# Metabolic Oscillations and Glycolytic Phenotypes of Cancer Cells

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Results

#### 3.1. Glycolytic Oscillations in Monolayers of HeLa Cervical and DU145 Prostate Cancer Cells

#### 3.2. Comparison of Spheroids and Monolayers of HeLa Cells

#### 3.3. Numerical Simulations of Glycolytic Oscillations in HeLa and DU145 Cells

#### 3.4. Effect of the Inhibitory Feedback Mechanism on the Glycolytic Oscillations

## 4. Discussion

^{+}exist in different subcellular compartments, including cytosol and mitochondria. Thus, to measure the NADH levels in cytosol and to assess glycolytic rates actually, we should use, for instance, a fluorescence resonance energy transfer (FRET) biosensor which can separately monitor glycolytic and mitochondrial NADH levels, as well as the NAD

^{+}/NADH ratio [58].

## 5. Materials and Methods

#### 5.1. Cultures and Starvation of Glucose for HeLa Cervical and DU145 Prostate Cancer Cells in Monolayers

_{2}for 3 d and then starved of glucose at 37 °C and 5% CO

_{2}for 24 h. The fluorescence microscopy was carried out in Dulbecco’s phosphate-buffered saline (DPBS; Sigma-Aldrich Co., LLC., Tokyo, Japan) at a pH of 6.9 at 25 °C (air-conditioned room temperature). DMEM was not used for the fluorescence measurement to avoid possible fluorescence from the amino acids and vitamins in it. The pH of 6.9 was selected from preliminary experiments in which the fraction of oscillating HeLa cells was relatively high.

#### 5.2. Cultures and Starvation of Glucose for HeLa Cervical Cancer Cells in Spheroids and Monolayers

_{2}for 30–48 h.

_{2}for 2 d. Then, the cells were incubated at 37 °C and 5% CO

_{2}for 0 to 48 h in 100% glucose-free DMEM. Following glucose starvation, the medium in the wells was replaced with DPBS (pH 6.90 for fluorescence microscopy analysis).

#### 5.3. Fluorescence Microscopy

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A mathematical model for the glycolytic oscillations in cancer cells. Variables: $G$, intracellular glucose; $X$, pool of intermediates following the PFK reaction; $Y$, lactate; ${A}_{3},$ ATP; ${G}_{\mathrm{e}\mathrm{x}}$, extracellular glucose; ${Y}_{\mathrm{e}\mathrm{x}}$, extracellular $Y$; ${J}_{\mathrm{i}\mathrm{n}}$, glucose flux into the extracellular solution; ${J}_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}}$, glucose transport through glucose transporter GLUT; ${J}_{\mathrm{P},\mathrm{Y}}$, flux of difference in $Y$ and ${Y}_{\mathrm{e}\mathrm{x}}$ through monocarboxylic transporter; ${v}_{1}$, reaction rate of PFK; ${v}_{2}$, reaction rate of PK; ${v}_{3}$, reaction rate of consumption of $Y$; ${v}_{4}$, reaction rate of consumption of ATP. PFK is allosterically activated by ${A}_{2}$ and inhibited by ${A}_{3}$ and $Y$. PK is allosterically inhibited by ${A}_{3}$. Abbreviations: PFK, phosphofructokinase; PK, pyruvate kinase; F1,6BP, fructose 1,6-bisphosphate; 1,3BPG, 1,3-bisphosphoglycerate.

**Figure 2.**Glycolytic oscillations in HeLa and DU145 cells in monolayers. A time series of NADH fluorescence signals from single cells (

**A**–

**C**) and their frequency distributions (

**D**,

**E**): HeLa cells with no starvation of glucose (

**A**), HeLa cells starved of glucose for 24 h (

**B**), DU145 cells starved of glucose for 24 h (

**C**), HeLa cells (N = 262 oscillatory cells/544 total cells) exhibiting median frequency of 0.034 Hz (

**D**), and DU145 cells (N = 172 oscillatory cells/651 total cells) exhibiting median frequency of 0.023 Hz (

**E**). Glucose of 20 mM was added to the cells at 30 s following each starving condition at 25 °C. Multiple oscillation curves in (

**A**–

**C**) are the typical examples of NADH fluorescence signals; different colors in the curves are ease of visibility to readers, the grey curves are original data, and the colored curves are their average. The color columns in panels (

**D**,

**E**) indicate the difference in the frequency ranges of oscillations.

**Figure 3.**Comparison of glycolytic oscillations in spheroids and monolayers of HeLa cells

^{a}. A time series of NADH fluorescence signals from single cells in spheroids (

**A**), and in monolayers (

**B**); and frequency distribution of oscillations in HeLa cells in spheroids (

**C**) and in monolayers (

**D**). Glucose (25 mM) was added to the cells at 30 s following starvation of both glucose and FBS for 24 h in spheroids (

**A**) and for 2 h in monolayers (

**B**) at 37 °C. The frequency distributions were calculated by ROIs (N = 289 oscillatory ROIs/563,162 total ROIs) from 74 spheroids starved of glucose and FBS for 0–52 h, exhibiting median frequency of 0.070 Hz (

**B**), and by HeLa cells (N = 49 oscillatory cells/4948 total cells) from 22 monolayers starved of glucose and FBS for 0–2 h, exhibiting median frequency of 0.031 Hz (

**D**), respectively. Multiple oscillation curves in (

**A**,

**B**) are the typical examples of NADH fluorescence signals; different colors in the curves are ease of visibility to readers, the grey curves are original data, and the colored curves are their average. The color columns in panels (

**C**,

**D**) indicate the difference in the frequency ranges of oscillations. (

^{a}Modified from [23] according to the permission procedure of the Wiley journal Content).

**Figure 4.**Numerical calculations of the mathematical model. (

**A**) Phase diagram spanned by the enzymatic activity $\alpha $ (Equation (1)) and the GLUT activity $\beta $ (Equation (2)). White area and blue-green-yellow gradation area indicate nonoscillatory and oscillatory states, respectively. The marks ⋆a$-$⋆d indicate points used for numerical simulation. (

**B**) Time series of simulated oscillations in $X$ for HeLa cells exhibiting frequency of 0.035 Hz (panel

**a**), and DU145 cells exhibiting frequency of 0.024 Hz (panel

**b**) in monolayers at 25 °C (Figure 2) calculated with $\alpha =0.37,\beta =1.0$ (⋆a in panel

**A**), and $\alpha =0.27,\beta =1$.0 (⋆b in panel

**A**), respectively; and for HeLa cells in spheroids exhibiting frequency of 0.072 Hz (panel

**c**), and in monolayers exhibiting frequency of 0.034 Hz (panel

**d**) at 37 °C (Figure 3) calculated with $\alpha =0.68,\beta =2$.0 (⋆c in panel

**A**), and $\alpha =0.37,\beta =2$.0 (⋆d in panel

**A**), respectively. The other parameter values are listed in Table 2.

**Figure 5.**Effect of the feedback inhibition of PFK by lactate ($Y$) on the oscillatory behaviors in the model. Calculated time series in (

**a**) $X$, (

**b**) ${G}_{\mathrm{e}\mathrm{x}}$, (

**c**) $Y$, and (

**d**) ${Y}_{\mathrm{e}\mathrm{x}}$ with the inhibition of PFK by $Y$ with $\alpha =0.37,\beta =1.0$ (

**A**) and without the inhibition of PFK by $Y$ with $\alpha =0.49,\beta =1.0$ (

**B**). The frequencies in $X$ are 0.035 Hz (

**A**), and 0.036 Hz (

**B**). Glucose of 25 mM was added at 30 s.

Rate laws |

$\frac{\mathrm{d}G}{\mathrm{d}t}={J}_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}}-{v}_{1}$ $\frac{\mathrm{d}X}{\mathrm{d}t}={v}_{1}-{v}_{2}$ $\frac{\mathrm{d}Y}{\mathrm{d}t}=2{v}_{2}-{v}_{3}-{J}_{\mathrm{P},\mathrm{Y}}$ $\frac{\mathrm{d}{A}_{3}}{\mathrm{d}t}=-2{v}_{1}+4{v}_{2}-{v}_{4}$ $\frac{\mathrm{d}{G}_{\mathrm{e}\mathrm{x}}}{\mathrm{d}t}={J}_{\mathrm{i}\mathrm{n}}-\phi {J}_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}}$ $\frac{\mathrm{d}{Y}_{\mathrm{ex}}}{\mathrm{d}t}=\phi {J}_{\mathrm{P},\mathrm{Y}}$ |

Transport kinetics |

${J}_{\mathrm{i}\mathrm{n}}=\left\{\begin{array}{c}\frac{{G}_{\mathrm{i}\mathrm{n}}}{{t}_{2}-{t}_{1}},{t}_{1}\le t\le {t}_{2}\\ 0,\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\end{array}\right.$ ${J}_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}}=\beta \frac{{G}_{\mathrm{e}\mathrm{x}}-\frac{G}{{K}_{\mathrm{e}\mathrm{q}}}}{{K}_{\mathrm{o}\mathrm{u}\mathrm{t}}\left(1+\frac{G}{{K}_{\mathrm{i}\mathrm{n}}}\right)+{G}_{\mathrm{e}\mathrm{x}}}$ ${J}_{\mathrm{P},\mathrm{Y}}=\kappa (Y-{Y}_{\mathrm{e}\mathrm{x}})$ |

Reaction rates |

${v}_{1}={\alpha k}_{1}G{A}_{3}f\left(G,{A}_{3},Y\right),$ $\hspace{1em}\hspace{1em}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}f\left(G,{A}_{3},Y\right)=\frac{{({A}_{0}-{A}_{3})}^{m}}{\left[1+{({A}_{0}-{A}_{3})}^{m}\left(\frac{1}{{K}_{1}}+\frac{G}{{K}_{1}{K}_{3}}+\frac{{A}_{3}}{{K}_{1}{K}_{4}}\right)+\frac{{A}_{3}^{m}}{{K}_{2}}+\frac{{Y}^{m}}{{K}_{8}}\right]}$ ${v}_{2}={\alpha k}_{2}X\left({A}_{0}-{A}_{3}\right)g(X,{A}_{3})$, $\hspace{1em}\hspace{1em}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}g\left(X,{A}_{3}\right)=\frac{1}{\left[1+\frac{{A}_{3}^{n}}{{K}_{5}}+\frac{X}{{K}_{6}}+\frac{({A}_{0}-{A}_{3})}{{K}_{7}}\right]}$ ${v}_{3}={\alpha k}_{3}Y$ ${v}_{4}={\alpha k}_{4}{A}_{3}$ |

Model-Step | Parameters | Meaning | Values | Sources |
---|---|---|---|---|

${J}_{\mathrm{i}\mathrm{n}}$ | ${G}_{\mathrm{i}\mathrm{n}}$ | $\mathrm{External}\mathrm{glucose}\mathrm{addition}\phantom{\rule{0ex}{0ex}}\mathrm{for}30\mathrm{s}\le t\le 32\mathrm{s}$ | 20 or 25 mM | Experiments |

${J}_{\mathrm{G}\mathrm{L}\mathrm{U}\mathrm{T}}$ | $\beta $ | Normalized maximum glucose uptake | 1.0 or 2.0 | Equation (2) |

$\phi $ | Ratio of cellular volume to extracellular volume | 0.1 | [23,29] | |

${K}_{\mathrm{i}\mathrm{n}}$ | Michaelis constant of GLUT for intracellular glucose | 12 mM | [23,29] | |

${K}_{\mathrm{o}\mathrm{u}\mathrm{t}}$ | Michaelis constant of GLUT for extracellular glucose | 10 mM | [23,29] | |

${K}_{\mathrm{e}\mathrm{q}}$ | Equilibrium constant | 1.0 | [23,29] | |

${J}_{\mathrm{P},\mathrm{Y}}$ | $\kappa $ | Transport constant of MCT | $0.1{\mathrm{s}}^{-1}$ | [23,29] |

${v}_{1}$ | ${k}_{1}$ | Rate constant of PFK reaction | $\alpha {a}_{1}$ | Equation (1) |

$\alpha $ | A common parameter for the four rate constants | 0.27–0.68 | This work | |

${a}_{1}$ | $\mathrm{A}\mathrm{constant}\mathrm{value}\mathrm{for}{k}_{1}$ | $1.0{\mathrm{m}\mathrm{M}}^{-(m+1)}\xb7{\mathrm{s}}^{-1}$ | [23,29] | |

$m$ | The number of substrate molecules bound to PFK | 4 | [23,29] | |

${K}_{1}$ | Dissociation constant for free PFK and m-molecules of ADP | $1.0{\mathrm{m}\mathrm{M}}^{m}$ | [23,29] | |

${K}_{2}$ | Dissociation constant for free PFK and m-molecules of ATP | $1.0{\mathrm{m}\mathrm{M}}^{m}$ | [23,29] | |

${K}_{3}$ | Dissociation constant for ADP-activated PFK and glucose | 1.0 mM | [23,29] | |

${K}_{4}$ | Dissociation constant for ADP-activated PFK and ATP | 1.0 mM | [23,29] | |

${K}_{8}$ | $\mathrm{Dissociation}\mathrm{constant}\mathrm{for}\mathrm{free}\mathrm{PFK}\mathrm{and}m-\mathrm{molecule}\mathrm{of}Y$ (lactate) | $2.0{\mathrm{m}\mathrm{M}}^{m}$ | This work | |

${v}_{2}$ | ${k}_{2}$ | Rate constant of PK reaction | $\alpha {a}_{2}$ | Equation (1) |

$\alpha $ | A common parameter for the four rate constants | 0.27–0.68 | This work | |

${a}_{2}$ | $\mathrm{A}\mathrm{constant}\mathrm{value}\mathrm{for}{k}_{2}$ | $0.5{\mathrm{m}\mathrm{M}}^{-1}\xb7{\mathrm{s}}^{-1}$ | [23,29] | |

$n$ | The number of substrate molecules Boud to PK | 4 | [23,29] | |

${K}_{5}$ | Dissociation constant for free PK and n-molecule of ATP | $20{\mathrm{m}\mathrm{M}}^{n}$ | [23,29] | |

${K}_{6}$ | $\mathrm{Dissociation}\mathrm{constant}\mathrm{for}\mathrm{free}\mathrm{PK}\mathrm{and}X$ (pool of intermediates) | 20 mM | [23,29] | |

${K}_{7}$ | Dissociation constant for free PK and ADP | 20 mM | [23,29] | |

${v}_{3}$ | ${k}_{3}$ | $\mathrm{Rate}\mathrm{constant}\mathrm{of}\mathrm{consumption}\mathrm{of}Y$ (lactate) | $\alpha {a}_{3}$ | Equation (1) |

$\alpha $ | A common parameter for the four rate constants | 0.27–0.68 | This work | |

${a}_{3}$ | $\mathrm{A}\mathrm{constant}\mathrm{value}\mathrm{for}{k}_{3}$ | $0.09{\mathrm{s}}^{-1}$ | [23,29] | |

${v}_{4}$ | ${k}_{4}$ | Rate constant of consumption of ATP | $\alpha {a}_{4}$ | Equation (1) |

$\alpha $ | A common parameter for the four rate constants | 0.27–0.68 | This work | |

${a}_{4}$ | $\mathrm{A}\mathrm{constant}\mathrm{value}\mathrm{for}{k}_{4}$ | $0.15{\mathrm{s}}^{-1}$ | [23,29] |

**Table 3.**Initial concentrations and total concentration (${A}_{0}$) of ATP and ADP for the calculations.

Initial Concentrations | Sources | |
---|---|---|

Variables | Values | |

${G}_{0}$ | 0.30 mM | [39] |

${X}_{0}$ | 0.30 mM | [23,29] |

${Y}_{0}$ | 0.30 mM | [23,29] |

${G}_{\mathrm{e}\mathrm{x},0}$ | 0 mM | [23,29] |

${Y}_{\mathrm{e}\mathrm{x},0}$ | 0 mM | [23,29] |

Total concentration of ATP and ADP | ||

Constant | Value | |

${A}_{0}$ | 3.0 mM | [28] |

$\mathbf{Cells}\mathbf{in}\mathbf{Monolayers}$ (at 25 °C) | $\mathbf{HeLa}\mathbf{Cells}\mathbf{in}\mathbf{Different}\mathbf{Morphology}$ (at 37 °C) | ||||||
---|---|---|---|---|---|---|---|

HeLa Cells | DU145 Cells | Spheroids | Monolayers | ||||

Exp. (Hz) | Sim. (Hz) | Exp. (Hz) | Sim. (Hz) | Exp. (Hz) | Sim. (Hz) | Exp. (Hz) | Sim. (Hz) |

0.034 | 0.035 | 0.023 | 0.024 | 0.070 | 0.072 | 0.031 | 0.034 |

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**MDPI and ACS Style**

Amemiya, T.; Shibata, K.; Yamaguchi, T.
Metabolic Oscillations and Glycolytic Phenotypes of Cancer Cells. *Int. J. Mol. Sci.* **2023**, *24*, 11914.
https://doi.org/10.3390/ijms241511914

**AMA Style**

Amemiya T, Shibata K, Yamaguchi T.
Metabolic Oscillations and Glycolytic Phenotypes of Cancer Cells. *International Journal of Molecular Sciences*. 2023; 24(15):11914.
https://doi.org/10.3390/ijms241511914

**Chicago/Turabian Style**

Amemiya, Takashi, Kenichi Shibata, and Tomohiko Yamaguchi.
2023. "Metabolic Oscillations and Glycolytic Phenotypes of Cancer Cells" *International Journal of Molecular Sciences* 24, no. 15: 11914.
https://doi.org/10.3390/ijms241511914