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

Abstract

The Michaelis-Menten (MM) rate law has been a fundamental tool in describing enzyme-catalyzed reactions for over a century. When substrates and enzymes are homogeneously distributed, the validity of the MM rate law can be easily assessed based on relative concentrations: the substrate is in large excess over the enzyme-substrate complex. However, the applicability of this conventional criterion remains unclear when species exhibit spatial heterogeneity, a prevailing scenario in biological systems. Here, we explore the MM rate law's applicability under spatial heterogeneity by using partial differential equations. In this study, molecules diffuse very slowly, allowing them to locally reach quasi-steady states. We find that the conventional criterion for the validity of the MM rate law cannot be readily extended to heterogeneous environments solely through spatial averages of molecular concentrations. That is, even when the conventional criterion for the spatial averages is satisfied, the MM rate law fails to capture the enzyme catalytic rate under spatial heterogeneity. In contrast, a slightly modified form of the MM rate law, based on the total quasi-steady state approximation (tQSSA), is accurate. Specifically, the tQSSA-based modified form, but not the original MM rate law, accurately predicts the drug clearance via cytochrome P450 enzymes and the ultrasensitive phosphorylation in heterogeneous environments. Our findings shed light on how to simplify spatiotemporal models for enzyme-catalyzed reactions in the right context, ensuring accurate conclusions and avoiding misinterpretations in in silico simulations.

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 sQSSA_o (Eqs 5 and 11) and the sQSSA_p (Eq 11) or the tQSSA_o (Eq 8) and the tQSSA_p (Eq 13). Both approximations of QSS are defined in terms of S or $\widehat{S}=S+C$.

Fig 2. The accuracy of the sQSSA_p and the tQSSA_p in heterogeneous environments.

(A) Homogeneous ICs that satisfy the validity condition of the sQSSA_o throughout the domain (blue region). E₀ ≡ 2μM, S₀ ≡ 2μM and K_M = 18.5μM. (B-C) Both the sQSSA_p and the tQSSA_p accurately capture the production of P throughout the domain (B) and the spatial average $\overline{P}$ (C). (D) The ICs in (A) were heterogenized so that the validity condition of the sQSSA_o 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 tQSSA_p model, but not the sQSSA_p model, is accurate. (G) The homogeneous ICs that do not satisfy the validity condition of the sQSSA_o through the domain. Here, E₀ ≡ 400μM, S₀ ≡ 300μM, and K_M = 18.5μM. (H-I) The tQSSA_p model, but not the sQSSA_p model, is accurate. (J) The ICs in (G) were heterogenized so that the validity condition of the sQSSA_o 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 sQSSA_p and tQSSA_p models are accurate. For all simulations, C₀ ≡ 0μM, P₀ ≡ 0μM and k_f = 3.4 ⋅ 10⁶M⁻¹s⁻¹, k_b = 60s⁻¹, k_cat = 3.2s⁻¹, D_E = D_S = D_C = D_P = D_* = 0.2μm²/s.

Fig 3. The sQSSA_p, but not the tQSSA_p, 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 sQSSA_o 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 sQSSA_p model, the tQSSA_p model accurately captures the production of P throughout the domain (C) and its spatial average $\overline{P}$ (D). (E) The ICs in (B) were homogenized so that the validity condition of the sQSSA_o is satisfied throughout the domain while maintaining the spatial average concentration. Specifically, S₀ ≡ 39 μM and E₀ ≡ 5μM were used. (F-G) Both the sQSSA_p and tQSSA_p 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 tQSSA_p model, but not the sQSSA_p model, accurately captures the initial velocity (I). For all simulations, we used k_f = 6.7 ⋅ 10⁵M⁻¹s⁻¹, k_b = 0.53s⁻¹, k_cat = 0.13s⁻¹, K_M = 1μM, and the diffusion coefficients: D_E = D_C = 0μm²/s, and D_P = D_S = 0.2μm²/s. Some parts of Fig 3 were retrieved from Biorender.

Fig 4. The sQSSA_p 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 DS_p with either the sQSSA_p (Eq 15) or the tQSSA_p (Eq 16). (B) The heterogeneous ICs where the sQSSA_o 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 tQSSA_p (ii) and sQSSA_p models (iii). On the other hand, in y < 15μm, the sQSSA_p model generates artificial ultrasensitivity (iii) unlike the full (i) and tQSSA_p models (ii). Here, ${\widehat{S}}_p/S_T=S_p/S_T$ for the sQSSA_p model, because DS_p is assumed to be negligible. (D) Complex ICs where S_T/D_T exhibits horizontal stripes and E_T/D_T exhibits vertical stripes. These ICs do not satisfy the validity condition of the sQSSA_o 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 tQSSA_p model (ii) exhibit horizontally striped patterns. In contrast, the sQSSA_p 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, S_p,0 ≡ 0 μM, ES₀ ≡ 0μM, and DS_p,0 ≡ 0 μM were used. In addition, we used k_fe = k_fd = 2.22 ⋅ 10⁶M⁻¹s⁻¹, k_be = k_bd = 1.84s⁻¹, k_e = k_d = 0.38s⁻¹, K_ME = K_MD = 1μM, and ${\delta }_S={\delta }_{S_p}={\delta }_{ES}={\delta }_{D{S}_p}=0.2\mu m^2/s$.