Conceptualizing Biogeochemical Reactions With an Ohm's Law Analogy
Abstract
In studying problems like plantsoilmicrobe 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 substrateconsumer relationships, many substrate kinetics and growth rules have been developed, including the famous Monod kinetics for singlesubstratebased growth and Liebig's law of the minimum for multiplenutrientcolimited 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 MichaelisMenten 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) multiplenutrientcolimited 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.
Key Points

Ohm's law is proposed to formulate biogeochemical reactions

Ohm's law successfully represents multiplenutrientcolimited growth

Ohm's law may help improve and unify biogeochemical models
Plain Language Summary
Currently, scientists often use ad hoc or empirical approaches to conceptualize and formulate biogeochemical processes encountered in environmental sciences. Here, we propose that many biogeochemical processes can be coherently conceptualized and formulated using an analogy based on Ohm's law, a mathematical theory that is widely used to model electric circuits, and the landatmosphere exchange of water and energy. We show that this Ohm's law analogy is able to explain observations such as why microbial growth often follows Monod kinetics, how fermentation can sometimes dominate aerobic respiration when glucose is plentiful, and how plants and microbes grow under multiplenutrientcolimitation. Since this Ohm's law analogy unifies the mathematical foundation of biogeophysics and biogeochemistry, we believe that it can potentially lead to more robust land ecosystem models for projecting the climate change.
1 Introduction
In earth system modeling, biogeochemistry strongly affects mass and energy exchanges between ecosystems and the physical climate system (Heinze et al., 2019). Morphologically, biogeochemistry has three pillars: biology, geophysics, and chemistry. In the context of mathematical modeling, geophysics and chemistry generally have much stronger theoretical foundations than biology (Brutsaert, 2005; Stumm & Morgan, 1996; Vallis, 2006), even though all three are macroscale responses that emerge from atomic interactions, which in an ideal (but impractical) scenario can be predicted by solving the Schrödinger equation of all atoms together (so that arguably they all are subtopics of physics; Feynman et al., 2011c).
In seeking a better understanding of ecological dynamics, for example, competition and symbiosis, mathematical formulations of the substrateconsumer relationship (e.g., the interactions between many microbes as consumers and their diverse substrates) are essential for theoretical modeling and interpreting empirical experiments, such as phytoplankton population dynamics (Tilman, 1982), plantmicrobial competition of nutrients (Zhu et al., 2017), and microbial decomposition of organic matter (Tang & Riley, 2013b; Yu 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 PierreFrancois 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, MichaelisMenten 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, 2017), with the renewal theory even being able to show that MichaelisMenten kinetics is the statistical mean of the stochastic description of a singleenzyme 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, theorybased 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 MichaelisMenten 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 theorybased approach has been used to derive the temperature response function of microbial activity (Ghosh & Dill, 2010) and to explain why MichaelisMenten kinetics are more appropriate for microbial uptake of small molecules, while reverse MichaelisMenten 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 substrateconsumer relationships. Similar analogies have been widely used by land models to represent the gradientdriven landatmosphere 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 (Tang & Riley, 2017). 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.
2 Methods
2.1 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).
To simplify the presentation, we henceforth assume that all variables are properly defined as in the international system of units.
As another analogy, Ohm's law has also been used to represent soil evaporation (Bonan, 2019; Tang & Riley, 2013a), where voltage is calculated as the difference between atmospheric and soil water vapor concentrations, resistance is the sum of atmospheric and soil resistance, and current is the evaporation flux.
2.2 MichaelisMenten Kinetics Interpreted With Ohm's Law
That the resistance in Equation 8 is of the unit time has also motivated some researchers to apply the timebudget idea to derive predatorprey relationships (e.g., Holling, 1959; Murdoch, 1973), where is referred as the mean time a predator spends on handling its prey, and is the mean time for a predator to encounter its prey. Further, McAdams and Shapiro (1995) noticed that the circuit analogy can be used to interpret and model genetic networks. However, few studies have pointed out the linkage between the timebudget analysis and Ohm's law, except, based on a suggestion by Thomsen et al. (1994), Almeida et al. (1997) made an analogy of the membrane electron transport chain to an electric circuit, and successfully used it to model denitrification. Later, Murkin (2015) suggested that the Ohm's law may be used to help students better understand enzyme kinetics in teaching biochemistry. Recently, this method has been used by DomingoFelez and Smets (2020) to build the Activated Sludge ModelElectron Competition (ASMEC) model, which demonstrated the efficacy of this analogy in constructing robust biogeochemical models. Further, the molecular biology of membrane electron transport chains and redox reactions are quite similar to the working principles of chemical batteries (Frederiksen & Andresen, 2008; SchmidtRohr, 2018), thereby motivating us to explore more extensively the applicability of Ohm's law analogy below.
In the Ohm's law analogy, kinetic interactions between an enzyme and its substrate molecules can be summarized as the batteryresistor relationship shown in Figure 1a, where the battery potential is enzyme concentration , and the battery's resistance is , while the resistor (i.e., substrate) has resistance . However, we note that this analogy is accurate only when the substrate is nonlimiting for the enzymes (i.e., when MichaelisMenten kinetics are more appropriate, Tang & Riley, 2019b). For cases when substrate is limiting, the reverse MichaelisMenten kinetics are more appropriate (Tang, 2015), and the roles of substrate and enzyme in the analogy are reversed (see Equation 9). We also note that the ECA kinetics are able to more accurately handle the wide range of substrate abundances with respect to enzymes (Tang, 2015). We next show how the Ohm's law analogy can help formulate biogeochemical kinetics for various situations.
3 Applications
3.1 Series Resistor CircuitBased Formulation of ChainLike Enzyme Reactions
From Equation 11, we assert that an enzyme chain is equivalent to a functional enzyme unit with kinetic traits and . Moreover, from Equations 13 and 14, we infer that increasing the chain length decreases the overall reaction rate (which is even slower than the slowest step ) and the half saturation coefficient of the enzyme chain.
Several interesting inferences can be drawn from Equations 1014 that will provide us with a better understanding of the tradeoffs 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 chainlike metabolic pathway as a whole can be represented similarly with the MichaelisMenten kinetics (e.g., Equation 12), there are tradeoffs 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 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 tradeoff 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 chemostatbased 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 9, 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 and , 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).
Therefore, if , the temperature dependence of will be approximately like that in Equation 15.
The temperature dependence of is determined by the temperature dependencies of and . Inside the microbial cytoplasm and cell membrane (and also for whole microbial cells in most natural environments, and chloroplasts in mesophyll cells), is positively related to diffusivity (Madigan et al., 2009). Thus, according to the StokesEinstein equation of translational diffusivity (, where is the Boltzmann constant, is the dynamic viscosity, and is the radius of the spherical particle; Feynman et al., 2011a), can be approximated with a linear temperature dependence divided by the temperature sensitivity of the dynamic viscosity (which is , where and are empirical parameters, according to the semiempirical VogelFulcherTammanHesse equation, GarciaColin et al., 1989). When the temperature dependence of is combined with the Eyringtype temperature dependence of , one may infer that the temperature sensitivity of (=) is of the Arrhenius type (because of the dynamic viscosity is very similar to the Arrhenius equation, and the linear temperature dependence of cancels out the linear part of the temperature dependence of ). Once again, if , will have an Arrheniustype temperature sensitivity as well.
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 plantsoilmicrobe 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 MichaelisMenten kinetics.
3.2 Series ResistorBased Formulation of EnzymeCatalyzed Redox Reactions
Compared to Equation 25, the SU model (i.e., Equation 26) is numerically more accurate (in approximating the law of mass action, the standard method that deals with biogeochemical reactions, Koudriavstev et al., 2001). Equations 25 and 26 differ by the term that accounts for the coexistence of schemas in Figures 2a and 2b.
Additionally, we note that Equation 27 can be extended (using a mixed seriesparallel circuit; see Figure 6b and Section 8) 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., 1990, 1992; Farquhar 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 plantsoilmicrobe interactions.
3.3 Parallel ResistorBased 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., MasclauxDaubresse 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.
3.4 Mixed Series and Parallel ResistorBased 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 (Tang & Riley, 2017). Below, we show that it can also be formulated using the schema of mixed series and parallel resistors in the Ohm's law framework.
In summary, the examples in Sections 58 show that the Ohm's law analogy can formulate both plant and microbial biogeochemistry in the same framework.
3.5 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.
Therefore, as illustrated above, the Ohm's law analogy enables us to quickly and vividly infer that increasing the substrate concentration 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 (i.e., the JacobMonod 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 substrategrowth 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 circuitnetwork 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 . When this 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 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, 2017), but the flux balance models are cumbersome to apply in dynamic environments.
3.6 Limitations of the Ohm's Law Analogy
Now the Ohm's law analogy will still work if is used to defined the substratedependent resistance. Moreover, Equation 46 suggests that mineral surfaces may slow the microbial uptake of substrate by effectively reducing the substrate delivery rate toward the microbes.
However, when the sizes of substrates and competitors are similar (e.g., in some predatorprey 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.
4 Conclusions
By exploiting the mathematical similarity between the Ohm's law and MichaelisMenten 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.
Acknowledgments
This research was supported by the Director, Office of Science, Office of Biological and Environmental Research of the US Department of Energy under contract no. DEAC0205CH11231 as part of the Next Generation Ecosystem ExperimentArctic project and the Energy Exascale Earth System Model (E3SM) project for J. Y. Tang and W. J. Riley, the TES Soil Warming SFA for W. J. Riley, and the Department of Energy, Office of Biological and Environmental Research, Genomic Sciences Program through the LLNL Microbes Persist Science Focus Area for GLM. E. L. Brodie was supported by funding from the Department of Energy, Office of Biological and Environmental Research, Subsurface Biogeochemical Research Program through the LBNL Watershed Function Science Focus Area. Financial support does not constitute an endorsement by the Department of Energy of the views expressed in this study.
Conflict of Interest
The authors declare no conflicts of interest relevant to this study.