362 8.4  Reaction, Diffusion, and Flow

slow slippage in a microscope stage, or thermal expansion effects, which are obviously then

simply experimental artifacts and need to be corrected prior to diffusion analysis.

In the case of inhomogeneous fluid environments, for example, the mean square displacement

of a particle after time interval τ as ~ τ^α (see Equation 4.18) where a is the anomalous diffusion

coefficient and 0 < α < 1, which depends on factors such as the crowding density of obstacles to

diffusion in the fluid environment. Confined diffusion has a typically asymptotic shape with τ.

In practice, however, real experimental tracking data of diffusing particles are intrinsically

stochastic in nature and also have additional source of measurement noise, which complicates

the problem of inferring the mode of diffusion from the shape of the mean square displace­

ment relation with τ. Some experimental assays involve good sampling of the position of

tracked diffusing biological particles. An example of this is a protein labeled with a nanogold

particle in an in vitro assay involving a similar viscosity environment using a tissue mimic

such as collagen to that found in the living tissue. The position of the nanogold particle can

be monitored as a function of time using laser dark-​field microscopy (see Chapter 3). The

scatter signal from the gold tag does not photobleach, and so they can be tracked for long

durations allowing a good proportion of the analytical mean square displacement relation to

be sampled thus facilitating inference of the type of underlying diffusion mode.

However, the issue of using such nanogold particles is that they are relatively large (tens of

nm diameter, an order of magnitude larger than typical proteins investigated) as a probe and

so potentially interfere sterically with the diffusion process, in addition to complications with

extending this assay into living cells and tissue due to problems of specific tagging of the right

protein and its efficient delivery into cells. A more common approach for live-​cell assays is

to use fluorescence microscopy, but the primary issue here is that the associated tracks can

often be relatively truncated due to photobleaching of the dye tag and/​or diffusing beyond

the focal plane resulting in tracked particles going out of focus.

Early methods of mean square displacement analysis relied simply on determining a

metric for linearity of the fit with respect to time interval, with deviations from this indi­

cative of diffusion modes other than regular Brownian. To overcome issues associated with

noise on truncated tracks, however, improved analytical methods now often involve aspects

of Bayesian inference.

KEY POINT 8.8

The principle of Bayesian inference is to quantify the present state of knowledge and

refine this on the basis of new data, underpinned by Bayes’ theorem, emerging from the

definition of conditional probabilities. It is one of the most useful statistical theorems

in science.

In words, Bayes’ theorem is simply posterior =​ (likelihood × prior)/​evidence.

The definitions of these terms are as follows, which utilizes statistical nomenclature of the

conditional probability “P(A|B)” meaning “the probability of A occurring given that B has

occurred.” The joint probability for both events A and B occurring is written as P(A∩B). This

can be written as the probability of A occurring given that B has occurred:

(8.93)

P A

B

P B

P A B

∩

(

) =

( )⋅(

| )

But similarly, this is the same as the probability of B occurring given that A has occurred:

(8.94)

P A

B

P A

P B A

∩

(

) =

( )⋅( |

)

Thus, we can arrive at a more mathematical description of Bayes’ theorem, which is

(8.95)

P A B

P B A P A

P B

(

| )

( |

)

=

⋅( )

( )