WO2009048533A2 - Reactions within microfluidic channels - Google Patents

Abstract

The present invention generally relates to microfluidics and, in particular, to microfluidic systems useful for determining reactions. In one aspect, two or more fluids are introduced into a microfluidic channel, and are allowed to come into contact. At or near the interface between the fluids, reactants contained within the fluids may react. By determining the reaction rate between the reactants with respect to position within the microfluidic channel, information about the reaction between the reactants can be obtained. The reaction rate may be determined, for instance, by measuring the rate of reaction at two or more points within the channel. In some cases, e.g., if the reaction rate can be determined optically or visually, the channel may be imaged and the image analyzed to determine the reaction rate. As a non-limiting example of a reaction, the reactants may be an enzyme and a substrate, and by determining reaction rates within the channel, Michaelis-Menten kinetics (or other reaction kinetics) of the enzymatic reaction may be determined. It should be understood, however, that other reactions besides Michaelis-Menten kinetics and/or enzymatic reactions may also be determined, including other catalytic or chemical reactions, and the like. The reaction profiles may be linear or non-linear, e.g., second order, third order, etc.

Description

REACTIONS WITfflN MICROFLUIDIC CHANNELS

RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Patent Application Serial No. 60/997,996, filed October 5, 2007, entitled "Reactions Within Microfluidic Channels," by Stone, et al, incorporated herein by reference.

FIELD OF INVENTION

The present invention generally relates to microfluidics and, in particular, to microfluidic systems useful for determining reactions.

BACKGROUND

Microfluidics is an area of technology involving the control of fluid flow at a very small scale. Microfluidic devices typically include relatively small channels, within which fluid can flow, which can be branched in some cases, or otherwise arranged, to allow fluids to be combined with each other, to divert fluids to different locations, to cause laminar flow to occur, to dilute fluids, and the like. Significant effort has been directed toward "lab-on-a-chip" microfluidic technology, in which researchers seek to carry out chemical or biological reactions on a small scale on a "chip," or a microfluidic device. Additionally, new techniques, not necessarily known on the macro scale, are being developed using microfluidics. Examples of techniques being investigated or developed at the microfluidic scale include high-throughput screening, drug delivery, chemical kinetics measurements, combinatorial chemistry (where rapid testing of chemical reactions, chemical affinity, and micro structure formation are desired), as well as the study of fundamental questions in the fields of physics, chemistry, and engineering.

SUMMARY OF THE INVENTION

The present invention generally relates to microfluidics and, in particular, to microfluidic systems useful for determining reactions. The subject matter of the present invention involves, in some cases, interrelated products, alternative solutions to a particular problem, and/or a plurality of different uses of one or more systems and/or articles.

In one aspect, the present invention is directed to a method of determining a reaction. In one set of embodiments, the method includes acts of flowing a first fluid containing a first reactant into a microfluidic channel, flowing a second fluid containing a second reactant into the microfluidic channel, forming an interface between the two fluids, causing a non-linear reaction between the first reactant and the second reactant that yields a product within the channel, determining a rate of product formation at a first point in the channel, and determining a rate of product formation at a second point in the channel.

In another set of embodiments, the method includes acts of flowing a first fluid containing a first reactant having a first diffusion coefficient into a microfluidic channel, flowing a second fluid containing a second reactant having a second diffusion coefficient into the microfluidic channel, wherein the second diffusion coefficient is larger than the first diffusion coefficient by at least two orders of magnitude, forming an interface between the two fluids, causing a reaction between the first reactant and the second reactant that yields a product within the channel, determining the rate of product formation at a first point in the channel, and determining the rate of product formation at a second point in the channel. In another aspect, the present invention is directed to a method of making one or more of the embodiments described herein. In another aspect, the present invention is directed to a method of using one or more of the embodiments described herein. Other advantages and novel features of the present invention will become apparent from the following detailed description of various non-limiting embodiments of the invention when considered in conjunction with the accompanying figures. In cases where the present specification and a document incorporated by reference include conflicting and/or inconsistent disclosure, the present specification shall control. If two or more documents incorporated by reference include conflicting and/or inconsistent disclosure with respect to each other, then the document having the later effective date shall control. BRIEF DESCRIPTION OF THE DRAWINGS

Non-limiting embodiments of the present invention will be described by way of example with reference to the accompanying figures, which are schematic and are not intended to be drawn to scale. In the figures, each identical or nearly identical component illustrated is typically represented by a single numeral. For purposes of clarity, not every component is labeled in every figure, nor is every component of each embodiment of the invention shown where illustration is not necessary to allow those of ordinary skill in the art to understand the invention.

Fig. 1 is a schematic diagram illustrating a microfluidic channel according to one set of embodiments;

Figs. 2A-2D include contour plots of scaled concentration as a function of scaled x and y distances, according to another set of embodiments;

Figs. 3A-3B include plots of numerical calculations of integral product concentration (with units of moles, scaled on the initial moles of substrate concentration) as a function of downstream position in the channel, according to yet another set of embodiments;

Figs. 4A-4C include plots of numerical calculations of integral product concentration (with units of moles, scaled on the initial moles of substrate concentration) as a function of downstream position in the channel, according to still another set of embodiments; and

Fig. 5 includes a plot of experimental measurements of the light emitted by the reaction between luciferase and ATP as a function of position in the microchannel, in another set of embodiments.

DETAILED DESCRIPTION

The present invention generally relates to microfluidics and, in particular, to microfluidic systems useful for determining reactions. In one aspect, two or more fluids are introduced into a microfluidic channel, and are allowed to come into contact. At or near the interface between the fluids, reactants contained within the fluids may react. By determining the reaction rate between the reactants with respect to position within the microfluidic channel, information about the reaction between the reactants can be obtained. The reaction rate may be determined, for instance, by measuring the rate of reaction at two or more points within the channel. In some cases, e.g., if the reaction rate can be determined optically or visually, the channel may be imaged and the image analyzed to determine the reaction rate. As a non-limiting example of a reaction, the reactants may be an enzyme and a substrate, and by determining reaction rates within the channel, Michaelis-Menten kinetics (or other reaction kinetics) of the enzymatic reaction may be determined. It should be understood, however, that other reactions besides Michaelis-Menten kinetics and/or enzymatic reactions may also be determined, including other catalytic or chemical reactions, and the like. The reaction profiles may be linear or non-linear, e.g., second order, third order, etc. As mentioned, one aspect of the invention is generally directed to determining reactions, such as chemical, biochemical, and/or biological reactions, within a channel such as a microfluidic channel. Typically, two or more fluids are introduced into a channel, some or all of which contain reactants, and the reactants are allowed to come into contact and/or react within the channel. As used herein, the term "fluid" generally refers to a substance that tends to flow and to conform to the outline of its container, i.e., a liquid, a gas, a viscoelastic fluid, etc. Typically, fluids are materials that are unable to withstand a static shear stress, and when a shear stress is applied, the fluid experiences a continuing and permanent distortion. The fluid may have any suitable viscosity that permits flow. If two or more fluids are present, each fluid may be independently selected among essentially any fluids (liquids, gases, and the like) by those of ordinary skill in the art, by considering the relationship between the fluids. The fluids may each be miscible or immiscible. Where the portions remain liquid for a significant period of time, then the fluids may be chosen to be at least substantially immiscible. Those of ordinary skill in the art can select suitable miscible or immiscible fluids, using contact angle measurements or the like, to carry out the techniques of the invention. As used herein, two fluids are immiscible, or not miscible, with each other when one is not soluble in the other to a level of at least 10% by weight under ambient conditions.

In certain cases, a fluidic stream may be produced on the microscale, for example, in a microchannel. Thus, in some, but not all embodiments, at least some of the components of the systems and methods described herein using terms such as "microfluidic" or "microscale." As used herein, "microfluidic," "microscopic," "microscale," the "micro-" prefix (for example, as in "microchannel"), and the like generally refers to elements or articles having widths or diameters of less than about 2 mm or about 1 mm, and less than about 100 micrometers in some cases. In some cases, an element or article includes a channel through which a fluid can flow. In all embodiments, specified widths can be a smallest width (i.e. a width as specified where, at that location, the article can have a larger width in a different dimension), or a largest width (i.e. where, at that location, the article has a width that is no wider than as specified, but can have a length that is greater). Thus, for example, a fluidic stream may be produced on the microscale, e.g., using a microfluidic channel. For instance, the fluidic stream may have an average cross-sectional dimension of less than about 1 mm, less than about 500 microns, less than about 300 microns, or less than about 100 microns. In some cases, the fluidic stream may have an average diameter of less than about 60 microns, less than about 50 microns, less than about 40 microns, less than about 30 microns, less than about 25 microns, less than about 10 microns, less than about 5 microns, less than about 3 microns, or less than about 1 micron. A "channel," as used herein, means a feature on or in an article (e.g., a substrate) that at least partially directs the flow of a fluid. In some cases, the channel may be formed, at least in part, by a single component, e.g. an etched substrate or molded unit. The channel can have any cross-sectional shape, for example, circular, oval, triangular, irregular, square or rectangular (having any aspect ratio), or the like, and can be covered or uncovered (i.e., open to the external environment surrounding the channel). In embodiments where the channel is completely covered, at least one portion of the channel can have a cross-section that is completely enclosed, and/or the entire channel may be completely enclosed along its entire length with the exception of its inlet and outlet. A channel may have an aspect ratio (length to average cross-sectional dimension) of at least 2:1, more typically at least 3: 1, 5:1, 10:1, 30:1, 100:1, 300:1, 1000:1, etc. As used herein, a "cross-sectional dimension," in reference to a fluidic or microfluidic channel, is measured in a direction generally perpendicular to fluid flow within the channel. An open channel generally will include characteristics that facilitate control over fluid transport, e.g., structural characteristics (an elongated indentation) and/or physical or chemical characteristics (hydrophobicity vs. hydrophilicity) and/or other characteristics that can exert a force (e.g., a containing force) on a fluid. The fluid within the channel may partially or completely fill the channel. In some cases the fluid may be held or confined within the channel or a portion of the channel in some fashion, for example, using surface tension (e.g., such that the fluid is held within the channel within a meniscus, such as a concave or convex meniscus). In an article or substrate, some (or all) of the channels may be of a particular size or less, for example, having a largest dimension perpendicular to fluid flow of less than about 5 mm, less than about 2 mm, less than about 1 mm, less than about 500 microns, less than about 200 microns, less than about 100 microns, less than about 60 microns, less than about 50 microns, less than about 40 microns, less than about 30 microns, less than about 25 microns, less than about 10 microns, less than about 3 microns, less than about 1 micron, less than about 300 nm, less than about 100 nm, less than about 30 nm, or less than about 10 nm or less in some cases. In one embodiment, the channel is a capillary. Of course, in some cases, larger channels, tubes, etc. can be used to store fluids in bulk and/or deliver a fluid to the channel. According to a first aspect of the invention, two or more fluids may be introduced into a microfluidic channel, and reactants contained within the fluids may be allowed to come into contact and/or react. The first and second reactants may be any suitable species that can react with each other, for example, an enzyme or a catalyst and its substrate, two compounds that can react (either spontaneously, or under action of a catalyst, for instance, a chemical catalyst, light, heat, electricity, etc.), or the like. As a non-limiting example, the catalyst may be in the form of nanoparticles contained within the fluid, for instance, platinum nanoparticles. It should be understood, however, that in some cases, the fluid itself may be a reactant. Thus, in such situations, the fluid is synonymous with the reactant. In some embodiments, when the two (or more) fluids come into contact, an interface (e.g., a fluidic interface) is formed between the fluids, e.g., if flow of one or more of the fluids within the channel is substantially laminar. In some cases, the interface may extend along the entire length of the channel where the fluids remain in contact. In other cases, however, the interface may not necessarily extend along the entire length of the channel, e.g., if mixing between the fluids occurs. Reactants contained in one fluid may diffuse or otherwise be transported across the interface to the other fluid, depending on their diffusivities. The two (or more) fluids may be introduced into a microfluidic channel using any suitable technique. For instance, a first fluid may be contained within a first channel and a second fluid may be contained within a second channel, where each of the first and second channels intersect to form a microfluidic channel, e.g., in a "Y" junction as is shown in Fig. 1, in a "T" junction, or the like. In some cases, the fluids may enter the microfluidic channel at the same location (e.g., as is shown in Fig. 1); in other cases, the fluids may enter the channel on a staggered basis. In some cases, more than two entry channels may be present.

The reactants are allowed to react within the microfluidic channel, and the reaction may be linear or non-linear in some cases. As used herein, a "non-linear reaction" is a reaction in which the rate is not a linear function of concentration, i.e., the reaction is not zeroth (no dependence on) or first order (linearly proportional to concentration). For instance, the reaction may be a Michaelis-Menten-type reaction (e.g., between an enzyme and a substrate), a second-order reaction, a third-order reaction, or the like, or the reaction may have other forms. Such reaction orders and/or a kinetic parameters may be determined using the systems and the methods as described herein. In embodiments with a linear reaction, the flow may be treated in some cases as laminar and the reactants may slowly diffuse toward one another (e.g., transverse to the flow) to react, e.g., to form one or more products. The kinetic parameters may then determined, for instance, by measuring the product as a function of position downstream from the junction. However, in a non-linear reaction system, the calculation of rate with respect to position within respect to length is often more complicated, and has not been studied in the past due to the increased complexity.

Of course, it should be understood that the present system is not limited to two reactants and/or two fluids. For example, in one set of embodiments, three, four, or more reactants may be introduced into a microfluidic channel, and the reactants may be contained within two fluids, or within three, four, or more fluids. The reactants may all independently be introduced into the microfluidic channel at the same time and/or at different times, depending on the particular reaction and its application. For instance, in one embodiment, a first reactant may be an enzyme or a catalyst, a second reactant may be a substrate that the enzyme or catalyst is able to recognize, and the third reactant may be an inhibitor that inhibits the reaction, e.g., a catalytic poison or a molecular inhibitor. The inhibition may be competitive, uncompetitive, noncompetitive, mixed, etc. As another non-limiting example, the third reactant may be a co-factor or a promoter for the reaction between the enzyme or catalyst and the substrate. As yet another non-limiting example, any of the reactants may comprise a nanoparticle (e.g., a platinum nanoparticle). In some cases, one or more of an inhibitor, promoter, co-factor, nanoparticle, enzyme, and/or catalyst, etc. may be provided in one or more fluids. The third reactant may be contained within the first and/or second fluids (containing, respectively, the first and/or second reactants), and/or the third reactant may be contained within a third fluid. Such systems may also be expanded, in some embodiments, to four reactants, five reactants, etc. In some cases, two, three, four, or more reaction products may be formed in the channel. Any of the reactant products formed in the system may also be used, in some cases, as a reactant in subsequent reactions in the channel. For example, a first reactant may react with a second reactant to form an intermediate product, and the intermediate product may react (e.g., sequentially or simultaneously) with a third reactant to form a final product.

In some cases, the diffusivity of one reactant may be substantially greater than that of another reactant. For instance, a first reactant may have a diffusion coefficient at least one order of magnitude (1Ox) greater than the diffusion coefficient of a second reactant, and in some cases, at least two orders of magnitude (10²x), at least three orders of magnitude (10³x), at least four orders of magnitude (10⁴x), at least five orders of magnitude (10⁵x), at least six orders of magnitude (10⁶x), etc. Examples of such systems include, but are not limited to, enzyme-substrate systems, e.g., where the enzyme has a molecular weight substantially greater than that of its substrate, in some cases by several orders of magnitude. In some instances, the diffusion of the reactants between the fluids containing the reactants may occur in substantially one direction, e.g., due to differences in diffusion coefficient. For instance, if a first fluid contains an enzyme and a second fluid contains a substrate, there may be significantly more transport of the substrate across the interface into the first fluid than of the enzyme across the interface into the second fluid. The reaction rate may be determined using any suitable technique. The reaction rate may be determined, for instance, by determining the rate at which a product forms, at the rate at which a reactant disappears, or the like. In one set of embodiments, the reaction rate may be determined at at least a first point in the channel and a second point in the channel. More than two points may also be used in some embodiments. In some cases, two or more reactions may take place in the channel. In such cases, the reaction rate of one or more of the additional reactions may be determined at at least a first point in the channel and a second point in the channel.

The rates may be determined sequentially or simultaneously, as discussed below. The rate of reaction can be determined in any fashion, either directly or indirectly. For instance, in some cases, a reactant and/or a product may produce a determinable signal. For instance, the reactant and/or the product may be absorptive, fluorescent, phosphorescent, radioactive, etc. In another set of embodiments, the rate of reaction itself is indirectly determined, e.g., via reaction of a reactant and/or a product with a second entity (e.g., a "signaling entity") in order to effect determination; for example, coupling or reaction of the reactant and/or product with the signaling entity may result in a determinable signal. As a specific non-limiting example, an enzyme such as luciferase may produce light, e.g., by reaction with a substrate such as luciferin. An example of this is discussed in the Examples, below. In some instances, reaction rates (e.g., rates of product formation) may be determined by an algorithm that calculates reaction order and/or a kinetic parameter. As used herein, "determining" refers to the detection and/or analysis of an entity, either quantitatively or qualitatively. Determination of an entity may include determination of the presence or absence of the entity, and/or a measurement of the amount or degree of the entity, e.g., the concentration of the entity, the density of the entity, etc. In some cases, the location of an entity may be determined, for example, within a microfluidic channel.

In one set of embodiments, the microfluidic channel may be imaged to determine the rate of reaction. For instance, the microfluidic channel may be imaged (e.g., optically or fluorescently, etc.), and the rate of reaction determined from the images, e.g., after steady state has been reached in the channel. Such techniques can allow a single determination of the channel to be used to determine reaction kinetics. Any suitable method may be used to acquire an image, e.g., by using a video camera such as a CCD camera. In some cases, a series of images are acquired with respect to time, e.g., to determine how a reaction changes with respect to time. A variety of materials and methods, according to certain aspects of the invention, can be used to form the fluidic or microfluidic system. For example, various components of the invention can be formed from solid materials, in which the channels can be formed via micromachining, film deposition processes such as spin coating and chemical vapor deposition, laser fabrication, photolithographic techniques, etching methods including wet chemical or plasma processes, and the like. See, for example, Scientific American, 248:44-55, 1983 (Angell, et al).

In one set of embodiments, at least a portion of the fluidic system is formed of silicon by etching features in a silicon chip. Technologies for precise and efficient fabrication of various fluidic systems and devices of the invention from silicon are known. In another embodiment, various components of the systems and devices of the invention can be formed of a polymer, for example, an elastomeric polymer such as polydimethylsiloxane ("PDMS"), polytetrafluoroethylene ("PTFE" or Teflon^®), or the like. For instance, according to one embodiment, system 10 shown in Fig. 1 may be implemented by fabricating the fluidic system separately using PDMS or other soft lithography techniques (details of soft lithography techniques suitable for this embodiment are discussed in the references entitled "Soft Lithography," by Younan Xia and George M. Whitesides, published in the Annual Review of Material Science, 1998, Vol. 28, pages 153-184, and "Soft Lithography in Biology and Biochemistry," by George M. Whitesides, Emanuele Ostuni, Shuichi Takayama, Xingyu Jiang and Donald E. Ingber, published in the Annual Review of Biomedical Engineering, 2001, Vol. 3, pages 335-373; each of these references is incorporated herein by reference).

Different components can be fabricated of different materials. For example, a base portion including a bottom wall and side walls can be fabricated from an opaque material such as silicon or PDMS, and a top portion can be fabricated from a transparent or at least partially transparent material, such as glass or a transparent polymer, for observation and/or control of the fluidic process. Components can be coated so as to expose a desired chemical functionality to fluids that contact interior channel walls, where the base supporting material does not have a precise, desired functionality. For example, components can be fabricated as illustrated, with interior channel walls coated with another material. Material used to fabricate various components of the systems and devices of the invention, e.g., materials used to coat interior walls of fluid channels, may desirably be selected from among those materials that will not adversely affect or be affected by fluid flowing through the fluidic system, e.g., material(s) that is chemically inert in the presence of fluids to be used within the device.

In some embodiments, various components of the invention are fabricated from polymeric and/or flexible and/or elastomeric materials, and can be conveniently formed of a hardenable fluid, facilitating fabrication via molding (e.g. replica molding, injection molding, cast molding, etc.). The hardenable fluid can be essentially any fluid that can be induced to solidify, or that spontaneously solidifies, into a solid capable of containing and/or transporting fluids contemplated for use in and with the fluidic network. In one embodiment, the hardenable fluid comprises a polymeric liquid or a liquid polymeric precursor (i.e. a "prepolymer"). Suitable polymeric liquids can include, for example, thermoplastic polymers, thermoset polymers, or mixture of such polymers heated above their melting point. As another example, a suitable polymeric liquid may include a solution of one or more polymers in a suitable solvent, which solution forms a solid polymeric material upon removal of the solvent, for example, by evaporation. Such polymeric materials, which can be solidified from, for example, a melt state or by solvent evaporation, are well known to those of ordinary skill in the art. A variety of polymeric materials, many of which are elastomeric, are suitable, and are also suitable for forming molds or mold masters, for embodiments where one or both of the mold masters is composed of an elastomeric material. A non-limiting list of examples of such polymers includes polymers of the general classes of silicone polymers, epoxy polymers, and acrylate polymers. Epoxy polymers are characterized by the presence of a three- membered cyclic ether group commonly referred to as an epoxy group, 1,2-epoxide, or oxirane. For example, diglycidyl ethers of bisphenol A can be used, in addition to compounds based on aromatic amine, triazine, and cycloaliphatic backbones. Another example includes the well-known Novolac polymers. Non-limiting examples of silicone elastomers suitable for use according to the invention include those formed from precursors including the chlorosilanes such as methylchlorosilanes, ethylchlorosilanes, phenylchlorosilanes, etc. Silicone polymers are used in certain embodiments, for example, the silicone elastomer polydimethylsiloxane. Non-limiting examples of PDMS polymers include those sold under the trademark Sylgard by Dow Chemical Co., Midland, MI, and particularly Sylgard 182, Sylgard 184, and Sylgard 186. Silicone polymers including PDMS have several beneficial properties simplifying fabrication of the microfluidic structures of the invention. For instance, such materials are inexpensive, readily available, and can be solidified from a prepolymeric liquid via curing with heat. For example, PDMSs are typically curable by exposure of the prepolymeric liquid to temperatures of about, for example, about 65 ⁰C to about 75 ⁰C for exposure times of, for example, about an hour. Also, silicone polymers, such as PDMS, can be elastomeric and thus may be useful for forming very small features with relatively high aspect ratios, necessary in certain embodiments of the invention. Flexible (e.g., elastomeric) molds or masters can be advantageous in this regard.

One advantage of forming structures such as microfluidic structures of the invention from silicone polymers, such as PDMS, is the ability of such polymers to be oxidized, for example by exposure to an oxygen-containing plasma such as an air plasma, so that the oxidized structures contain, at their surface, chemical groups capable of cross-linking to other oxidized silicone polymer surfaces or to the oxidized surfaces of a variety of other polymeric and non-polymeric materials. Thus, components can be fabricated and then oxidized and essentially irreversibly sealed to other silicone polymer surfaces, or to the surfaces of other substrates reactive with the oxidized silicone polymer surfaces, without the need for separate adhesives or other sealing means. In most cases, sealing can be completed simply by contacting an oxidized silicone surface to another surface without the need to apply auxiliary pressure to form the seal. That is, the pre- oxidized silicone surface acts as a contact adhesive against suitable mating surfaces. Specifically, in addition to being irreversibly sealable to itself, oxidized silicone such as oxidized PDMS can also be sealed irreversibly to a range of oxidized materials other than itself including, for example, glass, silicon, silicon oxide, quartz, silicon nitride, polyethylene, polystyrene, glassy carbon, and epoxy polymers, which have been oxidized in a similar fashion to the PDMS surface (for example, via exposure to an oxygen-containing plasma). Oxidation and sealing methods useful in the context of the present invention, as well as overall molding techniques, are described in the art, for example, in an article entitled "Rapid Prototyping of Microfluidic Systems and

Polydimethylsiloxane," Anal. Chem., 70:474-480, 1998 (Duffy et al), incorporated herein by reference. Another advantage to forming microfluidic structures of the invention (or interior, fluid-contacting surfaces) from oxidized silicone polymers is that these surfaces can be much more hydrophilic than the surfaces of typical elastomeric polymers (where a hydrophilic interior surface is desired). Such hydrophilic channel surfaces can thus be more easily filled and wetted with aqueous solutions than can structures comprised of typical, unoxidized elastomeric polymers or other hydrophobic materials.

In one embodiment, a bottom wall is formed of a material different from one or more side walls or a top wall, or other components. For example, the interior surface of a bottom wall can comprise the surface of a silicon wafer or microchip, or other substrate. Other components can, as described above, be sealed to such alternative substrates. Where it is desired to seal a component comprising a silicone polymer (e.g. PDMS) to a substrate (bottom wall) of different material, the substrate may be selected from the group of materials to which oxidized silicone polymer is able to irreversibly seal (e.g., glass, silicon, silicon oxide, quartz, silicon nitride, polyethylene, polystyrene, epoxy polymers, and glassy carbon surfaces which have been oxidized). Alternatively, other sealing techniques can be used, as would be apparent to those of ordinary skill in the art, including, but not limited to, the use of separate adhesives, thermal bonding, solvent bonding, ultrasonic welding, etc.

The following are each incorporated herein by reference: U.S. Patent Application Serial No. 11/246,911, filed October 7, 2005, entitled "Formation and Control of Fluidic Species," published as U.S. Patent Application Publication No. 2006/0163385 on July 27, 2006; U.S. Patent Application Serial No. 11/024,228, filed December 28, 2004, entitled "Method and Apparatus for Fluid Dispersion," published as U.S. Patent Application Publication No. 2005/0172476 on August 11, 2005; U.S. Patent Application Serial No. 11/360,845, filed February 23, 2006, entitled "Electronic Control of Fluidic Species," published as U.S. Patent Application Publication No. 2007/000342 on January 4, 2007; International Patent Application No. PCT/US2006/007772, filed March 3, 2006, entitled "Method and Apparatus for Forming Multiple Emulsions," published as WO 2006/096571 on September 14, 2006; U.S. Patent Application Serial No. 11/368,263, filed March 3, 2006, entitled "Systems and Methods of Forming Particles," published as U.S. Patent Application Publication No. 2007/0054119 on March 8, 2007; U.S. Provisional Patent Application Serial No. 60/920,574, filed March 28, 2007, entitled "Multiple Emulsions and Techniques for Formation"; and International Patent Application No. PCT/US2006/001938, filed January 20, 2006, entitled "Systems and Methods for Forming Fluidic Droplets Encapsulated in Particles Such as Colloidal Particles," published as WO 2006/078841 on July 27, 2006. The following are also incorporated herein by reference: U.S. Patent Application

Serial No. 09/954,978, filed September 18, 2001, entitled "Differential Treatment of Selected Parts of a Single Cell with Different Fluid Components," by Takayama, et al., now U.S. Patent No. 6,653,089, issued November 25, 2003; U.S. Patent Application Serial No. 09/954,710, filed September 18, 2001, entitled "Method and Apparatus for Gradient Generation," by Jeon, et al, now U.S. Patent No. 6,705,357, issued March 16, 2004; U.S. Provisional Patent Application Serial No. 60/997,996, filed October 5, 2007, entitled "Reactions Within Microfluidic Channels," by Stone, et al; and U.S. Provisional Patent Application Serial No. 60/997,994, filed October 5, 2007, entitled "Formation of Particles for Ultrasound Application, Drug Release, and Other Uses, and Microfluidic Methods of Preparation," by Stone, et al; and an International Patent Application filed on even data herewith, entitled "Formation of Particles for Ultrasound Application, Drug Release, and Other Uses, and Microfluidic Methods of Preparation," by Stone, et al. The following examples are intended to illustrate certain embodiments of the present invention, but do not exemplify the full scope of the invention.

EXAMPLE l

This example describes an embodiment of the invention useful for determining an enzymatic reaction that follows Michaelis-Menten kinetics. Enzymatic reactions, where one reacting species serves as a catalyst for converting another species into a desired product, are ubiquitous in biological systems. Accordingly, accurate measurements of the rate constants associated with specific enzymatic reactions are crucial for applications in biochemistry, medicine, and biochemical engineering. Many enzymatic reactions are characterized by the Michaelis-Menten reaction scheme

E + S^ZZl E - S—^→E + P (1) where E , S , and P represent the enzyme, substrate and product, respectively. This reaction was first investigated in detail by Michaelis and Menten, who showed that, in a well-mixed system, the initial rate of product formation (i.e., the reaction 'velocity') is

provided that the concentration of intermediate species E-S is quasisteady. Here the subscript '0' denotes the initial concentration or reaction rate, and K_n, ≡ (k_cat +R₂)Ik_x is the Michaelis constant, with units of concentration. The constant K_n, serves as an important indicator of whether the reaction rate is limited by the amount of substrate (i.e., [S]₀ <s: K_1n) or by the enzyme being saturated ([S]₀ » K_n ). Consequently, determination of K_n, is a primary objective of kinetic analyses on many enzymatic reactions. The conventional approach to determine K_1n is to measure the initial reaction rate for many different initial substrate concentrations and to fit the data to Eq. (2). Such plots, called Michaelis-Menten plots (or Lineweaver-Burke plots in linearized form) require multiple separate experiments to yield accurate measurements of K_n, . The present example illustrates an approach based on a co-flow microfluidic device, which yields the enzymatic rate constants ki and k_cat with a single experiment. This example illustrates reactions involving enzymes, which are typically large compared to the substrate and consequently have a significantly smaller diffusivity, i.e. D_E « D_s . However, it should be noted that other species besides enzymes may be used as well, in other embodiments of the invention. This example demonstrates that, with this approach, the governing equations accounting for convection, diffusion, and reaction are simplified and under appropriate conditions a simple power-law solution is obtained. Analytical and numerical calculations, as discussed below show that the product concentration ([P]) scales in the downstream direction as x⁵'² in this particular example, with prefactor proportional to k_xk_cat . Further downstream the species may be treated as being well-mixed, and for sufficiently high initial substrate concentrations the product concentration may scale linearly in x with the slope generally proportional to K_m • Thus, measuring the product as a function of position may be used to determine the desired rate constants, without necessitating multiple experiments or comparison to numerical calculations. Moreover, the necessary measurements can be accomplished in a single microfluidic experiment, as shown in this example. Reaction-Diffusion Model

At the outset, it should be understood that the model explained in detail below is for explanatory purposes only, and is not to be taken as limiting the scope of the invention. In this example, a pressure-driven flow along the x-axis of a microfluidic system with width w and depth h is considered. The flow profile inside a microfluidic channel, although laminar, depends in general on both y and z (Fig. 1). In microchannels with large aspect ratio (w/h » 1), however, the flow is mainly uniform in the y -direction and parabolic in the z -direction. Because of the parabolic profile, in general the species concentrations are not uniform in the z-direction. The concentration profiles near the top and bottom of the channel (where the velocity is lowest) are different compared to the center of the channel. Most importantly, if the height h is sufficiently small, the species concentrations are approximately homogeneous in the z - direction. As a first approximation, the parabolic profile complexity may be neglected and attention focused on the behavior near the center of the channel (z ~ h/2), where the species concentrations are relatively uniform. In this case, the reaction-diffusion process may be modeled as being two-dimensional. Moreover, it can be assumed that convective transport by a mean fluid velocity u dominates over diffusive transport in the downstream x -direction, which is valid if the Peclet number Pe = uhl D is relatively large, as is representative of most microfluidic conditions (e.g., for a system with u ~ 10^"2 m/s. h ~ 10^"4 m, and D ~ 10^'9 m²/s, the Peclet number is 103). In this situation, the governing equations for the concentrations of each species involved in the enzymatic reaction are

Substrate: u^ = D_s ^-^β- Ic₁[S][E^ k₂[ES] (3) dx dy

Enzyme: u (4)

Intermediate: _u%^ = D_tM] ^p- ₊ k_l[S][E]-(k₂ + ^E-S] (5)

Product: _u^ = D_p ^ ₊ k_cβl[E-S] (6) ox oy Here D₁ is the diffiisivities of species j , and u is the mean fluid velocity in the channel.

This model is essentially the basis of the Michaelis-Menten model, with the additional complexity of the transverse diffusive terms. At the entrance of a Y-shaped junction in which the substrate and the enzyme are introduced, e.g., as is shown in Fig. 1, looking in the x-direction, the substrate is introduced on the right-hand side (y < 0) and the enzyme on the left (y > 0), while the intermediate and product concentrations on both sides are assumed to be zero. Thus, the initial conditions are

[S] = (I-HCy)) [S],

[E] = H(y) [E],

= 0 : [E-S] = 0

[P] = O where the subscript / denotes the initial concentration and H(y) is the unit step (Ηeaviside) function, i.e., H{y < 0) = 0 and H{y > 0) = 1.

Equations (3)-(6) can be made dimensionless by normalizing the substrate and product concentrations on [S]₁ and the other concentrations on [E]₁ . Choosing the characteristic length scale D_s Iu for both x and y then yields the dimensionless system

Substrate: M = ^gI -K₁[S][E] + K₂Q[ES] (8) dX dY²

Enzyme: (9)

a2 r

Intermediate: ^p- = δ_lES] ^f?- + K₁Q-¹ [S] [E]-(K₂ + K_cat IBS] (10)

d[P] _ ₅ d²[P]

Product: - ^υLΛ = _δp ^A _{+ Kcal}Q[ES] (11) dX ^P dY¹

Here the following dimensionless parameters may be defined: K - ^k^^E]' ^Ds K - ^k2°s _κ - Ka,D_{s O} _ [E], _({2, u u u [S]₁ and δ_j is the diffusivity of species j normalized on D₅.

This system of equations is readily solved numerically by standard methods, but for ease of experimental interpretation a compact analytical solution is desirable. Before examining the detailed numerical solution, this example first demonstrates that under appropriate conditions simple scaling laws exist for the total amount of product obtained as a function of distance down the channel. Two assumptions used in this example are (i) a small enzyme diffusivity and (ii) a small reaction rate. Perturbation solution

As a first step, situations where the enzyme diffusivity is small compared to that of the substrate are examined, i.e., δ_E « 1 . Most enzymes are significantly larger than their respective substrates. For example, the molecular weight of firefly luciferase is 61 kDa while ATP is only 0.5 kDa, yielding D_EID_S ~ 0.2 (δ_E « 10^"2 ). Since the intermediate species is larger than the enzyme itself, it follows that the intermediate diffusivity δ_ES is likewise small. Thus, when the enzyme and substrate are brought together at a Y-junction, the substrate rapidly diffuses in the y -direction toward the enzyme, while the enzyme remains relatively confined to its original side of the channel. Close to the junction, then, the transverse diffusion of the enzyme and intermediate may be neglected. Also, the concentration of intermediate species is approximately governed by a balance between convection and the forward reaction, i.e., the first and third terms in equation (9). Note that the back reaction is negligible for small values of x because the initial concentration of E-S is small. This represents a useful simplification, since equations (9) and (10) are reduced from partial to ordinary differential equations. This approximation requires that the concentration of intermediate species varies with position as:

[_E . S] _x ^[S][E]_{x (12B)} u

To make further progress, estimates of [S] and [E] as functions of x are required.

As a second simplification, attention is focused on systems where the rate of depletion of substrate and enzyme by chemical reaction rate is "small" compared to the influence of diffusion and convection. A precise definition of small is obtained via a formal perturbation analysis, which indicates that the reaction terms in equations (8) and (9) are negligible if it can also be assumed that the dimensionless rate constant K₁ is small, K_{χ ≡} hMΑ _<< \ ₍₁3) Given the representative values D₅ « 10^~9 m²/s and u « 10^~2 m/s, and a practical upper bound of [E]₁ < 10^~3 M, equation (13) is satisfied if £, « 10⁸ IVT¹

which covers a wide variety of enzymatic reactions. For systems that satisfy this constraint, the reaction term in equation (8) is negligible compared to the diffusive and convection terms. In this situation, it is convenient to treat K_x as a perturbation parameter, i.e. ε ≡ K₁ , and to define the expansion solution

[S] _≡ [S₀] + s[S₁] + *² [S₂] + - " (14) with comparable expressions for the other three species. Here [S₀] represents the zeroth order concentration in the absence of any reaction, while [S₁] represents the first order correction due to the reaction. Substitution of the perturbation expansion into equations (8) - (11) and recalling that the enzyme and intermediate diffusivities are negligible, thus, the zeroth order governing equations are

Substrate: %M ₌ QM (I₅) dx dY²

Others species: MJ _{= 0}, M^-U ₀, Sl = O (16) dX dX dX Given the initial conditions specified in equation (7), the zeroth order intermediate and product concentrations are zero for all X and the enzyme concentration is simply H(Y) . If it is assumed assume that the transverse width w of the channel is much greater than the characteristic width D₈ Iu , then for small X the walls of the channel may be assumed to be sufficiently far from the diffusion front that their influence is negligible. In this situation, equation (15) has the classic similarity solution

where erfc denotes the complimentary error function.

With the zeroth-order solutions in hand, for the order ε system of equations,

Substrate: (18)

Enzyme: W- = -β^'1 [S₀][S₀] (19) oX Intermediate: §i^L = Q^₀][E₀] (20) oX

Product: ^ 3tøU] _ == δ_δ *_p, ^ 9²IP₁) ₊ K_calQ[E-SJ (21) dx dY²

Equation (18) is soluble in theory using an involved procedure based on Green's functions. However, inspection of equation (20) indicates that a detailed solution for [S₁] is not necessary to obtain the leading order solution for [ES₁] . Indeed, equation (20) is a first-order ODE with respect to X , so a general solution for arbitrary X and Y is obtainable. The resulting solution involves imaginary exponential integrals.

Instead, it is noted that from an experimental point of view, measuring the total amount of product formed at any specific value of X is desired, i.e., the integral of [P] with respect to Y . Thus, an integral solution, defined by integrating over both the width and height of the channel, is

(E-S₁ ) ≡ HJJiE-S₁] dY (22) where h is the scaled height of the channel. Note that this quantity, when converted back into dimensional terms, has units of moles per unit length. A comparable definition applies for (P₁) . Integrating equation (20) over Y and making use of the result

^J- 2 UVJJ the leading order concentration of intermediate is obtained

Likewise, after noting the diffusive flux of product must be zero at infinity (i.e., 3(P₁)ZdY = 0 at Y = ±∞), the leading order product concentration is obtained

/M =IM- _X^ ₍₂5)

Recasting this last result into dimensional terms (and recalling that [P] = K₁[P₁] ), the initial product concentration varies as

^ _ 4MΛJ5],[£],Z)f _{χ5;2 (26)} It should be understood that this result represents the average amount of product, with units of moles per length, at a specific value of x, for this particular system. Note that the product increases linearly with the concentrations of substrate and enzyme, consistent with the reaction scheme in equation (1). Likewise, to leading order the product scales linearly with the rate constants k_x and k_cat . The reverse binding rate constant k₂ does not affect the product to leading order, however, because the influence of the reverse reaction is negligible at early times. Not surprisingly, the amount of product depends on the substrate diffusivity (which controls how quickly the substrate diffuses into the enzyme) and is sensitive to the average velocity. Increasing the velocity decreases the amount of product at a given value of x because the reactants are pushed further downstream before they are able to react.

The linear dependence on the concentrations revealed in equation (26) suggests that this microfluidic technique could be used to determine unknown substrate or enzyme concentrations, provided all the other parameters are known a priori. Most frequently, however, the rate constants are unknown. Experimentally, measurements of the amount of product versus x^s'² should yield a straight line with slope proportional to ^ k₀₀₁ . This measurement thus provides a bound on possible values of the rate constants.

Downstream solution

The previous analysis focused on the initial creation of product close to the junction where enzyme and substrate are brought together. In this region, the species are not well mixed and the diffusivity of the substrate limits the reaction. Sufficiently far downstream, however, all of the species are completely mixed by diffusion. In this region, all of the diffusive terms in equations (8)-(l 1) vanish, so the governing equations reduce to

Substrate: u^ = -It₁[S][E] + It₂[E-S] (27) dx

Enzyme: u^ = -*,[S][£] + (*₂ +^E-S] (28) ox

Intermediate: u (29)

Product: u^ = k_cal[ES] (30) dx This is exactly the same system of equations studied by Michaelis and Menten, with the time derivatives recast as spatial derivatives in the moving reference frame. Following Michaelis and Menten, using the usual assumption that the concentration of intermediate species is quasi-steady, i.e., d[ES] /dx = 0 , yields the result

which has the same form as the classic Michaelis-Menten result. Here, however, the reaction rate far downstream is of interest, so unlike the classic result the reaction rate and substrate concentrations are not the initial values. Generally speaking, the reaction rate will decrease as the reaction progresses since [S] necessarily decreases. Because [S] is unknown, equation (31) is not particularly helpful for predicting the reaction rate for arbitrary substrate concentrations.

Nonetheless, if the substrate concentration far downstream is still large compared to K_1n , then equation (31) simplifies to d[P] _ k_cal[E]_{0 (32)} dx u In this situation, the reaction rate is constant with magnitude proportional to k_cal ; if the initial enzyme concentration and velocity are known, then k_cal is readily determined. Once k_cal is known, k_x is then obtained by fitting the upstream concentrations to x⁵'² . Under what conditions will [S] be large enough, even far downstream, for this approach to be valid? The product must not be depleted by subsequent reactions (as is the case with luciferase, see below) nor by any reverse reaction of product back to an intermediate state. Furthermore, the amount of substrate consumed in the upstream part of the channel should be small. To estimate the amount of substrate consumed, a mixing length L_ma is defined which describes the length of channel required for all species to be well mixed across the entire width of the channel. Since the substrate and product diffusivities are much larger than the enzyme diffusivity, the mixing length is governed by the enzyme and an estimate of the required mixing length is

^Lmιx ^"TT^" V> ^J ) Thus, with a sufficiently high initial substrate concentration and in the absence of other mixing forces, a constant reaction rate will pertain for JC » L_mx . To test whether a significant fraction of substrate is consumed prior to reaching x = L_mx , the amount of consumed substrate can be estimated by integrating equation (32) over x from 0 to L_mιx .

It should be understood that this calculation yields an overestimate of the substrate consumed, because it assumes that the substrate and enzyme are well-mixed even at x = 0. Nonetheless, it serves as a useful upper bound on how much substrate is consumed. Integration yields

[S]_a consumed (34)

Thus, if the inequality

is satisfied, then equation (32) may be safely used to determine k_cal . Of course, if k_cat is not known a priori, then equation (35) serves as a post facto consistency check.

At even larger distances downstream, eventually the substrate concentration will decrease to values such that [S] ~ K_m and equation (32) no longer applies, i.e., the rate of product formation will decrease with JC . Furthermore, a more accurate gauge of when the initial x⁵'² scaling changes to linear is desirable. Numerical Calculations

Equations (3) through (6) were solved numerically in Matlab using standard methods, with the initial conditions at x = 0 given by equation (7). No-flux boundary conditions were applied at fixed values of y = ±w/2 , where wul D₅ = 50 was chosen as a representative width; changes in w did not qualitatively affect the results.

Representative values of the initial concentrations, rate constants and diffusivities were chosen based on reported values for the reaction between ATP and firefly luciferase and are listed in Table 1. For each series of numerical calculations, all parameters were fixed at the values specified in Table 1 except for the systematically varied parameter. Representative numerically calculated contour plots of the scaled concentration of each species in the enzymatic reaction near the entrance of a Y-j unction microfiuidic channel as a function of xand y are presented in Fig. 2. Blue is zero concentration, red is high concentration. Flow is in the positive x-direction (left to right). The substrate diffuses rapidly compared to the enzyme, so the intermediate and product species are produced almost entirely on the enzyme side of the channel. The parameters used in these figures are: [S]₁ = 10^"2 M, [E]₁ = 1(T⁶ M, ki = 10⁴ NT¹S^"1, k₂ = 1 s^"1, k_cat = 10 s^"1, D_s = Dp = 10^"9 m²/s, D_E = D_ES ⁼ 10^"" m²/s. From this perspective, the substrate is entering from the bottom (i.e., y < 0 ) while the enzyme enters along the top ( y > 0 ). Due to its higher diffusivity, the substrate rapidly spreads outward in the transverse direction (Fig. 2A), while the slower enzyme stays relatively confined to its original side of the channel (Fig. 2B). Because the rate of binding between substrate and enzyme is sufficiently slow and the initial substrate concentration is sufficiently large, comparatively little of the substrate and enzyme are consumed by reactions. Thus, to good approximation both the substrate and enzyme concentrations are (at these early stages) governed entirely by diffusion.

In contrast, the intermediate species E-S can only form where the concentrations of enzyme and substrate are both nonzero. Thus, E S appears only on the enzyme side of the channel in regions where the substrate concentration has appreciably increased by diffusion (Fig. 2C). The profile of the resulting E S contours is consequently asymmetric, growing in the positive y-direction as the substrate penetrates further into the enzyme side of the channel. The concentration of product is similarly asymmetric, since it is only produced wherever E S is formed (Fig. 2D). Because the diffusivity of product is much larger, however, P spreads out noticeably by diffusion whereas the concentration profile oϊ E S is relatively sharp.

To compare the numerical calculations with the scaling predictions of the previous section, the numerically calculated concentrations of each species were integrated from y = -w/2 to y = +w/2 using a standard quadrature routine. This example focuses on the product concentrations, since these are the most relevant experimentally. Fig. 3 A shows the influence of the initial substrate concentration (with all other parameters fixed) in logarithmic coordinates. In Fig. 3A, the initial substrate concentration is varied. The symbols in Fig. 3A are as follows: squares, 10^"7 M; triangles, 10^"4 M; circles, 10-¹ M. The squares and triangles are not differentiable at this scale. The inset in Fig. 3A is a plot of the integral production concentration at a fixed value of x_u/D_s = 10⁶ versus the initial substrate concentration. For small values of x, the amount of product (scaled on the initial substrate concentration) is linear on a log-log plot and increases as x^sl2 , in accord with the perturbation theory. Note the scaled magnitude is invariant with initial substrate concentration at small x ; although the total amount of product is increased for larger [S]_11n, , the relative proportion is unchanged.

The x^5/2 dependence persists for values of xul D_s up to approximately 10⁴ , at which point the curve begins to bend over and the slope approaches 1, consistent with equation (32). For small values of [S]_ml , the scaled downstream product concentration is similarly independent of [S]_1n,, , as the curves collapse onto one another. For sufficiently high initial substrate concentration, however, the relative proportion of product begins to decrease. Physically, the relative decrease occurs because so much substrate is present that there is insufficient enzyme to yield an equivalent reaction rate. This behavior is captured in the inset of Fig. 3 A, which shows the scaled amount of product at a fixed value of x for different initial substrate concentrations. For small [S]₁₁₁₁, , the scaled amount of product is invariant, but above a critical concentration the relative amount of product decreases. In regard to extracting the rate constant from the downstream product concentration, it is clear from Fig. 3A that a high value of [S]_1n,, is preferable in terms of yielding a long linear regime. The effect of initial enzyme concentration, with all other parameters fixed, is shown in Fig. 3B. In Fig. 3B, the initial enzyme concentration is varied. The symbols in Fig. 3B, in order from triangles to circles, represent initial enzyme concentrations of 10^"9, 10^~8, 10^"7, ... to 10^"3 M, respectively. Increases in enzyme concentration increase the amount of product formed at given values of x . Regardless of the enzyme concentration, however, the initial x⁵¹² scaling is observed, further corroborating the perturbation theory. The downstream linear scaling is also observed, but in contrast to the effect of initial substrate concentration, the linear regime is most robust for very small values of [E]₁₁₁₁₁. Physically, if the initial enzyme concentration is high, a larger proportion of substrate is consumed near the entrance of the device before the species are well mixed. Thus, smaller enzyme concentrations are favorable for extracting kinetic parameters using the approach described here.

The influence of each kinetic rate constant is explored in Fig. 4. In Fig. 4A, the binding rate constant ki is varied. The symbols in Fig. 4A, in order from triangle to circle, represent, respectively, ki = 10¹, 10², ... to 10⁷ M^'V. In Fig. 4B, the catalytic rate constant k_cat is varied. The symbols in Fig. 4B, in order from triangle to circle, represent, respectively, k_ca, = 10^"2, 10^"1, ... to 10⁴ s^"1. In Fig. 4C, the reverse binding rate constant fø is varied. The symbols in Fig. 4C, in order from triangle to circle, represent, respectively, fo ⁼ 10°, 10^"2, ... to 10³ s^"1. The general trends of initial x⁵¹² scaling followed by a transition to linear growth are found in each case. For small x , increases in £, (Fig. 4A) and k_cat (Fig. 4B) both proportionally increase the amount of product at a given value of x , consistent with equation (26). At large x , however, increases in k_λ (or k_cal ) eventually fail to raise the reaction rate, since the other reaction rate constant limits the overall rate of the reaction. In other words, the reaction rate saturates for sufficiently high values of either ^₁ or k_cal . In contrast, the reverse binding constant k₂ has no effect on the initial rate of product formation at small x (Fig. 4C). For sufficiently large values of k₂ , the amount of product decreases, since the reverse reaction becomes favored. Experiments

To test the x⁵ⁿ scaling, a series of experiments were performed in a Y-j unction microfluidic channel using the bioluminescent reaction between adenosine triphosphate (ATP) and firefly luciferase/luciferin as a model system. Because of its high quantum yield and sensitivity to ATP, this reaction is widely used in biology to measure the concentration of ATP in solution. The chemical reaction is described by

ATP + LH₂ + E 7^"^^" E LH₂ - ATP *'"' > E ■ LH₂ ^■ AMP + PP,

E - LH₂ - AMP + O₂ ^k-^Λ > E L + AMP + CO₂ + hv where E represents luciferase, LH₂ is D-luciferin, L is dehydroluciferin, AMP is adenosine monophosphate, and PP₁ represents inorganic pyrophosphates. Subsequent reactions after the light-emitting step ultimately release the enzyme. This reaction is more complicated than the standard Michaelis-Menten reaction given in equation (1) because the light-emitting species is depleted by subsequent reactions. It does share the feature, however, that an enzyme and substrate form an intermediate which subsequently gives off light. Moreover, by mixing luciferin and luciferase together prior to addition of ATP, the two effectively form a single enzymatic complex due to their high affinity for each other. In this situation, the reaction scheme can be simplified as

S + rf -^→ Ϊ - S *-^■' > P⁽¹⁾ *-'² > P⁽²⁾ + /?i/ (37)

where S is ATP, E^* is the luciferin/luficerase complex, and /^>(1) and P^m are the products described in equation (36). Thus, the rate of light production is given by

^ = ^(I)] (38)

In the context of the microfluidic device, by the preceding analysis (cf. equation 26), at small values of x that the concentration P⁽¹⁾ will grow as

^⁽¹⁾ ~ *,*-.i*^5/2 (39)

Thus, the rate of photon emission is expected to initially increase as x⁵'². Therefore, counting photons versus position can be used here.

ATP, D-luciferin and firefly luciferase were purchased from Aldrich-Sigma. Physiological salt solution (PSS) was prepared as following (in mM): 4.7 KCl, 2.0 CaCl₂, 1.2 MgSO₄, 140.5 NaCl, 21.0 tris(hydroxymethyl)aminomethane, and 11.1 dextrose with 5 bovine serum albumin, pH adjusted to 7.4. The luciferase/luciferin solution was prepared by adding 100 microliters (μl) of 1 mg/ml luciferase and 2.5 mg D-luciferin into 5 mL PSS buffer. ATP solution was prepared by adding 3 mg of ATP into 10 mL distilled, deionized 18 M water (DDW) and then diluted with PSS buffer to 420 micromolar (μM). The luciferase/luciferin and ATP solution were always prepared on the day of use. Microfluidic chips were fabricated in poly(dimethylsiloxane) (PDMS) using standard soft photolithography techniques. The system was a Y-shaped microchannel, as depicted in Figure 1. The height and width of the main channel was 38 and 100 micrometers, respectively. The angle between the two arms of the chip is 90° (arbitrary). The ATP and preincubated luciferase/luciferin solution were injected at the same constant flow rate (3 microliters/min) into the microdevice using a single syringe pump (Kd Scientific, KDSlOl). The flow rate in the main channel was thus 6 microliters/min, corresponding to a mean velocity of 4 cm/s and a Reynolds number of approximately Re = uhlv ~ 2 (where v is the kinematic viscosity of the liquid), indicating the flow was laminar in this configuration. The bioluminescent signal resulting from the luciferase reaction was measured in different downstream positions of the channel by translating a 10Ox objective (NA = 0.75) on a microscope (Leica DMIRB, Bannockburn, IL) along the length of the channel. The light signal was recorded by a photon-counting photomultiplier tube (Hamamatsu, Model R1527P, Japan) installed in a housing with high voltage power supply (Photon Technology International, Model 814, Birmingham, NJ) and attached to the microscope side port. The detected number of incident photons was recorded using a national Instruments data acquisition board and Labview software. Photons were counted for 30 seconds at each location and repeated five times before moving to the next detection area. Experiments were performed at room temperature (~25 ⁰C) in a dark room; calibration trials in the absence of enzyme yielded a background intensity of 8 photons per second.

Results

Representative results from two separate experiments conducted on the same day with the same batch of enzyme are presented in Fig. 5. In Fig. 5, the squares and circles represent two different experiments under identical conditions. For the first 500 microns (log jc < 2.7), any signal resulting from the reaction is indistinguishable from the background intensity; the corrected photon intensity (i.e., signal minus background) fluctuates close to zero. Beginning around x = 500 microns, however, the amount of detected light increases. When plotted logarithmically, the rate of photon emission clearly scales as x^5/2 for x < 3000 microns (i.e., log x < 3.5). Several other experiments yielded similar x^5/2 scaling, although the magnitude of the photon emission rate differed. These differences are presumably due to varying degrees of enzyme viability from one batch to another, as well as the lack of precise temperature control during the experiment.

The key point, however, is that the experimental results strongly corroborate the scaling predictions and the detailed numerical calculations. The rate of photon emission scaled as x⁵¹² for small x . This result corroborates the predictions of the perturbation theory and the detailed numerical calculations (cf. equation (39)). For larger x , the rate began to decrease and appears to be approaching a slope of 1, qualitatively in agreement with the behavior depicted in Figs. 3 and 4. Not too much significance should be attached to this observation, however, since the true reaction [equation (36)] involved subsequent reactions that deplete the amount of the light emitting species. Indeed, other experiments at even greater x showed that the rate of light emission decreased for sufficiently large x . This result is consistent with the more complicated reaction scheme given in equation (36). However, the concentration of light-emitting species scaled as x⁵¹² close to the entrance of the channel, in accord with the theory.

The results as described above are applicable to other reaction systems. For instance, for a linear reaction (A + B --> P), the concentration of P would increase as x^3/2, when the diffusivity of A is significantly greater than the diffusitivity of B, e.g., by at least two orders of magnitude. More generally, in a reaction where A + B — > A+B — > P₁ ^„ > p₂ ._> p₃ _->... p_m the concentration of species n in this particular system will vary as x^(1/2+n).

EXAMPLE 2

The theoretical calculations outlined in Example 1 can also be carried out using a simplified analysis. This example describes a theoretical analysis of the system starting from equation (15) in Example 1. Equation (15) has a well-known analytical solution, which serves as the basis of the formal perturbation analysis in Example 1. Here, the extent of diffusion in the y-direction is described by the classic convective-diffusive length scale l_D = JD_sx/u . In other words, for sufficiently small x the length scale l_D provides an estimate of how far the substrate has diffused into the enzyme rich side of the channel (cf. Figure 1).

To estimate the reaction rate, however, an estimate of how much substrate has diffused into the enzyme-rich side is needed, rather than how far. In the absence of significant depletion by chemical reaction, conservation of mass requires that the integral amount of substrate be conserved. An integral average concentration (S) can be defined, given by integrating the volumetric concentration over both the width and height of the channel. Note (S) has dimensions of moles per length of channel and is a function of x; comparable definitions apply for the other species. In terms of a scaling estimate for the enzyme side of the channel {y > 0):

Here, IQ represents the width of the region with nonzero substrate concentration. This estimate is valid for small x where the applicable values of y (i.e., where the concentration is nonzero) are small compared to the width of the channel. Substitution of this estimate for (S) into equation (12b) yields the scaling expression

This intermediate concentration may be substituted into equation (6). After integration with respect to x:

1/2

<P>

«^V2 (42) The same result (with a different numerical prefactor) is obtained via a formal perturbation analysis as outlined in Example 1. Equation (42) represents the average amount of product, with dimensions of moles per length, at a specific ^-position along the channel. Note that according to equation (42), the concentration of product increases linearly with the initial concentrations of substrate and enzyme, consistent with the reaction scheme in equation (1). Likewise, the product concentration scales linearly with the rate constants ki and k_cat- The reverse binding rate constant fø does not affect the product concentration, however, because the influence of the reverse reaction is negligible at early times. The amount of product depends on the substrate diffusivity, which controls how quickly the substrate diffuses into the enzyme, and is quite sensitive to the average fluid velocity. Increasing the velocity decreases the amount of product at a given value of x because the reactants are pushed further downstream before they are able to react.

The immediately preceding analysis focused on the formation of product close to the junction where enzyme and substrate are first brought together. In this region, the species are not well mixed and the diffusivity of the substrate limits the reaction. Sufficiently far downstream, however, all of the species may be completely mixed by diffusion. In this region, all of the diffusive terms in equations (3)-(6) vanish, so the governing equations reduce to those used in the classic Michaelis-Menten analysis, with the time derivatives recast as spatial derivatives in the moving reference frame. Hence, the assumption that the concentration of intermediate species is quasi-steady may be used, i.e., d[E -S]IdX = Q ; since [E-S] = [E]₁ - [E]:

This equation has the same form as the Michaelis-Menten result (cf. eq (2)). However, the reaction rate far downstream is of interest here, so unlike in the classic result, the relevant substrate concentrations in equation (43) are not the initial values. Generally speaking, the reaction rate will decrease as the reaction progresses since [S] necessarily decreases. Because [S] is unknown, equation (43) is not particularly helpful for predicting the reaction rate for arbitrary substrate concentrations. Nonetheless, if the substrate concentration far downstream is still large compared to K_m, then from equation (43) the reaction rate no longer depends on the specific value of [S], and the product concentration scales linearly with x, viz., k [E] [P] « J≤H — i_x> large _{x an(}j [S] » K_m « ^" (44)

This result is analogous to the "maximum velocity" found in the classic Michaelis-Menten analysis for very high substrate concentrations. If the initial enzyme concentration and velocity in the microfiuidic channel are known, then k_cal is readily determined. Once k_cat is known, kj is then obtained by fitting the upstream concentrations to x^5/2 using equation (42). While several embodiments of the present invention have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and/or structures for performing the functions and/or obtaining the results and/or one or more of the advantages described herein, and each of such variations and/or modifications is deemed to be within the scope of the present invention. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and/or configurations will depend upon the specific application or applications for which the teachings of the present invention is/are used. Those skilled in the art will recognize, or be able to ascertain using no more than routine experimentation, many equivalents to the specific embodiments of the invention described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, the invention may be practiced otherwise than as specifically described and claimed. The present invention is directed to each individual feature, system, article, material, kit, and/or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and/or methods, if such features, systems, articles, materials, kits, and/or methods are not mutually inconsistent, is included within the scope of the present invention.

All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and/or ordinary meanings of the defined terms.

The indefinite articles "a" and "an," as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean "at least one."

The phrase "and/or," as used herein in the specification and in the claims, should be understood to mean "either or both" of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with "and/or" should be construed in the same fashion, i.e., "one or more" of the elements so conjoined. Other elements may optionally be present other than the elements specifically identified by the "and/or" clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to "A and/or B", when used in conjunction with open-ended language such as "comprising" can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

As used herein in the specification and in the claims, "or" should be understood to have the same meaning as "and/or" as defined above. For example, when separating items in a list, "or" or "and/or" shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as "only one of or "exactly one of," or, when used in the claims, "consisting of," will refer to the inclusion of exactly one element of a number or list of elements. In general, the term "or" as used herein shall only be interpreted as indicating exclusive alternatives (i.e. "one or the other but not both") when preceded by terms of exclusivity, such as

"either," "one of," "only one of," or "exactly one of." "Consisting essentially of," when used in the claims, shall have its ordinary meaning as used in the field of patent law.

As used herein in the specification and in the claims, the phrase "at least one," in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase "at least one" refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, "at least one of A and B" (or, equivalently, "at least one of A or B," or, equivalently "at least one of A and/or B") can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.

It should also be understood that, unless clearly indicated to the contrary, in any methods claimed herein that include more than one step or act, the order of the steps or acts of the method is not necessarily limited to the order in which the steps or acts of the method are recited.

In the claims, as well as in the specification above, all transitional phrases such as "comprising," "including," "carrying," "having," "containing," "involving," "holding," "composed of," and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases "consisting of and "consisting essentially of shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.

What is claimed is:

Claims

1. A method of determining a reaction comprising: flowing a first fluid containing a first reactant into a microfluidic channel; flowing a second fluid containing a second reactant into the microfluidic channel; forming an interface between the two fluids; causing a non-linear reaction between the first reactant and the second reactant that yields a product within the channel; determining a rate of product formation at a first point in the channel; and determining a rate of product formation at a second point in the channel.

2. The method of claim 1, comprising imaging at least a portion of the channel to determine the rate of product formation at the first point in the channel and determine the rate of product formation at the second point in the channel.

3. The method of claim 2, wherein imaging comprises optical imaging.

4. The method of claim 2, wherein imaging comprises fluorescent imaging.

5. The method of claim 1, wherein the interface extends a length of the channel.

6. The method of claim 1, wherein the rates of product formation at the first point and the second point are determined simultaneously.

7. The method of claim 1, wherein the rates of product formation are determined by an algorithm that calculates reaction order and/or a kinetic parameter.

8. The method of claim 1, further comprising: using the reaction product as a third reactant; flowing a third fluid containing a fourth reactant into the microfluidic channel; causing a reaction between the third reactant and the fourth reactant that yields a second product within the channel; determining a rate of second product formation at a third point in the channel; and determining a rate of second product formation at a fourth point in the channel.

9. The method of claim 1, wherein the first fluid is a liquid, and the second fluid is a liquid.

10. The method of claim 1, wherein the first reactant is an enzyme, and the second reactant is a substrate that the enzyme is able to bind.

11. The method of claim 10, further comprising providing an inhibitor that can at least partially inhibit reaction between the enzyme and the substrate.

12. The method of claim 11, wherein the inhibition is competitive.

13. The method of claim 11, wherein the inhibition is noncompetitive.

14. The method of claim 11, wherein the inhibition is uncompetitive.

15. The method of claim 11, wherein the inhibitor is contained within the first fluid.

16. The method of claim 11, wherein the inhibitor is contained within the second fluid.

17. The method of claim 10, further comprising providing a co-factor for the enzyme.

18. The method of claim 17, wherein the co-factor is contained within the first fluid.

19. The method of claim 17, wherein the co-factor is contained within the second fluid.

20. The method of claim 1, wherein the first reactant is a catalyst, and the second reactant is a substrate that the catalyst is able to recognize.

21. The method of claim 1, wherein the first reactant is a nanoparticle.

22. The method of claim 1, wherein the first reactant comprises platinum.

23. The method of claim 1, wherein fluid flow within the channel is substantially laminar.

24. The method of claim 1, wherein the first fluid and the second fluid are introduced into the microfluidic channel through first and second channels, respectively, that intersect the microfluidic channel.

25. The method of claim 24, wherein the first and second channels form a "Y" junction with the microfluidic channel.

26. The method of claim 24, wherein the first and second channels form a "T" junction with the microfluidic channel.

27. The method of claim 1, wherein the microfluidic channel is defined by a substrate formed by soft lithography.

28. The method of claim 1, wherein the non-linear reaction between the first reactant and the second reactant follows Michaelis-Menten kinetics.

29. The method of claim 1, wherein the non-linear reaction is at least a second-order reaction.

30. The method of claim 1, wherein the first fluid is the first reactant.

31. A method of determining a reaction comprising: flowing a first fluid containing a first reactant having a first diffusion coefficient into a microfluidic channel; flowing a second fluid containing a second reactant having a second diffusion coefficient into the microfluidic channel, wherein the second diffusion coefficient is larger than the first diffusion coefficient by at least two orders of magnitude; forming an interface between the two fluids; causing a reaction between the first reactant and the second reactant that yields a product within the channel; determining the rate of product formation at a first point in the channel; and determining the rate of product formation at a second point in the channel.

32. The method of claim 31, comprising imaging at least a portion of the channel to determine the rate of product formation at the first point in the channel and determine the rate of product formation at the second point in the channel.

33. The method of claim 32, wherein imaging comprises optical imaging.

34. The method of claim 32, wherein imaging comprises fluorescent imaging.

35. The method of claim 31, wherein the interface extends a length of the channel.

36. The method of claim 31, wherein the rates of product formation at the first point and the second point are determined simultaneously.

37. The method of claim 31, wherein the rates of product formation are determined by an algorithm that calculates reaction order and/or a kinetic parameter.

38. The method of claim 31, further comprising: using the reaction product as a third reactant; flowing a third fluid containing a fourth reactant into the microfluidic channel; causing a reaction between the third reactant and the fourth reactant that yields a second product within the channel; determining a rate of second product formation at a third point in the channel; and determining a rate of second product formation at a fourth point in the channel.

39. The method of claim 31, wherein the second diffusion coefficient is larger than the first diffusion coefficient by at least three orders of magnitude.

40. The method of claim 31, wherein the second diffusion coefficient is larger than the first diffusion coefficient by at least four orders of magnitude.

41. The method of claim 31, wherein the second diffusion coefficient is larger than the first diffusion coefficient by at least five orders of magnitude. i

42. The method of claim 31, wherein the first fluid is a liquid, and the second fluid is a liquid.

43. The method of claim 31, wherein the first reactant is an enzyme, and the second reactant is a substrate that the enzyme is able to bind.

44. The method of claim 42, further comprising providing an inhibitor that can at least partially inhibit reaction between the enzyme and the substrate.

45. The method of claim 44, wherein the inhibition is competitive.

46. The method of claim 44, wherein the inhibition is noncompetitive.

47. The method of claim 44, wherein the inhibition is uncompetitive.

48. The method of claim 44, wherein the inhibitor is contained within the first fluid.

49. The method of claim 44, wherein the inhibitor is contained within the second fluid.

50. The method of claim 42, further comprising providing a co-factor for the enzyme.

51. The method of claim 50, wherein the co-factor is contained within the first fluid.

52. The method of claim 50, wherein the co-factor is contained within the second fluid.

53. The method of claim 31, wherein the first reactant is a catalyst, and the second reactant is a substrate that the catalyst is able to recognize.

54. The method of claim 31, wherein the first reactant is a nanoparticle.

55. The method of claim 31, wherein the first reactant comprises platinum.

56. The method of claim 31, wherein fluid flow within the channel is substantially laminar.

57. The method of claim 31, wherein the first fluid and the second fluid are introduced into the microfluidic channel through first and second channels, respectively, that intersect the microfluidic channel.

58. The method of claim 57, wherein the first and second channels form a "Y" junction with the microfluidic channel.

59. The method of claim 57, wherein the first and second channels form a "T" junction with the microfiuidic channel.

60. The method of claim 31, wherein the microfluidic channel is defined by a substrate formed by soft lithography.

61. The method of claim 31, wherein the reaction between the first reactant and the second reactant follows Michaelis-Menten kinetics.

62. The method of claim 31, wherein the reaction is at least a second-order reaction.

63. The method of claim 31, wherein the first fluid is the first reactant.