The Physics of Chromosomes: Loops and Entropy, that's what it's … · 2017-03-26 · stfq_tfaq...
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The Physics of Chromosomes: Loops and Entropy, that's what it's all about Dieter W. Heermann Heidelberg University
Transcript of The Physics of Chromosomes: Loops and Entropy, that's what it's … · 2017-03-26 · stfq_tfaq...
The Physics of Chromosomes: Loops and Entropy, that's
what it's all aboutDieter W. Heermann
Heidelberg University
Heermann, Heidelberg University, 2015
Derive a chromosome model that works across species, across phases, hence that is able to capture the underlying principles of chromosome folding.
Can explain data from various sources simultaneously.
Can predict biological as well as mechanical properties.
Key Problem
Heermann, Heidelberg University, 2015
Yes, we can!
Key Problem
Heermann, Heidelberg University, 2015
Human chromosomes (interphase and metaphase)
Escherichia coli
Yeast
Key Problem
Heermann, Heidelberg University, 2015
How can we obtain information on
the structure?
Heermann, Heidelberg University, 2015
Key Experiments: Eukaryotes
Label (for example fluorescence in situ hybridization (FISH)) specific sites along the chromosome.
Heermann, Heidelberg University, 2015
Localization microscopy reveals expression-dependent parameters of chromatin nanostructures Manfred Bohn, Philipp Diesinger, Rainer Kaufmann, Yanina Weiland, Patrick Müller, Manuel Gunkel, Alexa von Ketteler, Paul Lemmer, Michael Hausmann, Dieter W. Heermann and Christoph CremerBiophysical Journal, Volume 99, Issue 5, 1358-1367, 8 September 2010
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
0 50 100 150 200 250 300
g(r)
r [nm]
Fibroblast CellsHeLa cells (strain I)HeLa cells (strain II)
Key Experiments: Eukaryotes
Heermann, Heidelberg University, 2015
T.Cremer and C. Cremer, Nature Reviews Genetics vol. 2, no. 4, pp. 292-301 (April, 2001)
Chromosome Territories
Key Experiments: Eukaryotes
Staining of the entire chromosome
Heermann, Heidelberg University, 2015
W. de Laat, Current Opinion in Cell Biology 2007, 19:317–320
Key Experiments: Eukaryotes
Heermann, Heidelberg University, 2015
Comprehensive Mapping of Long-Range Interactions Reveals Folding Principles of the Human Genome E. Lieberman-Aiden et. al., Science 2010
Key Experiments: Eukaryotes
Heermann, Heidelberg University, 2015
Key Experiments
Spatial Information
Measure two, few or many physical positions
Topological Information
Measure contacts without spatial information
Heermann, Heidelberg University, 2015
The Model
Heermann, Heidelberg University, 2015
coarse grained description
The Model
Heermann, Heidelberg University, 2015
The Model
Basic assumptions:The backbone of the chain is given by a simple self-avoiding walk.
Heermann, Heidelberg University, 2015
A B C
chromosome chromosome chromosome
chro
mos
ome
The effect of loops on the conformation
The Model
Heermann, Heidelberg University, 2015
The Model
Basic assumptions:The backbone of the chain is given by a simple self-avoiding walk. Two parts of the polymer form loops with a certain probability. Loops are not static but can change in the course of time; their size and position are chosen from a broad range.
Diffusion-Driven Looping Provides a Consistent Framework for Chromatin Organization Manfred Bohn and Dieter W. HeermannPLoS ONE 5(8): e12218. doi:10.1371/journal.pone.0012218 (2010)
Heermann, Heidelberg University, 2015
Influence of the catenation constraint on elongation and segregation of ring polymers Manfred Bohn, Dieter W. Heermann, Odilon Lourenço, Claudette CordeiroMacromolecules, 43 (5), 2564–2573 (2010)
Topological interactions between ring polymers: Implications for chromatin loops Manfred Bohn and Dieter W. HeermannJ. Chem. Phys. 132, 044904 (2010)
Loops
Heermann, Heidelberg University, 2015
Loops
0
1
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0 0.5 1 1.5 2 2.5 3
effective
potentialUring(r)
r/Rg
A B
0.8
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1
1.05
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
exp(�U(0)/N)
1/N
N=64
N=128
N=256
N=384
N=512
N=1024
N=1536
N=2048
Heermann, Heidelberg University, 2015
Loops
0
1
2
3
4
5
6
0 0.5 1 1.5 2 2.5 3
effective
potentialUring(r)
r/Rg
A BN=64
N=128
N=256
N=384
N=512
N=1024
N=1536
N=2048So loops repel each other!
Let‘s look at prokaryotes.
Heermann, Heidelberg University, 2015
e.coli
oriC
ter
I2
C1 C4
D3
C10
E10
F10
A7
D8 E4 G2
B8
B9
D9
A11
A
B
IL05IL06
IL05IL06
IL01
t
IL01
t
Source: Willenbrock and Ussery Genome Biology 2004 5:252 doi:10.1186/gb-2004-5-12-252
Curved DNA
Plectonemic supercoils
Toroidal supercoil
RNA
RNA polymerase
Heermann, Heidelberg University, 2015
Loops: One More Thing
Transcription Factor Induced DNA Domain Formation Structures the E. coli Chromosome M. Fritsche, S. Li, P. Wiggins and D.W. Heermann, Nucleic Acids Res. 2012 Feb;40(3):972-80.
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bio
Ansatz: Dynamic Loop Model + genes that are co-regulated by a set of same or similar transcription factors, might stay in physical proximity in order to guarantee the efficiency of gene regulation and expression.
Heermann, Heidelberg University, 2015
Loops: One More Thing
Transcription Factor Induced DNA Domain Formation Structures the E. coli Chromosome M. Fritsche, S. Li, P. Wiggins and D.W. Heermann, Nucleic Acids Res. 2012 Feb;40(3):972-80.
Heermann, Heidelberg University, 2015
e.coliCurved DNA
Plectonemic supercoils
Toroidal supercoil
RNA
RNA polymerase
Looped Star Polymers Show Conformational Transition from Spherical to Flat Toroidal Shapes Pascal Reiß, Miriam Fritsche, and Dieter W. Heermann, Phys. Rev. E 84, 051910 (2011)
Heermann, Heidelberg University, 2015
e.coliCurved DNA
Plectonemic supercoils
Toroidal supercoil
RNA
RNA polymerase
Looped Star Polymers Show Conformational Transition from Spherical to Flat Toroidal Shapes Pascal Reiß, Miriam Fritsche, and Dieter W. Heermann, Phys. Rev. E 84, 051910 (2011)
Entropic repulsion between rings leads to an effective bending rigidity.
Heermann, Heidelberg University, 2015
Curved DNA
Plectonemic supercoils
Toroidal supercoil
RNA
RNA polymerase
e.coli
Heermann, Heidelberg University, 2015
Curved DNA
Plectonemic supercoils
Toroidal supercoil
RNA
RNA polymerase
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
Heermann, Heidelberg University, 2015
e.coli
0 0.25 0.5 0.75 10
200
400
600
800
1000
1200
Num
ber o
f Cel
ls
Locus Position (Cell Lengths)
oriC C4
ter
lac
A
C
B
10
0 1
1
Long Axis (Cell Lengths)
Shor
t Axi
s
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.2 0.4 0.6 0.8 1Frequency
Relative Position
oriCC4
lac
Heermann, Heidelberg University, 2015
Mo
nte
Ca
rlo
ste
ps
Chromosome segregation by the Escherichia coli Min systemBarbara Di Ventura, Benoit Knecht, Helena Andreas, William J. Godinez, Miriam Fritsche, Karl Rohr, Walter Nickel, Dieter W Heermann, Victor Sourjik Molecular Systems Biology 9 Article number: 686 doi:10.1038/msb.2013.44, 2014
0 0.5 10
0.05
0.1
0.15
0 1 2 3 4 5
0.2
0 1 2 3 4 5
0.2
00 1 2 3 4 5
0 1 2 3 4 5
200
400
600
0
200
400
600
Distance (μm)
Gra
y va
lue
eulav yarG
Distance (μm)
A
C
0
0.3
0.6MinD MinDR219D
0
1
2
3 MinD MinDR219D
4.4 μm 5.6 μm
4.4 μm 5.6 μmCell length
Cell length
gmax
lmingmin
gmax –lmin
gmax –gminR =
R
D
D (μ
m)
E
F
G
MSD
(μm
2 )
Δt (s)
t = 0 s
t = 1 s
Instantaneous velocity(μm/s)
Freq
uenc
y
IH
ΔminB ΔminC
D
Freq
uenc
y of
anuc
leat
e ce
lls
ΔminB ΔminC
B
0
0.2
0.4
0.6
e.coli
Heermann, Heidelberg University, 2015
entropic forces alone are not sufficient to achieve and maintain full separation of chromosomes
e.coli
Chromosome segregation by the Escherichia coli Min systemBarbara Di Ventura, Benoit Knecht, Helena Andreas, William J. Godinez, Miriam Fritsche, Karl Rohr, Walter Nickel, Dieter W Heermann, Victor Sourjik Molecular Systems Biology 9 Article number: 686 doi:10.1038/msb.2013.44, 2014
Heermann, Heidelberg University, 2015
e.coli
0 0.25 0.5 0.75 10
200
400
600
800
1000
1200
Num
ber o
f Cel
ls
Locus Position (Cell Lengths)
oriC C4
ter
lac
A
C
B
10
0 1
1
Long Axis (Cell Lengths)
Shor
t Axi
s
• Assumption of spatial proximity (dynamic loop model + transcriptional network) leads to a „looped-star“.
• Loop structure + entropic repulsion + confinement induces ordering.
• Can explain recent experiments.
Heermann, Heidelberg University, 2015
Yeast
Transcriptional Regulatory Network Shapes the Genome Structure of Saccharomyces cerevisiae Songling Li and Dieter W. Heermann, Nucleus. 2013 May-Jun;4(3):216-28
Spindle Pole Body (SPB) loci
rDNA gene cluster
Heermann, Heidelberg University, 2015
Yeast
Transcriptional Regulatory Network Shapes the Genome Structure of Saccharomyces cerevisiae Songling Li and Dieter W. Heermann, Nucleus. 2013 May-Jun;4(3):216-28.
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bio
Ansatz: Dynamic Loop Model + genes that are co-regulated by a set of same or similar transcription factors, might stay in physical proximity in order to guarantee the efficiency of gene regulation and expression.
Heermann, Heidelberg University, 2015
Yeast
Therefore, our approach provided quantitative measurementsand visualizations of subnuclear territories that agree with estab-lished notions of nuclear organization. Themapswere considerablyless accurate and noisier when cell number was reduced to 100(Fig. 2f–j). Similarly, compartmentalization was lost in the absenceof nucleolar alignment (Fig. 2k–o). This underscores the impor-tance ofl arge samples and nuclear landmark alignment, two keyfeatures of our method.
Remodeling of gene territories upon activationTo demonstrate the method’s potential for functional studies ofnuclear organization, we revisited the repositioning of galactosegenes upon transcriptional activation10–12(Fig. 3). We first analyzedthe gene clusterGAL7-GAL10-GAL1 (hereafter referred to asGAL1)and found that in presence of glucose, where the gene is repressed,GAL1 concentrates in a small 0.58mm3 territory close to the nuclearcenter (Fig. 3a), whereas in presence of galactose, whereGAL1 isactive, its territory expands to 0.9mm3 and is enriched near thenuclear periphery (1Dw2-test on signed distances:P1 o 10 4; 2Dw2-test on point distributions:P2 o 10 4; Supplementary Fig. 2a,band Supplementary Note 7 online), confirming publishedresults11,12. Notably, the territory clearly split into two regions:one close to the nuclear center, as in the repressed state, anotherone close to the SPB (Fig. 3b). Combined with the earlier observa-tion thatGAL1 mRNA is detected when the locus is preferentiallyperipheral11, this result supports a model where the on/offstates oftranscription correspond to twolocation states. Note that thebimodality disappeared in the absence of nucleolar alignment(Fig. 3c,d) and was therefore not apparent from previous staticanalyses. This result lends strongstatistical support to earlierdynamic data suggesting two spatial states forGAL1 (ref. 11).
An alternative possibility is that the switch in growth mediumaltered the cell cycle, which in turnmight affect locus positioning25.However the proportion of cells exhibiting peripheral versus centralGAL1 was similar in G1 and S phase (data not shown). Thenucleolar volume in the presence of galactose was roughly halvedcompared to that in glucose, while nuclear volumes were reducedonly slightly (Table 1). To examine whether relocalization resultedfrom medium-dependent alterations of nucleolar morphology, wemappedURA3 , whose expression is not affected by the switch fromglucose to galactose, and which occupied a territory similar toGAL1 in glucose (Figs. 1f and 3a). No significant change inURA3localization was visible in galactose (P1 ¼ 0.12; P2 ¼ 0.26),confirming the activation-dependent nature ofGAL1 territoryremodeling (Supplementary Fig. 4a,b online).We next examined GAL2, which is activated by the same
upstream activated sequences and binding factors asGAL110.Like GAL1, GAL2 is known to relocalize toward the peripheryupon activation10, suggesting that both genes are recruited tonuclear pores where they possibly share the same transcriptionmachinery3,26. Probability maps showed thatGAL2 was confinedto a small (0.86 mm3) territory between the nucleolus and thenuclear center when repressed (Fig. 3e). Upon galactose induction,the GAL2 territory slightly shifted toward the nuclear periphery(P1 o 10 4), consistent with earlier findings10 (Fig. 3f,m).However, whereas activatedGAL1 partly concentrated near thenuclear periphery and the SPB, activatedGAL2 accumulatednear the nucleolus. Thus,GAL2 was juxtaposed less frequentlywith the nuclear envelope thanGAL1 but more often withthe nucleolus. Furthermore, activatedGAL1 and GAL2 occupiedlargely distinct territories and are thus unlikely to be transcribed bythe same transcription machinery (Fig. 3b,f ).
Pmax = 2.1n = 1,913
a SPB
0
Pmax
Pmax = 1.14 Pmax = 0.93 Pmax = 0.44n = 1,395n = 657n = 2,663 Probability density (μm
–3)n = 2,000 Pmax = 0.3
n = 100
b c d e
n = 100 n = 100 n = 100 n = 100Pmax = 2.08 Pmax = 1.91
n = 1,395n = 657n = 2,663n = 1,913 n = 2,000
Pmax = 2.32 Pmax = 0.73 Pmax = 1.52
Pmax = 0.53 Pmax = 0.38 Pmax = 0.47 Pmax = 0.44 Pmax = 0.31
rDNA CEN Tel VIIL Random
g h i jf
l m n ok
Subsampled Subsampled Subsampled Subsampled Subsampled
Not aligned Not aligned Not aligned Not aligned Not aligned
Figure 2 | Validation of mapping method for different loci and the SPB. (a–o) Probability maps as inFigure 1f (a–d, simulated map ine), maps obtained usingonly 100 cells (f–j) and maps obtained without alignment of the nucleolar centroid (k–o), of the SPB protein Spc29 (a,f ,k), rDNA locus (b,g, l), centromericplasmid (c,h,m), subtelomere on the left arm of chromosome 7 (d,i ,n) and a simulated point randomly located within the nuclear volume (e, j,o). The color barindicates probability density from 0 toPmax. Scale bar, 1mm.
1034 | VOL.5 NO.12| DECEMBER 2008| NATURE METHODS
ARTICLES
Gene TerritoriesData taken fromA. B. Berger, G. G. Cabal, E. Fabre, T. Duong, H. Buc, U. Nehrbass, et al. High-resolution statistical mapping reveals gene territories in live yeast. Nat Methods, 5(12):1031–7, 2008
Transcriptional Regulatory Network Shapes the Genome Structure of Saccharomyces cerevisiae Songling Li and Dieter W. Heermann, Nucleus. 2013 May-Jun;4(3):216-28.
Heermann, Heidelberg University, 2015
Yeast
10-8
10-6
10-4
10-2
100
1 10 100 1000
Freq
uenc
y
Genomic Distance [kb]
slope = -1.55
ExperimentFitted Line
Data: Justin O‘Sullivan
10-8
10-6
10-4
10-2
100
1 10 100 1000
Freq
uenc
y
Genomic Distance [kb]
slope = -1.55
Gene Prox ModelFitted Line
Transcriptional Regulatory Network Shapes the Genome Structure of Saccharomyces cerevisiae Songling Li and Dieter W. Heermann, Nucleus. 2013 May-Jun;4(3):216-28.
Heermann, Heidelberg University, 2015
Human Chromosomes
Diffusion-Driven Looping Provides a Consistent Framework for Chromatin Organization Manfred Bohn and Dieter W. HeermannPLoS ONE 5(8): e12218. doi:10.1371/journal.pone.0012218 (2010)
Ansatz: Dynamic Loop Model + gene expression.
Human Chromosomes
60 70 80genomic position [Mb]
Chromosome 11
media
n tra
nscr
iptio
n 90 100 110 120 130
0
100
200
60 70 80
Heermann, Heidelberg University, 2015
Chromosome 11
genomic distance [Mb]
6
4
2
0 10 3020 50 60 7040
Chromosome 1
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0 10 3020genomic distance [Mb]
0
8
6 10
60 70 80genomic position [Mb]
Chromosome 11
med
ian tr
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90 100 110 120 1300
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200
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Chromosome 1
Chromosome 11
med
ian tr
ansc
riptio
n
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0
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3
0 1 2 3
genomic distance [Mb]
mea
n sq
. disp
lace
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t R
2 [
µm2]
0
4
6
0 2 4 6 8 10 12
genomic distance [Mb]
A
B
C
2
4 8
Chromosome 1
mea
n sq
. disp
lace
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µm2]
mea
n sq
. disp
lace
men
t R
2 [
µm2]
mea
n sq
. disp
lace
men
t R
2 [
µm2]
Human Chromosomes
R
Heermann, Heidelberg University, 2015
2d
d1
Diffusion-Driven Looping Provides a Consistent Framework for Chromatin Organization Manfred Bohn and Dieter W. HeermannPLoS ONE 5(8): e12218. doi:10.1371/journal.pone.0012218 (2010)
Human Chromosomes
Heermann, Heidelberg University, 2015
Chromosome 11
genomic distance [Mb]
6
4
2
0 10 3020 50 60 7040
Chromosome 1
0
2
4
0 10 3020genomic distance [Mb]
0
8
6 10
60 70 80genomic position [Mb]
Chromosome 11
med
ian tr
ansc
riptio
n
90 100 110 120 1300
100
200
050 60 70 80
genomic position [Mb]
Chromosome 1
Chromosome 11
med
ian tr
ansc
riptio
n
100
200
0
1
2
3
0 1 2 3
genomic distance [Mb]
mea
n sq
. disp
lace
men
t R
2 [
µm2]
0
4
6
0 2 4 6 8 10 12
genomic distance [Mb]
A
B
C
2
4 8
Chromosome 1
mea
n sq
. disp
lace
men
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2 [
µm2]
mea
n sq
. disp
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2 [
µm2]
mea
n sq
. disp
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The dynamic loop model very well explains differences between ridges and anti-ridges by different local looping probabilities.
0
2
4
6
8
10
0 2 4 6 8 10 12
mean
square
displacem
ent
⌦ R2↵
genomic distance [Mb]
chr 11 ridge
chr 11 anti-ridge
model in ridge region
model in anti-ridge region
Human Chromosomes
Heermann, Heidelberg University, 2015
Chromosome 11
genomic distance [Mb]
6
4
2
0 10 3020 50 60 7040
Chromosome 1
0
2
4
0 10 3020genomic distance [Mb]
0
8
6 10
60 70 80genomic position [Mb]
Chromosome 11
med
ian tr
ansc
riptio
n
90 100 110 120 1300
100
200
050 60 70 80
genomic position [Mb]
Chromosome 1
Chromosome 11
med
ian tr
ansc
riptio
n
100
200
0
1
2
3
0 1 2 3
genomic distance [Mb]
mea
n sq
. disp
lace
men
t R
2 [
µm2]
0
4
6
0 2 4 6 8 10 12
genomic distance [Mb]
A
B
C
2
4 8
Chromosome 1
mea
n sq
. disp
lace
men
t R
2 [
µm2]
mea
n sq
. disp
lace
men
t R
2 [
µm2]
mea
n sq
. disp
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The dynamic loop model also reproduces the experimental findings on the scale of a complete chromsome arm.
0
2
4
6
8
10
12
0 10 20 30 40 50 60 70 80
mean
square
displacem
ent
⌦ R2↵
genomic distance [Mb]
chr 11 long distance
RL model for long distance measurements
Human Chromosomes
Heermann, Heidelberg University, 2015
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
10 100
relative
abundanceh(l)
size of contact l
⇠ l�2.4
⇠ l�2
⇠ l�0.81
0 50 100 150 200 250
average loop number per conformation
2d
d1
Human Chromosomes
Heermann, Heidelberg University, 2015
Agrees with Sandra Goetze et. al. MOLECULAR AND CELLULAR BIOLOGY, June 2007, p. 4475–4487
Human Chromosomes
Heermann, Heidelberg University, 2015
We understand the folding pattern of chromosomes in interphase:
Folding is governed by loops on all scales mediated by proteins (transcptional hubs, ...)
Human Chromosomes
Heermann, Heidelberg University, 2015
T.Cremer and C. Cremer, Nature Reviews Genetics vol. 2, no. 4, pp. 292-301 (April, 2001)
Why do chromosomes not mix?
Human Chromosomes
Heermann, Heidelberg University, 2015
r
REPLUSIVE FORCE
radius of gyration
overlap between
chromosomes
Human Chromosomes
Heermann, Heidelberg University, 2015
A. Linear chains B. 45 loops per chain C. 92 loops per chain
0512
1024
bead number
0 512 1024
0512
1024
bead number
0 512 1024
0512
1024
bead number
0 512 1024
Human Chromosomes
Heermann, Heidelberg University, 2015
Human Chromosomes
Ansatz: Dynamic Loop Model + gene expression.
Chromosomes in Metaphase
Zhang Y, Heermann DW (2011) Loops Determine the Mechanical Properties of Mitotic Chromosomes. PLoS ONE 6(12): e29225. doi:10.1371/journal.pone.0029225
60 70 80genomic position [Mb]
Chromosome 11
media
n tra
nscr
iptio
n
90 100 110 120 1300
100
200
60 70 80
Heermann, Heidelberg University, 2015
Chromosomes in Metaphase
Zhang Y, Heermann DW (2011) Loops Determine the Mechanical Properties of Mitotic Chromosomes. PLoS ONE 6(12): e29225. doi:10.1371/journal.pone.0029225
Heermann, Heidelberg University, 2015
Chromosomes in Metaphase
Reversible and Irreversible Unfolding of Mitotic Newt Chromosomes by Applied ForceMichael Poirier, Sertac Eroglu Didier Chatenay, and John F. Marko Mol Biol Cell. 2000 January; 11(1): 269–276.
Heermann, Heidelberg University, 2015
Chromosomes in Metaphase
Structural changes of Xenopus sperm nuclei in mitotic egg extract; control sperm nuclei (a), decondensed sperm after 10 min of incubation in the extract (b), chromosomal structures (c–g) found after 30, 60, 90, 120, and 150 min, respectively.
Houchmandzadeh B , Dimitrov S J Cell Biol 1999;145:215-223
Heermann, Heidelberg University, 2015
Chromosomes in Metaphase
Heermann, Heidelberg University, 2015
Summary
Human Chromosomes
• FISH experiments on single chromosomes
• Bio-chemical experiments (4c, Hi-C)
• Partial genome staining experiments
• Whole genome staining experiments
• Mechanical data (metaphase)
Escherichia coli Yeast
Dynamic Loop Model + Biological Input:
Heermann, Heidelberg University, 2015
The Physical Architecture of the Genome: Common Principles in
Prokaryotes and Eukaryotes
The principles are loops, entropy and confinment
Unified Model
Heermann, Heidelberg University, 2015
Manfred BohnPhilipp Diesinger
Roel van DrielSandra GötzeMariliis Tark
Hans-Jörg JerabekSongling Li
Miriam FritscheChristoph CremerLindsay Shopland
Paul WigginsJörg Bewersdorf
Heidelberg Graduate School of
Mathematical and Computational
Methods for the Sciences
Spatio/Temporal Probabilistic Graphical
Models and Applications in Image Analysis
Center for Interdisciplinary Scientific Computing,
Heidelberg
Heermann, Heidelberg University, 2015
Thank you for your attention!
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