On the stationary state analysis of reaction-diffusion mechanisms for biological pattern formation

We present a biologically plausible two-variable reaction-diffusion model for the developing vertebrate limb, for which we postulate the existence of a stationary solution. A consequence of this assumption is that the stationary state depends on only a single concentration-variable. Under these circumstances, features of potential biological significance, such as the dependence of the steady-state concentration profile of this variable on parameters such as tissue size and shape, can be studied without detailed information about the rate functions. As the existence and stability of stationary solutions, which must be assumed for any biochemical system governing morphogenesis, cannot be investigated without such information, an analysis is made of the minimal requirements for stable, stationary non-uniform solutions in a general class of reaction-diffusion systems. We discuss the strategy of studying stationary-state properties of systems that are incompletely specified. Where abrupt transitions between successive compartment-sizes occur, as in the developing limb, we argue that it is reasonable to model pattern reorganization as a sequence of independent stationary states.     

Implicit-explicit methods for reaction-diffusion problems in pattern formation

Reaction-diffusion mechanisms have been used to explain pattern formation in developmental biology and in experimental chemical systems. To approximate the corresponding spatially discretized models, an explicit scheme can be used for the reaction term and an implicit scheme for the diffusion term. Such implicit-explicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods in fluid flow problems.In this work, we analyze the performance of several of the best known linear multistep IMEX schemes for reaction-diffusion problems in pattern formation. For the linearized two chemical system, the growth properties exhibited by IMEX schemes are examined. Schemes which accurately represent the growth of the linearized problem for large time steps are identified. Numerical experiments show that first order accurate schemes, as well as schemes which produce only a weak decay of high frequency spatial error may yield plausible results which are nonetheless qualitatively incorrect. For such schemes, computations using refinements in the time step are likely to produce essentially the same (erroneous) results. Higher order schemes which produce a strong decay of high frequency errors are proposed instead.Our findings are demonstrated on several examples.The work of the author was partially supported by an NSERC Postgraduate Scholarship     

Speed of pattern appearance in reaction-diffusion models: Implications in the pattern formation of limb bud mesenchyme cells

It has been postulated that fibroblast growth factor (FGF) treatment of cultured limb bud mesenchyme cells reinforces the lateral inhibitory effect, but the cells also show accelerated pattern appearance. In the present study, we analyze how a small change in a specific parameter affects the speed of pattern appearance in a Turing reaction-diffusion system using linear stability analysis. It is shown that the sign of the change in appearance speed is qualitatively decided if the system is under the diffusion-driven instability condition, and this is confirmed by numerical simulations. Numerical simulations also show that a small change in parameter value induced easily detectable differences in the appearance speed of patterns. Analysis of the Gierer-Meinhardt model revealed that a change in a single parameter can explain two effects of FGF on limb mesenchyme cells—reinforcement of lateral inhibition and earlier appearance of pattern. These qualitative properties and easy detectability make this feature a promising tool to elucidate the underlying mechanisms of biological pattern formationwhere the quantitative parameters are difficult to obtain.     

Gene networks capable of pattern formation: from induction to reaction-diffusion

One of the main aims of developmental biology is to understand how a single and apparently homogeneous egg cell achieves the intricate complexity of the adult. Here we present two models to explain the generation of developmental patterns through interactions at the gene level. One model considers direct-contact induction between cells while the other takes into account diffusion of hormones. We show that sets of cells involving identical gene networks and communicating through hormones spontaneously exhibit ordered patterns. We have characterized these patterns and the specific networks responsible for them. The models allow to (i) compare diffusion and direct-contact induction processes as mechanisms of pattern generation; (ii) identify the possible range of behaviour of real gene networks and (iii) suggest causal mechanisms to generate known patterns. The evolutionary implications are discussed.     

A reaction-diffusion theory of morphogenesis with inherent pattern invariance under scale variations

In the framework of reaction-diffusion theory we deal with the problem of pattern regulation in morphogenesis. A generic model is proposed where the kinetic terms follow constraints imposed by scale invariance considerations. These constraints allow a class of kinetic schemes to be formulated so that, starting with an initially homogeneous morphogen distribution in the field, a stable gradient is established of the form: S(chi,L) = Lpf(chi/L). Here L is the length of the morphogenetic field, chi is the position variable and f(chi/L) is some monotonic function of the relative distance. With this distribution a scale invariant gradient can be constructed which leads to pattern regulation. A linear stability analysis of the model permits the definition of the parameter values enabling the system to abandon the homogeneous state spontaneously. Simulations of the evolution of the system towards its final stable state result in approximate pattern invariance for different field lengths. The accuracy of this invariance is in agreement with some recent quantitative experimental findings in both developing and regenerating systems.     

Resource heterogeneity generated by shrubs and topography on coastal sand dunes

In early stages of primary succession, colonizing plants can create resource patches that influence the abundance and distribution of other species. To test whether different colonizing shrubs generate contrasting patches on coastal sand dunes, we compared soil characteristics and light availability under the nitrogen-fixing shrub Lupinus arboreus, under the non-nitrogen-fixing shrub Artemisia pycnocephala, and between shrubs on dunes at a site in northern California. Concentrations of inorganic nitrogen and net nitrogen mineralization rates were generally 1–10 times greater in soil under Lupinus than under Artemisia or between shrubs. Soil water content was mostly lower under shrubs. Mean photon flux density near ground level was reduced by at least 80% at 35 cm inside shrub canopies. Topography appeared to have more effect on soil moisture but less direct effect on nitrogen availability than did Lupinus. However, Lupinus probably increases nitrogen levels more on higher, drier dunes. Microhabitats under and between nitrogen-fixing shrubs constitute a mosaic of individually poor but complementary patches in which high levels of light and moderate levels of soil nitrogen are present but tend not to occur together.     

The pulse current pattern generated by developing fucoid eggs

Using a newly developed extracellular vibrating electrode, we have made the first study of the spatial distribution of the growth currents around a single developing egg. This pattern was studied during the current pulses wihic traverse two-celled Pelvetia embryos. These pulses can be stimulated to occur with a periodicity of 70 min by mild acidification of the dea water medium. Current enters only at the growing rhizoid's tip while leaving both the base of the rhizoid cell and the whole outer membrane of the thallus cell. The field in front of the rhizoid cell falls off as the inverse cube of the distance from the rhizoid cell's center in the manner of a dipole field. The total inward and outward currents are equal, agreeing with theory. The current density at the rhizoid cell's base is twice that at the top of the thallus cell and this probably represents a change in the outer membrane's properties. There are no significant differences in the durrent density over the thallus cell. These results suggest a model in which the pulse current leaks in through newly opened channels in the growing tip and leaks out elsewhere due to the resultant fall in the membrane potential.     

On the cause and nature of C9-related heterogeneity of terminal complement complexes generated on target erythrocytes through the action of whole serum

The binding of C8 and C9 from human serum to target erythrocytes was quantified, and the molecular stoichiometries of C9:C8 within terminal C5b-9(m) complexes were determined. Low doses of serum generated terminal complexes with mean C9:C8 ratios of 2 to 3:1, whereas complexes generated by highest serum doses harbored an average of six to eight C9/C8 molecules. From the collective biochemical and ultrastructural data, we concluded that heterogeneous populations of terminal complexes regularly form on target membranes; those containing high numbers of C9 molecules (greater than or equal to six to eight) exhibit the structure of the classical "lesion", whereas those containing low numbers of C9 do not exhibit this typical structure, although they probably still function as small pores. A major cause for this heterogeneity of the lesions derives from shortage of C9, which is naturally present in a 2 to 1 molar ratio relative to C8 in serum. Generation of terminal complexes harboring high numbers of C9 on erythrocyte membranes is possible in spite of this natural shortage because SC5b-9 does not form in the fluid phase to compete for C9 binding. If interrupted, the process of C9-C9 oligomerization cannot be recontinued, and "incomplete" C5b-9 complexes are unable to bind additional C9 upon reincubation with this component. The demonstrated heterogeneity of terminal complexes with respect to their C9 content may explain the functional heterogeneity of complement lesions observed previously by other investigators.     

Spatial and temporal litterfall heterogeneity generated by woody species in the Central Monte desert

In arid and semiarid environments, the presence of woody species generates a series of environmental gradients that increase spatial heterogeneity and modify the pattern of distribution of the other species. We postulate that the temporal and spatial variability in litter input generated by woody species is a relevant factor in the generation of edaphic heterogeneity by redistribution of nutrients and the physical effects of litter. The objective of this study was to determine the temporal and spatial variability in the amount of litter input under the canopy of dominant woody plants (Prosopis flexuosa and Larrea divaricata) and in exposed areas at the Ñacuñán Reserve, in the central zone of the Monte desert. Litterfall was collected during 2 years from 30-cm-diameter litter traps distributed at three microsites: under P. flexuosa canopy, under L. divaricata canopy, and in exposed areas. Microhabitats beneath Prosopis showed the highest litter input per m² (between 320 and 527 g/m²), and, consequently, more than 50% of it fell to the soil beneath the canopy of P. flexuosa. Only 10% fell on exposed areas, which exhibited an annual input rate per m² of a lower order of magnitude than the sites under Prosopis. Litterfall presented a peak in summer as a consequence of convective storms, and a second one in autumn due to phenological shedding. Our results suggest that woody species have a central importance in the dynamics of nutrients in arid lands by both the increase of total productivity and litterfall, and the spatial and temporal regulation of litter input.     

Structural complexity of arthropod guilds is affected by the agricultural landscape heterogeneity generated by fencerows

Intensive farming imposes harsh conditions impeding the persistence of most arthropod species within crop fields. Hence, arthropods surviving the unfavourable conditions prevailing within crop fields may disperse towards nearby uncropped margins, such as fencerows. Here, we evaluate the influence of landscape heterogeneity on the abundance of different guilds, particularly herbivores and their natural enemies. Said heterogeneity mostly derives from fencerow network density. Hence, we developed an approach based on fitting linear‐mixed models to elucidate the effects of landscape heterogeneity and field position (fencerows and crop interiors) on arthropod diversity. Mixed models were fitted to arthropod data obtained by pitfall trap samplings in 36 crop fields. Arthropod communities were structurally and functionally more complex along fencerows than within nearby crop interiors. Arthropods abundance was modulated by landscape heterogeneity, increasing the abundance of natural enemies as the landscape heterogeneity increased. On the contrary, herbivores abundance decreased as landscape heterogeneity increased. Consequently, the ratio between herbivores and natural enemies also decreased as landscapes became more heterogeneous. Natural enemies with larger body sizes, mostly carabid beetles, were more sensitive to landscape homogenisation. Our study reveals that, despite the coarse‐grained landscapes in the Rolling Pampa, fencerow density appears as a key factor for structuring complex arthropod guilds in intensively farmed agricultural mosaics. In landscapes with higher density of fencerows, arthropods tend to concentrate along them, thus increasing the community structural complexity as well as the predation pressure over herbivores. This structural complexity of upper trophic levels enhances the ‘top‐down’ regulation of herbivore populations, consequently decreasing the probability of pest outbreaks within crop fields.     

Correlation equations for endemic diseases: externally imposed and internally generated heterogeneity

The simple susceptible–infectious–recovered (SIR) model has provided many insights into the behaviour of a single epidemic. However, most of epidemiology is concerned with endemic infections, and for this to occur fresh susceptibles need to be generated. This is usually provided by individuals becoming susceptible soon after birth or by recruitment to cohorts at risk. This paper develops a correlation model, predicting the behaviour of connected pairs of individuals, which includes demographic processes as well as the basic epidemiology. In addition to the local spatial correlations, we consider three other forms of heterogeneity: internally generated heterogeneity in terms of stochasticity or imposed heterogeneity in terms of non-uniform vaccination or age-structure.     

放牧和刈割条件下草山草坡群落空间异质性分析

采用变异矩分析和分形方法,研究了草山草坡群落在放牧和刈割条件下的空间异质性及空间自相关性,结果表明,群落空间格局有尺度依赖性,刈割条件下空物异质性及空间相关性弱,多样性梯度即β多样性小,放牧消除地形引起的样地差异,因而使空间异质性简单化。     

Oviposition pattern of phytophagous insects: on the importance of host population heterogeneity

Frequency distributions of insect immatures per host are often fitted to contagious distributions, such as the negative binomial, to deduce oviposition pattern. However, different mechanisms can be involved for each theoretical distribution and additional biological information is needed to correctly interpret the fits. We chose the chestnut weevil Curculio elephas, a pest of the European chestnut Castanea sativa, as a model to illustrate the difficulties of inferring oviposition pattern from fits to theoretical distributions and from the variance/mean ratio. From field studies over 13–16 years, we show that 20 out of the 31 yearly distributions available fit a negative binomial and 25 a zero-inflated Poisson (ZIP). No distribution fits a Poisson distribution. The ZIP distribution assumes heterogeneity within the fruit population. There are two categories of host: the first comprises chestnuts unsuitable for weevil oviposition or in excess relative to the number of weevil females, and the second comprises suitable fruits in which oviposition behavior is random. Our results confirm this host heterogeneity. According to the ZIP distribution, the first category of hosts includes on average 74% of the chestnuts. A negative binomial distribution may be generated by either true or false contagion. We show that neither interference between weevil females, nor spatial variation in the infestation rate exist. Consequently, the observed distributions of immatures are not the result of false contagion. Nevertheless, we cannot totally exlude true contagion of immatures. In this paper we discuss the difficulty of testing true contagion in natural conditions. These results show that we cannot systematically conclude in favour of contagion when fitting a distribution such as the negative binomial or when a variance/mean ratio is higher than unity. Received: 22 September 1997 / Accepted: 15 December 1997