Silvan Türkcan

A Bayesian Inference Scheme to Extract Diffusivity and Potential Fields from Confined Single-Molecule Trajectories

Currently used techniques for the analysis of single-molecule trajectories only exploit a small part of the available information stored in the data. Here, we apply a Bayesian inference scheme to trajectories of confined receptors that are targeted by pore-forming toxins to extract the two-dimensional confining potential that restricts the motion of the receptor. The receptor motion is modeled by the overdamped Langevin equation of motion. The method uses most of the information stored in the trajectory and converges quickly onto inferred values, while providing the uncertainty on the determined values. The inference is performed on the polynomial development of the potential and on the diffusivities that have been discretized on a mesh. Numerical simulations are used to test the scheme and quantify the convergence toward the input values for forces, potential, and diffusivity. Furthermore, we show that the technique outperforms the classical mean-square-displacement technique when forces act on confined molecules because the typical mean-square-displacement analysis does not account for them. We also show that the inferred potential better represents input potentials than the potential extracted from the position distribution based on Boltzmann statistics that assumes statistical equilibrium.

Inferring maps of forces inside cell membrane microdomains.

Mapping of the forces on biomolecules in cell membranes has spurred the development of effective labels, e.g., organic fluorophores and nanoparticles, to track trajectories of single biomolecules. Standard methods use particular statistics, namely the mean square displacement, to analyze the underlying dynamics. Here, we introduce general inference methods to fully exploit information in the experimental trajectories, providing sharp estimates of the forces and the diffusion coefficients in membrane microdomains. Rapid and reliable convergence of the inference scheme is demonstrated on trajectories generated numerically. The method is then applied to infer forces and potentials acting on the receptor of the toxin labeled by lanthanide-ion nanoparticles. Our scheme is applicable to any labeled biomolecule and results show its general relevance for membrane compartmentation.

Observing the Confinement Potential of Bacterial Pore-Forming Toxin Receptors Inside Rafts with Nonblinking Eu3+-Doped Oxide Nanoparticles

We track single toxin receptors on the apical cell membrane of MDCK cells with Eu-doped oxide nanoparticles coupled to two toxins of the pore-forming toxin family: α-toxin of Clostridium septicum and ε-toxin of Clostridium perfringens. These nonblinking and photostable labels do not perturb the motion of the toxin receptors and yield long uninterrupted trajectories with mean localization precision of 30 nm for acquisition times of 51.3 ms. We were thus able to study the toxin-cell interaction at the single-molecule level. Toxins bind to receptors that are confined within zones of mean area 0.40 ± 0.05 μm(2). Assuming that the receptors move according to the Langevin equation of motion and using Bayesian inference, we determined mean diffusion coefficients of 0.16 ± 0.01 μm(2)/s for both toxin receptors. Moreover, application of this approach revealed a force field within the domain generated by a springlike confining potential. Both toxin receptors were found to experience forces characterized by a mean spring constant of 0.30 ± 0.03 pN/μm at 37°C. Furthermore, both toxin receptors showed similar distributions of diffusion coefficient, domain area, and spring constant. Control experiments before and after incubation with cholesterol oxidase and sphingomyelinase show that these two enzymes disrupt the confinement domains and lead to quasi-free motion of the toxin receptors. Our control data showing cholesterol and sphingomyelin dependence as well as independence of actin depolymerization and microtubule disruption lead us to attribute the confinement of both receptors to lipid rafts. These toxins require oligomerization to develop their toxic activity. The confined nature of the toxin receptors leads to a local enhancement of the toxin monomer concentration and may thus explain the virulence of this toxin family.

Probing membrane protein interactions with their lipid raft environment using single-molecule tracking and Bayesian inference analysis.

The statistical properties of membrane protein random walks reveal information on the interactions between the proteins and their environments. These interactions can be included in an overdamped Langevin equation framework where they are injected in either or both the friction field and the potential field. Using a Bayesian inference scheme, both the friction and potential fields acting on the ε-toxin receptor in its lipid raft have been measured. Two types of events were used to probe these interactions. First, active events, the removal of cholesterol and sphingolipid molecules, were used to measure the time evolution of confining potentials and diffusion fields. Second, passive rare events, de-confinement of the receptors from one raft and transition to an adjacent one, were used to measure hopping energies. Lipid interactions with the ε-toxin receptor are found to be an essential source of confinement. ε-toxin receptor confinement is due to both the friction and potential field induced by cholesterol and sphingolipids. Finally, the statistics of hopping energies reveal sub-structures of potentials in the rafts, characterized by small hopping energies, and the difference of solubilization energy between the inner and outer raft area, characterized by higher hopping energies.

Receptor Displacement in the Cell Membrane by Hydrodynamic Force Amplification through Nanoparticles

We introduce an intrinsically multiplexed and easy to implement method to apply an external force to a biomolecule and thus probe its interaction with a second biomolecule or, more generally, its environment (for example, the cell membrane). We take advantage of the hydrodynamic interaction with a controlled fluid flow within a microfluidic channel to apply a force. By labeling the biomolecule with a nanoparticle that acts as a kite and increases the hydrodynamic interaction with the fluid, the drag induced by convection becomes important. We use this approach to track the motion of single membrane receptors, the Clostridium perfringens ε-toxin (CPεT) receptors that are confined in lipid raft platforms, and probe their interaction with the environment. Under external force, we observe displacements over distances up to 10 times the confining domain diameter due to elastic deformation of a barrier and return to the initial position after the flow is stopped. Receptors can also jump over such barriers. Analysis of the receptor motion characteristics before, during, and after a force is applied via the flow indicates that the receptors are displaced together with their confining raft platform. Experiments before and after incubation with latrunculin B reveal that the barriers are part of the actin cytoskeleton and have an average spring constant of 2.5 ± 0.6 pN/μm before vs. 0.6 ± 0.2 pN/μm after partial actin depolymerization. Our data, in combination with our previous work demonstrating that the ε-toxin receptor confinement is not influenced by the cytoskeleton, imply that it is the raft platform and its constituents rather than the receptor itself that encounters and deforms the barriers formed by the actin cytoskeleton.

Calibrating optical tweezers with Bayesian inference

We present a new method for calibrating an optical-tweezer setup that does not depend on input parameters and is less affected by systematic errors like drift of the setup. It is based on an inference approach that uses Bayesian probability to infer the diffusion coefficient and the potential felt by a bead trapped in an optical or magnetic trap. It exploits a much larger amount of the information stored in the recorded bead trajectory than standard calibration approaches. We demonstrate that this method outperforms the equipartition method and the power-spectrum method in input information required (bead radius and trajectory length) and in output accuracy.

Optical tweezers calibration with Bayesian inference

We present a new method for calibrating an optical-tweezer setup that is based on Bayesian inference1. This method employs an algorithm previously used to analyze the confined trajectories of receptors within lipid rafts2,3. The main advantages of this method are that it does not require input parameters and is insensitive to systematic errors like the drift of the setup. Additionally, it exploits a much larger amount of the information stored in the recorded bead trajectory than standard calibration approaches. The additional information can be used to detect deviations from the perfect harmonic potential or detect environmental influences on the bead. The algorithm infers the diffusion coefficient and the potential felt by a trapped bead, and only requires the bead trajectory as input. We demonstrate that this method outperforms the equipartition method and the power-spectrum method in input information required (bead radius and trajectory length) and in output accuracy. Furthermore, by inferring a higher order potential our method can reveal deviations from the assumed second-order potential. More generally, this method can also be used for magnetic-tweezer calibration.

Single Molecule Tracking Under an External Force Field Created by Amplifying Hydrodynamic Drag with a Nano-Parachute

The membrane architecture plays a crucial role in many cellular processes. In this work we track the motion of two membrane receptors in an induced external force field that distorts the thermal equilibrium and gives access to the receptor interaction with the environment. We introduce a scheme that is easier to use and cheaper than optical tweezers and allows multiplexed measurements.Lanthanide doped nanoparticles (Y0.6Eu0.4VO4) are coupled to two different bacterial pore-forming toxins. Single-molecule tracking (SMT) of receptor-bound labeled toxins in the membrane of MDCK cells reveals that the receptors undergo confined diffusion in stable domains. We take advantage of the hydrodynamic interaction of labeled receptors with a controlled fluid flow within a microfluidic channel to apply a force on the receptors (Figure 1A). The nanoparticle label acts as a parachute and increases the hydrodynamic interaction with the fluid, so that drag induced by convection becomes important. In B), SMT shows mainly elastic displacements of the receptor over distances up to 10 times the confining domain diameter. Once the flow stops, the receptors return to their initial position indicating attachment to the cytoskeleton.View Large Image | View Hi-Res Image | Download PowerPoint Slide