Interestingly, the constraints define polytopes in a high-dimensional space, which is usually called the flux space. One can easily prove that polytopes are convex sets, which is a property that makes them amenable to several manipulations, such as random sampling based on artificially centered hit and run algorithms [5]. These polytopes are usually represented as follows:

$$ S\cdot v = 0 $$ $$ v_{min} \leq v \leq v_{max} $$

where $S$ is the stoichiometric matrix, $v_{min},v_{max}$ describe lower and upper bounds of the metabolic fluxes.

As this polytope is high dimensional, methods have to be applied in order to determine probable flux distributions or to compare flux spaces corresponding to different cells or states. The usual procedure describes the maximization of a linear objective function $c^T\ v$ subjected to the constraints defined in ~\ref{eq:steadyState}. This concept is based on the assumption that biological systems have evolved in order to maximize a certain objective for example the growth rate of the organism. To convert this principle into linear programming the following algorithm is formulated: Maximize $c^T\ v$ in respect to the above constraints. This method is usually called Flux Balance Analysis (FBA) and the linear objective maximized is in most cases the so-called biomass reaction, though it can also take other forms, such as ethanol production or ATP maximization, depending on the context.

Although linear programs can be solved quickly with an optimal objective function value returned, there can exist many alternate optimal solutions, of which available solvers return only one. In order to capture this alternate solution space, flux variability analysis (FVA) can be employed [6]. In FVA, for each reaction $i$, the flux $v_i$ is maximized, then minimized, subject to \ref{eq:steadyState} and $c^Tv \geq s \cdot \max(c^Tv)$ with $s \in [0,1]$ and usually equal to 1.

Most of the constraint based modeling approaches have also been implemented in diverse software packages. Here, the very popular COBRA toolbox for Matlab [7] was used.

Metabolic model of Delftibactin producing E.coli

As explained in the previous section, to simulate the metabolism of a cell using flux balance analysis, the stoichiometric representation of the underlying metabolic network has to be available. As the biochemistry of E.coli has been extensively studied, there also exist comprehensive metabolic reconstructions. Here we used the iAF1260 model [8], which captures 2077 reactions of K-12 MG1655 E.coli corresponding to 1260 genes. Of course, wild type E.coli is not able to produce Delftibactin. Thus the reactions corresponding to the gene products we are trying to introduce into our TOP10 cells (LINK delftibactin project) were appended to the aforementioned model.

In particular, wild type E.coli can produce all the monomers necessary for Delftibactin production, with the exception of methylmalonyl-CoA, which is required for the PKS part of the synthetase. Thus, a reaction converting propanoyl-CoA to methylmalonyl-CoA with the following stoichiometry was added:

$$ 1 \: ppcoa[c] \: + \: 1 atp[c] \: + \: 1 \: hco3[c] \: \to :\ 1 :\ mmcoa-S[c] :\ + :\ 1 :\ pi[c] \: + \; 1 adp[c] \: + \: + 1 h[c] $$

Subsequently, the main delftibactin production reaction was added. The stoichiometry is governed by the amino acids monomers which are combined under usage of 1 ATP each and the Claisen condensation of methyl-malonyl CoA. This core reaction of the synthetase was accompanied with the subsequent modification enzymes [9] which are also necessary for the functional activity of Delftibactin. These comprise the aspartic acid dioxygenase (encoded by the gene DelD), the N5-hydroxyornithine formyltransferase (DelP) and finally the lysine/ornithine N-monooxygenase (DelL). The stoichiometry of this lumped reaction is the following:

1 ala-L[c] + 2 ser-L[c] + 1 asp-L[c] + 2 thr-L[c] + 1 gly[c] + 2 orn[c] + 1 arg-L[c] + 10 atp[c] + 1 akg[c] + 2 o2[c] + 1 10fthf[c] + 1 nadph[c] + 1 h[c] + 1 mmcoa-S[c] \to 10 amp[c] + 20 pi[c] + 1 co2[c] + 1 succ[c] + 1 thf[c] + 1 nadp[c] + 1 h2o[c] + 1 coa[c] + 1 co2[c] + 1 delftibactin

Considering constraints of the metabolic model, it was assumed that E.coli grows on minimal glucose medium under aerobic conditions. The maximal glucose uptake rate was set to 10.5 $ \frac{mmol}{g_{dw}h} $ and the oxygen uptake rate to 15 $ \frac{mmol}{g_{dw}h} $ and the ATP maintenance flux (which represents the non-growth associated energy required for maintaining the biological processes of the cells) was set to 7.6 $ \frac{mmol}{g_{dw}h} $. Those fluxes have been previously measured for aerobically growing E.coli [10].

As we were interested in the optimal case, that is the maximal possible delftibactin production based on the stoichiometrically imposed constraints, we initially used FBA with delftibactin production as the objective function. The resulting flux was 1.2105 $ \frac{mmol}{g_{dw}h} $. However, the corresponding specific growth rate was 0. Thus, the maximally possible specific growth rate in dependence of the delftibactin synthesis rate was determined. The polytope was split into 150 intervals. For each of these intervals, the growth rate was fixed and FBA was simulated again. Fig. shows the resulting production envelope, which in turn shows the optimal delftibactin production rate as a function of the specific growth rate (FIGURE). As could have been expected, the delftibactin synthesis drains many resources which are also necessary for bacterial growth (ATP, amino acids) and thus these two objectives represent a natural trade-off. The maximal growth rate, when the bacteria are not producing delftibactin at all is 0.8093 $ \frac{1}{h} $.

Bioreactor Simulations

Next, we estimated the cost of delftibactin production using a fed batch process. A flux balance analysis model only gives an estimate for steady-state fluxes and does not capture dynamic behaviors, such as fermentation processes. This is solved by dynamic FBA (dFBA) frameworks [11], which essentially discretize time into small steps. Flux and growth rates are calculated by FBA and subsequently included into appropriate differential equations. The results are used to constrain the metabolic network in next time step, etc.

A fed-batch process with an exponential feeding strategy, in which the glucose concentration, remains constant and delftibactin is produced, was modelled by dFBA using the DyMMM (Dynamic Multispecies Metabolic Modeling framework ) MATLAB framework [12] with the following equations adapted from Zhuang et al. [13].

$$ \frac{dV}{dt}=\left\{\begin{matrix}\frac{v_{glc}XV}{S_{glc}-S_{glc}^{feed}}, if \ V < V_{max} \\ 0, if \ V \geq V_{max} \end{matrix}\right.$$ $$ \frac{dX}{dt}=\mu X $$ $$ \frac{dS_{glc}}{dt} = v_{glc}X + \frac{dV}{dt}\frac{S_{glc}^{feed}-S_{glc}}{V} $$ $$ \frac{dS_{delftibactin}}{dt} = v_{delftibactin}X $$ $$ |v_{glc}| \leq |v_{glc}^{max}| \frac{S_{glc}}{S_{glc} + K_m} $$

(figure : cost/productivity/yield vs different parameters)

where X [g/L] is the E.coli biomass, V, Vmax [L] are the currently and maximally filled volume of the bioreactor, $S_{glc}$, $S_{glc}^{feed}$, $S_{delftibactin}$ [mmol/L] are the concentrations of glucose and delftibactin in the reactor or the feed, $K_m$ [mmol/L] the Michaelis-Menten constant for glucose uptake, $ \mu $ [1/h] the growth rate of the bacteria and finally $ v_{glc},v_{delftibactin} $ $ \frac{mmol{g_{dw}h} $ the flux rates of the corresponding reactions with $ v_{glc}^{max} $ being the maximal possible flux.

The starting conditions were set to a volume of 1 L, a biomass of 0.1 g/L and a glucose concentration of 20 mmol/L, as in [13]. The other bioreactor parameters were set as follows: The concentration of glucose in the feed was equal to 1 mol/L, $K_m$ equal to 1 mmol/L [13] and finally, the maximal glucose uptake rate was set to 10.5 \frac{mmol}{g_{dw}h} [10]. The time step was set to 0.1 h.

Simulations with those conditions were performed for 150 equally spaced growth rate values on the production envelope, as shown previously in Fig.X. Fig. Y shows the resulting trajectory for the biomass concentration and delftibactin concentration for 3 representative points from the envelope. As expected, higher growth rates result faster convergence towards the steady-state. Reaching the steady state implies the end of the fermentation process. This is expected, as in the model, the glucose feed is stopped as soon as the 10 L of the bioreactor have been filled. Also, the final concentration of delftibactin ranges from approximately 40 to 80 \frac{mmol}{L}, with higher titers achieved for lower growth rates, while dry biomass (10 to 50 \frac{g}{L}) titer is higher in the case of high growth rates.

The above observations actually illustrate another aspect of the growth rate - production rate tradeoff, namely the economic aspect. Lower growth rates actually lead to higher final delftibactin concentration in the bioreactor, which is of course wanted, but on the other hand the process takes a lot longer to finish, thus requiring more operational costs. Exactly this trade-off will be explored in the next session.

Delftibactin Production Cost

The ultimate goal of the aforementioned simulations was the optimistic calculation of the cost required for production of delftibactin. Thus, for each of the simulated points, we calculated the costs by starting with the following assumptions:

• The cost of the growth medium of the E.coli is equal to the cost of glucose spent.
• The operation of the 10 L bioreactor requires 1 full time technician with a wage of 2000 euros per month.
• After finishing the operation of the bioreactor, the next cycle can start after a time offset of 3 hours.
• The electricity cost of the bioreactor is calculated based on its power rate and assuming constant operation.
• Delftibactin has been secreted to the supernatant and it does not have to be purified in order to selectively precipitate gold, as shown in our experiments. Instead hydrogen peroxide has to be used to remove reducing agents except delftibactin from the medium.

For these calculations, the following prices were looked up and used:

In particular, let $n_{delftibactin}$ [mol] denote the number of delftibactin mol produced, $n_{glc}$ [mol] the glucose consumed and $t_f$ [h] the time at which the fermentation ends. All of these values can be can be easily calculated based on the results of the previous simulations.

$$ n_{delftibactin} = S_{delftibactin}V_f $$ $$ n_{glc} = S_{glc}^{start}V_0 + S_{glc}^{feed}(V_f-V_0) $$

and $t_f$ is just the time after which $\frac{dS_{delftibactin}}{dt} = 0 $.

Now also let

$t_{offset}$ [h] be the time until the next fermentation cycle can start, $P_{glc}$ [euros/mol glucose] be the price of glucose and p_{reactor}, the cost of operating the reactor per time (equal to the wage for 1 technician per time and the electricity cost). Then the final cost $P_{delftibactin}$ per mol of delftibactin is equal to:

$$ P_{delftibactin} = \frac{n_{glc}P_{glc} + (t_f+t_{offset})p_{reactor}}{n_{delfti}} $$

Based on the above equation equation, for each simulated point of the production envelope, the cost per mmol of delftibactin was calculated (Fig.). In the figure the trade-off between growth rate and delftibactin production immediately becomes obvious. In fact, the relationship of price to growth rate appears to be parabolic and actually has a global minimum. This global minimum of 0.1755 Euro per mmol of Delftibactin achieved to a growth rate $ \mu = 0.37 h^{-1}$ (total process time of 29.7 hours, final delftibactin titer of 61 mmol/L) is the cost we will assume in the next section.

Cost estimation for whole procedure

Having estimated the cost for produced delftibactin, it is now possible to also calculate the cost for the whole gold recovery process. Similarly, to the cost calculation for the gold procedure and using the aforementioned costs for aqua regia and NaOH, we calculated that for recovery of 1 mol Gold approximately 5 Euros are required in addition to the necessary delftibactin. Under the assumptions that 1 molecule of delftibactin binds and reduces exactly one gold ion, one can then quickly calculate the cost for the whole process (175.5 Euros for 1 mol Deftibactin, hence 180.5 Euros total per mol Gold). The current gold price, as well as the price for recovery from electronic trash with the different methods has been summarized in the table below.

In summary, our engineered bacteria, assuming optimality of delftibactin production and a laboratory scale bioreactor can recover gold from trash at similar costs with state of the arts methods and significantly cheaper compared to gold price.

Discussion

Proposed workflow

In order to recover metals from electronic waste, several different methods have been developed and are industrially utilized. In some of the common processes, the trash is incinerates, thus releasing toxic dioxins [15]. Pyrometallurgical processing, which is based on smelting, is a very common process, but also one that is often considered to be non-selective, requiring huge energy inputs and releasing harmful products in an uncontrolled way [16].

Another commonly used method is hydrometallurgical processing, in which diverse chemicals are used for the leaching of the trash in order to dissolve the metals in aqueous solutions. Traditionally, cyanide has been used for this, but due to environmental accidents, it is increasingly avoid [17]. Still, hydrometallurgical is often considered to be highly selective and have a good recovery.

Aqua regia used in our proposed pipeline also falls into the category of hydrometallurgical processing. Note that aqua regia is not necessarily the best choice of leaching agent to be used at industrial scale, for example the high cost of the necessary equipment (special stainless steel) [18]. There’s also toxic by-products, such as chlorine gas, especially if plastic is not separated from the metals in a initial step. Thus, the environmental impact could be improved for example by using bacteria in a first step in order to recycle the plastic, as has been done in several iGEM projects.

It is important to note again that the method proposed here only requires the existence of Gold ions in a fairly neutral solution. Thus, new developments in ecologically friendly leaching agents or even bioleaching agents [17] could be immediately coupled to the delftibactin process. In fact, it could easily replace the gold extraction step in most used in most industrially used processes in a plug-and-play fashion, whereas the chemical isolation steps would have to be extensively readjusted to the new conditions.

Finally, it has to be mentioned that efficient industrial processes consist of closed-loop systems [19] and pipelines, which try to isolate as many metals as possible, e.g. by iterative steps of electrolysis at different voltages so at to precipitate and reduce the different metals. Such processes of course are best from an economic perspective, but are difficult to compare to our suggestion in a quantitative manner. Of course, further developments could allow the creation of similar pipelines based almost exclusively on bioengineering, which could then be compared.

Constraint-based modeling

One of the main assumptions that went into the FBA based modeling was that the E.coli bacteria have been engineered in such a way, so as to maximize the delftibactin production rate. In fact, the simulated points lie on the pareto frontier (product envelope) and they place an upper bound to what is theoretically possible, simply due to stoichiometric considerations.

In order to get an insight to the resulting numbers, it could be useful to compare with two other recent approaches that modelled NRPS/PKS systems in a similar way. Huang et al. [20] used metabolic flux analysis in order to simulate the production of the antibiotic Daptomycin by Streptomyces roseosporus. Here the maximum daptomycin production was approximately equal to 0.06 $\frac{mmol}{g_{dw}h}$ for a glucose uptake rate of 0.7 $\frac{mmol}{g_{dw}h}$. Thus, the yield of daptomycin per glucose is appropriately equal to 0.086. Compare this to our simulations, in which the yield was $ \frac{v_{delftibactin}}{v_{glucose}} = \frac{0.68}{10.5} = 0.065 $. As daptomycin and delftibactin have similar molecular mass, this shows that the metabolic capatibilities for production of NRPS/PKS products of these two organisms, according to current flux balance models, are fairly similar.

In the above publication, the experimentally measured titer of daptomycin in fed-batch culture was 581.5 mg/L. Our modelled fed-batch process titer of 63 g/L (61 mmol/L and delftibactin has a molecular mass of 1033 dalton). This factor of 2 shows two things: For one, natural systems will have to be significantly engineered in order to achieve NRP production rates close to the Pareto frontier. Yet, this was one of the main reasons we started the experiments for the introduction of the D.acidovorans genes into E.coli: The latter can be manipulated and used for metabolic engineering a lot easier. On the other hand, this factor also shows that our calculation might be very optimistic for the time being and that significant further research will have to be done until such as a system can applied at industrial scale.

Bioreactor simulation

In general, there exist three main methods for the cultivation of bacteria in bioreactors: The most classical one is the batch method, in which all of the initial medium is inoculated at the beginning point. In contrast, in the continuous fermentation, new medium keeps flowing into the reactor while also being withdrawn at the same rate. Continuous fermentation of course requires a lot less labour costs, provides higher yields and is more controllable than the batch process. On the other hand, in real processes, the microorganisms often mutate to non producing variants, thus making continuous processes problematic [21]. In between these methods lies the fed-batch, which was chosen here, in which medium keeps being added to the reactor, but is not withdrawn.

This method, was simulated using dynamic FBA (dFBA), which quickly proved to be a very valuable method for combining dynamic processes with stoichiometric metabolic models. One disadvantage are the rather long simulation times, but recently methods have been developed in order to efficiently numerically simulate differential equations of which the right hand side actually requires the optimization of a linear program [22]. Applying such methods could have allowed a greater coverage of the parameter space and in a quicker way.

In particular, one of the problems that appear in aerobic E.coli fermentations is the fact that at high growth rates (almost surely for growth rates above 0.35 1/h) a lot of acetate is produced which actually suppresses growth [23]. This is something that was not considered in the simulations here, but could definitely be easily added to the simulations ran here, by adding an appropriate inhibition term to the differential equations and also tracking the acetate concentration in the dFBA model.

Final judgment

In total, under optimal conditions, our method achieves similar cost to more traditional chemical methods. Of course, optimal production is only theoretically possible and compared to actual measurements in Streptomyces (see above), the final titer is 2 orders of magnitude higher. Still, it has to be also considered that here the fermentation process was assumed to take place in a small scale 10 L bioreactor with high operation costs. Production at industral scale with 10 m^3 bioreactors would of course reduce operational costs.

The chemical method to which we compared ours might also be more expensive in reality, as operation costs and costs for ion-exchange chromatography were neglected. This could also be balanced though by the material costs, which might be cheaper when bought in bulk at industrial scale.

Finally, it also has to be considered that there’s still 1 order of magnitude difference in regards to the profitability (cost of Gold compared to price for gold recovery with our method) with the gold price steadily rising, so that with improved NRPS and metabolic engineering, our system could indeed turn out to be usable.

The fact that it is based on a biological system makes it possible to incorporate it into diverse processes in a plug-and-play fashion and it is also more environment friendly compared to the chemical and big energy resources needed for other methods.

Outlook

• engineering e.g. using NRPSDesigner of molecules similar to delftibactin with specificity for other metals

• gold nanoparticles [24]

• pipeline possible to completely recover electronic waste using NRPS as well as previous igem projects (e.g. plastic recycling projects)…