Theory and analysis of generalized mixing and dilution of biochemical fluids using digital microfluidic biochips

Abstract

Digital microfluidic (DMF) biochips are recently being advocated for fast on-chip implementation of biochemical laboratory assays or protocols, and several algorithms for diluting and mixing of reagents have been reported. However, all methods for such automatic sample preparation suffer from a drawback that they assume the availability of input fluids in pure form, that is, each with an extreme concentration factor ( CF ) of 100%. In many real-life scenarios, the stock solutions consist of samples/reagents with multiple CF s. No algorithm is yet known for preparing a target mixture of fluids with a given ratio when its constituents are supplied with random concentrations. An intriguing question is whether or not a given target ratio is feasible to produce from such a general input condition. In this article, we first study the feasibility properties for the generalized mixing problem under the (1:1) mix-split model with an allowable error in the target CF s not exceeding 1 2d, where the integer d is user specified and denotes the desired accuracy level of CF . Next, an algorithm is proposed which produces the desired target ratio of N reagents in ONd mix-split steps, where N ( ≥ 3) denotes the number of constituent fluids in the mixture. The feasibility analysis also leads to the characterization of the total space of input stock solutions from which a given target mixture can be derived, and conversely, the space of all target ratios, which are derivable from a given set of input reagents with arbitrary CF s. Finally, we present a generalized algorithm for diluting a sample S in minimum (1:1) mix-split steps when two or more arbitrary concentrations of S (diluted with the same buffer) are supplied as inputs. These results settle several open questions in droplet-based algorithmic microfluidics and offer efficient solutions for a wider class of on-chip sample preparation problems.