Biometrics 61, 199–207 March 2005 A Stochastic Model to Analyze Clonal Data on Multi-Type Cell Populations Ollivier Hyrien, 1, ∗ Margot Mayer-Pr¨ oschel, 2 Mark Noble, 2 and Andrei Yakovlev 1 1 Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, New York 14641, U.S.A. 2 Department of Biomedical Genetics, University of Rochester Medical Center, Rochester, New York 14641, U.S.A. ∗ email: Ollivier Hyrien@urmc.rochester.edu Summary. This article presents a stochastic model designed to analyze experimental data on the devel- opment of cell clones composed of two (or more) distinct types of cells. The proposed model is an extension of the traditional multi-type Bellman–Harris branching stochastic process allowing for nonidentical time-to- transformation distributions defined for different cell types. A simulated pseudo likelihood method has been developed for the parametric statistical inference from experimental data on cell clones under the proposed model. The method uses simulation-based approximations of the means and the variance–covariance matri- ces of cell counts. The proposed estimator for the vector of unknown parameters is strongly consistent and asymptotically normal under mild regularity conditions, while its variance–covariance matrix is estimated by the parametric bootstrap. A Monte Carlo Wald test is proposed for the test of hypotheses. Finite sample properties of the estimator have been studied by computer simulations. The model and associated methods of parametric inference have been applied to the analysis of proliferation and differentiation of cultured O-2A progenitor cells that play a key role in the development of the central nervous system. It follows from this analysis that the time to division of the progenitor cell and the time to its differentiation (into an oligodendrocyte) are not identically distributed. This biological finding suggests that a molecular event determining the type of cell transformation is more likely to occur at the start rather than at the end of the mitotic cycle. Key words: Branching stochastic process; Monte Carlo test; Multi-type cell clones; Oligodendrocytes; Pseudo likelihood; Simulation-based inference. 1. Introduction Over the years, branching stochastic processes have become firmly established as a theoretical means that is adequate for the analysis of clonal growth of cultured cells. Because the choice of a particular model is frequently determined by its tractability, the Bellman–Harris branching process and its minor modifications have been traditionally considered as a fairly general framework for such studies. The multi-type ver- sion of this process is defined as follows. Let Z (i) k (t), i, k = 1, ... , K, be the number of cells of the kth type at time t given the clonal growth starts with a single (initiator) cell of type i at time t = 0. The vector Z (i) (t)=(Z (i) 1 (t), ... , Z (i) K (t)) is said to be a Bellman–Harris branching stochastic process with K types of cells if the following conditions are met. Each cell of type k,1 ≤ k ≤ K, transforms into j 1 , ... , j K , daughter cells of types 1, ... , K, respectively, with probability p k (j 1 , ... , j K ). The time to transformation is a nonnegative random variable (r.v.) with cumulative distribution function (c.d.f.) F k (·). The cells pertaining to the same clone, as well as different cell clones, evolve independently of one another. Various examples of such branching processes and their biological applications can be found in Jagers (1969, 1975), Nedelman, Downs, and Pharr (1985), Yakovlev and Yanev (1989), Yakovlev, Mayer- Pr¨oschel, and Noble (1998), von Collani et al. (1999), Zorin et al. (2000), Boucher et al. (1999, 2001), and Hyrien et al. (2001), to name a few. In the above model, each cell has a lifespan distribution that depends on its type but not on the types of its progeny. A natural extension of this model would require introducing several lifespan distributions (one for each type of transforma- tion) for each cell type. One example that motivates such an extension is given by the phenomenon of generation of oligo- dendrocytes, the myelin-forming cells of the central nervous system (CNS), from dividing oligodendrocyte-type-2 astro- cyte (O-2A) progenitor cells under in vitro conditions (Raff, Miller, and Noble, 1983; Noble, Fok Seang, and Cohen, 1984). These bipotential precursor cells can be purified from optic nerves and other regions of the embryonic, postnatal, and adult rat CNS (Power et al., 2002). When stimulated to di- vide by platelet-derived growth factor or by purified cortical astrocytes, O-2A progenitor cells generate oligodendrocytes in tissue culture (Noble et al., 1984, 1988). Oligodendrocytes are terminally differentiated cells that never divide. There is no “resting” stage in the life cycle of progenitor cells, and 199