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Summary

                          

                          
                        

                        

                          

                            
                            

                              
Dynamic Metabolic Model


This simulator integrates the nonlinear ODE system proposed by
Mader (2003) for cellular energy metabolism, extended by the
two-compartment lactate model of Heck, Bartmus & Grabow (2022).
The state vector comprises five coupled variables — phosphocreatine
[PCr], oxygen uptake V̇O2, muscle lactate
[Lam], blood lactate [Lab], and muscle
glycogen [Gly] — whose dynamics are governed by the creatine-kinase /
adenylate-kinase equilibrium (Mader 2003, Eq. 1–6) and
Michaelis–Menten-type rate equations for oxidative phosphorylation,
anaerobic glycolysis, and glycogen consumption.


State equations (Heck et al. 2022, §4.6.5; Heck 2021, Ch. 4.7):


d[PCr]/dt = νATP.VO₂ + νATP.La.pH − νATP.demand − νATP.rest


dV̇O2/dt = (V̇O2,ss − V̇O2) / τVO₂


d[La]m/dt = Volrel⁻¹ · (νLa.ss.pH − νLa.ox.m) + K1 · (Lab − Lam)


d[La]b/dt = Vrel · (K1 · (Lam − Lab) − νLa.ox.b − νLa.res.b)


dGly/dt = −νLa · Volrel · costgly


CHEP Equilibrium (Eq. 4.1–4.5)


The coupled creatine-kinase and adenylate-kinase equilibria
determine [ATP], [ADP], [AMP], and [Pi] from [PCr] and pH.
[ATP]/[ADP] = [H⁺] · M₂ · [PCr]/[Pi],
with M₂ = 1.66·10⁹ (Veech et al. 1979).


Oxidative Phosphorylation (Eq. 4.12)


V̇O2,ss = V̇O2max / (1 + Ks₁ / [ADP]²)


Hill-type activation with n = 2. Half-maximum at [ADP] = 0.035 mmol/kg
(Ks₁ = 0.035² = 0.001225). First-order kinetics with τVO₂ = 10 s.


Glycolysis (Eq. 4.14–4.16)


νLass,pH = VLamax / ((1 + [H⁺]³/Ks₃) · (1 + Ks₂/[ADP]³))


ADP³-dependent activation (half-max at [ADP] = 0.15 mmol/kg) with
pH-dependent PFK inhibition (50% activity at pH ≈ 6.7).


Two-Compartment Lactate Model (Eq. 4.36–4.38)


Muscle and blood lactate are coupled by concentration-dependent
MCT diffusion: K1 = Kdif · Lab−1.4.
Elimination via PDH-limited oxidation (Eq. 19: KLaO₂)
and gluconeogenesis (Eq. 39/40: Cori cycle).


Glycogen (Heck 2021, Ch. 4.7)


Muscle glycogen is tracked as the fifth state variable.
Two effects of glycogen depletion:


• VLamax × f(Gly): Linear (surface-proportional) reduction of
maximal glycolytic rate with decreasing glycogen (Heck 2021, Fig. 29).


• V̇O2max × g(Gly): Fourth-root reduction
g(r) = a + (1−a) · r1/4,
where a = 0.5–0.8 is the residual oxidative capacity from fat oxidation
alone (Heck 2021, Fig. 30; Hargreaves 2006; McArdle's disease data).


Exhaustion


Simulation terminates when [PCr] < 1.0 mmol/kgm (Heck 2021).




References:
Mader A (2003) Eur J Appl Physiol 88:317–338 ·
Mader A, Heck H (1986) Int J Sports Med 7(S1):45–65 ·
Mader A, Heck H (1994) BSW 8(2):124–162 ·
Heck H, Bartmus U, Grabow V (2022) Ch. 4, Springer ·
Heck H (2021) Simulation Supplement



                            

                          

                        

                      

                      

                        
                        

                          

                            
                            

                              
Energy System Contributions

                              
Three parallel ATP sources power skeletal muscle: oxidative phosphorylation, anaerobic glycolysis, and PCr hydrolysis via creatine kinase. PCr acts as a temporal buffer — it covers the instantaneous deficit when aerobic + glycolytic ATP production cannot yet meet demand (Mader 2003; Heck et al. 2022).

                              

                                Oxidative PhosphorylationPCr Buffer (CK)Anaerobic Glycolysis
                              

                            

                          

                          

                            Simulation Plot
                          

                          

                            

                          

                          

                            Pathway Details
                          

                          

                            
                              Model Theory & Equations
                            
                            

                              

                                

                                  
Oxidative Phosphorylation

                                  
Aerobic ATP via mitochondria. Driven by [ADP] (Hill n=2). Slow onset (τ=10s), high capacity.

                                  

                                    V̇O2 = V̇O2max × [ADP]2 / (Ks1 + [ADP]2)
                                  

                                

                                

                                  
Anaerobic Glycolysis

                                  
ATP from glycogen via PFK. Activated by [ADP], inhibited by [H⁺].

                                  

                                    V̇La = V̇Lamax × f(ADP) × f(pH)
                                  

                                

                                

                                  
PCr Buffer (Creatine Kinase)

                                  
Near-instantaneous ATP from PCr via CK (Keq≈166). Finite store (≈23 mmol/kg).

                                  

                                    dPCr/dt = ATPox + ATPgly − ATPdemand
                                  

                                

                              

                            

                          

                        

                      

                    

                  

                

                
                
                
              

            
          
        

        

          

            

              

                
Steady-State Analysis

                

                  

                

                

                  Variables
                  

                    
                    

                      
                      
                      
                      
                      
                      
                      
                      
                    

                  

                

                
              

              

                

                  
                  

                    

                      

                        

                          1-compartment steady-state model: The algebraic curves (vLass, vLaox, PD, CLass) describe the isolated muscle at constant pH = 7.0 (Heck et al. 2022, §4.3). Gross lactate formation vLass follows a Hill-type ADP³-activation (Eq. 5); maximal oxidative elimination vLaox scales linearly with VO₂ss (Eq. 8). The pyruvate deficit PD = vLaox − vLass represents oxidative reserve (fat oxidation capacity). The Crossing Point where PD = 0 defines the “theoretical maxLass” (Mader & Heck 1986, Eq. 27) — the algebraic boundary between steady-state and non-steady-state lactate behaviour.
                        

                        

                          La(b) — simulated step test: Blood lactate is computed by running a cumulative step test through the full dynamic 2-compartment ODE model (Mader 2003; Heck et al. 2022, §4.6). This integrates all five state variables (PCr, VO₂, Lam, Lab, Glycogen) including non-linear MCT diffusion K1 = Kdif · Lab−1.4 (Eq. 4.36), pH-dependent glycolysis, and VO₂ kinetics (τ = 10 s). The step duration determines how closely each step approaches its true steady state.
                        

                        

                          Dual MLSS concept: The 1-compartment Crossing Point (algebraic) yields the “theoretical maxLass”. The dynamic 2-compartment simulation (30-min constant-load binary search, ΔLab < 1.0 mmol/L criterion) yields the “practical maxLass” — typically ~10 W lower, because the muscle–blood gradient and non-linear diffusion slow lactate equilibration (Heck 2021).
                        

                        

                          pH correction (Eq. 18): PFK activity declines with rising [H⁺]: vLamaxpH = vLamax / (1 + [H⁺]³ / Ks3) with Ks3 = 10−20.2 (Mader & Heck 1994). Effect: vLass is reduced at high intensities → PD increases, MLSS shifts ~5 W higher.
                        

                        

                          Glycogen effects (Heck 2021, Ch. 4.7): VLamax scales linearly with glycogen fill (surface-proportional glycogenolysis, Fig. 29). VO₂max is attenuated via a fourth-root function: g(r) = a + (1−a) · r1/4, where a = 50–80% is the residual capacity from fat oxidation alone (Fig. 30).
                        

                        

                          References: Mader A (2003) Eur J Appl Physiol 88:317–338 · Mader A, Heck H (1986) Int J Sports Med 7(S1):45–65 · Mader A, Heck H (1994) BSW 8(2):124–162 · Heck H, Bartmus U, Grabow V (2022) Ch. 4, Springer · Heck H (2021) Simulation Supplement
                        

                      

                    

                  

                

              

            

            
            
            
          

        

        

          

            

              

                

                  
                

                

                  

                    

                      

                        

                          
Oxidation and Glycolysis Characteristics

                        

                        

                          

                        

                        

                          

                            

                              

                                
                                

                                  
                                  
                                  
                                  
                                

                              

                            

                            

                              

                                
                                

                                  
                                

                              

                            

                            

                              

                                

                                  
                                

                              

                            

                          

                        

                        
                      

                    

                    

                      

                        

                          
CHEP Equilibrium System and −ΔG_ATP

                        

                        

                          

                        

                        

                          

                            

                              

                                
                                

                                  
                                  
                                  
                                  
                                  
                                

                              

                            

                            

                              

                                
                                

                                  
                                

                              

                            

                            

                              

                                

                                  
                                

                              

                            

                            

                              

                                

                                  
                                

                              

                            

                          

                        

                        
                      

                    

                    

                      

                        

                          
High-Energy Phosphates vs. Metabolites

                        

                        

                          

                        

                        

                          

                            

                              

                                
                                

                                  
                                  
                                  
                                  
                                

                              

                            

                            

                              

                                
                                

                                  
                                

                              

                            

                            

                              

                                

                                  
                                

                              

                            

                          

                        

                        
                      

                    

                  

                

                
              

              

                

                  

                    
Current Parameters

                    

                      

                    

                    
                  

                

                

                  

                    
Reference Values

                    

                      

                    

                    
                  

                

              

              

                

                  
                  

                    

                      

                        

                          Activation Curves: Shows ADP-dependent activation of oxidative phosphorylation (VO₂ss) and glycolysis (vLass). The vLass curves shift toward higher ADP concentrations at lower pH due to pH inhibition of phosphofructokinase (PFK).
                        

                        

                          CHEP Equilibrium: The coupled adenylate-creatine phosphate system displays thermodynamically linked concentrations of PCr, ATP, ADP, and AMP, along with the Gibbs free energy of ATP hydrolysis (−ΔG_ATP).
                        

                        

                          Energy State: Alternative representation with high-energy phosphates (PCr + 2·ATP + ADP) on the X-axis. Zones indicate 'Rest' and 'Contractile Insufficiency' states.
                        

                      

                    

                  

                

              

            

            
            
            
          

        

        

          
          

            

              
                

                  

                    
Editable Parameters

                    

                      

                        
                      

                      

                        
                      

                      

                        
                      

                      

                        
                      

                      

                      Simulation Settings
                      

                        
                      

                      

                      
                        

                          
                        

                        

                          
                        

                      
                    

                    
                  

                

                

                  

                    
Complete Initial State

                    

                      

                    

                    
                  

                

              
            

            

              
              

                

                  
                    

                      

                        
Thermodynamics

                        

                          

                            
                          

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
Phosphate Pools

                        

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
CHEP Equilibrium

                        

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
Oxidative Phosphorylation

                        

                          

                            
                          

                          

                            
                          

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
Glycolysis

                        

                          

                            
                          

                          

                            
                          

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
Lactate Resynthesis

                        

                          

                            
                          

                          

                            
                          

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
Lactate Kinetics

                        

                          

                            
                          

                          

                            
                          

                          

                            
                          

                          

                            
                          

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
Buffering & Thresholds

                        

                          

                            
                          

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                    

                      

                        
ATP Demand Mapping

                        

                          

                            
                          

                          

                            
                          

                        

                        
                      

                    

                  
                  

                  

                    
Apply or Reset

                    

                      
                        

                          
                        

                        

                          
                        

                      
                      

                        

                      

                    

                    
                  

                

                

                  

                    
Scientific Descriptions of Model Constants

                    

                      

                    

                    
                  

                

              

            

          

        

      

      

        
        

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