Model and theory schematics. (a) Activation of the membrane-anchored substrate is enhanced by membrane-associated enzyme molecules (E), which act upon inactive substrate with second-order rate constant k_act . Substrate inactivation occurs with first-order rate constant k_i. The finite lifetime of the enzyme at the membrane is considered. (b) Dilute enzyme limit. The model considers only one enzyme on the membrane, and the density of activated substrate falls to approximately zero far away from the enzyme. The influence of substrate depletion in the vicinity of the enzyme is inferred from the value of the effective enzymatic rate constant (Eq. (1)). (c) High-density approximation. The model implicitly takes into account the influence of neighboring enzyme molecules, whose activities are homogenized using a mean-field theory.

Comparison of theory and BD simulation results: long-lived enzymes at low density. The effective enzymatic rate constant is shown as a function of the true reaction rate constant (κ = k_act D). Da is the scaled inactivation rate constant (Da = k_is² D). The dilute enzyme limit (Eq. (9)) is shown by solid curves; simulation results are represented by symbols. Simulations were performed for a periodic domain with one enzyme at sufficiently low density (η_E = n_E s² = 10⁻⁵ << Da).

Comparison of theory and simulation results: single enzyme with finite lifetime. The effective enzymatic rate constant was computed as a function of the mean lifetime of the enzyme on the membrane, τ_on = Dt_on /s² , in the diffusion-limited regime (κ = 10⁶) and at low enzyme density (η_E = 10⁻⁵). The transient solution given by Eq. (4)24 and BD simulation results with deterministic enzyme lifetimes are shown by solid curves and closed symbols, respectively. The probabilistic lifetime solution given by Eq. (11) and corresponding BD results are shown by dashed curves and open symbols, respectively.

Brownian dynamics algorithm. (a) In the bulk membrane, the next position of the particle is taken to be random on the concentric circle boundary touching the closest enzyme (shown by a gray disk). An inactive particle (shown by a small open circle) can be activated in a thin activation layer next to the enzyme boundary. The activated particle (shown by a small filled circle) is converted back to the inactive state while making diffusion steps. (b) In the activation layer, a new rectangular coordinate system χ, ξ is defined, neglecting curvature of the boundary (the size of the activation layer and thus the curvature is exaggerated here for the purpose of illustration). The next particle position (χ_n, ξ_n) is defined by displacements Δχ, Δξ sampled from the Gaussian distribution and assuming that the enzyme boundary is reflective. Particle activation is modeled as a one-dimensional problem in χ. (c) Distribution density function g_act describing the number of activation events (with κ = 10⁶) as a function of the initial distance to the reactive boundary, with s = 3.5 nm and Δt_AL = 10⁻⁹ s. The varied parameter is D (in nm²/s).

Evaluation of the mean-field approximation for higher enzyme densities. The effective enzymatic rate constant is shown as a function of enzyme density (η_E = n_E s²) in the diffusion limit (κ = 10⁶). Solid curves are the theoretical predictions (Eq. (9), with Da^* = Da + αη_E substituted for Da); simulation results, with a random array of 200 enzymes, are represented by filled symbols. Theory and simulation results for regular enzyme arrays, arranged on a triangular lattice, are represented by dashed curves and open symbols, respectively.

Random receptor array with spontaneous enzyme association/dissociation. Constant parameter values are taken from Shea et al.13:k_i = 0.1 s⁻¹, s = 3.5 nm, and η_E = 2.333 × 10⁻⁵; k_r = 1/t_on is the dissociation rate constant of the enzyme, and conditions are such that the duration of the enzyme-off state is t_off = 20t_on . BD computations were performed for a random array of 200 receptors with spontaneous association and dissociation of enzymes. (a) Relaxation of α toward steady-state behavior in long-run BD simulations. The relative diffusion coefficient D is varied as: (1) 2x10⁻¹¹, (2) 2x10⁻¹⁰, or (3) 2x10⁻⁹ cm²/s. (b) Effective rate constant versus k_r. Closed symbols denote BD results (averaged over the fluctuations in the steady-state regime), and open symbols denote values reported by Shea et al.31: circles, D = 2x10⁻¹¹; squares, D = 2x10⁻¹⁰; triangles, D = 2x10⁻⁹. Solid curves show the mean-field theory predictions (accounting for the enzyme density), and dashed curves show the dilute enzyme limit given by Eq. (11).