Supplemental Information
A. Expression and purification of EGFP
Expression of Enhanced Green Fluorescent Protein (EGFP) uses the pET plasmid in E.
coli C41 grown in yeast extract medium and induced with 2.25 mM Isopropyl-β-D-1thiogalactopyranoside.
Initial growth medium consists of 3 g/L [NaPO3]6, 1.6 mL/L 2M
MgSO4.7H2O, 1.9 g/L Citrate.H2O, 2.1 g/L NH4Cl, 0.1 g/L FeSO4.7H2O, 15 g/L Yeast Extract,
20 g/L dextrose, 0.2 µg/L MnCl2.4H2O, and 0.4 mL/L antifoam 204. Post- sugar-depletion feed
medium consists of 3 g/L [NaPO3]6, 10.5 g/L MgSO4.7H2O, 2.6 g/L Na3Citrate.2H2O, 3 g/L
(NH4)2CO3, 0.1 g/L FeSO4.7H2O, 210 g/L Yeast Extract, 275 g/L dextrose, and 2.4 µg/L
MnCl2.4H2O. Purification of EGFP begins by harvesting the cells at a density of 70 g/L dry cell
weight as measured by optical density using centrifugation followed by resuspension of the cell
pellet at 70 g/L dry cell weight in 20 mM Tris buffer pH 8.0. Lysis of the cells proceeds with a
high-pressure homogenizer using three passes at 20,000 psi peak pressure. Centrifugation of the
homogenized suspension removes the insoluble protein and cell debris. The final step consists of
filtering the supernatant, which contains the EGFP, using a 0.2-µm filter.
B. Numerical validation of the 3-D multiphysics model
Numerical validation of the 3-D Comsol Multiphysics model uses two approximate
methods for computing the cross-stream diffusion in the extending microchannel: 3-D uni-axial
flow and 2-D plug flow. Unlike the model presented in the main text, which computes the entire
volume of the device using a conjoint mesh, the validation efforts perform the computation in
two stages. The first stage computes the hydrodynamic focusing region only using the 3-D
multiphysics model in COMSOL, and the second stage propagates this solution to distances
downstream using one or the other approximate technique.
The alternative computations
computations require less RAM and therefore take place on a desktop personal computer with 8GB of RAM
1. 3-D Finite Element Uni-axial Flow Computations
Uni-axial flow computations in FLEXPDE 6.2 first propagate the analytically known
velocity distribution along the length of the channel and then solve the advection-diffusion
equation using 0, the known solution at the output of the hydrodynamic focusing region for the
inlet boundary condition, c(x,y)|z=0. Use of FLEXPDE for the second stage of computation allows
for explicit definition of the governing equations allowing for computation of the uni-axial
advection-diffusion equation, which is computationally simpler than the 3-D advection-diffusion
equation. Specifically, the uni-axial flow model considers only the extending microchannel
portion of the model and therefore solves the following equation whose form is simpler and
requires less memory to solve than the full 3-D convection-diffusion equation used by the
Comsol model presented in the main text.
uz
 2
c
2 
 D 2  2 c
z
y 
 x
[S-1]
The analytical liquid velocity in the axial direction, uz(x,y), follows from the known analytical
solution to the Stoke’s equation for a channel of width w and height h,
u z ( x, y) 
Q  cosh( n x w0.5 ) 
1 
 sin( n hy )

3
w

cosh(
n

)
n
n odd
2h


[S-2]
,
for flow directed in the z-direction with x and y normalized between 0 and 1 and normalization
constant Q determined by setting ∫∫uzdxdy equal to the total volumetric flow-rate.
The FlexPDE computation uses a free tetrahedral finite element mesh with adaptive
meshing giving a total of 5-million elements corresponding to a characteristic element size of 4
µm. This particular computation requires 1,000 minutes of computer time using a personal
computer equipped with 8-GB RAM operating under Windows 7.
2. 2-D Finite Difference Plug Flow Computations
As a speedy alternative to 3-D finite elements, the present work develops a 2-D
computation scheme which approximates the velocity field as transverse-uniform, uz(x,y) = uAVG,
essentially following Kamholz et al, implemented here in Matlab.
The numerical scheme
marches a 2-D solution along the direction of flow using march step δz = uAVGδx2/4D, where
uAVG represents the mean fluid velocity. This corresponds to the fully explicit Forward-Time
Centered-Space scheme over a 2-D square lattice with equal node spacing, δx, in the x- and ydirections,
c ik, j 1
 c ik, j

z 4 D 1 k
1
1
1
( c i 1, j  c ik1, j  c ik, j 1  c ik, j 1  c ik, j )
2
u z x 4
4
4
4
,
[4]
the solution for which is accurate at distances large compared with δz provided δz ≤ uzδx2/4D for
numerical stability. In practice, the maximum march step δz = uzδx2/4D is often chosen in order
to minimize computation time. This computation requires less than 1 minutes of computer time
using a personal computer equipped with 8-GB RAM running Windows 7.
3. Results of the 3-D uni-axial and 2-D plug flow computations
3-D uni-axial flow computations appear nearly indistinguishable from the full 3-D
computations presented in the main text as shown in Figure S-1. This result confirms the meshindependence of the conjoint mesh used for the computations shown in the main text.
A subtle difference from the full 3-D computations occurs near the entrance of the
extending microchannel because the 3-D uni-axial computations compute the hydrodynamic
focusing region separately from the diffusion-mixing region. As a result, they can perform this
first stage of computation with finer resolution than achievable using the full 3-D model by use
of adaptive regridding of the finite element mesh. The main text does not apply adaptive
regridding to the conjoint mesh because Comsol regrids the mesh for the entire length of the
extending microchannel leading to intractable computation times and out-of-memory errors.
2-D computation leads to several inaccuracies which manifest themselves in Figure S-1 and
Figure S-2 as described in the methods section of the main text.
Figure S-1. Contour plots show experimental and computed concentration distributions at axial
distances along the length of the extending microfluidic channel according to the full 3-D
computations shown in the main text as well as alternative computations described here in the
supplemental information for purposes of numerical validation. Plots show profiles acquired
horizontally across the experimental distributions and simulations. The model assumes diffusion
coefficient 6 x 10-10 m2/s for fluorescein.
Figure S-2. Width of core based on full-width-half-max of horizontal profiles across the
concentration distributions. Solid and dashed curves 3-D and 2-D computations, respectively,
using assumed diffusion coefficients DFluor = 6 x 10-10 m2/s, and DEGFP = 9 x 10-11 m2/s.