Figure 1.
Elemental biochemical processes involved in the energy status of cells.
The synthesis sources of ATP are coupled to energy-consumption processes through a network of enzymatic reactions which, interconverting ATP, ADP and AMP, shapes a permanent cycle of synthesis-degradation for the adenine nucleotides. This dynamic functional structure defines the elemental processes of the adenylate energy network, a thermodynamically open system able to accept, store, and supply energy to cells.
Figure 2.
Oxidative phosphorylation and substrate-level phosphorylation generate ATP which is degraded by kinases (also ATPases) and ligases yielding ADP and AMP, respectively. The three adenine nucleotides are catalytically interconverted by adenylate kinase according to the needs of the metabolic system. AMP is also subjected to processes of synthesis-degradation, some AMP molecules are de novo biosynthesized, and a part of AMP is hydrolyzed. According to experimental observations, a very small number of ATP molecules may not remain in the adenylate reactive structure. The system (thermodynamically open) needs a permanent input of nutrients as primary energy source and a consequent output of metabolic waste. The biochemical energy system depicted in the figure represents some key essential parts of the adenylate energy system.
Table 1.
Values of the kinetic parameters used to simulate some of the dynamics of the adenylate energy system.
Figure 3.
Numerical analysis for the model of the adenylate energy system.
a–c: (cf. Scenario I in text) In y-axis we are plotting the max and the min of the different variables α, β and γ. For situations with no oscillations (stable fixed point colored in solid black lines) the max and the min are coincident. For situations with oscillations, the max and the min of the oscillations are plotted separately; in blue we are coloring the max of the oscillation, in red, its minimum value. is the control parameter. The numerical integration shows simple solutions. For small
values (
) the adenine nucleotide concentrations present different stable steady states which lose stability at a Hopf bifurcation at
∼1. For
, the attractor is a stable limit cycle. d–f: (Scenario II) The delay r2 is the control parameter. The numerical bifurcation analysis reveals that the temporal structure is complex, emerging 5 Hopf bifurcations as well as a secondary bifurcation of Neimark-Sacker type. Two pairs of Hopf bifurcations are connected in the parameter space. A third supercritical Hopf bifurcation occurs at r2∼71.94, rapidly followed by another Hopf bifurcation, subcritical, at r2∼72.83. This marks the beginning of the region where the system is multi-stable. The last Hopf bifurcation, born at r2∼72.83, which is subcritical exhibiting the presence of several Torus bifurcations, occurs on a branch of limit cycles when a pair of complex-conjugated Floquet multipliers, leave the unit circle. Branches of stable (resp. unstable) steady states are represented by solid (resp. dashed) black lines; branches of stable (resp. unstable) limit cycles are represented by the max of the oscillation in blue and the minimum in red and by solid (resp. dashed). Hopf bifurcation points are black dots labeled H; Torus bifurcation points are blue dots labeled TR. The bifurcation parameters
(Scenario I) and r2 (Scenario II) are represented on the horizontal axis. The max and min values of each variable are represented on the vertical axis.
Figure 4.
Dynamical solutions of Scenario I.
For = 1.02 (normal activity for the ATP synthesis), periodic oscillations emerge. (a) ATP concentrations. (b) ADP concentrations. (c) AMP concentrations. (d) The Gibbs free energy change for ATP hydrolysis to ADP. (e) The total adenine nucleotide (TAN) pool. It can be observed that ATP and ADP oscillate in anti-phase (the ATP maximum concentration corresponds to the ADP minimum concentration). Likewise, it is noted that the total adenine nucleotide pool shows very small amplitude of only 0.27
and a period around 65 s. (f) ATP transitions between different periodic oscillations and a steady state pattern for several values of
(0.97, 1.08, 1.02, 0.97). Maxima and minima values per oscillation are shown in y-axis.
Figure 5.
Emergence of oscillations in the AEC (Scenario I).
Different oscillatory behavior appears when varying , the modifying factor for the ATP synthesis. (a) For
= 1.02 (normal activity of ATP synthesis) the AEC periodically oscillates with a very low relative amplitude of 0.090. (b) At higher values of ATP synthesis (
= 1.08) large oscillations emerge with a amplitude of 0.199. (c) AEC transitions between different periodic oscillations and steady state patterns for several values of
(0.92, 1, 1.03, 1.06, 1.04, 1, 1.03, 1.02, 0.9).
Figure 6.
Robustness analysis for the adenylate energy charge (AEC) across different modeling conditions.
In y-axis we have plotted the max and the min of the AEC. For situations with no oscillations stable fixed are point colored in solid black lines. In x-axis we have plotted the control parameter, which models the energy level stored in the proton gradient generated by the enzymatic oxidation of input nutrients. From left to right, we can see that the system has a fixed point solution which is stable for
<1 (black solid line) and becomes unstable for
>1 (black dashed line), i.e., there is a Hopf bifurcation (H) at
∼1. For
>1, the limit cycle solution becomes stable, in magenta (blue) we have colored the max (min) of the oscillations. In red, we are coloring the average AEC value of the oscillations. For
<1, the AEC values range from 0.752 and 0.779, and for
>1, the AEC average value between the maximum and minimum per period range from 0.768 to 0.756. At very small
values, for
the AEC exhibits values below 0.6 (Figure 7). The AEC does not substantially change during the simulations indicating that it is strongly buffered against the changes of the main control parameter of the system.
Figure 7.
AEC dynamics under low production of ATP.
AEC values as a function of time. At very small values (
), which represents a strong reduction of the ATP synthesis due to low substrate intake, the dynamic of the adenylate energy system shows a steady state behavior that slowly starts to descend, in a monotone way, up to reach the lowest energy values (AEC ∼0.59) at which the steady state loses stability and oscillatory patterns emerge with a decreasing trend. Finally, when the maximum of the energy charge oscillations reaches a very small value (AEC ∼0.28) the adenylate system suddenly collapses after 12,000 seconds of temporal evolution.
Figure 8.
Experimental vs numerical results of AEC oscillations.
Figure 7a illustrates a classical study of the intracellular adenine nucleotides in a population of intact cells belonging to the yeast Saccharomyces cerevisiae [30] which exhibits AEC rhythms, with max = 0.9, min = 0.6 and a period around 50 s. The authors fitted the experimental points to a sinusoidal curve. Figure 7b shows AEC oscillations belonging to our model at high values of ATP synthesis ( = 1.1), with max = 0.873, min = 0.656 and a period of 65 s.
Figure 9.
Emergence of oscillations in the AEC (Scenario II).
Different oscillatory behavior appears when varying r2, controlling the ADP time delays. (a) For r2 = 37 s the AEC periodically oscillates with a very low relative amplitude of 0.045. (b–c) Existence of complex AEC oscillatory patterns for: (b) r2 = 72 s and (c) r2 = 94 s. (d–e) AEC transitions between different oscillatory behavior and steady state patterns for several r2 values. (d) 50 s, 27 s, 30 s, 32 s, 33 s, 72 s, 52 s. (e) 50 s, 27 s, 30 s, 32 s, 34 s, 36 s, 33 s, 36 s, 38 s, 40 s.