Theoretical background

In order to analyse the clustering of particles, the second reduced moment K function and the pair-correlation function (PCF) were used (Stoyan et al., 1995). The K function is the number of gold particles (or points) at distances shorter than a given distance from a typical particle divided by the average density of these particles (see Fig. 1). PCF is a ratio of the density of gold particles at a given distance from a typical particle to the average density of these particles. Thus it is in principle possible to obtain PCF from the K function by differentiation using the following formula (Stoyan et al., 1995):

The clustering causes an increase in the density of particles in the neighbourhood of a typical particle and, therefore, an increase of the values of both the PCF and K function. In order to analyse the colocalization, we used the cross-K function, i.e. the number of gold particles of the 1st type at distances shorter than a given distance from a typical particle of the 2nd type divided by the average density of the particles of the 1st type (see Fig. 1).

Fig. 1
FIG. 1. A graphical representation of the evaluation of K and cross K functions. K function is the ratio of the number of gold particles (points) at distances shorter than a given distance r (i.e. the points which are inside the circle with the radius r) from a typical particle of the same type (large point in the center of the circle) to the average density of these particles (here - the total number of particles in the frame divided by the area of this frame). In the case of the cross K function, the large point in the center of the circle is a typical particle of the 2nd type while the remaining points are the particles of the 1st type.

Similarly, we evaluated the pair-crosscorrelation function (PCCF) as the ratio of the density of particles of the 1st type at the given distance from a typical particle of the second type to the average density of the particles of the 1st type. For PCCF and cross-K function the same relationship holds as for PCF and K function (see formula above). The pair- (cross)correlation functions are useful for exploratory analysis, while the K functions - for statistical testing (Stoyan et al., 1995). The exploratory analysis provides one with information if the clustering or colocalization is present, and if so at what distances. This information can be then used in more detailed confirmatory analysis, especially for setting the appropriate ranges of distances that are to be tested. Inappropriately low range might cause an omission of significant clustering/colocalization at distances longer than set. In contrast, too high range - much larger than the typical size of clusters - would cause a low spatial resolution of the test.

The above functions were calculated from pooled data from all images of nucleoplasm in each experimental group. Let N be the number of images of size a x b. Let n,i be the number of 10 nm particles in the image number i; similarly, let n,i be the number of 5 nm particles in image number i; i=1..N. Then density of j-th labelling (j=1 or j=2 in case of 10 nm or 5 nm gold particles, respectively) can be estimated as

where A=ab is the area of one image. The functions were then estimated by sampling distances d(x,y) between points x and y which correspond to pairs of particles in each window (one window corresponds to one image). However, the sampling of long distances is negatively biased, because the probability that one particle from the pair is situated outside of the window (image) increases with an increase of the distance between gold particles (Fig. 2A). This negative bias, called boundary effect may be corrected using the geometric covariogram of the window (Ripley, 1988). In order to obtain this parameter, a rectangle with sides a and b (which corresponds to an image) was shifted by distance r in all directions from 0 to 2, and the area of intersections between the original rectangle and the shifted ones was evaluated (see Fig. 2B).

Fig. 2
FIG. 2. Correction of the boundary effect using geometric window covariogram. (A) The probability that one gold particle from the pair is situated outside of the window (image) increases with an increase of the distance r at which the colocalization/clustering is evaluated. (B) This negative influence, called boundary effect, may be corrected using the geometric covariogram of the window. To obtain this parameter, a rectangle with sides a and b is shifted by distance r in all directions ( angle ranges from 0 to 2), and the areas of intersections between the original rectangle and the shifted ones (grey-coloured area) were measured. The geometric window covariogram (r) equals to the average area of the intersections; a and b equal to the length of the sides of the window.

The average area of intersections which corresponds to the geometric window covariogram (r) can be then calculated for r < min (a,b) accordingly:

Let Xj,i be a set of labels of type j in i-th image and let r''> r' 0. Then the bottom-censored K function for points x and y will be estimated similarly to the classical K function (in the case of r'=0, Ripley, 1980):

A graphical representation of the bottom-censored K function evaluation is given in Fig. 3. Also the bottom censored cross K function for 10 nm particles (j=1) and 5 nm particles (j=2) will be estimated as:

Condition r'd(x,y)< r'' means that point y is lying inside the sector created with circles of the radius r' and r'' which have center in the point x (see Fig. 3).

Fig. 3
FIG. 3. Illustration for the estimation of the bottom censored K and cross K functions for particles of one or two types, respectively. Let x, y and y will be the particles of the same type. In both cases d(x,y) and d(x,y) (distances between points x and y and x and y) fulfil the condition r'd(x,y)< r''. Any point yn will be taken into account for the evaluation of the bottom censored K function only if it is laying inside the sector, delimited by the circles of the radius r' and r'' with a common center in the point x. In case of the evaluation of the bottom censored cross K function, the point x will be a gold particle of one type while points yn will be the gold particles of the other type. Thus the colocalization/clustering at a specified range of distances can be evaluated by comparing the densities of particles yn inside the sector to the average density of these particles.

Thus the expression "1[condition]" is for indicator function; it has value 1 if the condition is fulfilled and 0 otherwise. Similarly, we estimate and which are the integrals (from r' to r'') of pair-correlation and pair-crosscorrelation functions (PCF and PCCF, Stoyan et al., 1995) respectively as:

          Histograms of PCFs and of PCCFs with delimiting points r0, r1, r2.... defined as 0, dr, 2dr.. were constructed using estimates of mean values of PCF and PCCF (which correspond to the heights of histogram bars). These mean values were estimated on the intervals (ri, ri+1....) and defined as

and

respectively. The width of histogram bars was calculated from the number of images and particle density for PCF histograms as:

and also for PCCF histograms as:

          Monte Carlo estimates (Barnard, 1963) of two-sided 95% confidence intervals for histogram bar heights and one-sided 1% and 5% tests for clustering or colocalization were carried out using 999 simulations of N realizations (N is equal to the number of evaluated images) of the binomial process with number of simulated points equal to the observed points numbers nj,i. The tests characteristics (values of K(r, r') functions which were defined above) were calculated. For the estimation of histogram confidence intervals the calculated 25th extreme value of the bar height was used as critical value for two-sided 5% tests. Similarly, calculated critical values were used for verifying the clustering and/or colocalization of the particles: for the one-sided tests the 50th value from the maximum was used at 5% confidence level, and the 10th value from the maximum - at 1% confidence level.


References

Barnard, G. (1963) Contribution to discussion of Bartlett. Journal of Royal Statistical Society B25, 294.

Ripley, B. D. (1980) Spatial statistics. John Wiley & Sons, New York, Chichester, Brisbane, Toronto

Ripley, B. D. (1988) Statistical inference for spatial processes. Cambridge University Press, Cambridge, New York, New Rochelle, Melbourne, Sydney

Stoyan, D., Kendall, W. S., and Mecke, J. (1995) Stochastic geometry and its applications. John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore