Team:Nottingham/Model

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Mathematical Modelling

Please Note . . .
The following report is an abridged version of our full modelling paper. This is available for download here, along with several accompanying files.

Highlights

  1. A refined structural model of C. sporogenes is presented.
  2. (R)-3-hydroxybutanoate (DBHB) is produced in 84% of all possible modes.
  3. Allowing acid inflow to the model increases the flux to DBHB.
  4. DBHB is prioritised over alcohol production in the model, ethanol knockouts yield no benefit to DBHB yield.
  5. DBHB is produced in all the highest ATP yielding elementary modes.
  6. A kinetic model is constructed, based on the ABE fermentation pathway in C. sporogenes.
  7. An administration strategy for our biotherapeutic is given.

Introduction

In this project, our challenge required an out-the-box thinking to help people suffering from neurodegenerative diseases.

Our new approach involves modifying C. sporogenes to produce a ketone body, DBHB. As this ketone has been linked to linked to neuroprotection through reduced neuronal oxidative stress [1].

With an absence of experimental data and opportunities due to COVID-19, significant emphasis is placed on the modelling. We were required to investigate the different mutants suggested by our colleagues to asses which would be most beneficial to our project. Our structural modelling performed the task of choosing the preferred mutant, whilst the kinetic modelling would suggest a feasible strategy for long-term therapy. The different modelling approaches and objectives are given here.

In the structural model:

  1. To improve the model from The University of Nottingham's 2019 iGEM team:
    1. By adding consistent biomass formation equations.
    2. Adding reaction formulas and improving the mass balance of reactions.
    3. Adding missing reactions.
    4. Standardising the models coded structure.
  2. Investigate the different mutants for their feasibility and capability to produce DBHB through Flux Balance Analysis (FBA) and Elementary Mode Analysis (EMA).
  3. Develop strategies to improve the production of DBHB via:
    1. Ethanol knockout
    2. Acid uptake analysis

In the dynamic model:


  1. Convert the existing dynamic model for batch culture such that it represents the conditions in the gut by allowing for
    1. Time dependent inflow of nutrients
    2. Outflow of all compounds
  2. Add DBHB Formation to the model using the best pathway as identified by the structural modelling and improve flux towards acetoacetyl-CoA.
  3. Investigate:
    1. Frequency of drug administration
    2. Initial spore concentration
    3. Dosage

DBHB Metabolism in C. sporogenes

We assessed two candidate pathways that could produce DBHB suggested by our colleagues, please see the modelling report, included above, for details about the enzymes involved, and for a more in-depth explanation of the enzymes used and a description about the plasmids used, please see our Design page.

It was important to have no side products being produced as a result of DBHB production, and the pathways we have investigated produce DBHB without other, potentially harmful products being produced and released into the human body.



Figure 1. Pathway DBHBA. The genes encoding for the enzymes which were added to the metabolic network are shown in blue.


Figure 2. Pathway DBHBB. The genes encoding for the enzymes which were added to the metabolic network are shown in blue.

Figures 1 and 2 showcase the two pathways we were investigating with the structural model. The first item of note about the two pathways is that for the DBHBB pathway we need to add (3R)-3-hydroxybutanoyl-CoA thioesterase to convert acetoacetyl-CoA to (3R)-3-hydroxybutanoyl-CoA since the hydroxybutanoyl dehydrogenase native to the acetone-butanol-ethanol fermentation pathway we are using converts acetoacetyl-CoA to (S)-3-hydroxybutanoyl-CoA. This distinction is very important as only the (R)-3-hydroxybutanoate form induces the production of the DBHB that we want, and this can only come from (3R)-3-hydroxybutanoyl-CoA. It was also brought to our attention, during a meeting with Professor Kieran Clarke from Oxford University as part of our Human practices, that the (S)-form of DBHB has no health benefits and is potentially linked to other serious diseases.

Constraint Based Modelling

Model Adjustments

We have improved the metabolic network of C. sporogenes from last year’s Nottingham iGEM team, which was originally based on a previously published model [2]. The adjustments to the model have been made using COBRApy which is a constraint based modelling package for Python [3]. An elementary mode analysis (EMA) was conducted in MATLAB R2020a using the Cell Net Analyzer tool, version 2020.2 [4], however, elementary mode analysis is not a model adjustment.

The model was improved significantly by the addition of metabolite formulas and by the standardisation of the COBRApy model. The equation used for biomass has been obtained from a single paper [5], an improvement over the previous model which used a theoretical biomass equation.

Results

Using FBA, we were able to confirm that a steady-state solution existed for both mutants that produces DBHB whilst supporting growth. From EMA, we were able to identify that 84% of possible modes produced DBHB for then DBHBA mutant, with 66% of all modes allocating a significant amount of flux towards DBHB production. This was significantly higher than the equivalent for the DBHBB mutant. Through further use of EMA, we were able to discover that the DBHBA pathway has a higher productivity of DBHB and ATP when compared to the DBHBB pathway.

When acid excretion is blocked in the model, DBHB becomes the optimal product with flux being prioritised to DBHB over biomass creation. Allowing more acid into the model whilst limiting the culture size directly increases the amount of DBHB produced by the model up to a limit. Which is crucial and suggests the biological validity of this observation, since DBHB production levels are not increasing towards infinity, they are instead capped by biological constraints.

Additionally, we found that knocking out ethanol production does not appear to yield any further improvement to the yield of DBHB.

Figure 3. DBHB per glucose for both pathways with highlighted max yield and maximum mode numbers

With the higher maximum theoretical yield, the greater productivity of DBHB and ATP, and a higher proportion of modes producing DBHB. Pathway DBHBA comes out on top as the strain we want to focus on further in this analysis. Dynamic modelling will now consider only the DBHBA mutant.


Dynamic Modelling

For our dynamic model, we built on top of some of the last year's project and "Kinetic modelling and sensitivity analysis of acetone-butanol-ethanol production” by Shinto et al. [6]. We adapted the model of the ABE pathway presented in that paper for our purposes by, first, fit it to the metabolic activity found in C. sporogenes and, second, adding DBHB production to the network, see Figure 4.



Figure 4. ABE Fermentation Pathway in the DBHBA of C. sporogenes. The network shown in black represents the native pathway, in red the added formation of acetoacetate and in green of DBHB. Ri is the reaction identifier used in the kinetic model for the corresponding reaction, see the modelling report for further details.

Kinetic Model

Following the approach for C. saccharobutylicum presented in [6], we employed the Michaelis-Menten equation for each enzymatic reaction in the metabolic network of C. sporogenes. We used kinetic parameters as determined in the last year’s Nottingham iGEM project [7]. Additionally, thiolase activity was increased in the DBHBA mutant to direct more flow towards DBHB production. To model the metabolic network (Figure 4), we adapted the rate equations representing metabolic reactions from [6] to our C. sporogenes model with DBHB production.

Batch culture

Starting with a batch culture model, we simulated the behaviour of the wildtype and the DBHBA mutant by solving the system of differential equations numerically. The results were produced in MATLAB 2020a using solvers ode45 for the wildtype and ode15s for the mutant, Figure 5.

Figure 5. Product formation in the wildtype (left) and the DBHBA mutant (right) as a function of time.

In the wildtype, no DBHB is produced, however, in the DBHB mutant we can observe the levels of DBHB growing from the 29th hour until they reach the concentration of 4.6 mmol. Additionally, a significant increase in butyrate formation is found in the mutant. This is caused by the increased thiolase activity forcing more flux towards DBHB and butyrate and, thus, away from growth, acetate and ethanol. Consequently, the mutant produces significantly less acetate, less ethanol and grows slower than the wildtype.

Continuous culture

In the batch culture model, there is an initial amount of nutrients available which wil deplete over time. This only tells us what happens after a singular dose. To model reoccurring administration of the drug we adapted this model as a continuous one.

Accordingly, we introduced a periodic inflow of nutrients, as well as an outflow of all compounds. As for the outflow, a term of the form −kX was added to the rate equation of each component X. On the other hand, the inflow of glucose is described by a periodic step function.

Glucose will be present with every meal/snack the patient has. However, to enable control over the culture size and, thus, DBHB production, we have assumed that growth is coupled to an inducer for glucose uptake or by coupling it to another necessary nutrient, e.g. an artificial amino acids, delivered by the capsule.

This would tackle two issues:

  1. avoiding ketoacidosis, and
  2. preventing growth outside the gut where the couple is not fulfilled.

Administration

Our aim was to investigate possible frequencies of administration of the drug using the continuous culture model we have developed. Taking into account that some patients on this treatment would be the elderly and the drug would possibly be administered to them by carers or nurses, we wanted to allow for less frequent administration and something relatively simple to keep track of. This was recommended to us by Human Practices, in particular, a meeting we had with Dr Morrant, specialist in Parkinson's disease, and Nicola Cook, acting senior carer. Accordingly, we looked into administration every 24 and 48 hours as well as weekly.

We found that administration every 24 or 48 hours was feasible, as for each of these the average DBHB levels were 4.6 and 4.8 mmol and even they would fall and rise, the oscillations were moderate and, thus, should not cause unforeseen problems. The results for the second option of administration are shown on the left in Figure 6. On the other hand, weekly administration was not feasible due to too the large oscillations of DBHB levels.

Figure 6. Product formation in the mutant with the drug administered every 48 hours (left) and with and without an activation switch implemented (right).

Finally, we decided that administration every other day was a good compromise between the numerical results and the advice we received about the issues with some patients who might be reluctant to take their medication.

An interesting finding was that the best way to administer the drug would be to activate the DBHB pathway, when the culture is established, see Figure 6 (right). In the case of administration every other day, the trigger would be delivered starting with the second dose.

Initial spore concentration and dosage

The initial amount of spores and the dosage are important parameters for a therapeutic application of our drug. Unfortunately, the current COVID-19-related restrictions have denied us to study the effect of these parameters on the growth of the culture. As a consequence, we performed our analysis by varying the relevant parameters.

For our investigation on the initial spore concentration we varied the initial biomass, see Figure 7.

Figure 7. DBHB levels when the DBHB mutant is administered every 48 hours, with initial amounts of 0.05, 0.1 and 0.4 mmol.

Our key findings were:

  1. The initial amount of spores determines how long it takes to approach a stable oscillating growth and product formation of the DBHBA mutant.
  2. The final level of biomass and product is unaffected by the initial amount of spores.
  3. The system seems to approach a stable limit cycle.

This is a very desirable behaviour for our drug, because it means that slight changes would not disrupt the effectiveness of the drug. This allows for expected variations such as the time a patient might take the medication, how they take it (e.g. before or after a meal), their diet and similar.

The dosage was analysed by varying the height h of our periodic step function describing the inflow of glucose (growth controlling nutrients), see Figure 8.

Figure 8. DBHB levels when h = 1.5, 2, 3.5 mmol/hour, respectively, in the periodic step function f(t) for the inflow of glucose, see also our modelling report.

Irrespective of the chosen dosage of the growth controlling nutrient, the time it takes for the culture to establish is the same, and overall, DBHB levels behave in very similar ways in the three cases. Importantly, the level of produced DBHB strongly depencds on the dosage and increases when the dosage increases. This finding shows that the level of DBHB could be regulated by the dosage of the growth controlling nutrient in the capsules and, thus, adapted to the patients needs.


Our full report, also available from the download link towards the top of the page:

The excel file linked here has a brief list of the reactions in the 2019 model and 2020 model. Some reactions may be missing from this list but are documented elsewhere in the report or the wiki. Download the excel file here.


References

1. Dominic P. D’Agostino et al. “Therapeutic ketosis with ketone ester delays central nervous system oxygen toxicity seizures in rats”. American Journal of Physiology - Regulatory Integrative and Comparative Physiology 304, 10 (2013). doi: 10.1152/ajpregu.00506.2012.

2. Mehak Kaushal et al. “Understanding regulation in substrate dependent modulation of growth and production of alcohols in C. sporogenes NCIM 2918 through metabolic network reconstruction and flux balance analysis”. Bioresource Technology 249 (2018), pp. 767–776. doi: 10.1016/j.biortech.2017.10.080.32.

3. COBRApy. 2020. url: https://opencobra.github.io/cobrapy/.

4. Cell Net Analyzer. 2020. url: https://www2.mpi-magdeburg.mpg.de/projects/cna/cna.html.

5. Author Manuscript and Metabolic Network Resolution. “NIH Public Access”. In: 101.5 (2009), pp. 1036–1052. doi: 10.1002/bit.22010.Genome-Scale.

6. Hideaki Shinto et al. “Kinetic modeling and sensitivity analysis of acetone-butanol-ethanol production”. Journal of Biotechnology 131.1 (2007), pp. 45–56. doi: 10.1016/j.jbiotec.2007.05.005.

7. Daniel Vaughan and James Abbott. “Dynamic and Structural models of C. sporogenes wildtype and mutant", Nottingham iGEM team 2019


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