Traveling waves in a model of influenza A drift

Between major pandemics, the influenza A virus changes its antigenic properties by accumulating point mutations (drift) mainly in the RNA genes that code for the surface proteins hemagglutinin (HA) and neuraminidase (NA). The successful strain (variant) that will cause the next epidemic is selected from a reduced number of progenies that possess relatively high transmissibility and the ability to escape from the immune surveillance of the host. In this paper, we analyse a one-dimensional model of influenza A drift (Z. Angew. Math. Mech. 76 (2) (1996) 421) that generalizes the classical SIR model by including mutation as a diffusion process in a phenotype space of variants. The model exhibits traveling wave solutions with an asymptotic wave speed that matches well those obtained from numerical simulations. As exact solutions for these waves are not available, asymptotic estimates for the amplitudes of infected and recovered classes are provided through an exponential approximation based on the smallness of the diffusion constant. Through this approximation, we find simple scaling properties to several parameters of relevance to the epidemiology of the disease.     

Entropic particle transport in periodic channels

The dynamics of Brownian motion has widespread applications extending from transport in designed micro-channels up to its prominent role for inducing transport in molecular motors and Brownian motors. Here, Brownian transport is studied in micro-sized, two-dimensional periodic channels, exhibiting periodically varying cross-sections. The particles in addition are subjected to an external force acting alongside the direction of the longitudinal channel axis. For a fixed channel geometry, the dynamics of the two-dimensional problem is characterized by a single dimensionless parameter which is proportional to the ratio of the applied force and the temperature of the particle environment. In such structures entropic effects may play a dominant role. Under certain conditions the two-dimensional dynamics can be approximated by an effective one-dimensional motion of the particle in the longitudinal direction. The Langevin equation describing this reduced, one-dimensional process is of the type of the Fick-Jacobs equation. It contains an entropic potential determined by the varying extension of the eliminated channel direction, and a correction to the diffusion constant that introduces a space dependent diffusion. Different forms of this correction term have been suggested before, which we here compare for a particular class of models. We analyze the regime of validity of the Fick-Jacobs equation, both by means of analytical estimates and the comparisons with numerical results for the full two-dimensional stochastic dynamics. For the nonlinear mobility we find a temperature dependence which is opposite to that known for particle transport in periodic potentials. The influence of entropic effects is discussed for both, the nonlinear mobility and the effective diffusion constant.     

Stochastic Optimal Foraging: Tuning Intensive and Extensive Dynamics in Random Searches

Recent theoretical developments had laid down the proper mathematical means to understand how the structural complexity of search patterns may improve foraging efficiency. Under information-deprived scenarios and specific landscape configurations, Lévy walks and flights are known to lead to high search efficiencies. Based on a one-dimensional comparative analysis we show a mechanism by which, at random, a searcher can optimize the encounter with close and distant targets. The mechanism consists of combining an optimal diffusivity (optimally enhanced diffusion) with a minimal diffusion constant. In such a way the search dynamics adequately balances the tension between finding close and distant targets, while, at the same time, shifts the optimal balance towards relatively larger close-to-distant target encounter ratios. We find that introducing a multiscale set of reorientations ensures both a thorough local space exploration without oversampling and a fast spreading dynamics at the large scale. Lévy reorientation patterns account for these properties but other reorientation strategies providing similar statistical signatures can mimic or achieve comparable efficiencies. Hence, the present work unveils general mechanisms underlying efficient random search, beyond the Lévy model. Our results suggest that animals could tune key statistical movement properties (e.g. enhanced diffusivity, minimal diffusion constant) to cope with the very general problem of balancing out intensive and extensive random searching. We believe that theoretical developments to mechanistically understand stochastic search strategies, such as the one here proposed, are crucial to develop an empirically verifiable and comprehensive animal foraging theory.     

Simulations of NMR-detected diffusion in suspensions of red cells: the "signatures" in q-space plots of various lattice arrangements

Coherence effects from pulsed field-gradient spin-echo (PGSE) nuclear magnetic resonance diffusion experiments have been observed and characterized for diffusants in many heterogeneous systems, ranging from porous materials to cell suspensions. The resulting coherence patterns appear in plots of the normalized PGSE signal intensities as a function of the spatial wave vector Q in a so-called q-space plot. The origin of these phenomena and their mathematical and physical underpinnings are now well established. We have conducted a number of studies of diffusion-coherence phenomena in suspensions of red blood cells and have made extensive use of computer simulations of molecular diffusion in virtual lattices of cells to aid in the interpretation and analysis of experimental data. In the current work we extended the canonical model used in these studies to investigate the effect that varying the packing arrangement of cells in the suspension has on the coherence patterns, as seen in q-space plots. We show that changes in the packing arrangement of cells are reflected in the q-space plots and in the results of diffusion tensor analysis and thus we speculate upon the possible clinical importance of these findings.     

Travelling wave solutions of diffusive Lotka-Volterra equations

We establish the existence of travelling wave solutions for two reaction diffusion systems based on the Lotka-Volterra model for predator and prey interactions. For simplicity, we consider only 1 space dimension. The waves are of transition front type, analogous to the travelling wave solutions discussed by Fisher and Kolmogorov et al. for a scalar reaction diffusion equation. The waves discussed here are not necessarily monotone. For any speed c there is a travelling wave solution of transition front type. For one of the systems discussed here, there is a distinguished speed c^* dividing the waves into two types, waves of speed c < c^* being one type, waves of speed c ? c^* being of the other type. We present numerical evidence that for this system the wave of speed c^* is stable, and that c^* is an asymptotic speed of propagation in some sense. For the other system, waves of all speeds are in some sense stable. The proof of existence uses a shooting argument and a Lyapunov function. We also discuss some possible biological implications of the existence of these waves.     

Phase and amplitude instability in delay-diffusion population models

We analyze three simple population models that include generation times in their growth dynamics. In the presence of a slow one-dimensional diffusion process it is shown that for sufficiently small wave numbers, the amplitude of the homogeneous limit cycle solution is unstable and the phase of the bifurcating diffusion wave obeys a Burgers' type equation with a negative coefficient in the diffusion term. Numerical solutions for the phase and amplitude in the post-critical regime display a turbulent like behavior in space and time when the size of the system is larger than some critical value. This result follows from the coupling between the delay and diffusion terms.     

One- and three-dimensional pathways for proteins to reach specific DNA sites

Proteins that interact with specific DNA sites bind to DNA at random and then translocate to the target site. This may occur by one-dimensional diffusion along the DNA, or through three-dimensional space via multiple dissociation/re-associations. To distinguish these routes, reactions of the ECO:RV endonuclease were studied on substrates with two ECO:RV sites separated by varied distances. The fraction of encounters between the DNA and the protein that resulted in the processive cleavage of both sites decreased as the length of intervening DNA was increased, but not in the manner demanded for one-dimensional diffusion. The variation in processivity with inter-site spacing shows instead that protein moves from one site to another through three-dimensional space, by successive dissociation/re-associations, though each re-association to a new site is followed by a search of the DNA immediately adjacent to that site. Although DNA-binding proteins are usually thought to find their target sites by one-dimensional pathways, three-dimensional routes may be more common than previously anticipated.     

Coinfection and superinfection in RNA virus populations: a selection-mutation model

In this paper, we present a general selection-mutation model of evolution on a one-dimensional continuous fitness space. The formulation of our model includes both the classical diffusion approach to mutation process as well as an alternative approach based on an integral operator with a mutation kernel. We show that both approaches produce fundamentally equivalent results. To illustrate the suitability of our model, we focus its analytical study into its application to recent experimental studies of in vitro viral evolution. More specifically, these experiments were designed to test previous theoretical predictions regarding the effects of multiple infection dynamics (i.e., coinfection and superinfection) on the virulence of evolving viral populations. The results of these experiments, however, did not match with previous theory. By contrast, the model we present here helps to understand the underlying viral dynamics on these experiments and makes new testable predictions about the role of parameters such the time between successive infections and the growth rates of resident and invading populations.     

Kinetics of target site localization of a protein on DNA: a stochastic approach

It is widely recognized that the cleaving rate of a restriction enzyme on target DNA sequences is several orders-of-magnitude faster than the maximal one calculated from the diffusion-limited theory. It was therefore commonly assumed that the target site interaction of a restriction enzyme with DNA has to occur via two steps: one-dimensional diffusion along a DNA segment, and long-range jumps coming from association-dissociation events. We propose here a stochastic model for this reaction which comprises a series of one-dimensional diffusions of a restriction enzyme on nonspecific DNA sequences interrupted by three-dimensional excursions in the solution until the target sequence is reached. This model provides an optimal finding strategy which explains the fast association rate. Modeling the excursions by uncorrelated random jumps, we recover the expression of the mean time required for target site association to occur given by Berg et al. in 1981, and we explicitly give several physical quantities describing the stochastic pathway of the enzyme. For competitive target sites we calculate two quantities: processivity and preference. By comparing these theoretical expressions to recent experimental data obtained for EcoRV-DNA interaction, we quantify: 1), the mean residence time per binding event of EcoRV on DNA for a representative one-dimensional diffusion coefficient; 2), the average lengths of DNA scanned during the one-dimensional diffusion (during one binding event and during the overall process); and 3), the mean time and the mean number of visits needed to go from one target site to the other. Further, we evaluate the dynamics of DNA cleavage with regard to the probability for the restriction enzyme to perform another one-dimensional diffusion on the same DNA substrate following a three-dimensional excursion.     

Sharp changes in resource availability may induce spatial nearly periodic population abundances

For many years, scientists have tried to understand the ubiquitous discrete nature of traits. As the emergence of nonuniform patterns in space via instability of the uniform pattern to spatial perturbations is well-understood in reaction–diffusion systems, several studies have suggested that a similar instability underlies discrete distributions of traits. In contrast, here we suggest that a different mechanism may underly species’ discrete distributions of trait values. We show that a point where niche availability changes sharply along the continuous niche axis promotes the discretized distribution of trait values even far from that point. In certain cases, this mechanism may apply also to patterns of population densities over space, such as patterns that were observed in vegetation biomass, as locations where environment changes sharply may promote spatially, nearly periodic stationary patterns.     

A Study of Early Afterdepolarizations in a Model for Human Ventricular Tissue

Sudden cardiac death is often caused by cardiac arrhythmias. Recently, special attention has been given to a certain arrhythmogenic condition, the long-QT syndrome, which occurs as a result of genetic mutations or drug toxicity. The underlying mechanisms of arrhythmias, caused by the long-QT syndrome, are not fully understood. However, arrhythmias are often connected to special excitations of cardiac cells, called early afterdepolarizations (EADs), which are depolarizations during the repolarizing phase of the action potential. So far, EADs have been studied mainly in isolated cardiac cells. However, the question on how EADs at the single-cell level can result in fibrillation at the tissue level, especially in human cell models, has not been widely studied yet. In this paper, we study wave patterns that result from single-cell EAD dynamics in a mathematical model for human ventricular cardiac tissue. We induce EADs by modeling experimental conditions which have been shown to evoke EADs at a single-cell level: by an increase of L-type Ca currents and a decrease of the delayed rectifier potassium currents. We show that, at the tissue level and depending on these parameters, three types of abnormal wave patterns emerge. We classify them into two types of spiral fibrillation and one type of oscillatory dynamics. Moreover, we find that the emergent wave patterns can be driven by calcium or sodium currents and we find phase waves in the oscillatory excitation regime. From our simulations we predict that arrhythmias caused by EADs can occur during normal wave propagation and do not require tissue heterogeneities. Experimental verification of our results is possible for experiments at the cell-culture level, where EADs can be induced by an increase of the L-type calcium conductance and by the application of I blockers, and the properties of the emergent patterns can be studied by optical mapping of the voltage and calcium.     

Modeling of ATP-mediated signal transduction and wave propagation in astrocytic cellular networks

Astrocytes, a special type of glial cells, were considered to have supporting role in information processing in the brain. However, several recent studies have shown that they can be chemically stimulated by neurotransmitters and use a form of signaling, in which ATP acts as an extracellular messenger. Pathological conditions, such as spreading depression, have been linked to abnormal range of wave propagation in astrocytic cellular networks. Nevertheless, the underlying intra- and inter-cellular signaling mechanisms remain unclear. Motivated by the above, we constructed a model to understand the relationship between single-cell signal transduction mechanisms and wave propagation and blocking in astrocytic networks. The model incorporates ATP-mediated IP3 production, the subsequent Ca2+ release from the ER through IP3R channels and ATP release into the extracellular space. For the latter, two hypotheses were tested: Ca2+- or IP3-dependent ATP release. In the first case, single astrocytes can exhibit excitable behavior and frequency-encoded oscillations. Homogeneous, one-dimensional astrocytic networks can propagate waves with infinite range, while in two dimensions, spiral waves can be generated. However, in the IP3-dependent ATP release case, the specific coupling of the driver ATP-IP3 system with the driven Ca2+ subsystem leads to one- and two-dimensional wave patterns with finite range of propagation.     

Analysis of Travelling Waves Associated with the Modelling of Aerosolised Skin Grafts

A previous model developed by the authors investigates the growth patterns of keratinocyte cell colonies after they have been applied to a burn site using a spray technique. In this paper, we investigate a simplified one-dimensional version of the model. This model yields travelling wave solutions and we analyse the behaviour of the travelling waves. Approximations for the rate of healing and maximum values for both the active healing and the healed cell densities are obtained. PACS 92B05     

Soliton behaviour in a bistable reaction diffusion model

We analyze a generic reaction-diffusion model that contains the important features of Turing systems and that has been extensively used in the past to model biological interesting patterns. This model presents various fixed points. Analysis of this model has been made in the past only in the case when there is only a single fixed point, and a phase diagram of all the possible instabilities shows that there is a place where a Turing-Hopf bifurcation occurs producing oscillating Turing patterns. In here we focus on the interesting situation of having several fixed points, particularly when one unstable point is in between two equally stable points. We show that the solutions of this bistable system are traveling front waves, or solitons. The predictions and results are tested by performing extensive numerical calculations in one and two dimensions. The dynamics of these solitons is governed by a well defined spatial scale, and collisions and interactions between solitons depend on this scale. In certain regions of parameter space the wave fronts can be stationary, forming a pattern resembling spatial chaos. The patterns in two dimensions are particularly interesting because they can present a coherent dynamics with pseudo spiral rotations that simulate the myocardial beat quite closely. We show that our simple model can produce complicated spatial patterns with many different properties, and could be used in applications in many different fields.      

Spatial patterns of human gene frequencies in Europe

The aims of this study of spatial patterns of human gene frequencies in Europe are twofold. One is to present new methodology developed for the analysis of such data. The other is to report on the diversity of spatial patterns observed in Europe and their interpretation as evidence of population processes. Spatial variation in 59 allele and haplotype frequencies (26 genetic systems) for polymorphisms in blood antigens, enzymes, and proteins is analyzed for an aggregate of 3,384 localities, using homogeneity tests, one-dimensional and directional spatial correlograms, and SYMAP interpolated surfaces. The data matrices are reduced to reveal the principal patterns by clustering techniques. The findings of this study can be summarized as follows: 1) There is significant heterogeneity in allele frequencies among the localities for all but one genetic system. 2) There are significant spatial patterns for most allele frequencies. 3) There is a substantial minority of clinal patterns in these populations. Clinal trends are found more frequently in HLA alleles than for other variables. North-south and northwest-southwest gradients predominate. 4) There is a strong decline in overall genetic similarity with geographic distance for most variables. 5) There are few, if any, appreciable correlations in pairs of allele frequencies over the continent, and there is little interesting correlation structure in the resulting correlation matrix. 6) Few spatial correlograms are markedly similar to each other, yet they form well-defined clusters. Spatial variation patterns, therefore, differ among allele frequencies. Patterns of human gene frequencies in modern Europe are diverse and complex. No single model suffices for interpretation of the observed genetic structure. Some clinal patterns reported here support the Neolithic demic-expansion hypothesis, others suggest latitudinal selection. Most of the clinal patterns are in HLA alleles, but there is also evidence from ABO for east-west migration diffusion. The majority of patterns are patchy, consistent with hypotheses of isolation by distance or of settlement of genetically differing, subsequently expanding ethnic groups. While undoubtedly there has been an ongoing stochastic process of differentiation consistent with the isolation-by-distance model, this has not obscured the directional patterns caused by migration (demic diffusion), and has perhaps only reinforced the contribution from settlement of ethnic units to patterns of genetic variation. However, the impact of the latter is most difficult to discern and requires further methodological developments.     

Random-walk models of cell dispersal included in mechanobiological simulations of tissue differentiation

Computational models have shown that biophysical stimuli can be correlated with observed patterns of tissue differentiation, and simulations have been performed that predict the time course of tissue differentiation in, for example, long bone fracture healing. Some simulations have used a diffusion model to simulate the migration and proliferation of cells with the differentiating tissue. However, despite the convenience of the diffusion model, diffusion is not the mechanism of cell dispersal: cells disperse by crawling or proliferation, or are transported in a moving fluid. In this paper, a random-walk model (i.e., a stochastic model), with and without a preferred direction, is studied as an approach to simulate cell proliferation/migration in differentiating tissues and it is compared with the diffusion model. A simulation of tissue differentiation of gap tissue in a two-dimensional model of a bone/implant interface was performed to demonstrate the differences between diffusion vs. random walk with a preferred direction. Results of diffusion and random-walk models are similar with respect to the change in the stiffness of the gap tissue but rather different results are obtained regarding tissue patterning in the differentiating tissues; the diffusion approach predicted continuous patterns of tissue differentiation whereas the random-walk model showed a more discontinuous pattern-histological results are not available that can unequivocally establish which is most similar to experimental observation. Comparing isotropic to anisotropic random walk (preferred direction of proliferation and cell migration), a more rapid reduction of the relative displacement between implant and bone is predicted. In conclusion, we have shown how random-walk models of cell dispersal and proliferation can be implemented, and shown where differences between them exist. Further study of the random-walk model is warranted, given the importance of cell seeding and cell dispersal/proliferation in many mechanobiological problems.     

Restricted Diffusion in Cellular Media: (1+1)-Dimensional Model

We consider the diffusion of molecules in a one-dimensional medium consisting of a large number of cells separated from the extra-cellular space by permeable membranes. The extra-cellular space is completely connected and allows unrestricted diffusion of the molecules. Furthermore, the molecules can diffuse within a given cell, i.e., the intra-cellular space; however, direct diffusion from one cell to another cell cannot occur. There is a movement of molecules across the permeable membranes between the intra- and extra-cellular spaces. Molecules from one cell can cross the permeable membrane into the extra-cellular space, then diffuse through the extra-cellular space, and eventually enter the intra-cellular space of a second cell. Here, we develop a simple set of model equations to describe this phenomenon and obtain the solutions using an eigenfunction expansion. We show that the solutions obtained using this method are particularly convenient for interpreting data from experiments that use techniques from nuclear magnetic resonance imaging.     

Calcium waves in agarose gel with cell organelles: implications of the velocity curvature relationship

Calcium oscillations and waves have been observed not only in several types of living cells but also in less complex systems of isolated cell organelles. Here we report the determination of apparent Ca2+ diffusion coefficients in a novel excitable medium of agarose gel with homogeneously distributed vesicles of skeletal sarcoplasmic reticulum. Spatiotemporal calcium patterns were visualized by confocal laser scanning fluorescence microscopy. To obtain characteristic parameters of the velocity curvature relationship, namely, apparent diffusion coefficient, velocity of plane calcium waves, and critical radius, positively and negatively curved wave fronts were analyzed. It is demonstrated that gel-immobilized cell organelles reveal features of an excitable medium. Apparent Ca2+ diffusion coefficients of the in vitro system, both in the absence or in the presence of mitochondria, were found to be higher than in cardiac myocytes and lower than in unbuffered agarose gel. Plane calcium waves propagated markedly slower in the in vitro system than in rat cardiac myocytes. Whereas mitochondria significantly reduced the apparent Ca2+ diffusion coefficient of the in vitro system, propagation velocity and critical size of calcium waves were found to be nearly unchanged. These results suggest that calcium wave propagation depends on the kinetics of calcium release rather than on diffusion.     

Dermatoglyphic variation in Europe

We describe the geographic variation patterns of 236 dermatoglyphic variables (118 for each sex) for 74 samples in Europe. Using principal components analysis and rotating to simple structure, we simplified these patterns to the first 20 axes, representing 74.2% of covariation. We then used heterogeneity tests, interpolated surfaces, one-dimensional and directional correlograms, and distances between correlograms to analyze the factor scores of these 20 axes. We also ordinated the 74 localities. The data are remarkable for showing little spatial autocorrelation, despite significant heterogeneity among localities. Only three factor axes exhibit consistently significant correlograms, indicating that there are few spatial patterns in the original variables in Europe. Almost all correlations between pairs of variables occur within serially homologous character sets and are thus developmentally determined. There is some support for demic diffusion from the southeast in finger patterns and ridge counts. We compare these results to those of previous studies and note that Lapps and Icelanders are outliers with respect to both genetics and finger tip variables, whereas Tatars are outliers with respect to craniometrics and dermatoglyphics. © 1996 Wiley-Liss, Inc.     

Extracellular space diffusion in central nervous system: anisotropic diffusion measured by elliptical surface photobleaching

Diffusion in the extracellular space (ECS) is crucial for normal central nervous system physiology. The determinants of ECS diffusion include viscous interactions with extracellular matrix/plasma membranes ("viscosity") and ECS geometry ("tortuosity"). To resolve viscosity versus tortuosity effects, we measured direction-dependent (anisotropic) diffusion in ECS in mouse spinal cord by photobleaching using an elliptical spot produced by a cylindrical lens in the excitation path. Anisotropic diffusion slowed fluorescence recovery when the long axis of the ellipse was parallel versus perpendicular to the direction of faster diffusion. A mathematical model was constructed to deduce diffusion coefficients (D(x), D(y)) from fluorescence recovery measured for parallel and perpendicular orientations of the long axis of the ellipse. Elliptical spot photobleaching was validated by photobleaching aqueous-phase fluorophores on a diffraction grating, where diffusion is one-dimensional. Measurement of the diffusion of 70 kDa FITC-dextran in spinal cord in living mice indicated that viscosity slows diffusion by approximately 1.8-fold compared with its diffusion in solution. ECS geometry hinders diffusion across (but not along) axonal fibers in spinal cord further by approximately fivefold. In cerebral cortex, however, approximately 50% of the hindrance to ECS diffusion comes from viscosity and approximately 50% from tortuosity. We suggest that the extracellular matrix might have evolved to facilitate rather than hinder diffusion even for large molecules.