Part 2 · Chapter 2.1

Glycolysis: The Embden-Meyerhof-Parnas Pathway

The universal cytosolic pathway that splits glucose into two pyruvate molecules, generating ATP by substrate-level phosphorylation and NADH for downstream oxidation. Ten enzymatic steps organize into an energy-investment phase (Steps 1–4) and an energy-payoff phase (Steps 5–10).

Learning Objectives

▸Enumerate and mechanistically describe each of the 10 glycolytic reactions and their enzymes
▸Distinguish hexokinase from glucokinase in terms of kinetics and tissue distribution
▸Apply the Monod–Wyman–Changeux (MWC) model to PFK-1 allosteric regulation
▸Derive net ATP and NADH yields and trace carbon flux
▸Contrast aerobic vs. anaerobic fate of pyruvate, and the Pasteur and Warburg effects

◆Overall Stoichiometry

Glycolysis is the most evolutionarily conserved metabolic pathway. Every step occurs in the cytosol, independent of mitochondria or oxygen. The pathway converts one hexose into two triose products:

\[ \text{Glucose} + 2\,\text{NAD}^+ + 2\,\text{ADP} + 2\,\text{P}_i \longrightarrow 2\,\text{Pyruvate} + 2\,\text{NADH} + 2\,\text{ATP} + 2\,\text{H}_2\text{O} + 2\,\text{H}^+ \]

The overall standard free energy change is \(\Delta G^{\circ\prime} \approx -96\;\text{kJ/mol}\), but under cellular conditions (low [ATP]/[ADP], depleted NADH) \(\Delta G \approx -74\;\text{kJ/mol}\), dominated by three essentially irreversible reactions: hexokinase, PFK-1, and pyruvate kinase. These are the three physiologically relevant control points.

◆Pathway Map

◆The Ten Reactions

Step 1 — Hexokinase / Glucokinase

Glucose is trapped inside the cell by phosphorylation at C6, producing glucose-6-phosphate (G6P). The reaction is strongly exergonic and essentially irreversible.

\[ \text{Glucose} + \text{ATP} \;\xrightarrow[\text{Mg}^{2+}]{\text{Hexokinase}}\; \text{G6P} + \text{ADP},\qquad \Delta G^{\circ\prime} = -16.7\;\text{kJ/mol} \]

Hexokinase (HK-I/II/III)

• Ubiquitous, high affinity \(K_m \approx 0.1\;\text{mM}\)
• Inhibited by its product G6P (feedback)
• Half-saturated at physiologic [glucose] (5 mM)

Glucokinase (HK-IV)

• Liver + pancreatic \(\beta\)-cells only
• Low affinity \(K_m \approx 10\;\text{mM}\)
• Sigmoidal kinetics, fructose-6P sensing
• Acts as the hepatic/islet glucose sensor

Step 2 — Phosphoglucose Isomerase (PGI)

An aldose→ketose isomerization converts G6P to fructose-6-phosphate (F6P) via an enediol intermediate. The reaction is near-equilibrium (\(\Delta G^{\circ\prime} = +1.7\;\text{kJ/mol}\), yet \(\Delta G \approx 0\) in the cell).

\[ \text{G6P} \;\rightleftharpoons\; \text{F6P} \]

Step 3 — Phosphofructokinase-1 (PFK-1) • Rate-Limiting

A second ATP investment commits the hexose to glycolysis. PFK-1 is the major regulatory enzyme of the pathway and is exquisitely controlled by allosteric effectors.

\[ \text{F6P} + \text{ATP} \;\xrightarrow{\text{PFK-1}}\; \text{F1,6BP} + \text{ADP},\qquad \Delta G^{\circ\prime} = -14.2\;\text{kJ/mol} \]

MWC Allosteric Model

PFK-1 is a tetramer of \(\alpha\beta\) subunits that toggles between a T (tense) and R (relaxed) state. The Monod–Wyman–Changeux formulation of the velocity is:

\[ v = V_{\max} \frac{\alpha(1+\alpha)^{n-1}}{(1+\alpha)^n + L\,(1+c\alpha)^n},\quad \alpha = [S]/K_m,\; L = \frac{[T]_0}{[R]_0} \]

Inhibitors (ATP, citrate) stabilize the T state (increase \(L\)); activators (AMP, fructose-2,6-bisphosphate) stabilize the R state (decrease \(L\)). The potent activator F2,6BP is synthesized by PFK-2 in response to insulin.

Step 4 — Aldolase (Triose Split)

Aldolase cleaves the 6-carbon F1,6BP into two 3-carbon triose phosphates via an aldol retro-condensation. Class I aldolases (animals) use a Schiff-base intermediate at an active-site lysine; Class II aldolases (microbes) use a Zn\(^{2+}\)cofactor.

\[ \text{F1,6BP} \;\rightleftharpoons\; \text{DHAP} + \text{GAP},\qquad \Delta G^{\circ\prime} = +23.8\;\text{kJ/mol} \]

The reaction is thermodynamically unfavorable at standard conditions but proceeds forward in cells because DHAP and GAP concentrations are kept very low by rapid downstream consumption (Le Chatelier’s principle).

Step 5 — Triose Phosphate Isomerase (TPI)

DHAP is interconverted to GAP (the only triose that continues forward). TPI is a textbook catalytically perfect enzyme with \(k_{\text{cat}}/K_m \approx 2\times 10^8\;\text{M}^{-1}\,\text{s}^{-1}\) — diffusion-limited.

Step 6 — GAPDH (Oxidation + Phosphorylation)

Glyceraldehyde-3-phosphate dehydrogenase couples an oxidation (GAP→1,3BPG) to capture of inorganic phosphate, producing a high-energy acyl-phosphate. An active-site cysteine forms a thioester intermediate with the aldehyde; NAD⁺ accepts hydride to become NADH.

\[ \text{GAP} + \text{NAD}^+ + \text{P}_i \;\rightleftharpoons\; \text{1,3-BPG} + \text{NADH} + \text{H}^+ \]

This is the only oxidative step of glycolysis; all 2 NADH per glucose arise here. In anaerobiosis, continued flux through this step requires NAD⁺ regeneration by lactate dehydrogenase.

Step 7 — Phosphoglycerate Kinase (PGK)

The first substrate-level phosphorylation: the acyl-phosphate bond of 1,3-BPG (\(\Delta G^{\circ\prime}_{\text{hyd}} = -49.3\) kJ/mol) is transferred to ADP (\(-30.5\) kJ/mol), yielding ATP with \(\Delta G^{\circ\prime} = -18.8\;\text{kJ/mol}\).

\[ \text{1,3-BPG} + \text{ADP} \;\rightleftharpoons\; \text{3-PG} + \text{ATP} \]

Step 8 — Phosphoglycerate Mutase

Relocates the phosphate from C3 to C2 via a 2,3-bisphosphoglycerate intermediate. Ready for dehydration.

Step 9 — Enolase

Dehydration produces phosphoenolpyruvate (PEP), whose enol–phosphate has an extraordinarily high phosphoryl-transfer potential (\(\Delta G^{\circ\prime}_{\text{hyd}} = -61.9\) kJ/mol). Enolase requires Mg\(^{2+}\); the antibiotic fluoride inhibits by chelating catalytic metals.

Step 10 — Pyruvate Kinase (PK)

The second substrate-level phosphorylation, and the last of the three irreversible control points. PEP phosphorylates ADP to ATP, producing enolpyruvate which tautomerizes to the more stable keto form of pyruvate.

\[ \text{PEP} + \text{ADP} \;\xrightarrow{\text{PK}}\; \text{Pyruvate} + \text{ATP},\qquad \Delta G^{\circ\prime} = -31.4\;\text{kJ/mol} \]

The L-isoform (liver) is allosterically activated by F1,6BP (feed-forward) and inhibited by phosphorylation via PKA (glucagon signaling). The M-isoform (muscle) is not phosphorylation-regulated; the M2-isoform (cancer cells) favors aerobic glycolysis (Warburg).

Net Energy Balance

Investment phase (per glucose)

− 1 ATP at Step 1 (hexokinase)
− 1 ATP at Step 3 (PFK-1)
Total: −2 ATP

Payoff phase (per 2 trioses)

+ 2 NADH at Step 6 (GAPDH)
+ 2 ATP at Step 7 (PGK)
+ 2 ATP at Step 10 (PK)
Total: +4 ATP, +2 NADH

\[ \text{Net per glucose: } +2\;\text{ATP},\; +2\;\text{NADH},\; 2\;\text{Pyruvate} \]

◆Simulation 1: Steady-State ODE Model of EMP

A coupled system of 15 ODEs tracks the concentrations of every glycolytic intermediate plus ATP, ADP, and NAD⁺/NADH. Rate laws are Michaelis–Menten with an MWC cooperativity term for PFK-1 and pyruvate kinase, and a simple load term (\(V_\text{ATPase}\)) representing cellular ATP demand. Integration reveals the characteristic rapid approach to a glycolytic steady state within ~30–60 s.

Python

script.py165 lines

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint

# Coupled ODE model of the 10-step Embden-Meyerhof-Parnas pathway
# State variables: concentrations (mM) of glycolytic intermediates
# Rate laws use simplified Michaelis-Menten with feedback regulation on PFK-1

# Initial concentrations (mM), typical for human erythrocyte
# Order: [GLC, G6P, F6P, F16BP, DHAP, GAP, 13BPG, 3PG, 2PG, PEP, PYR, ATP, ADP, NAD, NADH]
y0 = [5.0, 0.05, 0.02, 0.02, 0.16, 0.006, 0.001, 0.06, 0.012, 0.017, 0.05,
      1.85, 0.14, 0.06, 0.04]

# Kinetic parameters (Vmax in mM/s, Km in mM)
# Values adapted from Teusink et al. 2000 yeast glycolysis model
params = {
    'Vhex': 0.8,   'Km_hex_glc': 0.1,  'Km_hex_atp': 0.1,
    'Vpgi': 5.0,   'Km_pgi_g6p': 1.4,  'Keq_pgi': 0.3,
    'Vpfk': 0.4,   'Km_pfk_f6p': 0.1,  'Ki_pfk_atp': 0.65, 'n_pfk': 2.5,
    'Vald': 1.5,   'Km_ald_f16bp': 0.3,
    'Vtpi': 8.0,   'Km_tpi_dhap': 1.2, 'Keq_tpi': 0.045,
    'Vgapdh': 1.2, 'Km_gapdh_gap': 0.21, 'Km_gapdh_nad': 0.09,
    'Vpgk': 3.0,   'Km_pgk_13bpg': 0.003, 'Km_pgk_adp': 0.2,
    'Vpgm': 4.0,   'Km_pgm_3pg': 0.2,  'Keq_pgm': 0.19,
    'Veno': 3.0,   'Km_eno_2pg': 0.1,  'Keq_eno': 6.7,
    'Vpk':  2.5,   'Km_pk_pep': 0.14,  'Km_pk_adp': 0.3, 'n_pk': 1.8,
    'Vatpase': 0.6, 'Km_atpase_atp': 0.5,  # ATP demand load
    'Vldh': 0.3,   'Km_ldh_pyr': 0.4,  'Km_ldh_nadh': 0.02,
}
p = params

def mm(S, Km):
    return S/(Km+S)

def glycolysis(y, t):
    GLC,G6P,F6P,F16BP,DHAP,GAP,BPG,PG3,PG2,PEP,PYR,ATP,ADP,NAD,NADH = y

# Step 1: Hexokinase (irreversible, ATP-dependent)
    v1 = p['Vhex']*mm(GLC,p['Km_hex_glc'])*mm(ATP,p['Km_hex_atp'])
    # Step 2: Phosphoglucose isomerase (reversible)
    v2 = p['Vpgi']*(G6P - F6P/p['Keq_pgi'])/(p['Km_pgi_g6p']+G6P)
    # Step 3: PFK-1 (Monod-Wyman-Changeux allostery with ATP inhibition)
    L = (1 + ATP/p['Ki_pfk_atp'])**p['n_pfk']
    alpha = F6P/p['Km_pfk_f6p']
    v3 = p['Vpfk']*alpha*(1+alpha)**(p['n_pfk']-1)/((1+alpha)**p['n_pfk']+L)
    # Step 4: Aldolase (reversible triose split)
    v4 = p['Vald']*F16BP/(p['Km_ald_f16bp']+F16BP)
    # Step 5: Triose phosphate isomerase
    v5 = p['Vtpi']*(DHAP - GAP/p['Keq_tpi'])/(p['Km_tpi_dhap']+DHAP)
    # Step 6: GAPDH (oxidation + phosphorylation, NAD+-dependent)
    v6 = p['Vgapdh']*mm(GAP,p['Km_gapdh_gap'])*mm(NAD,p['Km_gapdh_nad'])
    # Step 7: PGK (substrate-level phosphorylation #1)
    v7 = p['Vpgk']*mm(BPG,p['Km_pgk_13bpg'])*mm(ADP,p['Km_pgk_adp'])
    # Step 8: Phosphoglycerate mutase
    v8 = p['Vpgm']*(PG3 - PG2/p['Keq_pgm'])/(p['Km_pgm_3pg']+PG3)
    # Step 9: Enolase (dehydration)
    v9 = p['Veno']*(PG2 - PEP/p['Keq_eno'])/(p['Km_eno_2pg']+PG2)
    # Step 10: Pyruvate kinase (substrate-level phosphorylation #2, allosteric)
    v10 = p['Vpk']*(PEP/p['Km_pk_pep'])**p['n_pk']/(1+(PEP/p['Km_pk_pep'])**p['n_pk'])           * mm(ADP,p['Km_pk_adp'])
    # ATP demand (cellular work)
    vatpase = p['Vatpase']*mm(ATP,p['Km_atpase_atp'])
    # LDH regenerates NAD+ from NADH (fermentation branch)
    vldh = p['Vldh']*mm(PYR,p['Km_ldh_pyr'])*mm(NADH,p['Km_ldh_nadh'])

dGLC   = -v1 + 0.01*(5.0-GLC)   # GLC resupply
    dG6P   = v1 - v2
    dF6P   = v2 - v3
    dF16BP = v3 - v4
    dDHAP  = v4 - v5
    dGAP   = v4 + v5 - v6
    dBPG   = v6 - v7
    dPG3   = v7 - v8
    dPG2   = v8 - v9
    dPEP   = v9 - v10
    dPYR   = v10 - vldh
    # ATP balance: -1 (hex) -1 (pfk) +1 (pgk) +1 (pk) - ATPase
    dATP   = -v1 - v3 + v7 + v10 - vatpase
    dADP   = -dATP
    # NAD+/NADH
    dNAD   = -v6 + vldh
    dNADH  = v6 - vldh
    return [dGLC,dG6P,dF6P,dF16BP,dDHAP,dGAP,dBPG,dPG3,dPG2,dPEP,dPYR,
            dATP,dADP,dNAD,dNADH]

t = np.linspace(0, 200, 4000)
sol = odeint(glycolysis, y0, t)

names = ['Glucose','G6P','F6P','F1,6BP','DHAP','GAP','1,3BPG','3PG',
         '2PG','PEP','Pyruvate','ATP','ADP','NAD+','NADH']
colors = ['#67e8f9','#7dd3fc','#60a5fa','#38bdf8','#818cf8','#a78bfa',
          '#c084fc','#e879f9','#f472b6','#fb7185','#fbbf24',
          '#22d3ee','#94a3b8','#86efac','#f87171']

fig, axes = plt.subplots(2, 2, figsize=(14, 9))

ax = axes[0,0]
for i,lbl in enumerate(['Glucose','G6P','F6P','F1,6BP']):
    ax.plot(t, sol[:,i], color=colors[i], lw=2, label=lbl)
ax.set_title('Hexose phase (Steps 1-4): energy investment',
             color='white', fontsize=12, fontweight='bold')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('[mM]', color='white')
ax.legend(facecolor='#0a0a1a', edgecolor='#67e8f9', labelcolor='white')
ax.set_facecolor('#0a0a1a'); ax.tick_params(colors='white')
ax.grid(alpha=0.2, color='#67e8f9')
for s in ax.spines.values(): s.set_color('#67e8f9')

ax = axes[0,1]
for i,lbl in zip([4,5,6,7,8,9,10], ['DHAP','GAP','1,3BPG','3PG','2PG','PEP','Pyruvate']):
    ax.plot(t, sol[:,i], color=colors[i], lw=2, label=lbl)
ax.set_title('Triose phase (Steps 5-10): energy payoff',
             color='white', fontsize=12, fontweight='bold')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('[mM]', color='white')
ax.legend(facecolor='#0a0a1a', edgecolor='#67e8f9', labelcolor='white', fontsize=8)
ax.set_facecolor('#0a0a1a'); ax.tick_params(colors='white')
ax.grid(alpha=0.2, color='#67e8f9')
for s in ax.spines.values(): s.set_color('#67e8f9')

ax = axes[1,0]
ax.plot(t, sol[:,11], color='#fbbf24', lw=2.5, label='ATP')
ax.plot(t, sol[:,12], color='#94a3b8', lw=2, label='ADP')
ax.axhline(1.85, color='#fbbf24', ls='--', alpha=0.3, label='Initial ATP')
ax.set_title('Adenine nucleotide pool (Steady state)',
             color='white', fontsize=12, fontweight='bold')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('[mM]', color='white')
ax.legend(facecolor='#0a0a1a', edgecolor='#fbbf24', labelcolor='white')
ax.set_facecolor('#0a0a1a'); ax.tick_params(colors='white')
ax.grid(alpha=0.2, color='#fbbf24')
for s in ax.spines.values(): s.set_color('#fbbf24')

ax = axes[1,1]
ax.plot(t, sol[:,13], color='#86efac', lw=2, label='NAD+')
ax.plot(t, sol[:,14], color='#f87171', lw=2, label='NADH')
ratio = sol[:,13]/np.maximum(sol[:,14],1e-6)
ax2 = ax.twinx()
ax2.plot(t, ratio, color='#60a5fa', lw=1.5, ls=':', label='NAD+/NADH')
ax2.set_ylabel('NAD+/NADH ratio', color='#60a5fa')
ax2.tick_params(colors='#60a5fa')
ax.set_title('Redox state balance (NAD+/NADH cycle)',
             color='white', fontsize=12, fontweight='bold')
ax.set_xlabel('Time (s)', color='white'); ax.set_ylabel('[mM]', color='white')
ax.legend(facecolor='#0a0a1a', edgecolor='#86efac', labelcolor='white', loc='upper left')
ax.set_facecolor('#0a0a1a'); ax.tick_params(colors='white')
ax.grid(alpha=0.2, color='#86efac')
for s in ax.spines.values(): s.set_color('#86efac')

fig.suptitle('Coupled ODE simulation of the EMP pathway',
             color='white', fontsize=15, fontweight='bold')
fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')

# Steady-state summary
ss = sol[-1]
flux = params['Vpk']*(ss[9]/params['Km_pk_pep'])**params['n_pk']     /(1+(ss[9]/params['Km_pk_pep'])**params['n_pk'])     *ss[12]/(params['Km_pk_adp']+ss[12])
print("=== Steady-state glycolytic intermediates (mM) ===")
for n,v in zip(names[:11], ss[:11]):
    print(f"  {n:12s} = {v:.4f}")
print()
print(f"  ATP = {ss[11]:.3f} mM   ADP = {ss[12]:.3f} mM")
print(f"  NAD+ = {ss[13]:.3f} mM   NADH = {ss[14]:.3f} mM")
print(f"  Steady-state flux through pyruvate kinase = {flux*1000:.2f} uM/s")
print(f"  Net ATP yield per glucose = 2  (2 consumed in steps 1,3;  4 produced in steps 7,10)")
print(f"  Net NADH yield per glucose = 2  (both at step 6, GAPDH)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

◆Regulation

Allosteric Effectors (PFK-1)

↑ Activate: AMP, ADP, F2,6BP, Pᵢ, F1,6BP
↓ Inhibit: ATP, citrate, H⁺, long-chain fatty acids
F2,6BP is a fructose-sparing second messenger synthesized by PFK-2 in response to insulin

Covalent / Transcriptional

Insulin → SREBP-1c → glucokinase, PFK-1 expression ↑
Glucagon → PKA → PFK-2 inhibited, FBPase-2 active → F2,6BP ↓ → glycolysis ↓
Pyruvate kinase (L isoform) is phosphorylated/inactivated by PKA (glucagon)
HIF-1\(\alpha\) under hypoxia upregulates nearly all glycolytic enzymes

The Pasteur vs. Warburg Effect

Louis Pasteur (1861) noticed that yeast fermentation slows in the presence of O₂ — oxygen suppresses glycolysis because OxPhos efficiently produces ATP, elevating [ATP] (PFK-1 inhibitor) and [citrate] (PFK-1 inhibitor).

Otto Warburg (1924) observed that many tumor cells maintain high glycolytic flux even in the presence of O₂ (“aerobic glycolysis”), converting most glucose to lactate. The Warburg effect is driven by PKM2 expression, HIF-1\(\alpha\)stabilization, oncogenic RAS/MYC, and loss of p53, and provides biosynthetic precursors (ribose, amino acids, lipids) for rapid proliferation.

◆Simulation 2: Pasteur / Warburg / MWC Regulation

Python

script.py76 lines

import numpy as np
import matplotlib.pyplot as plt

# Pasteur vs. Warburg effect: compare glycolytic flux under different conditions
# Normal cell (Pasteur effect): O2 present -> glycolysis slows, OxPhos dominates
# Cancer cell (Warburg effect): O2 present BUT glycolysis remains elevated

conditions = ['Normal\ncell\n(Pasteur: O2 low)',
              'Normal\ncell\n(Pasteur: O2 high)',
              'Cancer cell\n(Warburg: O2 high)']

glycolytic_flux = [8.5, 2.2, 9.8]       # relative units
oxphos_flux     = [0.4, 5.2, 2.1]       # relative units
lactate_output  = [7.9, 0.3, 7.6]       # relative units
atp_total       = [17.0, 33.0, 22.0]    # ATP/glucose

x = np.arange(len(conditions))
w = 0.22

fig, axes = plt.subplots(1, 2, figsize=(14, 5.5))

ax = axes[0]
ax.bar(x-1.5*w, glycolytic_flux, w, label='Glycolysis', color='#67e8f9', alpha=0.9)
ax.bar(x-0.5*w, oxphos_flux,     w, label='OxPhos',     color='#fbbf24', alpha=0.9)
ax.bar(x+0.5*w, lactate_output,  w, label='Lactate',    color='#f87171', alpha=0.9)
ax.bar(x+1.5*w, [a/4 for a in atp_total], w, label='ATP/glucose \u00f74', color='#a78bfa', alpha=0.9)
ax.set_xticks(x); ax.set_xticklabels(conditions, color='white', fontsize=9)
ax.set_ylabel('Relative flux', color='white')
ax.set_title('Pasteur vs. Warburg Effect',
             color='white', fontsize=13, fontweight='bold')
ax.legend(facecolor='#0a0a1a', edgecolor='#67e8f9', labelcolor='white')
ax.set_facecolor('#0a0a1a'); ax.tick_params(colors='white')
ax.grid(alpha=0.2, color='#67e8f9', axis='y')
for s in ax.spines.values(): s.set_color('#67e8f9')

# PFK-1 cooperativity curve: MWC model
ax = axes[1]
f6p = np.linspace(0, 5, 400)
Km = 0.1; n = 3.0
for i,(atp,col,lab) in enumerate([(0.5,'#86efac','Low ATP'),
                                    (2.0,'#fbbf24','Medium ATP'),
                                    (5.0,'#f87171','High ATP (inhibitory)')]):
    L = (1 + atp/0.65)**n
    alpha = f6p/Km
    v = alpha*(1+alpha)**(n-1)/((1+alpha)**n + L)
    ax.plot(f6p, v/v.max(), color=col, lw=2.5, label=lab)

ax.set_xlabel('[Fructose-6-phosphate] (mM)', color='white')
ax.set_ylabel('Relative PFK-1 velocity', color='white')
ax.set_title('PFK-1 Allosteric Regulation (MWC Model)',
             color='white', fontsize=13, fontweight='bold')
ax.legend(facecolor='#0a0a1a', edgecolor='#f87171', labelcolor='white')
ax.set_facecolor('#0a0a1a'); ax.tick_params(colors='white')
ax.grid(alpha=0.2, color='#f87171')
for s in ax.spines.values(): s.set_color('#f87171')

# Compute trapezoid integral as a cumulative work/flux proxy
f_norm = f6p/Km
area_low  = np.trapezoid(f_norm*(1+f_norm)**2/((1+f_norm)**3+(1+0.5/0.65)**3), f6p)
area_high = np.trapezoid(f_norm*(1+f_norm)**2/((1+f_norm)**3+(1+5.0/0.65)**3), f6p)

fig.patch.set_facecolor('#0a0a1a')
plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')

print("=== Hallmark comparisons ===")
print(f"Pasteur effect cell (aerobic): ~{atp_total[1]} ATP per glucose (efficient OxPhos).")
print(f"Warburg cell (aerobic glycolysis): ~{atp_total[2]} ATP per glucose "
      f"but {glycolytic_flux[2]/glycolytic_flux[1]:.1f}x faster glycolytic flux.")
print(f"Lactate output ratio (Warburg/Pasteur-aerobic) = {lactate_output[2]/lactate_output[1]:.1f}")
print()
print(f"PFK-1 activity integral (area under curve):")
print(f"  Low ATP : {area_low:.3f}")
print(f"  High ATP: {area_high:.3f}  (~{area_low/area_high:.1f}x inhibition)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

◆Fates of Pyruvate

Aerobic (most tissues)

Pyruvate enters mitochondrion via the MPC, oxidized by PDH to acetyl-CoA, feeds TCA + ETC. Yields 30–32 ATP per glucose.

Anaerobic (muscle, RBC)

Lactate dehydrogenase regenerates NAD⁺ so GAPDH can continue. Yields only 2 ATP per glucose but rapid.

Ethanolic (yeast)

Pyruvate decarboxylase + alcohol dehydrogenase yield CO₂ + ethanol while regenerating NAD⁺. Basis of brewing and biofuels.

Clinical Relevance

Pyruvate Kinase Deficiency

Most common glycolytic enzymopathy causing hereditary hemolytic anemia; RBCs cannot make enough ATP to maintain Na⁺/K⁺-ATPase.

FDG-PET

18F-fluorodeoxyglucose is phosphorylated by hexokinase but not further metabolized; tumors trap FDG due to elevated glycolysis (Warburg).

Arsenate Poisoning

As\(\text{O}_4^{3-}\) substitutes for Pᵢ at GAPDH, forming a labile 1-arsenato-3-phosphoglycerate that hydrolyzes spontaneously—uncoupling ATP synthesis at Step 7.

Lactic Acidosis

Sepsis, shock, metformin overdose, or thiamine deficiency impair OxPhos and drive lactate production; falling pH ↓ PFK-1 to protect against runaway acidosis.

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