Modeling Phosphorus Metabolism and Current Generation in Shewanella

Overview

To better understand the coupling relationship between phosphorus metabolism and electricity generation in Shewanella, we constructed an ODE model by focusing on four interrelated physiological processes: phosphorus metabolism, carbon metabolism and cellular activities, electricity generation, and plasmid vector gene expression.

This model describes the effects of the introduction of two PolyP hydrolases, PPK2 and NADK, on phosphorus and energy metabolism in Shewanella, revealing possible reasons for the increased capacity for electricity production and phosphorus aggregation.

This model helped us explore the possibility of introducing hydrolases for the enhancement of electricity production and phosphorus aggregation, which guided our wet experiments and made it possible to advance our project.

Assumptions

To make our model more reasonable, we make the following assumptions:

(A). All PolyP molecules are considered to have the same degree of polymerization, and the change in PolyP levels is measured by changes in concentration.

(B). Shewanella needs to form new electricity-related structures (such as nanowires) on the electrode when transferred from LB medium (growth) to M9 medium (electricity generation & phosphorus aggregation). Incomplete electricity-generating structures do not support the transfer of a large number of electrons to the electrode.

(C). Normal cellular activities require the consumption of ATP and NADH.

(D). During the electricity generation process, the total amount of ATP and ADP, GTP and GDP, NAD⁺ NADH NADP⁺ NADPH remains constant.

(E). No growth of Shewanella during electricity generation.

(F). Extracellular electron transport (EET) in Shewanella is complex and includes diffusion-based EET, conduction-based EET, and interactions between the two^[1]. We hypothesize that only diffusion-based EET is available and that the mediator are FMN/FMNH₂.

In the following sections, we will start by writing chemical reaction equations, derive the rate equations of the reactions, then construct ordinary differential equations with respect to time for each variable, and finally solve them in MATLAB.

1. Chemical Reaction Equations and Rate Equations

1.1 Phosphorus Metabolism

1.1.1 Absorption and Transport of Pi into the Cell

The absorption of phosphorus from the external environment into Shewanella cells is divided into three steps: first, the diffusion of phosphorus into the cell periplasm, and second, the transport of phosphorus from the periplasm into the cell cytoplasm through the Pit or Pst transport system^[2].

Pi diffuses from the medium to the cell periplasm:

$$ \begin{gather*} P i_{ {cul }} \rightleftharpoons P i_{ {sur }} \tag{1}\\ v_{m t , P i}=k_{m t , P i} \cdot S_{V} \cdot\left(\left[P i_{ {cul }}\right]-\left[P i_{ {sur }}\right]\right) \end{gather*} $$

Pi is transported from the periplasm into the cytoplasm via the Pit system:

$$ \begin{gather*} P i_{ {sur }} \xrightarrow{P i t_{ {in }}} P i_{ {in }} \tag{2}\\ v_{ {in }, p i t}=k_{ {in,pit }} \cdot \frac{\left[P i_{ {sur }}\right]}{K_{ {in,pit }}+\left[P i_{ {sur }}\right]} \end{gather*} $$

The Pit transporter is a low-affinity and high-velocity system^[2], so Pi can be transported out of the cytoplasm to the periplasm via the Pit system:

$$ \begin{gather*} P i_{ {in }} \xrightarrow{Pi t_{ {out }}} P i_{ {sur }} \tag{3} \end{gather*} $$

$$ \begin{gather*} v_{ {out }, p i t}=k_{ {out }, p i t} \cdot\left[P i_{ {in }}\right] \end{gather*} $$

So the net rate of Pi entering the cytoplasm through Pit system is:

$$ \begin{gather*} v_{pit}=v_{ {in }, p i t}-v_{ {out }, p i t} \end{gather*} $$

Pi can also be transported from the periplasm into the cytoplasm via the Pst system, which is a high-affinity and low-velocity unidirectional system^[2], so we ignore the reverse process.

We also take the effect of ATP on the Pst system into consideration and improve the rate equation of Van Dien et al^[2].

$$ \begin{gather*} A T P_{in}+P i_{ {sur }} \xrightarrow{P st} 2P i_{ {in }}+A D P_{in} \tag{4}\\ v_{ {in,pst }}=k_{ {in,pst }} \cdot [P s t] \cdot \frac{\left[P i_{ {sur }}\right]}{K_{ {in,pst } ,Pi}+\left[P i_{ {sur }}\right]} \cdot \frac{[A T P]}{K_{ {in,pst }, A T P}+[A T P]} \end{gather*} $$

1.1.2 Enzyme-Involved Phosphorus Metabolism Reactions

The Shewanella genome has basal expression of PPK1 (ATP-polyphosphate phosphotransferase), which catalyzes the following reaction:

$$ \begin{equation*} A T P+P o l y P_{n} \stackrel{P P K 1}{\rightleftharpoons} A D P+ { Poly } P_{n+1} \tag{5} \end{equation*} $$

Based on the Michaelis-Menten equation, we can write the rate equation for this reaction^[3] (similarly for others):

$$ v_{P P K 1}=[P P K 1] \cdot \frac{[P o l y P]}{K_{PPK1,PolyP}+[P o l y P]} \cdot\left(k_{P P K 1 , f} \cdot \frac{[A T P]}{K_{P P K 1 , A T P}+[A T P]}-k_{P P K 1 , r} \cdot \frac{[A D P]}{K_{P P K 1 , A D P}+[A D P]}\right) $$

This reaction is reversible, but in vivo, the forward reaction catalyzed by this enzyme is greater than the reverse reaction (i.e., ATP consumption).

If the plasmid-imported PPK2 is successfully expressed, it catalyzes the following reaction:

$$ \begin{gather*} & G D P+ { PolyP }_{n} \stackrel{P P K 2}{\rightleftharpoons} G T P+ { PolyP}_{n-1} \tag{6} \end{gather*} $$

$$ v_{P P K 2}=[P P K 2] \cdot \frac{[ { PolyP }]}{K_{PPK2 , PolyP}+[ { PolyP }]} \cdot\left(k_{P P K 2 , f} \cdot \frac{[G D P]}{K_{P K 2 , G D P}+[G D P]}-k_{P P K 2 , r} \cdot \frac{[G T P]}{K_{P P K 2 , GT P}+[G T P]}\right) $$

This reaction is reversible, but in vivo, the forward reaction catalyzed by this enzyme is greater than the reverse reaction (i.e., PolyP consumption).

If the plasmid-imported NADK (NAD⁺ kinase) is successfully expressed, it catalyzes the following reaction:

$$ \begin{gather*} N A D^{+}+ {Poly } P_{n} \stackrel{N A D K}{\rightleftharpoons} N A D P^{+} + PolyP_{n-1} \tag{7}\\ v_{N A D K}=[N A D K] \cdot \frac{[ { PolyP }]}{K_{NADK,PolyP}+[ { Poly } P]} \cdot (\frac{\left[N A D^{+}\right]}{K_{PPK2,NAD}+\left[N A D^{+}\right]}-\frac{\left[N A D P^{+}\right]}{K_{PPK2,NADP}+\left[N A D P^{+}\right]}) \end{gather*} $$

This reaction means NADK consumes PolyP to convert NAD⁺ into NADP⁺.

At the same time, we also consider the following reaction in cells that allows the synthesis of NADPH from NADP⁺:

$$ \begin{gather*} N A D P^{+} \rightleftharpoons N A D P H \tag{8}\\ v_{N A D P 2 N A D P H}=v_{N2N,f} \cdot \frac{\left[N A D P^{+}\right]}{K_{N2N,NADP}+\left[N A D P^{+}\right]}-v_{N2N,r} \cdot \frac{[N A D P H]}{K_{N2N,NADPH}+[N A D P H]} \end{gather*} $$

We also consider the following reaction in cells that allows mutual conversion between ATP and GTP:

$$ v_{G D P 2 G T P}=v_{G2G,f} \cdot \frac{[A T P]}{K_{G2G,ATP}+[A T P]} \cdot \frac{[G D P]}{K_{G2G,GDP}+[G D P]}-v_{G2G,r} \cdot \frac{[A D P]}{K_{G2G,ADP}+[A D P]} \cdot \frac{[G T P]}{K_{G2G,GTP}+[G T P]} $$

1.2 Carbon Metabolism and Cellular Activities

1.2.1 Cellular Activities and Electricity Consumption

Normal cellular activities require the consumption of ATP and NADH. We use $NGAR$ (non-growth rate dependent ATP requirement) to measure ATP consumption.

$$ N G A R=k_{ngar} \cdot \frac{[A T P]}{K_{ngar,ATP}+[A T P]} \cdot \frac{[N A D H]}{K_{ngar,NADH}+[N A D H]} $$

The carbon source we provide to bacteria is lactate. According to Kouzuma^[4], Shewanella oxidizes a molecule of lactate via two pathways: NADH-dependent pathway and Formate-dependent pathway. The NADH-dependent pathway can completely oxidize lactate to water and CO₂ but relies on the complete structure of the electron transport chain; the Formate-dependent pathway oxidizes lactate to acetate, not relying on the integrity of the electricity-generating structure.

The energy efficiency of lactate through the NADH-dependent pathway is higher, but it also generates more electrons. When Shewanella is transferred from LB medium (growth) to M9 medium (electricity generation), it needs to form new electricity-related structures (such as nanowires) on the electrode. Incomplete electricity-generating structures do not support the transfer of a large number of electrons to the electrode, so the NADH-dependent pathway tends to saturate, the efficiency of TCA cycle is low, and lactate can only go through the Formate-dependent pathway.

The equations for the competition between the two pathways for lactate are extremely complex and it is unrealistic to consider them in such detail in this model, so we simplify them and introduce $r$ to represent the proportion of lactate going through the NADH-dependent pathway. At the same time, we introduce $IEPS$ to indicate the integrity of the electricity production structure (a number between 0 and 1), and the synthesis of electricity-related structures requires ATP, GTP and NADPH.

The equations for $IEPS$ were proposed and improved by ourselves. In order to reduce the influence of irrelevant factors (i.e., $(1-IEPS)$ in the equation) on our model, and to ensure the proper functioning of the code, we added a cubic factor to the term in front of $(1-IEPS)$, which enlarges the effect of the substrate on the synthesis of the electricity-producing structure.

$$ r=r_k \cdot I E P S+r_b $$

$$ v_{I E S P}=k_{IESP} \cdot (\frac{[A T P]}{K_{IESP,ATP}+[A T P]} \cdot \frac{[N A D P H]}{K_{IESP,NADPH}+[N A D P H]} \cdot \frac{[G T P]}{K_{IESP,GTP}+[G T P]})^3 \cdot(1-I E P S) $$

1.2.2 Lactate Absorption

Lactate enters the cytoplasm from the medium via LldP^[5]

$$ \begin{equation*} { Lactate }_{ {cul }} \stackrel{ { LldP }}{\rightleftharpoons} { Lactate }_{ {in }} \tag{9} \end{equation*} $$

$$ v_{ {LldP }}=k_{ {in,LldP }} \cdot [L l d P] \cdot \frac{\left[ { Lactate }_{ {cul }}\right]}{K_{ {in,LldP,Lac }}+\left[ { Lactate }_{ {cul }}\right]}-k_{out,LldP} \cdot\left[ { Lactate }_{ {in }}\right] $$

1.2.3 Lactate Oxidation

According to Kouzuma^[4], the oxidation of lactate requires consumption of ADP, Pi, NAD⁺, and terminal electron acceptors:

Therefore, metabolites associated with phosphorus and electrogenesisthe are calculated as follows:

Total ATP production: \(\frac{11}{3}+5r\)

Total electron transfer: \(4+8 r\)

The overall reaction equation for lactate oxidation can be written in the following form:

$$ \begin{gather*} & { Lactate }+A D P+P i_{i n}+N A D^{+}+NADPH+Mo \xrightarrow{ { many \ steps }}\left(\frac{11}{3}+5 r\right) \cdot A T P+(4+8 r) \cdot e^{-} \tag{10}\\ & e^{-}=\frac{1}{2}NADH,~ \frac{1}{2}FMN_2 ~ { or } ~ \frac{1}{2}M r \\ \end{gather*} $$

$$ v_{ {Lac }}=k_{ {Lac }} \cdot \frac{\left[ { Lactate }_{ {in }}\right]}{K_{M , { Lac }}+\left[ { Lactate }_{ {in }}\right]} \cdot \frac{[A D P]}{K_{Lac,ADP}+[A D P]} \cdot \frac{\left[P i_{i n}\right]}{K_{Lac,Pi}+\left[P i_{i n}\right]} \cdot \frac{\left[N A D^{+}\right]}{K_{Lac,NAD}+\left[N A D^{+}\right]} \cdot \frac{\left[M_{o}\right]}{K_{Lac,Mo}+\left[M_{o}\right]} \cdot \frac{[G D P]}{K_{Lac,GDP}+[G D P]} \cdot \frac{[NADPH]}{K_{Lac,NADPH}+[NADPH]} $$

From the metabolic reaction process, we know that the rate of FMNH₂ produced during lactate oxidation is:

$$ v_{M,{r 1}}=n_{M,{r1}} \cdot v_{L a c} $$

The rate of FMNH₂ reduction by NADH (electron transport chain):

$$ v_{M,{r 2}}=k_{M,{r2}} \cdot[N A D H] \cdot\left[M_{o}\right] $$

The rate of ATP generation is composed of two parts, lactate oxidation and electron transport chain:

$$ v_{A T P}=(1-r) \cdot v_{L a c} +n_{M,{r2}} \cdot v_{M,{r 2}} $$

The rate of NADH generation is as follows:

$$ v_{N A D H}=(4 \cdot r) \cdot v_{L a c} $$

1.3 Electricity Generation

Our model's electricity generation part refers to the method in the reference ^[1]. Let \(a_1\) be the rate constant for FMNH₂ oxidation, and \(a_2\) be the rate constant for FMN reduction:

$$ \begin{aligned} & a_{1}=k_{0} \cdot e^{(1-\alpha) \cdot \frac{n_{e} F \cdot(\epsilon-E^0)}{R \cdot T}} \\ & a_{2}=k_{0} \cdot e^{(-\alpha) \cdot \frac{n_{e} F \cdot(\epsilon-E^0)}{R \cdot T}} \end{aligned} $$

The rate of FMNH₂ oxidation and the rate of FMN reduction:

$$ \begin{aligned} & v_{r o}=M_{r} \cdot a_{1} \\ & v_{r r}=M_{o} \cdot a_{2} \end{aligned} $$

Thus, the net generation rate of e^- at the electrode surface is given by:

$$ v_{E E T}=\left(v_{r o}-v_{r r}\right) \cdot n_e \cdot I E P S $$

And current generation is given by:

$$ \begin{equation*} I=\left(N \cdot V_{ {cell }} \cdot F \cdot v_{E E T}\right) \cdot I E P S \tag{11} \end{equation*} $$

1.4 Plasmid Gene Expression

According to the central dogma, we consider the genes involved in phosphorus metabolism going through four processes: DNA transcription, mRNA translation, mRNA degradation, and protein degradation:

$$ \begin{gather*} D N A_{X} \xrightarrow{k_{0,X}} m R N A_{X}+D N A_{X} \tag{12}\\ m R N A_{X} \xrightarrow{d_{0,X}} \varnothing \tag{13} \end{gather*} $$

$$ \begin{gather*} m R N A_{X} \xrightarrow{k_{1,X}} X+m R N A_{X} \tag{14}\\ X \xrightarrow{d_{1,X}} \varnothing \tag{15} \end{gather*} $$

$$ X=P P K 2 ~ { or } ~ N A D K $$

For specific descriptions of these equations, see 2.

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All parameters of the model and their corresponding justifications, implications, and references are available at parameters-and-references.pdf

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References

[1]: Renslow, R.; Babauta, J.; Kuprat, A.; Schenk, J.; Ivory, C.; Fredrickson, J.; Beyenal, H. Modeling Biofilms with Dual Extracellular Electron Transfer Mechanisms. Phys. Chem. Chem. Phys. 2013, 15 (44), 19262.

[2]:Van Dien, S. J.; Keasling, J. D. A Dynamic Model of the Escherichia Coli Phosphate-Starvation Response. Journal of Theoretical Biology 1998, 190 (1), 37–49.

[3]:Van Dien, S. J.; Keasling, J. D. Effect of Polyphosphate Metabolism on the Escherichia Coli Phosphate-Starvation Response. Biotechnol. Prog. 1999, 15 (4), 587–593.

[4]:Kouzuma, A. Molecular Mechanisms Regulating the Catabolic and Electrochemical Activities of Shewanella Oneidensis MR-1. Biosci Biotechnol Biochem 2021, 85 (7), 1572–1581.

[5]:Núñez, M. F.; Kwon, O.; Wilson, T. H.; Aguilar, J.; Baldoma, L.; Lin, E. C. C. Transport of -Lactate, -Lactate, and Glycolate by the LldP and GlcA Membrane Carriers of Escherichia Coli. Biochemical and Biophysical Research Communications 2002, 290 (2), 824–829.