Analytical solutions for distributions of chemotactic bacteria

This paper reports general and specialized results on analytical solutions to the governing phenomenological equations for chemotactic redistribution and population growth of motile bacteria. It is shown that the number of bacteria cells per unit volume,b, is proportional to a certain prescribed function ofs, the concentration of the critical substrate chemotactic agent, for steady-state solutions through an arbitrary spatial region with a boundary that is impermeable to bacteria cell transport. Moreover, it is demonstrated that the steady-state solution forb ands is unique for a prescribed total number of bacteria cells in the spatial region and a generic Robin boundary condition ons. The latter solution can be approximated to desired accuracy in terms of the Poisson-Green's function associated with the spatial region. Also, as shown by example, closed-form exact steady-state solutions are obtainable for certain consumption rate functions and geometrically symmetric spatial regions. A solutional procedure is formulated for the initialvalue problem in cases for which significant population growth is present and bacteria cell redistribution due to motility and chemotactic flow proceeds slowly relative to the diffusion of the chemoattractant substrate. Finally, a remarkably simple exact analytical solution is reported for a stradily propagating plane-wave which features motility, chemotactic motion and bacteria population growth regulated by substrate diffusion.     

An asymptotic solution for traveling waves of a nonlinear-diffusion Fisher's equation

We examine traveling-wave solutions for a generalized nonlinear-diffusion Fisher equation studied by Hayes [J. Math. Biol. 29, 531–537 (1991)]. The density-dependent diffusion coefficient used is motivated by certain polymer diffusion and population dispersal problems. Approximate solutions are constructed using asymptotic expansions. We find that the solution will have a corner layer (a shock in the derivative) as the diffusion coefficient approaches a step function. The corner layer at z = 0 is matched to an outer solution for z < 0 and a boundary layer for z > 0 to produce a complete solution. We show that this model also admits a new class of nonphysical solutions and obtain conditions that restrict the set of valid traveling-wave solutions. Supported by a National Science Foundation graduate fellowship. This work was performed under National Science Foundation grant DMS-9024963 and Air Force Office of Scientific Research grant AFOSR-F49620-94-1-0044.     

Model and analysis of chemotactic bacterial patterns in a liquid medium

 A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals. Received: 10 March 1998 / Revised version: 7 June 1998     

Numerical and exact solutions for continuum of alleles models

 Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution, and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that, depending on the mutation rate, interpolates between the ``House of Cards' approximation and the Gaussian approximation. The exact solution is also used to test the accuracy of the numerical method. Received: 7 November 2001 / Revised version: 5 September 2002 / Published online: 18 December 2002 Key words or phrases: Continuum of alleles – Numerical solution – Exact solution – Mutation selection balance – Stabilising selection     

Note on a case of nonlinear diffusion

When the functionQ in the equation ∇²c +Q(c) = 0 is positive and is of a specified kind, the equation admits of a centrally spherical solution such thatc is positive everywhere, tending to zero at infinity anddc/dr=0 atr=0. Physically this corresponds to a local concentration of the solute in an infinite medium without any membranes present. This result would indicate the possibility of the formation of spontaneous concentrations and non-uniformities in non-linear diffusion fields. Possible biological implications are mentioned. *** DIRECT SUPPORT *** A01E2041 00002     

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

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.     

Solitary waves in prestressed elastic tubes

In this work, we studied the propagation of non-linear waves in a pre-stressed thin elastic tube filled with an inviscid fluid. In the analysis, analogous to the physiological conditions of the arteries, the tube is assumed to be subject to a uniform pressureP ₀ and a constant axial stretch ratio λ_z. In the course of blood flow it is assumed that a large dynamic displacement is superimposed on this static field. Furthermore, assuming that the displacement gradient in the axial direction is small, the non-linear equation of motion of the tube is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. The result is discussed for some elastic materials existing in the literature.     

Computational Approaches to Solving Equations Arising from Wound Healing

In the wound healing process, the cell movement associated with chemotaxis generally outweighs the movement associated with random motion, leading to advection-dominated mathematical models of wound healing. The equations in these models must be solved with care, but often inappropriate approaches are adopted. Two one-dimensional test problems arising from advection-dominated models of wound healing are solved using four algorithms—MATLAB’s inbuilt routine pdepe.m, the Numerical Algorithms Group routine d03pcf.f, and two finite volume methods. The first finite volume method is based on a first-order upwinding treatment of chemotaxis terms and the second on a flux limiting approach. The first test problem admits an analytic solution which can be used to validate the numerical results by analyzing two measures of the error for each method: the average absolute difference and a mass balance error. These criteria as well as the visual comparison between the numerical methods and the exact solution lead us to conclude that flux limiting is the best approach to solving advection-dominated wound healing problems numerically in one dimension. The second test problem is a coupled nonlinear three species model of wound healing angiogenesis. Measurement of the mass balance error for this test problem further confirms our hypothesis that flux limiting is the most appropriate method for solving advection-dominated governing equations in wound healing models. We also consider two two-dimensional test problems arising from wound healing, one that admits an analytic solution and a more complicated problem of blood vessels growth into a devascularized wound bed. The results from the two-dimensional test problems also demonstrate that the flux limiting treatment of advective terms is ideal for an advection-dominated problem.     

Stability of periodic travelling wave solutions of a nerve conduction equation

Summary Nagumo's nerve conduction equation has travelling wave solutions of pulse type and periodic wave type. We consider the stability of the latter ones. We denote byL(c) the minimum spatial period of a periodic travelling wave solution whose propagation speed isc. It is shown that this travelling wave solution is unstable ifL′(c)<0.     

Perspective: Here's to Fisher,additive genetic variance,and the fundamental theorem of natural selection

Fisher's fundamental theorem of natural selection, that the rate of change of fitness is given by the additive genetic variance of fitness, has generated much discussion since its appearance in 1930. Fisher tried to capture in the formula the change in population fitness attributable to changes of allele frequencies, when all else is not included. Lessard's formulation comes closest to Fisher's intention, as well as this can be judged. Additional terms can be added to account for other changes. The "theorem" as stated by Fisher is not exact, and therefore not a theorem, but it does encapsulate a great deal of evolutionary meaning in a simple statement. I also discuss the effectiveness of reproductive-value weighting and the theorem in integrated form. Finally, an optimum principle, analogous to least action and Hamilton's principle in physics, is discussed.     

On periodic cohort solutions of a size-structured population model

 We consider a size-structured population model with discontinuous reproduction and feedback through the environmental variable ‘substrate’. The model admits solutions with finitely many cohorts and in that case the problem is described by a system of ODEs involving a bifurcation parameter β. Existence of nontrivial periodic n-cohort solutions is investigated. Moreover, we discuss the question whether n cohorts (n≧2) with small size differences will tend to a periodic one-cohort solution as t→∞. Received 16 March 1995; received in revised form 7 January 1997     

Antagonistic effects of vasotocin and isotocin on the upper esophageal sphincter muscle of the eel acclimated to seawater

The effects of isotocin (IT) and vasotocin (VT), which are fish analogues of mammalian oxytocin and vasopressin respectively, were examined in the isolated upper esophageal sphincter (UES) muscle. IT relaxed and VT constricted the UES muscle in a concentration-dependent manner. The relaxation by IT and the contraction by VT were completely blocked by H-9405 (an oxytocin receptor antagonist) and by H-5350 (a V₁-receptor antagonist), respectively, suggesting that the eel UES possesses both IT and VT receptors. Truncated fragments of VT did not show any significant effects, indicating that all nine residues are essential for the VT and IT actions. IT may relax the UES muscle through enhancing cAMP production, since similar relaxation was also observed after treatment with 3-isobutyl-1-methylxantine, forskolin and 8-bromoadenosine, 3′, 5′-cyclic mono-phosphate (8BrcAMP). Although 8-bromoguanosine, 3′, 5′-cyclic monophosphate also relaxed the UES, its effect was less than 1/3 of that 8BrcAMP, suggesting minor contribution of nitric oxide (NO) in the relaxation of the UES muscle. Both peptides seem to act directly on the UES muscle, not through release of other substances from the epithelial cells, since similar relaxation and contraction were observed even in the scraped UES preparations. When IT and VT were intravenously administrated (in vivo experiments), the drinking rate of the seawater eel was enhanced by IT and was inhibited by VT. These effects correspond to the in vitro results described above, relaxation by IT and contraction by VT in the UES muscle. The significance of the relaxing effect by IT is discussed with respect to controlling the drinking behavior of the eel.     

Effects of the V82A and I54V mutations on the dynamics and ligand binding properties of HIV-1 protease

A major problem in the antiretroviral treatment of HIV-infections with protease-inhibitors is the emergence of resistance, resulting from the occurrence of distinct mutations within the protease molecule. In the present work we investigated the structural properties of a triple mutant (I54V-V82A-L90M) and a double mutant (V82A-L90M) that both confer strong resistance to ritonavir (RTV), but not to amprenavir (APV). For the unliganded double mutant protease molecular dynamics simulations revealed a contraction of the ligand binding pocket, which is enhanced by the I54V mutation. The observed displacement of backbone atoms of the 80s loops (residues 80–85 and 80’–85’ of the dimer) was found to primarily affect binding of the larger RTV molecule. The pocket contraction detected for the unbound protease upon mutation is also observed in the presence of APV, but not of RTV. As a consequence, the protein-ligand contacts lost upon the V82A mutation are restored by 80s loop motions for the APV-bound, but not for the RTV-bound form. RTV binding is therefore both hampered in the initial recognition step due to the poor fit of the bulky inhibitor into the small pocket of the mutant free protease and by the loss of protein-ligand interactions in the RTV-bound protease. The synergistic nature of both effects offers an explanation for the high level of resistance observed. These findings demonstrate that large inhibitors, which tightly bind to wild-type protease, may nevertheless be prone to the emergence of resistance in the presence of particular patterns of mutations. This information should be helpful for the design of novel and more effective drugs, e.g., by targeting different residues or by developing allosteric inhibitors that are capable of regulating protease dynamics.     

Some exact solutions of a generalized Fisher equation related to the problem of biological invasion

The problem of biological invasion in a model single-species community is considered, the spatiotemporal dynamics of the system being described by a modified Fisher equation. For a special case, we obtain an exact solution describing self-similar growth of the initially inhabited domain. By comparison with numerical solutions, we show that this exact solution may be applicable to describe an early stage of a biological invasion preceding the propagation of the stationary travelling wave. Also, the exact solution is applied to the problem of critical aggregation to derive sufficient conditions of population extinction. Finally, we show that the solution we obtain is in agreement with some data from field observations.     

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.     

EPR Studies of Recombinant Horse L-Chain Apoferritin and its Mutant (E 53,56,57,60 Q) with Haemin

Structural similarities between ferritins and bacterioferritins have been extensively demonstrated. However, there is an essential difference between these two types of ferritins: whereas bacterioferritins bind haem, in-vivo, as Fe(II)-protoporphyrin IX (this haem is located in a hydrophobic pocket along the 2-fold symmetry axes and is liganded by two axial Met 52 residues), eukaryotic ferritins are non-haem iron proteins. However, in in-vivo studies, a cofactor has been isolated from horse spleen apoferritin similar to protoporphyrin IX; in in-vitro experiments, it has been shown that horse spleen apoferritin is able to interact with haemin (Fe(III)-protoporphyrin IX). Studies of haemin incorporation into horse spleen apoferritin have been carried out, which show that the metal free porphyrin is found in a pocket similar to that which binds haem in bacterioferritins (Précigoux et al. 1994 Acta Cryst D50, 739–743). A mechanism of demetallation of haemin by L-chain apoferritins was subsequently proposed (Crichton et al. 1997 Biochem 36, 15049–15054) which involved four Glu residues (E 53,56,57,60) situated at the entrance of the hydrophobic pocket and appeared to be favoured by acidic conditions. To verify this mechanism, these four Glu have been mutated to Gln in recombinant horse L-chain apoferritin. We report here the EPR spectra of recombinant horse L-chain apoferritin and its mutant with haemin in basic and acidic conditions. These studies confirm the ability of recombinant L-chain apoferritin and its mutant to incorporate and demetallate the haemin in acidic and basic conditions.     

Sensitivity analysis of a nonlinear lumped parameter model of HIV infection dynamics

A formal sensitivity analysis is performed on a delay differential equation model for the viral dynamics of an in vivo HIV infection during protease inhibitor therapy. We present results of both a differential analysis as well as a principle component based analysis and provide evidence that suggests the exact times at which specific parameters have the most influence over the solution. We offer insight into the pairwise mathematical relationships between the productively infected T-cell death rate δ, the viral plasma clearance rate c, and the time delay τ between infection and viral production as they relate to the viral dynamics. The results support the claim that the presence of a nonzero delay has a major impact on the model dynamics. Lastly, we comment upon the inadequacies of an alternative principle component based analysis.     

On the Forkker-Planck equation in the stochastic theory of mortality: I

This paper, consisting of two parts, gives all the mathematical details that were omitted in a previous work by G. A. Sacher and E. Trucco (“The Stochastic Theory of Mortality.”Ann. N. Y. Acad. Sci.,96, 985–1007, cited here as ST). We assume that the reader is familiar with ST, where the stochastic theory of mortality, originally proposed by Sacher, is discussed at length. We recall that the basic model presented there refers to an ensemble of particles performing Brownian motion in one dimension, with the added constraint of two absorbing barriers. These two points, collectively, are designated as the “lethal bound.” Part I (section 1 to 4) deals with the special case in which the two absorbing barriers are symmetrically located at a finite distance from the origin. The solution of the Fokker-Planck equation is obtained from the theory of eigenvalue problems. Quite generally, the eigenfunctions functions belong to the family of Kummer's confluent hypergeometric functions, but the symmetry condition imposed here results in considerable simplification and makes it possible to estimate the first few eigenvalues by a graphical procedure. In section 3 we show how perturbation theory can be applied in the limiting case of “weak homeostasis,” and section 4 deals with the opposite extreme of “strong homeostasis.” A rigorous proof is given for the result corresponding to equation (28) of ST (asymptotic or quasi-static approximation for the “force of mortality”). This work was performed under the auspices of the U.S. Atomic Energy Commission.     

Exact fundamental solutions of linear parabolic equations with spatially varying coefficients

The exact general solution is obtained to a linear second order ordinary differential equation which has quite general coefficients depending on an arbitrary function of the independent variable. From this, the exact fundamental solution is derived for the corresponding linear parabolic partial differential equation with coefficients depending on the single space coordinate. In a special case this latter equation reduces to one of the Fokker-Planck type. These coefficients are then generalised and the appropriate fundamental solution is obtained. Extensions are given to linear parabolic equations in two andn space dimensions. The paper provides a collection of basic examples which illustrate and develop the theory for the generation of the exact fundamental solutions. Reduction to, and the corresponding fundamental solutions of the Fokker-Planck equations is presented, where appropriate.