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 carbon 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.
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.
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.
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:
$$ \begin{gather*} P i_{ {sur }} \xrightarrow{P i t_{ {in }}} P i_{ {in }} \tag{2}\\ v_{ {in }, p i t}=k_{ {in,pit }} \cdot \frac{\left[P i_{ {sur }}\right]}{K_{ {in,pit }}+\left[P i_{ {sur }}\right]} \end{gather*} $$
Pi is transported out of the cytoplasm to the periplasm via the Pit system:
$$ \begin{gather*} P i_{ {in }} \xrightarrow{Pi t_{ {out }}} P i_{ {sur }} \tag{3} \end{gather*} $$
$$ \begin{gather*} v_{ {out }, p i t}=k_{ {out }, p i t} \cdot\left[P i_{ {in }}\right] \end{gather*} $$
$$ \begin{gather*} v_{pit}=v_{ {in }, p i t}-v_{ {out }, p i t} \end{gather*} $$
Pi is transported from the periplasm into the cytoplasm via the Pst system:
$$ \begin{gather*} A T P_{in}+P i_{ {sur }} \xrightarrow{P st} 2P i_{ {in }}+A D P_{in} \tag{4}\\ v_{ {in,pst }}=k_{ {in,pst }} \cdot [P s t] \cdot \frac{\left[P i_{ {sur }}\right]}{K_{ {in,pst } ,Pi}+\left[P i_{ {sur }}\right]} \cdot \frac{[A T P]}{K_{ {in,pst }, A T P}+[A T P]} \end{gather*} $$
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 (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 cells, 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 cells, 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:
$$ \begin{gather*} N A D^{+}+ {Poly } P_{n} \stackrel{N A D K}{\rightleftharpoons} N A D P^{+} + PolyP_{n-1} \tag{7}\\ v_{N A D K}=[N A D K] \cdot \frac{[ { PolyP }]}{K_{NADK,PolyP}+[ { Poly } P]} \cdot (\frac{\left[N A D^{+}\right]}{K_{PPK2,NAD}+\left[N A D^{+}\right]}-\frac{\left[N A D P^{+}\right]}{K_{PPK2,NADP}+\left[N A D P^{+}\right]}) \end{gather*} $$
The meaning of this reaction is that 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:
$$ v_{G D P 2 G T P}=v_{G2G,f} \cdot \frac{[A T P]}{K_{G2G,ATP}+[A T P]} \cdot \frac{[G D P]}{K_{G2G,GDP}+[G D P]}-v_{G2G,r} \cdot \frac{[A D P]}{K_{G2G,ADP}+[A D P]} \cdot \frac{[G T P]}{K_{G2G,GTP}+[G T P]} $$
Normal cellular activities require the consumption of ATP and NADH. We use NGAR (non-growth rate dependent ATP requirement) to measure the levels of these consumption-related compounds, i.e.:
$$ N G A R=k_{ngar} \cdot \frac{[A T P]}{K_{ngar,ATP}+[A T P]} \cdot \frac{[N A D H]}{K_{ngar,NADH}+[N A D H]} $$
The carbon source we provide to bacteria is lactate. According to the literature, 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 TCA pathway is higher, but it also generates (transfers) 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 is low, and lactate can only go through the Formate-dependent pathway. The synthesis of electricity-related structures requires the consumption of substances such as ATP.
To characterize the above facts, we introduce \(r\) to represent the proportion of lactate going through the NADH-dependent pathway, and introduce IESP to indicate the integrity of the electricity production structure (a number between 0 and 1):
$$ r=r_k \cdot I E P S+r_b $$
$$ v_{I E S P}=k_{IESP} \cdot (\frac{[A T P]}{K_{IESP,ATP}+[A T P]} \cdot \frac{[N A D P H]}{K_{IESP,NADPH}+[N A D P H]} \cdot \frac{[G T P]}{K_{IESP,GTP}+[G T P]})^3 \cdot(1-I E P S) $$
Lactate enters the cytoplasm from the medium via LldP:
$$ \begin{equation*} { Lactate }_{ {cul }} \stackrel{ { LldP }}{\rightleftharpoons} { Lactate }_{ {in }} \tag{9} \end{equation*} $$
$$ v_{ {LldP }}=k_{ {in,LldP }} \cdot [L l d P] \cdot \frac{\left[ { Lactate }_{ {cul }}\right]}{K_{ {in,LldP,Lac }}+\left[ { Lactate }_{ {cul }}\right]}-k_{out,LldP} \cdot\left[ { Lactate }_{ {in }}\right] $$
According to the literature, the oxidation process of lactate requires consumption of ADP, Pi, NAD+, and terminal electron acceptors:
Figure1: Two metabolic pathways of lactate molecules in Shewanella
Therefore, metabolites associated with phosphorus and electrogenesisthe are calculated as follows:
| NADH-dependent pathway | Procedure | Product | ATP | e- |
|---|---|---|---|---|
| Lactate $\rightarrow$ Pyruvate | QH2 | $$\frac{\mathbf{4}}{\mathbf{3}} \mathrm{r}$$ | 2 r | |
| Pyruvate $\rightarrow$ Acetyl-CoA | NADH | $$4r$$ | 2 r | |
| TCA cycle | 3 NAD(P)H | $$\frac{\mathbf{4 0}}{\mathbf{3}} \mathrm{r}$$ | 8 r | |
| Total | $$\frac{\mathbf{5 6}}{\mathbf{3}} \mathrm{r}$$ | 12 r |
| Formate-dependent pathway | Procedure | Product | ATP | e- |
|---|---|---|---|---|
| Lactate $\rightarrow$ Pyruvate | QH2 | $$\frac{\mathbf{4}}{\mathbf{3}} (\mathrm{1-r})$$ | 2(1-r) | |
| Pyruvate $\rightarrow$ Formate | MQH2 | $$\frac{\mathbf{4}}{\mathbf{3}} (\mathrm{1-r})$$ | 2(1-r) | |
| Acetyl-CoA $\rightarrow$ Acetyl-P | / | / | / | |
| Acetyl-P $\rightarrow$ Acetate | ATP | $$1-r$$ | / | |
| Total | $$\frac{\mathbf{11}}{\mathbf{3}} \mathrm{r}$$ | 4(1-r) |
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:
$$ v_{N A D H}=(4 \cdot r) \cdot v_{L a c} $$
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:
$$ \begin{aligned} & v_{r o}=M_{r} \cdot a_{1} \\ & v_{r r}=M_{o} \cdot a_{2} \end{aligned} $$
EET (extracellular electron transfer) rate (The net rate at which FMNH2 is oxidized at the electrode):
$$ v_{E E T}=\left(v_{r o}-v_{r r}\right) \cdot I E P S $$
Current generation:
$$ \begin{equation*} I=\left(N \cdot V_{ {cell }} \cdot n_e \cdot F \cdot v_{E E T}\right) \cdot I E P S \tag{11} \end{equation*} $$
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.
All parameters of the model and their corresponding justifications, implications, and references are available at parameters-and-references.pdf
| Variable Name | Meaning | Unit |
|---|---|---|
| $$P i_{ {cul }}$$ | Phosphate concentration in the medium | mM |
| $$P i_{ {sur }}$$ | Phosphate concentration in the cell periplasm | mM |
| $$P i_{ {in }}$$ | Phosphate concentration inside the cell | mM |
| $$ATP$$ | ATP concentration inside the cell | mM |
| $$ADP$$ | ADP concentration inside the cell | mM |
| $$PolyP$$ | PolyP concentration inside the cell | mM-Pi |
| $$GTP$$ | GTP concentration inside the cell | mM |
| $$GDP$$ | GDP concentration inside the cell | mM |
| $$NAD^{+}$$ | NADP+ concentration inside the cell | mM |
| $$N A D H$$ | NADH concentration inside the cell | mM |
| $$NADP^{+}$$ | NADP+ concentration inside the cell | mM |
| $$N A D P H$$ | NADPH concentration inside the cell | mM |
| $$N G A R$$ | Non-growth rate dependent ATP requirement | mM/min |
| $$I E P S$$ | Integrity of the electricity production structure | / |
| $$r$$ | Proportion of lactate in the NADH-dependent pathway | / |
| $$Lactate_{ {cul }}$$ | Lactate concentration in the cell periplasm | mM |
| $$Lactate_{ {in }}$$ | Lactate concentration inside the cell | mM |
| $$M_{o}$$ | Concentration of oxidized terminal electron acceptor | mM |
| $$M_{r}$$ | Concentration of reduced terminal electron acceptor | mM |
| $$I$$ | Current | A/m2 |
| $$m R N A_{X}$$ | mRNA concentration of enzyme X | mM |
| $$X$$ | Concentration of enzyme X | mM |
[1]: Renslow, Ryan, et al. "Modeling biofilms with dual extracellular electron transfer mechanisms." Physical Chemistry Chemical Physics 15.44 (2013): 19262-19283.