Fig 1.
Bone remodeling in multiple myeloma.
Multiple myeloma cells (MM) produce growth factors that activate osteoclasts (OC), which increase bone resorption, or that inhibit osteoblast (OB) differentiation. OC and OB secrete growth factors that affect each other and MM cells.
Table 1.
Multiplication factors for diffusible factors produced by osteoclasts (OC), osteoblasts (OB) and multiple myeloma cells (MM).
Table 2.
Multiplication factors for tumor-stroma interactions in multiple myeloma.
Fig 2.
Example of the dynamics for scenario 1.
In the presence of a small number of MM cells, the stable point on the OB-OC border becomes a saddle point and clonal selection leads to a stable coexistence of OC and MM cells. (N = 10, c3 = 1.4, c2 = 1.2, c1 = 1). The arrows show the direction of the dynamics, and the colors show its speed (the euclidean distance between the frequencies at time t and t+1).
Fig 3.
Effect of initial frequencies on the dynamics for scenario 1.
Different initial frequencies of OC, OB, and MM cells do not change the final state of the population. Parameters as in Fig 2.
Fig 4.
Fitness of the three cell types as a function of their contribution (cost) for scenario 1.
As the value of cost increases the fitness of OC and MM increases and the fitness of OB decreases. Same parameters as Fig 2.
Fig 5.
Dynamics with pairwise interactions in scenario 1.
The dynamics described in Fig 2 changes when N = 2, resulting in a new stable point between OC and OB, and a new polymorphic saddle point, in addition to the stable point between OC and MM. The arrows show the direction of the dynamics, and the colors show its speed (the euclidean distance between the frequencies at time t and t+1).
Fig 6.
Effect of group size in scenario 1.
The effect of group size N on the position of the fixed points on the OC-OB and OC-MM edges. Same parameters as Fig 2.
Fig 7.
Effect of the parameters in scenario 1.
The effect of a, b and d on the stable point in Fig 2. Changes in a and b, but not of d, change the stable point.
Fig 8.
Different types of dynamics for scenario 1.
(A) The game has one polymorphic stable point between OB and OC. In this case, clonal selection leads to the regular OC-OB balance and prevents invasion of MM cells. (B) The game has two polymorphic stable points. In this case, the final state of the game depends on the initial frequencies. The arrows show the direction of the dynamics, and the colors show its speed (the euclidean distance between the frequencies at time t and t+1).
Fig 9.
Examples of the dynamics for scenario 2.
When c1 = c2 = c3 = 1, the game has (A) one stable point on the OC-OB edge if a<b and a<2N/(N-1); (B) two stable points on the OC-OB and OC-MM edges if a<b and a<2N/(N-1), b+d>a and b<2N/(N-1); (C) one stable point on the OC-MM edges if b+d>a and b<2N/(N-1). When the costs are not equal (bottom panels: c1 = 1, c2 = 1.2, c3 = 1.4) the dynamics and the equilibria are different. N = 10. The arrows show the direction of the dynamics, and the colors show its speed (the euclidean distance between the frequencies at time t and t+1).
Fig 10.
Examples of the dynamics for scenario 3.
(A) For small group size (N = 10) the game has one stable point on the MM vertex. (B) If group size increases (N = 50) the game has two stable points. (c1 = 1, c2 = 1.2, c3 = 0.8). The arrows show the direction of the dynamics, and the colors show its speed (the euclidean distance between the frequencies at time t and t+1).
Fig 11.
Effect of reducing the fraction of MM cells.
(A) If a population is modified by reducing the fraction of MM cells (1, dotted arrow), it can be moved into the basin of attraction of the stable point on the OC-OB edge, and it will then evolve (2, continuous arrow) to the healthy OC-OB equilibrium. (B) Changes in frequencies over time corresponding to panel A, without therapy (dotted lines) or with therapy (continuous line) introduced at generation 300. (c1 = 1.2, c2 = 1, c3 = 1.4).
Fig 12.
Comparison with models with pairwise interactions.
A comparison of our model and the pairwise game of Dingli et al. [13] with b>1 (row A), b<1, b+d<1 (row B) or b<1, b+d>1 (row C). The arrows show the direction of the dynamics, and the colors show its speed (the euclidean distance between the frequencies at time t and t+1).