Introduction

It is very well known that endurance training can improve the athlete performance during various activities. For instance, it can speed up and thus shorten the time of the run on distances where oxidative ATP supply predominates, for example of the 1500 m run (see, e.g., Zoladz et al. 2022). However, the mechanisms responsible for this effect are not fully understood.

Endurance training can lead to an increase in total mitochondrial protein, amount/activities of enzymes involved in mitochondrial bioenergetics, skeletal muscle oxidative/respiratory capacity, OXPHOS activity in mitochondria, and mitochondrial volume density (Holloszy 1967; Baldwin et al. 1972; Krieger et al. 1980; Hoppeler et al. 1985; Wibom et al. 1992; Suter et al. 1995; Fernström et al. 2004; Tarnopolsky et al. 2007; Pesta et al. 2011; Jacobs et al. 2013; Zoladz et al. 2013, 2014, 2022; Scalzo et al. 2014).

Endurance training changes several kinetic properties of the working skeletal muscle and whole-body bioenergetic system. It increases \(\dot{V}\)O2max, increases critical power, decreases the characteristic transition time τp or t0.63 of the primary phase II of the pulmonary \(\dot{V}\)O2 on-kinetics and markedly reduces the slow component of the \(\dot{V}\)O2 on-kinetics (Hoppeler et al. 1985; Casaburi et al. 1987; Gaesser and Wilson 1988; Poole et al. 1990; Jenkins and Quigley 1992; Roca et al. 1992; Phillips et al. 1995; Suter et al. 1995; Womack et al. 1995; Carter et al. 2000; Fernström et al. 2004; Berger et al. 2006; Burnley and Jones 2007; Hottenrott et al. 2012; Zoladz et al. 2013; Zoladz et al. 2014).

Increased cytoplasmic Pi has a central role in muscle fatigue (Allen et al. 2008). Recently, the “Pi double-threshold” mechanism of muscle fatigue at the biochemical level was proposed (Korzeniewski and Rossiter 2020, 2021). This mechanism is based on three assumptions: (i) the additional ATP usage, which underlies the slow component of \(\dot{V}\)O2 and metabolite on-kinetics, is initiated when Pi exceeds a certain critical value, termed Picrit (Korzeniewski and Rossiter 2020); (ii) muscle work is terminated because of fatigue when Pi reaches another, higher, peak value (Pipeak) (Korzeniewski 2019); and (iii) Pi increase and additional ATP usage increase stimulate each other through a self-driving positive feedback mechanism (Korzeniewski and Rossiter 2020). In sufficiently intense exercise, this mechanism ultimately causes Pi to reach Pipeak (and \(\dot{V}\)O2 to reach \(\dot{V}\)O2max) and leads to exercise termination because of fatigue. The first threshold corresponding to Picrit (point i), the second threshold corresponding to Pipeak (point ii), and the positive feedback (point iii) were used to derive an abstract fatigue factor F representing various fatigue-related metabolites: H+, NH4+, IMP, AMP, ADP, Pi etc. (Korzeniewski and Zoladz 2003).

A recently described computer model based, the “Pi double-threshold” mechanism can predict various skeletal muscle properties: changes over time of several variables including muscle \(\dot{V}\)O2, cytosolic ADP, pH, PCr and Pi during rest-to-work transition in skeletal muscle; the end-exercise constancy of these variables at different power outputs above CP; the hyperbolic shape of the power-duration curve with CP as an asymptote; and the hypoxia/hyperoxia-induced decrease/increase in critical power and \(\dot{V}\)O2max, and increase/decrease of t0.63 (transition time of the primary phase II of the \(\dot{V}\)O2 on-kinetics, analogous to τp) (Korzeniewski and Rossiter 2020; Korzeniewski 2023a).

Finally, the “Pi double-threshold” mechanism is able to account for training-induced changes in \(\dot{V}\)O2max, CP and \(\dot{V}\)O2 on-kinetics (shortening of t0.63, decrease of the slow component) both in healthy individuals (Korzeniewski and Rossiter 2021; Korzeniewski 2023b) and MM patients (Korzeniewski 2022). Theoretical studies predict that muscle training causes an increase in OXPHOS activity and decrease in Pipeak, the latter under the assumption that the increase in OXPHOS activity in vivo corresponds quantitatively to the increase in mitochondria volume density and/or OXPHOS (enzymes) activity in vitro.

In a recent study, Hureau et al. (2022) postulated that Pi is the main metabolite related to peripheral fatigue, while H+ mediates in central fatigue through group III/IV muscle afferents. This finding decidedly supports the Pi double-threshold mechanism of muscle fatigue, which postulates that both working muscle and central nervous system somehow sense the metabolic state of working muscles (Korzeniewski 2019, 2023b; Korzeniewski and Rossiter 2020).

The present study is intended to test possible mechanisms at the biochemical level that are responsible for the endurance-training-induced increase of work intensity, increase in mean run speed and thus shortening of the time needed to run a certain distance. This concerns exercises where OXPHOS is the main source of ATP, in particular the 1500 m run. It is hypothesized that the endurance-training-induced marked increase in OXPHOS activity attenuates the increase in Pi during exercise of increased intensity. This in turn delays the reaching Pipeak and termination of exercise. This effect is likely to be accompanied by a decrease in Pipeak. It is expected that the latter effect increases the homeostasis of “bioenergetic” metabolites in the system: ADP, PCr, Pi and H+. Nevertheless, it is also possible that a moderate increase in OXPHOS activity takes place in the result of training in the absence of any change in Pipeak. Generally, it is supposed that the training-induced changes in the system cause that more intense exercise is terminated because of fatigue (Pi reaches Pipeak) at the moment of completing of the 1500 m run after a shorter time than before training. Therefore, the mean ATP turnover, power output and run speed are higher, so that the time of the 1500 m run is markedly shortened.

Theoretical methods

Ethical approval

This is a purely theoretic study that did not involve any experiments on humans or animals.

Computer model

The previously developed computer model of the skeletal muscle bioenergetic system, including detailed kinetic OXPHOS description, was used (Korzeniewski 1998, 2018a, b, 2023a, b; Korzeniewski and Zoladz 2001; Korzeniewski and Liguzinski 2004; Korzeniewski and Rossiter 2015, 2020). The model involves the each-step activation (ESA) (parallel activation) mechanism, according to which ATP usage, NADH supply, glycolysis/glycogenolysis and all OXPHOS complexes are directly activated by some cytosolic factor/mechanism (likely to involve cytosolic Ca2+ ions and proteins (de)phosphorylation) during rest-to-work or low-to-high-work transitions in skeletal muscle, heart and other tissues (Korzeniewski 1998, 2007, 2017). The complete model description is given in Korzeniewski (2019) and located on the web site: http://bernardkorzeniewski.pl.

Within the model, the “Pi double-threshold mechanism” of muscle fatigue is expressed by a fixed Picrit = 18 mM, Pipeak = 25 mM (in the standard version of the model for untrained physically active individuals) and kinetic equation for the additional ATP usage (for Pi > Picrit), in which the additional ATP usage flux is proportional to the current Pi-Picrit difference (Korzeniewski and Rossiter 2020; Korzeniewski 2023a). An astonishingly similar value of Picrit of 15 mM was found by Hureau et al. (2022) and related to Pi precipitation with Ca2+ within sarcoplasmic reticulum.

This model was widely tested and was demonstrated to be able to reproduce a broad range of apparently unrelated kinetic properties of the skeletal muscle bioenergetic system, and was used for numerous theoretical studies (see Korzeniewski 2017 for a review and Korzeniewski 2018a, b, 2019, 2021, 2022, 2023a, b; Korzeniewski and Rossiter 2020, 2021, 2022).

Computer simulations

The activity of ATP usage (AUT, proportional to power output) is scaled to 1 at rest. It is increased step-wise at the onset of exercise to a desired value. One AUT unit corresponds to about 3 W during whole-body exercise (e.g., cycling or running). This value may vary (between about 2–4 W), depending on, e.g., working muscle mass and type of exercise. Particular OXPHOS complexes, NADH supply block (TCA cycle, fatty acids β-oxidation, malate-aspartate shuttle) and glycolysis are activated with some delay in parallel with ATP usage at the onset of exercise through ESA (Korzeniewski 1998, 2017; Korzeniewski and Rossiter 2015).

Within the computer model, rate constants that appear in kinetic equations for all OXPHOS complexes (complex I, complex III, complex IV, ATP synthase, ATP/ADP carrier, Pi carrier) and NADH supply block (kC1, kC3, kC4, kSN, kEX, kPI, and kDH, respectively) can be grouped into a single rate constant of OXPHOS: kOX, which corresponds to OXPHOS activity. In the standard model version, corresponding to healthy physically active individuals, the relative kOX is scaled to 1. This corresponds to untrained physically active individuals in the present study.

It was assumed or estimated in the present study that the OXPHOS activity increases by 20 or 12.07%, from kOX = 1 (at Pipeak = 25.0 mM) to kOX = 1.2 (at Pipeak = 25.0 or 23.745 mM) or kOX = 1.1207 (at Pipeak = 25.0 mM) as the result of training.

It was assumed that the total work performed during the whole the 1500 m run is identical before and after training. Within the model, this means that the product of ATP usage activity minus one (AUT – 1) (AUT = 1 at rest) and time of run (trun) (in seconds): (AUT − 1) × trun is constant and equals 31,917 (a.u.). Therefore, as trun decreased 1.137 times as a result of training (from 389 to 342 s) (Zoladz et al. 2022), AUT-1 must increase 1.137 times. In the standard version of the model (untrained, physically active individuals) the exercise duration of 389 s corresponds (through power-duration dependence) to AUT = 83.05. Therefore, after training AUT = (83.05 – 1) × 1.137 + 1 = 94.33. It was assumed that AUT, corresponding to work intensity/power output, was constant throughout the whole run.

In order to obtain trun = 342 s in the trained state for kOX = 1.2, Pipeak was decreased from 25 to 23.745 mM (trun is very sensitive to parameter values and therefore exact parameter values are needed). A training-induced reduction of Pipeak was already postulated previously under the assumption that the training-induced increase in OXPHOS activity in working muscles in vivo corresponds to an increase in OXPHOS enzyme activity measured in vitro (Korzeniewski and Rossiter 2021; Korzeniewski 2022). In order to check what would happen when OXPHOS activity and Pipeak were not modified, but the run speed (exercise intensity) was increased, a simulation was carried out where only AUT was increased from 83.05 to 94.33 (untrained state with a speeded-up run). A simulation of the trained state was also made in which OXPHOS activity was increased by 20% and AUT was increased 1.137 times in relation to untrained state, but Pipeak = 25 0.0 mM was kept unchanged (trained state with markedly increased OXPHOS activity and unreduced Pipeak). The purpose of the simulation of this fictitious situation was to demonstrate the role of the training-induced reduction of Pipeak.

Finally, a simulation of the trained state was carried out where Pipeak was kept the same, as in untrained state (25.0 mM), but kOX was increased by only about 12% (to 1.1207), that is much less than the rise in mitochondrial enzymes amounts/activities, especially COX (most related to OXPHOS activity, see Zoladz et al. 2022), measured in vitro (trained state with moderately increased OXPHOS activity and unchanged Pipeak).

Summing up, five cases of the 1500 m run were simulated:

Case 1. Untrained state: kOX = 1, AUT = 83.05, Pipeak = 25.0 mM, resulting trun = 389 s.

Case 2. Untrained state with an increased ATP usage activity (mean run speed): kOX = 1, AUT = 94.33, Pipeak = 25.0 mM, resulting trun = undetermined.

Case 3. Fictitious trained state with a markedly higher OXPHOS activity and unchanged peak Pi in relation to the untrained state: kOX = 1.2, AUT = 94.33, Pipeak = 25.0 mM, resulting trun = undetermined.

Case 4. Trained state with a markedly higher OXPHOS activity and lowered peak Pi in relation to the untrained state: kOX = 1.2, AUT = 94.33, Pipeak = 23.745 mM, resulting trun = 342 s.

Case 5. Trained state with a moderately higher OXPHOS activity and unchanged peak Pi in relation to the untrained state: kOX = 1.1207, AUT = 94.33, Pipeak = 25.0 mM, resulting trun = 342 s.

Theoretical results

Muscle training, resulting in an increased OXPHOS activity (kOX) by 20.0 or 12.1% (from 1.0 to 1.2 or 1.1207) and decreased Pipeak from 25.0 to 23.745 mM or unchanged Pipeak (Case 4 or 5 vs. Case 1, respectively), resulted in a modeled increase in ATP usage activity (AUT) during the 1500 m run from 83.05 to 94.33 a.u. and markedly increase \(\dot{V}\)O2max from 13.3 mM min−1 to 14.8 mM min−1 (by 12%). This would not be possible without an increase in kOX (Case 2), as exercise would be terminated because of fatigue before completing the 1500 m run. On the other hand, the training-induced increase in \(\dot{V}\)O2max related to rise in AUT to 94.33 would be greater, to 16.0 mM min−1, and the distance covered much longer than 1500 m, if the decrease in Pipeak were not be associate with the kOX increase by 20% (Case 3). AUTcrit, corresponding to critical power, increased from 75 to 85 (by 13%), while t0.63 decreased from 24.4 s to 20.7 s (by 16%) in trained muscle in relation to untrained muscle (Case 4 vs. Case 1). These changes are well within the ranges encountered in experimental studies. When kOX was moderately increased by about 12% and Pipeak remained unchanged (Case 5), the increase in \(\dot{V}\)O2max was very similar, as in Case 4, while the increase in CP and decrease in t0.63 were smaller: by 11% and 11%, respectively. The theoretical results for particular cases are discussed in detail below.

After the onset of exercise in untrained muscle (Case 1), muscle \(\dot{V}\)O2 first rose quickly (primary phase II of the \(\dot{V}\)O2 on-kinetics) and next increased with a much lower pace (slow component of the \(\dot{V}\)O2 on-kinetics). In addition, PCr, Pi and pH first changed fast (pH exhibited an initial overshoot), and then the rate of the changes fell, but were still quite marked (slow component of metabolites). ADP was increasing markedly, after the initial very fast rise, throughout the whole run period. Termination of exercise because of fatigue took place after 6.48 min (389 s) in the moment when Pi reached Pipeak = 25 mM (and thus \(\dot{V}\)O2 reached \(\dot{V}\)O2max). This is equivalent to covering of the distance of 1500 m, as the total work produced was equivalent to 389 s × (83.05 − 1) = 31,917 (a.u.). The slow component of the \(\dot{V}\)O2 on-kinetics appeared after less than 1 min after the onset of exercise, when Pi exceeded Picrit and the additional ATP usage was initiated (Fig. 1). t0.63 equals 24.4 s. Generally, the difference between the end-exercise and rest metabolite concentrations was relatively high (Table 1). This case represents certainly the very heavy exercise intensity domain, as defined by Whipp (1996) (see also Korzeniewski and Rossiter 2022), or severe exercise intensity domain in other classifications, as no steady state is reached and exercise is ultimately terminated because of fatigue.

Fig. 1
figure 1

Simulated time courses of selected fluxes and metabolite concentrations in working muscles during the 1500 m run of untrained individuals (Case 1). Upper panel: \(\dot{V}\)O2, ADP and pH; middle panel: PCr, Pi and ATP; lower panel: ATP usage (vUT), ATP supply by OXPHOS (+ aerobic glycolysis) (vOX), creatine kinase (vCK) and anaerobic glycolysis (vGL). Exercise is terminated because of muscle fatigue at the end of the run (6.55 min)

Full size image
Table 1 Simulated rest and end-run metabolite concentrations in 1500 m run in untrained (Case 1) and trained (Case 4 and 5) muscle
Full size table

When a higher mean run speed/work intensity (AUT = 94.33) was applied without training (kOX = 1, Pipeak = 25.0, that is nothing was changed in the system) (Case 2), locomotory muscles became fatigued (Pi reached Pipeak and \(\dot{V}\)O2 reached \(\dot{V}\)O2max) and exercise was terminated after 2.67 min (160 s) of run (Fig. 2). The total work performed was 160 × (94.33 – 1) = 14,933. This means that the run was terminated after covering the distance of 14,933/31917 × 1500 m = 702 m. Therefore, in the absence of endurance training, an about 1.137-fold increase in work intensity (increase in mean run speed from 3.86 to 4.39 m s−1) would shorten the distance covered before the termination of exercise because of fatigue over twice. Notably, t0.63 = 24.1 s was almost identical, as in Case 1.

Fig. 2
figure 2

Simulated time courses of selected fluxes and metabolite concentrations in working muscles during the 1500 m run of untrained individuals with increased speed (work intensity) (Case 2). Upper panel: \(\dot{V}\)O2, ADP and pH; middle panel: PCr, Pi and ATP; lower panel: ATP usage (vUT), ATP supply by OXPHOS (+ aerobic glycolysis) (vOX), creatine kinase (vCK) and anaerobic glycolysis (vGL). Exercise is terminated because of muscle fatigue at the end of the run (2.67 min)

Full size image

In a fictitious case of trained muscle, where AUT, proportional to work intensity/speed, was increased from 83.05 to 94.33 and OXPHOS activity kOX was increased by 20% (from 1.0 to 1.2), but Pipeak was unchanged in relation to untrained state (25.0 mM) (Case 3), the system remained in the very heavy exercise intensity domain. However, the exercise (run) duration of the determined intensity (mean speed) was markedly longer: trun = 12.37 min (742 s). The total work performed was 742 × (94.33 – 1) = 69,251 (a.u.). This means that the run would be terminated after covering the distance of 69,251/31917 × 1500 m = 3256 m, that is over twice longer than 1500 m. Changes in metabolites were slower, than in untrained muscle (Case 1), but the end-exercise metabolite levels were identical, while end-exercise slow component of the \(\dot{V}\)O2 on-kinetics was markedly higher. t0.63 = 20.7 s was markedly shorter, than in untrained muscle. This is demonstrated in Fig. 3, where only the first 8 min of the exercise lasting totally 12.37 min are shown.

Fig. 3
figure 3

Simulated time courses of selected fluxes and metabolite concentrations in working muscles during the 1500 m run of trained individuals in a fictitious case where there is no training-induced reduction in Pipeak accompanying a marked OXPHOS activity increase (Case 3). Upper panel: \(\dot{V}\)O2, ADP and pH; middle panel: PCr, Pi and ATP; lower panel: ATP usage (vUT), ATP supply by OXPHOS (+ aerobic glycolysis) (vOX), creatine kinase (vCK) and anaerobic glycolysis (vGL). Exercise is terminated because of muscle fatigue at the end of the run after 12.36 min of simulation (only the first 8 min are shown)

Full size image

In the realistic case of trained muscle, where OXPHOS activity increased markedly from kOX = 1.0–1.2 and Pipeak decreased from 25 to 23.745 mM in relation to rest (Case 4), the duration of the 1500 m run was shorter (342 s) than in untrained muscle (389 s), and its intensity (mean speed) was higher: AUT = 94.33 vs. 83.05 (4.39 vs. 3.86 m s−1). As a result, the total work performed before the termination of exercise because of fatigue after covering the distance of 1500 m remained unchanged in relation to untrained individuals: 342 s × (94.33 – 1) = 31,917 (a.u.). The higher work intensity and OXPHOS activity in trained than in untrained muscle resulted in a higher \(\dot{V}\)O2, both total (\(\dot{V}\)O2max) (14.83 mM min−1 vs. 13.26 mM min−1) and the primary phase II component (11.22 mM min−1 vs. 9.72 mM min−1). The absolute magnitude of the slow component in trained muscle in Case 4 after 342 s (end-exercise) was slightly higher, 3.61 mM min−1, than in untrained muscle in Case 1 after 342 s, 3.21 mM min−1, and after 389 s (end-exercise), 3.54 mM min−1. t0.63 = 20.7 s was markedly shorter, than in untrained muscle in (Case 1). The metabolic homeostasis was markedly improved, that is the difference between the end-exercise and rest metabolites was markedly smaller in trained muscle in Case 4, than in untrained muscle in Case 1 (Fig. 4 and Table 1). The smaller changes in metabolites were caused by the decrease in Pipeak, as other metabolites changed in parallel with Pi (Korzeniewski and Rossiter 2020, 2021, 2022).

Fig. 4
figure 4

Simulated time courses of selected fluxes and metabolite concentrations in working muscles during the 1500 m run of trained individuals with marked OXPHOS stimulation and lowered Pipeak (Case 4). Upper panel: V̇O2, ADP and pH; middle panel: PCr, Pi and ATP; lower panel: ATP usage (vUT), ATP supply by OXPHOS (+ aerobic glycolysis) (vOX), creatine kinase (vCK) and anaerobic glycolysis (vGL). Exercise is terminated because of muscle fatigue at the end of the run (5.72 min)

Full size image

Endurance training shifted the power–duration curve in the system in Case 4 upward in relation to Case 1 and thus increased AUTcrit, corresponding to critical power (Fig. 5). Of course, only one point of this curve for untrained and trained muscle corresponds to the 1500 m run. Arrows in Fig. 5 indicate the transition from the untrained to trained state for the work intensity (AUT) and duration corresponding to the 1500 m run. The lower panel in Fig. 5 demonstrates that the curvature constant W’ remained essentially unaffected by training in Case 4 (the AUT – 1/t dependencies for untrained and trained muscle are parallel to each other). AUTcrit, corresponding to critical power, increased with training from 75 to 85 (by 13%).

Fig. 5
figure 5

Simulated power-duration dependence for the parameter values in untrained (Case 1) and trained (Case 4) muscle. Upper panel: AUT (analogous to power output)-time dependence; lower panel: AUT-1/time dependence

Full size image

In the case of trained muscle, where OXPHOS activity is increased moderately by about 12% (kOX = 1.1207), and Pipeak remained unchanged at 25.0 mM (Case 5), the duration of the 1500 m run was shorter (1.137 times) (342 s) than in untrained muscle (389 s), and its speed/intensity was higher (AUT = 94.33 vs. 82.05), as in Case 4. The metabolic homeostasis in Case 5 was the same, as in untrained muscle (Case 1), but poorer than in Case 4 (see Table 1 and Fig. 6). This is caused by the unchanged Pipeak. The power-duration curve in Case 5 was similar, as in Case 4 (not shown), whereas the training-induced increase in AUTcrit (CP) in relation to rest was smaller, from 75 to 83 (by 11%).

Fig. 6
figure 6

Simulated time courses of selected fluxes and metabolite concentrations in working muscles during the 1500 m run of trained individuals with moderate OXPHOS activity increase and unchanged Pipeak (Case 5). Upper panel: V̇O2, ADP and pH; middle panel: PCr, Pi and ATP; lower panel: ATP usage (vUT), ATP supply by OXPHOS (+ aerobic glycolysis) (vOX), creatine kinase (vCK) and anaerobic glycolysis (vGL). Exercise is terminated because of muscle fatigue at the end of the run (5.72 min)

Full size image

Discussion

Mechanism of training-induced shortening of the 1500 m run time

The present study is intended to propose a Pi-based biochemical mechanism that is responsible for the endurance-training-induced increase of the mean speed of run on distances where OXPHOS is the main ATP supplier, using the 1500 m run as an example. This effect of training is associated with a greater work intensity (power output) and thus speed during run and shortening of the run time, while the total work performed remains unchanged. In other words, the mechanism is to explain how to cause exercise termination because of fatigue at a higher work intensity just in the moment of completing of the 1500 m run, and not before (too early termination of run) or after (too slow run, too long run time). The computer model used involves the Pi double-threshold mechanism of muscle fatigue.

Computer simulations demonstrate that the encountered training-induced shortening of the 1500 m run time can be accounted for by a marked increase in OXPHOS activity (kOX) and decrease of Pipeak in trained vs. untrained individuals (Case 4 vs. Case 1) or by a moderate increase in OXPHOS activity at unchanged Pipeak (Case 5 vs. Case 1). These possibilities can be distinguished primarily by the fact that a marked improvement of metabolite homeostasis takes place in Case 4 vs. Case 1, but not in Case 5 vs. Case 1.

General discussion

It is repeated throughout the text above that exercise is terminated because of fatigue when Pi reaches Pipeak. Of course, this is a certain simplification, as people are not completely exhausted after finishing the 1500 m run and can move with a lower speed/work intensity.

Of course, the possible training-induced decrease in Pipeak in itself decreases muscle performance. It was introduced in Korzeniewski and Rossiter (2021) in order to account for the training-induced increase in \(\dot{V}\)O2max and CP observed in experimental studies at a typical increase in OXPHOS activity measured experimentally in vitro. Therefore, either Pipeak actually decreases (Case 4) or the real increase in OXPHOS activity in vivo is lower (Case 5). The advantage of the decrease in Pipeak is an improvement of metabolite homeostasis.

The parameter values used in the model, such as Pipeak, Picrit or additional ATP usage rate constant (kadd, co-determining the magnitude of the slow component) can differ between different muscles, individuals, types of exercise, training statuses and even conditions (for instance external temperature or humidity). “Standard” values of these parameters are used in the present study, but this can be only a semi-quantitative approximation. This study is intended to provide a reliable general mechanism, and not a strictly quantitative description.

A rather high activity of the additional ATP usage is used in this study. This leads to a relatively high intensity of the \(\dot{V}\)O2 and metabolites slow component. In many cases/individuals, these values can be lower.

The 1500 m run was used as an example in the present study. However, the postulated mechanism(s) of the endurance-training-induced improvement of running performance applies in principle to all run distances where OXPHOS predominates as the ATP-supplying process (apart from very long runs, e.g., marathon, where other factors, for example glycogen depletion, can contribute to muscle fatigue). In the 1500 m run, the regular (as opposed to additional) ATP usage activity, proportional to PO, is moderately above CP, so that exercise is terminated because of fatigue in the moment of reaching 1500 m. Namely, AUT = 83.05 and 94.33 vs. AUTcrit = 75 vs. 85 before and after training, respectively. In shorter runs, for instance the 800 m run, PO can be more above CP, as there is less time for Pi to accumulate before it reaches Pipeak. On the other hand, in longer runs PO must be less above CP, as Pi increases more slowly in order not to reach Pipeak before the end of the run.

Study limitations

The computer model used in the present study, as every model of this kind, constitutes only a simplification and approximation of the complex real skeletal muscle bioenergetic system it refers to.

The model is a one-compartment model, as it does not account differences between power-generating muscles (e.g., gluteus, quadriceps, biceps femoris, gastrocnemius and soleus) and various muscle fiber types within muscles. It involves parameters and variables (rate constants, activities, fluxes, and metabolite concentrations) averaged over the entire working muscles group and particular muscles. On the other hand, it is compared with “one-compartment” experimental data: muscle (or pulmonary) \(\dot{V}\)O2 and muscle PCr, Pi, ADP, ATP and H+ concentrations. When doing so, the model is able to account, at least semi-quantitatively, for a surprisingly wide range of different dynamic properties of the skeletal muscle bioenergetic system.

The “Pi double-threshold” mechanism involves explicitly only the total concentration of Pi as the main fatigue-related metabolite, which is supposed to be the most important fatigue-related factor in peripheral fatigue (Allen and Westerblad 2001; Allen et al. 2008; Hureau et al. 2022). Nevertheless, other metabolites, such as H+, ADP, NH4+, IMP and AMP, can also contribute to peripheral (and central) muscle fatigue (Allen et al. 2008; Hureau et al. 2022). On the other hand, the levels of these metabolites (at least H+ and ADP) change in parallel with Pi during exercise (Korzeniewski and Rossiter 2020, 2021). For this reason, Pi can be treated as a “representative” of the whole group of metabolites related to muscle fatigue. Some authors (Wilson et al. 1988; Sundberg et al. 2019) proposed that deprotonated form of Pi–H2PO4−, rather than Pi itself, is the factor that directly leads to muscle fatigue and exercise intolerance. H2PO4− seems an attractive candidate for the main peripheral fatigue factor as, first, its relative increase during rest-to-work transition is greater than that of Pi (Sundberg et al. 2019; Korzeniewski and Rossiter 2022) and, second, it represents both the increase in Pi and H+ (acidification increases the fraction of Pi being in the form of H2PO4−). Notably, a substitution within the computer model of Pi by H2PO4− gives similar general results (not shown). In addition, altered Ca2+ sensitivity and central fatigue were postulated to contribute to fatigue generation (Allen et al. 2008; Allen and Westerblad 2001). On the other hand, Pi can cause Ca2+ precipitation in sarcoplasmic reticulum (Allen and Westerblad 2001). In addition, as it was discussed in Korzeniewski (2019), Pi (and other related metabolites) can potentially mediate in central fatigue (the central nervous system can sense somehow the metabolic state of working muscle fibers, for instance through type III/IV afferents). Recently, Hureau et al. (2022) postulated that Pi is the main metabolite related to peripheral fatigue through precipitation with Ca2+ within sarcoplasmic reticulum, while H+ mediates in central fatigue through group III/IV muscle afferents, which decidedly supports this possibility.

The model involves explicitly only muscle \(\dot{V}\)O2, while pulmonary \(\dot{V}\)O2 is measured in most experimental studies. A dissociation of the pulmonary and muscle \(\dot{V}\)O2 kinetics can be expected under certain conditions, for instance during very intense exercise or off-transient (Poole and Jones, 2012; Krustrup et al. 2009).

The model used involves a constant capillary (extracellular) O2. This certainly constitutes a marked simplification. For instance, the possible impact of O2 perfusion/diffusion on the system (Wagner 2006) is not taken into account. On the other hand, O2 concentration stabilizes during exercise on an approximately constant level (Richardson et al. 1995; McDonough et al. 2005) and muscle fiber O2 depends little on the work rate at higher exercise intensities (Clanton et al. 2013). It was postulated by Poole and Jones (2005, 2012) that O2 delivery is not limiting for the system under normal conditions in healthy individuals working in upright position. In addition, t0.63, \(\dot{V}\)O2max and CP are little sensitive to O2 in normoxia, hyperoxia and even mild hypoxia (Korzeniewski 2023a), and therefore, a very low capillary/mitochondrial O2 at \(\dot{V}\)O2max would be required in order to markedly affect \(\dot{V}\)O2max. Nevertheless, even in such a case, potential O2 diffusion limitations would act through a fall in muscle fiber O2 and acceleration of the reaching of Pipeak (Korzeniewski 2023a).

In the simulations of the effect of training made in this study kOX was increased by either 20 or about 12%, while Pipeak was decreased from 25.0 to 23.745 or left unchanged, respectively. However, an increase in kOX by, say, 25 or 15% (and appropriate decrease of Pipeak) would work equally well. For this and other, already discussed reasons, the present work offers only a semi-quantitative mechanism of the training-induced speeding up and shortening of the time of the 1500 m run.

In the present computer simulations, it is assumed that: (i) the work intensity and run speed is constant during the whole run; (ii) endurance training does not change the working muscle mass; (iii) endurance training does not change the body weight; (iv) all muscles behave kinetically/metabolically in the same way during the whole run; and (v) endurance training does not change mechanical efficiency. Of course, all these assumptions certainly constitute only a rough approximations. Thus, the present computer model provides only an approximate, semi-quantitative representation of the reality. Nevertheless, this fact seems unlikely to affect markedly the general theoretical results and conclusions of the present study.

The present study is partly based on a limited set of empirical data from different studies, subjects, training protocols, etc. Some parameter values had to be adjusted. Therefore, the model predictions can be treated as only approximate and semi-quantitative. The role of in silico studies is to propose concrete mechanisms that can be verified or falsified in the experimental way and thus stimulate and direct future experimental studies.

Conclusions

The mechanism at the biochemical level responsible for the endurance-training-induced increased speed and shortening of the duration of the 1500 m run is an increase in OXPHOS activity and probably a decrease in the peak Pi value (Pipeak), that is Pi concentration at which exercise is terminated because of fatigue. The latter effect improves the metabolite homeostasis during run. The greater run speed/work intensity allowing to complete the 1500 m run in a shorter time is possible because of the attenuation of Pi increase. This delays the reaching of Pipeak by Pi at a given work intensity in the result of the rise in OXPHOS activity. Endurance training increases critical power and increases the whole power–duration curve, while the curvature constant W’ remains essentially unchanged for the parameters values used. The present study generates semi-quantitative predictions, and thus the postulated mechanism can be verified or falsified in the experimental way. Therefore, the present study can stimulate and direct further experimental studies. The proposed general mechanism applies also to the runs on other distances, in which OXPHOS constitutes the main source of ATP. A practical conclusion of the present study is that in order to optimize the run time, the average PO during the 1500 m run should be moderately above CP, more in shorter runs and less in longer runs.