The merging of network theory and microarray data analysis techniques has spawned a new field: gene coexpression network analysis. While network methods are increasingly used in biology, the network vocabulary of computational biologists tends to be far more limited than that of, say, social network theorists. Here we review and propose several potentially useful network concepts. We take advantage of the relationship between network theory and the field of microarray data analysis to clarify the meaning of and the relationship among network concepts in gene coexpression networks. Network theory offers a wealth of intuitive concepts for describing the pairwise relationships among genes, which are depicted in cluster trees and heat maps. Conversely, microarray data analysis techniques  can also be used to address difficult problems in network theory. We describe conditions when a close relationship exists between network analysis and microarray data analysis techniques, and provide a rough dictionary for translating between the two fields. Using the angular interpretation of correlations, we provide a geometric interpretation of network theoretic concepts and derive unexpected relationships among them. We use the singular value decomposition of module expression data to characterize approximately factorizable gene coexpression networks, i.e., adjacency matrices that factor into node specific contributions. High and low level views of coexpression networks allow us to study the relationships among modules and among module genes, respectively. We characterize coexpression networks where hub genes are significant with respect to a microarray sample trait and show that the network concept of intramodular connectivity can be interpreted as a fuzzy measure of module membership. We illustrate our results using human, mouse, and yeast microarray gene expression data. The unification of coexpression network methods with traditional data mining methods can inform the application and development of systems biologic methods. Similar to natural languages, network language is ever evolving. While some network terms (concepts) are widely used in gene coexpression network analysis, others still need to be developed to meet the ever increasing demand for describing the system of gene transcripts. There is a need to provide an intuitive geometric explanation of network concepts and to study their relationships. For example, we show that certain seemingly disparate network concepts turn out to be synonyms in the context of coexpression modules. We show how coexpression network language affects our understanding of biology. For example, there are geometric reasons why highly connected hub genes in important coexpression modules tend to be important, and why hub genes in one module cannot be hubs in another distinct module. We provide a short dictionary for translating between microarray data analysis language and network theory language to facilitate communication between the two fields. We describe several examples that illustrate how the two data analysis fields can inform each other. Many biological networks share topological properties. Common global properties include modular organization t-test of differential expression leads to a gene significance measure. Many traditional microarray data analysis methods focus on the relationship between the microarray sample trait and the gene expression data. For example, gene filtering methods aim to find a list of  genes that are significantly associated with the microarray sample trait; another example are microarray-based prediction methods that aim to accurately predict the sample trait on the basis of the gene expression data.One goal of this article is to describe existing and novel network concepts . Gene coexpression networks are built on the basis of a gene coexpression measure. The network nodes correspond to genes—or more precisely to gene expression profiles. The ija between gene expression profiles ix and jx can be defined by hard thresholding the coexpression similarity ijs as followsija = 1) if the absolute correlation between their expression profiles exceeds the (hard) threshold τ. Hard thresholding of the correlation leads to simple network concepts  but it may lead to a loss of information: if τ has been set to 0.8, there will be no link between two genes if their correlation equals 0.799. To preserve the continuous nature of the coexpression information, one could simply define a weighted adjacency matrix as the absolute value of the gene expression correlation matrix, i.e., [ija] = [ijs]. However, since microarray data can be noisy and the number of samples is often small, we and others have found it useful to emphasize strong correlations and to punish weak correlations. It is natural to define the adjacency between two genes as a power of the absolute value of the correlation coefficient β≥1. This soft thresholding approach leads to a weighted gene coexpression network. We present empirical results for weighted and unweighted networks in the main text, Using a thresholding procedure, this coexpression similarity is transformed into a measure of connection strength (adjacency). An unweighted network adjacency n individuals filled out an interest questionnaire, which was used to define a pairwise similarity score ijs. For convenience, we assume that the similarity measure takes on values between 0 and 1. Our definition of the affection network is based on the following assumption: the more similar the interests between two individuals, the more affection they feel for each other. More specifically, we assume that the affection (adjacency) ija between two individuals is proportional to their similarity on a logarithmic scale, i.e.,ija = ijβs (Equation 2). A soft threshold β = 2 implies that the affection ija equals 0.25 if the similarity ijs equals 0.5.Since humans are organized into social networks, social network analogies should be intuitive to many readers. Therefore, we will refer to the following ‘affection network’ throughout this article. Assume that GS that assigns a nonnegative number to each gene; the higher iGS the more biologically significant is gene i. We assume that the minimum gene significance is 0. For example, if a statistical significance level  is available for each gene, the gene significance of the ith gene can be defined as minus log of the p-value, i.e., iGS = −log(ip). In this article, we are particularly interested in gene significance measures that are based on a microarray sample trait, e.g., a clinical outcome. The microarray sample trait T =  may be quantitative  or binary . Since our goal is to provide a simple geometric interpretation of coexpression network analysis, we define the trait-based gene significance measure by raising the correlation between the ith gene expression profile ix and the clinical trait T to a power ββ could be used in Equation 4, we use the same power as in Equation 2 to facilitate a simple geometric interpretation.Many network applications use at least one gene significance measure. Abstractly speaking, we define a gene significance measure as a function iax+b, jcx+d) = cor, we can assume without loss of generality that the vectors ix have a mean 0 and are of the same length. In other words, they correspond to points on a hypersphere.We find it convenient to express network quantities in terms of correlation coefficients since the correlation between two vectors can be interpreted as the cosine of the angle between them (measured in radians) if the vectors are scaled to have a mean of 0. Since the correlation is scale-invariant, i.e., cor between the two points on the hypersphere. Soft thresholding methods (Equation 2) preserve the continuous nature of these distances. The higher the soft threshold β, the more weight is assigned to short geodesic distances compared to large distances.The network adjacency iGS = |cor|β, (Equation 4) is scale-invariant, the sample trait T can also be considered a point on the hypersphere. Analogous to the network adjacency, the smaller the geodesic distance between the ith gene expression profile and the trait T, the higher the gene significance of the ith gene. In other words, the smaller the angle between the sample trait and the expression profile, the more significant is the gene.Since the trait-based gene significance measure ija = |cor| between the genes. For each gender and tissue type ija. As can be seen from the color bar underneath the heat maps, red and green in the heat map indicate high and low absolute correlation, respectively. The genes in the rows and columns of each heat map are sorted by the corresponding cluster tree.As a motivational example, we study the pairwise correlations among 498 genes that had previously been found to form a sub-network related to mouse body weight. The microarray data measure the expression levels in multiple tissue samples  from male and female mice of an F2 intercross. Approximately 100 tissue samples are available for each gender/tissue combination. The biological significance of this subnetwork is described in ClusterCoef = 0.42, 0.43, 0.41, respectively). In contrast, the female adipose, male adipose, and male brain networks have lower mean clustering coefficients . Difference in module structure may reflect true biological differences or they may reflect noise .It is visually obvious that the heat maps and the cluster trees of different gender/tissue combinations can look quite different. Network theory offers a wealth of intuitive concepts for describing the pairwise relationships among genes that are depicted in cluster trees and heat maps. To illustrate this point, we describe several such concepts in the following. By visual inspection of As another example for the use of network concepts, compare the cluster tree of the female brain network with that of the male brain network. The cluster tree of the female network appears to be comprised of a single large branch, i.e., a highly connected hub gene at the tip of the branch forms the center in this network. In contrast, the cluster tree corresponding to the male brain network appears to split into multiple smaller branches, i.e., no single gene forms the center. To measure whether a highly connected hub gene forms the center in a network, one can use the concept of centralization (defined below). The female brain and male brain networks have centralization 0.34 and 0.21, respectively.i as the absolute correlation between the gene expression profile ix and body weight T, i.e., iGS = |cor|. HGS as slope of a regression line (through the origin) between gene significance and scaled connectivity. As can be seen from T.These examples illustrate that graph theory contains a wealth of network concepts that can be used to describe microarray data. But we will argue that microarray data analysis techniques can also be used to derive network theoretic results. For example, network theorists have long studied the relationship between gene significance and connectivity. Several network articles have pointed out that highly connected hub nodes are central to the network architecture A = [ija], where 1≤i,j≤n. We assume that the pairwise adjacency (connection strength) ija takes on values in the unit interval, i.e., 0≤ija≤1. For notational convenience, we set the diagonal elements to 1. In the NCF by evaluating a network concept function NCF on the adjacency matrix A and/or a corresponding gene significance measure GS. This abstract definition will be useful in defining intramodular network concepts  and eigengene-based analogs of network concepts . In the following, we describe several network concepts including the connectivity, the maximum adjacency ratio, the density, and the centralization.We define network concepts for (weighted) undirected networks that can be represented by a symmetric adjacency matrix connectivity  of the ith gene is defined byik equals the number of genes that are directly linked to gene i. In weighted networks, the connectivity equals the sum of connection weights between gene i and the other genes.The maximum connectivity is defined asiscaled connectivity K of the i-th gene is defined byiK≤1. Note that we distinguish the scaled from the unscaled connectivity by using an upper case “K” and a lower case “k”, respectively.The Social Network Interpretation of the Connectivity: For the aforementioned affection network (Equation 3), assume that the affection (adjacency) ija equals 1 if two individuals strongly like each other; it equals 0.5 if they are neutral towards each other, and it equals 0 if they strongly dislike each other. Then the scaled connectivity iK is a measure of relative popularity: high values of iK indicate that the ith person is well liked by many others.Potential Uses of the Connectivity: The connectivity is the most widely used concept for distinguishing the nodes of a network. As described in the motivational example and detailed below, intramodular connectivity can be used to define a systems biologic gene screening strategy that keeps track of module membership information maximum adjacency ratio of gene i as followsik = j≠iija>0. One can easily verify that 0≤ija≤1 implies 0≤iMAR≤1. Note that iMAR = 1 if all nonzero adjacencies take on their maximum value of 1, which justifies the name “maximum adjacency ratio.” By contrast, if all nonzero adjacencies take on a small (but constant) value ija = ε, then iMAR = ε will be small.ΣFor weighted networks, we define the iSocial Network Interpretation of the Maximum Adjacency Ratio: MAR = 1 suggests that the ith individual does not form neutral relationships; this individual either strongly likes or dislikes others. In contrast, iMAR = 0.5 suggests the ith individual forms less intense relationships with others.Potential Uses of the Maximum Adjacency Ratio: Since iMAR = 1 for all genes in an unweighted network, the maximum adjacency ratio is only useful for weighted networks. The MAR can be used to determine whether a hub gene forms moderate relationships with a lot of genes or very strong relationships with relatively few genes. To illustrate this point, we show in the following simple example that the MAR can be used to distinguish nodes that have the same connectivity. Assume a network (labeled by I) for which the adjacency between node 1 and every other node equals aj1,(I) = 1/(n−1). Then k1(I) = (n−1)/(n−1) = 1 and MAR1(I) = 1/(n−1). For a different network (labeled by II) where a1,2(II) = 1 and aj1,(II) = 0 for j≥3, the connectivity k1(II) still equals 1 but MAR1(II) = 1.iMAR is often highly correlated with the connectivity iK . As we demonstrate in iMAR is sometimes  superior to iK when it comes to identifying biologically important intramodular hub genes. As aside, we mention that a directed network analog of iMAR has been used in the analysis of metabolic fluxes In weighted coexpression networks, we find empirically that network density  denotes the vector of connectivities and the function vector v is defined by pS(v) = iipv.ΣThe Social Network Interpretation of the Density: The density measures the overall affection among individuals. A density close to 1 indicates that all individuals strongly like each other while a density of 0.5 suggests the presence of more ambiguous relationships.Potential Uses of the Density: The density of genes in a subnetwork  can be used to measure whether this sub-network is tight or cohesive. In our motivational mouse tissue example, we find that a network of genes has high density in liver tissue but low density in adipose tissue. The goal of many module detection methods is to find clusters of genes with high density.network centralization  across the individuals.Potential Uses of the Heterogeneity: Describing the reasons for and the meaning of the heterogeneity of complex networks has been the focus of considerable research in recent years clustering coefficient of gene i is a density measure of local connections, or “cliquishness” iClusterCoef equals 1 if and only if all neighbors of i are also linked to each other. For weighted networks, 0≤ija≤1 implies that 0≤ijClusterCoef≤1 The Social Network Interpretation of the Clustering Coefficient: The higher the clustering coefficient of an individual, the higher is the affection among his friends. The clustering coefficient is zero if all of his friends strongly dislike each other.Potential Uses of the Clustering Coefficient: As described in our motivational example, the mean clustering coefficient has been used to measure the extent of module structure present in a network. The relationship between the clustering coefficient and connectivity has been used to describe structural  properties of networks hub gene significance:iGS is proportional to the scaled connectivity (iGS = icK), the hub gene significance equals the constant of proportionality: HubGeneSignif = c. The hub gene significance equals the slope of the regression line between iGS and iK if the intercept term is set to 0  and grade point average.Potential Uses of the Hub Gene Significance: Several studies have shown that the relationship between connectivity and gene significance  carries important biological information. For example, in the analysis of yeast networks, highly connected hub genes were found to be essential for yeast survival and there is evidence that hub genes are preserved across species network significance measure as the average gene significance of the genes:We define the Social Network Interpretation of the Network Significance: The network significance simply measures the average grade point average among the individuals.Potential Uses of the Network Significance: We refer to the network significance of a module network as “module significance.” The module significance measure can be used to address a major goal of gene network analysis: the identification of biologically significant subnetworks or pathways.centroid significance as the gene significance of a suitably chosen representative node (centroid) in the network.i.centroid denotes the index associated with the centroid. A centroid can be defined in many different ways, e.g., based on connectivity or other centrality measures. In our applications, we define the centroid as the most highly connected gene in the network. If multiple genes attain the maximum connectivity, we define the centroid significance by their average gene significance.We define the centroid conformity of the ith gene as the adjacency between the centroid and the ith geneith gene.We define the Social Network Interpretation of the Centroid Conformity: In our affection network, we choose the most popular individual as centroid; then his or her grade point average is the centroid significance. The centroid conformity of the ith individual equals his or her affection (connection strength) with the most popular individual.Potential Uses of the Centroid Conformity: Below, we will characterize coexpression networks for which the adjacency ija can be approximated by a product of the centroid conformities: ija≈i CentroidConformityjCentroidConformity. We will use this insight to derive relationships among seemingly disparate network concepts. For example, the mean clustering coefficient (Equation 12), the density (Equation 9), and the heterogeneity (Equation 11) measure different network properties but we show that they satisfy a simple relationship (Equation 31) in coexpression modules. Further, we will use the centroid significance to derive a simple relationship (Equation 37) between module significance (Equation 14) and hub gene significance (Equation 13).One of the many biological applications of gene coexpression networks is the identification of pathways (modules) and centrally located genes (referred to as module centroids). In our applications, we define highly connected intramodular hub genes as module centroids. Weighted gene coexpression network analysis WGCNA,  can be ca priori defined gene sets. Instead, modules are constructed from the expression data by using a tight clustering procedure. Although it is advisable to relate the resulting modules to gene ontology information to assess their biological plausibility, it is not required. Because the modules may correspond to biological pathways, focusing the analysis on modules (and corresponding centroids) amounts to a biologically motivated data reduction method. Intramodular hub genes are centrally located in the module and thus lend themselves as candidates for biomarkers. Examples of biological studies that show the importance of intramodular hub genes can be found reported in r>0.90), typically dozens of candidates result. Although these candidates are statistically equivalent, they may differ in terms of biological plausibility or clinical utility.An outline of WGCNA is presented in Roughly speaking, we define network modules as groups of highly interconnected genes. As detailed in Q modules. We denote the adjacency matrix of the genes inside the qth module by Aq)( as the gene significance measure restricted to the module genes. Denote by nq) to denote quantities associated with the qth module. But for notational convenience, we sometimes omit (q) when the context is clear.In the following, we assume that a module detection method  has found NCF(Aq) by evaluating a network concept function NCF on the adjacency matrix Aq)( and/or a corresponding gene significance measure GSq) of a module network simply as the The high dimensionality of gene expression data has inspired two broad categories of data reduction techniques. The first category, often used by network theorists, is to reduce the gene coexpression networks into modules. Each module can be represented by a centroid, e.g., an intramodular hub gene. The second category, often used by microarray data analysts, reduces the gene expression data to a small number of components that capture the essential behavior of the expression profiles qth module, denote by Xq)( the nq)(×m matrix of nq)( gene expression profiles across m microarrays:ix denotes the gene expression vector of the ith gene.For the Xq)( is given by Xq)( = Uq)(Dq)((Vq)T, where Uq)( is an nq)( and Dq) of Xq)] whose entries correspond to the pairwise correlations between the rows (genes) of Xq)(| are arranged in decreasing order. Adapting terminology from Vq) and simply refer to E as the eigengene. The module eigengene can be used to summarize and represent the expression profiles of the module genes, see Eq)( and Ep).The module eigengenes of different modules can be highly correlated . DetectiUnlike traditional microarray data reduction methods that impose orthogonality  or independence , gene coexpression network analysis can be considered a pathway-based data reduction method that allows dependencies between the modules. When focusing on the use of module eigengenes, network analysis can be considered a variant of oblique factor analysis.While a high level view of modular gene coexpression networks can be viewed as a data reduction technique, many network analyses focus on the pairwise relationships of relatively few (hundreds) of correlated genes, i.e., genes that form a single module in a larger network. For example, the 498 genes of our motivational example were part of a body weight related module, which was found in a large gene coexpression network based on the female mouse liver samples empirical evidence that many module adjacency matrices, i.e., networks comprised of genes of a single module, are approximately factorizable i and j can approximately be factored into gene specific contributions, i.e., ijaq)(≈iCFq)(jCFq)( with iCFq)(jCFq)(]. The conformity vector CFq)≤1 and the close resemblance to the proportion of variance explained by the module eigengene (Equation 22). In the EF(Xq)≈1 implies that the correlation matrix factors as followsAn open theoretical research question is to characterize microarray data that lead to factorizable coexpression networks. Here we solve this problem for the case of modules in a gene coexpression network. Toward this end, we propose the following measure of (q)) is close to 1, the adjacencies of the weighted coexpression module network A(q) = |cor(X(q))|β and the trait-based gene significance measure GSi(q) = |cor(xi(q),T)|β can be factored as followsIf the eigengene factorizability EF. with respect to T, also denoted as EigengeneSignifAs described in EAq)( and gene significance measure EGSq)( as follows:The eigengene-based approximations on the right hand side of Equation 25 motivate us to define the eigengene-based adjacency matrix to 1 see .ija can be approximated by a product of the centroid conformities (Equation 16): ija≈i CentroidConformityjCentroidConformity.Abstractly speaking, Observation 1 allows us to characterize coexpression networks for which the adjacency EF(Xq)≈1 if the module gene expressions ixq)( = Eq)(. A rough geometric intuition of ijaq)( (Equation 25) is presented in Eq)( and the ith gene expression profile is denoted by iθ. The angle between gene expression profiles i and j is denoted by ijθ. In the ijθ≈|iθ±jθ| and sin(iθ) sin(jθ)≈0 imply approximate factorizability of the correlation matrix.In the eigengene-based network concepts as a step towards a geometric interpretation of network concepts. Analogous to the case of intramodular network concepts, we define eigengene-based network concepts by evaluating the network concept function NCF(EAq) on the eigengene-based adjacency matrix EAq)( (Equation 28) and the eigengene-based gene significance measure EGSq)( (Equation 29). One can easily derive the following formulas for eigengene-based network concepts:Aq)(≈EAq) this implies NCF(Aq)≈NCF(EAq). We summarize this observation as followsHere we define (q) = |cor(X(q))|β and the eigengene factorizability EF(X(q)) is close to 1, the network concepts can be approximated by their eigengene-based analogs.If AThis observation is illustrated in ija = ijβs (Equation 2) on the network heterogeneity. Using extensive simulation studies reported on our webpage, we found that for the vast majority of networks, the heterogeneity increases with the soft threshold β. Thus, for most coexpression networks, increasing β makes it easier to discern highly connected genes from less connected genes. However, one can construct networks for which increasing β leads to a lower heterogeneity. The situation is much simpler for the eigengene-based heterogeneity EHeterogeneityq)( (Equation 30). In the β. Thus, the heterogeneity will be an increasing function of β if it can be approximated by its eigengene based analog (Observation 2).It can be advantageous to replace network concepts by their eigengene-based analogs when studying theoretical properties. To illustrate this point, we briefly describe the effect of soft thresholding A major theoretical advantage of eigengene-based network concepts is that they reveal simple relationships amongst each other. For example, it is straightforward to derivemaximum eigengene conformity as ea,maxq), where e,jaq)( = |cor(jxq)|β (Equation 26). In most modules, we find genes that have very high correlations (r≈0.99) with the module eigengene. For a low power β, this implies that the maximum eigengene conformity is approximately equal to 1:To arrive at particular simple relationships among network concepts, we make use of the following terminology. We denote the maximum conformity assumption. With the results in the We refer to Equation 32 as the (q) = |cor(X(q))|β, EF(X(q))≈1 and the maximum conformity assumption applies, intramodular network concepts satisfy the following relationshipsIf A(q)) denotes the mean clustering coefficient, ClusterCoefmax(q) = maxj(ClusterCoefj(q)) and MARmax(q) = maxj(MARj(q)).where mean( = |cor(ixq)|β (Equation 33) facilitates a geometric interpretation of the intramodular connectivity: the smaller the angle iθ between the ith gene expression profile and the module eigengene, the larger is |cos(iθ)|β = e,iaq)( = |cos(ijθ)|β where ijθ denotes the angle between gene expression profiles i and j. By definition (Equation 9), the smaller the pairwise angles ijθ between the gene expression profiles, the higher is the module density. Equation 39 provides another interpretation: the smaller the angles iθ between the module gene expression profiles and the module eigengene, the higher is the density. Thus, the density can be interpreted as a measure of average closeness between the gene expression profiles and the module eigengene. By definition, coexpression module networks have a relatively high density (see We provide two geometric interpretations of the density. The first makes use of the relationship sity see .Eaq)( (Equation 29) if the eigengene has a small angle with the sample trait and iθ is small. Similarly, the geometric interpretation of the hub gene significance (Equation 13) is straightforward: the smaller the angle between the module eigengene and the sample trait, the higher is the hub gene significance (Equation 34).The We provide two geometric interpretations of the module significance (Equation 14). The first interpretation is based on the definition of the module significance as average gene significance; a module has high module significance if on average the angles between the module expression profiles and the sample trait tend to be small. The second interpretation of the module significance is based on Equation 37: a module has high significance if the module density is high and the angle between the module eigengene and the sample trait is small.Here we illustrate how the geometric interpretation of gene coexpression networks can be used to derive results, which may be interesting to microarray data analysts.Multiple approaches are conceivable for summarizing the expression profiles of the genes inside a single module. One approach (popular with statisticians) applies a singular value decomposition to the expression data and summarizes the module with the module eigengene. Another approach (popular with network theorists) is to construct a module network and to use the most highly connected hub gene as centroid. Since Equation 33 implies that hub genes are highly correlated with the module eigengene, we find that the two seemingly different approaches will lead to very similar results in practice .qth module eigengene gives rise to an eigengene-based scaled intramodular connectivity measureSince module construction is computationally intensive, one often restricts the module detection analysis to a subset of the original genes on the microarray, e.g., the most varying and/or the most connected genes. To counter this loss of information, generalizing the intramodular connectivity to extramodular genes, i.e. genes outside the module, is an important problem. Our solution is motivated by the relationship between the intramodular connectivity and its eigengene based analog (Equation 33). Specifically, the iKq)( over iKq)( (Equation 41). The fuzzy module membership measures Kicor,q) genes lie intermediate between the proper module genes.Module detection usually involves certain parameter choices. For some genes, it may be difficult to decide whether they belong to a particular module or whether they belong to more than one module. Instead of reporting a binary indicator of module membership, it can be advantageous to report a fuzzy measure of module membership, which takes on values in the unit interval . A natural choice for a fuzzy measure of module membership is the eigengene-based scaled intramodular connectivity measure In the following, we provide several examples that illustrate potential uses of the geometric interpretation.p-values) to several eigengene-based network concepts. For example, the eigengene-based hub gene significance is a monotonic function of the correlation between the eigengene and the sample trait (Equation 34). Thus, one can use a correlation test p-value p-value for assessing the statistical significance between Eq)( (Equation 41).While fundamental network concepts are defined as functions of the network adjacency matrix, their eigengene-based analogs are often simple monotonic functions of correlation coefficients. This insight can be used to attach significance levels .The relationship between centralization and density (Equation 40) is surprisingly simple for coexpression networks but it does not hold in general networks. For a general network, one can only derive an upper bound for the centralization in terms of the density i as hub gene in module 1 if its scaled connectivity iK(1) is very high (say larger than 0.9). Further, we refer to two modules as distinct if their respective eigengenes have a low correlation, say |cor(E(1),E(2))|<0.3. We refer to gene i as intermediate between modules 1 and 2 if it has a moderately high connectivity with both modules, say iK(1)>0.5 and iK(2)>0.5.The geometric interpretation of gene coexpression network analysis can be used to argue that a gene that lies “intermediate” between two distinct modules cannot be a highly connected intramodular hub gene in either module see . More prEquation 33 allows us to translate statements about the scaled intramodular connectivity into statements about the angles between genes and module eigengenes. A gene is an intermediate gene if it has a moderately small angle with both module eigengenes. If the eigengenes are distinct , the intermediate gene cannot have a very small angle with either module eigengene, i.e., it cannot be an intramodular hub gene in either module. A geometric interpretation of this example can be found in As an important caveat, we mention that intermediate network genes may well be highly connected “hub” genes if the factorizability property does not hold such as in the entire network comprised of multiple distinct modules.For a trait-based gene significance measure, the striking relationship between module significance and hub gene significance (Equation 37) suggests a positive relationship between connectivity and gene significance (high hub gene significance) in modules that are enriched with significant genes (high module significance).E and T2 is small which implies that the hub gene significance with regard to iGS2 = |cor| is high. By contrast, the angle between E and T1 is large, which implies that the hub gene significance with regard to iGS1 = |cor| is low.Further, Equation 34 shows that the hub gene significance of a module network is determined by the angle between the module eigengene and the sample trait. This allows us to describe situations when a module has high hub gene significance, i.e., when there is a strong positive relationship between gene significance and intramodular connectivity. In the example provided in To facilitate the communication between microarray data analysts and network theorists, we provide a short dictionary for translating between microarray data analysis and network theory terminology. More specifically, for a subset (module) of genes that have high expression factorizability, p-value) using a correlation or regression based test statistic.In general, eigengene-based concepts are no substitute for network concepts. It is natural to use network concepts when describing the pairwise relationships between genes and to use eigengene-based network concepts when relating the gene expression profiles to a module eigengene. Since eigengene-based network concepts tend to be relatively simple, they often simplify theoretical derivations. Further, many of them allow one to calculate a statistical significance level  patients. A detailed description of the data, modules, and biological implications can be found in T. We defined a gene significance measure as the absolute value of the correlation between T and the gene expression profiles (Equation 4). The module significance was defined as average gene significance (Equation 14). We used the patient survival time as microarray sample trait iGS to the scaled connectivity iK, we arrive at a hub gene significance measure (Equation 13). As illustrated in By relating the gene significance measure e,tabrown = |cor| = 0.202. An advantage of the eigengene-based hub gene significance (the eigengene significance) is that it allows one to compute a corresponding p-value. Using a correlation test, we find that the value of the eigengene significance e,tabrown is statistically insignificant (p = 0.30) in this dataset. However, when we combined these data with an additional data set, we found that the brown module eigengene is significantly related to survival time We defined the module eigengene significance (Equation 27) as the absolute value of the correlation between the module eigengene and patient survival time. The brown module eigengene also had the highest eigengene significance: EF(X)≥0.97 (EF(X) = 0.66).We visualize the gene expression profiles of module genes with a heat map plot  where ro(X)≥0.97 . By contFor each module, β = 1 is used for the weighted network construction (Equation 2). The analogous results for β = 6 are depicted in β = 1 and β = 6, respectively. For completeness, we also report the results for the grey, nonmodule genes in the figures. But since our theoretical results assume proper modules, we exclude the grey genes from the calculation of the squared correlation coefficient R2. The summary of a robustness analysis with regard to different soft thresholds β and hard thresholds τ is reported in R2>0.85), which confirm our theoretical results. Only the R2 values for the relationship between clustering coefficient and its eigengene-based analog is decreased if β>3.Our empirical results illustrate Observation 2 regarding the relationship between intramodular network concepts and their eigengene-based analogs. iKq)( is highly correlated (R2>0.99) with e,iaq)2×Density (Equation 31). This relationship is diminished for soft thresholds β>3 as can be seen from β  between network concepts and their eigengene-based analogs depend on the soft threshold β. Overall, we find that our theoretical results are highly robust in weighted networks. The relationship between the clustering coefficient and its eigengene-based analog is diminished (down to 0.44) for β>3. The relationship between heterogeneity and its eigengene-based analog is diminished (down to 0.54 when β<3).Here we focus on the female mouse liver tissues of the above-mentioned F2 mouse cross. Specifically, 135 female mice were used to construct a weighted network comprised of 3,400 highly connected genes. The biological significance and gene ontology enrichment analysis of the 12 modules in this large network is described in R2 value (down to 0.21) for low values of β≤3 but the other relationships among network concepts are highly robust with respect to β. For unweighted networks, the R2 values tend to be lower and several relationships show a marked dependency on the hard threshold τ  assumed a sample-trait based gene significance measure. Although this important assumption is violated for knock-out essentiality, it is striking that the relationship between hub gene significance and module significance can still be observed empirically  between network concepts and their eigengene-based analogs depend on the soft threshold β. Overall, we find that our theoretical results are highly robust in weighted networks. The relation β = 3 or 4 but the other relationships among network concepts are highly robust with respect to β. For unweighted networks, the R2 values tend to be lower and several relationships show a marked dependency on the hard threshold τ (eshold τ .ith gene is determined by the angle between the ith gene expression profile and the sample trait T (Equation 4); the scaled intramodular connectivity of the ith gene (Equation 33) is determined by the angle between the ith gene expression profile and the module eigengene; the hub gene significance (Equation 34) is determined by the angle between module eigengene and the sample trait.Network theoretic methods and concepts are increasingly used for the systems biologic analysis of microarray data. We illustrate how network concepts can be used for describing large correlation matrices and for arriving at biologically plausible data reduction techniques. Many alternative approaches for defining gene coexpression networks are possible, e.g., m-dimensional Euclidean space, network analysis can be considered a special case of directional statistics. When focusing on the use of module eigengenes, network analysis can be considered a variant of oblique factor analysis.The geometric interpretation of gene coexpression network analysis reveals a deep connection to other statistical methods. Since it projects the gene expressions profiles onto the hypersphere in an ijaq)(≈iCFq)(jCFq)(. We provide an intuitive formula of the conformity iCFq)(≈|cor(ixq)|β. Since the module eigengene Eq)|β measures how well gene i conforms to the overall module. This insight led us to coin the term “conformity”. Using the singular values, we propose a measure of eigengene factorizability (Equation 24) that is analogous to the proportion of variance explained by the module eigengene (Equation 22). We provide a geometric interpretation of network factorizability in A high level view of modules and their centroids (eigengenes) can be used to define eigengene networks ijaq)( = |cor(ixq))|β≈|cor(ixq)cor(jxq)|β = |cor(ixq)|β|cor(jxq)|β.The derivation of Observation 1 in the β.Using multiple different gene coexpression networks from mouse tissues, brain cancer, and yeast, we provide empirical evidence that coexpression modules tend to have high eigengene factorizability and that the maximum conformity assumption (Equation 32) is satisfied for low powers of ixq)|β. Algebraically, eigengene-based network concepts are closely related to “approximate conformity based” network concepts We propose eigengene-based analogs of network concepts (Equation 30). While network concepts are functions of the adjacency matrix, eigengene-based network concepts are analogous functions of the eigengene conformities |cor(EF(Xq)>0.95) and if the maximum conformity assumption (Equation 32) is satisfied. However, even if the maximum conformity assumption does not hold, one can still find simple relationships among the network concepts (Equation 49).We use the correspondence between intramodular network concepts and their eigengene-based analogs to provide a geometric interpretation of network concepts. Observation 2 states that network concepts in weighted gene coexpression module networks are approximately equal to their eigengene-based analogs. A major theoretical advantage of eigengene-based network concepts is that they reveal simple relationships. To arrive at particularly simple relationships, we make the maximum conformity assumption (Equation 32) for the results presented in the main text. The geometric interpretation of gene coexpression networks facilitates the derivation of several results that should be interesting to network theorists. For example, we argue that highly connected intramodular hub genes cannot be intermediate between two distinct coexpression modules . The geoβ and τ, respectively. Thus, weighted coexpression networks are preferable over unweighted networks when a geometric interpretation of network concepts is desirable.We provide a robustness analysis that shows that many of our theoretical results apply even if our underlying assumptions are not satisfied . We findThe correspondence between coexpression module networks and the singular value decomposition  can breaEF(Xq). To illustrate this point, While this article has a theoretical bent, we illustrate the results on three different microarray data sets  that are described in our online R software tutorials, in network concept function to be function of a square matrix M = [ijM]  and/or a corresponding vector G = . For example, M could be the adjacency matrix  and G could be a corresponding gene significance measure.Analogous to MB in the denominator of the clustering coefficient function are given by ijb = 1 if i≠j and iib = Ind(iim>0). Here the indicator function Ind(·) takes on the value 1 if the condition is satisfied and 0 otherwise.We make use of the following network concept functions:A–I (where I denotes the identity matrix) equal 0. Now we are ready to define the  network concepts that are studied in this article.According to our convention, the diagonal elements of the adjacency matrix are set to 1. Therefore, the diagonal elements of Definition of Fundamental Network Concepts:The fundamental network concepts of a network A are defined by evaluating the network functions (Equation 42) on A–I and the gene significance measure GS, i.e.,For example, the connectivity is given byintramodular network conceptNCF(Aq) by evaluating the network concept function on the restricted adjacency matrix Aq)( and the restricted gene significance measure GSq)(.We define an EAq)( = eaq)((eaq)T (Equation 28) and the eigengene-based gene significance measure E,iGSq)( (Equation 29), we define an eigengene-based network concept as NCF(EAq).We will now define eigengene-based network concepts. Using the eigengene-based adjacency matrix As example, consider the eigengene-based connectivity given byq) in the following, e.g., we will write EF(X) instead of EF(Xq). We will argue that if the eigengene factorizability EF(X) is close to 1, the adjacencies of the weighted coexpression module network A = |cor(X)|β and the trait-based gene significance measure iGS = |cor|β can be factored as followsHere we derive Observation 1, which characterizes approximately factorizable gene coexpression module networks. To simplify the presentation, we omit the superscripts ], which is scale-invariant, we can assume that the gene expression profiles have been scaled as follows: m is the number of microarray samples. Then one can derive the following relationshipsui1,|d1|2uj1,/m = corcor. Using the fact that U is an orthogonal matrix, it is straightforward to show thatSince our gene coexpression networks are defined with respect to the correlation matrix . For high values of EF(E), the correlation matrix can be factored as followsu1 denotes an eigenvector of length 1.An alternative, possibly more direct way of motivating the observation is based on the insight that the squared singular values ea,maxq)(<<1). It is straightforward to derive the following relationships among eigengene-based network concepts:Here we describe relationships among eigengene-based network concepts if the maximum conformity assumption does not hold  = |cor(X(q))|β and the eigengene factorizability is close to 1 (EF(X(q))≈1), the relationships among eigengene-based concepts approximately apply to their network analogs as well.If AFor example, we find thatq) in the following expressions. We denote by l,iθ the angle between the right singular vector lv (Equation 20) and the ith gene expression profile ix. The smaller the angle l,iθ, the bigger the correlation cor = cos. Using In the following we provide details on our geometric interpretation of the factorizability. To simplify the notation, we sometimes drop the superscript (EF(Xq)≈1 if the module gene expressions ix are approximately orthogonal ≈0) to the right singular vectors vl for l≥2, i.e., if on average the gene expression profiles point in the direction of the module eigengene v1 = E.Thus, ija≈e,iae,ja (Equation 25) depicted in iθ = θi1, the angle between the module eigengene E and the ith gene expression profile and by ijθ the angle between gene expression profiles i and j. Using the assumptions described in ijθ≈|iθ±jθ| and sin(iθ) sin(jθ)≈0, we find thatUnder this assumption, we provide a rough geometric intuition of β (Equation 2). Recall that ia = |cor| is a nonnegative number.Here we prove that the eigengene-based heterogeneity increases with the soft threshold β, it suffices to prove the followingTo prove that the heterogeneity increases with Proposition:i, i = 1,…,n} be a group of nonnegative number and β>1 then the following inequality holds Let {a:To prove the Proposition, we will make use of the followingLemma:i, i = 1,…,n} and {vi, i = 1,…,n} be groups of nonnegative numbers, and θ be a number 0≤θ<Let {u1. Then the following inequality holds:θ1 = β/(2β−1), iu = ia, and iv = i2βa to deriveθ2 = (2β−2)/(2β−1), iu = ia, and iv = i2βa to deriveθ1+θ2 = 2 and 3−(2θ1+θ2) = 1. The last inequality completes the proof since it is equivalent to the inequality in Equation 53.Text S1Robustness Analysis of the Brain Cancer Gene Coexpression Network. This supporting text provides a detailed analysis of the brain cancer gene coexpression network. The robustness analysis illustrates how the results change with regard to different network construction methods.(3.83 MB PDF)Click here for additional data file.Text S2Robustness Analysis of the Mouse Gene Coexpression Network. This supporting text provides a detailed analysis of the mouse tissue gene coexpression network. The robustness analysis illustrates how the results change with regard to different network construction methods.(3.76 MB PDF)Click here for additional data file.Text S3Robustness Analysis of the Yeast Gene Coexpression Network. This supporting text provides a detailed analysis of the yeast cell cycle gene coexpression network. The robustness analysis illustrates how the results change with regard to different network construction methods.(2.62 MB PDF)Click here for additional data file.Text S4Brain Cancer Network Comprised of 500 Prognostic Genes. Here we analyze a brain cancer network comprised of the 500 genes with highest absolute correlation with brain cancer survival time. The results illustrate that our theoretical results also apply to small networks comprised of sample trait related genes. The robustness analysis illustrates how the results change with regard to different network construction methods.(0.38 MB PDF)Click here for additional data file.


A cut-off at 410 μg l–1 gave 90% sensitivity with 90% specificity which means that the diagnostic efficiency of sCD26 is higher than that shown by other markers, particularly in patients at early stages. Moreover, sCD26 as a variable is not related with Dukes’ stage classification, age, gender, tumour location or degree of differentiation. With a follow-up of 2 years until recurrence, preliminary data show that sCD26 can be managed as a prognostic variable of early carcinoma patients. In addition, the origin of sCD26 is discussed. © 2000 Cancer Research CampaignCD26 is an ectoenzyme with dipeptidyl peptidase IV activity expressed on a variety of cell types. Although the function of the high concentration of serum-soluble CD26 (sCD26) is unknown, it may be related to the cleavage of biologically active polypeptides. As CD26 or enzymatic activity levels were previously associated with cancer, we examined the potential diagnostic and prognostic value of preoperative sCD26 measurements by ELISA in colorectal carcinoma patients. We found a highly significant difference between sCD26 levels in healthy donors (mean 559.7 ± 125.5 μg l


Acrocephalus arundinaceus) populations distributed over the vast Eurasian breeding range of the species, and revealed unexpected postglacial expansion patterns from two glacial refugia. There were 58 different haplotypes forming two major clades, A and B. Clade A dominated in Western Europe with declining frequencies towards Eastern Europe and the Middle East, but showed a surprising increase in frequency in Western and Central Asia. Clade B dominated in the Middle East, with declining frequencies towards north in Central and Eastern Europe and was absent from Western Europe and Central Asia. A parsimonious explanation for these patterns is independent postglacial expansions from two isolated refugia, and mismatch distribution analyses confirmed this suggestion. Gene flow analyses showed that clade A colonised both Europe and Asia from a refugium in Europe, and that clade B expanded much later and colonised parts of Europe from a refugium in the Middle East. Great reed warblers in the eastern parts of the range have slightly paler plumage than western birds  and our results suggest that the plumage diversification took place during the easterly expansion of clade A. This supports the postglacial expansion hypothesis proposing that postglacial expansions drive diversification in comparatively short time periods. However, there is no indication of any (strong) reproductive isolation between clades and our data show that the refugia populations became separated during the last glaciation. This is in line with the Pleistocene speciation hypothesis invoking that much longer periods of time in isolation are needed for speciation to occur.Pleistocene glacial cycles play a major role in diversification and speciation, although the relative importance of isolation and expansion in driving diversification remains debated. We analysed mitochondrial DNA sequence data from 15 great reed warbler ( Pleistocene glacial cycles cause repeated range expansions and contractions with important present-day demographic consequences for temporal and boreal populations and species in the Northern Hemisphere An alternative hypothesis, referred to as the “postglacial expansion hypothesis”, is that speciation takes place during expansion rather than during isolation Acrocephalus arundinaceus). This is a socially polygynous long-distance migratory passerine bird that breeds in reed marsh habitats in most of the Palaearctic between latitudes 35° and 60° N, east to North-Western Mongolia A. a. arundinaceus in the western parts, and A. a. zarudnyi in the eastern parts of the range Here we use molecular genetic data to evaluate postglacial colonization patterns in a bird with only slight phenotypic variation, the great reed warbler  by using mtDNA sequence data from a total of 15 populations distributed over most of the species' Eurasian breeding range now including also Eastern Europe, the Middle East and Western and Central Asia, and (ii) by using recently developed statistical tools that make it possible to test more sophisticated hypotheses regarding the divergence and expansion, e.g. inferring asymmetric gene flow between populations and time since divergence We sequenced 494 bps of the control region II in 281 great reed warblers from 15 Eurasian populations . This reA Neighbour-Joining analysis of the 281 sequences revealed two major clades (A and B) supported by a bootstrap value of 75% , and a mDXY, was 0.013 (±0.004 SE) and the number of net nucleotide substitutions per site between clades, DA, was 0.010 (±0.004 SE). Assuming a mutation rate of 15% (range 10–20%) per site Myr−1 for the control region DXY and DA, respectively.The average number of nucleotide substitutions per site between the clades, In FST varied substantially and reached its highest value between the Netherlands and Iran  or an expanding population . This supports the view that there have been expansions from two glacial refugia populations. When the two clades were tested separately, the mismatch distributions for both clades differed significantly from what would be expected in a stable population , whereas the distributions did not deviate from the expected in an expanding population   and 20 (15–30) kyr BP, respectively.When all haplotypes were tested simultaneously there were two clear peaks in the mismatch distribution at 1–2 bp and 6–7 bp, respectively . This di = 0.31) . The valaM<8 migrants per generation), strong bi-directional gene flow between almost all central European populations , a tendency for an asymmetric gene flow from eastern Europe to Russia  and pronounced asymmetric gene flow from Russia to Kazakhstan  than for clade A , and, if anything, there was weak asymmetric gene flow from southern to northern populations  has colonised much of Europe in only 150 years a method seem more plausible: 4 kyr BP for clade A and 1.5 kyr BP for clade B. Still, these estimates are sensitive to mutation rates and generation times, and assume a constant molecular clock and migration-drift equilibrium, thus should be interpreted cautiously. Although it may be very difficult to date the postglacial expansion in the great reed warbler, our data strongly suggest that the expansion of clade A preceded that of clade B. The level of gene flow is higher between populations for clade A, and clade A has colonised a much wider area and is now present all the way from Western Europe to Central Asia.The great reed warbler is highly philopatric in some populations, yet has a relatively high dispersal potential Speciation may be the most spectacular consequence of repeated glacial cycles, but there are also other important consequences of glacial population bottlenecking and postglacial expansions. For instance, geographical barriers may result in limits to present-day range distributions and failures to colonize seemingly suitable habitat Postglacial expansions may also leave more subtle traces in form of lower genetic variation in populations at the expanding range limits; referred to as the ‘leading edge hypothesis’ A study of the breeding ecology of the great reed warbler in Sweden has been ongoing for more that two decades Carduelis flammea; Motacilla flava; Dendroica coronata; Phylloscopus trochilus; Junco hyemalis; The Pleistocene speciation hypothesis suggests that speciation occurs in isolated refugia over one or more full glacial cycles A. a. zarudnyi) have paler plumage than western birds (A. a. arundinaceus) A. a. zarudnyi is also recognised in eastern parts of the Middle East As mentioned above, great reed warblers in the eastern parts of the range  pre- or post-zygotic isolation mechanisms between the great reed warbler clades  by a standard phenol-chloroform extraction protocol. The samples from the Netherlands consisted of dried contour feathers plucked from nestlings (different nests) and DNA was extracted by using 5% Chelex 100  and FTPH3 (5′-AAGGCTGGGAGAGTTGTTGA- 3′) following the procedures in Bensch and Hasselquist Phe gene The mitochondrial control region II in great reed warblers (GenBank: AF111791) was amplified and sequenced with the primers BCML4 . Before the cyclic reactions the samples were incubated at 94°C for 2 min, and after completion at 72°C for 10 min. The PCR product was precipitated (NH4Ac and ethanol) and then dissolved in 20 µL of water; 2–4 µL was then used for sequencing  in an ABI Prism 3100 capillary sequencer (Applied Biosystems).Polymerase chain reactions (PCRs) were performed in volumes of 25 µL and included 10–50 ng of total genomic DNA, 0.125 mM of each nucleotide, 1.5 mM MgClThe data set of 93 control region sequences from six populations gathered and analysed in Bensch and Hasselquist naSP 4.10.9 H) and nucleotide diversity (π) were calculated according to Nei Basic population statistics were calculated using the program DuzzlerBayes 3 rModeltest2Evolutionary relationships between haplotypes were assessed by the Neighbour-Joining method with MEGA 3.1 FST .We calculated the degree of genetic differentiation measured as DXY, and the number of net nucleotide substitutions per site between clades, DA, were calculated in DnaSP ×g, and t = (DA/2μ)×g, respectively, where μ is the rate of substitution per site Myr−1 and g is the generation time . We estimated the parameter τ, i.e. the date of the expansion measured in units of mutational time, which can be used to estimate the time since expansion with the expression t = (τ/2μk)×g, where μ and g are as above and k is the sequence length (i.e. 494 bp).We calculated mismatch distributions in Da, Isolation with Migration, analytic, version 7/13/2007 θ of all populations ; a value of 100 as the maximum migration rate between populations ; and 10 for the maximum time of population splitting. For each pair of populations we used a ‘burn-in’ period of 1×106 steps and a sampling period of 1×107 steps. The output parameters  were expressed in demographic units as follow: effective population size, NE = θ/4μkg; number of migrants per generation, M = 2NE×(m×2μk) = θ×m/2g; divergence time in years, T = tg/2μk; where μ, k and g are as above.We also inferred populations sizes, asymmetric gene flow and time since divergence between pairs of populations with demographic population genetic models implemented in the program IMTable S1ST-values between populations using data from (a) clade A and B haplotypes; (b) clade A haplotypes; and (c) clade B haplotypes.F(0.08 MB DOC)Click here for additional data file.Table S2Estimates of demographic parameters from two-population isolation with migration models implemented in the program IMa (0.08 MB DOC)Click here for additional data file.Figure S1Neighbour-Joining tree based on 281 mtDNA control region II sequences of great reed warblers from 15 breeding populations. Labels refer to population . The two major haplotype clades are shown, clade A and B. The dotted line indicates a relatively large uniform cluster of sequences within clade A only found in Russia and Kazakhstan (see text for details).(3.30 MB TIF)Click here for additional data file.


Hubbell's 2001 neutral theory unifies biodiversity and biogeography by modelling steady-state distributions of species richness and abundances across spatio-temporal scales. Accurate predictions have issued from its core premise that all species have identical vital rates. Yet no ecologist believes that species are identical in reality. Here I explain this paradox in terms of the ecological equivalence that species must achieve at their coexistence equilibrium, defined by zero net fitness for all regardless of intrinsic differences between them. I show that the distinction of realised from intrinsic vital rates is crucial to evaluating community resilience.An analysis of competitive interactions reveals how zero-sum patterns of abundance emerge for species with contrasting life-history traits as for identical species. I develop a stochastic model to simulate community assembly from a random drift of invasions sustaining the dynamics of recruitment following deaths and extinctions. Species are allocated identical intrinsic vital rates for neutral dynamics, or random intrinsic vital rates and competitive abilities for niche dynamics either on a continuous scale or between dominant-fugitive extremes. Resulting communities have steady-state distributions of the same type for more or less extremely differentiated species as for identical species. All produce negatively skewed log-normal distributions of species abundance, zero-sum relationships of total abundance to area, and Arrhenius relationships of species to area. Intrinsically identical species nevertheless support fewer total individuals, because their densities impact as strongly on each other as on themselves. Truly neutral communities have measurably lower abundance/area and higher species/abundance ratios.Neutral scenarios can be parameterized as null hypotheses for testing competitive release, which is a sure signal of niche dynamics. Ignoring the true strength of interactions between and within species risks a substantial misrepresentation of community resilience to habitat loss. Hubbell's 2001 neutral theory (HNT) unifies the disciplines of biodiversity and biogeography by modelling steady-state distributions of species richness and relative species abundance across spatio-temporal scales A neutral model of empirical relationships eliminates “the entire set of forces competing for a place in the explanation of the pattern” Theoretical studies have sought various ways to reconcile neutral patterns with niche concepts. Intrinsically similar species can coexist under niche theory S, abundance of individuals, N, and zero-sum dynamics as emergent properties  rates. The reality that species differ in their life history traits therefore underpins the assumption of ecological equivalence, which then permits fitting of intrinsically neutral models with vital rates set equal to the realised rates. In the next section, these predictions are extended to simulate the drift of species invasions that sustains the dynamics of recruitment following deaths and extinctions amongst multiple species of dominants and fugitives.Species characterized by extremely different intrinsic attributes can achieve ecological equivalence in a zero-sum game played out at dynamic equilibrium. Take for example a two-species community comprising a dominant competitor displacing the niche of a fugitive e.g., . The fugThe same principle of trade-offs in character traits conversely allows a sexually reproducing species to withstand invasion by highly fecund asexual mutants These local-scale dynamics apply equally at the regional scale of biogeography, reconfiguring individual death as local extinction, and birth as habitat colonization The above examples of dominant versus fugitive and sexual versus asexual were illustrated with models that gave identical realised rates of both birth and death at coexistence equilibrium. Fitness invariance and zero-sum dynamics, however, require only that species have identical net rates of realised birth minus death. The simulations in the next section show how neutral-like dynamics are realised for communities of coexisting species with trade-offs in realized as well as intrinsic vital rates.S-species systems under drift of limited immigration and new-species invasions , and an accelerating decline in rank abundances of rare species (cf. linear for Fisher log-series) that is significantly less precipitous than predicted by broken-stick models of randomly allocated abundances amongst fixed S and N; n-individual species. Further trials confirm that reduced dispersal limitation exacerbates the negative skew from the log-normal distribution, while sustaining a higher total density of individuals. The extinction-invasion imbalance sets the equilibrium species richness, S, as a power function of total population density, N. This can be expressed as the Arrhenius relationship: S =  zcK  = 0.45 for Lotka-Volterra, mean ratio of 0∶1 values = 58∶42 for dominant-fugitive). The zero-sum gradient of N against K is simply the equilibrium fraction of occupied habitat, which is 1–1/R for a closed neutral scenario, where R is per capita lifetime reproduction before density regulation (b/d in kF/K = (1–1/R)/α, where R and α are system averages. Further simulation trials show the slope increasing with immigration, for example by a factor of 1.9 between closed and fully open  Lotka-Volterra communities. Dispersal limitation therefore counterbalances effects of the net competitive release obtained in niche scenarios from ijα<1 , and consequently it maximizes species packing as expressed by the power function predicting S from N in F1,42 = 122.72, P<0.001). The lower coefficients of Lotka-Volterra and dominant-fugitive scenarios are further differentiated by competitive asymmetry . In effect, the neutral scenario has the lowest average abundance of individuals per species, n, for a community of size K with given average R, which is also reflected in the modal values in K = 1000 patches.The less crowded neutral scenario sustains a somewhat higher density of species than non-neutral scenarios . These interactions may be measurable directly from field data as inter-specific impacts of equal magnitude to intra-specific impacts; alternatively, Lotka-Volterra models of the sort described here can estimate average competition coefficients at an observed equilibrium N, given an average R . This distinction of intrinsically neutral from non-neutral dynamics has been masked in previous theory by the convention for neutral models either to fix Nijα = 0 (i≠j) allows indefinite expansion of S and hence also of N. A slightly less extreme neutral-niche community is modelled by setting all interspecific impacts to a common low value. Simulations at ijα = 0.1 for all i≠j give a zero-sum relation N = 4.026K, which has >4-fold steeper gradient than that for the Lotka-Volterra scenario  will underestimate R, and hence resilience, if the ijα are undervalued by setting all to zero.Modelling zero-sum ecological drift as an emergent property reveals a key distinguishing feature of truly neutral communities. Their intrinsically identical species self-regulate to a lower total density as a result of inter-specific impacts equalling intra-specific impacts. Any empirical test for competitive release is therefore also a test for niche structure. For example, removing habitat is predicted to give a relative or absolute advantage to species towards the fugitive end of a dominant-fugitive spectrum, which may be picked up in correlated life-history traits for winners or losers under habitat loss or degradation Ecological equivalence is a much more permissive requirement for neutrality than is currently acknowledged in theoretical developments on HNT. Coexistence equilibria largely achieve the neutrality-defining mission, to eliminate all of the forces competing for a place in explanations of pattern. It remains an open question whether they do so best amongst species with most or least competitive release in each others' presence e.g., . Models The following protocols apply to simulations of single and multi-niche communities with density-dependent recruitment and density-independent loss of individuals. They produce the outcomes illustrated in K equally accessible habitat patches within a wider meta-community of mK patches. The dynamics of individual births and deaths are modelled at each time step by species-specific probability b of each resident, immigrant, and individual of new invading species producing a propagule, and species-specific probability d of death for each patch resident. Recruitment to a patch is more or less suppressed from intrinsic rate b by the presence there of other species according to the value of ijα, the impact of species j on species i relative to i on itself, where the intraspecific impact iiα = 1 always. A patch can be occupied by only one individual of a species, and by only one species unless all its resident ijα<1. Conventional Lotka-Volterra competition is thus set in a metapopulation context by equating individual births and deaths to local colonisations and extinctions  over time t approximated by:ik = (1–i/bid)K. Coexistence of any two species to positive equilibrium n1, n2 requires them to have intrinsic differences such that k1>α12k2 and k2>α21k1.The community occupies a homogenous environment represented by a matrix of i has randomly set competitive impacts with respect to each other resident species j, of ijα received and jiα imposed. It has randomly set ib, and an intrinsic lifetime reproduction iR = i/dib that is stratified in direct proportion to its dominance rank amongst residents, obtained from its ranked mean α-received minus mean α-imposed. For example, an invader with higher dominance than all of three resident species will have random iR stratified in the bottom quartile of set limits Rmin to Rmax. Communities are thereby structured on a stochastic life-history trade-off between competitive dominance and population growth capacity. This competition-growth trade-off is a well-established feature of many real communities, which captures the fundamental life-history principle of costly adaptations b and R for all species and α = 1 for all.Each time-step in the simulation offers an opportunity for one individual of each of two new species to attempt invasion (regardless of the size of the meta-community). Each new species b of producing a propagule. Each propagule has small probability ν of speciation (following m/KK–1]in), assuming the same density in/K of each species i as in the sample community, and in proportion to their probability (mK/K) of landing within the sample community, and modified by a dispersal limitation parameter ω. In effect, for each resident species in the community, [(1–mK/K) in]ω1– external residents each produce an immigrating propagule with probability ib. Thus if mK≫K and ω = 0, a colonist is just as likely to be an immigrant from outside as produced from within the sample community  it either beats the odds on repulsion by all other propagules simultaneously attempting to colonise the patch, or benefits from the random chance of being the first arrival amongst them. Each pre-established resident risks death with species-specific probability id = i/Rib at each time step. Each patch has probability X of a catastrophic hazard at each time step that extirpates all its occupants. The model thus captures the principles of stochastic niche theory Each propagule lands on a random patch within the sample community and establishes there only if (a) its species is not already present, and (b) it beats each probability mK = 106, dispersal limitation parameter ω = 0.5, speciation probability per resident propagation event ν = 10−12, two invasion attempts per time-step (setting Hubbell's θ∼4 independently of mK), probability of catastrophe per patch X = 0.01. For neutral communities, all species take competition coefficients α = 1, individual intrinsic propagation probability b = 0.5, individual intrinsic lifetime reproduction R = 1.5 (so lifespan R/b = 3); for Lotka-Volterra communities, each species i takes random 0≤ijα≤1, random 0≤ib≤1, iR between 1.2 and 1.8 and proportional to dominance rank; dominant-fugitive communities are as Lotka-Volterra except for random binary ijα = 0 or 1. All scenarios are thereby sampled from a large meta-community with moderate dispersal limitation, low extrinsic mortality, and sufficient invasions to sustain a reasonably high asymptote of species richness from the starting point of two species each occupying 5 patches. Skew in the lognormal distribution of species abundances Click here for additional data file.


The aims of this analysis are to compare toxicity and survival according to age, performance status (PS), gender and prior radical pelvic radiotherapy (RT). The primary end points were overall survival and an irinotecan-specific toxicity composite end point (TCE) defined as the occurrence of grade 3 or 4 diarrhoea, neutropenia, febrile neutropenia, fever, infection or nausea and vomiting. Between 1997 and 2003, 339 eligible patients with advanced colorectal cancer (CRC) progressing on or within 24 weeks of completing fluoropyrimidine-based chemotherapy were prospectively registered in a multicentre randomised trial. All patients commenced irinotecan at 350 mg m−2 once every 3 weeks. There were no differences in proportions of patients developing TCE by age , PS  or prior RT . Males experienced more toxicity than females , but this was not significant after controlling for other co-variates (P=0.06). Patients aged ⩾70 had similar objective responses  and survival  compared to younger patients. Elderly patients derive the same benefit without experiencing more toxicity with second-line irinotecan treatment for advanced CRC. Our data do not support the recommendation to reduce the starting dose for the elderly patients.Elderly patients are recommended to have a reduced starting dose (300 mg m Elderly patients with colorectal cancer (CRC) have the same benefit from fluorouracil-based chemotherapy as younger patients in both adjuvant  performance status (PS) 2. This guideline is based on the two pivotal second-line studies – both recommended 300 mg m−2 for patients aged ⩾70 or PS 2 as these factors were previously recognised risk factors for developing toxicity (−2) for these particular patient populations. In addition, it is generally accepted that patients who had previous radical pelvic radiotherapy (RT) are also at risk of developing severe toxicity with irinotecan, especially diarrhoea. The aims of our analysis are to compare toxicity and survival according to age, PS, gender and prior radical pelvic RT in a group of patients treated within a prospective randomised controlled trial, all of whom commenced irinotecan at a dose of 350 mg m−2 given every 3 weeks.The dose adjustment guideline for irinotecan monotherapy, produced by the manufacturer which is in general circulation with clinicians, recommends a reduced starting dose of 300 mg mWe performed a phase III multicentre prospective randomised controlled trial recruiting patients from six oncology centres in the United Kingdom. The primary efficacy end point of this study has been reported previously  aged <70 vs ⩾70 years; (ii) PS 0–1 vs 2; (iii) male vs female and (iv) previous pelvic RT  vs no pelvic RT. The primary end points were irinotecan-specific toxicity composite end point (TCE) and overall survival (OS). TCE was defined as the occurrence of either grade 3 or 4 diarrhoea, neutropenia, febrile neutropenia, fever, infection or nausea and vomiting. These toxicities were components of the gastrointestinal syndrome that caused early 60-day mortality related to irinotecan therapy (vs ⩾70), PS (0–1 vs 2), gender , previous pelvic RT (yes vs no), number of metastatic sites (0 or 1 vs >1), baseline serum alkaline phosphatase  vs >ULN), bilirubin , haemoglobin (⩽11 g/dl vs >11g dl−1) and white blood cell (⩽11 × 109 l vs >11 × 109 l−1) levels. Apart from the a priori defined comparison groups, other factors were chosen because of their prognostic value in patients treated with 5-FU based chemotherapy for metastatic CRC  or continue until disease progression (n=25). The efficacy of the randomisation part of the study is the subject of a separate publication. In this current analysis, nine (2.6%) patients were excluded due to reduced starting dose , ineligible patient (n=1), death before starting treatment (n=1) and no clinical data (n=2); therefore, 339 patients treated with irinotecan 350 mg m−2 once every 3 weeks were analysed.Between November 1997 and May 2003, 348 patients were prospectively registered into this study. A total of 55 patients with responding or SD after eight cycles of irinotecan were randomised to stop irinotecan (P=0.0228), the incidences of infection, fever and febrile neutropenia were not significantly increased. Patients who had prior RT did not have a significantly higher incidence of diarrhoea compared to those who did not receive prior RT (P=0.0921). P>0.05). Time to occurrence of TCE was also similar by age , but this was not significant after controlling for other co-variates (P=0.06) in multivariate logistic regression modelling. Baseline bilirubin level was not significantly associated with TCE (P=0.149).P=0.74; Figure 3P=0.092) and gender (P=0.512) were not significant prognostic factors. There was no difference in survival between patients who developed the TCE and those who did not (P=0.317).For the whole group, the objective response rate was 9.4% : 6.3–12.6%). −2 of irinotecan once every 3 weeks and this represents the largest single study with second-line irinotecan monotherapy reported to our knowledge. In this study we have shown that patients aged 70 or over had similar benefit and toxicity to irinotecan as younger patients. Poor PS and previous pelvic RT did not influence the incidence of irinotecan-related severe toxicity in these patients.Our study included 339 fluoropyrimidine and thymidylate synthase inhibitor-resistant CRC patients, all treated uniformly at 350 mg mAlthough survival is the most important end point in evaluating new agents, insistence on its use as the only end point in clinical trials can result in the need for thousands of patients to be studied. Accordingly, composite end points have been increasingly used to increase the overall event rate and thereby reduce the number of patients needed to test specific hypotheses. The use of composite end points has resulted in widespread acceptance of therapies in heart failures and acute coronary syndrome , althoug−2 in patients aged ⩾70 years with PS 2. These guidelines have not, however, been incorporated into the Summaries of Product Characteristics. We have sought to use our independent data set to validate or refute these recommendations. Elderly patients are perceived to tolerate treatment more poorly and the benefits are less certain in elderly patients. In a systematic review of managing CRC in elderly patients, it was concluded that there is good evidence to support patients ⩽80 years of age having similar OS benefits with adjuvant 5-FU-based chemotherapy for colon cancer and with palliative first-line monotherapy for colorectal cancer, to younger patients  did not suffer higher frequency of grades 3–4 toxicity compared to those aged under 70 (n=101)  , although one has to note that our eligibility criteria would exclude patients with bilirubin level above 1.25 and 1.5 times the ULN in the absence and presence of liver metastasis respectively. A recent study has also found that baseline serum bilirubin did not reliably predict overall irinotecan-related toxicity in patients treated with irinotecan monotherapy within a phase III trial  SN-38 ratio compared to SN-38 glucuronide ratio (In our study, baseline serum bilirubin level did not influence the occurrence of TCE (In our study, we have confirmed that the prognostic importances of some clinical and biological factors found in 5-FU based chemotherapy (In conclusion, elderly and PS 2 patients derive the same benefit without experiencing more toxicity with second-line irinotecan treatment for advanced colorectal cancer. Pelvic RT did not result in additional toxicity. Our data do not support the recommendations to give a reduced starting dose to elderly and PS 2 patients.


Thin polydimethylsiloxane (PDMS) films are frequently used in “lab on a chip” devices as flexible membranes. The common solvent used to dilute the PDMS for thin films is hexane, but hexane can swell the underlying PDMS substrate. A better solvent would be one that dissolves uncured PDMS but doesn't swell the underlying substrate. Here, we present protocols and spin curves for two alternatives to hexane dilution: longer spin times and dilution in tert-butyl alcohol. The thickness of the PDMS membranes under different spin speeds, spin times, and PDMS concentrations was measured using an optical profilometer. The use of tert-butyl alcohol to spin thin PDMS films does not swell the underlying PDMS substrate, and we have used these films to construct multilayer PDMS devices. Polydimethylsiloxane (PDMS), an elastomer widely used in microdevice fabrication Uncured PDMS is often diluted in solvent in order to spin thin (<5 µm) films. Hexane is often used as a solvent for uncured PDMS Here we propose two alternatives to hexane dilution. The first alternative, longer spin times (>5 min), is an obvious solution, but PDMS film thickness data for long spin times is not widely available. The second alternative is the use of tert-butyl alcohol (TBA) as a solvent for uncured PDMS. TBA dilutes uncured PDMS, but does not swell cured PDMS membranes. We present thickness data for various concentrations of PDMS in TBA for various spin speeds.We tested 12 different solvents for their ability to: (1) dissolve uncured PDMS, and (2) not visibly warp a cured PDMS membrane when poured directly onto the membrane. Tert-butyl alcohol (TBA) was the only solvent that met both requirements. We also measured the swelling in the thickness of cured PDMS soaked in water, TBA, and hexane to dry PDMS . The sliω, and thickness, hk are the experimentally derived constants shown in t, angular velocity, ω, and thickness, h:h0 and c have been experimentally derived here to be h0 = 180 µm and c = 2.86×10−10 RPM−2 µm−2 s−1.The thickness of the PDMS film for longer spin times is shown in The PDMS film spun onto on a PDMS substrate was found to be thicker than the PDMS film spun onto a glass substrate. Biocompatibility of the film was verified when single yeast cells were grown in yeast growth media on 10 different TBA-diluted films for 8 hours under a microscope. Each time, a normal ∼90 minute budding cycle was achieved within 3 hours of placing the cells on the film.We have presented two alternatives to hexane dilution of PDMS: long spin times and dilution in tert-butyl alcohol (TBA). We have found that TBA is an excellent solvent for PDMS because it does not swell the underlying PDMS layer, and it dissolves uncured PDMS when it is mixed at 45°C. We have used dilution in TBA extensively in our lab to construct multilayer PDMS devices. For example, TBA-dissolved PDMS has served as a protective layer for single-cell magnetic trapping elements To measure swelling ratios, PDMS part A and part B  were mixed in a 10∶1 (weight∶weight) ratio and cured for three hours at 55°C in a 85 mm×125 mm×7 mm rectangular mold. 12 identical pieces were cut from the mold. The thickness (∼7 mm) of each piece was measured in three locations with a micrometer. Four pieces were placed into each solvent: deionized water at room temperature, TBA  at 45°C, and hexane (Sigma-Aldrich) at room temperature, and allowed to soak for two hours. The pieces were removed from the solvent, dried, and immediately measured with a micrometer as before.All spin coat tests were performed on one of two substrates: a 2″×3″ glass slide, or a 2″×3″ glass slide coated with PDMS. The bare glass was rinsed in ethanol and deionized (DI) water. The PDMS membrane was rinsed in DI water only. The substrate was blown dry using nitrogen, and then dried on a 60 C hotplate for 30 minutes. It was then allowed to cool to room temperature and placed on the spinner . PDMS part A and part B  were then mixed in a 10∶1 (weight∶weight) ratio in a 25 ml polystyrene beaker using a glass stirring rod for 2 minutes. The PDMS was placed in a vacuum desiccator until bubbles were no longer visible (10–13 minutes). The PDMS was then mixed with 45°C TBA in the appropriate concentration until fully dissolved (30–45 s). The TBA needs to be warmed to 45°C for better mixing with the PDMS; TBA is a solid at room temperature. The mixture was then poured onto the glass slide. The total time between combining PDMS parts and starting the spin was exactly 15 min for each sample. The spinner was then started and then run for the desired time. Finally, the PDMS was placed on a 60 C hotplate until fully cured (1–2 hours).To measure the thickness of the membrane, the area of the PDMS to be profiled was scraped away with a clean razor blade. Both the PDMS and the glass in the area of the removed PDMS were then briefly electrolessly plated in silver per manufacturer's instructions  Data analysis was performed using the R programming language. Estimates of the parameters in Equations (1) and (2) were found using the non-linear least squares (nls) function.