Beyond homogeneity: Assessing the validity of the Michaelis–Menten rate law in spatially heterogeneous environments
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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