Talking with Cells about Fluctuations
Analyses of Intra-Cellular Mechanics Using Optical Tweezers
- Talking with Cells about Fluctuations - Analyses of Intra-Cellular Mechanics Using Optical Tweezers
- Fig. 1: (A) Differential Interference Contrast (DIC) image of a bead brought into the vicinity of a cell by optical tweezers. (B) Scheme of Kelvin-Voigt model of an optically trapped bead interacting with a cell.
- Fig. 2: Viscous friction of a particle influenced by a nearby cell (A) Autocorrelation functions of the bead´s position fluctuations in x,y,z at a defined distance to the cell. The viscous drag coefficient gamma can be extracted from the slope of the high frequency part of the autocorrelation function. (B) Two-dimensional bead position trajectory averaged over 10 ms and color encoded by the x component of the viscous friction.
- Fig. 3: (A) Stiffness parameter κ and (B) friction parameter γ in the direction of filopodial extension during the binding process of the bead to the filopodial tip. Inset: Sketch of bead placed in vicinity of a filopodium.
- Formular 1
- Formular 2
All cellular processes are governed by Brownian motion. The Brownian motion of proteins and particles inside a cell as well as in close proximity to the cell is regulated by the physical properties of the cytoplasm, the cell membrane and the extracellular matrix. Optical tweezers allow to place a particle at a well-defined position close to or in contact with a cell, and the existing fast and precise tracking techniques enable to determine and analyze its fluctuations. Here, we use optical trapping combined with 3D back focal plane interferometric tracking  to investigate the fluctuations of the bead close to cell surfaces as well as in contact with cellular protrusions and the cell membrane.
Many features of life have been investigated by simply looking at cells under the microscope. However, in order to study mechanical and dynamical properties at the single cell level interactive methods such as optical tweezers, micropipette aspiration and atomic force microscopy are needed. In particular, optical tweezers have features that make them very suitable to study biophysical properties. They allow placing a particle at arbitrary places close to and in contact with cells. The precision of optical tweezers is in the nanometer range, thereby enabling displacement at molecular scales or measurements of optical forces in the range of piconewtons - forces that are comparable to those exerted by cells or even individual motor proteins. However, the statistical nature of any movement at the scale relevant for cells is often disregarded. At this scale, everything is governed by Brownian motion, which should not be considered as pure noise. The Brownian motion of a small object follows physical laws and is heavily influenced by its direct surroundings. The bead, which serves as a probe in an optical tweezers experiment is also subject to these thermal forces. The fluctuations of the bead depend on its local environment and, when connected to, e.g., a cellular membrane or a protein, on the properties of the molecular link and the mechanical characteristics of the entire system investigated. Fast tracking techniques allow determining the bead's fluctuations in a wide frequency range and enable to analyze the nature of the fluctuations on a cellular level.
New calibration methods even allow quantitative microrheological measurements in media with complex viscoelastic properties such as the cell interior , . In this study, we investigate the influence of a cell on the frictional properties of a nearby bead as well as the change of the viscoelasticity during binding processes to cellular protrusions.
Analysis of the Bead's Fluctuations Reveal Properties of its Surroundings
The stochastic movement of the bead in an optical trap affected by a cell can be described by the Langevin equation (see attached picture: formular 1) where r describes the bead's center position, γb is the frictional coefficient for the bead in the surrounding fluid, κopt (r(t) - rtrap(t)) is the optical force, which can be considered as linear with a stiffness κopt for small displacements from the trap center rtrap(t). The system is driven by a random thermal force Fth, which accounts for the Brownian motion of the particle. In principle, the force exertion of the cell on the bead can be arbitrarily complex and is also changing in time. However, for short timescales and in the high frequency range, respectively, one can argue, that the viscoelastic behavior of the cell can be approximated by a Kelvin-Voigt model (fig. 1 B) leading to (see attached picture: formular 2) and thus to linear friction and a harmonic binding potential.
The resulting Langevin equation is similar to the one of an unperturbed bead in a trap and thus can be treated analogously by analysis of the autocorrelation function, the power spectral density or the mean square displacement of the bead [4,5]. With these methods one can extract the total stiffness κ=κopt+κc and the total friction parameter γ=γb+γc and thus the influence of the cell on the bead, i.e., κc and γc.
Changing Viscosity During the Approach to the Cell Body
According to the analytical theory of Happel and Brenner , the proximity of a stiff, planar wall leads to changes in the friction parameter of the bead. Close to the membrane of a cell, the change in the hydrodynamics leads to an increased friction parameter, which is affected by both, the physical properties of the cell membrane and the influence of the pericellular and extracellular matrix .
Figure 2 A shows how the stiffness and the friction parameters can be derived from the high frequency part of the autocorrelation function of the bead's trajectory. On the timescale of µs, the autocorrelation function can be approximated to be linear. Measurements of time periods in the ms range are therefore sufficient to extract the friction parameter of the bead. By displacement of the trap or by using the motion of the bead in a weak, extended optical trap, position and time dependent friction maps can be obtained. In figure 2 B, the friction parameter in proximity of a cell is illustrated by a color coded friction map . The friction of the bead depends significantly on its distance to the cell membrane. The obtained friction maps might differ for various cell types and parts of the cell that the bead is approached to.
Analysis of Stiffness During the Binding Process
It has been shown that microbeads, uncoated or coated with specific receptor proteins, bind to cellular membranes. Some kinds of cells even engulf the beads, a process which is called phagocytosis. In this case, the bead serves as a model for a bacterium. In the example shown in figure 3, a bead has been placed close to a filopodium and its binding process to the filopodium has been analyzed by the methods described above. Since the binding process to such needle like cellular protrusions with a diameter of 100-300 nm, does not imply a displacement of the bead that can be resolved in the microscope images, it is often impossible to distinguish by analysis of the images whether or not the bead has bound to the filopodium. However, the binding to another object changes the fluctuations of the bead. The analysis of the stiffness parameter κ=κopt+κc reveals the exact point in time, at which the binding occurs. Due to the effects described in the previous paragraph, the friction parameter γ increases already prior to the binding. In addition, the fluctuation analysis provides a quantitative measure of the rigidity κc of the bond and the underlying structure (fig. 3).
Summary and Conclusions
Fluctuations are essential in the nanoscale world and play a crucial role for cellular processes. The stochastic nature of the Brownian motion, which follows physical laws, is important for all features of living systems, especially for driving reactions and movements of cells in different manners. However, the mobility of particles and proteins depends significantly on their surroundings. Cellular processes are adapted and optimized to the varying fluctuation properties, a fact which needs to be investigated in greater detail. The stochastic motions of a bead in the nanometer range contain information about the stiffness of, e.g., a molecular bond or a filopodium as well as information about the changing mobility close to cellular surfaces. Optical tweezers in combination with a fast bead tracking system are tools, which allow to apply forces in a range relevant for cellular processes and to measure the tiny displacements appearing at a subcellular level. On a molecular scale, 3D interferometric tracking also allows to analyze fluctuations governed by elastic, steric and frictional forces that appear during the interaction of the probe with a cell.
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Dr. Felix Kohler (corresponding author via Email request button below)
University of Oslo
Department of Physics
Prof. Dr. Alexander Rohrbach
University of Freiburg
Department of Microsystems Engineerings