Determining the persistence length of biopolymers and rod-like

macromolecular assemblies from electron microscope images and

deriving some of their mechanical properties

Shlomo Trachtenberg1, *, and Ilan Hammel

2

1Department of Microbiology and Molecular Genetics, IMRIC, The Hebrew University-Hadassah Medical School,

The Hebrew University of Jerusalem, Jerusalem 91120, Israel 2 Department of Pathology, Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv 69978, Israel

Persistence length, from which a polymer’s rigidity/flexibility and other parameters (e.g., Young’s modulus, bending force

constant, buckling persistence length, flexural deformation, and flexural time) are derived, can be determined, by simple

length measurements, from electron micrographs of isolated, negatively stained or vitrified polymers.

Keywords persistence length; rigidity; flexibility; Young’s modulus; bending force constant; buckling persistence length;

flexural deformation

1. General introduction and importance of parameters

Linear polymers and assemblies of them are ubiquitous in cellular (and viral) systems and constitute a fundamental

component of their structural foundation. The filamentous structures of eukaryotic (e.g., actin, tubulin, intermediate

filaments and neurofilaments) and prokaryotic (e.g., MreB, FtsZ, pili, and flagellar filaments) cytoskeletal proteins

underlie the determination, control and maintenance of cellular structure, shape and mechanical properties as well as

intracellular transport phenomena. Complexes of acto/myosin and tubulin underlie cell motility through the structure of

muscle, flagella and cilia, as well as cell division. In bacteria, flagellar filaments and pili, rod-like appendages, underlie

cell motility, communal behavior and pathogenesis. The mechanics of folding and packing of DNA, a linear

polyelectrolyte, is crucial in maintaining the informatics and genetic control of cells and viruses.

The analysis and understanding of these dynamic systems depends largely on determining their basic mechanical

properties. Since these (and many other) complexes are dimensionally amenable to electron microscopy imaging, their

quantitative morphological analysis is largely feasible through the processing of straightforward geometrical data

available in such images.

Analyzing electron micrographs has some distinct advantages over hydrodynamic, light scattering, and X-ray

scattering methods: (a) it is based on simple and accurate measurements of lengths; (b) it enables the handling of large

numbers of individual filaments rather than bulk specimens; and (c) parameters such as outer radii or radii of strong

intermolecular connectivity can be determined precisely from TEM three dimensional image reconstructions or STEM

generated radial mass reconstructions. The general morphology of negatively stained specimens, although air-dried on

carbon support films, does not differ from that of frozen hydrated and freeze-dried specimens.

2. Data collection, selection and measurements

Digital or analog images are recorded at the practically lowest magnification. Each electron micrograph is scanned at

the practically highest resolution and stored as a TIFF image. The digitized image is imported to ImageJ- a public

domain, Java-based image processing program (http://rsb.info.nih.gov/ij/). The contours of selected filaments are traced

and measured, as are the lengths of the straight lines connecting their free ends yielding the contour length, L, and the

end-to-end distance, R, respectively (see Fig. 1). Only filaments satisfying the following criteria were included in the

data set for analysis: (a) both ends of the filaments were visible; (b) no kinks or sharp bends are present; and (c) the

filament is not in touch with, or crossed over by, any other filament.

3. Data analysis

3.1 Assessing the filament’s flexibility: Persistence length

The flexibility parameter,λ, or the persistence length (1/λ) provides a measure of the distance along which the

polymer’s direction persists before changing its course. The lengths measured are R, L, and l (see Fig. 1). We determine

R (end-to-end distance) as the length of the straight line connecting the free ends of a curved filament; l is the contour

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length of a segment along the curved filament; and L is the contour length of the curved filament between its endpoints.

The relation between R, L, and λ is determined by Landau and Lifshitz [1].

R2 =

2

λ2

λl −1+ exp −λL( ) (1)

The flexibility parameter,λ, is inversely proportional to the elastic modulus of bending,ε, and linearly proportional to

the thermal energy, KBT [2].

ε =3KBT

4λ (2)

KB is the Boltzmann constant: 1.38 x 10 -16

erg. K-l).

If we denote for each i-th filament the observed end-to-end distance, R, and contour length, L, as Ri, and Li,

respectively (Fig. 1b), then we may assume [3] that for each filament the following approximation holds:

Ri λ( )≈ 2

λ2

λLi −1+ exp −λLi( ) (3)

To estimateλ, we calculate the minimum of the sum of the square of the difference between observed Ri and the

estimated length R(Li):

min S λ( )= Ri − Ri λ( ) 2

i∑ (4)

The calculatedλ, at that minimal value of the sum S (λ), is defined as the most probable value of λ. This approach

provides a statistical weight proportional to the length data of each filament.

3.2 Persistence Length: Estimating θ from R and L

In the second approach (Fig. 1b), the relation between the average cosine of an angle [θ (l)] between the tangents of

two points of distance l along the curved filament (with a contour length L) is determined [4] by:

cosθ l( ) = exp −λl( ) (5)

Measuring L and R are far easier and more accurate than measuring the angle,θ, between tangents to the curved line.

However, for rigid polymers, θ can be estimated from R and L and used as an alternative method for estimating λ as

Figure 1.

A schematic depicting the relation between

R, L, l, and θ. λ can be estimated for mildly

curved polymers in two ways as described

under Data Evaluation (Sections 3.1). The

angle (θ) is determined from one turn (1b) of

the complete filament (1a).

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derived from Eq. (1) (see Fig. 1). We assume that the relation holds for each i-th filament. For small angles, or mild

curvature:

cos θ( )= 2 cos2 θ

2

−1 (6)

cos θ( )≈ 2 Ri / Li( )2 −1;θ→ 0 (7)

cos θi( )= exp −λLi( ) (8)

λi =− ln 2 Ri / Li( )2 −1

Li

(9)

The mean value of n filaments that is calculated for each λi is the apparent value of λ, i.e.:

λ =1

n

λi

i=1

n

∑ (10)

The range of theoretical values, within the limits of an experimental system [5], is shown in Figs. 3a and 3b.

(Example of results obtained from real experimental data of bacterial flagellar filaments using those two approaches are

very similar and summarized in Figs. 4a-c).

Figure 3. The relation between either L, the filament’s contour length and end to end distance, R, (a) or the angle,θ, between

tangents (b). Filaments with a contour length of about 16 µm and a persistence length of (λ>0.2 µm) are flexible. Filaments of the

same length, but with a persistence length smaller than 0.01 µm, are rigid.

Figure 2

Scattergram representation of the relation

between the flexibility parameter,λ, and

the polymer's contour length, R.

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Figure 4. Scatter plots (R, L) of experimental data of bacterial flagellar filaments [5]. The dotted and dashed lines correspond to λ

values calculated according to Eqs. (10, apparent value) and (6-9, regression line), respectively. The points fall generally on a straight

line indicating that the structures are, overall, straight and highly rigid. The R. lupini H13-3 right-handed complex filaments [6] data

(a) and a left-handed (sjw1660) (b) and right-handed (sjw1655) (c) pair of straight S. typhimurium filaments [7] derived from the

same wild-type parent are presented.

3.3 Parameters Derived from the Persistence Length

3.3.1 Bending force constant.

The bending force constant, B, is a measure of the polymer’s resistance to bending [8]. It is linearly correlated with

the persistence length, at a given absolute temperature (T).

B=λKBT (11)

Thus, care should be taken to collect all the data at a constant temperature.

3.3.2 Young’s modulus.

Young’s modulus, Y, can be calculated [9], from the flexibility parameter,λ:

Y= 64λKBT/(πd4)

(12)

Young's modulus is the ratio of stress, which has units of pressure, to the dimensionless strain value; therefore,

Young's modulus has units of pressure. Young's modulus allows the behavior of a linear rode-like protein to be

calculated under tensile or compressive loads. For example, it can be used to estimate the amount a polymer will extend

under tension or buckle under compression.

3.3.3 Flexural deformation/modulus.

The flexural deformation (Y) is the root-mean-square excursion of the filament’s free end, when held fixed at the

opposite end, due to thermal fluctuation. It can be estimated [10] as:

Y ≈ Y2 = L

3/ (8λ) (13)

It represents the physical measurement of stiffness in a material, equaling the ratio of stress (due to applied load) to

the resultant deformation of the polymer (a high value indicates a stiff polymer).

3.3.4 Flexural time.

Bending time (also defined as flexural time, τF) can be estimated as:

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τ F =5.53×10

−3( )πηs L4

λKBT ln(L / d)[ ] (14)

Ookubo et al. [10] showed that the flexural time, is associated with the fundamental flexural deformation and is

related to the persistence length of semi-flexible rods.

3.3.5 Buckling.

The measure of the contraction a filament can suffer before it buckles (δ) can be estimated [11] as:

δ = 4π 4 (d / 2)2λ3 (15)

Buckling is the collapse under pressure or stress. Buckling occurs slightly before the theoretical buckling strength of

a structure, due to plasticity of the material. Thus, it is a "failure mode" characterized by a sudden collapse of a linear

structure subjected to high compressive stresses. When the compressive load is near buckling, the structure will bow

significantly and approach yield.

3.4 Error Analysis

Cumulative errors can be calculated according to Hammel [12]. Since most equations are multi-parametric, cumulative

errors can be large. Thus, particular care should be taken throughout data collection and measurements. In addition,

since some parameters are the result of averaging calculated values, it is recommended to use large data sets. We

determine the final result obtained by the combination of parameters (into an equation) as the desired result designated

F =(X1, X2, X3, . . ., Xn). Taylor expansion of this equation (in which only the first-power terms have been retained)

yields the following approximation [13; 14] :

∆F

F=

∂F

∂Xi

∆Xi

∑

F (16)

Equation (16) is used for estimating the propagation of errors introduced to functions having various parameters. It

assumes that the limits of errors for each experimental parameter have been estimated. Equation (16) defines the

propagation of errors as the uncertainty introduced into a formula by the different independent measurements and

observations.

3.5 Recent applications

For various and more recent applications see, e.g., references [15]-[24].

Acknowledgments Supported by funds from the Israel Science Foundation And the US-Israel Binational Science Foundation (ST).

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