An integrated multidisciplinary model describing initiation of cancer and the Warburg hypothesis
© Rietman et al.; licensee BioMed Central Ltd. 2013
Received: 13 February 2013
Accepted: 29 May 2013
Published: 10 June 2013
In this paper we propose a chemical physics mechanism for the initiation of the glycolytic switch commonly known as the Warburg hypothesis, whereby glycolytic activity terminating in lactate continues even in well-oxygenated cells. We show that this may result in cancer via mitotic failure, recasting the current conception of the Warburg effect as a metabolic dysregulation consequent to cancer, to a biophysical defect that may contribute to cancer initiation.
Our model is based on analogs of thermodynamic concepts that tie non-equilibrium fluid dynamics ultimately to metabolic imbalance, disrupted microtubule dynamics, and finally, genomic instability, from which cancers can arise. Specifically, we discuss how an analog of non-equilibrium Rayleigh-Benard convection can result in glycolytic oscillations and cause a cell to become locked into a higher-entropy state characteristic of cancer.
A quantitative model is presented that attributes the well-known Warburg effect to a biophysical mechanism driven by a convective disturbance in the cell. Contrary to current understanding, this effect may precipitate cancer development, rather than follow from it, providing new insights into carcinogenesis, cancer treatment, and prevention.
The metabolic shift from aerobic to anaerobic glucose biochemical energy processing by cells is strongly correlated with the transition to cancer, or as some have come to characterize the process, a reversion to a more primitive and competitive level of cellular existence (Warburg , Szent-Gyorgyi ), which may still possess some rudimentary cooperative elements e.g. early metazoans (Davies and Lineweaver ). Our focus in this manuscript is to develop a molecular physics model based on non-equilibrium thermodynamics to quantitatively describe that process. By better understanding this transition we should be able to not only address cancer more effectively but also other metabolic diseases including mitochondrial diseases (e.g. ) and diseases of proton pumps (e.g. ). This modeling approach may also shed some light on the relationship between the Warburg effect for cancer and the so-called inverse-Warburg effect  for neurological diseases e.g. Alzheimer’s disease .
where the symbols E, S, ES, P represent the enzyme, substrate, enzyme-substrate complex, and reaction product, respectively. The coefficients labeled by k’s represent forward, reverse and enzyme-substrate decomposition rate constants as indicated by their subscripts. Obviously, if there is a huge abundance of S and limited amount of E, the reaction is rate limited by the concentration of E. But if the cell is malfunctioning and producing an excess of E when an abundance of S is present, then the cell will increase the P concentration by massively parallel reactions. This is governed by the well-known Michaelis-Menten reaction kinetics of saturable chemical reactions . This effect is also described by the Le Chatelier principle of reaction dynamics . It states that chemical reactions move forward or backward so as to reduce excesses in the quantity of reactants or products, respectively, introduced into in the reaction vicinity.
We argue that non-equilibrium thermodynamics is a driving force for the ultimate transition of a cell from the normal to the cancer state, and that the first step in the process is an increased operation of the cytoplasmic glucose processing machinery due to a non-equilibrium mechanism analogous to Rayleigh-Benard convection (Figure 1) [16, 17]. Our proposed scheme that describes the development of the cancer phenotype at a cellular level is essentially a biophysics model, which should clearly be considered in parallel with more biochemical schemes [18–22]. We begin by reviewing some relevant physics of nonlinear dynamics and thermodynamics.
Most of the remainder of the paper will be devoted to present arguments and analyses aimed at supporting the use of Equation (4) in the context of the Warburg effect.
Several in vitro and in vivo experiments have already demonstrated that an increase in extracellular glucose [30, 31] or lactic acid  concentration can lead to increased tumorigenesis. Further, hypoxia has been linked to causing the metabolic shift to glycolysis, and to cause malignant progression, with much experimental evidence [33–38].
Goldbeter  discusses at length glycolytic oscillations, and uses a series of differential equations to model the dynamics for some of the chemical species involved in the reactions. The observed oscillations come about as a result of the fact that there is a delay in manufacture of the intermediate nicotinamide adenine dinucleotide (NADH) and the fact that there is a finite number of molecular components available for the glycolytic processing. The cycle time has been observed in S. carsbegensis to be about 5 minutes using fluorescence of the glycolytic intermediate, NADH . More interestingly, Hess et al.  measured the oscillation frequency for different doses of fructose or glucose as input. As shown in Figure 3, as the concentration of the glucose (or fructose) increases, the period of oscillation decreases. This can only come about from increased concentration in the number of enzymes to participate in the reactions. As the number of enzymes and the number of molecular components increase, so does the entropy because the increase in numbers enables more ways to dissipate free energy, in this case represented as chemical potential of a glucose gradient . As the concentration increases the cycle time decreases indicating a more efficient processing of glucose per time unit.
More recently, Aromolaran et al.  describe, and show experimentally, glycolytically generated adenosine triphosphate (ATP) and Ca2+ waves propagating through a cell from application of glycolytic inhibitors focally injected from a glass pipette with a 1.5 micron diameter tip. The authors discover that glycolytically generated ATP is likely a key modulator of Ca2+ homeostasis. Of course, this has direct effect on the permeability of the mitochondrial wall, as shown by Yang et al.  who describe glycolytic oscillation depolarizing the mitochondrial membrane. The authors describe a model based on the logistic function showing there is a region where the oscillations are “too rapid for observation.” Though they do not use the term ‘chaos’, this is likely a chaotic state observed in the logistic functions and other “chaotic dynamical functions.”
In the case of Rayleigh-Benard convection rolls, as shown in Figure 2, the rolls can be modeled with a sine-circle map, for example, θi+1 = f(θ i ) where the function is periodic in the angle [44, 45]. There are some theoretical arguments for the glucose oscillators being embedded in the cell membrane. Demetrius et al.  argue that the enzymes’ concentration would oscillate due to periodicities in the redox potential and the result could be modeled as harmonic oscillators. Further, Tyner et al.  measure electrical gradients in the cell, and we show that a relevant protein, glyceraldehyde 3-phosphate dehydrogenase (GAPDH) associated with glucose processing in the cytoplasm accumulate at the membrane thus showing experimental support for our hypothesis that molecular oscillators accumulate at boundaries. Lastly, Pokorny  suggests Duffing oscillators  as a potential model of oscillatory states of a cell. In the following subsections we first provide some experimental validation for the differences in GAPDH localization. Then we follow that with some simplistic analytical modeling to show phase locking of oscillators potentially resulting chaos and disruptions of mitochondria.
Experimental validation for GAPDH distribution
These predictions of GAPDH distribution were tested by using antibodies against GAPDH in normal human mammary epithelial cells (HMECs) under normal culture conditions, purchased from Life technologies.
These clear experimental results support our conjecture that there is an increase in the glycolytic processing as a result of an external glucose gradient. We now describe some analytical models outlining the phase locking of the glucose oscillators and the implications.
Sine circle Map modeling
There has been research showing a quasi-periodic route to chaos in biochemical systems [21, 39, 50, 51]. The biochemical system, in these cases, glycolysis, starts at some given frequency and as external glucose is pulsed, starting at a frequency similar to the glycolysis value, the biochemical reactions begin to phase lock with the external pulses and first results in increased amplitude followed by period doubling as the external glucose pulse frequency increases. Finally, at a stochastic external glucose pulse frequency, the observed internal pulses are only about 3–4 X higher than the original glucose oscillations, but the actual sine wave appears as if there is a frequency and/or amplitude modulation. The oscillations observed in the cell extracts are not chaotic. Analytical models of these biochemical systems [52, 53], based on differential equations, support a periodic-doubling route to chaos, though in the real biochemical systems chaos is rarely observed. This is likely due to the fact that a cell is not just “a bag of chemicals” but really a hybrid between a chemical network and molecular nanomachines. The equations of dynamics also apply to mechanical systems, yet real machines are not capable of being driven into chaotic state, because of friction and physical constraints in the machine. Similarly, nanomachines are not likely able to be driven into chaotic states because of secondary bonding effects, e.g. van der Waals forces, and steric hindrance and potential energy surfaces .
In the above, cell-free extract, experiments, the investigators monitored NADH to observe the oscillations in the range of 0.002 to 0.005 Hz. Yang, et al.  did experiments on tissue models of rabbit ventricular myocytes and observed adenosine diphosphate (ADP) oscillations. Of course, when combining these experimental observations it must be remembered that there are 2 NADH molecules and 4 molecules of ADP per glucose molecule in the glycolysis reaction. Yang et al.  observed oscillation increases from about 0.02 to 0.067 Hz; but like the cell-free experiments, they did not observe chaos. Since chaos has been observed in real-world chemical systems , the lack of observed chaos in these biochemical and tissue experiments can be explained as follows.
In the case of tissue-based experiments, the oscillators are operating at a fixed frequency all driven by chemical kinetics and Le Chatelier’s principle. The observation of increase in frequency is likely the mean-field effect from an increase in the numbers of oscillators and this would also account for the observed effect on decrease of energy-density-rate as the concentration increases. This means the cell is adapting to the excess external glucose by producing more glycolysis oscillators. As will be shown shortly, these oscillators can phase lock with each other and produce oscillations at a frequency about 2 or 3 times higher.
In the case of the cell-free extract, it is not likely that more oscillator-components are being produced on demand, so Le Chatelier’s principle will not be modulating the overall molecular network. Instead, the existing molecular components for oscillator construction are fixed, and more in situ oscillators may form because of the excess glucose. Again these oscillators can phase lock and produce the observed frequencies.
Kaneko  discusses many options of coupling the oscillators. In all cases the phase diagram contains regions of fixed points, oscillations, and undefined or chaos. This is to be expected from the bifurcation of the function as shown in Figure 6. The hard boundary in our phase diagram at κ = 2 is to be expected as shown in Figure 7. The fact that the oscillations begin prior to 2.5 is due to the fact that the thresholding and signal transfer changes the dynamics. The regions in the phase diagram labeled n are for n-cycle, or strange attractor .
In mapping this phase diagram to glycolytic oscillations we would not expect all threshold values to be valid. If we set the threshold to 0.5 then the bifurcation parameter represents the glucose dosage and we have 2 cycles over a small range until the 4-cycles followed by continued increase in the glucose (bifurcation parameter) results in n-cycles. A 4-cycle in this phase space would look like a doubled 2-cycle, amplitude modulated glycolytic oscillator; and an n-cycle would look like an amplitude or frequency modulated 2-cycle or 4-cycle. These types of modulations have been observed by Hess et al.  and von Klitzing and Betz .
We now turn our attention to the mitochondria and explore the possible implications from glycolytic oscillations on mitochondrial stability. It is known that the glycolytic oscillations result in similar pH oscillations (e.g. Hess et al. ). But these oscillations are about π/2 out of phase. This has implications on the polarization of the mitochondrial membrane. The cell contains buffer mechanisms to minimize pH imbalance, but too much polarization of the mitochondrial membrane will cause the mitochondria to break down. The phase lag in pH can lead to a potential problem.
The ratio of V f /K m shows an actual phase transition at pH 7.5. The authors did not discuss the significance of this phase transition. Later work by Aromolaran et al.  showed waves of Ca2+ ions traversing the cell as a result of a localized ATP perturbation. These waves are able to traverse the entire cell within 30 seconds – far faster than diffusion. Other work by Ramanujand and Herman  show a nonlinear scaling of glucose metabolism in normal and cancer cells, where the scaling exponent is different for both types of cells. This is analogous to our observed Φ variation as a function of glucose. Lastly, Aon et al.  describe experiments on percolation and criticality  in mitochondrial networks of a cell. The authors used a local perturbation induced by a two-photon laser excitation. They observed a cell-wide transition to take place within 4 seconds resulting in depolarization of the majority of the mitochondria in the cell.
This depolarization of the mitochondria membrane also accompanies an ATP depletion. This can in turn effect the following reactions :
D-glucose + ATP → D-glucose-6-phosphate + ADP
D-fructose-6-phosphate + ATP → D-fructose-1,6-bisphosphate + ADP
1,3-bisphosphoglycerate + ADP → 3-phosphoglycerate + ATP
phosphoenolpyruvate + ADP → pyruvate + ATP
ATP is needed to maintain Ca2+. A lag in production of ATP, as the above reactions compete with other reactants and products in the overall molecular network, could induce changes in the cytoskeleton via pH effects on the growth dynamics of the microtubules.
amongst other reactions. Using Le Chatelier’s principle we can argue that forcing this reaction in the reverse direction by using 3-bromopyruvate will deplete the ATP and thus induce the cancer cell to enter into an apoptotic state. This type of behavior has in fact been reported by Ko et al.  and Mathupala et al. , who reported significant reduction in tumor volume in mice by treatment with 3-bromopyruvate.
The metabolic shift to glycolysis leads to acidosis, which subsequently results in an acidic extracellular pH . Gillies has documented an acidic extracellular environment in numerous tumors . Several studies link acidosis to genomic instability. Morita et al. show low pH leads to sister-chromatin exchanges and chromosomal aberrations [71, 72]. Brusick, Cifone, and Cipollaro report that low pH environments (~pH 6.5) caused genomic toxicity [73–75]. Several studies link the tumor environment, with hypoxia and low pH as inducing genomic instability by DNA repair activity being reduced , and enrichment for mismatch-repair deficient cells .
Thus, convincing evidence exists that a hypoxic and acidic environment will lead to genomic instability due to impairment of DNA repair processes.
In addition, Suresh  has reviewed the biomechanics of normal and cancer cells and shows that transformed cells routinely have altered deformability. This effect is likely due to the differences in the actin microfilaments – the major structural element of a cell [79, 80].
The cytoskeleton also consists of microtubules that have a bending stiffness 2.6 X 10-23 N m2 which is about 1000 times stiffer than actin filaments . Microtubules, unless stabilized by ligands or microtubule-associated protein, are in a constant dynamic instability process being polymerized and depolymerized with a half-life of about 2 minutes . This dynamic instability phenomenon has been mathematically modeled in detail by Sept et al. , and Bolterauer et al.  among others. The ends of actin filaments and microtubules are caped with ATP and guanosine triphosphate (GTP), respectively. GTP is created by the citric acid cycle in the mitochondria. Obviously, a mitochondria failure may reduce the concentration of ATP and GTP, if other systems do not compensate for the mitochondrial failure. A reduction in GTP results in a higher rate of depolymerization of microtubules  and a decrease in ATP concentration reduces the rate of growth of actin filaments . Too few microtubules and/or microtubules being too short to participate in proper spindle pole formation can lead to mitotic catastrophe and/or potentially lead to chromosome instability. Bakhoum et al.  and Thompson et al.  describe the mechanism for this instability. Persistent mis-oriented attachment of chromosomes to the spindle microtubules leads to severe chromosome segregation defects.
We have described an integrated system model for the progression of a healthy cell to a cancer state and some of the implications. The potentially aberrant state of the cell may start by an excess glucose or other nutrient external to the cell impacting the cell or by internal defects leading to metabolic enzyme redistribution processes. This excess nutrient is essentially a chemical potential difference between inside and outside of the cell creating stress. Through a process analogous to Rayleigh-Benard convection, stable molecular oscillators accumulate in the cytoplasm to exploit this chemical gradient. The continued activity of these oscillators results in mitochondrial destabilization, which may occur as a non-equilibrium phase transition. Once the mitochondria begin to perform aberrantly there will be a chemical imbalance in key components for microtubule assembly/disassembly. This imbalance is driven by Le Chatelier’s principle. The disruption in microtubule lengths and/or microtubule count will lead to chromosomal instability via kinetochore-microtubule dynamics finally leading to mitotic failure. In unlucky cases this will result in chromosome mis-segregation and cancer if mitotic catastrophe does not occur.
The above synthesis of ideas surrounding the subject of Warburg initiation, or the transition from aerobic glycolysis to anaerobic suggests not only an avenue for treatment but also an avenue for prevention of cancer.
We hypothesize that excess of glucose and sugar-like energy sources or metabolic enzyme abnormalities, through a non-equilibrium phase transition (a symmetry breaking phenomenon) analogous to the Rayleigh-Benard convection, may cause a cell to prefer to process this energy source using substrate glycolysis. Continued excess substrate glycolysis will cause further phase transitions to disrupt the mitochondria via depolarization and also disrupt microtubule dynamics. When a cell then passes through mitosis, the chance of mitotic failure is increased. When a cell enters mitotic failure, it may undergo an aneuploidy event [88, 89]. All this suggests that a low glycemic diet would lower the incidence of cancer, and may suggest a mechanism why metformin, which lowers blood glucose levels, is associated with improved outcomes in diabetic cancer patients [90, 91] and reduced risk of pancreatic cancer .
The above synthesis of ideas also supports targeting cells that have made the glycolytic switch. For instance, the work of Pedersen  and his colleagues (Ko et al. [69, 94]; Mathupala et al. ) have used 3-bromopyruvate to inhibit glyceraldehyde 3-phosphate dehydrogenase (GAPDH), which effectively inhibits glycolysis . In addition, 3-bromopyruvate may force, via Le Chatelier’s principle, some reverse reactions to essentially deprive the cancer cell of substrate-created ATP. This leaves the cell little choice except to enter apoptosis. We further hypothesize that a Br derivative of 3-phosphoglycerate would similarly, though perhaps not as energetically, and perhaps not as toxically, facilitate via Le Chatelier’s principle, a reverse reaction to deprive a cancer cell of ATP.
Further, since microtubule dynamics are dysregulated by glucose oscillations and its associated pH oscillations, we speculate that metronomic dosing of microtubule poisons (e.g. nocodazol, taxol, vinblastine) would be an effective treatment strategy for cancers.
In our list of “prescriptions” above many of these are already known or in use. These current practices are essentially “prediction” of our theory and were strictly based on biophysics with little detailed biochemical or cellular biochemistry being considered.
We thank Heiko Enderling for helpful discussions. We thank Eric J. Chaisson for discussion on the Phi parameter. DEF acknowledges funding from Alberta Innovates Health Solutions and the Alberta Cancer Foundation. JAT acknowledges funding from NSERC, the Allard Foundation, Alberta Advanced Education and Technology, the Canadian Breast Cancer Foundation and the Alberta Cancer Foundation. EAR, PH, and LH acknowledge grant number U54CA149233 from the National Cancer Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Cancer Institute. We thank Philip Winter for computer and software support.
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