Fig 1.
QSSAs for an enzyme-catalyzed reaction.
A model describing an enzyme-catalyzed reaction based on mass-action kinetics (Eqs 2 and 9) can be simplified using either the sQSSAo (Eqs 5 and 11) and the sQSSAp (Eq 11) or the tQSSAo (Eq 8) and the tQSSAp (Eq 13). Both approximations of QSS are defined in terms of S or .
Fig 2.
The accuracy of the sQSSAp and the tQSSAp in heterogeneous environments.
(A) Homogeneous ICs that satisfy the validity condition of the sQSSAo throughout the domain (blue region). E0 ≡ 2μM, S0 ≡ 2μM and KM = 18.5μM. (B-C) Both the sQSSAp and the tQSSAp accurately capture the production of P throughout the domain (B) and the spatial average (C). (D) The ICs in (A) were heterogenized so that the validity condition of the sQSSAo model does not hold near x = 30μm (red region), unlike the other region (blue region). Here, E0(x) = S0(x) = 18.5(tanh(20x/3 − 190) + 1) + 10−4μM, which maintain the spatial average concentration with the ICs in (A), was used. (E-F) The tQSSAp model, but not the sQSSAp model, is accurate. (G) The homogeneous ICs that do not satisfy the validity condition of the sQSSAo through the domain. Here, E0 ≡ 400μM, S0 ≡ 300μM, and KM = 18.5μM. (H-I) The tQSSAp model, but not the sQSSAp model, is accurate. (J) The ICs in (G) were heterogenized so that the validity condition of the sQSSAo is satisfied in x < 15μm (blue region), but not in x ≥ 15μm (red region). Here, E0(x) = 300(tanh(2x/3 − 10) + 1) + 100μM and S0(x) = 290(tanh(10 − 2x/3) + 1) + 10μM, which keep the spatial average concentration with the ICs in (G), were used. (K-L) Both the sQSSAp and tQSSAp models are accurate. For all simulations, C0 ≡ 0μM, P0 ≡ 0μM and kf = 3.4 ⋅ 106M−1s−1, kb = 60s−1, kcat = 3.2s−1, DE = DS = DC = DP = D* = 0.2μm2/s.
Fig 3.
The sQSSAp, but not the tQSSAp, poses a risk when the enzyme is localized within cells, unlike in an in vitro experiment.
(A) Enzymes localized within cellular organelles. (B) The heterogeneous ICs where the sQSSAo are invalid near x = 15μm (red region) but valid in the other region (blue region). Here S0 ≡ 39 μM, E0(x) = 5 ⋅ f(x)μM, where f(x) is the normalized probability density function of the normal distribution with the mean of 15 μm and the standard deviation of 0.2 μm. (C-D) Unlike the sQSSAp model, the tQSSAp model accurately captures the production of P throughout the domain (C) and its spatial average (D). (E) The ICs in (B) were homogenized so that the validity condition of the sQSSAo is satisfied throughout the domain while maintaining the spatial average concentration. Specifically, S0 ≡ 39 μM and E0 ≡ 5μM were used. (F-G) Both the sQSSAp and tQSSAp models are accurate. (H) Enzyme distributions with varying heterogeneity were constructed using E0(x) = 5 ⋅ f(x|σ) μM, where f(x|σ) is the normalized probability density function of the normal distribution with the mean of 15μm and the standard deviation of σ. As σ decreases, the heterogeneity increases (see Method for details). (I) For the enzyme distribution with varying heterogeneity, the tQSSAp model, but not the sQSSAp model, accurately captures the initial velocity (I). For all simulations, we used kf = 6.7 ⋅ 105M−1s−1, kb = 0.53s−1, kcat = 0.13s−1, KM = 1μM, and the diffusion coefficients: DE = DC = 0μm2/s, and DP = DS = 0.2μm2/s. Some parts of Fig 3 were retrieved from Biorender.
Fig 4.
The sQSSAp generates false patterns due to artificial ultrasensitivity.
(A) The model diagram depicting the GK mechanism. The full model (Eq 14) based on mass-action kinetics can be reduced by replacing ES and DSp with either the sQSSAp (Eq 15) or the tQSSAp (Eq 16). (B) The heterogeneous ICs where the sQSSAo is valid in y > 15μm (black triangle), but invalid in y < 15μm (white triangle). For this, we used S0(x, y) = 500 tanh(3.3y − 50) + 520μM, E0(x, y) = 10 tanh(0.2x − 3) + 20μM, and D0 ≡ 20μM. (C) In y > 15μm, the ultrasensitivity of the full model (i) is accurately captured by both the tQSSAp (ii) and sQSSAp models (iii). On the other hand, in y < 15μm, the sQSSAp model generates artificial ultrasensitivity (iii) unlike the full (i) and tQSSAp models (ii). Here, for the sQSSAp model, because DSp is assumed to be negligible. (D) Complex ICs where ST/DT exhibits horizontal stripes and ET/DT exhibits vertical stripes. These ICs do not satisfy the validity condition of the sQSSAo throughout the domain. For this, we used S0(x, y) = 40 cos(2y/3) + 100 μM, E0(x, y) = 5 cos(2x/3) + 100μM, and D0 ≡ 100 μM. (E) The full model (i) and the tQSSAp model (ii) exhibit horizontally striped patterns. In contrast, the sQSSAp model (iii) results in a grid pattern due to the artificial ultrasensitivity. (C) and (E) were obtained when t = 37.5s. For initial conditions of other species, Sp,0 ≡ 0 μM, ES0 ≡ 0μM, and DSp,0 ≡ 0 μM were used. In addition, we used kfe = kfd = 2.22 ⋅ 106M−1s−1, kbe = kbd = 1.84s−1, ke = kd = 0.38s−1, KME = KMD = 1μM, and
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