Long non-coding RNAs (lncRNAs) can regulate gene expression. Some are essential for organismal development and physiology and can contribute to diseases including cancer. Whilst most lncRNAs exhibit little sequence similarity, conservation of lncRNA transcription relative to neighbouring protein-coding genes suggests potential functional significance. Most positionally equivalent lncRNAs are uncharacterized and it remains unclear whether they exert similar roles in distant species. Here, we identified syntenic melanoma-associated lncRNAs predicted to be components of the MITF gene regulatory network in human melanoma, with positionally equivalent transcripts in zebrafish. We prioritized Differentiation Antagonizing Non-Protein Coding RNA (DANCR), a cancer-associated lncRNA critical for maintaining somatic progenitor cells in human models, for functional investigation. DANCR is a multi-exonic, cytoplasmically-enriched lncRNA transcribed from syntenic regions in the human and zebrafish genomes. MITF and c-MYC, key melanoma transcription factors, regulate human DANCR expression and melanoma patients with high DANCR display significantly decreased survival. DANCR is a melanoma oncogene that controls cancer-associated gene expression networks and promotes human melanoma cell proliferation and migration. Zebrafish dancr is dynamically expressed across multiple different cell types in the developing embryo, regulates genes involved in cell death, and is essential for embryonic development. Our work suggests that cancer-critical lncRNAs such as DANCR, expressed from similar regions in vertebrate genomes, may regulate related genes and processes involved in both embryonic development and tumorigenesis across species. ### Competing Interest Statement The authors have declared no competing interest.

Linear thresholding systems have been used as a model of neural activation and more recently proposed as a model of gene regulation. Here we exhibit linear thresholding systems whose dynamics produce surprisingly long cycles.

Linear thresholding systems have been used as a model of neural activation and have more recently been proposed as a model of gene activation. Deterministic linear thresholding systems can be turned into non-deterministic systems by the introduction of noise. Under mild conditions on the noise, we show that the deterministic model can be deduced from the probabilities of the non-deterministic model.

Central nervous system-expressed long non-coding RNAs (lncRNAs) are often located in the genome close to protein coding genes involved in transcriptional control. Such lncRNA-protein coding gene pairs are frequently temporally and spatially co-expressed in the nervous system and are predicted to act together to regulate neuronal development and function. Although some of these lncRNAs also bind and modulate the activity of the encoded transcription factors, the regulatory mechanisms controlling co-expression of neighbouring lncRNA-protein coding genes remain unclear. Here, we used high resolution NG Capture-C to map the cis-regulatory interaction landscape of the key neuro-developmental Paupar-Pax6 lncRNA-mRNA locus. The results define chromatin architecture changes associated with high Paupar-Pax6 expression in neurons and identify both promoter selective as well as shared cis-regulatory-promoter interactions involved in regulating Paupar-Pax6 co-expression. We discovered that the TCF7L2 transcription factor, a regulator of chromatin architecture and major effector of the Wnt signalling pathway, binds to a subset of these candidate cis-regulatory elements to coordinate Paupar and Pax6 co-expression. We describe distinct roles for Paupar in Pax6 expression control and show that the Paupar DNA locus contains a TCF7L2 bound transcriptional silencer whilst the Paupar transcript can act as an activator of Pax6. Our work provides important insights into the chromatin interactions, signalling pathways and transcription factors controlling co-expression of adjacent lncRNAs and protein coding genes in the brain.

AbstractCentral nervous system-expressed long non-coding RNAs (lncRNAs) are often located in the genome close to protein coding genes involved in transcriptional control. Such lncRNA-protein coding gene pairs are frequently temporally and spatially co-expressed in the nervous system and are predicted to act together to regulate neuronal development and function. Although some of these lncRNAs also bind and modulate the activity of the encoded transcription factors, the regulatory mechanisms controlling co-expression of neighbouring lncRNA-protein coding genes remain unclear. Here, we used high resolution NG Capture-C to map thecis-regulatory interaction landscape of the key neuro-developmentalPaupar-Pax6lncRNA-mRNA locus. The results defined chromatin architecture changes associated with highPaupar-Pax6expression in neurons and identified both promoter selective as well as sharedcis-regulatory interactions with thePauparandPax6promoters involved in regulatingPaupar-Pax6co-expression in neuronal cells. The TCF7L2 transcription factor, a major regulator of chromatin architecture and effector of the Wnt signalling pathway, binds to a subset of these candidatecis-regulatory elements to coordinatePauparandPax6co-expression. We identify a functional TCF7L2 boundcis-regulatory element within thePaupargene, suggesting that thePauparDNA locus itself regulatesPax6expression in addition to its previously described transcriptdependent modes of action. Our work provides important insights into the chromatin interactions, signalling pathways and transcription factors controlling co-expression of adjacent lncRNAs and protein coding genes in the brain.

The MITF and SOX10 transcription factors regulate the expression of genes important for melanoma proliferation, invasion and metastasis. Despite growing evidence of the contribution of long noncoding RNAs (lncRNAs) in cancer, including melanoma, their functions within MITF-SOX10 transcriptional programmes remain poorly investigated. Here we identify 245 candidate melanoma associated lncRNAs whose loci are co-occupied by MITF-SOX10 and that are enriched at active enhancer-like regions. Our work suggests that one of these, Disrupted In Renal Carcinoma 3 (DIRC3), may be a clinically important MITF-SOX10 regulated tumour suppressor. DIRC3 depletion in human melanoma cells leads to increased anchorage-independent growth, a hallmark of malignant transformation, whilst melanoma patients classified by low DIRC3 expression have decreased survival. DIRC3 is a nuclear lncRNA that activates expression of its neighbouring IGFBP5 tumour suppressor through modulating chromatin structure and suppressing SOX10 binding to putative regulatory elements within the DIRC3 locus. In turn, DIRC3 dependent regulation of IGFBP5 impacts the expression of genes involved in cancer associated processes and is needed for DIRC3 control of anchorage-independent growth. Our work indicates that lncRNA components of MITF-SOX10 networks are an important new class of melanoma regulators and candidate therapeutic targets that can act not only as downstream mediators of MITF-SOX10 function but as feedback regulators of MITF-SOX10 activity.

We study the HNN extension of $\mathbb{Z}^m$ given by the cubing endomorphism $g\mapsto g^3$, and prove that such groups have rational growth. To do so, we describe a method of computing the subgroup growth series of the horocyclic subgroup $\mathbb{Z}^m$ in this family of examples, prove that for all $m$ the group has rational growth. In the appendix, the subgroup growth series has been computed for all $m \leq 10$.

We study the HNN extension of $mathbb{Z}^m$ given by the cubing endomorphism $gmapsto g^3$, and prove that such groups have rational horocyclic growth. To do so, we describe a method of computing the subgroup growth series of the horocyclic subgroup $mathbb{Z}^m$ in this family of examples, prove that for all $m$ the subgroup has rational growth. In the appendix, the subgroup growth series has been computed for all $m leq 10$.

We show that there exists an algorithm to decide any single equation in the Heisenberg group in finite time. The method works for all two-step nilpotent groups with rank-one commutator, which includes the higher Heisenberg groups. We also prove that the decision problem for systems of equations is unsolvable in all non-abelian free nilpotent groups.

A group presentation is said to have rational growth if the generating series associated to its growth function represents a rational function. A long-standing open question asks whether the Heisenberg group has rational growth for all finite generating sets, and we settle this question affirmatively. We also establish almost-convexity for all finite generating sets. Previously, both of these properties were known to hold for hyperbolic groups and virtually abelian groups, and there were no further examples in either case. Our main method is a close description of the relationship between word metrics and associated Carnot-Caratheodory Finsler metrics on the ambient Lie group. We provide (non-regular) languages in any word metric that suffice to represent all group elements.

Previous analysis of Epstein-Barr virus (EBV) persistent infection has involved biological and immunological studies to identify and quantify infected cell populations and the immune response to them. This led to a biological model whereby EBV infects and activates naive B-cells, which then transit through the germinal center to become resting memory B-cells where the virus resides quiescently. Occasionally the virus reactivates from these memory cells to produce infectious virions. Some of this virus infects new naive B-cells, completing a cycle of infection. What has been lacking is an understanding of the dynamic interactions between these components and how their regulation by the immune response produces the observed pattern of viral persistence. We have recently provided a mathematical analysis of a pathogen which, like EBV, has a cycle of infected stages. In this paper we have developed biologically credible values for all of the parameters governing this model and show that with these values, it successfully recapitulates persistent EBV infection with remarkable accuracy. This includes correctly predicting the observed patterns of cytotoxic T-cell regulation (which and by how much each infected population is regulated by the immune response) and the size of the infected germinal center and memory populations. Furthermore, we find that viral quiescence in the memory compartment dictates the pattern of regulation but is not required for persistence; it is the cycle of infection that explains persistence and provides the stability that allows EBV to persist at extremely low levels. This shifts the focus away from a single infected stage, the memory B-cell, to the whole cycle of infection. We conclude that the mathematical description of the biological model of EBV persistence provides a sound basis for quantitative analysis of viral persistence and provides testable predictions about the nature of EBV-associated diseases and how to curb or prevent them.

Epstein-Barr virus (EBV) maintains a lifelong infection. According to the germinal center model (GCM), latently infected B cells transit the germinal center (GC) to become resting memory cells. Here, the virus resides quiescently occasionally reactivating to infect new B cells, completing the cycle of infection. The GCM remains the only model that explains EBV biology and the pathogenesis of lymphoma. Recent work suggests modifications to the model notably that the virus contributes only modestly to the GC process and predictions from mathematical models that quiescence within memory B cells shapes the overall structure of viral infection but is not essential for persistence. Rather, it is the cycle of infection which allows viral persistence at the very low levels observed.

In previous work, we studied host response to a pathogen which uses a cycle of immunologically distinct stages to establish and maintain infection. We showed that for generic parameter values, the system has a unique biologically meaningful stable fixed point. That paper used a simplified model of T-cell activation, making proliferation depend linearly on antigen-T-cell encounters. Here we generalize the way in which T-cell proliferation depends on the sizes of the antigenic populations. In particular, we allow this response to become saturated at high levels of antigen. As a result, we show that this family of generalized models shares the same steady-state behavior properties with the simpler model contemplated in our previous work.

In this paper we consider the question of sensor network coverage for a two-dimensional domain. We seek to compute the probability that a set of sensors fails to cover given only non-metric, local (who is talking to whom) information and a probability distribution of failure of each node. This builds on the work of de Silva and Ghrist who analyzed this problem in the deterministic situation. We first show that it is part of a slightly larger class of problems which is #P-hard, and thus fast algorithms likely do not exist unless P = NP. The question of whether the specific problem is, in fact, #P-hard remains open. We then give a deterministic algorithm which is feasible in the case of a small set of sensors, and give a dynamic algorithm for an arbitrary set of sensors failing over time which utilizes a new criterion for coverage to give an early warning of potential failure. These algorithms build on the theory of topological persistence.

In this paper, we introduce a new class of epidemics on networks which we call SI(S/I). SI(S/I) networks differ from SIS networks in allowing an infected individual to become reinfected without first passing to the susceptible state. We use a covering-graph construction to compare SIR, SIS, and SI(S/I) networks. Like the SIR networks that cover them, SI(S/I) networks exhibit infection probabilities that are monotone with respect to both transmission probabilities and the initial set of infectives. The same covering-graph construction allows us to characterize the recurrent states in an SIS or SI(S/I) network with reinfection.

It is the purpose of this article to review results that have long been known to communications network engineers and have direct application to epidemiology on networks. A common approach in epidemiology is to study the transmission of a disease in a population where each individual is initially susceptible (S), may become infective (I) and then removed or recovered (R) and plays no further epidemiological role. Much of the recent work gives explicit consideration to the network of social interactions or disease-transmitting contacts and attendant probability of transmission for each interacting pair. The state of such a network is an assignment of the values {S,I,R} to its members. Given such a network, an initial state and a particular susceptible individual, we would like to compute their probability of becoming infected in the course of an epidemic. It turns out that this and related problems are NP-hard. In particular, it belongs in a class of problems for which no efficient algorithms for their solution are known. Moreover, finding an efficient algorithm for the solution of any problem in this class would entail a major breakthrough in computer science.

We model the immune surveillance of a pathogen which passes through n immunologically distinct stages. The biological parameters of this system induce a partial order on the stages, and this, in turn, determines which stages will be subject to immune regulation. This corresponds to the system’s unique asymptotically stable fixed point.

We have performed the first extensive profiling of Epstein-Barr virus (EBV) miRNAs on in vivo derived normal and neoplastic infected tissues. We describe a unique pattern of viral miRNA expression by normal infected cells in vivo expressing restricted viral latency programs (germinal center: Latency II and memory B: Latency I/0). This includes the complete absence of 15 of the 34 miRNAs profiled. These consist of 12 BART miRNAs (including approximately half of Cluster 2) and 3 of the 4 BHRF1 miRNAs. All but 2 of these absent miRNAs become expressed during EBV driven growth (Latency III). Furthermore, EBV driven growth is accompanied by a 5-10 fold down regulation in the level of the BART miRNAs expressed in germinal center and memory B cells. Therefore, Latency III also expresses a unique pattern of viral miRNAs. We refer to the miRNAs that are specifically expressed in EBV driven growth as the Latency III associated miRNAs. In EBV associated tumors that employ Latency I or II (Burkitt's lymphoma, Hodgkin's disease, nasopharyngeal carcinoma and gastric carcinoma), the Latency III associated BART but not BHRF1 miRNAs are up regulated. Thus BART miRNA expression is deregulated in the EBV associated tumors. This is the first demonstration that Latency III specific genes (the Latency III associated BARTs) can be expressed in these tumors. The EBV associated tumors demonstrate very similar patterns of miRNA expression yet were readily distinguished when the expression data were analyzed either by heat-map/clustering or principal component analysis. Systematic analysis revealed that the information distinguishing the tumor types was redundant and distributed across all the miRNAs. This resembles "secret sharing" algorithms where information can be distributed among a large number of recipients in such a way that any combination of a small number of recipients is able to understand the message. Biologically, this may be a consequence of functional redundancy between the miRNAs.

Following the definition of asynchronous automatic structures in [3], we define non-deterministic asynchronous automatic structures and characterize these in terms of the asynchronous fellow traveller property. We show that any group with a non-deterministic asynchronous automatic structure has an asynchronous automatic structure. Non-deterministic asynchronous automatic structures are a labor saving method of showing that a group has an asynchronous automatic structure. They also allow one to define an equivalence relation on the class of non-deterministic asynchronous automatic structures which descends to the subclasses of deterministic asynchronous automatic structures and synchronous automatic structures.