The influence of gene expression time delays on Gierer-Meinhardt pattern formation systems.

There are numerous examples of morphogen gradients controlling long range signalling in developmental and cellular systems. The prospect of two such interacting morphogens instigating long range self-organisation in biological systems via a Turing bifurcation has been explored, postulated, or implicated in the context of numerous developmental processes. However, modelling investigations of cellular systems typically neglect the influence of gene expression on such dynamics, even though transcription and translation are observed to be important in morphogenetic systems. In particular, the influence of gene expression on a large class of Turing bifurcation models, namely those with pure kinetics such as the Gierer-Meinhardt system, is unexplored. Our investigations demonstrate that the behaviour of the Gierer-Meinhardt model profoundly changes on the inclusion of gene expression dynamics and is sensitive to the sub-cellular details of gene expression. Features such as concentration blow up, morphogen oscillations and radical sensitivities to the duration of gene expression are observed and, at best, severely restrict the possible parameter spaces for feasible biological behaviour. These results also indicate that the behaviour of Turing pattern formation systems on the inclusion of gene expression time delays may provide a means of distinguishing between possible forms of interaction kinetics. Finally, this study also emphasises that sub-cellular and gene expression dynamics should not be simply neglected in models of long range biological pattern formation via morphogens.

Modelling and analysis of planar cell polarity.

Planar cell polarity (PCP) occurs in the epithelia of many animals and can lead to the alignment of hairs, bristles, and feathers. Here, we present two approaches to modelling this phenomenon. The aim is to discover the basic mechanisms that drive PCP, while keeping the models mathematically tractable. We present a feedback and diffusion model, in which adjacent cell sides of neighbouring cells are coupled by a negative feedback loop and diffusion acts within the cell. This approach can give rise to polarity, but also to period two patterns. Polarisation arises via an instability provided a sufficiently strong feedback and sufficiently weak diffusion. Moreover, we discuss a conservative model in which proteins within a cell are redistributed depending on the amount of proteins in the neighbouring cells, coupled with intracellular diffusion. In this case, polarity can arise from weakly polarised initial conditions or via a wave provided the diffusion is weak enough. Both models can overcome small anomalies in the initial conditions. Furthermore, the range of the effects of groups of cells with different properties than the surrounding cells depends on the strength of the initial global cue and the intracellular diffusion.

Gene expression time delays and Turing pattern formation systems.

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.

Unravelling Nature's networks.

Dramatic progress has been made recently in determining the genetic and molecular composition of cells. This has prompted the development of new approaches to the challenge of understanding how basic cellular mechanisms are coordinated to produce the dazzling complexity of living systems. To face this challenge fully, it is critical not only to know what genes and proteins are expressed in cells, but also to understand the spatiotemporal dynamics of their networks of interactions. The sheer scale and complexity of cellular interaction networks necessitates a multi-disciplinary effort in which sophisticated experimental techniques are employed in combination with computational analysis and mathematical modelling. Such approaches are beginning to provide insight into basic structures and mechanisms, and promise to become critical to the post-genomic mission of understanding the cell as a complex dynamical system.

On the structure of protein-protein interaction networks.

We present a simple model for the underlying structure of protein-protein pairwise interaction graphs that is based on the way in which proteins attach to each other in experiments such as yeast two-hybrid assays. We show that data on the interactions of human proteins lend support to this model. The frequency of the number of connections per protein under this model does not follow a power law, in contrast to the reported behaviour of data from large-scale yeast two-hybrid screens of yeast protein-protein interactions. Sampling sub-graphs from the underlying graphs generated with our model, in a way analogous to the sampling performed in large-scale yeast two-hybrid searches, gives degree distributions that differ subtly from the power law and that fit the observed data better than the power law itself. Our results show that the observation of approximate power law behaviour in a sampled sub-graph does not imply that the underlying graph follows a power law.

A model of primitive streak initiation in the chick embryo.

Initiation of the primitive streak in avian embryos provides a well-studied example of a pattern-forming event that displays a striking capacity for regulation. The mechanisms underlying the regulative properties are, however, poorly understood and are not easily accounted for by traditional models of pattern formation, such as reaction-diffusion models. In this paper, we propose a new activator-inhibitor model for streak initiation. We show that the model is consistent with experimental observations, both in its pattern-forming properties and in its ability to form these patterns on the correct time-scales for biologically realistic parameter values. A key component of the model is a travelling wave of inhibition. We present a mathematical analysis of the speed of such waves in both diffusive and juxtacrine relay systems. We use the streak initiation model to make testable predictions. By varying parameters of the model, two very different types of patterning can be obtained, suggesting that our model may be applicable to other processes in addition to streak initiation.

Elegant hypothesis and inelegant fact in developmental biology.

The relationship between theoretical and experimental approaches to the problem of pattern generation during embryonic development has often been uneasy. This stems at least in part from the different emphases that have typically been used in the two approaches. The spectacular success of modern genetic techniques in uncovering developmental mechanisms has led to a widespread belief that theory is no longer very relevant. However, recent examples of data-driven modelling point to new roles for theoretical approaches in exploring important issues such as the robustness and evolution of pattern-forming mechanisms.

Restricted-range gradients and travelling fronts in a model of juxtacrine cell relay.

Patterning events in development often depend on the transmission over a range of several cell diameters of signals emanating from a localized source. Experimental studies of such long-range signalling by members of the TGF-beta family of growth factors suggests that a cell-relay mechanism in which cells signal only with their immediate neighbours (i.e., juxtacrine signalling) may be operating in some tissues. Here, this possibility is investigated through the analysis of a model of juxtacrine signalling. Depending on the strength of the signal relay between cells, a localized signal source can generate either stable gradients or travelling fronts of cell activation. Both of these behaviors could in principle be involved in the long-range transmission of signals and patterning of cell fates by cell relays. There are significant and surprising differences between the gradients generated by the mechanism studied here, and those generated by the diffusion of a morphogen. In particular, there is an upper limit on the distance over which any given level of cell activation can be attained in a relay-mediated gradient. irrespective of the strength of signal source.

Cell communities and robustness in development.

The robustness of patterning events in development is a key feature that must be accounted for in proposed models of these events. When considering explicitly cellular systems, robustness can be exhibited at different levels of organization. Consideration of two widespread patterning mechanisms suggests that robustness at the level of cell communities can result from variable development at the level of individual cells; models of these mechanisms show how interactions between participating cells guarantee community-level robustness. Cooperative interactions enhance homogeneity within communities of like cells and the sharpness of boundaries between communities of distinct cells, while competitive interactions amplify small inhomogeneities within communities of initially equivalent cells, resulting in fine-grained patterns of cell specialization.

The community effect and ectoderm-mesoderm interaction in Xenopus muscle differentiation.

Community effects are believed to play an important role in the patterning of many tissues during development. They involve an interaction between neighbouring equivalent cells that is necessary for them to proceed to their fully differentiated state. However, the mechanisms underlying these effects remain unclear. In this paper, diffusion-based mathematical models are constructed and analysed in order to study possible mechanisms for the community effect in Xenopus muscle differentiation. These models differ from each other in the assumptions that are made about the nature of an inhibitory effect that ectodermal tissue has been observed to have on muscle differentiation. It is possible to construct consistent models based on all the forms of inhibition considered. However, each model requires the diffusible factors on which it is based to have different properties. The current data from tissues reaggregate experiments are insufficient to determine the mechanisms underlying the community effect; the work presented here suggests that quantitative analysis of a further series of reaggregate experiments will make it possible to distinguish between the proposed mechanisms.

Pattern formation by lateral inhibition with feedback: a mathematical model of delta-notch intercellular signalling.

In many developing tissues, adjacent cells diverge in character so as to create a fine-grained pattern of cells in contrasting states of differentiation. It has been proposed that such patterns can be generated through lateral inhibition--a type of cell-cell interaction whereby a cell that adopts a particular fate inhibits its immediate neighbors from doing likewise. Lateral inhibition is well documented in flies, worms and vertebrates. In all of these organisms, the transmembrane proteins Notch and Delta (or their homologues) have been identified as mediators of the interaction--Notch as receptor, Delta as its ligand on adjacent cells. However, it is not clear under precisely what conditions the Delta-Notch mechanism of lateral inhibition can generate the observed types of pattern, or indeed whether this mechanism is capable of generating such patterns by itself. Here we construct and analyse a simple and general mathematical model of such contact-mediated lateral inhibition. In accordance with experimental data, the model postulates that receipt of inhibition (i.e. activation of Notch) diminished the ability to deliver inhibition (i.e. to produce active Delta). This gives rise to a feedback loop that can amplify differences between adjacent cells. We investigate the pattern-forming potential and temporal behaviour of this model both analytically and through numerical simulation. Inhomogeneities are self-amplifying and develop without need of any other machinery, provided the feedback is sufficiently strong. For a wide range of initial and boundary conditions, the model generates fine-grained patterns similar to those observed in living systems.

The phenotype in placental trisomy 7.

Trisomy 7, in mosaic state, was identified at chorionic villus sampling. The pregnancy was closely followed, and proceeded uneventfully. Mosaic trisomy 7 was confirmed in the term placenta, the organ having no structural abnormalities; the karyotype of the phenotypically normal baby was 46,XY. Trisomy 7, mosaic or nonmosaic, detected at chorionic villus sampling in an ultrasonographically normal pregnancy, appears typically to be associated with a normal fetal karyotype, and placental growth, structure, and function are not discernibly compromised.

Early prenatal diagnosis of genetic abnormality by chorion villus sampling.

We describe a series of 100 cases of prenatal genetic diagnosis using the technique of chorion villus sampling. The advantage of chorion villus sampling in terms of earlier diagnosis, and its disadvantage of a higher incidence of false results, compared with amniocentesis, are noted. The comparative risks for pregnancy loss are discussed.

Trisomy 16 detected at chorion villus sampling.

Trisomy 16 detected at chorion villus sampling (CVS) may reflect the placental but not the fetal karyotype. We describe a case in which the pregnancy continued until intrauterine death at 37 weeks. Cytogenetic study of two placental samples showed 47, + 16 and 46,XX; the fetus was morphologically grossly normal, but fetal tissue culture was unsuccessful. Conservative management may be appropriate when trisomy 16 is detected at CVS and the pregnancy is normal ultrasonographically.

Ring 21 chromosome: the mild end of the phenotypic spectrum.

The case is reported of a child with the karyotype 46,XY,r(21), who presented with linear growth retardation but who appears, at age 2 years 8 months, to be developing normally mentally. There is a small number of reports of mildly affected cases of r(21), and of some with an apparently completely normal phenotype. We presume a structural and functional cytogenetic heterogeneity underlies the observed phenotypic heterogeneity in the ring 21 spectrum.