First capture success in two dimensions: The search for prey by a random walk predator in a comprehensive space of random walks

Random walk models are an important tool used for understanding how complex organisms redistribute themselves through space and time in search of targets such as food, shelter, or mates. These walks are easily studied with agent-based models, which can be used to ask which search strategy is best according to some efficiency metric. Current studies however, generally do not consider the full range of potential random walks, success metrics, and constraints on the walker, and implementation details vary widely. It is therefore difficult to compare results across studies. In this paper, we investigate predator search behaviour in a comprehensive space of key movement variables that allows the predator to select from a continuum of random walks ranging from Brownian walks (BWs) to correlated random walks (CorRWs) which include directional persistence, to composite random walks (ComRWs) which feature intensive and extensive search modes (ISMs and ESMs), and finally to more complex correlated composite random walks (CCRWs). We specifically focus on the search behaviour of a predator between the initiation of a search for a prey item and the first successful acquisition of a prey target: we call this interval the “search-to-capture” event. We measure the predator's success against three metrics of energetic cost: (1) the time elapsed, (2) the distance travelled, and (3) an equally weighted combination of time and distance. In addition, we explore the effect of three different constraints on the predator: (1) hunting success in the extensive search mode, (2) detection radius when in the extensive search mode, and (3) prey density. Our work confirms the broadly held notion that CCRW movement patterns should always outperform BWs, but find instructive cases where other walks are superior. We also show that, within the CCRW category, there is a wide range of possible walks and rank these according to measures of energetic cost. Our work also offers insights into the evolutionary pressures surrounding the “search-to-capture” event, and suggests that CCRW predators with low hunting success in one movement mode experience higher evolutionary pressures and are thus more likely to adopt a nearly optimal random walk. Our work highlights the need for comprehensive studies that examine several aspects of random walks simultaneously.     

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     

Directed cell migration on fibronectin gradients: effect of gradient slope

The migration of human microvascular endothelial cells (hMEC) was measured on a range of fibronectin gradient slopes. hMEC drift speed increased with increasing gradient slope with no concurrent change in cellular persistence time or random cell speed. The frequency of discrete cellular motion in the gradient direction increased with gradient slope. Morphological polarization of cells on the gradients is also characterized and correlated with cellular drift speed. These experiments present the first demonstration of cellular response to changing haptotactic gradient slope using an in vitro system for the quantitative study of cell migration.     

Kinetic studies of cell migration in a modified Boyden chamber: dependence on cell concentration and effects of the chymotrypsin-like cationic protein of human granulocytes.

Migration of PMN cells was measure kinetically by the "leading front" method in a modified Boyden chamber. In some experiments, release of lactoferrin and of chymotrypsin-like cationic protein were measured simultaneously. Migration, both directed and random, was increased by increased cell concentration. Kinetically, directed migration consisted of two phases. The second phase was more influenced by the cell concentration. Preincubation of the cells with low concentrations of chymotrypsin-like cationic protein stimulated cell migration in the same way as did high cell concentrations, i.e., by primarily affecting the second phase. The effect of chymotrypsin-like cationic protein was time and dose dependent and dependent on the chymotrypsin-like activity of the protein. At high concentrations of protein, the cells were immobilized. Release of granular proteins did not take place during the first phase of migration but was parallel to the second phase. It is concluded that chymotrypsin-like cationic protein might be one of the substances responsible for the effect of cell concentration seen on migration in vitro.     

Evolution of cell motility in an individual-based model of tumour growth

Tumour invasion is driven by proliferation and importantly migration into the surrounding tissue. Cancer cell motility is also critical in the formation of metastases and is therefore a fundamental issue in cancer research. In this paper we investigate the emergence of cancer cell motility in an evolving tumour population using an individual-based modelling approach. In this model of tumour growth each cell is equipped with a micro-environment response network that determines the behaviour or phenotype of the cell based on the local environment. The response network is modelled using a feed-forward neural network, which is subject to mutations when the cells divide. With this model we have investigated the impact of the micro-environment on the emergence of a motile invasive phenotype. The results show that when a motile phenotype emerges the dynamics of the model are radically changed and we observe faster growing tumours exhibiting diffuse morphologies. Further we observe that the emergence of a motile subclone can occur in a wide range of micro-environmental growth conditions. Iterated simulations showed that in identical growth conditions the evolutionary dynamics either converge to a proliferating or migratory phenotype, which suggests that the introduction of cell motility into the model changes the shape of fitness landscape on which the cancer cell population evolves and that it now contains several local maxima. This could have important implications for cancer treatments which focus on the gene level, as our results show that several distinct genotypes and critically distinct phenotypes can emerge and become dominant in the same micro-environment.     

A chemotactic model of trunk neural crest cell migration

Trunk neural crest cells follow a common ventral migratory pathway but are distributed into two distinct locations to form discrete sympathetic and dorsal root ganglia along the vertebrate axis. Although fluorescent cell labeling and time‐lapse studies have recorded complex trunk neural crest cell migratory behaviors, the signals that underlie this dynamic patterning remain unclear. The absence of molecular information has led to a number of mechanistic hypotheses for trunk neural crest cell migration. Here, we review recent data in support of three distinct mechanisms of trunk neural crest cell migration and develop and simulate a computational model based on chemotactic signaling. We show that by integrating the timing and spatial location of multiple chemotactic signals, trunk neural crest cells may be accurately positioned into two distinct targets that correspond to the sympathetic and dorsal root ganglia. In doing so, we honor the contributions of Wilhelm His to his identification of the neural crest and extend the observations of His and others to better understand a complex question in neural crest cell biology.     

The effect of seasonal host birth rates on population dynamics: the importance of resonance

Many of the simple mathematical models currently in use often fail to capture important biological factors. Here we extend current models of insect-pathogen interactions to include seasonality in the birth rate. In particular, we consider the SIR model with self-regulation when applied to specific cases--rabbit haemorrhagic disease and fox rabies. In this paper, we briefly summarize the results of the model with a constant time-independent birth rate, a, which we then replace with the time dependent birth rate a(t), to investigate how this effects the dynamics of the host population. We can split parameter space into an area in which the model without seasonality has no oscillations, in which case a simple averaging rule predicts the behaviour. Alternatively, in the area where oscillations to the equilibrium do occur in the non-seasonal model, disease persistence is more complicated and we get more complex dynamical behaviour in this case. We apply resonance techniques to discover the structure of the subharmonic modes of the SIR model with self-regulation. We then look at whether many biological systems are likely to display these "resonant" dynamics and find that we would expect them to be widespread.     

Investigations into the design principles in the chemotactic behavior of Escherichia coli

Inspired by the recent studies on the analysis of biased random walk behavior of Escherichia coli[Passino, K.M., 2002. Biomimicry of bacterial foraging for distributed optimization and control. IEEE Control Syst. Mag. 22 (3), 52-67; Passino, K.M., 2005. Biomimicry for Optimization, Control and Automation. Springer-Verlag, pp. 768-798; Liu, Y., Passino, K.M., 2002. Biomimicry of social foraging bacteria for distributed optimization: models, principles, and emergent behaviors. J. Optim. Theory Appl. 115 (3), 603-628], we have developed a model describing the motile behavior of E. coli by specifying some simple rules on the chemotaxis. Based on this model, we have analyzed the role of some key parameters involved in the chemotactic behavior to unravel the underlying design principles. By investigating the target tracking capability of E. coli in a maze through computer simulations, we found that E. coli clusters can be controlled as target trackers in a complex micro-scale-environment. In addition, we have explored the dynamical characteristics of this target tracking mechanism through perturbation of parameters under noisy environments. It turns out that the E. coli chemotaxis mechanism might be designed such that it is sensitive enough to efficiently track the target and also robust enough to overcome environmental noises.     

Stochastic model of leukocyte chemosensory movement

Journal of Mathematical Biology - We propose a hypothesis for a unified understanding of the persistent and biased random walk behavior of leukocytes exhibiting random motility and chemotaxis,...     

Cell sorting by differential cell motility: a model for pattern formation in Dictyostelium

In the slug stage of the cellular slime mold Dictyostelium discoideum, prespore cells and four types of prestalk cells show a well-defined spatial distribution in a migrating slug. We have developed a continuous mathematical model for the distribution pattern of these cell types based on the balance of force in individual cells. In the model, cell types are assumed to have different properties in cell motility, i.e. different motive force, the rate of resistance against cell movement, and diffusion coefficient. Analysis of the stationary solution of the model shows that combination of these parameters and slug speed determines the three-dimensional shape of a slug and cell distribution pattern within it. Based on experimental data of slug motive force and velocity measurements, appropriate sets of parameters were chosen so that the cell-type distribution at stationary state matches the distribution in real slugs. With these parameters, we performed numerical calculation of the model in two-dimensional space using a moving particle method. The results reproduced many of the basic features of slug morphogenesis, i.e. cell sorting, translocation of the prestalk region, elongation of the slug, and its steady migration.     

Modelling the movement of a soil insect

We use a linear autoregressive model to describe the movement of a soil-living insect, Protaphorura armata (Collembola). Models of this kind can be viewed as extensions of a random walk, but unlike a correlated random walk, in which the speed and turning angles are independent, our model identifies and expresses the correlations between the turning angles and a variable speed. Our model uses data in x- and y-coordinates rather than in polar coordinates, which is useful for situations in which the resolution of the observations is limited. The movement of the insect was characterized by (i) looping behaviour due to autocorrelation and cross correlation in the velocity process and (ii) occurrence of periods of inactivity, which we describe with a Poisson random effects model. We also introduce obstacles to the environment to add structural heterogeneity to the movement process. We compare aspects such as loop shape, inter-loop time, holding angles at obstacles, net squared displacement, number, and duration of inactive periods between observed and predicted movement. The comparison demonstrates that our approach is relevant as a starting-point to predict behaviourally complex moving, e.g. systematic searching, in a heterogeneous landscape.     

Microtubule-dependent migration of the cell nucleus toward a future leading edge in amoebae ofPhysarum polycephalum

Summary In several cell types, an intriguing correlation exists between the position of the centrosome and the direction of cell locomotion. The centrosome is positioned between the leading edge pseudopod and the nucleus. This suggests that the polarized distribution of organelles in the cytoplasm is coupled spatially with structural and functional polarity in the cell cortex. To study cellular polarization with special interest in the roles of microtubules, we have analyzed the effects of microtubule-disrupting reagents and local laser irradiation on behaviors of both the nucleus and the centrosome in living amoebae ofPhysarum polycephalum. Physarum cells often have 2–3 pseudopods. One of the pseudopods keeps extending to become a stable leading edge while the rest retracts, a crucial step that reorients cells during locomotion. The nucleus, together with the centrosome, moves specifically toward the pseudopod that will become the leading edge. Disruption of microtubules with nocodazole randomizes positions of the nucleus, indicating the involvement of microtubules in the directional migration of the nucleus toward a specific pseudopod. The migration direction of the nucleus is reversed immediately after the UV laser is irradiated at regions between the nucleus and the future leading pseudopod. In contrast, irradiation at regions between the future tail and the nucleus does not affect nuclear migration. By immunofluorescence, we confirmed fragmentation of microtubules specifically in the irradiated region. These results suggest that the nucleus is pulled together with the centrosome toward the future leading-edge pseudopod in a microtubule-dependent manner. Microtubules seem to exert the pulling force generated in the cell cortex on the centrosome. They may serve as a mediator of shape changes initiated in the cell cortex to the organelle geometry in the endoplasm.     

The complex effect of granulocyte colony-stimulating factor on human granulopoiesis analyzed by a new physiologically-based mathematical model

Neutropenia, frequently a side effect of chemo- and radiotherapy, increases susceptibility to microbial infections and is a life-threatening condition. For realistically predicting drug treatment effects on granulopoiesis, we have constructed a new mathematical model of granulopoiesis in the bone marrow and in the peripheral blood, featuring cell cycle phase transition and detailed granulocyte-colony stimulating factor (G-CSF) pharmacokinetics (PK) and pharmacodynamics (PD), including intracellular second messenger. Using this model, in conjunction with clinical results, we evaluated the system parameters, implemented those in the model and successfully retrieved the results of several independent clinical experiments under a wide range of G-CSF regimens. Our results show that the introduction of G-CSF-controlled intracellular second messenger is indispensable for precise retrieval of the clinical results, and suggest that the half-life of this messenger varies between a single and multiple G-CSF administration schedules. In addition, our model provided reliable steady-state, as well as dynamic, estimations of human granulopoiesis parameters. These included an estimation of apoptosis index in the post-mitotic compartment, which corroborates previous results. At present the model is used for suggesting improved drug regimens.     

The effect of intrinsic stochasticity on transmitted HIV drug resistance patterns

Estimates of transmitted HIV drug-resistance prevalence vary widely among and within epidemiological surveys. Interpretation of trends from available survey data is therefore difficult. Because the emergence of drug-resistance involves small populations of infected drug-resistant individuals, the role of stochasticity (chance events) is likely to be important. The question addressed here is: how much variability in transmitted HIV drug-resistance prevalence patterns arises due to intrinsic stochasticity alone, i.e., if all starting conditions in the different epidemics surveyed were identical? This ‘thought experiment’ gives insight into the minimum expected variabilities within and among epidemics. A simple stochastic mathematical model was implemented. Our results show that stochasticity alone can generate a significant degree of variability and that this depends on the size and variation of the pool of new infections when drug treatment is first introduced. The variability in transmitted drug-resistance prevalence within an epidemic (i.e., the temporal variability) is large when the annual pool of all new infections is small (fewer than 200, typical of the HIV epidemics in Central European and Scandinavian countries) but diminishes rapidly as that pool grows. Epidemiological surveys involving hundreds of new infections annually are therefore needed to allow meaningful interpretation of temporal trends in transmitted drug-resistance prevalence within individual epidemics. The stochastic variability among epidemics shows a similar dependence on the pool of new infections if treatment is introduced after endemic equilibrium is established, but can persist even when there are more than 10,000 new infections annually if drug therapy is introduced earlier. Stochastic models may therefore have an important role to play in interpreting differences in transmitted drug-resistance prevalence trends among epidemiological surveys.     

The effect of positive assortative mating on genetic parameters in a simulated beef cattle population

Summary Two simulated data sets, representing random mating and positive assortative mating in a beef cattle population over 10 rounds of mating, were each composed of 100 replicates. Three correlated traits were considered; calving ease (CE), 200 day weight (WW) and postweaning gain (PG). All selection practiced in the simulation was random. Positive assortative mating, which was based on parental WW phenotypic records, increased the progeny additive genetic variance of WW. The absolute values of genetic covariances and correlations between WW with CE and PG were also increased by positive assortative mating. Variances or covariances did not reach their expected equilibrium values due to overlapping generations, low replacement rates and only 10 rounds of mating.The financial assistance of Agriculture Canada and the Natural Sciences and Engineering Research Council of Canada are gratefully acknowledged     

Naringenin is a novel inhibitor of Dictyostelium cell proliferation and cell migration

Naringenin is a flavanone compound that alters critical cellular processes such as cell multiplication, glucose uptake, and mitochondrial activity. In this study, we used the social amoeba, Dictyostelium discoideum, as a model system for examining the cellular processes and signaling pathways affected by naringenin. We found that naringenin inhibited Dictyostelium cell division in a dose-dependent manner (IC(50) approximately 20 microM). Assays of Dictyostelium chemotaxis and multicellular development revealed that naringenin possesses a previously unrecognized ability to suppress amoeboid cell motility. We also found that naringenin, which is known to inhibit phosphatidylinositol 3-kinase activity, had no apparent effect on phosphatidylinositol 3,4,5-trisphosphate synthesis in live Dictyostelium cells; suggesting that this compound suppresses cell growth and migration via alternative signaling pathways. In another context, the discoveries described here highlight the value of using the Dictyostelium model system for identifying and characterizing the mechanisms by which naringenin, and related compounds, exert their effects on eukaryotic cells.     

Analysis of a one-dimensional random walk with irreversible losses at each step: applications for protein movement on DNA

We analysed a one-dimensional random walk between two points when the migrating particle could be irreversibly lost (dissociated) from the system at each step of the process. We show that in the case of losses at each step the average number of steps made by the particle that reaches the final point does not obey quadratic dependence on the distance between the starting and the final points: for long distances this dependence is linear. This is because losses "select" for shorter pathways between the starting and the final points. We applied this analysis to protein translocations within long DNA molecules.     

Multi-scale modeling of a wound-healing cell migration assay

A continuum model and a discrete model are developed to capture the population-scale and cell-scale behavior in a wound-healing cell migration assay created from a scrape wound in a confluent cell monolayer. During wound closure, the cell population forms a sustained traveling wave, with close contact between cells behind the wavefront. Cells exhibit contact inhibition of migration and contact-limited proliferation. The continuum model includes the two dominant mechanisms and characteristics of cell migration and proliferation, using a cell diffusivity function that decreases with cell density and a logistic proliferative growth term. The discrete model arises naturally from the continuum model. Individual cells are simulated as continuous-time random walkers with nearest-neighbor transitions, together with a birth/death process. The migration and proliferation parameters are determined by analysing individual mice 3T3 fibroblast cell trajectories obtained during the development of a confluent cell monolayer and in a wound healing assay. The population-scale model successfully predicts the shape and speed of the traveling wave, while the discrete model is also successful in capturing the contact inhibition of migration effects.     

Hybrid cellular automaton modeling of nutrient modulated cell growth in tissue engineering constructs

Mathematic models help interpret experimental results and accelerate tissue engineering developments. We develop in this paper a hybrid cellular automata model that combines the differential nutrient transport equation to investigate the nutrient limited cell construct development for cartilage tissue engineering. Individual cell behaviors of migration, contact inhibition and cell collision, coupled with the cell proliferation regulated by oxygen concentration were carefully studied. Simplified two-dimensional simulations were performed. Using this model, we investigated the influence of cell migration speed on the overall cell growth within in vitro cell scaffolds. It was found that intense cell motility can enhance initial cell growth rates. However, since cell growth is also significantly modulated by the nutrient contents, intense cell motility with conventional uniform cell seeding method may lead to declined cell growth in the final time because concentrated cell population has been growing around the scaffold periphery to block the nutrient transport from outside culture media. Therefore, homogeneous cell seeding may not be a good way of gaining large and uniform cell densities for the final results. We then compared cell growth in scaffolds with various seeding modes, and proposed a seeding mode with cells initially residing in the middle area of the scaffold that may efficiently reduce the nutrient blockage and result in a better cell amount and uniform cell distribution for tissue engineering construct developments.