Slow and fast invasion waves in a model of acid-mediated tumour growth

This work is concerned with a reaction-diffusion system that has been proposed as a model to describe acid-mediated cancer invasion. More precisely, we consider the properties of travelling waves that can be supported by such a system, and show that a rich variety of wave propagation dynamics, both fast and slow, is compatible with the model. In particular, asymptotic formulae for admissible wave profiles and bounds on their wave speeds are provided.     

Anomalous spreading speeds of cooperative recursion systems

This work presents an example of a cooperative system of truncated linear recursions in which the interaction between species causes one of the species to have an anomalous spreading speed. By this we mean that this species spreads at a speed which is strictly greater than its spreading speed in isolation from the other species and the speeds at which all the other species actually spread. An ecological implication of this example is discussed in Sect. 5. Our example shows that the formula for the fastest spreading speed given in Lemma 2.3 of our paper (Weinberger et al. in J Math Biol 45:183–218, 2002) is incorrect. However, we find an extra hypothesis under which the formula for the faster spreading speed given in (Weinberger et al. in J Math Biol 45:183–218, 2002) is valid. We also show that the hypotheses of all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) whose proofs rely on Lemma 2.3 imply this extra hypothesis, so that all but one of the theorems of (Weinberger et al. in J Math Biol 45:183–218, 2002) and all the examples given there are valid as they stand.     

Spreading speeds as slowest wave speeds for cooperative systems

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c(*) for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c(*) and a fastest speed c(f)(*). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c(*) is always characterized as the slowest speed of a class of traveling wave solutions.     

A mathematical model for indirectly transmitted diseases

We consider a mathematical model for the indirect transmission via a contaminated environment of a microparasite between two spatially distributed host populations having non-coincident spatial domains. The parasite is benign in a first population and lethal in the second one. Global existence results are given for the resulting reaction-diffusion system coupled with an ordinary differential equation. Then, invasion and persistence of the parasite are studied. A simplified model for the transmission of a hantavirus from bank vole to human populations is then analysed.     

Dynamics of a simple regulatory switch

We consider the dynamics of a model toggle switch abstracted from the genetic interactions operative in a fungal stress response circuit. The switch transduces an external signal and propagates it forward by mediating the transport between compartments of two interacting gene products. The transport between compartments is assumed to be related to the degree of association between the interacting proteins, a fact for which there exists a wealth of biological evidence. The ubiquity and modularity of this cellular control mechanism warrants a detailed study of the dynamics entailed by various modelling assumptions. Specifically, we consider a general gate model in which both of the associating proteins are freely transportable between compartments. A more restrictive, but biologically supported model, is considered in which only one of the two proteins undergoes transport. Under the strong assumption that the disassociation of the interacting proteins is unidirectional we show that the qualitative dynamics of the two models are similar; that is they both converge to unique periodic orbits. From a biophysical perspective the assumption of unidirectional dissociation is unrealistic. We show that the same result holds for the more restrictive model when one weakens the assumption of unidirectional binding or disassociation. We speculate that this is not true for the more general model. This difference in dynamics may have important biological implications and certainly points to promising avenues of research.     

The structure of optimal time- and age-dependent harvesting in the Lotka-McKendrik population model

The paper analyzes optimal harvesting of age-structured populations described by the Lotka-McKendrik model. It is shown that the optimal time- and age-dependent harvesting control involves only one age at natural conditions. This result leads to a new optimization problem with the time-dependent harvesting age as an unknown control. The integral Lotka model is employed to explicitly describe the time-varying age of harvesting. It is proven that in the case of the exponential discounting and infinite horizon the optimal strategy is a stationary solution with a constant harvesting age. A numeric example on optimal forest management illustrates the theoretical findings. Discussion and interpretation of the results are provided.     

A stochastic model for prostate-specific antigen levels

We introduce a continuous stochastic model for the prostate-specific antigen (PSA) levels following radiotherapy and derive solutions for the associated partial differential (Kolmogorov-Chapman) equation. The solutions describe the evolution of the time-dependent density for PSA levels which take into account an absorbing condition along the boundary and various initial conditions. We include implications for single-dose and multi-dose radiation treatment regimens and discuss parameter estimation and sensitivity issues.     

An age-and-cyclin-structured cell population model for healthy and tumoral tissues

We present a nonlinear model of the dynamics of a cell population divided into proliferative and quiescent compartments. The proliferative phase represents the complete cell cycle (G ₁−S−G ₂−M) of a population committed to divide at its end. The model is structured by the time spent by a cell in the proliferative phase, and by the amount of Cyclin D/(CDK4 or 6) complexes. Cells can transit from one compartment to the other, following transition rules which differ according to the tissue state: healthy or tumoral. The asymptotic behaviour of solutions of the nonlinear model is analysed in two cases, exhibiting tissue homeostasis or tumour exponential growth. The model is simulated and its analytic predictions are confirmed numerically.     

Persistence in reaction diffusion models with weak allee effect

We study the positive steady state distributions and dynamical behavior of reaction-diffusion equation with weak Allee effect type growth, in which the growth rate per capita is not monotonic as in logistic type, and the habitat is assumed to be a heterogeneous bounded region. The existence of multiple steady states is shown, and the global bifurcation diagrams are obtained. Results are applied to a reaction-diffusion model with type II functional response, and also a model with density-dependent diffusion of animal aggregation. J. S. is partially supported by United States NSF grants DMS-0314736 and EF-0436318, College of William and Mary summer grants, and a grant from Science Council of Heilongjiang Province, China.     

Mathematical modeling and qualitative analysis of insulin therapies

Several insulin therapies are widely in clinical use with the basic strategy that mimics insulin secretion in a normal glucose-insulin endocrine metabolic regulatory system. In this paper, we model the insulin therapies using a delay differential equation model. We study the dynamics of the model both qualitatively and quantitatively. The analytical results show the existence and uniqueness of a stable periodic solution that corresponds to ultradian insulin secretion oscillations. Numerically we simulate the insulin administration based on our model. The numerical simulation results are in agreement with findings of clinical studies.     

Characterization of a cAMP-stimulated cAMP phosphodiesterase in Dictyostelium discoideum

A cyclic nucleotide phosphodiesterase, PdeE, that harbors two cyclic nucleotide binding motifs and a binuclear Zn(2+)-binding domain was characterized in Dictyostelium. In other eukaryotes, the Dictyostelium domain shows greatest homology to the 73-kDa subunit of the pre-mRNA cleavage and polyadenylation specificity factor. The Dictyostelium PdeE gene is expressed at its highest levels during aggregation, and its disruption causes the loss of a cAMP-phosphodiesterase activity. The pdeE null mutants show a normal cAMP-induced cGMP response and a 1.5-fold increase of cAMP-induced cAMP relay. Overexpression of a PdeE-yellow fluorescent protein (YFP) fusion construct causes inhibition of aggregation and loss of the cAMP relay response, but the cells can aggregate in synergy with wild-type cells. The PdeE-YFP fusion protein was partially purified by immunoprecipitation and biochemically characterized. PdeE and its Dictyostelium ortholog, PdeD, are both maximally active at pH 7.0. Both enzymes require bivalent cations for activity. The common cofactors Zn(2+) and Mg(2+) activated PdeE and PdeD maximally at 10 mm, whereas Mn(2+) activated the enzymes to 4-fold higher levels, with half-maximal activation between 10 and 100 microm. PdeE is an allosteric enzyme, which is approximately 4-fold activated by cAMP, with half-maximal activation occurring at about 10 microm and an apparent K(m) of approximately 1 mm. cGMP is degraded at a 6-fold lower rate than cAMP. Neither cGMP nor 8-Br-cAMP are efficient activators of PdeE activity.     

Spreading speeds of spatially periodic integro-difference models for populations with nonmonotone recruitment functions

An idea used by Thieme (J. Math. Biol. 8, 173-187, 1979) is extended to show that a class of integro-difference models for a periodically varying habitat has a spreading speed and a formula for it, even when the recruitment function R(u, x) is not nondecreasing in u, so that overcompensation occurs. Numerical simulations illustrate the behavior of solutions of the recursion whose initial values vanish outside a bounded set.     

Intracellular localization of secretable cAMP in relaying Dictyostelium discoideum cells

Dictyostelium discoideum cells synthesize and secrete the chemoattractant cAMP within minutes after chemotactic stimulation. During development, this signal-relay process is instrumental in cell aggregation, pattern formation, and differentiation. Cyclic AMP is known to accumulate inside the cell before secretion. In this study we investigated the subcellular localization of the nascent cAMP. After chemotactic stimulation at 0 degrees C and subsequent accumulation of intracellular cAMP, the newly synthesized chemoattractant could be released by gently opening cells in two different ways. Both methods make the cytosolic compartment accessible, whereas intracellular compartments surrounded by a membrane remain largely intact. The first method involved rapid lysis by forced passage through a 5-micron pore-size Nuclepore filter. The second technique was electropermeabilization under carefully controlled conditions that ensured the formation of small, stable pores in the plasma membrane. These pores allowed the passage of small molecules, such as cAMP, but not of macromolecules. To confirm the selectivity for the plasma membrane of both methods, we showed that a typical vesicular cell compartment, the lysosome, remained intact. Both procedures immediately released all intracellularly accumulated cAMP. We interpret our results as strong evidence for accumulation of nascent cAMP in the cytosolic compartment rather than in a vesicular compartment before it is secreted. This implies that cAMP secretion takes place via a trans-membrane transport mechanism, rather than by exocytosis.     

On the stability of separable solutions of a sexual age-structured population dynamics model

The Sharpe-Lotka-McKendrick-von Foerster equations for non-dispersing age-sex-structured populations with a harmonic mean type mating law are considered and their separable solutions are analysed. For certain forms of the demographic rates the underlying evolution equations are reduced to systems of ODEs, the long time behavior of their solutions is studied, and the stability of separable solutions is discussed. It is found that for the constant death rates and constant sex ratio of newborns with stationary birth rates this model admits one one-parameter class of separable solutions, two such classes (repeated or different) or no such ones. In the case of special forms of age-dependent birth rates, solutions of one of these two different classes corresponding to the greater root of the characteristic equation are locally stable, solutions of the other one corresponding to the smaller root are unstable, and the population dies out if the model does not admit separable solutions or if initial densities of newborns are small enough in the case of the existence of separable solutions. In the case of constant vital rates, the model has no separable solutions or admits only one class of such ones that are globally stable.     

Cell-type-specific responsiveness to cAMP in cell differentiation of Dictyostelium discoideum.

In Dictyostelium discoideum, both prespore and prestalk differentiation require extracellular cAMP. We investigated the difference in inducibility of the two cell types by cAMP. Previous studies indicate that cAMP added in the early stage of development inhibits prespore differentiation, and this was confirmed using three species of prespore specific mRNAs. By contrast, early treatment with cAMP did not inhibit, but induced the expression of prestalk-specific mRNA. These results indicate that differentiation pathways of the two cell types have different processes in the early stage of development.     

Specific photoaffinity labeling of the cAMP surface receptor in Dictyostelium discoideum

The recent observation that ammonium sulfate stabilizes cell-surface [3H]cyclic AMP binding in Dictyostelium discoideum (Van Haastert, P., and Kien, E. (1983) J. Biol. Chem. 258, 9636-9642) led us to attempt to identify the surface cAMP receptor by photoaffinity labeling with 8-azido-[32P]cAMP using this stabilization technique. 8-azido-[32P]cAMP specifically labeled a polypeptide which migrates as a closely spaced doublet (Mr = 40,000 to 43,000) on sodium dodecyl sulfate-polyacrylamide gel electrophoresis. Greater than 60% of the labeled polypeptide was found associated with membranes. This protein was distinguished from the cytosolic regulatory subunit of the cAMP-dependent protein kinase (Mr = 41,000) by differences in developmental regulation, specificity, and subcellular localization. No kinase regulatory subunit was detected in membranes by western blot analysis. Our preliminary observations show that labeling of this doublet correlates closely with cAMP-binding activity, suggesting that it is the surface receptor which mediates chemotaxis and cAMP signaling.     

cAMP regulation of cell differentiation in Dictyostelium discoideum and the role of the cAMP receptor

DNA polymerases and DNA ligases have been studied during development of the amphibian, axolotl. Three forms of DNA polymerase, I, II, and III, with sedimentation coefficients in sucrose of 9, 6, and 3.1 S, respectively, have been found in the axolotl egg. The activity of these three DNA polymerases is unchanged during early embryonic development. The activity of DNA polymerase III then increases significantly, beginning at the tailbud stage, while the activity of DNA polymerase II increases at the larval stage. DNA polymerase I does not show significant variations during this time. On the basis of their catalytic properties, it appears that DNA polymerases I and II are α-type DNA polymerases whereas DNA polymerase III is a β-type enzyme. Two different DNA ligases are found in the axolotl, one showing a sedimentation coefficient in sucrose of 8.2 S (heavy form) and the other, 6 S (light form). The 6 S enzyme is the major DNA ligase activity found in the egg before and after fertilization. Its activity then decreases during embryonic development. It can be observed again, as the only DNA ligase activity, in some adult tissues. The 8.2 S enzyme appears during the first division cycle of the fertilized egg, is present at all stages of embryonic development, and is absent from the adult tissues tested. Properties of the two DNA ligases at different stages of embryonic development have also been compared.     

Intracellular cAMP in Dictyostelium discoideum: Characterization of an aggregation-deficient mutant with elevated cAMP pools

Forty aggregation-deficient mutants of Dictyostelium discoideum were screened for changes in intracellular cAMP during the first 10 hr of starvation. The pools in 39 of the mutants remained low and relatively static during this period. However, amoebae of one mutant, strain HC151, exhibited significantly elevated levels of intracellular cAMP during vegetative growth and for several hours after starvation. A more detailed analysis of this mutant indicated that the elevated cAMP pools in these cells are a consequence of the premature appearance and partial activation of an adenylate cyclase. The mutation(s) altering adenylate cyclase regulation in this strain appears to map in linkage group IV. Complementation tests between strain HC151 and another mutant, HH201, which has recently been shown to produce an adenylate cyclase activity precociously [1], indicated that the mutations affecting adenylate cyclase activity in these strains map at different loci. Although both of these mutations behave recessively in heterozygous diploids with respect to gross development, an examination of early cAMP metabolism and terminal spore differentiation in these diploids suggest that these mutations are at least partially expressed during some stage(s) of the developmental cycle.     

A model of intracellular transport of particles in an axon

In this paper we develop a model of intracellular transport of cell organelles and vesicles along the axon of a nerve cell. These particles are moving alternately by processive motion along a microtubule with the aid of motor proteins, and by diffusion. The model involves a degenerate system of diffusion equations. We prove a maximum principle and establish existence and behavior of a unique solution. Numerical results show how the transportation of mass depends on the relevant parameters of the model.