Measuring Fast and Slow Enzyme Kinetics in Stationary Droplets

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Measuring fast and slow enzyme kinetics in stationary droplets Etienne Fradet, Christopher D Bayer, Florian Hollfelder, and Charles N. Baroud Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.5b03567 • Publication Date (Web): 02 Nov 2015 Downloaded from http://pubs.acs.org on November 2, 2015

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Analytical Chemistry

Measuring fast and slow enzyme kinetics in stationary droplets Etienne Fradet,† Christopher Bayer,‡ Florian Hollfelder,‡ and Charles N. Baroud∗,† Laboratoire d’Hydrodynamique (LadHyX) and Department of Mechanics, Ecole Polytechnique, CNRS, 91128, Palaiseau, France., and Department of Biochemistry, University of Cambridge, 80 Tennis Court Road, Cambridge, UK CB2 1GA. E-mail: [email protected]

∗

To whom correspondence should be addressed Laboratoire d’Hydrodynamique (LadHyX) and Department of Mechanics, Ecole Polytechnique, CNRS, 91128, Palaiseau, France. ‡ Department of Biochemistry, University of Cambridge, 80 Tennis Court Road, Cambridge, UK CB2 1GA. †

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Abstract We present a new microfluidic platform for the study of enzymatic reactions using static droplets on demand. This allows us to monitor both fast and slow reactions with the same device and minute amounts of reagents. The droplets are produced and displaced using confinement gradients, which allows the experiments to be performed without having any mean flow of the external phase. Our device is used to produce six different pairs of drops, which are placed side by side in the same microfluidic chamber. A laser pulse is then used to trigger the fusion of each pair, thus initiating a chemcial reaction. Imaging is used to monitor the time evolution of enzymatic reactions. In the case of slow reactions, the reagents are completely mixed before any reaction is detected. This allows us to use standard Michaelis-Menten theory to analyze the time evolution. In the case of fast reactions, the time evolution takes place through a reaction-diffusion process, for which we develop a model that incorporates enzymatic reactions in the reaction terms. The theoretical predictions from this model are then compared to experiments in order to provide measurements of the chemical kinetics. The approach of producing droplets through confinement gradients and analyzing reactions within stationary drops provides an ultra-low consumption platform. The physical principles are simple and robust, which suggests that the platform can be automated to reach large throughput analyses of enzymes.


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Measuring the rate of biochemical reactions is a critical step in the experimental characterization of dynamic biological processes. Steady-state kinetics are widely used to characterise enzymes and benchmark them by measuring Michaelis-Menten parameters. 1–4 The number of enzymes to be characterised is steadily rising, as more enzymes are discovered in functional metagenomic studies 5,6 or by large scale sequencing efforts 7 followed by bioinformatic predictions. Databases that combine sequence, structure and functional data will play a large role in recording and mapping molecular biodiversity and have to be supplied with steady-state data. 8 Also rational protein design 9 and directed evolution 10 create large numbers of mutants that have to be quantitatively characterised to assess the success of the protein engineering approach. Finally mechanistic studies focus increasingly on the catalytic effects of interactions between networks of residues, requiring characterisation of substantial numbers of mutants. 11,12 Practical limitations of such investigations include the time resolution, the sample consumption and the detection limit of the experimental technique in use, as sketched in Fig. 1. Slow reactions are usually studied in titer plates, consuming typically 1-100 µL of sample per well and involving reaction times ranging from few tens of seconds up to few days. Fast and pre-steady state reactions are studied in stopped or quenched flow machines, in which the reagents are injected in a cuvette where turbulent flow ensures a good mixing within a few ms, which comes at the price of throughput as they allow for only one reaction to be studied at a time. In the last two decades, microfluidic devices have shown much promise as analytical tools for chemical and biochemical reactions, as their micrometric size ensures drastically reduced sample consumption, especially when droplets are used to compartmentalise reagents into femto-to-nanoliter volumes. 13 Fast chaotic mixing of the drop contents was achieved using winding channels, reducing the time resolution to a few ms at high injection rates. 14 While the residence time of moving droplets on chip is limited, formats exist that hold droplets stationary for long-term observation (e.g. parking lots 15,16 or dead-end channels 17 ), in wide


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using a standard flow focusing geometry39,42 and deposited in the integrated trap structures shown in Fig. 1. Two oil streams (from Feed 1) merge perpendicularly to the aqueous stream (from Feed 2). An oil stream is introduced from Feed 4 to prevent any droplets from entering the array during the initial stabilization stage. Once droplets are of uniform size, the oil flow from Feed 4 is reversed resulting in the entrapment of droplets in the array. A rapid channel expansion at the entrance of the array ensures a fast and homogenous proliferation of droplets into the trap array. To ensure that only one droplet enters each trap an exhaust channel was structured through the centre (Fig. 2). Accordingly, when a trap is empty, oil is able to flow through this exhaust channel. However, as soon as a droplet is captured it blocks the exit resulting in the termination of liquid flow within the trap and preventing a second droplet from entering the trap. Since most droplets (>90%) pass through the array chamber without being trapped, the structure is ideal for randomly sampling a defined subset of a large droplet population. This concept is analogous to the approach reported by DiCarlo et al. for trapping populations of single cells in a microfluidic array albeit without using emulsion droplets.7 Once each array is filled with droplets (as shown in Fig. 2), all inlets except Feed 1 are closed to maintain this arrangement. This procedure ensures reliable control of oil flow from Inlet 1 throughout the trap array into Feed 4. After the trapped droplets have been incubated for a given period of time they can be released by supplying a fresh oil flow from Feed 4. After flow reversal, droplets occasionally got stuck at the rear side of the adjacent trap. Therefore the shape of the trap backside (characterized by the angle alpha, shown in Fig. 2A) was slightly

Time resolution Single drops

Array of drops

Cell encapsulation Immobilizing aqueous droplets in the described manner allows facile monitoring of encapsulated cells over extended periods of time. In the current studies bacterial cells (1 to 2 mm in diameter) were encapsulated within such trapped droplets, and their interactions and movement were monitored over a period of 15 minutes. For easy visualization E. coli cells expressing red fluorescent protein (from plasmid mRFP1)41 were encapsulated within droplets. Fig. 3 illustrates a trapped droplet containing seven cells. It can be seen that the encapsulated cells begin to aggregate after a few minutes. Control experiments with fewer cells per droplet (two to three cells) and fluorescent beads did not show any aggregation events (data not shown). These observations suggest that aggregation is favoured at higher densities of E. coli cells and is likely a result of oxygen depletion. This effect has been described previously in the literature.43 The combination of the ability to tailor the number of cells in a droplet by dilution36 or to change variables such as pH and salt concentration with the ability to subsequently monitor a cell’s response will be highly desirable in facilitating an understanding of basic phenomena at the single cell level and the behaviour of populations of cells.30 The encapsulation of single cells has been demonstrated previously,18,29,44 but here we show that such a set-up allows information about the morphological behaviour of a population to be obtained simultaneously with the optical readout from each single bacterial cell.


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Droplet stability and long-term droplet storage When conducting reactions in a defined reactor it is important to be able to maintain the reaction volume or to be able to

Microtiter plates

Anal. Chem. 1999, 71, 5340-5347

Quantitative Analysis of Molecular Interaction in a Microfluidic Channel: The T-Sensor Andrew Evan Kamholz,*,† Bernhard H. Weigl,†,‡ Bruce A. Finlayson,§ and Paul Yager|

Department of Bioengineering and Department of Chemical Engineering, University of Washington, Seattle Washington 98195

10 s

The T-sensor is a recently developed microfluidic chemical measurement device that exploits the low Reynolds Fig. 2 Droplet trapping arrays. (A) Design of an individual trap (withinamicrofabricated scale bar indicating the dimensions) The rear of the trap was drawn to give number flow conditions channels. The rate of droplets that couldchemical be retrieved from an of array to above 90%. (B) Schematic flow profile for a 110� angle (a). This improved the percentage interdiffusion and resulting interaction comdroplets approaching a trap. Liquid enters through the opening in the the trap. Schematic ponents from two or centre more of input fluid(C) streams canflow be profile when a droplet is residing in the trap. monitored optically, allowing of analyte As soon as a trap is occupied by a droplet, additional droplets pass outside themeasurement trap. This mechanism ensures that only a single droplet is trapped per concentrations on a continuous basis. In a simple form feature. (D) Image of an array containing uniformly trapped aqueous droplets. (E)–(G) Brightfield images of droplet release over a 2 second time period. T-sensor, concentration a target analyteAqueous is The direction of flow is indicated by the whiteof arrows. Figuresthe A–C are not drawn of to scale. Conditions: droplets were formed from buffer (PBS determined by measuring fluorescence intensity in a at pH 7.4) in light mineral oil and 1.5% (w/w) Span 80. Scale bars indicated in the bottom right corner of each image correspond to a distance of 75 mm. region where the analyte and a fluorescent indicator have interdiffused. An analytical model has been developed that 694 | Lab Chip, 2009, 9, 692–698 journal is ª The Royal Society of Chemistry 2009 predicts device behavior from the diffusionThis coefficients of the analyte, indicator, and analyte-indicator complex and from the kinetics of the complex formation. Diffusion coefficients depend on the local viscosity which, in turn, depends on local concentrations of all analytes. These relationships, as well as reaction equilibria, are often unknown. A rapid method for determining these unknown parameters by interpreting T-sensor experiments through the model is presented.

Train of drops

T sensor

Stopped-flow

Microfluidics is becoming a prevalent tool for a broad range of applications that include cell separations,1 flow injection reaction analysis,2,3 cell patterning,4 DNA analysis,5 and cell manipulation.6 The T-sensor is a microfabricated fluidic device that, due to its small dimensions and typically low volumetric flow rate, is generally operated at Reynolds numbers of less than 1.7 Inspired in part by the concept of field-flow fractionation,8 its simplest embodiment (Figure 1a) involves two fluids entering through separate inlet ports and merging to flow adjacently. The microscale conditions induce laminar flow, meaning that there is no convective mixing across the two input streams. Thus, small molecules that can diffuse significant distances during the average residence

1 ms 10 nL

* Corresponding author: (e-mail) [email protected]; (fax) (206) 616 1984. † Department of Bioengineering, Box 352141. ‡ Present address: Micronics, Inc., 8717 148th Ave. NE, Redmond WA 98052. § Department of Chemical Engineering, Box 351750. | Department of Bioengineering, Box 352255. (1) Yang, J.; Huang, Y.; Wang, X. B.; Becker, F. F.; Gascoyne, P. R. Anal. Chem. 1999, 71, 911-8. (2) Hodder, P. S.; Blankenstein, G.; Ruzicka, J. Analyst 1997, 122, 883-7. (3) Bo ¨kenkamp, D.; Desai, A.; Yang, X.; Tai, Y.; Marzluff, E.; Mayo, S. Anal. Chem. 1998, 70, 232-6. (4) Kenis, P. J.; Ismagilov, R. F.; Whitesides, G. M. Science 1999, 285, 83-5. (5) Simpson, P. C.; Roach, D.; Woolley, A. T.; Thorsen, T.; Johnston, R.; Sensabaugh, G. F.; Mathies, R. A. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 2256-61. (6) Li, P. C.; Harrison, D. J. Anal. Chem. 1997, 69, 1564-8. (7) Weigl, B. H.; Holl, M. R.; Schutte, D.; Brody, J. P.; Yager, P. Analytical Methods and Instrumentation, MicroTAS 96 special edition, 1996. (8) Giddings, J. C. Science 1993, 260, 1456-65.

Figure 1. (a) Photograph of a silicon microfabricated device. For operation as a T-sensor, two inputs and one output are used. Both outputs are active when the device is used as an H-filter.33 (b) Schematic representation of flow in the T-sensor with two input fluids, each containing one diffusing species. The flow is steady state, projecting the interdiffusion along the length of the channel. The asymmetric development of the interdiffusion region (relative to the center of the channel at 1/2d) is due to the difference in diffusion coefficients between the two diffusing species.

1 µL

5340 Analytical Chemistry, Vol. 71, No. 23, December 1, 1999

time of the device will redistribute between streams (Figure 1b). Large molecules or particles that do not diffuse significantly during the same interval will not move appreciably from their original stream unless an external field is applied (such as gravity or an electric field). The critical dimension that governs the extent of 10.1021/ac990504j CCC: $18.00

100 µL

Sample consumption

© 1999 American Chemical Society Published on Web 10/28/1999

Figure 1: Approaches to measurement of kinetic data and their timescales. Currently most kinetic studies are carried out in microtiter plates (often involving manual pipetting or robotic liquid handling systems). Sample consumption can be brought down 1 µL with this system while the observed rections last at least 10 s. For faster reactions, stopped-flow machines are used with dead-times down to 1 ms and sample consumption of a 100 µL per run. The last decade has brought microfluidic approaches to the fore. First, reaction-diffusion patterns were used to study fast chemical reaction using minute amounts of reagents, thereby bringing the sample consumption down to 10 µL. Next, droplets were suggested as chemical vessels, either arrange along a train of drops for fast reactions, or trapped against obstacles for longterm observations. In both cases, drops allowed to bring the sample consumption down to 10 nL, or smaller depending on the drop size. This work introduces a set-up for kinetic measurements in which slow and fast reactions can be monitored using minimal amounts of reagents, only one drop of each reagent. microchannels by placing obstacles on the path of droplets, 18 or by etching pockets in the channel roof to anchor them. 19 The sequence of droplets can be controlled in organized 2D arrays of droplets, 20,21 or by droplet-on-demand systems. 22,23 Different microfluidic approaches have been developed to achieve the objectives of ultra-low sample consumption per reaction, multiplexing of parallel reactions on chip, or scan different reaction timescales. Scaling down the sample size is not trivial, as the dominant physical processes governing flows at the micrometer scale differ greatly from macroscale flows: Interfacial effects and viscosity dominate in smaller formats, and mixing by turbulence is difficult to achieve. An


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t=0s Separated reagents

t ~ 100 ms Flow

a)

Diffusion

b) 0

t ~ 1 min Well-mixed

c)

t

d)

x t < 0s

500 µm

t = 2 ms

t=1s

Fast reactions Reaction-diffusion Initially separated reagents No-flux boundary conditions

t = 250 s

Slow reactions Standard kinetics

Figure 2: Timeline of the fusion and mixing of two drops : The two regimes for the analysis of a reaction with initially separated reagents. a) Two equally-sized droplets, one containing dye, are stored in our device using capillary traps. The trap is the goggle like pattern where the height of the channel is slightly increased. After fusion, the trap still holds the merged droplet in place. The axis labelled x denotes the distance from the original interface between the two resting droplets. An x-value of zero refers to the position of the original interface. b) Just after fusion, a strong flow takes place but the dye remains well separated from pure water. c) After the flow has abated the dye diffuses from high to low concentrations. Reactions faster than the diffusion time of the front are therefore modeled using a reactiondiffusion model along the droplet. d) Once the diffusion front has reach the edge of the drop, the merged drop becomes well-mixed. Reactions taking place on time scales much longer than the diffusion time can therefore be modeled using standard kinetics. alternative is to leverage the dominant phenomena and rely on diffusion of species, rather on active mixing, to allow the reactions to take place. 24 In the present work we demonstrate a new set-up for measuring enzymatic steady state (Michaelis-Menten) kinetics and outline the formal treatment of the data emerging from this system. Reactions are initiated by fusion of adjacent droplets and reaction progress takes place via reaction and diffusion. As shown in Fig. 2, the processes that must be modeled will depend on the rate of the reaction compared with the rate at which the molecules diffuse within the droplets. Therefore by adapting the analysis method to the type of reaction, the platform that we present here can be used to study both slow and very fast reactions without modification of the experimental protocols. Below we demonstrate the device and show how to obtain kinetics data for both the slow and fast reaction regimes.



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Experimental setup a)

B3

B4

B2 B1

B5

A

B6

oil

b)

exit

A

B1 B2 B3 B4 B5 B6

h (µm)

c)

180 160 140 120 100 80 60 40 20 0

Figure 3: (a) Global design of the device, showing the central chamber connected to the nine microchannels: One input for the oil, seven inputs for the aqueous reagents, and one exit channel. (b) Topography of the rectangular test section (4 × 15 mm wide). Height modulations are etched into the ceiling to produce, transport, and anchor droplets. The arrows correspond to the different fluids connected to the device. (c) Color micrograph of six pairs of droplets paired with a common (light blue) sample. Each droplet has a diameter of nearly 500 µm. The microfluidic device has been described previously, 22 where it was used to measure the evolution of a simple chemical reaction. It consists of a rectangular test section that is connected to nine microchannels for the different fluids, as shown in Fig. 3a: one channel serves as an input for the external oil phase, seven channels serve to introduce the aqueous reagents, and one channel is used as an exit. The ceiling of the test section is patterned with topographical features that determine the device functionality (Fig. 3b for dimensions). These features allow the production, transport, and pairing of droplets of the aqueous reagents in a passive manner, once the chamber is filled with oil. These “anchors and rails” 19 therefore 6 ACS Paragon Plus Environment

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allow the droplet operations to take place without relying on the flow of the external oil phase, which is at rest for most of the experimental procedure. The droplets are first formed, at the junction between the inlet channels and the main chamber, by step emulsification. 25 Since the step to inlet height ratio determines the drop size, 26 the value of the height ratio is fixed at 0.5 for all of the reagent channels, in order to obtain drops of equal sizes (diameter ' 500 µm, volume ' 30 nl). To propel a drop once it is formed, a V-shaped rail, with larger depth than the rest of the chamber, is placed directly in front of each junction. As a result, the droplet spontaneously moves toward the wider part of the groove as it experiences a gradual deconfinement, in order to lower its surface energy. 27 In this way, droplets of type B are produced and migrate until they meet the circular anchor where they become trapped. Droplets of type A, on the other hand, reach a flat region once produced. For this reason they need to be pushed by an oil flow to reach the bottom row of the array section. The device priming protocol is demonstrated in SI movie 1. In some cases a laser is used to help guide a particular drop into a trap, as described previously. 20 While this involves some manual control currently, the anchors can be aligned diagonally in future versions of the device, in order to force the drops into them passively. Altogether, the device allows us to form six pairs of droplets in an array format. As shown on Fig. 3c, the droplets from the inlet A are all formed from the same solution and are paired with six droplets coming from six different solutions B1-B6. The physical principles behind the design are not limiting and the number of pairs can in principle be increased indefinitely, since the interactions between droplets are very weak. 19 However the practical limitation is that each independent inlet must be connected to a different syringe and the instrumentation around the microfluidic device can become overwhelming. Such issues can be solved in principle by using multiplexed control from pressure sources or preloaded devices, but we have not implemented such protocols at this stage.



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Slow reaction: well-mixed analysis Before turning to the more complex case of fast reactions, as defined in Fig. 2, we set out to show that slow reactions can be measured in our devices. For this we used the enzyme β-Dglucosidase from sweet almond and the model substrate, 4-nitrophenyl β-D-glucopyranoside, as a test case. 28 The reaction is modeled by a standard Michaelis-Menten formalism, in which enzyme E binds to the substrate S and then produces the product P, kon

k

cat − * E+S− ) − − E · S −−→ E + P

(1)

koff

with kon , koff and kcat being the rate constants for association, dissociation and chemical catalysis, respectively. 29 We wish to measure the two constants kcat and KM = (kcat +koff )/kon for our reaction. Here the parallelized device provides a way to obtain six data points simultaneously on the same chip. As shown in Fig. 4, six droplets, containing the substrate at different concentration [S]0 , are first produced from six different inlets to fill the top row of our array. Then, droplets with a fixed enzyme concentration [E]0 are produced and pushed to the remaining sites using an outer oil flow. This outer flow is maintained to mix the droplet contents once they have merged and, after a few minutes, a clear difference in light absorption between the drops is observed (Fig. 4a). For each of the merged droplets, a progress curve was generated by measuring the optical absorption during 10 minutes, and converting this signal to a concentration of the product P, as shown in Fig. 4b. The dose response could thus be obtained by measuring the rate of the initial reaction V0 with increasing concentrations of the substrate [S]0 . We observed that an increase in [S]0 yielded a larger reaction rate (Fig. 4c). KM and kcat were obtained after determination of inital rates and by fitting these rates to the classic Michaelis-Menten model, which provided the values KM = 9.3 ± 3.1 mM and kcat = 0.7 ± 0.09 s−1 . These were in good agreement with plate reader measurements (KM = 9.0 ± 1.0 mM and kcat = 0.50 ± 8 ACS Paragon Plus Environment

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Figure 4: : Reaction kinetics of β-D-glucosidase a) Steady state assay: the enzyme concentration is [E]0 = 5 µM and the substrate concentration increases from left to right [S]0 = 0-31 mM. The gray levels are measured in the colored boxes. The scale bar is 500 µm. b) Time courses of the product concentration [P](t). The colored dots are extract from (a) and the solid lines are best fits with [P](t) = V0 t. c) Variations of the initial reaction rate with the substrate concentration. The colored dots are experimental data points and the solid black line a best fit on kcat and KM . d) Inhibition assay: the enzyme concentration is [E]0 = 5 µM, the substrate concentration is [S]0 = 23 mM and the inhibitor concentration is increased from left to right [D]0 = 0-1500 nM. The gray levels are measured in the colored boxes. The scale bar is 500 µm. e) Time courses of the product concentration [P](t). The values for V0 are extracted from d) and the solid lines are best fits with to the linear equation [P](t) = V0 t. f) The coloration indicates the corresponding concentrations. Variations of the initial reaction rate with the inhibition concentration. (f) The colored dots are experimental data points and the solid black line a best fit. 0.02 s−1 ), as well as with previous experiments by Gielen et al. 28 who obtained KM = 11.4 ± 2 mM and kcat = 0.9 ± 0.2 s−1 . In addition to measurement of the steady state parameters, KM and kcat , we also addressed the determination of the inhibition constant KI for the competitive inhibitor 1deoxynojirimycin hydrochloride (denoted D), whose action is described by the following 9 ACS Paragon Plus Environment


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reaction:

KI

kon

k

cat − * − * DE + S − ) − − E · S −−→ E + P ) − −E+S+D−

(2)

koff

In this experiment, droplets from the enzyme solution at a fixed concentration (10 µM) were produced and paired with droplets containing the substrate with increasing concentration of inhibitor [D]0 , while [S]0 was kept constant ([S]0 = 46 mM). Again a clear difference between the drops was observed after a few minutes as shown in Fig. 4d. The resulting initial rates V0 decreased as the inihibitor concentration increased (Fig. 4e), which yielded an redinhibitor concentration for a 50% reduction of activity IC50 = 180±58 nM and KI =51±16 nM, consistent with a plate reader measurement (IC50 = 240 ± 13 nM and KI =67 ± 4 nM) and previous measurements ( IC50 = 110 ± 40 µM and KI = 36 ± 13 nM). 28

Fast reaction: reaction-diffusion analysis In order to analyse fast reactions, we must re-visit the Michaelis-Menten formalism in the context reaction-diffusion systems. We begin by writing the reaction-diffusion equations, following Ristenpart et al., 30 then make simplifications that are relevant to our experiments in order to obtain an analytical approximation of the reaction evolution.

Michaelis-Menten kinetics for initially separated species In traditional experimental conditions, the substrate and enzyme are well mixed and the initial rate of the reaction V0 is measured under conditions where (i) the substrate concentration exceeds the enzyme concentration ([S]0 [E]0 ), (ii) the enzyme is not consumed by the reaction so that, at all times, [E]0 =[E]+[E·S], and (iii) the intermediate complex forms as fast as it degrades so that, at all times, [E][S]=KM [E·S]. By then writing the initial rate of the reaction as 10 ACS Paragon Plus Environment

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V0 =

kcat [E]0 [S]0 KM + [S]0

,

(3)

the two parameters (kcat ) and (KM ) are obtained from measurements of V0 at different values of [E]0 and [S]0 . Conversely, when the reagents E and S are initially separated, they must diffuse towards the region where they will encounter each other and react. As a result, the overall progress of the reaction will be determined by an interplay between reaction and diffusion. This replaces the ordinary differential equations (ODEs) that describe the well-mixed system by partial differential equations (PDEs) that are both functions of time and space. In our formalism we will keep only one spatial dimension, which corresponds to the axis labelled x on Fig. 2a. Indeed, the chemical species transport mostly takes place along this direction since it corresponds to the highest concentration gradients. We can therefore write this reaction-diffusion system as a set of PDEs in time t and one spatial direction x: ∂[E] ∂t ∂[S] ∂t ∂[E·S] ∂t ∂[P] ∂t

∂ 2 [E] − kon [E][S] + koff [E·S] + kcat [E·S] ∂x2 ∂ 2 [S] = DS − kon [E][S] + koff [E·S] ∂x2 ∂ 2 [E·S] = DE + kon [E][S] − koff [E·S] − kcat [E·S] ∂x2 ∂ 2 [P] = DS + kcat [E·S], ∂x2 = DE

(4a) (4b) (4c) (4d)

with DE and DS the diffusion coefficients of E and S. Here we have distinguished the large molecules (E and E·S) from the small ones (S and P), and assumed a common diffusion coefficient for each type: DE for both large species, and DS for the small. The size contrast between the enzyme molecules and their substrates and products justifies this simplification. This system of equations is subject to initial conditions on all concentrations, which we impose by setting the initial distributions of E and S as two Heaviside step functions H, i.e.



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[S]=1 for x < 0 and [S]=0 for x > 0, while [E]= 0 for x < 0 (Figure 2). In addition, the initial conditions for intermediate complex [E·S] and the product [P] are taken as zero for all positions x. This gives the following initial conditions for the problem:

[S](x, 0) = [S]0 H(x ≤ 0),

[E](x, 0) = [E]0 H(x ≥ 0),

[E·S](x, 0) = [P](x, 0) = 0.

(5)

The mathematical description of the different concentration fields is finally closed by assuming no-flux boundary conditions at the edges of the droplet x = ± l. The system of equations (4a)-(4d) is readily solved numerically using finite differences. Such a procedure however fails to provide physical insight into the underlying processes, in addition to being impractical for routine measurements. A complementary approach would be to derive an approximate analytical solution to the equations that reduces the analysis to a measurement of a single variable in time. This is what is done below by revisiting the assumptions that underly the standard Michaelis-Menten theory in a reaction-diffusion framework.

Analytical solution Comparing the experimental measurements with the theoretical model consists of comparing the profile of the reaction product as a function of space and time. This process is described by Eq. (4d), which is a diffusion equation for [P] with a source term that depends on [E·S]. By estimating this source term we can therefore provide an analytical approximation of [P](x, t), which we will do below. By applying the three Michaelis-Menten hypotheses at a local level, we can reduce the set of equations as follows: (i) When [S]0 [E]0 , we assume that the substrate distribution is nearly insensitive to the chemical reaction so that the reaction terms in the substrate mass balance (4b) can be neglected. Then, the substrate concentration field is solely given by its


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diffusion: [S]0 [S](x, t) = erfc 2

x √ 4DS t

,

(6)

with erfc the complementary error function. (ii) The overall conservation of the enzyme implies that the total amount of enzyme [E]tot = [E] + [E·S] is constant. Indeed, summing Eqs. (4a) and (4c) yields a diffusion equation on Etot , whose solution is [E]0 x . [E]tot (x, t) = [E](x, t) + [E·S](x, t) = erfc − √ 2 4DE t

(7)

(iii) Finally, we assume that the steady state situation that governs the formation and degradation of the intermediate complex E·S applies at every instant and position. This translates into a local equilibrium between the concentrations of the enzyme, the substrate and the intermediate complex:

[E](x, t)[S](x, t) = KM [E·S](x, t).

(8)

Now combining Eqs. (7) and (8) yields the relation [E·S]=([E]tot [S])/(KM +[S]), which is the source term we were searching for in Eq. (4d). It is written in terms of two error functions [Eqs. (6) and (7)] that correspond to two diffusing but non-reacting species. Putting all this x x √ √ together, and using the shorthand notation ξS = erfc 4DS t and ξE = erfc − 4DE t , the mass balance of Eq. (4d) becomes: ∂[P] ∂[P] kcat [E]0 ξE · ξS . = DS 2 + 2KM ∂t ∂x 2 + ξS S0

(9)

Through these simplifications, the system of four coupled nonlinear equations [Eqs. (4a)(4d)] has been replaced by a single diffusion equation with a known source term [Eq. (9)]. The spatio-temporal dependence of this source term is limited to a dependance on a diffusive √ similarity variable η = x/ 4DS t and on the ratio of the diffusion coefficents ρ = (DS /DE )1/2 . 13 ACS Paragon Plus Environment


Its amplitude depends on the chemistry through the chemical constants kcat and KM .

a) 0.8

[S]0 /KM

b)

0.4 0.2 0.0 -2

0

10

-1

max(p)

0.6

p

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10

-2

-3

10

-4

-1

10 -3 10

0

1 2 3 η = x/ 4DS t

1

10

1 -1

10

1

10

3

10

5

10

[S]0 /KM

Figure 5: Simulations of p based on the analytical model of Eq. (10). These curves are obtained by solving Eq. (10). (a) Self-similar form p as a function of the similarity variable η, for varying KM/[S]0 . b) Variations of the maximum of p as a function of KM/[S]0 . This motivates us to search for a general solution to Eq. (9) of the form [P](x, t) = A(t)·p(η), where p(η) has a self-similar shape that does not vary in time. The time-dependent amplitude A(t) is expected, from scaling analysis, to have the form A(t) = kcat E0 t. Injecting this form into Eq. (9), and writing k = 2KM /[S]0 , yields the following equation for p: p00 (η) + 2ηp0 (η) − 4p(η) + 2

erfc (−ρη) erfc (η) = 0, k + erfc (η)

(10)

subject to the boundary conditions p(±∞) = 0. Equation (10) must be solved numerically, which yields a series of bell-shaped curves p(η) that depend on ρ = (DS /DE )1/2 and k = 2KM /[S]0 . Focusing on the variations of p with k, we find that the shape of p is preserved but that its amplitude pm = max(p) shows a strong dependence on k, as shown in Fig. 5. The above mathematical analysis served to replace the PDE for [P] by an ODE for p. In practice however, the measured concentration of the product in an actual experiment would depend on both A and p, such that the overall amplitude of the measuremed signal would grow linearly in time with a growth rate as V0 =[E]0 kcat pm . In this sense, the plot of Fig. 5 closely resembles a Michaelis-Menten curve for the well-mixed case: the growth rate V0 increases linearly with [S]0 for [S]0 KM , before saturating at [E]0 kcat as [S]0 becomes larger than KM .


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Comparing experiment and theory The model described above can be compared with measurements in droplets, after switching to a fast enzymatic reaction that takes place faster than diffusion can mix the contents. Here we choose to study the hydrolysis of 4-methylumbelliferyl phosphate (4-MUP) catalyzed by alkaline phosphatase (AP) as a model for an enzymatic reaction well known to be fast. 31,32 The reaction product 4-methylumbelliferone (denoted P ) was was detected by fluorescence, using a standard epi-fluorescence setup and DAPI filters. A basic experiment involving merging two droplets in our reaction chamber was set up: starting with one droplet containing the substrate 4-MUP (left, Fig. 6a), the other droplet containing the enzyme AP (right). After merging, the fluorescent product P forms in the zone between the reservoirs of fresh reagents, so that a fluorescent strip emerges and widens along the merged droplet. The evolution of this strip is monitored along the axis labeled with the x coordinate, during 50 s, and translated to product concentration through an intensity calibration. The spatio-temporal evolution thus measured is first fitted with the profiles obtained from a full simulation of the RD system of Eqs.(4a)-(4d). We find that the simulated profiles are in very good agreement with the measured profiles for the initial moments of the reaction (up to 6 seconds in this case, Fig. 6b). However, this fitting procedure depends on four unknown parameters, KM , kcat , DE , and DS . It is therefore not expected to be very selective on the parameter values that are obtained. Nevertheless, the best fit between experiment and simulation is obtained for KM = 4 µM, kcat = 85 s−1 , in very good agreement with plate measurements for KM (KM = 4 µM in plate), but overestimates kcat by a factor of 8 (kcat = 11 s−1 in plate). The next step is to verify that the analytical solution indeed agrees with the experimental data. For this, the self-similarity and the scaling of the analytical solution, predicted by Eq. (10), are first tested. We begin by verifying that the amplitude of the product concentration (hp in Fig. 6b) indeed increases linearly with time, for early times (Fig. 7), 15 ACS Paragon Plus Environment


a)

S (4-MUP)

E (AP)

x

0 P b)

6

t=0s t = 0.7 s t = 1.4 s t = 2.0 s t = 2.7 s t = 3.4 s t = 4.1 s t = 4.7 s t = 5.4 s t = 6.1 s

5 [P] (µM)

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4

hp

3 2 1 0 -200

-100

0

100 x (µm)

200

300

Figure 6: Evolution of a fast enzymatic reaction: (a) A droplet containing a solution of MUP at 100 µM (left hand side) has been merged with a droplet containing a solution of AP at 25 nM (right hand side). The fluorescence image is taken at t = 16 s. The scale bar is 500 µm. (b) Profile of the product concentration [P] as a function of x at several time points. The solid lines are simulated from the RD system of Eqs. (4a)-(4d). hp denotes the height of the concentration curve at each time step. which confirms the expected scaling of the amplitude A. Moreover, the self-similarity is also well confirmed, by plotting the profile of [P], normalised by hp at every time step, vs. similarity variable η. The curves collapse onto a master curve, which corresponds to the time-independent shape function that characterises a given experiment. These measurements are repeated for several values of [S]0 and [E]0 , and the value of the initial rate is plotted, as shown in Figs. 7c, d. As expected for a standard Michaelis-Menten analysis, we find that the evolution of V0 with the substrate initial concentration yields a saturation curve, while the dependence on the enzyme initial concentration shows a linear


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a) 16

b) 1.0

E0 = 25 nM

[P]/hp

hp (µM)

8

V0

0.4

2

4

6

0.0 -3

8 10 12 14 16 18 20 t (s)

5

d) 5

4

4 V0 (µM/s)

c)

0.6

0.2

4 0 0

3 2 1 0

t =1.5 s t =2.7 s t =3.9 s t =5.0 s t =6.2 s

0.8

12

V0 (µM/s)

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-2

-1 0 1 η = x/ 4DS t

2

3

3 2 1

0

20

40

60 [S]0 (µM)

80

100

0

0

20

40 60 [E]0 (nM)

80

100

Figure 7: Reaction-diffusion experiments: (a) Product amplitude hp as a function of time, for the experiment shown in Fig. 6. The points are experimental and the solid line extracted from the definition of V0 = kcat [E]0 pm , as defined in previous section. (b) Product amplitude, scaled by its maximum at each time step, vs. the similarity variable η. Solid line is the solution of Eq. (10). The collapse of the experimental points confirms the self-similarity of the shape function. Here, E0 = 25 nM and S0 = 100 µM. (c) and (d) Variations of the product amplitdue initial rate V0 with the substrate and enzyme initial concentrations, respectively. The black curves are obtained from simulations of the RD system. In all subfigures, the values of the kinetic and diffusion coefficients were obtained from a single fit of the experiment of Fig. 6 by the RD model. increase.



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Conclusions Here we present a format to perform on-demand reactions between the contents of two droplets. The stationary droplet format allows the observaion of either fast or slow reactions in the same device, without any modifications of the experimental protocol. However, while the analysis of the slow reactions is straight-forward and can follow the well established models for well-mixed vessels, the analysis of faster reactions must take into account the full reaction-diffusion problem. For simple reactions, we have previously shown that comparing the experiments with numerical solutions to the reaction-diffusion front could provide a very good estimate of the reaction rate. 22 In the case of enzymatc reactions however, one must obtain the Michaelis-Menten parameteres KM and kcat from different elements of the experiment/numerics confrontation. In the slow reaction case, the measurements we obtain from the droplet format reproduce the values obtained using different techniques. The same is also true of the value of the binding constant KM for the case of fast reactions. However, we observe a large deviation in the value of the catalysis constant in the droplets, compared with the micro-titer plates. We attribute this to the fact that the water-oil interface may interfere with the reaction, particularly because AP has been shown to adsorb onto the water-oil interface. 33 Such an exchange between the interface and the bulk has also been shown to modify the kinetics of a reaction by reducing the system’s entropy and providing different routes for the reaction to take place. 34 We therefore conjecture that similar phenomena are operating in this case. In spite of this potential artefact, the measurements that we obtain from our experiments remain semi-quantitative and we expect that their precision can be improved, for example by screening the interfacial effects using bovine serum albumin 35 or other inert proteins. Compared with other microfluidic techniques, 14,30 the current approach requires a single droplet to perform a measurement. Most other droplet-based approaches rely on average measurements on very large number of droplets and co-flow measurements require relatively large volumes. In contrast, the droplets in our devices are produced in small numbers by 18 ACS Paragon Plus Environment

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relying on the unique microchannel geometry for their production and transport. This implies that the total reagent consumption remains small. Indeed, the six parallel measurements of Fig. 4 are performed using below 200 nl of enzyme solution and the device requires a few minutes to be loaded. This reduction of the sample consumption and time to obtain measurements suggests that a large number of successive screens can be performed by an automated system. For example, a robot can be used to form successive aqueous plugs of enzyme in oil, 28,36,37 and lead them into the reaction chamber. There, the plugs will break into equally sized droplets, by the action of the step junctions, and can be transported into the anchors by an external flow. Once the drops are merged with their counter-drops and the reaction observed, they can be evacuated for the next plug to be introduced and analyzed. Finally, the microfluidic platform is not limited to kinetic measurements. Other applications can include simple measurements to dose chemicals in an environmental or clinical sample, or even the production of well-controlled micro batches of compounds on demand. The robustness of the droplet manipulation by confinement gradients makes this type of approach well suited for industrialisation and for field or bedside applications.

Methods Device fabrication: All devices were made of a PDMS block (Dow Corning Sylgard 184) sealed onto a glass slide by plasma bonding. Molds for PDMS were fabricated using dry film photoresist soft lithography techniques. 38 The multilayer masters were etched in stacks of Eternal Laminar negative films of thicknesses 15 µm, 35 µm and 50 µm depending on the desired local thickness. Successive layers of photoresist were deposited using a PEAK Photo Laminator (PS320) at a temperature T = 100 ◦ C and exposed to UV (Hamamatsu Lightningcure LC8) using a succession of masks that determine the features at each height. Once all layers were deposited and exposed, the complete device was developed by immersion



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in an aqueous bath of carbonate potassium at 1 % (w:w) to reveal the mold. The geometry was then verified using an optical profilometer (Zygo NewView 7100). The mold was used to produce a PDMS channel. After plasma bonding on a glass slide, the internal channel surface was made hydrophobic using a dilute solution of 1H,1H,2H,2Hperfluorodecyltrichlorosilane (Sigma-Aldrich) in FC40 oil (3M Fluorinert) (20 mL in 1 mL of FC40) for approximately 5 min. The channel was then rinsed with pure FC40 to remove the residue chemicals remaining in the bulk.

Device operation: The top row of traps is first filled with a single drop of type B reagents which originate from different syringes outside the chip. To this end, the flow rate of the reagent solutions is fixed at 0.5 µl/min, stopped once a drop has detached and then fixed to -1 µl/min in order to remove the thread from the step junction. This is performed using a Nemesys (Cetoni) syringe pump. Once the top row is occupied, six drops of the type A reagent are generated and pushed toward the bottom row of the array with an oil flow. As droplets are pushed towards the array section, a laser is focused on the water/oil interface to help guide a given drop into a trap. 20 This leads to all six positions being occupied by six different pairs of drops as illustrated on Fig. 3c with dyed colored droplets. Adjacent droplets trapped in our device do not coalesce as their surface is covered with surfactant molecules. Nonetheless, a laser pulse on the droplet/droplet interface can merge two touching drops, 39 the fusion starting once the laser is removed. By doing so, droplets fusion can be triggered at will in our static array of paired drops.

Imaging and optics: Imaging was performed using an inverted microscope (Nikon TE2000) equipped with epifluorescent illumination (Exfo X-cite 6210C). Two cameras were connected to the microscope side ports (Photron Fastcam 1024 PCI and Spot Insight). To manipulate droplets by laser heating, a 4f conjugate lens system was built to focus a 1480 nm continuous wave infrared laser source (Fitel Furukawa FOL1424) in the microscope focal plane. 40 Two galvanometric mirrors (Cambridge Technologies 6210H) placed on the laser path permitted 20 ACS Paragon Plus Environment

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beam positioning in the microchannels to be controlled by a mouse click using in-house Labview programs.

Determining IC-50 for the inhinitor: The increase in absorbance was recorded for 10 min and the experiment was repeated three times. Conditions : T = 25 ◦C, [PBS]= 50 mM, pH = 7. Enymes were obtained from Sigma-Aldrich. A value for KI was extracted by fitting the normalized reaction rates V0 /Vmax against the initial inhibitor concentration [I]0 via the Cheng-Prusoff relation: V0 V0max

1 = D0 1 + IC50

S0 IC50 = 1 + KI KM

with


(11)


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Acknowledgement The authors acknowledge the help of Caroline Frot with microfabrication, useful discussions with Liisa van Vliet, Fabrice Gielen, Stéphanie Descroix and Anais Ali Cherif. CB and FH ar ERC Starting Investigators. The research leading to these results received funding from the European Research Council (ERC) Grant Agreement 278248 Multicell. CDB was supported by the Cambridge european Trust and the BBSRC. We acknowledge funding from the EPSRC (Engineering and Physical Sciences Research Council).


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(40) Cordero, M.; Rolfsnes, H.; Burnham, D.; Campbell, P.; McGloin, D.; Baroud, C. New J. Phys. 2009, 11, 075033.


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Measuring Fast and Slow Enzyme Kinetics in Stationary Droplets

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