The lack of holes due to the high activation energy of acceptors greatly limits the performance and practical applicability of wide-bandgap semiconductor devices. Here, we study a novel effect which allows homojunction semiconductor devices, such as p-i-n diodes, to operate well above the limit imposed by doping of the p-type material. Using a rigorous numerical approach, we show that the density of injected holes can exceed the density of holes in the p-type injection layer by up to four orders of magnitude depending on the semiconductor material, dopant, and temperature, which gives the possibility to significantly overcome the doping problem. We present a clear physical explanation of this unexpected feature of wide-bandgap semiconductor p-i-n diodes and closely examine it in 4H-SiC, 3C-SiC, AlN, and ZnS structures. The predicted effect can be exploited to develop bright-light-emitting devices, especially electrically driven nonclassical light sources based on color centers in SiC, AlN, ZnO, and other wide-bandgap semiconductors.Electrically driven light sources are essential in a wide range of applications, from indication and display technologies to high-speed data communication and quantum information processing. Wide-bandgap semiconductors promise to advance solid-state lighting by delivering novel light sources. However, electrical pumping of these devices is still a challenging problem. Many wide-bandgap semiconductor materials, such as SiC, GaN, AlN, ZnS, and Ga The possibility to create a high density of nonequilibrium charge carriers in the active region of semiconductor optoelectronic devices is essential for a wide range of optoelectronic applications, from light emitting diodes (LEDs)  and injeThe traditional solution to this problem is to use the superinjection effect in double heterostructures ,18,19. AFor decades, it was believed that the superinjection effect was a unique feature of semiconductor heterostructures, while the maximum density of injected electrons and holes in homojunction structures could not exceed the free carrier densities in the n-type and p-type injection layers, respectively . HoweverkBT300K, which limits the density of holes in the p-type doped material to less than 1010 cm−3. However, in this work, we show that at high forward bias voltages, it is possible to significantly overcome this doping limit using the superinjection effect in homojunction p-i-n diodes and inject far more holes into the i-region of the p-i-n diode than the doping of the p-type layer provides.In this work, using a rigorous numerical approach, we investigate for the first time the possibility to observe and exploit the superinjection effect in homojunction diodes based on wide-bandgap semiconductor materials beyond diamond. We discuss in detail how to improve the efficiency of hole injection in SiC p-i-n diodes, which recently demonstrated great promise for quantum applications as single-photon sources ,24,25. INd = 1018 cm−3. The activation energy of donors is 0.06 eV [ηn = 5%, which provides an electron density of neqn = 5.5 × 1017 cm−3. The p-type injection layer is doped with boron. The acceptor compensation ratio is also equal to ηp = 5% [Na = 1018 cm−3 and their activation energy equals 0.32 eV [peqp = 4.3 × 1014 cm−3 in equilibrium. The electron and hole mobilities in the p-, i-, and n-type regions are calculated using the theory from [ 0.06 eV . The nit ηp = 5% . The con 0.32 eV , which pory from . The sizory from . Both meJ = 6 A/cm2 (V > 2.8 V), the maximum densities of both electrons and holes in the i-region of the p-i-n diode are found near the i–n junction, which is counterintuitive, since one expects to find the maximum density of electrons in the proximity of the i–n junction and the maximum density of holes in the proximity of the p–i junction, especially in the presence of recombination. Even more surprising is the fact that at a current density of 380 A/cm2, the hole density near the i–n junction exceeds the hole density in the p-type injection layer by a factor of 44 , the band bending is very strong in the p-type region, i.e., the electric field E is high. At the same time, the bands are almost flat, i.e., E ≈ 0, in the n-type region. The reason for this is that the activation energy of acceptors is much higher than the activation energy of donors, and accordingly, the density of free carriers in the n-type region is three orders of magnitude higher than in the p-type region. Therefore, at high bias voltages, the drift hole transport dominates in the p-region, while the diffusion hole transport dominates in the n-region, which can be seen in Jp can be expressed as p is the hole density, E is the electric field, μp is the hole mobility, and q is the electron charge. On the contrary, in the n-type region near the i–n junction,Dp is the hole diffusion coefficient, Lp is the hole diffusion length in the n-type layer, and d is the size of the i-region. The hole current at the p–i junction and the hole current at the i–n junction are connected to each other via the current continuity equation:R(z) is the net recombination rate. We can rewrite this equation as K represents the hole current reduction factor due to recombination in the i-region. If the recombination rate is low, K ≈ 1 and Jp|zd+0= ≈ Jp|z= −0. Using Equations (1)–(4), we obtain a relation connecting the hole densities at the p–i and i–n junctions:kB is the Boltzmann constant and T is the device temperature.To understand the observed superinjection of holes in the 4H-SiC p-i-n diode, let us look at K is not very high), the hole density p|zd+0=  in the n-region in the vicinity of the i–n junction can exceed the hole density in the p-type layer. Moreover, the hole diffusion current at the i–n junction is positive, i.e., the gradient of the hole density at z = d is negative. This means that the hole density in the i-region in the vicinity of the i–n junction is even higher than p|zd+0= .Equation (5) shows that if the electric field in the p-type layer is strong and the recombination rate in the i-region is low or moderate  accurately predicts that at high forward bias voltages, the hole density at the i–n interface increases above the hole density in the p-type injection layer. If the recombination in the i-region is strong, we should take into account the bias-dependent hole current reduction factor K. Nevertheless, we find that at a bias voltage of 4 V, p|z=d+0 = 7.0 × 1014 cm−3, which is 1.6 times higher than the hole density in the p-type injection layer. This value agrees with that obtained in the self-consistent simulations . The hjunction d,e, whicThe above explanation shows that the superinjection effect in the 4H-SiC p-i-n diode arises at high bias voltages due to the high contrast in the hole transport mechanisms of the p-type and n-type injection layers, which is naturally provided by the very high activation energy of acceptors. The potential well for holes near the i–n junction, which is formed at high bias voltages and where holes are accumulated, is a consequence rather than a cause of the superinjection effect. The real reason for the superinjection effect is the requirement for the current continuity, which can be ensured only by an area with a high hole density between two regions (the p-type and n-type regions of the diode) with different (drift and diffusion) hole transport mechanisms. Therefore, the superinjection of holes can be observed in many wide-bandgap semiconductor p-i-n diodes that feature a similar asymmetry in the hole conduction mechanisms of the n-type and p-type layers. The key requirement for such an asymmetry is a low density of free carriers in the p-type layer and a high density of free carriers in the n-type layer.pmax increases with the injection current. At a current density of 15 kA/cm2, pmax reaches 2.1 × 1017 cm−3, which is 500 times higher than the hole density in the p-type injection layer. Although such an injection current seems to be too high for practical applications, a significant improvement of hole injection can be achieved at current densities as low as ~100 A/cm2 peqp=Na is the acceptor concentration, gA is the degeneracy factor of the acceptor level, and Nv is the effective density of states in the valence band. ηp = 5%. At EA = 0, the density of holes in the p-type region peqp is as high 8.2 × 1017 cm−3. However, at EA = 0.23 eV, peqp decreases by a factor of 70, to 1.14 × 1016 cm−3. This activation energy corresponds to aluminum, which is the most shallow acceptor in 4H-SiC [EA = 0.5 eV, peqp is as low as 4.1 × 1011 cm−3, i.e., six orders of magnitude lower than the concentration of acceptors. However, the lower the density of holes in the p-type layer, the higher the asymmetry in the conduction properties of the p-type and n-type regions, and therefore the stronger the superinjection effect in the p-i-n diode.Here, n 4H-SiC . At EA =EA = 0.23 eV), the maximum density of injected holes can exceed the hole density in the p-type layer by more than tenfold  and (7) show that the densities of free carriers in the p-type and n-type injection layers greatly depend on the device temperature. Since the activation energy of acceptors is very high, even a small decrease in temperature significantly decreases the hole density. If the p-type layer of the diode is doped with aluminum, the density of holes decreases by a factor of 160 as the temperature drops from 300 K to 200 K a. In theActivation energies of acceptors and other material parameters in 6H-SiC are roughly the same as in 4H-SiC . AccordiThe activation energies of acceptors in 3C-SiC are very similar to those in 4H-SiC: 0.26 eV for Al  and 0.3414–1015 cm−3, the p-type doping problem is even more pronounced in such materials as gallium nitride, aluminum nitride, and zinc sulfide. If in the case of GaN , one cacceptors , the ext[10 cm−3 . Therefo[10 cm−3 ). The la10 cm−3. Therefore, it was proposed not to use p–n and p-i-n structures and try to implement the hole injection mechanism in metal/insulator/semiconductor structures [9 cm−3 until the current density exceeds 0.02 A/cm2. Above 0.02 A/cm2, the maximum density of injected holes rapidly increases with the injection current and can reach 7.8 × 1012 cm−3, which is three orders of magnitude higher than the hole density in the p-type layer. Another example of a wide-bandgap semiconductor, which can hardly be used as an active layer of the double heterostructure, is zinc sulfide . At the ructures . Figure We have numerically demonstrated the superinjection of holes in homojunction p-i-n diodes based on different wide-bandgap semiconductors. This effect gives the possibility to create a high density of nonequilibrium holes in the i-region of the p-i-n diode at high forward bias voltages. The created density of holes exceeds the hole density in the p-type injection layer by more than two orders of magnitude, which can be exploited to improve the performance of light-emitting devices. The highest hole density is found near the i–n junction, which contradicts the common belief about p-i-n diodes, since in the i-region, one could expect to find the highest hole density near the p–i junction. However, we have provided a clear physical explanation for the observed effect, which is based on the asymmetry in the free carrier densities in the n-type and p-type layers. In SiC, GaN, AlN, ZnS, and many other wide-bandgap semiconductors, this asymmetry is naturally provided by the high activation energy of acceptors, which greatly limits the density of holes in the p-type layer of the diode. Hence, for optoelectronic devices, it is crucially important to overcome this doping limit, especially if it is not possible to use heterostructures. We have demonstrated that the superinjection of holes can be observed at high forward bias voltages. At the same time, the superinjection effect does not show itself at low and moderate injection currents (bias voltages). The higher the density of holes in the p-type layer, the higher the threshold current see b. This iEA, as it determines the hole density in the p-type region. We found that the superinjection of holes can hardly be detected at doping levels above 1017 cm−3 if EA is lower than ~0.15 eV. On the other hand, at high activation energies (EA ≳ 0.4 eV), it is strong even at relatively low injection currents. Assuming the density of electrons to be high in the n-type region of the p-i-n diode (neqn ≳ 1017 cm−3), the results of the numerical simulations and Equation (5) show that the higher the activation energy of acceptors, the stronger the effect. However, we should emphasize that the absolute value of the density of injected holes in a diode with a high activation energy of acceptors in the p-type layer  cannot exceed the absolute value of the density of injected holes in a diode with a low activation energy of acceptors in the p-type layer   see . TherefoTo directly measure the theoretically predicted strength of the superinjection effect, one should study two single color centers ,25,46,472 for the ZnS diode and more than ~1 kW/cm2 for other considered diodes. This should also be taken into account in practical applications.A disadvantage of the superinjection effect is that when a very high density of injected holes is achieved, the electron current is at least an order of magnitude higher than the hole current. Therefore, the energy efficiency of LEDs based on the superinjection effect in homojunction p-i-n diodes will be lower than that of those based on double heterostructures. A strong effect also requires high bias voltages (>6 V for 4H-SiC diodes and >30 V for AlN) that can create a high electric field in the p-type region of the diode (see Equation (5)). Accordingly, the heat generation rate is considerably high: more than 100 W/cmpeqp, but also provides it in the vicinity of the i–n junction, where the density of injected electrons is maximal (see We should emphasize that the superinjection effect not only gives a unique possibility to create a density of nonequilibrium holes which is orders of magnitude higher than the doping limit imal see c,d. Thus


Most existing algorithms for modeling and analyzing molecular networks assume a static or time-invariant network topology. Such view, however, does not render the temporal evolution of the underlying biological process as molecular networks are typically “re-wired” over time in response to cellular development and environmental changes. In our previous work, we formulated the inference of time-varying or dynamic networks as a tracking problem, where the target state is the ensemble of edges in the network. We used the Kalman filter to track the network topology over time. Unfortunately, the output of the Kalman filter does not reflect known properties of molecular networks, such as sparsity.Drosophila Melanogaster.To address the problem of inferring sparse time-varying networks from a set of under-sampled measurements, we propose the Approximate Kernel RecONstruction (AKRON) Kalman filter. AKRON supersedes the Lasso regularization by starting from the Lasso-Kalman inferred network and judiciously searching the space for a sparser solution. We derive theoretical bounds for the optimality of AKRON. We evaluate our approach against the Lasso-Kalman filter on synthetic data. The results show that not only does AKRON-Kalman provide better reconstruction errors, but it is also better at identifying if edges exist within a network. Furthermore, we perform a real-world benchmark on the lifecycle  of the Drosophila melanogaster than the Lasso-Kalman filter. Finally, all of the code reported in this contribution will be publicly available.We show that the networks inferred by the AKRON-Kalman filter are sparse and can detect more known gene-to-gene interactions for the  Understanding the dynamical behavior of living cells from their complex genomic regulatory networks is a challenge posed in systems biology; yet it is one of critical importance . Gene expression data can be used to infer, or reverse-engineer, the underlying genomic network to analyze the interactions between the molecules. Unfortunately, most of the existing work on reverse-engineering genomic regulatory networks estimates one single static network from all available data, which is often collected during different cellular functions or developmental epochs. The idea that molecular networks are remodeled as a function of time and stage is well understood; this conclusion is supported by the developmental networks of sea urchin embryos . Throughn large p problem . In particular, the system is under-determined. However, exploiting the fact that molecular networks are sparse, one can use compressive sensing to find a solution. Compressive sensing is concerned with the optimal reconstruction of a sparse signal from an under-determined linear system  controls the tradeoff between the Kalman estimate and sparsity. An α close to zero will result in a solution that is close to the Kalman estimate, but that may not be sparse. The opposite happens when α is close 1, which will produce a sparser solution, but may be far from the Kalman estimate. This is achieved by minimizing the reconstruction error .l0-optimization problem, which finds the optimal sparsest solution in a linear under-determined system. Consider the following 0 is the l0-norm, which is defined as the support of the vector, p≫n and denote in the sequel s=p−n. Without loss of generality, Φ is assumed to be full-rank. Compressive sensing theory [Φ, the l1-norm solution is equivalent to the l0-norm solution. Unfortunately, it is impossible to check if the RIP condition is satisfied for a given matrix. Despite this strict condition, l1 has been routinely used to find a sparse solution in systems of the form  is an approximation to computationally complex Kernel RecONstruction (KRON) problems . KRON isl1 regularization in  starts off from the l1-regularized Kalman estimate in  find the indices with the (s+δ) smallest magnitudes of the l1 solution, (ii) set exactly s of these indices to zero, (iii) re-solve the system Φx=y. All the possible l1 approximation to make it closer to the l0-norm. The size of the neighborhood δ is tunable depending on the computational power available, and vary from 0 (l1-approximation) to n .Following this reasoning, AKRON finds a sparser solution by exploring ution of  are evalExample: To understand the AKRON algorithm and illustrate the importance of the δ-neighborhoods, we present a simple numerical example. Consider the following randomly generated noiseless system as in  indices in Θ are such that x1−x∗∥2≤ε. □Let l0-solution are bounded from below. We first need the following Lemma.The following proposition derives an upper bound for l0-solution x∗ and approximate l1-solution x1. Denote by Θ and x∗, respectively. Assume that ∥x1−x∗∥2≤ε. Let R be the number of indices 1)j|≤ε. If δ=R, then AKRON yields the optimal solution.Consider the system in  with thel0-solution, it is sufficient to choose s zeros in “correct places”, i.e., with indices in Θ. Recall that AKRON sets s out of the smallest-magnitude (s+δ) entries in x1 to zero. Therefore, AKRON will yield the optimal solution if out of these (s+δ) smallest-magnitude entries, there are at least s entries from Θ. But all entries from Θ have |(x1)j|≤ε, for otherwise the assumption ∥x1−x∗∥2≤ε would be violated. This means that only those entries from x1)j|≤ε, could be chosen by AKRON, and there are only R of them. Hence δ=R will yield the optimal l0-solution.To obtain the sparsest δ for optimality of AKRON, namely, if δ=R, then we are guaranteed the optimal l0-solution. Since this condition is not necessary, we could reach optimality with δ≤R. □Lemma 1 provides a sufficient condition on l0-solution x∗ and approximate l1-solution x1. Let x∗. Assume that the non-zero entries of x∗ are bounded from below, i.e., x1−x∗∥2≤ε. If ε<η, then l0-solution. In particular, if δ=0 suffices.Consider the system in  with thex∗)j|≥η for δ≤R from Lemma 1. Thus, Since |: This algorithm is the output of the Kalman filter with the l1 projection applied to the state vector. The (S) indicates whether the smoother was implemented.l1-KF(S) to seed AKRON. It is also implemented with and without the smoother.AKRON-KF(S): This is the proposed approach using the output of Our experiments make use of the following algorithms for a sparse reconstruction of a time-varying network: TP), true negative (TN), false positive (FP) and false negative (FN) rates. These rates are summarized through accuracy (acc), sensitivity (sen), specificity (spe), and Matthew’s correlation coefficient (mcc), which are defined below. Both of the above algorithms can reconstruct a network that represents the interactions between genes. We compute the true positive .Matthew’s correlation coefficient provides a more balanced statistic for examining the overall trade-offs between the different rates  on data that we have complete control over. All results in this section are presented as the average over 25 monte carlo simulations. Averaging is performed because there could be a large degree of variation in the time-varying networks that are randomly generated.α in . Given these results, we choose α=0.2 for the remainder of the experiments since this value provides a reasonable trade-off between the different statistics that were assessed. Figure First, we evaluate the impact of α in  on the en  benefit significantly from a larger value of δ. For example, consider a network that is 9×50×4. δ=1 provides a little reduction in error; however, increasing δ to 3, significantly reduces the error and increases the other statistics. Finally, we observe that smoothing improves the error of the system; however, we should note that this improvement comes at a computational cost.Second, we expand the synthetic experiments to evaluate the impact of α=0.2 based on the experiments in the previous section for l1-KF and AKRON-KF.The application of interest is the inference of the time-varying wing-muscle genomic network of the Drosophila Melanogaster (fruit fly). The Drosophila’s microarray dataset originally consists of 4028 genes taken over 66 different time points . The datIn this application, we considered a list of 11 genes that are responsible for the wing muscle development, which has been considered by many researchers before –9, 20. Tl1-KF and AKRON-KF are shown in Figs. l1-KF. AKRON-KF was able to find all the connections reported in Flybase:  appears in the embryonic, larval, and pupal stages,  appears in all four stages,  appears in the embryonic, larval, and pupal stages,  appears in the embryonic, larval, and pupal stages and  appears in the embryonic and larval stages. The other two connections appear through one medium gene  appears in the embryonic, pupal and adulthood phases through one additional gene and  appears through one or more additional genes only in the embryonic and larval phases.The networks reconstruction using the l1-KF and AKRON-KF with smoothers are shown in Figs. l1-KF.The networks reconstruction using the l1-KF [l1-KF are the only algorithms able to recover all known interactions and specify the developmental stage where these interactions occur. Although the l1-KF also finds all reported interactions, the networks are denser (less sparse) than the AKRON-KF. With regard to the MDL results in Table Table l1-KF , SMURC [l1-KF  and Dynal1-KF  considerl1-KF , random l1-KF , and nonl1-KF ) assumedIn this work, we addressed the problem of inferring time-varying molecular networks as a tracking problem that can be solved using the Kalman filter. The major difficulty, however, is that there is not a sufficient number of observations at each time point, which makes the state-space model unobservable and the tracking senseless. Fortunately, molecular networks are known to be sparse because the dynamics of every gene are governed by only a small number of genes. By incorporating the sparsity condition, we show that the tracking problem becomes feasible.l1-KF). Our proposed approach leverages the AKRON algorithm to find a sparser solution that is more representative of the ground truth. The proposed tracker/smoother first computes the output of l1-KF; then explores growing neighborhoods of the l1-projection to look for sparser solutions, eventually reaching the optimal sparsest estimate. The size of these neighborhoods is a tunable parameter that depends on the computational power available. AKRON-KF was benchmarked on synthetic and real-world data against l1-KF. The results demonstrate that the proposed approach is better at recovering sparse time-varying networks than l1-KF. Not only was the reconstruction error of the proposed approach lower than l1-KF, but it was also better at detecting whether an edge exists in a network. AKRON-KF tracker was applied to infer the wing muscle gene regulatory network of the Drosophila Melanogaster during four developmental phases of its life cycle, and successfully identified all seven known interactions reported in Flybase. We should also note that our proposed approach will work for time-series networks that have more than four times steps and sparsity levelsWe presented the AKRON Kalman filter, which builds on our previous work on the Lasso-Kalman filter (Our future work includes applying the AKRON-KF to other types of data, particularly data related to different types of cancers to create a predictive network biomarker for clinical outcome. These ideas have applicability in translational clinical cancer research, basic cancer research, and in network-based drug discovery.


Scientific Reports 10.1038/s41598-019-49900-9, published online 23 September 2019Correction to: The original version of this Article contained an error in the title of the paper, where the word “graphene” was incorrectly given as “grapheme”. This has now been corrected in the PDF and HTML versions of the Article.


Subcellular localization prediction of protein is an important component of bioinformatics, which has great importance for drug design and other applications. A multitude of computational tools for proteins subcellular location have been developed in the recent decades, however, existing methods differ in the protein sequence representation techniques and classification algorithms adopted.In this paper, we firstly introduce two kinds of protein sequences encoding schemes: dipeptide information with space and Gapped k-mer information. Then, the Gapped k-mer calculation method which is based on quad-tree is also introduced.>From the prediction results, this method not only reduces the dimension, but also improves the prediction precision of protein subcellular localization. Proteins must be in a particular area in the cell for the participation in normal life activities . However, a multitude of protein sequences are increasingly identified into public biology databanks in post genome era, which results from the development of high-throughput technology. Therefore, it is indispensable to develop an automated method for fast and accurately annotating the subcellular attributes proteins. Predicting the subcellular locations of proteins can provide useful hints about the function of proteins, increase our understanding of the mechanisms of certain diseases, and ultimately help develop new drugs . Such, mApoptosis refers to the orderly death of genetically controlled cells in order to maintain internal environmental stability. Apoptosis is not only a special cell death type, but also has important biological significance and complex molecular biology mechanism , 7. Therk-peptide information. Compared to AAC, k-peptide information could be get a better predict results. But when k is larger, there exist dimension disaster, bad effects and some other problems. When doing the homologous sequence alignment, we will consider space penalty. By this, the best match between two sequences will be found. When extracting k-peptide information of sequences, we also take k-peptide information extraction with the space into consideration. The detailed description of the novel representation will be shown in the following section.The amino acid composition of protein sequence is the simplest sequence statistics. However, AAC only take the information of the whole protein sequences into consideration, and ignore the partial order information. Therefore, some scholars consider the dipeptide composition information, and someone also proposed a prediction method which is based on We first use two benchmark datasets constructed by the previous investigators. The ZD98 dataset, constructed by Zhou and Doctor , which hDipeptide information with the blank space Before we start to discuss peptide information with spaces, we first study the simple dipeptide information with spaces. The basic form is “XDY”, where X and Y are one of amino acids, D is a certain number of spaces. D is reduced to d, 0 ≤ d ≤ l − 2. The l is the length of protein sequence. It indicates the common dipeptide information when d = 0. Obviously, the d value will affect the results of prediction directly.k-mer and with spaces method was abbreviated as Gapped k-mer. In brief, with a space instead of some amino acids in dipeptide information method, then statistics all the different types of the peptides frequency.Peptide information extraction method without spaces was abbreviated as k-peptide information of 20 kinds of amino acids directly, then it will get 20k and 21k dimension vector for k-mer and Gapped k-mer, respectively. The dimension of feature vector is very high when k > 3. This case will result in time-consuming classification forecast operation and cause too much redundant information. So in order to reduce the dimension, we adopted the following two kinds of measures:k-peptide information, we only consider that kind of k-peptide, which was used in the training dataset, instead of all the different types of k-peptide. If we take all kinds of k-peptide, it will produce many zero vectors which have no effect on classification and cause the large dimension. Take the ZD98 dataset as test, if we consider all of the different 4-peptide when k = 4, k-mer will get 204 =160,000 dimension feature vector. If only statistical all 4-peptide of ZD98 datasets, it will produce 22,265-dimensional feature vector. Compared to the former, dimension was reduced by 86.09%. Though dimension is still very big, it reduces the dimension feature vector in a certain extent. What’s more, the larger the k, the greater the proportion of reduced.In statistical The amino acid reduced. 20 kinds of amino acids in a certain way were reduced into a few classes, which can greatly reduce the feature vector dimensions. Here we use a reduced solution which was used by a lot of researchers. According to the physical and chemical properties of amino acids, 20 kinds of amino acid were reduced to classify. Table If statistics k-mer and Gapped k-mer, the dimensions of the feature vector will be greatly reduced. Such as extracting k-mer characteristic information with k-mer in ZD98 dataset, the dimension has been reduced to 1071. The dimension was reduced by 99.33%.Through these two measures, when extracting characteristic information with k-mer was used to statistic k-frequency of protein sequence. For example, a protein sequence S can be represented as follows:n is the varity of k-peptide, vi is the frequency of i-th k-peptide in protein sequence S.The k-mer is statistics frequency with space k-peptide in the protein sequence S:m is the varity of k-peptide with g spaces, vi is the frequency of i-th k-peptide with g spaces in protein sequence S.The meaning of Gapped k-mer method. The number of spaces will be considered by Gapped 3-mer. If the number of spaces is 1, then the types of Gapped 3-mer are “_BB, _BW, _WW, _WG, L_B, B_W, W_G, B_B, LB_, BB_, BW_, WW_”. There are 12 kinds of types. Thus, a 12-dimension vector can be obtained by Gapped k-mer.Such as, a piece of protein sequence “RIAVYYPG”. According to the reduced solution of Table k-mer can be thought of a special case for Gapped k-mer .Here we should note that the “BBB” and “BBW” of 3-mer statistics in the above example will be regarded as belonging to the same type “BB_”. t classes. So the dimension of feature vector is tk for k-mer method. However, the dimension of feature vector is k-mer method (the number of space is g). In general, when k is small, feature vector dimensions of Gapped k-mer is lower than k-mer.For a long enough protein sequence, according to some kind of reduced solution, the sequence can be reduced into k and spaces (g).In the classified prediction model of training and testing, we need to consider selection problem of parameters k-mer calculation method, which is based on multi-tree. For a scheme reducing amino acids into n classes, k-mer statistics of k peptide can be regarded as the kinds of statistics, i.e. a full depth k of n types leaf nodes in the tree. The first amino acid of n peptide is the head node, the second is the second straton node and so on. Finally the k layer is the leaf node. By comparing all k peptide in protein sequence to traverse the full n fork tree, and accumulating number of occurrences corresponding to a leaf node at the same time. Finally, the frequency of the leaf node corresponds to frequency of k-peptide. For Gapped k-mer, we need to consider the location of the space appears. If spaces appear in the m location of k peptide, the m layer of the original k peptide would be removed. Then merging the lower subtree, summing the frequency of the corresponding leaf node, and the frequency of gapped k-mer is obtained. It’s similar for the multiple spaces. The following is a simple example.Here we first introduce a simple Gapped A short sequence which has been reduced, Seq = BBBWWBBWB. Through the statistics of binary tree 3 peptide, frequency is shown in Fig. k-mer. If the second position is space, such as the 3-peptides frequency of “X_Y” type. At this time you only need to delete the second floor. Merging its lower subtree, as shown in the Fig. Statistics the frequency of type 3-peptides with 1 space in Gapped The frequency of all three peptides, such as “X_Y” is obtained by this method. The frequency of “B_B” types of the peptides is 2.When using the same method to statistic form, such as “_XY”, the frequency of all 3 peptides, we just eliminating the head node, and merging the lower two subtrees.k-mer multi-tree. Then we can get all Gapped k-mer characteristic information vector, which is a simple statistical method.Through the tree structure, we only need to construct the k-classification problem, k*(k-1)/2 classifiers need to be constructed. Meanwhile, the radial basis function (RBF) was selected as the kernel function due to that it outperforms the other kinds of kernel functions. Then a simple grid search strategy over C and γ values based on 10-fold cross-validation for each dataset was selected, where C and γ were allowed to take the values only between 2−5 to 25.Among these classifier, SVM exhibits quite promising results . SupportTP, TN, FP and FN were the number of true positives, true negatives, false positives and false negatives, respectively; N is the total number of locative proteins and k is the class number.As is well known, independent dataset test, K-fold (such as five-fold or ten-fold) cross-validation (subsampling test), and jackknife test ) are often used to examine a predictor’s effectiveness in practical application. Because considerable arbitrariness exists in the independent dataset test and K-fold cross-validation, and only the jackknife test is the least arbitrary that can always yield a unique result for a given benchmark dataset. Accordingly, the jackknife test is adopted here to examine the quality of the present predictor. For comprehensive evaluation, sensitivity (Sn), specificity (Sp) and Matthew’s correlation coefficient (MCC), as well as the overall accuracy (OA) over the entire dataset are reported. These parameters were detailed in the following equations:d are shown in Fig. We first use dipeptide information with the blank space in ZD98 and CL317 datasets. By the form of “XDY”, a 400-dimension vector was gotten, and then was put into SVM to predict. The predicted results with the change of the parameter d don’t have very big effect on the prediction classification. In CL317 dataset, with the increasing of d, the prediction result tends to decline. It shows that dipeptide information with the blank space is not very good.From Fig. k-mer, the dimension of eigenvectors and overall accuracy in different k value and g value are shown in Tables By the extraction algorithm based on Gapped k-mer polypeptide Gapped has been reduced a lot, Note that a blank line in the table 0 is the results of k-mer feature method, the result show that Gapped k-mer are better than k-mer. Moreover, the number of space is better not too lager, best one or two spaces, which can be considered as in the protein evolution, mutations occur only in a few location of protein sequences, while the majority of other position without mutations, by matching with a few space, we can match the similarity of two protein sequences better, as above, “BBB” and “BBW”, they are regarded as the same type “BB_” in statistics with a space of 3-peptide information. Namely the space can matching any character, so in the process of practice, use the “BB_” to regular matching the original sequence, statistic the frequency. Conversely, “BBW” can belong to “BB_”, but to “_BW” and “B_W”, namely a variety of types of Gapped k-mer can matching to a k-peptide in the sequence, compared to the precise k-mer matching, the fuzzy matching can extract more similar information.From Tables k-mer and Gapped k-mer on two data sets, respectively. The results show that with the k value increase, the accuracy of the two methods is generally improved, when extracting the same k-peptide information, the accuracy of Gapped k-mer is higher than k-mer in whole. The accuracy of Gapped k-mer reached a maximum value both in ZD98 and CL317. But accurate did not improve when k = 6, that meaning unable to extract more information at this time. So the effect is best when k = 5 in using the Gapped k-mer method to extract feature information.Figure k-mer method, we compare it with other approaches that have been reported. In recent years, mostly through a variety of characteristics of method of information fusion in the protein subcellular location prediction research. In order to facilitate comparison, we also incorporate other information. At present, many scholars use GO information or GO information and it features information fusion used for protein subcellular localization prediction. Here, we fusing GO information with Gapped k-mer , SVM as classifier, compare with other method, the result in Table To illustrate the effectiveness of Gapped -mer k = , g = 1, k-mer. Extracting reduced k-peptide component information with the space k-peptide representation. Then multi-tree calculating Gapped k-mer was also introduced. The new feature representation are used to construct a model and combined with dimension reduction to make the subcellular localization prediction of proteins. Meanwhile, we also discussed the effects of different parameters on the experimental results. The influence of parameters k and g on the experiment are discussed. Prediction accuracy got the highest when k = 5, g = 1. Compared to k-mer, our method not only reduces the dimension, but also improves the prediction precision, as shown in Table In this paper, we proposed a new method to predict protein subcellular localization based on Gapped k-mer and GO information, no matter overall accuracy or absolute true overall accuracy, the achieved results by the proposed method are much higher than Li and Yu.Li and Yu  applied


Background: Healthcare-associated infections (HAIs) are a serious public health problem. They can be associated with morbidity and mortality and are responsible for the increase in patient hospitalization. Antimicrobial resistance among pathogens causing HAI has increased at alarming levels. In this paper, a robust method for analyzing genome-scale metabolic networks of bacteria is proposed in order to identify potential therapeutic targets, along with its corresponding web implementation, dubbed FindTargetsWEB. The proposed method assumes that every metabolic network presents fragile genes whose blockade will impair one or more metabolic functions, such as biomass accumulation. FindTargetsWEB automates the process of identification of such fragile genes using flux balance analysis (FBA), flux variability analysis (FVA), extended Systems Biology Markup Language (SBML) file parsing, and queries to three public repositories, i.e., KEGG, UniProt, and DrugBank. The web application was developed in Python using COBRApy and Django.Results: The proposed method was demonstrated to be robust enough to process even non-curated, incomplete, or imprecise metabolic networks, in addition to integrated host-pathogen models. A list of potential therapeutic targets and their putative inhibitors was generated as a result of the analysis of Pseudomonas aeruginosa metabolic networks available in the literature and a curated version of the metabolic network of a multidrug-resistant P. aeruginosa strain belonging to a clone endemic in Brazil (P. aeruginosa ST277). Genome-scale metabolic networks of other gram-positive and gram-negative bacteria, such as Staphylococcus aureus, Klebsiella pneumoniae, and Haemophilus influenzae, were also analyzed using FindTargetsWEB. Multiple potential targets have been found using the proposed method in all metabolic networks, including some overlapping between two or more pathogens. Among the potential targets, several have been previously reported in the literature as targets for antimicrobial development, and many targets have approved drugs. Despite similarities in the metabolic network structure for closely related bacteria, we show that the method is able to selectively identify targets in pathogenic versus non-pathogenic organisms.Conclusions: This new computational system can give insights into the identification of new candidate therapeutic targets for pathogenic bacteria and discovery of new antimicrobial drugs through genome-scale metabolic network analysis and heterogeneous data integration, even for non-curated or incomplete networks. Acinetobacter baumannii, Pseudomonas aeruginosa, and Enterobacteriaceae family), and R&D on new antibiotics against these is considered to be of critical priority , previously called hospital infections, are a serious public health problem and can develop either as a direct result of medical or surgical treatment or from being in contact with a healthcare setting. These infections include central line-associated bloodstream infections, catheter-associated urinary tract infections, ventilator-associated pneumonia (VAP), and surgical site infections. Among the pathogens related to HAI, the group of bacteria is the one that stands out. More than 2 million HAIs occur each year in the USA , with 50priority . In humafibrosis  and is aGiven the potential severity of multidrug-resistant bacteria and the lack of treatment options, the identification and implementation of effective strategies to prevent such infections are urgent priorities.The integration of mathematical, statistical, and computational methods for biological data analysis to enable the discovery of new therapeutic targets for any bacteria is extremely relevant. The combination of bioinformatics, system modeling, and heterogeneous data integration can be a powerful tool for this purpose.Several strategies have been proposed to search for drug targets from genome-scale models of bacterial metabolism. More often, essential genes are identified from single virtual knockouts where flux balance analysis (FBA)  is used The construction of genome-scale metabolic network is a laborious endeavor. It combines automated steps with manual curation. The most used protocol, proposed by Tier 1 corresponds to PGDBs that have received at least 1 year of manual curation and are updated continuously. Tier 2 includes PGDBs that have received moderate (less than a year) amounts of review and are usually not updated on an ongoing basis. Finally, Tier 3 refers to PGDBs that were created computationally and received no subsequent manual review or updating.The BioCyc database  classifidraft metabolic reconstructions are considered Tier 3 models. Published curated metabolic models are classified as Tier 2, unless the model is identified in the literature as Tier 1.In this work, the same classification for genome-scale metabolic network models is adopted. The focus here is on metabolic network models that can be classified as Tier 2 and Tier 3, according to the BioCyc database classification. In this manuscript, FindTargetsWEB. The proposed method is computationally efficient, user-friendly, and robust to errors in reconstructed genome-scale metabolic networks, which are more frequent in Tier 3 (draft) metabolic networks. The web interface of the application is straightforward, and results are sent directly to an email address informed by the user. To demonstrate the flexibility of FindTargetsWEB, 10 genomic-scale metabolic networks of bacterial strains are analyzed in this paper. Nine of the 10 networks are available in the literature, all classified as Tier 2 models in this work: P. aeruginosa PAO1—version 2008   . Both reThe web application proposed in this work combines FBA, flux variability analysis (FVA) , extendeCOnstraint-Based Reconstruction and Analysis Toolbox (COBRA)   methods  (COBRA)  is a Pyt (COBRA) Figure 1Target identification is carried out through a computational workflow that runs the metabolic network analysis and pinpoints genes whose virtual knockout interrupts the generation of biomass. Therefore, the minimum level of curation required for a metabolic network model to be processed by FindTargetsWEB is to have a biomass reaction flux greater than zero. The list of potential targets is filtered using FVA (if the user decides to do so), and the workflow retrieves possible inhibitors for the identified genes, verify if such inhibitors are available as approved drugs, and evaluate their toxicity to humans by querying several repositories.https://www.gnu.org/software/glpk/), which is the COBRApy default solver that is easily deployable on Linux platforms. The system is deployed in an Ubuntu v18.04 server with 64GB RAM. Prior to processing, when needed, SBML files were converted to the SBML level 3 format using the command cobra.io.sbml3.write_sbml_model from COBRA. The SBML files processed in this manuscript were retrieved from the BioModels repository  relations. In the same way as the previous step, if the value of the generation of biomass has zeroed, the corresponding gene information is stored in a second list. It is worth noting that one gene can be associated with more than one reaction, and one reaction may require the expression of several genes.5. Consolidation/unification of knockouts results—In this step, both lists generated in the previous steps are unified, i.e., the list of reactions generated in step 3 and the gene list generated in step 4. In order to a gene to be included in the final list, it should be included in the list of step 4 and be associated with at least one reaction stored in step 3 (see Algorithm 1). These are the candidate genes that the workflow is going to consider in the following steps. It should be highlighted that the final list is filtered according to the FVA processing performed in step 2, if the option FBA+FVA is selected by the user.Algorithm 1: Consolidation of knockout results (SBML with mapped genes)Input: List of genes from knocked-out reactions/list of knocked-out genesOutput: Unified list of target genes in a text fileprocedure UnificationTargetsList	2:			read targetGeneListFromReact	3:			read targetGeneList	4:			open file “targetgenes.txt”	5:			for all targetgene in targetGeneListFromReact do	6:				if targetgene in targetGeneList then	7:				write targetgene in file “targetgenes.txt”	8:				end if	9:			end for	10:			close file “targetgenes.txt”	11:	end procedure	1:	6a. Search for EC numbers of consolidated genes. In this step, the system queries the KEGG repository to obtain the EC number of each gene included in the final gene list obtained in the previous step (file “targetgenes.txt”). KEGG (Kyoto Encyclopedia of Genes and Genomes) is a knowledge base for systematic analysis of gene functions, linking genomic information with higher order functional information  metabolic network models, EC numbers are retrieved from KEGG based on reaction information. This step is particularly important for incomplete metabolic reconstructions that do not include GPR relations and is directly related to the application’s requirement of robustness to incompleteness on metabolic network data. The KEGG search is performed using all the compounds involved in the corresponding reaction. See Algorithm 2 for a detailed description of the processing related to this step. It is worth emphasizing that this step is executed only for incomplete descriptions of genome-scale metabolic networks. The complexity of Algorithm 2 is O(C), where C is the number of compounds included in the SBML file.Algorithm 2: Search for EC numbers using reaction information (SBML without mapped genes)Input: List of chemical compounds of reactionOutput: List of EC numbers foundread listCompoundFromSBML	4:	5:	# Instance of biomodels python module	6:	k <- KEGG instance	7:	8:	# setting timeout in seconds	9:	k.timeout <- 200000	10:	11:	# All compounds in SBML file. Ex.: 2 A –> B	12:	for all compound in listCompoundFromSBML do	13:	14:			# find all stoichiometric values with regex method in compound. Ex.: A –> B	15:			compound_no_stoich <- remove all stoichiometric values in compound	16:			param_splt <- empty	17:	18:			# Verify if reaction is reversible or irreversible	19:			if “< = >“ in compound_no_stoich then	20:				param_splt <- “< = >“	21:			else	22:				param_splt <- “–>“	23:			end if	24:	25:			# Separate compounds in reactant and product. Ex.: compound_splt = 	26:			compound_splt <- compound_no_stoich.split(param_splt)	27:	28:			# If compound belongs to a transport reaction (influx or eflux), jump to next iteration	29:			if compound_splt.length < 2 then	30:				continue	31:			end if	32:	33:			list_ec_number_0 <- initialize empty list	34:			list_ec_number_1 <- initialize empty list	35:		36:			# Start iterating compound_splt list with reactant and product	37:			for  do	38:	39:				# Get reactant or product in this variable	40:				item_compound <- compound_splt[x] without spaces	41:			list_id_cpd_KEGG <- initialize empty list	42:			43:			# If reactant or product contains “+”, find ID in KEGG for components.	44:			# Else, find ID in KEGG for only one component.	45:			if “ + “ in item_compound then	46:				item_compound_splt <- item_compound.split(“ + “)	47:				for all cpd_item in item_compound_splt do	48:					# find id compound in KEGG for cpd_item	49:					# and insert in ids list	50:					result_id_cpd <- k.find	51:					insert result_id_cpd in list_id_cpd_KEGG	52:				end for	53:			else	54:				result_id_cpd <- k.find	55:				insert item_compound in list_id_cpd_KEGG	56:			end if	57:			58:			# Here, all list_id_cpd_KEGG are concatenated	59:			# found to search the reaction in KEGG.	60:			# In Python, if list_id_cpd_KEGG length is less than 2,	61:			# don’t put the “+” in end of string.	62:			str_item_compound_in_cpd <- list_id_cpd_KEGG concat with “+”	63:			64:			# find all reactions in KEGG with IDs of compounds	65:			result_link_reactions_cpd <- k.link	66:			67:			# All results of result_link_reactions_cpd are inserted here	68:			set_id_reaction_KEGG <- insert all reactions found in KEGG.	69:			70:			# find all EC numbers in KEGG with reactions IDs	71:			# in set_id_reaction_KEGG and insert in result_list_ec	72:			result_list_ec = k.link	73:			if x = 0 then	74:				insert result_list_ec in list_ec_number_0	75:			else	76:				insert result_list_ec in list_ec_number_1	77:			end if	78:	79:		end for	80:	81:		list_ec_number_intersect <- initialize empty list	82:		txt_file <- initialize txt file	83:		84:		# Starts to iterate the list of ECs to identify intersections	85:		# If found, related EC numbers are written in a text file	86:		for all ec_number_0 in list_ec_number_0 do	87:			if ec_number_0 in list_ec_number_1 then	88:				record ec_number_0 in a txt_file	89:			end if	90:		end for	91:		92:	end for	93:	94: end procedure	1:	procedure alternativeStepToGetECNumberWithoutGenes(listCompoundFromSBML)	2:	# file with all compounds in SBML.	3:	7. Search for EC numbers on DrugBank—With the EC numbers obtained in the previous steps, the system queries the DrugBank repository to verify if this database has any record of the listed EC numbers. The DrugBank database is a repository that combines detailed drug data with comprehensive drug target information  is necessary to confirm whether the protein retrieved has a homologue in the target bacterium. Clearly, exact matches are also possible. In any case, the retrieved data is validated in the next step.8. Search for homologues on UniProt—Finally, the system searches for sequence similarity between the proteins described by UniProt IDs retrieved in the last step and the proteins encoded by the genome of the target bacterium using the BLAST  (ch tool)  applicatIn this step, the homology between the target protein and human proteins is also considered. If the sequence similarity with a human protein is greater than the similarity with the target bacterium, the protein under analysis is discarded, since the inhibition of that protein could be harmful to the host. Otherwise, several data are stored, such as metabolic pathway, function, and catalytic activity, among others. This step of the workflow is the most time-consuming, since BLAST is executed for all proteins identified in the previous step.9. Search for existing inhibitors—The last step is to query the DrugBank repository, using the stored UniProt IDs, in order to retrieve known inhibitors, if available. After this last step, the system generates spreadsheets containing all results that are sent to the user in a compressed file.This method presents as results candidate genes that, when knocked-out, will cease the biomass production of the microorganism. Candidate genes must be associated with potential drug targets in DrugBank, and their sequence similarity to human proteins is also checked. The application then identifies available ligands, most often inhibitors, to the selected genes.Results of FindTargetsWEB’s analysis are sent to the user as a compressed file, to the e-mail address informed at the start of execution. Five spreadsheets are included in the compressed file:08-filter_ECNumbers_DrugBank—This spreadsheet contains the EC number of putative targets, along with product, organism name, UniProt ID, and DrugBank ID- 11- hits_Uniprot—This spreadsheet contains additional information related to UniProt queries, such as percentage of sequence similarity, BLAST e-value, gene name, pathway, function, and catalytic activity.- 13-list_inhibitors_per_target—This spreadsheet lists all inhibitors found for all targets. Included information are drug name, drug group , and drug action.- 14-list_inhibitors_approved—This spreadsheet lists all inhibitors with approved drugs found for all targets. Included information are drug name, drug group (approved), and drug action.- model_data—This spreadsheet lists data related to the input SBML file, such as gene IDs and associated reactions. The complete information of which reactions are associated with each gene in the metabolic network model is included in this file.- summary_results—This spreadsheet contains a summary of data included in the previous files. Included fields are EC numbers, product, organism name, gene name, pathway, function, catalytic activity, drug name, drug group, and drug action.- P. aeruginosa, K. pneumoniae, H. influenzae, S. aureus, P. putida, and a host-pathogen genome-scale reconstruction based on the M. tuberculosis metabolic network are presented. It should be highlighted that FindTargetsWEB can carry out analysis for other bacterial species, as indicated by the list box on the initial web page of the application. Indeed, even this list can be easily expanded to include additional species of interest, through a user request to FindTargetsWEB support team.In this section, analysis results for several strains of P. aeruginosa is discussed. A survey of the literature is also presented to confirm the feasibility of the candidate genes as antibacterial drug targets. Gene function and related pathways are also considered in the evaluation of results.To evaluate the accuracy of results for several metabolic networks, initially, the analysis of four metabolic networks of P. aeruginosa strains analyzed by FindTargetWEB were: PAO1 version 2008—iMO1056 (BioModels ID 1507180020)  , PAO1 ve7180020) , PA14—iP7180020) , and a cCCBH4851 . The SBMP. aeruginosa spp., which may vary between 0.3 and 0.8 h-1, depending on cultivation conditions  and murB (EC 1.3.1.98) genes encode enzymes involved in bacterial cell wall synthesis and have been identified as essential in both Pseudomonas spp. and Escherichia coli (folP gene product (EC 2.5.1.15) is important for folic acid biosynthesis, which is fundamental for bacterial growth and reproduction (folA gene product (EC 1.5.1.3) is related to the biosynthesis of cofactors, being an important intermediary of folate metabolism. It is considered the key enzyme of this process and essential for microbial growth (aroE gene (EC 1.1.1.25), which has been described as a potential therapeutic target of both P. putida and E. coli  and lysine. Lysine is utilized for protein synthesis in bacteria and forms part of the peptidoglycan cross-link structure in the cell wall of most gram-positive species, whilst meso-DAP is the peptidoglycan cross-linking moiety in the cell wall of gram-negative bacteria   are favorable candidate drug targets and several inhibitors have been already reported  . Becausereported . SpecififabA participates in fatty acid synthesis (FAS) processes, which includes also fabB, fabD, fabI, and fabH. The proteins encoded by these genes have an essential role during the synthesis of bacterial phospholipid membranes, lipopolysaccharide (LPS), and lipoproteins, thus representing attractive targets due to the structural differences between the human and bacterial proteins and the essentiality of FAS  of the overall method. The growth rate of the Tier 3 version of the P. aeruginosa CCBH4851 network is 1.757 h-1, which is less consistent to the biology of P. aeruginosa spp. than the growth rate obtained by the Tier 2 version of the network. The processing of this Tier 3 network generated 32 targets in the FBA+FVA analysis, and 48 targets using the FBA-only method. It is remarkable that this less curated version of P. aeruginosa CCBH4851 network generated more potential targets in the FBA+FVA analysis than the corresponding Tier 2 network.To evaluate the robustness of FindTargetsWEB regarding Tier 3 networks, which generally are networks generated automatically without manual curation, FindTargetsWEB processed a preliminary version of the metabolic network model of dapA is present as a potential target in the Tier 3 network.Among targets identified using the FVA+FBA method, 10 targets are common between the Tier 2 and Tier 3 networks. For the FBA-only analysis, 21 targets are common between the two versions. It is worth mentioning that many targets found in Tier 2 networks are present in the analysis of the CCBH4851 Tier 3 network, which corroborates the relevance of the targets found even in draft versions of metabolic networks. This comparison also highlights the importance of careful curation of automatically generated metabolic networks. For instance, from the targets discussed in the previous subsection, only P. aeruginosa were also processed using FindTargetsWEB. In the previous subsections, results for P. aeruginosa metabolic network models were presented, but it is also possible to analyze networks of other species of bacteria. In this subsection, FindTargetsWEB results for a metabolic network reconstruction of K. pneumoniae MGH78578—iYL1228 (BioModels ID 1507180054) (H. influenzae—iCS400 (BioModels ID 1507180053) (Metabolic networks of bacteria other than 7180054)  and H. i7180053)  are pres7180053) Table 4.K. pneumoniae, a total of 45 unique potential targets were found using the FBA+FVA method and also 45 for the FBA-only method. Some of the more representative targets are listed in For lpxA gene is involved in the initial steps of lipid A production through the Raetz pathway. As stated in the previous subsection, fabA, fabB, and fabF participate in FAS processes and represent attractive targets due to the structural differences between the human and bacterial proteins and the essentiality of FAS. The cytoplasmic protein N-acetylglutamate (NAG) kinase (encoded by argB), which promotes phosphorylation of NAG in a rate-limiting step of bacterial L-arginine production, occurs through acetylated intermediates, unlike mammals which use non-acetylated intermediates, and for this reason, it was previously considered a candidate drug target  and P. aeruginosa (folA). Both methods, FBA+FVA and FBA-only, generate exactly the same results. Therefore, the FVA ranges for all targets in It is worth mentioning that the genes M. tuberculosis H37Rv model, iNJ661 , which integrates a cell-specific alveolar macrophage model, iAB-AMØ-1410, from the global human metabolic reconstruction, with an , iNJ661 . The intM. tuberculosis survival   , reconstmurB, aroC), while others such as mvaA , tkt , and dfrA  are defined as essential for S. aureus in both minimal and rich medias  networks, pointing out that such metabolic network models are very common and are the only models available for some organisms. The system is deployed as a web application and is asynchronous: the user is notified when results are available. The performance of the system is optimized, since the COBRApy framework can make use of multiple cores available in the host machine, and it is able to process the metabolic network of various bacteria, as described in the previous section. The only requirement is the availability of an SBML level 3 file describing the corresponding genome-scale metabolic network. The user interface is straightforward see 2, and tE. coli and Bacillus subtilis, without pinpointing any potential drug. K. pneumoniae. However, their method is not automated, and it was not applied to other species of bacteria. None of these works go as far as FindTargetsWEB, which can process metabolic network models of several species of bacteria, identify potential targets, confirm homology with the analyzed gene, and identify all available drugs for the potential target in a fully automated manner.Other proposals for the analysis of metabolic networks at genomic scale are available in the literature. FBA+FVA and FBA-only, one can conclude that the FBA+FVA method represents a way to prioritize the targets identified by the FBA-only method, since the set of targets identified by FBA+FVA is a proper subset of the set of targets identified by FBA-only. However, as stated in the detailed description of the targets of the Results section, potential targets that are associated with the FBA-only method and do not appear as results of the FBA+FVA method should not be disconsidered. Many important targets described in the literature have a FVA range greater than zero, and a careful analysis of both sets of potential targets is advised.Regarding the options to identify potential targets, i.e., P. aeruginosa and other bacteria and can be effective against non multidrug-resistant strains. As expected, for the multidrug-resistant strain, most of the approved drugs are not effective. For instance, it is known that P. aeruginosa can be resistant to both to trimethoprim and sulfamethoxazole  is both essential to metabolic growth and biofilm formation, according to the FUNCTION field returned by FindTargetsWEB and literature sources License: Not ApplicableAny restriction to use by non-academics: Not ApplicableThe user must provide a SBML level 3 file describing the metabolic network reconstruction and an e-mail address to which the results will be forwarded.All datasets generated for this study are included in the manuscript and the supplementary files.TCM, FPSJ, and FABS designed the system. TCM was the main programmer. MWC and ADC-A tested the system and evaluated its correctness. All authors have equally participated in the writing of this paper.The authors would like to thank CAPES, FAPERJ, CNPq and FIOCRUZ (INOVA-FIOCRUZ VPPCB-007-FIO-18-2-29) for financial support.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Scientific Reports 10.1038/s41598-018-24002-0, published online 11 April 2018Correction to: The original version of this Article contained typographical errors in title of the paper, where ‘QRS’ was incorrectly given as “Qrs” and a colon was omitted. This has now been corrected in the PDF and HTML versions of the Article and in the accompanying Supplementary Information file.


Moreover, an example is given to support one of our results, and an application is proposed.In this paper, we introduce the concept of  In this paper, b-metric-like spaces which is considered to be an interesting generalization of metric spaces, b-metric spaces [b-metric-like spaces.The Banach contraction mapping principle in metric spaces is an important tool in nonlinear analysis, many authors have been devoting in generalizing metric spaces and the Banach contraction mapping principle. And then, many generalized metric spaces were introduced. In 2013, Alghamdi et al. [c spaces –8. Firstb-metric-like on a nonempty set X is a function if  The pair b-metric-like space with coefficient s.A b-metric-like spaces were introduced as follows.Some concepts in b-metric-like b on X generalizes a topology X whose base is the family of open b-balls Each b-metric-like space A sequence b-metric-like space A sequence b-metric-like space is called to be complete if every Cauchy sequence X converges with respect to A β:Let On the other hand, in , GeraghtLetbe a complete metric space andbe a mapping. IfTsatisfiesfor anywherethenThas a unique fixed point.Recently, many papers about generalization of Geraghty contraction appeared –13. In 2paces in , and theLetbe a complete metric-like space andbe a mapping. If there existssuch thatfor allwherethenThas a unique fixed point.In this paper, we define the new concept of In this section, we begin with the following definitions.X be a nonempty set, f and g be self-mappings on X and f and g are called to be weakly compatible if x is called to be a coincidence of f and g, and w is called to be a point of coincidence of f and g.Let b-metric-like space with coefficient Let Letbe a b-metric-like space, Tandgbe self-mappings onXsuch thatis acontraction of Geraghty type. Ifis a point of coincidence ofTandg, thenT and g, then there exists Suppose that Letbe ab-metric-like space with coefficientbe two mappings withandgXis complete. If the pairis acontraction of Geraghty type, thenTandghave a unique point of coincidence. In addition, ifTandgare weakly compatible, thenTandghave a unique common fixed point.T and g have a point of coincidence. If there exists some T and g, so T and g. We assume that For an arbitrary mentclass2pt{minimmentclass2pt{minimCase I: Case of Applying , 10), a, a\documApplying , we obtamentclasspt{minimaCase II: Case of (k)). By , taking }s=1, by  and 11)\documentyn(k))|. , 15) an an\documgX. Since gX is complete, then there exist From the above discussions, we get that  holds. TBy , we haveNext, we prove Case I. }n→+∞ in , applyin}n→+∞ in , we obtaT and g have a coincidence point u, and v is a point of coincidence of T and g. By Lemma T and g, then we can find T and g have a unique point of coincidence. Moreover, T and g are weakly compatible, then we have T and g have a unique common fixed point. □Case II. }n→+∞ in , and takmentclass2pt{minimmentclass2pt{minimLetting Letbe a complete b-metric-like space with coefficientandbe a mapping. If there existssuch thatfor anywherethenThas a unique fixed point.Taking Letbe a complete metric-like space andbe a mapping. If there existssuch thatfor anywherethenThas a unique fixed point.Corollary Taking Letbe a metric-like space andbe two mappings withandgXis complete. Suppose that there existssuch thatwhereThenTandghave a unique point of coincidence. In addition, ifTandgare weakly compatible, thenTandghave a unique common fixed point.Now, we use an example to illustrate the validity of our main result.Let Case 1: Case 2: We get Case 3: We get Case 4: We get T has a unique fixed point, 0 is the unique fixed point of T.From the above discussions, we know that X with the b-metric-like:Consider the following integral equation:f.Let ution to  is equivNow, we prove the following result.Suppose that the following hypotheses hold: (i)is continuous;(ii)For allthere exists continuoussuch that(iii)There existssuch thatThen the integral equation (has a unique solutionequation  has a unFrom  and 29)29), for f has a unique fixed point x is the unique solution for the integral equation (From the above, we can see that all the conditions of Corollary equation . □In this paper, we introduce a new concept of m 2.1 in . At the


A solid k-mer is a k-mer frequently occurring in NGS reads. The other k-mers are called weak k-mers. A solid k-mer does not likely contain errors, while a weak k-mer most likely contains errors. An intensively investigated problem is to find a good frequency cutoff f0 to balance the numbers of solid and weak k-mers. Once the cutoff is determined, a more challenging but less-studied problem is to: (i) remove a small subset of solid k-mers that are likely to contain errors, and (ii) add a small subset of weak k-mers, that are likely to contain no errors, into the remaining set of solid k-mers. Identification of these two subsets of k-mers can improve the correction performance.NGS data contains many machine-induced errors. The most advanced methods for the error correction heavily depend on the selection of solid k-mers and a mixture of Gaussian distributions to model correct k-mers, and combine them to determine f0. To identify the two special subsets of k-mers, we use the z-score of k-mers which measures the number of standard deviations a k-mer’s frequency is from the mean. Then these statistically-solid k-mers are used to construct a Bloom filter for error correction. Our method is markedly superior to the state-of-art methods, tested on both real and synthetic NGS data sets.We propose to use a Gamma distribution to model the frequencies of erroneous z-score is adequate to distinguish solid k-mers from weak k-mers, particularly useful for pinpointing out solid k-mers having very low frequency. Applying z-score on k-mer can markedly improve the error correction accuracy.The  The massively parallel next-generation sequencing (NGS) technology is revolutionizing a wide range of medical and biological research areas as well as their application domains, such as medical diagnosis, biotechnologies, virology, etc . It has https://www.ncbi.nlm.nih.gov/sra/ERR091571/) is 6.92 billion; however, this number can be reduced to only 1.98 billion after error correction. This reduction significantly alleviates the burden of graph manipulation.The information contained in NGS data is deep and broad, but the raw data is still error prone. Various kinds of errors exist in the raw sequencing data, including substitution, insertion and deletion. The substitution error rate can be as high as 1 to 2.5% for the data produced by the Illumina platform ; and thek-spectrum-based approach, including Quake [k-spectrum-based. It provides a very good scalability and competitive performance. Scalability is crucial for NGS data analysis since the input volume is usually huge.Owing to the importance of error correction, dozens of approaches have been proposed to cope with various types of errors. Depending on the key ideas that have been used, existing approaches can be categorized into three major approaches: (i) the ng Quake , Reptileng Quake , DecGPU ng Quake , SGA [11ng Quake , Musket ng Quake , Lighterng Quake , Blue [1ng Quake , BFC [16ng Quake , MECAT [ng Quake  (ii) theng Quake , HSHREC ng Quake , HiTEC [ng Quake , Fiona [ng Quake  and;  add a small subset of weak k-mers that are likely to contain no errors, into the set of solid k-mers. This is achieved by using f0 as well as z-score of k-mer, z(κ). With the purified set of solid k-mers, the correction performance can be much improved.In this research, we focus on a more challenging but less-studied problem: (i) remove a small subset of solid k-mer frequencies by using KMC2 [z-scores of k-mers. Later, the statistically-solid k-mers are mined by considering both frequency and z-score. After that, the Bloom filter is constructed by the statistically-solid k-mers, and the weak k-mers are corrected. The newly proposed approach is named as ZEC, short for z-score-based error corrector.Our approach starts with counting ing KMC2 , then cak-mer is conventionally defined as a k-mer which occurs in a data set of NGS reads with high frequency. A solid k-mer is usually considered error-free, and taken as the template for error correction. If a k-mer is not solid, then it is defined as a weak k-mer considered as error-containing. Existing k-mer-based approaches use a frequency cutoff, f0, to identify solid and weak k-mers from NGS reads, e.g., BLESS2 [f0 is determined.A solid , BLESS2 , Musket , BLESS2 , and BFC, BLESS2 . The maik-mer is not definitely error-free. Sometimes, it may contain errors with a small chance. It is also true for the weak k-mers — a weak k-mer can be absolutely error-free. The reason that a solid k-mer is not always correct is that the coverage is not under uniform distribution. Thus the cutoff f0 itself is unable to perfectly distinct correct k-mers from erroneous k-mers; cf. the part labeled as α and β in Fig. k-mers as many as possible.In fact, a solid k-mer set as well as the weak k-mer set, so that more correct k-mers can be identified.In this study, we present a time and memory efficient algorithm to purify the solid R be the input set of NGS reads, and K be the set of k-mers contained in R. To determine whether a k-mer, say κ, of K is correct or not, the following metrics are examined: f(κ), the frequency of κ;z(κ), the z-score of κ.Let f(κ) is as follows: (i) scan each read r of R from the beginning to the end; (ii) sum over the occurrence that κ appears. Then the summation is f(κ). This approach works for one k-mer, but it cannot be applied to all the k-mers simultaneously as the number of k-mers can be very large, demanding a huge size of memory.The straightforward approach to determine k,x)-mer; and (iii) it applies the minimizer idea to deal with k-mer.In this study, we make use of the k-mer counting algorithm, KMC2 , to solvk-mer κ, we define the neighbor of κ, N(κ), as D is the edit distance between κ and κ′, and the d0 is the predefined maximum distance. The default value of d0 is 1 as used in this study, but user can adjust this value to any reasonable integer.Given a k-mer cluster centered at κ is defined as k-mers is defined as The z-score of κ, z(κ), is computed by μ is the averaged frequency of F(κ) and σ is the standard deviation of F(κ).The z-score of each k-mer given the frequency of the k-mer as well as that of its neighbor that have been determined by the aforementioned approach.It is straightforward to calculate the k-mers based on their frequency only, we examine their z-scores as well.Unlike existing approaches that determining solid f0 is used to distinct weak and solid k-mers, which is determined as the count minimizing misclassification rates , and those correct ones by a mixture of Gaussian distributions PN(X). A Gamma distribution is defined as: k accounts for the shape of the distribution, θ is for the scale of the distribution, i.e., how the data spread out, Γ(k) is the Gamma function evaluated at k; cf. the dash sky-blue line in Fig. πi is the mixture parameter, μi and σi represent the mean and standard deviation of the component i, and K is the number of Gaussian components. In this study, K is set as 2, with one accounting for k-mers that are from GC rich or poor regions, and the other for the rest correct k-mers.Traditionally, an optimal mentclass2pt{minimk-mers. An example of the two distributions are shown in Fig. f0, such that it can minimize the area marked as α and β. Note that, the threshold f0 determined in this way may not be the intersection point of the two density functions.The two distributions are estimated by using EM algorithm based on the frequencies of f0 cannot perfectly distinct the solid k-mers from the weak k-mers. Taking Fig. k-mers marked by α will be wrongly corrected although they do not have errors but just because their frequencies are lower than f0; likely, the ones marked by β will keep unchanged although they have errors because they have high frequency. To further refine the purity as well as the completeness of solid k-mers, we borrow the statistical idea of using z-score to solve the problem. The purity is defined as It is clear that the optimal pcorrect is the proportion of correct k-mers in the solid k-mers, and perroneous is the proportion of erroneous k-mers in the solid k-mers. The completeness is calculated as where k-mers in the solid k-mers, and Ncorrect is the total number of correct k-mers.where z-score as well as the frequency are collectively incorporated into solid k-mer identification through the following two situations: f(κ)<f0 and z(κ)≥z0, then κ is removed from the weak k-mers and added to the solid k-mers, i.e., increases the completeness.If f(κ)≥f0 and κ is removed from the solid k-mers and added to the weak k-mers, i.e., improves the purity.If The f0 is the minimum frequency that has been determined, while the z0 and z-score and minimum z-score for weak k-mers and solid k-mers, respectively.The z0 and z-score distribution automatically. To obtain the optimal z0, the z-scores of the k-mers having frequency less than f0 are collected. Later, the distribution of these z-scores is estimated and z0 is set as the value having the lowest density between two peaks . Analogously, z-scores of k-mers having frequency greater than f0.The k-mers and; (ii) correct errors in weak k-mers by the Bloom filter.Our error correction model contains two main steps: (i) build Bloom filter from solid h, the false positive rate p, the size of the bit vector n, and the actual number of elements m is Bloom filter  is a prom is the number of solid k-mers that have been determined from all the k-mers by means of the aforementioned algorithm. Per existing approaches, p is set to 1%. One can also tune p, h and n to fit the real hardware limitations.In our study, k-mers that are used to construct Bloom filter, which directly improves the error correction performance in theory. Note that, the solid k-mers play the key role in error correction, as all the rest k-mers (viz. the weak k-mers) are to be corrected based on the solid ones.It has been reported that the Bloom filter has been successfully used to correct NGS errors, such as BLESS2  and BFC Figure k-mer of the read from the beginning to the end sequentially. (ii) partition the k-mers into groups that each group contains only solid k-mers or weak k-mers, deemed as solid group Gs or weak group Gw, respectively. The order of the groups is kept according to their appearance in the read. (iii) correct the errors causing the weak group Gw according to the following situations: Gw is the first group and there exists a successive group Gs that is solid, we iteratively change the first base of each k-mer of Gw to its alternatives and check the existence of the k-mers against the Bloom filter. Once there exist a solution that makes all the weak k-mers solid, the amendment of the bases is accepted, thus the correction of the error. This process is applied to the k-mers of Gw from the last one to the first one. In case the number of k-mers contained in Gw is less than a predefined value, say τ, the processive solid k-mers that are extended from the corrected k-mers will be generated until the total number of k-mers in Gw is τ. If this criterion cannot be satisfied, the solution is abandoned. On the other hand, if Gs does not exist, we will alter the bases to their alternatives of all the k-mers iteratively until a solution that make all the k-mers solid can be found.If Gw has a solid processive group Gs and a solid successive group k-mer in Gw by its alternatives from the first k-mer to the last k-mer, namely the forward search. Solutions that make all the k-mers solid till the current substitution are recorded. Similarly, the backward search is conducted on the first base of the k-mers from the last one to the first one. A solution is accepted if the forward search and the backward search meet and the k-mers contained in both of them are solid. In case the number of k-mers in Gw is less than k, we will only alter the last base of the first k-mer.If Gw is the last group and there exists a solid processive group Gs, we will apply the backward search to obtain the solution. Analogously to the first situation, if the number of k-mers of Gw is less than τ, we will extend the k-mers toward their downstream until the number is satisfied. In case Gs does not exist, it is the same as the second part of the first situation, thus the same approach is applied.If By using Bloom filter, the errors contained in each read can be correct as follows: (i) check the existence of each We collected six data sets to test the performance of our proposed method in comparison with the state-of-art methods. Four of the six data sets are the NGS reads produced by the Illumina platform, including Staphylococcus aureus (S. aureus), Rhodobacter sphaeroides (R. sphaeroides), Human Chromosome 14 (H. chromosome 14) and Bombus impatiens (B. impatiens). These data sets are the gold standards used by GAGE  for NGS gain, recall, precision and per base error rate (pber). Gain is defined as (TP−FP)/(TP+FN), recall is TP/(TP+FN), precision is TP/(TP+FP) and pber is Ne/N, where TP stands for the number of corrected bases that are truly erroneous bases, FP represents the number of corrected bases that are not sequencing errors intrinsically, FN is the number of erroneous bases that remain untouched, Ne is the number of erroneous bases and N is the total number of bases. Among these metrics, gain is the most informative.The error correction performance is evaluated through the widely accepted procedure implemented by . MetricsAll experiments are carried out on a cluster having eight Intel Xeon E7 CPUs and 1Tb RAM. Each CPU has eight cores.Overall Performance of ZEC. The experimental results of ZEC are presented in Table Relation with GC-content. A previous study by Ross et al. [k-mers obtained from the reads sequenced from these regions are more likely to be treated as weak. Figure k-mer frequency derived from H. chromosome 14. It can be seen that the k-mers having a low frequency can spread out wider than those having a high frequency, and the wide range is coincident with the GC content. This result is in accordance with the performance shown in Table k-mers is necessary, particularly for the subset of k-mers that have a low frequency. More importantly, it empirically supports our idea of using mixture model to treat solid k-mers and weak k-mers separately.s et al.  shows thComparison with State-of-the-art. The performance of ZEC is much superior to the state-of-the-art methods, including Lighter [gain. For instance, on the dataset R4, the gain of ZEC is 0.746, while the best performance produced by the other methods is 0.705. For the synthetic datasets, ZEC also has higher gain than other methods. For example, on the dataset S2, the gain of ZEC is 0.853, while the best and worst gain generated by the other methods are 0.849 and 0.058, respectively. The lowest average per-base error rate of ZEC also consolidates its effectiveness. Lighter , Racer [ Lighter , BLESS2  Lighter , Musket  Lighter , SGA [11 Lighter . See Tabz-score for identifying the two special subsets of k-mers from the sets of solid k-mers and weak k-mers. An example of z-score distribution pertaining to k-mer frequency is shown in Fig. k-mers shown in the figure have relatively low frequencies—less than 9, while the z-scores are pretty high—greater than 1. Interestingly, almost all the solid k-mers (the top right region) have the similar level of z-scores comparing to these highlighted ones. These observations indicate that the highlighted k-mers are very likely to be correct k-mers instead of erroneous k-mers although their frequencies are very low. The z-score distribution pertaining to the other three real data sets has similar patterns compared to the one shown here.The key to the performance improvement is the idea of using k-mers that can be refined comparing to the solely frequency determined k-mers are 12.3%, 14.2%, 11.4%, 7.1% for the real data R1, R2, R3 and R4, respectively; see Fig. By exploring the four real data sets, we found that the proportion of k-mers is not trivial for very large data sets, as the k-mers and their frequencies are usually too large to be hold by a main memory of a moderate computer. We designed a novel algorithm and solved this problem. The efficiency of the algorithm in terms of the memory usage and running speed are studied.Calculating z-score of k-mers as well as their frequencies) that can be held by only one bit vector. The memory saving ratio is calibrated as the ratio between the real memory allocation and the input data volume. For instance, the ratio of 0.01 pertaining to the data R4 means that the allocated memory is one percent of the input size of R4. That is, 70Mb memory is allocated for holding the 6.97Gb data. It is promising that, with one percent memory allocation, around 22 percent input data can be hold by only one bit vector. When the memory allocation increased to 2.5 percent, 30 percent input data and even more can be held by one bit vector. Obviously, keep increasing the size of allocated memory does not guarantee the linear scale of holding the input data. Based on the experiments, we set the memory allocation ratio to 2.5 percent through the whole study. Typically, three bit vectors are constructed for holding all the input. Note that, the size of bit vector decreases along with the reduced size of input. The ratios between the input and the total allocated memory are 20.0, 13.4, 7.0 and 7.9 for the four real datasets, respectively.Figure k-mer in a bit vector is O(1) pertaining to time complexity by using hash, this algorithm is pretty fast. For instance, based on our computing power, it only takes 387 s to construct the bit vectors and calculate the z-scores of all the k-mers of R4—the largest data set.Regarding the running speed, this algorithm is linearly scaled. Since locating each z-score of a k-mer different from its genuine value. However, the false positive rate is pretty small, usually less than 1%, thus this impact can be neglected.Since a Bloom Filter has false positives, this may cause the k-mers having low frequency, achieving an improvement of 11.25% on weak k-mers. However, some issues still remain further exploration, including neighbor inclusion and neighbor retrieval.Our model effectively pinpoints out correct k-mers are determined given a k-mer of interest, say κ. Our current approach takes k-mers having edit distance of 1 as neighbors of κ, but there still has a small chance that a true neighbor having edit distance larger than 1. Suppose the error rate is e, the probability of a k-mer having exactly one error is k·e(1−e)k−1/k·e=(1−e)k−1. When e=1% and k=1, the probability is (1−0.01)31−1=73.97%. That been said, about 26% real neighbors are excluded. However, even extending the minimum edit distance from 1 to 2 significantly elongates running time. This is because the number of candidate k-mers increases from 3∗k to 3∗k∗3∗(k−1).Neighbor inclusion means how neighbor k-mers is too large to fit into a main memory. Hence, a more sophisticated approach is required to solve this problem. We use Bloom Filter to overcome the limitation. For k-mers having small count, say 5, we use classical Bloom Filters to save them, each Bloom Filter saves k-mers having the same count. For k-mers having large count, we use coupled-Bloom Filter to save them. One Bloom Filter for k-mer encoding, while the other is for count representation. This approach significantly reduces memory usage while achieving constant time complexity of k-mer retrieval. However, it may cause false positives although the probability is small. Hence, more effort is required to handle this problem.Neighbor retrieval is another issue to be considered. Usually, the size of counted k-mers to construct the Bloom filter. These k-mers are mined from all the k-mers of a NGS data set by considering both their frequency and z-score, particular the latter one that can effectively fishing out the solid k-mers having low frequency. Pinpointing out such k-mers has been a very challenging problem. The experimental results show that our approach markedly outperforms the existing state-of-the-art methods in terms of error correction performance.We have proposed a novel method for correcting the NGS errors. The novel idea is the use of statistically-solid


The development of high-throughput sequencing technologies for 16S rRNA gene profiling provides higher quality compositional data for microbe communities. Inferring the direct interaction network under a specific condition and understanding how the network structure changes between two different environmental or genetic conditions are two important topics in biological studies. However, the compositional nature and high dimensionality of the data are challenging in the context of network and differential network recovery. To address this problem in the present paper, we proposed two new loss functions to incorporate the data transformations developed for compositional data analysis into D-trace loss for network and differential network estimation, respectively. The sparse matrix estimators are defined as the minimizer of the corresponding lasso penalized loss. Our method is characterized by its straightforward application based on the ADMM algorithm for numerical solution. Simulations show that the proposed method outperforms other state-of-the-art methods in network and differential network inference under different scenarios. Finally, as an illustration, our method is applied to a mouse skin microbiome data. However, a more reasonable assumption is that the difference between the precision matrices are sparse, not that both matrices are sparse, since direct interactions may not be sparse while the changes under different conditions are often sparse . To be specific, θij is replaced with θijsij, where l, u) = ,  and (0.0.2) separately to study the performance of CDTr loss when the exchangeable condition is satisfied by different degrees. These scenarios are named as Band-exact (Band-e), Band-approx1 (Band-a1), Band-approx2 (Band-a2) and Cluster-exact (Cluster-e), Cluster-approx1 (Cluster-a1), Cluster-approx2 (Cluster-a2), respectively. To obtain a positive definite precision matrix Θ, we first compute the smallest eigenvalue of Θ (denoted by e); then the diagonal elements of Θ are set as |e| + 0.3. The deviation to the exchangeable condition is measured with dev = ‖GΣ − ΣG‖F. The deviations under the aforementioned six scenarios are listed in n = 50, 100, 150, 200, a total of 100 datasets are generated and used to recover the network structure. Four state-of-the-art methods for network recovery are investigated, including gCoda  ∪ , if |i − j| = m, m = 1, 2, 3, 4.Neighbor graph: Select p points from Scale-free graph: A scale-free graph is produced, following the B-A algorithm  ∪ .lgorithm . The iniHub graph: Partition the nodes into 3 disjoint groups evenly and select a node as hub for each group. The hubs are connected with the non-hubs in the same group with strength uniformly distributed in  ∪ .Block graph: Divide p nodes into 5 blocks evenly. Connect pairs in the same block with probability 0.3 and pairs in different blocks with probability 0.1. The strength between connected nodes is uniformly distributed in  ∪ .We further conducted several experiments on the following six representative network structures, without considering the exchangeable condition.e| + 0.3, where e is the smallest eigenvalue of Θ. The deviations from the exchangeable condition of these networks are listed in Similarly, the diagonal elements of Θ are set as |GΣxln G and assumes that the inverse of GΣxln G also approximates the inverse of Σ. However, as Fang et al.  ∪ .Band graph: Connect pair  with strength uniformly distributed in  ∪ , if |i − j| = m, m = 1, 2, 3, 4.Neighbor graph: Select p points from Scale-free graph: The scale-free graph is generated with the B-A algorithm  ∪ .Block graph: Divide p nodes into 5 blocks evenly. Connect pairs in the same block with probability 0.5 and pairs in different blocks with probability 0.3. The strength between connected nodes is uniformly distributed in  ∪ .We investigate the performance of DCDTr loss with some experiments in this section. The first precision matrix Θ is generated as follows:e| + 0.3, where e is the smallest eigenvalue of Θ or Θ*, respectively. The deviations from the exchangeable condition of Θ and Θ* are listed in ℓ1-M, are used to estimate the true differential matrix Δ. Similarly, we evaluate the true positive rate and true negative rate at different tuning parameters and then compute the area under the ROC curve (AUC). We take the sample size n = 100, 200, 300, 400 and repeat this procedure 100 times.Then 10% of the connected pairs in Θ will change to an unconnected state, while the same number of unconnected pairs in Θ will change to a connected state, such that we get another precision matrix Θ*. For scale-free and hub graph, the ratio of change is 40% based on the sparsity of the two graphs. The diagonal elements of Θ and Θ* are set as |p = 77 OTUs and n = 232 samples . We use Bayesian-multiplicative replacement + used in Algorithm 1 and Algorithm 2 for the numerical solutions of lasso penalized CDTr and DCDTr loss are presented in this Supplementary. We also demonstrate the relationship between D-trace loss and CDTr loss, as well as the relationship between DTL loss and DCDTr loss. The detailed formulas of ℓ1-minimization method and joint graphical lasso  are listed in this Supplementary.The matrix operators (PDF)Click here for additional data file.


Nature Communications 10.1038/ncomms9025, published online 13 August 2018.Correction to: 13).’ This should say ‘C-terminus’ instead of ‘N-terminus’. The error has not been fixed in the PDF or HTML versions of the Article.This Article contains an error in the first sentence of the ‘KIF1A purification and labelling’ section of the Methods, which incorrectly reads ‘A construct containing the first 382 residues of KIF1A with a His-tag and a Cys residue in the N-terminal, was kindly provided by N. Hirokawa (University of Tokyo, Japan; see ref.


Nature Communications 10.1038/s41467-019-10278-x, published online 03 June 2019.Correction to: AppNL-G-F’ mice at 24 months  was inadvertently duplicated from the image to its left, the representative image for the ‘WT’. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in Fig. 2b, in which the representative image for ‘


Scientific Reports 10.1038/s41598-018-37122-4, published online 29 January 2019Correction to: The original version of this Article contained an error in the title of the paper, where the word “metalloproteinases” was incorrectly given as “metallorproteinases”. This has now been corrected in the PDF and HTML versions of the Article.


C. albicans,” panel F. Please see the complete, correct There is an error in the caption for


Correction to:npj Digital Medicine 10.1038/s41746-019-0099-8, Published online 10 April 2019The original version of the published Article contained an error in the spelling of the first Author’s name. “Paisan Raumviboonsuk” has been changed to “Paisan Ruamviboonsuk”. This has been corrected in the HTML and PDF version of the Article.


Nature Communications; 10.1038/s41467-018-07599-8; published online 10 December 2018Correction to: The original PDF version of this Article contained an error in Fig. 2, in which only the central part of the figure, between ‘Mass loading’ and ‘Gravimetric performance’, was visible. This has been corrected in the PDF version of the Article. The HTML version was correct from the time of publication.