Modelling delta-notch perturbations during zebrafish somitogenesis

The discovery over the last 15 years of molecular clocks and gradients in the pre-somitic mesoderm of numerous vertebrate species has added significant weight to Cooke and Zeeman's ‘clock and wavefront’ model of somitogenesis, in which a travelling wavefront determines the spatial position of somite formation and the somitogenesis clock controls periodicity (Cooke and Zeeman, 1976). However, recent high-throughput measurements of spatiotemporal patterns of gene expression in different zebrafish mutant backgrounds allow further quantitative evaluation of the clock and wavefront hypothesis. In this study we describe how our recently proposed model, in which oscillator coupling drives the propagation of an emergent wavefront, can be used to provide mechanistic and testable explanations for the following observed phenomena in zebrafish embryos: (a) the variation in somite measurements across a number of zebrafish mutants; (b) the delayed formation of somites and the formation of ‘salt and pepper’ patterns of gene expression upon disruption of oscillator coupling; and (c) spatial correlations in the ‘salt and pepper’ patterns in Delta-Notch mutants. In light of our results, we propose a number of plausible experiments that could be used to further test the model

Macroscopic limits of individual-based models for motile cell populations with volume exclusion

Partial differential equation models are ubiquitous in studies of motile cell populations, giving a phenomenological description of events which can be analyzed and simulated using a wide range of existing tools. However, these models are seldom derived from individual cell behaviors and so it is difficult to accurately include biological hypotheses on this spatial scale. Moreover, studies which do attempt to link individual- and population-level behavior generally employ lattice-based frameworks in which the artifacts of lattice choice at the population level are unclear. In this work we derive limiting population-level descriptions of a motile cell population from an off-lattice, individual-based model (IBM) and investigate the effects of volume exclusion on the population-level dynamics. While motility with excluded volume in on-lattice IBMs can be accurately described by Fickian diffusion, we demonstrate that this is not the case off lattice. We show that the balance between two key parameters in the IBM (the distance moved in one step and the radius of an individual) determines whether volume exclusion results in enhanced or slowed diffusion. The magnitude of this effect is shown to increase with the number of cells and the rate of their movement. The method we describe is extendable to higher-dimensional and more complex systems and thereby provides a framework for deriving biologically realistic, continuum descriptions of motile populations

Distinguishing graded & ultrasensitive signalling cascade kinetics by the shape of morphogen gradients in Drosophila

Recently, signalling gradients in cascades of two-state reaction–diffusion systems were described as a model for understanding key biochemical mechanisms that underlie development and differentiation processes in the Drosophila embryo. Diffusion-trapping at the exterior of the cell membrane triggers the mitogen-activated protein kinase (MAPK) cascade to relay an appropriate signal from the membrane to the inner part of the cytosol, whereupon another diffusion-trapping mechanism involving the nucleus reads out this signal to trigger appropriate changes in gene expression. Proposed mathematical models exhibit equilibrium distributions consistent with experimental measurements of key spatial gradients in these processes. A significant property of the formulation is that the signal is assumed to be relayed from one system to the next in a linear fashion. However, the MAPK cascade often exhibits nonlinear dose–response properties and the final remark of Berezhkovskii et al. (2009) is that this assumption remains an important property to be tested experimentally, perhaps via a new quantitative assay across multiple genetic backgrounds. In anticipation of the need to be able to sensibly interpret data from such experiments, here we provide a complementary analysis that recovers existing formulae as a special case but is also capable of handling nonlinear functional forms. Predictions of linear and nonlinear signal relays and, in particular, graded and ultrasensitive MAPK kinetics, are compared

The Taxonomy of Algebraic Surfaces

The problems of taxonomy are problems of order. Any discrete set can be arranged in linear order but it does not follow that any linear order is satisfactory. The separation of natural neighbors may be inevitable. Examples are Linnaeus\u27 botanical classification, or the arrangement of logical classes (abed, abcd, ----) where a natural arrangement applies in general surfaces of connectivity greater than one, or n-dimensional space. Any number of interrelations of a discrete finite set can be indicated by a three dimensional model where the elements are points and the relations say colored lines, as in a Cayley color group abstracted from a surface

Power spectra methods for a stochastic description of diffusion on deterministically growing domains

A central challenge in developmental biology is understanding the creation of robust spatiotemporal heterogeneity. Generally, the mathematical treatments of biological systems have used continuum, mean-field hypotheses for their constituent parts, which ignores any sources of intrinsic stochastic effects. In this paper we consider a stochastic space-jump process as a description of diffusion, i.e., particles are able to undergo a random walk on a discretized domain. By developing analytical Fourier methods we are able to probe this probabilistic framework, which gives us insight into the patterning potential of diffusive systems. Further, an alternative description of domain growth is introduced, with which we are able to rigorously link the mean-field and stochastic descriptions. Finally, through combining these ideas, it is shown that such stochastic descriptions of diffusion on a deterministically growing domain are able to support the nucleation of states that are far removed from the deterministic mean-field steady state

Stochastic reaction & diffusion on growing domains: understanding the breakdown of robust pattern formation

Many biological patterns, from population densities to animal coat markings, can be thought of as heterogeneous spatiotemporal distributions of mobile agents. Many mathematical models have been proposed to account for the emergence of this complexity, but, in general, they have consisted of deterministic systems of differential equations, which do not take into account the stochastic nature of population interactions. One particular, pertinent criticism of these deterministic systems is that the exhibited patterns can often be highly sensitive to changes in initial conditions, domain geometry, parameter values, etc. Due to this sensitivity, we seek to understand the effects of stochasticity and growth on paradigm biological patterning models. In this paper, we extend spatial Fourier analysis and growing domain mapping techniques to encompass stochastic Turing systems. Through this we find that the stochastic systems are able to realize much richer dynamics than their deterministic counterparts, in that patterns are able to exist outside the standard Turing parameter range. Further, it is seen that the inherent stochasticity in the reactions appears to be more important than the noise generated by growth, when considering which wave modes are excited. Finally, although growth is able to generate robust pattern sequences in the deterministic case, we see that stochastic effects destroy this mechanism for conferring robustness. However, through Fourier analysis we are able to suggest a reason behind this lack of robustness and identify possible mechanisms by which to reclaim it