A microscale mathematical model for metabolic symbiosis: Investigating the effects of metabolic inhibition on ATP turnover in tumors

Highlights

• We model concentrations of metabolites in cancer tissue surrounding a blood vessel.

• The model captures metabolic symbiosis between glycolytic and respiring cells.

• The Warburg effect leads to higher ATP yields in a hypoxic tumor microenvironment.

• Optimized metabolic behavior includes lactate production and consumption.

• Glycolytic and lactate dehydrogenase inhibition can diminish energy production.

Abstract

Cancer cells are notorious for their metabolic adaptations to hypoxic and acidic conditions, and especially for highly elevated glycolytic rates in tumor tissues. An end product of glycolysis is lactate, a molecule that cells can utilize instead of glucose to fuel respiration in the presence of oxygen. This could be beneficial to those cells that do not have sufficient oxygen as it conserves glucose for glycolysis. To better quantify this phenomenon we develop a diffusion-reaction mathematical model for nutrient concentrations in cancerous tissue surrounding a single cylindrical microvessel. We use our model to analyze the interdependence between cell populations׳ metabolic behaviors on a microscopic scale, specifically the emerging paradigm of metabolic symbiosis that exists between aerobic and glycolytic cells. The ATP turnover rates are calculated as a function of distance from the blood vessel, which exhibit a lactate-consuming population at intermediate distances from the vessel. We also consider the ramifications of the Warburg effect where cells utilize aerobic glycolysis along with this lactate-consuming respiration. We also investigate the effect of inhibiting metabolic pathways on cancer cells since insufficient ATP can trigger cell apoptosis. Effects that could be induced by metabolic inhibitors are analyzed by calculating the total ATP turnover in a unit tissue annulus in various parameter regimes that correspond to treatment conditions where specific metabolic pathways are knocked out. We conclude that therapies that target glycolysis, e.g. lactate dehydrogenase inhibitors or glycolytic enzyme inhibition, are the keys to successful metabolic repression.

Introduction

The extensive metabolic requirements for cancer cell proliferation coupled with the harsh microenvironment in solid tumors culminate in a highly adaptive and complex network for cellular energy production. The genetically altered metabolic behavior of cancer cells has led to a number of emerging metabolic paradigms, in addition to those that are universally exhibited in both cancerous and normal cells. We will investigate this complex metabolic behavior by formulating a minimal mathematical model that includes the essential metabolites of glucose, lactate and oxygen in the tissue surrounding a microvessel. The cylindrical geometry used here has been used in a similar context to consider interactions between metabolites and tumor cells with treatment effects in a simplified setting (e.g. Bertuzzi et al., 2000, Bertuzzi et al., 2007a). The model presented here will enable the quantification of various behaviors, such as the symbiotic relationship that exists between lactate-producing glycolytic cells and lactate-consuming respiratory cells, and the analysis of metabolic dependence on various physiological conditions such as hypoxia and induced metabolic inhibition. Metabolic inhibition including glycolytic inhibitors among many others targets could be very important for cancer treatment since an ATP deficit can induce apoptosis (Izyumov et al., 2004). The key consideration for addressing this problem with mathematics is the formulation of nutrient consumption rates that encompass the various primary facets of cancer cell metabolism and their corresponding ATP yields.

In normal well-oxygenated tissues the primary source of ATP is the process of cellular respiration. The complete conversion of glucose to carbon dioxide and water has an ideal yield of about 29 ATP, although realistically the yield is substantially lower (Brand, 2005). The preliminary stage of cellular respiration is glycolysis, the conversion of glucose to pyruvate; this process directly produces 2 ATP. In hypoxic conditions this pyruvate is preferentially converted into lactate via the enzyme lactate dehydrogenase (LDH) to regenerate the essential cofactor NAD⁺. In oxygenated conditions this pyruvate is transported across the inner mitochondrial matrix where it is decarboxylated and enters the citric acid cycle; the citric acid cycle directly generates 2 more ATP per glucose. The primary energy payoff is a result of cofactor oxidization that enables the electron transport chain to establish a proton gradient across the inner mitochondrial matrix. ATP synthase utilizes this electrochemical gradient to drive the phosphorylation of approximately 25 additional ATP per glucose molecule.

The aforementioned universal traits that cancer cells and normal cells share include cellular responses to various levels of oxygen, lactate or glucose. Examples include a Crabtree-like effect and a Pasteur-like effect (Casciari et al., 1992a). The Crabtree-like effect is when oxygen consumption decreases as glucose concentration increases. This can be explained by an increasing reliance on glycolysis for ATP when hyperglycaemic conditions are encountered. The Pasteur-like effect is decreased glucose consumption as oxygen increases. This is due primarily to the inhibition of various metabolic steps by the presence of elevated ATP and other intermediaries. However, cancer cells are unique in that they preferentially utilize glycolysis, even in the presence of oxygen, coined aerobic glycolysis. This phenomenon is generally referred to as the Warburg effect whereby cells rely primarily on glycolysis even in the presence of sufficient oxygen to perform respiration (Warburg, 1956). There is a perceived inefficiency of this metabolic strategy, namely the dramatically reduced ATP yield, just 2 per glucose instead of 29, however, it has the benefits of faster ATP production and it is likely that much of this glucose is being consumed for proliferative (Vander Heiden et al., 2009) (e.g. by the pentose phosphate pathway) purposes. In addition to the typical glycolytic phenotype exhibited in many cancers, there is also a developing story of a co-operative relationship existing between aerobic and anaerobic cancer cells. The lactate necessarily produced by glycolytic cells is being pushed back into the respiratory cycle by being converted into pyruvate (summarized in Feron, 2009, Nakajima and Van Houten, 2012); this spatial relationship is shown in Fig. 1. Lactate consumption has been observed in vitro in various models (Bouzier et al., 1998, Katz et al., 1974) as well as in vivo as early as the early 1980s (Sauer et al., 1982). However, a renewed interest in the topic was piqued when Sonveaux et al. (2008) showed that reducing lactate uptake by cancer cells led to hypoxic cell death, a particularly difficult subpopulation to target using traditional methods.

Metabolic phenomena have been studied in great detail by mathematical models, but models of tumor metabolism rarely include the interaction of the transport mechanisms of microvessels with the localized metabolic behavior of cells (with one recent exception McGillen et al., 2013). In the section to follow, we will develop a mathematical model that describes the concentrations of molecules that are important to cellular metabolism in the tissue around a single three-dimensional vessel that exhibits diffusion-dominated interstitial transport. We will then use this model to demonstrate how the properties of the tumor cell population, such as glucose, lactate and oxygen consumption rates, affect tumor hypoxia and ATP production around a single vessel. The effects of metabolic inhibitors will be investigated by parameter changes that could be elicited by the application of glycolysis inhibitors, lactate dehydrogenase (LDH) inhibitors or respiratory inhibitors. We are interested in those metabolic inhibitors that could cripple the cells׳ ability to produce ATP. Furthermore, these micrometer scale predictions give an indication of local shifts in cell metabolism, which could aid in developing combination treatments that can simultaneously hinder multiple metabolic pathways in tumors. Our simulations show that treatments targeting glycolysis via glycolytic enzyme inhibition or LDH inhibition, which have been thoroughly investigated (Pelicano et al., 2006, Gatenby and Gillies, 2007, Granchi and Minutolo, 2012), could be the most successful metabolic suppression strategy.