ATP-dependent mechanics of red blood cells

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ATP-dependent mechanics of red blood cells, Timo Betz, Martin Lenz, Jean-Francois Joanny, and Cecile Sykes., PNAS vol.106 no.36 (2009) [1]

Keywords

Equipartition theorem, Cell membrane, Cell mechanics

Summary

The fluctuations of the membrane of red blood cells is studied using an optical deflection method similar to that used to detect the position of trapped colloids in an optical trap. Membrane fluctuations observed in the absence of ATP are regarded as occurring passively - that is they are thermal in nature. The frequency dependence of these thermal fluctiations is compared with membrane fluctuations occurring in the presence of APT, and a significant deviation from equilibrium statistics os observed.


Soft Matter

Figure 1

The basic experimental setup for this study is shown is Figure 1. A tightly focused laser beam is incident on the edge of a red blood cell immobilized on a wall of a well formed by two cover slips. The scattered light is picked up (in "transmission mode") with a quadrant photodiode (QPD). This is same manner in which the position of optically trapped microparticles is usually measured. The QPD consists of 4 separate photodiodes, but really only 2 are needed for this experiment. The relative amount of signal in one "side" of the QPD is proportional to the position of the scattering object, within some linear regime which is found from a simple calibration. Though the authors do not say explicitly that they take the differential signal of the two sides of the QPD, this is how it is generally done in an optical tweezer setup. It is possible that the QPD signal used is simply a sum over all four quadrants. Whatever the case, the important result is that the position of the membrane can be measured to sub-nanometer precision and with temporal resolution down to 10 <math>\mu s</math>.

Figure 2

The power spectral density (PSD = <math>|\tilde{x}|^2</math>, where <math>\tilde{x}</math> in the fourier transform of a time series of membrane fluctuations) of the membrane fluctuations is the basic type of data obtained using this setup. An example of such a curve is shown in figure 2. The data agree with the high frequency power law fall off predicted by theory <math>PSD \propto f^{-5/3}</math> as detailed in the appendix. The theory also relates the PSD to physical properties of the system: the membrane bending rigidity, surface tension and effective viscosity of the cell. This is a nice result, especially in the context of possibly extending this method to a diagnostic type of implementation. The authors tabulate these parameters from their fits and compare to values obtained by other well established studies and find very good agreement.

The nature of the membrane fluctuations is investigated next. The authors take a simple but insightful approach to quantifying the extent to which the membrane fluctuations are non-thermal. First, the PSD of ATP-depleted cells is measured; since ATP is the "energy carrier" in cells, these fluctuations are taken to be independent of any active process going on in the cell. In other words the fluctuations in these cell membranes are assumed to be thermally driven so that their energy is <math>k_B T</math> as can be seen from the equipartition theorem. The PSD of membrane fluctuations by cells with access to ATP are then compared to this thermal baseline by defining an effective energy <math>\frac{E_{eff}}{k_B T} = \frac{PSD^{ATP}}{PSD^{ATP-depl}} \times \frac{g^{ATP-depl}}{g^{ATP}}</math>, where the function g (defined in the supplementary information) takes into account the different physical parameters of each type of cell. Interestingly, there is a departure from <math>E_{eff} = 1</math> at low frequencies meaning that active processes inside the cell become resolvable at these long time scales (>0.1 s or <10Hz). A plot of this effective energy as a function of fluctuation frequency is shown in figure 3, for a PKC-activated cell (the significance of this plot lies in the difference between the PSD of cells with and without ATP and not in the specific nature of what PKC does to the cell). Qualitatively, we can interpret this as saying that the cell can interact with its surroundings actively through its membrane at a speed of up to 10Hz or so. This corroborates an in-vitro study done on cytoskeletal networks (reference 21 [2]) which finds a deviation from equilibrium response at roughly the same frequency.

This study is very interesting because it investigates deviations form thermal equilibrium in a living system - something which is quite to do since a lot of methods for probing the microscale dynamics rely on thermodynamics equilibrium (passive microrheology, optical tweezer calibration, etc..). This is a natural aspect of living systems to want to study and speaks of the fundamental nature of what it means for a collection of chemicals to be "living". Exaclty how does a living entity such as a cell stave off the second law of thermodynamics by resisting overwhelming accumulations of entropy over long time scales. Experiments such as this one (and reference 21 [3]) are beginning to shed light on this deep question.

Figure 3