Numerical simulations of microtubule self-organisation by reaction and diffusion

This article addresses the physical chemical processes underlying biological self-organisation by which a homogenous solution of reacting chemicals spontaneously self-organises. Theoreticians have predicted that self-organisation can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence of an external field, such as gravity, at a critical moment early in the process may determine the morphology that subsequently develops. The formation, in-vitro, of microtubules, a constituent of the cellular skeleton, shows this type of behaviour. The preparations spontaneously self-organise by reaction-diffusion and the morphology that develops depends upon the presence of gravity at a critical bifurcation time early in the process. Here, we present numerical simulations of a population of microtubules that reproduce this behaviour. Microtubules can grow from one end whilst shrinking from the other. The shrinking end leaves behind a chemical trail of high tubulin concentration. Neighbouring microtubules preferentially grow into these regions, whilst avoiding regions of low tubulin concentration. The chemical trails produced by individual microtubules thus activate and inhibit the formation of neighbouring microtubules and this progressively leads to self-organisation. Gravity acts by way of its directional interaction with the macroscopic density fluctuations present in the solution arising from microtubule disassembly.     

Gravity dependence of microtubule preparations

The mechanisms by which biological processes are effected by gravity are not understood. Theoreticians have proposed that gravitational effects could come about from the bifurcation properties of certain types of non-linear chemical reactions that self-organise by reaction and diffusion. We have found that in-vitro preparations of microtubules, an important element of the cellular skeleton, show this type of behaviour. They self-organise by reaction and diffusion and the morphology that arises depend upon the presence of gravity, at a critical moment or bifurcation time, early in the process. At a molecular level this behaviour results from an interaction of gravity with macroscopic concentration and density fluctuations created by microtubule contraction and elongation. Numerical simulations predict macroscopic self-organisation in qualitative agreement with experiment. It is plausible that microtubule organisation by these processes occurs in-vivo.     

Historical and conceptual background of self-organization by reactive processes

Order, form, pattern and organization are properties central to much living matter. The physicochemical processes by which an initially homogeneous solution of reacting chemicals or biochemicals might self-organize is hence a question of fundamental biological importance. In most cases, solutions of reacting chemicals in a test-tube do not self-organize. Because of this, for many years, it was not thought possible that reactive processes could result in self-organization. However, progressively over the last hundred years, it has been shown that this is not always the case, and under certain conditions, the combination of reaction with molecular diffusion can lead to macroscopic self-organization. In 'complex' systems comprised of populations of strongly coupled elements, new 'emergent' properties, such as self-organization, arise by way of the dynamics of the system. Self-organizing reaction-diffusion systems form a specific type of complex system. Here, I will give a personal overview of the conceptual and historical background to this approach with an emphasis on biological self-organization.     

Microtubules viewed as molecular ant colonies

Populations of ants and other social insects self-organize and develop 'emergent' properties through stigmergy in which individual ants communicate with one another via chemical trails of pheromones that attract or repulse other ants. In this way, sophisticated properties and functions develop. Under appropriate conditions, in vitro microtubule preparations, initially comprised of only tubulin and GTP, behave in a similar manner. They self-organize and develop other higher-level emergent phenomena by a process where individual microtubules are coupled together by the chemical trails they produce by their own reactive growing and shrinking. This behaviour is described and compared with the behaviour of ant colonies. Viewing microtubules as populations of molecular ants may provide new insights as to how the cytoskeleton may spontaneously develop high-level functions. It is plausible that such processes occur during the early stages of embryogenesis and in cells.     

Self-Organization in Living Cells: Networks of Protein Machines and Nonequilibrium Soft Matter

Microscopic self-organization phenomena inside a living cell should not represent merely a reduced copy of self-organization in macroscopic systems. A cell is populated by active protein machines that communicate via small molecules diffusing through the cytoplasm. Mutual synchronization of machine cycles can spontaneously develop in such networks – an effect which is similar to coherent laser generation. On the other hand, an interplay between reactions, diffusion and phase transitions in biological soft matter may lead to the formation of stationary or traveling nonequilibrium nanoscale structures.     

Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species' densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.     

Nematic Ordering Pattern Formation in the Process of Self-Organization of Microtubules in a Gravitational Field

Papaseit et al. (Proc. Natl. Acad. Sci. U.S.A. 97, 8364, 2000) showed the decisive role of gravity in the formation of patterns by assemblies of microtubules in vitro. By virtue of a functional scaling, the free energy for MT systems in a gravitational field was constructed. The influence of the gravitational field on MT’s self-organization process, that can lead to the isotropic to nematic phase transition, is the focus of this paper. A coupling of a concentration gradient with orientational order characteristic of nematic ordering pattern formation is the new feature emerging in the presence of gravity. The concentration range corresponding to a phase coexistence region increases with increasing g or MT concentration. Gravity facilitates the isotropic to nematic phase transition leading to a significantly broader transition region. The phase transition represents the interplay between the growth in the isotropic phase and the precipitation into the nematic phase. We also present and discuss the numerical results obtained for local MT concentration change with the height of the vessel, order parameter and phase transition properties. An erratum to this article can be found at     

Homogenization techniques for population dynamics in strongly heterogeneous landscapes

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.     

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns.. Received: 10 January 1996/Revised version: 3 July 1996     

Pigmentation pattern formation in butterflies: experiments and models

Butterfly pigmentation patterns are one of the most spectacular and vivid examples of pattern formation in biology. They have attracted much attention from experimentalists and theoreticians, who have tried to understand the underlying genetic, chemical and physical processes that lead to patterning. In this paper, we present a brief review of this field by first considering the generation of the localised, eyespot, patterns and then the formation of more globally controlled patterns. We present some new results applied to pattern formation on the wing of the mimetic butterfly Papilio dardanus.     

Spatio-temporal pattern formation in Rosenzweig–MacArthur model: Effect of nonlocal interactions

Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.     

Brief exposure to high magnetic fields determines microtubule self-organisation by reaction-diffusion processes

A frequent feature of microtubule organisation in living systems is that it can be triggered by a variety of biochemical or physical factors. Under appropriate conditions, in vitro microtubule preparations self-organise by a reaction-diffusion process in which self-organisation depends upon, and can be triggered by, weak external physical factors such as gravity. Here, we show that self-organisation is also strongly dependent upon the presence of a high magnetic field, for a brief critical period early in the process, and before any self-organised pattern is visible. These results provide evidence that external physical factors trigger self-organisation by way of an orientational bias that breaks the symmetry of the reaction-diffusion process. As microtubule organisation is central to many cell functions, this behaviour provides a mechanism by which strong magnetic fields can intervene in biological processes.     

整体大于部分之和: 生态自组织斑图及其涌现属性

近30年来, 自组织理论已经发展成为解释生态系统呈现规则空间格局的有效理论。伴随着生态系统自发有序空间格局的生成, 自组织过程产生一系列的涌现属性, 这些特征对生态系统功能至关重要。在此, 我们将介绍这一正蓬勃发展的研究领域的主要理论进展。首先, 叙述了自组织这一概念的发展历程与定义, 详细阐述了自组织理论的两个经典理论框架: 图灵原理与相分离原理。然后, 根据几个典型的生态自组织研究案例, 描述了图灵原理与相分离原理在不同生态系统中的具体数学模型表达形式。接着, 分别阐述了图灵原理的涌现属性对生态系统功能以及相分离原理的涌现属性对细胞功能的作用。最后, 从多尺度自组织斑图、瞬态斑图和生物个体行为自组织3个方面对未来生态自组织理论发展方向进行了探讨。自组织研究在生态学与生物学研究中方兴未艾, 希望更多的学者在未来关注与参与该领域的发展。     

Microtubule Dynamics may Embody a Stationary Bipolarity Forming Mechanism Related to the Prokaryotic Division Site Mechanism (Pole-to-Pole Oscillations)

Cell division mechanisms in eukaryotes and prokaryotes have until recently been seen as being widely different. However, pole-to-pole oscillations of proteins like MinE in prokaryotes are now known to determine the division plane. These protein waves arise through spontaneous pattern forming reaction—diffusion mechanisms, based on cooperative binding of the proteins to a quasistationary matrix (like the cell membrane or DNA). Rather than waves, stationary bipolar pattern formation may arise as well. Some of the involved proteins have eukaryotic homologs (e.g. FtsZ and tubulin), pointing to a possible ancient shared mechanism. Tubulin polymerizes to microtubules in the spindle. Mitotic microtubules are in a highly dynamical state, frequently undergoing rapid shortening (catastrophe), and fragments formed from the microtubule ends are inferred to enhance the destabilization. Here, we show that cooperative binding of such fragments to microtubules may set up a similar pattern forming mechanism as seen in prokaryotes. The result is a spontaneously formed, well controllable, bipolar state of microtubule dynamics in the cell, which may contribute to defining the bipolar spindle.     

Effect of weightlessness on colloidal particle transport and segregation in self-organising microtubule preparations

Weightlessness is known to effect cellular functions by as yet undetermined processes. Many experiments indicate a role of the cytoskeleton and microtubules. Under appropriate conditions in vitro microtubule preparations behave as a complex system that self-organises by a combination of reaction and diffusion. This process also results in the collective transport and organisation of any colloidal particles present. In large centimetre-sized samples, self-organisation does not occur when samples are exposed to a brief early period of weightlessness. Here, we report both space-flight and ground-based (clinorotation) experiments on the effect of weightlessness on the transport and segregation of colloidal particles and chromosomes. In centimetre-sized containers, both methods show that a brief initial period of weightlessness strongly inhibits particle transport. In miniature cell-sized containers under normal gravity conditions, the particle transport that self-organisation causes results in their accumulation into segregated regions of high and low particle density. The gravity dependence of this behaviour is strongly shape dependent. In square wells, neither self-organisation nor particle transport and segregation occur under conditions of weightlessness. On the contrary, in rectangular canals, both phenomena are largely unaffected by weightlessness. These observations suggest, depending on factors such as cell and embryo shape, that major biological functions associated with microtubule driven particle transport and organisation might be strongly perturbed by weightlessness.     

Avoiding toxic prey may promote harmful algal blooms

Blooms of freshwater cyanobacteria are a worldwide spread environmental issue. Despite toxin producing planktonic species are generally expected to be poor competitors for resources, dense blooms of toxic cyanobacteria, such as Microcystis, do often occur in nature. Experimental results suggest that the formation of such blooms is promoted by the predatory activity of zooplankton. In fact, such predator grazes on both the nontoxic and toxic species despite the toxicity of the latter actually inhibits its growth. We model this phenomenon through a Lotka–Volterra reaction–diffusion system. Our goal is to investigate the coupled role of toxicity and zooplankton's predation in the persistence of the toxic prey and to study the mechanisms behind the formation of spatially local toxic blooms. It is known that the classical Lotka-Volterra system consisting of one prey and one predator never exhibits pattern formation. In this paper, we show that the introduction of a toxic prey may destabilize the spatially homogeneous coexistence and trigger spatial pattern formation. We also show that local blooms more likely occur when predators avoid the toxic prey and when the strength of the toxicity is of an intermediate level.     

Secondary cell wall patterning—connecting the dots,pits and helices

All plant cells are encased in primary cell walls that determine plant morphology, but also protect the cells against the environment. Certain cells also produce a secondary wall that supports mechanically demanding processes, such as maintaining plant body stature and water transport inside plants. Both these walls are primarily composed of polysaccharides that are arranged in certain patterns to support cell functions. A key requisite for patterned cell walls is the arrangement of cortical microtubules that may direct the delivery of wall polymers and/or cell wall producing enzymes to certain plasma membrane locations. Microtubules also steer the synthesis of cellulose—the load-bearing structure in cell walls—at the plasma membrane. The organization and behaviour of the microtubule array are thus of fundamental importance to cell wall patterns. These aspects are controlled by the coordinated effort of small GTPases that probably coordinate a Turing''s reaction–diffusion mechanism to drive microtubule patterns. Here, we give an overview on how wall patterns form in the water-transporting xylem vessels of plants. We discuss systems that have been used to dissect mechanisms that underpin the xylem wall patterns, emphasizing the VND6 and VND7 inducible systems, and outline challenges that lay ahead in this field.     

Gravitational Symmetry Breaking Leads to a Polar Liquid Crystal Phase of Microtubules In Vitro

Recent space-flight experiments performed by Tabony's team provided further evidence that a microgravity environment strongly affects the spatio-temporal organization of microtubule assemblies. Characteristic time and length scales were found that govern the organization of oriented bundles under Earth's gravitational field (GF). No such organization has been observed in a microgravity environment. This paper discusses physical mechanisms resulting in pattern formation under gravity and its disappearance in microgravity. The subtle interplay between chemical kinetics, diffusion, gravitational drift, thermal fluctuations, electrostatic interactions and liquid crystalline characteristics provides a plausible scenario.     

Gravitational effects on biological systems

The possible effects of the earth's gravitational field on biological systems have been studied from a quantitative point of view, focusing the attention to a very simple system, a solution containing proteins, which biochemists might use in experiments. Gravity has been compared with other forces which are known to influence protein activity, including thermic agitation, weak electrostatic interactions, Van der Waals forces and viscous dissipation. Comparisons have been described in terms of the energy of the interaction per mole, referring to some physically simple cases and substances of biological interest. From this study it is evident that the earth's gravitational energy should be taken into account when considering the chemical behaviour of solutions containing substances that have high molecular weight, such as a typical protein, since its value is comparable to other weak interactions. Moreover, since solutions represent the basis of much more complex biological processes taking place inside cells, the influence of gravity should extend also to cellular biochemical behaviour, especially in presence of altered gravity, both in microgravity (such as on satellites orbiting around the earth), and in macrogravity (such as in a centrifugating biological system).