The Influence of Gene Expression Time Delays on Gierer–Meinhardt Pattern Formation Systems

Real-time monitoring of gene expression dynamics and population levels in a multispecies microbial community could enable the study of the role of changing gene expression patterns on eco-evolutionary outcomes. Here we report the design and validation of a unique experimental platform with an in situ fluorescence measurement system that has high dynamic range and temporal resolution and is capable of monitoring multiple fluorophores for long-term gene expression and population dynamics experiments. We demonstrate the capability of our system to capture gene expression dynamics in response to external perturbations in two synthetic genetic systems: a simple inducible genetic circuit and a bistable toggle switch. Finally, in exploring the population dynamics of a two species microbial community, we show that our system can capture the switch between competitive exclusion and long-term coexistence in response to different nutrient conditions.

Course 11 Pattern formation in visual cortex

Turing reaction–diffusion systems have been used to model pattern formation in several areas of developmental biology. Previous biomathematical Turing system models employed static domains which failed to incorporate the growth that inherently occurs as an organism develops. To address this shortcoming, we incorporate an exponentially growing domain into a Turing system, allowing one to more realistically model biological pattern formation. This Turing system can generate patterns on an exponentially growing domain in any of the eleven coordinate systems in which the Helmholtz equation is separable, making the system incredibly flexible and giving one the capability to mathematically model pattern formation on a geometrically diverse group of domains. Linear stability analysis is employed to generate mathematical conditions which ensure such a system can generate patterns. We apply the exponentially growing Turing system to a prolate spheroidal domain and conduct numerical simulations to investigate the system’s pattern-generating behavior. We find that the addition of growth to a Turing system causes a significant change in the pattern-generating behavior of the system. While a static domain Turing system converges to a final pattern, an exponentially growing domain Turing system produces transient patterns that continually evolve and increase in complexity over time.

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

Stochastic evolutionary growth and pattern formation models are treated in a unified way in terms of algorithmic models of nonlinear dynamic systems with feedback built of a standard set of signal processing units. A number of concrete models is described and illustrated by numerous examples of artificially generated patterns that closely imitate wide variety of patterns found in the nature.

Examples of biological pattern formation are life cycles of bacteria and social amoebae, embryonic tissue formation, wound healing or tumour growth. Thereby, development of a particular spatio‐temporal "multi‐cellular" pattern may be interpreted as cooperative phenomenon emerging from an intricate interplay of local (e.g. by adhesion) and non‐local (e.g. via diffusing signals) cell interactions. Mathematical models are required for the analysis of cooperative phenomena. Typical modeling attempts focus on a macroscopic perspective, i.e. the models (e.g. partial differential equations) describe the spatio‐temporal dynamics of cell concentrations. More recently, cell‐based models have been suggested in which the fate of each individual cell can be tracked. Cellular automata are discrete dynamical systems and may be utilized as cell‐based models. Here, we analyze spatio‐temporal pattern formation in cellular automaton models of interacting discrete cells. In particular, we introduce lattice‐gas cellular automata. In the same spirit as cellular automata, lattice‐gas cellular automaton (LGCA) and lattice Boltzmann (LB) models are promising tools for studying transport and interaction processes in biological systems. In these models the update rule is split into two parts which are called collision (interaction) and propagation, respectively. The collision rule of LGCA can be compared with the update rule for CA in that it assigns new states to each cell based on the states of the sites in a local neighborhood. After the collision step the state of each node is propagated to a neighboring node. This split of the update rule guarantees propagation of quantities while keeping the rules simple. The desired behaviour, e.g. spatio‐temporal pattern formation of a LGCA shows up in the macroscopic limit which can be derived from a theory of statistical mechanics on a lattice. Instead of discrete particles, LB models deal with continuous distribution functions which interact locally and which propagate after "collision" to the next neighbour node. LB models can be interpreted as mean‐field approximations of LGCA. Model applications are bacterial pattern formation and tumour growth.Ref.:A. Deutsch, S. Dormann (2005) Cellular automaton modelling of biological pattern formation, Birkhauser, Boston

Quantum chaos in Rydberg atoms.- Crystallization of electrons and holes.- Natural patterns in nonequilibrium systems.- Linear aspects of the Faraday instability.- On curved cellular flames.- Nonlinear flow behavior and shear-induced structure of fluids.- Experimental study of horizontally shaken granular matter - The swelling effect.- On relaxational granular compaction.- From microscopic to macroscopic traffic models.- The modelling concept of sociodynamics.- The morphogenesis of dictyostelium discoideum - Pattern formation in a biological excitable system.- Path optimization in chemical and biological systems on the basis of excitation waves.- BSE viewed dynamically: A possible early cure based on passive immunization against PrPSc.- Molecular semiotic structures in the cellular immune system: Key to dynamics and spatial patterning?.- Molecular evolutionary dynamics.- The retinal spreading depression: A model for nonlinear behavior of the brain.- An analytically solvable model of collective excitation patterns in cortical tissue.- Interaction of meandering spiral waves in active media.- Spatiotemporal patterns in a passivating electrochemical system.- Instabilities of pollutant concentrations in the troposphere due to chemical reactions.- Hydrodynamic singularities.- Disordered structures analyzed by the theory of Markov processes.- Transition to turbulence in shear flows.- Deformation of charge density waves in quasi-one-dimensional semiconductors visualized by scanning electron microscopy.- Probing nonlinear carrier transport in semi- and superconductors via low-temperature scanning laser microscopy.- Nonlinear spatio-temporal emission dynamics of broad area laser diodes.- Gravitational slowing down of clocks implies proportional size increase.

We consider the Gierer-Meinhardt system (1.1), shown below, on a bounded smooth domain $\Omega\subset\mathbb{R}^n$ ($n\ge1$) with a homogeneous Neumann boundary condition. For suitable exponents $a$, $b$, $c$ and $d$, we establish certain sufficient conditions for global existence. Theorem 1.1 here, combined with Theorem 1.2 of [6], implies a classical phenomenonon the effect of the initial data on global existence and finite time blow-up. This work is a continuation of our earlier result [6] for the Gierer-Meinhardt system.  &nbspThe Gierer-Meinhardt system was introduced in [1] to model activator-inhibitor systems in pattern formation in ecological systems.

Self-organized pattern dynamics of somitogenesis model in embryos

While it is somewhat well known that spatial PDE extensions of the Rosenzweig-MacArthur predator-prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing-Hopf mechanisms, in two distinct spatial Rosenzweig-MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator-prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator-prey-quarry-resource-scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing-Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig-MacArthur predator-prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig-MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations.

The purpose of the study is to identify and analyze patterns of the diurnal dynamics of the expression of genes that differ in the shape of the curve. It can be expected that the similarity of the patterns of daily expression (shape of the curve) of genes is a reflection of the synchronization of gene expression by common external and internal signals or participation in similar biological processes. Different signals that have daily dynamics (light, activity, nutrition, stress, temperature, etc.) can affect different levels of expression regulation, which can be manifested in various forms of patterns of daily gene expression. In our research, we used experimental data on gene expression at the level of translation (ribosome profling) in the liver and kidney of a mouse (GSE67305 and GSE81283). To identify genes with a daily rhythm of expression, we used a oneway analysis of variance. To identify similar­in­shape curves of the daily dynamics of gene expression, we propose an approach based on cluster analysis. The distance between the genes was calculated by aligning the phases and fnding the maximum cross­correlation between the patterns of the daily expression of these genes by the cyclic shift. This approach allowed us to identify genes that have not only expression patterns with a single maximum (sinusoidal, asymmetrical, shifted to the left or right, pulsed), but also complex composite signals with several extremes. As a result, the groups of genes united by the similarity of the shape of the daily expression curve without regard to their phase characteristics were identifed. GO enrichment analysis of groups of genes with sharply different patterns of daily expression (sinusoidal and pulsed) in the mouse kidneys and liver showed that the group of genes with a sinusoidal pattern was more associated with regulation of circadian rhythm and metabolism. The group of genes with a pulsed pattern is largely associated with the protective functions of the organism, which require the quick response. Thus, our studies have confrmed the effectiveness of the proposed approach to the analysis of the diurnal dynamics of gene expression. The identifed dynamic patterns of diurnal expression are important for the further study of complex circadian regulation, synchronization and interaction of biological processes with diurnal dynamics in mammals.

The Gierer-Meinhardt system characterizes the fundamental dynamics of pattern formation through the interaction of two variables. Currently, research on the spatiotemporal evolution of this system at home and abroad has been limited to Turing instability and patterns, and research on composite patterns formed by non-uniform cross-diffusion distributions is not yet in-depth. This article presents a composite pattern of the system based on linear, periodic, and radial distributions of cross diffusion. We conduct a comprehensive study of the two-dimensional spatiotemporal dynamics of the Gierer-Meinhardt model, treating the cross-diffusion coefficients as bifurcation parameters. Using multiscale analysis, the amplitude equation at the Turing threshold is derived. Subsequently, the effects of the proportional-derivative controller, fractional diffusion orders, and anisotropy on system stability, pattern formation, and evolution speed are systematically investigated. Numerical simulations are provided to validate the conclusions.

Pattern formation in the thiourea–iodate–sulfite system: Spatial bistability, waves, and stationary patterns

Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities

Global attractivity of equilibrium in Gierer–Meinhardt system with activator production saturation and gene expression time delays