Model

A Suppressed-Active-Dead(SAD) Model

Overview

To better analyze the effect of silver ions on Shewanella, we constructed a Suppressed-Active-Dead (SAD) model according to the research of Haque et al. This model describes the inhibition of E. coli by silver ions or silver nanoparticles(AgNPs). We validated this model for both S. aureus and Shewanella using other collected literature data. After demonstrating that the model was widely available, we used it to estimate the usage of silver ions in the pre-experiment and help us design future experiments. This model helped us better understand our work and could also help future teams using silver ions in their projects.

Description

Logistic growth model is widely used to describe the growth of bacteria. In the classic Logistic growth model: $$ \frac{\mathrm{d} n}{\mathrm{d} t} = k_0N(1-\frac{N}{A}) $$ \( n \) is the number of bacteria, \( k_0 \) is the growth rate, and \( A \) is the environmental capacity.

The SAD model extends the logistic growth model. It takes into account the effect of silver ions/silver nanoparticles on cells. It uses three differential equations to describe the growth of bacteria under silver ion/silver nanoparticle stress. These three equations represent the three states of bacteria, respectively.

$$ \left \{ \begin{align} \frac{\mathrm{d} N_a}{\mathrm{d} t} &=\beta N_s+k_0N_a(1-\frac{N_a}{A-N_d-N_s})\\ \frac{\mathrm{d} N_s}{\mathrm{d}t }&=-\alpha N_s-\beta N_s\\ \frac{\mathrm{d} N_d}{\mathrm{d}t }&=\alpha N_s\\ \end{align} \right. $$

The model assumes that bacteria are first in the S state in the presence of silver, and the cell density is expressed as \( N_s \). Cells in the S state cannot grow normally. Some cells in the S state have recovered their growth ability and enter the A state, which is represented by \( Na \). The parameter controls this process: the activation rate \( \beta \). The bacteria of the A state can grow normally in the manner described by the logistic model. Some cells in the S state die and enter the D state, represented by \( N_d \). The parameter that controls this process is the death rate \( \alpha \).

Assumptions

The model makes additional assumptions based on the experimental results:

1. \( k_0 \) and \( A \) are not affected by silver ions/AgNPs
2. \( \alpha \) is related to silver ion, but not to silver ion/AgNP concentration
3. \( \beta \) is related to silver ion/AgNP concentration
4. OD value is equal to \( N = N_a + N_s + N_d \)

Validation

We first verified that this model was usable. We assumed \( k_0=0.5\mathrm{h}^{-1} \), \( A=2 \), \( \alpha=0.7\mathrm{h}^{-1}\). Using different \( \beta \)s, we use Python's Scipy to solve the ODE model and use Matplotlib to draw the following graph.

It can be found that a change in the \( \beta \) will shift the curve to the right. The change in the curve indicates that the growth of the cells is delayed. This change reflects the negative effect of silver on bacterial growth.

We then verified that this model is still usable for other bacteria. We compared the performance of this model under S. aureus and Shewanella using data from the Lu et al and Suresh et al studies, respectively. The curve_fit function in Scipy can automatically calculate suitable parameters based on data points. We used it to estimate approximate \( k_0 \), \( A \), \( \alpha \) and \( \beta \) parameters. We compared the difference between the real and fitted data using this formula: $$ R^2=1-\frac{\sum(N-N_{fitted})^2}{\sum(N-N_{real})^2} $$

The larger the \( R^2 \), the better the fitting effect. The fitting results of the two bacteria are shown in the figures below.

These results demonstrate that the model is broadly applicable and can be used in Shewanella.

Analysis

We further analyzed the parameters of the model. We analyzed the effect of parameter \( \beta \) and parameter \( \alpha \) changes on the model. For the growth curves of S. aureus under 6.25 μg/mL AgNP, we used \( k_0 \) and \( A \) measured without AgNP addition as fixed parameters and different \( \alpha \) and \( \beta \) as variable parameters. Calculating the difference between the real and fitted data for different parameters yields this result.

The red part in the figure represents the 95% degree of fit. It can be found that \( \alpha \) and \( \beta \) do not have strict value ranges. There seems to be some kind of positive correlation between the two parameters. This result suggests to us that \(\alpha \) and \( \beta \) cannot be estimated simply from experimental OD data. A separate experiment to measure \( \alpha \) is necessary. Once the \( \alpha \) is determined, the corresponding optimal \( \beta \) can be found.

Application

We not only performed a more detailed analysis of the existing SAD model but also used it in our project. The fsolve function in the Scipy package can solve multivariate equations. For the standard logistic growth model, the formula can be obtained from the ODE: $$ N = A\frac{\mathrm{e}^{k_0t+C} }{1+\mathrm{e}^{k_0t+C} } $$

\( C \) is a constant.

From this formula, we derived three equations using the growth data of Shewanella in the absence of silver from our experiment. We found \( k_0 \) and \( A \) using the fsolve function in Scipy. Next, we determined the value of \( \alpha \) based on the experimental data of Suresh et al.

To calculate the silver ion concentration to be used, we need to find the relationship between beta and silver ion concentration. According to the experimental results of Haque et al., different silver ion/AgNP concentrations are linearly related to the log value of \( \beta \). The leastsq function in Scipy can calculate a linear relationship between data. Therefore, we use the leastsq function to get the following relationship.

We get the following parameter list:

Using these parameters, we calculated the growth curves of Shewanella for silver ion concentrations ranging from 1 μM to 40 μM. The curves coincide. The results showed that within this range, silver ions had no significant effect on the growth of Shewanella.

We experimented according to the simulation results, and the experimental results were not as expected (learn more in Results). We confirmed that the experiment was correct. This implies that \( \beta \) and \( \alpha \) may be affected by other factors besides silver ions.

We revised the model parameters according to the new experimental data, and obtained simulation results consistent with the experiments:

Summarize

This model describes three bacterial state transitions. Our project uses silver nanoparticles to boost the power generation of Shewanella, but silver harms bacterial growth. This model helps us understand how silver acts as an inhibitor of growth and can help us design preliminary experiments to determine the usage of silver ions. At the same time, the model is general and can be used by other teams to assess the effect of silver ions on the bacteria they use.

MFC Model

Abstract

Due to time constraints, we did not build a complete MFC for experiments. Therefore, at the suggestion of Zhiqi Cong, we referred to the relevant literatures and develop a two-chamber MFC mathematical model. Using parameters from the literature, we use this model to calculate the voltage versus time of the MFC under a 100 ohm load. We speculate that the synthesis of silver nanoparticles affects the \(Y_{ac}\) parameter of bacteria, and we observed an increase in output voltage by changing this parameter. This model proves that our concept is feasible and provides a reference for our subsequent wet experiments.

Introduction

Considering that it is difficult and infeasible to calculate the electricity production model of Shewanella from the perspective of biological metabolism, we hope to establish a model that can describe the working process of MFC based on the mass balance equations during the working process of the fuel cell. So we develop a two-chamber MFC model.

Assumptions

We have the following assumptions:

• The MFC contains three parts: bulk liquid in the anode chamber, biofilm attached to the electrode of anode and bulk liquid in the cathode chamber.
• The electrical resistances of electrodes has been neglected.
• The pH is assumed to be constant(pH = 7) in the bulk liquid of cathode and anode chambers.
• The temperature is 30 °C.

Chemical reactions

We assume that the substrate for the anodic reaction is lactic acid and the substrate for the cathodic reaction is oxygen. The overall reactions are assumed as follows.

Half reaction in the anode:

$$\left. \text{C}_{3}\text{H}_{6}\text{O}_{3} + 3\text{H}_{2}\text{O}\rightarrow 3\text{C}\text{O}_{2} + 12\text{H}^{+} + 12e^{-} \right.$$

Half reaction in the cathode:

$$\left. \text{O}_{2} + 4\text{H}^{+} + 4e^{-}\rightarrow 2\text{H}_{2}\text{O} \right.$$

Kinetic equations

In the MFC, Bacteria oxidize substrates to generate electrons. According to Monod equation, we have:

$$r_{s}(t) = r_{max}\frac{C_S}{K_S+C_S} \phi_{a}$$

where \(r{_s}\) is the substrate consumption rate in the anode chamber, \(r_{max}\) is the reaction rate constant, \(C_S\) is the substrate concentration, \(K_S\) is the substrate saturation constant, \(\phi_a\) is the volume fraction of the active biomass.

It is assumed that the substrate is diffused from the bulk liquid to the biofilm and then oxidized by the active biomass. The produced \(\ce{CO2}\) and \(\ce{H+}\) are diffused back to the bulk liquid. We also assumed that the only species changes in the cathode chamber are the dissolved oxygen.

According to Esfandyari et al., We learn that the mass balance of the active biomass can be described by:

$${\frac{\mathrm{d} }{\mathrm{d}t}\left( {A_{m}\rho\phi_{a}(t)L(t)} \right) = \text{growth\ rate} - \text{inactivation\ rate} + \text{detachment\ rate} }$$

So,

$$\frac{\mathrm{d} }{\mathrm{d}t}\left( {\phi_{a}(t)} \right) = Y_{ac}r_{s}(t) - b_{ina}\phi_{a}(t) + \frac{\phi_{a}(t)}{L(t)}\delta(t) - \frac{\phi_{a}(t)}{L(t)}\frac{\mathrm{d} }{\mathrm{d}t}\left( {L(t)} \right)$$

where \(L(t)\) is the thickness of biofilm, \(Y_{ac}\) is the bacterial yield, \(A_{m}\) is the area of biofilm interface, \(\rho\) is the biomass density, \(b_{ina}\) is the inactivation coefficient and \(\delta(t)\) is the detachment rate which can be calculated by \(\delta(t) = - b_{det}L(t)\) where \(b_{det}\) is the detachment coefficient.

By mathematical transformation, we can get:

$$\frac{\mathrm{d}L(t)}{\mathrm{d}t} = Y_{ac}r_{s}(t)L(t) + \delta(t)$$

According to our previous assumption, the mass balance of the substrate in the biofilm can be described by:

$${\frac{\mathrm{d} }{\mathrm{d}t}\left( {A_{m}C_{S}(t)L(t)} \right) = \text{diffusion\ rate} - \text{consumption\ rate} }$$

So,

$$\frac{\mathrm{d} }{\mathrm{d}t}\left( {C_{S}(t)} \right) = \frac{D_{s} }{L_{l}L(t)}\left( {C_{S}{}_{b}(t) - C_{S}(t)} \right) - \rho r_{s}(t) - \frac{C_{S}(t)}{L(t)}\frac{\mathrm{d} }{\mathrm{d}t}\left( {L(t)} \right)$$

Correspondingly, the following formulas can be obtained:

$$\frac{\mathrm{d} }{\mathrm{d}t}\left( {C_{\text{C}\text{O}_{2} }(t)} \right) = \frac{D_{\text{C}\text{O}_{2} } }{L_{l}L(t)}\left( {C_{\text{C}\text{O}_{2} }{}_{b}(t) - C_{\text{C}\text{O}_{2} }(t)} \right) + 3\rho r_{s}(t) - \frac{C_{\text{C}\text{O}_{2} }(t)}{L(t)}\frac{\mathrm{d} }{\mathrm{d}t}\left( {L(t)} \right)$$

$$\frac{\mathrm{d} }{\mathrm{d}t}\left( {C_{\text{H} }(t)} \right) = \frac{D_{\text{H} } }{L_{l}L(t)}\left( {C_{\text{H} }{}_{b} - C_{\text{H} }(t)} \right) + 12\rho r_{s}(t) - \frac{C_{\text{H} }(t)}{L(t)}\frac{\mathrm{d} }{\mathrm{d}t}\left( {L(t)} \right)$$

where \(D_{\ce{CO2} }\) is the effective diffusivity of carbon dioxide, \(C_{\ce{CO2} }(t)\) is the concentration of carbon dioxide in the biofilm, \(C_{\ce{CO2}b}(t)\) is the concentration of carbon dioxide in the bulk liquid, \(D_H\) is the effective diffusivity of \(\ce{H+}\), \(C_H(t)\) is the concentration of \(\ce{H+}\) in the biofilm and \(C_{Hb}\) is the concentration of \(\ce{H+}\) in the bulk liquid.

The volume changes of the bulk liquid are related to the rate of thickness change of the biofilm. So,

$$\frac{dV_{L}(t)}{dt} = - A_{m}\frac{dL(t)}{dt}$$

In the bulk liquid, according to the previous formula, it can be known accordingly:

$${\frac{d}{dt}\left( {C_{S}{}_{b}(t)V_{L}(t)} \right) = - \text{rate\ of\ the\ biofilm\ growth} - \text{diffusion\ rate} }$$

$$\frac{d}{dt}\left( {C_{S}{}_{b}(t)} \right) = \frac{1}{V_{L}(t)}\left( {- \frac{A_{m}D_{s} }{L_{l} }\left( {C_{S}{}_{b}(t) - C_{S}(t)} \right)} \right)$$

So, correspondingly, the following formulas can be obtained:

$$\frac{d}{dt}\left( {C_{\text{C}\text{O}_{2} }{}_{b}(t)} \right) = \frac{1}{V_{L}(t)}\left( {- \frac{A_{m}D_{\text{C}\text{O}_{2} } }{L_{l} }\left( {C_{\text{C}\text{O}_{2} }{}_{b}(t) - C_{\text{C}\text{O}_{2} }(t)} \right)} \right)$$

In cathode chamber, because we have assumed that the pH is constant. According to Santos et al., there is,

$$\frac{dC_{\text{O}_{2} } }{dt} = k_{La}\left( { {C_{\text{O}_{2} } }^{*} - C_{\text{O}_{2} } } \right) - q_{\text{O}_{2} }C_{\text{O}_{2} }$$

where \(C_{\ce{O2} }\) is the dissolved oxygen concentration, \({C_{\ce{O2} } }^{*} \) is the equilibrium concentration of the dissolved oxygen, \( K_{La} \) is the overall volumetric oxygen mass transfer coefficient and \(q_{O_2}\) is the specific uptake rate of oxygen.

Electrochemical equations

According to Katuri and Scott, $$\eta_{act} = \frac{b}{2.303}\text{sin}h^{- 1}\left\lbrack \frac{i}{2i_{0,ref}C_{S} } \right\rbrack$$

where \(\eta_{act}\) is the activation overpotential which is the potential difference of the equilibrium value required to generate the current, \(b\) is the Tafel coefficient, \(i\) is the current density and \(i_{0,ref}\) is the exchange current density in reference conditions.

We assumed that the cathode voltage is constant. According to the electrical formula,

$$E_{output} = V_C - E_{anode} - \eta_{ohm} - \eta_{act}$$

where \(E_{output}\) is the output voltage, \(V_C\) is the cathode voltage, \(\eta_{ohm}\) is the potential loss due to internal resistance. By the Nernst equation,

$$E_{anode} = E_{0,anode} - \frac{RT}{12F}\ln\left( \frac{C_{\text{C}\text{O}_{2} }^{3}C_{\text{H} }^{11} }{C_{S} } \right)$$

where \(E_{0,anode}\) is the standard cell electromotive force of anode, \(F\) is Faraday constant, \(R\) is the universal constant of gases, and \(T\) is the temperature.

By Ohm law,

$$\eta_{ohm} = \left( {\frac{d_{m} }{k_{m} } + \frac{d_{cell} }{k_{aq} } } \right)i$$

$$E_{output}=iA_mR_{ext},$$

where \(d_m\) is the membrane thickness, \(d_{cell}\) is the distance between electrodes, \(k_m\) is the membrane conductivity, \(R_{ext}\) is the resistance value of the external load, and \(k_{aq}\) is the solution conductivity.

Results

Collecting relevant literatures, we got such a parameter table.

Based on previous experimental results (see more in Results), we observed that Shewanella with Atox1 formed thicker biofilm. According to the formula, $$\frac{\mathrm{d}L(t)}{\mathrm{d}t} = Y_{ac}r_{s}(t)L(t) + \delta(t)$$ we speculate that it is the change of \(Y_{ac}\). So we set a different \(Y_{ac}\). Based on these parameters, we use Python's Scipy to solve the equations. We got the following results.

The results confirmed our conjecture. This model not only helps us further prove our concept, but also helps us carry out the following wet experiments. At the same time, this model can also help other teams making MFCs to make experimental predictions smoothly.

Download model code and data HERE

References

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3. Lu, M. et al. Synergistic bactericidal activity of chlorhexidine-loaded, silver-decorated mesoporous silica nanoparticles. IJN 12, 3577–3589 (2017).

4. Suresh, A. K. et al. Silver Nanocrystallites: Biofabrication using Shewanella oneidensis, and an Evaluation of Their Comparative Toxicity on Gram-negative and Gram-positive Bacteria. Environ. Sci. Technol. 44, 5210–5215 (2010).

5. Esfandyari, M., Fanaei, M. A., Gheshlaghi, R. & Akhavan Mahdavi, M. Mathematical modeling of two-chamber batch microbial fuel cell with pure culture of Shewanella. Chemical Engineering Research and Design 117, 34–42 (2017).

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7. Sedaqatvand, R. & Nasr Esfahany, M. Comparison of Conduction Based and Mediator Based Models for Microbial Fuel Cells. Journal of Petroleum Science and Technology 1, 24–29 (2011).

8. Katuri, K. P. & Scott, K. On the dynamic response of the anode in microbial fuel cells. Enzyme and Microbial Technology 48, 351–358 (2011).

9. Zeng, Y., Choo, Y. F., Kim, B.-H. & Wu, P. Modelling and simulation of two-chamber microbial fuel cell. Journal of Power Sources 195, 79–89 (2010).

10. Tang, Y. J., Meadows, A. L. & Keasling, J. D. A kinetic model describing Shewanella oneidensis MR-1 growth, substrate consumption, and product secretion. Biotechnology and Bioengineering 96, 125–133 (2007).

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13. Santos, V. E., Galdeano, C., Gomez, E., Alcon, A. & Garcia-Ochoa, F. Oxygen uptake rate measurements both by the dynamic method and during the process growth of Rhodococcus erythropolis IGTS8: Modelling and difference in results. Biochemical Engineering Journal 32, 198–204 (2006).