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.     

Comparison of reaction-diffusion simulations with experiment in self-organised microtubule solutions

This article deals with the physical chemical processes underlying biological self-organization by which an initially homogenous solution of reacting chemicals spontaneously self-organizes so as to give rise to a preparation of macroscopic order and form. Theoreticians have predicted that self-organization can arise from a coupling of reactive processes with molecular diffusion. In addition, the presence or absence of an external field, such as gravity, at a critical moment early in the self-organizing process may determine the morphology that subsequently develops. We have found that the formation in vitro of microtubules, a major element of the cellular skeleton, show this type of behaviour. The microtubule preparations spontaneously self-organise by way of reaction and diffusion, and the morphology of the state that forms depends on the presence of gravity at a critical moment early in the process. We have developed a numerical reaction-diffusion scheme, based on the chemical dynamics of a population of microtubules, which simulates the experimental self-organisation. In this article we outline the main features of these simulations and discuss the manner by which a permanent dialogue with experiment has helped develop a microscopic understanding of the collective behaviour.     

Ground-based methods reproduce space-flight experiments and show that weak vibrations trigger microtubule self-organisation

The effect of weightlessness on physical and biological systems is frequently studied by experiments in space. However, on the ground, gravity effects may also be strongly attenuated using methods such as magnetic levitation and clinorotation. Under suitable conditions, in vitro preparations of microtubules, a major element of the cytoskeleton, self-organise by a process of reaction–diffusion: self-organisation is triggered by gravity and samples prepared in space do not self-organise. Here, we report experiments carried out with ground-based methods of clinorotation and magnetic levitation. The behaviour observed closely resembles that of the space-flight experiment and suggests that many space experiments could be carried out equally well on the ground. Using clinorotation, we find that weak vibrations also trigger microtubule self-organisation and have an effect similar to gravity. Thus, in some in vitro biological systems, vibrations are a countermeasure to weightlessness.     

Microtubule self-organisation by reaction-diffusion processes causes collective transport and organisation of cellular particles

Background  

The transport of intra-cellular particles by microtubules is a major biological function. Under appropriate in vitro conditions, microtubule preparations behave as a 'complex' system and show 'emergent' phenomena. In particular, they form dissipative structures that self-organise over macroscopic distances by a combination of reaction and diffusion.     

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.     

Reaction-diffusion microtubule concentration patterns occur during biological morphogenesis.

Reaction-diffusion processes can lead to a macroscopic concentration pattern from an initially homogeneous solution, and thus provide a physical-chemical mechanism for biological pattern formation and morphogenesis. The central prediction of reaction-diffusion theory is that the patterns contain periodic concentration variations in some of the reactives. Microtubules assembled in vitro spontaneously self-organise and form stationary striped macroscopic structures. In agreement with reaction-diffusion theory. Here we show, in agreement with reaction-diffusion theory, that these preparations contain substantial microtubule concentration variations. Similar striped microtubule patterns arise during Drosophila embryogenesis. A characteristic of these patterns is their dependence on sample dimensions. In Drosophila eggs shortened by ligation, we found that the microtubule pattern varied with egg fragment length in the same way as the in vitro microtubule pattern varied with sample length, and as expected from theory. This is evidence that reaction-diffusion structures occur during Drosophila morphogenesis.     

Microtubule self-organisation by reaction-diffusion processes in miniature cell-sized containers and phospholipid vesicles

Under appropriate conditions, in vitro microtubule preparations self-organise over macroscopic distances by a process of reaction and diffusion. To investigate whether such self-organisation can also occur in objects as small as a cell or an embryo we carried out experiments in miniature containers of cellular dimension. When assembled under self-organising conditions in wells of 120–500 μm, microtubules developed organised structures. Self-organisation is strongly affected by shape, being highly favoured by elongated forms. In wells of more complex shape, geometrical factors may either oppose or strengthen one another and so inhibit or reinforce self-organisation. Microtubules were also assembled within phospholipid vesicles of 2–5 μm diameter. Under self-organising conditions, we observed large shape changes from spheroids to long tubes (50–100 μm) and intertwined coils. We conclude that self-organisation of microtubules by reaction–diffusion processes can occur in containers of cellular dimensions and is capable of strongly deforming the cellular membrane.     

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.     

Application of Quasi-Steady-State Methods to Nonlinear Models of Intracellular Transport by Molecular Motors

Molecular motors such as kinesin and dynein are responsible for transporting material along microtubule networks in cells. In many contexts, motor dynamics can be modelled by a system of reaction–advection–diffusion partial differential equations (PDEs). Recently, quasi-steady-state (QSS) methods have been applied to models with linear reactions to approximate the behaviour of the full PDE system. Here, we extend this QSS reduction methodology to certain nonlinear reaction models. The QSS method relies on the assumption that the nonlinear binding and unbinding interactions of the cellular motors occur on a faster timescale than the spatial diffusion and advection processes. The full system dynamics are shown to be well approximated by the dynamics on the slow manifold. The slow manifold is parametrized by a single scalar quantity that satisfies a scalar nonlinear PDE, called the QSS PDE. We apply the QSS method to several specific nonlinear models for the binding and unbinding of molecular motors, and we use the resulting approximations to draw conclusions regarding the parameter dependence of the spatial distribution of motors for these models.     

Diffusion approximation of the stochastic process of microtubule assembly

Microtubules are protein polymers that guide intracellular motility. Stochastic switching of a microtubule between states of elongation, shortening, and pause is described in detail by the dynamic instability (DI) model. Recently we have described the dynamics of microtubules phenomenologically as generalized diffusion of their ends. Genesis of the diffusion dynamics and accuracy of diffusion model are studied in this work. It is shown that wandering of the end of a microtubule undergoing DI asymptotically approaches the Wiener diffusion process. Accuracy of the diffusion approximation is evaluated by comparing its predictions with results of simulation of DI. Stationary distributions of microtubule length and lifetime that are predicted by both models differ qualitatively between two cell types considered. However, predictions of the diffusion model are in each case practically identical to predictions of the DI model being also consistent with experimental data. The peculiar stochastic process of microtubule assembly thus converges at cell scale to a kind of widespread-in-nature diffusion process. This result is considered an example of qualitative change in dynamical properties in transition from the molecular to cellular level of biological organization. Additionally, it suggests employment of diffusion process theory in studying functions of microtubules in the cell.     

Collision induced spatial organization of microtubules

The dynamic behavior of microtubules in solution can be strongly modified by interactions with walls or other structures. We examine here a microtubule growth model where the increase in size of the plus-end is perturbed by collisions with other microtubules. We show that such a simple mechanism of constrained growth can induce ordered structures and patterns from an initially isotropic and homogeneous suspension. We find that microtubules self-organize locally in randomly oriented domains that grow and compete with each other. A weak orientation bias, similar to the one induced by gravity or cellular boundaries is enough to influence the domain growth direction, eventually leading to a macroscopic sample orientation.     

Bistability and oscillations in chemical reaction networks

Bifurcation theory is one of the most widely used approaches for analysis of dynamical behaviour of chemical and biochemical reaction networks. Some of the interesting qualitative behaviour that are analyzed are oscillations and bistability (a situation where a system has at least two coexisting stable equilibria). Both phenomena have been identified as central features of many biological and biochemical systems. This paper, using the theory of stoichiometric network analysis (SNA) and notions from algebraic geometry, presents sufficient conditions for a reaction network to display bifurcations associated with these phenomena. The advantage of these conditions is that they impose fewer algebraic conditions on model parameters than conditions associated with standard bifurcation theorems. To derive the new conditions, a coordinate transformation will be made that will guarantee the existence of branches of positive equilibria in the system. This is particularly useful in mathematical biology, where only positive variable values are considered to be meaningful. The first part of the paper will be an extended introduction to SNA and algebraic geometry-related methods which are used in the coordinate transformation and set up of the theorems. In the second part of the paper we will focus on the derivation of bifurcation conditions using SNA and algebraic geometry. Conditions will be derived for three bifurcations: the saddle-node bifurcation, a simple branching point, both linked to bistability, and a simple Hopf bifurcation. The latter is linked to oscillatory behaviour. The conditions derived are sufficient and they extend earlier results from stoichiometric network analysis as can be found in (Aguda and Clarke in J Chem Phys 87:3461–3470, 1987; Clarke and Jiang in J Chem Phys 99:4464–4476, 1993; Gatermann et al. in J Symb Comput 40:1361–1382, 2005). In these papers some necessary conditions for two of these bifurcations were given. A set of examples will illustrate that algebraic conditions arising from given sufficient bifurcation conditions are not more difficult to interpret nor harder to calculate than those arising from necessary bifurcation conditions. Hence an increasing amount of information is gained at no extra computational cost. The theory can also be used in a second step for a systematic bifurcation analysis of larger reaction networks. We have added a dedication of the paper to K. Gatermann.     

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.     

The assembly of microtubule protein in vitro. The kinetic role in microtubule elongation of oligomeric fragments containing microtubule-associated proteins.

The kinetics of assembly were studied for bovine and pig microtubule protein in vitro over a range of conditions of pH, temperature, nucleotide and protein concentration. The kinetics are in general biphasic with two major processes of similar amplitude but separated in rate by one order of magnitude. Rates and amplitudes are complex functions of solution conditions. The rates of the fast phase and the slow phase attain limiting values as a function of increasing protein concentration, and are more stringently limited at pH 6.5 than pH 6.95. Such behaviour indicates that mechanisms other than the condensation polymerization of tubulin dimer become rate-limiting at higher protein concentration. The constancy of the wavelength-dependence of light-scattering and ultrastructural criteria indicate that microtubules of normal morphology are formed in both phases of the assembly process. Electrophoretic analysis of assembling microtubule protein shows that MAP- (microtubule-associated-protein-)rich microtubules are formed during the fast phase. The rate of dissociation of oligomeric species on dilution of microtubule protein closely parallels the fast-phase rate in magnitude and temperature-dependence. We propose that the rate of this process constitutes an upper limit to the rate of the fast phase of assembly. The kinetics of redistribution of MAPs from MAP-rich microtubules may be a factor limiting the slow-phase rate. A working model is derived for the self-assembly of microtubule protein incorporating the dissociation and redistribution mechanisms that impose upper limits to the rates of assembly attainable by bimolecular addition reactions. Key roles are assigned to MAP-containing fragments in both phases of microtubule elongation. Variations in kinetic behaviour with solution conditions are inferred to derive from the nature and properties of fragments formed from oligomeric species after the rapid temperature jump. The model accounts for the limiting rate behaviour and indicates experimental criteria to be applied in evaluating the relative contributions of alternative pathways.     

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     

Nonequilibrium self-assembly of linear fibers: microscopic treatment of growth, decay, catastrophe and rescue

Many of the large structures of cells are constructed from fibers. These fibers self-assemble from individual proteins in a far-from-equilibrium fashion. Nonequilibrium self-assembly results in a highly dynamic process at the subcellular level that can be regulated and tuned to carry out many of the biological functions of the cell: growth, division and locomotion. We construct and analyze a nonequilibrium model of the dynamic end of a biological fiber that possesses site-resolved resolution. We solve for the steady states of this nonequilibrium system using a variational method. The results are compared to exact numerical solutions for systems with modest size. Using an effective reaction coordinate, we construct an effective potential from the steady-state distribution. The stochastic transitions of the system can be analyzed in this representation. We then apply this method to model microtubule systems. Predictions for macroscopic catastrophe, rescue and dynamic instability in the steady states are analyzed. We find that the length of the cap of the microtubule is small. The relations between the catastrophe/rescue rate and the growth rate are also discussed.     

Polyamine Sharing between Tubulin Dimers Favours Microtubule Nucleation and Elongation via Facilitated Diffusion

We suggest for the first time that the action of multivalent cations on microtubule dynamics can result from facilitated diffusion of GTP-tubulin to the microtubule ends. Facilitated diffusion can promote microtubule assembly, because, upon encountering a growing nucleus or the microtubule wall, random GTP-tubulin sliding on their surfaces will increase the probability of association to the target sites (nucleation sites or MT ends). This is an original explanation for understanding the apparent discrepancy between the high rate of microtubule elongation and the low rate of tubulin association at the microtubule ends in the viscous cytoplasm. The mechanism of facilitated diffusion requires an attraction force between two tubulins, which can result from the sharing of multivalent counterions. Natural polyamines (putrescine, spermidine, and spermine) are present in all living cells and are potent agents to trigger tubulin self-attraction. By using an analytical model, we analyze the implication of facilitated diffusion mediated by polyamines on nucleation and elongation of microtubules. In vitro experiments using pure tubulin indicate that the promotion of microtubule assembly by polyamines is typical of facilitated diffusion. The results presented here show that polyamines can be of particular importance for the regulation of the microtubule network in vivo and provide the basis for further investigations into the effects of facilitated diffusion on cytoskeleton dynamics.     

Cadherin-mediated regulation of microtubule dynamics

Epithelial polarization and neuronal outgrowth require the assembly of microtubule arrays that are not associated with centrosomes. As these processes generally involve contact interactions mediated by cadherins, we investigated the potential role of cadherin signalling in the stabilization of non-centrosomal microtubules. Here we show that expression of cadherins in centrosome-free cytoplasts increases levels of microtubule polymer and changes the behaviour of microtubules from treadmilling to dynamic instability. This effect is not a result of cadherin expression per se but depends on the formation of cell-cell contacts. The effect of cell-cell contacts is mimicked by application of beads coated with stimulatory anti-cadherin antibody and is suppressed by overexpression of the cytoplasmic cadherin tail. We therefore propose that cadherins initiate a signalling pathway that alters microtubule organization by stabilizing microtubule ends.     

Dissipative particle dynamics simulation of the relaxation behaviour of a triblock copolymer supramolecular network

Since block copolymers self-assemble into various nanostructures and these are widely used in soft materials such as cosmetics and paints, these continue to be the subjects of fundamental studies that progress new technologies. ABA triblock copolymers self-organise into supramolecular networks in which crosslinked structures change upon heating and other external stimulation. Supramolecular networks that consist of ABA triblock copolymers have three components: bridges, loops and dangles. These supramolecular networks have structural regions (clusters) in which end blocks of polymer chains are aggregated and connected by the internal blocks of the polymer chains. Despite of the importance of relaxation behaviour during the measurement and control of polymer materials, the molecular-level behaviour of these systems has not been addressed. We observed pull-out phenomenon that the ends of these polymers detach from clusters, and estimated characteristic detachment times by counting the number of detachments and compared it with the time of longest relaxation in that system.     

Stability and Hopf bifurcation for a prey-predator model with prey-stage structure and diffusion

In this paper, we first propose a prey-predator model with prey-stage structure and diffusion. Then we discuss the following three problems: (1) stability of non-negative constant steady states for the reduced ODE system and the corresponding reaction diffusion system with homogeneous Neumann boundary conditions; (2) Hopf bifurcation for the ODE system; (3) Hopf bifurcation created by diffusion.