Motivation
Our project rests on the premise that our genetically modified organism — Escherichia coli, is able to replicate and survive in the sugarcane system. Fortunately, we were able to perform the experiments required to support these claims in a laboratory setting. As it would have been rather difficult to measure the growth of E. coli directly inside the sugarcane matrix, we extracted the juice from a sugarcane plant and cultured our bacteria in this medium and performed optical density (OD at 600nm) measurements on the liquid at one hour intervals of time. This experiment was performed three times, and therefore, three sets of optical density measurements were obtained.
Experimental Data
The optical density was subsequently converted to population size (in the units cfu/mL) using the conversion ratio \( 1 \text{ OD} \) = \( 10^{-6} \text{cfu/mL} \). Then, the data was converted to the semi-log domain by calculating the logarithm (base 10) of the optical density data while retaining the time series data as it is.
Figure 1: A scatterplot showing the variation of OD versus time (three runs)
We calculated the average of the OD data for the three runs and used that data for further analysis.
Mathematical Model
We selected a simple logistic equation to model the growth of the bacteria. More sophisticated models that model bacterial growth such as Gompertz, Stannerd or Richards exist Zwietering et. al, 1990, but we felt that a logistic model best suited our data and purpose.
Developed in 1838 by Belgian mathematician Pierre-François Verhulst, a logistic model makes the assumption that the rate of population growth is limited and is dependent on the size of the population at any particular instant of time. Specifically, the rate of growth \( \mu \) is equal to
$$ \mu = \mu_{\text{max}}\left(1 - \frac{N}{N_{\text{max}}}\right) $$
Since the rate of population growth is directly proportional to the population size at that given time Verhulst, 1838, we obtain the following differential equation governing the growth.
$$ \frac{dN}{dt} = \mu N $$
$$ \frac{dN}{dt} = \mu_{\text{max}} N \left(1 - \frac{N}{N_{\text{max}}}\right) $$
Here, \( \mu_{\text{max}} \) is the maximum possible rate of population growth and \( N_{\text{max}} \) is the carrying capacity of the medium. Solving the above ODE, we obtain the following closed-form solution for the population size as a function of time:
$$ N(t) = \frac{N_0N_{\text{max}}}{N_0 + (N_{\text{max}} - N_0)\cdot exp(-\mu_{\text{max}} t)} $$
We initially solve the ODE using SciPy's solve_ivp method with an initial guess of the parameter \( \mu_{\text{max}} \) set to 1. We do not consider the carrying capacity \( N_{\text{max}} \) to be a variable parameter and set it to \( 1 + 10^{-6} \) times the maximum population size in the data series obtained from experiment.
Once a solution is obtained with these parameter values, we run an optimiser using SciPy's minimize method to modify the values of the parameter \( \mu_{\text{max}} \). The cost function used to minimise the values of these parameters was the least-squares sum between the predictions and the actual population size (also known as the \(\ell^2 \) norm). This process is performed to fit our growth curve to the data as neatly as possible. Once a minimum value for the parameter \( \mu_{\text{max}} \) is obtained, we plot the final curve. Fitting the data to our curve, we obtain the following growth function:
$$ N(t) = \frac{5.1644}{1 + 8.8751\cdot exp(-1.2017t)} $$
Figure 2: A graph of the function that models bacterial growth in the sugarcane.
Insights
Optical density (OD) is used by us to estimate the density of cells our culture, however it cannot be compared between instruments without a standardized calibration protocol and is challenging to relate to actual cell count.
To this end, in Phase 2 we plan on calibrating OD to estimated cell count using serial dilution of silica microspheres, which produces highly precise calibration (95.5% of residuals <1.2-fold), is easily assessed for quality control, also assesses instrument effective linear range, and can be combined with fluorescence calibration to obtain units of Molecules of Equivalent Fluorescein (MEFL) per cell, allowing direct comparison and data fusion with flow cytometry measurements as outlined in Beal et al.
With this model (and experiment), we have shown that our organism is able to survive and replicate within the sugarcane matrix with a nontrivial growth rate such that it reaches a sufficiently large population size. We have also obtained a closed-form growth function that models the growth of our bacteria in the sugarcane system.