Hydrodynamics of quartz crystal microbalance experiments with liposome-DNA complexes

The quartz crystal microbalance (QCM) is widely used to study surface adsorbed molecules, often of biological significance. However, the relation between raw acoustic response (frequency shift $\Delta f$ and dissipation factor $\Delta D$) and mechanical properties of the macromolecules still needs to be deciphered, particularly in the case of suspended discrete particles. We study the QCM response of suspended liposomes tethered to the resonator wall by double stranded DNA, with the other end attached to surface-adsorbed neutravidin through a biotin linker. Liposome radius and dsDNA contour length are comparable to the wave penetration depth ($\delta\sim 100\ \mathrm{nm}$). Simulations, based on the immersed boundary method and an elastic network model for the liposome-DNA complex, are in good agreement with experimental results for POPC liposomes. We find that the added stress at the resonator surface, i.e. the impedance Z sensed by QCM, is dominated by the flow-induced liposome surface-stress, which propagates towards the resonator by viscous forces. QCM signals are extremely sensitive to the liposome's height distribution P(y) which depends on the actual number and mechanical properties of the tethers, in addition to the usual local attractive/repulsive chemical forces. Our approach helps in deciphering the role of hydrodynamics in acoustic sensing and revealing the role of parameters hitherto largely unexplored. A practical consequence would be the design of improved biosensors and detection schemes.

involving 3D unstationary flow patterns resulting from the viscous propagation of the fluidinduced forces acting on the solutes. Numerical studies of QCM hydrodynamics of discrete particles qualitatively reproduce experimental observations, such the coverage dependence of ∆D/∆f , providing preliminary answers to a large list of still unexplained phenomena [2].
These studies [5,9] carried out in 2D using the commercial package COMSOL and more recently in 3D using lattice Boltzmann solvers [19,20], considered adsorbed rigid particles at fixed positions (obstacles). However, fixing the particle position introduces ad-hoc forces into the fluid (those required to keep the obstacle fixed), which alter the measurement of the system impedance. A correct representation of QCM hydrodynamics requires solving the dynamics of the analytes, including the fluid traction acting on them and other interactions (inter-particle forces, surface or contact forces and advection due to a mean flow, if required); evidently, this issue is particularly important in the case of suspended particles. In this Letter we present an experimental and theoretical study of the QCM response of individual liposome-DNA complexes. Simulations are carried out with a finite volume fluctuating hydrodynamic solver equipped with the immersed boundary method, which describes the particle dynamics and uses an elastic network model to reproduce molecular mechanical properties (e.g. bending rigidity). The good agreement with experiments (carried out with POPC liposomes) allows us to affirm that the analyte impedance is strongly dominated by the hydrodynamic perturbation created by the liposome, which is suspended in the liquid and tethered by a DNA strand (Fig. 1).
Experiments. Liposome-DNA (LDNA) complexes were formed by sequential injections of neutravidin (NAv), DNA, and liposomes; more details in Supplementary Information (SI). dsDNA with 21, 50 and 157 base pairs (bp) having lengths L DNA 7, 17 and 53nm, respectively, were used. Liposomes of radius of 15, 25, 50 and 100 nm were considered. In order to be captured by the NAv layer previously formed on the surface, DNA fragments bear a biotin at one end. In addition, a cholesterol was incorporated to the opposite DNA end for subsequent liposome binding due to its strong affinity for the lipid membrane. Ringdown QCM experiments were performed with an E4-Qsense (Biolin, Sweeden) device at T = 25 o C under continuous flow velocity of 60 µL/min. Measurements of ∆f and ∆D based on the ring-down approach have been described elsewhere [21]. Briefly, after excitation pulses separated by milliseconds, a crystal sensor resonator performs underdamped oscillations described by , where x 0 ≈ 2nm, and Γ and f depend on the acoustic response related to the sensor loading. The decay rate Γ is often expressed in terms of the "dissipation factor", D = 2Γ/f n . The fundamental frequency of the particular cut of the quartz crystal is f 0 = 5MHz and here we report experimental results for the seveth harmonic f 7 = 35MHz. QCM experiments monitor the time evolution of the acoustic signal registering the changes in frequency (∆f ) and dissipation (∆D) upon successive sample injections and surface binding of NAv, DNA and liposomes (Fig. S1).
These shifts increase with the amount of deposited material. To get an intensive quantity the procedure consists in plotting the acoustic ratio −∆D/∆f against ∆f and extrapolating it to the limit of an infinitesimally small load: this defines the so called dissipation capacity DC = − lim ∆f →0 ∆D/∆f . The minus sign is convenient because an extra load usually implies ∆f < 0 and ∆D > 0 which, following the analogy with an overdamped spring [2], are commonly intrepreted as extra "mass" (∆f ) and "dissipation" (∆D).
Impedance analysis. Our numerical analysis is based on the well established small load approximation (SLA) [2] which relates the impedance, Z =σ/v 0 (ratio of the wall stress and the surface velocity) to the complex frequency shift ∆f = ∆f + i∆Γ measured in ring-down experiments. The SLA is valid if the resonator's mass per unit area is much larger than the load, which is a safe approximation in our case (where ∆f /f 0 ∼ 10 −5 or even less). The impedance of the complex load is expressed as the sum of Z = Z (Q) +Z (0) +Z (DNA) +Z (LDNA) , where the impedances correspond, respectively, to the clean quartz resonator Z (Q) , the (unloaded) Newtonian solvent Z (0) [11] the DNA strand without a liposome Z (DNA) and, the LDNA impedance Z (LDNA) . For any contribution (different from Q), the SLA yields [2] ∆f (a) /f 0 = iZ (a) /(πZ (Q) ), where Z (Q) = 8.8 × Simulations. Our mesoscopic model is based on a fluctuating hydrodynamic solver for compressible unsteady flow equipped with the immersed boundary method to couple fluid and structure dynamics [22][23][24]. It is implemented in the GPU code FLUAM (FLuid And Matter interaction), a second-order accurate finite volume scheme on a staggered grid [25] of side h. The liposome and dsDNA are represented using beads of radius h ( Fig. 1) connected by harmonic springs and/or bending potentials (see SI). An elastic network is used to model the membrane of the (hollow) liposome by connecting nearest neighbours of the network with harmonic springs: the bonding force F ij = −k L (r ij − r 0 ) includes the equilibrium distance r 0 ≈ h and the spring constant k L determining the liposome rigidy. Here we consider the rigid limit (high k L ) to focus on the leading impedance contribution. The double stranded DNA (dsDNA) is modelled by a bead model for semiflexible polymers, with bending energy extracted from the DNA persistence length at room temperature (50 nm). We shall use the term link to denote the DNA-wall force F link .
The geometry is illustrated in Fig. 1. The simulation box is periodic in the resonator plane x − z, with dimensions L × L y × L. Rigid no-slip walls at y = 0 and y = L y are imposed via explicit boundary conditions [22,26]). The top wall is kept at rest while the bottom wall at y = 0 moves in the x direction with velocity v wall = v 0 cos(2πf t) with v 0 set to fix a small wall displacement x 0 < h. To achieve the required numerical convergence we used a spatial resolution of h = 3.958 nm (see SI). The code units map the density and kinematic viscosity of water at T = 25 o C (see SI). The sound velocity c was set to match the experimental value of the group f 7 δ/c ∼ 3 × 10 −3 , whose smallness indicates a minor effect of fluid-compressibility [27]. Moreover, the large time scale separation between liposome diffusion time and the QCM oscillation period (6πηR 3 f /k B T > 10 4 ) makes it possible to neglect thermal fluctuations over the short simulation runs we used to evaluate the impedance (about 10 periods). We stress, however, that in our simulations the liposome is free to move according to the flow traction; this is essential to obtain unbiased results, particularly because the liposomes are not adsorbed but suspended. where the prefactor Z S = (20π/3)ηR 3 /(L 2 δ 2 ) is taken to be consistent with the stresslet of a sphere under steady shear [36]. Close to the resonator, the ambient flow includes a significant contribution from the wall reaction field [32] and the impedance becomes a decreasing function of the surface-to-sphere distance y − R. This reasoning leads us to the following ansatz, . (1) Using A ≈ 1.40(R/δ) 2 , B ≈ 1.5 − 0.03 exp(2.5 R/δ) and C ≈ 0.01 (see SI), Eq. 1 fits our numerical results for y/δ > 0.04 [37] within less than 5% error (see Fig. 3).
The contribution of the DNA can now be estimated as Z where Z (LDNA) (y) = Z (L) (y)+Z . Relation 2 is extremely useful because it decomposes the analyte and anchor contributions, allowing for a fast evaluation of non-trivial effects.
P (y) encodes important microscopic information about the anchor: bending rigidity, linker-DNA tilt energy [33] or large values of the DNA coverage [34] which could lead to multiple anchors connected to the liposome [28]). In general, P (y) can also introduce information into Eq. 2 about physico-chemical forces between the analyte and the wall (solvation, dispersion, electrostatic forces) as well as the effect of advection under a strong Poiseulle flow. All these effects are known to alter the acoustic ratio and their relevance can be tested using Eq. 2, by pre-evaluating P (y), either theoretically or from Monte Carlo simulations (MC). We use MC sampling to obtain P (y) for an electrically neutral liposome (POPC) anchored by a single DNA chain. The inset of Fig. 3 shows P (y), which applied to Eq. 2 (fed with Z (L) in Eq.
1 and Z (DNA) DNA ) predicts values of DC quite close to experimental results (crossed symbols in Fig. 2).
We have shown that the acoustic response of suspended particles in QCM is mainly determined by the hydrodynamic response of the analyte, which strongly depends on its height distribution P (y). The hydrodynamic response of liposomes also depends on their bending rigidity κ and membrane fluidity [30]. Soft liposomes, with smaller κ and higher fluidity, experimentally yield slightly smaller DC [28]. A recent analytical study [35] indicates that both effects might be antagonistic (DC decreases with κ but increases with fluidity).
Disentangling all this subtle information hidden beneath the complex QCM hydrodynamic fields requires a combined effort of experiments, simulations and hydrodynamic theory. This work represents a step in this direction.