Chapter 8 – Theoretical Biophysics  379

of size n of the putative step are consistent being sampled from the same underlying distribu­

tion in light of the variance values on either side of the putative step, σ², by estimating the

equivalent t statistic:

(8.126)

t

x

x

n

post

post

post

post

2

=

〈

〉−〈

〉

+

σ

σ

2

/

This can then be converted into an equivalent probability, and thus the step rejected if, at

some preset probability confidence limit, it is consistent with the pre-​ and post-​means being

sampled from the same distribution. Improved methods of step detection involve incorp­

orating model-​dependent features into step acceptance, for example, involving Bayesian

inference. The Fourier spectral methods discussed earlier for determining the brightness of

a single dye molecule on a photobleach time series improve the robustness of the step size

estimate compared with direct step detection methods in real space, since they utilize all of

the information included in the whole time series, but sacrifice information as to the precise

time at which any individual step event occurs. These Fourier spectral methods can also be

used for determining the step size of the translocation of molecular machines on tracks, for

example, using optical tweezer methods (see Chapter 6), and in fact were originally utilized

for such purposes.

8.6  RIGID-​BODY AND SEMIRIGID-​BODY BIOMECHANICS

Rigid-​body biomechanics is concerned with applying methods of classical mechanics to bio­

logical systems. These can involve both continuum and discrete mathematical approaches.

Biomechanics analysis crosses multiple length scales from the level of whole animals through

to tissues, cells, subcellular structures, and molecules. Semirigid-​body biomechanics

FIGURE 8.10  Filtering steppy data. Four examples of simulated photobleach steps for a

tetramer complex. Raw, noise-​free simulated data are shown (line), with noise added (dots),

applied with either Chung–​Kennedy, median, or a polynomial fit filter. The latter is not edge

preserving and so blurs out the distinct step edges.

KEY BIOLOGICAL

APPLICATIONS: IN SILICO

IMAGE ANALYSIS TOOLS

Molecular colocalization deter­

mination; Copy number estima­

tion; Molecular stoichiometry

quantitation of complexes.