Correction: Randomized, Controlled Intervention Trial of Male Circumcision for Reduction of HIV Infection Risk: The ANRS 1265 Trial

Background: Quantifying cell division and death is central to many studies in the biological sciences. The fluorescent dye CFSE allows the tracking of cell division in vitro and in vivo and provides a rich source of information with which to test models of cell kinetics. Cell division and death have a stochastic component at the single-cell level, and the probabilities of these occurring in any given time interval may also undergo systematic variation at a population level. This gives rise to heterogeneity in proliferating cell populations. Branching processes provide a natural means of describing this behaviour. Results: We present a likelihood-based method for estimating the parameters of branching process models of cell kinetics using CFSE-labeling experiments, and demonstrate its validity using synthetic and experimental datasets. Performing inference and model comparison with real CFSE data presents some statistical problems and we suggest methods of dealing with them. Conclusion: The approach we describe here can be used to recover the (potentially variable) division and death rates of any cell population for which division tracking information is available. Background Quantifying the dynamics of cell populations involves measuring rates of division and death. On a practical level, knowledge of these rates can be important for the clinical assessment of diseases characterised by dysregulated cell populations such as neoplasias. Perhaps more fundamentally, quantifying cell dynamics is important for testing hypotheses regarding the population biology of cells. Studies of cell proliferation have benefited in recent years from the development of a method to measure the number of divisions single cells have undergone using CFSE (Carboxy Fluoroscein Succinimidyl Ester), a fluorescent and cell-membrane impermeable dye. CFSE is now used widely in immunology to study lymphocyte dynamics [1] but also in oncology [2], stem cell research [3,4] and to study the kinetics of bacterial division [5]. Briefly, the procedure is as follows. A population of cells is stained with CFSE, and the dye contained in each cell is shared approximately equally among daughter cells upon division. The fluorescence intensities of the population of CFSE-labeled cells can then be measured at a later time using flow cytometry. Cohorts of cells that have underPublished: 12 June 2007 BMC Bioinformatics 2007, 8:196 doi:10.1186/1471-2105-8-196 Received: 25 September 2006 Accepted: 12 June 2007 This article is available from: http://www.biomedcentral.com/1471-2105/8/196 © 2007 Yates et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Page 1 of 20 (page number not for citation purposes) BMC Bioinformatics 2007, 8:196 http://www.biomedcentral.com/1471-2105/8/196 gone the same number of divisions are usually observed to have approximately log-normally distributed intensities, with median decreasing roughly two-fold with each division. Analysis of CFSE profiles allows the estimation of the proportions of cells in culture that are in each generation. These proportions can indicate the extent of division in a population, but CFSE information can also be used to simultaneously quantify division and death if the total numbers of live cells in each generation are known at two or more timepoints. In in vitro experiments, these can be estimated by adding known numbers of fluorescent beads to the culture, sampling from it, counting both cells and beads in the sample using flow cytometry and scaling the generation proportions appropriately. The information CFSE provides regarding this generational structure augments methods of pulse-labelling with markers such as BrDU (5-bromo-2'-deoxyuridine) or tritiated thymidine, which have traditionally been used to quantify proliferation. These compounds are taken up during DNA synthesis and allow the measurement of the proportion of the population undergoing mitosis during the labelling period. This technique has been used in conjuction with mathematical models to quantify the turnover of populations that are essentially homogeneous (see, for example, [6]). Models have been used to quantify turnover from CFSE data in similar situations [7-11]. In these studies, all cells are considered to be identical, and death or entry into division are represented as Poisson processes. ODEs are usually used, providing the expected numbers of cells in each division. While these models are useful as a starting point, in their simplest form they allow for arbitrarily short inter-division times. This is a biologically unrealistic artifact which can lead to difficulties in the interpretation of estimates of average division and death rates [12]. Other CFSE modeling studies have overcome this by turning to the classic Smith-Martin model of the cell cycle [13]. In this model cells are assumed to spend exponentially-distributed times in a quiescent Aphase before progressing deterministically through an 'actively dividing' B-phase (roughly corresponding to DNA synthesis and mitosis) of finite duration. However, if different susceptibilities to death are allowed in the two phases, as might reasonably be expected given the metabolic differences between quiescence and mitosis, it has been shown that CFSE data alone is not sufficient to identify all parameters of the general Smith-Martin model [9,10], and additional information (such as the proportion of cells in each generation that are in the Aand Bphases) is required. As a further complication, it has increasingly been recognised that rates of division and death are usually not homogeneous, and that it is essential to consider this if CFSE is to be used as a practical tool for studying cell dynamics in any depth. Rates of division and death typically vary systematically at a population level. This variation might occur with the number of divisions a cell has undergone; with time, for example as the availability of nutrients, inter-cellular signalling molecules or proor anti-apoptotic factors changes over the course of an experiment; or both. Some of these issues were tackled in a series of elegant studies by Gett and Hodgkin [14], Deenick et al. [15], and the subsequent extension of their analysis by de Boer and colleagues [12,16]. They quantified the kinetics of in vitro stimulation of CFSE-labeled T cells, using a hybrid model in which entry into the first division is stochastic and subsequent divisions are deterministic. They discuss the estimation of the distribution of entry times into the first division, and showed a significant improvement in fit using a division-dependent death rate. Towards a more general approach, Leon et al. [17] proposed a framework for modeling asynchronous division with CFSE data and used this to determine the parameters of probability distributions of inter-division times, allowing for heterogeneity in cell kinetics with respect to division history. However, their analytic approach and the lack of treatment of the sources of discrepancy between model and data make the fitting and comparison of models difficult, and so limits its practical usefulness. In this paper we present a distinct and complementary method of modeling CFSE data. We use discrete-time branching processes to describe heterogeneous cell kinetics and suggest a likelihood-based method of inference. Branching processes have been applied successfully to model cell growth in many areas in biology [18-22]. In such models, cells are considered to act independently and divide and die according to probabilistic rules. In a discrete-time process a cell is assumed to either divide once, die or survive undivided in each discrete time interval (Figure 1). The method we present here has at least two advantages over existing approaches. Firstly, in many cases even timeseries of CFSE data may be insufficient to identify the parameters of more detailed models of cell division, and in some cases (as in the general Smith-Martin model discussed above) unique identification of all parameters with CFSE alone is not possible. In contrast, branching processes make minimal assumptions regarding the cell cycle – essentially, the finite timestep imposes a lower bound on the time required to complete a division – and in general all of their parameters are identifiable. In particular this allows useful dynamical information to be recovered even from limited CFSE datasets, such as a single timepoint. Secondly, the inference procedure we propose provides a statistically sound basis for model fitting. Many studies (implicitly) ascribe the discrepancies between the model and the counts of cells in each generation recovPage 2 of 20 (page number not for citation purposes) BMC Bioinformatics 2007, 8:196 http://www.biomedcentral.com/1471-2105/8/196 ered from CFSE profiles as measurement error terms of constant variance. In this paper we challenge this assumption and use a standard stochastic description of cell population dynamics, along with a more realistic treatment of the sources of discrepancy between model and data, to provide the appropriate weighting to each observation when fitting models. Specifically, when estimating parameters of stochastic models from data it is important to assess the relative contributions of fluctuations arising from the intrinsically probabilistic nature of cell dynamics and measurement error or other forms of experimental noise. In this paper we describe two frameworks for parameter estimation; one when fluctuations are the most important form of discrepancy between model and data, and the other when other forms of measurement error dominate. In the latter case, the procedure we describe in this paper can be applied to any model used to describe CFSE data that provides the expected cell counts in each generation. Using a likelihood-based estimation method requires calculating the probability (likelihood) of a set of observations arising given a model. The generating-function approach we describe allows us in principle to write an exact likelihood given a specification of a branching process model, initial cell numbers, and experimentally observed cell counts at one or more timepoints. However, this method becomes impractical when used with more than a few cells or one or two cell divisions, and is essentially impossible to apply to experimental situations which involve typically tens of thousands of cells. We propose a solution to this problem with the use of a QuasiLikelihood estimation method. This requires only the first two moments of the probability distribution of the total numbers of cells in each generation – that is, their expectation values and their variance-covariance matrix. We will show that this key simplification allows the model parameters to be inferred from CFSE information. Results In Section 1 we describe the theory underlying the parameter estimation and in Section 2 we validate it using synthetic datasets. In Section 3 we describe how to deal with statistical issues that may arise with the application of the method to experimental data, and illustrate this with an analysis of data from an in vitro T cell proliferation experiment. 1. Cell kinetics as a branching process Calculating the probability distribution of cell counts To apply a maximum likelihood method to estimate parameters of a stochastic model of cell division and death from CFSE data, we need to characterise the probability distribution of cell counts predicted by the model. In this section we outline this calculation for a general branching process model in discrete time, or a GaltonWatson process [23]. In these models, during each timestep a cell can do one of the following: divide, with probability γ; survive without dividing, with probability δ; or die, with probability 1 γ δ (Figure 1). A particular model of the kinetics of a cell population specifies these probabilities, which in the simplest case might be assumed to be constant. In general they may depend on either the number of divisions the cell has undergone (which we refer to as the generation number), explicitly on time, or both. The key assumptions are that all cells act independently, their offspring generate their own branching processes according to the same rules, and that cells retain no memory of events in previous timesteps other than the total number of divisions they have undergone. The parameters of biological interest are usually γ and α (the probabilities of division and death). However, in the formalism we use here it proves simpler to work with the quantities γ and δ (the probability of survival without A simple branching process in discrete time Figure 1 A simple branching process in discrete time. A schematic representation of a branching process. The numbers in the circles denote the generation of the cell or the number of divisions it has undergone since being labeled with CFSE. We begin with a population of undivided cells at time 0. In each timestep, each cell divides with probability γ, survives without dividing with probability δ and dies with probability α = 1 γ δ. At a later timestep t, sorting cells according to their CFSE content allows the numbers of cells in each generation to be estimated. The formalism we describe in this paper allows us to calculate the moments of the probability distribution of these counts at one timestep given knowledge of the number of cells in each generation at an earlier time. 0 1