A phase space model of hemopoiesis and the concept of stem cell renewal  Mark Alexander Kirkland  Experimental Hematology  Volume 32, Issue 6, Pages 511-519 (June 2004) DOI: 10.1016/j.exphem.2004.02.013

Figure 1 The standard model of hemopoiesis, showing self-renewing stem cells at the left of the diagram, with a hierarchy of cells of more limited potential to the right. Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)

Figure 2 A probability density function of stem cell renewal, showing the probability of the outcome of a single division of a cell indicated by the arrow. “Stemness” is an arbitrary measurement of the stem cell potential of the parental cell and its progeny. (A): Renewing system where the probability density function includes some progeny with greater stemness than the parent cell. (B): Nonrenewing system: if no de-differentiation is allowed in the system then all progeny will have less stemness than the parent cell, and the system will run down over time. Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)

Figure 3 (A): A simplified phase space model of hemopoiesis. Cells close to the terminal classes (a) have a very low probability of de-differentiating, and a very high probability of moving toward a terminal class. Cells further out (b) will have intermediate characteristics, while cells close to the origin (c) will have maximal “stemness” and a low cell cycle rate. However, cells at (a) and (b) have shorter cell cycle times and are present in larger numbers, and so, globally, make a significant contribution to the total “stemness” of the population. (B): The role of attractors in a phase space model. A cell at (a) will have a high probability of differentiation toward attractor 1 and a low probability of differentiation toward attractor 2, a cell at (b) the reverse. However, if only attractor 1 is present, then a cell at (b) might give rise to progeny that ultimately undergo differentiation toward that fate. A cell at (c), on the other hand, will have a very low (though not zero) probability of differentiating toward either attractor under any conditions. Note that these attractors might represent different hemopoietic growth factors, but might equally represent hemopoietic and nonhemopoietic growth factors. Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)

Figure 4 A regionalized phase space model can be derived from the model shown in Figure 3A. The length of the arrows indicates the probability of transition between regions. Lighter-colored regions at the center have the highest probability of de-differentiation (indicated by arrows directed toward the center). The point at which the probability of differentiation (outward pointing arrows) is only toward the terminal classes indicates the point of full commitment, and therefore the boundary of the stem cell pool (thick white line). If a single region encompassing all regions within this boundary is considered then this region is the equivalent of current stem cell models—for each cell division within that region there will be, on average, one cell leaving the stem cell pool and one cell remaining within it. Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)

Figure 5 Output of a simple Microsoft Excel spreadsheet model based on the regionalized phase space shown in Figure 4. Layer 1 is the central region (white in Fig. 4) while layers 2, 3, and 4 are the progressively more differentiated regions (progressively darker gray in Fig. 4). “Output” represents the cells leaving the stem cell compartment (bounded by thick white line in Fig. 4). The model was seeded with 1000 cells in layer 1 and utilized the parameters shown in Table 1. The output is essentially stable after 50 iterations, though with oscillations. The number of cells in layer 1 falls to around half the starting number, then stabilizes, even though the total stemness of the system is the same at the end of 200 iterations as it was at the beginning. Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)

Figure 6 Differential output of individual seeded cells. When the relative contribution of individual cells seeded in the model shown in Figure 5 is analyzed, it is evident that many of the seeded cells are lost with time. Of 30 cells analyzed, only 18 remain after 25 time intervals, falling to 11 after 50 and 10 after 75 time intervals. Some clones that were major contributors to the output between intervals 50 and 75 (e.g., clone 10) then fall away, to be replaced by others. Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)

Figure 7 Cumulative distribution of total cells produced from individual seeded stem cells (triangles) closely matches a gamma distribution (solid gray line, derived using the GAMMADIST function in Microsoft Excel with parameters α=0.45, β=36, cumulative=true). Experimental Hematology 2004 32, 511-519DOI: (10.1016/j.exphem.2004.02.013)