Pattern formation in morphogenesis

We first treat the Gierer-Meinhardt equations by linear stability analysis to determine the critical parameter, at which the homogeneous distributions of activator and inhibitor concentrations become unstable. We find two types of instabilities: one leading to spatial pattern formation and another one leading to temporal oscillations. We consider the case where two instabilities are present. Using the method of generalized Ginzburg-Landau equations introduced earlier we then analyze the nonlinear equations. As we are mainly interested in spatial pattern formation on a sphere we consider the problem under an appropriate constraint. Combining the two occurring solutions we find patterns well-known in biology, such as a gradient system and temporal oscillations.     

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.     

A four-variable model for the pattern-forming mechanism in Hydra

A generalized Gierer-Meinhardt model has been used to account for the transplantation experiments in Hydra. In this model, a cross inhibition between the two organizing centres (namely, head and foot) are assumed to be the only mode of interaction in setting up a stable morphogen distribution for the pattern formation in Hydra.     

A pre-pattern formation mechanism for the spiral-type patterns of the sunflower head

We propose a reaction-diffusion system for the laying down of the spacing patterns of the sunflower head. A detailed analysis shows that spatially varying diffusion coefficients lead to a spiral-type pre-pattern. The relationship between this pre-pattern and the fructification pattern of the sunflower head is discussed. The reaction kinetics chosen is of the Gierer-Meinhardt type. Furthermore, it is suggested that a pre-pattern formation mechanism of this type may also play a role in phyllotaxis.     

Analytical treatment of pattern formation in the Gierer-Meinhardt model of morphogenesis

Summary We first perform a linear stability analysis of the Gierer-Meinhardt model to determine the critical parameters where the homogeneous distribution of activator and inhibitor concentrations becomes unstable. There are two kinds of instabilities, namely, one leading to spatial patterns and another one leading to temporal oscillations. Focussing our attention on spatial pattern formation we solve the corresponding nonlinear equations by means of our previously introduced method of generalized Ginzburg-Landau equations. We explicitly consider the two-dimensional case and find both rolls and hexagon-like structures. The impact of different boundary conditions on the resulting patterns is also discussed. The occurrence of the new patterns has all the features of nonequilibrium phase transitions.     

Application of reaction-diffusion models to cell patterning in Xenopus retina. Initiation of patterns and their biological stability

We have examined the behavior of two reaction-diffusion models, originally proposed by Gierer & Meinhardt (1972) and by Kauffman, Shymko & Trabert (1978), for biological pattern formation. Calculations are presented for pattern formation on a disc (approximating the geometry of a number of embryonic anlagen including the frog eye rudiment), emphasizing the sensitivity of patterns to changes in initial conditions and to perturbations in the geometry of the morphogen-producing space. Analysis of the linearized equations from the models enabled us to select appropriate parameters and disc size for pattern growth. A computer-implemented finite element method was used to solve the non-linear model equations reiteratively. For the Gierer-Meinhardt model, initial activation (varying in size over two orders of magnitude) of one point on the disc's edge was sufficient to generate the primary gradient. Various parts of the disc were removed (remaining only as diffusible space) from the morphogen-producing cycle to investigate the effects of cells dropping out of the cycle due to cell death or malfunction (single point removed) or differentiation (center removed), as occur in the Xenopus eye rudiment. The resulting patterns had the same general shape and amplitude as normal gradients. Nor did a two-fold increase in disc size affect the pattern-generating ability of the model. Disc fragments bearing their primary gradient patterns were fused (with gradients in opposite directions, but each parallel to the fusion line). The resulting patterns generated by the model showed many similarities to results of "compound eye" experiments in Xenopus. Similar patterns were obtained with the model of Kauffman's group (1978), but we found less stability of the pattern subject to simulations of central differentiation. However, removal of a single point from the morphogen cycle (cell death) did not result in any change. The sensitivity of the Kauffman et al. model to shape perturbations is not surprising since the model was originally designed to use shape and increasing size during growth to generate a sequence of transient patterns. However, the Gierer-Meinhardt model is remarkably stable even when subjected to a wide range of perturbations in the diffusible space, thus allowing it to cope with normal biological variability, and offering an exciting range of possibilities for reaction-diffusion models as mechanisms underlying the spatial patterns of tissue structures.     

Numerical simulations of the effects of retinoids on pattern formation in a hydroid

Summary Retinoids have been shown to influence pattern formation in hydroid polyps (Hydractinia echinata) at various levels. These effects are counteracted by an inhibitor isolated from Hydra. The present study provides a theoretical attempt to elucidate the role of retinoids by computer simulations based on a simple model of pattern formation in Hydractinia. The elements of this model are morphogens of the Gierer-Meinhardt type, namely a long-range inhibitor and a short-range activator. From the calcualtions, a reducing effect of retinoids on the propagation of the inhibitor seems most probable.     

Bayesian ranking of biochemical system models

MOTIVATION: There often are many alternative models of a biochemical system. Distinguishing models and finding the most suitable ones is an important challenge in Systems Biology, as such model ranking, by experimental evidence, will help to judge the support of the working hypotheses forming each model. Bayes factors are employed as a measure of evidential preference for one model over another. Marginal likelihood is a key component of Bayes factors, however computing the marginal likelihood is a difficult problem, as it involves integration of nonlinear functions in multidimensional space. There are a number of methods available to compute the marginal likelihood approximately. A detailed investigation of such methods is required to find ones that perform appropriately for biochemical modelling. RESULTS: We assess four methods for estimation of the marginal likelihoods required for computing Bayes factors. The Prior Arithmetic Mean estimator, the Posterior Harmonic Mean estimator, the Annealed Importance Sampling and the Annealing-Melting Integration methods are investigated and compared on a typical case study in Systems Biology. This allows us to understand the stability of the analysis results and make reliable judgements in uncertain context. We investigate the variance of Bayes factor estimates, and highlight the stability of the Annealed Importance Sampling and the Annealing-Melting Integration methods for the purposes of comparing nonlinear models. AVAILABILITY: Models used in this study are available in SBML format as the supplementary material to this article.     

Why are proteins marginally stable?

Most globular proteins are marginally stable regardless of size or activity. The most common interpretation is that proteins must be marginally stable in order to function, and so marginal stability represents the results of positive selection. We consider the issue of marginal stability directly using model proteins and the dynamical aspects of protein evolution in populations. We find that the marginal stability of proteins is an inherent property of proteins due to the high dimensionality of the sequence space, without regard to protein function. In this way, marginal stability can result from neutral, non-adaptive evolution. By allowing evolving protein sub-populations with different stability requirements for functionality to complete, we find that marginally stable populations of proteins tend to dominate. Our results show that functionalities consistent with marginal stability have a strong evolutionary advantage, and might arise because of the natural tendency of proteins towards marginal stability.     

Reformation of the marginal band of avian erythrocytes in vitro using calf-brain tubulin: peripheral determinants of microtubule form

The microtubules of nucleated erythrocytes form an extraordinary structure: they are organized into a marginal band at the periphery of the cell. This unusual organelle, recurring in detail in each cell, provides an excellent opportunity to study the determinants of microtubule form. We have been able to reform the marginal band, using detergent-extracted erythrocytes that have been depleted of microtubules in vivo and phosphocellulose-purified tubulin from calf brain. We find that detergent-extracted cytoskeletons incubated under these conditions again have microtubules, and that the pattern of these microtubules recapitulates several features of the intact marginal band. In particular, most of the microtubules after regrowth are located in a band at the periphery of the cell, and curve to form an ellipse. These results support the hypothesis that the specification of microtubule location and shape in these cells is governed by determinants that reside at the periphery of the cell.     

Temporal organization in a morphogenetic field

We have investigated the possibility of self-sustained oscillations of chemical concentrations during morphogenetic processes. In particular, we have shown that, in the case of the Gierer-Meinhardt model, self-sustained oscillations are possible when an order parameter, connected with the decay constants of morphogenes, crosses a critical value. The analytical results are in agreement with the numerical results of Gierer and Meinhardt.     

Relation between protein stability, evolution and structure, as probed by carboxylic acid mutations

Native proteins are marginally stable. Low thermodynamic stability may actually be advantageous, although the accumulation of neutral, destabilizing mutations may have also contributed to it. In any case, once marginal stability has been reached, it appears plausible that mutations at non-constrained positions become fixed in the course of evolution (due to random drift) with frequencies that roughly reflect the mutation effects on stability ("pseudo-equilibrium hypothesis"). We have found that all glutamate-->aspartate mutations in wild-type Escherichia coli thioredoxin are destabilizing, as well as most of the aspartate-->glutamate mutations. Furthermore, the effect of these mutations on thioredoxin thermodynamic stability shows a robust correlation with the frequencies of occurrence of the involved residues in several-hundred sequence alignments derived from a BLAST search. These results provide direct and quantitative experimental evidence for the pseudo-equilibrium hypothesis and should have general consequences for the interpretation of mutation effects on protein stability, as they suggest that residue environments in proteins may be optimized for stabilizing interactions to a remarkable degree of specificity. We also provide evidence that such stabilizing interactions may be detected in sequence alignments, and briefly discuss the implications of this possibility for the derivation of structural information (on native and denatured states) from comparative sequence analyses.     

Cross-trait prediction accuracy of summary statistics in genome-wide association studies

In the era of big data, univariate models have widely been used as a workhorse tool for quickly producing marginal estimators; and this is true even when in a high-dimensional dense setting, in which many features are “true,” but weak signals. Genome-wide association studies (GWAS) epitomize this type of setting. Although the GWAS marginal estimator is popular, it has long been criticized for ignoring the correlation structure of genetic variants (i.e., the linkage disequilibrium [LD] pattern). In this paper, we study the effects of LD pattern on the GWAS marginal estimator and investigate whether or not additionally accounting for the LD can improve the prediction accuracy of complex traits. We consider a general high-dimensional dense setting for GWAS and study a class of ridge-type estimators, including the popular marginal estimator and the best linear unbiased prediction (BLUP) estimator as two special cases. We show that the performance of GWAS marginal estimator depends on the LD pattern through the first three moments of its eigenvalue distribution. Furthermore, we uncover that the relative performance of GWAS marginal and BLUP estimators highly depends on the ratio of GWAS sample size over the number of genetic variants. Particularly, our finding reveals that the marginal estimator can easily become near-optimal within this class when the sample size is relatively small, even though it ignores the LD pattern. On the other hand, BLUP estimator has substantially better performance than the marginal estimator as the sample size increases toward the number of genetic variants, which is typically in millions. Therefore, adjusting for the LD (such as in the BLUP) is most needed when GWAS sample size is large. We illustrate the importance of our results by using the simulated data and real GWAS.     

Significance of butterfly eyespots as an anti-predator device in ground-based and aerial attacks

Many butterfly genera are characterised by the presence of marginal eyespots on their wings. One hypothesis to account for an occurrence of eyespots is that these wing pattern elements are partly the outcome of visual selection by predators. Bicyclus anynana (Satyrinae) has underside spotting on its wings but there is also a seasonal form in which the eyespots are reduced in size or totally absent. This natural variation gives us a useful tool to test the hypothesis that marginal eyespot patterns can decoy the attacking predator by, at least sometimes, diverting attack from vital body parts to the edges of the wings. We used lizards, Anolis carolinensis , and pied flycatchers, Ficedula hypoleuca , as predators for living spotted and spotless B. anynana . The presence of eyespots did not increase the escape probability of resting butterflies once captured (even a form with enlarged eyespots did not add to effective deflection of attacks). There was also no evidence that eyespots influenced the location of strikes by the predators. This study thus provides no support that marginal eyespot patterns can act as an effective deflection mechanism to avoid lizard or avian predation.     

Multiple two-sample testing under arbitrary covariance dependency with an application in imaging mass spectrometry

Large-scale hypothesis testing has become a ubiquitous problem in high-dimensional statistical inference, with broad applications in various scientific disciplines. One relevant application is constituted by imaging mass spectrometry (IMS) association studies, where a large number of tests are performed simultaneously in order to identify molecular masses that are associated with a particular phenotype, for example, a cancer subtype. Mass spectra obtained from matrix-assisted laser desorption/ionization (MALDI) experiments are dependent, when considered as statistical quantities. False discovery proportion (FDP) estimation and  control under arbitrary dependency structure among test statistics is an active topic in modern multiple testing research. In this context, we are concerned with the evaluation of associations between the binary outcome variable (describing the phenotype) and multiple predictors derived from MALDI measurements. We propose an inference procedure in which the correlation matrix of the test statistics is utilized. The approach is based on multiple marginal models. Specifically, we fit a marginal logistic regression model for each predictor individually. Asymptotic joint normality of the stacked vector of the marginal regression coefficients is established under standard regularity assumptions, and their (limiting) correlation matrix is estimated. The proposed method extracts common factors from the resulting empirical correlation matrix. Finally, we estimate the realized FDP of a thresholding procedure for the marginal p-values. We demonstrate a practical application of the proposed workflow to MALDI IMS data in an oncological context.     

Marginal tests with sliced average variance estimation

We present a new computationally feasible test for the dimensionof the central subspace in a regression problem based on slicedaverage variance estimation. We also provide a marginal coordinatetest. Under the null hypothesis, both the test of dimensionand the marginal coordinate test involve test statistics thatasymptotically have chi-squared distributions given normallydistributed predictors, and have a distribution that is a linearcombination of chi-squared distributions in general.     

Conservative patterns

Conservative patterns dominate the pattern formation of systems governed by competition and limited resources. Using a simple mechanical model the principle of marginal stability, period doubling in space, the existence of conserved values and relations to biological pattern formation are discussed.Presented at the 9th Cybernetics-Congress, Göttingen, March 1986Part of Ph.D.-Thesis     

Stochastic amplification of spatial modes in a system with one diffusing species

The problem of pattern formation in a generic two species reaction–diffusion model is studied, under the hypothesis that only one species can diffuse. For such a system, the classical Turing instability cannot take place. At variance, by working in the generalized setting of a stochastic formulation to the inspected problem, spatially organized patterns can develop, seeded by finite size corrections. General conditions are given for the stochastic patterns to occur. The predictions of the theory are tested for a specific case study.     

Dynamic membrane structure induces temporal pattern formation

The understanding of temporal pattern formation in biological systems is essential for insights into regulatory processes of cells. Concerning this problem, the present work introduces a model to explain the attachment/detachment cycle of MARCKS and PKC at the cell membrane, which is crucial for signal transduction processes. Our model is novel with regard to its driving mechanism: Structural changes within the membrane fuel an activator–inhibitor based global density oscillation of membrane related proteins. Based on simulated results of our model, phase diagrams were generated to illustrate the interplay of MARCKS and PKC. They predict the oscillatory behavior in the form of the number of peaks, the periodic time, and the damping constant depending on the amounts of MARCKS and PKC, respectively. The investigation of the phase space also revealed an unexpected intermediate state prior to the oscillations for high amounts of MARCKS in the system. The validation of the obtained results was carried out by stability analysis, which also accounts for further enhanced understanding of the studied system. It was shown, that the occurrence of the oscillating behavior is independent of the diffusion and the consumption of the reactants. The diffusion terms in the used reaction–diffusion equations only act as modulating terms and are not required for the oscillation. The hypothesis of our work suggests a new mechanism of temporal pattern formation in biological systems. This mechanism includes a classical activator–inhibitor system, but is based on the modifications of the membrane structure, rather than a reaction–diffusion system.     

Small mammal species richness and turnover along elevational gradient in Yulong Mountain,Yunnan, Southwest China

Understanding the species diversity patterns along elevational gradients is critical for biodiversity conservation in mountainous regions. We examined the elevational patterns of species richness and turnover, and evaluated the effects of spatial and environmental factors on nonvolant small mammals (hereafter “small mammal”) predicted a priori by alternative hypotheses (mid‐domain effect [MDE], species–area relationship [SAR], energy, environmental stability, and habitat complexity]) proposed to explain the variation of diversity. We designed a standardized sampling scheme to trap small mammals at ten elevational bands across the entire elevational gradient on Yulong Mountain, southwest China. A total of 1,808 small mammals representing 23 species were trapped. We observed the hump‐shaped distribution pattern of the overall species richness along elevational gradient. Insectivores, rodents, large‐ranged species, and endemic species richness showed the general hump‐shaped pattern but peaked at different elevations, whereas the small‐ranged species and endemic species favored the decreasing richness pattern. The MDE and the energy hypothesis were supported, whereas little support was found for the SAR, the environmental stability hypothesis, and the habitat complexity. However, the primary driver(s) for richness patterns differed among the partitioning groups, with NDVI (the normalized difference vegetation index) and MDE being the most important variables for the total richness pattern. Species turnover for all small mammal groups increased with elevation, and it supported a decrease in community similarity with elevational distance. Our results emphasized for increased conservation efforts in the higher elevation regions of the Yulong Mountain.