The Warburg effect as an adaptation of cancer cells to rapid fluctuations in energy demand

Abstract

To maintain optimal fitness, a cell must balance the risk of inadequate energy reserve for response to a potentially fatal perturbation against the long-term cost of maintaining high concentrations of ATP to meet occasional spikes in demand. Here we apply a game theoretic approach to address the dynamics of energy production and expenditure in eukaryotic cells. Conventionally, glucose metabolism is viewed as a function of oxygen concentrations in which the more efficient oxidation of glucose to CO2 and H2O produces all or nearly all ATP except under hypoxic conditions when less efficient (2 ATP/ glucose vs. about 36ATP/glucose) anaerobic metabolism of glucose to lactic acid provides an emergency backup. We propose an alternative in which energy production is governed by the complex temporal and spatial dynamics of intracellular ATP demand. In the short term, a cell must provide energy for constant baseline needs but also maintain capacity to rapidly respond to fluxes in demand particularly due to external perturbations on the cell membrane. Similarly, longer-term dynamics require a trade-off between the cost of maintaining high metabolic capacity to meet uncommon spikes in demand versus the risk of unsuccessfully responding to threats or opportunities. Here we develop a model and computationally explore the cell's optimal mix of glycolytic and oxidative capacity. We find the Warburg effect, high glycolytic metabolism even under normoxic conditions, is represents a metabolic strategy that allow cancer cells to optimally meet energy demands posed by stochastic or fluctuating tumor environments.

Fig 1. Schematic representation of the demand-driven metabolic model.

ATP demand is composed of two types of demand; (dark) slow changing base-load demand, primarily for macromolecules synthesis and (light) peak demand, that rapidly changes primarily to support membrane transporters. In this model oxidative phosphorylation (high efficiency, slow response time) supplies base-load demand, and glycolytic metabolism (less efficient, fast response time) supplies peak ATP demand. The later is less efficient but is more highly responsive.

Fig 2. Typical dynamics of the system for different values of glycolytic capacity, c_g.

The optimal level of glycolytic capacity occurs where the curves in Panels E, F and G become flat. The left panels show the temporal dynamics of (A) ATP demand, d(t), (B) glycolytic ATP production, c_gP_g, (C) oxidative ATP production, c_op_o, and (D) ATP level U_ATP. Dash line denotes the minimum ATP level that is allowed in the system, U^*_ATP. Higher glycolytic capacity allows more glycolytic ATP to supply peak demand and reduces the maximum ATP level that is needed to ensure that U_ATP≥U^*_ATP. The right panels show (E) total glycolytic ATP production over one period, P_g^total, (F) total oxidative ATP production over one period, P_o^total, and (G) maximum ATP level over the period max(U_ATP). Parameters used in this simulation: Demand parameters: f = 10Hz, period = 4, BL = 5, k_m = 50, k_g = 10000, U^*_ATP = 0.745, P_g^max = 5 and c_m = 1.

Fig 3. Normalized cost terms.

(A) Normalized excessive ATP, A^norm, and (B) normalized glucose consumption, F^norm, as function of normalized glycolytic capacity, c_g^norm, for efficiency ration of f_g/o = 2. Dots are overlay of all values between PL = 1, 1¼, 1½, …, 5. Red lines are quadratic polynomial fit, which is used for calculating production cost for any value of peak-demand and glycolytic capacity. Whether presented as panel (A) or (B), the optimal level of glycolytic capacity that minimizes costs occurs at the point where normalized excess ATP or normalized glucose consumption reaches 0.

Fig 4. Optimal ATP production mode for peak-demand.

The effects of the costs of excess ATP, a, glycolytic capacity, c, glucose, f, and the production efficiency ratio, f_g/o, on whether it is optimal for the cell to use oxidative phosphorylation (o) or glycolysis (g) to meet peak ATP demand. As cost parameters change, there are abrupt shifts from one metabolic strategy to the other. The lines separating regions of oxidative versus glycolytic metabolism are called “isolegs” and their slopes and intercepts indicate how the cost parameters interact. Increasing the production efficiency ratio, f_g/o, decreases the region over which glycolysis is optimal. Decreasing a or c generally favors glycolysis. The effect of f on the optimal metabolism for meeting peak demand depends on the relative glucose efficiency of oxidation versus glycolysis, f_g/o. The values for all other parameters are the same as those in Fig 2.

Fig 5. Influence of peak-demand parameters on the steady-state population distribution.

Mean of population distribution as function power of the amplitude distribution, p, (bottom) and the mean time between events of demand, λ (left). The graph demonstrates that short time between bursts of peak-demand and high probability of high-amplitude demand results with a selection of population with high glycolytic capacity. Simulation parameters: k_drift = 0.05, Α_haz = 1, Α_opp = 1, Α_fix = 1 and m = 0.01. Detail description of the simulation process can be found in S3 Appendix.

Fig 6. Influence of opportunistic and hazardous coefficient on the steady-state population distribution.

Mean capacity that is selected as function of the opportunistic coefficient, α_opp, and the hazardous coefficient, α_haz for fixed-cost coefficient, α_fix = 1 (a) and α_fix = 0.5 (b). The graphs demonstrate that hazard is more influential in selection of high glycolytic capacity than opportunity