DNA stretching in RB-PCR

The RB-PCR is a technique to duplicate DNA at a natural convection flow field. As shown in Fig. 10, the DNA replication includes three steps which are called denaturation,

annealing and extension and performed at T = 367, 328 and 345 K, respectively. In order to generate those three thermal environments, a Rayleigh-B´enard flow is performed at an enclosed reactor with the top thermal boundary at 298 K and bottom boundary at 383 K.

While a DNA chain is at the reactor, it passes through those three thermal environments continuously. Consequently, the DNA replication can be implemented at the RB-PCR reactor. The quantity and preciseness of the replicated DNA chain are investigated using the gel electrophoresis. However, the investigation of the extension and movement of the DNA chain at the RB-PCR reactor are unknown. In order to predict the behavior of the DNA chain in the RB-PCR, a simulation of the DNA chain of L=44.0 µm contained at the RB-PCR reactor is performed using the proposed numerical model. The fluid properties of the simulation are shown in Table 2 and 3 where ν are varied with T . The dimensions of the computational domain and boundary conditions of the simulation are shown in Fig. 11. In the present study, the computational domain is a square enclosure with the dimension of 300 µm and can be regarded as a microreactor in a lab-on-chip. Since the dimension of the computational domain is very small, the convectional intensity of the fluid flow in the RB-PCR reactor is not strong enough to induce the Rayleigh-B´enard flow. So long as the RB-PCR reactor is 5^o inclined to the vertical direction, a buoyancy-driven flow is generated in the reactor. The buoyancy-buoyancy-driven flow which is similar to the Rayleigh-B´enard flow can drive the DNA chain to pass through the three thermal environments in the RB-PCR reactor. In practice, the microreactor is manufactured by etching processes producing a shallow channel for the DNA replication. Therefore, the simulation is performed in the 2D computational domain. Moreover, all the boundaries are defined as stationary solid walls. Subsequently, the left and right boundaries are set as adiabatic walls. At the beginning of the simulation, a steady buoyancy-driven flow without the DNA chain is obtained. Fig. 12 displays the velocity magnitude contours

and the streamline starting from the specific location. It is observed that the magnitude of the flow velocity in the streamline ranges from 2.60E-5 to 4.20E-5 m/s. Therefore, the DNA chain would be stretched by the fluid flow that the flow velocities within the range implemented as the inlet conditions mentioned in section 3.1. Subsequently, the DNA chain is put into the specific location adjacent to the hot bottom and the bead-spring model is used to predict the DNA extension while the DNA chain is transported by the buoyancy-driven flow. In our in-house code for the bead-spring model, both ends of the DNA chain are free. Therefore, the DNA chain is able to move with the fluid flow at the RB-PCR reactor.

Fig. 13 shows the position of the DNA chain in the buoyancy-driven flow and the behavior of the DNA chain. According to Fig. 13, it is observed that the velocity contours of the fluid flow dose not change a lot as the DNA chain moves. As shown in Fig. 13(a), the DNA chain is placed at the selected high-speed area of the fluid flow in the beginning of the simulation. The beads of the DNA chain are horizontally arranged. Moreover, the first bead is regarded as the right end of the DNA chain and denoted as B1. Fig. 13(b) depicts the behavior of the DNA chain at the left high-speed area of the fluid flow after 4 s. It is observed that the DNA chain is retracted and rotated by the fluid flow. Also, the DNA chain is not parallel to the velocity of the ambient fluids. It may be because the velocity of the beads adjacent to the center of the reactor is slower than the beads apart from the reactor. Since the DNA chain is driven by the slow flow velocity at 3.00E-5 m/s at the top from 4.0 s to 7.8 s, the DNA chain is observed the similar behavior as shown in Fig. 13(b) and (c). Fig. 13(d) depicts the behavior of the DNA chain at the right high-speed area of the fluid flow after 12.5 s. The axial direction of the DNA chain is similar to the ambient flow direction. It is observed that the DNA chain is almost vertical.

In the end of the cycle as shown in Fig. 13(e), the length of the DNA chain is shorter

than the beginning of the simulation. The direction of the DNA chain is reversed and not parallel to the flow direction due to the velocity gradient along the y-direction at the bottom.

Fig. 14 revels the time history of the extension of the DNA chain at the RB-PCR reactor. We can observe that the extension of the DNA chain is greatly reduced from 0 to 4 s. Meanwhile, the DNA chain passes through the thermal environment at 368 K.

The DNA chain is accelerated by the ambient fluids from a rest state. Subsequently, the extension of the DNA chain is slowly decreased from 6 to 12.5 s. It is because the DNA chain at the top area of the RB-PCR reactor is driven by the fluid flow with the stable velocity at 3.00E-5 m/s. After 12.5 s, the DNA chain is moved into the right high-speed area of the fluid flow and the extension is greatly decreased. However, the retraction of the DNA chain is not stronger than the beginning of the simulation. As shown in the inset figure, the extension of the DNA chain is reduced to 1.5 % of L at the end of the simulation. It is found that the DNA chain is retracted in the entire simulation and the extension of the DNA chain can be regarded as with three stages.

Fig. 15 revels the time history of the rotating angle θ between the tangential direction at the middle point B₅ of the DNA chain. In the beginning, θ is equal to 0^owhile the DNA chain is placed horizontally. Subsequently, the DNA chain is transported and clockwise rotated by the fluid flow. From t = 0 to 7.8 s, θ varies from 0 to 45^o while the DNA chain moves from the left side to the top side of the RB-PCR reactor. From 7.8 to 12.5 s, θ is greatly increased while the DNA chain moves from the top to the right side of the reactor.

θ is almost 90^o at t = 12.5 s. After 12.5 s, θ is greatly increased to 150^o while the DNA chain moves from the right side to the bottom. It should be noted that the DNA chain is reversed but θ does not reach 180^o.

Fig. 16 revels the time history of the length of the springs which describe the behavior

of the DNA chain in the simulation. It is observed that all |Sⁱ| are retracted in the same trend just as the time history of the extension of the DNA chain. Furthermore, because the bonding force of the DNA chain is so large that the hydrodynamic drag produced by the RB-PCR reactor can not elongate the DNA chain. Nevertheless, the springs at the ends of the DNA chain, |S¹| and |S⁹|, are shorter than the others in the entire simulation.

Eventually, all |Sⁱ| are less than 15 % of the R^S at the end of the cycle.

Fig. 17 displays the time history of the hydrodynamic drag exerted on each bead of the DNA chain. It is observed that the DNA chain is driven by the small |F^dragi | which is about 10⁻¹⁸ N. In addition, the ends of the DNA chain are driven by the larger hydrodynamic drag, |F^drag1 | and |F^drag10 |, than the others. From 0 to 4 s, |F^drag1 | and

|F^drag10 | are significantly increased to 100 aN and then decreased to 20 aN. However, the others |F^dragi | change slightly. After 4 s, all the |F^dragi | are very small and close to 10 aN . Hence, according to Eq. 2.10, it is observed that the relative velocity between the DNA chain and the ambient fluid is retained in a constant value. As a result, we can say that the DNA chain are moved at the speed close to the speed of the ambient fluids after 4 s.

Fig. 18 displays the time history of the velocity of each bead of the DNA chain. It is found that | ˙ri| oscillates with respect to time. Furthermore, at each time, all | ˙ri| are similar. So long as the DNA chain passes through the high-speed areas of the fluid flow at 4, 8, 12 and 17 s, the | ˙rⁱ| are faster than the others. As a result, the magnitude of the velocity of the DNA chain is proportional to the speed of the ambient fluids. Also, the spring force in the bead-spring model are not obvious for determining the movement of the DNA chain at the RB-PCR reactor.

CONCLUSIONS AND FUTURE WORK

4.1 Conclusions

The bead-spring model coupled with the numerical flow model has been established to predict the behavior of the DNA chain in the flow field. First, a continuous real DNA chain is transferred into a sectional bead-spring model using the coarse-graining method. Since the psychical parameters of the bead-spring model are obtained, the hydrodynamic drag and spring force exerted on a bead of the bead-spring model can be derived. Subsequently, those force are implemented in the Langevin equation to predict the motion of the bead.

Furthermore, two time increment are utilized in this study. A larger time increment dt is used for solving the fluid flow. On the other hand, a smaller time increment dtsub is implemented for predicting the motion of the bead. In particular, dtsub is determined by the estimation which simplifies the Langevin equation as an O.D.E. and then solves the equation. As a result, dtsub should be less than m/ξ to keep the stability of the bead-spring model. In order to predict the motion of the bead of the bead-bead-spring model, the

semi-implicit Euler method is performed. Since the semi-implicit Euler method is used to calculate the Langevin equation with dtsub, ˙ri and ri can be obtained. To improve the efficiency of the simulation, the flow field is regarded as being ”frozen” during dt and the bead-spring model is implemented by increasing dt_sub.

In order to verify the spring model, a benchmark test using the proposed bead-spring model to predict the extension of the DNA chain in a uniform flow is performed.

Subsequently, the predicted extensions of the DNA chains are compared with Perkins et al.’s [1] experimental results and show good agreements. So long as the inlet velocities are faster than 2.00E-5 m/s, the difference percentage between the predicted results and Perkins et al.’s results are less than 10 %. Moreover, we can observe that when the DNA chain is stretched by the fast uniform flow, the DNA chain can be well extended very soon.

Also, a short DNA chain can be well extended sooner than a long DNA chain at the same inlet flow. In terms of the benchmark test, it turns out that the proposed bead-spring model is able to predict the extension of the DNA chain in the flow field. Furthermore, the proposed bead-spring model is used to predict the behavior of the DNA chain of L=44.0 µm at the micro RB-PCR reactor which is used to replicate DNA in natural convection.

The predicted DNA chain can move freely with the ambient fluids at the RB-PCR reactor.

As a result, the extension and rotating angle of the DNA chain can be determined by the proposed model. Also, the length of the springs, hydrodynamic drag exerted on the beads and magnitude of the velocity of the beads of the bead-spring model are investigated. It is observed that the DNA chain rotates, shrinks and is deformed while it moves. The extension of the DNA chain is reduced to 1.5 % of L in the end of the simulation. In addition, the DNA chain is clockwise rotated from 0 to 150^o by the fluid flow. Therefore, the DNA chain is almost reversed after one cycle. All |Sⁱ| are reduced which are less than 15 % of the RS in the end of the simulation. Meanwhile, it is observed that the reduction

of |Sⁱ| is large at two ends of the DNA chain. Nevertheless, the hydrodynamic drag at two ends of the DNA chain are larger than the middle bead. However, all |F^dragi | are close to 10 aN after 4 s of the beginning of the simulation. According to those results, the bead-spring model of the DNA chain would provide useful information to investigate the behavior of the DNA chain at the RB-PCR reactor.