Fig. 3: (Modeling results) Fick’s law of diffusion and Michaelis-Menten kinetics captures the diffusion-like mixing of active and inactive fluids.

a The simulated distribution of ATP concentrations starts as a step function (black, \(t=\) 0 h) and then develops into a smoothed hill function (red, \(t=\) 30 h) as ATP evolves from a one-sided distribution to a homogeneous state. b The model converts the ATP distribution into the speed distribution of active fluid via Michaelis-Menten kinetics: \(\bar{v}={\bar{v}}_{{{{{{\rm{m}}}}}}}[C/(C+K)\,]\), where \({\bar{v}}_{{{{{{\rm{m}}}}}}}\) \(=\) 6.2 µm/s and \(K=\) 270 µM (based on our previous studies⁴⁸). The corresponding mean speed distribution of active fluid evolves from a step function distribution (black, \(t=\) 0 h) to a near-constant function (red, \(t=\) 30 h) (Supplementary Movie 2). Inset: The plot of the Michaelis-Menten equation (Eq. 4). c In the simulation, the diffusion-driven mixing process leads the squared interface displacement to be proportional to time, regardless of initial ATP concentration \({C}_{0}\) (see Supplementary Note 3 for derivation of \(\Delta {x}^{2}\propto t\)). Inset: Interface displacement increases rapidly with time initially, followed by a gradual deceleration similar to the experimental observation (Fig. 2a inset). d In the simulation, the interface progression coefficient \({P}_{{{{{{\rm{I}}}}}}}\) is determined by fitting the \(\Delta {x}^{2}\) vs. \(t\) data (Panel c) to \(\Delta {x}^{2}=2{P}_{{{{{{\rm{I}}}}}}}t\) with \({P}_{{{{{{\rm{I}}}}}}}\) as fitting parameter. The model \({P}_{{{{{{\rm{I}}}}}}}\) increases with the initial concentration of ATP, \({C}_{0}\) (red dots), similarly to how the experimentally analyzed \({P}_{{{{{{\rm{I}}}}}}}\) varies with caged ATP concentration (blue dots; each error bar represents the standard deviation of \(\ge\)3 trials). The model \({P}_{{{{{{\rm{I}}}}}}}\) and experimental \({P}_{{{{{{\rm{I}}}}}}}\) differ by only ~10%. The magenta curve shows the analytical solution, \({P}_{{{{{{\rm{I}}}}}}}\left({C}_{0}\right)\) (Supplementary Equation 7), which reproduces the numerical results (red dots). (See Supplementary Note 3 for derivation of \({P}_{{{{{{\rm{I}}}}}}}\) as a function of \({C}_{0}\)).