Fig 1.
Multi-scale modeling of cancer metabolism.
(A) Flux Balance Analysis (FBA). The arrows represent fluxes of species within a reduced representation of cell metabolism and cell growth; the detailed network used in FBA is presented in S1 Fig. Key steps associated with three hypotheses are labeled: ① Warburg effect, ② Reverse Warburg and ③ Glutamine addiction. The uptake and production rates are qi/n [g/g-DW-hr] for the ith metabolite and the nth metabolic phenotype. We impose the maximum growth rate, μm,n [hr-1], for a given metabolic phenotype of each cell type as an objective function within the FBA. (B) Cell: Biomass (Xm [g]) growth of each cell type is modeled as a Monod-like process parameterized by the same maximum growth rates used in FBA that are modulated by functions of metabolite concentrations, fn({Cj})Monod. The change in Volume of the cell (Vm [L]) is calculated from biomass growth by applying a constant density of the cell (ρ[g-DW/L]). Yield coefficients (Yi/n [g-DW/g]) for each metabolite (i) and corresponding metabolic phenotype (n) are defined in terms of the uptake and production rates (qi/n) obtained from FBA. Extracellular space: Species balances for each explicit metabolite follow reaction-diffusion kinetics and govern the concentration profiles of metabolite at the multicellular scale. These equations (Eqs 1–4 in text) are integrated into and solved within an agent-based model (ABM—iDynoMiCS). (C) ABM simulations: i) Radial, two-dimensional growth: Tumor cells grow radially out from an initial cluster of cells with metabolites supplied at the edge of the cell mass such that radial gradients of concentration emerge (color map–red is high and blue is low concentration). Two phenotypes are displayed (red: tumor cells and blue: stromal cells). As the tumor grows, concentration gradients of metabolites become significant, making the tumor growth a diffusion-limited process that can result in different growth dynamics as well as distinct spatial distribution of cell subpopulations. ii) Axial, one-dimensional growth: Layers of tumor cells (red) and stromal cells (blue) are initiated near a blood vessel that supplies metabolites (from the top), such as glucose and oxygen in the blood stream. Growth pushes cells deeper into the tissue, away from the vessel, such that strong gradients of metabolite can again occur. iii) Krogh length calculation: To evaluate the impact of diffusion limitations in a simple model, we treat cells as continuum with uniform, zeroth order kinetics of metabolite consumption to calculate the distance over which the concentration of limiting metabolites falls to zero within the tumor mass; we refer to this distance as the Krogh length of a given metabolite.
Fig 2.
Metabolic profiles of various cell types in different hypotheses.
Values of metabolic fluxes [mmol/g-DW-hr] under different metabolic phenotypes obtained from the FBA are shown in boxes. Blue: normoxic phenotype. Orange: hypoglycemic phenotype. Black: hypoxic phenotype. (A) Representation of the Warburg Effect hypothesis includes: (i) Stromal cell: Normoxic stromal cells are quiescent and aerobic (use mainly OXPHOS to generate ATP for maintenance) (blue values). Hypoxic stromal cells are quiescent and anaerobic (use primarily glycolysis to generate ATP for maintenance) (black values). (ii) Warburg tumor cell: Normoxic Warburg tumor cells are highly proliferative and aerobically glycolytic (use mainly glycolysis to generate ATP to grow—blue values). The values of flux shown represent the metabolic phenotype of Warburg Number, WN = 2. Hypoxic Warburg tumor cells are quiescent and anaerobically glycolytic (use glycolysis to generate ATP for maintenance—black values). (B) Representation of the Reverse Warburg effect includes: (i) Hijacked stromal cell: Normoxic hijacked stromal cells are quiescent and undergo aerobic glycolysis (use mainly glycolysis to generate ATP for maintenance—blue values). Hypoxic hijacked stromal cells are quiescent and anaerobic (use mainly glycolysis to generate ATP for maintenance—black values). (ii) Reverse Warburg tumor cell: Normoxic reverse Warburg tumor cells are highly proliferative and uptake lactate aerobically; however they utilize OXPHOS to generate ATP to grow, fueled by lactate and oxygen instead of undergoing glycolysis using glucose (blue values). Hypoglycemic reverse Warburg tumor cells are quiescent and consume lactate to fuel mitochondria for maintenance (orange values). Hypoxic reverse Warburg tumor cells are quiescent and undergo anaerobic glycolysis to produce lactate (black values). Note the different directions of arrows for lactate fluxes. (C) Representation of Glutamine Addiction includes: (i) Stromal cell: Normoxic stromal cells are quiescent and aerobic (use mainly OXPHOS to generate ATP for maintenance—blue values). Hypoxic stromal cells are quiescent and anaerobic (use primarily glycolysis to generate ATP for maintenance—black values. (ii) “Glutamine-addicted” tumor cell: Normoxic “glutamine-addicted” tumor cells are highly proliferative and aerobic; instead of utilizing glucose in glycolysis, they undergo OXPHOS to generate ATP to grow, fueled by glutamine and oxygen (blue values). Hypoglycemic “glutamine-addicted” tumor cells are quiescent and consume glutamine to fuel mitochondria for maintenance (orange values). Hypoxic “glutamine-addicted” tumor cells do not consume glutamine. They are quiescent and undergo anaerobic glycolysis (black values). Note the different directions of arrows for glutamine flux.
Table 1.
Summary of uptake/production rates and yield coefficients of metabolites under different cellular metabolic phenotypes.
Fig 3.
Distribution of cells in radial simulations.
Initial conditions (t = 0) and end points (t = 100 days) are shown for the three hypotheses with cells seeded initially with 1, 3 and 5 layers of stromal cells surrounding the tumor cells.
Fig 4.
Axial agent-based model of growth of Warburg tumor cells in a perivascular tissue.
(A) Snapshots of predicted cellular structure and concentration fields of metabolites at different times. Tumor cells with one, three, and five of layers of healthy stromal cells separating them from the source of nutrients (top, representing interface with blood). Color bars present concentrations of oxygen and glucose in g/L. (B) Growth trajectories of tumor cells from simulations in three cases in (A). For each case, the trajectories for 11 independent simulations are shown. Initial positions of cells were randomly generated within the corresponding stromal or tumor compartment (see Methods). Empty circle: one layer of stromal cells; time = 0 day. Filled circle: one layer of stromal cells, t = 25 days. Empty diamond: three layers of stromal cells; t = 25 days. Filled diamond: three layers of stromal cells; t = 50 days. Empty square: five layers of stromal cells; time = 50 days. Filled square: five layer of stromal cells; t = 200 days.
Fig 5.
(A-C) Comparisons of growth curves from axial simulations as in Fig 4A for Warburg tumor cells with Warburg Number, WN = 0 (Control), 2, 10, and 34 with 1, 3 and 5 layers of stromal cells on top. Each time point represents the average of 11 simulations; error bars represent standard deviation. Note differences in vertical scales on plots. (D) Comparison of growth rate of tumor cells at early and late times (see Methods), extracted from the average growth curves in (A-C). (E) Concentration fields of metabolites in the case of 5 layers of stromal cells at t = 150 days. (F) Krogh lengths of oxygen and glucose based on consumption of tumor cells vs. Warburg Number. Solid line: Oxygen. Dotted line: Glucose.
Fig 6.
(A-C) Comparison of growth curves from axial simulations of tumor cells between the Warburg effect (WN = 2) and Reverse Warburg effect when 1, 3 and 5 layers of hijacked stromal cells are seeded between the source and tumor cells. Each time point represents the average of 11 simulations; error bars represent standard deviation. Note differences in vertical scales on plots. (D) Comparison of growth rate of tumor cells at early and late times, extracted from the average growth curves in (A-C). (E) Concentration fields of metabolites in the case of 5 layers of stromal cells at t = 0.
Fig 7.
(A—C) Comparison of growth curves from axial simulations of tumor cells between the Warburg effect and Glutamine addiction when 1, 3 and 5 layers of stromal cells are imposed in between the source and tumor cells, respectively. Note differences in vertical scales on plots. Each time point represents the average of 11 simulations; error bars represent standard deviation. (D) Comparison of growth rate of tumor cells at early and late times (see Methods). (E) Concentration fields of metabolites in the case of 5 layers of stromal cells at t = 0.