Correlation in complex networks follows a linear relation between the degree of a node and the total degrees of its neighbors for six different classes of real networks. This general linear relation is an extension of the Aboav-Weaire law in two-dimensional cellular structures and provides a simple and different perspective on the correlation in complex networks, which is complementary to an existing description using Pearson correlation coefficients and a power law fit. Analytical expression for this linear relation for three standard models of complex networks: the Erdos-Renyi, Watts-Strogatz, and Barabasi-Albert networks is provided. The slope and intercept of this linear relation are described by a single parameter a together with the first and second moment of the degree distribution of the network. The assortivity of the network can be related to the sign of the intercept. Complex networks are a convenient model for studying the topological and structural property of complex systems ͓1,2͔. It consists of nodes which interact among themselves via their connections. Mathematical quantities such as the degree distribution, cluster coefficient, and the average shortest path length are the standard properties in a preliminary characterization of networks. In this paper, we focus on the neighbor connectivity that relates to the degree correlation among the nodes. By extending the Aboav-Weaire Law ͓3,4͔, which was well studied for two-dimensional cellular patterns such as soap froth ͓5,6͔, we find that it is also a good measure to describe the correlation in a wide variety of real and artificial networks in higher dimensions. The complex networks that we have checked are the neural networks ͓7͔, food webs ͓8͔, word co-occurrence ͓9,10͔, scientist collaboration ͓11͔, internet ͓12͔, and yeast protein interaction ͓13͔.We also report here the Aboav's parameters for several standard models: the Erdos-Renyi ͓14͔, Watts-Strogatz ͓15͔, and Barabasi-Albert networks ͓16͔. We find that the results on the assortivity of the network using the Aboav-Wearie law are consistent with conclusions based on the analysis of the Pearson correlation coefficient.Originally, the Aboav-Weaire law was discovered from the empirical analysis of two-dimensional cellular structures in metal grains and later extended to a variety of cellular structures in two and three dimensions. The findings of Aboav provide a linear relation between the total degrees of nearest neighbor nM͑n͒ with the degree of the cell nM͑n͒ =5n + 8, where M͑n͒ is the mean of number of edges of neighboring cells surrounding a cell with n edges ͓3͔. Weaire generalizes Aboav's observation and restates this observation in terms of the variance 2 of the degree distribution of the cellular network ͓4͔:withThis form for the expression of M͑n͒ is usually tested empirically by a plot of nM͑n͒ vs n, which should be linear with slope A and intercept B. As the Aboav-Weaire law can be understood as a statement on the topological correlation of the cellular network, we attempt to general...

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CO 2 -derived methanol is an attractive raw material for biobased production of value-added chemicals. Here, we investigated the native methylotrophButyribacterium methylotrophicum, which could synchronously assimilate methanol and CO 2 to butyric acid anaerobically. Supplementation with an approximate amount of bicarbonate could improve methanol metabolism of B. methylotrophicum, and 2.04 g/L butyric acid was finally obtained from 100 mM methanol and 20 mM bicarbonate. The genes involved in methanol metabolism were further identified through homologous alignment and transcriptome analysis. The methyltransferase cluster along with genes of the carbonyl branch of the Wood−Ljungdahl pathway (WLP) was found to be transcriptionally activated for the assimilation of methanol and CO 2 . To engineer B. methylotrophicum, an efficient electrotransformation protocol and several functional promoters were subsequently developed. Following a systematic investigation of various parameters, the electrotransformation efficiency was increased to 3.2 × 10 3 transformants/μg DNA. The activities of four heterologous promoters including P thl , P araE , P ptb , and P adc were comparatively determined. With these genetic toolkits, transformants overexpressing genes associated with methyltransferase system or butyric acid synthesis were obtained, where methanol consumption was increased by 16.9 and 14%, and butyric acid production was increased by 13.8 and 28.6%, respectively, in methanol and CO 2 medium. These results exhibit the great potential of B. methylotrophicum as a chassis for C1 bioconversion.

BackgroundMethanol has attracted increased attention as a non-food alternative carbon source to sugar for biological production of chemicals and fuels. Moreover, the high degree of reduction of methanol offers some advantages in increasing the production yields of NAD(P)H-dependent metabolites. Here, we demonstrate an example of methanol bioconversion with the aim of improving production of NAD(P)H-dependent chemicals in synthetic methylotrophic Escherichia coli.ResultsA synthetic methylotrophic E. coli was engineered with a nicotinamide adenine dinucleotide (NAD+)-dependent methanol dehydrogenase (MDH) and ribulose monophosphate (RuMP) pathway. Regarding the limited MDH activity, the role of activator proteins in vivo was investigated, and the NudF protein was identified capable of improving MDH activity and triggering increased methanol metabolism. Using 13C-methanol-labeling experiments, we confirmed methanol assimilation in the methylotrophic E. coli. A cycling RuMP pathway for methanol assimilation was also demonstrated by detecting multiple labeled carbons for several compounds. Finally, using the NAD(P)H-dependent metabolite lysine as a test, the potential of methanol bioconversion to generate value-added metabolites was determined. To further characterize the benefit of methanol as the carbon source, extra NADH from methanol oxidation was engineered to generate NADPH to improve lysine biosynthesis by expression of the POS5 gene from Saccharomyces cerevisiae, which resulted in a twofold improvement of lysine production. Moreover, this new sink further pulled upstream methanol utilization.ConclusionThrough engineering methanol metabolism, lysine biosynthesis, and NADPH regeneration pathway from NADH, the bioconversion of methanol to improve chemical synthesis was successfully achieved in methylotrophic E. coli.Electronic supplementary materialThe online version of this article (10.1186/s13068-019-1356-4) contains supplementary material, which is available to authorized users.