Universal Microfluidic Gradient Generator

Transcription

1 Universal Microfluidic Gradient Generator Daniel Iriia 1, Dan A Geba 2, Mehet Toner 1 1 BioMEMS Resource enter, enter for Engineering in Medicine and Surgical Services, Massachusetts General Hospital, Shriners Hospital for hildre and Harvard Medical School, Bosto MA Departent of Matheatics, University of alifornia, Berkeley, A orresponding author: Mehet Toner, BioMEMS Resource enter, Building 114, 16 th St, harlestow MA Eail: toner@hs.harvard.edu Keywords: icrofluidics, gradient, odel

2 Essential for our understanding of cellular responses to biocheical heterogeneities in their icroenvironent is our ability to consistently replicate and systeatically alter cheical concentrations around cells in vitro. While icrofluidic devices have shown unatched capability in generating linear stable cheical gradients, no systeatic ethod exists today for replicating ore coplex gradients. Here we deonstrate a universal approach for generating stable gradients of any profile, starting fro only two input concentrations, by the use of atheatical theory and icrofluidic principles. Precise coplex gradients are directly relevant for bacteria and eukaryotic cell adaptation and igration. Spatial cheical gradients are also iportant for iune cells igrating against intruders 1,2, axons regenerating towards their connection target 3, and ebryonic cells differentiating in response to orphogens 4. We restrict the diffusion between two parallel streas of distinct concentrations flowing at low Reynolds nuber (lainar flow) by using a series of dividers in the longitudinal direction of the channel. At steady state, the target concentration profile is generated at the outlet, in direction transversal to the channel, by the parallel flow of several sub-streas (Figure 1). The underlying working principle is the coplete ixing of controlled fractions of adjacent streas. A new concentration ( = 1, n ) is generated by the ixing of precise fractions fro streas of concentrations n-1,-1 and n-1,. In order for such ixing to occur regularly at any positions across one level, a set of constrains for the position of dividers at two consecutive levels has to be considered: 0 < u0 < un-1,0 < u1 < un-1,1 < < u-1 < un-1,-1 < u < un-1, < < un-1,n-1 < u n (Eq.1) Using ass conservation equations: (u - u -1 ) = n-1,-1 (u n-1,-1 - u -1) + n-1, (u - u n-1,-1 ) = 1, n (Eq.2) we deonstrate that for any given set of i and u i, there exists at least one set of n-1,i and u n-1,i ( i = 0, n ) that verifies the copatibility condition fro Equation 1 (see suppleentary inforation). By using atheatical induction principles, this is also proof that for any array of N,i ( i = 0, N+ 1) concentrations at the output, a coplete set of concentrations and divider

3 positions can be found for generating these concentrations, starting fro two distinct input concentrations. One interesting reark eerging fro the atheatical deonstration is that an infinite nuber of design solutions satisfying the divider position constrains and ixing equations are possible. These solutions can be systeatically represented by the introduction of a set of coefficients α ( 0 α < 1) at our disposal (see suppleentary inforation). < ij Based on these principles we developed a coputer algorith and built icrochannels with dividers for generating four gradients, the equivalent of power, exponential, error, and cubic root functions (see suppleentary inforation). oparisons between the experiental and predicted concentration profiles showed differences less than 0.05 between the noralized experiental and theoretical concentrations for the icrofluidic devices in which the output is reconstituted fro N=10 unequal sub-streas (Figure 1b-e). The predicted error of gradient reproduction was calculated for the particular choice of divider position to be less than 0.1. In conclusio a systeatic approach for designer gradients at icroscale is presented. A atheatical apparatus was developed consisting of two parts, one for deonstrating the universal applicability of our approach and one for extracting the design details starting fro any chosen output. 1. Wilkinso P.. Assays of leukocyte locootion and cheotaxis. Journal of Iunological Methods 216, (1998). 2. Jeo N. L. et al. Neutrophil cheotaxis in linear and coplex gradients of interleukin- 8 fored in a icrofabricated device. Nature Biotechnology 20, (2002). 3. Goodhill, G. J. Matheatical guidance for axons. Trends in Neurosciences 21, (1998). 4. Gurdo J. B. & Bourillot, P. Y. Morphogen gradient interpretation. Nature 413, (2001).

4 a Level n-1 Level n Level N Input 00 u n-1,-1 n-1,-1 u -1-1 u u N,0 N,0 u N,1 N,1 Output n-1, 01 Flow u n-1, n-1,+1 +1 u u N,N N,2 N,N N,N+1 b c A A (x) = x 5 2x e (x)= e 2x 1 = e Noralized oncentration d (x) = erf(x) e (x) = x1/ Noralized Distance Figure 1. Universal gradient generator. a, The sub-strea concentrations and position of the dividers at an interediate level inside the icrochannel. b-e, The resulting icrofluidic structures, iages of steady state fluorescent gradients inside the devices, and coparisons between experiental and theoretical noralized concentration profiles for fifth power (b), exponential (c), error (d), and cubic root functions (e). Fluorescence intensity was easured at level AA for all channels. Scale bar is 500µ.

5 Suppleentary inforation

6 Input Level 0 Level 1 Level n-1 Level n Level N Output u N,0 N, n-1,-1 u n-1,-1-1 u -1 u N,1 N,1 u 10 N,2 u n-1, u n-1, u +1 u 11 n-1,+1 u Flow u N,N N,N N,N+1 Suppleentary figure 1. Scheatics of the dividers and concentrations in the icrochannel for generating a concentration gradient at the output. Propositon 1. onsidering a set of concentrations i ( i = 0, n + 1) and dividers positions u i ( i = 0, n ) known (suppleentary figure 1), with 0 0 < 1 < n+ 1 and 0 < U 0 < U1 < U n, the ass conservation equations 1 (u - u-1) = n-1,-1 (un-1,-1 - u -1) + n-1, (u - un-1, - ) = 1,n Eq.S1 have a unique solution u n-1,i ( i = 0,n -1 ) that verifies the condition 0 u0 < un-1,0 < u1 < un-1,1 < < u -1 < un-1,-1 < u < un-1, < < un-1,n-1 < u n < Eq.S2 if and only if the following inequality holds: 0 0 n-1,0 1 n-1,1 n-1,n n+ 1. Eq.S3 Proof: We start with the first equation that can be written in the for u 1 ( u u 1 ) + n 1, u n 1, 1 u 1 = Eq.S4 1

7 By plugging u n-1,-1 into the copatibility condition u reductions < u u we obtain after a couple of -1 n-1,-1 < u u ) ( u u ) ( u u ) Eq.S5 1 ( which iplies, considering u -1 <u that 1 n 1,. Eq.S6 It can be seen iediately that the arguent is reversible, therefore we obtain the equivalence stated in proposition 1. The restrictions we apply on the order of these concentrations do not affect the universality of the approach, because any continuous function can be decoposed into sipler functions on intervals. Several streas of different concentration profiles can then be cobined to reproduce a ore coplex function 1. Proposition 2. One way to write the copleentary equations to the ass conservation equations is the following for: α α Eq.S7 n 1, = + ( 1 ) + 1 where α ( 0 α < 1 ) would be coefficients at our disposal. < n Proof: This equation contains iplicitly the condition fro Equation S6 and assures us that we can solve the syste at each level obtaining positions for dividers which verify the copatibility condition. The restrictions for particular α values are dictated ainly by practical considerations, like the inial distance between dividers that can still be resolved by the icrofabrication techniques. Proposition 3. The iniu nuber of sub-streas with concentrations N,,i which would reconstitute the target concentration profile with necessary precision ε can also be calculated Proof : Due to the fact that the doain of the function f is the copact interval [ 00, 01 ], ƒ is ore than continuous, naely unifor continuous. Thus, for any ε we could find a particular N such that for any u N,N u N, 0 x 1,x 2 in the interval [u N,0,u N,N ], with x1 x2 <, then f ( x1 ) f ( x2 ) < ε. In other words, for N a particular desired precision ε of the approxiation of a concentration profile, we can deterine the iniu nuber of icrofluidic streas necessary to reconstitute that concentration profile. In practice narrower streas are used in the regions of higher gradient steepness. The experiental results usually show better precision than predictions as a result of gradient soothing by diffusion after the last set of dividers. 1 Dertinger, S. K. W., hiu, D. T., Jeo N. L. & Whitesides, G. M. Generation of gradients having coplex shapes using icrofluidic networks. Analytical heistry 73, (2001).

8 Scheatic protocol for calculation of the position of the dividers inside the ain channel. Output function alculate the position of the last set of dividers onsider one set of paraeters alpha u 1 = alculate the concentrations and divider positions at the precedent level n 1, = α + ( 1 α ) + 1 ( u u 1 ) + 1 u 1 u 1 Is this the first level? Write the positions of all dividers in the channel Suppleentary figure 2. Logic diagra for calculating the position of the dividers in the channel for a desired output concentration gradient. The length of the dividers in the longitudinal direction parallel to the flow of the dividers was calculated to atch the velocity of the streas and the tie required for 99% coplete ixing by diffusion of the erged streas at each level.

9 Experiental ethod. Standard icrofabrication technology has been used to fabricate 10µ wide dividers in 400µ wide and 20 µ high channels in polydiethylsiloxane (PDMS) on glass. Briefly, a 20µ layer of SU8 (Microche, Newto MA) was spun on a silicon wafer and photopatterned according to anufacturer s instructions. PDMS (Sylgard 184; Dow orning, Midland, MI) was prepared according to the anufacturer's instructions and cast over the photoresist old to create copleentary icrochannels in PDMS. Through holes, defining the inlets and outlets, were punched using a beveled 25-gauge needle. The bonding surfaces of the PDMS and a regular glass slide (1 3 inches Fisher Scientific, Pittsburgh, PA) were treated with oxygen plasa (150 Torr, 50 W, 20 s) produced in the parallel plate plasa asher (March Inc., oncord, A). A good seal between the PDMS and glass was achieved by heating the assebly at 75º for 10 inutes on a hotplate. oncentration profiles have been tested in four icrofluidic devices using Fluorescein isothiocynate (FIT) dyes and distilled water as starting solutions. Gravitational flow was eployed to aintain a slow (10 µ/s), controlled fluid velocity inside the icrochannel. The fluorescence intensity was quantified in icrographs at 100 µ downstrea fro the last set of dividers (section AA) and equivalent concentration calculated based on a linear dependence between concentrations and fluorescence intensities. Acknowledgeents: This work was supported by the National Institute of Bioedical Iaging and Bioengineering (BioMEMS Resource enter, P41 EB002503)