Figure 1.
Schematic representation of the multiscale model.
The “energetic scale” submodel [equations (1) through (10)] governs dynamics of adenylate (AMP, ADP and ATP). Interconversions among the three species occur via maintenance metabolism (e.g., biosynthesis, volume control), chemical energy to support proliferation and angiogenesis signal production, glycolysis and, most importantly, the adenylate kinase reaction, among others. Sources for adenylate include de novo synthesis of AMP and salvage from nucleic acid catabolism. Adenylate sinks include AMP destruction by deaminases and nucleotidases and ATP loss to nucleic acid synthesis. Clonal expansion or regression at the tissue scale [model (11)] depends primarily on mean tumor microvessel density, which is controlled in part by angiogenic factor secretion by existing clones. Blood vessels grow from existing vasculature via chemotaxis and maturation of vascular endothelial cell precursors in and near the tumor. At the evolutionary scale, angiogenic and proliferative potential varies among clones (different colored cell subpopulations) as they compete for resources delivered by microvessels. Evolutionary scale dynamics are handled in the simulation (see “Simulation Methods” above).
Table 1.
Dependent variables studied in this model.
Table 2.
Parameters and default values representing a resting cell (from [17]).
Figure 2.
Evolutionarily “dominant” proliferation commitment () in 1000 simulations for each of two mutation rates:
= 0.1 and 0.02 (plotted as mean interarrival time between mutations 10 and 50 hours, respectively).
All parameters except were set to the defaults in [17]. Shown are distributions of the histopathologist's “dominant” clone (clone with the most mass; blue) and the evolutionary biologist's dominant clone (clone with the largest per capita growth rate; black).
Figure 3.
Change over time in weighted mean proliferation effort, (see equation (13)).
Blue curves in both panels represent the first 20 of the 1000 simulations plotted in Fig. 2. Solid black curves are the ensemble averages of all 1000 runs, dashed black curves mark the inner 95th percentile range for all runs, and dotted black lines represent the ESS predicted by adaptive dynamics theory ( min−1). Dashed red lines represent mean time of the final mutation (
. (A)
; (B)
.
Figure 4.
Dynamics of the Shannon diversity index, , of individual tumors evolving in proliferation commitment,
.
Purple curves in both panels represent the first 20 of the 1000 simulations plotted in Fig. 2. Solid black curves are the ensemble averages of all 1000 runs. Dashed red lines represent mean time of the final mutation (). (A)
; (B)
.
Figure 5.
Evolutionarily dominant tumor angiogenesis factor commitment () in 1000 simulations for two values of proliferative effort (
min−1 and 3.65 min−1).
In both cases, . All other parameters except
were set to defaults from [17]. (Compare Fig. 2.)
Figure 6.
Change over time in weighted mean angiogenesis effort, .
Blue curves in both panels represent the first 20 of 1000 simulations plotted in Fig. 5. Solid black curves are ensemble averages of all 1000 runs, dashed black curves are the inner 95th percentile of all runs, and dashed red lines are mean time of the last mutation. (Compare Fig. 3.) (A) min−1; dotted horizontal line is the ESS from adaptive dynamics (
min−1). (B)
min−1; the ESS value of
min−1 is not shown.
Figure 7.
Per capita growth rate of various strategies against tumor vascularization.
Gray, dashed horizontal line represents zero growth rate. Solid lines represent clones with high proliferation commitment ( = 3.65 min−1), while dashed lines represent clones with low proliferation commitment (
= 1.085 min−1). Black lines represent the lowest angiogenic clones (
= 0 min−1), red lines; the highest angiogenic clones (
= 4 min−1), which encompasses the possible curves of all intermediate angiogenic clones.
Figure 8.
Dynamics of the Shannon diversity index, , of individual tumors evolving in angiogenic commitment (
) for two different constant proliferation commitments: (A)
min−1; (B)
min−1.
Purple curves in both panels represent the first 20 of the 1000 simulations plotted in Fig. 5. Solid black curves are the ensemble averages of all 1000 runs. Dashed red lines represent mean time of the final mutation (), with
.