Pattern Formation In Biological Systems Research Articles

Approximate Solution of PHI-Four and Allen–Cahn Equations Using Non-Polynomial Spline Technique

The aim of this work is to use an efficient and accurate numerical technique based on non-polynomial spline for the solution of the PHI-Four and Allen–Cahn equations. A recent discovery suggests that the PHI-Four equation focuses on its implications for particle physics and the behavior of scalar fields in the quantum realm. In materials science, ongoing research involves using the Allen–Cahn equation to understand and predict the evolution of microstructures in various materials as well as in biophysics. It depicts pattern formation in biological systems and the dynamics of spatial organization in tissues. To obtain an approximate solution of both equations, this technique uses forward differences for the time and cubic non-polynomial spline function for spatial descretization. The stability of the suggested technique is addressed using the von Neumann technique. Convergence test is carried out theoretically to show the order of convergence of the scheme. Some numerical tests are carried out to confirm accuracy and efficiency in terms of absolute error LR. Convergence rates for different test problems are also computed numerically. Numerical results and simulations obtained are compared with the existing methods.

Patterns in soft and biological matters.

The issue is devoted to theoretical, computer and experimental studies of internal heterogeneous patterns, their morphology and evolution in various soft physical systems-organic and inorganic materials (e.g. alloys, polymers, cell cultures, biological tissues as well as metastable and composite materials). The importance of these studies is determined by the significant role of internal structures on the macroscopic properties and behaviour of natural and manufactured tissues and materials. Modern methods of computer modelling, statistical physics, heat and mass transfer, statistical hydrodynamics, nonlinear dynamics and experimental methods are presented and discussed. Non-equilibrium patterns which appear during macroscopic transport and hydrodynamic flow, chemical reactions, external physical fields (magnetic, electrical, thermal and hydrodynamic) and the impact of external noise on pattern evolution are the foci of this issue. Special attention is paid to pattern formation in biological systems (such as drug transport, hydrodynamic patterns in blood and pattern dynamics in protein and insulin crystals) and to the development of a scientific background for progressive methods of cancer and insult therapy (magnetic hyperthermia for cancer therapy; magnetically induced drug delivery in thrombosed blood vessels). The present issue includes works on pattern growth and their evolution in systems with complex internal structures, including stochastic dynamics, and the influence of internal structures on the external static, dynamic magnetic and mechanical properties of these systems. This article is part of the theme issue 'Patterns in soft and biological matters'.

Fast solvers for optimal control problems from pattern formation

The modeling of pattern formation in biological systems using various models of reaction–diffusion type has been an active research topic for many years. We here look at a parameter identification (or PDE-constrained optimization) problem where the Schnakenberg and Gierer–Meinhardt equations, two well-known pattern formation models, form the constraints to an objective function. Our main focus is on the efficient solution of the associated nonlinear programming problems via a Lagrange–Newton scheme. In particular we focus on the fast and robust solution of the resulting large linear systems, which are of saddle point form. We illustrate this by considering several two- and three-dimensional setups for both models. Additionally, we discuss an image-driven formulation that allows us to identify parameters of the model to match an observed quantity obtained from an image.

Dynamic membrane structure induces temporal pattern formation

The understanding of temporal pattern formation in biological systems is essential for insights into regulatory processes of cells. Concerning this problem, the present work introduces a model to explain the attachment/detachment cycle of MARCKS and PKC at the cell membrane, which is crucial for signal transduction processes. Our model is novel with regard to its driving mechanism: Structural changes within the membrane fuel an activator–inhibitor based global density oscillation of membrane related proteins. Based on simulated results of our model, phase diagrams were generated to illustrate the interplay of MARCKS and PKC. They predict the oscillatory behavior in the form of the number of peaks, the periodic time, and the damping constant depending on the amounts of MARCKS and PKC, respectively. The investigation of the phase space also revealed an unexpected intermediate state prior to the oscillations for high amounts of MARCKS in the system. The validation of the obtained results was carried out by stability analysis, which also accounts for further enhanced understanding of the studied system. It was shown, that the occurrence of the oscillating behavior is independent of the diffusion and the consumption of the reactants. The diffusion terms in the used reaction–diffusion equations only act as modulating terms and are not required for the oscillation. The hypothesis of our work suggests a new mechanism of temporal pattern formation in biological systems. This mechanism includes a classical activator–inhibitor system, but is based on the modifications of the membrane structure, rather than a reaction–diffusion system.

An Efficient, Nonlinear Stability Analysis for Detecting Pattern Formation in Reaction Diffusion Systems

Reaction diffusion systems are often used to study pattern formation in biological systems. However, most methods for understanding their behavior are challenging and can rarely be applied to complex systems common in biological applications. I present a relatively simple and efficient, nonlinear stability technique that greatly aids such analysis when rates of diffusion are substantially different. This technique reduces a system of reaction diffusion equations to a system of ordinary differential equations tracking the evolution of a large amplitude, spatially localized perturbation of a homogeneous steady state. Stability properties of this system, determined using standard bifurcation techniques and software, describe both linear and nonlinear patterning regimes of the reaction diffusion system. I describe the class of systems this method can be applied to and demonstrate its application. Analysis of Schnakenberg and substrate inhibition models is performed to demonstrate the methods capabilities in simplified settings and show that even these simple models have nonlinear patterning regimes not previously detected. The real power of this technique, however, is its simplicity and applicability to larger complex systems where other nonlinear methods become intractable. This is demonstrated through analysis of a chemotaxis regulatory network comprised of interacting proteins and phospholipids. In each case, predictions of this method are verified against results of numerical simulation, linear stability, asymptotic, and/or full PDE bifurcation analyses.

Noise Reduction in the Intracellular Pom1p Gradient by a Dynamic Clustering Mechanism

Chemical gradients can generate pattern formation in biological systems. In the fission yeast Schizosaccharomyces pombe, a cortical gradient of pom1p (a DYRK-type protein kinase) functions to position sites of cytokinesis and cell polarity and to control cell length. Here, using quantitative imaging, fluorescence correlation spectroscopy, and mathematical modeling, we study how its gradient distribution is formed. Pom1p gradients exhibit large cell-to-cell variability, as well as dynamic fluctuations in each individual gradient. Our data lead to a two-state model for gradient formation in which pom1p molecules associate with the plasma membrane at cell tips and then diffuse on the membrane while aggregating into and fragmenting from clusters, before disassociating from the membrane. In contrast to a classical one-component gradient, this two-state gradient buffers against cell-to-cell variations in protein concentration. This buffering mechanism, together with time averaging to reduce intrinsic noise, allows the pom1p gradient to specify positional information in a robust manner.

Interactions Between Pattern Formation and Domain Growth

In this paper we develop a theoretical framework for investigating pattern formation in biological systems for which the tissue on which the spatial pattern resides is growing at a rate which is itself regulated by the diffusible chemicals that establish the spatial pattern. We present numerical simulations for two cases of interest, namely exponential domain growth and chemically controlled growth. Our analysis reveals that for domains undergoing rapid exponential growth dilution effects associated with domain growth influence both the spatial patterns that emerge and the concentration of chemicals present in the domain. In the latter case, there is complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of these interactions is revealed by a weakly nonlinear analysis of the full system. This yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude, the particular functional forms used to model the local tissue growth rate and the kinetics of the two diffusible species dictating the resulting behaviour.

Morphological Diagram of Bacterial Colony Pattern Simulated Using a Revised Mimura–Sakaguchi–Matsushita Model

In this Note we report the attempt of improving the Mimura–Sakaguchi–Matsushita (MSM) model of bacterial colony formation by revising the nonlinear diffusion term in the reaction diffusion equation. Using this revised MSM model, all five typical colony patterns are reproduced and the experimental morphological diagram of colony patterns are completely simulated. During the last few decades, much attention has been paid to self-assembly pattern formation in various fields. Among them, pattern formation in biological systems is the most sophisticated and interesting. Due to the complexity of biological systems, recoganizing the mechanism of pattern formation in biological systems is difficult. Therefore, investigation of simple biological objects, which is dominated by purely physical conditions, is a good choice. Here we focus on bacterial colony formation, which is one of the simplest biological objects. It is known that some kinds of bacteria such as Bacillus subtilis (B. subtilis) exhibit various colony patterns on an agar medium by changing the incubation conditions, i.e., the concentration of agar Ca and that of nutrient Cn in the medium. Cn determines the rate of division of active bacteria, and Ca controls the motivebility of bacteria. Ohgiwari et al. completed an experimental morphological diagram (see Fig. 1). It can be seen from region A (high Ca and low Cn) of Fig. 1 that the colony pattern exhibits tip-splitting growth with diffusion-limitted aggregation (DLA)-like ramified structures. Keeping a high Ca and increasing Cn (region B),the pattern becomes round with Eden-like rough interface. If Cn is high and Ca is low (region D), the pattern looks macroscopically disklike. The colony growth in region D is consistent with the solution of the two-dimentional Fisher equation. In region E, which is between regions A and D, the pattern shows highly branched structures similar to the dense-branching morphology (DBM). When Cn is high and Ca takes an appropriate intermediate value (region C), colony exhibits concentricring patterns which are produced alternately in the active state and inactive state of the bacteria. In order to explain these observations, various reaction diffusion models have been proposed. The MSM model is one of them, which is based on the assumption that there exist two types of bacteria; active ones that move, grow and divide, and inactive ones that do nothing at all. Using the MSM model with linear diffusion patterns A, C, D and E are reproduced. Using MSM model with nonlinear diffusion, pattern B is also generated. In the MSM model, the relation among the population density of active bacteria u, the density of the inactive bacteria w, and the nutrient concentration v is expressed by a set of equations as

Cell behavior and cell–cell communication during fruiting body morphogenesis in Myxococcus xanthus

Formation of spatial patterns of cells from a mass of initially identical cells is a recurring theme in developmental biology. The dynamics that direct pattern formation in biological systems often involve morphogenetic cell movements. An example is fruiting body formation in the gliding bacterium Myxococcus xanthus in which an unstructured population of identical cells rearranges into an asymmetric, stable pattern of multicellular fruiting bodies in response to starvation. Fruiting body formation depends on changes in organized cell movements from swarming to aggregation. The aggregation process is induced and orchestrated by the cell-surface associated 17 kDa C-signal protein. C-signal transmission depends on direct contact between cells. Evidence suggests that C-signal transmission is geometrically constrained to cell ends and that productive C-signal transmission only occurs when cells engage in end-to-end contacts. Here, we review recent progress in the understanding of the pattern formation process that leads to fruiting body formation. Gliding motility in M. xanthus involves two polarly localized gliding machines, the S-machine depends on type IV pili and the A-machine seems to involve a slime extrusion mechanism. Using time-lapse video microscopy the gliding motility parameters controlled by the C-signal have been identified. The C-signal induces cells to move with increased gliding speeds, in longer gliding intervals and with decreased stop and reversal frequencies. The combined effect of the C-signal dependent changes in gliding motility behaviour is an increase in the net-distance travelled by a cell per minute. The identification of the motility parameters controlled by the C-signal in combination with the contact-dependent C-signal transmission mechanism have allowed the generation of a qualitative model for C-signal induced aggregation. In this model, the directive properties of the C-signal are a direct consequence of the contact-dependent signal-transmission mechanism, which is a local event involving direct contact between cells that results in a global organization of cells. This pattern formation process does not depend on a diffusible substance. Rather it depends on a cell-surface associated signal to direct the cells appropriately.

Pattern formation in reaction diffusion systems: a Galerkin model

Reaction diffusion systems are extremely useful for studying pattern formation in biological systems. We carry out a Lorenz like few mode truncation of a reaction diffusion system and show that it not only gives the same qualitative behaviour as the more complicated systems but also indicates of the existence of a Hopf-bifurcation in the turing region.

Robust symmetric patterns in the Faraday experiment

Symmetric patterns were recently obtained in the famous Faraday experiment by using a special liquid. A model is developed to explain the robustness of the observed patterns. A dissipation mechanism is introduced phenomenologically and shown to be responsible for the stabilization of patterns with several symmetries. The resulting system of nonlinear equations resembles the well-known Turing equations used to study pattern formation in biological systems.

A mathematical model for pattern formation in biological systems

A system of reaction-diffusion equations with applications in biological problems is presented and studied theoretically and numerically. Solutions demonstrating the following types of behaviour have been obtained: (i) time constant, spatially inhomogeneous, (ii) time periodic, spatially homogeneous, and (iii) time periodic, travelling wave-like. Some applications in embryological problems are discussed.