DeDoc2 Identifies and Characterizes the Hierarchy and Dynamics of Chromatin TAD-Like Domains in the Single Cells.

Topologically associating domains (TADs) are functional chromatin units with hierarchical structure. However, the existence, prevalence, and dynamics of such hierarchy in single cells remain unexplored. Here, a new generation TAD-like domain (TLD) detection algorithm, named deDoc2, to decode the hierarchy of TLDs in single cells, is reported. With dynamic programming, deDoc2 seeks genome partitions with global minimal structure entropy for both whole and local contact matrix. Notably, deDoc2 outperforms state-of-the-art tools and is one of only two tools able to identify the hierarchy of TLDs in single cells. By applying deDoc2, it is showed that the hierarchy of TLDs in single cells is highly dynamic during cell cycle, as well as among human brain cortex cells, and that it is associated with cellular identity and functions. Thus, the results demonstrate the abundance of information potentially encoded by TLD hierarchy for functional regulation. The deDoc2 can be freely accessed at https://github.com/zengguangjie/deDoc2.

DeTOKI identifies and characterizes the dynamics of chromatin TAD-like domains in a single cell.

Topologically associating domains (TAD) are a key structure of the 3D mammalian genomes. However, the prevalence and dynamics of TAD-like domains in single cells remain elusive. Here we develop a new algorithm, named deTOKI, to decode TAD-like domains with single-cell Hi-C data. By non-negative matrix factorization, deTOKI seeks regions that insulate the genome into blocks with minimal chance of clustering. deTOKI outperforms competing tools and reliably identifies TAD-like domains in single cells. Finally, we find that TAD-like domains are not only prevalent, but also subject to tight regulation in single cells.

The idemetric property: when most distances are (almost) the same.

We introduce the idemetric property, which formalizes the idea that most nodes in a graph have similar distances between them, and which turns out to be quite standard amongst small-world network models. Modulo reasonable sparsity assumptions, we are then able to show that a strong form of idemetricity is actually equivalent to a very weak expander condition (PUMP). This provides a direct way of providing short proofs that small-world network models such as the Watts-Strogatz model are strongly idemetric (for a wide range of parameters), and also provides further evidence that being idemetric is a common property. We then consider how satisfaction of the idemetric property is relevant to algorithm design. For idemetric graphs, we observe, for example, that a single breadth-first search provides a solution to the all-pairs shortest paths problem, so long as one is prepared to accept paths which are of stretch close to 2 with high probability. Since we are able to show that Kleinberg's model is idemetric, these results contrast nicely with the well known negative results of Kleinberg concerning efficient decentralized algorithms for finding short paths: for precisely the same model as Kleinberg's negative results hold, we are able to show that very efficient (and decentralized) algorithms exist if one allows for reasonable preprocessing. For deterministic distributed routing algorithms we are also able to obtain results proving that less routing information is required for idemetric graphs than in the worst case in order to achieve stretch less than 3 with high probability: while Ω(n 2) routing information is required in the worst case for stretch strictly less than 3 on almost all pairs, for idemetric graphs the total routing information required is O(nlog(n)).

Decoding topologically associating domains with ultra-low resolution Hi-C data by graph structural entropy.

Submegabase-size topologically associating domains (TAD) have been observed in high-throughput chromatin interaction data (Hi-C). However, accurate detection of TADs depends on ultra-deep sequencing and sophisticated normalization procedures. Here we propose a fast and normalization-free method to decode the domains of chromosomes (deDoc) that utilizes structural information theory. By treating Hi-C contact matrix as a representation of a graph, deDoc partitions the graph into segments with minimal structural entropy. We show that structural entropy can also be used to determine the proper bin size of the Hi-C data. By applying deDoc to pooled Hi-C data from 10 single cells, we detect megabase-size TAD-like domains. This result implies that the modular structure of the genome spatial organization may be fundamental to even a small cohort of single cells. Our algorithms may facilitate systematic investigations of chromosomal domains on a larger scale than hitherto have been possible.

Resistance and Security Index of Networks: Structural Information Perspective of Network Security.

Recently, Li and Pan defined the metric of the K-dimensional structure entropy of a structured noisy dataset G to be the information that controls the formation of the K-dimensional structure of G that is evolved by the rules, order and laws of G, excluding the random variations that occur in G. Here, we propose the notion of resistance of networks based on the one- and two-dimensional structural information of graphs. Given a graph G, we define the resistance of G, written , as the greatest overall number of bits required to determine the code of the module that is accessible via random walks with stationary distribution in G, from which the random walks cannot escape. We show that the resistance of networks follows the resistance law of networks, that is, for a network G, the resistance of G is , where and are the one- and two-dimensional structure entropies of G, respectively. Based on the resistance law, we define the security index of a network G to be the normalised resistance of G, that is, . We show that the resistance and security index are both well-defined measures for the security of the networks.

Dynamics, morphogenesis and convergence of evolutionary quantum Prisoner's Dilemma games on networks.

The authors proposed a quantum Prisoner's Dilemma (PD) game as a natural extension of the classic PD game to resolve the dilemma. Here, we establish a new Nash equilibrium principle of the game, propose the notion of convergence and discover the convergence and phase-transition phenomena of the evolutionary games on networks. We investigate the many-body extension of the game or evolutionary games in networks. For homogeneous networks, we show that entanglement guarantees a quick convergence of super cooperation, that there is a phase transition from the convergence of defection to the convergence of super cooperation, and that the threshold for the phase transitions is principally determined by the Nash equilibrium principle of the game, with an accompanying perturbation by the variations of structures of networks. For heterogeneous networks, we show that the equilibrium frequencies of super-cooperators are divergent, that entanglement guarantees emergence of super-cooperation and that there is a phase transition of the emergence with the threshold determined by the Nash equilibrium principle, accompanied by a perturbation by the variations of structures of networks. Our results explore systematically, for the first time, the dynamics, morphogenesis and convergence of evolutionary games in interacting and competing systems.

Three-Dimensional Gene Map of Cancer Cell Types: Structural Entropy Minimisation Principle for Defining Tumour Subtypes.

In this study, we propose a method for constructing cell sample networks from gene expression profiles, and a structural entropy minimisation principle for detecting natural structure of networks and for identifying cancer cell subtypes. Our method establishes a three-dimensional gene map of cancer cell types and subtypes. The identified subtypes are defined by a unique gene expression pattern, and a three-dimensional gene map is established by defining the unique gene expression pattern for each identified subtype for cancers, including acute leukaemia, lymphoma, multi-tissue, lung cancer and healthy tissue. Our three-dimensional gene map demonstrates that a true tumour type may be divided into subtypes, each defined by a unique gene expression pattern. Clinical data analyses demonstrate that most cell samples of an identified subtype share similar survival times, survival indicators and International Prognostic Index (IPI) scores and indicate that distinct subtypes identified by our algorithms exhibit different overall survival times, survival ratios and IPI scores. Our three-dimensional gene map establishes a high-definition, one-to-one map between the biologically and medically meaningful tumour subtypes and the gene expression patterns, and identifies remarkable cells that form singleton submodules.

Homophyly/Kinship Model: Naturally Evolving Networks.

It has been a challenge to understand the formation and roles of social groups or natural communities in the evolution of species, societies and real world networks. Here, we propose the hypothesis that homophyly/kinship is the intrinsic mechanism of natural communities, introduce the notion of the affinity exponent and propose the homophyly/kinship model of networks. We demonstrate that the networks of our model satisfy a number of topological, probabilistic and combinatorial properties and, in particular, that the robustness and stability of natural communities increase as the affinity exponent increases and that the reciprocity of the networks in our model decreases as the affinity exponent increases. We show that both homophyly/kinship and reciprocity are essential to the emergence of cooperation in evolutionary games and that the homophyly/kinship and reciprocity determined by the appropriate affinity exponent guarantee the emergence of cooperation in evolutionary games, verifying Darwin's proposal that kinship and reciprocity are the means of individual fitness. We propose the new principle of structure entropy minimisation for detecting natural communities of networks and verify the functional module property and characteristic properties by a healthy tissue cell network, a citation network, some metabolic networks and a protein interaction network.

Emergence of super cooperation of prisoner's dilemma games on scale-free networks.

Recently, the authors proposed a quantum prisoner's dilemma game based on the spatial game of Nowak and May, and showed that the game can be played classically. By using this idea, we proposed three generalized prisoner's dilemma (GPD, for short) games based on the weak Prisoner's dilemma game, the full prisoner's dilemma game and the normalized Prisoner's dilemma game, written by GPDW, GPDF and GPDN respectively. Our games consist of two players, each of which has three strategies: cooperator (C), defector (D) and super cooperator (denoted by Q), and have a parameter γ to measure the entangled relationship between the two players. We found that our generalised prisoner's dilemma games have new Nash equilibrium principles, that entanglement is the principle of emergence and convergence (i.e., guaranteed emergence) of super cooperation in evolutions of our generalised prisoner's dilemma games on scale-free networks, that entanglement provides a threshold for a phase transition of super cooperation in evolutions of our generalised prisoner's dilemma games on scale-free networks, that the role of heterogeneity of the scale-free networks in cooperations and super cooperations is very limited, and that well-defined structures of scale-free networks allow coexistence of cooperators and super cooperators in the evolutions of the weak version of our generalised prisoner's dilemma games.

Entanglement guarantees emergence of cooperation in quantum prisoner's dilemma games on networks.

It was known that cooperation of evolutionary prisoner's dilemma games fails to emerge in homogenous networks such as random graphs. Here we proposed a quantum prisoner's dilemma game. The game consists of two players, in which each player has three choices of strategy: cooperator (C), defector (D) and super cooperator (denoted by Q). We found that quantum entanglement guarantees emergence of a new cooperation, the super cooperation of the quantum prisoner's dilemma games, and that entanglement is the mechanism of guaranteed emergence of cooperation of evolutionary prisoner's dilemma games on networks. We showed that for a game with temptation b, there exists a threshold arccos √b/b for a measurement of entanglement, beyond which, (super) cooperation of evolutionary quantum prisoner's dilemma games is guaranteed to quickly emerge, giving rise to stochastic convergence of the cooperations, that if the entanglement degree γ is less than the threshold arccos √b/b, then the equilibrium frequency of cooperations of the games is positively correlated to the entanglement degree γ, and that if γ is less than arccos √b/b and b is beyond some boundary, then the equilibrium frequency of cooperations of the games on random graphs decreases as the average degree of the graphs increases.