Cell division, cell cycle times

While trying to understand the dynamics of affinity maturation of B cells during the germinal center reaction, Tom Kepler and I decided to look at some simple models of cell culture growth. We started by assuming that all genotypes are selectively neutral, and wanted to estimate the mutation rate in the culture from the proportion of mutants found at the end of the culture period. I wrote a simulation of such a growing culture, based on the assumption that each cell has a certain life time, at the end of which it divides, and each of the two progeny has a probability $\mu$ of being a mutant. We soon realized that the currently used method for mutation rate estimation, the fluctuation analysis of Luria and Delbrück (1943), gives an incorrect estimate of the mutation rate in any realistic culture of cells. The reason is that the Luria-Delbrück (L-D) distribution assumes that all cells in the culture have a constant probability of dividing at all times, which amounts to assuming that their cell cycle time is exponentially distributed. This is clearly wrong for any type of cell. This assumption, however, is necessary to make the distribution mathematically tractable. The bias, assuming a gamma-distributed cell cycle time, can be as high as 30%. That the L-D probability function can be in significant disagreement with the data for more realistic cell-cycle time distributions was first pointed out by Kendall (1952). His rigorous treatment of the problem, however, led to intractable coupled nonlinear integral equations.

What the distribution of cell cycle times is for a particular type of cell is generally unknown, and clearly depends on a variety of external circumstances. Starved cells may persist without dividing for long periods of time (1996), while cells that are placed in a chemostat, with abundant supply of nutrients, continue to divide for long periods of time (1996). There are two classes of cell-cycle time distributions that have so far been considered in modeling the experimental data. Most of the distributions can be obtained from the gamma distribution, using different parameterizations. Early studies of bacterial cell growth, for example, make this assumption (1932), and possible interpretations of it are discussed by Kendall (1952). More recent models of the cell cycle arrive at different distributions of interdivision times. Smith and Martin (1973), for example, introduced a 2-phase model of mammalian cell cycle. According to this model, cells in G1 phase of the cell cycle are viewed as being in a state A, from which they have a constant rate per unit time, $\lambda$, of transition to phase B. Phase B corresponds to the replication phase of the cell cycle, and is assumed to take a constant time, T_B. Another transition point has been later incorporated in this model (1980), and variants of it with a variable B phase have also been proposed (1981). In my simulations, I explored both the case of gamma-distributed cell cycle times, and the case of the 2-phase cell cycle.

In this chapter I will present improved methods for estimating mutation rates and constructing confidence intervals, that take into account the cell-cycle time distribution. These methods are valid for the parameter regime of $\mu N$, the product of the mutation rate and culture size, being larger than 0, while $\mu \log(N)$, the probability of an individual cell being mutated, being much smaller than 1. This parameter regime covers a large range of experimental systems, while not being addressed by the extant methods of mutation rate estimation. In particular, it covers the germinal center reaction. Although we do not have a general form of the mutant distribution for any type of cell-cycle time distribution, we have reached a number of important goals:

• We have a 0^th order estimate of the mutation rate using the mean proportion of mutants in a set of parallel cultures. This method can be used for all the cell-cycle time distributions that we encountered.
• We have a continuum approximation for the Luria-Delbrück distribution which can be calculated more easily than the discrete distribution. The experiments that are designed for mutation rate estimation seem to fall largely in the parameter range for which the continuum approximation holds (see for example Lea and Coulson (1949)).
• We found a way to parameterize the continuum Luria-Delbrück distribution for cultures of cells that have a 2-phase cell cycle. The parameters depend only on the cell-cycle time distribution, and are independent of the mutation rate and cell culture size. This allows us to design a general method for constructing confidence intervals for the mutation rate in this type of cell cultures.
• We found that, if the cell cycle time is gamma-distributed with a given shape parameter, the 5 and 95 percentile values of the proportion of mutants in cultures of known size scales linearly with the mutation rate. I estimated the parameters for the linear fit for cell culture sizes in the range of 10⁴-10⁶ cells. This is below the experimental range of bacterial culture sizes, but it approaches the culture size for eukaryotic cells. I show how to construct the confidence interval for the mutation rate in this parameter range. However, we are continuing this work, with the goal of finding a culture-size independent method of mutation rate estimation for cultures of cells with gamma-distributed cell cycle times.

Given these results, we are in the position of improving the methodology of mutation rate estimation.

I will first describe the computational model that I designed for testing our theoretical predictions, and for fitting the parameters of the generalized continuum Luria-Delbrück distribution. I will then outline the derivation of the mean proportion of mutants in a culture of a given size, and I will show that the theoretical predictions are well fitted by simulation data.

I will next introduce the continuum approximation of the Luria-Delbrück distribution, due to T. Kepler (Kepler & Oprea, in preparation). This represented the basis for most of our further explorations. I will describe the parameterization that we designed for extending this distribution to fit the simulation data for 2-phase cell cycle times.

I will then show that for gamma-distributed cell cycle times, the 5 and 95 percentile values of the distribution of the proportion of mutants scales linearly with the mutation rate. Thus, even for a culture of cells with gamma-distributed cell cycle, as long as the culture size is not larger than 10⁶, we can still construct confidence intervals for the mean.

Finally, I conclude with a discussion of other issues that arise in estimating mutation rates in bacterial cultures and germinal centers, and I propose ways to circumvent these problems.