Team:Vanderbilt/Modeling/Populations
Population Dynamics
The metabolic load of a genetic circuit is of importance for optimizing said circuit. The more load that a circuit has on it, the more likely it is to mutate, and make all work done to optimize and create that circuit null. Therefore it is of importance to see how a circuit with a reduced metabolic load will function in population of cells. Take a population of cells that have been engineered to produce a specific protein, and these cells are under a very heavy metabolic load. Now, introduce a small separate population of cells into this original population, and these cells have a reduced metabolic load. It is generally safe to make the assumption that metabolic load and cell fitness are inversely correlated, i.e. as load increases, fitness decreases and vice-versa. One would assume, from basic cellular biology, and logical reasoning that this smaller population of cells, with a higher level of fitness, could quickly become the dominant cell type in the population. But how quickly does this happen? What factors does it depend on?
To model cell type population dynamics, the Price equation was implemented. The Price equation is a recursive model based on four factors (for our model); cell type 1 population size, cell type 2-population size, cell type 1 fitness level, and cell type 2 fitness level. With these four inputs, the recursive Price equation can predict the percentage of the cell population taken over by the mutant cell type. For our model, we use a simplified version of the Price equation that keeps fitness levels constant from generation to generation. This is motivated in the thought that while there are selective pressures in the environment, those selective pressures were the exact cause of the increased fitness level (reduced metabolic load).
In the model, the mutant cell type has a slightly higher fitness level than the control population, on the level of we approximate one and five percent. The crux of the Price equation is that the fitness level is proportional to how efficiently a cell will reproduce. For example, a cell type with fitness level of f = 1 would reproduce at a rate of 1 cell per generation. Ratio of cell deaths to births is equal, and the total population stagnates at a certain equilibrium level. This is commonly called the carrying capacity of the ecosystem. However, introduce a mutant cell with fitness level f = 1.5, and this cell type will quickly grow past the equilibrium level of the ecosystem, as it is able to more efficiently use its resources, and thus divide faster. The mutant population will be at a population level at generation n of P0 * f (n-1). Therefore the growth model is exponential, and once the mutant cell type gains steam it can easily take over the population.
Figure 1 shows the effect of a density dependent scaling factor on the Price Equation. We included a scaling factor that is a ratio of the two cell populations. It was unclear to us whether or not the scaling factor would make a difference, but as you can see the solid line(with scaling factor) out grows the dashed line (without scaling factor) once it reaches a certain population density.
In the competition graph models we see the downward sloping curve represents the population of the wild type, and the upward sloping curve represents the population of the mutant type. As you can see, there is a point where the two lines intersect each other. We call this point the break-even point, and it represents the point at which the mutant population and the wild type population are equal. In most cases, the number of generations it takes to go from the break-even point to the point at which the mutant cell line takes over is on the order of between 10 and 20 generations, or roughly 1-2 days. This is of utmost importance to notice. Empirical data gathered by our team suggests that this is consistent with the model, in that the time from break-even to total take over is roughly on the order of 1-2 days (for an RFP expressing circuit). Furthermore, this suggests that the fitness advantage gained by losing the functional RFP gene is near or around 5%. This is of utmost importance, in that it gives us a model to approximate what the level of fitness gained would be for the loss of a certain circuit or gene sequence.
Figure 2 shows both populations simultaneously, wild and mutant type. As should be plainly obvious, the populations grow inversely to each other i.e. for every increase in the mutant population there is a corresponding decrease in the wild type population. The point at which the wild type and mutant type are equal is size is what we are calling the break-even point. The various colors represent different levels of fitness advantage. The higher a fitness advantage, the less time it takes a cell population to reach the break-even point.
Figure 3 is a more zoomed in version of the previous figure. It shows how quickly a cell population can go from the break even point to completely dominating the population. Within the scale of between 10 and 20 generations a mutant cell population can go from the break even point to overwhelming the wild type. So on the order of magnitude of days, the makeup of a cell population can change.
Figure 4 shows the relationship between fitness advantage and the number of generations it takes for the mutant cell type to overtake the wild cell type. Each data point represents the time at which a mutant type cell with that level of fitness advantage will overwhelm the wild cell type. This graph shows that there is a very clear exponential decay equation governing this relation. An exponential fit was used to determine the exact parameters of this equation, and it was found to be f(x) = (4*10^16) * e^(-31.45x). And this equation has an R^2 VALUE OF 0.9372.
