Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.     

Ordering Dynamics in Neuron Activity Pattern Model: An Insight to Brain Functionality

We study the domain ordering kinetics in d = 2 ferromagnets which corresponds to populated neuron activities with both long-ranged interactions, V(r) ∼ r ⁻ⁿ and short-ranged interactions. We present the results from comprehensive Monte Carlo (MC) simulations for the nonconserved Ising model with n ≥ 2, interaction range considering near and far neighbors. Our model results could represent the long-ranged neuron kinetics (n ≤ 4) in consistent with the same dynamical behaviour of short-ranged case (n ≥ 4) at far below and near criticality. We found that emergence of fast and slow kinetics of long and short ranged case could imitate the formation of connections among near and distant neurons. The calculated characteristic length scale in long-ranged interaction is found to be n independent (L(t) ∼ t ^1/(n−2)), whereas short-ranged interaction follows L(t) ∼ t ^1/2 law and approximately preserve universality in domain kinetics. Further, we did the comparative study of phase ordering near the critical temperature which follows different behaviours of domain ordering near and far critical temperature but follows universal scaling law.     

Response Characteristics of Soil Fractal Features to Different Land Uses in Typical Purple Soil Watershed

As a fundamental characteristic of soil physical properties, the soil Particle Size Distribution (PSD) is important in the research on soil moisture migration, solution transformation, and soil erosion. In this research, the PSD characteristics with distinct methods in different land uses are analyzed. The results show that the upper bound of the volume domain of the clay domain ranges from 5.743μm to 5.749μm for all land-use types. For the silt domain of purple soil, the value ranges among 286.852~286.966 μm. For all purple soil land-use types, the order of the volume domain fractal dimensions is D_clay<D_silt<D_sand. However, the values of D_silt and D_sand in the Pinus massoniana Lamb, Robinia pseudoacacia L and Ipomoea batatas are all higher than the corresponding values in the Citrus reticulate Blanco and Setaria viridis. Moreover, in all the land-use types, all of the parameters in volume domain fractal dimension (D_vi) are higher than the corresponding parameter values from the United States Department of Agriculture (D_vi(U)). The correlation study between the volume domain fractal dimension and the soil properties shows that the intensity of correlation to the soil texture and soil organic matter has the order as: D_silt>D_silt(U)>D_sand (U)>D_sand and D_silt>D_silt(U)>D_sand>D_sand(U), respectively. As it is compared with all D_vi, the D_silt has the most significant correlativity to the soil texture and organic matter in different land uses of the typical purple soil watersheds. Therefore, D_silt will be a potential indictor for evaluating the proportion of fine particles in the PSD, as well as a key measurement in soil quality and productivity studies.     

Reduced heat shock response in human mononuclear cells during aging and its association with polymorphisms in HSP70 genes

Age-dependent changes in heat shock response (HSR) were studied in mononuclear cells (monocytes and lymphocytes) collected from young (mean age = 22.6 ± 1.7 years) and middle-aged (mean age = 56.3 ± 4.7 years) subjects after 1 hour of heat shock at 42°C. Genotype-specific HSR was measured by genotyping the subjects for 3 single nucleotide polymorphisms, HSPA1A(A-110C), HSPA1B(A1267G), and HSPA1L(T2437C), 1 each in the 3 HSP70 genes. A significant age-related decrease in the induction of Hsp70 occurred after heat shock in both monocytes and lymphocytes. The noninducible and inducible forms of Hsp70 decreased 1.3-fold (P < 0.001) and 1.4-fold (P < 0.001), respectively, in the monocytes with age. In the young subjects, a positive association was found between HSPA1L(T2437C) polymorphism and HSR. CC carriers had a significantly lower induction than TT carriers in both monocytes (P = 0.015) and lymphocytes (P = 0.044). This polymorphism, which is present in the coding region of HSPA1L gene, can affect the chaperoning function of Hsp70. These data consolidate our other observations that the CC genotype is unfavorable for human longevity and provide a functional explanation in terms of variations in HSR.     

Structural Changes and Associated Reduction of Hydraulic Conductance in Roots of Sorghum bicolor L. following Exposure to Water Deficit

The effects of a severe water deficit on total root (L_t) and axial (L_x) hydraulic conductances and on the development of the hypodermis, endodermis, and xylem were studied in sorghum (Sorghum bicolor L.). Water deficit was imposed in the upper rooting zone while the lower zones were kept moist. L_t and L_x were based on water flow rates obtained by applying suction to proximal xylem ends of excised roots. The development of the hypodermis, endodermis, and other tissues were examined by staining with fluorescent berberine hemisulfate and phloroglucinol-HCl. The L_t value (x 10⁻⁸ meters per second per megapascal) for unstressed control roots was 22.0 and only 5.9 for stressed roots. The low L_t in stressed roots was attributed, in part, to accelerated deposition of lignin and suberin in the hypodermis and endodermis. Calcofluor, an apoplastic tracer that binds to cellulose, was blocked in stressed roots at the lignified and suberized outer tangential walls of the hypodermis but readily penetrated the cortical walls of similar root regions in controls where the casparian band was not developed. L_x per unit root length was about 100 times lower in stressed roots than in controls because of the persistence of late metaxylem cross-walls and the smaller diameter and lower number of conductive protoxylem and early metaxylem vessels.     

Photoreactions in Phycomyces : Responses to the stimulation of narrow test areas with ultraviolet light

Equipment has been developed for ultraviolet illumination of sharply bounded test areas of the growing zone of sporangiophores of Phycomyces. The growing zone is opaque for this light and the tropic responses are negative. Periodic short narrow stimuli on alternating sides produce periodic tropic responses when applied at x > 0.5 mm, but none for x < 0.5 mm, where x is the distance below the sporangium. Sustained tropic stimuli, applied at constant x, produce tropic responses for any x > 0.1 mm. The lag between stimulus and response is 3.3 min. for any x > 0.5 mm. For smaller x the lag increases progressively. In all cases the tropic bend occurs at values of x > 0.5 mm. Sustained tropic stimuli, applied at constant height relative to ground, produce relatively sharp tropic bends. The center of the bend is at all times close to the simultaneous position of the stimulated area. The boundaries of a light-adapted zone move less than 0.1 mm in 10 min. relative to the sporangium. It is concluded that the receiving and adapting structures do not move relative to the sporangium, and that the responding system does not move relative to ground. The two systems move relative to each other with the speed of growth. The responding system does not extend above x = 0.5 mm.     

Intracellular Transport Dynamics of Endosomes Containing DNA Polyplexes along the Microtubule Network

We have explored the transport of DNA polyplexes enclosed in endosomes within the cellular environment by multiple particle tracking (MPT). The polyplex-loaded endosomes demonstrate enhanced diffusion at short timescales (t < 7 s) with their mean-square displacement (MSD) 〈Δx(t)²〉 scaling as t^1.25. For longer time intervals they exhibit subdiffusive transport and have an MSD scaling as t^0.7. This crossover from an enhanced diffusion to a subdiffusive regime can be explained by considering the action of motor proteins that actively transport these endosomes along the cellular microtubule network and the thermal bending modes of the microtubule network itself.     

The rate of allelism of lethal genes in a geographically structured population

The expected rate of allelism, E[I(x)], of lethal genes between two colonies with distance x in a structured population is studied by using one- and two-dimensional stepping-stone models. It is shown that E[I(x)] depends on the magnitude of selection in heterozygous condition (h), the rate of migration among adjacent colonies (m), the number of loci which produce lethal mutations (n) and the effective population size of each colony (N).——E[I(x)] always decreases with distance x. The rate of decrease is affected strongly by the magnitude of m. The rate of decrease is faster when m is small. E[I(x)] also decreases with increasing N and n. The effect of h on E[I(x)] is somewhat complicated. However, E[I(0)] is always smaller when h is small than when it is large.——For large x, the following approximate formulae may be obtained: (see PDF) where q and Var (q) are the mean and the variance of gene frequencies in each colony, t is approximated as t=h, (see PDF), -h for the partially recessive, completely recessive, and overdominant lethals, respectively, and C₀ is a function of m and t. It is clear that E[I(x)] declines exponentially with x in a one-dimensional habitat. The decrease E[I(x)] is faster in a two-dimensional habitat than in a one-dimensional habitat. The present result is applied to some of the existing data and the estimation of population parameters is also discussed.     

Cumulative Damage Survival Models for the Randomized Complete Block Design

A continuous time discrete state cumulative damage process {X(t), t ≥ 0} is considered, based on a non‐homogeneous Poisson hit‐count process and discrete distribution of damage per hit, which can be negative binomial, Neyman type A, Polya‐Aeppli or Lagrangian Poisson. Intensity functions considered for the Poisson process comprise a flexible three‐parameter family. The survival function is S(t) = P(X(t) ≤ L) where L is fixed. Individual variation is accounted for within the construction for the initial damage distribution {P(X(0) = x) | x = 0, 1, …,}. This distribution has an essential cut‐off before x = L and the distribution of L – X(0) may be considered a tolerance distribution. A multivariate extension appropriate for the randomized complete block design is developed by constructing dependence in the initial damage distributions. Our multivariate model is applied (via maximum likelihood) to litter‐matched tumorigenesis data for rats. The litter effect accounts for 5.9 percent of the variance of the individual effect. Cumulative damage hazard functions are compared to nonparametric hazard functions and to hazard functions obtained from the PVF‐Weibull frailty model. The cumulative damage model has greater dimensionality for interpretation compared to other models, owing principally to the intensity function part of the model.     

Equilibrium studies of lecithin-cholesterol interactions I. Stoichiometry of lecithin-cholesterol complexes in bulk systems

The maximum molar ratio of lecithin:cholesterol in aqueous dispersions has been reported to be 2:1, 1:1, or 1:2. The source of the desparate results has been examined in this study by analyzing (a) the phase relations in anhydrous mixtures (from which most dispersions are prepared) and (b) various methods of preparing aqueous dispersions, with the purpose of avoiding the formation of metastable states that may be responsible for the variability of the lecithin-cholesterol stoichiometry. Temperature-composition phase diagrams for anhydrous mixtures of cholesterol (CHOL) with dimyristoyl (DML) and with dipalmitoyl (DPL) lecithin were obtained by differential scanning calorimetry (DSC). Complexes form with molar ratios for lecithin:CHOL of 2:1 and 1:2; they are stable up to 70°C. When x(CHOL) < 0.33, two phases coexist: complex (2:1) plus pure lecithin; when 0.33 < x(CHOL) < 0.67 complexes (2:1) and (1:2) coexist as separate phases. The corresponding phase diagram in water for these mixtures was determined by DSC and isopycnic centrifugation in D₂O-H₂O gradients. Aqueous dispersions were prepared by various methods (vortexing, dialysis, sonication) yielding identical results except as noted below. The data presented supports the following phase relations. When x(CHOL) < 0.33, two lipid phases coexist: pure lecithin plus complex (2:1) where the properties of the lecithin phase are determined by whether the temperature is below or above T_c, the gel-liquid crystal transition temperature. Therefore, complex (2:1) will coexist with gel state below T_c and with liquid crystal above T_c. The densities follow in the order gel > complex (2:1) > liquid crystal. The density of complex (2:1) is less sensitive to temperature in the range 5°-45°C compared to the temperature dependence for DML and DPL where large changes in density occur at T_c. When x(CHOL) > 0.33, CHOL phase coexists with complex (2:1); anhydrous complex (1:2) is apparently not stable in H₂O. The results are independent of the method and temperature used for preparing the lipid dispersions. However, when dispersions are prepared by sonication or with solvents at T > T_c, an apparent 1:1 complex is formed. Evidence suggests the 1:1 complex is metastable.     

A Property of Second Order Differential Equations with Application to Formal Kinetics

It is shown that if x(t) is the solution of a second order differential equation, with real negative characteristic roots (not necessarily distinct), which exhibits an extremum at t = T, then T|x(T)|/|A| [UNK] 1/e where A is the area under the x(t) curve. This result is compared to a special case previously derived by M. Morales and applications of the theorem to formal kinetic problems are discussed.     

Asymptotic Distributions of Coalescence Times and Ancestral Lineage Numbers for Populations with Temporally Varying Size

The distributions of coalescence times and ancestral lineage numbers play an essential role in coalescent modeling and ancestral inference. Both exact distributions of coalescence times and ancestral lineage numbers are expressed as the sum of alternating series, and the terms in the series become numerically intractable for large samples. More computationally attractive are their asymptotic distributions, which were derived in Griffiths (1984) for populations with constant size. In this article, we derive the asymptotic distributions of coalescence times and ancestral lineage numbers for populations with temporally varying size. For a sample of size n, denote by T_m the mth coalescent time, when m + 1 lineages coalesce into m lineages, and A_n(t) the number of ancestral lineages at time t back from the current generation. Similar to the results in Griffiths (1984), the number of ancestral lineages, A_n(t), and the coalescence times, T_m, are asymptotically normal, with the mean and variance of these distributions depending on the population size function, N(t). At the very early stage of the coalescent, when t → 0, the number of coalesced lineages n − A_n(t) follows a Poisson distribution, and as m → n, n(n ? 1)T_m/2N(0) follows a gamma distribution. We demonstrate the accuracy of the asymptotic approximations by comparing to both exact distributions and coalescent simulations. Several applications of the theoretical results are also shown: deriving statistics related to the properties of gene genealogies, such as the time to the most recent common ancestor (TMRCA) and the total branch length (TBL) of the genealogy, and deriving the allele frequency spectrum for large genealogies. With the advent of genomic-level sequencing data for large samples, the asymptotic distributions are expected to have wide applications in theoretical and methodological development for population genetic inference.     

The mathematical theory of group selection. I. Full solution of a nonlinear Levins E = E(x) model

The first complete overtime solution is obtained for a group selection model of Levins E = E(x) type with recolonization but no other gene flow between islands. Assuming a subdivided population at carrying capacity, the model describes selection at a biallelic locus (A, a) where a is opposed by Mendelian selection but is favored by a lower rate of extinction of demes having high a frequency. By contrast to the linear diffusion equations encountered in classical mathematical genetics, the PDE governing the dynamics is now nonlinear in the metapopulation gene frequency distribution φ(x, t); furthermore, the initial conditions now heavily influence the equilibrium distribution φ_∞(x). A fully explicit formula (20) expressing this dependence is derived. The results indicate that a fixation is never reached, but (A, a) polymorphism in the metapopulation will result if , where s 1 parametrizes the strength of Mendelian selection, E(x) is the Levins extinction operator, h (typically in the open interval (0, 1)) is the dominance of a, and B is a parameter measuring the flatness of the initial distribution f(x) in the x → 1 limit.     

Different Effects of Three Selected Lactobacillus Strains in Dextran Sulfate Sodium-Induced Colitis in BALB/c Mice

Aim

To analyze the changes of different Lactobacillus species in ulcerative colitis patients and to further assess the therapeutic effects of selected Lactobacillus strains on dextran sulfate sodium (DSS)-induced experimental colitis in BALB/c mice.

Methods

Forty-five active ulcerative colitis (UC) patients and 45 population-based healthy controls were enrolled. Polymerase chain reaction (PCR) amplification and real-time PCR were performed for qualitative and quantitative analyses, respectively, of the Lactobacillus species in UC patients. Three Lactobacillus strains from three species were selected to assess the therapeutic effects on experimental colitis. Sixty 8-week-old BALB/c mice were divided into six groups. The five groups that had received DSS were administered normal saline, mesalazine, L. fermentum CCTCC M206110 strain, L. crispatus CCTCC M206119 strain, or L. plantarum NCIMB8826 strain. We assessed the severity of colitis based on disease activity index (DAI), body weight loss, colon length, and histologic damage.

Results

The detection rate of four of the 11 Lactobacillus species decreased significantly (P < 0.05), and the detection rate of two of the 11 Lactobacillus species increased significantly (P < 0.05) in UC patients. Relative quantitative analysis revealed that eight Lactobacillus species declined significantly in UC patients (P < 0.05), while three Lactobacillus species increased significantly (P < 0.05). The CCTCC M206110 treatment group had less weight loss and colon length shortening, lower DAI scores, and lower histologic scores (P < 0.05), while the CCTCC M206119 treatment group had greater weight loss and colon length shortening, higher histologic scores, and more severe inflammatory infiltration (P < 0.05). NCIMB8826 improved weight loss and colon length shortening (P < 0.05) with no significant influence on DAI and histologic damage in the colitis model.

Conclusions

Administration of an L. crispatus CCTCC M206119 supplement aggravated DSS-induced colitis. L. fermentum CCTCC M206110 proved to be effective at attenuating DSS-induced colitis. The potential probiotic effect of L. plantarum NCIMB8826 on UC has yet to be assessed.     

Exercise Tolerance Can Be Enhanced through a Change in Work Rate within the Severe Intensity Domain: Work above Critical Power Is Not Constant

Introduction

The characterization of the hyperbolic power-time (P-t _lim) relationship using a two-parameter model implies that exercise tolerance above the asymptote (Critical Power; CP), i.e. within the severe intensity domain, is determined by the curvature (W’) of the relationship.

Purposes

The purposes of this study were (1) to test whether the amount of work above CP (W>CP) remains constant for varied work rate experiments of high volatility change and (2) to ascertain whether W’ determines exercise tolerance within the severe intensity domain.

Methods

Following estimation of CP (208 ± 19 W) and W’ (21.4 ± 4.2 kJ), 14 male participants (age: 26 ± 3; peak V˙O2: 3708 ± 389 ml.min^-1) performed two experimental trials where the work rate was initially set to exhaust 70% of W’ in 3 (‘THREE’) or 10 minutes (‘TEN’) before being subsequently dropped to CP plus 10 W.

Results

W>CP for TEN (104 ± 22% W’) and W’ were not significantly different (P>0.05) but lower than W>CP for THREE (119 ± 17% W’, P<0.05). For both THREE (r = 0.71, P<0.01) and TEN (r = 0.64, P<0.01), a significant bivariate correlation was found between W’ and t _lim.

Conclusion

W>CP and t _lim can be greater than predicted by the P-t _lim relationship when a decrement in the work rate of high-volatility is applied. Exercise tolerance can be enhanced through a change in work rate within the severe intensity domain. W>CP is not constant.     

Length-dependent structure formation in Friedreich ataxia (GAA)n*(TTC)n repeats at neutral pH

More than 15 human genetic diseases have been associated with the expansion of trinucleotide DNA repeats, which may involve the formation of non-duplex DNA structures. The slipped-strand nucleation of duplex DNA within GC-rich trinucleotide repeats may result in the changes of repeat length; however, such a mechanism seems less likely for the AT-rich (GAA)_n·(TTC)_n repeats. Using two-dimensional agarose gels, chemical probing and atomic force microscopy, we characterized the formation of non-B-DNA structures in the Friedreich ataxia-associated (GAA)_n·(TTC)_n repeats from the FRDA gene that were cloned with flanking genomic sequences into plasmids. For the normal genomic repeat length (n = 9) our data are consistent with the formation of a very stable protonated intramolecular triplex (H-DNA). Its stability at pH 7.4 is likely due to the high proportion of the T·A·T triads which form within the repeats as well as in the immediately adjacent AT-rich sequences with a homopurine· homopyrimidine bias. At the long normal repeat length (n = 23), a family of H-DNAs of slightly different sizes has been detected. At the premutation repeat length (n = 42) and higher negative supercoiling, the formation of a single H-DNA structure becomes less favorable and the data are consistent with the formation of a bi-triplex structure.     

Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθ ⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.     

Cross-Correlation Functions for a Neuronal Model

Cross-correlation functions, R_XY(t,τ), are obtained for a neuron model which is characterized by constant threshold θ, by resetting to resting level after an output, and by membrane potential U(t) which results from linear summation of excitatory postsynaptic potentials h(t). The results show that: (1) Near time lag τ = 0, R_XY(t,τ) = f_U [θ-h(τ), t + τ] {h′(τ) + E_U [u′(t + τ)]} for positive values of this quantity, where f_U(u,t) is the probability density function of U(t) and E_U [u′(t + τ)] is the mean value function of U′(t + τ). (2) Minima may appear in R_XY(t,τ) for a neuron subjected only to excitation. (3) For large τ, R_XY(t,τ) is given approximately by the convolution of the input autocorrelation function with the functional of point (1). (4) R_XY(t,τ) is a biased estimator of the shape of h(t), generally over-estimating both its time to peak and its rise time.