Constructing confidence intervals for the mean mutation rate in cultures of cells that have a gamma-distributed cell cycle time

Mutant distributions in cultures in which the cells have a gamma-distributed cell cycle time are not amenable to the same type of parameterization that I described in section . Thus, at the moment we do not have an expression for these distributions that we could use in estimating the mutation rate using the above method. We have, however, explored the behavior of the mutant distributions that we obtain from our simulations. We found some properties that allow us to construct a confidence interval based on the observed mean proportion of mutants. The approach is the following.

We construct the mutant distribution empirically, through simulation. The limiting factors are the running time and the memory taken up by the cell objects. Constructing a culture of size N requires N-1 division events. The problem for large culture sizes is two-fold. Not only does it take longer to simulate the culture growth, but also the number of independent runs that we would have to do to obtain an accurate distribution becomes larger.^6.1 Cultures of size 10⁴-10⁵ can, nonetheless, be simulated on the currently available workstations. Thus, for the germinal center reaction, in which the number of cells does not surpass $2\times10^4$, we can still simulate the growth of the cell population.

 

As I mentioned before, when the cell cycle time is gamma-distributed, only the order parameter of the gamma distribution determines the relative probability of realizing different genealogical trees. I denoted this parameter by q. For a given q, I generated 10⁴ independent runs for each value of N (either 10⁴ or 10⁵ cells) and each value of the mutation rate. I then investigated the behavior of the quantiles of the distribution of the proportion of mutants. Let x_0.05 and x_0.95 denote the 5 and 95 quantiles, respectively, and $\bar{x}$ the mean proportion of mutants. It turns out that the quantities $\Delta_L = \bar{x} - x_{0.05}$ and $\Delta_U = \bar{x} - x_{0.95}$ are quite similar for the two values of N. Furthermore, they are well approximated by linear functions of $\mu$, for a given order of the cell-cycle time distribution, q.

 

Table 6.4: Linear regression of $\Delta_L$ and $\Delta_U$ as functions of the mutation probability.

Variable	Slope(S.E.)	Intercept(S.E.)	Correlation
			coefficient
$\Delta_L$	9.181502 (0.1760939)	0.0006655711 (0.0002685078)	0.997435
$\Delta_U$	23.63253 (0.1867707)	-0.0001423919 (0.0002847877)	0.9995631

 

The information on the linear regression is given in Table . How do we use these findings to construct the 1-a confidence interval for the mutation rate? If we write our linear fits as

$$\Delta_U(\mu) \delta_U \mu + O(\mu^2)$$ = (6.39)

$$\Delta_L(\mu) \delta_L \mu + O(\mu^2)$$ = (6.40)

then the 1-a confidence interval on $\mu$ is given approximately by the bounds

$$\mu_L = \frac{x}{c \log(N) +\delta_U}$$ (6.41)

and

$$\mu_U = \frac{x}{c \log(N) -\delta_L},$$ (6.42)

where x is the observed proportion of mutants in the culture and c is the correction factor for the mean, given by the cell cycle parameters.