Device fabricationFabrication started with \(\sim 500 \, \upmu {{\rm{m}}}\) thick silicon wafers (p-type, \(10\!-\!20 \, \Omega \, {{{{\rm{cm}}}}}\), \(\left\langle 100\right\rangle\)) and borofloat glass wafers \(\sim 550 \, \upmu {{\rm{m}}}\) thick that are \(100 \, {{\rm{mm}}}\) in diameter.The silicon wafer was etched in 5 layers to define the (1) alignment marks, (2) droplet generation microchannel, (3) reaction chamber (4) and (5) ports for external fluidic and electrical connections (Supplementary Fig. 1a–c). The first four layers were defined by reactive ion etching using SF6 gas. In the 4th layer the ports were etched almost through the wafer from the backside. In the 5th layer, the ports were completed by laser drilling through the ports from the front side. The photoresist for each layer was patterned using a standard photolithographic approach with a maskless exposure system (Heidelberg MLA 150).Eight hundred nanometer \((800 \, {{{{nm}}}})\) of ITO was sputter deposited on piranha cleaned borofloat glass wafers at LGA Thin Films. This was followed by 2 layers of photolithographic processing: (1) the electrodes and alignment marks were defined by RIE of ITO from the entire wafer (barring the electrodes) using CH4 and H2 gases and (2) the contact pads were defined by evaporative deposition of \(10 \, {{{{\rm{nm}}}}}\) and \(200 \, {{{{\rm{nm}}}}}\) Gold followed by metal liftoff. After each layer, the wafer was wet cleaned using \(5:1:1::{{\rm{{H}}}_{2}{O}:{{H}}_{2}{{O}}_{2}:N{H}_{4}OH}\) at \(70\) °C for 1 h (Supplementary Fig. 1d).The glass and silicon wafers were then aligned (Supplementary Fig. 1e) and bonded anodically at 350 °C by applying a voltage of 350 V for 6 min. The wafers are then diced into 25 mm × 30 mm chips using a laser cutter (Supplementary Fig. 1f, g, and Fig. 1b). The chips were silanized using vapor phase deposition of Dimethyldichlorosilane (DDMS) at Integrated Surface Technologies to make the device interior surface hydrophobic for the formation of water droplets. The silane on the exterior surface of the chip was stripped using UV ozone treatment to stick fluidic connectors using Loctite 401 adhesives (Supplementary Fig. 1h). Electrical leads are soldered onto the gold contact pads through the electrical ports in the silicon wafer (Supplementary Fig. 1h).Experimental setupThe experimental setup consists of the device holder interfaced with the fluidic, electrical, and optical subsystems.Device sample holderA 3D printed plastic sample holder was used to mount the device on the experimental setup. Copper pins hold the device in place with a \(0.17\,{{\rm{mm}}}\) glass coverslip (\(24 \, {{\rm{mm}}}\times 40 \, {{\rm{mm}}}\)) from SPI below it. The sample holder was screwed onto a X-Y stage from Newport (Model#-406) mounted on a modified Nikon TE2000U inverted microscope (Supplementary Fig. 2a, b).Fluidic subsystemA piezo driven pressure controller (OB1 MK3+ from Elveflow) with a \(30 \, {{\rm{{psi}}}}\) input from the house nitrogen supply and a maximum output of \(2000 \, {{\rm{mbar}}}\) was connected to the input of a fluidic tank (\(15 \, {{\rm{ml}}}\) plastic tube) using a \(10 \, {{\rm{mm}}}\) OD tubing (Supplementary Fig. 2c, d). The output of the tank flows into the device via a \(1/16 \, {{\rm{inch}}}\) OD and \(1/32 \, {{\rm{inch}}}\) ID polytetrafluoroethylene (PTFE) tubing from Masterflex (item#-EW06407-41) which was plugged into the fluidic port of the device using the Nanoport Assemby from IDEX Health & Science (part# – N-333).Electrical subsystemAn A.C. high voltage amplifier (A. A. Lab Systems Ltd. Model#-A-303) amplifies the signal from a function generator (Hewlett Packard Model#-8116A) to generate a maximum output amplitude of \(200 \, {{\rm{V}}}\) (Supplementary Fig. 2e, f). The amplified output was connected to the electrical leads of the device through a single pole double throw (SPDT) switch which connects the supply across either of the electrodes (E1 or E2) and the ground pad.Optical subsystemThe optical subsystem (Supplementary Fig. 2g, h) was built around a modified Nikon inverted TE2000U microscope. A blue LED (SOLIS-445C from Thorlabs, \(445 \, {{\rm{nm}}}\) and \(5.4 \, {{\rm{W}}}\) min) with a band pass excitation filter (D480/30x from Chroma) images the bead (\(6 \, \upmu {{\rm{m}}}\) diameter fluorescent green streptavidin coated polystyrene beads with excitation maxima at \(441 \, {{\rm{nm}}}\) and emission maxima at \(486 \, {{\rm{nm}}}\), Catalog#-24157) and the fluid flow in the device onto a sCMOS camera from Thorlabs (Part#-CS2100M-USB). The experiments were recorded at 33 frames per second. A red LED (M625L4 from Thorlabs, \(625 \, {{\rm{nm}}}\) and \(700 \, {{\rm{mW}}}\)) excites the Alexa-647 to detect nucleotides labeled with the fluorophore (dCTP-AF647) in the reagent droplet. A sCMOS camera from PCO (PCO edge 5.5) was used to capture the low light intensity levels emanating from the nucleotides coupled to the initiator strands on the beads at \(2 \, {{\rm{s}}}\) integration time. Appropriate bandpass excitation (Item#-86-988 from Edmund Optics, \(640 \, {{\rm{nm}}}\) center wavelength, \(14 \, {{\rm{nm}}}\) bandwidth, OD – 6) and emission (Item#-86-987 from Edmund Optics, \(676 \, {{\rm{nm}}}\) center wavelength, \(29 \, {{\rm{nm}}}\) bandwidth, optical density – 6) filters were used to ensure non-overlap of the excitation and emission spectrum. A Nikon objective (ELWD-20, 20x mag, 0.45 NA) with a correction collar for spherical aberration correction (set at \(0.7 \, {{\rm{mm}}}\) which is \(0.17 \, {{\rm{mm}}}\) thick glass coverslip \(+0.53 \, {{\rm{mm}}}\) thick borofloat glass of the device) was used for imaging.Sample preparationPreparing the oil solution by adding surfactant\(4 \, {{\rm{ml}}}\) of Span 80 (S6760 from Sigma Aldrich) was added to \(200 \, {{\rm{ml}}}\) of \(1 \, {{\rm{cSt}}}\) silicone oil (PSF – \(1 \, {{\rm{cSt}}}\) from Clearco Products) to make a \(2.5\%w/w\) solution. It was sonicated for \(30 \, {{\rm{mins}}}\) to ensure complete dissolution of the surfactant.Attaching initiator strand to beadsThe initiator strand, which was a biotinylated oligomer with \(25\) bases (T25mer, 5′ biotin, IDT) was attached to the \(6\upmu {{\rm{m}}}\) diameter (Rb = 3 μm) streptavidin coated green-fluorescent polystyrene beads using the strong biotin-streptavidin hydrogen bond. The reaction was carried out for \(60\min\) at 2 °C, 14 RPM. The approximate starting yield (140 attomoles per bead (T25mer bound)) was determined by measuring the optical density at \(260\) nm (Nanodrop) before and after initial binding, then subtracting the supernatant and wash OD values from the starting yield. Binding/ wash buffer: \(20 \, {{\rm{mM}}}\) Tris pH \(7.5\), \(1 \, {{\rm{M}}}\) NaCl, \(1 \, {{\rm{{mM}}}}\) EDTA, \(0.0005\%\) Triton-X 100 (45 μl (plus 5 μl 100 μM T25) for binding reaction, and 500 μl for wash steps). Based on a particle concentration of \(1.4\%\), the number of beads was ~867,000 per 25 μl reaction (accounting for a \(20\%\) loss due to mixing and washing steps). Beads with initiator strands were then spun down using an Eppendorf Minispin (Catalog#-022620100) to remove the supernatant and were segregated into two parts (i) for Dielectrophoretic Bead-Droplet Reaction in the fabricated chip, and (ii) for benchtop synthesis in columns.Suspending initiated beads in oil solutionThe spun down initiated beads are suspended in the oil solution by sonication. The concentration of the beads in the silicone oil solution were tuned to ensure mostly a single bead floats in the vicinity of the electrodes within the field of view of the objective.Preparing reagent solutionThe reagent solution was prepared by mixing \(25 \, \upmu {{\rm{l}}}\) of reagents consisting of the fluorescently labeled base (dCTP-AF647) in a buffer solution of \(50 \, {{\rm{mM}}}\) Potassium Acetate, \(20 \, {{\rm{mM}}}\) Tris-acetate, \(10\,{{\rm{mM}}}\) Magnesium Acetate, and \(0.25 \, {{\rm{mM}}}\) Cobalt Chloride with the enzyme (TdT) solution consisting of \(3 \, \upmu {{\rm{l}}}\) of \(50 \, {{\rm{mM}}}\) KPO4, \(100 \, {{\rm{mM}}}\) Sodium Chloride, \(1.43 \, {{\rm{mM}}}\) β-ME, \(50\%\) glycerol, and \(0.1\%\) Triton X-100 solution in an Eppendorf tube. This reagent solution was formulated by initial benchtop experiments as described in the subsequent experimental procedure section. A trace amount of sodium salt of fluorescein (F6377 from Sigma Aldrich) was added to the reagent using a toothpick to discriminate it from the continuous phase inside the microfluidic device.Filling device with oil solution as continuous phaseThe device was completely immersed in \(50 \, {{\rm{ml}}}\) of the \(2.5\%\, {{\rm{w/w}}}\) solution of Span80 in \(1 \, {{\rm{cSt}}}\) silicone oil contained in a glass jar inside a vacuum desiccator. As the desiccator was evacuated the air inside the device was drawn out. When the desiccator is refilled with air, the silicone oil solution gushes into the device to fill it completely without any trapped air bubbles.Mounting device on sample stageThe device was then removed from the glass jar, its outer surface was cleaned by thoroughly wiping with isopropanol, and then mounted on the sample holder (Supplementary Fig. 3a). The objective was focused on the output of the droplet generation channel and the ITO electrodes.Making electrical and fluidic connections to device portsElectrical connections are made from the output of the amplifier to the ground pad and to the trap electrodes through the SPDT switch (Supplementary Fig. 3a). The fluidic tank was filled with 15 ml of the above oil solution. Pressure was applied using the pressure controller to fill the output PTFE tubing from the tanker with oil solution which is dipped at the other end inside the 1.5 ml tube containing the reagent. Just before oil starts dripping from the tubing into the reagent tube, the height of the PTFE tubing and reagent tube were raised to suck the reagent into the tubing. Then the tubing was lowered again into another Eppendorf tube containing the oil solution. As the reagent solution started dripping, the height was raised again to fill the PTFE tubing with the oil solution while ensuring there are no trapped air bubbles. The tubing was then connected to the device inlet while pushing out the oil solution at the bottom to ensure no air gaps and fluidic continuity (the oil solution inside the device and at the bottom of the tubing are the same). This approach prevented immediate flow of the reagent solution through the droplet generation channel as soon as the PTFE tubing was connected thus allowing time for experimental setup and control (Supplementary Fig. 3b).Experimental procedure for bead-droplet interactionEncapsulation and ejection of bead from dropletBeads suspended in the oil solution were introduced into the device through the oil inlet (Supplementary Fig. 1a). The voltage supply was switched on and set to around \(40 \, {{\rm{V}}}\) amplitude at \(200 \, {{\rm{Hz}}}\) to dielectrophoretically trap a bead floating near the top electrode. Then the voltage supply was switched off. Following this a pressure of \(60 \, {{\rm{mbar}}}\) was applied on the pressure controller to drive reagent flow in the device. As the reagent approached the entrance of the microfluidic channel, a sudden pressure pulse of approximately \(10 \, {{\rm{mbar}}}\) for approximately \(300 \, {{\rm{ms}}}\) was exerted to dispense a single droplet into the reaction chamber (Supplementary Movie 5). A higher-pressure pulse or a longer pulse duration would lead to the generation of multiple droplets (Supplementary Movie 6). Then the voltage supply was switched on again at \(20 \, {{\rm{V}}}\) amplitude and \(200 \, {{\rm{Hz}}}\) to trap the droplet adjacent to the bead on the top electrode.At this point the supply voltage was gradually increased to \(\sim \, 120 \, {{\rm{V}}}\) amplitude. The droplet moves toward the electrode to encapsulate the bead. Subsequently the voltage was reduced to \(0.1 \, {{\rm{V}}}\) amplitude and the bead was ejected out of the droplet. Supplementary Movie 1 depicts the encapsulation and ejection process. The bead can undergo multiple encapsulations and ejection (Supplementary Movie 7) as well by switching the voltage between high and low. The images extracted from these videos were suitably processes for enhanced clarity without altering the data and results.Estimating interfacial tension and contact angleThe interfacial tension values reported in the main text were measured using the standard Wilhelmy plate method. The contact angles were measured by capturing the droplet shape on a streptavidin coated glass slide (GS-SV-5 from Nanocs) and silanized glass surface and then estimating the angle it forms on the surface through shape fitting.Estimating electrical conductivity of reagentThe conductivity of the reagent droplet was measured using an Orion 3 Star Conductivity Portable.Modeling of bead-droplet InteractionCoupled electrohydrodynamic simulationsThe electric field driven encapsulation of the microbead into the microdroplet and its ejection out of it was modeled by using coupled electrohydrodynamic simulations in COMSOL Multiphysics (Supplementary Fig. 4). The fluid flow (aqueous droplet motion in silicone oil medium) was modeled as two-phase fluid flow using the Navier-Stokes equation45. The phase variable \(\phi\) tracked the fluidic interface between the droplet and the suspension medium using the phase field method83. \(\phi=-1\) within the silicone oil medium and \(\phi=1\) in the droplet. It transitions from \(-1\) to \(1\) at the interface of the droplet and the suspension medium. To model the driving electric force, the electric charge continuity equation45,51,84 was used to solve for the non-uniform electric field distribution when an AC voltage was supplied across a pair of electrodes of different dimensions. The electric force density on the fluids was evaluated by using \(\vec{\nabla }\cdot {T}^{\leftrightarrow}\). Here, \({T}^{\leftrightarrow}\) is the Maxwell Stress Tensor44,51,84,85,86,87. This force sets the fluid flow in motion. As the droplet moves, the boundary condition of the electric charge continuity equation changes. This changes the electric field distribution and \(\vec{\nabla }\cdot {T}^{\leftrightarrow}\) in turn. This underlines the basic coupling between the Navier Stokes equation and the charge continuity equation. In the simulations, the bead is modeled as a stationary polarizable dielectric particle with a fixed contact angle (\(\theta=145^{\circ}\)) that the droplet forms on its surface in the silicone oil medium. This leads to an additional capillary force acting on the bead-droplet system when the bead and the droplet are in contact. For simplicity, axis symmetric simulations were adopted. This helps focus on the essential physical interaction without getting into the nuances of device design.The encapsulation and ejection process can be modeled as a balance between the electric and capillary force. To understand the scaling laws underlying these forces, we derived approximate analytical equations for the electric and capillary forces.Dielectrophoretic forceThe dielectrophoretic force44,51,84,85,86,87 on a generic charge neutral polarizable particle (of radius \({R}_{p}\)) can be evaluated by expressing its polarizability and the non-uniform electric field as an infinite series of multipolar expansions87. We base our approximate analysis on the dipolar/first (\(n=1\)) term of this series. The approximate dielectrophoretic force (\({\vec{F}}_{{DEP}}\)) then scales as \({\vec{F}}_{{DEP}}\propto {K}_{1}{R}_{p}^{3}\). Here \({K}_{1}\) is the familiar Classius-Mossotti factor and \({R}_{p}\) is the radius of the polarizable particle which can be the droplet or the bead. \({K}_{1}\) is given by \(\frac{{\tilde{\varepsilon }}_{p}-{\tilde{\varepsilon }}_{o}}{{\tilde{\varepsilon }}_{p}+2{\tilde{\varepsilon }}_{o}}\). Here \({\tilde{\varepsilon }}_{p/o}\) is the complex permittivity of the particle or the suspension oil medium which is a function of their respective relative permittivities (\({\varepsilon }_{p/o}\)) and conductivities (\({\sigma }_{p/o}\)). Therefore, the water droplet experiences a much larger force than the bead due to its larger permittivity contrast with the oil medium as well as much larger size (Supplementary Fig. 5). So, if multiple droplets are suspended in the reaction chamber, the primary electric field driven effect is the merger of the droplets.Capillary forceThe capillary45,51,52 force on the bead-droplet system arises due to the change in the total interfacial energy of the system as the bead is encapsulated/ejected within the droplet. For our case in which \({R}_{b} \; \ll \; {R}_{d}\), the change in total interfacial energy is given as \(\triangle {U}_{{IT}}\approx -{\gamma }_{{ow}}\cos \theta \triangle {A}_{{ws}}\). Here, \({\gamma }_{{ow}}\) is the oil-water interfacial tension and \({A}_{{ws}}\) is the interfacial area of the water/aqueous droplet and the solid bead surface (Supplementary Fig. 6). From this the scaling of the interfacial capillary force can be approximated as \({\vec{F}}_{{IT}}\propto {R}_{b}{\gamma }_{{ow}}\cos \theta\).Experimental procedure for chemical couplingChemical coupling of base and control on the deviceThe above physical process was used (with a 300 s encapsulation time) for the enzymatic coupling of fluorescently tagged nucleotides onto the initiator strand on the bead with a few additional intermediate images of the area around the reaction zone captured as enlisted below.
Before loading the beads into the device an image to estimate the background noise under red illumination (Supplementary Fig. 7a, b).
After the bead was dielectrophoretically trapped on the top electrode an image each with the blue and red excitation are captured to measure the level of the red fluorescence signal from the site of the bead just prior to the reaction (Supplementary Fig. 7c, d).
Finally, after the encapsulation and ejection process another set of images under blue and red excitation were captured (Supplementary Fig. 7e, f).
These images were captured as 16-bit Tiff files. Supplementary Fig. 7c, f are used in Fig. 4a of the main text. These steps were repeated to see the repeatability of the chemical coupling reaction on our platform. The three different reacted beads under red excitation are depicted separately in Supplementary Fig. 7g and are also used in Fig. 4 of the main text.The control experiment was repeated using the above process but with beads without initiator strands and reagent droplets without the enzymes (Supplementary Fig. 7h–m). Supplementary Fig. 7h, m are used in Fig. 4b of the main text.Column synthesisFirstly, free solution reactions are implemented to develop optimal room temperature protocol (Supplementary Fig. 8a, b) for translation into DBDR. Results were analyzed using reverse-phase high performance liquid chromatography (HPLC) (Supplementary Fig. 8b).To simulate an enzymatic synthesis reaction using a column, an open-top nylon syringe filter (Omicron SFNY04XB, \(4 \, {{\rm{mm}}}\), \(0.45 \, \upmu {{\rm{m}}}\)) was used. To the bottom filter (\(0.45 \, \upmu {{\rm{m}}}\) pore size) which was held in place by a plastic ring, \(15 \, \upmu {{\rm{l}}}\) of beads (\(1.2 \, {{\rm{M}}}\)) were added. A top filter was then positioned above the reagent bed. Between the filters the reaction volume was about \(15 \, \upmu {{\rm{l}}}\). A \(1 \, {{\rm{ml}}}\) syringe was used to push the bead medium (\(10 \, {{\rm{mM}}}\) HCL, \(2 \, {{\rm{M}}}\) NaCl, \(1 \, {{\rm{mM}}}\) EDTA, \(0.0005\%\) Triton-X 100, pH \(7.3\)) passed the bottom filter until it completely exited the drip director. \(50\,\upmu {{\rm{l}}}\) (\(6 \, \upmu {{\rm{l}}}\) TdT (\(20\,{{\rm{U}}}/\upmu {{\rm{l}}}\)), \(5 \, \upmu {{\rm{l}}}\) \(10x\) TdT buffer (\(50 \, {{\rm{mM}}}\) potassium acetate, \(50 \, {{\rm{mM}}}\) Tris-acetate, \(10\,{{\rm{mM}}}\) magnesium acetate, pH 7.9 @ 25 °C), 5 μl 10× (2.5 mM) solution of \({{\rm{Co}}{{Cl}}_{2}}\), 0.25 μl 1 mM Alexa Fluor™ 647-aha-dCTP, 33.75 μl water) were added to the top filter, and an empty 1 ml syringe was used to push the reagent passed the top filter (flow rate at \(50 \, \upmu {{\rm{l}}}\) per second), into the reaction area until the reagent could be seen inside the drip director. The column was kept upright for \(5 \, {{\rm{mins}}}\). Once the reaction was completed, the empty 1 ml syringe was used to push the spent reagents through the column until the drip director was clear. Afterwards, \(1 \, {{\rm{ml}}}\) of bead suspension medium (described above) was used to wash the beads. These beads were then taken in a \(1.5 \, {{\rm{ml}}}\) Eppendorf tube for analysis using fluoroscopy.As a control, synthesis was performed on beads without initiators and reagents without TdT and the reaction was analyzed through fluorescence measurements.Measuring fluorescence from beads reacted in columnsAbout 3 µl of the reacted bead suspension in the buffer was taken in an Eppendorf tube and was diluted to ensure the bead concentration was small enough to prevent signal interference from beads in different planes while being large enough to have ample beads within the field of view to get a statistically significant inference about fluorescence intensity distribution. The beads were introduced into the device filled with MilliQ water. The same chips were used for fluorescence measurements to ensure identical optical environment for comparison between on-chip experiments with their column counterparts. Once the beads settled down (imaged using blue excitation) the excitation was switched to red to image the fluorescence intensity of the beads. Many such frames of red fluorescent beads were collected with a large number of beads (\(285\)). Some are shown in Supplementary Fig. 9a. A few representative beads spanning the entire range of fluorescence intensities are used in Fig. 4d of the main text.The control experiments implemented using DBDR (Fig. 4b of main text) were reiterated on the columns. The fluorescence of these beads was measured following the same procedure as discussed in the previous paragraph (Supplementary Fig. 9b).Image processing and data analysisEncapsulation and ejection of bead from dropletThe recorded video (Supplementary Movie 1) of the encapsulation and ejection process was analyzed frame by frame using ImageJ and snapshots that best represent the processes were selected and labeled for Fig. 2 of the main text.Establishing enzymatic coupling of base to the initiator strands on the bead in DBDRMaintaining the same scale of \(250-3300\) across the red fluorescence images, the difference in brightness of the bead with enzymatic coupling (Fig. 4a main text and Supplementary Fig. 7f) and the control bead (Fig. 4b main text and Supplementary Fig. 7m) was obvious.Analyzing fluorescence intensity distributionEach frame in Supplementary Fig. 9a was analyzed using predefined image processing functions in Matlab to detect the beads (Supplementary Fig. 10), binarize them, evaluate their mean fluorescence intensity, and evaluate fluorescence intensity distribution across a horizontal line passing through the bead center. Average fluorescence values across frames were collected to plot a histogram of the fluorescence intensity distribution of all beads reacted on the benchtop using synthesis columns.The average fluorescence intensity of the three beads reacted using DBDR (Supplementary Fig. 7g) were found to be \(2184.5\), \(2171.7\), \(1816.2\) (same unit as was used for column fluorescence data). These values are higher than the fluorescence intensity (represented in Supplementary Fig. 10d) of all the \(285\) beads reacted in columns whose fluorescence data was collected. This indicates that the reaction fidelity using DBDR is higher than synthesis columns.Analyzing statistical distribution of dataTo establish the statistical significance of our fluorescence comparison-based claim that the solid-phase synthesis reaction fidelity achieved using DBDR is higher than synthesis columns we resort to statistical hypothesis testing. We seek to establish that the mean fluorescence intensity of beads reacted using DBDR is higher than the mean fluorescence intensity of beads reacted using synthesis columns at significance level of \(0.05\) or a confidence level of \(95\%\). The t-test which tests for the null hypothesis of equivalence of sample means for both the synthesis methods using the following test statistic88,89,90,91,92 (\(t\) value) is appropriate for our purpose.$$t=\frac{{\mu }_{{DBDR}}-{\mu }_{{Column}}}{\sqrt{\frac{{\sigma }_{{DBDR}}^{2}}{{N}_{{DBDR}}}+\frac{{\sigma }_{{Column}}^{2}}{{N}_{{Column}}}}}$$
(1)
Here, \({\mu }_{{DBDR}/{Column}}\) is the sample mean of the respective synthesis methods, \({\sigma }_{{DBDR}/{Column}}\) is the sample standard deviation of the respective synthesis method, and \({N}_{{DBDR}/{Column}}\) is the number of bead samples over which the mean and the standard deviation were evaluated in the respective synthesis methods. For columns, \({N}_{{column}}=285\). This is the total number of beads that were accounted for in the histogram in Supplementary Fig. 10d. For DBDR, \({N}_{{DBDR}}=3\). These are the 3 beads represented in Supplementary Fig. 7g. The relevant values are summarized in Supplementary Table 2a. As the standard deviations and the number of bead samples are unequal in the two synthesis methods, we use the Welch’s t-test for the statistical significance analysis89,90,93. The degree of freedom for the Welch t-test which is given by the Welch-Satterthwaite equation is90,94,95:$${df}=\frac{{\left(\frac{{\sigma }_{{DBDR}}^{2}}{{N}_{{DBDR}}}+\frac{{\sigma }_{{Column}}^{2}}{{N}_{{Column}}}\right)}^{2}}{\frac{{\left(\frac{{\sigma }_{{DBDR}}^{2}}{{N}_{{DBDR}}}\right)}^{2}}{{N}_{{DBDR}}-1}+\frac{{\left(\frac{{\sigma }_{{Column}}^{2}}{{N}_{{Column}}}\right)}^{2}}{{N}_{{Column}}-1}}$$
(2)
The values of \(t\) and \({df}\) evaluated using Eq. 1 and eq. 2 are \(11.65\) and \(2.14\) (summarized in Supplementary Table 2b). Therefore, \(2 \, < \,{df}=2.14 \, < \, 3\). Using a standard t-test table for two-tailed testing we see that if \({df}=2\) for a two-tailed significance level (\(\alpha\)) of \(0.01\) the critical t-value (\({t}^{*}\)) is \(9.925 \, < \, t\) and for a two-tailed significance level (\(\alpha\)) of \(0.002\) the critical t-value (\({t}^{*}\)) is \(22.327 \, > \,t\). On the other hand, if \({df}=3\) for a two-tailed significance level (\(\alpha\)) of \(0.002\) the critical t-value (\({t}^{*}\)) is \(10.215 \, < \, t\) and for a two-tailed significance level (\(\alpha\)) of \(0.001\) the critical t-value (\({t}^{*}\)) is \(12.924\, > \,t\). Thus, we can safely say that our null hypothesis can be rejected at significant level of \(\alpha=0.01\) or at a confidence level of \(99\%\). Hence, our result is definitely significant at \(\alpha=0.05\) or a confidence level of \(95\%\). This was confirmed using the inbuilt ttest2 function in Matlab for Welch’s t-test which rejected the null hypothesis. To evaluate if the sample sizes (\({N}_{{DBDR}}\) and \({N}_{{column}}\)) were sufficient for statistical testing, we calculate the power of the statistical test96 using the inbuilt sampsizepwr function in Matlab for \(\alpha=0.05\). We obtain a statistical power of almost \(1\) (a power of \(0.8\) at \(\alpha=0.05\) is generally considered adequate96). Therefore, our sample size suffices for statistical testing.The Welch’s t-test, which is a parametric test is generally robust for normal distributions with unequal sample sizes and standard deviations89,90,93. For deviations from normal distributions (Supplementary Fig. 10d), nonparametric tests (which do not assume any specific distribution profile) operating on the ranks of the experimentally observed values rather than the actual values themselves are more robust93,97. It is established in statistical literature that a rank transformation on the conventional Welch’s t-test would counter the combined effects of unequal standard deviations as well as non-normal distributions93. Therefore, we apply the above statistical testing procedure to the combined ranks of the average fluorescence intensities of beads reacted using the synthesis column and DBDR. The \(3\) beads reacted using DBDR have higher fluorescence intensities than the \(285\) beads reacted in synthesis columns. So, the beads reacted in columns have ranks from \(1\) to \(285\). While the beads reacted using DBDR have ranks from \(286\) to \(288\). The respective means (\({\mu }_{{DBDR}/{column}}\)) and standard deviations (\({\sigma }_{{DBDR}/{column}}\)) are summarized in Supplementary Table 3a. Using Eqs. 1 and 2, we evaluate \(t=29.41\) and \({df}=285.79\). We see that at \(\alpha=0.05\), \({t}^{*} \,=\, 1.9683 \; < \;t=29.41\) for \({df}=285\) or \(286\). Therefore, the null hypothesis of equivalence of means of DBDR and columns can be safely rejected at the confidence level of \(95\%\). The inbuilt ttest2 function in matlab confirms this. Furthermore, a statistical power of 0.9729 is obtained which confirms that the sample size suffices for the statistical inference. The results are summarized in Supplementary Table 3b.Modeling of bead stacking in column and comparison of reagent access with bead-droplet reactorAnalytical modeling of ideal case of beads stacked in perfect latticesTo understand how stacking of beads in synthesis columns limits reagent access to the bead surfaces, we begin by considering the ideal case scenario of stacking of beads into perfect lattices. In such a case, effectively one bead is enclosed in a unit cell and access the reagents within the void of the unit cell. The simple cubic is one of the most loosely packed lattices while the rhombohedral is one of the most tightly packed67. The volume of a bead of radius \({R}_{b}\) is \(\frac{4}{3}\pi {R}_{b}^{3}\) (Supplementary Fig. 11). The volume of a simple cubic unit cell is \(8{R}_{b}^{3}\) while that of a rhombohedral unit cell is \(4\sqrt{2}{R}_{b}^{3}\). Therefore. the void volume in these unit cells is \(4\left(2-\frac{\pi }{3}\right){R}_{b}^{3} \, \approx \, 103\,{{\rm{{fl}}}}\) and \(4\left(\sqrt{2}-\frac{\pi }{3}\right){R}_{b}^{3} \, \approx \,40\,{{\rm{fl}}}\) respectively69,70. The reagent solution has 5 μM concnetration of nucleotides (2.5 × 10−7 mmoles of nucleotides were added to 50 μl of reagent solution). Therefore, within the void volume of the simple cubic and rhombohedral unit cells there would be around \(103 \, {{\rm{{fl}}}}\times 5\,\upmu {{\rm{M}}}=0.52 \, {{\rm{attomoles}}}\), and \(40\,{{\rm{fl}}}\times 5\,\upmu {{\rm{M}}}=0.2\,{{\rm{attomoles}}}\) of fluorescently labeled nucleotides respectively which is \(\approx \, 280\) and \(\approx \, 700\) times more than the number of nucleotides in the voids of the simple cubic and rhombohedral lattice respectively. On the other hand, the volume of the droplet (\({R}_{d}=25\mu m\)) which forms a contact angle (\(\beta\)) of \(138^{\circ}\) on the device surface is \(\frac{(2-3\cos \beta+{\cos }^{3}\beta )}{3}\pi {R}_{d}^{3}=65 \, {{\rm{pl}}}\)51,52. At the same \(5 \, \upmu {{\rm{M}}}\) concentration of fluorescently labeled nucleotides, the nearly \(100 \, {{\rm{attomoles}}}\) of initiator strands will have access to \(65 \, {{\rm{pl}}}\times 5 \, \upmu {{\rm{M}}}=325 \, {{\rm{attomoles}}}\) of fluorescently labeled nucleotides within the droplet.Particle tracking modeling for imperfect tracksFor simplicity of modeling as well as to reduce memory and time requirements, a 2D mirror symmetric simulation was set up combining turbulent fluid flow and particle tracking simulations. The simulation space mimicked the dimensions of the reaction column. A rhombohedral stack of beads was defined at the bottom of the simulation space. Fluid flow equations for turbulent flow68 were simulated to mimic fluid flow in the column when reagents are introduced into it. This fluid flow exerted a drag force on the particles which set them in motion within the column dispersing the perfect stack in the process. The beads have increased and varying access to reagents. Once the reagent inflow subsides, the beads eventually settle down under the influence of gravity into imperfect stacks. Further modeling details can be found in Supplementary Note 5 and Supplementary Fig. 12.Reagent diffusionThe synthesis columns used in the reaction are 2.28 mm long whereas the microdroplets used in DBDR are \(50 \, \upmu m\) (\(5\times {10}^{-5}{{\rm{m}}}\)) in diameter (Supplementary Fig. 13). The diffusion time is approximated as \({t}_{D}\approx \frac{{l}^{2}}{2D}\)45. Here, \(D\) is the well-known diffusion coefficient which is \(\approx 1.3\times {10}^{-5}{{\rm{cm}}}^{2}/{{\rm{s}}}\)71,72 for single nucleotides. Using this we obtain \({t}_{D,\,{col}} \, \approx \, 2000\, {{\rm{s}}}\) and \({t}_{D,\,{DBDR}}\,\approx \,1\,{{\rm{s}}}\).Electric field driven enhancement of reagent concentration in DBDRThe reagent droplet in DBDR consists of many positive and negatively charged species which are necessary for the enzymatic coupling of nucleotides to the initiator strands. The applied AC electric field for dielectrophoretic trapping of the droplet and bead also drives ion migration closer to and further away from the encapsulated bead during alternate phases of the AC cycle. This changes the time averaged concentration of ions that the initiators on the bead are exposed to over an entire AC cycle. This migration of ions is modeled using the Nernst-Planck equation45,98 in COMSOL Multiphysics. We model the effect of the AC supply voltage amplitude and frequency. Details can be found in Supplementary Note 6, Supplementary Table 4, and Supplementary Figs. 14–17.