Prior Percutaneous Coronary Intervention is Associated with Low Health-related Quality of Life after Coronary Artery Bypass Graft

The forces acting on a DNA molecule tethered to a bead subjected to a magnetic stretching force on the z axis, (Fm) and a drag force on the x axis due to the liquid flow (Fd) are reppresented in Figure C.1a. The drag force acting on the bead (Fd = F_dˆi) as a result of the applied flow is given by Stokes’ law:

Fd= 6πηcRv (C.1)

where ηc is the viscous coefficient of the buffer corrected for the distance to the surface, R is the diameter of the bead and v is the velocity of the fluid. Therefore, the resulting force vector Ftacting on the bead is given by Ft = 6πηRvˆi + F_mˆk. In order to determine the drag force it is first necessary to calculate the viscous coefficient ηc. The viscosity near surfaces can be significantly larger than the bulk viscosity η when the distance between the surface and the bead, z, is comparable to R. This situation is common in MT experiments, where the molecular extension of the DNA l0is a few micrometers. Then, for the bead-DNA system, z = l0+ R. The viscosity at a distance z to the surface can be found considering Faxen’s corrections [Happel and Brenner, 1983]:

ηc(z) = η

1 −₁₆⁹ _R^z
+¹_{8 R}^z
3

−₂₅₆⁴⁵ _R^z
4

−₁₆¹ _R^z
5 (C.2) The extension of the molecule l0in absence of drag (Fd= 0) can be found from the Moroz and Nelson interpolation formula for the worm-like chain (WLC) described

165

166 C.2. Model

Figure C.1:Combined stretching and drag forces in MT experiments. a) Diagram of forces acting on the bead and related quantities. b) Laminar flow profile of velocities for a circular pipe. Similar profiles are obtained in rectangular MT microfluidics cells.

in section 2.4.2, including an elastic component:

l₀(F_m) = L

where ξ is the persistence lenght of the DNA and S is the stretching modulus.

In addition, to extract the exerted force by the flow drag, it is necessary to deter-mine the velocity inside the liquid chamber from the flow profile as a function of the distance to the surface v(z) [Brewer and Bianco, 2008]. The control parameter in the syringe pump used for the exchange of buffers is the flow rate, Q. Then, the average velocity inside the liquid chamber is ¯v = Q/A, where A is the cross-sectional area.

For our liquid chambers the width, w = 7 mm and the height, h, of a single-parafilm layer chamber is 0.125 mm or 0.250 mm for a double parafilm layer. Although the theory of laminar flows is well defined for circular pipes, it is necessary to apply empirical approximations for the determination of v(z) in other geometries [Çen-gel and Cimbala, 2013]. This is the case of the rectangular duct of the microfluidics liquid chamber employed in our MT setup. The assumption of laminar flow can be justified attending to the Reynolds number (Re) given by

Re = 2ρ¯vr_h

η (C.4)

where ρ is the density of the fluid, η is the viscosity and rh is the hydraulic radius.

For a rectangular pipe rh = hw/(h + w), obtaining Re ≈ 9.5 for the maximum flow rate of our syringe pump (Q = 999 µl/min). Since Re  2000, we can assume laminar flow conditions.

The flow velocity profile through the center of a rectangular pipe (y = w/2) and at a certain distance to the surface z can be obtained following the approximations described in Shah [1978] as

Figure C.2: Characterization of combined magnetic and drag stretching forces on DNA molecules. Results for the drag force (Fd) and the measured extension in the MT setup (zm) for double-parafilm layer cells (a) and for single parafilm layer cells (b). The increase of the drag force with the stretching force is the result of the increase of the velocity with the distance to the surface. Parameters used: Contour length L = 2 µm, persistence length ξ

=45 nm, bead radius R = 500 nm, stretch modulus S = 1200 pN, and bulk viscosity η = 1 mPa·s.

In the previous expresion m is an empirically-determined constant given by m = 1.7 + 1

2α^−1.4 (C.6)

being α = h/w the aspect ratio of the chamber. The design of the microfluidic cells used in MT experiments –where h  w– results in parabolic profiles similar to that described for circular pipes (Figure C.1b). Finally, it is possible to determine the drag force as Fd(z(F ), v(z)) = 6πη_c(z)Rv(z).

The sum of the magnetic force and the drag force results in a total force Ftwhich stretches the molecule with an angle θ with respect to vertical axis. The stretching of the DNA by Ftincreases the end-to-end distance of the molecule l(Ft), and this can be found from the WLC model (equation C.3) substituting the magnetic force, Fm, by |Ft| =q

F_m² + F_d².

The angle with the vertical axis θ is obtained from cos θ = Fm|F_t|⁻¹ and the z axis component of the extension is determined as zd = l cos θ. Nevertheless, the

168 C.2. Model pendulation of the bead results in an understimation of the measured extension of the bead due the hidden extension by the radius of the bead (see sections 3.4.2 and 3.7.2). Considering south-pole attachment of the DNA, the measured extension in the MT software is obtained by subtracting the contribution of the bead radius as zm= zd− R(1 − cos θ). For simplicity, this analysis considers DNA attachment to the south pole of the bead.

The solution to the previous equations for a characteristic 2 µm DNA molecules used in MT experiments is represented in Figure C.2. As expected, the lower veloci-ties for a given flow in cells with two parafilm layer (Figure C.2a) compared with cells made with one parafilm layer (Figure C.2b) results in lower drag forces. Importantly, the non-linear dependency of the velocity with the height of the cell leads to much larger forces in cells fabricated with a single-parafilm layer than when two parafilm layers are used for its assembly. Consequently, shorter extensions are measured for the same flow in single-parafilm layer cells. Moreover, the non-linear mechanical response of the DNA results in a more dramatic reduction of the extension with the applied flow at low magnetic forces than at higher forces (F > 1 pN), where bending fluctuations are suppressed by the magnetic stretching force.

Remarkably, there is a plateau region where the flow barely affects the extension measured zm. For instance, at 1 pN the flow in double-parafilm layer cells can go up to ∼100 µl/min before the extension l is significantly affected by the drag force (Figure C.2a). It is important to note that therefore, the lack of reduction of extension cannot be used as a hallmark of a negligible drag compared with the stretching force.

For instance, at a stretching force of 1 pN, and using double-parafilm layer cells, the extension is reduced only a 4% for a ∼100 µl/min flow, but the drag force equals 0.26 pN. Special care must be taken to avoid exert drag forces comparable (or higher) than the magnetic force, since this can affect the determination of the mechanical prop-erties of the molecule or prevent the binding of proteins. The experiments described in this thesis used a flow rate of 18 µl/min at most. This results in drag forces of 0.04 pN for DNA molecules stretched with a magnetic force of 0.35 pN, or to a drag forces of ∼ 0.02 pN in supercoiled or condensed molecules (for the same magnetic force).

In summary, the method here described can be used as a reference of the applied force on the DNA during buffer exchange and as its consequences for the measured coordinates during data acquisition.

C.3 References

Brewer, L. R. and Bianco P. R. Laminar flow cells for single-molecule studies of DNA-protein interactions. Nat Methods, 5(6):517-25, 2008.

Çengel, Y. A. and Cimbala, J. M. Fluid mechanics: fundamentals and applications.

McGraw-Hill, 3rd edition, 2013.

Happel, J. and Brenner, H. Low Reynolds number hydrodynamics. Martinus Nijhoff Publishers. 2nd edition, 1983.

Shah, R. K., London, A. L., Irvine, T. F., and Hartnett, J. P. Laminar flow forced con-vection in ducts. A source book for compact heat exchanger analytical data. Academic Press, 1st edition, 1978.

Implementation of C/C++ libraries to run under LabVIEW

D.1 Introduction

LabVIEW is a widely-used programming language in the scientific community for machine control and data analysis. Its intuitive environment combines an easy pro-gramming with a large variety of built-in functions and a manifold of tools for cre-ating graphical user interfaces (GUI). However, it is restricted by its slowness in processing large amounts of data or to carry complex mathematical calculations and this is intrinsic to its interpreted language condition.

In contrast, compiled programming languages, e.g. C or Fortran, are the best op-tion for simulaop-tions and numerical evaluaop-tion. Unfortunately, these programming languages involve a deeper understanding of the internal architecture of the com-puter and the creation of GUI is neither fast nor straightforward.

The inclusion of compiled libraries in LabVIEW is a beneficial solution to combine the power of both approximations. This appendix illustrates the basic guidelines for the creation of a C-written dynamic library (DLL in the Windows systems) and its calling under the LabVIEW environment. For didactic purposes, the explanation takes as example the generation of a step fitting routine. In the example, the Lab-VIEW environment passes a two-dimensional array containing the data to fit to the C library, and this returns to LabVIEW the result of the fitting in the form of a two dimensions array. The objective is to show a complex situation that can be used as a reference for the rational to follow for other purposes.