Kinetic cycles of the simplified three species enzyme kinetics. Comparing to the closed system of cycle (a), cycle (b) brings in two more substrates D and E which can break the detailed balance to generate a non-equilibrium steady state flux, with the reaction: by keeping a non-equilibrium concentration ratio [D]/[E]. By neglecting the fluctuation of concentrations of D and E, cycle (b) can be represented by cycle (a) in terms of the pseudo-first-order rate constants: and .
Schematic diagram of the three-species kinetic cycle
Scheme for the reversible Schnakenberg model. The dash curve in the box uncovers the self-catalytic mechanism of species X, and the dash box shows the key part that generates oscillation while B and A are energy input and output through the non-equilibrium concentrations.
The stationary solution y 0 of Eqs. (31) in three different regions of energy pump B/A. The unstable fixed states are marked as the dash line within 0.1 < B < 0.9, that is 2 < B/A < 18, which refers to a limit cycle around the unstable fixed point. The star on the curve is noted as the Hopf bifurcation point. All parameters are set as in Eqs. (32).
The amplitudes of the concentration of species X and Y vary with energy pump. Both curves are drawn inside the parameter range of the limit cycle, where A = 0.06 and B ∈ (0.15, 0.85). All other parameters are the same as Eqs. (32).
The non-equilibrium chemical potential difference ΔG can generate the non-zero deterministic flux J SS . The parameters we use are all shown in Eqs. (32), while B is chosen within (0, 0.1), and with these parameters this model shows one monostable steady state and leads to a steady state deterministic flux J SS .
Flux and its direction flowing on the landscape and along the deterministic trajectory. The red arrows in (a), (c), (e), and (g) are vectors of probabilistic flux for different energy pump strengths, and (b), (d), (f), and (h) show their directions. The black solid line refers to the deterministic trajectory. The gradual change color from red to blue represents hill to valley of the potential landscape, which is similar as in Ref. 17.
Sketch for two different kinds of decomposition of the probabilistic flux vectors. (a) shows how the vector is mapped to the deterministic trajectory and (b) shows that for all the flux vectors that pass through the cut line, their normal component will be summed into current. The start point O is set as the coordinate origin in the phase space.
The relationship of deterministic effective flux and loop integration of probability flux. Flux Ave represents effective deterministic flux and Flux IntTra is loop averaged probability flux which can be calculated by the definition equation (33), and ΔG means chemical potential difference and also Gibbs free energy. (a) The two fluxes show the similar behavior with changing of ΔG. While the variation range ΔG ∈ (10.3, 12.05) is corresponding to the limit cycle range B ∈ (0.15, 0.85). (b) We can obtain the direct correlation between probability flux and averaged effective deterministic flux. In inset figure, we perform the linear regression analysis with origin software on two variables, in which R − Square(COD) = 0.53222.
The total current passing through the Poincare line OP in Fig. 8(b) versus non-equilibrium potential difference ΔG. The inse figure comes from linear fitness with R − Square(COD) = 0.93679.
The relationship of chemical driving force and probability flux under loop integration along with the deterministic trajectory. Flux IntTra is the loop averaged probability flux. ∇μ IntTra is loop averaged driving force that calculated in the same way as Flux IntTra . (a) The probabilistic flux and its driving force ∇μ have the similar behavior with respect to the Gibbs free energy ΔG, and the variation range of ΔG is same as Fig. 9. (b) The probability flux correlates with the chemical potential driving force. The inset figure is the linear regression analysis to two variables, and the coefficient of determination is R − Square = 0.87872, which means strong linear relation between two variables.
(a) The period of oscillation X varies with energy pump. (b) Comparison between the period and the deterministic effective flux Flux Ave (same as figures above).
Phase coherence varies with fluctuation and Gibbs free energy. (a) Phase coherence versus system volume. The axis of 1/V is set as logarithmic coordinate. We choose the variation range of 1/V as (0.0008, 0.2) which includes the value 0.005 that we used for all above figures. Here B = 0.5, corresponding to ΔG = 11.5, can ensure that the Schnakenberg model behaves as stochastic oscillation. (b) Phase coherence versus the Gibbs free energy ΔG. Here 1/V = 0.005 is fixed, and the variation region of ΔG is (10.2, 12.2), same as above, which keeps the model behaving as oscillation. Both vertical coordinates of the two curves use the same scale.
Landscape barrier height versus 1/V and Gibbs free energy. Barrier height of the potential landscape is calculated from the landscape top (red center or edge) to the bottom of the oscillation ring (blue valley) shown in Fig. 7. (a) is drawn when 1/V ∈ (0.0008, 0.01) with fixed energy pump B = 0.5, corresponding to ΔG = 11.5. Based on the logarithm data of 1/V, we obtain the linear fitness curve with R − Square(COD) = 0.86391. (b) is drawn when Gibbs free energy ΔG ∈ (10.2, 12.2) with 1/V = 0.05, with linear fitness R − Square(COD) = 0.65715. Both vertical coordinates of the two curves use the same scale.
Ratio of diffusion force over probability flux force versus 1/V and Gibbs free energy ΔG. The ratio is defined by the ratio of the most probable magnitude of the diffusion force and the most probable probability magnitude of the flux force . (a) The ratio is drawn against the fluctuation magnitude in the loglog coordinate with 1/V ∈ (0.0008, 0.01) and B = 0.5. Based on the double logarithm data of 1/V, we obtain the linear fitness curve with high score R − Square(COD) = 0.97935. (b) The curve is drawn versus Gibbs free energy with the same variation as above and the fluctuation magnitude is fixed at 1/V = 0.005, which also shows good linear trend with the linear fitness score R − Square(COD) = 0.96207.
Sketch map for the definition of phase coherence.
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