Oligodendrocytes produce the myelin sheaths that enwraps axons in the central nervous system. They are involved in signal propagation along the nerves and are particularly essential for normal impulse conduction. Some demyelinating diseases, such as multiple sclerosis, have been shown to be associated with injury or dysfunction of the oligodendrocyte population (
Blakemore, 2008;
Korn, 2008). The biology of oligodendrocytes and their lineage is well documented in the literature (e.g.
Franklin and Ffrench-Constant, 2008). Oligodendrocytes are assumed to be nonproliferating, terminally differentiated cells, and are generated from their immediate precursor cells, the O-2A progenitor cells. These O-2A progenitor cells have the ability to self-renew by dividing, to differentiate into oligodendrocytes, or to die. Due to the critical role of oligodendrocytes, the proper regulation of the differentiation of O-2A progenitor cells into oligodendrocytes is therefore particularly important for the normal functioning of the central nervous system. O-2A progenitor cells have been shown to respond to the presence of various environmental signals or factors that may modulate their ability to self-renew and differentiate, and one of the primary objectives of the research performed on the oligodendrocyte lineage is to identify factors that could restore the normal balance in the generation of oligodendrocytes.
NT-3 is one member of a family of growth factors known as neurotrophins. Althaus
and others (2008) gave an overview on neurotrophins and their effects on oligodendroglial cells. It has been demonstrated that, in rodents and humans at least, NT-3 may be synthesized by oligodendrocytes. Although NT-3 has been shown to be involved in the growth, differentiation, and survival of neurons (
Zhou and Rush, 1996;
Kalb, 2005), its role (and that of other neurotrophins as well) on the regulation of nonneuronal cells of the central nervous system is still unclear.
In an attempt to better understand the effect of NT-3 on the regulation of the processes of division and differentiation of O-2A progenitor cells, we conducted the longitudinal clonal experiments presented in Section 2.2. In particular, the objectives of this experiment were to assess the hypothesis that the presence of NT-3 alone (i.e. without coexposure to PDGF) in pure cultures has no effect on the proliferation and differentiation of O-2A progenitor cells, as suggested by
Barres and others, 1994, and to gain some quantitative insight into how NT-3 may alter division and differentiation of O-2A progenitor cells.
Histograms for the number of O-2A progenitor cells and for the number of oligodendrocytes per clone with and without NT-3 suggested a striking difference between these 2 experimental conditions in respect to the number of O-2A progenitor cells. When cells were cultured without NT-3, the number of O-2A progenitor cells appeared to increase over time, suggesting that these cells maintained some capacity for self-renewal after 6 days of culture even in the absence of PDGF. By day 6, more than 60% of the clones still contained at least one O-2A progenitor cell, and the clones contained on average 1.9 O-2A progenitor cells and 2.1 oligodendrocytes. In contrast, in clones grown in the presence of NT-3, O-2A progenitor cells underwent division first, just as when they were cultured without NT-3, and then appeared to differentiate massively. By day 6, less than 20% of the clones still contained at least one O-2A progenitor cell. The mean numbers of oligodendrocytes per clone (~eq1.9) was not much different than that without NT-3, but the number of O-2A progenitor cells was considerably lower (~eq0.8).
We investigate further the effects of NT-3 on the processes of division and differentiation of O-2A progenitor cells to determine whether NT-3 affected the decision of O-2A progenitor cells to divide or differentiate the timing of cellular division or the timing of cellular differentiation, or all of the above. Changes in cell fate would be seen through either an increase or a decrease of the probability of division of O-2A progenitor cells, whereas a modification of the time it takes for O-2A progenitor cells to divide or to differentiate would be reflected by changes in the parameters of the distribution of the time to division and/or of the distribution of the time to differentiation of these cells.
We fitted our branching process model using the proposed simulated composite likelihood approach. The ultimate parameter estimates were obtained using 15 000 simulated clones to approximate the distribution of the number of cells of each type per clone under the above model. We started from multiple initial values and let optimization run for more than a day each time. We used the conditioning sets
I_{ij} =
,{1},{1,
j∧2},…,{1,2,…,
j∧5}). In the last case, we obtain a simulated maximum likelihood estimator. The model was fitted separately to the data obtained with and without NT-3 because none of the model parameters were common to both experimental conditions.
In our experiments, the follow-up period was long enough so all initiator precursor cells had either divided or differentiated by the time the experiment ended. By determining the composition of a clone, it is possible to decipher whether the corresponding initiator cell divided or differentiated (in the latter case, the clone always contains a single oligodendrocyte). Thus, let N_{div} and N_{diff} = n − N_{div} denote the numbers of initiator cells having divided and differentiated by the end of the experiment, respectively. For every 0 ≤ j_{1} < j_{2}, let N_{div}(j_{1},j_{2}) denote the number of initiator cells having divided between day j_{1} and j_{2}. Define the vector N_{div} = {N_{div}(t − 1,t);t = 1,…,6}. Conditional on N_{div} = N, N_{div} follows a multinomial distribution with parameters N and {p_{1}(m_{1},σ_{1}),…,p_{6}(m_{1},σ_{1})}, where p_{t}(m,σ) = G_{1}(t;m,σ) − G_{1}(t − 1;m,σ) denotes the probability that any initiator cell that will ultimately divide, completes its cycle between day t − 1 and t. The estimates for m_{1} and σ_{1} can therefore be computed by maximizing the multinomial likelihood function associated with the vector of counts N_{div}. The mean and variance of the time to differentiation of O-2A progenitor cells can be estimated in a similar fashion for clones starting with an initiator cell that ended up differentiating. We used these maximum likelihood estimates as starting values in our composite likelihood analyses.
reports parameter estimates resulting from these analyses. Also reported are the total numbers of mismatches for the final parameter estimates. In our analyses of cell clones cultured without NT-3, the number of mismatches remained low (0-1) until we started to condition upon the most recent 3 (or more) observations. For cell clones exposed to NT-3, the number of mismatches increased substantially when we started to condition upon the most recent observations. In either case, the largest number of mismatches we encountered ranged between 6 and 9, which is relatively high compared to the number of cell clones observed for each experimental condition (
n = 20). Alternatively, we could have increased the number of simulations to reduce the number of mismatches, but each analysis required between 1 and 2 days. We therefore discuss results obtained with the marginal composite likelihood (no conditioning) for cells exposed to NT-3, and with the composite likelihood conditioning upon the most recent 2 observations for control (unexposed) cells. Standard errors for the corresponding parameter estimates were obtained using a parametric bootstrap approach. We compared parameter estimates across treatment groups using a Wald test. The fitted marginal distribution for the number of O-2A progenitor cells and for the number of oligodendrocytes per clone is displayed in as solid lines. The estimated probabilities of division of O-2A progenitor cells are presented in (panels A and B) as a function of the cell generation for each culture condition. These analyses pointed out the following conclusions:
- (a)In each experiment, the probability of division of O-2A progenitor cells was found to decrease when the number of divisions increased. This is consistent with the results of previously analyzed data sets (Yakovlev, Boucher, and others, 1998, Yakovlev, Mayer-Pröschel, and others, 1998, Yakovlev and others, 2000; von Collani and others, 1999; Boucher and others, 1999, Boucher and others, 2001; Zorin and others, 2000; Hyrien and others, 2005a, Hyrien and others, 2005b, Hyrien and others, 2006).
- (b)Exposure to NT-3 decreased the probability of division of O-2A progenitor cells. For instance, cells of generation 3 divided with a probability estimated as 0.35 (vs. 0.53) when cells were exposed (vs. unexposed) to NT-3. This may explain that the numbers of O-2A progenitor cells were much smaller in the presence of NT-3. The difference between the probabilities of division across treatment groups was almost significant (p = 0.07).
- (c)In both culture conditions, the mean time to division was shorter for initiator O-2A progenitor cells than for O-2A progenitor cells of subsequent generations (40 vs. 50 h with NT3, and 31.5 vs. 44 h without NT3). This is consistent with results from previous studies based on time-lapse experiments (Hyrien and others, 2006).
- (d)Exposure to NT-3 increased the mean time to division of O-2A progenitor cells (e.g. 50 vs. 44 h for cells of generation greater than or equal to 2) and decreased their mean time to differentiation (36 vs. 44 h). When comparing the distribution of the time to division across treatment groups using the Wald test, we found that the difference was significant (p < 0.0001). The effect of NT-3 on the time to differentiation was almost significant (p = 0.05).
| Table 3.Estimates of the model parameters (means and standard deviations expressed in hours). See text for explanations. CL_{k} = simulated composite likelihood with I_{ij} = {1, 2…, j ∧ k}; ML = simulated maximum likelihood; s.e. = standard errors |
The overall conclusion of our analysis is that NT-3 may modulate the proliferation of O-2A progenitor cells and their differentiation into oligodendrocytes, and that it does not act only in concert with PDGF. In particular, we found that NT-3 induces differentiation of O-2A progenitor cells into oligodendrocytes by decreasing the probability of division, by increasing the mean time to division, and by decreasing the mean time to differentiation. The results of our analysis are tentative and should be confirmed in other experiments.
The effect of NT-3 may be perceived as subtle when expressed as a change of the probability of division, of the distribution of the time to division, and of the distribution of the time to differentiation. To better appreciate the impact of such changes on the composition of cell clones over time, we simulated the fitted branching process over 20 days and calculated the mean number of O-2A progenitor cells and the mean number of oligodendrocytes cultured with and without NT-3 as a function of time (). While the number of O-2A progenitor cells declines under either culture conditions (as expected from the theory of branching processes since lim_{k→∞}p_{k} < 0.5), the number of oligodendrocytes differed substantially. For instance, by day 20, the number of oligodendrocytes was 2-fold larger in untreated cells. This seems to contradict our conclusion that NT-3 induces differentiation, but it does not actually. Indeed the best strategy to increase the number of oligodendrocytes over a medium-term period is to first increase the pool of progenitor cells by self-renewal, and then have some of these progenitor cells differentiate into oligodendrocytes. This seems to be what happened when cells were not exposed to NT-3 compared to when they were.
The mean numbers of oligodendrocytes in clones exposed and unexposed to NT-3 drifted apart from each other after day 6 only. This would explain why our permutation test did not detect any significant difference between day 0 and day 6 in counts of oligodendrocytes.