Modeling Stem Cell Development by Retrospective Analysis of Gene Expression Profiles in Single Progenitor-Derived Colonies

Authors

  • Neal Madras,

    1. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada
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  • A. L. Gibbs,

    1. Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada
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  • Y. Zhou,

    1. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada
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  • P. W. Zandstra,

    1. Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
    2. Department of Anatomy and Cell Biology, University of Toronto, Toronto, Ontario, Canada
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  • Jane E. Aubin Ph.D.

    Corresponding author
    1. Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
    2. Department of Anatomy and Cell Biology, University of Toronto, Toronto, Ontario, Canada
    • Department of Anatomy and Cell Biology, Faculty of Medicine, University of Toronto, Room 6255 Medical Sciences Building, One King's College Circle, Toronto, Ontario M5S 1A8, Canada. Telephone: 416-978-4220; Fax: 416-978-3954
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Abstract

The process of development of various cell types is often based on a linear or deterministic paradigm. This is true, for example, for osteoblast development, a process that occurs through the differentiation of a subset of primitive fibroblast progenitors called colony-forming unit-osteoblasts (CFU-Os). CFU-O differentiation has been subdivided into three stages: proliferation, extracellular matrix development and maturation, and mineralization, with characteristic changes in gene expression at each stage. Few analyses have asked whether CFU-O differentiation, or indeed stem cell differentiation in general, may follow more complex and nondeterministic paths, a possibility that may underlie the substantial number of discrepancies in published reports of progenitor cell developmental sequences. We analyzed 99 single colonies of osteoblast stem/primitive progenitor cells cultured under identical conditions. The colonies were analyzed by global amplification poly(A) polymerase chain reaction to determine which of nine genes had been expressed. We used the expression profiles to develop a statistically rigorous map of the cell fate decisions that occur during osteoprogenitor differentiation and show that different developmental routes can be taken to achieve the same end point phenotype. These routes appear to involve both developmental “dead ends” (leading to the expression of genes not correlated with osteoblast-associated genes or the mature osteoblast phenotype) and developmental flexibility (the existence of multiple gene expression routes to the same developmental end point). Our results provide new insight into the biology of primitive progenitor cell differentiation and introduce a powerful new quantitative method for stem cell lineage analysis that should be applicable to a wide variety of stem cell systems.

Introduction

Most current models of mesenchymal stem cell and osteoprogenitor differentiation assume a hierarchical arrangement, with an expanding series of cells leading from a small number of stem cells through a larger but finite number of committed progenitors and a still larger number of mature cells with a finite lifespan [1–, 4]. Although recently the role of plasticity or transdifferentiation events of cells usually considered terminally differentiated, has been the subject of intense interest [5,, 6], there is an apparently restricted osteoprogenitor detectable in colony assays performed on cultures of bone or bone marrow cell isolates under particular standardized growth conditions. The ability to identify and isolate osteoprogenitors as pure populations at discrete and identifiable stages has been difficult and lags behind what is possible in certain other lineages (e.g., hematopoietic). Thus, a retrospective assay based on differentiated cell function/phenotype—a colony assay—combined with other biochemical and molecular assays has become widely used and has contributed substantially to the understanding of osteoblast development [5]. Briefly, bone nodule or bone colony formation represents the end-stage of a proliferation-differentiation sequence of a low frequency of cells within unfractionated primary populations of bone-(e.g., rat calvaria [RC]) or bone marrow-derived cell cultures [7–, 12].

Morphologically and molecularly recognizable osteoblasts and bone nodules form in such cultures at predictable and reproducible times after plating [5,, 13]. The process is often subdivided into three stages: A) proliferation; B) extracellular matrix development and maturation, and C) mineralization, with characteristic changes in gene expression at each stage [14]. Expression of osteoblast-associated genes (e.g., type I collagen [COLL I], alkaline phosphatase [ALP], osteopontin [OPN], bone sialoprotein [BSP], osteocalcin [OCN], parathyroid hormone/parathyroid hormone-related protein [PTH/PTHrP] receptor [PTH1R] and others) is asynchronously acquired and/or lost as the progenitor cells differentiate and matrix matures and mineralizes. In general, ALP is thought to increase then decrease when mineralization is well progressed, OPN is high during both proliferation and differentiation stages and is upregulated prior to certain other matrix proteins, including BSP and OCN, with OCN first appearing in postproliferative osteoblasts approximately concomitant with mineralization [9,, 15] (Fig. 1A). However, many of these conclusions are based, in large part, on models in which differentiation and bone nodule formation are not synchronous throughout the population, and many of the available osteoblast markers used are not restricted to osteoblasts but may be expressed at substantial levels by other cells (e.g., fibroblasts) in the population, leading to discrepancies in the reported osteoblast developmental sequence in the published literature [5,, 16]. In addition, although limiting dilution analysis of bone-derived cell populations, such as those used here (i.e., the RC cell model), has suggested that osteoprogenitor cell differentiation/bone colony formation is a single-hit phenomenon with only one cell type (likely the osteoprogenitor cell) limiting [7], we have also reported evidence that osteoblast differentiation in this model appears to have a stochastic component [9,, 15,, 17].

Figure Figure 1..

Models of the osteoblast developmental sequence.A) The established model of osteoblast development follows a linear deterministic sequence in which expression of osteoblast-associated genes (numbered 1-9; see text for definitions) is acquired sequentially as differentiation proceeds. B) A new model based on our analysis. Each shaded circle represents an event in the development of a typical osteoblast colony. A circle containing an unbracketed number denotes the activation of the gene with that label (Table1). An arrow from event i to event j indicates that event i always precedes event j. The arrow is dashed if there is some evidence for precedence, but it is statistically inconclusive; shorter dashes correspond to less statistical evidence. Bracketed numbers denote genes that must have been activated previously. A circle's horizontal (left to right) position roughly corresponds to the average time at which its event occurs. The circle marked with x represents an early event, perhaps related to expression of gene 1. Genes 3, 4, and 6 are activated independently (and “exchangeably”) after this early event. Genes 2 and 5 are activated independently of all other genes analyzed.

Table Table 1.. The nine genes studied in this paper with their reference numbers
  1. a

    *n of colonies (out of 99) in which each gene's expression was detected.

  2. b

    †Average normalized level of expression (see Materials and Methods) among those colonies for which the expression was detected.

LabelGenenof occurrences*Average level of expression†
1OPN994.385
2PDGFRα750.030
3FGFR1740.120
4PTH-R760.057
5PTHrP800.041
6COLL750.948
7ALP510.138
8BSP641.565
9OCN300.257

To clarify the osteoblast developmental sequence, we have been applying molecular methods to single osteoprogenitor cell colonies at different stages of their differentiation and maturational sequences [8,, 15]. Our results suggest that the number of molecularly recognizable transitions through which an osteoprogenitor progresses as it matures is greater than the three stages summarized above (a minimum of seven stages is evident [5,, 8,, 15,, 17]). To date, however, we have analyzed these single-colony gene expression profiles using accepted developmental paradigms that include a unilinear or deterministic sequence of gene expression patterns. In this paper, we reassessed osteoblast development using an approach that does not require the application of established developmental paradigms. We analyzed 99 isolated colonies of osteoblast stem/primitive progenitor cells cultured for a fixed length of time under identical conditions by a combination of replica plating [18] and global amplification poly(A) polymerase chain reaction [19] to determine which of nine genes had been expressed. Our goal was to use these measurements to elucidate the dynamics of gene expression during clonal osteoprogenitor development and to construct a model for the developmental history of osteoblast colonies. Individual variations in the rates of development among the clones allowed us to include a stochastic component in a model that otherwise used a single end point. We have developed a statistically rigorous map of the cell fate decisions that take place during osteoprogenitor differentiation and show that different developmental routes can be taken to achieve the same end point phenotype (Fig. 1B). Our results provide new insight into the biology of primitive osteoprogenitor cell differentiation and introduce a powerful new method for stem cell lineage analysis that should be applicable to a wide variety of stem cell systems.

Materials and Methods

Cell Culture, Osteoblastic Colony Isolation, and cDNA Preparation

Cells were enzymatically isolated from the calvariae of 21-day Wistar rat fetuses, by sequential digestion with collagenase, and plated at low density as described [8]. Osteoblast colonies for analysis were chosen either based on their morphology [8,, 20] or from a replica plating technique [15, Liu and Aubin, submitted]. In either case, dishes were rinsed with phosphate-buffered saline, a cloning ring was placed around selected colonies, and the cells were released with 0.01% trypsin (when matrix lacked mineral) or a 1:1 mixture of 0.01% trypsin and collagenase (when osteoid was mineralizing); the enzyme(s) were neutralized by adding minimal essential medium (α-MEM) containing 15% fetal bovine serum. Total RNA was extracted using a miniguanidine thiocyanate method, and cDNA was synthesized by oligo(dT) priming, poly(A)-tailed, and amplified by PCR with oligo(dT) primer as described [8,, 21]. The colonies analyzed were collected from 12 independent cell isolations (each resulting from a pool of at least 25 calvariae) and replica plating experiments.

Southern Blots and Hybridization

Amplified cDNA (5 μl) was run on 1.5% agarose gels, transferred onto 0.2 μm-pore-size nylon membrane (ICN; Costa Mesa, CA), and immobilized by baking at 80°C for 2 hours. Prehybridization, hybridization with labeled probes for mRNAs of interest (Table 1), and probe signal quantification were performed as described [8]. The signal intensity for each probe was standardized against total cDNA [8,, 20]. For the analysis reported here, each colony is reported as expressing (+) or not expressing (–) a particular gene of interest relative to background; the actual level of expression is not considered (Fig. 2).

Figure Figure 2..

A schematic of the Southern lineage blot describing expression of nine genes in 99 colonies on a + (expressed; black boxes; value 1) or – (not expressed; white boxes; value 0) scale.

Mathematical Analysis

Mathematical calculations and statistical analyses were carried out using SAS, Splus, Maple, and custom programs written in Fortran and C.

Results

Development of the Analytical Framework

Expression levels of the nine genes listed and numerically labeled in Table 1 were measured from 99 osteoprogenitor/osteoblast colonies. Our analysis focused on genes expressed above a background level (Fig. 2); we assumed that it was harder to measure magnitudes of expression levels accurately than to decide which genes were expressed at all. This information is encoded in an array of binary variables Bk(i) (i = 1,…,99, k = 1,…,9), defined to be 1 (respectively, 0) if expression of gene k is (respectively, is not) observed in the i-th colony. Time (days) is indexed so that the experiment begins at time 0 and ends at time τ. We define the variable Tk(i) to be the time at which gene k is first expressed in the i-th colony (i = 1,…,99, k = 1,…,9). These times, Tk(i), are not observed directly; we can only observe whether or not they occur before the end of the experiment. That is, if there were no errors in detecting gene expression, then for each i and k we could say that Tk(i) ≤ τ if and only if Bk(i) = 1. To allow for detection errors, we define qk to be the “false negative” probability that we do not detect the expression of gene k in a particular colony when in fact that gene is expressed there. In terms of conditional probabilities, this says that qk = Pr{Bk(i) = 0 | Tk(i) ≤ τ} = 1 - Pr{Bk(i) = 1 | Tk(i) ≤ τ}. We assume that there are no “false positive” errors, i.e., Pr{Bk(i) = 0 | Tk(i) > τ} = 1.

The overriding tenet of our model is that the 99 colonies are stochastically independent and governed by the same probability law. Formally, for each i = 1,…,99, we define the vector V(i) = (B1(i),…,B9(i),T1(i),…,T9(i)) consisting of the i-th colony's 18 associated variables. Then V(1),…,V(99) is a sequence of 99 independent random vectors drawn from some 18-dimensional multivariate probability distribution. Our goal is to describe the salient features of this multivariate distribution. When discussing the distribution (as opposed to the 99 observations drawn from it) we omit the superscripts and use the random vector V = (B1,…,B9,T1,…,T9).

A key aspect of our data is displayed in the 9 × 9 matrix E shown in Figure 3. For each pair of genes k and l, let Ekl be the number of colonies in which we detect the expression of gene k but do not detect the expression of the gene l. Formally, Ekl is the number of i's such that Bk(i) = 1 and Bl(i) = 0. Observe that Ekl is 0 and E1k is large for every k = 2,3,…,9. This strongly suggests that gene 1 (OPN) is expressed before every other gene, i.e., Pr{T1 < Tk} = 1 for every k = 2,…,9. In contrast, E32 = 19 and E23 = 20, so it seems unlikely that one of genes 2 or 3 must always precede the other. This paper shows how to analyze such claims statistically. For the most part, our methods are novel adaptations of standard techniques in statistics and probability, which may be found in [22] and many other general statistics references.

Figure Figure 3..

The matrix (Ekl).The 9×9 matrix E summarizes observed precedences: each entry, Eklis the number of colonies in which expression is observed for gene k but not for gene l (genes are numbered as in Table1).

Assessment of Experimental Error

Our approach begins by establishing a statistically rigorous upper bound for each “false negative” probability qk (k = 1,…,9). First, observe that, since gene 1 (OPN) was detected in every colony, no errors were made in detecting gene 1 in 99 cases. This has probability (1 - q1)99 and we would consider it statistically unusual if this were less than 0.05. Hence, we are reasonably sure that (1 - q1)99 ≥ 0.05, which is equivalent to q1 ≤ 1 - (0.05)1/99 = 0.030 (with 95% confidence).

It is reasonable to assume that qk increases as the average expression level decreases, and that genes with similar expression levels should have similar qk's. Therefore, from Table 1, we obtain

equation image(1)

In Section A.1 of the Appendix, we generalize the method used above for bounding q1 to obtain the bounds

equation image(2)

each with (at least) 95% confidence. Importantly, these error bounds were obtained by purely statistical means. We emphasize that they are (conservative) bounds rather than estimates; the true qk values could be significantly smaller.

Can Bone Development Be Explained by a Linear/ Deterministic Expression of Genes?

Although some aspects of the order of expression events are clear from the data (e.g., OPN is expressed early and OCN late), many others are not (e.g., does platelet-derived growth factor receptor alpha [PDGFRα] precede fibroblast growth factor receptor 1 [FGFR1] or vice versa?). We first asked whether the data are consistent with a deterministic linear order of the expression events, with the apparent inconsistency being due to observational errors.

To answer this, we suppose that the events T1,…,T9 always occur in the same order. For each k = 1,…,9, write σ(k) for the label of the gene (as listed in Table 1) that is always k th in this order. For example, if COLL always occurs second, then σ(2) would be 6. Then, we would have

equation image(3)

The sequence, σ(1),…σ(9), is a permutation of 1,2,…,9. We ask if there is any permutation such that the data are statistically consistent with Equation 3.

Each permutation would imply certain false negatives in the data. For example, suppose that Equation 3 holds, and that in colony 14 we observe Bσ(3)(14) = 0 and Bσ(7)(14)= 1. Since gene σ(3) is always expressed before gene σ(7), and since there are no false positives, the observation Bσ(3)(14) = 0 must be a false negative error. For each permutation, we computed the minimum number of false negatives that would be necessary in order to explain the data if Equation 3 were true (this is the number of values of i and k such that B(i)σ(k) = 0 and there exists a value l > k such that B(i)σ(l) = 1). The fewest such errors is 85, attained by two different permutations: 1,2,5,4,3,6,8,7,9 (i.e., OPN, PDGFRα, PTHrP, PTH-R, FGFR1, COLL, BSP, ALP, OCN) and 1,5,2,4,3,6,8,7,9. In Appendix A.2, we show that this many errors is very unlikely, which in turn provides strong evidence against Equation 3 and leads us to reject the possibility that there is a linear sequence of genes that accounts for the development of a bone nodule colony.

Is the Expression of One Gene Independent of the Expression of Other Genes?

We next determined, for each pair of genes k and m, whether their expressions Bk and Bm were independent. (The error probabilities qk and qm do not play a role here, since all errors are assumed to be independent.) We excluded gene 1 from this analysis, because B1(i) = 1 for every colony i, which prevents any meaningful testing of independence. To illustrate the method, consider the data for PDGFRα and FGFR1 (k = 2 and m = 3) in Table 2. Testing it for independence using Fisher's exact test (two-tailed) [23] gives a p value of 0.788. Table 3 lists the results for all pairs. Any p value below 0.05 is sufficient to reject the independence hypothesis on any single test. However, since we are testing 28 different pairs, we can reasonably expect to see one or two p values below 0.05 purely by chance, even when the corresponding pair of variables is independent. Thus, we need to be more cautious unless the p value is very small indeed. See Section A.3 of the Appendix for further discussion of this matter. In particular, that section shows that any p value below 0.0018 can be understood as a clear case for rejecting independence, while values between 0.0018 and 0.05 should be regarded as inconclusive indications of dependence. However, of the five “inconclusive” values that appear in Table 3, we expect all but at most one or two to be truly caused by dependence.

Table Table 2.. The number of events where both FGFR1 and PDGFRα were detected (1,1), either FGFR1 or PDGFRα were detected (0,1 and 1,0), and neither were detected (0,0).
  FGFR1
  01
PDGFRα0519
 12055
Table Table 3.. Results of Fisher's exact test for independence of each pair, Bk and Bm, for each pair of genes k and m. A p value below 0.05 indicates evidence against independence; smaller p values indicate stronger evidence.
 23456789
2 0.7880.0240.1471.0000.8181.0000.127
3  0.0300.2420.0010.0020.0550.024
4   0.3710.0010.0010.08010–4
5    0.3901.0000.7941.000
6     10–90.01310–4
7      10–910–11
8       10–7

Considering the test results among the group of genes 2, 3, 4, 5, and 6 (the “early” genes, as suggested by Table 1), we see that 3, 4, and 6 are mutually pairwise dependent (although the test result between genes 3 and 4 lies in the “inconclusive” range), while genes 2 and 5 are independent of each of the others (except for an “inconclusive” indication of dependence between genes 2 and 4). Notably, genes 2 and 5 also pass the tests for independence with the later genes 7, 8, and 9. In contrast, each of 3, 4, and 6 clearly fails the independence test with each of 7, 8, and 9 (except that 3 and 4 marginally pass the test with 8, and the results for 3 with 8 and 9 lie in the “inconclusive” region). Importantly, every failure of independence is due to strong positive correlation. In the context of Table 2, this means that the diagonal entries are larger than what they would be if the two random variables were independent. This result is also reflected in the sample covariances (data not shown). Positive correlations indicate, for example, that if we detect gene 3, then we are more likely to detect gene 4. This makes intuitive sense, since detection of a given gene indicates that the development of the cell has not been too slow, making it more likely that other genes are also expressed.

The following model illustrates how positive correlations could arise. Consider three events (C, D, and E) which occur at the random times TC, TD, and TE. Event C appears first. Once C occurs, then D and E each occur after an additional random (and independent) amount of time. Letting XCD be the duration of time between events C and D, and XCE the time between C and E, then TD = TC + XCD and TE = TC + XCE. Assume that XCD, XCE, and TC are independent. Then TC, TD, and TE are mutually positively correlated (in fact, Cov(Ti, Tj) = Var(TC) > 0 when ij). Moreover, consider the binary variables BC, BD,and BE (where Bk is 1 if Tk ≤ τ and is 0 otherwise). Then each of Cov(BC, BD), Cov(BC, BE), and Cov(BD, BE) is nonnegative, by the FKG Inequality [24].

Our analysis rules out situations in which negative correlations could arise. One such scenario has two alternative developmental pathways, with gene D occurring only on one pathway and gene E occurring only on the other. Then no colony would have BD = 1 = BE, and hence Cov(BD, BE) < 0 (since E(BDBE) = 0).

Can Genes Replace Each Other in the Same Developmental Program (Exchangeability)?

We showed above that genes 3, 4, and 6 are mutually dependent, in fact, positively correlated. We can determine if there is a preferred order in which these genes need to be expressed to result in bone nodule development. In Figure 3, observe that the values of Eij for i, j ∈{3,4,6} are all approximately equal (all between 11 and 15). This suggests no preferred order among these three genes. We formalize this as follows.

Hypothesis A. The genes 3, 4, and 6 occur in random order. More precisely, as soon as some (early) event occurs (say at a random time T*), each of genes 3, 4, and 6 waits for a random amount of time before it is expressed. Moreover, these three waiting times are i.i.d. (independent and identically distributed).

That is, there exist i.i.d. random variables X3, X4, and X6, all independent of T*, such that Tk = T* + Xk for k = 3,4,6. The event at time T* could be the expression of gene 1 (i.e., T* = T1), the expression of some other gene not examined in this experiment, or some other event.

Under Hypothesis A, T3, T4, and T6 are not independent, but they are identically distributed. In fact, T3, T4, and T6 are exchangeable random variables under Hypothesis A. This means that the probability law of the vector (T3,T4,T6) is the same as that of (T4,T6,T3) and of its four other permutations. For example, exchangeability implies that Pr{B3 = 1, B4 = 0, B6 = 1} = Pr{B4 = 1, B6 = 0, B3 = 1} and similarly for every other permutation.

In Section A.4 of the Appendix, we use this property to devise a statistical test of Hypothesis A. As described there, the tests show that the data are clearly consistent with the hypothesis of exchangeability implied by Hypothesis A. Due to the nature of statistical hypothesis testing, we cannot say that Hypothesis A has been accepted, but at most we can say that our data does not refute it. (Anything more would require information about the power of the test, which is beyond the scope of this paper; indeed, there is no obvious choice of alternative distributions for which one should do power computations.) That is, during osteoblast development, we hypothesize that there is a set of genes necessary for phenotypic maturation, but within this set there is flexibility in the order in which the genes are expressed.

In Section A.4, we also test exchangeability of the random variables {B2, B3, B4, B5, B6 } corresponding to the five “early” genes. That is, we tested whether we could extend Hypothesis A to the group of five genes 2,3,4,5,6. As explained in Section A.4, this was rejected.

Other Precedence Constraints

In this section, we consider the three “late” genes: ALP, BSP, and OCN. First, for gene 7 (ALP), observe from Figure 3 that E7k is much smaller than Ek7 for each of k = 2,3,4,5,6. Therefore, ALP typically is expressed after genes 2 through 6. In particular, E76 = 0, which suggests

Hypothesis B. Gene 6 (COLL) is always expressed before gene 7 (ALP). Moreover, none of genes 2, 3, 4, or 5 are a prerequisite for the expression of gene 7.

By our tests of independence, genes 2 and 5 are clearly not prerequisites for gene 7. Recall that we accepted Hypothesis A, that B3, B4, and B6 are exchangeable among themselves. If gene 6 has a special relationship with gene 7 that genes 3 and 4 do not (in particular, if Hypothesis B is true), then we would expect the following to be false:

Hypothesis B*. The variables B3, B4, and B6 are exchangeable with respect to the joint distribution of genes 3, 4, 6, and 7.

As described in Section A.5 of the Appendix, our tests indicate that Hypothesis B* should be rejected, but we do not reject the exchangeability of B3 and B6 with respect to the joint distribution of genes 3, 4, and 7. This is consistent with Hypothesis B. Further data are needed before one can conclusively accept Hypothesis B, however.

Finally, we considered genes 8 (BSP) and 9 (OCN). In particular, we looked at the possibilities that Pr{T8 < T7} = 1 and that Pr{Tk < T9} = 1 for every k ≤ 8, as suggested by the matrix E of Figure 3. We found that the statistical evidence was not strong enough to either confirm or reject these possibilities. See Section A.5 of the Appendix for details.

Summary of Expression Analysis

Combining the results of the statistical analyses, we have constructed a new model of osteoblast differentiation that is consistent with the observed relationships of expression among particular genes or subsets of genes (Fig. 1B). Our analysis indicates that the expression of genes 2 (PDGFRα) and 5 (PTHrP) may occur independently of other aspects of osteoblast development. Conversely, gene 1 (OPN), which precedes the expression of all of the other bone-related genes, may initiate the expression of genes 3 (FGFR1), 4 (PTH-R), or 6 (COLL). Once all the genes in this group are expressed, further maturation depends upon the expression of gene 7 (ALP), which may be contingent upon gene 8 (BSP) being independently activated. Finally, when genes 6, 7, and 8 are expressed, gene 9 (OCN) is expressed and the gene cascade associated with the clonal development of osteoblast phenotype is accomplished.

Discussion

In this paper, we readdressed the established linear/deterministic view of osteoblast development (Fig. 1A) by a retrospective analysis of gene expression profiles in individual osteoprogenitor clones that ultimately gave rise to mature osteoblasts making mineralized osteoid, i.e., the bone nodule colony-forming assay. The analysis provides an analytical framework of osteoprogenitor differentiation and is unique because it is based solely on establishing expression relationships among genes at different stages of development and not on any preconceived notion of their expression order. A primary advantage of this approach is that it allows identification of developmental gene expression patterns for genes with both known and unknown expression profiles. It also allows elucidation of the developmental history of a colony without an exhaustive examination of each cell fate division that takes place during its development [24]. The principles established, while tested for osteoblasts, should be widely applicable to other lineages.

The calculated gene expression profiles of individual clones were compared with those based on established osteoprogenitor developmental paradigms (Figs. 1A and 1B, Figure 1.). We first asked whether there was a predetermined order to the expression of a set of nine genes associated with osteoblast development. The analysis, which depended on a reduction of sequential expression errors to levels below those due to observational error, did not support a linear or deterministic sequence. Instead, although all the tested colonies eventually developed into fully mineralized bone nodules, the range of expression profiles at different stages of development suggested that it may be possible for osteoprogenitors to differentiate into mature osteoblasts using different developmental routes (Fig. 1B). To confirm and extend this possibility, we examined the correlation between the expression of a particular gene and any other individual gene. In some cases, strong positive correlations were found between particular pairs of genes, e.g., expression of gene 7 (ALP) required expression of gene 6 (COLL). In other cases, such as genes 2 (PDGFRα) and 5 (PTHrP), expression appeared independent of any other gene tested. One hypothesis is that acquisition of expression of these latter genes occurs at very early times in osteoblast development, i.e., at a stage prior to the earliest replica plating data points we have available. Alternatively, the expression profiles may be indicative of, or a prefacing of, potential development along pathways other than osteoblastic (e.g., chondrocytic or adipocytic) as may occur from multipotent progenitors (i.e., bipotential cells or mesenchymal stem cells) present in the population [1,, 5], but these may not progress under the culture conditions in which osteoblast development predominates (and was thus selected for analysis).

Our revised hierarchy of clonogenic osteoprogenitor differentiation (Fig. 1B) is consistent with two hypotheses of developmental history underlying osteoblast formation. One is that there is more than one pattern of sequential gene expression profiles that results in the same type of mature cell (developmental flexibility hypothesis). The other is that, as each colony matures from an undifferentiated progenitor cell, some divisions result in pathways that are either not supported by the environment (leading to lineage extinction) or give rise to other cell types that may exist in the final colony (i.e., cells expressing gene 2 (PDGFRα) or gene 5 (PTHrP)) but do not notably contribute to its overall phenotype under conditions where osteoblast development predominates (also, lineage extinction). At present, our data do not allow us to select between these two possibilities and it is possible that both occur. There is, for example, some evidence for mixed lineages in a small proportion of osteoblast colonies (e.g., adipocyte-osteoblast colonies, [25,, 26]). It is also interesting to note that colony size appears to have no (detectable) impact on expression profiles obtained, i.e., as raised above, there is a stochastic component underlying CFU-O differentiation, such that a small bone colony can be as “mature” as a big colony [9,, 15]. In addition, the variability seen in the rate of colony development is such that the gene expression “snapshot” taken during replica plating should give us an adequate description of all potential cell fate decisions during colony development (with the possible exception of very early genes such as gene 1, OPN). Taken together, our results argue against the existence of “rogue” differentiation routes, i.e., the expression of genes 2 and 5, or expression of genes 3 and 4, as deadends in the differentiation process, and for the possibility that their expression is required earlier during osteoblast development than was measured using our replica plating strategy. However, our results also suggest that some gene combinations are exchangeable and that higher order interactions among genes exist during osteoblast development. They support the hypothesis that great complexity underlies the interactions among groups of genes, and that further analysis will require larger gene (e.g., microarray technologies) and cellular (e.g., substantially more progenitor colonies) data sets. Interestingly, in contrast to the gene expression programs that appear to underlie early events in osteoblast development, global late gene expression profiles appear to be more fixed (Fig. 1B). This is consistent with other well-defined gene expression sequences during the maturation of committed progenitor cells (including, e.g., erythrocytes) and may be usual for cells that have committed to a particular mature state and lost their developmental flexibility. They also support the view that different developmental routes may contribute to and/or account for the substantial molecular heterogeneity seen amongst mature osteoblasts in vitro [20] and in vivo [27] (as predicted in [12]).

Although a stochastic component defines their development, osteoblast differentiation appears to be a default pathway for progenitor cells in RC populations cultured under the conditions we describe here [5]. Whether survival of particular cell types results from a microenvironment effect, or commitment to particular lineages is initiated by the exposure of an uncommitted cell to a particular signal that, upon exit of the cell from the stem cell state, stabilizes the subsequent identity of the progeny, is not yet known. This so-called directive mechanism clearly depends on the interactions between the cellular sensing systems—typically cell surface receptors—and the cellular microenvironment. Substantial evidence exists that both these mechanisms can occur and that the particular mechanism used may be dependent on the tissue system/cellular microenvironment. For example, in hematopoiesis, several studies suggest that exposure to growth factors may not be obligatory for the differentiation of primitive cells and that, at least under certain conditions, the identity of the differentiated cell population may be intrinsically determined [28]. Particularly interesting, in this regard, is the recent demonstration that coexpression of multiple lineage-restricted genes precedes commitment in multipotent progenitors [29–, 31]. This multilineage “priming” process [30] is consistent with the flexibility in the gene expression profiles we have seen during osteoprogenitor development and implies that the commitment of a multipotent cell to a particular pathway may reflect the stabilization of a particular subset of a group of expressed molecules. The stabilization process may occur in a stochastic manner in the absence of a particular instructive signal. Conversely, upon commitment of an undifferentiated cell, an instructive signal may stabilize a particular set or subset of expressed transcription factors, resulting in the production of specific cell types.

Conclusions

We have presented a novel and powerful way to analyze gene expression relationships during clonal stem or progenitor cell differentiation. This approach has not only given us insight into the cascade of genes expressed during osteoprogenitor development, but has, for the first time, provided quantitative support to observed developmental flexibility during stem and primitive progenitor cell differentiation. Our approach addresses the general need for an analytical/quantitative way to examine stem cell differentiation programs, should be useful to address questions of stem or progenitor cell commitment in response to changes in microenvironmental stimuli, and is a powerful way of analyzing the large amount of information created in genetic databases of stem cell developmental programs [32].

Appendix

A.1 Derivation of Bounds on Experimental Error

To get bounds on q2,…,q9, we look at the 0 entries in matrix E (Fig. 3). First, consider E94, which is 0. It turns out that there were 30 colonies in which both genes 4 and 9 were detected. Let A94 be the number of colonies in which genes 4 and 9 were both expressed. Since there are no false positives, we know that A94 ≥ 30. The probability of detecting 9 and missing 4 in one of these A94 colonies is q4 (1 – q9). But there were no misclassifications of this kind (since E94 = 0), and the probability of this (given A94) is (1 – q4(1 – q9))A94. As above, we are 95% confident that

equation image

, i.e., that

equation image(4)

We believe q9q4, so we deduce that q9 (1 – q9) ≤ 0.095, and thus q9 ≤ 0.106 with 95% confidence. Inserting this bound into Equation A1 gives q4 ≤ 0.106.

We repeat these calculations for E96, E97, and E98, which are all 0. Each of these cases has 30 colonies in which both genes are detected, implying that qk ≤ 0.106 with high confidence for k = 6,7,8. Alternatively, we could apply q4 ≤ 0.106 to Equation 1 and deduce qk ≤ 0.106 for every k, with high confidence. An analogous calculation for E76 yields q6 ≤ 0.064. By Equation 1, the same bound should hold for q8. In summary, we have shown

equation image(5)

each with (at least) 95% confidence.

A.2 Linear/Deterministic Expression of Genes

As described in part 3 of the Results section, we ask if there can be a permutation σ such that the data are statistically consistent with Equation 3, i.e.,

equation image

Recall that each permutation implies certain false negatives in the data. The fewest such errors is 85, attained by two different permutations: 1,2,5,4,3,6,8,7,9 and 1,5,2,4,3,6,8,7,9. We shall show that this many errors is very unlikely for any permutation, which in turn provides strong evidence against Equation 3. We split the 99 × 9 observations Bk(i) into three groups: Group A contains B1(i), i = 1,…,99 (i.e., all OPN data); Group B contains B6(i), B8(i), i = 1,…,99 (all COLL and BSP data); and Group C contains Bk(i), k = 2,3,4,5,7,9, i = 1,…,99 (all other data). Recall that our worst-case bounds on the error probabilities for groups A, B, and C, respectively, are qA = 0.030, qB = 0.064, and qC = 0.106. Let MA, MB, and MC denote the true number of expression events {Tk(i) ≤ τ} that occurred (whether observed or not) in Groups A, B, and C, respectively. We observed 99 of these events in Group A, 139 in Group B, and 386 in Group C. Hence, MA = 99, MB ≥ 139, and MC ≥ 386. Now let FA, FB, and FC be the number of false negatives in the three groups, i.e., FA = MA – 99, FB = MB – 139, FC = MC – 386. If Equation 3 holds, then 85 ≤ FA + FB + FC = MA + MB + MC – 624. For each y = A,B,C, let Gy, be a binomially distributed random variable with parameters n = My and p = qy. Then Gy is stochastically greater than Fy. It turns out that Pr{GA + GB + GCMA + MB + MC – 624} is about 10–3 or smaller for all values of MA, MB, and MC such that MA = 99, MB ≥ 139, and MC ≥ 386, and MA + MB + MC ≥ 624 + 85. (This can be calculated using the normal approximation for (GA + GB + GC – μ) / σ, where μ = qAMA + qBMB + qCMC and σ2 = qA (1 - qA)MA + qB (1 – qB)MB + qC (1 – qC) MC). That is, the probability of seeing enough errors is negligible. Hence we reject the possibility that there is a linear sequence of genes that accounts for the development of a bone nodule colony.

A.3 Testing Independence

In part 4 of the Results section, we performed Fisher's exact test on each pair Bk and Bm (for 2 ≤ k < m ≤ 9), which is 28 tests. For a single pair, one would normally reject independence when the p value is less than 0.05. However, in 28 tests, there is a reasonable chance of seeing one or more p values below 0.05 by chance alone, even when independence holds. To be cautious, we can choose a threshold, z, such that Pr{min{p1, K, p28} < z} is at most 0.05, where pi is the p value from the i-th test under the assumption that the null hypothesis is true. Since Pr{pi < z} = z (up to discretization error), we can choose z = 0.05/28 = 0.0018 (by Bonferroni's inequality). Therefore, any p value between 0.05 and 0.0018 should be regarded with some caution as being inconclusive. Any p value below 0.0018 may be safely used to reject independence. There are nine such values in Table 3. There are five values between 0.0018 and 0.05 in Table 3 that must be regarded with caution. However, it seems highly unlikely that all five of these values are due to chance resulting from five null hypotheses being true. Indeed, there are at most 28 – 9 = 19 cases where the null hypothesis is not conclusively rejected. If we assume that the null hypothesis holds in all 19 of these cases, and if we further assume that the p values are approximately independent, then we can say that the number of p values below 0.05 in these 19 cases has an approximate Poisson distribution with mean 19 × 0.05 = 0.95. This implies that the approximate probability of seeing j or more p values below 0.05 is 0.246 for j = 1, 0.071 for j = 2, and 0.016 for j = 3. We conclude that all but at most two of the inconclusive p values are likely due to the falsity of the independence null hypothesis.

A.4 Exchangeability

We now develop the statistical test of Hypothesis A based on exchangeability, as described in part 5 of the Results section. First, for each colony i = 1,K, 99, let S346(i)= {r ∈ {3,4,6}: Br(i) = 1}, which is the random subset of {3,4,6} corresponding to the genes that have been detected in the i-th colony by time τ. Next, let N346(i) = B3(i) + B4(i) + B6(i), which is the size of S346(i). The number of colonies in which N346(i) was 0 (respectively, 1, 2, 3) was 8 (respectively, 10, 28, 53). For each subset C of {3,4,6}, let n[C] be the number of colonies i such that S346(i) = C.

For each k, exchangeability implies that, conditioned on the event that N346 = k, the random set S346 is uniformly distributed among all k-element subsets of {3,4,6}. We test this uniformity using the classical goodness-of-fit statistic GF = Σ3r=1 (n[Cr] – E(n[Cr]))2 / E(n[Cr]), where the Crs are the k-element subsets of {3,4,6} and E(·) denotes expected value. For k = 2, the observed values (n[{3,4}] = 8, n[{3,6}] = 10, and n[{4,6}] = 10) give E(n[Cr]) = 28/3 and GF = 0.29. Under Hypothesis A, the approximate distribution of GF is x22 (chi-squared with 2 degrees of freedom). A chi-square table shows that 0.29 has a p value near 0.10. For k = 1, the data n[{3}] = 3, n[{4}] = 5, and n[{6}] = 2 give E(n[Cr]) = 10/3 and GF = 1.40. In this case, there are too few observations for the chi-square approximation to be valid, so we used a simple Monte Carlo simulation to compute Pr{GF ≥ 1.40} = 0.63 under Hypothesis A. That is, we simulated a million realizations of the test statistic GF under Hypothesis A and found that 63% of simulated values were greater than 1.40.

The k = 1 and k = 2 tests show that the data are clearly consistent with the hypothesis of exchangeability implied by Hypothesis A. We also used this method to test exchangeability of the random variables {B2, B3, B4, B5, B6}, corresponding to the five “early” genes. We rejected uniformity of S23456 for k = 3 (with p value 0.007), marginally accepted it for k = 1, and accepted it for k = 2 and k = 4. Overall, then, the data are not consistent with exchangeability of all five of {B2, B3, B4, B5, B6}.

A.5 Precedence Constraints for Later Genes

Here we describe the methods referred to in part 6 of the Results section. Hypothesis B* from that section implies that (conditioned on the event {N346 = 2}) the expression of gene 7 is independent of the set S346. We therefore test independence of B7 and S346 conditioned on {N346 = 2}, using the data in Table 4. Fisher's exact test gives the p value 0.045, which marginally rejects Hypothesis B*. The analogous test for independence of B7 and S346 conditioned on {N346 = 1} gives little information since the row sum of the B7 = 1 row is 1.

Table Table 4.. The number of occurrences of B7 and S346 in the cases where {N346 = 2}
  S346   
  {3,4}{3,6}{4,6}Row sum
B7086519
 10459
 Column sum81010 

Next, we tested independence of B7 and S34 conditioned on {N34 = 1}, obtaining the p value 1.000 from Fisher's exact test. This contrasted with the value 0.014 (respectively, 0.015) for the independence of B7 and S34 (respectively, S46) conditioned on {N36 = 1} (respectively, {N46 = 1}). Taken together, these tests provide strong evidence against Hypothesis B*, indicating that while the pair of genes {3,4} may be exchangeable when considering gene 7 as well, the trio {3,4,6} is not.

We now turn to gene 8 (BSP). For each of k = 2,3,4,5,6, E8k is considerably smaller than Ek8, but the reverse is true for k = 7. This suggests that BSP usually gets expressed after genes 2 through 6 and before gene 7. We observed 47 colonies in which both 7 and 8 were expressed, and four colonies in which 7 was expressed but 8 was not. Could the latter four all be errors? The number of errors of this type would have a binomial distribution that is approximately Poisson with mean 50q8(1 - q7). The probability that such a distribution has a value of four (or more) is less than 0.05 only if q8 < 0.03. Our bound, q8 ≤ 0.064, is too weak for this, so we conclude that there is no strong evidence that Pr{T8 < T7} = 1.

Finally, we consider gene 9 (OCN). For every k ≤ 8, Ek9 is large and E9k is small. Apparently OCN is usually expressed after the other genes. In fact, E9k = 0 for k = 1,4,6,7,8, but it is not clear whether these zeroes are due to strict precedence constraints or simply to late expression times of OCN. Nor can we reject the possibility that Pr{Tk < T9} = 1 for every k ≤ 8, with the nonzero values E92, E93, and E95 due to observational errors. Although the case for strict precedence of gene 9 by the two later genes (7 and 8) is stronger than for the earlier genes, we cannot confirm this statistically without more data or smaller bounds on the qks.

Acknowledgements

Supported by operating grants (P.Z. and N.M.), a Postdoctoral Fellowship (A.G.), and an Undergraduate Research Award (Y.Z.) from the Natural Sciences and Engineering Research Council of Canada and an operating grant from the Canadian Institutes of Health Research (J.E.A.; MT-12390).

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