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Statistical analysis of spatial pointpatterns.
Applications to 3D heterochromatin structures ininterphase nuclei and to root systems of pure
stands of European beech and Norwegian spruce
Frank Fleischer
University of Ulm
Department of Applied Information Processing
Department of StochasticsKarlsruher Stochastik-Tage 2004, Frank Fleischer 1
Overview
1. Overview
2. Spatial Point Processes and their Characteristics
3. Examination of Heterochromatin Structures inInterphase Nuclei
4. Analysis of the 2D-Distribution of Tree Roots
5. Summary and Outlook
Karlsruher Stochastik-Tage 2004, Frank Fleischer 2
OverviewHeterochromatin structures in interphase nuclei
3D-reconstruction of NB4 cell nuclei (DNA shown in graylevels) and centromere distributions (shown in red)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 3
OverviewAnalysis of the 2D-Distribution of Tree Roots
Root systems of a) spruce [Kiefer] and b) beech [Buche]
Karlsruher Stochastik-Tage 2004, Frank Fleischer 4
Spatial Point Processes and their Characteristics
Let X = {X1, X2, ...} be a point process in Rd, whered ∈ {2, 3}
Let X(B) = #{n : Xn ∈ B} denote the number ofpoints Xn of X located in a set B
The intensity measure Λ is defined as
Λ(B) = EX(B)
In the stationary case
Λ(B) = λ|B|
Karlsruher Stochastik-Tage 2004, Frank Fleischer 5
Spatial Point Processes and their Characteristics
The pair correlation function g(r) is given as
g(r) =ρ(2)(r)
λ2,
where ρ(2)(r) is the second product density function
Let C1 and C2 be two spheres with infinitesimalvolumes dV1 and dV2 and midpoints x1 and x2
respectively
The probability for having in each sphere at least onepoint of X is approximately equal to ρ(2)(x1, x2)dV1dV2
Karlsruher Stochastik-Tage 2004, Frank Fleischer 6
Spatial Point Processes and their Characteristics
In the case of complete spatial randomness gPoi(r) ≡ 1
Therefore g(r) > 1 (g(r) < 1) indicates that there is aclustering (repulsion) effect for point pairs of such adistance
Karlsruher Stochastik-Tage 2004, Frank Fleischer 7
Spatial Point Processes and their Characteristics
The K-Function is defined by
λK(r) = E∑
Xn∈B
X(b(Xn, r)) − 1
λ|B|
In the Poisson case KPoi(r) = bdrd, where bd is the
volume of the d-dimensional unit sphere
Karlsruher Stochastik-Tage 2004, Frank Fleischer 8
Spatial Point Processes and their Characteristics
Estimation of λK(r). Edge effects are occuring
Karlsruher Stochastik-Tage 2004, Frank Fleischer 9
Spatial Point Processes and their Characteristics
The L-function L(r) is defined as
L(r) = d
√K(r)
bd
In the Poisson case L(r) = r
Often the slope of L(r) − r is regarded
Karlsruher Stochastik-Tage 2004, Frank Fleischer 10
Spatial Point Processes and their CharacteristicsOther Characteristics
Let δ(Xi) be the distance of a point Xi ∈ X to itsnearest neighbor in X and let λ(r) be the intensity ofthe point process consisting of those points Xi ∈ X
with δ(Xi) ≤ r
The nearest-neighbor distance distribution D(r) isdefined by
D(r) =λ(r)
λ
Karlsruher Stochastik-Tage 2004, Frank Fleischer 11
Spatial Point Processes and their CharacteristicsOther Characteristics
The spherical contact distribution function Hs(r) isgiven by
Hs(r) = 1 − P (X(b(0, r)) = 0)
The J-function is defined by
J(r) =1 − Hs(r)
1 − D(r)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 12
Examination of Interphase Nuclei
A leukemia cell line NB4 has been observed beforeand after differentiation, resulting in 3D confocalmicroscopy images
Position of Centromeres in interphase nuclei have beendetected by automated segmentation methods
Data is given as 3D-coordinates of the centromeremidpoints
28 cell nuclei from undifferentiated cells with 68.25chromocenters on average
27 cell nuclei from differentiated cells with 57.26chromocenters on average
Karlsruher Stochastik-Tage 2004, Frank Fleischer 13
Examination of Interphase Nuclei
3D-reconstruction of NB4 cell nuclei (DNA shown in graylevels) and centromere distributions (shown in red)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 14
Examination of Interphase Nuclei
Projections of the 3D chromocenter distributions of anundifferentiated (left) NB4 cell and a differentiated (right)
NB4 cell onto the xy-plane
Karlsruher Stochastik-Tage 2004, Frank Fleischer 15
Examination of Interphase Nuclei
g(r)
r
0.5
1
1.5
2
500 1000 1500 2000
Averaged estimated pair correlation functions forundifferentiated cells and for differentiated cells
Karlsruher Stochastik-Tage 2004, Frank Fleischer 16
Examination of Interphase Nuclei
L(r)-r
r
–300
–200
–100
0
100
200
500 1000 1500 2000
Averaged estimated functions L̂(r) − r for undifferentiatedcells and for differentiated cells
Karlsruher Stochastik-Tage 2004, Frank Fleischer 17
Examination of Interphase NucleiResults
Number of chromocenters is significantly decreasedduring differentiation
Hardcore effects for both groups, no point pairdistances of less than 350 nm. Reasons are both theapplied detection algorithm as well as the natural sizeof the chromocenters
Karlsruher Stochastik-Tage 2004, Frank Fleischer 18
Examination of Interphase NucleiResults
Especially the number of point pairs with a distancebetween 350 nm and 700 nm is reduced. While the cellsbefore differentiation show clustering effects, the cellsafter differentiation show even slight signs of repulsion.
Centromeres are clumping during differentiation, theirdistance is reduced below the detectable range(350 nm)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 19
Analysis of the 2D-Distribution of Tree Roots
Adjoining pure stands of European beech andNorwegian spruce are examined
Two-dimensional vertical profile walls with an area of2 m × 1 m have been obtained
The position of small roots with diameters between2 mm and 5 mm have been marked
20 profile walls of European beech with 52.7 points onaverage
16 profile walls of Norwegian spruce with 81.19 pointson average
Karlsruher Stochastik-Tage 2004, Frank Fleischer 20
Analysis of the 2D-Distribution of Tree Roots
a) b)
Samples of small root patterns for a) European beech andb) Norwegian spruce
Karlsruher Stochastik-Tage 2004, Frank Fleischer 21
Analysis of the 2D-Distribution of Tree Roots
Data is inhomogeneous with respect to the vertical axis
A vertical transformation has to be performed
For beeches (spruces) the depth coordinates havebeen assumed to be gamma (exponentially) distributed
The transformed depth htran is given as
htran =F ∗(horig)
F ∗(htot)htot,
where horig is the original depth, htot is the total depthof the sampling window and F ∗(x) is the assumeddepth distribution
Karlsruher Stochastik-Tage 2004, Frank Fleischer 22
Analysis of the 2D-Distribution of Tree Roots
Depth distribution of a) beech and b) spruce
Karlsruher Stochastik-Tage 2004, Frank Fleischer 23
Analysis of the 2D-Distribution of Tree Roots
a) b)
Samples of small root patterns for European beech,a) original, b) transformed
c) d)
Samples of small root patterns for Norwegian spruce,c) original, d) transformed
Karlsruher Stochastik-Tage 2004, Frank Fleischer 24
Analysis of the 2D-Distribution of Tree Roots
1
2
3
4
5
6
7
10 20 30 40 50
Radius r
Averaged estimated pair correlation functions forNorwegian spruce (·) and European beech (♦)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 25
Analysis of the 2D-Distribution of Tree Roots
0
0.5
1.0
1.5
2.0
2.5
10 20 30 40 50
Radius r
Averaged estimated functions L̂(r) − r for Norwegianspruce (·) and European beech (♦)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 26
Analysis of the 2D-Distribution of Tree Roots
Both tree root systems show clear signs of clusteringfor short (less than ≈ 10 cm) point pair distances
The clustering effect in the case of the Norwegianspruces (European beeches) seems to be stronger(weaker) in a smaller (larger) region of custering
A Matern-cluster model has been fitted to thetransformed data
Karlsruher Stochastik-Tage 2004, Frank Fleischer 27
Analysis of the 2D-Distribution of Tree RootsMatern-cluster model
A Matern-cluster process XMC is based on a Poissonprocess with intensity λp, whose points are calledparent points
Around each parent point a disc with fixed radius R istaken, in which the points of XMC are scattereduniformly
The number of points in such a disc is Poissondistributed with parameter πR2λd, where λd is the meannumber of points per unit area generated by a singleparent point in a disc of radius R
Karlsruher Stochastik-Tage 2004, Frank Fleischer 28
Analysis of the 2D-Distribution of Tree RootsMatern-cluster model
Parameters of the Matern-cluster processes have beenestimated using minimum-contrast methods applied tothe theoretical and estimated pair correlation function
Results are λsprucep = 0.00690, λbeech
p = 0.00603,λ
spruced = 0.00774, λbeech
d = 0.00253, Rspruce = 4.9 cm andRbeech = 7.4 cm
Afterwards the models have been validated bysimulation-based goodness-of-fit tests for differentpoint process characteristics
Karlsruher Stochastik-Tage 2004, Frank Fleischer 29
Analysis of the 2D-Distribution of Tree Roots
A retransformation of the data has to be performed
The inverse transformation is given by
horig = (F ∗)−1(F ∗(htot)
htothtran)
The inhomogeneous Matern-cluster models arisinghave inhomogeneous Poisson processes as parentprocesses with intensity functions
λp(x, y) = λp(y) = λpf∗(y)
F ∗(htot)htot,
where f∗(x) is the density function of the suitabledistribution function F ∗(x)
Karlsruher Stochastik-Tage 2004, Frank Fleischer 30
Analysis of the 2D-Distribution of Tree Roots
The cluster regions are given by
{(x, y) : (x − xp)2 + (F ∗(y) − F ∗(yp))
2(htot
F ∗(htot))2 ≤ R2}
They are no longer circles, but the images of thesecircles under the inverse transformation
Karlsruher Stochastik-Tage 2004, Frank Fleischer 31
Analysis of the 2D-Distribution of Tree Roots
Realizations with cluster regions of the homogeneousMatern-cluster model and of the retransformed
inhomogeneous Matern-cluster model for Norwegianspruce
Karlsruher Stochastik-Tage 2004, Frank Fleischer 32
Summary and Outlook
For the interphase nuclei clear differences between thetwo examined groups have been detected
Next steps are a fitting and validating of an appropriatepoint process model
For the tree root patterns a Matern-cluster model hasbeen fitted to the data
Mixed stands have already been examined, too.Confirmation of the results for pure standsInteresting results for effects between different treespecies
Karlsruher Stochastik-Tage 2004, Frank Fleischer 33
Cooperation Partners
We want to thank our cooperation partners
Dr. Dr. M. Beil, Department of Internal Medicine I,University of Ulm
Prof. Dr. M. Kazda, Department Systematic Botanyand Ecology, University of Ulm
Karlsruher Stochastik-Tage 2004, Frank Fleischer 34
Literature
Beil, M., Fleischer, F., Paschke, S. and Schmidt, V.,Statistical analysis of 3D centromeric heterochromatin structure ininterphase nuclei. Preprint
Fleischer, F., Eckel, S., Kazda, M., Schmid, I. andSchmidt, V., Statistical analysis of the spatial distribution of treeroots in pure stands of Fagus sylvatica and Picea abies. Preprint(both available at www.geostoch.de)
Stoyan, D., Kendall, W. S. und Mecke, J., StochasticGeometry and its Applications. J. Wiley & Sons, Chichester,2nd edition, 1995
Karlsruher Stochastik-Tage 2004, Frank Fleischer 35