Some equivalent compartmental models

Compartmental models with strongly connected digraphs always have a stable equilibrium solution. Each such model may be reduced by a sequence of matrix operations to a mammilary system with the same equilibrium solution. It may also be reduced by a sequence of operations on the digraph. It may then be further reduced to a model whose digraph is a 2-cycle with the same mean first passage time m. If this m is taken as an indicator of the relative stability, then the latter is independent of the complexity as measured by the number of arcs per vertex. In a number of special cases the index m is related to the eigenvalues of the matrix. It is shown to have a simple relation to other time parameters as well.     

Scaling behaviors of weighted food webs as energy transportation networks

Food webs can be regarded as energy transporting networks in which the weight of each edge denotes the energy flux between two species. By investigating 21 empirical weighted food webs as energy flow networks, we found several ubiquitous scaling behaviors. Two random variables A_i and C_i defined for each vertex i, representing the total flux (also called vertex intensity) and total indirect effect or energy store of i, were found to follow power law distributions with the exponents α≈1.32 and β≈1.33, respectively. Another scaling behavior is the power law relationship, , where η≈1.02. This is known as the allometric scaling power law relationship because A_i can be treated as metabolism and C_i as the body mass of the sub-network rooted from the vertex i, according to the algorithm presented in this paper. Finally, a simple relationship among these power law exponents, η=(α−1)/(β−1), was mathematically derived and tested by the empirical food webs.     

Properties of Normal Phylogenetic Networks

A phylogenetic network is a rooted acyclic digraph with vertices corresponding to taxa. Let X denote a set of vertices containing the root, the leaves, and all vertices of outdegree 1. Regard X as the set of vertices on which measurements such as DNA can be made. A vertex is called normal if it has one parent, and hybrid if it has more than one parent. The network is called normal if it has no redundant arcs and also from every vertex there is a directed path to a member of X such that all vertices after the first are normal. This paper studies properties of normal networks. Under a simple model of inheritance that allows homoplasies only at hybrid vertices, there is essentially unique determination of the genomes at all vertices by the genomes at members of X if and only if the network is normal. This model is a limiting case of more standard models of inheritance when the substitution rate is sufficiently low. Various mathematical properties of normal networks are described. These properties include that the number of vertices grows at most quadratically with the number of leaves and that the number of hybrid vertices grows at most linearly with the number of leaves.     

A Simple Algorithm for Finding All k-Edge-Connected Components

The problem of finding k-edge-connected components is a fundamental problem in computer science. Given a graph G = (V, E), the problem is to partition the vertex set V into {V ₁, V ₂,…, V _h}, where each V _i is maximized, such that for any two vertices x and y in V _i, there are k edge-disjoint paths connecting them. In this paper, we present an algorithm to solve this problem for all k. The algorithm preprocesses the input graph to construct an Auxiliary Graph to store information concerning edge-connectivity among every vertex pair in O(Fn) time, where F is the time complexity to find the maximum flow between two vertices in graph G and n = ∣V∣. For any value of k, the k-edge-connected components can then be determined by traversing the auxiliary graph in O(n) time. The input graph can be a directed or undirected, simple graph or multigraph. Previous works on this problem mainly focus on fixed value of k.     

Reconstruction of certain phylogenetic networks from the genomes at their leaves

A network N is a rooted acyclic digraph. A base-set X for N is a subset of vertices including the root (or outgroup), all leaves, and all vertices of outdegree 1. A simple model of evolution is considered in which all characters are binary and in which back-mutations occur only at hybrid vertices. It is assumed that the genome is known for each member of the base-set X. If the network is known and is assumed to be “normal,” then it is proved that the genome of every vertex is uniquely determined and can be explicitly reconstructed. Under additional hypotheses involving time-consistency and separation of the hybrid vertices, the network itself can also be reconstructed from the genomes of all members of X. An explicit polynomial-time procedure is described for performing the reconstruction.     

A circular loop of the 16-residue repeating unit in ice nucleation protein

Ice nucleation protein (INP) from Gram-negative bacteria promotes the freezing of supercooled water. The central domain of INPs with 1034-1567 residues consists of 58-81 tandem repeats with the 16-residue consensus sequence of AxxxSxLTAGYGSTxT. This highly repetitive domain can also be represented by tandem repeats of 8-residues or 48-residues. In order to elucidate the structure of the tandem repeats, NMR measurements were made for three synthetic peptides including QTARKGSDLTTGYGSTS corresponding to a section of the repetitive domains in Xanthomonas campestris INP. One remarkable observation is a long-range NOE between the side chains of Tyr(i) and Ala(i-10) in the 17-residue peptide. Medium-range NOEs between the side chains of Tyr(i) and Leu(i-4), Thr(i-3) or Thr(i-2) were also observed. These side chain-side chain interactions can be ascribed to CH/π interaction. Structure calculation reveals that the 17-residue peptide forms a circular loop incorporating the 11-residue segment ARKGSDLTTGY.     

Uprooted Phylogenetic Networks

The need for structures capable of accommodating complex evolutionary signals such as those found in, for example, wheat has fueled research into phylogenetic networks. Such structures generalize the standard model of a phylogenetic tree by also allowing for cycles and have been introduced in rooted and unrooted form. In contrast to phylogenetic trees or their unrooted versions, rooted phylogenetic networks are notoriously difficult to understand. To help alleviate this, recent work on them has also centered on their “uprooted” versions. By focusing on such graphs and the combinatorial concept of a split system which underpins an unrooted phylogenetic network, we show that not only can a so-called (uprooted) 1-nested network N be obtained from the Buneman graph (sometimes also called a median network) associated with the split system \(\Sigma (N)\) induced on the set of leaves of N but also that that graph is, in a well-defined sense, optimal. Along the way, we establish the 1-nested analogue of the fundamental “splits equivalence theorem” for phylogenetic trees and characterize maximal circular split systems.     

Quarnet Inference Rules for Level-1 Networks

An important problem in phylogenetics is the construction of phylogenetic trees. One way to approach this problem, known as the supertree method, involves inferring a phylogenetic tree with leaves consisting of a set X of species from a collection of trees, each having leaf-set some subset of X. In the 1980s, Colonius and Schulze gave certain inference rules for deciding when a collection of 4-leaved trees, one for each 4-element subset of X, can be simultaneously displayed by a single supertree with leaf-set X. Recently, it has become of interest to extend this and related results to phylogenetic networks. These are a generalization of phylogenetic trees which can be used to represent reticulate evolution (where species can come together to form a new species). It has recently been shown that a certain type of phylogenetic network, called a (unrooted) level-1 network, can essentially be constructed from 4-leaved trees. However, the problem of providing appropriate inference rules for such networks remains unresolved. Here, we show that by considering 4-leaved networks, called quarnets, as opposed to 4-leaved trees, it is possible to provide such rules. In particular, we show that these rules can be used to characterize when a collection of quarnets, one for each 4-element subset of X, can all be simultaneously displayed by a level-1 network with leaf-set X. The rules are an intriguing mixture of tree inference rules, and an inference rule for building up a cyclic ordering of X from orderings on subsets of X of size 4. This opens up several new directions of research for inferring phylogenetic networks from smaller ones, which could yield new algorithms for solving the supernetwork problem in phylogenetics.     

An analysis of learned kin recognition in hymenoptera

A model is developed to explain data on the probability with which a strange conspecific hymenopteran female will be accepted into a group of sisters. The analysis is based on a genetic labeling system (primarily odors) of m loci and n_i equally frequent alleles at the i-th locus (i = 1,…, m). Three recognition mechanisms are considered (viz: genotype recognition; foreign-label rejection; and habituated-label acceptance) where all three mechanisms depend on individuals learning the labels represented in their group. The probability with which non-kin will be accepted into large sibling group is calculated for a number of different labeling systems. These different labeling systems are compared and a comparision is also made between the three recognition mechanisms mentioned above. A general expression is then derived, in terms of the number of loci and alleles in the labeling system and the size of the sibling group, for the probability with which “strang” sisters are accepted into a group of sisters with whom they have had no prior contact. These results are applied to existing data on the primitively eusocial sweat bee Lasioglossum zephyrum and the present indication is that recognition in L. zephyrum can be modeled by foreign-label rejection with a genetic labeling system of four or five loci with eight to ten alleles in total (i.e. each locus will have two or at most three alleles).     

A Hierarchical Approach to Cooperativity in Macromolecular and Self-Assembling Binding Systems

The study of complex macromolecular binding systems reveals that a high number of states and processes are involved in their mechanism of action, as has become more apparent with the sophistication of the experimental techniques used. The resulting information is often difficult to interpret because of the complexity of the scheme (large size and profuse interactions, including cooperative and self-assembling interactions) and the lack of transparency that this complexity introduces into the interpretation of the indexes traditionally used to describe the binding properties. In particular, cooperative behaviour can be attributed to very different causes, such as direct chemical modification of the binding sites, conformational changes in the whole structure of the macromolecule, aggregation processes between different subunits, etc. In this paper, we propose a novel approach for the analysis of the binding properties of complex macromolecular and self-assembling systems. To quantify the binding behaviour, we use the global association quotient defined as K _c = [occupied sites]/([free sites] L), L being the free ligand concentration. K _c can be easily related to other measures of cooperativity (such as the Hill number or the Scatchard plot) and to the free energies involved in the binding processes at each ligand concentration. In a previous work, it was shown that K_c could be decomposed as an average of equilibrium constants in two ways: intrinsic constants for Adair binding systems and elementary constants for the general case. In this study, we show that these two decompositions are particular cases of a more general expression, where the average is over partial association quotients, associated with subsystems from which the system is composed. We also show that if the system is split into different subsystems according to a binding hierarchy that starts from the lower, microscopic level and ends at the higher, aggregation level, the global association quotient can be decomposed following the hierarchical levels of macromolecular organisation. In this process, the partial association quotients of one level are expressed, in a recursive way, as a function of the partial quotients of the level that is immediately below, until the microscopic level is reached. As a result, the binding properties of very complex macromolecular systems can be analysed in detail, making the mechanistic explanation of their behaviour transparent. In addition, our approach provides a model-independent interpretation of the intrinsic equilibrium constants in terms of the elementary ones.     

A systematic study of labelling an α-helix in a protein with a lanthanide using IDA-SH or NTA-SH tags

The previously published IDA-SH and NTA-SH tags are small synthetic lanthanide-binding tags derived from cysteine, which afford site-specific lanthanide labelling by disulfide-bond formation with a cysteine residue of the target protein. Following attachment to a single cysteine in an α-helix, sizeable pseudocontact shifts (PCS) can be observed, if the lanthanide is immobilized by additional coordination to a negatively charged amino-acid side chain that is located in a neighboring turn of the helix. To identify the best labelling strategy for PCS measurements, we performed a systematic study, where IDA-SH or NTA-SH tags were ligated to a cysteine residue in position i of an α-helix, and aspartate or glutamate residues were placed in the positions i ? 4 or i + 4. The largest anisotropy components of the magnetic susceptibility tensor were observed for an NTA-SH tag in position i with a glutamate residue in position i ? 4. While the NTA-SH tag produced sizeable PCSs regardless of the presence of nearby carboxyl groups of the protein, the IDA-SH tag generated a good lanthanide binding site only if an aspartate was placed in position i + 4. The findings provide a firm basis for the design of site-directed mutants that are suitable for the reliable generation of PCSs in proteins with paramagnetic lanthanides.     

Entropy and the complexity of graphs: II. The information content of digraphs and infinite graphs

In a previous paper (Mowshowitz, 1968), a measureI _g (X) of the structural information content of an (undirected) graphX was defined, and its properties explored. The class of graphs on whichI _g is defined is here enlarged to included directed graphs (digraphs). Most of the properties ofI _g observed in the undirected case are seen to hold for digraphs. The greater generality of digraphs allows for a construction which shows that there exists a digraph having information content equal to the entropy of an arbitrary partition of a given positive integer. The measureI _g is also extended to a measure defined on infinite (undirected) graphs. The properties of this extension are discussed, and its applicability to the problem of measuring the complexity of algorithms is considered.     

Functions of Multivector Variables

As is well known, the common elementary functions defined over the real numbers can be generalized to act not only over the complex number field but also over the skew (non-commuting) field of the quaternions. In this paper, we detail a number of elementary functions extended to act over the skew field of Clifford multivectors, in both two and three dimensions. Complex numbers, quaternions and Cartesian vectors can be described by the various components within a Clifford multivector and from our results we are able to demonstrate new inter-relationships between these algebraic systems. One key relationship that we discover is that a complex number raised to a vector power produces a quaternion thus combining these systems within a single equation. We also find a single formula that produces the square root, amplitude and inverse of a multivector over one, two and three dimensions. Finally, comparing the functions over different dimension we observe that C?(?³) provides a particularly versatile algebraic framework.     

Naphthalene-hydrophobized β-1,3-glucan nanogel for doxorubicin delivery to immunocytes

A water soluble β-1,3-glucan schizophyllan (SPG) can be recognized by an immunocyte receptor called dectin-1. When we introduced naphthalene into the side chain of SPG (nSPG), it formed nanogel by physical cross-link and gained capability to ingest hydrophobic compounds such as doxorubicin. Our in vitro assay revealed that this nanogel can be used as specific delivery of anti-cancer drugs to immunocytes.     

Uptake of disaccharides by the aerobic yeast Rhodotorula glutinis

Sucrose (and raffinose), trehalose, maltose, cellobiose, and lactose were examined for their transport into Rhodotorula glutinis. Melibiose and lactose were found not to be transported at all. Sucrose, raffinose and trehalose are split by periplasmic hydrolases prior to the penetration of their monosaccharide components into cells, the hydrolysis being the rate-limiting factor for the uptake process. Maltose and cellobiose appear to use specific uptake systems. Experiments with protoplasts of Rhodotorula glutinis support the conclusions that sucrose and trehalose are not consumed in the absence of exoenzymes.     

A scaling law of multilevel evolution: how the balance between within- and among-collective evolution is determined

Numerous living systems are hierarchically organized, whereby replicating components are grouped into reproducing collectives—e.g., organelles are grouped into cells, and cells are grouped into multicellular organisms. In such systems, evolution can operate at two levels: evolution among collectives, which tends to promote selfless cooperation among components within collectives (called altruism), and evolution within collectives, which tends to promote cheating among components within collectives. The balance between within- and among-collective evolution thus exerts profound impacts on the fitness of these systems. Here, we investigate how this balance depends on the size of a collective (denoted by N) and the mutation rate of components (m) through mathematical analyses and computer simulations of multiple population genetics models. We first confirm a previous result that increasing N or m accelerates within-collective evolution relative to among-collective evolution, thus promoting the evolution of cheating. Moreover, we show that when within- and among-collective evolution exactly balance each other out, the following scaling relation generally holds: Nmα is a constant, where scaling exponent α depends on multiple parameters, such as the strength of selection and whether altruism is a binary or quantitative trait. This relation indicates that although N and m have quantitatively distinct impacts on the balance between within- and among-collective evolution, their impacts become identical if m is scaled with a proper exponent. Our results thus provide a novel insight into conditions under which cheating or altruism evolves in hierarchically organized replicating systems.     

Determination of monosaturated fatty acid double-bond position and geometry for microbial monocultures and complex consortia by capillary GC-MS of their dimethyl disulphide adducts

Monounsaturated fatty acid double-bond position and geometry have been determined for microbial monocultures and complex microbial consortia by capillary GC-MS of their dimethyl disulphide (DMDS) adducts. The technique has permitted (i) chromatographic separation and positive identification of adducts derived fromcis/trans isomers, (ii) characterization of long-chain monounsaturated components (up to 26:1), and (iii) the identification of a wide range of monounsaturated components derived from methanotrophic soil material. The methanotrophic soil sample contained a high relative proportion of the novel phospholipid ester-linked fatty acid 18:1Δ 10c. The DMDS procedure offers a simple and rapid approach that can be routinely applied to microbial fatty acids derived from environmental samples and monocultures.