Why are cell populations maintained via multiple compartments?

Abstract

We consider the maintenance of 'product' cell populations from 'progenitor' cells via a sequence of one or more cell types, or compartments, where each cell's fate is chosen stochastically. If there is only one compartment then large amplification, that is, a large ratio of product cells to progenitors comes with disadvantages. The product cell population is dominated by large families (cells descended from the same progenitor) and many generations separate, on average, product cells from progenitors. These disadvantages are avoided using suitably constructed sequences of compartments: the amplification factor of a sequence is the product of the amplification factors of each compartment, while the average number of generations is a sum over contributions from each compartment. Passing through multiple compartments is, in fact, an efficient way to maintain a product cell population from a small flux of progenitors, avoiding excessive clonality and minimizing the number of rounds of division en route. We use division, exit and death rates, estimated from measurements of single-positive thymocytes, to choose illustrative parameter values in the single-compartment case. We also consider a five-compartment model of thymocyte differentiation, from double-negative precursors to single-positive product cells.

Keywords: branching process; cell fate; clonality; generation; probability generating function; progenitor cell.

Figure 1.

The one-compartment system. A single progenitor cell (shown on the left, green) is the founder of the population. In the compartment (represented by the dashed box), each cell (shown as a blue filled circle), independently, may die, divide or ‘exit’. An exit event is the differentiation of a cell to product cell type (shown as a red empty circle). The random variable R is the number of product cells when no cells remain in the compartment. We count the product cells as a cumulative total and do not consider any death or division events of product cells. The quantity N = IE(R) is the ‘amplification factor’: the mean number of product cells per progenitor.

Figure 2.

The multiple-compartment system. A single progenitor cell (shown on the left, green) is the founder of the population. Each cell in compartment c, independently, may die, divide or transit from compartment c to compartment c + 1, where c = 1, …, C − 1. Cells that exit compartment C are product cells (shown in red). The overall amplification factor N is the mean number of product cells per progenitor, which is the product of the amplification factors in each compartment.

Figure 3.

We classify the set of product (red) cells according to generation (number of divisions from the progenitor cell). The progenitor cell is said to be in generation 0. Whenever a cell in generation n divides, the result is two daughter cells in generation n + 1. The final state of the process is a population of red cells, each having made the transition at a different time and each with its own generation number. The case C = 1 is illustrated here. If C > 1 then the mean number of divisions in the product population is the sum of the mean numbers of divisions in each compartment.

Figure 4.

The quantity q_k is the probability that k cells exit a compartment, descended from one progenitor cell. Results, using (2.12), are shown for two different choices of p_b and p_e. (a) We use the estimates of Sawicka et al. [14]: p_b = 0.4004 and p_d = 0.0885 for SP4 thymocytes. (b) Their estimates for SP8 thymocytes: p_b = 0.2449 and p_d = 0.3170.

Figure 5.

(a) The probability, q_k (using (2.12) and (2.16)), that the number of product cells is k, logarithmic scales, with and without death. The dashed line is the power law q_k = k^−3/2. (b) k^3/2q_k in the same two cases. The vertical dotted lines, at k = 6N²/(1 − 2p_d), indicate where the power law ceases to be an accurate approximation. The parameter values, calculated using (2.13) so that N = 2.57 in both cases, are p_d = 0, p_b = 0.455, p_e = 0.545, and p_d = 0.0885, p_b = 0.4004, p_e = 0.5111. The latter set of values corresponds to those of SP4 thymocytes, as discussed above.

Figure 6.

Plot of k^3/2Q_k(C) as a function of k, with logarithmic scales, for C = 1, C = 2 and C = 10. The distribution of R narrows as the number of compartments increases. The solid lines are the exact results, computed using (2.16) and (A 10). The dots are averages obtained from Gillespie realizations. Parameter values, chosen using (2.13) with N = 25, are C = 1: p_d = 0, p_b = 0.4898; C = 2: p_d(1) = p_d(2) = 0, p_b(1) = p_b(2) = 0.4444 and N₁ = N₂ = 5; C = 10: p_d(c) = 0, p_b(c) = 0.2158 and N_c = 1.38 for each c = 1, …, 10.

Figure 7.

One realization with C = 1, showing generation numbers from left to right, with Z₀ = 1. Cyan cells divide, red cells exit and black cells die. In this realization Y₀ = 0, Y₁ = 1, Y₂ = 0, Y₃ = 1, Y₄ = 2 and Y₅ = 2. Thus, we have R = 6. The parameter values are p_b = 0.45 and p_d = 0.15.

Figure 8.

Lines of constant D (red) and lines of constant N (blue) in the part of the plane representing possible parameter values. The two quantities characterizing the population of cells exiting a compartment, as functions of p_b and p_d, (2.7) and (4.5). Each blue line is the set of pairs (p_b, p_d) corresponding to the indicated value of N. Each red line is the set of pairs (p_b, p_d) corresponding to the indicated value of D. The triangular part of parameter space corresponding to N > 1 is at bottom right. The green dots are the estimates of Sawicka et al. [14], for SP4 and SP8 thymocytes.

Figure 9.

One realization with C = 2, showing generation numbers from left to right. Cells in the first compartment are shown as circles, and cells in the second compartment as squares. Cyan cells divide, red cells exit and black cells die. Arrows indicate a transition from the first to the second compartment. In this realization Y₀(1) = 0, Y₁(1) = 0, Y₂(1) = 1, Y₃(1) = 2, Y₄(1) = 1 and Y₅(1) = 0; Y₀(2) = 0, Y₁(2) = 0, Y₂(2) = 0, Y₃(2) = 1, Y₄(2) = 3 and Y₅(2) = 0. Thus, we have R = 4. The parameter values are C = 2, p_b(1) = p_b(2) = 0.45 and p_d(1) = p_d(2) = 0.15.

Figure 10.

Average generation number of product cells, as a function of the mean number of exiting cells. (a) Plot for the case C = 1. (b) Plot for the case C = 2, with parameters chosen so that N₁ = N₂. Given a value of N, D is lower when C = 2 (proportional to $\sqrt{N}$ as N → +∞) than when C = 1 (proportional to N as N → +∞).

Figure 11.

The probability distribution of the random variable G, the generation number in the product cell population. One, two and three compartments have been shown. In all cases, N = 100, and all compartments are identical. Solid lines correspond to p_d = 0 and dotted lines to p_d = 0.05.

Figure 12.

(a) Mathematical model of T cell development from the DN3a to the SP stage [81]. (b,c) Numerical results for two cases of the five-compartment thymus model. The histograms show the distributions of family sizes and of cell generation number in the population of product cells. The difference between the two cases is the first compartment, where only death and asymmetric division have non-zero probabilities. Table 1 gives the probabilities for all five compartments, and quantities derived from them.

Figure 13.

The standard deviation of R as a function of the mean of R, N, for different values of C. The lines use the formula (B 1), and each line corresponds to one value of C. The dots are obtained as averages over numerical realizations. Parameter values have been chosen so that N_c is independent of c, p_d(c) = 0, and thus, N_c = N^1/c, p_e(c) = 1 − p_b(c) and p_b(c) = (N_c − 1)/(2N_c − 1), for all c = 1, …, C.

Figure 14.

The distribution of R, with and without asymmetric division, when C = 1. In red, the symmetric case (2.12), p_a = 0, and in blue, the purely asymmetric case, p_e = 0, generated using (C 9). In both cases, we have chosen N = 25 and p_d = 0.25.

Figure 15.

Lines of constant N (blue) and curves of constant D (red) in the part of the plane representing possible parameter values when p_a = 0.2. Each blue line is the set of pairs (p_b, p_d) corresponding to the indicated value of N. Each red curve is the set of pairs (p_b, p_d) corresponding to the indicated value of D.