Self-mixing in microtubule-kinesin active fluid from nonuniform to uniform distribution of activity

Active fluids have applications in micromixing, but little is known about the mixing kinematics of systems with spatiotemporally-varying activity. To investigate, UV-activated caged ATP is used to activate controlled regions of microtubule-kinesin active fluid and the mixing process is observed with fluorescent tracers and molecular dyes. At low Péclet numbers (diffusive transport), the active-inactive interface progresses toward the inactive area in a diffusion-like manner that is described by a simple model combining diffusion with Michaelis-Menten kinetics. At high Péclet numbers (convective transport), the active-inactive interface progresses in a superdiffusion-like manner that is qualitatively captured by an active-fluid hydrodynamic model coupled to ATP transport. Results show that active fluid mixing involves complex coupling between distribution of active stress and active transport of ATP and reduces mixing time for suspended components with decreased impact of initial component distribution. This work will inform application of active fluids to promote micromixing in microfluidic devices.


Introduction
Miniaturization enhances production efficiency in chemical engineering, biological engineering, and pharmaceutical manufacturing. 1 For example, microreactors-millimeter-scale devices with channels to mix chemicals and induce chemical reactions-are used to synthesize materials, 2 test enzymes, 3 and analyze protein conformations. 4 These devices require mixing to homogenize reactants, which is challenging because fluid dynamics at the micron scale are dominated by laminar flow.Mixing at a macroscopic scale is achieved by turbulence-induced advection repeatedly stretching and folding components until a uniform state is reached, 5 but at a microscopic scale, turbulence is inhibited (Reynolds number ) and mixing is dominated by molecular diffusion, which is slow and difficult to control.Approaches such as serpentine design 6 and vibrating bubbles 7 have been developed to enhance micromixing, but these are driven by external energy sources and thus require external components that limit miniaturization. 1,8tive fluids-fluids with microscopic constituents that consume local fuel to generate movement [9][10][11][12][13][14][15][16][17][18] have the potential to enhance mixing at the micron scale.Active fluids self-organize into chaotic turbulencelike flows [19][20][21][22][23] that promote micromixing by repeatedly stretching and folding fluid, even at low Reynolds numbers. 24Prior work on active mixing has focused on active systems with uniform activity distribution. 24- 26However, mixing processes often start from a state of nonuniformity.8][29][30][31][32][33][34] Spatiotemporal patterns of activity that are prescribed from an external source 31,32 or emerge as an additional dynamical variable that coevolves with system [27][28][29] have been studied.However, the effect of nonuniform distributions of activity on mixing has not been elucidated.
Here, we studied the mixing dynamics of a microtubule-kinesin suspension whose activity is governed by the transport of ATP, the system's energy source.We controlled the initial distribution of ATP by using caged ATP that can only fuel the fluid after exposure to ultraviolet (UV) light.This allowed us to repeatedly observe the transient dynamics that carry the system from heterogeneous activity to homogeneous activity.We explored mixing dynamics ranging from diffusion-dominated to convection-dominated by varying the ATP concentration, composition of kinesin motors, flow cell geometry, and initial distribution of ATP.We contextualized the results with models at two levels of complexity.A simple model captured the mixing dynamics in a diffusion-limited regime, whereas a more complex model that included active-fluid hydrodynamics reproduced aspects of observed enhanced transport and activity-dependent progression of the active-inactive interface.

Results
Active and inactive fluids self-mix into a homogeneous active fluid.2][43][44] In microtubule-kinesin active fluid, microtubules self-assemble by depletion into bundles that extend spontaneously, driving chaotic vortical flows.The extension is driven by kinesin motor dimers that hydrolyze ATP to "walk" along pairs of antiparallel microtubules and force them in opposite directions (Fig. 1a). 14We augmented the microtubule-kinesin system with UV light-activated chemistry that allowed us to create distinct patterns of activity.In this light-activated system, the ATP is "caged"-its terminal phosphate is esterified with a blocking group (Fig. 1b)-such that it cannot be hydrolyzed by kinesin motors until the blocking group is removed by exposure to UV light. 45,46In this system, the activation of the fluid is irreversible.After the fluid is activated, the action of the kinesin motors causes the microtubule network to become a 3D selfrearranging isotropic active gel consisting of extensile microtubule bundles that buckle and anneal repeatedly until the ATP is exhausted. 14To quantify the evolution of the activity distribution, we suspended fluorescent tracers in the solvent and monitored the tracer motion to extract the speed distribution of active fluid flows (Fig. 1d).To observe the structure of the active suspension, we labelled microtubules with Alexa 647 (Fig. 1c).
When the fluid was in its inactive state, before it had been activated by UV light, the kinesin motor dimers were bound to microtubules, creating a quiescent crosslinked microtubule network that behaved like an elastic gel (Fig. 1c, top panel).The inactive gel was essentially isotropic, but after the fluid was loaded into a rectangular flow cell (20 × 4 × 0.1 mm 3 ) we observed some alignment of the bundles near the boundary (Supplementary Fig. 1). 47To create an active-inactive interface, we used a mask to apply UV light to one side of the sample, which released the ATP and activated the fluid on that side only (Fig. 1c, second panel; Supplementary Video 1).The spatial pattern of activity evolved from a sharp interface to become increasingly diffuse as the initially active region invaded the inactive region (Fig. 1d, second and third panels).We quantified this evolution of activity distribution with the normalized speed profile (Fig. 1e), which shows how the interface between regions widened and shifted as the active and inactive parts of the microtubule system blended over a period of hours.
When active fluid mixes with inactive fluid, are the mixing dynamics governed by a superdiffusionlike process?Active fluids enhance the motion of suspended tracers from diffusive (having a mean squared displacement [MSD] proportional to time lapse: MSD ~ with 1) to superdiffusive ( ). 14,26,35 The progression of the active-inactive interface can also be described as diffusion-like or superdiffusionlike as follows: Suppose the displacement of the active-inactive interface is and the squared interface displacement increases with time as with the interface progression coefficient and the interface progression exponent .If 1, the progression of the active-inactive interface is defined as diffusion-like; if 1, the progression of the active-inactive interface is defined as superdiffusion-like.
Because active fluids enhance microscale transport, we hypothesized that the active-inactive interface would progress in a superdiffusion-like manner ( 1).To test this hypothesis, we quantified the displacement of the active-inactive fluid interface as a function of time (Fig. 2a inset) and found that motion of the interface decelerated as the active fluid mixed with the inactive fluid such that the squared interface displacement progressed as with an interface progression exponent (Fig. 2a).We repeated the measurement for caged ATP concentrations from 0.5 to 8 mM (0.5 mM is enough to maximize the flow speed of active fluid 48 ) and consistently found that 1 across this range (Fig. 2b).These results invalidated our hypothesis and suggested that the progression of active-inactive interface is diffusion-like.Notably, while the diffusive time-scaling remained consistent, the prefactor exhibited a monotonic but nonlinear dependence on caged ATP concentration (Fig. 2c).
Modeling results showed that the mixing process of active and inactive fluids is governed by diffusionlike processes of ATP at the active-inactive interface.In the experiments, the interface progression coefficient 1, which suggested that diffusion dominated the dynamics of the active-inactive interface.To contextualize our observations, we constructed a minimal model that combines diffusion of ATP with a previously measured relation between ATP concentration and local fluid velocity. 48Herein, we modeled ATP's dispersion using Fick's law of diffusion: 1 where represents the spatial distribution of ATP concentrations at time and is the diffusion coefficient of ATP in active fluid.We chose 140 µm 2 /s, which is one-fifth the diffusion coefficient of ATP in water, 49 because the crosslinked microtubule network makes the fluid more viscous than water 50 and our measurement on diffusion coefficient of suspended fluorescein was one-fifth of its reported value in aqueous solution (Supplementary Discussion 1).To simplify the modeling, we considered a 1D active fluid system confined in a segment, 0 -, where 20 mm is the segment length, and applied noflux boundary conditions 2 To mimic the UV-activation process (Fig. 1c), we initiated the ATP concentration with a step function 3 where 0.5-8 mM, the initial ATP concentrations in the activated region; is the complementary error function; and 10 mm is the initial position of the active-inactive interface.We chose 0.001 to generate a sharp concentration transition at the interface.We numerically solved Eqs.1-3 to determine the spatial and temporal distribution of ATP concentrations (Fig. 3a; Supplementary Video 2).To relate the evolving ATP distribution to local flow speed, we leverage previous experimental results 48 that found the average velocity in bulk samples follows Michaelis-Menten kinetics where is the mean speed of active fluid, 6.2 µm/s is the saturated mean speed, and 270 µM is the ATP concentration that leads to half of the saturated mean speed, .(Model selection is described in Supplementary Discussion 2.) The mean speed distributions (Fig. 3b) showed that initially one side of the sample was activated (black) and then the sample evolved toward a more uniformly activated state (red).The squared interface displacement of the active-inactive interface increased linearly with time, (Fig. 3c; see Supplementary Discussion 3 for derivation of ), which matched our experimental observation of 1 (Figs.2a & 2b).Further, we compared the dependency of the interface progression coefficient (determined by fitting vs. data to with as the fitting parameter; Figs.2c & 3c) on initial ATP concentrations between experiment and model and also found excellent agreement (differed by ~10%; Fig. 3d).Taken together, the agreement between simulation and experiment on the scaling of the dynamics (Figs.2b & 3c) and dependency on initial ATP concentration (Fig. 3d) indicated that the dispersion of ATP is dominated by diffusion and that Michaelis-Menten kinetics are appropriate for a coarse-grained model to connect ATP concentration with local flow speed of active fluid, 48 without the need to introduce a more complex hydrodynamic model. 42,51perdiffusion-like progression of the active-inactive interface only emerges when active transport is dominated by convection.The success of the diffusion-limited model suggests that the active transport in the active fluid systems studied above was dominated by diffusion.This inspired us to question whether the progression of active-inactive interface would become superdiffusion-like when the active transport becomes convection-like. 27,28To answer this question, we varied experimental parameters to explore a wider range of fluid flow speeds.To achieve lower flow speeds, we altered the composition of motor proteins by replacing a fraction of the processive motors (K401), which exert force on microtubules continuously, with nonprocessive motors (K365) that detach after each force application.The reduced number of processive motors has the net effect of driving the extensile motion of microtubules more slowly (Fig. 4a inset left). 36,48To achieve higher flow speeds, we increased the height of the sample container to decrease hydrodynamic drag (Fig. 4a inset right). 48,52Throughout these experiments, we kept the caged ATP concentration constant (5 mM).
As in the previous experiments, we analyzed the spatiotemporal progression of activity to find the interface progression exponent as a function of the average flow speed in the bulk of the initially activated area, (Fig. 4a).Because changing channel geometry alters the characteristic size of vortices in active fluids, 39 we unified our datasets by plotting as a function of the Péclet number, (Fig. 4b), defined as where is the correlation length of flow velocity (see Supplementary Discussion 4) and 140 µm 2 /s is our estimate of ATP diffusion in the system (see Supplementary Discussion 1). 53,54The Péclet number is a dimensionless quantity representing the ratio of convective transport rate to diffusive transport rate.A larger Péclet number (typically of order 10 or above) indicates convection-dominated active transport, and a smaller Péclet number (typically of order 1 or below) indicates diffusion-dominated active transport.Our data showed that for 3, the interface progression exponent remained 1 (Fig. 4b), which corresponded to the regime captured by our model (Fig. 3).Then as increased to greater than , grew monotonically (Fig. 4b).For the largest explored in our experiments ( 16), reached ~1.7, which indicated that convective processes were beginning to emerge and dominate the active transport.Overall, our data suggested that as the active transport transitioned from diffusion-dominated ( 3) to convection-dominated ( 3) regimes, the progression of active-inactive interfaces transitioned from diffusion-like ( 1) to superdiffusion-like ( ).
Active fluid flows reduce the mixing time of UV-activated fluorescent dyes.To this point, we had characterized the mixing of active and inactive fluids by the progression of the interface between them; however, like milk blending into coffee, the mixing process often involves dispersion of suspended components.To characterize how suspended components disperse during the progression of the activeinactive interface, we designed another series of experiments with suspended components that were initially nonuniform.We doped inactive fluid with suspended UV-activated fluorescent dyes and exposed one side of the sample container to UV light, which simultaneously activated the fluid and the fluorescent dye.We found that in an inactive sample ( 0), where dyes dispersed only by molecular diffusion, the dye barely dispersed, whereas in a sample where one side was activated ( 8.2 µm/s), the dyes were transported by active fluid flows and almost completely dispersed through the sample in 4 hours (Fig. 5a).To quantify the dispersion rate, we adopted Saintillan and Shelley's method 25 to analyze the normalized multiscale norm of dye brightness as a function of time: , where 5 is the Fourier coefficient at wave vector in a Fourier expansion of the dye brightness and 4.84 µm is the pixel size of the micrographs.We found that the normalized multiscale norm decayed faster as increased from 0 to 8.2 µm/s (Fig. 5b).In light of reports that the norm decays exponentially, 25 we quantified the decay rate by fitting the first hour data to with (mixing time) as the fitting parameter (Fig. 5b inset) and found that the mixing time decreased with flow speed of active fluid (Fig. 5c  inset).When the fluid was inactive ( 0), dye dispersion was dominated by molecular diffusion and the mixing time was 24 hours; slightly activating the fluid ( 2 µm/s) reduced the mixing time to 8 hours, which demonstrated that active fluid flows enhanced the mixing process of suspended components. 26 reveal how the mechanism of active transport (i.e., diffusion-dominated or convection-dominated) altered the mixing time, we analyzed the mixing time as a function of the Péclet number and found that the mixing time monotonically decreased as the active transport became more convection-like (Fig. 5c).Notably, there was no discernible transition in mixing time as the active transport transitioned from diffusion-dominated to convection-dominated, although there was a transition in the progression of activeinactive interfaces (Fig. 4b).This dependence of mixing time on the Péclet number in active-inactive fluid systems was similar to that in an activity-uniform active fluid system (Supplementary Discussion 5 and Supplementary Fig. 6b), which showed that Péclet number was the controlling parameter for mixing time of suspended components in active fluid systems, regardless of the distribution of activity.
A continuous active fluid model captures the mixing of nonuniform active fluid systems.Our experimental data showed that as the active transport became more convection-like, the active-inactive interface progression transitioned from diffusion-like to superdiffusion-like (Fig. 4b) and the mixing time of suspended components decreased monotonically (Fig. 5c).To determine whether this complex mixing process could be modeled with an existing active fluid model, we adopted Varghese et al.'s model 51 because it successfully describes the transition from coherent to chaotic flow in 3D microtubule-kinesin active fluid systems. 52The model describes microtubules as self-elongating rods whose nematic order, , is subject to spontaneous decay due to the rods' rotational molecular diffusion and reorientation by solvent flow.Thus, the dimensionless kinetic equation for can be written as: 6 where is the dimensionless time, is the dimensionless spatial gradient operator, is the dimensionless Laplacian operator, is the dimensionless vorticity tensor, is the dimensionless strain rate tensor, is the flow alignment coefficient, and is the system dimensionality.The dimensionless flow field is governed by the Stokes equation ), where is the dimensionless pressure and is the dimensionless active stress exerted by self-elongating rods with a dimensionless activity coefficient . 55ecause the activity coefficient increases with ATP concentration, 56 we selected an -ATP relation 57

8
where is the dimensionless activity level, is the ATP concentration, and 270 µM. 48We selected this relation because it captures the dynamics of microtubule bundle extension and kinesin kinetics (Michaelis-Menten), which play critical roles in the activity of microtubule-kinesin active fluid systems. 58,59inally, given that ATP diffused as a result of thermal fluctuation as well as flowed with the active fluid, we modeled ATP dispersion with a convection-diffusion equation: 9where is the dimensionless ATP molecular diffusion coefficient.To simplify modeling, we considered a 2D active fluid system ( ) 51 confined in a 112 × 22 rectangular boundary with no-slip boundary condition for flows ( ) and no-flux boundary condition for rods ( where represents a unit vector normal to boundaries).To solve the equations for , , and (Eqs.6, 7, & 9), we determined the initial conditions as quiescent solvent ( under uniform pressure ( ) with the rods in an isotropic state [  and  , where is a spatial uniform random number between −1 and +1] and 5 mM of ATP distributed on only one side of the system.Then we evolved the fluid flows and ATP distributions for 200 units of dimensionless time ( 0-200) with the finite element method. 60r modeling results (Supplementary Video 4) showed that in an inactive system ( 0; 16; Fig. 6a, left column), ATP dispersed only by molecular diffusion, but when one side of the sample was activated ( 25; 16; middle column), the system developed chaotic turbulence-like mixing flows that actively transported the ATP toward the inactive region.In a third simulation where the ATP molecular diffusion rate was increased ( 25; 64; right column), the mixing process sped up.These simulation results showed that the process of ATP dispersion was controlled by both molecular diffusion of ATP and active fluid-induced convection.
To quantify the efficacy of ATP mixing by active fluid, we analyzed the normalized multiscale norm of ATP concentrations as a function of time 25 for 0-25 and = 1-128 (Eq. 5 with 1).We found that the norms decayed exponentially with time: , where is the dimensionless mixing time (Fig. 6b), which was consistent with results reported by Saintillan and Shelley. 25We analyzed mixing time as a function of dimensionless activity level, , for each dimensionless molecular diffusion coefficient (Fig. 6c) and found that when ATP diffused slowly ( 16; blue to light green curves), mixing time decreased with increasing activity level or faster active transport (Fig. 6b inset), which was consistent with our experimental observation (Fig. 5c).Our simulation also showed that as ATP diffused sufficiently fast ( 32; olive and red curves), the mixing time was nearly independent of activity level.Overall, increasing both the molecular diffusion coefficient and the activity level decreased mixing time.Thus, our simulation showed that both molecular diffusion (represented as ) and active fluid-induced convection (related to ) played important roles to disperse and homogenize the suspended components; which mechanism dominated the dispersion depended upon the competition between these two mechanisms.
To demonstrate how the competition of these two mechanisms affected the progression of active-inactive interfaces, we analyzed the interface progression exponent as a function of for various diffusion coefficients (Fig. 6c inset) and found that when the diffusion mechanism was relatively weak ( 2; dark blue curve), the convection mechanism dominated the interface progression, leading it to progress in a superdiffusion-like, or more precisely, ballistic-like manner (  2).Contrarily, as the diffusion mechanism became relatively strong ( 8; light green curve), diffusion mechanisms dominated the interface progression, leading it to progress in a diffusion-like manner ( 1).Interestingly, we found that in an intermediate strength of diffusion mechanism ( 4; dark green curve), increasing activity level transitioned the interface progression from diffusion-like to superdiffusion-like, which is consistent with our experimental observation (Fig. 4).Overall, our active-fluid hydrodynamic model qualitatively captures the mixing dynamics of active and inactive fluid systems in terms of active-inactive interface progression and dispersion of suspended components.

Fluid activity enhances mixing of suspended components and reduces impact of nonuniformity of component distribution on mixing time.
Up until this point we had only explored one configuration of nonuniform active fluid systems: an activated bulk on one side of a channel adjacent to an inactive bulk on the other side.To explore how other spatial configurations of activity affect mixing, we used a checkerboard pattern of UV light to split the activated region into cells.As in previous experiments, 50% of the total fluid was activated.Fluid activated in a checkerboard pattern evolved to a homogeneous state more quickly than fluid that was activated on one side only (1 hour vs. 10 hours; Fig. 7a).UV-activated fluorescent dyes showed that the mixing time decreased as the grid size decreased from 3 mm to 1 mm (Fig. 7b).
To elucidate how checkerboard mixing driven by active fluid differed from that driven by molecular diffusion alone, we applied our established active-fluid hydrodynamic model for both active ( 25) and inactive ( 0) fluid systems (Fig. 8a; Supplementary Video 6).As expected, we found that the mixing time increased monotonically with grid size for both active and inactive fluid systems (Fig. 8b), with the active fluid system (red curve) having a shorter mixing time than the inactive fluid system (black curve).Interestingly, we found that when the grid size was sufficiently small ( 5), the active and inactive fluids had the same simulated mixing time.We also found that as the grid size increased from 5 to 22, the mixing time of the inactive fluid increased more than the mixing time of the active fluid (40× vs. 3×).

Discussion
The self-mixing process of microtubule-kinesin active fluid with nonuniform activity was driven by active transport at the active-inactive interface.We estimated the contributions of diffusive and convective transport using the Péclet number, .We found that when the active transport was dominated by the diffusion mechanism ( 3), the active-inactive interface progressed in a diffusion-like manner ( 1; Fig. 2).These dynamics were quantitatively captured by a Fick's law-based model that quasi-statically related local activity to the local concentration of ATP by using a previously measured ATP-velocity relation (Fig. 3). 48 we raised the Péclet number ( 3) by increasing both the local fluid velocity and mixing length scale, we found that the active-inactive interface concomitantly progressed in a more superdiffusion-like manner ( 1; Fig. 4).We observed experimentally that increasing the Péclet number decreased the mixing time of suspended fluorescent dyes (Fig. 5c), which demonstrated that more convective flow mixed the suspended components faster.These results, along with the progression of the active-inactive interfaces, were qualitatively captured by an active-fluid hydrodynamic model (Fig. 6c) that coupled active stressinduced fluid flow and transport of ATP molecules (Eqs.6-9).Interestingly, while our hydrodynamic model predicted interface progression exponent 2 for high activity levels (Fig. 6c inset), in our experiments appeared to plateau at 1.7 (Fig. 4b).Our model may have overestimated because the microtubule network in the inactive portion of the sample is crosslinked by immobile kinesin motor dimers that cause it to behave like an elastic gel.When the ATP molecules arrive at the active-inactive interface, they fuel the motor dimers, which fluidizes the network.Crucially, this fluidizing/melting process takes time to develop. 47,50Thus, for the interface to progress, not only does ATP need to be transported to the inactive fluid region, but the ATP-fueled motors also needs time to melt the gel-like microtubule network into a fluid.Such melting dynamics could delay the progression of the active-inactive interface and lower .In the simulation, the melting dynamics were absent; the network melted almost instantly as soon as ATP arrived at the inactive fluid, and would only depend on active transport of ATP.Our additional studies (Supplementary Discussion 6) support the idea of a network melting mechanism by showing that the progression of the active-inactive interface fell behind the progression of ATP molecules (Supplementary Fig. 7e), whereas in the simulation the fronts of both coincided (Supplementary Fig. 8c).Future research to elucidate the network melting dynamics could involve monitoring dyes, tracers, and microtubules simultaneously to reveal the correlations among ATP dispersion, active fluid flows, and microtubule network structure (melting).The process could be modeled with the active-fluid hydrodynamic model used herein, modified to include ATP-dependent rheological constants and additional relevant dynamic processes to represent the melting process of the gel-like network at the interface.
We also found that the distribution of activity had a significant effect on mixing time.Systems consisting of more, smaller active areas (checkerboard pattern; high uniformity [Fig.7]) evolved to a homogeneous state faster than systems with the same total active area distributed as one piece (one side active; low uniformity [Fig.1c]).This is likely because the smaller grid size increased the active-inactive interface area, which allowed the active fluid to interact with inactive fluid more efficiently.Interestingly, our active-fluid hydrodynamic model showed that when the grid size was sufficiently small, the mixing times of active and inactive fluids were indistinguishable (Fig. 8b).This may be because the active fluid needs time to "warm up" from an initial quiescent state before reaching its steady activity state. 47In experiments, the system had a warm-up time that may have been caused by network melting (Supplementary Discussion 6).Although a network melting mechanism was not included in the model, the simulated active fluid flow took dimensionless time to rise because the onset of the flows was triggered by the initial activity-driven instability in extensile field, which took finite dimensionless time to develop (~1 dimensionless time in this case; Supplementary Video 6). 25,47Thus, in cases where the grid size was sufficiently small, the model showed molecular diffusion completing the mixing before emergence of active fluid flows.We also found that mixing time in an active fluid system was less sensitive to initial distribution of activity than that in an inactive system (Fig. 8b), which suggests that introducing active fluid to a microfluidic system could drastically reduce the impact of the initial condition on mixing efficacy.
This study has limitations.Observations on the mixing of active microtubule-kinesin fluid and inactive microtubule-kinesin fluid may not be generalizable to cases in which active fluid mixes with other types of fluid.Also, we did not characterize the degree of chaos in the system, such as by measuring Lyapunov exponents and topological entropies. 24Future research could track tracers in 3D and measure how these quantities change in the 3D isotropic active microtubule network at different strengths of active transport (i.e., Lyapunov exponent vs. Péclet number and topological entropy vs. Péclet number).
Another limitation of this study was that our results for interface progression transitioning from diffusionlike to superdiffusion-like (Fig. 4) were based on large length-scale data that we analyzed considering the interface as one piece with a specific position coordinate (Fig. 2a).However, the interface is the region where ATP concentration decays from saturation ( ; see Eq. 4) to 0, and according to previous studies 14,35 tracer motion in this region should transition from superdiffusion-like to diffusion-like behaviors.Directly measuring the mean squared displacement of tracers across the active-inactive interface would elucidate the transition of the interface progression behaviors on the microscopic scale.Such measurements were not practical in our system because the active-inactive interface changed position and width with time (Fig. 1e); tracers initially in the diffusive zone could later be in the superdiffusive zone as the interface passed by, and it would be difficult to distinguish between the diffusive and superdiffusive data.Future research to elucidate tracer behaviors at active-inactive interfaces could utilize fluid that is only active when it is exposed to light 31,32 to provide a stable activity gradient and thus obtain a reliable mean squared displacement of tracers at different parts of the interface.
Overall, this work demonstrated that mixing in nonuniform active fluid systems is fundamentally different from mixing in uniform active fluids.Mixing in nonuniform active fluid systems involves complex interplays among spatial distribution of ATP, active transport of ATP (which can be either diffusion-like or convection-like, depending on Péclet number), and a fluid-gel transition of the microtubule network at the interface. 47This work paves the path to the design of microfluidic devices that use active fluid to promote or optimize the micromixing process 8 to enhance production efficiency in chemical and biological engineering and pharmaceutical development. 1The results may also provide insight into intracellular mixing processes, because the cytoplasmic streaming that supports organelles within cells is powered by cytoskeletal filaments and motor proteins that function similarly to microtubule-kinesin active fluid. 61Methods Polymerize microtubules.Microtubules constitute the underlying network of microtubule-kinesin active fluid.Microtubules were polymerized from bovine brain -and -tubulin dimers purified by three cycles of polymerization and depolymerization. 62,63The microtubules (8 mg/mL) were then stabilized with 600 µM guanosine-5′[(α,β)-methyleno]triphosphate (GMPCPP, Jena Biosciences, NU-4056) and 1 mM dithiothreitol (DTT, Fisher Scientific, AC165680050) in microtubule buffer (80 mM PIPES, 2 mM MgCl2, 1 mM ethylene glycol-bis(β-aminoethyl ether)-N,N,N',N'-tetraacetic acid, pH 6.8) and polymerized by a 30-minute incubation at 37 °C and a subsequent 6-hour annealing at room temperature before being snap frozen with liquid nitrogen and stored at 80 °C.The microtubules were then labeled with Alexa Fluor 647 (excitation: 650 nm; emission: 671 nm; Invitrogen, A-20006) and mixed with unlabeled microtubules at 3% labeling fraction during polymerization to image microtubules for non-fluorescein experiments (Figs. 1, 2, & 4).For fluorescein experiments, the microtubules were unlabeled (Figs. 5 & 7).
Dimerize kinesin motor proteins.Kinesin motor proteins power the extensile motion of sliding microtubule bundle pairs in active fluid by forming a dimer and walking on adjacent antiparallel microtubules to force them in opposite directions (Fig. 1a). 14,64We expressed kinesin in the Escherichia coli derivative Rosetta 2 (DE3) pLysS cells (Novagen, 71403), which we transformed with DNA plasmids from Drosophila melanogaster kinesin (DMK) genes. 65For most experiments in this paper, we used processive motors that include DMK's first 401 N-terminal DNA codons (K401). 66To explore the effect of low mean speed of active fluid bulk on the interface progression exponent , we mixed in fractions of nonprocessive motors whose plasmid included DMK's first 365 codons (K365, Fig. 4 inset left). 36,48,67The kinesin motors were tagged with 6 histidines enabling purification via immobilized metal ion affinity chromatography with gravity nickel columns (GE Healthcare, 11003399).To slide adjacent microtubule bundle pairs, kinesin motors needed to be dimerized, so the kinesin motors were tagged with a biotin carboxyl carrier protein at their N terminals, which allowed the kinesins to be bound with biotin molecules (Alfa Aesar, A14207). 14,62To dimerize the kinesin, we mixed either 1.5 µM K401 processive motors or 5.4 µM K365 nonprocessive motors with 1.8 µM streptavidin (Invitrogen, S-888) and 120 µM DTT in microtubule buffer, incubated them for 30 minutes at 4 °C, and then snap froze them with liquid nitrogen and stored them at 80 °C.
Prepare microtubule-kinesin active fluid with caged ATP and caged fluorescein.To prepare the active fluid, we mixed 1.3 mg/mL microtubules with 120 nM kinesin motor dimers and 0.8% polyethylene glycol (Sigma 81300), which acted as a depleting agent to bundle microtubules (Fig. 1a). 14Kinesin steps from the minus to the plus end of microtubules by hydrolyzing ATP and producing adenosine diphosphate. 59To control the initial spatial distribution of ATP and thus the activity distribution of active fluid, we used 0.5 to 8 mM caged ATP (adenosine 5'-triphosphate, P3-(1-(4,5-dimethoxy-2-nitrophenyl)ethyl) ester, disodium salt and DMNPE-caged ATP, Fisher Scientific, A1049), which is ATP whose terminal phosphate is esterified with a blocking group rendering it nonhydrolyzable by kinesin motors unless exposed to 360-nm UV light.Exposure to UV light removes the blocking group (Fig. 1b) and allows the kinesin motors to hydrolyze the ATP into ADP and activate the active fluid. 45,46The ATP hydrolyzation decreased ATP concentrations, which slowed down the kinesin stepping rate and thus decreased active fluid flow speed. 14,35,48,58,59To maintain ATP concentrations so as to stabilize the activity level of the active fluid bulk over the course of our experiments, we included 2.8% v/v pyruvate kinase/lactate dehydrogenase (Sigma, P-0294), which converted ADP back to ATP. 14,68 To feed the pyruvate kinase enzyme, we added 26 mM phosphenol pyruvate (BeanTown Chemical, 129745).We imaged the active fluid samples with fluorescent microscopy for 1 to 16 hours, which could bleach the fluorescent dyes and thus decrease the image quality over the course of experiments.To reduce the photobleaching effect, we included 2 mM trolox (Sigma, 238813) and oxygen-scavenging enzymes consisting of 0.038 mg/mL catalase (Sigma, C40) and 0.22 mg/mL glucose oxidase (Sigma, G2133) and fed the enzymes with 3.3 mg/mL glucose (Sigma, G7528). 14o stabilize proteins in our active fluid system, we added 5.5 mM DTT.To track the fluid flows, we doped the active fluid with 0.0016% v/v fluorescent tracer particles (Alexa 488-labeled [excitation: 499 nm; emission: 520 nm] 3-µm polystyrene microspheres, Polyscience, 18861).To test how active fluid could mix suspended components, we introduced 0.5 to 6 µM caged, UV-activated fluorescent dyes (fluorescein bis-(5-carboxymethoxy-2-nitrobenzyl) ether, dipotassium salt; CMNB-caged fluorescein, ThermoFisher Scientific, F7103), which are colorless and nonfluorescent until exposed to 360-nm UV light. 45The dye concentration was chosen to maintain a sufficient signal-to-noise ratio while avoiding brightness saturation in micrographs.Upon UV exposure, the fluorescein was uncaged and thus became fluorescent and could be observed with fluorescent microscopy.Because the fluorescent spectrum of the fluorescein overlapped with our Alexa 488 tracers, for our experiments with caged fluorescein (Fig. 5) we replaced the tracers with Flash Red-labeled 2-µm polystyrene microspheres (Bangs Laboratories, FSFR005) and used unlabeled microtubules (0% labeling fraction) to prevent fluorescent interference from microtubules while imaging tracers.
Prepare active-inactive fluid systems.To prepare the active-inactive fluid system, we loaded the inactive microtubule-kinesin fluid with caged ATP to a polyacrylamide-coated glass flow cell (20 × 4 × 0.1 mm 3 ) with Parafilm (Cole-Parmer, EW-06720-40) as a spacer sandwiched between a cover slip (VWR, 48366-227) and slide (VWR, 75799-268) 36 and sealed the channel with epoxy (Bob Smith Industries, BSI-201).Then we masked one side of the sample with a removable mask of opaque black tape (McMaster-Carr, 76455A21) attached to a transparent plastic sheet (Supplementary Fig. 9a) and shined UV light on the sample for 5 minutes before removing the mask (Supplementary Fig. 10).In the unmasked region, the UV light released the ATP from the blocking group and activated the fluid by allowing the ATP to fuel the local kinesin motors; in the masked region, the fluid remained quiescent (Fig. 1c; Supplementary Video 1). 45To explore how the progression exponent changed with active fluid bulk mean speed, we accelerated fluid flows by making the flow cell taller by stacking layers of Parafilm to decrease hydrodynamic resistance (Fig. 4a inset right). 48,52To explore how the spatial nonuniformity of activity influenced the mixing efficacy of the active-inactive fluid system, we masked the sample with checkerboard-patterned masks (FineLine Imaging, Fig. 7a).
Performing these analyses required monitoring at least two components in two fluorescent channels in each sample; for example, the dye dispersion experiments (Fig. 5) required analyzing fluorescent dyes (excitation: 490 nm; emission: 525nm) and Flash Red-labeled tracers (excitation: 660 nm; emission: 690 nm) simultaneously.This could have been accomplished by programming a microscope to rapidly switch back and forth between filter cubes, but this would have quickly worn down the turret motor and the time required to switch filter cubes ( 4 s) and move the stage to capture adjacent images and stitch them (~3 s) would have made the minimum time interval between frames 10 s, which would have prevented us from tracking high-density tracers (1000 mm -3 with a mean separation of 5 µm in a 0.1-mm-thick sample) whose speeds were 1 to 10 µm/s, even with a predictive Lagrangian tracking algorithm. 69To overcome this technical challenge in imaging our samples, we established a dual-channel imaging system that consisted of a multiband pass filter cube (Multi LED set, Chroma, 89402-ET) and voltage trigger (Nikon) placed between the light source (pE-300 ultra , CoolLED, BU0080) and camera (Andor Zyla, Nikon, ZYLA5.5-USB3).Instead of changing filter cubes, the multiband pass filter cube allowed us to switch between multiple emission and excitation bands by switching between channels with the same filter cube (Supplementary Fig. 11).We alternatively activated the blue (401-500 nm) and red (500-700 nm) LEDs to excite and observe the fluorescent dyes and tracers almost simultaneously.The LED light source communicated directly with the camera via voltage triggering to coordinate LED activation time and bypass computer control to further boost the light switching rate.This technique shortened our channel switching time to 3 to 5 µs; thus the time interval between image acquisitions of different fluorescent channels was only limited by exposure times of each channel.This setup allowed us to image two fluorescent channels almost simultaneously (within milliseconds) and thus enabled us to monitor two fluorescent components side-byside, such as microtubules and tracers (Fig. 1c; Supplementary Video 1), caged fluorescent dyes and tracers (Fig. 5c), and caged fluorescent dyes and microtubules (Supplementary Video 5).

Analyze positions of active-inactive fluid interfaces.
We characterized the mixing kinematics of active and inactive fluids by analyzing the interface progression exponents and coefficients (Figs.2-4).These analyses required identification of the interface positions.We determined the interface positions by first tracking tracers in sequential images with the Lagrangian tracking algorithm, 69 which revealed the tracers' trajectories and corresponding instantaneous velocities .Then we analyzed the speed profile of tracers by binning the tracer speed across the width of the channel where was the horizontal coordinate of the th bin and the represented the average speed of tracers in the th bin.Then we normalized the speed profile by rescaling the speed profile to be 1 in the active zone and 0 in the inactive zone: , where is the average of speed profiles in the active zone and is the average of speed profiles in the inactive zone (Fig. 1e).Then we defined the width of the active-inactive fluid interface as where the normalized speed profile is between 0.2 and 0.8 (dashed lines in Fig. 1e inset) and the position of the interface as where the normalized speed profile is 0.5 (solid line).The interface position was then analyzed for each frame, which allowed us to determine the interface displacement vs. time (Fig. 2a inset) and the squared interface displacement vs. time (Fig. 2a).To determine the interface progression exponent, , we fit vs.
data to , with and as fitting parameters (Figs.2b  & 4).To determine the interface progression coefficient, , we assumed and fit vs. data to with as the fitting parameter (Figs.2c, 3c, & 3d).
Generate flow speed map of active-inactive fluid system.To visualize the activity distribution in our active-inactive fluid system, we analyzed the distribution of flow speed to generate flow speed maps (Fig. 1d).To complete this analysis, we analyzed the flow velocities of fluids by analyzing the tracer motions in sequential images with a particle image velocimetry algorithm, 70 which revealed the flow velocity field and associated distribution of flow speed in each frame.A heat color bar (Fig. 1d) was used to plot the speed distributions into color maps to reveal the evolution of speed distribution from the pre-activated state (black) to the homogeneously activated state (red/yellow).
Numerically solve the Fick's law equations.We modeled diffusion-dominated active-inactive fluid mixing with the Fick's law equation, which required us to solve for the concentration of ATP (Eq.1).To simplify the modeling, we considered a one-dimensional active fluid system confined in a segment 0where 20 mm, the length of our experimental sample.Given that ATP is confined in the segment, we applied no-flux boundary conditions to the ATP concentrations (Eq.2).In the experiment, we exposed the left side of the sample to UV light to release ATP, so in our model the ATP concentration has a step function as its initial condition (Eq.3).With the initial condition and boundary conditions, we solved the Fick's law equation to determine the spatial and temporal distribution of ATP.We used Mathematica 13.0 to numerically solve this differential equation with the NDSolveValue function by feeding Eqs.1-3 into the function followed by specifying the spatial and temporal domains, which output the numerical solution of ATP concentration and allowed us to determine the evolution of ATP distribution (Fig. 3a).Then we converted the ATP distribution to mean speed distribution of active fluid by the Michaelis-Menten kinetics (Eq.4; Fig. 3b inset), which showed a uniform mean speed in the left side of the sample followed by gradual activation of the right side of the system until a uniform state was reached (Fig. 3b).Then we defined the position of the active-inactive fluid interface as where the mean speed decayed to a half (Fig. 3c inset), which allowed us to plot the squared interface displacement as a function of time (Fig. 3c).The plot in log-log axes exhibited a line with a unit slope, which suggested that the squared interface displacement is linearly proportional to time.By assuming this linear relation, we determined the interface progression coefficient by fitting with as the fitting parameter.Finally, we repeated the calculation with different initial ATP concentrations in the left side of the system, , and plotted the interface progression coefficient, , as a function of .This plot allowed us to compare the simulation results with the experimental measurements to examine the validity of our Fick's law-based model (Fig. 3d).
Numerically solve the active nematohydrodynamic equations in weak forms.To model the mixing of active and inactive fluids, we adopted Varghese et al.'s active fluid model to include the dynamics of the ATP concentration field. 22Our model had four main components: (1) the kinetic equation describing the kinematics of self-elongating rods that flow and orient with the solvent as well as diffuse translationally and rotationally (Eq.6), ( 2) the Stokes equation describing how the solvent was driven by the active stress exerted by the self-elongating rods (Eq.7), (3) the relation between and ATP that describes how the active stress depended on the nonuniform ATP distribution (Eq.8), and ( 4) the continuity equation of ATP transport that describes how ATP diffuses as well as flows with the solvent (Eq.9).We numerically solved these coupled equations with appropriate boundary and initial conditions using the finite element method by first converting them to their weak forms and then implementing them symbolically in COMSOL Multiphysics TM . 60We show below the weak form of the convection-diffusion equation governing the evolution of ATP concentration field: 10 where is the test function and represents the system spatial domain.After solving these equations, we determined the spatial and temporal distributions of ATP concentrations and active fluid flow speeds (Fig. 6a and Supplementary Video 4), which allowed us to explore how the activity level of active fluid and molecular diffusion of ATP influenced the mixing process of ATP in nonuniform active fluid systems (Figs.6b & c).
Simulate dispersion of checkerboard-patterned ATP in active and inactive fluids.We applied our established hydrodynamic model to simulate how the initially checkerboard-patterned ATP would disperse in active fluid and inactive fluid (Fig. 8).In this exploration, we used the same equations for , and (Eqs.6-9), along with their initial and boundary conditions, except that was initialized in a checkerboard pattern in a 45 × 45 simulation box.We used two different checkerboard patterns for different trials.One has the -axis origin in the center of a grid square: mod mod ceil mod ceil 11 where 5 mM is the initial ATP concentration, represents the dimensionless grid size of the checkerboard pattern, mod represents the remainder of divided by , and ceil represents the rounding of toward positive infinity (e.g., Fig. 8a with  22).The other checkerboard pattern had the -axis origin at a vertex of the grid: 12The simulation was performed for the dimensionless grid size ranging from 2 to 22 for both active ( 25) and inactive ( 0) systems (Fig. 8a).We analyzed the corresponding mixing time (averaged over 2 trials) as a function of (Fig. 8b) to reveal how the -dependence of mixing times differed in active and inactive fluid systems.Depletants force microtubules into bundles where the microtubules can be bridged by kinesin motor dimers.The kinesin motors "walk" along the microtubules, forcing them to slide apart.(b) To develop an experimental system where we could create a distinct boundary between active and inactive fluid, we synthesized microtubule-kinesin active fluid with caged ATP.The caged ATP is not hydrolysable by kinesin motors, and thus cannot power the active fluid, until it is released by exposure to ultraviolet light. 45,46This process is not reversible.(c) We exposed only one side of the sample to ultraviolet light, which released the ATP and activated the microtubule-kinesin mixture on that side of the channel.The released ATP dispersed toward the unexposed region, which activated the inactive fluid and expanded the active region until the system reached an activity-homogeneous state (Supplementary Video 1).Because of the limited speed of multi-position imaging, only one-quarter of the active region was imaged., where is the average of speed profiles in the active zone and is the average of speed profiles in the inactive zone.Inset: The interface of the active and inactive fluids is determined as the region where the normalized speed profile is between 0.2 and 0.8 (dashed blue lines).The position of the interface is determined as where the normalized speed profile was 0.5 (solid blue line).c) Selected examples of squared interface displacement versus time for four caged ATP concentrations from 1 mM (black) to 8 mM (red).The progression rate of the interface was characterized with an interface progression coefficient determined by fitting the vs. data to with as the fitting parameter (colored lines).Increasing caged ATP concentrations increased the interface progression coefficient (steeper fit lines from black to red), which indicates that the interface progressed more quickly at higher caged ATP concentrations., where 6.2 µm/s and 270 µM (based on our previous studies 48 ).The corresponding mean speed distribution of active fluid evolved from a step function distribution (black, 0 h) to a near-constant function (red, 30 h) (Supplementary Video 2).Inset: The plot of the Michaelis-Menten equation (Eq.4).(c) In the simulation, the diffusion-driven mixing process led the squared interface displacement to be proportional to time, regardless of initial ATP concentration (see Supplementary Discussion 3 for derivation of ).Inset: Interface displacement increased 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 was determined by fitting the vs. data (Panel c) to with as fitting parameter.The model increased with the initial concentration of ATP, (red dots), similarly to how the experimentally analyzed varied with caged ATP concentration (blue dots; each error bar represents the standard deviation of 3 trials).The model and experimental differed by only ~10%.The magenta curve shows the analytical solution, (Eq.S7), which reproduced the numerical results (red dots).(See Supplementary Discussion 3 for derivation of as a function of .)  3) to convectiondominated ( 3).(a) The active-inactive interface progression exponent ( ) increased with the flow-speed level of the active fluid ( ).Shown are data from experiments with low ATP concentration (0.5 mM, black dots), high ATP concentration (8 mM, red dots), decreased flow speeds (from nonprocessive motors partially replacing processive motors; blue dots), increased flow speeds (from increased sample height; green dots), 48 and both nonprocessive motors and increased sample height (purple dot).The magenta curve represents the moving average of .Although the analyzed from each experiment was noisy, the moving averaged exhibited an overall monotonic increase with the flow-speed level of active fluid . Each dot represents one experimental measurement.Each error bar in represents the slope fitting error in (Fig. 2a), and each error bar in represents the standard deviation of flow speeds in the active region.(b) The same data as in Panel a, plotted as a function of Péclet number, , where is the correlation length of flow velocity in active fluid deduced from sample container height (inset) and is the diffusion coefficient of ATP.Each error bar in is the same as in Panel a, and each error bar in represents propagated uncertainties from in Panel a and in inset.Inset: Correlation length of flow velocity in active fluid increased monotonically with sample container height . 39The red dashed line represents the line interpolation of blue dots.The error bars represent the standard deviations of two trials.(See Supplementary Discussion 4 for measurements and analyses of .)We developed the Fick's law-based model (Fig. 3) to describe the experimentally observed mixing of active and inactive fluids (Fig. 2).Our model used Michaelis-Menten kinetics to convert distribution of ATP to flow speed (Eq.1][12] Our previous work showed that Michaelis-Menten kinetics reasonably connect flow speed of active fluid with ATP concentrations when the ATP concentration is above 100 µM. 13 Below this concentration, inactive kinesin motor dimers start to act as a crosslinkers in the microtubule network, causing the network to behave more like an elastic gel, and Michaelis-Menten kinetics fail to describe the flow speed because Michaelis-Menten kinetics is an enzyme-based model that does not consider network rheology. As such, our adoption of Michaelis-Menten kinetics to convert ATP concentration to flow speed is an approximation for active fluid with high ATP concentrations ( 100 µM), and it is unclear how the results of our Fick's law-based model would change if we chose a different relation between ATP concentration and flow speed.Here we considered that the flow speed is connected to ATP concentration via a positive power-law exponent, 0, of the Michaelis-Menten relation: S3 where 6.2 µm/s is the saturated mean speed of active fluid and 270 µM is the ATP concentration that leads to half of the saturated mean speed, (Supplementary Fig. 3a). 13Then we explored how would affect the predicted active-inactive interface progression in terms of progression exponent and coefficient .
To find vs. and vs. , we first considered the case of initial ATP concentration 8 mM and solved the 1D Fick's law equation with the same boundary and initial conditions (Eqs.1-3) for .Then we converted ATP concentration to flow speed using our power-lawed Michaelis-Menten equation (Eq.S3), which showed that increasing power law exponents led to a slower progression of the active-inactive interface (Supplementary Fig. 3b).However, such an -induced variation in interface progression appeared not to interfere with the relation of interface displacement with time; we found that the squared interface displacement increased linearly with time across our explored power-law exponents ( 0.5-10; Supplementary Fig. 3c), which suggests that interface progression exponent will equal 1 regardless of the value of the power-law exponent in our explored range of exponents (Supplementary Fig. 3d inset).Contrarily, we found that the interface progression coefficient decreased with increasing power-law exponents (Supplementary Fig. 3d).Overall, our exploration revealed that the active-inactive interface progression exponent being 1 was a consequence of the diffusion-like process of ATP dispersion.This result was insensitive to the choice of flow speed-ATP model (Eq.S3), but the interface progression coefficient varied rapidly with the model choice.Increasing the power-law exponent from 0.5 to 10 decreased the interface progression coefficient from 740 to 34 µm 2 /s.Given that our experimentally measured interface progression coefficient 451 8 µm 2 /s for 8 mM caged ATP concentration (Fig. 3d), this also suggested that selecting 1 (or slightly larger than 1) would best match our model with the experimental results (Supplementary Fig. 3d) and that Michaelis-Menten kinetics (Eq.4) is a good approximation to connect ATP concentrations with the local flow speed of active fluid.
The calculations in our Fick's law-based model revealed that the interface progression coefficient for a given depends on the selected flow speed-ATP relation (Supplementary Fig. 3a), which implies that the ATP dependence of the interface progression coefficient should change with the selected flow speed-ATP relation as well.To investigate, we repeated the above calculation and determined the interface progression coefficients, , as a function of initial ATP concentration, , for power-law exponents ranging from 0.5 to 10 (dots in Supplementary Fig. 4a).Our analysis revealed that increasing the powerlaw exponents decreased as a function of (Supplementary Fig. 4a) in a similar way that it did for flow speed (Supplementary Fig. 3a).Inspired by this observation, we fit each vs. to their corresponding power-lawed Michaelis-Menten equation with the same power-law exponent : S4 with and as fitting parameters (curves in Supplementary Fig. 4a).The resulting data fit well to the equations, with overall goodness of fit 0.99 (Supplementary Fig. 4b).This suggests that the ATP dependence in the flow speed could pass to the resulting interface progression coefficient .This analysis also showed that the consistency between the model and experimentally measured vs. (Fig. 3d) was under the condition of 1, which reenforced our assertion that adopting the Michaelis-Menten equation to convert ATP concentration to active fluid flow speed was an appropriate approach for building a coarse-grained model that matches the experimental results.The network melting mechanism was absent in our active-fluid hydrodynamic model; the model assumed that the network could melt almost instantly upon arrival of ATP (with negligible warm-up time from the initial development of activity-driven instability in the extensile field 1,18 ), so we expected that the profile discrepancies observed in the experiment (Supplementary Fig. 7e) would not exist in our model.To examine the validity of our expectation, we analyzed the profiles both directly from flow speed distribution and as calculated on the basis of ATP distribution.In the simulation, we could directly access the ATP distribution (Supplementary Fig. 8a), which allowed us to determine the corresponding ATP concentration profile (Supplementary Fig. 8c inset).Then we converted the concentration profile to speed profile by the Michaelis-Menten equation (Eq.4) followed by normalization to extract the normalized speed profile (blue curve in Supplementary Fig. 8c).To compare this ATP-based profile with the profile from flow speed distribution, we considered the speed map at the same time (Supplementary Fig. 8b), averaged the speed distribution vertically to get the speed profile, and normalizing the profile to get the normalized speed profile (red curve in Supplementary Fig. 8c).The modeling results showed that the ATP-based speed profile and the flow speed-based speed profile nearly overlapped across the active-inactive interface, which means that, in the simulation, ATP and activity progressed at the same pace.This is not consistent with experimental observations that the progression of activity fell behind ATP (Supplementary Fig. 7e).This mismatch between experiments and model results supported the existence of a network melting mechanism-in which the network needed to undergo a melting process before it could become fluidizedwhich was absent in the model.(left axis).(e) The profile of ATP concentrations was converted to the profile of flow speed by the Michaelis-Menten equation (Eq.4).The speed profile was normalized (blue curve) as in Fig. 1e and the normalized profile was then compared with the profile measured from the velocimetry of tracers (red curve).The profile analyzed from the velocimetry fell behind the profile deduced from fluorescein brightness, which suggests that the melting mechanism of the crosslinked microtubule network slowed the progression of the active-inactive interface.

Fig. 1 :
Fig. 1: (Experimental results) Mixing of activated and inactive microtubule-kinesin fluid.(a) Microscopic dynamics in microtubule-kinesin active fluid.Depletants force microtubules into bundles where the microtubules can be bridged by kinesin motor dimers.The kinesin motors "walk" along the microtubules, forcing them to slide apart.The collective sliding dynamics cause the microtubules to form an extensile microtubule network that stirs the surrounding solvents and causes millimeter-scale chaotic flows.14(b)  To develop an experimental system where we could create a distinct boundary between active and inactive fluid, we synthesized microtubule-kinesin active fluid with caged ATP.The caged ATP is not hydrolysable by kinesin motors, and thus cannot power the active fluid, until it is released by exposure to ultraviolet light.45,46This process is not reversible.(c) We exposed only one side of the sample to ultraviolet light, which released the ATP and activated the microtubule-kinesin mixture on that side of the channel.The released ATP dispersed toward the unexposed region, which activated the inactive fluid and expanded the active region until the system reached an activity-homogeneous state (Supplementary Video 1).Because of the limited speed of multi-position imaging, only one-quarter of the active region was imaged.(d) Tracking tracer particles revealed the speed distribution of fluid flows, showing the activation of the left-hand side by UV light and the expansion of the active region into the inactive region.(e) Binning the same-time speeds vertically across the interface of active and inactive fluids revealed the speed profile which is normalized as, where is the average of speed profiles in the active zone and is the average of speed profiles in the inactive zone.Inset: The interface of the active and inactive fluids is determined as the region where the normalized speed profile is between 0.2 and 0.8 (dashed blue lines).The position of the interface is determined as where the normalized speed profile was 0.5 (solid blue line).
Fig. 1: (Experimental results) Mixing of activated and inactive microtubule-kinesin fluid.(a) Microscopic dynamics in microtubule-kinesin active fluid.Depletants force microtubules into bundles where the microtubules can be bridged by kinesin motor dimers.The kinesin motors "walk" along the microtubules, forcing them to slide apart.The collective sliding dynamics cause the microtubules to form an extensile microtubule network that stirs the surrounding solvents and causes millimeter-scale chaotic flows.14(b)  To develop an experimental system where we could create a distinct boundary between active and inactive fluid, we synthesized microtubule-kinesin active fluid with caged ATP.The caged ATP is not hydrolysable by kinesin motors, and thus cannot power the active fluid, until it is released by exposure to ultraviolet light.45,46This process is not reversible.(c) We exposed only one side of the sample to ultraviolet light, which released the ATP and activated the microtubule-kinesin mixture on that side of the channel.The released ATP dispersed toward the unexposed region, which activated the inactive fluid and expanded the active region until the system reached an activity-homogeneous state (Supplementary Video 1).Because of the limited speed of multi-position imaging, only one-quarter of the active region was imaged.(d) Tracking tracer particles revealed the speed distribution of fluid flows, showing the activation of the left-hand side by UV light and the expansion of the active region into the inactive region.(e) Binning the same-time speeds vertically across the interface of active and inactive fluids revealed the speed profile which is normalized as, where is the average of speed profiles in the active zone and is the average of speed profiles in the inactive zone.Inset: The interface of the active and inactive fluids is determined as the region where the normalized speed profile is between 0.2 and 0.8 (dashed blue lines).The position of the interface is determined as where the normalized speed profile was 0.5 (solid blue line).

Fig. 2 :
Fig. 2: (Experimental results) The progression of the active-inactive interface was governed by a diffusion-like process of ATP at the interface, regardless of caged ATP concentration.(a) Squared interface displacement ( ) as a function of time ( ) revealed a long-term ( 200 s) linear relation: with the interface progression exponent .Inset: The interface displacement versus time showed that the interface moved rapidly initially and then gradually slowed down.(b) The interface progression exponent was on average and was independent of the caged ATP concentration.Each error bar represents the standard deviation of 3 trials.(c) Selected examples of squared interface displacement versus time for four caged ATP concentrations from 1 mM (black) to 8 mM (red).The progression rate of the interface was characterized with an interface progression coefficient determined by fitting the vs. data to with as the fitting parameter (colored lines).Increasing caged ATP concentrations increased the interface progression coefficient (steeper fit lines from black to red), which indicates that the interface progressed more quickly at higher caged ATP concentrations.

Fig. 3 :
Fig. 3: (Modeling results) Fick's law of diffusion and Michaelis-Menten kinetics captured the diffusion-like mixing of active and inactive fluids.(a) The simulated distribution of ATP concentrations started as a step function (black, 0 h) and then developed into a smoothed hill function (red, 30 h) as ATP evolved from a one-sided distributed to a homogeneous state.(b) The model converted the ATP distribution into the speed distribution of active fluid via Michaelis-Menten kinetics:, where 6.2 µm/s and 270 µM (based on our previous studies 48 ).The corresponding mean speed distribution of active fluid evolved from a step function distribution (black, 0 h) to a near-constant function (red, 30 h) (Supplementary Video 2).Inset: The plot of the Michaelis-Menten equation (Eq.4).(c) In the simulation, the diffusion-driven mixing process led the squared interface displacement to be proportional to time, regardless of initial ATP concentration (see Supplementary Discussion 3 for derivation of ).Inset: Interface displacement increased 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 was determined by fitting the vs. data (Panel c) to with as fitting parameter.The model increased with the initial concentration of ATP, (red dots), similarly to how the experimentally analyzed varied with caged ATP concentration (blue dots; each error bar represents the standard deviation of 3 trials).The model and experimental differed by only ~10%.The magenta curve shows the analytical solution, (Eq.S7), which reproduced the numerical results (red dots).(See Supplementary Discussion 3 for derivation of as a function of .)

Fig. 4 :
Fig. 4: (Experimental results) Progression of the active-inactive interface transitioned from diffusion-like ( 1) to superdiffusion-like ( 1) as the active transport changed from diffusion-dominated ( 3) to convectiondominated (3).(a) The active-inactive interface progression exponent ( ) increased with the flow-speed level of the active fluid ( ).Shown are data from experiments with low ATP concentration (0.5 mM, black dots), high ATP concentration (8 mM, red dots), decreased flow speeds (from nonprocessive motors partially replacing processive motors; blue dots), increased flow speeds (from increased sample height; green dots),48  and both nonprocessive motors and increased sample height (purple dot).The magenta curve represents the moving average of .Although the analyzed from each experiment was noisy, the moving averaged exhibited an overall monotonic increase with the flow-speed level of active fluid.Each dot represents one experimental measurement.Each error bar in represents the slope fitting error in (Fig.2a), and each error bar in represents the standard deviation of flow speeds in the active region.Inset: The flow-speed level of the active fluid was tuned by replacing processive motors (K401) with nonprocessive motors (K365) with the same overall motor concentrations (120 nM) (left)36,48  or by altering the sample height (right).48,52(b)  The same data as in Panel a, plotted as a function of Péclet number,, where is the correlation length of flow velocity in active fluid deduced from sample container height (inset) and is the diffusion coefficient of ATP.Each error bar in is the same as in Panel a, and each error bar in represents propagated uncertainties from in Panel a and in inset.Inset: Correlation length of flow velocity in active fluid increased monotonically with sample container height .39The red dashed line represents the line interpolation of blue dots.The error bars represent the standard deviations of two trials.(See Supplementary Discussion 4 for measurements and analyses of .)

Fig. 5 :Fig. 6 :
Fig. 5: (Experimental results) Active fluid flows promoted mixing of UV-activated fluorescent dyes, which were initially activated in the left-hand side of the container only.(a) Dispersion of UV-activated fluorescent dyes (magenta) in inactive ( 0; left column) and active ( 8.2 µm/s; right column) microtubule-kinesin fluid.Active fluid flows actively transported fluorescent dyes and enhanced their dispersion.Time stamps are hour:minute:second.(See also Supplementary Video 3.) (b) Selected examples of normalized multiscale norm vs. time for different active bulk flow speeds, .Normalized multiscale norm decreased faster in a faster-flowing active fluid system.Inset: The normalized multiscale norm, , in log-linear axes behaved as a straight line, which suggests that the norm decayed exponentially with time.The decay time scale (or mixing time) was determined by fitting the normalized multiscale norm versus time data to with as the fitting parameter (dashed blue line).(c) The mixing time decreased monotonically with Péclet number, which demonstrated that a stronger convection mechanism led to faster mixing of suspended components.Each dot represents one experimental measurement.Each error bar in represents the slope fitting error in (Panel b inset), and each error bar in (defined as ) represents propagated uncertainties from (see inset) and (see Supplementary Fig. 5d).Inset: Mixing time, , as a function of active bulk mean speed, .Each error bar in is the same as in Panel c, and each error bar in represents the standard deviation of flow speeds in the active region.Notably, the mixing time of the inactive fluid system ( 0) was 24 hours (top-left dot); minimally activating the fluid ( 2 µm/s) reduced the mixing time to 8 hours.

Fig. 7 :Fig. 8 :b 2 : 2 :
Fig. 7: (Experimental results) Fluid activated in a checkerboard pattern mixed faster when the checkerboard grid was smaller.(a) Checkerboard-patterned UV lights were used to activate active fluid and caged fluorescent dyes.(Supplementary Video 5).(b) The mixing time of the checkerboard-activated fluid increased with grid size, which demonstrated that the mixing efficacy of active fluid depended on distribution of activity: more nonuniform active fluid mixed the system more slowly.Each error bar in represents the standard deviation of two trials.

Supplementary Fig. 3 :
(Modeling results) The Fick's law-based model showed that the power-law exponent, , in flow speed-ATP relation , influenced the interface progression coefficients but not the interface progression exponents.(a) Plot of flow speed of active fluid as a function of ATP concentration, , with different power-law exponents , where 6.2 µm/s and 270 µM. 13 (b) The distribution of flow speed at time 6 hours in the active fluid system for different power-law exponents.Each system had an initial ATP concentration of 8 mM.The curves are colored based on the color bar in Panel a. (c) The corresponding squared interface displacement, , increased linearly with time, , for each explored powerlaw exponent .The curves were the fitting of with the interface progression coefficient, , and interface progression exponent, , as fitting parameters.Inset: The corresponding interface displacement as a function of time for different power-law exponents.The dots and curves are colored based on the color bar in Panel a.(d) The corresponding interface progression coefficient, , decreased with power-law exponent .The red dashed line represents the experimentally measured 451 8 µm 2 /s for 8 mM caged ATP concentration (Fig. 3d).The error bars represent the fitting error in Panel c. Inset: The interface progression exponents remained 1 across explored power-law exponents 0.5-10.The error bars represent the fitting error in Panel c.

4 :
(Modeling results) The ATP dependence of interface progression coefficient is inherent from the ATP dependence of the corresponding flow speed of active fluid .(a) Interface progression coefficient, , as a function of initial ATP concentration, , for different power-law exponents .The curves are the fitting to with and as fitting parameters.(b) The goodness of fit, , for power-law exponent data in Panel a.

Supplementary Fig. 7 :
(Experimental results) The active-inactive interface progressed more slowly than expected from distribution of ATP.(a) Micrograph of fluorescein (magenta) in the active-inactive sample with an initial ATP concentration of 3.2 mM.The fluorescein was initially caged and thus did not fluoresce; exposure of the left side of the sample to ultraviolet light both activated the microtubule-kinesin fluid and uncaged the fluorescein, allowing it to fluoresce.The white dashed rectangle is the region of interest in the analysis in Panels c-e.The time stamp represents hour:minute:second.(b) The corresponding map of flow speed of active fluid deduced from tracking the motion of tracers.(c) Profile of gray values of the fluorescein micrograph in Panel a.The profile was normalized with a baseline model 17 that included an upper baseline determined by a line fitting to the gray value profile in the active bulk (the top-left magenta portion of the profile), where 46 mm -1 and 1,920, and a lower baseline determined by a line fitting to the gray value profile in the inactive bulk (the bottom-right magenta portion of the profile), where 91 mm -1 and 1,190.These two baselines served as the upper and lower references for profile normalization.(d) The gray values were normalized by the baseline model (right axis).The profile of ATP concentrations was deduced by scaling the normalized profile of gray values by the initial concentrations of ATP,