Chemical waves in open flows of active media: their relevance to axial segmentation in biology.

The boundary forcing of open flows of active media can lead to a variety of spatiotemporal structures, depending on the local kinetics of the medium and on the characteristics of the forcing. Here, we demonstrate that regardless of the local kinetics, the combination of flow and boundary forcing is a powerful method for replacing intrinsic modes with extrinsic ones. This entrainment of dynamics has important implications for biological morphogenesis. During early embryonic development it is frequently observed that stripes of gene expression and segments arise one after the other along a growth-axis. We show that axial growth can be viewed as an open flow of cells away from a growth zone. Based on this realisation, we demonstrate using three generic reaction-diffusion-advection schemes how a space-periodic structure is induced, one "segment" at a time along the growth/flow axis, by a segmental clock that is synchronised within the growth zone. The schemes are investigated in the context of an abrupt and a gradual change in the properties of the segmental clock. Experimental observations provide evidence that the latter is involved in the early development of many vertebrates.

Self-organized localization of macrostates by additive noise in a fast-slow dynamical system: effect of the slow nonreactive mode on the barrier crossing rate of the fast, bistable mode

We have studied the stochastic dynamics of a two-dimensional gradient system composed of a fast, bistable mode and a slow, monotonically decaying mode. The coupling is bidirectional and cooperative. Additive white noise acts on the fast mode only. We find that the noise intensity controls the location of macrostates (shape of the probability density function), the appearance of bimodality in the slow-mode probability distribution and, together with the coupling strength, the rate of fast-mode barrier crossing. These features arise from the interplay of noise, widely separated time scales, and bidirectional, excitatory coupling. They are believed to be generic.

A chemical flow system mimics waves of gene expression during segmentation.

The early vertebrate developmental process of somitogenesis involves bands of gene expression that form periodically at the posterior end of the presomitic mesoderm (PSM) and traverse it with decreasing width and velocity. We have constructed a chemical flow system that, based on the novel flow-distributed oscillator (FDO) mechanism of wave pattern formation, reproduces key physical features of the PSM and observe concentration waves having similar spatio-temporal behavior. This suggests that the gene expression waves can be understood qualitatively in terms of phase dynamics in an open flow of a self-oscillating medium and that chemical flow systems can be used to mimic and model biological pattern formation during axial growth. In fact, expressions for wavelength and wave velocity derived from phase dynamics are found to be in quantitative agreement with measurements from both the biological and the chemical systems. This indicates that they, despite their significant differences, have common dynamics.

Pulsating wave propagation in reactive flows: flow-distributed oscillations

Stationary waves in a reactive flow with equal transport coefficients were recently generated by passing the oscillating Belousov-Zhabotinsky reaction medium through a tubular packed bed reactor while keeping the concentrations constant at the inflow boundary [M. Kaern and M. Menzinger, Phys. Rev. E 60, 3471 (1999)]. Here we study the effects of oscillatory boundary conditions, and observe traveling wave fronts that propagate in either a pulsating or a steady manner. The present experiment is isothermal and conditions are such that all species have identical transport properties. This excludes rapid thermal or activator diffusion and the wave pulsation appears to be induced by an essentially kinematic mechanism. Our experimental findings are supported by numerical simulations of the full reaction-diffusion-advection system using a realistic kinetic model. Finally, the kinematic essence of the mechanism inducing wave pulsation is captured in an iterative one-variable map.

Non-turing stationary patterns in flow-distributed oscillators with general diffusion and flow rates

An analytical prediction [P. Andresen et al., Phys. Rev. E 60, 297 (1999)] and its experimental confirmation [M. Kaern et al., Phys. Rev. E 60, 3471 (1999)] establish a mechanism for forming stationary, space-periodic structures in a reactive flow (reaction-diffusion-convection system) with equal diffusion and flow rates. In this paper we generalize the analysis to systems with unequal diffusion and flow rates. Interestingly, stationary waves also exist outside the oscillatory Hopf domain of the batch system-hence the parameter space in which these structures exist is bigger than that initially predicted [P. Andresen et al., Phys. Rev. E. 60, 297 (1999)] (for equal diffusion and flow rates). On the other hand, we find that these stationary waves exist only for parameter values outside of and up to the Turing regime. We clarify the nature of the instability in terms of a boundary-forcing problem, whereby a time-periodic pattern is carried over the whole domain by the flow while the phase is fixed at the inflow boundary.

Reply to "Comment on 'Flow-distributed oscillations: stationary chemical waves in a reacting flow' "

Stationary waves (wavelength lambda) are the necessary result of spatial recurrence of phase in the open flow (rate v) of an oscillating medium with fixed inflow boundary conditions. Any nonlinear dependence lambda(v) on flow is the result of dispersion that is implicit in lambda=v/omega(v), where omega(v) is the oscillation frequency in a reference frame moving with the flow. The flow-distributed oscillator mechanism extends thus from the purely kinematic limit omega=const to the case where a nonlinear dependence lambda(v) is subsumed by the dispersion relationship omega(v).

Segmentation and somitogenesis derived from phase dynamics in growing oscillatory media.

The formation of spatially repetitive structures along the growth axis of a developing embryo is a common theme in developmental biology. Here we apply the novel flow-distributed oscillator (FDO) mechanism of wave pattern formation to the problem of axial segmentation in general and to somitogenesis in particular. We argue that the conditions for formation of FDO waves are satisfied during somitogenesis in the chick and mouse and that the waves of gene expression observed in these species arise from phase dynamics in a growing oscillatory medium. We substantiate this claim by showing that the FDO mechanism allows the waves to be mimicked by an inorganic experiment and that it predicts a wavelength that coincides with that observed experimentally. To see whether the FDO mechanism is compatible with other aspects of somitogenesis, we construct an FDO-based model of somitogenesis and successfully test it against a number of experimental observations, including the effect of heat shock. Our analysis provides a rigorous physical basis for the hypothesis that the phase dynamics of a segmental clock controls important stages of segmentation during somitogenesis in the chick and mouse as well as in other organisms that undergo segmentation during their axial growth.

Statistical methods for assessing the dimensions of synaptic vesicles in nerve terminals.

Chemical transmission between neurons occurs by the release of neurotransmitter packaged within vesicles of the presynaptic neuron onto a postsynaptic target. The amount of transmitter contained within a vesicle is in part regulated by the size of the vesicle. Thus, it is of general interest to quantify the dimension of vesicles in understanding the basic principles of chemical synaptic transmission. These vesicles can only be measured by electron microscopic techniques. Obtaining the true dimensions of synaptic structures is therefore complicated by stereological considerations. In this study, we suggest improved methods for determining the distributions (and mean sizes) for populations of vesicle diameters by mathematical processes involving (1) an implicit inversion of the empirical data distribution, (2) an explicit inversion approach, and (3) an approach based on substituting the empirical distribution into the inversion formula and then isotonizing using an iterated convex minorant algorithm. These procedures provide distributions that better represent the true population distributions (and means) for comparisons with other data sets of vesicle diameter measures.

Computational dynamics of gradient bistable networks.

We describe a neural-like, homogeneous network consisting of coupled bistable elements and we study its abilities of learning, pattern recognition and computation. The technique allows new possibilities of pattern recognition, including the memorization and perfect recall of several memory patterns, without interference from spurious states. When the coupling strength between elements exceeds a critical value, the network readily converges to a unique attractor. Below this critical value one could perfectly recall all memorized patterns.

Turing instabilities in general systems.

We present necessary and sufficient conditions on the stability matrix of a general n(> or = 2)-dimensional reaction-diffusion system which guarantee that its uniform steady state can undergo a Turing bifurcation. The necessary (kinetic) condition, requiring that the system be composed of an unstable (or activator) and a stable (or inhibitor) subsystem, and the sufficient condition of sufficiently rapid inhibitor diffusion relative to the activator subsystem are established in three theorems which form the core of our results. Given the possibility that the unstable (activator) subsystem involves several species (dimensions), we present a classification of the analytically deduced Turing bifurcations into p (1 < or = p < or = (n - 1)) different classes. For n = 3 dimensions we illustrate numerically that two types of steady Turing pattern arise in one spatial dimension in a generic reaction-diffusion system. The results confirm the validity of an earlier conjecture [12] and they also characterise the class of so-called strongly stable, matrices for which only necessary conditions have been known before [23, 24]. One of the main consequences of the present work is that biological morphogens, which have so far been expected to be single chemical species [1-9], may instead be composed of two or more interacting species forming an unstable subsystem.

Flow-distributed oscillations: stationary chemical waves in a reacting flow.

A recent prediction of stationary waves in open, reacting flows is experimentally verified. We show that stationary waves are generated by a mechanism whereby the flow carries a time-oscillating subelement, behaving like a batch reactor, through space while a fixed boundary condition at the inflow locks the phase of the oscillation. This mechanism can generate stationary patterns when all diffusion coefficients are equal. The experimental system is the ferroin-catalyzed Belousov-Zhabotinsky reaction in a tubular reactor, fed by the outflow of a continuous flow stirred tank reactor (CSTR). Parameter conditions are such that the concentrations are constant in the CSTR while they oscillate in the flow tube.

Dynamics of analog-to-frequency transduction in sensory receptors.

We analyze the analog-to-frequency transduction by sensory neurons in vivo. Spatially extended neural models are made to fire by applying the stimulus to the membrane potential at the boundary of the trigger zone. The membrane property that provides a broad dynamical range of frequency is a prolonged hyperpolarizing afterpotential. In support of this PDE model, we study the stimulus-dependent location of pulse-initiation in crayfish stretch receptors Ringham (1971).

Antiparasitic activity of diallyl trisulfide (Dasuansu) on human and animal pathogenic protozoa (Trypanosoma sp., Entamoeba histolytica and Giardia lamblia) in vitro.

Garlic (Allium sativum L.) and one of its major components, allicin, have been known to have antibacterial and antifungal activity for a long time. Diallyl trisulfide is a chemically stable final transformation product of allicin which was synthesized in 1981 in China and used for treatment of bacterial, fungal and parasitic infections in man. The activity of diallyl trisulfide was investigated in several important protozoan parasites in vitro. The IC50 (concentration which inhibits metabolism or growth of parasites by 50%) for Trypanosoma brucei brucei, T.b. rhodesiense, T.b. gambiense, T. evansi, T. congolense and T. equiperdum was in the range of 0.8-5.5 micrograms/ml. IC50 values were 59 micrograms/ml for Entamoeba histolytica and 14 micrograms/ml for Giardia lamblia. The cytotoxicity of the compound was evaluated on two fibroblast cell lines (MASEF, Mastomys natalensis embryo fibroblast and HEFL-12, human embryo fibroblast) in vitro. The maximum tolerated concentration for both cell lines was 25 micrograms/ml. The results indicate that the compound has potential to be used for treatment of several human and animal parasitic diseases.