Spatial patterns produced by a reaction-diffusion system in primary hair follicles

This paper is the third in a series examining the role of a reaction-diffusion (RD) system as the principal mechanism providing spatial information for cell differentiation during hair follicle initiation and development and hair fibre formation. A theoretical mechanism is described by which the RD system supplies positional information during hair follicle development. Solutions of the RD system within the primordial follicle are described as well as the sequence of spatial patterns provides the follicle/epidermis boundary conditions required to account for the density and grouping of follicles during initiation. At the same time the spatial patterns are also shown to be capable of providing the positional information which determines various geometrical aspects of follicle development; in particular the development of follicles at an angle to the skin surface and the initiation and location of sweat glands and sebaceous glands on the follicle.     

A non-linear analysis for spatial structure in a reaction-diffusion model

A non-linear stability analysis using a multi-scale perturbation procedure is carried out on the practical Thomas reaction-diffusion mechanism which exhibits bifurcation to non-uniform states. The analytical results compare favourably with the numerical solutions. The sequential patterns generated by this model by variations in a parameter related to the reaction-diffusion domain indicate its capacity to represent certain key morphogenetic features required in a recent model by Kauffman for pattern formation in theDrosophila embryo.     

Dissipative structures associated to certain reaction-diffusion equations in mathematical ecology

Sufficient conditions are determined for the existence of stable spatially heterogeneous solutions of the Lotka-Volterra many species equations extended to include spatial diffusion. Some properties of such solutions are obtained. A general definition is proposed for the concept of a dissipative structure associated with a given reaction-diffusion equation. Finally, an approximate solution is presented for two interacting species in one spatial dimension, although the question of stability for this example is left open. Supported in part by NSERC A-7667 to P. L. Antonelli and by a University of Alberta President's Fund grant to J. R. Royce.     

Spatial distribution and dose-response relationship for different operation modes in a reaction-diffusion model of the MAPK cascade

The mitogen-activated protein kinase (MAPK) cascade plays a critical role in the control of cell growth. Deregulation of this pathway contributes to the development of many cancers. To better understand its signal transduction, we constructed a reaction-diffusion model for the MAPK pathway. We modeled the three layers of phosphorylation-dephosphorylation reactions and diffusion processes from the cell membrane to the nucleus. Based on different types of feedback in the MAPK cascade, four operation modes are introduced. For each of the four modes, spatial distributions and dose-response curves of active kinases (i.e. ppMAPK) are explored by numerical simulation. The effects of propagation length, diffusion coefficient and feedback strength on the pathway dynamics are investigated. We found that intrinsic bistability in the MAPK cascade can generate a traveling wave of ppMAPK with constant amplitude when the propagation length is short. ppMAPK in this mode of intrinsic bistability decays more slowly than it does in all other modes as the propagation length increases. Moreover, we examined the global and local responses to Ras-GTP of these four modes, and demonstrated how the shapes of these dose-response curves change as the propagation length increases. Also, we found that larger diffusion constant gives a higher response level on the zero-order regime and makes the ppMAPK profiles flatter under strong Ras-GTP stimulus. Furthermore, we observed that spatial responses of ppMAPK are more sensitive to negative feedback than to positive feedback in the broader signal range. Finally, we showed how oscillatory signals pass through the kinase cascade, and found that high frequency signals are damped faster than low frequency ones.     

Spatial regulation of Fus3 MAP kinase activity through a reaction-diffusion mechanism in yeast pheromone signalling

Signal transduction through mitogen-activated protein kinase (MAPK) cascades is thought to occur through the assembly of macromolecular complexes. We quantified the abundance of complexes in the cytoplasm among the MAPKs Ste11, Ste7, Fus3 and the scaffold protein Ste5 in yeast pheromone signalling using fluorescence cross-correlation spectroscopy (FCCS). Significant complex concentrations were observed that remained unchanged on pheromone stimulation, demonstrating that global changes in complex abundances do not contribute to the transmission of signal through the cytoplasm. On the other hand, investigation of the distribution of active Fus3 (Fus3(PP)) across the cytoplasm using fluorescence lifetime imaging microscopy (FLIM) revealed a gradient of Fus3(PP) activity emanating from the tip of the mating projection. Spatial partitioning of Fus3 activating kinases to this site and deactivating phosphatases in the cytoplasm maintain this Fus3(PP)-activity distribution. Propagation of signalling from the shmoo is, therefore, spatially constrained by a gradient-generating reaction-diffusion mechanism.     

Distance of the reduced Brusselator from equilibrium

The new concept of a nonequilibrium parameter P is applied to a reduced Brusselator, considered as a kinetic model for cellular processes. The reduction corresponds to omitting a monomolecular reaction. The deviation from equilibrium is due to a fixed nonequilibrium value of an extracellular concentration B, responsible for energizing and determining the steady-state value of the overall chemical affinity ?. The value of ? is insensitive to different steady states possible for a given set of rate constants. In contrast, the parameter P is state dependent. In particular, it may jump together with state variables. Two limiting cases of high ? are investigated, B→ 0 and B→ 1. In the first case P grows monotonically with ?. In the second case there is always a steady state solution with P→ 0. The physical interpretation of this effect of “equilibrium far from equilibrium” reveals the real predictive power of the parameter P. Relaxation regimes are investigated for a doubly reduced Brusselator. Both P and ? are in general time dependent and have jumps in their time derivatives. The canonical form of P is compared with the noncanonical one in the context of robustness of the new concept with respect to incomplete information about the system studied. These forms of P are different in relaxation to a nonequilibrium state and coincide in relaxation to an equilibrium state.     

Steady state bifurcation analysis of reaction-diffusion equations-A critique

This note supplies missing details and corrects some mathematical errors in the recent significant paper by Auchmuty and Nicolis (1975).     

Osmoelastic coupling in biological structures: a comprehensive thermodynamic analysis of the osmotic response of phospholipid vesicles and a reevaluation of the "dehydration force" theory

A comprehensive thermodynamic analysis of the osmotic response of phospholipid vesicles is presented, using the Gibbs free energy of a vesicle suspension including the elastic contribution of the bilayer membrane. The results indicate that, in addition to the hydrostatic pressure difference across the membrane and the interbilayer pressure due to electrostatic repulsion, the elastic pressure arising from the coupling between the osmotic stress and the elasticity of the membrane (osmoelastic coupling) should participate in the osmotic response of phospholipid vesicles. The data of Cowley et al. [Cowley, A. C., Fuller, N. L., Rand, R. P., & Parsegian, V. A. (1978) Biochemistry 17, 3163-3168] and of Parsegian et al. [Parsegian, V. A., Fuller, N., & Rand, R. P. (1979) Proc. Natl. Acad. Sci. U.S.A. 76, 2750-2754] on the osmotic shrinkage of multilayer vesicles are discussed in terms of the elastic pressure and the interbilayer pressure, and the proposed "dehydration force" theory is reevaluated from the viewpoint of the present analysis.     

Spatial organization of the lacunar-canalicular system in the structures of bone lamellae

By means of light microscopical techniques and scanning electron microscopy spatial organization of the lacunar-canalicular system (LCS) has been studied in structures of a mature lamellar bone. A method for making corrosive casts of osseous lacunae and canaliculi is suggested, owing to which their spatial organization can be analysed. Certain data on interconnections of the osseous lacunae with each other and with vascular canals and natural surfaces of the bone are presented. The role of LCS as a component of the microcirculatory bed of the lamellar bone is discussed.     

Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains

By using asymptotic theory, we generalise the Turing diffusively-driven instability conditions for reaction-diffusion systems with slow, isotropic domain growth. There are two fundamental biological differences between the Turing conditions on fixed and growing domains, namely: (i) we need not enforce cross nor pure kinetic conditions and (ii) the restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Our theoretical findings are confirmed and reinforced by numerical simulations for the special cases of isotropic linear, exponential and logistic growth profiles. In particular we illustrate an example of a reaction-diffusion system which cannot exhibit a diffusively-driven instability on a fixed domain but is unstable in the presence of slow growth.     

Between order and disorder in protein structures: analysis of "dual personality" fragments in proteins

In their natural environment, three-dimensional structures of proteins undergo significant fluctuations and are often partially or completely disordered. This phenomenon recently became the focus of much attention, as many proteins, especially from higher organisms, were shown to contain large intrinsically disordered regions. Such disordered regions may become ordered only under very specific circumstances, if at all, and can be recognized by specific amino acid composition and sequence signatures. Here, we suggest that the balance between order and disorder is much more subtle in that many regions are very close to the order/disorder boundary. Specifically, analysis of redundant sets of experimental models of protein structures, where emphasis is put on comparison of structures of identical proteins solved in different conditions and functional states, shows hundreds of fragments captured in two states: ordered and disordered. We show that such fragments, which we call here "dual personality" (DP) fragments, have distinctive features that differentiate them from both regularly folded and intrinsically disordered fragments. We hypothesize, and show on several examples, that such fragments are often targets of regulation, either by allostery or posttranslational modifications.     

Spatial analysis of a subalpine heath woodland

The horizontal spatial patterns of heath species have implications in dynamic community modelling, fire behaviour modelling and ecological interpretation. The distribution pattern of the dominant woody species in a subalpine heath woodland near Kiandra, New South Wales was analysed as spatial point processes. The species analysed were Bossiaea foliosa, Grevillea australis, Hakea lissosperma and Oxylobium ellipticum, and both large (>2m) and small (<2 m) snowgums (Eucalyptus pauciflora) They all exhibited significant aggregation at scales ranging from 1 to 15 m. Bivariate spatial analyses of these species revealed significant negative association between G. australis and H. lissosperma, between G. australis and O. ellipticum, and between O. ellipticum and H. lissosperma, B. foliosa was independently distributed with respect to other shrub species. There was some evidence for negative association between small snowgums and shrub species, though small snowgums were positively associated with large snowgums. The joint spatial distribution of the individuals of all shrub species was also aggregated. A Poisson cluster process was developed and tested to model the joint spatial pattern of the shrub stratum.     

"Bridge" structures between nucleic acid molecules fixed in the structure of a liquid crystal

The binding of daunomycin and copper ions to poly(I).poly(C) molecules fixed in a particle of a liquid-crystalline dispersion was studied. A thermodynamic model of adsorption was developed, which makes it possible to describe the formation of complexes of a particular kind, "bridges" that connect adjacent nucleic acid molecules fixed in a liquid crystal. The bridges represent chelate complexes, which incorporate the molecules of the antibiotic daunomycin and copper ions. Equations describing the dependence of the concentration of these bridges in solution on the concentration of their constituents were derived. The family of dependences of experimental amplitudes of bands in CD spectra typical of "bridge" structures on the concentration of copper ions represents a set of S-shaped curves, and, as the concentration of daunomycin in solution increases, the level of saturation of these curves increases. The analysis of experimental data with the use of this model suggests that the structures of this type compete with daunomycin molecules for the binding sites on poly(I).poly(C). By using this model, the energies of formation of bridge structures were calculated.     

The role of a reaction-diffusion system in the formation of hair fibres

Epithelial cells destined to form the hair fibre begin to differentiate while still in the hair follicle bulb. The fibre cells continue to differentiate as they migrate out of the bulb and up the follicle towards the skin surface. The anatomy of the hair follicle and the different cell types observed within the follicle are briefly reviewed. A theoretical scheme for cell differentiation, capable of producing all the observed mature cell types, is presented. A major component of the scheme is a reaction-diffusion system of morphogens similar to that originally proposed by Turing (1952). The mathematical solution of the equations defining the reaction-diffusion system within the follicle bulb is discussed. The sequence of patterns in the spatial distribution of the morphogens expected in the hair follicle bulb is calculated and found to be in good agreement with the sequence of patterns of orthocortical and paracortical cells in the fibre cross section as the diameter of the fibre increases. The spatial patterns of the morphogens are also compared with the shape of the fibre cross section. It is concluded that a reaction-diffusion system may play a major role in the morphogenesis of hair fibres.     

Automated microsequence analysis in the presence of a synthetic "carrier"

When small amounts of protein are subjected to automated sequence analysis, significant material washes out during the solvent wash steps and prevents extended analysis. Inclusion of a synthetic “carrier,” succinylated poly-ornithine, with the protein to be sequenced significantly reduces protein washout and permits extended automated microsequence analysis. This carrier also permits microquantities of protein to be sequenced in the presence of the detergent, sodium dodecyl sulfate.     

Traveling waves in a piecewise-linear reaction-diffusion model of excitable medium

One-dimensional autowaves (traveling waves) in excitable medium described by a piecewise-linear reaction-diffusion system have been investigated. Two main types of wave have been considered: a single pulse and a periodic sequence of pulses (wave trains). In a two-component system, oscillations are due to the second component of the reaction-diffusion system, while in a one-component system, they are caused by external periodic excitation (forcing). Using semianalytical solutions for the wave profile, the shape and velocity of autowaves have been found. It is shown that the dispersion relation for oscillating sequences of pulses has an anomalous character.     

Spatial structures in mitochondrial suspension induced by cation efflux

The formation of spatial structures in a thin unstirred layer of a mitochondrial suspension has been studied. It is shown that the structure formation depends on the state of the ion-transporting systems of mitochondria and that pattern development coincides with the activation of cation efflux from preloaded mitochondria. Spatial structure formation is an energy-dependent process and is suppressed by respiratory chain inhibitors. Patterning is also inhibited by EGTA, EDTA and ruthenium red, reflecting the requirement for divalent cation translocation in mitochondria for the studied phenomenon.