Convective flow reversal in self-powered enzyme micropumps

Significance

Surface-bound enzymes act as pumps in the presence of their specific substrates or promoters, thereby combining sensing and fluidic pumping into a single self-powered microdevice. Using a combination of theory and experiments, we have elucidated the mechanism of the prototypical urease-based pump. We find that even simple enzymatic reactions can drive complex, time-dependent flows whose direction and speed depend critically on the relative diffusivities and expansion coefficients of the reactants and products. Our approach allows us to accurately predict the behavior of new pump designs under different conditions.

Abstract

Surface-bound enzymes can act as pumps that drive large-scale fluid flows in the presence of their substrates or promoters. Thus, enzymatic catalysis can be harnessed for “on demand” pumping in nano- and microfluidic devices powered by an intrinsic energy source. The mechanisms controlling the pumping have not, however, been completely elucidated. Herein, we combine theory and experiments to demonstrate a previously unreported spatiotemporal variation in pumping behavior in urease-based pumps and uncover the mechanisms behind these dynamics. We developed a theoretical model for the transduction of chemical energy into mechanical fluid flow in these systems, capturing buoyancy effects due to the solution containing nonuniform concentrations of substrate and product. We find that the qualitative features of the flow depend on the ratios of diffusivities $\delta =D_P/D_S$ and expansion coefficients $\beta ={\beta }_P/{\beta }_S$ of the reaction substrate (S) and product (P). If $\delta >1$ and $\delta >\beta$ (or if $\delta <1$ and $\delta <\beta$), an unexpected phenomenon arises: the flow direction reverses with time and distance from the pump. Our experimental results are in qualitative agreement with the model and show that both the speed and direction of fluid pumping (i) depend on the enzyme activity and coverage, (ii) vary with the distance from the pump, and (iii) evolve with time. These findings permit the rational design of enzymatic pumps that accurately control the direction and speed of fluid flow without external power sources, enabling effective, self-powered fluidic devices.

Nonmechanical nano/microscale pumps that provide precise control over fluid flow without an external power source and are capable of turning on in response to specific analytes in solution are needed for the next generation of smart micro- and nanoscale devices. Specific applications involve neutralization of toxins (1), on-demand drug or antidote delivery (2), and the directed focusing of targeted analytes for optimal sensing (3). Recent experiments have shown that surface-bound enzymes act as pumps in the presence of their specific substrates or promoters, driving large-scale fluid flows (2, 4). Furthermore, fluid velocity increases with increasing reaction rate. Therefore, enzymatic catalysis provides an intrinsic energy source for fluid movement and thus can be harnessed to overcome a significant obstacle in nano- and microfluidics: the need to use pressure-driven pumps to push fluids through devices. To fully exploit this behavior, it is vital to develop a fundamental understanding of how the chemical energy is transduced into mechanical fluid flow. One possible mechanism involves thermal buoyancy effects: An exothermic reaction catalyzed by the enzyme lowers the solution density, causing fluid to be drawn in and rise above the pump (Fig. 1A). Not all observed pumping, however, can be explained by this mechanism. In several instances (4), the thermal input from the enzymatic reaction is far lower than that required to generate the observed pumping velocity. Even more remarkable is the observation that urease-based pumps generate flows that are in the opposite direction to that expected for thermally driven movement (2). One possible explanation involves a reaction-induced net increase in solution density, through changes in solutal composition, rather than a net decrease due to heat release (Fig. 1B). Herein, we examine the anomalous flow pattern for the urease pump and, by developing a theoretical model, we show that the flow pattern depends on the relative diffusivities and expansion coefficients of the reactants and products. Furthermore, we isolate conditions where the system exhibits a previously unobserved phenomenon: a pronounced reversal in the flow direction. Our experiments confirm the latter prediction and demonstrate that both the speed and direction of fluid pumping (i) depend on the enzyme activity and coverage, (ii) vary with the distance from the pump, and (iii) evolve with time. These findings allow us to rationally control fluid flow for the directed delivery of payloads to designated locations in microchambers.

Fig. 1.

Possible mechanisms of fluid convection in enzyme-powered micropumps. The fluid flow observed in enzyme-powered micropumps in the presence of substrate has been attributed to two factors: (A) thermal buoyancy and (B) solutal buoyancy. Exothermic enzymatic reactions heat the fluid, lowering solution density and causing fluid to be drawn in and rise above the pump (A). Solutal buoyancy effects arise from the differences in density between the reactants and the products of an enzymatic reaction. Converting reactants to products can either decrease the density of the solution at the pattern surface, which drives flows as in A, or increase the density, which drives convective flows in the opposite direction as in B.

Results and Discussion

Modeling and Simulation.

Reaction-induced mechanisms for driving fluid flow.

To ascertain how chemical reactions at the enzyme pump contribute to the fluid flows, we assess the relative strengths of the following effects: diffusioosmosis, thermal buoyancy, solutal buoyancy, and thermodiffusion (Soret effect).

Diffusioosmosis is the flow of water along a charged, fixed surface caused by tangential gradients in concentrations of either charged or uncharged solute (5, 6). Due to the diffusioosmotic effect, concentration gradients around dissolving salt particles (7, 8) can drive the radial motion of tracer particles at speeds of ∼10 μm/s, which is similar in order of magnitude to the flows observed from enzyme pumps (2). Notably, the motion of a particle due to solute concentration gradients near a wall is a combination of the diffusioosmotic fluid flow $u^{\text{DO}}$ along the fixed wall and diffusiophoretic motion of the particle relative to the fluid flow, $v^{\text{DP}}$, both of which are proportional to the quantity $\nabla C/C$, where C is the solute concentration (5, 7). Near an impermeable wall, the concentration gradients are parallel to the wall and, due to radial symmetry of the enzyme patch, would cause tracer particles to move either toward or away from the pump. Inverting the pump setup (so that the enzyme patch is now on the top wall of the chamber) does not affect the direction of lateral concentration gradients so the direction of tracer motion should remain unchanged. In experiments, however, tracer particles near the surface bearing the enzyme patch moved in the opposite direction relative to the patch when the setup was inverted (2). Thus, we conclude that diffusioosmosis is not the dominant driving mechanism for the flow in these pump systems (2).

The observation that the direction of pumping was reversed when the chamber was inverted is strongly indicative of buoyancy effects, which depend on the direction of the gravitational force. Sengupta et al. (2) numerically showed that temperature gradients generated by the reaction (assuming the reaction rate $F={10}^{-7}$ mol⋅s⁻¹ and enthalpy of reaction $\Delta H=-100$ kJ⋅mol⁻¹) could lead to convective flow speeds of around 1 μm/s toward the patch in the normal setup; this value is similar in order of magnitude for several different enzymes. An exception, however, is the urease-based pump, which catalyzes an exothermic reaction yet generates flows in the opposite direction, i.e., away from the patch in the normal setup, consistent with a reaction-induced increase of fluid density at the pump (2). To explain this behavior, it was hypothesized that the reaction product was denser than the initial urea solution (Fig. 1B) and this solutal buoyancy effect was larger than the density decrease due to exothermicity.

Indeed, a rough calculation shows that solutal density effects dominate over thermal effects. For small changes in solute concentration S and temperature T, the fluid density can be approximated by (9) $\rho ={\rho }_0\left(1+{\beta }_S\left(S-S_0\right)-{\beta }_T\left(T-T_0\right)\right)$, where ${\rho }_0$ is the density at a reference point with temperature T₀ and solute concentration S₀. The volumetric expansion coefficients ${\beta }_S$ and ${\beta }_T$ measure the sensitivity of the fluid density to changes in concentration and temperature, respectively. For a localized (point-like) reaction occurring at rate F, the steady-state temperature and substrate concentration fields in an unbounded 3D environment are respectively $T=T_{\infty }-(1/4\pi r)F\mathrm{\Delta }H/D_T{c}_p{\rho }_0$ and $S=S_{\infty }-(1/4\pi r)F/D_S$, where r is the distance from the reaction, $D_T$ is the heat diffusivity, $D_S$ is the solute diffusivity, $\mathrm{\Delta }H$ is the enthalpy of the reaction (negative for exothermic reactions), and $c_p$ is the constant-pressure specific heat capacity of the fluid. The ratio of temperature to concentration gradients is then ${\partial }_{r}T/{\partial }_{r}S=D_S\mathrm{\Delta }H/D_T{c}_p{\rho }_0$. The notation ${\partial }_r=\partial /\partial r$ signifies the spatial derivative in the radial direction. The influence of thermal buoyancy relative to solutal buoyancy is indicated by the ratio of their Rayleigh numbers,

$$\frac{{\text{Ra}}_T}{{\text{Ra}}_S}=\frac{{\beta }_T{\partial }_{r}T}{{\beta }_S{\partial }_{r}S}=\frac{{\beta }_T{D}_S\mathrm{\Delta }H}{{\beta }_S{D}_T{c}_p{\rho }_0}.$$ [1]

We consider the solute to be urea and the reaction to be the hydrolysis of urea by urease ($\Delta H=-59.6\hspace{0.17em}\text{kJ}\cdot {\text{mol}}^{-1}$) (10). Using known physical properties of water and urea (11), we obtain ${\text{Ra}}_T{\text{/Ra}}_S\approx 0.002$ and thus the small thermal effects can be neglected for the urease-based pump. This suggests that solutal buoyancy plays a dominant role, and factors like the differential diffusion of substrate and product species should be considered.

With respect to thermodiffusion (the Soret effect), it is known that temperature gradients can contribute to the diffusive flux of solute, which for dilute solutions is (12)

$$J_r=-D_S\left[{\partial }_{r}S+\left(\sigma +{\beta }_T\right)S\cdot {\partial }_{r}T\right],$$ [2]

where $\sigma$ is the Soret coefficient and other variables are as defined above. In the absence of temperature gradients, this expression yields the usual Fickian diffusive flux. The magnitude and sign of the Soret coefficient depend on factors such as the solute size, background salt concentration, and temperature (13, 14). The Soret coefficient is typically positive, resulting in solute migration toward lower temperatures. Hence, solute becomes depleted in hot regions, driving fluid convection. This combination of repulsion and convection can be harnessed to trap DNA by laser heating (13). We assess the relative influence of the Soret effect in our system by the ratio of contributions from thermal gradients and solutal gradients in Eq. 2, $J_r^T/J_r^S=\left(\sigma +{\beta }_T\right)S\cdot {\partial }_{r}T/{\partial }_{r}S$. Using estimates for temperature and concentration gradients from above and the Soret coefficient $\sigma =8.1\times {10}^{-4}{\text{K}}^{-1}$ (for urea at concentration $S=0.417\hspace{0.17em}\text{M}$) (14), this ratio is $J_r^T/J_r^S\approx 6\times {10}^{-8}<<1$. Hence, we neglect thermodiffusion for our model of the urease pump.

Model for double diffusive solutal convection pumps.

Below, we argue that the interplay of the relative diffusivities and expansion coefficients of the reaction substrate S and product P gives rise to temporally and spatially complex fluid flow. In our model, we consider a shallow, fully enclosed chamber of width 2L in the horizontal (x and z) directions and height H in the y direction (Fig. 2). Enzymes are bound to a circular patch of radius R at the center of the bottom wall and occupy a volume of fluid of depth h above the patch; in this volume, the reaction $S\to P$ converts substrate to product with a reaction rate per unit volume f. We consider only constant reaction rates; this assumption is valid if the reaction rate is low enough that substrate depletion is insignificant over time scales of interest.

Fig. 2.

Geometry for enzyme-powered microfluidic pumps used in simulations. Volume occupied by enzymes tethered to a circular patch at center of lower wall is marked in yellow.

For nondimensionalization, we define the characteristic length scale $l*=R$, time scale $t*=R^2/D_S$, and concentration scale $S*=P*=f t*h/H$, where $D_S$ is the diffusion coefficient of the substrate. For the urease-based pumps in our experiments, the length and time scales correspond to $l*\approx 3\hspace{0.17em}\text{mm}$ and $t*\approx 100\text{\hspace{0.17em}min}$, respectively. The dimensionless advection–reaction–diffusion equations are

$${\partial }_{t}S+\mathbf{u}\cdot \nabla S={\nabla }^{2}S-{\chi }_E,\hspace{1em}{\partial }_{t}P+\mathbf{u}\cdot \nabla P=\delta \hspace{0.17em}{\nabla }^{2}P+{\chi }_E,$$ [3]

where ${\partial }_t=\partial /\partial t$, the parameter $\delta =D_P/D_S$ is the ratio of the product diffusion coefficient $D_P$ to the substrate diffusion coefficient $D_S$, and ${\chi }_E$ is the characteristic function, which takes the value 1 inside the volume occupied by enzymes and the value 0 outside. No-flux boundary conditions are used for the substrate and product. The chamber is initially filled uniformly with the enzyme’s substrate at concentration $S\left(t=0\right)=S_0$ and contains no product, $P\left(t=0\right)=0$.

Writing the change in substrate concentration (relative to the initial, homogeneous value) $\mathrm{\Delta }S=S-S_0$ and the change in product concentration $\mathrm{\Delta }P=P-P_0=P$, we use the linear model for the dimensional fluid density (9), $\rho ={\rho }_0\left(1+{\beta }_S\mathrm{\Delta }S+{\beta }_P\mathrm{\Delta }P\right)={\rho }_0\left(1+\mathrm{\Delta }\rho \right)$, where ${\rho }_0$ is the initial, uniform fluid density and $\mathrm{\Delta }\rho ={\beta }_S\mathrm{\Delta }S+{\beta }_P\mathrm{\Delta }P$ is the fractional change in density due to changes in solute composition. The constants ${\beta }_S$ and ${\beta }_P$ are the volumetric coefficients of expansion due to the substrate and product, respectively. Positive values of ${\beta }_S$ and ${\beta }_P$ indicate that fluid density increases with solute concentration, as is commonly the case. Defining the parameter $\beta ={\beta }_P/{\beta }_S$ and rescaling the density by $\rho *={\rho }_0{\beta }_{S}S*$, the dimensionless equation for density is $\rho ={\rho }_0+\mathrm{\Delta }\rho$, $\mathrm{\Delta }\rho =\mathrm{\Delta }S+\beta \mathrm{\Delta }P$.

Following the Boussinesq approximation (15), the fluid flow $\mathbf{u}$ is governed by the incompressible Navier–Stokes equations with a buoyancy force term. We use the length and time scales as above and define the velocity scale $u*=l*/t*$ and pressure scale $p\ast =\mu /t\ast$, where $\mu$ is the dynamic viscosity of the fluid. The dimensionless Navier–Stokes equations driven by buoyancy forces are

$${\text{Sc}}^{-1}\left({\partial }_t\mathbf{u}+\mathbf{u}\cdot \nabla \mathbf{u}\right)=-\nabla p+{\nabla }^2\mathbf{u}-\text{Ra}\cdot \mathrm{\Delta }\rho \hspace{0.17em}{\mathbf{e}}_y,\hspace{1em}\nabla \cdot \mathbf{u}=0,$$ [4]

where $\text{Sc}=\mu /{\rho }_0{D}_S\sim \mathrm{1,000}$ is the Schmidt number. The buoyancy force term in Eq. 4 is proportional to the rescaled density change $\mathrm{\Delta }\rho =\mathrm{\Delta }S+\beta \mathrm{\Delta }P$, and the characteristic strength of convection is determined by the Rayleigh number, $\text{Ra}={\rho }_{0}g\hspace{0.17em}{\beta }_S{R}^{5}f\hspace{0.17em}h\hspace{0.17em}{\left(\mu \hspace{0.17em}{D}_S^{2}H\right)}^{-1}$, where g is the acceleration due to gravity. In Eq. 4, we have omitted the uniform buoyancy force proportional to ${\rho }_0$ because any constant force can be incorporated into the pressure gradient term by considering a modified pressure field. We impose no-slip boundary conditions for the fluid and assume that the fluid is initially at rest, $\mathbf{u}\left(\mathbf{x},t=0\right)=\mathbf{0}$.

Our modeling approach is similar to that in prior reaction–diffusion–convection studies (16–18), except that we confine the reaction to a finite region containing bound enzymes. To the best of our knowledge, the latter setup has not been treated theoretically, but has clear relevance to the function of enzyme-driven pumps, as well as mixing in microfluidic reaction chambers.

In general, we expect buoyancy effects to cause fluid to either rise or sink above the enzyme patch, driving toroidal convection rolls. Such flow patterns were described in previous experiments with enzyme pumps (2, 4) and have also been induced by localized heating in thin layers of oil (19). In the case of sinking fluid, resulting from a local increase in density, fluid is pushed out radially away from the pump at the bottom of the chamber and drawn in radially at the top of the chamber. We refer to this as “outward” pumping. Conversely, “inward” flow moves toward the pump along the bottom and spreads radially outward at the top of the chamber. To quantify the characteristics of the flow, we define the outward flow at a point in the chamber,

$$u_{\text{out}}\left(x,y,z\right)=\{\begin{array}{c}\hspace{0.5em}\hspace{0.5em}\mathbf{u}\left(x,y,z\right)\cdot {\mathbf{e}}_r,\hspace{1em}\text{if\hspace{0.17em}}y<H/2,\\ -\mathbf{u}\left(x,y,z\right)\cdot {\mathbf{e}}_r,\hspace{1em}\text{if\hspace{0.17em}}y>H/2,\end{array}$$ [5]

where ${\mathbf{e}}_r={\left(x,0,z\right)}^{\text{T}}/\sqrt{x^2+z^2}$ is a unit vector in the radial direction. We also define the vertically averaged outward radial flow,

$${\overline{u}}_{\text{out}}\left(x,z,t\right)=\frac{1}{H}{\displaystyle \underset{0}{\overset{H}{\int }}{u}_{\text{out}}}\left(x,y,z,t\right)\text{d}y,$$ [6]

which is positive for flow in the outward sense and negative for inward flow, as described above. Hence, inward flow could be quantified by ${\overline{u}}_{\text{in}}=-{\overline{u}}_{\text{out}}$.

Because the Schmidt number is large, the inertial terms in the Navier–Stokes equations (Eq. 4) may be neglected, resulting in Stokes flow. The fundamental solution of Stokes flow driven by a point force acting between infinite, parallel plates is known (20). Each small volume element of fluid exerts a point-like buoyancy force, driving a toroidal flow (Fig. S1). This clearly demonstrates that horizontal (as well as vertical) fluid flows are generated by vertical forces. Summing up contributions from each volume element, we find that the net radial flow produced by the pump is approximately proportional to radial changes in fluid density in the chamber, ${\overline{u}}_{\text{out}}\sim -\left(\partial \rho /\partial r\right)$ (SI Materials and Methods). By deriving an expression for the density field (treating the pump as a point source of product and point sink of substrate—see SI Materials and Methods for details), we can use the above relationship to predict the flow field near the pump.

Fig. S1.

Stokes flow driven by a vertical force between no-slip parallel plates. A downward force (white arrow) is exerted at a point in the fluid. The two no-slip plane boundaries are at $y=0$ and $y=1$, respectively. The fluid flow is obtained numerically using the boundary element method (29) and is approximately equal to the Green’s function solution for a point force between infinite, parallel plates (20). The color map represents the magnitude of flow in the cross-section through the particle; red indicates high speed, and blue indicates low speed.

Analysis revealed unexpectedly rich behavior of the flow field that depends on $\delta$ and $\beta$: The direction of flow can reverse over time and depends on the distance from the pump. Assuming $\delta =D_P/D_S>1$ (as for the urease pump), this behavior occurs if $\delta >\beta \left(={\beta }_P/{\beta }_S\right)$. Here, a wave of high-density fluid propagates radially from the pump; the flow near the pump will be inward and the flow far from the pump (beyond the peak in density) will be outward (Fig. 3A). Flow reversal is also predicted for $\delta <1$ if $\delta <\beta$, but now flow is in the opposite direction, driven by a wave of low-density fluid. For all other combinations of $\delta$ and $\beta$, the fluid flow direction (outward if $\beta >1$, inward if $\beta <1$) will not vary with time or position. Density profiles for various combinations of $\delta$ and $\beta$ are depicted in Fig. S2.

Fig. 3.

Calculated evolution of the density gradient over time and resulting simulated flow field for two combinations of δ and β. (A and B, Top) Side view of flow fields showing streamlines at time $t=20$ (all variables in dimensionless units) computed using the full 3D Navier–Stokes–Boussinesq model with chamber height $H=2$ and large width $2L=80$ (approximating a horizontally infinite domain). The color map indicates the local outward radial flow, normalized by a characteristic speed U* (defined in SI Materials and Methods). The enzyme reaction occurs in a thin layer above the yellow boxes at the bottom left of the cross-sections. (A and B, Bottom) Evolution of density gradient field $\left(-\left(\partial /\partial r\right)\mathrm{\Delta }\rho \right)$ over time using the simplified, point source model. The dashed green curve in A tracks the position of the density peak over time (see SI Materials and Methods for a formula for this curve). Because the point source approximation is not accurate at short distances, the region $r<1$ is omitted. Parameters are (A) $\delta =2$, $\beta =1.7$ and (B) $\delta =1.7$, $\beta =2$.

Fig. S2.

Theoretical concentration and density profiles at time t = 5 with substrate S and product P respectively consumed and released at equal, constant rate at r = 0. This model considers 2D diffusion in an unbounded domain. (A) Product concentration decreases at small r and increases at large r as diffusivity ratio $\delta =D_P/D_S$ increases. (B–D) Density changes due to the corresponding concentration profiles from A when solutal expansion ratio $\beta ={\beta }_P/{\beta }_S$ is varied from 1 to 3.

Fig. 3 shows the temporal evolution of the density gradient $\left(-\left(\partial /\partial r\right)\mathrm{\Delta }\rho \right)$ computed from the point source model in an infinite 2D domain and the flow structure obtained from full 3D simulations of the Navier–Stokes–Boussinesq model. The plots support the predicted correlation between the density gradient and the flow field (Fig. S3). Fig. 3A shows the predicted flow reversal when $\delta >1$ and $\delta >\beta$; the peak in the density lies at a finite distance from the pump, giving rise to inward flow on the left and outward flow on the right. In contrast, when $\delta <\beta$ (Fig. 3B), the fluid density is greatest at the enzyme patch ($r=0$) and decreases monotonically with distance, leading to outward flow everywhere.

Fig. S3.

Comparison of simulated flow speeds and density gradients. (A and B, Left) Contour maps of pumping direction and flow speed as a function of distance from the center of the pump and time. Flow fields are obtained from 3D simulation of the Navier–Stokes–Boussinesq equations. (A and B, Center) Evolution of vertically averaged density gradient field using simulation data. (A and B, Right) Evolution of vertically averaged density gradient field using the theoretical point source model. The region $x<1$ is excluded because the point source approximation is singular and not valid close to the source. Parameters are identical to those in Fig. 3: (A) $\delta =2,\beta =1.7$ and (B) $\delta =1.7,\beta =2$. Green, dashed curves in A are the zero-level contours.

Having uncovered the fundamental process of flow reversal, we performed simulations with the patch size and chamber aspect ratio approximating the experimental setup (Experimental Observations) for $\delta >1$ and $\delta >\beta$. Fig. 4 shows three distinct periods in the flow evolution. At early times, flow is in the outward direction throughout the chamber. At late times, flow is in the inward direction throughout. At intermediate times, there is an expanding inner region of inward flow and a coexisting outer region with outward flow.

Fig. 4.

Simulated spatiotemporal flow field evolution illustrating flow reversal in an enclosed chamber. (A) Outward flow ${\overline{u}}_{\text{out}}$ (Eq. 6) as a function of time and distance from the pump center. The scaling factor U* is defined in SI Materials and Methods. (B) Top (x–z) view of the chamber with snapshots at different times in each quadrant, as labeled. Arrows indicate flow in the bottom half of the chamber. The perimeter of the enzyme patch is shown by the yellow circle. (C) Side views of the flow at the same times as in B, increasing from top to bottom. The cross-sections are at $z=0$ and only the right half ($x\ge 0$) is shown. Parameters are $L=5$, $H=0.5$, $\delta =2$, and $\beta =1.7$. Movie S1 shows the flow field in the bottom half of the chamber.

We also performed simulations taking into account the chemical advection terms in Eq. 3. These nonlinear effects are expected to be negligible if the Péclet number, which is proportional to the Rayleigh number, is small, but may become significant if the reaction rate is high, for instance. With the simulation parameters $\beta =1.08$ and $\delta =1.1$ (Fig. 5A), we found that increasing the Péclet number delayed the onset of the outward-to-inward flow reversal at positions close to the pump. Fig. 5B (Pe = 15, red curve) demonstrates that at high Pe, the emergence of additional flow direction transitions is possible; a small region of inward flow transiently appears close to the side wall.

Fig. 5.

Advection–diffusion–reaction simulations for transitions from outward to inward pumping at various Péclet numbers (Pe). We calculate Pe in these simulations according to Eq. S11 (SI Materials and Methods). For each value of Pe, the region below the curve exhibits outward flow and the region above the curve exhibits inward flow. The Pe = 15 (red) case in B has an additional region of inward flow, as labeled. The dashed curves (underlying the Pe = 0.2 curves) correspond to the Pe = 0 limit (chemical advection disabled in simulations). The expansion coefficient ratios are (A) $\beta =1.08$ and (B) $\beta =1.05$. Other simulation parameters are: L = 3.5, H = 0.5, and $\delta =1.1$.

The examples in Fig. 5 indicate that the effects of increasing Pe depend on model parameters. In these examples, the dynamics are particularly sensitive to parameter values because $\beta \approx \delta$, which means the system is near the critical point for flow reversal. The time of first appearance of inward flow varies from $t\approx 0.15$ (Fig. 5B, Pe = 15) to $t\approx 2$ (Fig. 5A, Pe = 5), corresponding to 15–200 min. The nonlinear dependence of the flow on reaction rate (via Pe) and physical parameters suggests further possibilities for designing complex flow patterns using enzymatic pumps. By choosing reactions with suitable values of $\delta$ and $\beta$ and controlling the rate of reaction, it is possible to tailor the flow characteristics.

Experimental Observations.

Fabrication of enzyme pumps.

Enzyme-powered pumps were fabricated as follows (Fig. S4). A gold (Au) patch (6 mm diameter) was patterned on a polyethylene glycol (PEG)-coated glass surface. A biotin thiol linker was then added to form a self-assembled monolayer (SAM) on the Au patch, followed by the addition of streptavidin. Enzyme attachment on the streptavidin-functionalized surface was then attained via a protein–ligand linkage, facilitated through a biotin tag. Enzyme coverage on the patch was varied by changing the enzyme concentration in the solution used to prepare the pattern. The fabrication was completed by placing a spacer (20 mm diameter, 1.3 mm height) on top of the enzyme-patterned surface to seal the chamber and create a closed system. A buffered solution of substrate with suspended tracer particles (2 µm in size) was injected into the chamber and the fluid flow was monitored with an optical microscope.

Fig. S4.

Schematic of the fabrication process for enzyme-powered micropumps and triggered fluid pumping by the presence of substrate. Au was patterned on a PEG-coated glass surface, using an e-beam evaporator. The patterned surface was functionalized with a biotin thiol linker, which forms a SAM on the Au surface. A streptavidin solution was then added, followed by the addition of biotin-tagged enzymes, achieving enzyme attachment through the use of a protein–ligand linkage. To create a closed system, a hybridization chamber was placed on top of the enzyme pump, and solutions of substrate with tracer particles were added to the chamber to track fluid motion. In the presence of substrate, the enzyme micropumps pump fluid and particles due to the enzymatic reaction. In the absence of substrate, no fluid pumping is observed. This process can be generalized for enzyme attachment to other surfaces, using different attachment strategies or other enzyme–substrate combinations.

Experimental results.

Experiments were performed on urease micropumps to test the predictions that the system exhibited an initial outward flow, which was later replaced by an inward flow region that developed at the vicinity of the pump and spread farther away with time. Movies of tracer particles were analyzed to measure the pumping speeds induced by the pump as a function of distance at different time intervals from the introduction of urea solution (Fig. 6). The urease concentration used for soaking the functionalized gold patch was 3 × 10⁻⁹ M. As shown, initially (during the first time interval) there was inward pumping close to the pump, whereas farther away a region of outward pumping was seen. With time, the region of inward pumping gradually extended farther away from the pump. The observations are in close agreement with the theoretical model, confirming the role of solutal buoyancy in the pumping mechanisms and validating the theoretical approach. We also note that a small region of inward flow is observed transiently at the perimeter of the chamber (Fig. 6 A and B). This structure is not predicted by our linear theory but was observed in simulations with large Pe effects (Fig. 5B).

Fig. 6.

Experimentally obtained spatiotemporal pumping behavior of urease-powered pumps at low enzyme coverages. The pumping speeds of urease-powered pumps containing low enzyme coverage were measured as a function of distance at different time intervals. The urease enzyme concentration used for soaking the functionalized gold patch was 3 × 10⁻⁹ M. (A) Initially there was inward pumping close to the pump, whereas farther away, a region of outward pumping was observed (B–F). With time, the region of inward pumping extended farther away from the pump, as the outward pumping regime was diminished. The means and SDs are calculated for 30 tracer particles. See Table S1 for data. See Fig. S5 for pumping speeds as a function of time at different distances away from the enzyme pump. See Table S2 for data.

Table S1.

Pumping speeds as a function of distance at different time intervals for a case of low enzyme coverage ([E] = 3.0 x 10⁻⁹ M; [urea] = 500 mM)

Distance away from the pump, mm	Pumping speed, μm/s	Fluid flow direction
t = 10–16 min
0	0.631 ± 0.086	Inward (−)
1	0.221 ± 0.064	Inward (−)
2	0.105 ± 0.054	Outward (+)
3	0.208 ± 0.059	Outward (+)
4	0.208 ± 0.060	Outward (+)
5	0.119 ± 0.044	Outward (+)
6	0.164 ± 0.055	Inward (−)
t = 17–23 min
0	0.670 ± 0.077	Inward (−)
1	0.268 ± 0.053	Inward (−)
2	0.0971 ± 0.0427	Outward (+)
3	0.186 ± 0.055	Outward (+)
4	0.152 ± 0.048	Outward (+)
5	0.103 ± 0.042	Outward (+)
6	0.0932 ± 0.0425	Inward (−)
t = 24–30 min
0	0.659 ± 0.077	Inward (−)
1	0.359 ± 0.059	Inward (−)
2	0.171 ± 0.051	Inward (−)
3	0.160 ± 0.050	Outward (+)
4	0.158 ± 0.043	Outward (+)
5	0.137 ± 0.041	Outward (+)
6	0.0921 ± 0.0395	Outward (+)
t = 31–37 min
0	0.643 ± 0.074	Inward (−)
1	0.397 ± 0.059	Inward (−)
2	0.132 ± 0.041	Inward (−)
3	0.0977 ± 0.0411	Outward (+)
4	0.160 ± 0.042	Outward (+)
5	0.168 ± 0.044	Outward (+)
6	0.0972 ± 0.047	Outward (+)
t = 38–44 min
0	0.545 ± 0.074	Inward (−)
1	0.360 ± 0.055	Inward (−)
2	0.237 ± 0.055	Inward (−)
3	0.0881 ± 0.0424	Outward (+)
4	0.108 ± 0.050	Outward (+)
5	0.200 ± 0.049	Outward (+)
6	0.131 ± 0.046	Outward (+)
t = 45–51 min
0	0.562 ± 0.074	Inward (−)
1	0.404 ± 0.053	Inward (−)
2	0.260 ± 0.054	Inward (−)
3	0.140 ± 0.047	Inward (−)
4	0.0996 ± 0.0375	Outward (+)
5	0.119 ± 0.040	Outward (+)
6	0.123 ± 0.049	Outward (+)

Fig. S5.

Pumping speeds as a function of time at different distances away from the enzyme pump for low enzyme coverage. The pumping speeds of urease-powered pumps containing low enzyme coverage were studied as a function of time at different distances away from the pump, in the presence of 500 mM urea. For this purpose, a functionalized gold patch was first exposed to a 3 × 10⁻⁹ M solution of urease, and once the pump was set up, 1-min movies were taken 50 µm away from the patch surface on every side to compare speeds and fluid direction, thereby ruling out extraneous flows. The microscope was then focused on one side of the patch, and 1-min movies were taken close to the pump and at increasing distances (1 mm apart) away from the Au patch until the wall was reached. The whole process was repeated five times. As shown in A, close to the pump the fluid direction starts outward at early times in the experiment and then becomes strongly inward. This inward pumping region spreads with time and distance as can be seen in B–D. Far from the pump (E and F), the fluid flow is outward throughout the experiment, but the speeds seem to decrease slowly with time. (G) At 6 mm away from the pump, fluid flow reversal is also observed, from inward to outward pumping, as the outward pumping region slowly spreads away from the Au patch. The means and SDs shown are calculated for 30 tracer particles.

Table S2.

Pumping speeds as a function of time at different distances away from the enzyme pump for a case of low enzyme coverage ([E] = 3.0 x 10⁻⁹ M; [urea] = 500 mM)

Time, min	Pumping speed, μm/s	Fluid flow direction
d = 0 mm
5	0.0954 ± 0.0498	Outward (+)
10	0.631 ± 0.086	Inward (−)
17	0.670 ± 0.079	Inward (−)
24	0.659 ± 0.077	Inward (−)
31	0.643 ± 0.074	Inward (−)
38	0.545 ± 0.074	Inward (−)
45	0.562 ± 0.074	Inward (−)
52	0.501 ± 0.066	Inward (−)
d = 1 mm
11	0.221 ± 0.064	Inward (−)
18	0.268 ± 0.053	Inward (−)
25	0.359 ± 0.059	Inward (−)
32	0.397 ± 0.059	Inward (−)
39	0.360 ± 0.055	Inward (−)
46	0.404 ± 0.053	Inward (−)
d = 2 mm
12	0.105 ± 0.054	Outward (+)
19	0.0971 ± 0.0427	Outward (+)
26	0.171 ± 0.051	Inward (−)
33	0.132 ± 0.041	Inward (−)
40	0.237 ± 0.055	Inward (−)
47	0.260 ± 0.054	Inward (−)
d = 3 mm
13	0.208 ± 0.059	Outward (+)
20	0.186 ± 0.055	Outward (+)
27	0.160 ± 0.050	Outward (+)
34	0.0977 ± 0.0411	Outward (+)
41	0.0881 ± 0.0424	Outward (+)
48	0.140 ± 0.047	Inwards (−)
d = 4 mm
14	0.208 ± 0.060	Outward (+)
21	0.152 ± 0.048	Outward (+)
28	0.158 ± 0.043	Outward (+)
35	0.160 ± 0.042	Outward (+)
42	0.108 ± 0.050	Outward (+)
49	0.0996 ± 0.0375	Outward (+)
d = 5 mm
15	0.119 ± 0.044	Outward (+)
22	0.103 ± 0.042	Outward (+)
29	0.137 ± 0.041	Outward (+)
36	0.168 ± 0.044	Outward (+)
43	0.200 ± 0.049	Outward (+)
50	0.119 ± 0.040	Outward (+)
d = 6 mm
16	0.164 ± 0.055	Inward (−)
23	0.0932 ± 0.0425	Inward (−)
30	0.0921 ± 0.0395	Outward (+)
37	0.0972 ± 0.0465	Outward (+)
44	0.131 ± 0.046	Outward (+)
51	0.123 ± 0.049	Outward (+)

To investigate the pumping behavior as a function of enzyme coverage, biotin–streptavidin functionalized Au surfaces were exposed to solutions containing different concentrations of biotin-tagged urease. At higher enzyme coverages, only outward pumping (i.e., away from the patch) was observed close to the pump after 20 min (Fig. 7). As the enzyme coverage was lowered, resulting in lower reaction rates, the fluid flow direction reversed, and inward pumping was observed. This fluid flow reversal occurs gradually as a function of enzyme concentration. The maximum observed inward pumping speed was around half the magnitude of the maximum observed outward speed. Further decreasing the enzyme coverage resulted in a reduction in magnitude of inward pumping, consistent with the scaling argument that, for low reaction rates, the flow speed is proportional to the rate of reaction (SI Materials and Methods). Flow reversal with time was verified over an intermediate range of enzyme coverage (Fig. 7, Inset): Outward pumping was observed initially, but was replaced by inward pumping over time.

Fig. 7.

Pumping speeds of urease-powered micropumps as a function of enzyme coverage. The pumping speeds of urease-powered pumps were measured as a function of the concentration of biotin-tagged urease in the solution used for soaking the biotin–streptavidin functionalized Au patch. Lower concentrations of soaking solution result in less enzyme immobilized on the patch. The speed and direction of fluid flow near the pump were monitored after 20 min of pump activation in the presence of 500 mM urea in 10 mM PBS. At higher enzyme coverages, only outward pumping (i.e., away from the patch) was observed. As the enzyme coverage was lowered, the fluid flow direction was reversed, and inward pumping was observed. (Inset) Pumping speeds over time for an intermediate enzyme coverage show outward pumping at early times and inward pumping at late times, as predicted by the theoretical model. The means and SDs shown are calculated for 30 tracer particles (Tables S3 and S4).

Table S3.

Enzyme concentrations used for soaking streptavidin-functionalized Au patterns, corresponding pumping speeds obtained in the presence of substrate, and fluid flow direction (enzyme = urease; k_cat = 23,000 s⁻¹; K_M = 0.0013 M; active sites = 6; [urea] = 500 mM) (30)

Enzyme concentration used for pattern soaking, M	Pumping speed, μm/s	Fluid flow direction
6.0 × 10−6	2.70 ± 0.12	Outward (+)
3.0 × 10−6	1.43 ± 0.11	Outward (+)
3.0 × 10−7	0.405 ± 0.069	Outward (+)
3.0 × 10−8	0.258 ± 0.080	Inward (−)
3.0 × 10−9	0.336 ± 0.075	Inward (−)
3.0 × 10−10	0.407 ± 0.058	Inward (−)
3.0 × 10−12	0.460 ± 0.078	Inward (−)
3.0 × 10−13	0.597 ± 0.088	Inward (−)
3.0 × 10−15	0.870 ± 0.080	Inward (−)
3.0 × 10−16	1.18 ± 0.09	Inward (−)
3.0 × 10−18	1.42 ± 0.14	Inward (−)
3.0 × 10−19	0.776 ± 0.106	Inward (−)
3.0 × 10−20	0.413 ± 0.075	Inward (−)

Table S4.

Pumping speeds and fluid flow direction as a function of time for a case of intermediate enzyme coverage ([E] = 3.0 x 10⁻⁸ M; [urea] = 500 mM)

Time, mins	Pumping speed, μm/s	Fluid flow direction
5	0.258 ± 0.059	Outward (+)
10	0.197 ± 0.049	Inward (−)
15	0.402 ± 0.080	Inward (−)
20	0.460 ± 0.078	Inward (−)

The observed dependence of pumping direction (at a fixed position and time) on enzyme coverage is consistent with simulation results in Fig. 5A for flows at different Péclet numbers. The transition from outward to inward flows near the pump was delayed as Pe (approximate reaction rate) was increased. Hence, at a fixed time it is possible that a low-enzyme pump has transitioned to inward pumping whereas a high-enzyme pump is still producing an outward flow, as we observe experimentally.

Experimental parameter estimates.

To quantify the change in solution density due to the enzyme catalyzed reaction, we compared the densities of a solution of urease and urea before and after reaction (details in SI Materials and Methods and Table S5). The estimated ratio of expansion coefficients was $\beta \approx 1.19\pm 0.08$. To estimate $\delta$, we consider the conversion of urea, CO(NH₂)₂, into ammonium bicarbonate (NH₄⁺ and HCO₃⁻ ions) and obtain diffusion coefficients for the relevant species from the literature (11). The substrate diffusivity is D_CO(NH2)2 = 1.38 × 10⁻⁵ cm²/s and the product diffusivity is D_{NH4 HCO3} = 1.48 × 10⁻⁵ cm²/s (based on the effective diffusivity for a salt with D_NH4+ = 1.96 × 10⁻⁵ cm²/s and D_HCO3− = 1.19 × 10⁻⁵ cm²/s). From these values, the ratio of diffusivities is $\delta =1.07$.

Table S5.

Density measurements of a solution containing the enzyme plus substrate before and after reaction and the percentage of difference in density

Solutions	Density, g/mL*
Urea (500 mM) + urease (3.0 x 10−7 M) before reaction, ${\rho }_{\text{initial}}$	1.07 ± 0.04
Urea (500 mM) + urease (3.0 x 10−7 M) after reaction, ${\rho }_{\text{final}}$	1.08 ± 0.04

^*Average values from n = 6 trials. The expansion coefficient ratio (β) was calculated for each trial, and the average was 1.19 ± 0.08.

The estimated values of $\delta$ and $\beta$ are similar in magnitude. Although they do not satisfy the theoretical condition for flow reversal ($\delta >\beta$, given $\delta =1.07$), plausible errors in the parameter estimates could mean that the true values do satisfy this condition. The expansion coefficient ratio is difficult to determine accurately because the relative increase in solution density due to the reaction is small (∼1%). Diffusion coefficients reported in the literature are also often in poor agreement (11).

SI Materials and Methods

Fabrication of Enzyme-Powered Pumps.

Biotinylation of enzymes (27).

An enzyme solution (2 mg/mL) was prepared with 100 mM PBS as the buffer. Before completing the total volume of the enzyme solution, 10 µL of a 0.1-mM 3-(N-maleimidylpropionyl) biocytin (Santa Cruz Biotechnology) biotin solution was added per mL of enzyme solution to get a ratio of 4:1 enzyme:biotin. This was to ensure that each enzyme molecule had at least one biotin molecule. For urease [urease from Canavalia ensiformis (Jack bean)] (Sigma Aldrich), the biotin linker used was 3-(N-maleimidylpropionyl) biocytin (Santa Cruz Biotechnology) for targeting the cysteine residues in the enzyme. The mixture was reacted for 2 h at room temperature. Dialysis of the final solution was carried out at 3803.4 × g (4,500 rpm; Thermo Scientific Sorvall ST16; TX-400 4 × 400 mL rotor) for 5 min, followed by washes with 10 mM PBS. The enzyme concentration of the solution obtained at the end was determined using the Beer–Lambert law,

$$A=\epsilon bC,$$

where A is the absorbance of the solution; ε is the molar absorptivity; b is the length of the light path, which is equal to the width of the cuvette; and C is the concentration of the analyte under study in solution. For a urease solution, the equation was solved for C, measuring the UV-VIS absorbance of the solution at 280 nm, and using b = 1 cm (width of the cuvette) and ε₂₈₀ = 75,592 M⁻¹⋅cm⁻¹, which was calculated experimentally. The concentrated enzyme solution was used later for the preparation of enzyme solutions of different concentrations, using serial dilutions. All of the enzyme solutions were stored at 4 °C until needed.

Preparation of Au patterns.

Using an e-beam evaporator, Au was patterned on a PEG-coated glass surface (MicroSurfaces). An electron beam was used to evaporate a thickness of 90 nm of Au on the PEG-functionalized surface, with a 10-nm adhesion layer of Cr. The radius of the gold pattern was 3 mm. The surface was cleaned thoroughly with isopropanol followed by acetone and dried by blowing nitrogen.

Biotinylation of Au patterns and enzyme immobilization.

Biotinylation of the Au patterns was achieved following a procedure reported previously for the biotinylation of Au nanoparticles, through the formation of a self-assembled monolayer (SAM), using a biotin–thiol linker (28). For the preparation of the linker, a sulfhydryl-reactive biotinylating agent, Biotin HPDP (Apexbio) (1 mg per three patterns), was dissolved in dimethylformamide (DMF) (Acros Organics) (0.13 mg/mL) through sonication for ∼6 min at 45 °C, and a tributyl phosphine solution (Sigma Aldrich) (5 μL/mg of biotin HPDP) was added to form a thiol end group. The reaction mixture was allowed to react for 30 min at 45 °C, after which it was dissolved in a solvent mixture of 1:1 H₂O:ethanol (8 mL of solution per milligram of biotin). This was the soaking solution for the Au patterns. The patterns were then incubated overnight at room temperature. After incubation, the patterns were washed several times with deionized water, followed by two washes with 10 mM PBS buffer. The SAM-modified surfaces were then incubated in a streptavidin (ProZyme) solution (9 μM in 10 mM PBS) for 3 h, after which the surfaces were washed with 10 mM PBS buffer. Incubation of streptavidin-containing surfaces with enzyme–biotin solutions was performed for 3–4 h prior the experiments. The enzyme-functionalized surfaces were thoroughly washed with 10 mM PBS to remove any unbound enzyme molecules from the PEG-coated glass surface.

Urease-Pump Experiments with Different Enzyme Coverage.

Streptavidin-functionalized patterns were incubated for 3–4 h with enzyme solutions of different concentrations that were prepared previously by serial dilutions from a stock enzyme solution. The enzyme concentrations prepared were as follows: 6.0 × 10⁻⁶ M, 3.0 × 10⁻⁶ M, 3.0 × 10⁻⁷ M, 3.0 × 10⁻⁸ M, 3.0 × 10⁻⁹ M, 3.0 × 10⁻¹⁰ M, 3.0 × 10⁻¹² M, 3.0 × 10⁻¹³ M, 3.0 × 10⁻¹⁵ M, 3.0 × 10⁻¹⁶ M, 3.0 × 10⁻¹⁸ M, 3.0 × 10⁻¹⁹ M, and 3.0 × 10⁻²⁰ M. Each solution was used to incubate one functionalized pattern. After enzyme immobilization, the patterns were washed with 10 mM PBS buffer, and secure-seal hybridization chambers (Electron Microscopy Sciences) with dimensions of 20 mm diameter and 1.3 mm height were used to form a closed-system device, to which 500 mM urea in 10 mM PBS buffer solution was added. To monitor the fluid flow in all our experiments, sulfate-functionalized polystyrene microspheres (Polysciences Inc.), 2 µm in size, were introduced as tracers suspended in the substrate solution. Movies were captured using an optical setup composed of an inverted microscope (Zeiss Axiovert 200 MAT) with a halogen lamp (12 V max, 100 W). Excitation light was focused into the sample through a 20× objective (EC Epiplan-NEOFLUAR 20×/0.55 HD DIC ∞/0; Zeiss). Emission light was collected by the objective, passed through interference filters, and finally detected by a high-sensitivity Flea 3 USB 3 digital camera (FL3-U3-32S2C; Point Gray Research), with a resolution of 2,080 × 1,552 pixels at 60 frames per second. One-minute movies were recorded using this camera, attached to the optical microscope. For the experiments comparing pumping speeds as a function of enzyme coverage, movies were recorded at a distance of 50 μm from the enzyme pump at four different locations around it. The reported pumping speeds correspond to the speeds obtained after 20 min. To compare pumping speeds as a function of time for intermediate enzyme coverage, movies were taken every 5 min at a distance of 50 μm away from the enzyme pump, focusing on one side of the patch only. For the spatiotemporal analysis of the pumping behavior of urease pumps at low enzyme coverage, 1-min movies were taken 50 μm away from the patch surface on four sides to compare speeds and fluid direction, thereby ruling out extraneous flows. The microscope was then focused on one side of the patch, and 1-min movies were taken close to the pump and at increasing distances (1 mm apart) away from the Au patch until the wall was reached. The whole process was then repeated five times to monitor the behavior at different time intervals. To measure fluid pumping speed in each experiment, 30 tracer particles were tracked for time intervals of 12–46 s, depending on the system, using Tracker, a video analysis and modeling software (Open Source Physics and comPADRE). Each experiment was repeated five times to ensure repeatability.

Supplementary Analysis of Theoretical Model.

The fundamental origin of the observed time-dependent flow structure is a competition between substrate- and product-driven flows. As the reaction proceeds, substrate is locally depleted at the enzyme patch, lowering the fluid density and driving an inward convective flow. Concurrently, the reaction product is formed at the enzyme patch, increasing the density of the solution and driving convection outward. The net flow is determined by the balance between these two opposing effects and depends on differences in the diffusivities and expansion coefficients between the substrate and product species. We now analyze the model to uncover qualitative features of the pumping behavior and, moreover, determine conditions on the diffusivities and expansion coefficients that lead to reversal of pumping direction over time.

The full system of equations for the evolution of the chemical fields (Eq. 3) and fluid flow (Eq. 4) cannot be solved analytically. We therefore use numerical simulations to obtain the solutions in the most general cases. To provide a qualitative understanding of the possible outcomes and predict when flow reversal is possible, we consider a limiting case for which the problem becomes mathematically tractable; namely, we neglect advection terms in the transport equation for the chemicals (substrate and product). This simplification is reasonable when flow speeds are low, corresponding to pumping at low reaction rates. With this approximation, our theoretical analysis proceeds as follows:

• i)We develop a simple model yielding expressions for the evolution of chemical concentration fields and the fluid density field.
• ii)We use the fundamental solution of the Stokes flow equations to formulate a relationship between horizontal gradients in the fluid density and the speed of flows driven by buoyancy forces.
• iii)We combine the qualitative features of the density evolution obtained in part i with the result of part ii to predict the structure of the flow field generated by the pump.
• iv)We discuss the scaling properties and dependence of the flow field on geometrical parameters, defining an empirically derived characteristic flow speed scale and Péclet number.

Theoretical analysis of density evolution in unbounded domains.

To obtain a simple expression for the density field, we consider the patch of enzyme to act as a point sink for the substrate and a point source for the product. If the chamber height H is small compared with its half width L, then we can neglect variations in the vertical (y) direction and consider a 2D problem for the concentration fields. Neglecting the advection terms in Eq. 4, we simplify the equations governing the evolution of substrate and product concentrations to

$$\begin{array}{l}{\partial }_{t}S={\nabla }_{\left|\right|}^{2}S-4\pi {\delta }^{\text{Dirac}},\\ {\partial }_{t}P=\delta \hspace{0.17em}{\nabla }_{\left|\right|}^{2}P+4\pi {\delta }^{\text{Dirac}},\end{array}$$ [S1]

where ${\nabla }_{\left|\right|}^2={\partial }^2/\partial x^2+{\partial }^2/\partial z^2$ and ${\delta }^{\text{Dirac}}$ denotes the Dirac delta function. The factors of $4\pi$ reflect the total sink and source strength for a circular patch of unit radius. Because the chamber half width L is assumed to be large compared with its height H, we may consider the domain to be unbounded in the x and z directions. The latter simplification is valid far from the chamber walls at early times (before significant diffusion from the point source to the walls takes place). Because the unbounded system has radial symmetry, we define $r=\sqrt{x^2+z^2}$ as the radial distance from the source. Solving the diffusion equations yields expressions for the changes in substrate and product concentrations (with respect to initial values),

$$\Delta S\left(r,t\right)=-E_1\left(\xi \right),\Delta P\left(r,t\right)=\frac{1}{\delta }{E}_1\left(\frac{\xi }{\delta }\right),$$ [S2]

where $\xi =r^2/4t$ and $E_1\left(\xi \right)={\displaystyle {\int }_0^1{s}^{-1}\exp \left(-\xi /s\right)ds}$ denotes the exponential integral function. The function $E_1$ decreases monotonically to zero, which is as expected for the concentration of a substance diffusing from a point source. Note that $E_1\left(\xi \right)$ diverges at $\xi \to 0$. This unphysical behavior is due to assuming that the reaction occurs at a single point in space; this approximation is accurate only far from the source.

Using the concentration profiles in Eq. S2, the nondimensionalized density change is

$$\Delta \rho \left(r,t\right)=\Delta S+\beta \Delta P=\frac{\beta }{\delta }\cdot E_1\left(\frac{\xi }{\delta }\right)-E_1\left(\xi \right).$$ [S3]

The plots in Fig. S2 show the spatial distribution of the concentrations $\Delta S$ and $\Delta P$ and fluid density $\Delta \rho$ for various values of $\delta$ and $\beta$ obtained using Eq. S3. Note that $\Delta \rho$ needs not be monotonic because $\Delta S$ and $\Delta P$ are of opposite signs. If there is a local extremum in the density as a function of distance, then $\left(d/d\xi \right)\Delta \rho =0$ at this point. The solution to this condition is ${\xi }_{\text{peak}}=\ln \left(\delta /\beta \right)/\left(1-1/\delta \right)$. Assuming (as is the case for urease pumps) that $\delta >1$, a physically valid solution exists only if $\delta >\beta$. This is consistent with the examples shown in Fig. S2 B–D. The fluid density attains a maximum (peak) in a ring around the enzyme. The radius of this ring expands with time, as described by

$$r_{\mathrm{peak}}=2\sqrt{{\xi }_{\mathrm{peak}}t},{\xi }_{\mathrm{peak}}=\frac{\ln \left(\delta /\beta \right)}{1-1/\delta }.$$ [S4]

Within the ring, at distances $r<r_{\text{peak}}$, the fluid density increases with r. Outside the ring, the fluid density decreases with r. If the condition $\delta >\beta$ is not satisfied, then the fluid density is greatest at the enzyme patch ($r=0$) and decreases monotonically with distance. Similarly, if $\delta <1$, then a local extremum occurs (also described by Eq. S4) if $\delta <\beta$ but this is a local minimum (trough).

The peak expansion speed ${\xi }_{\text{peak}}$ is largest for high values of $\delta /\beta$ and for values of $\delta$ close to 1. Notably, ${\xi }_{\text{peak}}\to 0$ as $\delta /\beta \to 1$ so it is possible for changes in the density field to occur on time scales much slower than the diffusion time.

Flow driven by horizontal density gradients.

For sufficiently slow fluid flows, the Navier–Stokes equations (Eq. 4) reduce to the equations of Stokes flow,

$$\begin{array}{l}\nabla p-{\nabla }^2\mathbf{u}=-\text{Ra}\cdot \Delta \rho \hspace{0.17em}{\mathbf{e}}_y,\\ \nabla \cdot \mathbf{u}=0,\end{array}$$ [S5]

driven by a density field $\Delta \rho$ that is decoupled from the flow field. Due to the linearity of the Stokes flow equations, the flow field can be written as a linear superposition of contributions from forces acting in the fluid. Consider an arbitrary force $\mathbf{F}$ acting on the fluid at the point ${\mathbf{x}}_0$. In the Stokes flow regime, the flow field due to such a point force can be written in the form $\mathbf{u}\left(\mathbf{x}\right)=\mathbf{F}\cdot \mathbf{J}(\mathbf{x},{\mathbf{x}}_0)$, where Green’s function $\mathbf{J}(\mathbf{x},{\mathbf{x}}_0)$ is a second-rank tensor field. The pressure field can also be expressed in a similar manner.

Green’s function for flow between infinite, parallel plates with no-slip boundary conditions was determined by Liron and Mochon (20). The solution was given in the form of a lengthy expression involving infinite series and the far field behaviors of different components of the tensor field were discussed. In our system, the walls are horizontal and only vertical forces due to gravity are present so the flow is radially symmetric around the vertical (y) axis. In Fig. S1, we plot the flow field due to a downward point-like force between parallel, horizontal walls. From Fig. S1, it can be seen that the force drives outward radial flow, i.e., away from the origin along the bottom surface and toward the origin along the top surface. Note that horizontal (radial) flows are generated despite the absence of horizontal forces. This is because horizontal and vertical flows are coupled through the incompressibility constraint, $\nabla \cdot \mathbf{u}=0$. For flow due to a vertical point force $\mathbf{F}=F_y\hspace{0.17em}{\mathbf{e}}_y$, we can write the velocity field as a sum of horizontal and vertical components,

$$\mathbf{u}\left(\mathbf{x}\right)={\mathbf{u}}_{\left|\right|}\left(\mathbf{x}\right)+{\mathbf{u}}_{\perp }\left(\mathbf{x}\right),\hspace{1em}{\mathbf{u}}_{\left|\right|}\left(\mathbf{x}\right)=F_y J_r\left(\mathbf{x},{\mathbf{x}}_0\right){\mathbf{e}}_r,\hspace{1em}{\mathbf{u}}_{\perp }\left(\mathbf{x}\right)=F_y J_y\left(\mathbf{x},{\mathbf{x}}_0\right){\mathbf{e}}_y,$$ [S6]

where ${\mathbf{e}}_r={\left(x-x_0,0,z-z_0\right)}^{\mathrm{T}}/\Vert {\left(x-x_0,0,z-z_0\right)}^{\mathrm{T}}\Vert$ is the unit vector in the radial direction. For the purpose of pumping fluid horizontally, we are interested only in the radial component of the flow. This component of flow decays exponentially with the radial distance $r=\sqrt{{\left(x-x_0\right)}^2+{\left(z-z_0\right)}^2}$ from the force (20).

The flow due to a continuous distribution of buoyancy forces, $\mathbf{f}\left(\mathbf{x}\right)=-\text{Ra}\hspace{0.17em}\Delta \rho \left(\mathbf{x}\right)\hspace{0.17em}{\mathbf{e}}_y$, is the convolution of this force field with Green’s function; i.e., $\mathbf{u}\left(\mathbf{x}\right)={\displaystyle \int \mathbf{f}\left({\mathbf{x}}_0\right)\cdot \mathbf{J}\left(\mathbf{x},{\mathbf{x}}_0\right)}\hspace{0.17em}\text{d}{\mathbf{x}}_0$. For purely vertical force distributions, the component of this flow field parallel to the walls is

$${\mathbf{u}}_{\left|\right|}\left(\mathbf{x}\right)={\displaystyle \int f_y\left({\mathbf{x}}_0\right)\hspace{0.17em}{J}_r\left(\mathbf{x},{\mathbf{x}}_0\right)}{\mathbf{e}}_r\hspace{0.17em}\text{d}{\mathbf{x}}_0.$$ [S7]

In the preceding section on the theoretical analysis of the density field evolution, we assumed that density variations in the vertical direction were negligible in a shallow enclosure. This assumption leads to the approximation $f_y\left({\mathbf{x}}_0\right)\approx f_y\left(x_0,z_0\right)$. The exponentially decaying nature of $J_r$ with distance means that only forces close to the evaluation point $\mathbf{x}$ are important to the local flow field. In the vicinity of $\mathbf{x}$, it is reasonable to approximate the force field with its first-order Taylor expansion, $f_y\left({\mathbf{x}}_0\right)\approx f_y\left(x,z\right)+\left(x_0-x\right)\left(\partial f_y/\partial x\right)\left(x,z\right)+\left(z_0-z\right)\left(\partial f_y/\partial z\right)\left(x,z\right)$. Substituting this into Eq. S7, we obtain

$${\mathbf{u}}_{\left|\right|}\left(\mathbf{x}\right)=f_y\left(\mathbf{x}\right){\displaystyle \int J_r\left(\mathbf{x},{\mathbf{x}}_0\right)}{\mathbf{e}}_r\hspace{0.17em}\mathrm{d}{\mathbf{x}}_0-\left({\displaystyle \int J_r\left(\mathbf{x},{\mathbf{x}}_0\right)}{\mathbf{e}}_r\hspace{0.17em}\otimes \left(\mathbf{x}-{\mathbf{x}}_0\right)\mathrm{d}{\mathbf{x}}_0\right){\nabla }_{\left|\right|}{f}_y\left(\mathbf{x}\right),$$ [S8]

where ${\nabla }_{\left|\right|}={\left(\partial /\partial x,0,\partial /\partial z\right)}^{\text{T}}$. Due to the radial symmetry of $J_r$, the first integral on the right-hand side of Eq. S8 vanishes whereas the second integral (in parentheses) is a diagonal matrix with diagonal elements that are, in general, finite. Hence, the horizontal components of the flow field can be approximately related to the local gradient of the density field, ${\mathbf{u}}_{\left|\right|}\left(\mathbf{x}\right)\sim {\nabla }_{\left|\right|}{f}_y\left(\mathbf{x}\right)\sim {\nabla }_{\left|\right|}\hspace{0.17em}\Delta \rho ={\nabla }_{\left|\right|} \rho$. The same scaling should apply to the derived quantities, $u_{\text{out}}$ and ${\overline{u}}_{\text{out}}$ (defined in Eqs. 5 and 6, respectively).

In Fig. S3, we compare the evolution of the vertically averaged radial flow field obtained from 3D simulation of the Navier–Stokes–Boussinesq equations with the vertically averaged density field and the approximate density field predicted by the point source model. There is a good correlation among these three quantities, verifying our theoretical analysis and the approximate result, ${\overline{u}}_{\text{out}}\sim -\left(\partial \rho /\partial r\right)$.

Theoretical predictions for pumping flow directions.

From the results of the preceding two sections, it follows that the pumping flow direction changes with distance from the pump if $\delta >1$ and $\delta >\beta$ (alternatively, if $\delta <1$ and $\delta <\beta$). The position of the transition point $r_{\text{peak}}$ evolves with time according to Eq. S4. At distances $r<r_{\text{peak}}$, the fluid flow is in the inward direction and at distances $r>r_{\text{peak}}$, flow is in the outward direction (alternatively, outward for $r<r_{\text{peak}}$ and inward for $r>r_{\text{peak}}$). Note that the density peak (and consequently the flow transition point) is not accurately described by Eq. S4 close to the pump or close to the side walls of the chamber. Deviations can be noted by comparing Fig. S3A (simulations in large domain and theoretical model in unbounded domain) with Fig. 4 (simulations in relatively small domain).

Flow field scaling.

For a given enzyme pump, dimensional analysis shows that the (dimensionless) flow field can be expressed as

$$\mathbf{u}=\text{Ra}\hspace{0.17em}\mathbf{U}\left(h/H,H/R,R/L\right),$$ [S9]

where the time-dependent vector field $\mathbf{U}$ characterizes the flow pattern and depends only on geometrical parameters of the system, namely, h/H, H/R, and R/L, and the constant Ra is defined in the main text as $\text{Ra}={\rho }_{0}g\hspace{0.17em}{\beta }_S{R}^{5}f\hspace{0.17em}h{\left(\mu \hspace{0.17em}{D}_S^{2}H\right)}^{-1}$. A similar formulation was considered by Sengupta et al. (2). Guided by the experimental pump geometry, we specifically consider the regime in which $h/H<<1$, $H/R<1$, and $H/L<<1$. At early times (before diffusion reaches the sides of the chambers), simulations with a range of parameter values in this regime demonstrated the scaling $\left|\mathbf{U}\right|\sim {(H/R)}^3$. This motivated the definition of a characteristic dimensionless flow speed,

$$U*={10}^{-4}\hspace{0.17em}{\left(H/R\right)}^3\text{Ra}={10}^{-4}{\rho }_{0}g\hspace{0.17em}{\beta }_S{H}^2{R}^{2}f\hspace{0.17em}h{\left(\mu \hspace{0.17em}{D}_S^2\right)}^{-1}.$$ [S10]

The numerical factor of ${10}^{-4}$ in this definition was empirically chosen to rescale the simulated flow fields to order unity. All flow speeds obtained by simulations presented in figures in this article are rescaled by U*.

We emphasize that U* is a generic flow velocity scale that does not reveal information about the spatiotemporal evolution of the flow nor indicate the dependence of flow speeds on $\delta$ and $\beta$. Hence, to define a characteristic Péclet number for a particular pump, we perform a simulation without chemical advection and determine the maximum vertically averaged radial flow speed, which we denote ${\text{Pe}}^{\left(0\right)}={\max }_{x,z,t}\left({\overline{u}}_{\text{out}}\right)$. The Rayleigh number ${\text{Ra}}^{\left(0\right)}$ used for this simulation is arbitrary because in the absence of advection, the flow speed is proportional to the Rayleigh number. Because we used the velocity scale $u*=D_S/R$ to nondimensionalize the model equations, the dimensionless speed ${\text{Pe}}^{\left(0\right)}$ is, in fact, the Péclet number associated with the substrate diffusivity at the length scale set by the patch radius R. For subsequent simulations, with chemical advection included, we define the characteristic Péclet number

$$\text{Pe}={\text{Pe}}^{\left(0\right)}\hspace{0.17em}\text{Ra}/{\text{Ra}}^{\left(0\right)}.$$ [S11]

This definition of Pe is used to describe simulations presented in Fig. 5.

Numerical Parameters.

Simulations of the pump setup shown in Fig. 2 were carried out on a 3D computational grid with typical (e.g., for results in Fig. 4) grid spacing $\Delta x=R/32$ and overall size $320\Delta x\times 16\Delta x\times 320\Delta x$. Convergence tests were performed to confirm that grid refinement produced negligible differences in flow and concentration fields. The radius and height of the circular enzyme patch was $R=32\Delta x$ and $h=\Delta x$, respectively, reflecting the thin surface layer to which the enzymes attach.

Quantitative Analysis of the Density Differences Between Reactants and Products in the Urease–Urea System.

The existence of a solutal density gradient in the urease–urea system was confirmed by measuring quantitatively the density difference between solutions containing urea and urease, freshly prepared and after reaction (2 d after preparation). The substrate concentration was 500 mM in 10 mM PBS buffer. The enzyme concentration was the lowest enzyme concentration used for pattern soaking for which the fluid flow direction was still outward ([E] = 3.0 × 10⁻⁷ M; high enzyme coverage). The values obtained are specified in Table S5. Taking the density of pure water to be ${\rho }_{\text{water}}=0.998\hspace{0.5em}\text{g}/\text{mL}$, we then estimate the expansion coefficient ratio as $\beta =\left({\rho }_{\text{final}}-{\rho }_{\text{water}}\right)/\left({\rho }_{\text{initial}}-{\rho }_{\text{water}}\right)\approx 1.19\pm 0.08$. This estimate treats the buffer, which is altered as the reaction proceeds, as part of the “substrate” and “product” solute.

Conclusions

We used a combination of theory and experiments to uncover the mechanism behind the remarkable spatiotemporal variations in pumping in urease-based pumps. Having ruled out several mechanisms, we formulated a model that accounts for buoyancy effects due to nonuniform concentrations of substrate and product that have different diffusivities and volumetric expansion coefficients. We find that the qualitative features of the flow depend on the ratios of diffusivities $\delta =D_P/D_S$ and expansion coefficients $\beta ={\beta }_P/{\beta }_S$ of the reaction substrate and product. If $\delta >1$ and $\delta >\beta$ (or if $\delta <1$ and $\delta <\beta$), then the flow direction reverses with time and distance from the pump. At any given position, the flow direction at later times can be opposite to that at earlier times, and within a certain range of time, the flow direction near the pump can be opposite to that far from the pump. Numerical simulations showed that chemical advection at large Péclet numbers could delay the reversal or otherwise alter the flow pattern. Experimental results clearly confirmed the validity of our model, providing an example where the flow near the pump is initially in the outward direction but changes to inward pumping over time. The region of inward pumping then gradually spreads farther out in space, replacing outward pumping in the entire system. These findings are unexpected and surprising as previous experiments with several enzyme micropumps have seen flow speeds only decay over time as substrate is consumed (2). Here, however, we show that the flow direction can change over tens of minutes.

Our work clearly illustrates that a simple chemical reaction of the form $S\to P$ catalyzed by a patch of immobilized enzymes can drive a complex, time-dependent flow. The balance between inward convective flow, driven by substrate depletion at the pump, and outward flow, driven by product formation at the pump, can result in flow reversal if the substrate and product have different diffusivities and different expansion coefficients satisfying the conditions above. As demonstrated in experiments, this flow reversal is observable even with small differences in diffusivity and expansion coefficients. We therefore expect that many other reaction systems can exhibit similar behavior.

We showed that, for low reaction rates, the flow induced by these enzymatic pumps is roughly proportional to the local horizontal gradients in fluid density and could therefore be predicted from a simple 2D model for the density evolution. Such a density field can be computed quickly, using fundamental solutions of the diffusion equation, whereas simulation of diffusion and advection in 3D domains is computationally demanding. Hence, our approach enables rapid evaluation of the behavior of new pump designs under different conditions. For example, flows due to multiple patches with differing enzyme loadings and arranged arbitrarily in the chamber can be predicted with our technique. Moreover, thermal effects can be incorporated in a similar manner, treating heat as one of the reaction products.

More accurate quantitative predictions of pumping can be achieved by extending the model to include the effects of substrate depletion on the rate of reaction (e.g., Michaelis–Menten kinetics) and cross-diffusion terms (the flux of a chemical species due to concentration gradients of another species; this flux can be significant in some systems) (21). Because the overall flow is sensitive to small changes in substrate and product concentration profiles, these nonlinear terms may affect the overall pumping behavior if the reaction rate is sufficiently high. We can also model more complex situations, such as oscillatory or sequential activation of pumps that could be engineered with multiple enzyme types interconnected by reaction pathways.

Materials and Methods

Numerical Methods.

The finite difference method was used to numerically solve the advection–diffusion–reaction equations governing concentration fields (Eq. 3), with second-order finite differences used for the Laplace operator and the first-order upwind scheme used for the advective term when we investigated the effects of finite Péclet number. The reaction term was implemented as a source contribution in the grid cells occupied by the patch. Because our model assumes a constant reaction rate, it is possible for substrate concentrations to become negative over the course of a simulation. We therefore verified in all simulations that the concentration changes remained small, justifying our assumption of constant reaction rates. The fluid flow was simulated using the lattice Boltzmann method (22, 23) with the D3Q19 scheme. The buoyancy force, defined in Eq. 4, was implemented as a body force acting at each grid point (24). Bounce-back rules were adopted to enforce the no-slip boundary conditions. This combination of finite difference and lattice Boltzmann methods has proved to be effective in modeling advection–diffusion processes (25, 26).

Biotinylation of Enzymes (27).

A urease [Canavalia ensiformis (Jack bean)] (Sigma Aldrich) solution (2 mg/mL) was prepared with 100 mM PBS as the buffer. Before completing the total volume of the enzyme solution, 10 µL of a 0.1-mM 3-(N-maleimidylpropionyl) biocytin (Santa Cruz Biotechnology) biotin solution was added per mL of enzyme solution to get a ratio of 4:1 enzyme:biotin. The mixture was stirred for 2 h at room temperature, followed by dialysis of the final solution and washes with 10 mM PBS (details in SI Materials and Methods). The concentrated enzyme solution (stored at 4 °C) was used later for the preparation of enzyme solutions of different concentrations, using serial dilutions.

Biotinylation of Au Patterns, Enzyme Immobilization, and Particle Tracking.

Au (90 nm) was patterned on a PEG-functionalized glass surface (MicroSurfaces), using an electron beam, with a 10-nm adhesion layer of Cr. Biotinylation of the circular Au patterns (radius = 3 mm) was achieved through the formation of a SAM, using a biotin–thiol linker (28) (preparation details in SI Materials and Methods). A solution of the linker in a solvent mixture of 1:1 H₂O:ethanol (8 mL of solution per milligram of biotin) was used for overnight incubation of the patterns at room temperature. After incubation, the patterns were washed several times with deionized water, followed by washes with 10 mM PBS buffer. The SAM-modified surfaces were then incubated in a streptavidin (ProZyme) solution (9 μM in 10 mM PBS) for 3 h, followed by incubation with enzyme–biotin solutions for 3–4 h before the experiments. The functionalized surfaces were washed with 10 mM PBS buffer after each incubation period to remove unbound material. Details of the experimental modifications of this procedure to control enzyme coverage are given in SI Materials and Methods.

Secure-seal hybridization chambers (Electron Microscopy Sciences) with dimensions of 20 mm diameter and 1.3 mm height were used to form closed-system devices, to which solutions of 500 mM urea in 10 mM PBS buffer solution were added. To monitor the fluid flow in all our experiments, 2 µm sulfate functionalized polystyrene microspheres (Polysciences Inc.) were introduced as tracers suspended in the substrate solution. Movies were captured using an optical setup composed of an inverted microscope (Zeiss Axiovert 200 MAT) with a halogen lamp (12 V max, 100 W) and a high-sensitivity Flea 3 USB 3 Digital Camera (FL3-U3-32S2C; Point Gray Research). To measure fluid pumping speed in each experiment, 30 tracer particles were tracked using Tracker (Open Source Physics and comPADRE). See SI Materials and Methods for tables of pumping speeds.