Mathematical models for cellular systems the von Foerster equation. Part I

P. B. M. Walker (1954) and H. C. Longuet-Higgins (quoted by Walker), as well as O. Scherbaum and G. Rasch (1957), made the first attempts towards a mathematical study of the age distribution in a cellular population. It was H. Von Foerster (1959), however, who derived the complete differential equation for the age density function,n(t, a). His equation is obtained from an analysis of the infinitesimal changes occurring during a time elementdt in a group of cells with ages betweena anda+da. The behavior of the population is determined by a quantity λ which we call the loss function. In this paper a rigorous discussion of the Von Foerster equation is presented, and a solution is given for the special case when λ depends, ont (time) anda (age) but not on other variables (such asn itself). It is also shown that the age density,n(t, a), is completely known only if the birth rate,α(t), and the initial age distribution, β(a), are given as boundary conditions. In Section II the steady state solution and some plausible forms of intrinsic loss functions (depending ona only) are discussed in view of later applications.     

On the derivation of a mean growth equation for cell cultures

IfN(t) is the expected number of cells in a culture at timet, the corresponding time derivative, andf(t−τ)dt the probability that a cell of aget−τ at timet will divide in the succeeding time intervaldt, then according to Hirsch and Engelberg (this issue) there obtains the integral equation for describing the dynamics of the cell population. It is the purpose of this note to give two alternative derivations of this equation, one based on the age density equation of Von Foerster, and the other based on a generalized form of the Harris-Bellman equation describing the first moment of an age dependent, branching process. In addition, a probability model is posed from which the Von Foerster equation and, hence, the Hirsch-Engelberg equation readily follows.     

Theory of transport in linear biological systems: I. Fundamental integral equation

The conditions under which the output,γ _b (t), of a biological system is related to the input,γ _a (t), by an integral equation of the typeγ _b (t) = ∫ ₀ ^t γ _a (ω)w(t−ω)dω, where ω(t) is a transport functioncharacteristic of the system, are analyzed in detail. Methods of solving this type of integral equation are briefly discussed. The theory is then applied to problems in tracer kinetics in which input and output are sums of exponentials, and explicit formulae, which are applicable whether or not the pool is uniformly mixed, are derived for “turnover time” and “pool” size.     

An urn model study of variability within a compartment

Formulas are derived for the mean and variance of the number of radioactive atoms present in a compartment (or urn). Initally,n ₁ radioactive atoms andb stable atoms are placed in the urn; and subsequently,r stable atoms are added and an equal number,r, of a random mixture of stable and radioactive atoms is removed per unit time. The expected number of radioactive atoms,E(t), present at timet is, as expected,n ₁ e^−λt where λ=(r/Δt)/(b+r+n ₁). The relative variance, σ²(t)/n ₁ ² , vanishes to zero forr=1, atoms per unit time and for a large number ofn ₁ radioactive atoms; but for a large number of bothr andn ₁ atoms the relative variance is ∼e ^−λt , equal to the fractional retention, fort>1/λ. Thus in studies where radionuclides are injected into animals and a single compartment represents the data, if a large variance is observed it might be due to the fact that large numbers of atoms are transferred out in unit time. When a small variance is observed, this is probably due to the fact that few atoms are transferred in smaller units of time (such that λ is the same in both cases). Research sponsored by the Energy Research and Development Administration under contract with Union Carbide Corporation.     

Multiregional Periodic Matrix for Modeling the Population Dynamics of Sardine (<Emphasis Type="Italic">Sardina pilchardus</Emphasis>) Along the Moroccan Atlantic Coast: Management Elements for Fisheries

In this paper, we present a deterministic time discrete mathematical model based on multiregional periodic matrices to describe the dynamics of Sardina pilchardus in the Central Atlantic area of the Moroccan coast. This model deals with two stages (immature and mature) and three spatial zones where sardines are supposed to migrate from one zone to another. The population dynamics is described by an autonomous recurrence equation N(t + 1) = A.N(t), where A is a positive matrix whose entries are estimated using data collected during biannual acoustic surveys carried out from 2001 to 2003 onboard the Norwegian research vessel “Dr Fridtjof Nansen”. The dominant eigenvalue λ of A that gives the long-term growth rate of fish population is smaller than one. This agrees with the stock decrease observed in the data collected. We show that λ is highly sensitive to the recruitment rate and much less sensitive to the reproduction rate. These results can clearly be used to define an efficient scenario in order to fight for instance against a stock decrease.     

Age and growth of the nototheniid fishTrematomus bernacchii Boulenger from Terra Nova Bay,Antarctica

Age and growth of the nototheniid fishTrematomus bernacchii Boulenger 1902 were estimated by reading the sagittal otoliths of 457 adult specimens caught off Terra Nova Bay (Ross Sea) in the austral summer 1990–1991. Annuli in ground and polished otoliths were examined using a dissecting microscope under reflected light. The Von Bertalanffy growth equation was L_t=273.5 [1 − e^{−0.109(t+2.10)}] for males (n=122) and L_t=422.2 [1 − e^{−0.055(t+1.92)}] for females (n=211) where L is total length in millimetres. Maximum estimated age was 21 years for females and 16 years for males. This is in agreement with the hypothesis that considers slow growth and old age as a typical feature of Antarctic fishes.     

Bounds on the total particle number for species governed by reaction-diffusion equations in the infinite spatial domain

Differential inequality methods are developed for establishing upper and lower bounds on the total particle numberN(t)=∫θ(x,t) d³ x associated with solutions to nonlinear reaction-diffusion equations of the form ∂θ/∂t=D∇²θ+fθ-gθ ⁿ⁺¹ , whereD(>0),n(>0),f andg are constant parameters. If finite in a neighborhood oft=0,N(t) is bounded below for allt≥0 by a certain derived function oft for equations withg≥0. An upper bound onN(t) is obtained for equations withn=1,f<0 andg<0. These results provide general preservation and extinction criteria for the total particle number.     

HIV spread in the San Francisco cohort: Scaling of the effective logistic rate for seropositivity

A simple scaling (semigroup) property is manifest in the functional form of the effective logistic rate for the increase in the HIV seropositive fraction in the San Francisco (City Clinic) cohort. Witht _i=4.5 years, this scaling property—r→λ^-2r undert→[λt+(λ−1)t _i] for all parameter values λ≧1—encapsulates the effects of relevant biological and sociological changes in the key epidemiological variables during the 8-year seropositive rise period, 1978–1985 inclusive.     

A mathematical model for drug administration by using the phagocytosis of red blood cells

 A mathematical model for the delivery of drug directly to the macrophages by using the phagocytosis of senescent red blood cells is proposed. The model is based on the following assumption: At time t=0 a preassigned red blood cell population n(0, a)=φ(a), a>0, loaded by the drug, is injected in the blood circulation. Among the cells of that population only those with an age a≧ā (ā=120 days) will be phagocytosed by macrophages. Of course, the lifetime of the drug must be higher than ā. Within the red blood cells it cannot be metabolized, neither can it diffuse through their membranes. The emphasis of the paper is on the mathematical properties and on the formulation of the control problem. Received 15 December 1994; received in revised form 20 July 1995     

Development of optical criteria to discriminate various types of highly turbid lake waters

Our aim was to refine the optical classification of turbid waters in order to improve the performance of water color algorithms. Bio-optical measurements and sampling of optically active substances were performed in highly turbid lakes Taihu, Chaohu, and Dianchi, and in Three Gorges reservoir in China. Based on strong correlations between trough depths of remote sensing reflectance (R _rs(λ)) near 680 nm (denoted as TD680) and the ratios of inorganic suspended matter (ISM) to total suspended matter (TSM) concentrations, an empirical model was developed for water classification. In the 400–900 nm spectral range, different correlations between R _rs(λ), TSM and chlorophyll a (Chla) concentrations indicate discrepancies among the following ISM/TSM ranges: ISM/TSM ≤ 0.5, 0.5 < ISM/TSM < 0.8, and ISM/TSM ≥ 0.8. Corresponding findings support an important conclusion that only high ISM/TSM ratios, usually above 0.5, and using the more sensitive and stable near infrared spectral range (730–820 nm), can assure good performances of the TSM remote sensing algorithms. Meanwhile, the particulate absorption a _p(λ) and scattering b _p(λ) were strongly influenced by different ranges of ISM/TSM ratios. Typically the a _p(675) peaks became more and more vague as ISM/TSM increased, and the b _p(λ) values first decreased and then increased with a marked inflexion at ISM/TSM ≈ 0.5. The TD680 threshold values were derived to discriminate three types of turbid waters, i.e., Type 1 (TD680 ≥ 0.0082 sr⁻¹), Type 2 (0.0082 sr⁻¹ > TD680 > 0 sr⁻¹), and Type 3 (TD680 ≤ 0 sr⁻¹). This study provides a promising tool for identifying various types of highly turbid waters, and is significant for the development of class-based algorithms of water color remote sensing.     

Age and growth of Lestrolepis japonica (Aulopiformes: Paralepididae) in Kagoshima Bay,southern Japan

 Age and growth of a paralepidid, Lestrolepis japonica, were determined from sagittal otoliths of specimens collected from April 2001 to February 2002 in Kagoshima Bay, southern Japan. Marginal growth increments indicated that the annulus was formed once a year, in August and September. Four groups were recognized, having none, one, two, or three annuli, respectively. Age in months was assigned to each individual on the basis of August as the birth month. The maximum recorded age was 48 months. Growth was expressed by von Bertalanffy's equation, L _t  = 171.91{1 − exp[−0.148(t + 0.403)]}, t being the age in months and L _t the standard length (mm) at age t. Received: July 15, 2002 / Revised: November 18, 2002 / Accepted: December 10, 2002     

Time delays produced by essential nonlinearity in population growth models

It is pointed out that the asymptotic general solution to the ϕ-model equation for a periodic carrying capacityK(t) andt≳r ⁻¹ is identical in form to the generalized logistic equation solution with a built-in developmental time delay τ(≲r ⁻¹) and associated parameter ranges of primary biological interest. In the case of the ϕ-model equation, the time delay is a purely dynamical consequence of the nonlinear form featured by the population growth rate.     

Stability of the analytical solution of Penna model of biological aging

There are some analytical solutions of the Penna model of biological aging; here, we discuss the approach by Coe et al. (Phys. Rev. Lett. 89, 288103, 2002), based on the concept of self-consistent solution of a master equation representing the Penna model. The equation describes transition of the population distribution at time t to next time step (t + 1). For the steady state, the population n(a, l, t) at age a and for given genome length l becomes time-independent. In this paper we discuss the stability of the analytical solution at various ranges of the model parameters—the birth rate b or mutation rate m. The map for the transition from n(a, l, t) to the next time step population distribution n(a + 1, l, t + 1) is constructed. Then the fix point (the steady state solution) brings recovery of Coe et al. results. From the analysis of the stability matrix, the Lyapunov coefficients, indicative of the stability of the solutions, are extracted. The results lead to phase diagram of the stable solutions in the space of model parameters (b, m, h), where h is the hunt rate. With increasing birth rate b, we observe critical b ₀ below which population is extinct, followed by non-zero stable single solution. Further increase in b leads to typical series of bifurcations with the cycle doubling until the chaos is reached at some b _c. Limiting cases such as those leading to the logistic model are also discussed.     

Aecidium dispori is the aecial anamorph ofPuccinia albispora, sp. nov. (Uredinales)

Aecidium dispori forms spermogonium and aecium onDisporum sessile andD. smilacinum, which are distributed in East Asia. TheAecidium species is found to be an aecial anamorph of aPuccinia fungus, with its uredinial-telial stage being formed onCarex conica, C. dolichostachya subsp.multifolia, C. pisiformis subsp.alterniflora andC. rugata. Urediniospores of this fungus are large, colorless, thick-walled with 4–5 equatorial germ pores. The morphological characteristics of urediniospores and the spermogonial-aecial host do not fit to any set of circumscribing characters of previously described species. We consider the fungus to be a new species and propose a new name,Puccinia albispora, for the fungus.     

Restriction analysis of lambda EMBL3 background recombinants: occurrence of lambda phages carrying a head to tail oriented left arm DNA sequence

Summary Eight representative recombinant background clones of λEMBL3 were analysed usingKpnI,BamHI,SalI,EcoRI andHindIII digestion. We found that λEMBL3 carries its own left arm in theBamHI cloning site. In this way, recombinant molecules were found to be generated which can grow onEscherichia coli strain NM539. In all cases analysed, the left arm DNA was inserted in a head to tail orientation. Seven clones carried a restoredBamHI site at thecos site-BamHI site connection. In the region where the inserted left arm and the right arm were ligated,BamHI cloning produces a large palindromic sequence consisting of two polylinkers. ThisBamHI site was incompletely cleaved in all cases analysed. We assume that a part of the λ DNA molecule in this region shows a cruciform structure prohibiting recognition or cleavage of this site by restriction endonucleaseBamHI.     

Studies on the relationship between cyanide-resistant respiration and expression of alternative oxidase in mung bean using antibodies prepared by synthetic polypeptide

Twelve peptides, including eight conservative amino acid residues in the amino acid sequence of hydrophilic S helix of the alternative oxidase (AOX), were synthesized by solid-phase method. The polypeptide was coupled with α-chymotrypsinogen, and the antibodies were obtained through immunizing domestic rabbit by injecting this complex. By using these antibodies, which were raised to immunoreact with total proteins of purified mitochondria from different organs of mung bean seedlings, we find that there are two hybridizable AOX bands in mitochondria. Their molecular weights are about 35 and 38 ku, respectively. Moreover, the respiratory parameters of hypocotyl, true leaf and cotyledon of mung bean seedlings show that true leaf has the highest total respiration (V_t), alternative pathway (AP) capacity (V_alt) and the activity of AP (ρV_alt) among the three organs. V_t andρV _alt of cotyledon ranked the second. Hypocotyl has the lowest V_t andρV _alt, but its V_alt is higher than that of cotyledon. These results are consistent with the analysis of Western blotting of expression of AOX. The highest V_t andρV _alt in true leaf are accompanied two hybridizable polypeptides of AOX protein, 35 ku and 38 ku respectively. The next is cotyledon V_t andρV _alt with only one 38 ku hybridizable polypeptide of AOX protein. HypocotylV _t andρV _alt is the lowest and its immunoblotting band is similar to that of cotyledon, but the expressive amount of 38 ku protein is less than that of cotyledon. The results suggest that the 35 ku AOX may contribute mainly to true leafρV _alt.     

Studies on the relationship between cyanide-resistant respiration and expression of alternative oxidase in mung bean using antibodies prepared by synthetic polypeptide

Twelve peptides, including eight conservative amino acid residues in the amino acid sequence of hydrophilic S helix of the alternative oxidase (AOX), were synthesized by solid-phase method. The polypeptide was coupled with α-chymotrypsinogen, and the antibodies were obtained through immunizing domestic rabbit by injecting this complex. By using these antibodies, which were raised to immunoreact with total proteins of purified mitochondria from different organs of mung bean seedlings, we find that there are two hybridizable AOX bands in mitochondria. Their molecular weights are about 35 and 38 ku, respectively. Moreover, the respiratory parameters of hypocotyl, true leaf and cotyledon of mung bean seedlings show that true leaf has the highest total respiration (Vt), alternative pathway (AP) capacity (Valt) and the activity of AP (ρValt) among the three organs. Vt and ρValt of cotyledon ranked the second. Hypocotyl has the lowest V, and ρValt, but its Valt is higher than that of cotyledon. These result     

Specific absorption coefficient and the phytoplankton package effect in Lake Taihu,China

In order to investigate variations of absorption and total chlorophyll-a (TChl-a)-specific absorption coefficient of phytoplankton in Lake Taihu, 57 water samples obtained from Lake Taihu during November 8–22, 2007 were used in this study. Package effect and accessory pigments’ influences on the absorption spectra were also examined. Phytoplankton absorption was measured by quality filter technical, and TChl-a concentration was measured by “hot ethanol” method. Results yielded significant variations in phytoplankton absorption and TChl-a-specific absorption coefficient. Phytoplankton absorption coefficient at 675 nm is highly correlated to TChl-a concentration, while absorption at 440 nm is less correlated to TChl-a concentration because of great package effect and accessory pigments’ influence. There was an inverse relationship between a _ph*(λ) and TChl-a concentration. Four types of absorption spectra are identified by normalizing a _ph*(λ) to a _ph*(440). The a _ph*(λ) variation is mainly due to accessory pigments and package effect, whose influence at 675 nm ranges from 83.2% to 28%, with an average of 65.3%. Meanwhile, the wide varying ratio of a _ph*(440) to a _ph*(675) indicates a large variation range in the ratio of accessory pigment to TChl-a concentration. Those findings are significant to estimate Chl-a concentration based on bio-optical model, estimate primary production from remote sensing, and plan further ecological restoration measures for Lake Taihu. Handling editor: Luigi Naselli-Flores     

Longevity and growth efficiency of two deep-dwelling Arctic zoarcids and comparison with eight other zoarcid species from different climatic regions

Our basic knowledge of the ecology, especially the age and growth of polar deep-sea biota is still scarce. This study provides first data about the age and growth of the two abundant Arctic fish species Lycodes frigidus and Lycodes squamiventer (Zoarcidae). Lycodes frigidus was caught at the deeper parts (1,546–3,576 m depth) of the HAUSGARTEN observatory (HG), west of Svalbard. The congener Lycodes squamiventer was caught at two HG stations (1,273–1,546 m) and at the H?kon Mosby Mud Volcano (HMMV, ~1,250 m), a cold seep in the southwestern Barents Sea. Age was determined by sagittal otolith increment analysis. Growth performance was assessed by fitting age–length data to a von Bertalanffy growth equation. Our data suggest that L. frigidus and L. squamiventer attain maximum ages of 33 and 21 years, respectively. Lycodes squamiventer from the HMMV had significantly higher growth rates and their maximum age and length was slightly lower compared to conspecifics from the shallow HG stations. Von Bertalanffy growth equations were L _t  = 58.9 ∗ (1 − e^(−0.042*t)) for L. frigidus, and L _t  = 25.3 ∗ (1 − e^(−0.074*t)) and L _t  = 24.2 ∗ (1 − e^{(−0.099 * t)}) for L. squamiventer from HG and the HMMV, respectively. A comparison of these data with those of eight other zoarcids indicates that growth performances are correlated with temperature: the higher the annual mean temperatures experienced, the higher the growth rates. However, maximum ages decrease with increasing temperatures.     

Population growth rates forAmblyomma americanum (Acari: Ixodidae) onBos indicus,B. taurus andB. indicus x B. taurus cattle

Densities ofAmblyomma americanum (L.) onBos indicus, B. taurus andB. indicus x B. taurus cattle are compared over a 3-year period, and the growth rate (rate of increase or decrease) of parasitic tick populations on each cattle genotype is estimated.Average log₁₀ densities of parasiticA. americanum larvae are significantly (P=0.05) lower onB. indicus cattle than onB. taurus andB. indicus x B. taurus cattle. Average log densities of nymphal and adult ticks onB. taurus cattle are significantly higher than onB. indicus cattle but neither cattle genotype differs in this regard fromB. indicus x B. taurus cattle.Estimated annual tick population growth rates (log₁₀) for parasiticA. americanum are positive onB. taurus cattle (+0.84 larvae, +0.09 nymphs, +0.22 adults calf^–1 year^–1), but are negative onB. indicus (–0.18 nymphs, –0.14 adults calf^–1 year^–1) andB. indicus x B. taurus cattle (–0.45 larvae, –0.24 nymphs, –0.14 adults calf^–1 year^–1). Populations of parasitic larvae were not detected onB. indicus cattle.