Separation of DNA by length in rotational flow: Lattice-Boltzmann-based simulations     

Faihan Alfahani  Michael Antonelli  Jennifer Kreft Pearce 《Biomicrofluidics》2015,9(4)

We use a lattice-Boltzmann based Brownian dynamics simulation to investigate the separation of different lengths of DNA through the combination of a trapping force and the microflow created by counter-rotating vortices. We can separate most long DNA molecules from shorter chains that have lengths differing by as little as 30%. The sensitivity of this technique is determined by the flow rate, size of the trapping region, and the trapping strength. We expect that this technique can be used in microfluidic devices to separate long DNA fragments that result from techniques such as restriction enzyme digests of genomic DNA.The development of novel methods for manipulating biopolymers such as DNA is required for the continued advancement of microfluidic devices. Techniques such as restriction enzyme digests for genomic sequencing rely on the detection of DNA that differ in length by sometimes thousands of base pairs.¹ Methods that separate DNA strands with resolutions on the order of kilobase pairs are required to analyze the products of this technique. To gain an insight into possible techniques to separate polymers, it can be helpful to review the methods to separate particles in microfluidic devices. Experimental work has shown how hydrodynamic mechanisms can lead to separation of particles based on size and deformability.² Eddies, microvortices, and hydrodynamic tweezers have been used to trap and sort particles. The mechanism of the trapping and sorting arises from the differences between interactions of the particles with the fluid.^2–8 In particular, counter-rotating vortices have been used to sort particles and manipulate biopolymers. They have been used to deposit DNA precisely across electrodes⁹ and trap DNA.^10,11 Vortex flow may therefore be a good basis for a technique for sorting DNA by length.Streaming flow has been used in experiments to separate colloids of different size.^3,4 Particles are passed through a channel with a flow field driven by oscillating bubbles and pressure. The flow field becomes a combination of closed and open streamlines. The vortex flow is controlled by the accoustic driving of the bubbles while pressure controls the net flow of the fluid. Larger particles are trapped in the closed vortex flow created by the bubbles, while smaller particles can escape the neighborhood of a bubble in the open streamlines. This leads to efficient separation of particles with size differences as small as 1 μm.Previous work on DNA has shown that counter-rotating vortices can be used to trap DNA dynamically. Long strands of DNA have been observed to stretch between the centers of two counter-rotating vortices. The polymer stays trapped in this state for significant amounts of time.¹² In a different experiment, the vortices were used to thermally cycle the polymer and allow replication via the polymerase chain reaction (PCR). The DNA is also trapped against one wall by a thermophoretic force in these experiments.¹⁰ The strength of the trap is controlled by the gradient in temperature created by a focused infrared laser beam.Trapping DNA at one wall by counter-rotating vortices has also been explored in simulation and found to depend on the Peclet number, Pe = u_maxL/D_m, where u_max is the maximum speed of the vortex, L is the box size, and D_m is the diffusion coefficient of one bead in the polymer chain.¹¹ The trapping rate of the DNA was shown to depend on the competition between the flow compressing the DNA into the trap region and the diffusion of the DNA out of the trap. For the work presented here, Pe ≅ 2000, similar to the previous work done with the same simulation.We extend the previous work to investigate if counter-rotating vortices can be used to separate DNA of different lengths. We use the same type of simulation outlined in Refs. ¹¹ and ^13–17, based on the lattice-Boltzmann method. The simulation method has successfully modeled systems as diverse as thermophoresis of DNA,¹⁴ migration of DNA in a microchannel,¹⁶ and translocation of DNA through a micropore.^17,18 Using this method, the fluid is broken into a lattice with size, ΔL, chosen to be 0.5 μm, and is coupled to a worm-like chain model with Brownian dynamics for the polymer.^19,20 The fluid velocity distribution function, n_i(r, t), describes the fraction of fluid particles with a discretized velocity, c_i, at each lattice site.^21–24 A discrete velocity scheme with nineteen different velocities in three dimensions is used. The velocity distributions will evolve according to ni(r+ciΔτ,t+Δτ)=ni(r,t)+Lij[nj(r,t)−njeq(r,t)],(1)where L is a collision operator such that the fluid relaxes to the equilibrium distribution, nieq given by a second-order expansion of the Maxwell-Boltzmann distribution nieq=ρaci[1+(ci·u)/cs2+uu:(cici−cs2I)/(2cs4)],(2)where cs=1/3ΔLΔτ is the speed of sound. Δτ is the time step for the fluid in the simulation, Δτ = 8.8 × 10⁻⁵. The coefficients a^c_i are determined by satisfying a local isotropy condition ∑iaciciαciβciγciδ=cs4(δαβδγδ+δαγδbetaγ+δαδδβγ).(3)To simplify computation, the velocity distributions are transformed into moment space. The density ρ, momentum density j, and momentum flux density Π are some of the hydrodynamic moments of n_i(r, t). The equilibrium conditions for these three moments are given by ρ=∑nieq,(4) j=∑ci·nieq,(5) Π=∑nieq·cici.(6)L has eigenvalues τ0−1,τ1−1,…,τ18−1, which are the characteristic relaxation times of the moments. The Bhatanagar-Gross-Krook model is used to determine L:²⁵ the non-conserved moments have a single relaxation time, τ_s = 1.0. The conserved moments are density and momentum; for these, τ⁻¹= 0. Fluctuations are added to the fluid stress as in the method of Ladd.²⁴ We have also compared simulations with lattice sizes of 1 μm and 0.25 μm and found no significant differences in the results.The DNA used in the simulation is represented by a worm-like chain model parameterized to capture the dynamics of YOYO-stained λ DNA in bulk solution at room temperature.^15,16,26 Long, flexible DNA is modeled since techniques to separate long DNA molecules with kilobase pair resolution are necessary to complete techniques such as genomic level sequencing using restriction enzyme digests.¹ In addition, such DNA is often used in experiment. Its properties are similar to unstained DNA or DNA stained by other methods.²⁷ Each molecule is represented by N_b beads and N_b − 1 springs. A chain composed of N_b − 1 springs will have a contour length of (N_b − 1) × 2.1 μm. The forces acting on each monomer include: an excluded volume force, a non-linear spring force, the viscous drag force, a random force that produces Brownian motion, a repulsive force from the container walls, and an attractive trapping force only at one wall as shown in Fig. ​Fig.11.¹³ The excluded volume interaction between beads i and j located at r_i and r_j is modeled using the following potential: Uijev=12kBTνNks2(34πSs2)exp(−3|ri−rj|24Ss2),(7)where ν=σk3 is the excluded volume parameter with σ_k = 0.105 μm, the length of one Kuhn segment, N_ks = 19.8 is the number of Kuhn segments per spring, and Ss2=Nks/6)σk2 is the characteristic size of the bead. This excluded volume potential reproduces self avoiding walk statistics. The non-linear spring force is based on force-extension curves from experiments and is given by fijS=kBT2σk[(1−|rj−ri|Nksσk)−2+4|rj−ri|nKσk−1]rj−ri|rj−ri|,(8)which applies when N_ks ≫ 1.Open in a separate windowFIG. 1.Simulation set-up. Arrows indicate direction of fluid flow. The region where the trapping force is active is shaded, and its width (X_stick) is shown. The region used to determine the trapping rate is indicated by the area labeled trap region. Figure is not to scale, the trap region and X_stick are smaller than shown.The beads are modeled as freely draining but subject to a drag force given by F_f = ?6πηa(u_p ? u_f).(9)The beads are also subjected to a random forcing term that is drawn from a Gaussian distribution with zero mean and a variance σ_v = 2k_BTζΔt.(10)The random force reproduces Brownian motion. To conserve total momentum, the momentum change imparted to the beads through their interactions with the fluid is balanced by a momentum change in the fluid. The momentum change is distributed to the three closest fluid lattice sites using a linear interpolation scheme based on the proximity of the lattice site to the polymer beads. Through this momentum transfer, hydrodynamic interactions between the beads occur.The beads are repelled from the walls with a force of magnitude Fwall=250kBTσk3(xbead−xwall)2, xbead>(xwall−1),(11)where the repulsion range is 1ΔL. Each monomer will also be attracted to the top wall by a force with magnitude Fstick=KstickkBTσk3(xbead−xwall+10)2, xbead>(xwall−Xstick)(12)and range X_stick (see Fig. ​Fig.1).1). The sticking force is turned off every one out of one hundred time steps of the polymer (1% of the simulation time steps). We vary both X_stick and K_stick to achieve separation of the polymers.In previous experiments, DNA has been trapped against one wall by using thermophoresis,¹⁰ dielectrophoresis,²⁸ and nanoplasmonic tweezers.²⁹ In the case of thermophoresis, the trap strength (K_stick) can be controlled by tuning the intensity of the temperature gradient and the trap extension (X_stick) can be controlled through the area over which the gradient extends. Both of these are set through focusing of the laser used to produce local heating. Similarly, the trap parameters can be controlled when using plasmonic tweezers by controlling the laser beam exciting the nanoplasmonic structures. In dielectrophoresis, the DNA is trapped by an AC electric field and can be controlled by tuning the frequency and amplitude of the field.In this work, the number of polymers, N_p, is 10 unless otherwise noted, and the container size is 25 ΔL × 50 ΔL × 2 ΔL. The time step for the fluid is Δτ = 8.8 × 10⁻⁵ s, and for the polymer is Δt = 3.7 × 10⁻⁶ s. The total simulation time is over 100 chain relaxation times, allowing sufficient independent samples to perform statistical analysis.Two counter-rotating vortices, shown in 1, are produced by introducing external forces to the fluid bound by walls in the x-direction and periodic in the y and z. Two forces of equal magnitude push on the fluid in the upper x region (12ΔL < x < 25ΔL): one in the +y-direction along y = 10ΔL, and one in the –y-direction along y = 40ΔL. Such counter-rotating vortices can be produced in microfluidic channels using acoustically driven bubbles,^3,4,30 local heating,¹⁰ or plasmonic nanostructures.⁵ The flow speed is controlled by very different external mechanisms in each case. We therefore choose a simple model to produce fluid flow that is not specific to one mechanism.The simulations are started using random initial conditions, and therefore, both lengths of polymer are dispersed throughout the channel. Within a few minutes, the steady state configurations pictured in Figs. ​Figs.22 and ​and33 are reached. We define the steady state as when the number of polymer chains in the trap changes by less than one chain (10 beads) per 1000 polymer time steps. Intermittently, some polymers may still escape and re-enter the trap even in the steady state. Three final configurations are possible: Both the lengths of DNA have become trapped, both lengths continue to rotate freely, or the shorter strand has become trapped while the longer rotates freely. Two of these states leave the polymers mixed; in the third, the strands have separated by size.Open in a separate windowFIG. 2.Snapshots at t = 0Δt (left) and t = 2500Δt (right) showing the separation of 15-bead strands (grey) from 10-bead strands (black) of DNA. For these simulations, K_stick = 55 and Y_stick = 0.7ΔL.Open in a separate windowFIG. 3.Snapshots at t = 0Δt and t = 2500Δt showing the separation of 13-bead strands (grey) from 10-bead strands (black) of DNA. For these simulations, K_stick = 55 and Y_stick = 0.7ΔL as in Fig. ​Fig.2.2. Note that one long polymer is trapped, as well as all of the shorter polymers.By tuning the attractive wall force parameters and fluid flow, the separated steady state can be realized. We first set the flow parameters that allow the larger chains to rotate freely at the center of the vortices while the shorter chains rotate closer to the wall. The trap strength, K_stick, and extension, X_stick, are changed until the shorter polymers do not leave the trap. The same parameters were used to separate 10-bead chains from 15-bead and 13-bead chains.As shown in Fig. ​Fig.2,2, we have been able to separate shorter 10-bead chains from longer 15-bead chains. In the steady state, 97% of the rotating polymers were long polymers averaged over twenty simulations initialized with different random starting conditions. For three simulations, one small polymer would intermittently leave the trap region. In two of these simulations, one long polymer became stably trapped in the steady state. In another simulation, one 15-bead chain was intermittently trapped. On average, the trapped polymers were 5% 15-bead chains and 95% 10-bead chains. Again, 97% of the rotating polymers were 15-bead chains.Simulations conducted with 10-bead and 13-bead chains also showed significant separation of the two sizes as can be seen in Fig. ​Fig.3.3. In the steady state, 30% of the trapped polymers are 13-bead chains and 70% are 10-bead chains, averaged over twenty different random initial starting conditions and 1000 polymer time steps. Only 14.8% of the shorter polymers were not trapped, leading to 85.2% of the freely rotating chains being 13-bead chains. This is therefore a viable test to detect the presence of these longer chains.We have also separated 20-bead chains from 10-bead chains with all of the shorter chains trapped and all of the longer chains freely rotating in the steady-state. These results do not change for twenty different random initial starting conditions and 1000 polymer time steps. None of the longer polymers intermittently enter the trap region nor do any of the shorter polymers intermittently escape.The separation is achieved by tuning the trapping force and flow rate. Strong flows will push all the DNA molecules into the trap. The final state is mixed, with both short and long strands trapped. For flows that are too weak, the short molecules are not sufficiently compressed by the flow and therefore do not enter the trap region. The end state is mixed, with all polymers freely rotating. Separation is achieved when the flow rate is tuned so that the short strands are compressed against the channel wall while the long polymers rotate near the center of the vortices. The trap strength must then be set sufficiently high enough to prevent the short strands from being pulled by the hydrodynamic drag force out of the trap.The mechanism of the separation depends on the differences in the steady state configurations of the polymers and chances of a polymer escaping the trap. As shown in Fig. ​Fig.4,4, both longer and shorter chains are pulled into the trap region by the flow. However, the longer chains have a larger chance of a bead escaping into a region of the flow where the fluid velocity is sufficient to pull the entire strand out of the trap. As shown in Ref. ¹¹, the trapping rate depends on diffusion in a polymer depleted region near the trap, in agreement with classical theory which neglects bead-wall interactions. In addition, the theory depends on the single bead diffusion rate and does not take into account the elastic force holding the beads together. Diffusion becomes as significant as convection in the polymer depleted region leading to dependence on the Peclet number. Since longer polymers have more beads; they have more chances of a single bead diffusing through this layer into the region where convection is again more important. Thus, they are pulled out of the trap at a faster rate than the shorter chains.Open in a separate windowFIG. 4.N, number of beads in the trap region, versus time for 15-bead DNA strands (solid line) and 10-bead DNA strands (dashed line). Here, ΔT = 10000Δt. The simulation parameters are the same as in Fig. ​Fig.22.In addition, longer chains have a second trap resulting from the microflow. As shown in Ref. ¹², DNA in counter-rotating vortices can tumble at the center of one vortex or be stretched between the centers of the two vortices. We have observed both these conformations for the longer polymer strand. They are a stable trajectory for the longer polymer that remains outside of the trapping region. As seen in Fig. ​Fig.4,4, few monomers of the longer chains enter the trap region once the steady state has been reached. However, the shorter polymer rotates at a larger radius than the longer polymer as seen in Fig. ​Fig.5.5. The shorter polymers therefore are pushed back into the trap while the longer strands rotate stably outside the trapping region.Open in a separate windowFIG. 5.Trajectories of 15-bead DNA (grey) and 10-bead DNA (black). The position of each monomer is plotted for 100 consecutive time steps. Note that the longer polymers rotate in the center of the channel while the shorter polymers rotate at the edges. Simulation parameters are the same as in Fig. ​Fig.22.This mechanism is similar to the one proposed for the separation of colloids by size in Refs. ³ and ⁴. In that experimental work, the smaller colloidal particles rotated at larger radii. This allowed the smaller beads to be pushed out of the vicinity of the vortices by the streaming flow, while the larger beads continued to circle. However, in our simulations, we have the additional mechanism of separation based on the increased chance of a longer polymer escaping the trap region. This mechanism is important for maintaining the separation. Long polymers initially in the trap region or which diffuse into the trap would not be able to escape without it.We expect that this technique could be used to detect the sizes of DNA fragments on the order of thousands of base pairs. It relies on the flexibility of the molecule and its interaction with the flow. Common lab procedures such as restriction enzyme digests for DNA fingerprinting can produce these long fragments. Current techniques such as gel electrophoresis require significant time to separate the long strands that move more slowly through the matrix. This effect could therefore be a good candidate for developing a microfluidic analysis that is significantly faster than traditional procedures. Our separation occurs in minutes rather than in hours as for gel electrophoresis.As pointed out in Ref. ², hydrodynamic effects have been shown to be important for microfluidic devices for separation. We have demonstrated, in simulation, a novel hydrodynamic mechanism for separating polymers by length. We hope that these promising calculations will inspire experiments to verify these results.  相似文献   

Enhancement of biosensing performance in a droplet-based bioreactor by in situ microstreaming     

Olivier Ducloux  Elisabeth Galopin  Farzam Zoueshtiagh  Alain Merlen    Vincent Thomy 《Biomicrofluidics》2010,4(1)

A droplet-based micro-total-analysis system involving biosensor performance enhancement by integrated surface-acoustic-wave (SAW) microstreaming is shown. The bioreactor consists of an encapsulated droplet with a biosensor on its periphery, with in situ streaming induced by SAW. This paper highlights the characterization by particle image tracking of the speed distribution inside the droplet. The analyte-biosensor interaction is then evaluated by finite element simulation with different streaming conditions. Calculation of the biosensing enhancement shows an optimum in the biosensor response. These results confirm that the evaluation of the Damköhler and Peclet numbers is of primary importance when designing biosensors enhanced by streaming.It has been pointed out that biosensing performances can be limited by the diffusion of the analytes near the sensing surface.¹ In the case of low Peclet number hydrodynamic flows, typical of microfluidic systems, molecule displacements are mainly governed by diffusive effects that affect time scales and sensitivity. To overcome this problem, the enhancement of biosensor performance by electrothermal stirring within microchannels was first reported by Meinhart et al.² Other authors^{3, 4} numerically studied the analyte transport as a function of the position of a nanowire-based sensor inside a microchannel, stressing on the fact that the challenge for nanobiosensors is not the sensor itself but the fluidic system that delivers the sample. Addressing this problem, Squires et al.⁵ developed a simple model applicable to biosensors embedded in microchannels. However, the presented model is limited to the case of a steady flow. The use of surface-acoustic waves (SAWs) for stirring in biomicrofluidic and chemical systems is becoming a popular investigation field,^{6, 7, 8, 9} especially to overcome problems linked to steady flows by enhancing the liquid∕surface interaction.^{1, 10, 11} The main challenges that need to be addressed when using SAW-induced stirring are the complexity of the flow and its poor reproducibility. However, some technical solutions were proposed to yield a simplified microstreaming. Yeo et al. presented a centrifugation system based on SAW that produces the rotation of the liquid in a droplet in a reproducible way by playing on the configuration of the transducers and reflectors,¹² and presented a comprehensive experimental study of the three-dimensional (3D) flow that causes particle concentration in SAW-stirred droplets,¹³ revealing the presence of an azimuthal secondary flow in addition to the main vortexlike circular flow present in acoustically stirred droplets. The efficiency of SAW stirring in microdroplets to favorably cope with mass transport issues was finally shown by Galopin et al.,¹⁴ but the effect of the stirring on the analyte∕biosensor interaction was not studied. It is expected to overcome mass transport limitations by bringing fresh analytes from the bulk solution to the sensing surface.The studied system, described in Fig. ​Fig.1,1, consists of a microliter droplet microchamber squeezed between a hydrophobic piezoelectric substrate and a hydrophobic glass cover. Rayleigh SAWs are generated using interdigitated transducers (interdigital spacing of 50 μm) laid on an X-cut LiNbO₃ substrate.^{1, 15, 16} The hydrophobicity of the substrate and the cover are obtained by grafting octadecyltrichlorosilane (OTS) self-assembled monolayers (contact angle of 108° and hysteresis of 9°). To do so, the surface is first hydroxylized using oxygen plasma (150 W, 100 mT, and 30 sccm³ O₂) during 1 min and then immersed for 3 h into a 1 mM OTS solution with n-hexane as a solvent.Open in a separate windowFigure 1(a) General view of the considered system. (b) Mean value of the measured speeds within the droplet as a function of the inlet power before amplification.When Rayleigh waves are radiated toward one-half of the microchamber, a vortex is created in the liquid around an axis orthogonal to the substrate due to the momentum transfer between the solid and the liquid. This wave is generated under the Rayleigh angle into the liquid.Speed cartographies of the flow induced in the droplet are realized using the particle image tracking technique for different SAW generation powers. To do so, instantaneous images of the flow are taken with a high-speed video camera at 200 frames∕s and an aperture time of 500 μs on a 0.25 μl droplet containing 1 μm diameter fluorescent particles. Figure ​Figure11 shows the mean speed measured in the droplet as a function of the inlet power. The great dependence of the induced mean speed with the SAW power enables a large range of flow speeds in the stirred droplet. Moreover, the flow was visualized with a low depth of field objective. It was found to be circular and two dimensional (2D) in a large thickness range of the droplet.The binding of analytes to immobilized ligands on a biosensor is a two step process, including the mass transport of the analyte to the surface, followed by a complexation step,Abulk↔kmAsurface+B↔ka,kdAB(1)with k_m as the constant rate for mass transport from and to the sensor, and k_a and k_d as the constant rates of association and dissociation of the complex.At the biosensor surface, the reaction kinetics consumes analytes but their transport is limited by diffusive effects. In this case, the Damköhler number brings valuable information by comparing these two effects. Calling the characteristic time of reaction and diffusion, respectively, τ_C and τ_M, the mixing time in diffusion regime can be approximated by τ_M≈h²∕D with D as the diffusion coefficient and h a characteristic length of the microchannel. Calling R_T the ligand concentration on the surface in mole∕m², the Damköhler number (D_a) can be written asDa=τMτC=kaRThD.(2)Depending on the type of reaction, the calculation of D_a helps determine if a specific biointeraction will benefit from a mass SAW-based microstreaming. If the Damköhler number is low, the reaction is slow compared to mass transport and the reaction will not significantly benefit from microstirring. For example, the hybridization of 19 base single stranded DNA in a microfluidic system with a characteristic length of 500 μm is characterized by a Damköhler number of 0.07 and is therefore not significantly influenced by mass transport. On the contrary, the binding of biotin to immobilized streptavidin is characterized by a D_a number of approximately 10⁴. In this case, the stirring solution will significantly improve the reaction rate.COMSOL numerical simulations were carried out to study the efficiency of the SAW stirring in the case of a droplet-based microbioreactor with a diameter of 1 mm. Assuming a 2D flow, the simulated model takes into account the convective and diffusive effects in the analyte-carrying fluid and the binding kinetics on the biosensor surface. This approach was thoroughly developed by Meinhart et al.²On the biosensor surface, the following equations are solved:∂B∂t=kacs(RT−B)−kdB,(3)∂B∂t=D|∂c∂y|y=0(4)with c as the local concentration of analytes in the droplet and B as the surface concentration of bound analytes on the biosensor surface. Simulation results show that a depleted zone is formed near the biosensor in the case of an interaction without stirring. This zone is characterized by a low concentration of analytes and results from the trapping of analytes on the biosensor surface, thus creating a concentration gradient on the vicinity of the biosensor. When stirring is applied, the geometry of the depleted zone is modified, as it is pushed in the direction of the flow. The geometry of the depleted zone then depends on many parameters, among which the diffusion coefficient D, the speed distribution of the flow (not only near the biosensor but also in the whole microfluidic system), and the reaction kinetics on the biosensor. In our case, which is assimilated to a simple circular flow, the depleted zone reaches a permanent state consisting of an analyte-poor layer situated in the exterior perimeter of the stirred droplet. The diffusion of analytes is then limited again by diffusion from the inner part of the droplet toward its exterior perimeter (see Fig. ​Fig.22).Open in a separate windowFigure 2(a) Mean concentration of bound analytes vs time for different mean flow speeds. (b) The obtained concentration profiles with and without circular stirring, t=10 000 s.The initial analyte and receptor concentrations are, respectively, 0.1 nM in the solution and 3.3×10⁻³ nM m on the biosensor surface, the diffusion coefficient is D=10⁻¹¹ m² s⁻¹, and the reaction constants are k_a=10⁶ M⁻¹ s⁻¹ and k_d=10⁻³ s⁻¹. Simulations show that the mean concentration of bound analytes highly increases with the flow speed, improving the efficiency of the biosensing device. To evaluate the benefits of in situ microstreaming with SAW, the same simulations were conducted for D_a numbers ranging from 10⁴ to 10⁸ M⁻¹∕s, by ranging the diffusion coefficient from 4×10⁻¹² to 4×10⁻⁹ m²∕s, and the association coefficient k_a from 10⁴ to 10⁸ M⁻¹∕s. The enhancement factor of analyte capture, defined as the ratio of the binding rate with streaming B and the binding rate without streaming B0, is plotted in Fig. ​Fig.33 for different values of D_a. Calculations are done in the case of a mean flow speed of 0.5 mm∕s.Open in a separate windowFigure 3(a) Enhancement factor (defined as the ratio between binding rate with streaming B and binding rate without streaming B0) for different Damkhöler numbers and (b) normalized enhancement factor for different Peclet numbers.One can notice the saturation of the enhancement factor curve for large value of D_a to the value of 3.5 for high D_a. This can be explained by the fact that for large k_a∕D_a ratios, the analytes, which normally require penetration in the depleted zone by diffusion, do not have time to interact with the biosensor when they pass in the vicinity of its surface. The efficiency of the streaming is then reduced for large values of D_a. In the case of our specific flow configuration, the enhancement factor reaches 3.2 for the interaction of streptavidin on immobilized biotin (D_a=10³).The reported simulation results can be compared to an experimental value obtained using the droplet-based surface plasmon resonance sensor streamed in situ using SAW reported by Yeo et al.¹² By monitoring the streptavidin∕biotin binding interaction on an activated gold slide, they showed that SAW stirring brings an improvement factor of more than 2. This difference can be accounted to the high complexity of the induced 3D flow, which was modeled in a simple manner in our calculations.Other factors must be taken into account when optimizing the improvement factor, such as the flow velocity and the characteristic length of the mixing. To do so, the Peclet number allows the comparison of the convective and diffusive effects.¹⁷ For δC a typical variation in concentration on the distance h, the Peclet number is given byPe=UhD.(5)A significantly high Peclet number causes a decrease in biosensing efficiency as the analytes do not have enough time to interact with the biosensing surface by diffusion through the analyte-poor layer. On the contrary, the case of a low Peclet number corresponds to the diffusion-limited problem. Therefore, for each Damköhler number, there is a Peclet number optimizing this factor. To illustrate this fact, Fig. ​Fig.3b3b shows the calculation of the enhancement factor as a function of the Peclet number for a given D_a.In this paper, we showed that surface loading of typical analytes on a droplet-based biosensor can be highly increased by SAW microstirring. The system permits the enhancement of the biosensing performances by the continuous renewal of the analyte-carrying fluid near the sensing surface. Thanks to mean flow speeds measured up to 1800 μm∕s, the SAW microstreaming can be beneficial to the biosensing of a large range of analyte∕ligand interactions. In addition to the biosensing performance improvement, such a method can be easily integrated in micro-micro-total-analysis systems, which makes it a convenient tool for liquid handling in future biochips.  相似文献   

Angular region: a measure of underdamped behavior     

F.M. Reza 《Journal of The Franklin Institute》1982,314(3):191-202

The natural modes of an underdamped dynamical system are given by the characteristic numbers of the quadratic operator pencil where the operator A depends on the dissipative and reactive elements of the system, while B depends solely on the reactive elements. The operator P(s) for every applied stimulus vector signal x must satisfy: A measure of underdamped behaviour is suggested by predetermining an angular region |φ| containing all natural modes of the system, When a comparison between positive operators A and B is available, say B²=KA, then The paper is motivated by Duffin-Krein-Gohberg's earlier mathematical contributions.  相似文献   

Gauss quadrature formula: An extension via interpolating orthogonal polynomials     

M.A. Bokhari 《Journal of The Franklin Institute》2007,344(5):637-645

The n-point Gauss quadrature rule states that