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MM and AL conceived, developed, and applied the computational algorithm and associated models and generated the computation results. MS, YL, RP, and EP developed the biological system and conceived, designed, and executed the biological experiments and associated mathematical analysis. MS, MM, and AL wrote the paper.

The authors have declared that no competing interests exist.

Tissue development is regulated by signaling networks that control developmental rate and determine ultimate tissue mass. Here we present a novel computational algorithm used to identify regulatory feedback and feedforward interactions between progenitors in developing erythroid tissue. The algorithm makes use of dynamic measurements of red cell progenitors between embryonic days 12 and 15 in the mouse. It selects for intercellular interactions that reproduce the erythroid developmental process and endow it with robustness to external perturbations. This analysis predicts that negative autoregulatory interactions arise between early erythroblasts of similar maturation stage. By studying embryos mutant for the death receptor FAS, or for its ligand, FASL, and by measuring the rate of FAS-mediated apoptosis in vivo, we show that FAS and FASL are pivotal negative regulators of fetal erythropoiesis, in the manner predicted by the computational model. We suggest that apoptosis in erythroid development mediates robust homeostasis regulating the number of red blood cells reaching maturity.

During development, progenitors undergo orderly differentiation through a series of maturation steps. The resulting number of fully differentiated progeny is precisely regulated to match physiologic and developmental needs, and is relatively resistant to environmental or gene-dose fluctuations [

An efficient identification of the main components of a regulatory developmental network requires quantitative measurements in developing tissue, in conjunction with mathematical modeling. The quantitative study of erythropoiesis was made possible recently by the development of a flow-cytometric assay that defines differentiation-stage-specific erythroblast subsets in erythropoietic tissue in vivo [

A variety of mathematical tools have been employed recently to reconstruct regulatory intracellular networks from experimental datasets. These include statistical correlation techniques, such as Bayesian inference, used to model gene expression (e.g., in [

In this study, we developed a novel algorithm that focuses on identifying the principal intercellular interactions between erythroblasts, without necessarily determining the complete regulatory developmental network. Rather than ranking models representing the whole network, we ranked the network links: individual feedforward or feedback interactions between erythroblasts arising in developmental time course. To operate under relative scarcity of the initial experimental data and to increase the confidence in the ultimate results, interactions were ranked based on several criteria, including their ability to endow the network with robustness to small perturbations of the strength of the network links. We selected robustness as one of the ranking criteria since many biological processes are found to be relatively resistant to small exogenous perturbations [

Our proposed algorithm contains several steps. The first consists of the acquisition of biological data describing the developmental process, in terms of time-dependent changes in developmental markers. In the second step the developmental process is defined in terms of a series of discrete states. In the third step, a generalized model is constructed, describing how regulatory interactions between cells in different states may be responsible for the time-dependent changes that are observed experimentally. In the fourth step, different model networks, or topologies, are generated, based on the generalized model developed in the previous step. In the fifth step, the goodness of fit and robustness properties associated with each model network are characterized, as well as the values of model parameters and the ability of each parameter to influence the model fit. In the sixth step, likely feedback and feedforward interactions are identified, based on whether they occur in the model networks that are more robust to parameter variation and that show better fit to the experimental data. In the seventh step, we identify candidate molecules that may regulate the biological process and mediate the interactions predicted by the algorithm. In the final step we examine quantitatively the role of these candidate molecules during the developmental process in wild-type mice and in relevant mutant mice, in order to ascertain that their function indeed matches that predicted by the algorithm.

Application of this algorithm to fetal erythropoiesis identified a negative autoregulatory interaction, between erythroblasts of similar maturation stage, as highly significant in the homeostatic regulation of erythroid development. We show that this interaction is exerted through the death receptor FAS and its ligand, FASL.

Fetal liver between E12 and E15 is primarily an erythropoietic tissue, although it also contains a small minority of non-erythroid cells. We divided all fetal liver cells, both erythroid and non-erythroid, into four developmental states, and then measured the relative frequencies of cells in each state as a function of time in the following manner. We used a recently developed flow-cytometric assay that makes use of the cell-surface markers CD71 (also known as TFRC) and Ter119 (also known as LY76) to identify differentiation-stage-specific erythroblasts in erythropoietic tissue [^{med}Ter119^{neg}). State 1 contains the earliest erythroid precursors. It also contains all the non-erythroid cells in fetal liver, which constitute 10% or less of all state 1 cells [^{high}Ter119^{low}, state 3 cells are CD71^{high}Ter119^{high}, and state 4 cells are CD71^{med/low}Ter119^{high}. All cells in states 2, 3, and 4 are in the erythroid lineage [

(A) CD71/Ter119 flow-cytometric histograms of fetal liver cells derived from embryos between E12 and E14.5. Developmental states 1 to 4 are noted.

(B) Cytospin preparations of E14.5 fetal liver cells sorted from each of states 1 to 4 show increasingly differentiated erythroid progenitors. Cells were stained for hemoglobin expression (brown) with diaminobenzidine and counterstained with Giemsa. Scale bar = 20 μm.

Developmental states 1 to 4 are defined as described in the text and in

Analysis of fetal liver progenitors on successive days of development (E12–E15.5) reveals a “developmental wave,” with a large number of progenitors appearing first in states 1 and 2, and progressing into states 3 and 4 with developmental age (

We next constructed a generalized model that describes how regulatory intercellular interactions may be responsible for generating the experimentally observed erythropoietic developmental wave in _{i}_{i}

We assumed that, within the time scale considered here, the ODE system is driven by its initial conditions, with no additional influx of cells to state 1 nor efflux from state 4. We also assumed that no “de-differentiation” occurs and that cells transition unidirectionally from state 1 to 2, 2 to 3, and 3 to 4. Therefore, each transition rate _{i}_{,i+1} from state _{i}

The transition functions _{i}_{,i+1} govern the rates of transition of cells from the “source” state, _{i}_{i+}_{1}_{12}_{23}, and _{34}, represent the rates of transition per cell due to this process from states 1 to 2, 2 to 3, and 3 to 4, respectively. The transition function _{i}_{,i+1} is also influenced by a second type of interaction, due to feedback or feedforward regulation of cells in the source state, _{i}_{j}_{i}x_{j}_{i}^{2}_{k}_{ij}_{i}_{i}_{,i+1} incorporates all these potential interactions:

The model contained in equations 1 and 2 is an abstract approximation of what are in reality much more complex processes. Specifically, the transition functions _{i−1}_{,i} and _{i}_{,i+1} combine processes of cell differentiation, proliferation, and death, all of which regulate the fractional number of cells in state

We generated different model networks, or topologies, based on the general model described above. Erythroblast differentiation due to EPO action, represented by the first-order state dependency term and the parameter _{ij}

(A) The relative numbers of cells in four states over the time course of the erythropoietic wave shown in

(B) The model topologies were generated by allowing up to three independent feedback and feedforward interactions between different states, so that the fractions of cells in some of the states might affect the transition probabilities between pairs of successive states. Each model topology was then fitted to the data shown in (A). Two examples of the possible 298 topologies and the corresponding best fits are shown.

(C) For each model topology, the parameters determined during the fitting process were then varied within a 5% range, resulting in 200 predictions for the corresponding model topology, each with its associated SSR. The distributions of SSR values for two such model topologies are shown. The mean SSR value, _{M}_{M}_{j}_{j}

(A and B) ANXA5 binding in each of the four fetal liver states (B) and the corresponding frequency of each state in the same fetal livers (A) during E11–E15.5. Each data point is the mean ± standard error of the mean for embryos in one litter. _{1}, _{2}, and _{3} are the fraction of cells in states 1, 2, and 3, respectively. Av^{+}_{2} is the fraction of cells in state 2 that bind ANXA5. Individual embryo data for the same dataset are presented in

(C and D) The fraction of cells expressing FAS in each state during E11–E15.5. Each data point in (C) is mean ± standard deviation for a litter of embryos. Individual embryos in the same dataset are shown in (D); each symbol denotes an independent experiment of one or more litters from the same embryonic age. Further analysis of embryos in this dataset, with the same symbols, is shown in _{1}, Φ_{2}, and Φ_{3} are the fraction of cells expressing FAS in states 1, 2, and 3, respectively. All curves are second-order polynomials. In order to avoid overlapping data points in (D), the data for some litters were slightly displaced along the horizontal axis (e.g., E13.5 was plotted as E13.45).

(E) Fraction of cells that are ANXA5^{+} and FAS^{+} in states 1 to 3, measured in the same fetal livers (E12.5–E15.5) or in the same litters (E11.5, due to the small number of cells in fetal liver at this age). Each point represents a single embryo (E12.5–E15.5) or a single experiment (E11.5, mean of four litters). Each symbol denotes an independent experiment of one or more litters. Curve is a second-order polynomial.

(F) Fraction of cells that are FASL^{+} and FAS^{+} in states 1 to 3, measured in the same fetal livers (E12.5–E15.5). Each color denotes an independent experiment of one or more litters.

(A) The fraction of FAS^{+} cells in state 2 (Φ_{2}) is inversely proportional to the fraction of all fetal liver cells that are in state 2 (_{2}). Each point represents an individual embryo from the same set of embryos shown in ^{−}^{1.009} (^{2} = 0.63).

(B and C) The fraction of ANXA5^{+} cells in state 2 (Av^{+}_{2}) is inversely proportional to _{2} between E12.5 and E15.5 (B), but not in earlier E11.5 embryos (C). Each point represents an individual embryo from the same set of embryos and with the same symbols as in ^{−}^{0.6} (^{2} = 0.4).

(D) Four E11.5 embryos from the same litter, showing the distribution of cells in states 1 (_{1}) and 2 (_{2}) in the top panels, and a fraction of state 2 cells that are FAS^{+} (Φ_{2}) in the lower panels. Staining control for FAS shown in the leftmost panels. See additional data on these in

(E) Potential time course of Φ, the fraction of cells that are FAS^{+}, in a cohort of erythroblasts traversing state 2. _{st2} is the length of state 2. Φ_{1} and Φ_{3} are used as estimates for Φ when cells enter and exit state 2, respectively. The dashed red line represents a constant rate of decline in Φ throughout state 2. The solid red curve represents an alternative time course, with an initial rapid decline in Φ that slows down due to depletion of FAS^{+} cells.

(F) Comparison of Φ_{2} with
Φ_{1,3}, the arithmetic mean of Φ_{1} and Φ_{3}, during E12.5 and E15.5. Data calculated for each embryo from values shown in _{1,3}/Φ_{2} would be expected to be equal to one. Since
Φ_{1,3}/Φ_{2} is greater than one for embryos younger than E14.5, the rate of decline in Φ in these embryos resembles the solid red curve in (E).

(G) Approximate method for relating Φ_{2} to
Φ_{1,3} when the rate of decline in Φ is nonlinear. Φ_{2} equals the area under the red Φ curve (shaded red, panel i) divided by _{st2}, and may be represented as the height of a rectangle (black dashed lines, panel ii) of height Φ_{2} and base _{st2}. The shaded red area is approximately equal to the area of a triangle (shaded blue, panel ii) of base _{a}, where _{a} is the length of time within state 2 when the rate of decline in Φ is relatively constant. The area of the blue triangle, given by
Φ_{1,3}_{a}, is therefore approximately equal to the area of the black dashed rectangle: Φ_{2} _{st2} =
Φ_{1,3}_{a}

(H) Graphical representation of Φ_{2} and how it relates to the rate of decline in Φ. At E11.5, Φ declines rapidly as the first cohort of erythroblasts progresses through state 2, as suggested by data in (A) and (D). At E12.5–E13.5, Φ declines sufficiently rapidly so that it plateaus prior to completion of state 2, as suggested by (F). By E14.5, the rate of decline has slowed down and remains constant throughout state 2. Throughout, Φ_{2} is directly proportional to _{a} (from [G]), and both are inversely related to _{2} (from [A], and see text).

Each model topology was described as an ODE system corresponding to a unique set of feedback and feedforward interactions (_{i}_{k}_{i}_{k},

Rather than describe a specific erythroid intercellular interaction network, our aim was to identify individual, significant feedback or feedforward interactions that are critical to the network's regulation, regardless of the network's final, complete form. With this aim in mind, we developed a comprehensive analysis of the 298 model topologies. In particular, rather than rank specific model topologies, we ranked the 12 feedback and feedforward interactions that constitute the models. Each of the 12 feedback/feedforward interactions participates in 67 of the 298 models we tested, either by itself, or in combination with one or two other interactions. We devised four criteria, or metrics, that test the performance of each interaction within the context of each of the 67 model topologies in which it participates.

We assumed that an individual interaction, if important, would contribute substantially to the dynamic output of the network, as well as to network robustness. This assumption was based on observations suggesting that single feedback or feedforward interactions may significantly influence the dynamic output of biological networks, such as oscillations or switch-like behavior [_{k}_{k}

To examine these properties for each potential feedforward and feedback interaction, we carried out multivariate analysis on each of the 298 models (_{M},_{M},

We used the results of this multivariate analysis to evaluate the role of each of the potential feedforward and feedback interactions in the network. We first asked whether certain interactions occur more frequently in models with better fitness and/or higher robustness. We examined this by calculating, for each interaction

(A) All possible feedforward and feedback interactions are rank-ordered according to whether they are mostly present in the models of higher fit to the data in _{j}_{M}_{j}

(B) All possible feedforward and feedback interactions are rank-ordered according to whether they are mostly present in the models of higher robustness. Specifically, the robustness to parameter variation metric _{j}_{M}_{j}

(C) The consistency of parameter values corresponding to particular feedforward and feedback interactions is evaluated by plotting the parameter values for each model

(D) A correlation matrix relating the control exercised by each feedback or feedforward interaction in each model in which it is present. Each row in the matrix corresponds to a single parameter, describing a single type of interaction. Each column represents one specific model topology. Models are arranged from left to right in descending order of their best fit to the experimental dataset. The color bar indicates the value of the control metric, defined as the correlation coefficient between the parameter values and their associated SSR values, obtained during the multivariate analysis illustrated in

Of the twelve potential feedback and feedforward interactions we considered during the implementation of the algorithm, only two, ff23 and ff34, satisfied all the required criteria. Both assume a relatively narrow range of positive values and score high on the fitness, robustness, and control criteria (

We tested these conclusions by repeating the above analysis on altered datasets (

Both ff23 and ff34 represent a special case of feedforward interactions, where the source state regulates itself (ff[

The fb43 interaction has a negative parameter value, and its effect is therefore to decrease the transition from state 3 to state 4, in proportion to the product of the relative number of cells in states 3 and 4 (

To begin to evaluate the biological plausibility of the predictions made by the computational algorithm, we first considered tissue architecture within fetal liver. Erythroblasts in tissue are found within anatomical units known as erythroblastic islands, where they form concentric rings around a central macrophage. We found that, at E15.5, 50% and 30% of all state 2 and 3 cells, respectively, are adjacent to cells of the same state (

Next, we considered candidate molecules that might mediate the ff23 or ff34 interactions. As indicated above, a likely underlying process described by these interactions is apoptosis arising from cell–cell interactions within the same developmental state. FAS is a cell-surface receptor of the tumor necrosis factor (TNF) receptor family. It triggers apoptosis when activated by its ligand, FASL, expressed on the surface of adjacent cells [

We examined the possibility that FAS-mediated apoptosis, triggered by FASL on neighboring cells, may form the basis for the negative autoregulatory interactions predicted by our modeling. We measured FAS and FASL expression as well as an early marker of apoptosis, ANXA5 (Annexin V) binding, in fetal livers freshly isolated from embryos between E11.5 and E15.5. ^{+} cells (Av_{2}^{+}; ^{+} cells (Φ_{2}; _{2}^{+} and Φ_{2} at E12.5, coinciding with a peak in the relative number of cells in state 2 (_{2}; _{2}^{+} and Φ_{2} to be positively correlated (^{+} or FAS^{+} cells in states 1, 3, or 4 did not vary significantly with development (

Glossary of Symbols Used in the Text

We assessed the role of FAS further, by measuring ANXA5 binding in embryos that carry an inactivating mutation in either FAS _{2} and Av_{2}^{+} are relatively unaffected by small variations in the true age of embryos. We found a 2.5-fold decrease in Av_{2}^{+} in _{2}, in

(A) Fraction of cells that are ANXA5^{+} in E14.5

(B) State 2 ANXA5 binding data shown in (A), plotted together with the ANXA5 binding data for the entire E11–E15.5 period from

(C) Distribution of cells in the four fetal liver states shows an increase in state 2 cells in E14.5 and E13.5 _{2} in

(D) Median FAS expression per cell in the FAS^{+} cell population in states 1 and 2. Each point represents data from an individual embryo. Two experiments are shown; in each, two litters of different embryonic ages were examined simultaneously. No normalization was performed on the fluorescence data.

Since ANXA5 binding in state 2 is largely due to FAS-mediated apoptosis, the trough in Av_{2}^{+} at E12.5 is likely to be due to the trough in Φ_{2} at this time (_{2}. FAS^{+} cells may be lost as a result of FAS-mediated apoptosis. Alternatively, a signaling event may suppress FAS expression in state 2 cells. In the latter case, the decrease in the number of FAS^{+} cells might be expected to be associated with a decline in the mean level of FAS expression per cell. However, we found that FAS expression per cell within the FAS^{+} population remained constant in spite of large variations in the size of this population (^{+} cells at E12.5 is likely to be due to their loss through apoptosis, consistent with results obtained above that suggest the presence of FAS-mediated apoptosis in state 2.

Apoptosis of FAS^{+} cells in state 2 could, in principle, be triggered by any FASL^{+} cells within fetal liver, whether erythroid cells in any of states 1 to 4, or non-erythroid cells that form up to 10% of state 1. To identify the source of FASL^{+} cells responsible for triggering FAS-mediated apoptosis in state 2, we examined the rate of this process. If FAS-mediated apoptosis is due to an ff23-type, autoregulatory interaction, where both FAS^{+} and FASL^{+} cells are within state 2, its rate should be proportional to _{2}^{2} (see equation 2 above, and table in ^{+} and FASL^{+} cells should, by mass action, be proportional to the product of their respective numbers, or ([FAS^{+}] × _{2}) × ([FASL^{+}] × _{2}) = [FAS^{+}] × [FASL^{+}] × _{2}^{2}, where [FAS^{+}] and [FASL^{+}] are the proportions of state 2 cells expressing FAS and FASL, respectively.

Our data show that, between E11.5 and E15.5, _{2} varies over a 5-fold range (_{2} is inversely proportional to _{2} (_{2}^{+} is also an inverse function of _{2} throughout most of development (_{2}^{+} is higher, for a given _{2}, than later in development. This effect at E11.5 may be due to a slower clearance of apoptotic cells in very early fetal livers.

The analysis below shows that the inverse relationship between Φ_{2} and _{2} is a consequence of a FAS-mediated cell loss whose rate is proportional to _{2}^{2}, strongly supporting an autoregulatory interaction between state 2 cells as the cause of their apoptosis.

To assess the rate of FAS^{+} cell loss, we considered how the proportion of FAS^{+} cells, Φ, decreases within a given cohort of cells that have entered state 2 together and are advancing through the state. (Erythroblasts are presumed to enter state 2 from state 1 continuously. We define a “cohort” as all erythroblasts that enter state 2 together at a given point in time.) The first such cohort of differentiating erythroblasts enters state 2 at ≈E11.5. The wide range of _{2} values found in E11.5 embryos (_{2}, the first cohort of cells has advanced further in state 2 than in embryos where _{2} is low (_{2} and Φ_{2} in E11.5 embryos (^{+} cells are lost rapidly as this first cohort of erythroblasts advances through the state (^{+} cells (Φ) within the cohort. In the ensuing E12.5–E15.5 period, it is likely that a similar process continues for each cohort of cells entering and traversing state 2. For each such cohort, Φ is highest at the time cells enter state 2, and decreases as the cells traverse state 2. Since there is little FAS-mediated apoptosis in states 1 and 3, we used the measured fractions of FAS^{+} cells in state 1 (Φ_{1}) and state 3 (Φ_{3}) as estimates of Φ when cells enter (_{st2}) state 2, respectively (

^{+}). To assess the rate with which Φ decreases, we compared Φ_{2}, the arithmetic mean of Φ for all cells in state 2, with Φ_{1} and Φ_{3}_{2} should equal the average of Φ_{1} and Φ_{3} (_{1}_{3}]/2). If, however, an initial rapid loss leads to depletion of FAS^{+} cells before state 2 is complete (solid red line, _{2}. We found that the ratio of _{2} alters with embryonic age in a manner similar to _{2} (_{2} is larger than one, suggesting that, for a given erythroblast cohort traversing state 2, an initial rapid decline in Φ slows down as FAS^{+} cells are depleted (

We assumed that prior to FAS^{+} cell depletion, Φ decreases at an approximately constant rate, for a period of length _{a}. The area under the curve described by Φ (shaded red, _{a} (shaded blue, _{a} (_{2} is the mean value for Φ throughout state 2, it is equivalent to the height of a rectangle (black dashed lines, _{st2}. The equivalence in the areas of the blue triangle and black rectangle in _{a}^{+} cell loss is constant, is directly proportional to Φ_{2}:
_{st2}/_{1}, Φ_{3}, and _{st2} are relatively constant). Equation 3 also shows the ratio _{2} to be equivalent to _{st2}/_{a}. From this and from _{a} is one-half to one-third of _{st2}, increasing with embryonic age (see

The direct proportionality between _{a} and Φ_{2} (equation 3) makes it possible to deduce that _{a} is also inversely proportional to _{2} (equation 5 below), since we found experimentally (_{2} is inversely proportional to _{2}:
_{st2}/_{.}

The rate of decline in Φ during the period _{a} is represented by the slope of the blue triangle's hypotenuse in _{1} _{3})/_{a}. The rate of loss of FAS^{+} cells, _{2} (since Φ represents the fraction of cells that are FAS^{+} during state 2):

Since _{a} is inversely proportional to _{2} (equation 5), it is possible to re-write equation 6, expressing _{2} , and giving the result that the rate of loss of FAS^{+} cells from state 2 is proportional to _{2}^{2}:
_{1} _{3}) _{st2}.

Of note, the proportionality constant, relating the rate of FAS^{+} cell loss to _{2}^{2}, contains the term Φ_{1}^{2} (equation 7). This term represents the product of the available fractions of FAS^{+} and FASL^{+} cells when cells enter state 2, since FAS and FASL expression are linearly related (^{+} cells in state 2 is found experimentally to be proportional to [FAS^{+}] × [FASL^{+}] × _{2}^{2}, as predicted for an autoregulatory ff23.

The inverse relationship we also found between Av_{2}^{+} and _{2} can be explained if, as is likely, ANXA5^{+} cells are cleared fast compared with the length of state 2. If so, as cells traverse state 2, the proportion of ANXA5^{+} cells will closely reflect Φ, and their mean, Av_{2}^{+}_{,} will relate to _{2} in the same way as Φ_{2}. The inverse relationship between Av_{2}^{+} and _{2} therefore provides an independent assay confirming that the rate of loss of FAS^{+} cells is inversely related to _{2}^{2}.

The ff23 interaction was selected based on criteria that included robustness of the erythropoietic network to perturbations in parameter values. It therefore follows that the absence of ff23 should result in increased sensitivity to such perturbations. The

Increased Variance in the Number of State 2 Cells in Embryos Mutant for FAS or FASL

Specifically, we examined the dataset of _{2}), where the ff23 interaction is exerted, for all embryos at a given embryonic age and on a given genetic background. Strikingly, the variance between

These findings provide an additional and more direct a posteriori justification for using robustness as one of the criteria for identification of new interactions in developmental networks.

The work we present here aims to tackle a key challenge. In many biological systems, multiple candidate regulators have been identified by bioinformatics and other methods. Given the complexity of biological networks, how can their principal regulatory features be identified? We were particularly interested in identifying interactions that endow a network with robustness to external and internal perturbations (represented by alterations of parameter values in the corresponding dynamical model), a property that is encountered frequently in developmental networks. We approached this problem by hypothesizing that, although a large number of interactions might be present in any given system, only a small subset is responsible for its robust regulatory behavior. This hypothesis guided us in developing a novel computational algorithm that identified a key negative autoregulatory interaction that controls the rapid growth phase of erythropoietic tissue in the developing embryo. We show that cells at an early erythroid differentiation stage, here termed state 2, undergo FAS-mediated apoptosis as a result of an intercellular interaction between FAS^{+} and FASL^{+} cells in the same state. Further, we find that apoptosis proceeds at a rate that is proportional to the square of the number of cells in state 2, a property that would buffer fluctuations in erythropoietic tissue growth and ensure it progresses close to its preprogrammed developmental trajectory.

The combined biological and computational approach we applied here could be used to identify homeostatic interactions regulating other tissues with rapid growth or turnover, such as skin, intestinal epithelium, or tumor metastasis.

An intrinsic and unique aspect of our developmental network reconstruction approach is that it explicitly selects for interactions that endow the developmental system with certain key regulatory properties. First, we explicitly assume that the underlying developmental process is robust to perturbations that might arise as a result of disease or fluctuations in gene dose, temperature, or nutrients. Robustness of developmental processes, though widely assumed and confirmed in multiple studies [

Requiring that putative interactions satisfy all of the above criteria can strongly constrain the number of these interactions. Indeed, we have found that only two interactions were consistently ranked high, with only one of them retaining the high ranking during additional perturbations of the original experimental dataset or the details of the computational analysis. The benefit of this highly selective ranking is that a relatively high confidence can be placed in the interactions chosen. On the other hand, by relaxing some of the criteria, one can expand the list, progressively testing for lower ranking interactions. In this process, different weights could be placed on different criteria. For example, by relaxing the fitness criterion, one can predict the potential importance of the fb43 interaction. The particular set of interactions identified is likely to be strongly dependent on the nature of the developmental process and, to a smaller degree, the particular dataset used for network reconstruction.

An important feature of the algorithm is that it deals with the relative number of cells in each of the developmental states, without explicitly breaking the rate of change in cell number down into its components, namely, the rates of cell death, cell division, and cell differentiation. Consequently, when the pivotal ff23 interaction was identified, the algorithm did not have the power to specifically predict which of these three processes is responsible for the interaction. The molecular candidates for mediating the ff23 interaction had to be identified subsequently, through reasonable assumptions about the system's biology. The level of abstraction at which we chose to develop the models underlying the algorithm was a result of the type of biological measurements that can be made reliably and reproducibly in erythropoietic and other developmental networks. There is no simple way to measure the rates of cell death, cell division, or cell differentiation in tissue in vivo at the present time. By contrast, the flow-cytometric datasets that we used could reliably and rapidly document the rate of change of the relative number of cells in each state during development. In the future, technological advances and increased knowledge of the erythropoietic system may allow the reconstruction of a more detailed mechanistic model. It is noteworthy, however, that in spite of these limitations, the algorithm we present had the power to provide a strong and precise prediction that successfully guided our subsequent biological investigation. We therefore suggest that dynamic measurements of cell frequency, which are relatively easy to obtain, may serve as the basis of predictive computational modeling in many other tissues.

Between E11.5 and E15.5, fetal liver mass increases over 100-fold, a rate ten times faster than overall embryonic growth in the same period (^{+} and FASL^{+} cells within state 2 leads to apoptosis of state 2 FAS^{+} cells at a rate that is proportional to the square of the relative number of cells within that state (_{2}^{2}). Therefore, FAS^{+} cells within state 2 constitute a reserve progenitor pool that can be rapidly tapped whenever _{2} is inappropriately low. Conversely, an inappropriately fast growth in the number of state 2 cells, or _{2}, would be damped rapidly by a sharp increase in apoptosis.

It was previously thought that, unlike adults, embryos do not possess an erythropoietic reserve, because of their higher vulnerability to mutations that lower erythropoietic rate [

The erythroblastic island architecture (_{2} (

To date, apoptosis had been implicated in the development of a unique set of tissues, where tissue function depends on selection—from amongst all available progenitors—of cells with specific characteristics, such as optimal cell connectivity in neural tissue, specific antigen receptors in lymphocytes, or the sculpturing of tissue architecture [

FAS and FASL have clear homeostatic functions in immune cells in the adult [

The details of the algorithm development and implementation are given in

C57BL6j and BalbCj

Wild-type embryos were each allotted an embryonic age equal to that calculated from the timing of gestation adjusted ±0.5 d. The addition or subtraction of 0.5 d was done based on morphological appearance of the embryo and on the relative developmental status of the erythropoietic system, judged from the CD71/Ter119 histograms. In the case of comparisons between

Freshly isolated fetal livers were mechanically dissociated and strained through a 70-μm strainer in the presence of phosphate-buffered saline and 5% fetal calf serum. Cells were immunostained at 4 °C in phosphate-buffered saline and 5% fetal calf serum in the presence of rabbit IgG (200 μg/ml, Jackson ImmunoResearch Laboratories,

Cell sorting of fetal liver cells was carried out on a DakoCytomation MoFlo (Dako,

Cells from freshly isolated fetal livers were kept at 4 °C and stained for TFRC and LY76. Cells from each of the states 1 to 4 were sorted. Cells from the indicated states were mixed at the indicated proportions, and plated in 96-well format in IMDM, in the presence of 4 μl/500 ml of 2-mercaptoethanol, 20% fetal calf serum, and 0.05 U/ml EPO (Amgen,

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embryonic day [number]

ordinary differential equation

sum of squared residuals

^{−/−}5b

^{−/−}mice due to decreased survival of early erythroblasts.

^{−/−}5b

^{−/−}mice: A direct role for Stat5 in bcl-X

_{L}induction.