Bulletin of Mathematical Biology

Mallet, D.; Pettet, G.

doi: 10.1007/s11538-005-9032-1pmid: 16794929

Haptotactic cell migration, a directed response to gradients of cell—extracellular matrix adhesion, is an important process in a number of biological phenomena such as wound healing and tumour cell invasion. Previously, mathematical models of haptotaxis have been developed on the premise that cells migrate in response to gradients in the density of the extracellular matrix. In this paper, we develop a novel mathematical model of haptotaxis which includes the adhesion receptors known as integrins and a description of their functional activation, local recruitment and protrusion as part of lamellipodia. Through the inclusion of integrins, the modelled cell matter is able to respond to a true gradient of cell–matrix adhesion, represented by functionally active integrins. We also show that previous matrix-mediated models are in fact a subset of the novel integrin-mediated models, characterised by specific choices of diffusion and haptotaxis coefficients in their model equations. Numerical solutions suggest the existence of travelling waves of cell migration that are confirmed via a phase plane analysis of a simplified model.

Castiglione, Filippo; Piccoli, Benedetto

doi: 10.1007/s11538-005-9014-3pmid: 16794930

We construct a population dynamics model of the competition among immune system cells and generic tumor cells. Then, we apply the theory of optimal control to find the optimal schedule of injection of autologous dendritic cells used as immunotherapeutic agent.

Vatov, Larisa; Kizner, Ziv; Ruppin, Eytan; Meilin, Sigal; Manor, Tamar; Mayevsky, Avraham

doi: 10.1007/s11538-005-9008-1pmid: 16794931

Mathematical modeling of brain function is an important tool needed for a better understanding of experimental results and clinical situations. In the present study, we are constructing and testing a mathematical model capable of simulating changes in brain energy metabolism that develop in real time under various pathophysiological conditions. The model incorporates the following parameters: cerebral blood flow, partial oxygen pressure, mitochondrial NADH redox state, and extracellular potassium. Accordingly, all the model variables are only time dependent (`point-model' approach). Numerical runs demonstrate the ability of the model to mimic pathological conditions, such as complete and partial ischemia, cortical spreading depression under normoxic and partial ischemic conditions. They also show that, when properly tuned, a model of this type permits the monitoring of only one or two crucial variables and the computation of the remaining variables in real time during clinical or experimental procedures.

Mukhopadhyay, B.; Bhattacharyya, R.

doi: 10.1007/s11538-005-9007-2pmid: 16794932

The paper is concerned with the effect of variable dispersal rates on Turing instability of a non-Lotka-Volterra reaction-diffusion system. In ecological applications, the dispersal rates of different species tends to oscillate in time. This oscillation is modeled by temporal variation in the diffusion coefficient with large as well as small periodicity. The case of large periodicity is analyzed using the theory of Floquet multipliers and that of the small periodicity by using Hill's equation. The effect of such variation on the resulting Turing space is studied. A comparative analysis of the Turing spaces with constant diffusivity and variable diffusivities is performed. Numerical simulations are carried out to support analytical findings.

Aittokallio, T.; Gyllenberg, M.; Polo, O.; Toivonen, J.; Virkki, A.

doi: 10.1007/s11538-005-9059-3pmid: 16794933

This work describes a comprehensive mathematical model of the human respiratory control system which incorporates the central mechanisms for predicting sleep-induced changes in chemical regulation of ventilation. The model integrates four individual compartments for gas storage and exchange, namely alveolar air, pulmonary blood, tissue capillary blood, body tissues, and gas transport between them. An essential mechanism in the carbon dioxide transport is its dissociation into bicarbonate and acid, where a buffering mechanism through hemoglobin is used to prevent harmfully low pH levels. In the current model, we assume high oxygen levels and consider intracellular hydrogen ion concentration as the principal respiratory control variable. The resulting system of delayed differential equations is solved numerically. With an appropriate choice of key parameters, such as velocity of blood flow and gain of a non-linear controller function, the model provides steady-state results consistent with our experimental observations measured in subjects across sleep onset. Dynamic predictions from the model give new insights into the behaviour of the system in subjects with different buffering capacities and suggest novel hypotheses for future experimental and clinical studies.

Santillán, Moisés; Zeronb, Eduardo

doi: 10.1007/s11538-005-9025-0pmid: 16794934

In this paper we study the stability of a previously introduced model for the tryptophan operon regulatory pathway. For this, we make use of the second Lyapunov's method. The results obtained for the wild-type and for a couple ofin-silico mutant bacterial strains allow a deeper understanding of the multiplicity of regulatory mechanisms in this operon. In particular, we confirm that enzyme inhibition and transcription attenuation strengthen the system stability, the effect of transcription attenuation being much shorter than that of enzyme inhibition. Furthermore, the analysis here presented provides some insights about how enzyme inhibition affects the system stability.

Song, Yun

doi: 10.1007/s11538-005-9045-9pmid: 16794935

In hidden Markov models, the probability of observing a set of strings can be computed using recursion relations. We construct a sufficient condition for simplifying the recursion relations for a certain class of hidden Markov models. If the condition is satisfied, then one can construct a reduced recursion where the dependence on Markov states completely disappears. We discuss a specific example—namely, statistical multiple alignment based on the TKF-model—in which the sufficient condition is satisfied.

Melkikh, A.; Seleznev, V.

doi: 10.1007/s11538-005-9035-ypmid: 16794936

Requirements on models of the active transport of ions in biomembranes have been formulated. The basic requirements include an explicit dependence of the resting potential and intracellular concentrations of ions on the difference of ATP-ADP chemical potentials, a consideration of the reversibility of the ionic pump operation, a correlation between theoretical and experimental data on the resting potential and intracellular concentrations of ions for different types of cells, the pump efficiency approaching 100%, and a tendency of the resting potential to the Donnan potential if the active transport is blocked. A model satisfying the aforementioned requirements has been proposed by the authors as an example.

Reluga, Timothy C.; Medlock, Jan; Galvani, Alison P.

doi: 10.1007/s11538-005-9027-ypmid: 16794937

One of the central goals of mathematical epidemiology is to predict disease transmission patterns in populations. Two models are commonly used to predict spatial spread of a disease. The first is the distributed-contacts model, often described by a contact distribution among stationary individuals. The second is the distributed-infectives model, often described by the diffusion of infected individuals. However, neither approach is ideal when individuals move within home ranges. This paper presents a unified modeling hypothesis, called the restricted-movement model. We use this model to predict spatial spread in settings where infected individuals move within overlapping home ranges. Using mathematical and computational approaches, we show that our restricted-movement model has three limits: the distributed-contacts model, the distributed-infectives model, and a third, less studied advective distributed-infectives limit. We also calculate approximate upper bounds for the rates of an epidemic's spatial spread. Guidelines are suggested for determining which limit is most appropriate for a specific disease.