Main narrative

The previous lecture in this topic, ‘Basics of cellular structures’, provided an overview of the structures within cells that relate to its mechanical properties, and how cells can move or exert forces. This lecture aims to provide a basic understanding of why a cell is as stiff as it is, and how this stiffness can be measured (Slide 1).

Note that these lectures are very qualitative and serve to give an overview of the structures that play the main mechanical role in a cell.

It is possible to develop a link between the simple model of rod elasticity and a more microscopic model of the filament, built as a polymer with constraints on the bending. See, for example, Biological Molecules Lecture 2 ‘Modelling DNA and RNA’, or the book by Nelson listed above. This would enable students to derive the expression of the force needed to stretch a molecule like DNA, within the freely jointed chain model. This is commonly used in single-molecule experiments and is one way of measuring the persistence length.

2.1 Measuring cellular mechanical properties

There are various ways, mostly developed in the early 21st century, to measure the mechanical properties of cells (see, for example, Kasza et al, which is the source of the following images and descriptions).

Bulk rheology (Slide 2) is most suitable for relatively large samples, such as ensembles of cells (i.e. bits of tissue, typically including the connecting extracellular material, secreted by many cell types), A sample is sheared between two plates using an oscillatory stress to probe the shear elastic modulus (G´, in phase) and the viscous modulus (G´´, out of phase).

Other methods are able to probe at the cellular or subcellular level. In magnetic bead cytometry (Slide 3), an external magnetic field applies a stress to a magnetic bead, the position of which is tracked to determine the response. In this diagram (and slides 3–7) the blue area is the nucleus and the red lines represent the cytoskeletal filaments.

Slide 4 shows traction force microscopy, where cell contractions deform a flexible substrate containing tracer beads. The forces are estimated from the beads’ displacements. The diagram in Slide 4 presents a top view, and the tracer beads are embedded in a slightly compliant material (e.g. the elastomer PDMS), under the cell.

In Slide 5, showing atomic force microscopy, a cantilever applies stress to the cell, and the cantilever deflection is measured by laser reflection. Depending on the tip size, this technique can be used to resolve finer or coarser detail.

Microrheology (Slide 6) involves surrounding a cell with tracer particles, the motion of which is monitored using either video or laser tracking techniques. The particle motion is either driven externally or is thermally induced and is interpreted to yield a viscoelastic modulus. Over long periods, the probe particles might be actively moved about by the cytoskeletal rearrangements.

Slide 7 shows whole cell stretching. Here, a cell is attached to two surfaces and force is applied while the plate separation is measured.

2.2 Elasticity

2.2.1 Introduction

Stiff molecules, such as actin filaments, microtubules or even double-strand DNA, can be imagined as thin elastic rods. Given that the essential mechanical components of the cell cytoskeleton are the tubulin and actin filaments, it is useful to consider first the general problem of bending a rod.

By analogy with a common Hookean spring, we are used to thinking of elasticity as arising from the stretching of ‘bonds’. This is indeed what happens microscopically when a material such as a metal sheet is flexed. This type of elasticity can be thought of as enthalpic, since it is the internal energy component of the system’s free energy that is being changed. This enthalpic elasticity is not the only type of elasticity.

It is well known that the free energy (Gibbs or Helmholtz) contains an entropic term, and in many systems, most notably ‘soft’ materials, the cost of bending or deforming is mainly due to a loss of entropy (the number of possible conformations compatible with the new constraints is reduced), rather than a strain of chemical bonds. The classic example is the shear modulus of rubber (see, for example, Jones 2002). For this class of entropy-dominated elasticity, the elastic constant is proportional to temperature T (see also Biological Molecules Lecture 2 ‘Modelling DNA and RNA’; a full treatment of this belongs in a polymer physics course; Jones (2002) is a good source of further detail).

2.2.2 Deformation

Considering a thin rod (Slide 8), there are three different types of deformation: stretch, bend and twist. The energy associated with a small deformation from equilibrium must be quadratic in the deformations (and must be a scalar, which limits the possible combinations).

Different modes of deformation may occur. We need different quantities to define the amount of deformation (the strain).

First we can define stretch, u(s), which is also known as extensional deformation:

Equation 1
Equation 1

Then bend β(s), which is a vector, measures how the rod’s tangent vector, t, changes as we move along it:

Equation 2
Equation 2

Finally, twist density ω(s), or torsional deformation, which is a scalar:

Equation 3
Equation 3

We also need to define the contour length, which is obtained by integrating over the unstretched length L0:

Equation 4
Equation 4

2.2.3 Energy

Using equations 1 to 3 and building all possible second-order (harmonic) terms, we can write the total stored elastic energy in each section of the rod (Slide 9) as:

Equation 5
Equation 5

The temperature term kT has been written explicitly as a factor in Equation 5, and the quantities A and C that arise this way are known as the bend and twist persistence lengths, respectively. It is most common to consider first the bending mode, and the bending persistence length is usually labelled as lp. If the elasticity is entropic then they are independent of T. Their physical meaning is that the ‘memory’ of direction (or rotation angle, for twist) is lost over the distance of A (or C, respectively).

In the case of a freely jointed and inextensible chain, then only the first term is non-zero and integrating gives the energy of a bent configuration:

Equation 6
Equation 6

If the chain contour length is much shorter than the persistence length, the chain will be essentially straight. If not, substantial bending can occur simply under thermal motion. For example, we can find the energy cost of a 90° bend, radius R. In this case (Slide 10) we have:

|β| = R–1.

Then from Equation 6:

Equation 7
Equation 7

Thus if R is large enough the elastic cost is negligible compared with kBT.

2.3 DNA and cells

DNA is a much more flexible molecule compared with the filaments discussed above. The distance between base pairs is 0.34 nm, and DNA carries negative charges. A single strand (ssDNA) has a persistence length of around 10 bases, and double strand (dsDNA) of around 100 bases. Thus, while flexible enough to fit within a cell (and within a nucleus), on the scale of a few bases (which is what a protein would experience) the molecule is quite straight, unless it is storing considerable elastic energy (relative to kBT).

The contour length of the DNA in a typical animal cell is about 3&mum. To fit this within a typical nuclear volume, a closed sphere of radius maybe 10 μm, requires a lot of bending; this is the challenge of DNA compaction. This is provided in great part by the adhesion or interaction with proteins that carry high electrostatic charge, and multivalent ions, which enable the DNA to adopt a tightly curved configuration.

Cell motility (Slide 11) is a complex process, even just from the mechanical and physical point of view (i.e. even simplifying the biochemical processes that take place). And yet it is very appealing to physicists because it is an important biological process for which it is possible to make quantitative models. The current thinking is that actin filaments are polymerised, and ‘push’ the leading edge of a cell. Monomers of actin flow towards the leading edge to sustain this. The polymerised filaments are dragged backwards by molecular motors that pull on them and cause them to become denser at the back of the cell. Here, actin filaments are degraded. This process has been shown to be physically plausible, and indeed many aspects of it have been directly imaged in living cells.

At the heart of cell motility – and constraining the rates of deformation, and the maximum forces that a cell can sustain – are the basic thermodynamic and mechanical properties of the cytoskeletal filaments. The diagrams in Slide 11 show the stages of protrusion, attachment and traction that take place at the leading edge. Actin filaments, polymerising, cross-linking with each other, and being actively pushed one relative to the other, are known to be at the heart of this process, but the details of how this works are still being debated.