Using three-dimensional microfluidic networks for solving computationally hard problems PNAS March 13, 2001 vol. 98 no. 6 2961~2966 Daniel T. Chiu, Elena Pezzoli, Hongkai Wu, Abraham D. Stroock, and George M. Whitesies Harvard University

Abstract Design of a parallel algorithm Nondeterministically polynomial complete problem : Maximal clique problem Algorithm parallel fabrication of the microfluidic system parallel searching of all potential solutions by using fluid flow parallel optical readout of all solutions

Algorithm Four steps 1) for every edge [i,j] of G, label (tag) every subgraph of G that contains vertices i and j 2) for every subgraph, count the number of tags 3) decide whether there are enough tags (edges) in each subgraph to be a clique 4) return the size and identity of the largest clique

Implementation of algorithm Fig A : schematic diagram of 3-vertex graph subgraph well edge reservoir 3D microfluidic system  to avoid the crossover  three vertices & four layers Quantitation of the connectivity  measure the flow from reservoir into well  use liquid containing a uniform suspension of fluorescent beads (filter membrane was used) connected by channel

putting a calibrated number of beads into each reservoir Step 1 putting a calibrated number of beads into each reservoir (parallel operation) spilts and flows simultaneously into both channels Step 2 exploit the optical systems to read out the relative amount of fluorescence in each well (parallel) Step 3 setting the appropriate optical detection threshold for each clique size Step 4 observing the position of the clique along the x and y axes in microfluidic device the size of a clique can be easily derived by knowing its relative displacement along the y axis

Schematic of the microfluidic device

Quntitation with fluorescence [1,2][1,3][2,3] introduced [1,2][2,3] introduced

Materials and Methods PDMS REM Small sizes of fluorescent beads (400nm or smaller) : no clogging in microfluidic system

Results and Discussion Needed layer (= edges layer + bottom layer) n(n-1)/2 for edges in six vertex problem, 15 + 1 = 16 layer Subgraphs with k vertices must have k(k-1)/2 units of fluorescence to be clique  threshold criterion More relaxed criterion to account for errors set a threshold halfway between the intensities expected for a k clique and a k-1 clique must have [(k-1)(k-2)/2 + (k-1)/2] fluorescence intensities

Even splitting the pressure drop along the two branches is identical cross section and the total length of each channel pathway from reservoir to waste are the same!  flow rate in each channel are indistinguishable

Experimental solution to a Three-Vertex Graph exactly three times!!

Experimental solution to a Six-Vertex Graph I

Experimental solution to a Six-Vertex Graph II

Discussion and Conclusions I Analog computation device Parallel operation Microfluidic channel Multi-layer structure Fluorescent beads strength high parallelism vs space-time tradeoff weakness exponential increase in its physical size

Discussion and Conclusions II Potential sources of error 1) biased splitting of fluorescent beads at each channel branching 2) misalighment between layers that results in error in the integrated fluorescence intensities  main cause of misalignment between layers is differential shrinkage of PDMS in different layers during fabrication To overcome errors 1) implement error-correction step before integrating the intensities from each layer (reset to zero) 2) use exactly the same procedures (using the same amount of catalyst and curing at the same temperature)

Discussion and Conclusions III Limitation largest graph : 20 vertices or 40 vertices impractical Advantages to using microfluidic system using parallel optical system different color beads (fluorescent) no requirement in power Personal thinking scale-up : both device and fluorescence