Optical detection of nanometric thermal fluctuations to measure the stiffness of rigid superparamagnetic microrods

Abstract

The rigidity of numerous biological filaments and crafted microrods has been conveniently deduced from the analysis of their thermal fluctuations. However, the difficulty of measuring nanometric displacements with an optical microscope has so far limited such studies to sufficiently flexible rods, of which the persistence length ( (1), the method that consists of measuring the dispersion of a thermostated system to assess its properties has brought a plethora of scientific successes. For instance, the tracking of Brownian or more complex diffusive particles immersed in a fluid opened up a new era in microrheology (2, 3). The calibration of numerous microtools such as optical tweezers (4), glass fibers (5), or atomic force microscope (AFM) cantilevers (6) also relies on the assessment of thermal motions. The principle of the method was also adapted to measure the intrinsic mechanical properties of nano-and microrods.This type of stiffness measurement, which is based on the monitoring of the rod-shape deformations caused by the thermal forces, was initially performed on biological polymers (7) and biological filaments of the cytoskeleton including actin (8-10), microtubules (9, 11-13), intermediate (14), and other filaments (15-17). It allowed in vivo measurements (13), although most studies were performed in vitro, in various boundary conditions (14). The method is also used in nanosciences, essentially to probe carbon nanotubes (18-20) that, despite their extremely large elastic modulus, have a sufficiently small radius to exhibit detectable fluctuations. Numerous other methods exist to measure the rod stiffness: Nanoindentation (21), AFM measurements (21, 22), and other techniques based on an external solicitation (21, 23) are commonly performed. However, they require relatively complex equipment. In contrast, the thermal method is neither invasive nor disruptive: It consists of acquiring a set of pictures of the rod, an observation that most often can be performed with an optical microscope only. For this purpose, superresolutive techniques have recently emerged and currently allow the optical observation of nanometric fluctuations, beyond the optical diffraction limit (24,25).If it is possible to take the anisotropic internal structure (such as the spontaneous curvature) of the rods into account (26), most analyses rely on the simple isotropic worm-like chain (WLC) model (27): For more than a decade, according to this model, the persistence length Lp = C k B T (where C is the bending modulus and kB T the thermal energy) was deduced from the experimental data after a decomposition of the filament shape in bending eigenmodes (Fig. S1), each of them being an independent probe of the filament flexural properties (10, 11, 13). However, this technique rapidly reaches its limit for stiff rods because the mode amplitude decays like n −2 (n being the mode number) and higher modes become quickly indistinguishable from the noise of the detection system (SI Text, Eigenmode Analysis of the Rod Fluctuations). To circumvent...

Extensive characterization of magnetic microrods observed using optical microscopy

“…S2) which shows deviation from linearity even at very low field (χ v varies by more than a factor of 2 between 2 and 10 mT). Consistently, previously detailed [11] viscous drag versus magnetic torque (VD/MT) experiments (in which the magnetically driven rotational kinetics of free rods in solution is analyzed), also showed that the rod material susceptibility χ depends on H and is proportional to χ v , the constant ratio χ/χ v being simply the ratio of the magnetic nanoparticle volume fraction within the rod and in the initial ferrofluid φ/φ…”

“…S2) of a ferrofluid suspension of negatively charged (citrated) nanoparticles (volume fraction φ v = 3.9%) obtained by VSM using a home-made apparatus, from which was deduced the log-normal distribution of the particle diameters (10-16 nm range, mean=13 nm) [16]. Following an already published protocol [11], the rods were then prepared by dialysis of a solution of the nanoparticles mixed with positively charged polymers (poly(diallyldimethylammonium chloride)) while being exposed to a ∼ 250 mT magnetic field. We used a microscopic setup equipped with magnetic tweezers designed to induce a uniform magnetic field on the sample (Fig.…”

“…1). We carefully calibrated the field, and checked that the influence of the gradient was negligible [11]. To prepare the samples, we flowed a 100x diluted rod solution into an observation chamber, so that some of the rods were found in a cantilevered configuration, i.e.…”

“…1C). Home made softwares were written to automatically pilot the magnetic tweezers and the camera, track noisy motion of the sample and digitize the rod shape with a resolution of 2-20 nm [11].…”

“…The mechanical characterization of the rod was previously fully described [11] and consisted of acquiring ∼900 images of the rod to analyze its thermal fluctuations, from which were deduced the rod persistence length Lp, its bending modulus C = L p /k B T (k B T is the thermal energy) and its Young modulus Y = C 4πr 4 . The magnetic susceptibility χ of the rod material (defined as χ = M H , where M is the magnetization and H the field inside the material) was deduced from magnetoelastic experiments in which we analyzed the deflection of the rod deformed by a uniform field.…”