Scheduling large-scale micro/nano biochemical testing: Exact and heuristic algorithms

Abstract

We consider a micro/nano fluidic toolbit that consists of a set of identical testing units, each of which contains a microchannel that has an array of equally spaced nanopores opened along it. In each microchannel, same equally spaced chemical liquid plugs shift back and forth under pneumatic pressure. Below each nanopore is a testing tube that accepts appropriate nanoscale chemical droplets from the microchannel above to perform biochemical tests. Each tube may require several different chemicals in sequence to get proper results. Liquid chemicals required in different tubes may be dropped simultaneously in a round if the liquid plug sequence in the microchannel above matches the chemical requirements in these tubes. The sizes of testing problems in terms of the numbers of tubes, liquid chemicals required in each tube and liquid plugs in the microchannel are large, efficient testing procedure requires careful “round” scheduling in order to shorten the testing time span. In this research, we model the biochemical test scheduling as the fixed plug sequence problem (FPSP), where the liquid plug layout in the microchannel is given. We show that the FPSP is NP-hard in general, and then develop both exact and heuristic algorithms. The computational performances of the proposed algorithms are provided and contrasted.

Introduction

The fears about another potential round of terrorist attacks by biological weapons have motivated researchers to find better ways to handle large-scale biochemical screening. Such efforts involve designing and implementing microfluidic devices, such as laboratory-on-a-chip devices. Besides deterring biological attacks, the technology of large-scale biochemical testing also facilitates rapid advances in gene discovery, genetic mapping and gene expression with broader applications ranging from infectious diseases, drug screening and cancer diagnostics to food quality and environmental evaluation.

In this work, we consider a micro/nano fluidic toolbit, which is conceptually illustrated in Fig. 1, which contains a large set of identical testing units. Each testing unit has a microchannel with an array of equally spaced nanopores opened along it. In the microchannel, equally spaced chemical liquid plugs shift back and forth under pneumatic pressure. Below each nanopore is a testing tube that accepts appropriate nanoscale chemical droplets from the microchannel above to perform biochemical tests. Each tube may require several different chemicals in sequence to get proper results. Liquid chemicals required in several different tubes may be dropped simultaneously in a round if the liquid plug sequence in the microchannel above matches the chemical requirements in these tubes.

In each testing unit, chemical liquid plugs are introduced to the microchannel in such a way that the center distance between any two adjacent liquid plugs equals the distance between any two adjacent nanopores. The train of equal-length chemical liquid plugs in the microchannel shifts back and forth under pneumatic pressure. When the liquid plug train comes to a proper alignment with the underlying testing tubes, the corresponding nanopores are opened and the required nanoscale chemical droplets are deposited into the corresponding tubes to finish a round of biochemical tests. In each test round, the liquid plug train needs to be precisely sensed and positioned by sensor arrays arranged near the nanopores [1]. Since the liquid plugs in the microchannel are at a micro scale and the droplets are at a much smaller nano scale, we assume that the size of each liquid plug remains unchanged after each test round. Another concern is the potential liquid contamination due to the direct contact of different liquid plugs in the microchannel. In the micro scale, there is no turbulent mixing but only slow diffusions at the direct liquid-liquid interface at a very low Reynold number [2], [3]. Ismagilov et al. [4] developed a technique that allows direct fluid–fluid contacts for combinatorial experiments and detection purposes. Every nano droplet comes out from the center of a liquid plug and therefore the impact of diffusion is negligible.

The numbers of tubes, liquid chemicals required in each tube and liquid plugs in the microchannel are large, and for the system to run efficiently and to achieve high testing throughput requires careful scheduling. This research is devoted to exploring good scheduling methods to shorten the total biochemical testing time span. Each testing unit in the studied micro/nano fluidic toolbit can be considered independently, so we focus on a single testing unit in the following discussion. Assuming each testing task takes the same amount of time to complete, our objective of minimizing the testing time span is equivalent to scheduling the test sequence in such a way that the number of test rounds is minimized.

In this work, we consider that the liquid plug layout in the microchannel is given, and so model the micro/nano biochemical test scheduling problem as the fixed plug sequence problem (FPSP). In Section 2, we establish the criteria for a scheduling solution to be feasible. Then the complexity of the FPSP is explored in Section 3. Mathematical programming models and exact algorithms are developed in Section 4. Several heuristics are provided and analyzed in Section 5. Solution qualities and performances of the proposed algorithms are computationally studied in Section 6. Section 7 concludes.