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Subnanometer optical precision
In order to better understand the information theoretical content of this profile, we fit its envelope and a single peak to Gaussian functions. These fits are shown in figure 1-8.a. From these experimental profiles we deduce, using the discretized version of equation 1.9, the Fisher information that they contain with respect to their translation as well as the maximum number of photons contained in one frame for a given camera well depth. Table 1.2 compares the number of photons and Fisher information that are contained in this PSF, as well as in a Gaussian fit of the envelope and in a single peak. Compared to these two cases, SDI permits to increase the number of photons contained in one frame without reducing drastically the information contained in one photon with regard to its horizontal translation (the direction that encodes for both the X and Z movement of the bead). Since the Y-profile is relatively well fitted by a Gaussian, the information contained in one photon concerning the translation along the axis Y can be computed from the Thomson-Webb formula (equation 1.10) and is also shown in table 1.2. It appears that it is smaller than in the X direction. This information could be increased by designing larger slits in the Y-direction (resulting in a thinner PSF at focus), but then, the PSF would be concentrated on a smaller camera area and the maximal number of photons received by frame would be smaller. Considering that the axial localization of the bead, whose information is contained in the X-direction of the PSF is the most critical for magnetic tweezers, we preferred to slightly sacrifice the precision in the Y-direction to optimize the frame-to-frame axial precision.
Stiffness and thermodynamics of a tethered molecule at constant force
Here we recall some generalities concerning the thermodynamics of a single molecule tethered at force F.
Indeed, the stiffness k of the energetic trap in which the bead is placed depends on the thermodynamics of the tethered molecule.
The probability density of a given polymer to have an end-to-end extension~l when a force Fz is applied in the direction z reads: rF(b;~l ) =C(b)eb(Flz−G(b;~l)) C(b) is a normalization constant and G(b;~l) is the partial free energy of the molecule: G(b;~l) = kBT ln(Z~l).
Hydrodynamic drag, stiffness and spatio-temporal resolution
We describe here to what extent the Brownian motion of the bead limits the detection of discrete extension steps of a tethered molecule. We are thus interested in the minimum observable step size s of the equilibrium position of the bead z0 (i.e the extension of the tethered molecule). The equipartition theorem states that the position of the bead dz = z−z0 verifies the following relation: sstat =< d2 z >= kBT kz (2.11).
This indicates the amplitude of the Brownian motion of the bead, but time does not participate in this relation. However, imagine that the bead goes through all the possible positions allowed by 2.11 within 1ms. Then, by averaging its position during 1ms, one could get a much better precision with regard to z0. This effect is explained in figure 2-4. A Brownian motion is simulated while increasing the equilibrium position by regular steps. Both simulations are performed with the same stiffness, but the drag, and thus the timescale of the Brownian motion changes. In figure 2-4.b, it is not possible to distinguish the individual steps of the position of the Brownian particle because its velocity is too small compared to the time interval between two steps. However, in figure 2-4.c, individual steps can be easily distinguished by averaging the position of the fast-moving bead over many frames. This shows that the spatial resolution of a setup cannot be decoupled of its temporal resolution.
Hydrodynamic drag close to a surface
Equation 2.13 shows that two solutions can be considered to limit the effect of the Brownian motion. First, the drag coefficient can be reduced by decreasing the size of the bead. Second, the stiffness of the substrate can be increased, in particular by taking shorter substrates. If we approximate that k is proportional to the number of monomers, we expect, following equation 2.12, that the spatial resolution decreases as the number of monomers 1. However, an antagonist effect takes place as the bead come closer to the surface. Indeed, the drag coefficient that opposes a bead movement close to plane surface increases as the bead comes closer to the surface. The correction to Stokes’ law in this configuration has been derived analytically by Brenner [Brenner, 1961]. This correction depends on the distance between the center of the bead and the surface h and is contained in a function l(hR ), where R is the radius of the bead. Thus, the corrected Stokes’ law reads, with z the extension of the molecule, i.e. the distance between the attachment point on the bead and the surface: g(z) = 6phRl( z+R R ).
Table of contents :
Remerciements
Introduction
Elements of context
1 3D-super resolution tracking with Stereo Darkfield Interferometry
1.1 Real-time tracking 3D super-resolution: fundamental concepts and challenges
1.2 PSF engineering: a state of the art
1.2.1 High resolution methods
1.2.2 Stereoscopic methods: the advantage of linearity
1.3 Stereo Darkfield Interferometry
1.3.1 Optical setup
1.3.2 Translation of the interferometric pattern and theoretical z-sensitivity
1.3.3 Characterization of the PSF of Stereo Darkfield Interferometry
1.3.4 Localization algorithm
1.3.5 Subnanometer optical precision
1.3.6 Linearity and field dependence
1.3.7 Impact of the size of the imaged particles
1.4 Conclusion
2 High-resolution magnetic tweezers
2.1 Physics of a tethered Brownian bead
2.1.1 Overdamped Langevin dynamics
2.1.2 Stiffness and thermodynamics of a tethered molecule at constant force
2.1.3 Hydrodynamic drag, stiffness and spatio-temporal resolution
2.1.4 Hydrodynamic drag close to a surface
2.1.5 Movement misalignment and post-correction
2.1.6 Discrepancy between this simple model and experimental noise values
2.1.7 Inclusion of hydrodynamic couplings and magnetic anisotropy
2.2 Two biophysical applications
2.2.1 Force spectroscopic measurement of the hybridization of short oligonucleotides with nanometric resolution
2.2.2 Helicase stepping: some preliminary results
2.3 Conclusion
3 Kinetic locking: a method to measure biomolecular interactions involving nucleic acids
3.1 An extremely stable DNA Fluctuating probe
3.1.1 The fluctuations of a 10-bp hairpin in the millisecond range
3.1.2 Correction of the impact of the acquisition frequency for rapid kinetic rates
3.1.3 Temporal stability
3.2 Kinetic locking
3.2.1 Kinetic model
3.2.2 Measurement of the binding kinetics in the case koff ≪k f and kon ≪k f
3.2.3 Binding thermodynamics of very short oligonucleotides
3.2.4 Influence of standard chemical modifiers and comparison with the nearest-neighbour model
3.2.5 Comment on the influence of the force on the thermodynamic measurements
3.3 An application of kinetic locking to the measurement of NA/protein interactions
3.3.1 RecQ detection through the kinetic locking of a fluctuating hairpin
3.3.2 Kinetic locking unveils the stabilization of the replication fork by RecQ
3.4 Conclusion
Conclusion
A Embedded SDI objective
B Brownian simulation of a tethered bead close to a surface
C Material and methods
C.1 Optics
C.2 Single-molecule sample preparation
C.3 Biochemistry
C.4 Single-molecule experiments
C.5 Data treatment
C.6 DNA sequences
C.7 Buffers
D Bibliography
