Vector-Borne Pathogen and Host Evolution in a Structured Immuno-Epidemiological System

Vector-borne disease transmission is a common dissemination mode used by many pathogens to spread in a host population. Similar to directly transmitted diseases, the within-host interaction of a vector-borne pathogen and a host’s immune system influences the pathogen’s transmission potential between hosts via vectors. Yet there are few theoretical studies on virulence–transmission trade-offs and evolution in vector-borne pathogen–host systems. Here, we consider an immuno-epidemiological model that links the within-host dynamics to between-host circulation of a vector-borne disease. On the immunological scale, the model mimics antibody-pathogen dynamics for arbovirus diseases, such as Rift Valley fever and West Nile virus. The within-host dynamics govern transmission and host mortality and recovery in an age-since-infection structured host-vector-borne pathogen epidemic model. By considering multiple pathogen strains and multiple competing host populations differing in their within-host replication rate and immune response parameters, respectively, we derive evolutionary optimization principles for both pathogen and host. Invasion analysis shows that the \({\mathcal {R}}_0\) maximization principle holds for the vector-borne pathogen. For the host, we prove that evolution favors minimizing case fatality ratio (CFR). These results are utilized to compute host and pathogen evolutionary trajectories and to determine how model parameters affect evolution outcomes. We find that increasing the vector inoculum size increases the pathogen \({\mathcal {R}}_0\), but can either increase or decrease the pathogen virulence (the host CFR), suggesting that vector inoculum size can contribute to virulence of vector-borne diseases in distinct ways.     

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.     

Quantifying interspecific spatial overlap in aquatic macrophyte communities

Levins’s asymmetrical α index quantifies between species overlap over resources more realistically than similar-purpose single-value indices. The associated community-wide \(\bar \alpha\) index expresses the degree of “species packing”. Both indices were formulated upon competing animal (i.e., mobile) organisms and are independent of population densities. However, overlap over resources for nonmobile organisms such as plants may have an impact even below carrying capacity. The proposed \(\hat \alpha\) index, based on Levins’s α index, quantifies spatial overlap for plants integrating information on species spatial distribution and crowding conditions. The \(\hat \alpha\) index is specifically designed for plant distribution data collected in discrete plots with density expressed as percent coverage (%cover) of substratum. We also propose a community-wide \({\hat \alpha_{\text{c}}}\) index, conceptually analogous to \(\bar \alpha\) , but furnished with a measure of dispersion (se \({\hat \alpha_{\text{c}}}\) ). Species importance within the community is inferred from comparisons of pairwise \(\hat \alpha\) ’s with \({\hat \alpha_{\text{c}}}\) . The \(\hat \alpha\) and \({\hat \alpha_{\text{c}}}\) indices correlate closely and exponentially with plant density, and correct apparent over- and underestimations of interaction intensity at low and very high crowding by Levins’s α and \(\bar \alpha\) , respectively. Index application to aquatic plant communities gave results consistent with within-community and general ecological patterns, suggesting a high potential of the proposed \(\hat \alpha\) and \({\hat \alpha_{\text{c}}}\) indices in basic and applied macrophyte ecological studies and management.     

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.     

The Basic Reproduction Number for Cellular SIR Networks

The basic reproduction number \(R_0\) is the average number of new infections produced by a typical infective individual in the early stage of an infectious disease, following the introduction of few infective individuals in a completely susceptible population. If \(R_0<1\) , then the disease dies, whereas for \(R_0>1\) the infection can invade the host population and persist. This threshold quantity is well studied for SIR compartmental or mean field models based on ordinary differential equations, and a general method for its computation has been proposed by van den Driessche and Watmough. We concentrate here on SIR epidemiological models that take into account the contact network N underlying the transmission of the disease. In this context, it is generally admitted that \(R_{0}\) can be approximated by the average number \(R_{2,3}\) of infective individuals of generation three produced by an infective of generation two. We give here a simple analytic formula of \(R_{2,3}\) for SIR cellular networks. Simulations on two-dimensional cellular networks with von Neumann and Moore neighborhoods show that \(R_{2,3}\) can be used to capture a threshold phenomenon related the dynamics of SIR cellular network and confirm the good quality of the simple approach proposed recently by Aparicio and Pascual for the particular case of Moore neighborhood.     

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.     

Laplacian Dynamics on General Graphs

In previous work, we have introduced a “linear framework” for time-scale separation in biochemical systems, which is based on a labelled, directed graph, G, and an associated linear differential equation, $dx/dt = \mathcal{L}(G)\cdot x$ , where $\mathcal{L}(G)$ is the Laplacian matrix of G. Biochemical nonlinearity is encoded in the graph labels. Many central results in molecular biology can be systematically derived within this framework, including those for enzyme kinetics, allosteric proteins, G-protein coupled receptors, ion channels, gene regulation at thermodynamic equilibrium, and protein post-translational modification. In the present paper, in response to new applications, which accommodate nonequilibrium mechanisms in eukaryotic gene regulation, we lay out the mathematical foundations of the framework. We show that, for any graph and any initial condition, the dynamics always reaches a steady state, which can be algorithmically calculated. If the graph is not strongly connected, which may occur in gene regulation, we show that the dynamics can exhibit flexible behavior that resembles multistability. We further reveal an unexpected equivalence between deterministic Laplacian dynamics and the master equations of continuous-time Markov processes, which allows rigorous treatment within the framework of stochastic, single-molecule mechanisms.     

Threshold dynamics of an infective disease model with a fixed latent period and non-local infections

In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction–diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number ${\mathcal{R}_0}$ for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and ${\mathcal{R}_0}$ . In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute ${\mathcal{R}_0}$ .     

Synchronicity of movement paths of barren-ground caribou and tundra wolves

Movement patterns of highly mobile animals can reveal life history strategies and ecological relationships. We hypothesized that wolves (Canis lupus) would display similar movement patterns as their prey, barren-ground caribou (Rangifer tarandus groenlandicus), and that movements of the two species would co-vary with season. We tested for interspecific movement dynamics using animal locations from wolves and caribou monitored concurrently from mid-October to June, across the Northwest Territories and Nunavut, Canada. We used a correlated random walk as a null model to test for pattern in movements and the bearing procedure to detect whether movements were consistently directional. There was a statistical difference between the movements of caribou and wolves (F _1,9 = 13.232, P = 0.005), when compared to a correlated random walk, and a significant interaction effect between season and species (F _1,9 = 6.815, P = 0.028). During winter, the movements of caribou were strongly correlated with the 80°–90° ( $\overline{X}$ X ¯ r = 0.859, SE = 0.065) and 270°–280° ( $\overline{X}$ X ¯ r = 0.875, SE = 0.059) bearing classes suggesting an east–west movement gradient. Wolf movements during winter showed large variation in direction, but were generally east to west. Peak mean correlation for caribou movements during spring was distinct at 40°–50° ( $\overline{X}$ X ¯ r = 0.978, SE = 0.006) revealing movement to the north-east calving grounds. During spring, wolf movements correlated with the 80°–90° ( $\overline{X}$ X ¯ r = 0.861, SE = 0.043) and 270°–280° ( $\overline{X}$ X ¯ r = 0.850, SE = 0.064) bearing class. Directionality of movement suggested that during winter, caribou and wolves had a similar distribution at the large spatial scales we tested. During spring migration, however, caribou and wolves employed asynchronous movement strategies. Our findings demonstrate the utility of the correlated random walk and bearing procedure for quantifying the movement patterns of co-occurring species.     

Phylogentic constraints,adaptive syndromes,and emergent properties: From individuals to population dynamics

The hypothesis is developed that there are causal linkages in evolved insect herbivore life histories and behaviors from phylogenetic constraints to adaptive syndromes to the emergent properties involving ecological interactions and population dynamics. Thus the argument is developed that the evolutionary biology of a species predetermines its current ecology.Phylogenetic Constraints refer to old characters in the phylogeny of a species and a group of species which set limits on the range of life history patterns and behaviors that can evolve. For example, a sawfly is commonly limited to oviposition in soft plant tissue, while plants are growing rapidly.Adaptive Syndromes are evolutionary responses to the phylogenetic constraints that minimize the limitations and maximize larval performance. Such syndromes commonly involve details of female ovipositional behavior and how individuals make choices for oviposition sites relative to plant quality variation which maximize larval survival. Syndromes also involve larval adaptations to the kinds of choices females make in oviposition. The evolutionary biology involved with phylogenetic constraints and adaptive syndromes commonly predetermines the ecological interactions of a species and its population dynamics. Therefore, these ecological interactions are calledEmergent Properties because they are natural consequences of evolved morphology, behavior, and physiology. They commonly strongly influence the three-trophic-level interactions among host plants, insect herbivores, and carnivores, and the relative forces of bottom-up and top-down influences in food webs. The arguments are supported using such examples as galling sawflies and other gallers, shoot-boring moths and beetles, budworms, and forest Macrolepidoptera. The contrasts between outbreak or eruptive species and uncommon and rare species with latent population dynamics are emphasized.     

Resource competition induces heterogeneity and can increase cohort survivorship: selection-event duration matters

Determining when resource competition increases survivorship can reveal processes underlying population dynamics and reinforce the importance of heterogeneity among individuals in conservation. We ran an experiment mimicking the effects of competition in a growing season on survivorship during a selection event (e.g., overwinter starvation, drought). Using a model fish species (Poecilia reticulata), we studied how food availability and competition affect mass in a treatment stage, and subsequently survivorship in a challenge stage of increased temperature and starvation. The post-treatment mean mass was strongly related to the mean time to mortality and mass at mortality at all levels of competition. However, competition increased variance in mass and extended the right tail of the survivorship curve, resulting in a greater number of individuals alive beyond a critical temporal threshold ( $T^{*}$ ) than without competition. To realize the benefits from previously experienced competition, the duration of the challenge ( $T_{c}$ ) following the competition must exceed the critical threshold $T^{*}$ (i.e., competition increases survivorship when $T_{c} > T^{*}$ ). Furthermore, this benefit was equivalent to increasing food availability by 20 % in a group without competition in our experiment. The relationship of $T^{*}$ to treatment and challenge conditions was modeled by characterizing mortality through mass loss in terms of the stochastic rate of loss of vitality (individual’s survival capacity). In essence, when the duration of a selection event exceeds $T^{*}$ , competition-induced heterogeneity buffers against mortality through overcompensation processes among individuals of a cohort. Overall, our study demonstrates an approach to quantify how early life stage heterogeneity affects survivorship.     

Multi-stage Vector-Borne Zoonoses Models: A Global Analysis

A class of models that describes the interactions between multiple host species and an arthropod vector is formulated and its dynamics investigated. A host-vector disease model where the host’s infection is structured into n stages is formulated and a complete global dynamics analysis is provided. The basic reproduction number acts as a sharp threshold, that is, the disease-free equilibrium is globally asymptotically stable (GAS) whenever \({\mathcal {R}}_0^2\le 1\) and that a unique interior endemic equilibrium exists and is GAS if \({\mathcal {R}}_0^2>1\). We proceed to extend this model with m host species, capturing a class of zoonoses where the cross-species bridge is an arthropod vector. The basic reproduction number of the multi-host-vector, \({\mathcal {R}}_0^2(m)\), is derived and shown to be the sum of basic reproduction numbers of the model when each host is isolated with an arthropod vector. It is shown that the disease will persist in all hosts as long as it persists in one host. Moreover, the overall basic reproduction number increases with respect to the host and that bringing the basic reproduction number of each isolated host below unity in each host is not sufficient to eradicate the disease in all hosts. This is a type of “amplification effect,” that is, for the considered vector-borne zoonoses, the increase in host diversity increases the basic reproduction number and therefore the disease burden.     

Interaction of ascorbate with photosystem I

Ascorbate is one of the key participants of the antioxidant defense in plants. In this work, we have investigated the interaction of ascorbate with the chloroplast electron transport chain and isolated photosystem I (PSI), using the EPR method for monitoring the oxidized centers \( {\text{P}}_{700}^{ + } \) and ascorbate free radicals. Inhibitor analysis of the light-induced redox transients of P₇₀₀ in spinach thylakoids has demonstrated that ascorbate efficiently donates electrons to \( {\text{P}}_{ 7 0 0}^{ + } \) via plastocyanin. Inhibitors (DCMU and stigmatellin), which block electron transport between photosystem II and Pc, did not disturb the ascorbate capacity for electron donation to \( {\text{P}}_{700}^{ + } \) . Otherwise, inactivation of Pc with CN^? ions inhibited electron flow from ascorbate to \( {\text{P}}_{700}^{ + } \) . This proves that the main route of electron flow from ascorbate to \( {\text{P}}_{700}^{ + } \) runs through Pc, bypassing the plastoquinone (PQ) pool and the cytochrome b ₆ f complex. In contrast to Pc-mediated pathway, direct donation of electrons from ascorbate to \( {\text{P}}_{700}^{ + } \) is a rather slow process. Oxidized ascorbate species act as alternative oxidants for PSI, which intercept electrons directly from the terminal electron acceptors of PSI, thereby stimulating photooxidation of P₇₀₀. We investigated the interaction of ascorbate with PSI complexes isolated from the wild type cells and the MenB deletion strain of cyanobacterium Synechocystis sp. PCC 6803. In the MenB mutant, PSI contains PQ in the quinone-binding A₁-site, which can be substituted by high-potential electron carrier 2,3-dichloro-1,4-naphthoquinone (Cl₂NQ). In PSI from the MenB mutant with Cl₂NQ in the A₁-site, the outflow of electrons from PSI is impeded due to the uphill electron transfer from A₁ to the iron-sulfur cluster F_X and further to the terminal clusters F_A/F_B, which manifests itself as a decrease in a steady-state level of \( {\text{P}}_{700}^{ + } \) . The addition of ascorbate promoted photooxidation of P₇₀₀ due to stimulation of electron outflow from PSI to oxidized ascorbate species. Thus, accepting electrons from PSI and donating them to \( {\text{P}}_{700}^{ + } \) , ascorbate can mediate cyclic electron transport around PSI. The physiological significance of ascorbate-mediated electron transport is discussed.     

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.     

Quasi equilibrium,variance effective size and fixation index for populations with substructure

In this paper, we develop a method for computing the variance effective size \(N_{eV}\) , the fixation index \(F_{ST}\) and the coefficient of gene differentiation \(G_{ST}\) of a structured population under equilibrium conditions. The subpopulation sizes are constant in time, with migration and reproduction schemes that can be chosen with great flexibility. Our quasi equilibrium approach is conditional on non-fixation of alleles. This is of relevance when migration rates are of a larger order of magnitude than the mutation rates, so that new mutations can be ignored before equilibrium balance between genetic drift and migration is obtained. The vector valued time series of subpopulation allele frequencies is divided into two parts; one corresponding to genetic drift of the whole population and one corresponding to differences in allele frequencies among subpopulations. We give conditions under which the first two moments of the latter, after a simple standardization, are well approximated by quantities that can be explicitly calculated. This enables us to compute approximations of the quasi equilibrium values of \(N_{eV}\) , \(F_{ST}\) and \(G_{ST}\) . Our findings are illustrated for several reproduction and migration scenarios, including the island model, stepping stone models and a model where one subpopulation acts as a demographic reservoir. We also make detailed comparisons with a backward approach based on coalescence probabilities.     

Interdependency and hierarchy of exact and approximate epidemic models on networks

Over the years numerous models of \(SIS\) (susceptible \(\rightarrow \) infected \(\rightarrow \) susceptible) disease dynamics unfolding on networks have been proposed. Here, we discuss the links between many of these models and how they can be viewed as more general motif-based models. We illustrate how the different models can be derived from one another and, where this is not possible, discuss extensions to established models that enables this derivation. We also derive a general result for the exact differential equations for the expected number of an arbitrary motif directly from the Kolmogorov/master equations and conclude with a comparison of the performance of the different closed systems of equations on networks of varying structure.     

Quantitative Genetics and Modularity in Cranial and Mandibular Morphology of Calomys expulsus

Patterns of genetic covariance between characters (represented by the covariance matrix \({\varvec{G}}\) ) play an important role in morphological evolution, since they interact with the evolutionary forces acting over populations. They are also expected to influence the patterns expressed in their phenotypic counterparts \(({\varvec{P}})\) , because of limits imposed by multiple developmental and functional restrictions on the genotype/phenotype map. We have investigated genetic covariances in the skull and mandible of the vesper mouse (Calomys expulsus) in order to estimate the degree of similarity between genetic and phenotypic covariances and its potential roots on developmental and functional factors shaping those integration patterns. We use a classic ad hoc analysis of morphological integration based on current state of art of developmental/functional factors during mammalian ontogeny and also applied a novel methodology that makes use of simulated evolutionary responses. We have obtained \({\varvec{P}}\) and \({\varvec{G}}\) that are strongly similar, for both skull and mandible; their similarity is achieved through the spatial and temporal organization of developmental and functional interactions, which are consistently recognized as hypothesis of trait associations in both matrices.     

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.     

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.