Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

A Generalised Age- and Phase-Structured Model of Human Tumour Cell Populations Both Unperturbed and Exposed to a Range of Cancer Therapies

We develop a general mathematical model for a population of cells differentiated by their position within the cell division cycle. A system of partial differential equations governs the kinetics of cell densities in certain phases of the cell division cycle dependent on time t (hours) and an age-like variable τ (hours) describing the time since arrival in a particular phase of the cell division cycle. Transition rate functions control the transfer of cells between phases. We first obtain a theoretical solution on the infinite domain −∞ < t < ∞. We then assume that age distributions at time t=0 are known and write our solution in terms of these age distributions on t=0. In practice, of course, these age distributions are unknown. All is not lost, however, because a cell line before treatment usually lies in a state of asynchronous balanced growth where the proportion of cells in each phase of the cell cycle remain constant. We assume that an unperturbed cell line has four distinct phases and that the rate of transition between phases is constant within a short period of observation (‘short’ relative to the whole history of the tumour growth) and we show that under certain conditions, this is equivalent to exponential growth or decline. We can then gain expressions for the age distributions. So, in short, our approach is to assume that we have an unperturbed cell line on t ≤ 0, and then, at t=0 the cell line is exposed to cancer therapy. This corresponds to a change in the transition rate functions and perhaps incorporation of additional phases of the cell cycle. We discuss a number of these cancer therapies and applications of the model.     

Dynamical machinery of a biochemical clock

Closed positive feedback loops of catalytic reactions between macromolecules, or hypercycles, provide a kinetic mechanism whereby each Species serves to catalyze selfreproduction of its successor in the loop. Hypercycles of five members or more evolve into limit cycles characteristic of a biochemical clock. Computer study of the coupled non-linear differential equations which describe these systems shows that the periodT _n of then-species limit cycle is given byT _n=nτ_n, where τ_n is an elemental repeat period reflecting translational time invariance. Analytic solutions of the equations are developed so that the time evolution of elementaryn-hypercycles can be traced in dynamical detail. It is shown that the magnitude of τ_n is, to good approximation, a linear function ofn. For a givenn, τ_n is a very sensitive function of the relative concentration a given member of the loop has at the time its predecessor dominates the state of the hypercycle. These concentrations decrease with increasingn. Aroundn=15 they become so small that elementary hypercycles become unstable against disruptive concentration fluctuations. Species concentrations for more realistic hypercycles tend not to be as small, so that the present estimate of a maximum number of components is a lower bound.     

Estimation of seasonal changes in translocation rates in leaves of a Japanese larch stand

To estimate the net rate of translocation in leaves of a larch stand, a new approach based on the summation method was proposed and given by a compartment model. The difference between the rate of translocation into and out of leaf biomass, namely, the net translocation rate (ΔT _r /Δt), was usually expressed by the difference between the growth rate of leaf biomass and the surplus production rate provided that the rate of leaf loss due to leaffall and grazing can be considered negligible. The rate, ΔT _r /Δt in a 19-yr-old larch stand, showed characteristic changes; it was positive from early April to late May, but after that it was negative until leaffall in late October. Our results confirmed that for the growth phase of positive ΔT _r /Δt translocation of assimilate stored in non-photosynthetic organs was indispensable for the growth. To quantify this, the ratio of ΔT _r /Δt to growth rate of leaf biomass was proposed.     

Interaction of maturation delay and nonlinear birth in population and epidemic models

 A population with birth rate function B(N) N and linear death rate for the adult stage is assumed to have a maturation delay T>0. Thus the growth equation N′(t)=B(N(t−T)) N(t−T) e⁻ d ₁ T−dN(t) governs the adult population, with the death rate in previous life stages d ₁≧0. Standard assumptions are made on B(N) so that a unique equilibrium N _e exists. When B(N) N is not monotone, the delay T can qualitatively change the dynamics. For some fixed values of the parameters with d ₁>0, as T increases the equilibrium N _e can switch from being stable to unstable (with numerically observed periodic solutions) and then back to stable. When disease that does not cause death is introduced into the population, a threshold parameter R ₀ is identified. When R ₀<1, the disease dies out; when R ₀>1, the disease remains endemic, either tending to an equilibrium value or oscillating about this value. Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay. Received: 2 November 1998 / Revised version: 26 February 1999     

Differential effects of caffeine on DNA damage and replication cell cycle checkpoints in the fission yeast Schizosaccharomyces pombe

Caffeine potentiates the lethal effects of ultraviolet and ionising radiation on wild-type Schizosaccharomyces pombe cells. In previous studies this was attributed to the inhibition by caffeine of a novel DNA repair pathway in S. pombe that was absent in the budding yeast Saccharomyces cerevisiae. Studies with radiation-sensitive S. pombe mutants suggested that this caffeine-sensitive pathway could repair ultraviolet radiation damage in the absence of nucleotide excision repair. The alternative pathway was thought to be recombinational and to operate in the G₂ phase of the cell cycle. However, in this study we show that cells held in G₁ of the cell cycle can remove ultraviolet-induced lesions in the absence of nucleotide excision repair. We also show that recombination-defective mutants, and those now known to define the alternative repair pathway, still exhibit the caffeine effect. Our observations suggest that the basis of the caffeine effect is not due to direct inhibition of recombinational repair. The mutants originally thought to be involved in a caffeine-sensitive recombinational repair process are now known to be defective in arresting the cell cycle in S and/or G₂ following DNA damage or incomplete replication. The gene products may also have an additional role in a DNA repair or damage tolerance pathway. The effect of caffeine could, therefore, be due to interference with DNA damage checkpoints, or inhibition of the DNA damage repair/tolerance pathway. Using a combination of flow cytometric analysis, mitotic index analysis and fluorescence microscopy we show that caffeine interferes with intra-S phase and G₂ DNA damage checkpoints, overcoming cell cycle delays associated with damaged DNA. In contrast, caffeine has no effect on the DNA replication S phase checkpoint in reponse to inhibition of DNA synthesis by hydroxyurea. Received: 16 June 1998 / Accepted: 13 July 1998     

Cell cycle control at the first restriction point and its effect on tissue growth

Cell cycle is controlled at two restriction points, R ₁ and R ₂. At both points the cell will commit apoptosis if it detects irreparable damage. But at R ₁ an undamaged cell also decides whether to proceed to the S phase or go into a quiescent mode, depending on the environmental conditions (e.g., overpopulation, hypoxia). We consider the effect of this decision at the population level in a spherical tissue {r < R(t)}. We prove that if the cells have full control at R ₁, they can manipulate the size of R(t) to ensure that 0 < c ≤ R(t) ≤ C < ∞; simulations further show that R(t) can be made nearly stationary. In the absence of such control, R(t) will either increase to ∞ or decrease to 0. The mathematical model and analysis involve a system of PDEs in {r < R(t)}.     

On the average cellular volume in synchronized cell populations

As was done by Sinclair and Ross (1969(, we consider a cellular population that consists initially (at time zero) ofN ₀ newborn cells, all with the same volumev _o. It is assumed that the occurrence of cell division is determined only by a cell’s age, and not by its volume. The frequency function of interdivision times, τ, is denoted byf(τ). If cell death is negligible, the expected number of cells,N(t), will increase according to the laws of a simple age-dependent branching process. The expression forN(t) is obtained as a sum over all generations; thevth term of this sum, in turn, is a multiple convolution integral, reflecting the life history ofvth generation cells (i.e., the lengths of thev successive interdivision periods plus the age of the cell at timet). Assuming that cell volume is a given function of cell age, e.g., linear or exponential, and that cellular volume is exactly halved at each division, it is possible to calculate the volume of a cell with a given life history, and thus the average cellular volume of the whole population as a function of time. If at time zero the volumes differ from cell to cell, the final equation must be modified by averaging over initial volumes. In the case of linear volume increase with age, a very simple asymptotic expression is found for the average cellular volume ast→∞. The case of exponential volume increase with age also leads to a simple asymptotic formula, but the resulting volume distribution is unstable. The mean cellular volume at birth and the second moment of the volume distribution can be calculated in a similar manner. Work supported by the U.S. Atomic Energy Commission.     

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.     

NMR relaxation study of water protons in Syrian hamster fetal cells at 300 MHz

TheT ₁ andT ₂ relaxation times of water protons in two cell types in culture derived from Syrian hamster fetuses (normal primary or secondary fetal cells vs BP6T tumor cells derived from the normal cells transformed by carcinogens) were measured at 7.05 Tesla magnetic field (proton frequency =300 MHz). TheT ₁/T ₂ ratios and the correlation time, τ _c , calculated from theT ₁/T ₂ ratio of cellular water protons, are significantly different in these two fibroblastic cell types of the same biological origin and with similar morphologies and growth rates in culture.     

A model for continuous production of thermally induced recombinant proteins

Summary General conditions for continuous expression of heterologous genes fromP _L promoter in two fermenters connected in series have been established. The induction time of the bacterial cells is calculated as a function of the retention time in the inducing reactor. Using this model, it is possible to adapt fermentation parameters to the particular behaviour of any specific recombinant clone.Nomenclature F(t) flow at timet [ml/min] - M _T (t) culture induced, at timeT of fermentation, during a period up tot [ml] - N _T (t) culture induced, at timeT of fermentation, during a period fromt tot+dt [ml] - p(t) product yield in a discontinuous culture [units/ml] - P(t) product yield at the outlet of the fermenter [units/ml] - v(t) volume of culture entered into the inducing reactor up to timet [ml] - V volume of the inducing reactor [ml] Greek letters retention time in the inducing reactor [min] - (t) average induction time at timet [min]     

Presteady-State Currents of the Rabbit Na+/Glucose Cotransporter (SGLT1)

The rabbit Na⁺/glucose cotransporter (SGLT1) exhibits a presteady-state current after step changes in membrane voltage in the absence of sugar. These currents reflect voltage-dependent processes involved in cotransport, and provide insight on the partial reactions of the transport cycle. SGLT1 presteady-state currents were studied as a function of external Na⁺, membrane voltage V _m , phlorizin and temperature. Step changes in membrane voltage—from the holding V _h to test values, elicited transient currents that rose rapidly to a peak (at 3–4 msec), before decaying to the steady state, with time constants τ≈4–20 msec, and were blocked by phlorizin (K _i ≈30 μm). The total charge Q was equal for the application of the voltage pulse and the subsequent removal, and was a function of V _m . The Q-V curves obeyed the Boltzmann relation: the maximal charge Q _max was 4–120 nC; V _0.5, the voltage for 50% Q _max was −5 to +30 mV; and z, the apparent valence of the moveable charge, was 1. Q _max and z were independent of V _h (between 0 and −100 mV) and temperature (20–30°C), while increasing temperature shifted V _0.5 towards more negative values. Decreasing [Na⁺] _o decreased Q _max, and shifted V _0.5 to more negative voltages 9by −100 mV per 10-fold decrease in [Na⁺] _o ). The time constant τ was voltage dependent: the τ-V relations were bell-shaped, with maximal τ_max 8–20 msec. Decreasing [Na⁺] _o decreased τ_max, and shifted the τ-V curves towards more negative voltages. Increasing temperature also shifted the τ-V curves, but did not affect τ_max. The maximum temperature coefficient Q ₁₀ for τ was 3–4, and corresponds to an activation energy of 25 kcal/mole. Simulations of a 6-state ordered kinetic model for rabbit Na⁺/glucose cotransport indicate that charge-movements are due to Na⁺-binding/dissociation and a conformational change of the empty transporter. The model predicts that (i) transient currents rise to a peak before decay to steady-state; (ii) the τ-V relations are bell-shaped, and shift towards more negative voltages as [Na⁺] _o is reduced; (iii) τ_max is decreased with decreasing [Na⁺] _o ; and (iv) the Q-V relations are shifted towards negative voltages as [Na⁺] _o is reduced. In general, the kinetic properties of the presteady-state currents are qualitatively predicted by the model. Received: 12 August 1996/Revised: 30 September 1996     

Body temperatures in free-flying pigeons

We examined the relationship between body temperature (T_b) of free flying pigeons and ambient water vapor pressure and temperature. Core or near core T_b of pigeons were measured using thermistors inserted into the cloaca and connected to small transmitters mounted on the tail feathers of free flying tippler pigeons (Columba livia). Wet and dry bulb temperatures were measured using modified transmitters mounted onto free-flying pigeons. These allowed calculation of relative humidity and hence water vapor pressure at flight altitudes. Mean T_b during flight was 42.0 ± 1.3 °C (n = 16). Paired comparisons of a subset of this data indicated that average in-flight T_b increased significantly by 1.2 ± 0.7 °C (n = 7) over that of birds at rest (t = −4.22, P < 0.05, n = 7) within the first 15 min of takeoff. In addition, there was a small but significant increase in T_b with increasing ambient air (T_a) when individuals on replicate flights (n = 35) were considered. Inclusion of water vapor pressure into the regression model did not improve the correlation between body temperature and ambient conditions. Flight T_b also increased a small (0.5 °C) but significant amount (t = 2.827, P < 0.05, n = 8) from the beginning to the end of a flight. The small response of T_b to changing flight conditions presumably reflects the efficiency of convection as a heat loss mechanism during sustained regular flight. The increase in T_b on landing that occurred in some birds was a probable consequence of a sudden reduction in convective heat loss. Accepted: 2 February 1999     

Distribution of cells in different stages of the cell cycle and some age-related physiological characteristics in dependence on the dilution rate in chemostat cultures ofCandida utilis

Position of cells in their cell cycle was determined microscopically in chemostat cultures ofCandida utilis. Proportion of cells in phase G₁ decreased in a linear manner from 86% to 58% with dilution rate. Proportion of cells in phase S increased in the same range ofD from 5.6 to 13.5% and in the (G₂+M) phase from 8.4 to 28.5%, again linearly. Differential centrifugation was used to separate chemostat cultures to mother and daughter cells. Analyses showed that, relative to mother cells, daughter cells contain 2.1–11.9% more protein and 25.5–34.6% more RNA in dry matter. Their mass is 34.4–5.6% lower and volume is 154–19% smaller.     

Maximal power and torque-velocity relationship on a cycle ergometer during the acceleration phase of a single all-out exercise

Seven subjects pedalled on a Monark cycle ergometer as fast as possible for approximately 7 s against four different resistances which corresponded to braking torques (T _B) equal to 19, 38, 57 and 76 N · m at the crank level. Exercise periods were separated by 5-min recovery periods. Pedal velocity was recorded every 50 ms by means of a disc with 360 slots fixed on the flywheel, passing in front of a photo-electric cell linked to a microcomputer which processed the data. Every 50 ms, the time necessary to perform half a pedal revolution (t _1/2) was computed by adding the 50-ms periods necessary to reach 669 slots (the number of slots corresponding to half a pedal revolution). To measuret _1/2 to an accuracy better than 50 ms, this time was computed by a linear interpolation of the time-slot number relationship. Power (P) was averaged duringt ₁₂ by adding the power dissipated against braking torque and the power necessary to accelerate the flywheel. The torque-velocity (T-) relationship was studied during the acceleration phase of a sprint against a single TB by computing every 50 ms the relationship between and T (N · m), equal to the sum ofT _B and the torque necessary to accelerate the flywheel at the same time. The T- relationships calculated from the acceleration phase of a single all-out exercise were linear and similar to the previously described relationships between peak velocity and braking force. These relationships can be expressed as follows: = _0,acc (1 –T/T _0,acc) where is pedal velocity,T the torque exerted on the crank andT _0,acc and _0,acc have the dimensions of maximal torque and maximal velocity respectively. Based on this model, maximal power (P _max,acc) is calculated as 0.257_{0, acc} T _{0, acc}. Maximal powerP _max,acc measured with the acceleration method was independent of braking torqueT _B and slightly higher thanP _max calculated from the relationship between peak velocity andT _B.     

Single-channel properties of a stretch-sensitive chloride channel in the human mast cell line HMC-1

A stretch-activated (SA) Cl⁻ channel in the plasma membrane of the human mast cell line HMC-1 was identified in outside-out patch-clamp experiments. SA currents, induced by pressure applied to the pipette, exhibited voltage dependence with strong outward rectification (55.1 pS at +100 mV and an about tenfold lower conductance at −100 mV). The probability of the SA channel being open (P _o) also showed steep outward rectification and pressure dependence. The open-time distribution was fitted with three components with time constants of τ_1o = 755.1 ms, τ_2o = 166.4 ms, and τ_3o = 16.5 ms at +60 mV. The closed-time distribution also required three components with time constants of τ_1c = 661.6 ms, τ_2c = 253.2 ms, and τ_3c = 5.6 ms at +60 mV. Lowering extracellular Cl⁻ concentration reduced the conductance, shifted the reversal potential toward chloride reversal potential, and decreased the P _o at positive potentials. The SA Cl⁻ currents were reversibly blocked by the chloride channel blocker 4,4′-diisothiocyanatostilbene-2,2′-disulfonic acid (DIDS) but not by (Z)-1-(p-dimethylaminoethoxyphenyl)-1,2-diphenyl-1-butene (tamoxifen). Furthermore, in HMC-1 cells swelling due to osmotic stress, DIDS could inhibit the increase in intracellular [Ca²⁺] and degranulation. We conclude that in the HMC-1 cell line, the SA outward currents are mediated by Cl⁻ influx. The SA Cl⁻ channel might contribute to mast cell degranulation caused by mechanical stimuli or accelerate membrane fusion during the degranulation process.     

Some comments on the conductance of slow inward current in cardiac muscle

Voltage clamp experiments, which determine the kinetic parameters of calcium conductance of cardiac muscle, (d _∞,f _∞, τ _d and τ _f ) are analyzed with a generally accepted expression for slow inward currentI _s=g _sdf (E-E _R). Activation (d) and inactivation (f) reach the final valuesd _∞ andf _∞ with time constants τ _d and τ _f respectively. The analysis indicates that the measuredf _∞ agrees with the theoreticalf _∞, but the measuredd _∞ differs from the theoreticald _∞ by a factor which depends on τ _d . The peak tension can be made to correlate closely with the theoreticald _∞ after a correction factor is applied to the raw measurements of activation. It can be shown that experiments designed to measure τ _f can also be used to determine τ _d with greater accuracy.     

Age and growth of lanternfishes, Symbolophorus californiensis and Ceratoscopelus warmingii (Myctophidae), in the Kuroshio–Oyashio Transition Zone

To estimate the age and growth of dominant lanternfishes in the Kuroshio–Oyashio Transition Zone, we examined the sagittal otolith microstructure of Symbolophorus californiensis (n = 30) and Ceratoscopelus warmingii (n = 93) collected from the western North Pacific during 1997–2003. Age of S. californiensis ranged from 81 to 541 days corresponding to postmetamorphosis stage (juveniles and adults), and the von Bertalanffy model was fitted: L _t  = 128[1 − exp{−0.003(t − 1.52)}], where L is the standard length (mm) and t is age in days. Age of C. warmingii ranged from 6 to 416 days, and growth before metamorphosis was linear (L _t  = 0.346t + 1.51), and the von Bertalanffy model was fitted to the postmetamorphosis stage: L _t  = 80.8 [1 − exp{−0.00769(t − 34.4)}]. Growth of these two lanternfishes was faster than that of other lanternfishes in previous studies but considerably slower than that of Japanese sardine (Sardinops melanostictus) and anchovy (Engraulis japonicus) distributed in the Kuroshio–Oyashio Transition Zone. Temperature and prey availability are discussed in relation to this difference in growth rate.     

Rat C-6 Glioma Cells Maintained In an Organ Culture System: A Study of Kinetic Parameters

Rat C-6 glioma cells were grown on a sponge foam matrix in an organ culture system and the cell cycle parameters, including the growth fraction (GF), were assessed after autoradiography. the zones of growth consisted of a compact upper layer (UL) at the gaseous interface, a central necrotic layer and a deeper lower layer (LL) which invaded the matrix. the fraction of continuously labeled mitoses (FCLM) was similar in both the UL and LL cells. the derivatives of the FCLM curves obtained in three experiments gave an average modal T_G2 of 5 hr. A mathematical model relating GF, T_G2, T_C and labeling index as a function of time, LI(t), was devised for cells in a steady state exposed continuously to tritiated thymidine and was applied to data obtained from UL cells. A mean GF of 9% (range: 8–10%) and a mean cell cycle time (T_C') of 27 hr (range: 13–47 hr) were obtained. the mean T_S was calculated to be 11 hr (range: 8–16 hr) by the method of grain counts per mitotic figure or grain index (GI). Knowledge of T_S permitted alternative calculation of the cell cycle time from the equation T_S/T_C= LI(0)/GF: this gave a mean cell cycle time (T_C) of 29 hr (range: 20–45 hr). Except for the GF, the cell kinetics were comparable to those of the same cell line grown in monolayer culture. the GF in the in vitro system described is in the lower range reported in some human malignant gliomas in vivo.