Chemical Engineering Fundamentals

Abstract

Kinetic rate constants for enzymatic reactions are typically measured with a series of experiments at different sub-strate concentrations in a well-mixed container. Here we demonstrate a microfluidic technique for measuring Michaelis-Menten rate constants with only a single experiment. Enzyme and substrate are brought together in a coflow microfluidic device, and we establish analytically and numerically that the initial concentration of product scales with the distance x along the channel as x 5/2. Measurements of the initial rate of product formation , combined with the quasi-steady rate of product formation further downstream, yield the rate constants. We corroborate the x 5/2 scaling result experimentally using the bioluminescent reaction between ATP and luciferase/luciferin as a model system. Enzymatic reactions, wherein 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, food science, and biochemical engineering. Many enzymatic reactions are characterized by the Michaelis-Menten 1 reaction scheme where E, S, and P represent, respectively, the enzyme, substrate, and product. 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 quasi-steady. 2 Here the subscript "i" denotes the initial concentration or reaction rate, and K m ≡ (k cat + k 2)/k 1 is the so-called Michaelis constant, with dimensions of concentration. The constant K m serves as an important indicator of whether the reaction rate is limited by the amount of substrate (i.e., [S] i , K m) or by the enzyme being saturated ([S] i. K m). Consequently, determination of K m is a primary objective of kinetic analyses on enzymatic reactions. The conventional method to determine K m is to measure the initial reaction rate for many different initial substrate concentrations [S] i and to fit the data to eq 2. This approach requires multiple separate experiments to yield accurate measurements of K m. In this work we propose a different approach, based on a coflow microfluidic device, which yields the enzymatic rate constants k 1 and k cat with a single experiment. The advantages and general features of microfluidic devices have been widely discussed; 3,4 a key advantage is the possibility of measuring rate constants with substantially reduced amounts of enzyme compared to standard techniques. Previous work on coflow microfluidic devices 5-11 has focused on reactions of the form A + B 7 98 K eq C, where A and B are brought together at a Y-shaped junction (cf. Figure 1). Provided the channel dimensions are sufficiently small, then the flow is laminar and the species slowly diffuse toward one another (transverse to the primary flow direction) and then react to form C. The kinetic parameters are determined by measuring [C] as a function of position downstream from the junction. For small molecules with comparable diffusivities, however, there is no straightforward procedure to extract rate constants from the experimental data; the rate constants must be treated as fitting parameters in numerical computations of the full set of reaction-diffusion equations. Reliance on numerical calculations is inconvenient for experimentalists, so a compact analytical solution is desirable. Here we demonstrate that under appropriate conditions the governing equations, accounting for convection, diffusion, and reaction, are simplified and a simple power-law solution is obtained for the spatial evolution of the product concentration. Specifically, analytical and numerical calculations show that the product Dubrocq, C.; Tabeling, P.; Charier, S.; Alcor, D.; Jullien, L.; Ferrage, F. Anal. Chem. 2005, 77, 3417. (10) Benninger, R. K. P.; Hofmann, O.; O ¨ nfelt, B.; Munro, I.; Dunsby, C.; Davis, D. M.; Neil, M. A. A.; French, P. M. W.; de Mello, A. J. Angew. Chem., Int. Ed. 2007, 46, 2228. (11) Matthews, S. M.; Elder, A. D.; Yunus, K.; Kaminski, C. F.; Brennan, C. M.; Fisher, A. C. Anal. Chem. 2007, 79, 4101. E + S { \} k 1 k 2 E‚S 98 k cat E + P (1) d[P] dt | i) k cat [E] i ([S] i K m + [S] i) (2)