Fig 1.
Fully (Scheme A) and partial (Scheme B) competitive inhibition schemes of Michaelis-Menten type enzyme kinetics.
In fully competitive inhibition, both substrate (S) and inhibitor (I) compete for the same active site of enzyme (E) to bind and form reversible complexes (ES, EI) which subsequently get converted into their respective products (P, Q). Whereas, in partial competitive inhibition, the reversibly formed enzyme-inhibitor (EI) is a dead-end complex. Here (e, s, i, x, y, p, q) are respectively the concentrations of enzyme, substrate, inhibitor, enzyme-substrate, enzyme-inhibitor, product of substrate and product of inhibitor. Further, k1 and ki are the respective forward rate constants, k-1 and k-i are the reverse rate constants and, k2 and k3 are the respective product formation rates.
Table 1.
Summary of variables and parameters used in the theory section.
Fig 2.
Occurrence of distinct steady state timescales with respect to enzyme-substrate (X) and enzyme-inhibitor (Y) complexes.
Here (S, I, E, X, Y, P, Q) are the dimensionless concentrations of substrate, inhibitor, enzyme, enzyme-substrate, enzyme-inhibitor, product of substrate and product of inhibitor. Trajectories are from numerical integration of Eqs 2.2.7–2.2.9 with the parameters ηS = 0.2, εS = 4.1, κS = 3.1, ρ = 10, ηI = 0.1, εI = 1.2, κI = 0.1 along with the initial conditions (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0. Further, upon fixing ρ, one finds that δ = 0.05, γ = 1.55 and σ = 0.17. Here V = εSX and U = ρεIY are the dimensionless reaction velocities corresponding to the conversion of the substrate and inhibitor into their respective products P and Q. A. The steady states corresponding to the enzyme-inhibitor and enzyme-substrate complexes occur at τCI = 0.03, τCS = 0.31 respectively. We should note that τCI is the time at which and τCS is the time at which
. Since τCS ≠ τCI with the current parameter settings, one cannot obtain a common steady state solution to Eqs 2.2.7–2.2.9. B. All the trajectories in the velocity-substrate-product (VPS) space fall within the plane V + P + S = 1. C. All the trajectories in the velocity-inhibitor-product (UQI) space fall within the plane U/ρ + Q + I = 1. D. Sample trajectories in the velocities-inhibitor-substrate (VIS, UIS) and velocities-products spaces (U, P, Q) and (V, P, Q).
Fig 3.
Trajectories of the enzyme kinetics with fully competitive inhibition at different values of δ.
The initial conditions for the simulation of Eqs 2.2.7–2.2.9 are set as (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0. A. Here the settings are ηS = 0.002, εS = 0.04, κS = 0.2, ηI = 0.01, εI = 0.06, κI = 0.1 and ρ = 3.333, σ = 1, δ = 0.1405, ϒ = 3. When δ < 1 and the steady state timescale of the enzyme-substrate complex is lower than the enzyme-inhibitor complex i.e., τCS < τCI, then the evolution of enzyme-substrate complex shows a bimodal type curve with respect to time. Particularly, when σ = 1 and δ > 1 or δ < 1, the temporal evolution of the enzyme-substrate and enzyme-inhibitor complexes show a complex behavior with multiple steady states. Single steady state with respect to Y occurs at (YCQ, QCQ, ICQ) and the corresponding non-steady state values in (V, P, S) space are (VCQ, PCQ, SCQ). B. Here the simulation settings are ηS = 0.02, εS = 0.06, κS = 0.1, ηI = 0.003, εI = 0.04, κI = 0.2 and ρ = 0.225, σ = 1, δ = 9, ϒ = 0.33. When δ > 1 and the steady state timescales of enzyme-substrate complex is higher than the enzyme-inhibitor complex i.e. τCS > τCI, then the evolution of enzyme-inhibitor complex shows a bimodal type curve with respect to time. The single steady state with respect to X occurs at (XCP, SCP, PCP) and the corresponding non-steady state values in the (U, I, Q) space are (UCP, ICP, QCP). C. Here the simulation settings are ηS = 0.02, εS = 0.06, κS = 8.1, ηI = 0.003, εI = 0.04, κI = 1.2 and ρ = 0.225, σ = 1, δ = 1.013, ϒ = 4.5.
Table 2.
Phase-space approximations of fully competitive inhibition scheme.
Table 3.
Phase-space approximations of the partial competitive inhibition scheme.
Fig 4.
Pre-steady state and post-steady state approximations of the enzyme kinetics with fully competitive inhibition in the velocity-substrate (V, S), velocity-inhibitor spaces (U, I) at different values of δ.
The phase-space trajectories start with V = 0 at S = 1 and U = 0 at I = 1, and end at V = 0 at S = 0 and U = 0 at I = 0 with maxima at the steady state. We considered the approximations of (V, U) under the conditions (ηS, ηI, εS, εI) → (0,0,0,0) which is the sQSSA in both (V, S) and (U, I) spaces (Eq 2.4.23, refined forms of sQSSA), (ηS, ηI, Q) → (0,0,0) (Eq 2.4.12) and (ηI, εS, εI) → (0,0,0) (Eq 2.6.10) corresponding to the post and pre-steady state regimes in the (V, S) space, (ηS, εS, εI) → (0,0,0) (Eq 2.6.23) and (ηS, ηI, P) → (0,0,0) (Eq 2.4.16) corresponding to the pre and post steady state regimes of the (U, I) space. Initial conditions for the numerical simulation of Eqs 2.2.7–2.2.9 are set as (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0. A-B. The simulation settings are ηS = 0.002, εS = 0.04, κS = 0.2, ηI = 0.01, εI = 0.06, κI = 0.1 and ρ = 3.333, σ = 1, δ = 0.1405, ϒ = 3. C-D. The simulation settings are ηS = 0.02, εS = 0.06, κS = 0.1, ηI = 0.003, εI = 0.04, κI = 0.2 and ρ = 0.225, σ = 1, δ = 9, ϒ = 0.33. E-F. Here the settings are ηS = 0.02, εS = 0.06, κS = 8.1, ηI = 0.003, εI = 0.04, κI = 1.2 and ρ = 0.225, σ = 1, δ = 1.013, ϒ = 4.5.
Fig 5.
Pre-steady state and post-steady state approximations of the enzyme kinetics with fully competitive inhibition in the velocity-product of substrate (V, P), velocity-product of inhibitor spaces (U, Q) at different values of δ.
The phase-space trajectories start at P = 0 and Q = 0, and end at P = 1 and Q = 1 with maxima at the steady state. We considered the approximations of (V, U) under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) which is the refined standard QSSA in both (V, P) and (U, Q) spaces (Eq 2.4.18), (ηS, ηI, Q) → (0,0,0) (Eq 2.4.12) and (ηI, εS, εI) → (0,0,0) (Eq 2.6.10) corresponding to the post and pre-steady state regimes in the (V, P) space, (ηS, εS, εI) → (0,0,0) (Eq 2.6.23) and (ηS, ηI, P) → (0,0,0) (Eq 2.4.16) corresponding to the pre- and post-steady state regimes of the (U, Q) space. Using the mass conservation laws V + P + S = 1 and U/ρ + Q + I = 1, V and P can be expressed in terms of S in a parametric form and U and Q can be expressed in in terms of I in a parametric form. Initial conditions for the simulation of Eqs 2.2.7–2.2.9 are (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0. A-B. Simulation settings are ηS = 0.002, εS = 0.04, κS = 0.2, ηI = 0.01, εI = 0.06, κI = 0.1 and ρ = 3.333, σ = 1, δ = 0.1405, ϒ = 3. C-D. Here the simulation settings are ηS = 0.02, εS = 0.06, κS = 0.1, ηI = 0.003, εI = 0.04, κI = 0.2 and ρ = 0.225, σ = 1, δ = 9, ϒ = 0.33. E-F. Here the simulation settings are ηS = 0.02, εS = 0.06, κS = 8.1, ηI = 0.003, εI = 0.04, κI = 1.2 and ρ = 0.225, σ = 1, δ = 1.013, ϒ = 4.5.
Fig 6.
Error associated with the sQSSA with stationary reactant assumption of the fully competitive enzyme kinetics over the velocity-substrate (V, S), velocity-inhibitor spaces (U, I) at different values of εS, εI.
Here δ will vary with respect to each iteration. We considered the error in the approximations of the reaction velocities (V, U) under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) and stationary reactant assumption as defined in Eq 2.4.2. The error was computed as error (%) = 100 |steady state velocities from simulation–approximated velocities| / steady state velocities from simulation. Here the simulation settings are ηS = 0.02, ηI = 0.01 and σ = 1. With these settings, upon fixing σ one finds that and
as defined in Eq 2.4.22. A1, B1, C1. Error % in the standard QSSA of V. A2, B2, C2. Error % in QSSA of U. A1-2. κS = 0.1, κI = 1. B1-2. κS = 1, κI = 0.1. C1-2. κS = 1, κI = 1.
Fig 7.
Error associated with the refined form of sQSSA with stationary reactant assumption of the fully competitive enzyme kinetics in the velocity-substrate (V, S), velocity-inhibitor spaces (U, I) at different values of εS, εI under the conditions that κS ≠ κI.
Here E1, E2 and δ will vary with respect to each iteration. We considered the error in the approximations of the reaction velocities (V, U) under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) various limiting conditions as defined in Eq 2.4.19 with (S, I) = (1, 1). The error was computed as error (%) = 100 |steady state velocities from simulation–approximated velocities| / steady state velocities from simulation. Here the simulation settings for A1-5 are ηS = 0.02, κS = 1, ηI = 0.01, κI = 0.1 and σ = 1. Simulation settings for B1-5 are ηS = 0.02, κS = 0.1, ηI = 0.01, κI = 1 and σ = 1. With these settings, upon fixing σ one finds that and
as defined in Eq 2.4.22 along with the inequality conditions E1, and E2 as defined in Eqs 2.7.5 and 2.7.6. A1, B1. Error % in the QSSA of V. A2, B2. Error % in the QSSA of U. A3, B3. E1 (Eq 2.7.5). A4, B4. E2 (Eq 2.7.6). A5, B5. δ as defined in Eq 2.4.22.
Fig 8.
Error associated with the refined form of sQSSA with stationary reactant assumption of the fully competitive enzyme kinetics in the velocity-substrate (V, S), velocity-inhibitor spaces (U, I) at different values of εS, εI under the condition that κS = κI.
Here E1, E2 and δ will vary with respect to each iteration. We considered the error in the approximations given in Eq 2.4.19 with (S, I) = (1, 1). The error was computed as error (%) = 100 |steady state velocities from simulation–approximated velocities| / steady state velocities from simulation. Here the simulation settings for A1-5 are ηS = 0.02, κS = κI = 1, ηI = 0.01 and σ = 1. Similar simulation settings for B1-5 with κS = κI = 0.1. With these settings, upon fixing the value of σ one finds that and
as defined in Eq 2.4.22 along with the inequality conditions E1, and E2 as defined in Eqs 2.7.5 and 2.7.6. A1, B1. Error % in QSSA of V. A2, B2. Error % in QSSA of U. A3, B3. E1 (Eq 2.7.5). A4, B4. E2 (Eq 2.7.6). A5, B5. δ as defined in Eq 2.4.22.
Fig 9.
Approximate solutions in the velocity-substrate-inhibitor spaces (V, I, S) and (U, I, S).
We considered the approximations of the reaction velocities (Z = V and U) under the conditions (ηS, ηI, εS, εI) → (0,0,0,0) which is the refined form of standard QSSA as given in Eqs 2.4.19 and 2.6.12 (for the relationship between S and I) and under the conditions (ϕI, ϕS) → (0,0) as given by the solutions of the coupled approximate linear ODEs Eqs 2.5.1.1 and 2.5.1.2 as given in Appendix A in S1 Appendix in a parametric form where τ acts as the parameter. Here the common initial conditions for the numerical simulation of Eqs 2.2.7–2.2.9 are (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0 and other simulation settings are ηS = 0.06, κS = 8.1, ηI = 0.03, κI = 1.2. σ = 1 for (A-D), σ = 0.1 for E and σ = 10 for F. A. εS = 0.08, εI = 0.04, ρ = 1, δ = 0.3083, ϒ = 3.375. B. εS = 13.8, εI = 0.4, ρ = 17.25, δ = 0.1485, ϒ = 0.1957. C. εS = 3.8, εI = 20.4, ρ = 0.093, δ = 3.617, ϒ = 36.24. D. εS = 33.8, εI = 20.4, ρ = 0.8284, δ = 1.031, ϒ = 4.074. E. εS = 33.8, εI = 20.4, ρ = 8.284, δ = 0.1031, ϒ = 0.4074. F. εS = 33.8, εI = 20.4, ρ = 0.0828, δ = 10.31, ϒ = 40.74.
Fig 10.
The φ-approximations of the fully competitive enzyme kinetics in the velocity-products spaces (V, P, Q) and (U, P, Q) at different values of δ.
The phase-space trajectories start at P = 0 and Q = 0, and end at P = 1 and Q = 1 with maxima at the steady state. We considered the φ-approximations of (V, U) which are the solutions of Eqs 2.5.1.1 and 2.5.1.2 as given in Appendix A in S1 Appendix in a parametric form where τ act as the parameter and standard QSSA solutions under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) in a parametric form where S acts as the parameter as given in Eqs 2.4.19 and 2.6.12. Common initial conditions for the numerical simulation of Eqs 2.2.7–2.2.9 are (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0 and other simulation settings are ηS = 0.06, κS = 8.1, ηI = 0.03, κI = 1.2. σ = 1 for (A-D), σ = 0.1 for E and σ = 10 for F. A. εS = 0.08, εI = 0.04, ρ = 1, δ = 0.3083, ϒ = 3.375. B. εS = 13.8, εI = 0.4, ρ = 17.25, δ = 0.1485, ϒ = 0.1957. C. εS = 3.8, εI = 20.4, ρ = 0.093, δ = 3.617, ϒ = 36.24. D. εS = 33.8, εI = 20.4, ρ = 0.8284, δ = 1.031, ϒ = 4.074. E. εS = 33.8, εI = 20.4, ρ = 8.284, δ = 0.1031, ϒ = 0.4074. F. εS = 33.8, εI = 20.4, ρ = 0.0828, δ = 10.31, ϒ = 40.74.
Fig 11.
The φ-approximations of the fully competitive enzyme kinetics in the velocity spaces (V, U) at different values of δ.
We considered the φ-approximations which are the solutions of Eqs 2.5.1.1 and 2.5.1.2 as given in Appendix A in S1 Appendix in a parametric form where τ act as the parameter and standard QSSA solutions for (V, U) under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) in a parametric form where S acts as the parameter as given in Eqs 2.4.19 and 2.4.22. The trajectory in the (V, U) space starts at (V, U) = (0, 0) and ends at (V, U) = (0, 0). Arrow in C indicates the direction of the trajectory evolution. Common initial conditions for the numerical simulation of Eqs 2.2.7–2.2.9 are (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) at τ = 0 and other simulation settings are ηS = 0.06, κS = 8.1, ηI = 0.03, κI = 1.2. σ = 1 for (A-D), σ = 0.1 for E and σ = 10 for F. A. εS = 0.08, εI = 0.04, ρ = 1, δ = 0.3083, ϒ = 3.375. B. εS = 13.8, εI = 0.4, ρ = 17.25, δ = 0.1485, ϒ = 0.1957. C. εS = 3.8, εI = 20.4, ρ = 0.093, δ = 3.617, ϒ = 36.24. D. εS = 33.8, εI = 20.4, ρ = 0.8284, δ = 1.031, ϒ = 4.074. E. εS = 33.8, εI = 20.4, ρ = 8.284, δ = 0.1031, ϒ = 0.4074. F. εS = 33.8, εI = 20.4, ρ = 0.0828, δ = 10.31, ϒ = 40.74.
Fig 12.
Pre- and post-steady state approximations of the enzyme kinetics with partial competitive inhibition.
Here the simulation settings are ηS = 0.02, εS = 0.1, κS = 0.1, χI = 0.03, εI = 0.06, κI = 0.5. Common initial conditions for the numerical simulation of Eqs 2.9.5–2.9.7 are (S, I, E, X, Y, P) = (1,1,1,0,0,0) at τ = 0. Post-steady state approximations were generated under the conditions that (ηS, χI) → 0 and the pre-steady state approximations were computed under the conditions that (εS, χI, εI) → (0,0,0). A. Simulation trajectories of (S, I, E, X, Y, P). Clearly, (I, E, Y) ends at the equilibrium states (I∞, E∞, Y∞) upon complete depletion of the substrate. When the steady state timescales of X and Y are different, then Y will exhibit a steady state where . B1-3, C and D. Simulated trajectories along with the in the pre- and post-steady state approximations. Post steady state approximations under the conditions that (ηS, χI, εS, εI) → (0,0,0,0) were generated using Eq 2.9.2.2. B1. (Y, S) space trajectory and approximations are computed using Eqs 2.9.3.10 and 2.9.4.9 for the pre and post-steady state regimes respectively. B2. (V, S) space trajectory with approximations using Eqs 2.9.3.8, 2.9.4.7 and 2.9.2.2 corresponding to the pre and post steady state regimes. B3. (S, I) space trajectory and approximations using the mass conservation law I = 1 − εIY (Eqs 2.9.3.10 and 2.9.4.9 with S ∈ [0,1] as the parameter). When
, then one finds that
, representing a local minimum in the (I, S) space. C. (V, S, I) space approximations (Table 3 for parametric representations). D. (V, P, S) space approximations (Table 3).
Fig 13.
Error associated with various steady state approximations along with the stationary reactant assumption corresponding to the enzyme kinetics with partial competitive inhibition in the velocity-substrate (V, S) space at different values of εS, εI.
Here E1 (A1, B1, C1 and D1, error in sQSSA given by Eq 2.9.10), E2 (A2, B2, C2 and D2, error in the refined form of sQSSA given in Eq 2.9.3.16) and E3 (A3, B3, C3 and D3, error in Eq 2.9.5.5) are logarithm of percentage errors. The computed error (%) = 100 |steady state velocities from simulation–approximated velocities| / steady state velocities from simulation. A1-3. ηS = 0.02, κS = 0.001, κI = 0.005, ηI = 0.01. B1-3. ηS = 0.02, κS = 1, κI = 5, ηI = 0.01. C1-3. ηS = 0.02, κS = 0.1, κI = 0.5, ηI = 0.01. D1-3. ηS = 0.02, κS = 0.1, κI = 0.1, ηI = 0.01.
Fig 14.
Pre- and post-steady state approximations in the inhibitor-substrate space for the enzyme kinetics with fully and partial competitive inhibition.
Common initial conditions for the simulation of fully competitive inhibition Eqs 2.2.7–2.2.9 are (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) and for partial competitive inhibition Eqs 2.9.5–2.9.7 are (S, I, E, X, Y, P) = (1,1,1,0,0,0) at τ = 0. For the fully competitive inhibition scheme (A, C, E where trajectories start at (I, S) = (1,1) and end at (I, S) = (0,0)), approximations were computed using Eq 2.4.22 which are valid under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) and the φ-approximations which are the solutions of Eqs 2.5.1.1 and 2.5.1.2 as given in Appendix A in S1 Appendix in a parametric form with τ as the parameter. For the partial competitive inhibition scheme (B, D, F where trajectories start at (I, S) = (1,1) and end at (I,S) = (I∞, 0)), post-steady state approximations were generated under the conditions that (ηS, χI) → 0 and the pre-steady state approximations were computed for (εS, χI, εI) → (0,0,0) using the conservation law I = 1 − εIY (using Eqs 2.9.3.10 and 2.9.4.9 respectively with S ∈ [0,1] as the parameter). Common simulation settings are ηS = 0.02, κS = 0.1, ηI = 0.03, χI = 0.03, κI = 0.5, σ = 1. A-B. εS = 5.5, εI = 0.06, ρ = 137.5, δ = 0.07, γ = 0.022. C-D. εS = 0.5, εI = 0.06, ρ = 12.5, δ = 0.634, γ = 0.024. E-F.εS = 0.05, εI = 0.06, ρ = 1.5, δ = 2.314, γ = 0.24.
Fig 15.
Approximation of the steady state substrate and inhibitor levels corresponding to fully (A, B) and partial competitive (C, D) schemes.
Common initial conditions for the simulation of fully competitive inhibition Eqs 2.2.7–2.2.9 are (S, I, E, X, Y, P, Q) = (1,1,1,0,0,0,0) and for partial competitive inhibition Eqs 2.9.5–2.9.7 are (S, I, E, X, Y, P) = (1,1,1,0,0,0) at τ = 0. A-D clearly show that (P, Q) ≅ (0,0) in the pre-steady state regime so that V ≅ 1 − S and U ≅ ρ(1 − I). These lines intersect the post-steady state approximations near the original steady state. In case of fully competitive inhibition, we considered the intersection (R1, R2) between the post steady state approximations under the conditions that (ηS, ηI, εS, εI) → (0,0,0,0) (Eq 2.4.23), (ηS, ηI, Q) → (0,0,0) (Eq 2.4.16), (ηS, ηI, P) → (0,0,0) (Eq 2.4.12) and the pre-steady state approximations under the conditions that (ηS, εS, εI) → (0,0,0) (Eq 2.6.10) and (ηI, εS, εI) → (0,0,0) (Eq 2.6.23) along with V ≅ 1 − S and U ≅ ρ(1 − I). In case of partial competitive inhibition, we considered the intersections (H1, H2) between the post-steady state approximations under the conditions that (ηS, χI, εS, εI) → (0,0,0,0) (Eq 2.9.2.2) and (ηS, χI) → (0,0) (Eq 2.9.4.7) and the pre-steady state approximations under the conditions that (χI, εS, εI) → (0,0,0,0) (Eq 2.9.3.8) along with V ≅ 1 − S. The settings are as follows. A-B. ηS = 0.02, κS = 1.1, κI = 0.2, ηI = 0.03, εS = 0.6, εS = 0.4, σ = 1, ρ = 2.25, δ = 0.244, ϒ = 3.67. C. ηS = 0.01, κS = 0.01, κI = 0.05, ηI = 0.02, εS = 0.9, εS = 0.5. D. ηS = 0.01, κS = 1, κI = 5, ηI = 0.02, εS = 0.9, εS = 0.5.