The goal of our project is to explore radiation resistance genes to find ways to increase UV tolerance. With a changing climate, organisms grow under increasing stress of DNA damage. The overall goal of our model is to evaluate the resistance of organisms to UV and compare the growth of resistant and non-resistant cells under different levels of UV. This modeling aims to assess the commercial advantage of resistant cells compared to normal cells by predicting their growth under UV radiation. To obtain useful parameters, our models use the well-described growth and radiation death of E. coli.
Modeling Cell Death under UV
To model cell death caused by UV radiation, we made the simplifying assumption that cell death from radiation was entirely due to DNA damage. Using differential equation models for bacterial death (Kowalski et al. 2002) and DNA lesion formation and repair (Krishna et al. 2007) we could make a connection from lesion formation to cell death. The importance of this was that reducing the number of lesions by DNA repair could be linked directly to increased cell survival.
Because lesion formation is linearly related to UV intensity when there is no lesion repair, the number of lesions could be treated as a measure of UV intensity and the lowered rate of lesion formation could instead be considered as lowering UV intensity over time.
Using a differential equation model for UV bactericide, we predicted the survival of cells under a given level of UV given their DNA lesion repair rate.
Our survival function took the survival function described by Kowalski et al. and rather than calculating survival from a constant UV dose, we calculated the derivative of the survival function, evaluating the effective intensity at each point and solving the differential equation to find the survival of cells over time. Additionally, we had to correct the mathematical model to prevent survival from going up when lesion repair began to overtake lesion formation
UV Resistance Parameter Estimization
We wrote an algorithm to determine the parameter for UV resistance, lambda, from laboratory data. The algorithm begins by calculating a survival curve for an overestimate and an underestimate of lambda. The algorithm then takes all non-outlier values of the measured survival percentage from laboratory experiments after 1, 2, 3, and 4 hours of UV exposure, computes the square of the vertical difference between these values and the model, and sums those values for each of the two estimates of lambda to find the total error for each estimate. The algorithm then calculates the difference of the two error values. If the error of the lower estimate of lambda is lower, the algorithm is run recursively using the same lower estimate of lambda and a new upper estimate between the previous upper estimate and the lower estimate. If the error of the upper estimate of lambda is lower, the recursion uses the same upper estimate of lambda and a new lower estimate between the previous lower estimate and the upper estimate. The algorithm returns when the absolute value of the difference between the error values of the upper and lower estimates of lambda is below a specified precision.
The above figure shows the process of the parameter estimation algorithm. The red points are the data recorded from laboratory experiments of Dsup-expressing cells under 0.03 mW/cm^2 of UVB. The blue asterisks show the average survival at each timepoint. Each curve is the calculation of the survival curve from the upper and lower lambda estimates. The dashed black line is the survival curve of the final calculated lambda.
Cell Growth under UV
The solution to the differential equation describing cell growth without UV. A modified version of this function using the survival curve calculated by our previous model was the ultimate goal of our modeling.
The core purpose of our model is predicting growth of cells under UV to assess the benefits of our radiation resistance genes. In commercial applications, we want to show how our resistant organisms will outperform non-resistant competitors.
The function starts from the point of UV exposure and traces forward by small time intervals. At each timepoint, the current population N is calculated. The first half of the equation for the current N is the previous N plus the rate of growth (d) of a population at the size of the previous N over the time interval. The second half of the equation describes the portion of the population lost to radiation damage. The dS matrix approximates the derivative of the survival curve of the cells, which is calculated based on their radiation resistance. Each time the population increases in this model (each d[N]) these new individuals are treated as a separate population that follows the survival curve until dS reaches 0 (once the population has reached its lowest percentage survival). The sum of new cells being created and old cells dying off leads to the overall growth equation. The implementation of this and all other functions can be found at our GitHub here.
- Mathematical modeling of ultraviolet germicidal irradiation for air disinfection (2002) Kowalski, W. Bahnfleth, W. Witham, D. Severin, B. Whittam, T. Quantitative Microbiology (2000)
- UV-induced mutagenesis in Escherichia coli SOS response: A quantitative model (2007) Krishna, S. Maslov, S. Sneppen, K. PLoS Computational Biology
- A new logistic model for Escherichia coli growth at constant and dynamic temperatures (2004) Fujikawa, H. Kai, A. Morozumi, S. Food Microbiology