Solutions to systems of nonlinear reaction-diffusion equations

General criteria which either preclude time-periodic dissipative structure solutions or imply asymptotically steady solutions are derived for generic systems of reaction-diffusion equations ∂c _i /∂t =D _i ∇² c _i +Q _i (c) subject to boundary conditions of practical interest, where the enumerator indexi runsl ton, c _i =c _i (x,t) denotes the concentration or density of theith participating molecular or biological species,D _i is the diffusivity constant for theith species, andQ _i (c), an algebraic function of then-tuplec=(c ₁,...,c _n ), expresses the local rate of production of theith species due to chemical reactions or biological interactions. It is demonstrated that certain functionals ofc which decrease monotonically with time can often be found, as exemplified here for Volterra and Verhulst-Volterran-species model systems, and thus time-periodic dissipative structure solutions are precluded for such systems of reaction-diffusion equations. It is shown that all solutions to a generic system of reaction-diffusion equations evolve dynamically to a unique steady state, $$\mathop {\lim }\limits_{t \to \infty } c_i (x, t) = \hat c_i (x)$$ , if the diffusivity constants are all sufficiently large in magnitude. A necessary condition for the existence of a periodic solution (either spatially uniform or non-uniform) is formulated in terms of the curl ofQ(c) inc-space. Finally, necessary and sufficient conditions are derived for the existence of time-periodic dissipative structure solutions in cases of “weak diffusion” with the reaction rate terms dominant in the governing equations.     

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.     

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     

Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations

 In this paper we study the existence of one-dimensional travelling wave solutions u(x, t)=φ(x−ct) for the non-linear degenerate (at u=0) reaction-diffusion equation u _t =[D(u)u _x ] _x +g(u) where g is a generalisation of the Nagumo equation arising in nerve conduction theory, as well as describing the Allee effect. We use a dynamical systems approach to prove: 1. the global bifurcation of a heteroclinic cycle (two monotone stationary front solutions), for c=0, 2. The existence of a unique value c ^*>0 of c for which φ(x−c ^* t) is a travelling wave solution of sharp type and 3. A continuum of monotone and oscillatory fronts for c≠c ^*. We present some numerical simulations of the phase portrait in travelling wave coordinates and on the full partial differential equation. Received 15 December 1995; received in revised form 14 May 1996     

The volumes of some compartment systems with sampling and loss from one compartment

If the concentrationc ₁(t)=∑ _i=1 ⁿ A _i exp (−α _i t) for one compartment, one presumes a linear catenaryn-compartment system without sinks and loss only from the same compartment, then the volumesV _i , rate constantsk _ij , and concentrationsc _i (t) in each compartment can be determined in terms of theA _i 's,A _i ′s, α _i ′s, the dose injectedD _o and the partition coefficientsr _ij =k _ij /k _ji . If the concentration would become uniform at equilibrium, then the total volume of distribution may be determined without knowledge ofr _ij or restriction to catenary configuration.     

Exploring optimal current stimuli that provide membrane voltage tracking in a neuron model

Studying neurons from an energy efficiency perspective has produced results in the research literature. This paper presents a method that enables computation of low energy input current stimuli that are able to drive a reduced Hodgkin–Huxley neuron model to approximate a prescribed time-varying reference membrane voltage. An optimal control technique is used to discover an input current that optimally minimizes a user selected balance between the square of the input stimulus current (input current ‘energy’) and the difference between the reference voltage and the membrane voltage (tracking error) over a stimulation period. Selecting reference signals to be membrane voltages produced by the neuron model in response to common types of input currents i(t) enables a comparison between i(t) and the determined optimal current stimulus i*(t). The intent is not to modify neuron dynamics, but through comparison of i(t) and i*(t) provide insight into neuron dynamics. Simulation results for four different bifurcation types demonstrate that this method consistently finds lower energy stimulus currents i*(t) that are able to approximate membrane voltages as produced by higher energy input currents i(t) in this neuron model.     

Mathematical interpretation of dose-response curves

The dose-response of an individual organism can be described by a step functions if the organism survives when the dose is below a certain lethal level and dies when this level is exceeded. If, in a population of organism, the lethal dose for an individual has a unimodal distribution, the latter's properties will determine the shape of the population's response in the following manner. If the distribution is symmetric the dose-response curve has a symmetric sigmoid shape when plotted on linear coordinates. The location of the inflection point and the curve's slope around it are determined by the distribution's mode and variance. When the distribution is skewed, the dose-response curve has an asymmetric sigmoid shape which becomes reminiscent of an exponential decay when the distribution is strongly skewed to the right. The population's dose-response curve can be constructed by integration of the step changes over the distribution range. The step function representing the dose-response of an individual organism can be approximate by a Fermi function, and the distribution of an lethal doses can be represented by the Weibull distribution function. When the two functions are combined, the resulting dose-response of the populationS(X)), which is the fraction of survivors after exposure to a doseX, is given by:S(X)=∫ ₀ ¹ [1/{+exp{(X-X _c (φ))/a _i ]}]dω whereX _c (ω)={(1/b)[-ln(1-ω)]}^(1/n),n andb being the constants of the Weibull distribution anda _i an arbitrarily small number, i.e.a _i ≪[X−X _c (ϕ)], whose actual magnitude is of little significance. This model can be used to determine the underlying distributions of experimental dose-response relationship. It was applied to published survival data of microorganisms exposed to pulsed electric field, X-ray radiation and ozone to show that the different observed shapes of the dose-response curve, and shifts between them, can be expressed in terms of the correponding distribution parameters, namely the mode, variance and skewness.     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

Construction of analytically solvable models for interacting species

By observing that the n-tuple of rate functionsQ(c) is orthogonal to the c-space gradients of each of the (n - 1) constants of the motion Φ _v (c), a generic canonical expression for the rate functions is given in terms of the exterior product of the gradients of the (n - 1) Φ _v 's. For models withQ so prescribed from the outset, an analytical general solution is obtainable directly for the system of autonomous ordinary differential equations dc/dt =Q(c). Thus, the generic canonical expression for the rate functions can be utilized to construct analytically solvable models for interacting biological species, as ilIus~rated by examples here.     

Note on psychophysical discrimination with more than two stimuli

H. D. Landahl's well-known theory of psychophysical discrimination between two stimuli (Psychometrica,3, 107–125, 1938) is generalized to the case ofn mutually inhibiting stimuli, such that all the corresponding reactions are mutually incompatible so that only one response at most can occur at a time. It is shown that while in the two-stimulus case a “no-response” situation does not necessarily need to occur, in the case ofn stimuli andn responses a “no-response” situation always occurs with finite probability. Therefore, there is a probabilityP _i of the occurrence of each responsei as well as a probabilityP _e of no response, with . The probabilitiesP _i andP _e are expressed in terms of the intensities of then stimuliS _i and in terms of then distribution functions of the fluctuations at then corresponding connections. The expressions are in the form of sums ofn-tuple integrals of the products of the distribution functions, the limits of integration being determined by the intensities of then stimuli.     

Inhibition by light of CO2 evolution from dark respiration: Comparison of two gas exchange methods

Two approaches to determine the fraction (μ) of mitochondrial respiration sustained during illumination by measuring CO₂ gas exchange are compared. In single leaves, the respiration rate in the light (`day respiration' rate R<sub>d</sub>) is determined as the ordinate of the intersection point of A–c<sub>i</sub> curves at various photon flux densities and compared with the CO<sub>2</sub> evolution rate in darkness (`night respiration' rate R_n). Alternatively, using leaves with varying values of CO₂ compensation concentration (Γ), intracellular resistance (r_i) and R_n, an average number for μ can be derived from the linear regression between Γ and the product r_iċR_n. Both methods also result in a number c* for that intercellular CO₂ concentration at which net CO₂ uptake rate is equal to –R_d. c* is an approximate value of the photocompensation point Γ* (Γ in the absence of mitochondrial respiration), which is related to the CO₂/O₂ specificity factor of Rubisco S_c/o. The presuppositions and limitations for application of both approaches are discussed. In leaves of Nicotiana tabacum, at 22 °C, single leaf measurements resulted in mean values of μ = 0.71 and c_* = 34 μmol mol⁻¹. At the photosynthetically active photon flux density of 960 μmol quanta m⁻² s⁻¹, nearly the same numbers were derived from the linear relationship between Γ and r_iċR_n. c* and R_d determined by single leaf measurements varied between 31 and 41 μmol mol⁻¹ and between 0.37 and 1.22 μmol m⁻² s⁻¹, respectively. A highly significant negative correlation between c* and R_d was found. From the regression equation we obtained estimates for Γ* (39 μmol mol⁻¹), S_c/o (96.5 mol mol⁻¹) and the mesophyll CO₂ transfer resistance (7.0 mol⁻¹ m² s). This revised version was published online in June 2006 with corrections to the Cover Date.     

Asymptotic theory for analyzing dose-response survival data with application to the low-dose extrapolation problem

A model of the form P(t, d)=1−;exp{-(t)σ^K_i=0q_idⁱ}, q_i 0, is proposed for analyzing dose-response survival data with right censoring. The q_i's in the dose polynomial are estimated by maximizing the Cox partial likelihood, and given these estimates. Λ(t) is estimated nonparametrically by an estimator proposed by Breslow. Large-sample properties of these estimators are established. Estimates and related large-sample properties are provided for the “virtually safe dose” and other parameters for assessing low-dose carcinogenic risk as a function of age, using data from animal carcinogenesis experiments.     

Control of the rate of phosphocreatine resynthesis after exercise in trained and untrained human quadriceps muscles

We examined the effect of differences in exercise intensity on the time constant (t _c) of phosphocreatine (PCr) resynthesis after exercise and the relationships betweent _c and maximal oxygen uptake (VO_2max) in endurance-trained runners (n = 5) and untrained controls (n = 7) (average VO_2max = 66.2 and 52.0 ml · min^–1 · kg^–1, respectively). To measure the metabolism of the quadriceps muscle using phosphorus nuclear magnetic resonance spectroscopy, we developed a device which allowed knee extension exercise inside a magnet. All the subjects performed four types of exercise: light, moderate, severe and exhausting. The end-exercise PCr: [PCr + inorganic phosphate (P_i)] ratio decreased significantly with the increase in the exercise intensity (P < 0.01). Although there was little difference in the end-exercise pH, adenosine diphosphate concentration ([ADP]) and the lowest intracellular pH during recovery between light and moderate exercise, significant changes were found at the two higher intensities (P < 0.01). These changes for runners were smaller than those for the controls (P < 0.05). The _c remained constant after light and moderate exercise and then lengthened in proportion to the increase in intensity (P < 0.05). The runners had a lowert _c at the same PCr and pH than the controls, particularly at the higher intensity (P < 0.05). There was a significant correlation betweent _c and [ADP] in light exercise and betweent _c and both end-exercise PCr and pH in severe and exhausting exercise (P < 0.05). The threshold of changes in pH andt _c was a PCr: (PCr + P_i) ratio of 0.5. There was a significant negative correlation between the VO_2max andt _c after all levels of exercise (P<0.05).However, in the controls a significant correlation was found in only light and moderate exercise (P < 0.05). These findings suggest the validity of the use oft _c at an end-exercise PCr:(PCr + P_i) ratio of more than 0.5 as a stable index of muscle oxidative capacity and the correlation between local and general aerobic capacity. Moreover, endurance-trained runners are characterized by the faster PCr resynthesis at the same PCr and intracellular pH.     

Random median sampling to enhance the precision of population estimates

We describe the characteristics of a sampling procedure called random median sampling that was proposed to enhance the precision of population estimates. In performing random median sampling, we first select a sampling item at random from the sampling area. We roughly compare the abundance of individuals in the selected item with that of the adjacent two items in order to identify the item that has median abundance, i.e., the item that has the second largest abundance among the three items. We count the number of individuals of the item having the median abundance. This procedure is repeated n times in the sampling area (i = 1, 2, ..., n). Let m _i be the ith median abundance. The estimates of the mean abundance per sampling item and the variance of estimates are given by Σm _i /n and Σ(m _i –Σm _i /n)²/n(n – 1), respectively. This method is a local application of the median ranked set sampling that was proposed by Muttlak (J Appl Stat Sci 6:245–255, 1997). Random median sampling is effective when the correlation coefficient between adjacent items is small. If the correlation coefficient is close to zero, random median sampling reduces the variance of estimates to 45 or 32% of that in simple random sampling when the distribution follows a normal distribution or a Laplace distribution, respectively. The sample size required to achieve a given precision of estimate decreases accordingly. The effectiveness of random median sampling, however, is small if the correlation coefficient is large. The condition in which random median sampling is superior to simple random sampling is also discussed.     

Three Types of Membrane Excitations in the Marine Diatom Coscinodiscus wailesii

Three types of electrical excitation have been investigated in the marine diatom Coscinodiscus wailesii. I: Depolarization-triggered, transient Cl⁻ conductance, G _Cl (t), followed by a transient, voltage-gated K⁺ conductance, G _K , with an active state a and two inactive states i ₁ and i ₂ in series (a-i ₁-i ₂). II: Similar G _Cl (t) as in Type-I but triggered by hyperpolarization; a subsequent increase of G _K in this type is indicated but not analyzed in detail. III: Hyperpolarization-induced transient of a voltage-gated activity of an electrogenic pump (i ₂-a-i ₂), followed by G _Cl (t) as in Type-II excitations. Type-III with pump gating is novel as such. G _Cl (t) in all types seems to reflect the mechanism of InsP⁻ ₃ and Ca²⁺-mediated G _Cl (t) in the action potential in Chara (Biskup et al., 1999). The nonlinear current-voltage-time relationships of Type-I and Type-III excitations have been recorded under voltage-clamp using single saw-tooth command voltages (voltage range: −200 to +50 mV, typical slope: ±1 Vs⁻¹). Fits of the corresponding models to the experimental data provided numerical values of the model parameters. The statistical significance of these solutions is investigated. We suggest that the original function of electrical excitability of biological membranes is related to osmoregulation which has persisted through evolution in plants, whereas the familiar and osmotically neutral action potentials in animals have evolved later towards the novel function of rapid transmission of information over long distances. Received: 2 December 1999/Revised: 3 March 2000     

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.     

Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration μ and the boundary concentration of the diffusing ion c ₀. Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k → ∞, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute μ and c ₀. The position of the advancing front ξ then provides, together with μ, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), μ and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.     

Kinetics of the electron transfer reaction of Cytochrome <Emphasis Type="Italic">c</Emphasis><Subscript>552</Subscript> adsorbed on biomimetic electrode studied by time-resolved surface-enhanced resonance Raman spectroscopy and electrochemistry

Cytochrome c ₅₅₂ (Cyt-c ₅₅₂) and its redox partner ba ₃ -oxidase from Thermus thermophilus possess structural differences compared with Horse heart cytochrome c (cyt-c)/cytochrome c oxidase (CcO) system, where the recognition between partners and the electron transfer (ET) process is initiated via electrostatic interactions. We demonstrated in a previous study by surface-enhanced resonance Raman (SERR) spectroscopy that roughened silver electrodes coated with uncharged mixed self-assembled monolayers HS–(CH₂) _n –CH₃/HS–(CH₂) _{n + 1}–OH 50/50, n = 5, 10 or 15, was a good model to mimic the Cyt-c ₅₅₂ redox partner. All the adsorbed molecules are well oriented on such biomimetic electrodes and transfer one electron during the redox process. The present work focuses on the kinetic part of the heterogeneous ET process of Cyt-c ₅₅₂ adsorbed onto electrodes coated with such mixed SAMs of different alkyl chain length. For that purpose, two complementary methods were combined. Firstly cyclic voltammetry shows that the ET between the adsorbed Cyt-c ₅₅₂ and the biomimetic electrode is direct and reversible. Furthermore, it allows the estimation of both the density surface coverage of adsorbed Cyt-c ₅₅₂ and the kinetic constants values. Secondly, time-resolved SERR (TR-SERR) spectroscopy showed that the ET process occurs without conformational change of the Cyt-c ₅₅₂ heme group and allows the determination of kinetic constants. Results show that the kinetic constant values obtained by TR-SERR spectroscopy could be compared to those obtained from cyclic voltammetry. They are estimated at 200, 150 and 40 s⁻¹ for the ET of Cyt-c ₅₅₂ adsorbed onto electrodes coated with mixed SAMs HS–(CH₂) _n –CH₃/HS–(CH₂) _{n + 1}–OH 50/50, n = 5, 10 or 15, respectively. Presented at the joint biannual meeting of the SFB-GEIMM-GRIP, Anglet France, 14–19 October, 2006.     

Characterizing conditions for generalized Verhulst logistic growth of a biological population

It is shown that a biological population or subpopulation composed ofN individuals can be governed by a generalized Verhulst logistic equation with time-dependent rate functions if and only if certain characterizing conditions are satisfied by ∂N/∂N ₀ whereN ₀ is the value ofN att=0.