## 3 Materials and Methods
### Model of the Study
#### 3.2.4 Gene Expression Value and Urea Hydrolysis Rate  
Using the RNAseq experiments, described above, three genes responding which exhibited significant changes in expression when the pressure was increased from zero to either 200 kPa or 400 kPa were identified and used as model data. The gene expression response data for these three genes are shown in Figure 3. Although there are relatively few data points, curves have been fitted and the dashed lines show a proposed continuation of the behaviour trend for each gene. These projections are not verified but give us a starting point to understand the likely effects of different expression profiles. The genes were selected because they showed a clear response to pressure changes, and each represents a different profile: increasing (Gene A) and decreasing (Gene C) in response to pressure and sensitivity to a particular pressure range (Gene B). Figure 2: The calculation process of the model used in the study. In our proposed engineered system, the pressure-sensing promoter gene will be linked to the production of urease, which is subsequentially responsible for inducing calcium carbonate formation in the soil. The calcium carbonate precipitation per unit time can be calculated from the urea hydrolysis rate of the bacteria. Here, to evaluate the influence of different gene expression relationships on the simulation results, we assume the maximum gene expression level as shown in Figure 3 (34.5) produces the maximum urease activity, which is measured in Section 3.1.2. This assumption allows the urea hydrolysis rate corresponding to different gene expressions to be calculated. Noting that here, the fitting curves may not reflect the real relationships due to the lack of data points for higher pressures, the proposed relationships are used for the preliminary exploration of the model. More investigations should be conducted in the future to quantify the relationship between gene expression and the proposed engineered system. The calcium carbonate precipitation rate is influenced not only by the rate of urea hydrolysis, which is controlled by the gene expression, but also by the bacteria cell distribution in the soil. It is assumed that the concentration of bacteria exhibits an exponential reduction along with soil depth [38]. The number of bacteria cells at the top of the soil has been measured in Section 3.1.1, and then by applying this relationship, the bacteria distribution in the soil will be calculated in-depth. In the horizontal direction, the bacteria distribution is assumed to be homogeneous. Previous studies on MICP modelling have shown that the simulation can be improved by experimental data [39]. Experiments with _S. pasteurii_ for soil improvement have been set up at a small scale in the lab allowing us to investigate the bacterial cell distribution and ureolysis rate. The number of bacteria cells at the top of the soil column has been measured in CFUs as \(9\times 10^{4}\) per gram of soil. The relationship mentioned in [38] is utilised here to calculate the bacteria distribution and Figure 4 shows the majority of bacteria only present in the top 10 m of soil. It is worth mentioning that here, the bacteria distribution in Soil 2 and Soil 3 are assumed in accordance with this relationship as well. Figure 3: Gene expression data and proposed behaviour for three pressure-responding genes (the x-axis represents the pressures applied to the samples and the y-axis is a relative unit compared to the ERCC ExFold RNA Spike baseline in sequencing control). To calculate the rate of urea hydrolysis, the peak rate--which occurs at the point of urease saturation--was used to calculate the rate which was 1310.96 mM per hour per OD\({}_{600}\). This was converted to a per-cell rate using a standard curve showing a relationship between CFUs and OD\({}_{600}\), to give a final urea hydrolysis rate of 1.03 \(\times\) 10\({}^{-6}\) mM per CFU per hour. In the model, we assume the maximum gene expression level produces the maximum urease activity.