Team:TU Delft-Leiden/Modeling/EET/Deterministic

Deterministic Model of EET Complex Assembly

One of the parts of our project is to enable cells to transport electrons to the extracellular environment, thus generating a current as a response to a signal, see the extracellular electron transport (EET) module. To do this, the cell needs EET-complexes (Extracellular Electron Transport complexes). The EET complex consists of three proteins: MtrA, a cytochrome on the inside of the outer membrane, MtrB, a β-barrel protein located in the outer membrane, and MtrC, another cytochrome, located on the cell surface. This complex enables the cell to transport electrons from the cytoplasm of the cell to the extracellular environment [1].

The assembly of the trans-membrane EET complex depends on many factors other than transcriptional and translational control, as it requires a large amount of post-translational modifications. In this section, we will set up a simplified model of this assembly process, largely based on section 1.3 of the thesis of Jensen [1]. Our goal is to predict how many EET complexes are formed under different initial conditions.

In our modelling of the assembly of the EET complex, we will, in addition to the assembly mechanism, also focus on the apparent reduced cell viability. Jensen proposes two possible explanations for this: the formation of cytosolic aggregates, and reduced membrane integrity due to the high amount of trans-membrane protein complexes. We speculate that the specific carbon source (L-lactate) needed to enable extracellular electron transport might also reduce cell viability. For more information on this, please refer to Flux Balance Analysis.

Extensive Model of EET Complex Assembly

Since localization is an important part of the assembly process, we will consider compounds in different areas of the cell as different species in our model. We will distinguish between the cytosol (abbreviated as cyt), the periplasm (peri), the inner membrane (mem1) and the outer membrane (mem2).

An important part of the cytochromes that form part of the Mtr complex are heme molecules. Heme molecules enable the MtrA and MtrC proteins to accept and donate electrons. Heme is formed in the cytosol in a series of steps from the compound δ-aminolevunlinic acid (δ-ALA). After the heme is formed, it is transported to the periplasm. This process is catalyzed by the Ccm cluster (CcmAH), which is located in the inner membrane [1]. These processes are described by reactions (1) and (2): $$ \delta\mbox{-}ALA(cyt) \ \xrightarrow{k_{1}} \ heme(cyt) \tag{1}$$ $$ heme(cyt) + \ CcmAH(mem1) \ \xrightarrow{k_{2}} \ heme(peri) + \ CcmAH(mem1) \tag{2}$$ As already mentioned, the Ccm cluster, bound to the inner membrane transports the heme molecules from the cytosol to the periplasm. Also, the cluster catalyzes the binding of MtrA and MtrC to the heme, thereby forming cytochromes. The CcmAH proteins are produced in the cytosol. From there, they are transported to the periplasm. When in the periplasm, the Ccm proteins will bind to the inner membrane [1]. To take into account the fact that the membrane is not capable of taking up an unlimited amount of Ccm clusters, or that a high amount of such membrane proteins might reduce cell viability, we introduce a limited amount of “membrane binding sites” (for the inner membrane those are called mem1bindingsite, for the outer membrane mem2bindingsite). The processes involving the assembly of the Ccm cluster are described by the reactions (3), (4) and (5). $$ \emptyset \ \xrightarrow{k_{3}} \ CcmAH(cyt) \tag{3}$$ $$ CcmAH(cyt) \ \xrightarrow{k_{4}} \ CcmAH(peri) \tag{4}$$ $$ CcmAH(peri) + \ mem1bindingsite \ \xrightarrow{k_{5}} \ CcmAH(mem1) \tag{5}$$ The Mtr complex consists of three proteins, MtrA, MtrB and MtrC, which are expressed from the same operon. To model the production and initial transport of these proteins, we will consider them as being one protein, MtrCAB. This is of course not a completely accurate description of reality, but, since the proteins are produced and transported in the same amount, this simplification does not change the outcome of deterministic modeling and reduces the number of reaction equations by three. The production of the MtrCAB proteins in the cytosol is described by equation (6). From the cytosol, the proteins are transported to the periplasm by a type II secretion system (equation (7)). $$ \emptyset \ \xrightarrow{k_{6}} \ MtrCAB(cyt) \tag{6}$$ $$ MtrCAB(cyt) \ \xrightarrow{k_{7}} \ MtrCAB(peri) \tag{7}$$ From the moment the mtr proteins enter the periplasm, the different proteins start to undergo different post-translational modifications and therefore need to be considered separately. To facilitate this, we enter reaction (8), which describes the MtrCAB separating in the three individual proteins MtrA, MtrB and MtrC. Since the fact that MtrCAB behaved as one protein is a model simplification and not occurring in real life, this reaction also has no physical meaning. Its only purpose is to reduce the amount of reactions. Therefore, the reaction rate (\(k_{8}\)) will be extremely high compared to the rates of the other reactions. In this way, this reaction will not influence the overall speed of the assembly process. $$ MtrCAB(peri) \ \xrightarrow{k_{8}} \ MtrA(peri) + \ MtrB(peri) + \ MtrC(peri) \tag{8}$$ In the periplasm, the MtrA protein will react with a heme molecule to form a MtrA-heme complex. This allows the protein to accept and donate electrons. This heme ligation reaction is catalyzed by the Ccm cluster. This process is described by equation (9): $$ MtrA(peri) + \ heme(peri) + \ CcmAH(mem1) \ \xrightarrow{k_{9}} \ MtrA\mbox{-}heme(peri) - \ heme(peri) + \ CcmAH(mem1) \tag{9}$$ The MtrC protein undergoes a similar heme ligation process, which is also catalyzed by CcmAH. This is described in equation (10). After heme ligation, the MtrC-heme complex has to undergo yet another post-translational modification, namely lipidization. In this reaction, a lipid is added to the MtrC-heme complex. This is described by equation (11). Since we have made the assumption that there will be an abundance of lipids, we do not consider the lipidization as a reaction between the protein and the lipid, but rather as a process that happens at a steady rate, independent of the lipid concentration. $$ MtrC(peri) + \ heme(peri) + \ CcmAH(mem1) \ \xrightarrow{k_{10}} \ MtrC\mbox{-}heme(peri) + \ CcmAH(mem1) \tag{10}$$ $$ MtrC\mbox{-}heme(peri) \ \xrightarrow{k_{11}} \ MtrC\mbox{-}heme\mbox{-}lipid(peri) \tag{11}$$ When the lipidized MtrC-heme complex is formed, it is translocated to the outer membrane, thereby filling one membrane binding site. The MtrB protein undergoes a similar process, also filling one membrane binding site. These processes are described by equations (12) and (13). $$ MtrC\mbox{-}heme\mbox{-}lipid(peri) + \ mem2bindingsite \ \xrightarrow{k_{12}} \ MtrC\mbox{-}heme\mbox{-}lipid(mem2) \tag{12}$$ $$ MtrB(peri) + \ mem2bindingsite \xrightarrow{k_{13}} \ MtrB(mem2) \tag{13}$$ Once the MtrB and lipidized MtrC-heme proteins are in place in the outer membrane, they can react with the MtrA-heme complex to form a transmembrane protein complex, which occupies one outer membrane binding site. This complex is capable of transporting electrons to the extracellular environment and will therefore be called EET (Extracellular Electron Transport complex). Since the MtrB and MtrC-heme complex both occupied one membrane binding site, this reaction frees one of those binding sites. This reaction is described by equation (14). $$ MtrA\mbox{-}heme(peri) + \ MtrB(mem2) + \ MtrC\mbox{-}heme\mbox{-}lipid(mem2) \ \xrightarrow{k_{14}} \ EET(mem2) + \ mem2bindingsite \tag{14}$$ The fourteen aforementioned reactions describe the assembly of the EET complex in quite some detail. However, it does not include the presumed formation of protein aggregates in the cytosol, which might be harmful for cell viability. Therefore, we included reaction (15), which describes the aggregation of the MtrCAB protein aggregates. $$ n \cdot MtrCAB(cyt) \ \xrightarrow{k_{15}} \ Aggregate(cyt) \tag{15}$$

This system of reactions can be translated to a system of coupled ODEs, via the procedure described in Deterministic Modeling Theory. The results are the following equations: $$ \frac{d}{dt} [\delta\mbox{-}ALA(cyt)] = \ -k_{1}[\delta\mbox{-}ALA(cyt)] \tag{16.1} $$ $$ \frac{d}{dt} [heme(cyt)] = \ k_{1}[\delta\mbox{-}ALA(cyt)] - \ k_{2}[heme(cyt)][CcmAH(mem1)] \tag{16.2} $$ $$ \frac{d}{dt} [heme(peri)] = \ k_{2}[heme(cyt)][CcmAH(mem1)] - \ k_{9}[MtrA(peri)][heme(peri)][CcmAH(mem1)] + \\ -k_{11}[MtrC(peri)][heme(peri)][CcmAH(mem1)] \tag{16.3} $$ $$ \frac{d}{dt} [CcmAH(cyt)] = \ k_{3} - \ k_{4}[CcmAH(cyt)] \tag{16.4} $$ $$ \frac{d}{dt} [CcmAH(peri)] = \ k_{4}[CcmAH(cyt)] - \ k_{5}[CcmAH(peri)][mem1bindingsite] \tag{16.5} $$ $$ \frac{d}{dt} [mem1bindingsite] = \ -k_{5}[CcmAH(peri)][mem1bindingsite] \tag{16.6} $$ $$ \frac{d}{dt} [CcmAH(mem1)] = \ k_{5}[CcmAH(peri)][mem1bindingsite] \tag{16.7} $$ $$ \frac{d}{dt} [MtrCAB(cyt)] = \ k_{6} - \ k_{7}[MtrCAB(cyt)]\exp(-k_{16}[MtrCAB(cyt)]) - \ k_{15}[MtrCAB(cyt)]^{n} \tag{16.8} $$ $$ \frac{d}{dt} [MtrCAB(peri)] = \ k_{7}[MtrCAB(cyt)]\exp(-k_{16}[MtrCAB(cyt)]) - \ k_{8}[MtrCAB(peri)] \tag{16.9} $$ $$ \frac{d}{dt} [MtrA(peri)] = \ k_{8}[MtrCAB(peri)] - \ k_{9}[MtrA(peri)][heme(peri)][CcmAH(mem1)] \tag{16.10} $$ $$ \frac{d}{dt} [MtrA\mbox{-}heme(peri)] = \ k_{9}[MtrA(peri)][heme(peri)][CcmAH(mem1)] + \\ -k_{14}[MtrA\mbox{-}heme(peri)][MtrB(mem2)][MtrC\mbox{-}heme\mbox{-}lipid(mem2)] \tag{16.11} $$ $$ \frac{d}{dt} [MtrB(peri)] = \ k_{8}[MtrCAB(peri)] - k_{13}[MtrB(peri)][mem2bindingsite] \tag{16.12} $$ $$ \frac{d}{dt} [MtrB(mem2)] = \ k_{13}[MtrB(peri)][mem2bindingsite] + \\ -k_{14}[MtrA\mbox{-}heme(peri)][MtrB(mem2)][MtrC\mbox{-}heme\mbox{-}lipid(mem2)] \tag{16.13} $$ $$ \frac{d}{dt} [MtrC(peri)] = \ k_{8}[MtrCAB(peri)] - \ k_{10}[MtrC(peri)][heme(peri)][CcmAH(mem1)] \tag{16.14} $$ $$ \frac{d}{dt} [MtrC\mbox{-}heme(peri)] = \ k_{10}[MtrC(peri)][heme(peri)][CcmAH(mem1)] - \ k_{11}[MtrC\mbox{-}heme(peri)] \tag{16.15} $$ $$ \frac{d}{dt} [MtrC\mbox{-}heme\mbox{-}lipid(peri)] = \ k_{11}[MtrC\mbox{-}heme(peri)] + \\ -k_{12}[MtrC\mbox{-}heme\mbox{-}lipid(peri)][mem2bindingsite] \tag{16.16} $$ $$ \frac{d}{dt} [MtrC\mbox{-}heme\mbox{-}lipid(mem2)] = \ k_{12}[MtrC\mbox{-}heme\mbox{-}lipid(peri)][mem2bindingsite] + \\ -k_{14}[MtrA\mbox{-}heme(peri)][MtrB(mem2)][MtrC\mbox{-}heme\mbox{-}lipid(mem2)] \tag{16.17} $$ $$ \frac{d}{dt} [EET] = \ k_{14}[MtrA\mbox{-}heme(peri)][MtrB(mem2)][MtrC\mbox{-}heme\mbox{-}lipid(mem2)] \tag{16.18} $$ $$ \frac{d}{dt} [mem2bindingsite] = \ -k_{13}[MtrB(peri)][mem2bindingsite] + \\ -k_{12}[MtrC\mbox{-}heme\mbox{-}lipid(peri)][mem2bindingsite] + \\ k_{14}[MtrA\mbox{-}heme(peri)][MtrB(mem2)][MtrC\mbox{-}heme\mbox{-}lipid(mem2)] \tag{16.19} $$ $$ \frac{d}{dt} [Aggregate(cyt)] = \ k_{15}[MtrCAB(cyt)]^{n} \tag{16.20} $$ This system of equations is exactly what you get when monitoring the flow of compounds from the reaction equations (1)-(15), with a small exception. In equations (16.8) and (16.9), there is a term describing the transport of the MtrCAB protein through the inner membrane through type II secretion. Jensen suggests [1] that the secretion pores might get clogged at high protein concentrations, thereby decreasing the transport rate. Therefore, instead of using \(k_{7}[MtrCAB(cyt)]\) as transport term, we have included a negative exponent, resulting in \(k_{7}[MtrCAB(cyt)]\exp(-k_{16}[MtrCAB(cyt)])\). This will result in a concentration-dependent transport curve with (approximately) the desired characteristics. This is visualized in figure 1.

This system of ODEs can be solved to obtain results that provide qualitative insight (see below) in the EET pathway assembly. However, since the system is depended on no less than sixteen different parameters, most of which we did not manage to find values for, despite extensive literature research, obtaining quantitative results is impossible in the limited time we have.

To obtain qualitative information from our model for the assembly of EET complex, we estimated the parameters and initial values. Since obtaining a numerically realistic solution is not feasible, and since very high or low values of parameters make the calculations very slow, we decided to take manageable values and estimate the ratios between the different variables. The used, but rather arbitrary, values can be found in tables 1 and 2.

Table 1: Parameters used to obtain qualitative results from the extensive ODE system describing the assembly of the EET pathway.

Parameters	Value	Physical meaning
\(\boldsymbol{k_{1}}\)	0.1	Heme production rate
\(\boldsymbol{k_{2}}\)	10	Rate of heme transport from cytosol to periplasm
\(\boldsymbol{k_{3}}\)	0.1	Ccm cluster protein production rate
\(\boldsymbol{k_{4}}\)	10	Rate of Ccm transport from cytosol to periplasm
\(\boldsymbol{k_{5}}\)	10	Rate of localization of Ccm in the inner membrane
\(\boldsymbol{k_{6}}\)	0.1	MtrCAB protein production rate
\(\boldsymbol{k_{7}}\)	10	Rate of transport of MtrCAB from cytosol to periplasm
\(\boldsymbol{k_{8}}\)	1000	Rate of seperation of MtrCAB into seperate part (not really happening, just to reduce the number of reactions)
\(\boldsymbol{k_{9}}\)	1	Rate of MtrA-heme complex formation
\(\boldsymbol{k_{10}}\)	1	Rate of MtrC-heme complex formation
\(\boldsymbol{k_{11}}\)	5	Rate of lipidization of MtrC
\(\boldsymbol{k_{12}}\)	10	Rate of MtrC localization in outer membrane
\(\boldsymbol{k_{13}}\)	10	Rate of MtrB localization in outer membrane
\(\boldsymbol{k_{14}}\)	1	Rate of EET complex formation
\(\boldsymbol{k_{15}}\)	10	Rate of aggregate formation
\(\boldsymbol{k_{16}}\)	4	Type II transport clogging parameter
\(\boldsymbol{n}\)	2	Number of proteins needed to form an aggregate

Table 2: Non-zero initial values of variables.

Variable	Initial Value
δ-ALA(cyt)	10
heme(cyt)	2
mem1bindingsite	10
mem2bindingsite	10

Solving the system with these parameters and initial values and plotting the most important variables (heme concentration, Ccm cluster concentration, EET complex concentration, and the concentration of cytosolic aggregates) yield the graphs in figure 2. A first thing that stands out is that the Ccm cluster is produced at a constant rate until it has filled all available binding sites, i.e. its concentration reaches the value 10. Also, it can be seen that initially, the production rate of heme is high. Later, the production is however halted due to depletion of δ-ALA and the produced heme is used up in the production of MtrC-heme and MtrA-heme complexes. At approximately the same time as the heme is depleted, the amount of EET complexes reaches a constant value. This is a strong indication that, for our choices of parameters and initial values, the amount of δ-ALA (and therefore heme) is rate limiting and not the amount of available binding sites. With our choice of parameters, aggregate formation does not seem to have a pronounced effect.

To investigate this possible limitation by the initial amount of δ-ALA, we calculated the steady state value of the amount of EET complexes for different initial δ-ALA conditions. The results are plotted in figure 3. Increasing the initial concentration of δ-ALA results in a higher concentration of EET complexes. However, this increase is very small and even with a 10,000-fold increase in the initial concentration, the EET concentration stays below the limit of 10 (the amount of binding sites). We therefore predict that adding extra δ-ALA to the cells will increase the amount of EET complexes, an effect which is also observed by Jensen [1].

Although the system of ODEs described in equation (16) has proven to be a valuable investigation of the mechanism which assembles the EET pathway, it is not suitable for the quantitative prediction of the amount of EET complexes. The most important reason for this is the large number of unknown parameters. Therefore we decided not to aim at enhancing this model, and rather set up a more simplified model (see below). Because of this decision, we halted our efforts to introduce relations describing reduced cell viability due to cytosolic protein aggregates and reduced membrane integrity. This reduced cell viability would provide negative feedback on various production rates, which would make the system even more complex.

Simplified Model of EET Complex Assembly

To sidestep the difficulties experienced with the previous model of the assembly of the EET complex, we reduced the system to a bare minimum. For this simplified model, we only included the production of MtrCAB (equation 1), the formation of cytosolic aggregates (equation 2), and the assembly of the EET complex (equation 3). $$ \emptyset \ \xrightarrow{k_{1}} \ MtrCAB \tag{1}$$ $$ n \cdot MtrCAB \ \xrightarrow{k_{2}} \ Aggregate \tag{2}$$ $$ MtrCAB \ \xrightarrow{k_{3}} \ EET \tag{3}$$ For equation (2), we assumed that it takes multiple (n) proteins to form an aggregate. To obtain a system of ODEs from these equations, we made some assumptions. Firstly, we assumed that the amount of EET complexes reduces the transcription and translation of the MtrCAB proteins due to reduced cell viability as a consequence of reduced membrane integrity. To include this, we added a negative exponential term to the rate \(k_{1}\). Also, the concentration of aggregates has a negative effect on the cell viability and is therefore also included in the negative exponent. Besides that, to include the clogging of the secretion system transporting the MtrCAB, we included another negative exponential term, similar to the one in equation (16.8) and (16.9) in the more detailed system. This results in the following system of ODEs: $$ \frac{d}{dt} [MtrCAB] = \ k_{1}\exp(-k_{4}[EET][Aggregate]) - \ k_{2}[MtrCAB]^{n} - \ k_{3}[MtrCAB]\exp(-k_{5}[MtrCAB]) \tag{4.1} $$ $$ \frac{d}{dt} [Aggregate] = \ k_{2}[MtrCAB]^{n} \tag{4.2} $$ $$ \frac{d}{dt} [EET] = \ k_{3}[MtrCAB]\exp(-k_{5}[MtrCAB]) \tag{4.3} $$ We are mostly interested in tuning the promoter strength (\(k_{1}\)) to reach an optimum between MtrCAB expression and cell viability, thereby obtaining the maximum amount of EET complexes. To find this optimum, we numerically solved the system of ODEs for different values of \(k_{1}\), and plotted the [EET] at our end time against \(k_{1}\). Since this system does not reach steady state, we decided to use the EET concentration after 100000 s (approx. 28 hours) to simulate an overnight culture. With really crude parameter estimates (see table 3), we obtained the graph displayed in figure 4.

Table 3: Parameters used to obtain qualitative results from the simplified ODE system describing the assembly of the EET pathway.

Parameters	Value	Physical meaning
\(\boldsymbol{k_{1}}\)	Varies between 0 and 1	MtrCAB production rate (promoter strength)
\(\boldsymbol{k_{2}}\)	1	Aggregation rate
\(\boldsymbol{k_{3}}\)	1	EET assembly rate
\(\boldsymbol{k_{4}}\)	1	Cell viability parameter
\(\boldsymbol{k_{5}}\)	1	Clogging parameter
\(\boldsymbol{n}\)	2	Number of proteins needed to form an aggregate

Despite the crude parameter estimates, this plot already shows (to a certain degree) the behavior experimentally found by [1] and [2]. A maximum at low promoter strength is clearly visible. This corroborates the statement in [2] that maximum promoter strength does not result in maximum EET concentration due to reduced cell viability.

Although our model is very much simplified and it does not represent all the actual mechanisms of the EET complex assembly process that happens in nature, the fact that it is able to match experimental data is a confirmation that our assumptions concerning cell viability might be correct. Therefore, we conclude that reduced cell viability because of the implementation of the EET pathway is the consequence of three molecular processes.
Firstly, the amount of EET complexes reduces the transcription and translation of the MtrCAB proteins as a consequence of reduced membrane integrity. To include this, we added a negative exponential term to the rate \(k_{1}\).
Secondly, the forming of MtrCAB aggregates. The concentration of MtrCAB aggregates has also negative effect on cell viability and is therefore also included in the negative exponential term to the rate \(k_{1}\).
Thirdly, the clogging of the secretion system transporting the MtrCAB complexes. To include this molecular process, we included a negative exponential term to \(k_{3}[MtrCAB]\).

References

[1] H.M. Jensen, “Engineering Escherichia coli for molecularly defined electron transfer to metal oxides and electrodes”, PhD Thesis Chemistry UC Berkeley, 2013.

[2] C.P. Goldbeck, H.M. Jensen et al., “Tuning Promoter Strengths for Improved Synthesis and Function of Electron Conduits in Escherichia coli”, ACS Synth. Biol. 2, 150-159, 2013.