Introducing bacterial growth

Thus far, all of our models have been considering single-celled dynamics, but that is not the case in real life since, in favourable environmental conditions, cells continuously undergo cell division.

To accurately include growth, we produced growth curves for E- coli and fed these back into the model. This allowed us to expand the use of the model and explore it further to predict total CaCO3 precipitation.

Aim: Quantify CaCO3 precipitation produced by a growing bacterial population.

Experimental support E. coli growth curve

Assumptions:

• Concentration of calcium ions and urea are evenly distributed across all cells
• There is no CaCO3 dissolution and urease catalysed reaction is much faster than its reverse
• Spontaneous urea hydrolysis is not included (has half-life of 40 years (Sigurdarson, Svane and Karring, 2018))

Model:

To accurately include growth in the model, we informed it with experiments. For this, we measured the E. coli growth curves (Fig 6). Looking towards additional model expansion, the growth curves measured were run at different pH levels. This is necessary as the process heavily influences pH and we need to understand how the cells deal with the changing environment.

Fig 6. E. coli growth curves in buffered Luria Broth at different pH. All buffers at 0.089 M concentration. pH 6, pH 7 and pH 8 in Tris-Cl buffer, pH 9 in bicine buffer, pH 10 and pH 11 in CAPS buffer.

The experimental growth curve at pH 7 was fitted with a logistic function (Eq. 4), which allowed us to extract the growth rate, k, of the bacteria and the maximum bacterial population reached, L. The logistic curve cell growth approximation used in the model the logistic function is plotted in Fig 7.

This logistic function was added to the model, such that it affected the rate of transcription of the enzymes, i.e., more cells will lead to more enzymes, which will precipitate more CaCO3.
This allowed us to benchmark CaCO3 precipitation to 7.1mg. Nonetheless, even models that have been informed by experiments require experimental confirmation that they are working as intended. For that reason, we developed a specialised assay to assess CaCO3 precipitation in our cells:

Aggregate Precipitation Assay

This assay measures the mass of precipitated compounds, making it an optimum assay to compare against our model. In the experiment, transformed and non-modified cells were cultured in an CaCL2-urea-LB medium (5% 1M CaCl2, 5% 1M urea). The resulting aggregate (after 15-hour culturing) was separated off by centrifugation dried and weighted. To evaluate the effect of the genetically engineered constructs, negative controls and blanks were run and the experimental value in Figure 8 represents the weight of engineered culture aggregate minus the control culture (for raw data please see the Proof-of-Concept page)

This assay yielded a precipitate weight of 3.95 mg, only differing from the value provided by our model by 3 mg.

This model is created with the help of experimentally measured growth curves and is then used in comparison with a calcium carbonate precipitation assay on which it is also validated. This result proves our model works, has predictive power and could be used to further explore the system, we have explored the dependence of calcium carbonate precipitation on initial concentrations of CO2, HCO3 and CO3. This investigation shows that the model creates CaCO3 with polynomial dependence on CO2 and HCO3 concentrations (Fig 9 a, b) and linearly on the CO3 concentrations (Fig 9c), this means that under pressurised conditions, when the amount of carbon dioxide dissolved in water increases the amount of calcium carbonate increases as well. This is great for our proposed implementation of including the bacteria into a building material, as the constrained environment will increase pressure.

Fig 9. Relationship between initial concentrations and final CaCO3 precipitation for a) CO2, where CaCO3 precipitation increases with polynomial dependence, b) HCO3, where CaCO3 precipitation also increases with polynomial dependence, but not as sharply and c) CO3, where CaCO3 precipitation increases linearly with initial CO3 concentration.

These differences in CaCO3 can be easily explained through the following points:

• As pH changes, CaCO3 will dissociate into calcium ions and carbonate, and the possibility of CaCO3 dissociation is not one that our model includes
• The model assumes that no carbon (CO2, HCO3, CO3, urea) is added to the system, however, in experiment there was aeration as well as CO2 production from the cell growth.
• We don’t consider the possibility of calcium binding to other ions, or of the carbonate ions binding to anything other than calcium

It would be interesting to model the CaCO3 precipitation including all of these parameters in the model, unfortunately, time restraints prevented us from doing so. Nonetheless, we encourage future iGEM teams to build on our model and include these parameters to explore its behaviour.

Introducing bacterial growth and pH feedback loop

Assumptions:

• pH changes instantaneously in the whole volume
• Cell maintains constant pH, only pH outside the cell is affected by any changes
• The volume in which the mRNA and proteins are scales with number of cells as ~𝑁𝑉𝑐𝑒𝑙𝑙
• Amount of urea in the cell is not a limiting factor, i.e., we assumed that urea is infinite
• Once CaCO₃ precipitates, it will not be lost
• Neglect H₂O

Experimental support: E. coli growth curves at different pHs

However, in reality, pH affects cell growth, protein production and action, and therefore number and overall activity. Bacterial strains are expected to grow differently at different pH levels and at non-physiological high/low pH the bacteria are expected to grow slower or die. This means, that the effect of our protein’s activity (net pH increase) might interfere with the bacterial growth which in turn affects protein production.

We have found a feedback loop! But how can we include it in our model?

We have to turn to experiments!

By measuring growth curves of our organisms (E. coli DH5a) at pH levels ranging from 6 to 11 (Fig 6), we were able to evaluate the effect of pH on bacterial growth. We fit the experimental growth curves with a logistic curve to extract their specific growth rates and maximum concentrations.

As a model of pH growth dependence, we choose a Gaussian distribution, a reasonable assumption (model needs to go to zero at large and low pH, be smooth and concave) which explains the experimental measurements. We fit these Gaussian distributions to the measured data, growth rates and maximum concentrations, respectively. The respective distributions are then used to scale growth rate µ and maximum concentration Nmax with pH during modelling of the logistic growth equation (Fig 10).

At this point, we were ready to include the feedback loop.

Figure 10. Involving pH dependant logistic cell growth into the model as number of genes of active genes.

Results

The bacterial growth fed back into pH change and vice-versa, as a result we saw delayed pH change and gradual calcium carbonate build up (Fig 8). The chemical system is bounded by urea and calcium ion concentrations, these two provide limits of calcium carbonate and of additional carbon dioxide that can be added to the system. Hence, it is clear that urea and calcium concentrations also affect the final pH. The more calcium carbonate is precipitated, the more the pH increases, until the moment that the cell culture perishes. However, we have shown that the culture survives long enough to utilise all calcium in the environment, hence for our purposes the overall reaction is self-sustainable.

This statement is confirmed by the fact that experimentally we observe significant calcite precipitation in both liquid and agar plate cultures

Figure 11: Feedback loop of changing environmental pH on end product calcium carbonate production.

One of the problems with the model combining growth and pH is that the ammonia hydration is instantaneous and lacks reverse reaction, inclusion of which would decrease the resulting pH. Further, the growth curves we used to fit pH growth dependence are of untransformed bacteria with no biomineralization activity. There might be major differences in the behaviour of the biomineralizing and non-biomineralizing bacteria as calcium carbonate can act as a pH buffer. And finally, the large changes in pH might influence the protein function, currently, this issue is side-stepped by assuming cells stay in homeostasis with constant internal pH, but in real conditions the protein action would be affected by pH.

As part of future work, we propose to establish a clearer link between the model and experimental system by directly measuring pH change over time. Further, we could evaluate the weight of calcium carbonate aggregate and compare it with experimental data.