On the basic reproduction number in a random environment

The concept of basic reproduction number $R_0$ in population dynamics is studied in the case of random environments. For simplicity the dependence between successive environments is supposed to follow a Markov chain. $R_0$ is the spectral radius of a next-generation operator. Its position with respect to 1 always determines population growth or decay in simulations, unlike another parameter suggested in a recent article (Hernandez-Suarez et al., Theor Popul Biol, doi:10.1016/j.tpb.2012.05.004, 2012). The position of the latter with respect to 1 determines growth or decay of the population’s expectation. $R_0$ is easily computed in the case of scalar population models without any structure. The main emphasis is on discrete-time models but continuous-time models are also considered.     

A network with tunable clustering, degree correlation and degree distribution, and an epidemic thereon

A random network model which allows for tunable, quite general forms of clustering, degree correlation and degree distribution is defined. The model is an extension of the configuration model, in which stubs (half-edges) are paired to form a network. Clustering is obtained by forming small completely connected subgroups, and positive (negative) degree correlation is obtained by connecting a fraction of the stubs with stubs of similar (dissimilar) degree. An SIR (Susceptible $\rightarrow $ Infective $\rightarrow $ Recovered) epidemic model is defined on this network. Asymptotic properties of both the network and the epidemic, as the population size tends to infinity, are derived: the degree distribution, degree correlation and clustering coefficient, as well as a reproduction number $R_*$ , the probability of a major outbreak and the relative size of such an outbreak. The theory is illustrated by Monte Carlo simulations and numerical examples. The main findings are that (1) clustering tends to decrease the spread of disease, (2) the effect of degree correlation is appreciably greater when the disease is close to threshold than when it is well above threshold and (3) disease spread broadly increases with degree correlation $\rho $ when $R_*$ is just above its threshold value of one and decreases with $\rho $ when $R_*$ is well above one.     

Pareto genealogies arising from a Poisson branching evolution model with selection

We study a class of coalescents derived from a sampling procedure out of $N$ i.i.d. Pareto $\left( \alpha \right) $ random variables, normalized by their sum, including $\beta $ –size-biasing on total length effects ( $\beta <\alpha $ ). Depending on the range of $\alpha ,$ we derive the large $N$ limit coalescents structure, leading either to a discrete-time Poisson-Dirichlet $ \left(\alpha ,-\beta \right) \Xi -$ coalescent ( $\alpha \in \left[ 0,1\right) $ ), or to a family of continuous-time Beta $\left( 2-\alpha ,\alpha -\beta \right) \Lambda -$ coalescents ( $\alpha \in \left[ 1,2\right) $ ), or to the Kingman coalescent ( $\alpha \ge 2$ ). We indicate that this class of coalescent processes (and their scaling limits) may be viewed as the genealogical processes of some forward in time evolving branching population models including selection effects. In such constant-size population models, the reproduction step, which is based on a fitness-dependent Poisson Point Process with scaling power-law $\left( \alpha \right) $ intensity, is coupled to a selection step consisting of sorting out the $N$ fittest individuals issued from the reproduction step.     

Necessary and sufficient conditions for R_{0} to be a sum of contributions of fertility loops

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341–1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio $R_0$ in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graph-reduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases $R_0$ allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single individual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of $R_{0}$ that may sometimes be used to good effect in cases where no simple formula for $R_{0}$ exists.     

Do calcium buffers always slow down the propagation of calcium waves?

Calcium buffers are large proteins that act as binding sites for free cytosolic calcium. Since a large fraction of cytosolic calcium is bound to calcium buffers, calcium waves are widely observed under the condition that free cytosolic calcium is heavily buffered. In addition, all physiological buffered excitable systems contain multiple buffers with different affinities. It is thus important to understand the properties of waves in excitable systems with the inclusion of buffers. There is an ongoing controversy about whether or not the addition of calcium buffers into the system always slows down the propagation of calcium waves. To solve this controversy, we incorporate the buffering effect into the generic excitable system, the FitzHugh–Nagumo model, to get the buffered FitzHugh–Nagumo model, and then to study the effect of the added buffer with large diffusivity on traveling waves of such a model in one spatial dimension. We can find a critical dissociation constant ( $K=K(a)$ ) characterized by system excitability parameter $a$ such that calcium buffers can be classified into two types: weak buffers ( $K\in (K(a),\infty )$ ) and strong buffers ( $K\in (0,K(a))$ ). We analytically show that the addition of weak buffers or strong buffers but with its total concentration $b_0^1$ below some critical total concentration $b_{0,c}^1$ into the system can generate a traveling wave of the resulting system which propagates faster than that of the origin system, provided that the diffusivity $D_1$ of the added buffers is sufficiently large. Further, the magnitude of the wave speed of traveling waves of the resulting system is proportional to $\sqrt{D_1}$ as $D_1\rightarrow \infty $ . In contrast, the addition of strong buffers with the total concentration $b_0^1>b_{0,c}^1$ into the system may not be able to support the formation of a biologically acceptable wave provided that the diffusivity $D_1$ of the added buffers is sufficiently large.     

Analytic Calculation of Finite-Population Reproductive Numbers for Direct- and Vector-Transmitted Diseases with Homogeneous Mixing

The basic reproductive number, $\mathcal {R}_{0}$ , provides a foundation for evaluating how various factors affect the incidence of infectious diseases. Recently, it has been suggested that, particularly for vector-transmitted diseases, $\mathcal {R}_{0}$ should be modified to account for the effects of finite host population within a single disease transmission generation. Here, we use a transmission factor approach to calculate such “finite-population reproductive numbers,” under the assumption of homogeneous mixing, for both vector-borne and directly transmitted diseases. In the case of vector-borne diseases, we estimate finitepopulation reproductive numbers for both host-to-host and vector-to-vector generations, assuming that the vector population is effectively infinite. We find simple, interpretable formulas for all three of these quantities. In the direct case, we find that finite-population reproductive numbers diverge from $\mathcal {R}_{0}$ before $\mathcal {R}_{0}$ reaches half of the population size. In the vector-transmitted case, we find that the host-to-host number diverges at even lower values of $\mathcal {R}_{0}$ , while the vector-to-vector number diverges very little over realistic parameter ranges.     

Sensory uncertainty and stick balancing at the fingertip

The effects of sensory input uncertainty, $\varepsilon $ , on the stability of time-delayed human motor control are investigated by calculating the minimum stick length, $\ell _\mathrm{crit}$ , that can be stabilized in the inverted position for a given time delay, $\tau $ . Five control strategies often discussed in the context of human motor control are examined: three time-invariant controllers [proportional–derivative, proportional–derivative–acceleration (PDA), model predictive (MP) controllers] and two time-varying controllers [act-and-wait (AAW) and intermittent predictive controllers]. The uncertainties of the sensory input are modeled as a multiplicative term in the system output. Estimates based on the variability of neural spike trains and neural population responses suggest that $\varepsilon \approx 7$ –13 %. It is found that for this range of uncertainty, a tapped delay-line type of MP controller is the most robust controller. In particular, this controller can stabilize inverted sticks of the length balanced by expert stick balancers (0.25–0.5 m when $\tau \approx 0.08$  s). However, a PDA controller becomes more effective when $\varepsilon > 15\,\%$ . A comparison between $\ell _\mathrm{crit}$ for human stick balancing at the fingertip and balancing on the rubberized surface of a table tennis racket suggest that friction likely plays a role in balance control. Measurements of $\ell _\mathrm{crit},\,\tau $ , and a variability of the fluctuations in the vertical displacement angle, an estimate of $\varepsilon $ , may make it possible to study the changes in control strategy as motor skill develops.     

A Bovine Babesiosis Model with Dispersion

Bovine Babesiosis (BB) is a tick borne parasitic disease with worldwide over 1.3 billion bovines at potential risk of being infected. The disease, also called tick fever, causes significant mortality from infection by the protozoa upon exposure to infected ticks. An important factor in the spread of the disease is the dispersion or migration of cattle as well as ticks. In this paper, we study the effect of this factor. We introduce a number, $\mathcal{P}$ , a “proliferation index,” which plays the same role as the basic reproduction number $\mathcal{R}_{0}$ with respect to the stability/instability of the disease-free equilibrium, and observe that $\mathcal{P}$ decreases as the dispersion coefficients increase. We prove, mathematically, that if $\mathcal{P}>1$ then the tick fever will remain endemic. We also consider the case where the birth rate of ticks undergoes seasonal oscillations. Based on data from Colombia, South Africa, and Brazil, we use the model to determine the effectiveness of several intervention schemes to control the progression of BB.     

Edge removal in random contact networks and the basic reproduction number

Understanding the effect of edge removal on the basic reproduction number ${\mathcal{R}_0}$ for disease spread on contact networks is important for disease management. The formula for the basic reproduction number ${\mathcal{R}_0}$ in random network SIR models of configuration type suggests that for degree distributions with large variance, a reduction of the average degree may actually increase ${\mathcal{R}_0}$ . To understand this phenomenon, we develop a dynamical model for the evolution of the degree distribution under random edge removal, and show that truly random removal always reduces ${\mathcal{R}_0}$ . The discrepancy implies that any increase in ${\mathcal{R}_0}$ must result from edge removal changing the network type, invalidating the use of the basic reproduction number formula for a random contact network. We further develop an epidemic model incorporating a contact network consisting of two groups of nodes with random intra- and inter-group connections, and derive its basic reproduction number. We then prove that random edge removal within either group, and between groups, always decreases the appropriately defined ${\mathcal{R}_0}$ . Our models also allow an estimation of the number of edges that need to be removed in order to curtail an epidemic.     

When does pathogen evolution maximize the basic reproductive number in well-mixed host–pathogen systems?

Pathogen evolution towards the largest basic reproductive number, $\mathcal R _0$ , has been observed in many theoretical models, but this conclusion does not hold universally. Previous studies of host–pathogen systems have defined general conditions under which $\mathcal R _0$ maximization occurs in terms of $\mathcal R _0$ itself. However, it is unclear what constraints these conditions impose on the functional forms of pathogen related processes (e.g. transmission, recover, or mortality) and how those constraints relate to the characteristics of natural systems. Here we focus on well-mixed SIR-type host–pathogen systems and, via a synthesis of results from the literature, we present a set of sufficient mathematical conditions under which evolution maximizes $\mathcal R _0$ . Our conditions are in terms of the functional responses of the system and yield three general biological constraints on when $\mathcal R _0$ maximization will occur. First, there are no genotype-by-environment interactions. Second, the pathogen utilizes a single transmission pathway (i.e. either horizontal, vertical, or vector transmission). Third, when mortality is density dependent: (i) there is a single infectious class that individuals cannot recover from, (ii) mortality in the infectious class is entirely density dependent, and (iii) the rates of recovery, infection progression, and mortality in the exposed classes are independent of the pathogen trait. We discuss how this approach identifies the biological mechanisms that increase the dimension of the environmental feedback and prevent $\mathcal R _0$ maximization.     

Studies on mass attenuation coefficient, effective atomic number and electron density of some vitamins

Mass attenuation coefficient, $ \mu_{m} $ , atomic cross-section, $ \sigma_{i} $ , electronic cross-section, $ \sigma_{e} $ , effective atomic number, $ Z_{\text{eff}} $ and effective electron density, $ N_{\text{el}} $ , were determined experimentally and theoretically for some vitamins (retinol, beta-carotene, thiamine, riboflavin, niacinamide, pantothenic acid, pyridoxine, biotin, folic acid, cyanocobalamin, ascorbic acid, cholecalciferol, alpha-tocopherol, ketamine, hesperidin) at 30.82, 59.54, 80.99, 356.61, 661.66 and 1,408.01?keV photon energies using a NaI(Tl) scintillation detector. The theoretical mass attenuation coefficients were estimated using mixture rules. The calculated values were compared with the experimental values for all vitamins.     

Bone refilling in cortical basic multicellular units: insights into tetracycline double labelling from a computational model

Bone remodelling is carried out by ‘bone multicellular units’ ( $\text{ BMU }$ s) in which active osteoclasts and active osteoblasts are spatially and temporally coupled. The refilling of new bone by osteoblasts towards the back of the $\text{ BMU }$ occurs at a rate that depends both on the number of osteoblasts and on their secretory activity. In cortical bone, a linear phenomenological relationship between matrix apposition rate and $\text{ BMU }$ cavity radius is found experimentally. How this relationship emerges from the combination of complex, nonlinear regulations of osteoblast number and secretory activity is unknown. Here, we extend our previous mathematical model of cell development within a single cortical $\text{ BMU }$ to investigate how osteoblast number and osteoblast secretory activity vary along the $\text{ BMU }$ ’s closing cone. The mathematical model is based on biochemical coupling between osteoclasts and osteoblasts of various maturity and includes the differentiation of osteoblasts into osteocytes and bone lining cells, as well as the influence of $\text{ BMU }$ cavity shrinkage on osteoblast development and activity. Matrix apposition rates predicted by the model are compared with data from tetracycline double labelling experiments. We find that the linear phenomenological relationship observed in these experiments between matrix apposition rate and $\text{ BMU }$ cavity radius holds for most of the refilling phase simulated by our model, but not near the start and end of refilling. This suggests that at a particular bone site undergoing remodelling, bone formation starts and ends rapidly, supporting the hypothesis that osteoblasts behave synchronously. Our model also suggests that part of the observed cross-sectional variability in tetracycline data may be due to different bone sites being refilled by $\text{ BMU }$ s at different stages of their lifetime. The different stages of a $\text{ BMU }$ ’s lifetime (such as initiation stage, progression stage, and termination stage) depend on whether the cell populations within the $\text{ BMU }$ are still developing or have reached a quasi-steady state whilst travelling through bone. We find that due to their longer lifespan, active osteoblasts reach a quasi-steady distribution more slowly than active osteoclasts. We suggest that this fact may locally enlarge the Haversian canal diameter (due to a local lack of osteoblasts compared to osteoclasts) near the $\text{ BMU }$ ’s point of origin.     

Oxidative power and intracellular distribution of mitochondria control cell oxygen regime when arterial hypoxemia occurs

The regulatory impact of the mitochondria spatial distribution and enlargement in their oxidative power $q_{O_2 } $ on tissue oxygenation of skeletal muscle during hypoxia were studied. Investigations were performed by mathematical modeling of 3D O₂ diffusion-reaction in muscle fiber. The oxygen consumption rate $V_{O_2 } $ and tissue $p_{O_2 } $ were analyzed in response to a decrease in arterial blood oxygen concentration from 19.5 to 10 vol % at moderate load. Cells with evenly (case 1) and unevenly (case 2) distributed mitochondria were considered. According to calculations, owing to a rise in mitochondria oxidative power from 3.5 to 6.5 mL/min per 100 g of tissue it is possible to maintain muscle oxygen $V_{O_2 } $ at a constant level of 3.5 mL/min per 100 g despite a decrease in O₂ delivery. The minimum value of tissue $p_{O_2 } $ was about 0 and an area of hypoxia appeared inside the cell in case 1. Whereas hypoxia disappeared and minimum value of $p_{O_2 } $ increased from 0 to 4 mmHg if mitochondria were distributed unevenly (case 2). The possibilities of such regulation depended on the relationship “the degree of hypoxemia — the level of oxygen delivery.” It was assumed that an increase in mitochondrial enzyme activity and their migration to places of the greatest oxygen consumption rate can improve the oxygen regime in the cell as it adapts to hypoxia.     

Towards a conceptualization of ETL and physical storage of semantic data warehouses as a service

The data warehouse technology has become the incontestable tool for businesses and organizations to make strategic decisions to ensure their competitively. The construction of a data warehouse ( $\mathcal{D}\mathcal{W}$ ) passes by selecting relevant information sources, extracting relevant data and loading them into the $\mathcal{D}\mathcal{W}$ . These processes require a precise expertise from designers related to logical and physical implementations of information sources, which is not usually an easy task. The diversity and heterogeneity of information sources makes the construction process of the $\mathcal{D}\mathcal{W}$ complex and time consuming. Domain ontologies have been proposed to reduce heterogeneity between sources, platforms, services, etc. They resolve syntax and semantic conflicts. The phenomenon of adopting domain ontologies by organizations creates a new type of databases, called semantic databases ( $\mathcal{S}\mathcal{D}\mathcal{B}$ ). As a consequence, they become a candidate for building the semantic $\mathcal{D}\mathcal{W}$ ( $\mathcal{S}\mathcal{D}\mathcal{W}$ ). To handle the diversity of information sources and hide the implementations aspects of sources, proposing a generic framework for constructing $\mathcal{D}\mathcal{W}$ becomes a necessity. In this paper, we first proposed an ontology-based approach for designing $\mathcal{S}\mathcal {D}\mathcal{B}$ . Secondly, ETL phases are defined at ontological level to hide the implementation details. Thirdly, a storage service for ontologies and its associated data is given. Finally, our proposal is validated through a case study and a tool.     

Release of non-exchangeable {}^{{{\text{15}}}}{\text{NH}}^{{\text{ + }}}_{{\text{4}}} from subgrade, decomposed granite substrates and uptake by non-mycorrhizal and mycorrhizal California native annual grass, Vulpia microstachys

Release rates of recently fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ from non-exchangeable interlayer sites in 2:1 silicate minerals were determined for decomposed granite (DG) saprolites from three locations in California, USA. Recently-fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ release from the DG substrate was quantified by extracting diffused $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ with H-resin, as well as a native, annual grass Vulpia microstachys. The $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ release data varied with via the method of extraction, which included H-resin pre-treatments (Na⁺ or H⁺) and V. microstachys uptake (mycorrhizal inoculated or uninoculated). After 6 weeks (1008 h), more $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ was recovered from fixed interlayer positions by the H-resins as compared to uptake by V. microstachys. The H⁺ treated H-resins recovered more released $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ (≈94 mg ${\text{NH}}^{{\text{ + }}}_{{\text{4}}} - {\text{N}}\;{\text{kg}}^{1} $ or (12%) of total fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ ) in two of the three DG samples as compared to the Na⁺ treated resins, (which recovered ≈70–78 mg ${\text{NH}}^{{\text{ + }}}_{{\text{4}}} - {\text{N}}\;{\text{kg}}^{{{\text{ - 1}}}} $ (or 9–10%) of the total fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ ). The V. microstachys assimilated 8–9% of the total fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ with mycorrhizal inoculum as compared to only 2% without a mycorrhizal inoculum, over the same time period. The fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ release kinetics from the H-resin experiments were most accurately described by first order and power function models, and can be characterized as biphasic using a heterogeneous diffusion model. Uptake of both the ¹⁵N and ambient, unlabelled N from the soils was closely related to plant biomass. There was no significant difference in percent of N per unit of biomass between the control and mycorrhizal treatments. The findings presented here indicate that observed, long-term $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ release rates from DG in studies utilizing resins, may overestimate the levels of fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ made available to plants and microorganisms. Additionally, the study suggested that mycorrhizae facilitate the acquisition and plant uptake of fixed $ {\text{NH}}^{{\text{ + }}}_{{\text{4}}} $ , resulting in markedly increased plant biomass production.     

Effects of collagen deposition on passive and active mechanical properties of large pulmonary arteries in hypoxic pulmonary hypertension

Proximal pulmonary artery (PA) stiffening is a strong predictor of mortality in pulmonary hypertension. Collagen accumulation is mainly responsible for PA stiffening in hypoxia-induced pulmonary hypertension (HPH) in mouse models. We hypothesized that collagen cross-linking and the type I isoform are the main determinants of large PA mechanical changes during HPH, which we tested by exposing mice that resist type I collagen degradation (Col1a1 $^\mathrm{R/R})$ and littermate controls (Col1a1 $^{+/+})$ to hypoxia for 10 days with or without $\beta $ -aminopropionitrile (BAPN) treatment to prevent cross-link formation. Static and dynamic mechanical tests were performed on isolated PAs with smooth muscle cells (SMC) in passive and active states. Percentages of type I and III collagen were quantified by histology; total collagen content and cross-linking were measured biochemically. In the SMC passive state, for both genotypes, hypoxia tended to increase PA stiffness and damping capacity, and BAPN treatment limited these increases. These changes were correlated with collagen cross-linking ( $p<0.05$ ). In the SMC active state, hypoxia increased PA dynamic stiffness and BAPN had no effect in Col1a1 $^{+/+}$ mice ( $p<0.05$ ). PA stiffness did not change in Col1a1 $^\mathrm{R/R}$ mice. Similarly, damping capacity did not change for either genotype. Type I collagen accumulated more in Col1a1 $^{+/+}$ mice, whereas type III collagen increased more in Col1a1 $^\mathrm{R/R}$ mice during HPH. In summary, PA passive mechanical properties (both static and dynamic) are related to collagen cross-linking. Type I collagen turnover is critical to large PA remodeling during HPH when collagen metabolism is not mutated and type III collagen may serve as a reserve.     

Coupling-induced population synchronization in an excitatory population of subthreshold Izhikevich neurons

We consider an excitatory population of subthreshold Izhikevich neurons which exhibit noise-induced firings. By varying the coupling strength J, we investigate population synchronization between the noise-induced firings which may be used for efficient cognitive processing such as sensory perception, multisensory binding, selective attention, and memory formation. As J is increased, rich types of population synchronization (e.g., spike, burst, and fast spike synchronization) are found to occur. Transitions between population synchronization and incoherence are well described in terms of an order parameter $\mathcal{O}$ . As a final step, the coupling induces oscillator death (quenching of noise-induced spikings) because each neuron is attracted to a noisy equilibrium state. The oscillator death leads to a transition from firing to non-firing states at the population level, which may be well described in terms of the time-averaged population spike rate $\overline{R}$ . In addition to the statistical-mechanical analysis using $\mathcal{O}$ and $\overline{R}$ , each population and individual state are also characterized by using the techniques of nonlinear dynamics such as the raster plot of neural spikes, the time series of the membrane potential, and the phase portrait. We note that population synchronization of noise-induced firings may lead to emergence of synchronous brain rhythms in a noisy environment, associated with diverse cognitive functions.     

Retention of Dissolved Inorganic Nitrogen by Foliage and Twigs of Four Temperate Tree Species

Nitrogen (N) retention by tree canopies is believed to be an important process for tree nutrient uptake, and its quantification is a key issue in determining the impact of atmospheric N deposition on forest ecosystems. Due to dry deposition and retention by other canopy elements, the actual uptake and assimilation by the tree canopy is often obscured in throughfall studies. In this study, ¹⁵N-labeled solutions ( $ ^{15} {\text{NH}}_{4}^{ + } $ and $ ^{15} {\text{NO}}_{3}^{ - } $ ) were used to assess dissolved inorganic N retention by leaves/needles and twigs of European beech, pedunculate oak, silver birch, and Scots pine saplings. The effects of N form, tree species, leaf phenology, and applied $ {\text{NO}}_{3}^{ - } $ to $ {\text{NH}}_{4}^{ + } $ ratio on the N retention were assessed. Retention patterns were mainly determined by foliar uptake, except for Scots pine. In twigs, a small but significant ¹⁵N enrichment was detected for $ {\text{NH}}_{4}^{ + } $ , which was found to be mainly due to physicochemical adsorption to the woody plant surface. The mean $ {{^{15} {\text{NH}}_{4}^{ + } } \mathord{\left/ {\vphantom {{^{15} {\text{NH}}_{4}^{ + } } {^{15} {\text{NO}}_{3}^{ - } }}} \right. \kern-0em} {^{15} {\text{NO}}_{3}^{ - } }} $ retention ratio varied considerably among species and phenological stadia, which indicates that the use of a fixed ratio in the canopy budget model could lead to an over- or underestimation of the total N retention. In addition, throughfall water under each branch was collected and analyzed for $ ^{15} {\text{NH}}_{4}^{ + } $ , $ ^{15} {\text{NO}}_{3}^{ - } $ , and all major ions. Net throughfall of $ ^{15} {\text{NH}}_{4}^{ + } $ was, on average, 20 times higher than the actual retention of $ ^{15} {\text{NH}}_{4}^{ + } $ by the plant material. This difference in $ ^{15} {\text{NH}}_{4}^{ + } $ retention could not be attributed to pools and fluxes measured in this study. The retention of $ ^{15} {\text{NH}}_{4}^{ + } $ was correlated with the net throughfall of K⁺, Mg²⁺, Ca²⁺, and weak acids during leaf development and the fully leafed period, while no significant relationships were found for $ ^{15} {\text{NO}}_{3}^{ - } $ retention. This suggests that the main driving factors for $ {\text{NH}}_{4}^{ + } $ retention might be ion exchange processes during the start and middle of the growing season and passive diffusion at leaf senescence. Actual assimilation or abiotic uptake of N through leaves and twigs was small in this study, for example, 1–5% of the applied dissolved ¹⁵N, indicating that the impact of canopy N retention from wet deposition on forest productivity and carbon sequestration is likely limited.     

Topological classification and enumeration of RNA structures by genus

To an RNA pseudoknot structure is naturally associated a topological surface, which has its associated genus, and structures can thus be classified by the genus. Based on earlier work of Harer–Zagier, we compute the generating function $\mathbf{D}_{g,\sigma }(z)=\sum _{n}\mathbf{d}_{g,\sigma }(n)z^n$ for the number $\mathbf{d}_{g,\sigma }(n)$ of those structures of fixed genus $g$ and minimum stack size $\sigma $ with $n$ nucleotides so that no two consecutive nucleotides are basepaired and show that $\mathbf{D}_{g,\sigma }(z)$ is algebraic. In particular, we prove that $\mathbf{d}_{g,2}(n)\sim k_g\,n^{3(g-\frac{1}{2})} \gamma _2^n$ , where $\gamma _2\approx 1.9685$ . Thus, for stack size at least two, the genus only enters through the sub-exponential factor, and the slow growth rate compared to the number of RNA molecules implies the existence of neutral networks of distinct molecules with the same structure of any genus. Certain RNA structures called shapes are shown to be in natural one-to-one correspondence with the cells in the Penner–Strebel decomposition of Riemann’s moduli space of a surface of genus $g$ with one boundary component, thus providing a link between RNA enumerative problems and the geometry of Riemann’s moduli space.     

On the page number of RNA secondary structures with pseudoknots

Let ${\mathcal {S}}$ denote the set of (possibly noncanonical) base pairs {i, j} of an RNA tertiary structure; i.e. ${\{i, j\} \in \mathcal {S}}$ if there is a hydrogen bond between the ith and jth nucleotide. The page number of ${\mathcal {S}}$ , denoted ${\pi(\mathcal {S})}$ , is the minimum number k such that ${\mathcal {S}}$ can be decomposed into a disjoint union of k secondary structures. Here, we show that computing the page number is NP-complete; we describe an exact computation of page number, using constraint programming, and determine the page number of a collection of RNA tertiary structures, for which the topological genus is known. We describe an approximation algorithm from which it follows that ${\omega(\mathcal {S}) \leq \pi(\mathcal {S}) \leq \omega(\mathcal {S}) \cdot \log n}$ , where the clique number of ${\mathcal {S}, \omega(\mathcal {S})}$ , denotes the maximum number of base pairs that pairwise cross each other.