Thermosiphon-Based PCR Reactor: Experiment and Modeling

Anal. Chem. 2004, 76, 3707-3715

Thermosiphon-Based PCR Reactor: Experiment and Modeling Zongyuan Chen,† Shizhi Qian,† William R. Abrams,‡ Daniel Malamud,‡ and Haim H. Bau*,†

Department of Mechanical Engineering and Applied Mechanics, School of Engineering and Applied Science, and Department of Biochemistry, School of Dental Medicine, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6315

The common thermal polymerase chain reaction (PCR) process requires the cycling of reagents through three distinct temperatures for denaturation (90-94 °C), annealing (50-55 °C), and extension (72 °C). In most benchtop PCR reactors, the sample is maintained stationary while the temperature is repetitively alternated, which makes it necessary to heat and cool both the reagents and the heating block. This process results in considerable thermal inertia, and it is relatively slow and energy-intensive. Someoftheseshortcomingscanbeovercomethroughminiaturization1-9 and the use of continuous-flow reactors.10-21 In continuous-flow

PCR reactors, the temperature of the three thermal zones is maintained fixed while the reagents are cycled continuously through these zones. Continuous-flow reactors allow for significantly shorter heating and cooling times with reduced energy consumption since it is not necessary to combat the thermal inertia of the apparatus. The continuous-flow PCR reactors can be classified into three categories: unidirectional,10-15 closed loop,16-20 and oscillatory.21 In the unidirectional reactor, a slug of fluid is propelled through a conduit that meanders through various heating zones. In the closed-loop reactor, fluid is circulated along a closed path. In the oscillatory PCR, the slug of fluid is moved back and forth among the various constant-temperature zones. The continuous-flow PCR reactors require, however, a means to propel the fluid. In recent years, various means for propelling the fluid, ranging from pressure10-15,21 to peristalsis16,17 to magnetohydrodynamics,18-20 have been proposed. Here, we put forward an intriguing alternative. The relatively large temperature variations needed for PCR induce significant variations in the fluid’s density, which under appropriate conditions can be used to generate fluid motion. Flow velocities on the order of millimeters per second are readily attainable. Recently, Krishnan et al.22 took advantage of the naturally occurring circulation in a cavity heated from below to 97 °C and cooled from above to 61 °C (the Rayleigh-Benard cell) to circulate reagents between two temperature zones. They demonstrated the amplification of a 295-base pair fragment of the human β-actin gene. In their device, however, the reactants were

* To whom correspondence should be addressed. E-mail: bau@seas. upenn.edu. † Department of Mechanical Engineering. ‡ Department of Biochemistry. (1) Kricka, L. J.; Wilding, P. Anal. Bioanal. Chem. 2003, 377, 820-825. (2) Belgrader, P.; Elkin, C. J.; Brown, S. B.; Nasarabadi, S. N.; Langlois, R. G.; Milanovich, F. P., Colston, B. W.; Marshall, G. D. Anal. Chem. 2003, 75, 3446-3450. (3) Shin, Y. S.; Cho, K.; Sun, H. L.; Chung, S.; Park S. J.; Chung, C.; Han D. H C.; Chang, J. K. J. Micromech. Microeng. 2003, 13, 768-774. (4) Koh, C. G.; Tan, W.; Zhao, M.; Ricco, A. J.; Fan, Z. H. Anal. Chem. 2003, 75, 4591-4598. (5) Yoon, D. S.; Lee, Y. S.; Lee, Y.; Cho, H. J.; Sung, S. W.; Oh, K. W.; Cha, J.; Lim, G. J. Micromech. Microeng. 2002, 12, 810-823. (6) Lagally, E. T.; Medintz, I.; Mathies, R. A. Anal. Chem. 2001, 73, 565-570. (7) Belgrader, P.; Benett, W.; Hadley, D.; Richards, J.; Stratton P.; Mariella R.; Milanovich, F. Science 1999, 284, 449-450. (8) Bruckner-Lea, C. J.; Tsukuda, T.; Dockendorff, B.; Follansbee, J. C.; Kingsley, M. T.; Ocampo, C.; Stults, J. R.; Chandler, D. P. Anal. Chim. Acta 2002, 469, 129-140. (9) Khandurina, J.; McKnight, T. E.; Jacobson, S. C.; Waters, L. C.; Foote, R. S.; Ramsey, J. M. Anal. Chem. 2000, 72, 2995-3000.

(10) Nakano, H.; Matsuda, K.; Yohda, M.; Nagamune, T.; Endo, I.; Yamane, T. Biosci. Biotechnol. Biochem. 1994, 58, 349-352. (11) Kopp, M. U.; Mello, A. J.; Manz, A. Science 1998, 280, 1046-1048. (12) Obeid, P. J.; Christopoulos, T. K.; Crabtree, H. J.; Backhouse, C. J. Anal. Chem. 2003, 75, 288-295. (13) Sun, K.; Yamaguchi, A.; Ishida, Y.; Matsuo, S.; Misawa, H. Sens. Actuators 2002, 84, 283-289. (14) Curcio, M.; Roeraade, J. Anal. Chem. 2003, 75, 1-7. (15) Park, N.; Kim, S.; Hahn, J. H. Anal. Chem. 2003, 75, 6029-6033. (16) Liu, J.; Enzelberger, M.; Quake, S. Electrophoresis 2002, 23, 1531-1536. (17) Chou, C. F.; Changrani, R.; Roberts, P.; Sadler, D.; Burdon, J.; Zenhausern, F.; Lin, S.; Mulholland, A.; Swami, N.; Terbrueggen, R. Microelectron. Eng. 2002, 61-62, 921-925. (18) Bau, H. H., IMECE 2001, MEMS 23884 Symposium Proceedings, New York, 2001. (19) Bau, H., H.; Zhu, J.; Qian, S.; Xiang, Y. Sens. Actuators, B 2003, 88, 205216. (20) West, J.; Karamata, B.; Lillis, B.; Gleeson, J. P.; Alderman, J.; Collins, J. K.; Lane, W.; Mathewson, A.; Berney, H. Lab Chip 2002, 2, 224-230. (21) Bu, M. Q.; Tracy, M.; Ensell, G.; Wilkinson, J. S.; Evans, A. G. R. J. Micromech. Microeng. 2003, 13, S125-S130. (22) Krishnan, M.; Victor, M. U.; Burns, M. A. Science 2002, 298, 793-793.

A self-actuated, flow-cycling polymerase chain reaction (PCR) reactor that takes advantage of buoyancy forces to continuously circulate reagents in a closed loop through various thermal zones has been constructed, tested, and modeled. The heating required for the PCR is advantageously used to induce fluid motion without the need for a pump. Flow velocities on the order of millimeters per second are readily attainable. In our preliminary prototype, we measured a cross-sectionally averaged velocity of 2.5 mm/s and a cycle time of 104 s. The flow velocity is nearly independent of the loop’s length, making the device readily scalable. Successful amplifications of 700and 305-bp fragments of Bacillus cereus genomic DNA have been demonstrated. Since the device does not require any moving parts, it is particularly suitable for miniature systems.

10.1021/ac049914k CCC: $27.50 Published on Web 05/28/2004

© 2004 American Chemical Society

Analytical Chemistry, Vol. 76, No. 13, July 1, 2004 3707

Figure 1. Schematic description of the general concept of the thermosiphon-based PCR device. The closed loop is filled with PCR reagents. The plane of the loop is inclined at an angle R (90° > R g 0°) with respect to the direction of the gravity. The loop is surrounded by three heaters that maintain different temperatures. The top section of the loop is cooled.

not subjected to uniform temperatures and flow conditions, and the reactants may have diffused away from the heated regions, thereby reducing the efficiency of the process and leading to nonspecific amplification. In more recent experiments, Braun et al.23 heated with a laser the center of a cylindrical cavity filled with PCR solution to induce a horizontal temperature gradient and flow circulation. They successfully amplified a 96-base pair segment of λ-DNA. In contrast, Wheeler et al.24 confined the fluid in a closed loop that was subject to two heating zones maintained at 94 and 57 °C. They successfully amplified both a 58- and a 160base pair segment of a multiple cloning site fragment of DNA. (We are grateful to an anonymous reviewer who brought this conference paper to out attention.) In this paper, we describe a loop-based reactor with three heating zones. We confine the reactants in a tube bent into a closed loop. There is a great amount of flexibility in selecting the geometry of the loop. Either planar or three-dimensional geometries can be used. Here, we focus only on planar loops. The plane of the loop is inclined at an angle R (90° > R g 0°) with respect to the direction of the gravity. See Figure 1 for an example. In our experiment, we adjusted the inclination angle (R) to control the flow rate in the loop. Various sections of the loop are subjected to different temperatures. When one vertical leg is subjected to a higher temperature than the other, the fluid in the hot leg (ascender) will be less dense than the fluid in the other leg (descender), thus generating a buoyancy force that induces fluid motion. Devices in which fluid circulation is driven by buoyancy are known as thermosiphons.25 These devices have been used in various energy conversion and cooling applications as passive means to circulate fluids without the need for a pump. In this paper, we describe the application of the thermosiphon for PCR. We refer to it as a self-actuated, flow-cycling (SAFC) PCR. The paper is organized as follows. The Experimental Section describes the experimental apparatus and procedure. DNA Amplification Experments summarizes our experimental observations. One-Dimensional Model provides a simple, one-dimensional (23) Braun, D.; Goddard, N. L.; Libchaber, A. Phys. Rev. Lett. 2003, 91, 1581031-4. (24) Wheeler, E. K.; Benett, B.; Stratton, P.; Richards, J.; Christian, A.; Chen, A.; Weisgraber, T.; Ness, K.; Ortega, J.; Milanovich, F. Proceedings of µTAS 2003 Seventh International Conference on Micro Total Analysis Systems. Squaw Valley, CA, October 2003. (25) Bau, H. H.; Torrance, K. E. J. Fluid Mech. 1981, 109, 417-433.

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Figure 2. Schematic depiction of the experimental setup. Three segments of the loop are surrounded by aluminum heating blocks. A thermocouple attached to point A is used to monitor the flow rate. Point B indicates the origin of the coordinate system.

model that predicts the flow rate in and the temperature distribution around the loop and provides the designer with essential guidelines. Three-Dimensional Model describes the results of three-dimensional numerical simulations that provide detailed information about the spatial temperature distribution. This section also examines the validity of the one-dimensional model. Discussion and Conclusions sets forth our conclusions and briefly compares the thermosiphon-based SAFC-PCR with other devices for DNA amplification. A list of abbreviations and symbols is provided at the end of the paper. EXPERIMENTAL SETUP To test the thermosiphon-based PCR reactor concept, we constructed a closed loop with Teflon tubing (inner diameter 764 µm, Daburn Electronics & Cable Corp.) bent to nearly the form of an isosceles right triangle (Figure 2). The length of the vertical leg was 76 mm. The total length and volume of the loop were, respectively, 260 mm and 119 µL. We demonstrate later that these dimensions can be significantly reduced. Once the sample was introduced into the tube, we closed the loop by tightly inserting the two ends of the tube into a short sleeve for a hermetic seal. Experience suggested that a tight seal would maintain pressurized conditions and prevent bubble formation. Bubble formation is a significant concern as the presence of bubbles in minute conduits can adversely affect or even block flow. It appears that surface wetability is closely related to bubble formation.26 We compared smooth hydrophobic and hydrophilic surfaces by immersing both Teflon and glass tubes filled with water in a bath at 95 °C under atmospheric pressure. We observed the formation of numerous bubbles in the nonwetting Teflon tube while we did not observe any bubbles in the glass tube. We surmise that the bubble nucleation in the Teflon tube was facilitated by the presence of a thin film of air next to the hydrophobic surface. Increases in the fluid’s pressure significantly reduced the rate of bubble formation and bubble size. By sealing the loop well, we in effect pressurized the solution, suppressing bubble formation. Heating was provided with three aluminum blocks that surrounded the tube. Each block was 60 mm in length and 25.4 mm in diameter and was equipped with three bores. One bore accommodated the Teflon tube, the second bore housed the (26) Thomas, O. C.; Cavicchi, R. E.; Tarlov, M. J. Langmuir 2003, 19, 61686177.

resistance heater (Omega, Cartridge Heater), and a T-type thermocouple was inserted in the third bore. The thermocouple was connected to a data acquisition system (HP 3497A and 3458A) that converted the analog signal to a digital one. A computer received the temperature signal via a PCI-GPIB interface (National Instruments) and determined the power input to the heater using a proportional-integral-differential controller. The control algorithm was programmed with LabVIEW. After introduction of the sample, all three heaters were programmed to maintain 95 °C for 6 min to ensure initial denaturation of the DNA. Subsequently, the vertical leg was heated to 94 °C (denaturation zone), the diagonal leg was maintained at 55 °C (annealing zone), and the horizontal leg was maintained at 72 °C (extension zone). The upper part of the tube was exposed to the ambient temperature to facilitate cooling from 94 to 55 °C. The other sections of the loop between the various heating zones were insulated. The arrangement of the heaters facilitated counterclockwise circulation in the loop. At the conclusion of the PCR amplification process, all the heaters were set to 72 °C for 7 min to facilitate final extension. A T-type thermocouple (Omega) was attached to the tube’s surface at point A (Figure 2). The corresponding temperature is denoted TA. This temperature reading was used to estimate the flow rate. As the hot fluid emerged from the heated vertical leg (ascender), it lost heat to the surroundings. The temperature at point A (TA) increased as the fluid velocity increased. Thus, by monitoring the temperature TA, it was possible to deduce the flow velocity. Neglecting axial conduction, a simple heat balance suggests that

πD2 dT FCpu j ) - πDh(T - T∞) 4 ds

(1)

The left- and right-hand sides of eq 1 represent, respectively, the rate of change of the convective heat flow and the thermal losses to the ambient. In the above, F and Cp are, respectively, the fluid density and specific heat, D is the inner diameter of the tube, T∞ is the ambient temperature, and h is the overall heat-transfer coefficient based on the inner diameter. This overall heat-transfer coefficient accounts for radial heat transfer from the fluid to the inner surface of the tube wall, conduction in the wall, and heat transfer to the ambient. To the first approximation, h is assumed to be temperature-independent. uj is the average fluid velocity in the loop. Upon integration, we obtain

(

ln

)

TH - T ∞ 4hl ) TA - T ∞ FCpu jD

(2)

where TH is the temperature of the fluid inside the ascender and l is the distance from the ascender’s exit to the point A. Setting θ ) (ln((TH - T∞)/(TA - T∞)))-1, eq 2 can be rewritten as

uj ) (4hl/FCpD)θ

(3)

To calibrate our thermal anemometer, we pumped liquid into the loop with a syringe pump (kd Scientific, PA) while maintaining the various heating blocks at their operating temperatures. Figure

Figure 3. Measured average velocity (u j , m/s) of the fluid in the loop as a function of θ ) (ln((TH - T∞)/(TA - T∞)))-1. During the calibration, the flow in the loop was driven by a syringe pump.

3 depicts the average velocity (uj ) as a function of θ. The symbols and solid line correspond, respectively, to the experimental data and a best-fit line. Regression analysis suggests uj ∼ 9.64 × 10-4 θ, where uj is expressed in meters per second. Assuming that the liquid properties correspond to water at 80 °C, h is found to be 75.15 W/(m2 K). The measurement of the flow velocity allowed us both to adjust the flow rate to a desired level and to monitor the flow conditions in the loop. The presence of bubbles and changes in the fluid’s viscosity, which inhibit the flow, could be detected by oscillatory or reduced temperature readings (TA) or both. When the device was positioned parallel to the gravity vector, the flow rate was too high. To reduce the fluid velocity, we inclined the device to an angle (R) of 30° with respect to the vertical. Under these conditions, using dye as a passive tracer and monitoring the dye progression with a video camera, we measured a flow velocity of ∼2.5 mm/s and a period of 104 s/cycle. The dye-based velocity measurement was consistent with the estimate obtained with our thermal anemometer. Since the fluid velocity depends sensitively on the tube’s diameter, we could have used a smaller diameter tube without sacrificing performance. In our prototype, we adjusted the flow velocity by tilting the loop’s plane with respect to the gravity vector. In more sophisticated prototypes, this function could be carried out with a valve. DNA AMPLIFICATION EXPERIMENTS PCR amplifications of 700- and 305-bp fragments of Bacillus cereus genomic DNA template were performed. In parallel, control runs were carried out in a standard benchtop PCR thermocycler (Techne Inc., Princeton, NJ). Both the control and SAFC PCR reactions were carried out with the same reagents: 50 mM Tris-HCl (pH 9.0), 20 mM (NH4)2SO4, 1.5-3.5 mM MgCl2, 200 µM dNTP, and 0.1 µg/µL BSA. Primers for the amplification of the 700- and 305-bp fragments of the B. cereus genomic DNA were, respectively, 0.24 (5′-GAA ACA ACA GTA TAC GAT TTT GAT-3′, 5′-TTT CGA AGA GCA ATC AGC TAA T-3′) and 1.5 µM (5′-TCT CGC TTC ACT ATT CCC AAG T-3′, 5′-AAG GTT CAA AAG ATG GTA TTC AGG-3′). Two different concentrations of the DNA template were prepared: 13.5 and 27 ng/µL. Two different concentrations of Taq were used: 0.05 and 0.1 unit/µL. The process began with a 6-min initial denaturation of the sample at 95 °C. Two different initial denaturation methods were Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

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Figure 4. Images of ethidium bromide-stained DNA products in a 1% agarose gel. (A) Initial denaturation of SAFC PCR was performed outside the SAFC loop. Lanes A1-5: 700-bp fragment. Lane A1: positive control I, 0.05 unit/µL Taq and 27 ng/µL DNA template. Lane A2: positive control II, 0.1 unit/µL Taq and 13.5 ng/µL DNA template. Lanes A3 and A4: SAFC PCR products using the same mixtures as for controls I and II, respectively. Lane A5: negative control using the same mixture as control II without any DNA. Lanes A6-10: 305-bp DNA fragment. Lane A6: positive control III, 0.05 unit/µL Taq and 27 ng/µL DNA template. Lane A7: positive control IV, 0.1 unit/µL Taq and 13.5 ng/µL DNA template. Lanes A8 and 9: SAFC PCR products using the same mixtures as for controls III and IV, respectively. Lane A10: negative control using the same mixture as control IV without any DNA. (B) Initial denaturation of SAFC PCR was performed in the SAFC loop with all three heating blocks maintained at 95 °C. Lane B1: positive control, 0.1 unit/µL Taq and 13.5 ng/µL DNA template. Lane B2: SAFC PCR products using the same mixture as the positive control.

used. In some earlier experiments, the sample was preheated prior to its introduction into the loop. In later experiments, the denaturation took place in the loop itself by maintaining all three heaters at 95 °C. Results were similar with both denaturation techniques. After the denaturation, the heaters were programmed to maintain three thermal zones: 94, 55, and 72 °C for the duration needed to complete 35 cycles (a total of 60 min). The process was completed with a final 7-min extension by maintaining the three heating zones at 72 °C for a total run time of 73 min. The device and the protocol were not optimized, and it is likely that the run time can be significantly reduced. The protocol used in the standard benchtop PCR thermocycler consisted of 5-min initial denaturation at 95 °C, 35 PCR cycles (1-min denaturation at 94 °C, 2-min annealing at 55 °C, and 1-min extension at 72 °C) and 5-min final extension at 72 °C for a total run time of 3 h including temperature ramp times. The temperature at point A (Figure 2) did not vary significantly during the amplification process, indicating that the flow rate remained nearly steady throughout the process. Hence, it appears that bubble formation and the changes of the solution’s viscosity due to the DNA amplification were not significant factors in our experiments. The PCR products were analyzed by agarose gel (1%) electrophoresis (Figure 4) and stained with ethidium bromide. The electrophoresis results confirm successful amplifications of the 700- and 305-bp fragments of B. cereus genomic DNA template. The experiments were repeated several times with excellent reproducibility. Panels A and B of Figure 4 report, respectively, results obtained when the preheating (initial denaturation) was 3710

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carried out outside and inside the loop. The curvature observed in the gel image may be attributed to sample, gel, and electric field nonuniformities. The amplification efficiency of the short DNA amplicon (305 bp) appeared higher than that of the longer amplicon (700 bp) using the same SAFC PCR protocol, probably due to the dependence of the reaction’s efficiency on the residence time. The strength of the signal obtained with the 305-bp DNA amplicon suggests that fewer than 35 cycles would be sufficient to detect the pathogen’s DNA. ONE-DIMENSIONAL MODEL In this section, we derive a simple, one-dimensional model that allows us to predict the fluid velocity in and the temperature distribution around the loop. For additional details of the derivation, see Bau and Torrance.25 We assume that the fluid is incompressible and that the density is constant everywhere except in the body force term in the momentum equation. The velocity profile is assumed to be parabolic (Poiseuille flow). Mass conservation requires that

u(s) ) u j

(4)

is independent of the axial coordinate (s). The momentum equation integrated around the loop results in a balance between the frictional and buoyancy forces.

32uj I

µ(s) D(s)2

ds ) IgF(s) cos γ(s) ds

(5)

In the above, s is a coordinate along the tube’s axis; µ(s) is the viscosity; F(s) is the density (that varies along the loop due to temperature variations); D(s) is the tube’s inner diameter (which may vary from one section to another in order to control the residence time in each heating zone); γ(s) is the angle between the direction of the gravitational acceleration (g ) 9.8 m/s2) and the tube’s axis; and I indicates integration around the loop. In our experimental setup, D was maintained constant (s-independent). To the first approximation, we assume that the density is a linear function of the temperature,

F(s) ) F0(1 - β(T(s) - T0))

(6)

where F0 and T0 are, respectively, the reference density and temperature. T(s) is the average temperature of the cross section located at s. β is thermal expansion coefficient. We will see in the next section that, due to the tube’s small diameter, the temperature variations within any cross section are relatively small. The temperature distribution around the loop can be obtained by solving the energy equation:

πD(s)2 dT F0Cpu ) - πD(s)h(T - Ti) j 4 ds

Figure 5. Cross-sectionally averaged temperature distribution as a function of the axial coordinate s. The origin of the coordinate s is at the point B (Figure 2), and it is directed in the counterclockwise direction. The dashed and solid lines correspond, respectively, to the predictions of the one- and three-dimensional models. The vertical thin dashed lines denote the locations of the heaters and the cooling zone.

(7)

In the above, the temperature Ti is the appropriate temperature of each of the heating sections. For example, in the denaturation, extension, and annealing zones, Ti equals, respectively, 94, 72, and 55 °C. Based on our experimental data, h ) 75.15 W/(m2 K) in the cooling section. Equations 5-7 were solved simultaneously with the finite element program Femlab to obtain the fluid velocity and the temperature distribution around the loop. The average velocity was calculated to be 2.65 mm/s. The one-dimensional model prediction of the axial temperature distribution is depicted (dashed line) in Figure 5. The velocity calculation can be further simplified by assuming that the loop is shaped like a triangle with each of the edges maintained at a different, constant temperature. This calculation was carried out by hand without any software tools. Like the more accurate calculation described above, it predicts an average velocity of 2.68 mm/s, which compares favorably with the measured velocity of 2.5 mm/s under similar conditions. The discrepancy between the estimated and measured values can be attributed to some of the simplifications that were made in the one-dimensional analysis on which we elaborate further in the next section. Close examination of eq 5 reveals that both the right- and lefthand sides of the equation are proportional in magnitude to the length of the loop. This suggests that the average velocity is nearly independent of the loop’s length. In other words, as long as thermal isolation can be maintained between the different heating zones, the loops depicted in Figures 1 and 2 can be scaled in length while maintaining nearly the same velocity. This assertion is confirmed in the next section, where we use a more comprehensive three-dimensional model to calculate the fluid’s velocity. The scaling down of the loop size will, however, adversely affect the residence time of the fluid in each of the heating sections.

Should it be desirable to maintain the same residence time while reducing the loop size, it would be necessary to simultaneously reduce the tube’s diameter. Equation 5 suggests that the fluid velocity scales like the diameter squared. Although the thermal expansion of the liquid results in density variations that, in turn, lead to differences in the flow rates among the various sections of the loop, these velocity variations are relatively small, and their effect on the residence time is neglected in the simplified analysis presented in this section. These variations will be accounted for, however, in the next section. THREE-DIMENSIONAL MODEL Further information on the detailed temperature and velocity distributions in the loop was obtained with three-dimensional numerical simulations. The three-dimensional simulations also served to verify the one-dimensional model’s predictions presented in the previous section. These simulations are helpful for future device design. Below, we introduce the various conservation laws using toroidal coordinates.25 The coordinate s is aligned with the loop’s axis. The coordinates r and φ are, respectively, radial and angular coordinates in the cross-sectional plane that is perpendicular to s. Accordingly, the fluid occupies the domain {0 < s < L, 0 < r < D/2, 0 < φ < 2π}, and the tube wall occupies the domain {0 < s < L, D/2 < r < D0/2, 0 < φ < 2π}, where D0 is the tube’s outer diameter. In our simulations, D and D0 were maintained fixed. One can, however, readily accommodate the more general case in which D and D0 are functions of s. The continuity, momentum, and energy equations in the fluid (0 < r < D/2) are, respectively,

div b u)0

(8)

u Fu b‚∇u b ) Fg b - ∇p + µ∇2b

(9)

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and

b u ‚∇T ) (k/FCp)∇2T

(10)

where b u(s,r,φ), T(s,r,φ), and p(s,r,φ) are, respectively, the velocity, temperature, and pressure. The thermophysical properties are temperature-dependent. The variation of the density as a function of temperature is calculated with eq 6. The temperature dependence of the viscosity (µ), specific heat (Cp), and thermal conductivity (k) were specified according to the correlations given in Gebhart et al.27 Within the tube wall (D/2 < r < D0/2), the temperature distribution is governed by the Laplace equation:

∇2T ) 0

(11)

On the outer surface of the tube, within the heating blocks, we use the isothermal boundary conditions.

T(s,D0/2,φ) ) Ti

(i ) 1, 2, 3)

(12)

where Ti is the temperature of heating block i. In the insulated sections, we specify

∇T‚eˆr ) 0

(13)

were eˆr is a unit vector in the radial direction. In the cooler section, we use the Robins-type boundary condition:

∂T(s,D0/2,φ) ) ha(T(s,D0/2,φ) - T∞) - ks ∂r

(14)

where ks is the thermal conductivity of the tube wall and ha is the convective heat-transfer coefficient between the outer tube wall and the environment. The temperature and heat flux are continuous at the interface between the solid wall and the fluid:

T(s,D-/2,φ) ) T(s,D+/2,φ)

(15)

∂T(s,D-/2,φ) ∂T(s,D+/2,φ) -k ) - ks ∂r ∂r

(16)

and

The fluid velocity satisfies the no-slip boundary condition at all solid surfaces:

u(s,D-/2,φ) ) 0

(17)

Finally, all the dependent variables satisfy the periodic conditions:

u(s + L,r,φ) ) u(s,r,φ)

(18)

T(s + L,r,φ) ) T(s,r,φ)

(19)

and

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The equivalent heat-transfer coefficient (h) accounts for the film resistance in the tube, the conductive wall resistance, and the thermal resistance between the tube’s outer surface and the ambient. That is,

h)

1 1 D0 1 1 D ln + + hwD 2ks D D0ha

(

)

(20)

In the calculation of ha, we used h) 75.15 W/(m2 K), the value that we measured in our experiment; hw ) 992 W/(m2 K), which is based on the heat-transfer coefficient for fluid flowing in a tube;28 and ks ) 0.26 W/(m K).29 Accordingly, ha ) 50.5 W/(m2 K). We solved the three-dimensional conjugate model with the Computational Fluid Dynamics (CFD) code CFD-ACE+ (CFDACE is a product of CFD Research Corp.). The simulations were carried out for the geometry and thermal conditions of our experimental setup. Figure 5 depicts the cross-sectionally averaged temperature T h (s) as a function of s. The origin of the coordinate s is fixed at point B in Figure 2, and the coordinate is directed counterclockwise. The solid and dashed lines correspond, respectively, to the three- and one-dimensional models’ predictions. Witness that following a relatively short development length, the temperatures assume uniform values in the tube segments embedded within the heating blocks. The figure also illustrates that, in the cooling zone, the temperature dips below 55 °C. This indicates that the cooling zone is unnecessarily long and can be shortened. Optimally, the fluid will enter the annealing section having the temperature of 55 °C. The one- and three-dimensional models’ predictions are in reasonable agreement. The one-dimensional model underestimates the thermal development length in the ascender. This underestimate may explain, in part, the onedimensional model’s overestimate of the fluid’s average velocity. Figure 6 depicts the temperature distribution in the cross sections of T(s, r, 0) and T(s, r, π) as functions of r at various s locations. The plane φ ) 0 and π coincides with the loop’s plane. The dashed rectangles indicate the locations of the various heating zones. The slight asymmetry in the temperature distribution is due to the buoyancy effects in each cross section. These buoyancy effects are relatively small due to the smallness of the tube’s diameter. Except at the bends and the cooling section, the figure illustrates that there are relatively small temperature variations within the various cross sections and that the reagents are subject to nearly uniform temperatures in each of the heating zones. Figure 7 depicts the velocity distributions u(s, r, 0) and u(s, r, π). Due to the relatively small diameter of the tube, the velocity profile is nearly parabolic. The nearly parabolic nature of the velocity profile explains the relatively good predictions obtained with the one-dimensional model presented above. The threedimensional theoretical calculations predict a mass flow rate of (27) Gebhart, B.; Jaluria, Y.; Mahajan, R. L.; Sammakia, B. Buoyancy Induced Flows and Transport; Hemisphere: New York, 1988. (28) Bird, R. B.; Steward, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons: New York, 1960; p 399. (29) Perry, R. H.; Green, D. W. Perry’s Chemical Engineers’ Handbook; 7th ed.; McGraw-Hill: New York, 1997; pp 2-337.

Figure 6. Temperature distributions within various cross sections along the closed loop. The locations of the heaters are indicated with dashed rectangles.

Figure 7. Velocity profile within various cross sections along the closed loop.

1.07 mg/s. Since the model accounts for the density variations as a function of the temperature, the actual average velocity of the fluid varies slightly from one cross section to another. The above value for the mass flow rate corresponds to a crosssectionally averaged velocity of 2.40 mm/s (flow rate of 1.1 µL/ s) at the ascender’s exit, which is in excellent agreement with the velocity of 2.5 mm/s measured experimentally. In the previous section, based on the one-dimensional model, we argued that the mass flow rate is nearly independent of the loop size as long as the relative lengths of the various heating zones are maintained the same. To confirm this assertion, we

carried out a three-dimensional simulation of a loop with a total length of 105 mm and with each heating zone having a length of 30 mm. The scaled-down loop had the same tube diameter as the larger loop. The mass flow rate predicted for the reduced-size loop was 0.95 mg/s. This is close to the value calculated for the larger loop (mass flow rate of 1.07 mg/s). This calculation illustrates that the loop size can be significantly reduced without adverse effect on the mass flow rate. The mass flow rate is not, however, the only factor of concern when the loop is being designed. The PCR extension process requires a certain residence time to complete. The amount of time required depends on the amplicon’s Analytical Chemistry, Vol. 76, No. 13, July 1, 2004

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length. A reduction in the loop size, without a corresponding alternation in the tube’s diameter, would result in a reduced residence time in each of the heating zones. The residence time can be controlled either by introducing additional resistance to the flow such as through the use of a valve or by reducing the tube’s diameter. Equation 5 suggests that the average velocity is proportional to the tube’s diameter squared. Reducing the tube’s diameter from 764 to 400 µm would result in a reduction of the mass flow rate from 0.95 to 0.09 mg/s and a reduction in the average velocity from 2.40 to 0.67 mm/s. DISCUSSION AND CONCLUSIONS A self-actuated, flow-cycling PCR device in which the flow is driven by the same thermal forces that are needed for the PCR reaction itself has been designed, constructed, modeled, and tested. The device does not require any pumps or temperature cycling to operate. During the operation, however, the device must be maintained in an upright position. The size of the device can be significantly reduced compared to the preliminary prototype described in the paper. Threedimensional calculations indicate that 2.5-fold reduction in the loop’s length will result in just an 11% reduction in the flow velocity. Sample introduction and removal could be readily automated. A simple one-dimensional model was introduced and verified through comparisons with three-dimensional calculations and experiments. This model can be used for device optimization. An optimized device could be integrated into a miniaturized system suitable for point-of-care detection of pathogen nucleic acids. Successful amplifications of 305- and 700-bp amplicons have been demonstrated. It is interesting to point out that most of the reported amplicons that have been used in conjunction with other continuous-flow PCR devices have been typically well below 300 bp. Based on the intensity of the gel images, it appears that the amplification efficiency of the short amplicon was significantly greater than that of the longer one. This suggests that we should be able to reduce the cycle time for the short amplicon without adverse effects. Another interesting potential application of the self-actuated device is its use in a rotating system such as a “laboratory on a computer disk (CD)”.30 In fact, thermosiphons have been used in rotating systems to enhance the cooling of turbine blades.31 In this case, the centripetal acceleration, ω2r(s), will replace the gravitational acceleration (g). In the above, ω is the angular rotation speed (s-1) and r(s) is the distance between the center of rotation and a point s along the loop. Since in the rotating system, very high accelerations (.9.8 m/s2) can be obtained, one can significantly reduce the size of the device. Furthermore, adjustment in the rotational speed would allow for control of the flow rate. The thermosiphon-based PCR device compares favorably with competing PCR devices aimed at microfluidic applications. In contrast to the unidirectional10-15 and oscillatory-flow21 PCR devices, the thermosiphon-based PCR machine does not require any pressure sources and may utilize a significantly shorter (30) Johnson, R. D.; Badr, I. H. A.; Barrett, G.; Lai, S.; Lu, Y.; Madou, M. J.; Bachas, L. G. Anal. Chem. 2001, 73, 3940-3946. (31) Japikse, D. Advances in Heat Transfer; Academic Press: New York; 1972; Vol. 9, pp 1-117, (32) Wa¨chtersha¨user, G. Proc. Natl. Acad. Sci. U.S.A. 1990, 87, 200-204.

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conduit, thus exposing the solution to a much smaller surface area and thereby reducing absorption to the conduit walls. By staining the double-stranded DNA with an appropriate marker such as the intercalating fluorescent marker SYBR-Green,32 one can readily monitor the amplification process on-line and in real time. Such real-time monitoring would allow one to control the number of amplification cycles with a feedback control. In contrast, in the flow-through PCRs, the number of amplification cycles is determined by device construction. The thermosiphon-based PCR allows easy adjustment of the number of amplification cycles. The flow rate in the thermosiphon-based PCR can be controlled by either adjusting the inclination angle of the device with respect to the gravity vector or using a valve to adjust the hydrodynamic resistance. As in other continuous-flow PCRs, the relative residence time in the various heating sections must be determined during the design stage and cannot be controlled readily during operation. In this respect, continuous-flow PCRs have a disadvantage compared to their stationary counterparts. GLOSSARY Cp

specific heat, J/(kg K)

D

inner diameter of the tube, m

D0

outer diameter of the tube, m

D+/D-

tube’s diameter D approached from above/below, m

g

gravitational acceleration, 9.8 m/s2

h

overall heat-transfer coefficient based on the tube’s inner diameter, W/(m2 K)

ha

convective heat-transfer coefficient between the outer tube surface and the environment, W/(m2 K)

k

heat conductivity of fluid, W/(m K)

ks

heat conductivity of the tube’s material, W/(m K)

l

distance from the ascender’s exit to the point A, m

L

total length of the loop, m

p

pressure, Pa

PCR

polymerase chain reaction

r

radial coordinate, m

s

axial coordinate, m

SAFC

self-actuated flow cycling

T

temperature, K

T0

reference temperature, K

T∞

ambient temperature, K

TA

temperature of the fluid at point A, K

TH

temperature of the fluid inside the ascender, K

Ti

temperature of each heater, K

T h

cross-sectionally averaged temperature, K

uj

average fluid velocity, m/s

b u

fluid velocity vector, m/s

R

angle between the direction of the gravity vector and the plane of the loop

β

thermal expansion coefficient, K-1

γ

angle between the direction of the gravity vector and the tube’s axis

φ

angular coordinate

θ

dimensionless temperature

µ

viscosity, kg/(m s)

F

fluid density, kg/m3

F0

fluid density at temperature T0, kg/m3

ω

angular rotation speed, s-1

ACKNOWLEDGMENT We are grateful to Drs. Z. Wu, J. Wang (University of Pennsylvania), and P. Corstjens (Leiden University) for helpful

discussions and to Ms. C. Davis and Mr. G. Tong (University of Pennsylvania) for their help with the PCR experiments. The experimental work was supported, in part, by NIH Grant U01DE014964 to the University of Pennsylvania. S.Q. and H.H.B. were also supported, in part, by DARPA through Grant N6600101-C-8056. Received for review January 14, 2004. Accepted April 20, 2004. AC049914K

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