Aberrant Behaviours of Reaction Diffusion Self-organisation Models on Growing Domains in the Presence of Gene Expression Time Delays

Turing’s pattern formation mechanism exhibits sensitivity to the details of the initial conditions suggesting that, in isolation, it cannot robustly generate pattern within noisy biological environments. Nonetheless, secondary aspects of developmental self-organisation, such as a growing domain, have been shown to ameliorate this aberrant model behaviour. Furthermore, while in-situ hybridisation reveals the presence of gene expression in developmental processes, the influence of such dynamics on Turing’s model has received limited attention. Here, we novelly focus on the Gierer–Meinhardt reaction diffusion system considering delays due the time taken for gene expression, while incorporating a number of different domain growth profiles to further explore the influence and interplay of domain growth and gene expression on Turing’s mechanism. We find extensive pathological model behaviour, exhibiting one or more of the following: temporal oscillations with no spatial structure, a failure of the Turing instability and an extreme sensitivity to the initial conditions, the growth profile and the duration of gene expression. This deviant behaviour is even more severe than observed in previous studies of Schnakenberg kinetics on exponentially growing domains in the presence of gene expression (Gaffney and Monk in Bull. Math. Biol. 68:99–130, 2006). Our results emphasise that gene expression dynamics induce unrealistic behaviour in Turing’s model for multiple choices of kinetics and thus such aberrant modelling predictions are likely to be generic. They also highlight that domain growth can no longer ameliorate the excessive sensitivity of Turing’s mechanism in the presence of gene expression time delays. The above, extensive, pathologies suggest that, in the presence of gene expression, Turing’s mechanism would generally require a novel and extensive secondary mechanism to control reaction diffusion patterning.     

Investigation of Turing instability for the Gierer–Meinhardt model

The dependence of the emergence of Turing instability for a distributed system of nonlinear differential equations that describe hydra morphogenesis based on the oscillatory properties of the corresponding trajectories of the system was investigated. The limits in the parameter space that provide diffusive instability were obtained. The frequency and amplitude dependences of the resulting spatial self oscillations on the values of the main parameters were investigated. Comparative analysis of the properties of the distributed system and corresponding trajectories of the system was carried out and the analytical conclusions were confirmed by the solutions of the system that were found using MATLAB.

     

The Influence of Hydrologic Residence Time on Lake Carbon Cycling Dynamics Following Extreme Precipitation Events

The frequency and magnitude of extreme events are expected to increase in the future, yet little is known about effects of such events on ecosystem structure and function. We examined how extreme precipitation events affect exports of terrestrial dissolved organic carbon (t-DOC) from watersheds to lakes as well as in-lake heterotrophy in three north-temperate lakes. Extreme precipitation events induced large influxes of t-DOC to our lakes, accounting for 45–58% of the seasonal t-DOC load. These large influxes of t-DOC influenced lake metabolism, resulting in lake net heterotrophy following 67% of the extreme precipitation events across all lakes. Hydrologic residence time (HRT) was negatively related to t-DOC load and heterotrophy; lakes with short HRT had higher t-DOC loads and greater net heterotrophy. The fraction of t-DOC mineralized within each lake following extreme precipitation events generally exhibited a positive relationship with lake HRT, similar to the previous studies of fractions mineralized at annual and supra-annual time scales. Event-associated turnover rate of t-DOC was higher than what is typically reported from laboratory studies and modeling exercises and was also negatively related to lake HRT. This study demonstrates that extreme precipitation events are ‘hot moments’ of carbon load, export, and turnover in lakes and that lake-specific characteristics (for example, HRT) interact with climatic patterns to set rates of important lake carbon fluxes.     

Turing Pattern Formation with Two Kinds of Cells and a Diffusive Chemical

We study a three-variable Turing system with two kinds of cells and a diffusive chemical, considering the formation and maintenance of fish skin patterns with multiple pigment cells. The two types of cells are produced from undifferentiated cells. They inhibit the supply rate of the other cell type, forming local clusters of the same cell type. In addition, the cells of one type can be maintained only in the presence of a diffusive chemical produced by the other cell type, resulting in the coexistence of two cell types in heterogeneous spatial patterns. We assume linear interaction among cells and the chemical, and cell supply rates constrained to be positive or zero. We derive the condition for diffusion-driven instability. In one-dimensional model, we examine how the wavelength of the periodic pattern depends on parameters. In the two-dimensional model, we study the condition for spot, stripe or reversed spot pattern to emerge (pattern selection). We discuss heuristic criteria for the pattern selection.     

The Influence of Fluid Flow on Modeling Quorum Sensing in Bacterial Biofilms

In this paper, we study quorum sensing in Pseudomonas aeruginosa biofilms. Quorum sensing is a process where bacteria monitor their population density through the release of extra-cellular signalling molecules. The presence of these molecules affects gene modulation leading to changes in behaviour such as the release of virulence factors. Here, we use numerical methods to approximate a 2-D model of quorum sensing. It is observed that the shape of the biofilm can have a profound effect on the onset of quorum sensing. This has serious repercussions for experimental observations since biofilms of the same biomass but different shapes can produce quite different results.     

Reaction–Diffusion Systems and External Morphogen Gradients: The Two-Dimensional Case,with an Application to Skeletal Pattern Formation

We investigate a reaction–diffusion system consisting of an activator and an inhibitor in a two-dimensional domain. There is a morphogen gradient in the domain. The production of the activator depends on the concentration of the morphogen. Mathematically, this leads to reaction–diffusion equations with explicitly space-dependent terms. It is well known that in the absence of an external morphogen, the system can produce either spots or stripes via the Turing bifurcation. We derive first-order expansions for the possible patterns in the presence of an external morphogen and show how both stripes and spots are affected. This work generalizes previous one-dimensional results to two dimensions. Specifically, we consider the quasi-one-dimensional case of a thin rectangular domain and the case of a square domain. We apply the results to a model of skeletal pattern formation in vertebrate limbs. In the framework of reaction–diffusion models, our results suggest a simple explanation for some recent experimental findings in the mouse limb which are much harder to explain in positional-information-type models.     

Host–Parasite Interactions and the Evolution of Gene Expression

Interactions between hosts and parasites provide an ongoing source of selection that promotes the evolution of a variety of features in the interacting species. Here, we use a genetically explicit mathematical model to explore how patterns of gene expression evolve at genetic loci responsible for host resistance and parasite infection. Our results reveal the striking yet intuitive conclusion that gene expression should evolve along very different trajectories in the two interacting species. Specifically, host resistance loci should frequently evolve to co-express alleles, whereas parasite infection loci should evolve to express only a single allele. This result arises because hosts that co-express resistance alleles are able to recognize and clear a greater diversity of parasite genotypes. By the same token, parasites that co-express antigen or elicitor alleles are more likely to be recognized and cleared by the host, and this favours the expression of only a single allele. Our model provides testable predictions that can help interpret accumulating data on expression levels for genes relevant to host−parasite interactions.     

The Dynamics of Turing Patterns for Morphogen-Regulated Growing Domains with Cellular Response Delays

Since its conception in 1952, the Turing paradigm for pattern formation has been the subject of numerous theoretical investigations. Experimentally, this mechanism was first demonstrated in chemical reactions over 20 years ago and, more recently, several examples of biological self-organisation have also been implicated as Turing systems. One criticism of the Turing model is its lack of robustness, not only with respect to fluctuations in the initial conditions, but also with respect to the inclusion of delays in critical feedback processes such as gene expression. In this work we investigate the possibilities for Turing patterns on growing domains where the morphogens additionally regulate domain growth, incorporating delays in the feedback between signalling and domain growth, as well as gene expression. We present results for the proto-typical Schnakenberg and Gierer–Meinhardt systems: exploring the dynamics of these systems suggests a reconsideration of the basic Turing mechanism for pattern formation on morphogen-regulated growing domains as well as highlighting when feedback delays on domain growth are important for pattern formation.     

A New Stochastic Individual-Based Model for Pattern Formation and its Application to Predator–Prey Systems

Reaction–diffusion theory has played a very important role in the study of pattern formation in biology. However, a group of individuals is described by a single state variable representing population density in reaction–diffusion models, and interaction between individuals can be included only phenomenologically. In this paper, we propose a new scheme that seamlessly combines individual-based models with elements of reaction–diffusion theory and apply it to predator–prey systems as a test of our scheme. In the model, starvation periods and the time to reproductive maturity are modeled for individual predators. Similarly, the life cycle and time to reproductive maturity of an individual prey are modeled. Furthermore, both predators and prey migrate through a two-dimensional space. To include animal migration in the model, we use a relationship between the diffusion and the random numbers generated according to a two-dimensional bivariate normal distribution. Despite the simplicity of this model, our scheme successfully produces logistic patterns and oscillations in the population size of both predator and prey. The peak for the predator population oscillation lags slightly behind the prey peak. The simplicity of this scheme will aid additional study of spatially distributed negative-feedback systems.     

Identification and Expression Pattern of DNA Polymerase α Gene in a Marine Diatom, Skeletonema costatum

To investigate the potential of DNA polymerase α as a marker for DNA replication in phytoplankton, two gene fragments that showed a high degree of similarity with eukaryotic DNA polymerase α were cloned from two strains of a diatom, Skeletonema costatum (Greville) Cleve. The gene fragments amplified with the polymerase chain reaction were 397 and 396 bp in length, respectively. The deduced amino acid sequences showed 44% to 61% similarity to the corresponding regions of DNA polymerase α sequences of eukaryotic organisms ranging from yeast to humans. The similarity was especially high in three evolutionarily conserved regions within the amplified fragments. Further, hybridization patterns from Southern blotting confirmed that the amplified fragments were an integral part on the genome of S. costatum. In batch cultures abundant messenger of DNA polymerase α appeared in the late exponential phase and the early stationary phase. This pattern suggests that DNA polymerase α expression is associated with actively dividing cells. Received October 27, 1998; accepted November 13, 1998.