Volume 13, Issue 10 e2021MS002469
Research Article
Open Access

Conceptualizing Biogeochemical Reactions With an Ohm's Law Analogy

Jinyun Tang

Corresponding Author

Jinyun Tang

Earth and Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Correspondence to:

J. Tang,

[email protected]

Contribution: Conceptualization, Formal analysis, ​Investigation, Writing - original draft, Funding acquisition

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William J. Riley

William J. Riley

Earth and Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Contribution: Writing - review & editing, Funding acquisition

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Gianna L. Marschmann

Gianna L. Marschmann

Earth and Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Contribution: Writing - review & editing

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Eoin L. Brodie

Eoin L. Brodie

Earth and Environmental Sciences Area, Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Contribution: Writing - review & editing, Funding acquisition

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First published: 05 October 2021
Citations: 1

Abstract

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.

Key Points

  • Ohm's law is proposed to formulate biogeochemical reactions

  • Ohm's law successfully represents multiple-nutrient-colimited 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 land-atmosphere 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 multiple-nutrient-colimitation. 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 substrate-consumer 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), plant-microbial 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 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, 2013b2017), 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 cm3)) 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 (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).

Ohm's law describes the relationship between voltage (urn:x-wiley:19422466:media:jame21460:jame21460-math-0001), electric current (urn:x-wiley:19422466:media:jame21460:jame21460-math-0002), and resistance (urn:x-wiley:19422466:media:jame21460:jame21460-math-0003) as
urn:x-wiley:19422466:media:jame21460:jame21460-math-0004(1)

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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0005, application of Ohm's law yields
urn:x-wiley:19422466:media:jame21460:jame21460-math-0006(2)
For a parallel concatenation of resistors urn:x-wiley:19422466:media:jame21460:jame21460-math-0007, application of Ohm's law leads to
urn:x-wiley:19422466:media:jame21460:jame21460-math-0008(3)
and the electric current through each resistor is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0009(4)
From Equations 3 and 4, we can further derive
urn:x-wiley:19422466:media:jame21460:jame21460-math-0010(5)
which states that when all other resistors are fixed, the fraction of current through urn:x-wiley:19422466:media:jame21460:jame21460-math-0011 increases with decreasing urn:x-wiley:19422466:media:jame21460:jame21460-math-0012. We will see later that this inference is very useful to explain shifts in metabolic pathways in biological organisms.

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 Michaelis-Menten Kinetics Interpreted With Ohm's Law

Michaelis-Menten kinetics represent the single-enzyme-catalyzed single-substrate reaction velocity urn:x-wiley:19422466:media:jame21460:jame21460-math-0013 as
urn:x-wiley:19422466:media:jame21460:jame21460-math-0014(6)
where, in the original application by Michaelis and Menten (1913), urn:x-wiley:19422466:media:jame21460:jame21460-math-0015 is the maximum specific catalytic rate enabled by the enzyme and urn:x-wiley:19422466:media:jame21460:jame21460-math-0016, urn:x-wiley:19422466:media:jame21460:jame21460-math-0017, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0018 are enzyme concentration, substrate concentration, and half saturation coefficient, respectively. We note that, for enzymes, urn:x-wiley:19422466:media:jame21460:jame21460-math-0019 also includes contributions from the dissociation process (e.g., Briggs & Haldane, 1925).
By defining urn:x-wiley:19422466:media:jame21460:jame21460-math-0020, Equation 6 can be rewritten as
urn:x-wiley:19422466:media:jame21460:jame21460-math-0021(7)
We then note that Equations 1 and 7 are mathematically of the same form. Therefore, for Michaelis-Menten kinetics, if we apply the Ohm's law analogy by regarding urn:x-wiley:19422466:media:jame21460:jame21460-math-0022 as voltage and urn:x-wiley:19422466:media:jame21460:jame21460-math-0023 as current, the corresponding resistance is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0024(8)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0025 represents the resistance as an intrinsic property (i.e., a kinetic trait) of the enzyme, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0026 represents the resistance introduced by the effective substrate delivery rate toward the enzyme (i.e., a kinetic trait of the substrate in the working environment of the enzyme). Further, urn:x-wiley:19422466:media:jame21460:jame21460-math-0027 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0028 are of the unit of time, where (in the renewal theory applied to enzyme-substrate interactions, e.g., Kooijman, 1998) urn:x-wiley:19422466:media:jame21460:jame21460-math-0029 is the mean time for the enzyme to convert the enzyme-bound substrate molecules into product molecules and urn:x-wiley:19422466:media:jame21460:jame21460-math-0030 is the mean time for the substrate molecules to approach the enzyme molecule and form enzyme-substrate complexes. Therefore, urn:x-wiley:19422466:media:jame21460:jame21460-math-0031 is the mean first passage time of the stochastic single-enzyme degradation of the substrate molecule (e.g., Kooijman, 1998; Ninio, 1987; Qian, 2008). In particular, in many reactions, urn:x-wiley:19422466:media:jame21460:jame21460-math-0032 is approximately proportional to the substrate diffusivity (Alberty & Hammes, 1958; Chou & Jiang, 1974), such that urn:x-wiley:19422466:media:jame21460:jame21460-math-0033 is the diffusive substrate flux sensed by enzyme molecules. We then observe that urn:x-wiley:19422466:media:jame21460:jame21460-math-0034 increases with the decrease of diffusive substrate flux, which can result from lower substrate concentration or lower diffusivity (due to tortuosity, adsorption, or lower moisture in porous media like soil). In Tang and Riley (2019a), the relationship between urn:x-wiley:19422466:media:jame21460:jame21460-math-0035 and diffusivity has enabled a way to parameterize how soil moisture affects microbial substrate uptake, which probably can also be used to parameterize how soil moisture affects plant root uptake of macronutrients.
Alternatively, for high-enzyme concentration systems (such as the hydrolysis of cellulose, Kari et al., 2017; Tang & Riley, 2019b), where the reverse Michaelis-Menten kinetics better describe the dynamics, we have
urn:x-wiley:19422466:media:jame21460:jame21460-math-0036(9)
where substrate urn:x-wiley:19422466:media:jame21460:jame21460-math-0037 instead plays the role of voltage, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0038 defines the resistance due to the effective deployment rate of enzyme urn:x-wiley:19422466:media:jame21460:jame21460-math-0039 to the substrate. Comparing Equations 8 and 9, we see that the roles of substrate and enzyme in the Ohm's law analogy are context dependent.

That the resistance urn:x-wiley:19422466:media:jame21460:jame21460-math-0040 in Equation 8 is of the unit time has also motivated some researchers to apply the time-budget idea to derive predator-prey relationships (e.g., Holling, 1959; Murdoch, 1973), where urn:x-wiley:19422466:media:jame21460:jame21460-math-0041 is referred as the mean time a predator spends on handling its prey, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0042 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 time-budget 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 Domingo-Felez and Smets (2020) to build the Activated Sludge Model-Electron Competition (ASM-EC) 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; Schmidt-Rohr, 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 battery-resistor relationship shown in Figure 1a, where the battery potential is enzyme concentration urn:x-wiley:19422466:media:jame21460:jame21460-math-0043, and the battery's resistance is urn:x-wiley:19422466:media:jame21460:jame21460-math-0044, while the resistor (i.e., substrate) has resistance urn:x-wiley:19422466:media:jame21460:jame21460-math-0045. However, we note that this analogy is accurate only when the substrate is nonlimiting for the enzymes (i.e., when Michaelis-Menten kinetics are more appropriate, Tang & Riley, 2019b). For cases when substrate is limiting, the reverse Michaelis-Menten 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.

Details are in the caption following the image

(a) Circuit schema for the Michaelis-Menten kinetics, with the example (in red box) depicting the conversion of pyruvate into acetyl-coA and CO2 by the enzyme complex pyruvate dehydrogenase complex; (b) Series resistor-based schema for an enzyme chain and its reaction on substrate urn:x-wiley:19422466:media:jame21460:jame21460-math-0046, where dotted lines indicate multiple resistors urn:x-wiley:19422466:media:jame21460:jame21460-math-0047 concatenated in series. Symbols are explained in the main text. The example for (b) depicts the metabolic pathway of glycolysis.

3 Applications

3.1 Series Resistor Circuit-Based Formulation of Chain-Like Enzyme Reactions

Many metabolic pathways consist of a chain of reactions. Examples include the Calvin-cycle (in photosynthesis), membrane electron transport chain, glycolysis (Figure 1b), and citric acid cycle, and note that most of these reaction pathways involve cofactors (Madigan et al., 2009; Taiz & Zeiger, 2006). Nonetheless, 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
urn:x-wiley:19422466:media:jame21460:jame21460-math-0048(10)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0049, such that the first right-hand side term is the total resistance represented by the maximum catalysis rate of the overall enzyme chain, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0050 is the resistance due to the incoming substrate flux to the first enzyme in the chain. For the overall chain, the specific reaction rate for substrate processing is then
urn:x-wiley:19422466:media:jame21460:jame21460-math-0051(11)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0052, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0053 represents the enzyme functional unit (e.g., for the glycolysis metabolic pathway in Figure 1b, urn:x-wiley:19422466:media:jame21460:jame21460-math-0054 could be the amount of enzyme hexokinase urn:x-wiley:19422466:media:jame21460:jame21460-math-0055, assuming that all other enzymes are highly regulated in forming the chain of enzymes catalyzing related biogeochemical reactions). Equation 11 can be simplified as
urn:x-wiley:19422466:media:jame21460:jame21460-math-0056(12)
with
urn:x-wiley:19422466:media:jame21460:jame21460-math-0057(13)
and
urn:x-wiley:19422466:media:jame21460:jame21460-math-0058(14)

From Equation 11, we assert that an enzyme chain is equivalent to a functional enzyme unit with kinetic traits urn:x-wiley:19422466:media:jame21460:jame21460-math-0059 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0060. Moreover, from Equations 13 and 14, we infer that increasing the chain length decreases the overall reaction rate urn:x-wiley:19422466:media:jame21460:jame21460-math-0061 (which is even slower than the slowest step urn:x-wiley:19422466:media:jame21460:jame21460-math-0062) and the half saturation coefficient urn:x-wiley:19422466:media:jame21460:jame21460-math-0063 of the enzyme chain.

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 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 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0064 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0065, 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, urn:x-wiley:19422466:media:jame21460:jame21460-math-0066 equals urn:x-wiley:19422466:media:jame21460:jame21460-math-0067, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0068 equals urn:x-wiley:19422466:media:jame21460:jame21460-math-0069. According to transition state theory (e.g., Eyring, 1935), urn:x-wiley:19422466:media:jame21460:jame21460-math-0070 would have the following temperature dependence,
urn:x-wiley:19422466:media:jame21460:jame21460-math-0071(15)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0072 is some reference reaction rate, urn:x-wiley:19422466:media:jame21460:jame21460-math-0073 is temperature, urn:x-wiley:19422466:media:jame21460:jame21460-math-0074 (>0) is the Gibbs energy of activation, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0075 is the universal gas constant. Similarly, for a reaction pathway consisting of a chain of enzymes, each urn:x-wiley:19422466:media:jame21460:jame21460-math-0076 will have a temperature dependence similar to that in Equation 15, that is,
urn:x-wiley:19422466:media:jame21460:jame21460-math-0077(16)
which when entered into Equation 13 leads to
urn:x-wiley:19422466:media:jame21460:jame21460-math-0078(17)

Therefore, if urn:x-wiley:19422466:media:jame21460:jame21460-math-0079, the temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0080 will be approximately like that in Equation 15.

The temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0081 is determined by the temperature dependencies of urn:x-wiley:19422466:media:jame21460:jame21460-math-0082 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0083. Inside the microbial cytoplasm and cell membrane (and also for whole microbial cells in most natural environments, and chloroplasts in mesophyll cells), urn:x-wiley:19422466:media:jame21460:jame21460-math-0084 is positively related to diffusivity (Madigan et al., 2009). Thus, according to the Stokes-Einstein equation of translational diffusivity (urn:x-wiley:19422466:media:jame21460:jame21460-math-0085, where urn:x-wiley:19422466:media:jame21460:jame21460-math-0086 is the Boltzmann constant, urn:x-wiley:19422466:media:jame21460:jame21460-math-0087 is the dynamic viscosity, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0088 is the radius of the spherical particle; Feynman et al., 2011a), urn:x-wiley:19422466:media:jame21460:jame21460-math-0089 can be approximated with a linear temperature dependence divided by the temperature sensitivity of the dynamic viscosity urn:x-wiley:19422466:media:jame21460:jame21460-math-0090 (which is urn:x-wiley:19422466:media:jame21460:jame21460-math-0091, where urn:x-wiley:19422466:media:jame21460:jame21460-math-0092 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0093 are empirical parameters, according to the semiempirical Vogel-Fulcher-Tamman-Hesse equation, Garcia-Colin et al., 1989). When the temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0094 is combined with the Eyring-type temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0095, one may infer that the temperature sensitivity of urn:x-wiley:19422466:media:jame21460:jame21460-math-0096 (=urn:x-wiley:19422466:media:jame21460:jame21460-math-0097) is of the Arrhenius type (because urn:x-wiley:19422466:media:jame21460:jame21460-math-0098 of the dynamic viscosity urn:x-wiley:19422466:media:jame21460:jame21460-math-0099 is very similar to the Arrhenius equation, and the linear temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0100 cancels out the linear part of the temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0101). Once again, if urn:x-wiley:19422466:media:jame21460:jame21460-math-0102, urn:x-wiley:19422466:media:jame21460:jame21460-math-0103 will have an Arrhenius-type temperature sensitivity as well.

When the above inferences are substituted to Equation 12, we can then infer the temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0104. From chemical thermodynamics, the temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0105 depends on chemical kinetics (as characterized by the Michaelis-Menten term, i.e., urn:x-wiley:19422466:media:jame21460:jame21460-math-0106 in this example) and thermodynamics (as a function of the Gibbs free energy) of the enzyme-catalyzed reaction (LaRowe et al., 2012). 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0107 (as measured by the number of amino acid residues) can be described as (Murphy et al., 1990)
urn:x-wiley:19422466:media:jame21460:jame21460-math-0108(18)
where
urn:x-wiley:19422466:media:jame21460:jame21460-math-0109(19)
and
urn:x-wiley:19422466:media:jame21460:jame21460-math-0110(20)
with heat capacity urn:x-wiley:19422466:media:jame21460:jame21460-math-0111 defined as the energy required to reorganize the water molecules surrounding the protein (Ratkowsky et al., 2005). urn:x-wiley:19422466:media:jame21460:jame21460-math-0112 increases with the nonpolar accessible area of the molecule, as measured by urn:x-wiley:19422466:media:jame21460:jame21460-math-0113, the average number of nonpolar hydrogen atoms per amino acid residue. urn:x-wiley:19422466:media:jame21460:jame21460-math-0114 also measures the hydrophobic contribution, with higher values implying higher hydrophobicity (and notice that greater urn:x-wiley:19422466:media:jame21460:jame21460-math-0115 implies higher hydrophobicity). Other parameters include urn:x-wiley:19422466:media:jame21460:jame21460-math-0116 as the enthalpy change at urn:x-wiley:19422466:media:jame21460:jame21460-math-0117 (the convergence temperature for entropy, i.e., the temperature at which the hydrophobic contributions to urn:x-wiley:19422466:media:jame21460:jame21460-math-0118 is zero) and urn:x-wiley:19422466:media:jame21460:jame21460-math-0119 as the enthalpy change at urn:x-wiley:19422466:media:jame21460:jame21460-math-0120 (the convergence temperature for enthalpy, i.e., the temperature at which the hydrophobic contributions to urn:x-wiley:19422466:media:jame21460:jame21460-math-0121 is zero), among which urn:x-wiley:19422466:media:jame21460:jame21460-math-0122, urn:x-wiley:19422466:media:jame21460:jame21460-math-0123, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0124 can be considered to be constant under environmental conditions, for example, Ratkowsky et al. (2005) took urn:x-wiley:19422466:media:jame21460:jame21460-math-0125 K, urn:x-wiley:19422466:media:jame21460:jame21460-math-0126 K, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0127 J K−1 (mol amino acid residue)−1, respectively. Assuming urn:x-wiley:19422466:media:jame21460:jame21460-math-0128, urn:x-wiley:19422466:media:jame21460:jame21460-math-0129, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0130 can be obtained from proteomic data for each type of enzyme (e.g., Sawle & Ghosh, 2011), we can then calculate urn:x-wiley:19422466:media:jame21460:jame21460-math-0131 for all enzymes involved in the chain. Therefore, putting together the kinetic, thermodynamic, and catalytically active enzyme fraction functions, we obtain
urn:x-wiley:19422466:media:jame21460:jame21460-math-0132(21)
where the thermodynamic temperature dependence of the reaction is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0133(22)
with urn:x-wiley:19422466:media:jame21460:jame21460-math-0134 (<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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0135 is an intrinsic property of the overall chemical reaction then allows the total fraction of active enzymes to be factored out as urn:x-wiley:19422466:media:jame21460:jame21460-math-0136. (Note that urn:x-wiley:19422466:media:jame21460:jame21460-math-0137 can be viewed as the probability for enzyme urn:x-wiley:19422466:media:jame21460:jame21460-math-0138 to be in the conformed state, thus, by the theory of conditional probability, the probability for the whole enzyme chain to be in active status is urn:x-wiley:19422466:media:jame21460:jame21460-math-0139.)
Unless Equation 21 is applied to organisms capable of growing on alternative electron acceptors or donors and the system is undergoing fast transition in redox status (e.g., heterotrophic microbes in the fluctuating zone of soil water table, Zhang & Furman, 2021), urn:x-wiley:19422466:media:jame21460:jame21460-math-0140 can be taken approximately as 1. Therefore, the temperature dependence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0141 is dominated by the kinetic term (i.e., the Michaelis-Menten term) and the temperature-dependent fraction of active enzymes (urn:x-wiley:19422466:media:jame21460:jame21460-math-0142). The kinetic term increases with temperature (see Equation 16), while the fraction of active enzymes first increases, then decreases with temperature (Equations 18-20). The overall temperature sensitivity of the reaction chain will then be of the form predicted by the macromolecular rate theory (MMRT; with fine tuning from substrate availability through the kinetic term, i.e., the denominator of the Michaelis-Menten term in Equation 21 which MMRT does not consider; 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0143):
urn:x-wiley:19422466:media:jame21460:jame21460-math-0144(23)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0145 is growth rate, urn:x-wiley:19422466:media:jame21460:jame21460-math-0146 is an empirical constant, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0147 is substrate-dependent activation energy. However, unlike previous assumptions that properties of some single control enzyme determine the overall growth (Johnson & Lewin, 1946), here urn:x-wiley:19422466:media:jame21460:jame21460-math-0148 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0149 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.

3.2 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 (urn:x-wiley:19422466:media:jame21460:jame21460-math-0150), and the other resistor (urn:x-wiley:19422466:media:jame21460:jame21460-math-0151) representing electron acceptors, with the enzyme (symbolically) being the battery. By applying the Ohm's law analogy, the reaction rate is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0152(24)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0153, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0154. When the two are combined, Equation 24 can be rewritten as
urn:x-wiley:19422466:media:jame21460:jame21460-math-0155(25)
with urn:x-wiley:19422466:media:jame21460:jame21460-math-0156, urn:x-wiley:19422466:media:jame21460:jame21460-math-0157, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0158. 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) below:
urn:x-wiley:19422466:media:jame21460:jame21460-math-0159(26)
Details are in the caption following the image

(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 O2 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. Type-1 and type-2 schema are equivalent and are not differentiated in the Ohm's law analogy based on resistance. Symbols are explained in the main text.

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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0160 that accounts for the coexistence of schemas in Figures 2a and 2b.

Equation 25 was derived as early as in Alberty (1953) and is called the additive model. It was found to be the superior formulation to model multiple nutrient limitations of microbial and plant growth in O’Neill et al. (1989) (where electron donors and acceptors are replaced with complementary nutrients, such as nitrogen and phosphorus). In particular, the additive model (Equation 25) can be extended to include an arbitrary number of nutrients:
urn:x-wiley:19422466:media:jame21460:jame21460-math-0161(27)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0162 are essential nutrients (e.g., carbon, nitrogen, phosphorus, potassium, and chloronium). Smith (19761979) 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 B12 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 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).
Details are in the caption following the image

(a) Comparison of the calibrated synthesizing unit (SU) model prediction for the algal growth rates data from Droop (1974); Panel (b) same as panel (a) but from the calibrated resistor-based additive model; (c) Contour of normalized growth rate as a function of normalized fluxes of substrates A (with urn:x-wiley:19422466:media:jame21460:jame21460-math-0163) and B (with urn:x-wiley:19422466:media:jame21460:jame21460-math-0164) for the SU model; Panel (d) same as panel (c) but for the resistor-based additive model. The additive model is presented as Equation 25, and the SU model is presented as Equation 26. Model parameters are in Table S1 in Supporting Information S1.

Details are in the caption following the image

(a) SU model predicted versus measured plant growth; (b) Additive model predicted versus measured plant growth. The data are from Shaver and Melillo (1984). Model parameters are in Table S1 in Supporting Information S1.

Details are in the caption following the image

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).

Additionally, we note that Equation 27 can be extended (using a mixed series-parallel 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., 19901992; 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 plant-soil-microbe interactions.

Details are in the caption following the image

(a) Mixed resistor circuit schema for redox reactions with alternative electron donors and acceptors, with example (in red box) depicting the use of acetate and methanol as electron donors, and nitrite and nitrate as electron acceptors during denitrification; (b) Circuit scheme for photosynthesis; (c) Circuit schema for the parallel fermentation and aerobic respiration pathways. Symbols are explained in the main text.

3.3 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.

We first formulate the competitive Michaelis-Menten kinetics using the schema in Figure 2c. For this case, the total resistance is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0165(28)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0166, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0167 is the resistance due to preprocessing of substrate urn:x-wiley:19422466:media:jame21460:jame21460-math-0168 before it is handed to the central enzyme urn:x-wiley:19422466:media:jame21460:jame21460-math-0169 (i.e., the enzyme that products of all substrates have to pass through), and urn:x-wiley:19422466:media:jame21460:jame21460-math-0170 is the resistance due to the maximum substrate processing rate of the central enzyme (which for redox reactions could be determined by the time spent on processing the electron donors if urn:x-wiley:19422466:media:jame21460:jame21460-math-0171 here are electron acceptors). If urn:x-wiley:19422466:media:jame21460:jame21460-math-0172, which is usually assumed for competitive Michaelis-Menten kinetics, the second term urn:x-wiley:19422466:media:jame21460:jame21460-math-0173 becomes urn:x-wiley:19422466:media:jame21460:jame21460-math-0174, and the reaction velocity is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0175(29)
and the corresponding flux through pathway urn:x-wiley:19422466:media:jame21460:jame21460-math-0176 is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0177(30)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0178. Therefore, urn:x-wiley:19422466:media:jame21460:jame21460-math-0179 is the reaction velocity computed from the competitive Michaelis-Menten kinetics. We note that Equation 30 is meaningful only when pathway urn:x-wiley:19422466:media:jame21460:jame21460-math-0180 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.

3.4 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 (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.

Based on the schema in Figure 6a, the total resistance is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0181(31)
where the resistance for electron donors is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0182(32)
and the resistance for electron acceptors is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0183(33)
Accordingly, the corresponding reaction flux through electron donor urn:x-wiley:19422466:media:jame21460:jame21460-math-0184 is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0185(34)
while the corresponding reaction flux through electron acceptor urn:x-wiley:19422466:media:jame21460:jame21460-math-0186 is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0187(35)
Now considering an application that involves two electron acceptors, for example, nitrate and nitrite in denitrification, we have
urn:x-wiley:19422466:media:jame21460:jame21460-math-0188(36)
which when combined with Equation 35 leads to
urn:x-wiley:19422466:media:jame21460:jame21460-math-0189(37)
and
urn:x-wiley:19422466:media:jame21460:jame21460-math-0190(38)
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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0191 as
urn:x-wiley:19422466:media:jame21460:jame21460-math-0192(39)
where
urn:x-wiley:19422466:media:jame21460:jame21460-math-0193(40)
with urn:x-wiley:19422466:media:jame21460:jame21460-math-0194 representing Rubisco enzyme, urn:x-wiley:19422466:media:jame21460:jame21460-math-0195 is due to light reaction, urn:x-wiley:19422466:media:jame21460:jame21460-math-0196 is due to RuBP flux, urn:x-wiley:19422466:media:jame21460:jame21460-math-0197 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0198 are associated with the carboxylation pathway, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0199 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0200 are associated with the oxygenation pathway. However, we will present detailed quantitative analysis elsewhere.

In summary, the examples in Sections 5-8 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.

According to the schema in Figure 6c, the specific ATP generation rate from the fermentation pathway is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0201(41)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0202 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), urn:x-wiley:19422466:media:jame21460:jame21460-math-0203 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), and urn:x-wiley:19422466:media:jame21460:jame21460-math-0204 is the ATP yield of fermentation. Similarly, the specific ATP generation rate from the aerobic respiration pathway is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0205(42)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0206 and urn:x-wiley:19422466:media:jame21460:jame21460-math-0207 are resistance associated with the citric acid cycle, and the electron transport chain, respectively, while urn:x-wiley:19422466:media:jame21460:jame21460-math-0208 is the incoming oxygen flux, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0209 is the ATP yield of aerobic respiration. Because the citric acid cycle involves many more enzyme-catalyzed steps than fermentation, urn:x-wiley:19422466:media:jame21460:jame21460-math-0210. Meanwhile, urn:x-wiley:19422466:media:jame21460:jame21460-math-0211 is about 20 times the value of urn:x-wiley:19422466:media:jame21460:jame21460-math-0212 (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., urn:x-wiley:19422466:media:jame21460:jame21460-math-0213), the following condition needs to be satisfied:
urn:x-wiley:19422466:media:jame21460:jame21460-math-0214(43)
where the term after the second “>” suggests that fermentation is more favorable only when oxygen is below a certain level of availability (note that urn:x-wiley:19422466:media:jame21460:jame21460-math-0215 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0216) 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0217 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0218) even under certain aerobic conditions. (For anaerobic conditions, urn:x-wiley:19422466:media:jame21460:jame21460-math-0219 approaches zero, and the inequality (Equation 43) is easily satisfied.)

Therefore, as illustrated above, the Ohm's law analogy enables us to quickly and vividly infer that increasing the substrate concentration urn:x-wiley:19422466:media:jame21460:jame21460-math-0220 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 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0221. When this urn:x-wiley:19422466:media:jame21460:jame21460-math-0222 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0223 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., 20142017), but the flux balance models are cumbersome to apply in dynamic environments.

3.6 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 (Tang & Riley, 2017) 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 urn:x-wiley:19422466:media:jame21460:jame21460-math-0224 under the influence of adsorption by mineral surface urn:x-wiley:19422466:media:jame21460:jame21460-math-0225 (with affinity parameter urn:x-wiley:19422466:media:jame21460:jame21460-math-0226) is
urn:x-wiley:19422466:media:jame21460:jame21460-math-0227(44)
where urn:x-wiley:19422466:media:jame21460:jame21460-math-0228 is the half saturation constant for the uptake of urn:x-wiley:19422466:media:jame21460:jame21460-math-0229 by microbe urn:x-wiley:19422466:media:jame21460:jame21460-math-0230 in the absence of urn:x-wiley:19422466:media:jame21460:jame21460-math-0231, and urn:x-wiley:19422466:media:jame21460:jame21460-math-0232 is the within-population competition effect introduced by ECA. Tang and Riley (2019b) showed that urn:x-wiley:19422466:media:jame21460:jame21460-math-0233 is negligible due to the large size contrast between microbes (and likewise fine roots) and substrate molecules. When urn:x-wiley:19422466:media:jame21460:jame21460-math-0234 is ignored, Equation 44 becomes
urn:x-wiley:19422466:media:jame21460:jame21460-math-0235(45)
with
urn:x-wiley:19422466:media:jame21460:jame21460-math-0236(46)

Now the Ohm's law analogy will still work if urn:x-wiley:19422466:media:jame21460:jame21460-math-0237 is used to defined the substrate-dependent resistance. Moreover, Equation 46 suggests that mineral surfaces may slow the microbial uptake of substrate urn:x-wiley:19422466:media:jame21460:jame21460-math-0238 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.

4 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.

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. DE-AC02-05CH11231 as part of the Next Generation Ecosystem Experiment-Arctic 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.

    Data Availability Statement

    Data are available through Shaver and Melillo (1984), Droop (1974), Franzluebbers (1999), and Doran et al. (1990).