Consider the random intervals I n = ω n + (0, n) (modulo 1) with their left points ω n independently and uniformly distributed over the interval [0, 1) = R/Z and with their lengths decreasing to zero. We prove that the Hausdorff dimension of the set lim n I n of points covered infinitely often is almost surely equal to 1/α when n = a/n α for some a > 0 and α > 1.
Respiratory syncytial virus (RSV) is the most important cause of lower respiratory tract infection in infants and young children. The vaccine-enhanced disease (VED) has greatly hindered the development of an RSV vaccine. Currently, there are no licensed vaccines for RSV. In this study, immunization of mice with hepatitis B virus core particles containing a conserved region of the G protein (HBc-tG) combined with interleukin-35 (IL-35) elicited a Th1-biased response and a high frequency of regulatory T (Treg) cells and increased the levels of IL-10, transforming growth factor β, and IL-35 production. Importantly, immunization with HBc-tG together with IL-35 protected mice against RSV infection without vaccine-enhanced immunopathology. To explore the mechanism of how IL-35 reduces lung inflammation at the gene expression level, transcription profiles were obtained from lung tissues of immunized mice after RSV infection by the Illumina sequencing technique and further analyzed by a systems biology method. In total, 2,644 differentially expressed genes (DEGs) were identified. Twelve high-influence modules (HIMs) were selected from these DEGs on the basis of the protein-protein interaction network. A detailed analysis of HIM10, involved in the immune response network, revealed that Il10 plays a key role in regulating the host response. The selected DEGs were consistently confirmed by quantitative real-time PCR (qRT-PCR). Our results demonstrate that IL-35 inhibits vaccine-enhanced immunopathology after RSV infection and has potential for development in novel therapeutic and prophylactic strategies. IMPORTANCE In the past few decades, respiratory syncytial virus (RSV) has still been a major health concern worldwide. The vaccine-enhance disease (VED) has hindered RSV vaccine development. A truncated hepatitis B virus core protein vaccine containing the conserved region (amino acids 144 to 204) of the RSV G protein (HBc-tG) had previously been shown to induce effective immune responses and confer protection against RSV infection in mice but to also lead to VED. In this study, we investigated the effect of IL-35 on the host response and immunopathology following RSV infection in vaccinated mice. Our results indicate that HBc-tG together with IL-35 elicited a balanced immune response and protected mice against RSV infection without vaccine-enhanced immunopathology. Applying a systems biology method, we identified Il10 to be the key regulator in reducing the excessive lung inflammation. Our study provides new insight into the function of IL-35 and its regulatory mechanism of VED at the network level.
Reconstructing large-scale gene regulatory networks (GRNs) is a challenging problem in the field of computational biology. Various methods for inferring GRNs have been developed, but they fail to accurately infer GRNs with a large number of genes. Additionally, the existing evaluation indexes for evaluating the constructed networks have obvious disadvantages because GRNs in most biological systems are sparse. In this paper, we develop a new method for inferring GRNs based on randomized singular value decomposition (RSVD) and ordinary differential equation (ODE)-based optimization, denoted as IGRSVD, from large-scale time series data with noise. The three major contributions of this paper are as follows. First, the IGRSVD algorithm uses the RSVD to handle the noise and reduce the original large-scale data into small-scale problems. Second, we propose two new evaluated indexes, the expected value accuracy (EVA) and the expected value error (EVE), to evaluate the performance of inferred networks by considering the sparse features in the network. Finally, the proposed IGRSVD algorithm is compared with the existing SVD algorithm and PCA_CMI algorithm using four subsets from E. coli and datasets from DREAM challenge. The experimental results demonstrate that the IGRSVD algorithm is effective and more suitable for reconstructing large-scale networks.
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