics (e.g., Equation 12), there are trade-offs between power and bioenergetic assimilation efficiency for various metabolic pathways of different lengths, which can be understood as follows. The function of an energy producing metabolic pathway is to harvest energy from substrate molecules, we thence can compare an ATP producing metabolic pathway to a thermal engine which also extracts energy from substrate molecules (i.e., fuels). The second law of thermodynamics suggests that a thermal engine has higher thermodynamic efficiency when it runs slower (and the highest efficiency can be achieved only when the system is in thermodynamic equilibrium, i.e., not running at all, Salamon et al., 2001). Equation 13 suggests that a longer reaction chain slows down the overall transformation rate from a given substrate to its final product, and thus its application to electron transport chains leads us to assert that a longer chain will likely be thermodynamically more efficient (this argument echoes the Ladder theorem in finite time thermodynamics, Salamon et al., 2017). In contrast, shorter electron transport chains imply faster substrate use even though they are less efficient in extracting Gibbs free energy from the substrate. For instance, by using a different electron transporter for each electron transported through a shorter chain, fewer protons are pumped across the membrane and thus fewer ATPs can be produced (Chen & Strous, 2013), or by using fewer intermediate electron carriers 6 of 19 such that fewer protons are pumped across the membrane for each transferred electron (if generating one ATP uses a fixed number of protons as is often observed), the same redox reaction will be faster but less efficient (Aledo & del Valle, 2002;Chen & Strous, 2013). Therefore, the length of electron transport chains can characterize the trade-off between substrate use rate and the corresponding bioenergetic assimilation efficiency, an important selection factor for organisms during their evolution. Since the structural information of electron transport chain can be inferred by genomic analysis (Lane & Martin, 2010), this insight from the Ohm's law formulation can then serve to better guide model parameterization of plant and microbial substrate uptake and use. Additionally, we note that in microbial modeling, the metabolic cost for constructing and maintaining the chain of enzymes is usually considered separately as part of the respiration for maintenance or structural biomass growth and is thus not part of the calculation of a substrate's bioenergetic assimilation efficiency (Kooijman, 2009). Indeed, in one chemostat-based study, Chen et al. (2017) found that Vibrionales bypass respiratory complex III to consume part of the oxygen using a cytochrome bd terminal oxidase to speed up growth, but the bioenergetic efficiency was reduced from ∼80% to ∼32% because of the longer canonical respiratory chain. Similarly, observations indicate that the less efficient fermentation pathway with fewer involved enzymes is faster than the aerobic respiration pathway that involves many more enzymes (and is thus longer and more efficient in extracting Gibbs free energy from substrate molecules, Madigan et al., 2009). In Section 3.5, we use the parallel circuit analogy to explain why such bypassing of more efficient pathways will occur under substrate abundant conditions.
The second inference to be made is about the temperature sensitivity of parameters , max chain E v and chain E K , two essential trait characteristics for biochemical modeling, whose mathematical parameterization (particularly for microbes) has been under intense debate (Allison et al., 2018;Davidson et al., 2012;Maggi et al., 2018).
In the simplest one-step case, which when entered into Equation 13 leads to However, because enzymes are proteins, their conformational states are also temperature dependent (Murphy et al., 1990). Thermodynamically, the undenatured (aka catalytically active) fraction of an enzyme population of length x E n (as measured by the number of amino acid residues) can be described as (Murphy et al., 1990) with heat capacity  p E C defined as the energy required to reorganize the water molecules surrounding the protein (Ratkowsky et al., 2005) f for all enzymes involved in the chain. Therefore, putting together the kinetic, thermodynamic, and catalytically active enzyme fraction functions, we obtain where the thermodynamic temperature dependence of the reaction is with  reac E G (<0) being the Gibbs free energy of the overall reaction being catalyzed, which is defined by the chemical activity of initial substrates and final products for the overall chemical reaction carried out by the chain of enzymes (e.g., Jin & Bethke, 2007). In Equation 21, we have taken the conventional assumption that the transition state theory description of the overall chemical reaction rate (as carried out by the chain of enzymes) is independent from the conformation status of the enzymes (Dill et al., 2011;Sawle & Ghosh, 2011). This assumption combined with the concept that T E F is an intrinsic property of the overall chemical reaction then allows the total fraction of active enzymes to be factored out as   Arcus et al., 2016;Schipper et al., 2014). Therefore, for a population of cells that are not under substrate limitation and are exponentially growing (so that one metabolic pathway dominates the metabolism), one should expect a MMRT-type temperature dependence of the metabolic rates. This result explains why Ratkowsky et al. (2005) were able to use the following equation to model bacterial growth rates under unlimited substrate supply (where the right-hand side of Equation 21 is reduced to where E g is growth rate, E c is an empirical constant, and  A E H is substrate-dependent activation energy. However, unlike previous assumptions that properties of some single control enzyme determine the overall growth (Johnson & Lewin, 1946), here E n and  p E C represent mean values of protein length and their thermal properties, under possible influences from other molecules, such as phospholipids (e.g., Mansy & Szostak, 2008).
In summary, for dynamic modeling of microbial substrate uptake and assimilation (and perhaps plant autotrophic respiration as well, e.g., Liang et al., 2018), we recommend representing the temperature sensitivity as in Equation 21 rather than using the MMRT directly. Additionally, we note that plant photosynthesis models have long represented carboxylation and oxygenation using a form similar to Equation 21 (e.g., Medlyn et al., 2002). Adopting a similar functional form for microbial biogeochemical reactions (and plant autotrophic respiration) may improve the coherence of coupled plant-soil-microbe interactions. Besides, the Ohm's law formulation above will further enable biogeochemical models to use proteomic information to inform their parameterization that is not possible with the Michaelis-Menten kinetics.

Series Resistor-Based Formulation of Enzyme-Catalyzed Redox Reactions
Many biogeochemical processes are of the redox type, for example, photosynthesis, aerobic respiration, nitrification, and anaerobic denitrification (Madigan et al., 2009;Taiz & Zeiger, 2006). Basically, enzyme-catalyzed redox reactions facilitate electron transfers from electron donors to electron acceptors. This process can be summarized with the schema in Figure 2a that has one resistor representing electron donors ( , S edon E r ), Figure 2. (a) Type-1 circuit schema for redox-type reactions where electron donor binds before the electron acceptor to the enzyme, with the example (in red box) depicting the binding of RuBP (ribulose 1,5-bisphosphate) and O 2 to Rubisco enzyme to produce PGA (3-phosphoglycerate) and Gly (glycine) in the oxygenation pathway of photosynthesis; (b) Type-2 circuit schema for redox-type reactions where electron acceptor binds before electron donor to the enzyme; (c) Circuit schema for parallel resistor-based representation of competitive enzymatic reactions, with an example of an organism (in the red box) building biomass from assimilating ammonia and nitrate as substitutable nitrogen sources.   1/ . We note that in this series resistor-based formulation, the total resistance (or mean first passage time) does not include the discount resulting from the concurrent binding of electron donors and acceptors to the enzyme (i.e., type-2 configuration in Figure 2b, where electron donor binds before electron acceptor to the enzyme, is as good as type-1 configuration in Figure 2a, where electron donor binds after electron acceptor to the enzyme; because these two configurations have the same resistance, they are not differentiated in the Ohm's law analogy). However, this discount can be incorporated by renewal theory (or law of mass action, where these two configurations are considered as different and allowed to occur concurrently, such that the total resistance is smaller), which leads to the synthesizing unit (SU) model (Kooijman, 1998) where j E S are essential nutrients (e.g., carbon, nitrogen, phosphorus, potassium, and chloronium). Smith (1976Smith ( , 1979 used Equation 27 to model plant growth and microbial growth under carbon, nitrogen, phosphorus, and potassium colimitation. Based on past successful applications (Franklin et al., 2011;Kooijman, 1998), the SU model (i.e., Equation 26) may be argued as mathematically more rigorous than the series resistor-based additive model (i.e., Equation 25 or 27). However, given the usually significant uncertainty of ecological data, the series resistor-based additive model may be equally good (even using the same parameters as in the SU model). Indeed, when we applied both the SU model and the resistor-based additive model to the measured algal growth rates under various levels of phosphorus and vitamin B 12 additions (Droop, 1974; this data set was also used by Kooijman (1998) when the SU model was first developed), both models can be satisfyingly calibrated with respect to the growth data (Figures 3a and 3b; albeit higher maximum growth rate is inferred for fast adapted algae by the SU model, Table S1 in Supporting Information S1). Nevertheless, when the normalized growth rates are plotted as a function of the normalized substrate fluxes, the SU and resistor-based additive models show very similar growth patterns (Figures 3c and 3d). The SU model and additive model also performed equally well for the plant growth data from Shaver and Melillo (1984) (Figure 4), and their parameter values are also quite comparable in magnitude ( Figure S1 and Table S1 in Supporting Information S1). Moreover, when the SU and additive models are used to model aerobic heterotrophic respiration using the parameterization from Tang and Riley (2019a), we once again find the two models driven by identical parameters resulted in very similar goodness of fit with respect to the measurements ( Figure 5 and Table S2 in Supporting Information S1). These lines of evidence suggest that one can probably use these two models alternatively (but more extensive studies are needed to quantify the resultant structural uncertainty in the broader context of biogeochemical modeling). In particular, both  Table S1 in Supporting Information S1.  Shaver and Melillo (1984). Model parameters are in Table S1 in Supporting Information S1. TANG ET AL. can be a substitute for Liebig's law of the minimum that is used by most existing biogeochemical models (Achat et al., 2016;Tang & Riley, 2021). However, the additive model (derived from the Ohm's law analogy) is computationally much simpler than the theoretically more accurate SU model for situations that involve many more complementary nutrients (Tang & Riley, 2021).
Additionally, we note that Equation 27 can be extended (using a mixed series-parallel circuit; see Figure 6b and Section 3.4) into a photosynthesis model to replace the Farquhar or Collatz model that is formulated based on Liebig's law of the minimum, which has to arbitrarily smooth the abrupt transitions from one limiting process to another (e.g., Collatz et al., 1990Collatz et al., , 1992Farquhar et al., 1980;Kirschbaum & Farquhar, 1984). Notably, the use of Liebig's law of the minimum and smoothing functions has been recently identified as one major source of uncertainty in modeling terrestrial ecosystem gross primary productivity (Walker et al., 2021). Taking these potential applications together, we contend that it is possible to use the same kinetics to formulate models of plant photosynthesis, microbial substrate dynamics, and biomass growth, a strategy that will likely enhance the mathematical coherence in modeling plant-soil-microbe interactions.

Parallel Resistor-Based Formulation of Competitive Kinetics
Many microorganisms can feed on multiple substrates. For example, Escherichia coli and yeasts are able to perform both aerobic and anaerobic respiration (e.g., Dashko et al., 2014;Unden & Bongaerts, 1997); some methanotrophic bacteria can oxidize methane, ammonia, and carbon monoxide (Bedard & Knowles, 1989); and some denitrifiers can consume oxygen, nitrate, nitrite, nitric oxide, and nitrous oxide while feeding on one carbon substrate (Chen & Strous, 2013). Plants can also use diverse mineral nitrogen forms to produce biomass (e.g., Masclaux-Daubresse et al., 2010;Tang & Riley, 2021). Moreover, some enzymes can react on different substrates. For example, the ribonuclease enzyme is able to degrade various RNA molecules (Etienne et al., 2020). One common feature shared by all these different biogeochemical processes is that the uptake of one substrate often competitively inhibits the uptake of others. Thus, it is meaningful for us to show that such problems can be formulated using the parallel circuit (plus one series resistor) using the Ohm's law analogy. Left panels are SU model-based prediction of respiration-soil-moisture relationship; right panels are based on the resistor-based additive model. The two models (described in Supporting Information S1) used identical parameters, which are detailed in Tang and Riley (2019a). The statistics for model-data fitting (in terms of linear regression and root mean square error) between two models are identical to 0.01 (see Table S2 in Supporting Information S1).

of 19
We first formulate the competitive Michaelis-Menten kinetics using the schema in Figure 2c. For this case, the total resistance is and the corresponding flux through pathway . Therefore, j E v is the reaction velocity computed from the competitive Michaelis-Menten kinetics. We note that Equation 30 is meaningful only when pathway E j produces new molecules. However, even for inhibitors, whose binding to enzymes does not produce new molecules, if we regard dissociation as a way of producing new molecules, then Equation 30 is still meaningfully representing competitive inhibition.

Mixed Series and Parallel Resistor-Based Formulation of Redox Reactions of Alternative Electron Donors and Acceptors
Many microorganisms (such as denitrifying bacteria that play an essential role in the Earth's nitrogen cycle; e.g., Robertson & Groffman, 2015) are able to grow on different electron donors and acceptors. Such processes can be modeled using the SUPECA kinetics . Below, we show that it can also be formulated using the schema of mixed series and parallel resistors in the Ohm's law framework.
Based on the schema in Figure 6a, the total resistance is , edon eacc l l edon j j eacc r r r r r where the resistance for electron donors is l edon E l edon S l edon max l edon f l edon l edon r r r v k S (32) and the resistance for electron acceptors is Now considering an application that involves two electron acceptors, for example, nitrate and nitrite in denitrification, we have   which are just Equation 10 in Almeida et al. (1997) that have been successfully used to fit the measurement of denitrification rates from Almeida et al. (1995). With proper number of resistors, the denitrifier model by Domingo-Felez and Smets (2020) can also be easily recovered from Equations 31-35.
Further, we note that the relationship between the light, Rubisco enzyme-catalyzed carboxylation and oxygenation reactions in photosynthesis can be formulated analogously in Figure 6b, from which we can obtain the gross carbon fixation rate g E A as In summary, the examples in Sections 3.1-3.4 show that the Ohm's law analogy can formulate both plant and microbial biogeochemistry in the same framework.

Other Potential Applications of the Ohm's Law Analogy
Besides the applications described above, we below derive some quite interesting results to further highlight the potential of the Ohm's law analogy in biogeochemical modeling.
First, we will explain why fermentation can occur even when there is still oxygen to support the energetically more efficient aerobic respiration. Such a phenomenon is called the Warburg effect (i.e., lactate producing aerobic fermentation) in proliferating mammalian cells (a phenomenon important to the understanding of cancer development), or the Crabtree effect (i.e., ethanol fermentation) of unicellular yeast Saccharomyces cerevisiae (e.g., de Alteriis et al., 2018). Escherichia coli have also been observed to shift to the seemingly bioenergetically less efficient yet faster metabolic pathways under high substrate concentrations (e.g., Flamholz et al., 2013;Labhsetwar et al., 2014). Depending on the details to be represented, we acknowledge that there are multiple ways to model such phenomenon even with the circuit analogy (Molenaar et al., 2009;Schuster et al., 2015), highlighting the challenge for a comprehensive and robust understanding of this biochemical phenomenon. We next present one plausible mathematical explanations to show that, under certain aerobic conditions, high glucose concentration makes fermentation more favorable.
According to the schema in Figure 6c, the specific ATP generation rate from the fermentation pathway is where S E f is the incoming flux of pyruvate (produced from glycolysis) sensed by the two metabolic pathways (and is proportional to the incoming glucose flux sensed by the organism under steady state), fm E r is the resistance associated with the conversion of pyruvate into fermentation products (which could be lactate, ethanol, or acetate depending on the organism, Madigan et al., 2009) (Madigan et al., 2009).
In a metabolically active organism, for fermentation to be more favorable than aerobic respiration (in terms of ATP production rate for the same amount of enzyme allocated, i.e.,  where the term after the second ">" suggests that fermentation is more favorable only when oxygen is below a certain level of availability (note that 2 O E f is approximately proportional to diffusion). When the oxygen availability is sufficiently low (even though the system is not qualified as anaerobic), higher substrate concentration (i.e., greater S E f ) will make fermentation more effective in generating ATP for the same amount of enzyme allocated for catabolic reaction. If we additionally consider that the fermentation pathway requires the organism to maintain a much smaller number of enzymes than required for the aerobic oxidation pathway (which is equivalent to increase the value of Y Y FM AO / further, making the inequality (Equation 43) even easier to be satisfied), we can expect fermentation to be preferred under high glucose supply (i.e., greater Therefore, as illustrated above, the Ohm's law analogy enables us to quickly and vividly infer that increasing the substrate concentration E S reduces the resistance faster for the fermentation pathway than for the aerobic respiration pathway. Since genomic expression usually follow the induction and then response paradigm 15 of 19 (i.e., the Jacob-Monod model, Tiwari et al., 1974), the microbes under consideration will metabolically shift toward fermentation even though oxygen is available and aerobic respiration yields more ATP per unit of carbon consumed (Causton et al., 2001). In contrast, models based on the flux balance method or law of mass action will be more sophisticated to formulate and understand for such metabolic shift (Kesten et al., 2015;Nilsson & Nielsen, 2016). Given the significance of metabolic shift in various contexts, including methane and hydrogen dynamics in environment and industrial biogeochemistry (Lu et al., 2009;Madigan et al., 2009), we expect to study this problem in a more quantitative and extensive way elsewhere.
Another very interesting application of the Ohm's law analogy is to qualitatively explain why the substrate-growth rate relationship of an exponentially growing bacterial population can be fitted with the Monod kinetics (Monod, 1949), whose validity is assumed implicitly in most existing studies of microbial growth on single substrate. For an exponentially growing bacterial population, the bacteria proteomes are approximately in steady state. Meanwhile, from the Ohm's law analogy, we know that any functioning circuit-network can be equivalently represented by a bulk resistor. Therefore, we contend that however complex the circuit representation of a bacterial metabolism would be, as a whole it can be equivalently represented by a constant resistance E E r . When this E E r is combined with the resistance associated with the incoming substrate flux (see Equation 7), we then say that the bacterial growth would very likely follow the Monod kinetics. However, when the bacteria are in transition from one metabolic state into another (e.g., from gluconate to succinate), extra resistors are introduced accompanying the change of proteomes, resulting in a dynamic E E r and thus Monod kinetics will fail for such situations (e.g., Erickson et al., 2017). This argument also explains why models based on flux balance analysis with proteomic constraints can simulate exponentially growing E. coli and yeast realistically (Labhsetwar et al., 2014(Labhsetwar et al., , 2017, but the flux balance models are cumbersome to apply in dynamic environments.

Limitations of the Ohm's Law Analogy
While the Ohm's law analogy can be used to model many challenging biogeochemical processes, it is not appropriate for all types of biogeochemical networks. For instance, it is not able to properly couple two or more consumers (i.e., two or more batteries) within a single circuit network, even though the electric circuit theory itself does not forbid such a configuration to occur (which can be solved with the Kirchhoff's law of voltage and current, e.g., Feynman et al., 2011b). Rather, the coupling can only be done by first representing the substrate dynamics of each consumer separately, and then coupling them together by differential equations. Such coupling could be critical when many consumers are competing for a limiting substrate, even though none of the consumers is substrate limited when other consumers are excluded (e.g., Etienne et al., 2020). The ECA kinetics (Tang & Riley, 2013b) and its progeny SUPECA kinetics  are more capable of resolving such situations. In soil biogeochemistry, one such situation is to model the interaction of a substrate molecule (e.g., ammonium, inorganic phosphorus, or dissolved organic carbon) that is simultaneously undergoing uptake by organisms and adsorption by mineral surfaces. Fortunately, a simple remedy is possible for the Ohm's law analogy from the ECA kinetics. In the ECA kinetics, microbial uptake of substrate E S under the influence of adsorption by mineral surface E M (with affinity parameter M E K ) is Now the Ohm's law analogy will still work if 1/k S f  is used to defined the substrate-dependent resistance.
Moreover, Equation 46 suggests that mineral surfaces may slow the microbial uptake of substrate E S by effectively reducing the substrate delivery rate toward the microbes.
However, when the sizes of substrates and competitors are similar (e.g., in some predator-prey relationships), the Ohm's law analogy will be too cumbersome to apply, and the ECA or SUPECA kinetics should be used. Nonetheless, it will be very interesting and helpful to construct and compare models for the same system using both the Ohm's law analogy and ECA (or SUPECA) kinetics.

Conclusions
By exploiting the mathematical similarity between the Ohm's law and Michaelis-Menten kinetics, we show that the electric circuit analogy can be used to derive many interesting results of biogeochemical kinetics. We show this approach reproduces many successful applications in the literature, including aerobic heterotrophic respiration, multinutrient colimited microbial (and plant) growth, and denitrification dynamics. This approach also sheds new insights on the temperature sensitivity of kinetic parameters in substrate uptake, the Warburg and Crabtree effect in prokaryotes and eukaryotes, and conceptually explains why the Monod relationship accurately represents the kinetics of exponentially growing bacterial populations, and why flux balance modeling constrained by proteomics is able to accurately model microbial growth. Based on these results, we expect that the Ohm's law analogy will help build a unified kinetic modeling framework of microbial and plant biogeochemistry to make more robust predictions.