Selection for migration modification

Simple models for the genetic control of the tendency to migrate are considered. It is shown that migration from a more favorable regime to a less favorable regime is selected against. The ramifications for general modifier theory are discussed.     

A model of hierarchical ecosystems with migration

A model of ecosystems with migration is proposed from the viewpoint of flow. This model explains the following two points: (1) How the density-dependent terms in population dynamics arise as a consequence of migration. (2) How the ecosystem exhibits a hierarchy in energy per unit biomass.     

A continuous migration model with stable demography

A probability model of a population undergoing migration, mutation, and mating in a geographic continuum R is constructed, and an integrodifferential equation is derived for the probability of genetic identity. The equation is solved in one case, and asymptotic analysis done in others. Individuals at x, y R in the model mate with probability V(x, y) dt in any time interval (t, t + dt). In two dimensions, if V(x,y) = V(x–y) where V(x) V(x/)/ ² approaches a delta function, the equilibrium probability of identity vanishes as 0. The asymptotic rate at which this occurs is discussed for mutation rates u u _o > 0 and for Cu , > 0, and u 0.Partially supported by NSF grant MCS79-03472Research was partially supported by Task Agreement No. DE-AT06-76EV71005 under Contract No. DE-AM06-76RL02225 between the U.S. Dept. Energy and the University of Washington     

A simple game-theoretic model for upstream fish migration

A simple game-theoretic model for upstream fish migration, which is a key element in life history of diadromous fishes, is proposed. Foundation of the model is a minimization problem on the cost of migration with the swimming speed and school size as the variables to be simultaneously optimized. Finding the optimizer ultimately reduces to solving a self-consistency equation without explicit solutions. Mathematical analytical results lead to the sufficient condition that the self-consistency equation has a unique solution, which turns out to be identified with the condition where the unique optimizer exists. Behavior of the optimizer is analyzed both mathematically and numerically to show its biophysical and ecological consequences. The analytical results demonstrate reasonable agreement between the present mathematical model and the theoretical and experimental results of upstream migration of fish schools reported in the past research.     

A discrete model with density dependent fast migration

The aim of this work is to develop an approximate aggregation method for certain non-linear discrete models. Approximate aggregation consists in describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete models with two different time scales, the slow one considered to be linear and the fast one non-linear because of its transition matrix depends on the global variables. In our discrete model the time unit is chosen to be the one associated to the slow dynamics, and then we approximate the effect of fast dynamics by using a sufficiently large power of its corresponding transition matrix. In a previous work the same system is treated in the case of fast dynamics considered to be linear, conservative in the global variables and inducing a stable frequency distribution of the state variables. A similar non-linear model has also been studied which uses as time unit the one associated to the fast dynamics and has the non-linearity in the slow part of the system. In the present work we transform the system to make the global variables explicit, and we justify the quick derivation of the aggregated system. The local asymptotic behaviour of the aggregated system entails that of the general system under certain conditions, for instance, if the aggregated system has a stable hyperbolic fixed point then the general system has one too. The method is applied to aggregate a multiregional Leslie model with density dependent migration rates.     

A steering model of endothelial sheet migration recapitulates monolayer integrity and directed collective migration

Cells in endothelial cell monolayers maintain a tight barrier between blood and tissue, but it is not well understood how endothelial cells move within monolayers, pass each other, migrate when stimulated with growth factor, and also retain monolayer integrity. Here, we develop a quantitative steering model based on functional classes of genes identified previously in a small interfering RNA (siRNA) screen to explain how cells locally coordinate their movement to maintain monolayer integrity and collectively migrate in response to growth factor. In the model, cells autonomously migrate within the monolayer and turn in response to mechanical cues resulting from adhesive, drag, repulsive, and directed steering interactions with neighboring cells. We show that lateral-drag steering explains the local coordination of cell movement and the maintenance of monolayer integrity by allowing closure of small lesions. We further demonstrate that directional steering of cells at monolayer boundaries, combined with adhesive steering of cells behind, can explain growth factor-triggered collective migration into open space. Together, this model provides a mechanistic explanation for the observed genetic modularity and a conceptual framework for how cells can dynamically maintain sheet integrity and undergo collective directed migration.     

A mechanobiological model of endothelial cell migration and proliferation

How angiogenesis is regulated by local environmental cues is still not fully understood despite its importance to many regenerative events. Although mechanics is known to influence angiogenesis, the specific cellular mechanisms influenced by mechanical loading are poorly understood. This study adopts a lattice-based modelling approach to simulate endothelial cell (EC) migration and proliferation in order to explore how mechanical stretch regulates their behaviour. The approach enables the explicit modelling of ECs and, in particular, their migration/proliferation (specifically, rate and directionality) in response to such mechanical cues. The model was first used to simulate previously reported experiments of EC migration and proliferation in an unloaded environment. Next, three potential effects (increased cell migration, increased cell proliferation and biased cellular migration) of mechanical stretch on EC behaviour were simulated using the model and the observed changes in cell population characteristics were compared to experimental findings. Combinations of these three potential drivers were also investigated. The model demonstrates that only by incorporating all three changes in cellular physiology (increased EC migration, increased EC proliferation and biased EC migration in the direction perpendicular to the applied strain) in response to dynamic loading, it is possible to successfully predict experimental findings. This provides support for the underlying model hypotheses for how mechanics regulates EC behaviour.     

Biased three-dimensional cell migration and collagen matrix modification

Various tumours can be resected even for cure with complete removal. Surgical excision with a wide margin to ensure complete removal has therefore been suggested as the primary treatment for such lesions. The histological examination of the three-dimensional (3D) excision margins (3D histology) constitutes an important part of the quality control mechanisms in tumour surgery. Understanding histologically relevant properties of the constituents of the microenvironment in tumours and the circumferential portion of non-lesional tissue is therefore critical.Accompanied by the increasing availability of high performance computers in recent decades, there has been a strong movement aiming at the development of mathematical models whose implementations allow in silico simulations of biological reaction networks. Due to its relevance in numerous biological and pathological processes, there have been various attempts to model biased migration of single cells. The model introduced in this paper plays a prominent role insofar as it covers the under-represented 3D case. Moreover, it is uniformly formulated for both two and three dimensions. The velocity of each cell is characterised by a generalised Langevin equation, a stochastic differential equation, where chemotaxis as well as contact guidance are considered to simulate selected aspects of interactions between carcinoma cell groups and fibroblast-like cells.Algorithmic and numeric aspects of the developed equations are detailed in this paper and the results of the simulations are illustrated in different manners to emphasise specific features. A simple test scenario as well as a geometry based on segmentation data of a real histological slide has served for verification of the software. The resulting morphologies closely resemble that of desmoplastic stromal reaction readily identifiable in histological slides of infiltrating carcinoma, and the images can be interpreted in the context of 3D histology.     

A receptor-electromigration-based model for cellular electrotactic sensing and migration

Directed cell migration in tissues mediates various physiological processes and is guided by complex cellular factors such as chemoattractant gradients and electric fields. Direct current (DC) electric fields can be generated in physiological settings and the electric field guided migration of various cell types (i.e., electrotaxis) has been demonstrated both in vitro and in vivo. Although several mechanisms have been proposed for electrotaxis, there are so far very few quantitative models. Furthermore, because chemoattractant gradients and electric fields co-exist in tissues, it is important to understand how chemotaxis and electrotaxis interact for mediating cell migration and trafficking. In this study, we developed a mathematical model to investigate the role of electromigration of cell surface chemoattractant receptors in mediating electrochemical sensing and migration of cells. Our results show that electromigration of chemoattractant receptors enables cell electrotactic sensing and migration in the presence of a uniform chemoattractant field. Furthermore, in the physiologically-relevant range of receptor electromigration rates, application of electric fields overcomes chemical guiding signals for directional sensing and migration of cells in co-existing competing electric fields and chemoattractant gradients.     

A novel model to study the dorsolateral migration of melanoblasts

Melanocytes derived from pluripotent neural crest cells migrate initially in the dorsolateral pathway between the ectoderm and dermomyotome. To understand the role of specific proteins involved in this cell migration, we looked for a cellular model that mimics the in vivo behavior of melanoblasts, and that allows functional studies of their migration. We report here that wild-type embryonic stem (ES) cells are able to follow the ventral and dorsolateral neural crest pathways after being grafted into chicken embryos. By contrast, a mutant ES cell line deficient for beta1 integrin subunits, proteins involved in cell-extracellular interactions, had a severely impaired migratory behavior. Interestingly, ES cells deficient for Kit, the tyrosine kinase receptor for the stem cell factor (SCF), behaved similarly to wild-type ES cells. Thus, grafting mouse ES cells into chicken embryos provides a new cellular system that allows both in vitro and in vivo studies of the molecular mechanisms controlling dorsolateral migration.     

A cellular automaton model for the migration of glioma cells

We present a study of in vitro cell migration in two dimensions as a first step towards understanding the mechanisms governing the motility of glioma cells. Our study is based on a cellular automaton model which aims at reproducing the kinetics of a lump of glioma cells deposited on a substrate of collagen. The dynamical effects of cell attraction and motion inertia are introduced through adequate automaton rules. We compare the density profiles given by the model to those obtained experimentally. The result of the best fit indicates a substantial cell-cell attraction due to cell-cell communication through gap junctions (or chemotaxis) and negligible inertia effects during migration. Tracking of individual migrating cells indicates highly convoluted cell trajectories.     

A note on the island model with sex dependent migration

Summary A theoretical calculation is presented which extends Wright's island model of drift and migration to differential migration between the two sexes. In this circumstance, local demes no longer have Hardy-Weinberg frequencies. There may be local heterozygote excess or deficiency depending, respectively, on whether migration occurs before or after mating. The magnitude of the local departure from Hardy-Weinberg is directly proportional to the difference between the migration parameters of the two sexes. These results could have important implications for studies where genetic markers are used for inferring population structure. An example from a study of Marmot colonies is cited.     

Posttranslational modification of human alphaA-crystallin: correlation with electrophoretic migration.

alphaA-crystallin is a major protein component of the human lens. It is known to undergo posttranslational modification. This study was done to further elucidate the temporal and spatial nature of these posttranslational modifications and to correlate the modified forms with electrophoretic migration. We dissected normal human lenses into concentric shells of fiber cells, separated the proteins by two-dimensional electrophoresis, and identified modified forms by mass spectrometry. We found that alphaA-crystallin migrated as a major spot and in over 20 additional protein spots. The extent of modification correlated with the age of the fiber cells and the depth within a lens. A correlation was also seen between these parameters and the concentration of modified forms that had full-length sequences but migrated at more acidic positions. These proteins were phosphorylated, acetylated, and/or deamidated. A few proteins migrated to a more basic position than the major form of alphaA-crystallin. The locations of several species that were truncated after C-terminal residues Ser172 and Ser162 were identified. Each of these species had intact N termini. The similarity of the C-terminal cleavage sites found in alphaA- and alphaB-crystallins was noted.     

A model for the study of epithelial migration in wound healing.

A model is described which enables the detailed study of epithelial regeneration in experimentally produced lesions in the common bile duct of the rabbit. The circular on slightly oval defect of 1 mm diameter produced by a specially developed apparatus has a perfectly smooth base. Epithelial migration in this model has been investigated using light microscopy of transverse sections and scanning electron microscopy of whole preparations. Typical changes in the border cells, characterised by the formation of tapered protusions, can be observed as early as two hours after the lesion has been made. Later the cells in the flattened edge of the moving border also show various types of protrusion which rest on the substratum. Mitotic activity in the surface epithelium and crypts in the surrounding region only increases after closure of the lesion, which usually takes place within 16--24 h.     

The island model with stochastic migration

The island model with stochastically variable migration rate and immigrant gene frequency is investigated. It is supposed that the migration rate and the immigrant gene frequency are independent of each other in each generation, and each of them is independently and identically distributed in every generation. The treatment is confined to a single diallelic locus without mutation. If the diploid population is infinite, selection is absent and the immigrant gene frequency is fixed, then the gene frequency on the island converges to the immigrant frequency, and the logarithm of the absolute value of its deviation from it is asymptotically normally distributed. Assuming only neutrality, the evolution of the exact mean and variance of the gene frequency are derived for an island with finite population. Selection is included in the diffusion approximation: if all evolutionary forces have comparable roles, the gene frequency will be normally distributed at all times. All results in the paper are completely explicit.     

Continuum model of cell adhesion and migration

The motility of cells crawling on a substratum has its origin in a thin cell organ called lamella. We present a 2-dimensional continuum model for the lamella dynamics of a slowly migrating cell, such as a human keratinocyte. The central components of the model are the dynamics of a viscous cytoskeleton capable to produce contractile and swelling stresses, and the formation of adhesive bonds in the plasma cell membrane between the lamella cytoskeleton and adhesion sites at the substratum. We will demonstrate that a simple mechanistic model, neglecting the complicated signaling pathways and regulation processes of a living cell, is able to capture the most prominent aspects of the lamella dynamics, such as quasi-periodic protrusions and retractions of the moving tip, retrograde flow of the cytoskeleton and the related accumulation of focal adhesion complexes in the leading edge of a migrating cell. The developed modeling framework consists of a nonlinearly coupled system of hyperbolic, parabolic and ordinary differential equations for the various molecular concentrations, two elliptic equations for cytoskeleton velocity and hydrodynamic pressure in a highly viscous two-phase flow, with appropriate boundary conditions including equalities and inequalities at the moving boundary. In order to analyse this hybrid continuum model by numerical simulations for different biophysical scenarios, we use suitable finite element and finite volume schemes on a fixed triangulation in combination with an adaptive level set method describing the free boundary dynamics.     

A generalized transport model for biased cell migration in an anisotropic environment

 A generalized transport model is derived for cell migration in an anisotropic environment and is applied to the specific cases of biased cell migration in a gradient of a stimulus (taxis; e.g., chemotaxis or haptotaxis) or along an axis of anisotropy (e.g., contact guidance). The model accounts for spatial or directional dependence of cell speed and cell turning behavior to predict a constitutive cell flux equation with drift velocity and diffusivity tensor (termed random motility tensor) that are explicit functions of the parameters of the underlying random walk model. This model provides the connection between cell locomotion and the resulting persistent random walk behavior to the observed cell migration on longer time scales, thus it provides a framework for interpreting cell migration data in terms of underlying motility mechanisms. Received: 8 April 1999