Dynamic permeability of the lacunar–canalicular system in human cortical bone

A new method for the experimental determination of the permeability of a small sample of a fluid-saturated hierarchically structured porous material is described and applied to the determination of the lacunar–canalicular permeability \((K_\mathrm{LC})\) in bone. The interest in the permeability of the lacunar–canalicular pore system (LCS) is due to the fact that the LCS is considered to be the site of bone mechanotransduction due to the loading-driven fluid flow over cellular structures. The permeability of this space has been estimated to be anywhere from \(10^{-17}\;\) to \(10^{-25}\; \hbox {m}^{2}\) . However, the vascular pore system and LCS are intertwined, rendering the permeability of the much smaller-dimensioned LCS challenging to measure. In this study, we report a combined experimental and analytical approach that allowed the accurate determination of the \(K_\mathrm{LC}\) to be on the order of \(10^{-22}\; \hbox {m}^{2}\) for human osteonal bone. It was found that the \(K_\mathrm{LC}\) has a linear dependence on loading frequency, decreasing at a rate of \(2 \times 10^{-24}\; \hbox {m}^{2}\) /Hz from 1 to 100 Hz, and using the proposed model, the porosity alone was able to explain 86 % of the \(K_\mathrm{LC}\) variability.     

A thermomechanical framework for reconciling the effects of ultraviolet radiation exposure time and wavelength on connective tissue elasticity

Augmentation of the mechanical properties of connective tissue using ultraviolet (UV) radiation—by targeting collagen cross-linking in the tissue at predetermined UV exposure time \((t)\) and wavelength \((\lambda )\) —has been proposed as a therapeutic method for supporting the treatment for structural-related injuries and pathologies. However, the effects of \(\lambda \) and \(t\) on the tissue elasticity, namely elastic modulus \((E)\) and modulus of resilience \((u_\mathrm{Y})\) , are not entirely clear. We present a thermomechanical framework to reconcile the \(t\) - and \(\lambda \) -related effects on \(E\) and \(u_\mathrm{Y}\) . The framework addresses (1) an energy transfer model to describe the dependence of the absorbed UV photon energy, \(\xi \) , per unit mass of the tissue on \(t\) and \(\lambda \) , (2) an intervening thermodynamic shear-related parameter, \(G\) , to quantify the extent of UV-induced cross-linking in the tissue, (3) a threshold model for the \(G\) versus \(\xi \) relationship, characterized by   \(t_\mathrm{C}\) —the critical \(t\) underpinning the association of \(\xi \) with \(G\) —and (4) the role of \(G\) in the tissue elasticity. We hypothesized that \(G\) regulates \(E\) (UV-stiffening hypothesis) and \(u_\mathrm{Y}\) (UV-resilience hypothesis). The framework was evaluated with the support from data derived from tensile testing on isolated ligament fascicles, treated with two levels of \(\lambda \) (365 and 254 nm) and three levels of \(t\) (15, 30 and 60 min). Predictions from the energy transfer model corroborated the findings from a two-factor analysis of variance of the effects of \(t\) and \(\lambda \) treatments. Student’s t test revealed positive change in \(E\) and \(u_\mathrm{Y}\) with increases in \(G\) —the findings lend support to the hypotheses, implicating the implicit dependence of UV-induced cross-links on \(t\) and \(\lambda \) for directing tissue stiffness and resilience. From a practical perspective, the study is a step in the direction to establish a UV irradiation treatment protocol for effective control of exogenous cross-linking in connective tissues.     

Individual-Based Model for Quorum Sensing with Background Flow

Quorum sensing is a wide-spread mode of cell–cell communication among bacteria in which cells release a signalling substance at a low rate. The concentration of this substance allows the bacteria to gain information about population size or spatial confinement. We consider a model for \(N\) cells which communicate with each other via a signalling substance in a diffusive medium with a background flow. The model consists of an initial boundary value problem for a parabolic PDE describing the exterior concentration \(u\) of the signalling substance, coupled with \(N\) ODEs for the masses \(a_i\) of the substance within each cell. The cells are balls of radius \(R\) in \(\mathbb {R} ^3\) , and under some scaling assumptions we formally derive an effective system of \(N\) ODEs describing the behaviour of the cells. The reduced system is then used to study the effect of flow on communication in general, and in particular for a number of geometric configurations.     

An approach to sequential NMR assignments of proteins: application to chemical shift restraint-based structure prediction

A procedure for the simultaneous acquisition of {HNCOCANH & HCCCONH} chemical shift correlation spectra employing sequential \(^{1}\hbox {H}\) data acquisition for moderately sized proteins is presented. The suitability of the approach for obtaining sequential resonance assignments, including complete \(^{15}\hbox {N}{},\, ^{1}\hbox {H}{}^{N},\, ^{13}\hbox {CO}{},\, ^{13}\hbox {C}^{\alpha }{},\, ^{13}\hbox {C}^{\beta }\) and \(^{1}\hbox {H}^{\alpha }\) chemical shift information, is demonstrated experimentally for a \(^{13}\hbox {C}\) and \(^{15}\hbox {N}\) labelled sample of the C-terminal winged helix (WH) domain of the minichromosome maintenance (MCM) complex of Sulfolobus solfataricus. The chemical shift information obtained was used to calculate the global fold of this winged helix domain via CS-Rosetta. This demonstrates that our procedure provides a reliable and straight-forward protocol for a quick global fold determination of moderately-sized proteins.     

N-site Phosphorylation Systems with 2N-1 Steady States

Multisite protein phosphorylation plays a prominent role in intracellular processes like signal transduction, cell-cycle control and nuclear signal integration. Many proteins are phosphorylated in a sequential and distributive way at more than one phosphorylation site. Mathematical models of \(n\) -site sequential distributive phosphorylation are therefore studied frequently. In particular, in Wang and Sontag (J Math Biol 57:29–52, 2008), it is shown that models of \(n\) -site sequential distributive phosphorylation admit at most \(2n-1\) steady states. Wang and Sontag furthermore conjecture that for odd \(n\) , there are at most \(n\) and that, for even \(n\) , there are at most \(n+1\) steady states. This, however, is not true: building on earlier work in Holstein et al. (Bull Math Biol 75(11):2028–2058, 2013), we present a scalar determining equation for multistationarity which will lead to parameter values where a \(3\) -site system has \(5\) steady states and parameter values where a \(4\) -site system has \(7\) steady states. Our results therefore are counterexamples to the conjecture of Wang and Sontag. We furthermore study the inherent geometric properties of multistationarity in \(n\) -site sequential distributive phosphorylation: the complete vector of steady state ratios is determined by the steady state ratios of free enzymes and unphosphorylated protein and there exists a linear relationship between steady state ratios of phosphorylated protein.     

Approximation of sojourn-times via maximal couplings: motif frequency distributions

Sojourn-times provide a versatile framework to assess the statistical significance of motifs in genome-wide searches even under non-Markovian background models. However, the large state spaces encountered in genomic sequence analyses make the exact calculation of sojourn-time distributions computationally intractable in long sequences. Here, we use coupling and analytic combinatoric techniques to approximate these distributions in the general setting of Polish state spaces, which encompass discrete state spaces. Our approximations are accompanied with explicit, easy to compute, error bounds for total variation distance. Broadly speaking, if \({\mathsf{T}}_n\) is the random number of times a Markov chain visits a certain subset \({\mathsf{T}}\) of states in its first \(n\) transitions, then we can usually approximate the distribution of \({\mathsf{T}}_n\) for \(n\) of order \((1-\alpha )^{-m}\) , where \(m\) is the largest integer for which the exact distribution of \({\mathsf{T}}_m\) is accessible and \(0\le \alpha \le 1\) is an ergodicity coefficient associated with the probability transition kernel of the chain. This gives access to approximations of sojourn-times in the intermediate regime where \(n\) is perhaps too large for exact calculations, but too small to rely on Normal approximations or stationarity assumptions underlying Poisson and compound Poisson approximations. As proof of concept, we approximate the distribution of the number of matches with a motif in promoter regions of C. elegans. Mathematical properties of the proposed ergodicity coefficients and connections with additive functionals of homogeneous Markov chains as well as ergodicity of non-homogeneous Markov chains are also explored.     

Elastic behavior of a red blood cell with the membrane’s nonuniform natural state: equilibrium shape,motion transition under shear flow,and elongation during tank-treading motion

Direct numerical simulations of the mechanics of a single red blood cell (RBC) were performed by considering the nonuniform natural state of the elastic membrane. A RBC was modeled as an incompressible viscous fluid encapsulated by an elastic membrane. The in-plane shear and area dilatation deformations of the membrane were modeled by Skalak constitutive equation, while out-of-plane bending deformation was formulated by the spring model. The natural state of the membrane with respect to in-plane shear deformation was modeled as a sphere ( \(\alpha =0\) ), biconcave disk shape ( \(\alpha =1\) ) and their intermediate shapes ( \(0<\alpha <1\) ) with the nonuniformity parameter \(\alpha \) , while the natural state with respect to out-of-plane bending deformation was modeled as a flat plane. According to the numerical simulations, at an experimentally measured in-plane shear modulus of \(2.5\times 10^{-6}\,\hbox {N}/\hbox {m}\) and an out-of-plane bending rigidity of \(2.0\times 10^{-19}\,\hbox {N}\cdot \hbox {m}\) of the cell membrane, the following results were obtained. (i) The RBC shape at equilibrium was biconcave discoid for \(\alpha >0.22\) and cupped otherwise; (ii) the experimentally measured fluid shear stress at the transition between tumbling and tank-treading motions under shear flow was reproduced for \(0.05<\alpha <0.34\) ; (iii) the elongation deformation of the RBC during tank-treading motion from the simulation was consistent with that from in vitro experiments, irrespective of the \(\alpha \) value. Based on our RBC modeling, the three phenomena (i), (ii), and (iii) were mechanically consistent for \(0.22<\alpha <0.34\) . The condition \(0.05<\alpha <0.22\) precludes a biconcave discoid shape at equilibrium (i); however, it gives appropriate fluid shear stress at the motion transition under shear flow (ii), suggesting that a combined effect of \(\alpha \) and the natural state with respect to out-of-plane bending deformation is necessary for understanding details of the RBC mechanics at equilibrium. Our numerical results demonstrate that moderate nonuniformity in a membrane’s natural state with respect to in-plane shear deformation plays a key role in RBC mechanics.     

A locally convergent rotationally invariant particle swarm optimization algorithm

Several well-studied issues in the particle swarm optimization algorithm are outlined and some earlier methods that address these issues are investigated from the theoretical and experimental points of view. These issues are the: stagnation of particles in some points in the search space, inability to change the value of one or more decision variables, poor performance when the swarm size is small, lack of guarantee to converge even to a local optimum (local optimizer), poor performance when the number of dimensions grows, and sensitivity of the algorithm to the rotation of the search space. The significance of each of these issues is discussed and it is argued that none of the particle swarm optimizers we are aware of can address all of these issues at the same time. To address all of these issues at the same time, a new general form of velocity update rule for the particle swarm optimization algorithm that contains a user-definable function \(f\) is proposed. It is proven that the proposed velocity update rule guarantees to address all of these issues if the function \(f\) satisfies the following two conditions: (i) the function \(f\) is designed in such a way that for any input vector \(\vec {y}\) in the search space, there exists a region \(A\) which contains \(\vec {y}\) and \( f\!\left( {\vec {y}} \right) \) can be located anywhere in \(A\) , and (ii) \(f\) is invariant under any affine transformation. An example of function \(f\) is provided that satisfies these conditions and its performance is examined through some experiments. The experiments confirm that the proposed algorithm (with an appropriate function \(f)\) can effectively address all of these issues at the same time. Also, comparisons with earlier methods show that the overall ability of the proposed method for solving benchmark functions is significantly better.     

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.     

Reconstructing a Phylogenetic Level-1 Network from Quartets

We describe a method that will reconstruct an unrooted binary phylogenetic level-1 network on \(n\) taxa from the set of all quartets containing a certain fixed taxon, in \(O(n^3)\) time. We also present a more general method which can handle more diverse quartet data, but which takes \(O(n^6)\) time. Both methods proceed by solving a certain system of linear equations over the two-element field \(\mathrm{GF}(2)\) . For a general dense quartet set, i.e. a set containing at least one quartet on every four taxa, our \(O(n^6)\) algorithm constructs a phylogenetic level-1 network consistent with the quartet set if such a network exists and returns an \(O(n^2)\) -sized certificate of inconsistency otherwise. This answers a question raised by Gambette, Berry and Paul regarding the complexity of reconstructing a level-1 network from a dense quartet set, and more particularly regarding the complexity of constructing a cyclic ordering of taxa consistent with a dense quartet set.     

Counting glycans revisited

We present an algorithm for counting glycan topologies of order \(n\) that improves on previously described algorithms by a factor \(n\) in both time and space. More generally, we provide such an algorithm for counting rooted or unrooted \(d\) -ary trees with labels or masses assigned to the vertices, and we give a “recipe” to estimate the asymptotic growth of the resulting sequences. We provide constants for the asymptotic growth of \(d\) -ary trees and labeled quaternary trees (glycan topologies). Finally, we show how a classical result from enumeration theory can be used to count glycan structures where edges are labeled by bond types. Our method also improves time bounds for counting alkanes.     

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.     

On linear birth-and-death processes in a random environment

We study the probability of extinction for single-type and multi-type continuous-time linear birth-and-death processes in a finite Markovian environment. The probability of extinction is equal to 1 almost surely if and only if the basic reproduction number \(R_0\) is \(\le 1\) , the key point being to identify a suitable definition of \(R_0\) for such a result to hold.     

Fishermen’s Profits Maximization: Case of Generalized Nash Equilibrium of a Non-symmetrical Game

In the present paper, we consider a bio-economic equilibrium model which describes the dynamics of a fish population fished by several fishermen seeking to maximize their profits. Each fisherman tries to find the fishing effort which maximizes his profit at biological equilibrium without any consultation with others, but all of them have to respect two constraints: (1) the sustainable management of the resources ; and (2) the preservation of the biodiversity. With all these considerations, our problem leads to a generalized Nash equilibrium problem. The objective is to show that even when a fisherman \(i\) provides a fishing effort equal to twice the fishing effort of a fisherman \(j\) , then the profit of fisherman \(i\) is not necessarily double that of fisherman \(j\) .     

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.     

Analysis of {}^{13}{\text{C}}^{{{\upalpha}}} and {}^{13}{\text{C}}^{{{\upbeta}}} chemical shifts of cysteine and cystine residues in proteins: a quantum chemical approach

Cysteines possess a unique property among the 20 naturally occurring amino acids: it can be present in proteins in either the reduced or oxidized form, and can regulate the activity of some proteins. Consequently, to augment our previous treatment of the other types of residues, the $ {}^{13}{\text{C}}^{{{\upalpha}}} $ and $ {}^{13}{\text{C}}^{{{\upbeta}}} $ chemical shifts of 837 cysteines in disulfide-bonded cystine from a set of seven non-redundant proteins, determined by X-ray crystallography and NMR spectroscopy, were computed at the DFT level of theory. Our results indicate that the errors between observed and computed $ {}^{13}{\text{C}}^{{{\upalpha}}} $ chemical shifts of such oxidized cysteines can be attributed to several effects such as: (a) the quality of the NMR-determined models, as evaluated by the conformational-average (ca) rmsd value; (b) the existence of high B-factor or crystal-packing effects for the X-ray-determined structures; (c) the dynamics of the disulfide bonds in solution; and (d) the differences in the experimental conditions under which the observed $ {}^{13}{\text{C}}^{{{\upalpha}}} $ chemical shifts and the protein models were determined by either X-ray crystallography or NMR-spectroscopy. These quantum-chemical-based calculations indicate the existence of two, almost non-overlapped, basins for the oxidized and reduced ?SH $ {}^{13}{\text{C}}^{{{\upbeta}}} $ , but not for the $ {}^{13}{\text{C}}^{{{\upalpha}}} $ , chemical shifts, in good agreement with the observation of 375 $ {}^{13}{\text{C}}^{{{\upalpha}}} $ and 337 $ {}^{13}{\text{C}}^{{{\upbeta}}} $ resonances from 132 proteins by Sharma and Rajarathnam (2000). Overall, our results indicate that explicit consideration of the disulfide bonds is a necessary condition for an accurate prediction of $ {}^{13}{\text{C}}^{{{\upalpha}}} $ and $ {}^{13}{\text{C}}^{{{\upbeta}}} $ chemical shifts of cysteines in cystines.     

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.     

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.     

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.