Colocalization of coregulated genes in human chromosome 19
Spatial macrodomains: comparison with data based on HiC maps
Chromosome entanglement, regulatory network properties and gene colocalizability
- A random-walk-like chromosome arrangement as shown and described in Fig. 4A.
- A mitotic-like spatial arrangement but with randomized gene pairings, see Fig. 4B. The chromosome spatial configuration is the same as in Fig. 1B, but the native 1,487 coregulatory pairings between the 412 selected genes have been randomly reshuffled. The number of pairings that each selected gene takes part to in the reshuffled network is the same as the native coregulatory network.
- A mitotic-like spatial arrangement but with randomized gene positions, see Fig. 4C. As in case 2 above, the chromosome spatial configuration is again the same as inFig. 1B, but the positions of the 412 genes involved in the native coregulatory network are randomly assigned along the chromosome (except for the centromeric region). The repositioned genes inherit the native coregulatory pairings.
Summary and conclusions
Materials and Methods
Coregulated gene pairs on Chr19
where  runs over the three coarse-grained expression levels for probe set . In Eq. 1, is the joint probability that, in a given experiment, the expression levels and are respectively observed for probe sets and , while the quantities and are the probabilities to observe expression level  for probe set  (marginal probabilities). The MI thus provides a statistically-founded measure of how the gene expression pattern for gene is predictable assuming the knowledge of another pattern (or, vice versa).
- Randomized pairings. The 1,487 native pairings between the considered set of probe sets were randomly reshuffled while preserving the native number of pairings for each gene. This alternative set of probe set pairs is obtained by applying the iterative randomization method described in ref. . The asymptotic fraction of randomized gene pairs matching the native ones is .
- Randomized positions. The set of native probe sets are randomly repositioned along the contour length of the chromosome, but the target gene pairings are kept the same as the native ones. Gene repositioning in the centromeric region (which is mostly void of genes) was disallowed.
The chromosome polymer model
where and run over the bead indices and the three terms correspond to the FENE chain-connectivity interaction , the bending energy, and the repulsive pairwise Lennard-Jones interaction. The three energy terms are parametrized as in previous studies of coarse-grained chromosomes ,. Specifically,
where is the distance of the centers of beads and , , and the thermal energy equals . ensures the connectivity of the chain, i.e. the centers of two consecutive beads must be at a distance about equal to their diameter. The bending energy has instead the standard Kratky-Porod form (discrete worm-like chain):
where . ensures that the chain of beads bends over contour lengths the size of the persistence length to model the experimental rigidity of the chromatin fiber .
This repulsive interaction controls the inter-chain excluded volume too:
where is the number of chains in solution and the index runs over the beads in chain . ensures that any two regions along the same chain or on different chains cannot pass through each other. In this way, intra- and inter-chain topology is preserved.
Steered Molecular Dynamics protocol.
where is the distance of the centers of mass of the chromosome stretches (mapped onto the discrete beads using the Affymetrix annotation table (http://www.affymetrix.com)) covered by the two genes. The stiffness of the harmonic constraint was controlled by the time-dependent parameter . The latter is ramped linearly in time from the initial value up to the value . The total duration of the steered dynamics was . This protocol favours the progressive reduction of the width of the distribution of probe set distances from the initially generous value of (see 5) down to . The simultaneous application of the constraints to each of the six chromosomes, which clearly are not necessarily compatible a priori, was implemented using the PLUMED plugin for LAMMPS . The protocol is sufficiently mild that no crossings of the chains should occur. This was checked by running the steering protocol on a circularized variants of the mitotic conformation shown in Fig. 1A, and checking that the initially unknotted topological state is maintained .
- The clustering coefficient, , is used to characterize connectivity properties of graphs. In the present case the graph of coregulation of pairs of genes. Each gene is represented by a node in the graph. Pairs of coregulated genes are represented by a link connecting the two corresponding nodes.
where is the number of neighbours of while is the number of distinct links between the neighbours of node . The clustering coefficient per node, , is clearly defined only for nodes with at least two neighbours. The clustering coefficient of the whole graph is obtained by averaging over all nodes with . The clustering coefficient provides a measure of the incidence of cliques of size (“triangular linkages”) in the graph.
Identification of spatial macrodomains
where is the contact map and , which is the domain representative, is the element belonging to the – interval for which is minimum. Consistently with intuition, the dissimilarity score, , takes on small or large values if respectively many or few domain members are in contact with the representative. For a given number of domains, the optimal domain partitioning is the one that minimizes the sum of the scores for the domains.
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