Feedback mechanisms control coexistence in a stem cell model of acute myeloid leukaemia

Abstract

Haematopoietic stem cell dynamics regulate healthy blood cell production and are disrupted during leukaemia. Competition models of cellular species help to elucidate stem cell dynamics in the bone marrow microenvironment (or niche), and to determine how these dynamics impact leukaemia progression. Here we develop two models that target acute myeloid leukaemia with particular focus on the mechanisms that control proliferation via feedback signalling. It is within regions of parameter space permissive of coexistence that the effects of competition are most subtle and the clinical outcome least certain. Steady state and linear stability analyses identify parameter regions that allow for coexistence to occur, and allow us to characterise behaviour near critical points. Where analytical expressions are no longer informative, we proceed statistically and sample parameter space over a coexistence region. We find that the rates of proliferation and differentiation of healthy progenitors exert key control over coexistence. We also show that inclusion of a regulatory feedback onto progenitor cells promotes healthy haematopoiesis at the expense of leukaemia, and that - somewhat paradoxically - within the coexistence region feedback increases the sensitivity of the system to dominance by one lineage over another.

Keywords: Acute myeloid leukaemia; Cancer; Dynamical systems; Haematopoietic stem cells; Stability analysis.

Fig. 1

Mechanistic model of the interacting species. Upper row: haematopoietic linage. Haematopoietic stem cell (S), progenitor blood cell (A) and differentiated blood cell (D). Lower row: leukaemia linage. Leukaemia stem cell (L) and mature leukaemia cell (T). Healthy progenitors and LSCs occupy and compete for the same ecological niche (rounded box) within the bone marrow. Differentiated blood- and mature leukaemia cells migrate into the bloodstream. Negative feedback control indicated in red. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 2

Steady states of Model I for species S, A, D, L and T over parameters ρ_S, δ_S, ρ_A and δ_A, respectively. Remaining parameters are held at fixed values. Asymptotic and critical values are indicated; shaded regions denote coexistence.

Fig. 3

Steady states of Model II for species S, A, D, L and T over parameters ρ_S, δ_S, ρ_A and δ_A, respectively. Two solutions exist for each species. Asymptotic or critical values are indicated when solutions are real numbers. Dotted lines denote boundaries where at least one species' steady state becomes complex: no species are plotted in the regions where their solutions are complex; and critical values that lie within these regions are marked but left unspecified. Shaded regions denote coexistence.

Fig. 4

Density plots correlating parameters ρ_S, ρ_A, δ_S and δ_A. Model I (A) and Model II (B). Each pair of values is represented by a two-dimensional bin with brightness corresponding to count: form zero (white) to a maximum (dark green): the darkest regions highlight where features are most frequently observed. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

Fig. 5

Histograms summarise the probability of observing three features of interest over a range of parameter values for ρ_S, ρ_A, δ_S and δ_A. $A^{*}\left(L^{*}\right)$ denotes the value of species $A\left(L\right)$ at steady state.

Fig. 6

Dominance of species within the coexistence region of parameter space.

Fig. 7

Dominance of species within the coexistence region of parameter space.