The softness of a material implies that it deforms easily when subjected to a stress. For a cell, an applied stress could arise from the cell’s environment, such as the action of a wave on water-borne cells or the pressure from a crowded region in a multicellular organism. The exchange of energy with the cell’s environment due to thermal fluctuations can also lead to deformations, although these may be stronger at the molecular level than the mesoscopic length scale of the cell proper. For example, Fig. 1.12 shows the fluctuations in shape of a synthetic vesicle whose membrane is a pure lipid bilayer that has low resistance to out-of-plane undulations because it is so thin. At fixed temperature, flexible systems may sample a variety of shapes, none of which need have the same energy because fixed temperature does not imply fixed energy. In this chapter, the kind of fluctuating ensembles of interest to cell mechanics are introduced in Section 2.1, followed up with a review of viscous fluids and their role in cell dynamics in Section 2.2. Many of the statistical concepts needed for describing fluctuating ensembles are then presented, using as illustrations random walks in Section 2.3 and diffusion in Section 2.4. Lastly, the subject of correlations is presented in Section 2.5, focusing on correlations within the shapes of long, sinuous filaments.

Fluctuations at the cellular scale

Among the common morphologies found among cyanobacteria (which trace their lineage back billions of years) are filamentous cells, two examples of which are displayed in Fig. 2.1. The images are shown at the same magnification, as indicated by the scale bars. The upper panel is the thin filament Geitlerinema PCC 7407, with a diameter of 1.5 ± 0.2 μm, while the lower panel is the much thicker filament Oscillatoria PCC 8973, with a diameter of 6.5 ± 0.7 μm (Boal and Ng, 2010). The filaments have been cultured in solution, then mildly stirred before imaging; clearly, the thinner filament has a more sinuous appearance than the thicker filament when seen at the same magnification. This is as expected: the resistance to bending possessed by a uniform solid cylinder grows like the fourth power of its diameter, so the thinner filament should have much less resistance to bending and hence appear more sinuous than the thicker one.