Welcome to the modeling page! On this page, we will introduce two mathematical approaches we developed in this project. The first model is a Genomic-Scale Metabolic Model (GEM) for solutions using open-source tools. This model helps us to define the key reactions in cellulose production, and also shows the theoretical limits for growth in different medium conditions. The second model is a kinetic representation of the growth and cellulose production, for the wild and for the modified chassis. This model can be adjusted when we have new data from the lab. Make than a powerful tool for process optimization.
More explanation of these models can be found in the sections below!
A genomic-scale metabolic model (GEM) contains all or most the metabolites and reactions of a microorganism. It can be used to predict the behavior of the microorganism under certain conditions and to understand the main reactions involved in the synthesis of the product of interest. It can also be used to understand the effect that different gene knockouts can have on the cell.
Building a metabolic network is a long and arduous process, for it involves the reconstruction of a complete reaction-by-reaction network. However, due to the joint efforts of several researchers, currently, there are collections with many networks and subnetworks available.
These initial steps create a model representing all the enzymes encoded by the organism's genome. With these enzymes and the use of collections of enzymatic reactions, it is possible to define the metabolites that these enzymes act on. Once the connection network between the metabolites is defined, these data need to be organized so that they can be analyzed mathematically. After having identified all the metabolites and the reactions, the latter needs to be balanced. With these data, a stoichiometric matrix of all the metabolites present in the model is constructed.
The model that was used in the present project was published by Rezazadeh et al. (2020) for Komagataeibacter xylinus, a species closely related to K. rhaeticus, allowing us to use the same model to gain insights into the metabolism of our strain.
After the model construction, it is necessary to check its consistency. The verification can be done manually for small models, but in the case of the K. xylinus’ model, it is unfeasible, as it has a total of 865 metabolites and 918 reactions. The way to do this check automatically is with the use of the software. We used the MEMOTE app, which is a free and open-source web application.
The results of the test it is summarized below.
After the consistency check, the model is ready to be used. With genomic-scale models (GEM), different types of analysis are possible, such as flux balance analysis, flux variability analysis, and parsimonious flux balance analysis.
In this work, the Cobrapy package was used for all analyses, and the notebooks are available in the supplementary material.
After a genomic scale model has been created, it is necessary to make a representation of it mathematically. For this, stoichiometric matrices are constructed, as shown in Figure 1 , where the lines represent the metabolites and the columns the reactions. Growth is symbolized by the insertion of a biomass reaction, which takes into account what the microorganism growth requires DNA, T3P, PEP, PYR, ATP and others. Exchange reactions between bacteria and medium are also inserted as transport reactions between the external medium and the cytoplasm.
After that, the matrix multiplication between the stoichiometric matrix and the column vector of reactions is set to zero.One of the functions can be chosen as an objective to be maximized respecting the constraints defined in the model. Maximizing this objective function using linear optimization will find the values of the fluxes.
One of the goals with the resolution of metabolic models is to find the shortest path, which is the walk with the smallest sum of flows in order to reach the maximum of the objective function. With the use of FBA as explained above, it finds only one path and optimizes only the objective function without considering the path.
With the use of pFBA it is possible to test different paths and choose the one that will provide the smallest sum of flows in order to find the shortest path to the objective function. This analysis can be used to find which reactions can be further optimized in order to make the path shorter or more efficient.
When the model is optimized using FBA, only the maximum value is sought. The use of FVA allows that, keeping the value obtained with FBA, it is possible to explore the maximum and minimum values that the chosen reactions can obtain.
First, it was necessary to validate the results of the metabolic model. The model published for K. xylinus previously ran in MATLAB with COBRA Toolbox. As we converted the model to Python, was tested it with cobrapy and the same results presented in the article were obtained, validating the use of a free tool for the simulations.
Our first objective was to understand the relationship between growth and cellulose production. FBA simulations were performed, limiting the growth rate between 0.0 and 0.7, which is its maximum rate, and setting the cellulose production as an objective function. It is possible to see in the simplified metabolism diagram that the reactions of cellulose and biomass production are concurrent (Figure 2).
The results obtained by the simulations confirm this analysis, showing that there is a linearly decreasing relationship between the production of biomass and the reduction of cellulose (Figure 3).
The result above was used to direct the strain engineering where we aim to decouple growth from cellulose production, prioritizing first the growth, and, after reaching the adequate biomass, induce cellulose production.
Using the GEM described above, it is possible to change the uptake conditions and use this to understand the minimum requirements for the culture media.
In collaboration with the Uioslo-Norway team, which goal is to use Komagataeibacter in co-culture to produce a copolymer containing cellulose and chitin. During the production of chitin, it is necessary to use nitrogen. So, to better understand the cultivation requirements, different cultivation conditions are simulated to suggest how much NH4 the Komagataeibacter needs for its development. We determined minimal NH4 for growth as well as oxygen requirements of the obligate aerobic bacteria. Thus, the model was used to test different media conditions changing the NH4 and O2 available to determine the minimum for cell growth and the minimum for cellulose production. Cellular uptake can be seen in the table below and in Figure 4.
In Figure 4, upper panel, it is possible to see that after a concentration of 40 mmol/l (horizontal axis), the increase in NH4 in the media doesn't lead to changes in growth; this means that 40 mmol/l is the threshold for NH4. In the vertical axis, after 5 mmol/l, the increase in oxygen available does not have an impact, which defines the oxygen limit for max growth. In the lower panel it is possible to see that after the concentration of 10 mmol/l of NH4, cellulose production no longer varies. This value uses cellulose production as an objective function.
With FVA analysis, it was possible to identify which reactions may have greater flux variations and still maintain the objective function at the stipulated value. To understand the flux variation, two cases were analyzed. First, keeping biomass production as the objective function. However, in order to have greater flexibility, the value of 90% of the theoretical maximum biomass was set. Second, an analysis was performed as an objective function of cellulose production (Figure 5).
Figure 5 shows changes in the ranges of various reactions within the microorganism. They illustrate that the flexibility of some reactions is affected when there is a change in the objective function. These reactions that had their ranges reduced are essential for the production of cellulose in Komagataeibacter.
Cellular growth has complex effects on cellulose yields, making modeling quite difficult. However, there are simplified mathematical models that can be used to get good insights into growth and BC production. One of the main parameters used is the metabolic yield coefficients which are the ratio between two variables of interest. eg biomass (x) and substrate(s):
The Yxs coefficient indicates how much of the substrate is converted into biomass. In the case of Komagataeibacter xylinus, 1.7 to 2.4% of the substrate is used for biomass.
The coefficient Yps indicates how much of the substrate (glucose), is converted into the product of interest (cellulose). In the case of Komagataeibacter xylinus, approximately 24% of the substrate is used for cellulose production.
The coefficient Ypx indicates the product of interest generated for each g of biomass. In the case of Komagataeibacter xylinus, for 1 g of biomass 10g of cellulose is producted.
In this work, the behavior was modeled only in a batch system. I was assuming perfect mixing and homogeneous conditions.
Growth in batch mode has different phases (lag, exponential, stationary, and death phase), the main responsible for this behavior is the concentration of substrate available in the medium. The substrate that causes this behavior is called the limiting substrate. The behavior of the growth rate of a cell considering its growth limited by a single substrate is analogous to the Michaelis-Menten model, but in the case of growth, it is called the Monod equation.
Modeling an ideal batch reactor for the culture.
where x is the cell concentration in g/L, u is the specific growth rate, and kd is the specific death rate. As a constant volume batch reaction is being considered, V = cte.
Whereas the growth rate is much higher than the specific cell death rate.
Wild-type Komagataeibacter has indirect product formation, but for modeling purposes, it will be considered associated with mixed-mode energy metabolism.
where Ypx, indicates how much product is being formed by each g of biomass, and mp is the product formed during cell maintenance itself. Commonly the terms are grouped as seen below:
How cellulose production was considered mixed with energy metabolism. The substrate will be consumed for growth, maintenance, and product formation, resulting in the equationbelow:
Organizing the complete model, and putting it in differential form, we obtain the ODE system below (batch mode of cellulose production):
where:
The parameters for the model were obtained from the literature for the wild type K. xylinus and estimated for the strain that will be modified.
Modeling results regarding biomass accumulation, cellulose production and substrate consumed can be seen in Figure 6.
With the results obtained in the FBA simulations, it was possible to determine the maximum specific rate, which was 0.7 [1/h], its rate growing in a static medium during the production of cellulose that consumes 24% of the substrate is 0.014, 2% of the maximum rate.
To simulate how the model can behave in batches, it was set that the cellulose production was knockout, so the specific rate could reach 0.14 growth, this is a value of a possible range since it is only 20% of the maximum (Figure 7).
The results in figure 7 show that the steady state would be reached 100 h earlier compared to a wild-type strain.
Subsequent induction of cellulose synthesis by returning cellulose production reactions greatly affects the behavior of the microorganism, therefore it is expected that this will lead to a change in the model, hence the induction model has its parameters changed.
With the model, it will be possible to predict the best moment to start the induction of cellulose production, as can be seen in Figure 9 below where different induction moments were tested.
To test the entire space of possibilities, the plots below were made in the form of a heatmap (Nils Gehlenborg and Bang Wong, 2012), Figure 10.
Computer simulations utilizing the published K. xylinus genome-scale metabolic model provided good insight into the enzymes that could have a greater impact on cellulose production when overexpressed or knocked down (Figure 5). Furthermore, it suggests that bimodal growth vs production regime might increase cellulose yields (Figure 9), and provide suggestions of ideal timing for the switch between biomass accumulation and BC synthesis stages (Figure 10).
These data can be used for the design and construction of suitable strains and bioreactors for testing and model refinement.
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