Bistability and Oscillations in the Huang-Ferrell Model of MAPK Signaling

Physicochemical models of signaling pathways are characterized by high levels of structural and parametric uncertainty, reflecting both incomplete knowledge about signal transduction and the intrinsic variability of cellular processes. As a result, these models try to predict the dynamics of systems with tens or even hundreds of free parameters. At this level of uncertainty, model analysis should emphasize statistics of systems-level properties, rather than the detailed structure of solutions or boundaries separating different dynamic regimes. Based on the combination of random parameter search and continuation algorithms, we developed a methodology for the statistical analysis of mechanistic signaling models. In applying it to the well-studied MAPK cascade model, we discovered a large region of oscillations and explained their emergence from single-stage bistability. The surprising abundance of strongly nonlinear (oscillatory and bistable) input/output maps revealed by our analysis may be one of the reasons why the MAPK cascade in vivo is embedded in more complex regulatory structures. We argue that this type of analysis should accompany nonlinear multiparameter studies of stationary as well as transient features in network dynamics

Time and length scales of autocrine signals in three dimensions

A model of autocrine signaling in cultures of suspended cells is developed on the basis of the effective medium approximation. The fraction of autocrine ligands, the mean and distribution of distances traveled by paracrine ligands before binding, as well as the mean and distribution of the ligand lifetime are derived. Interferon signaling by dendritic immune cells is considered as an illustration.Comment: 15 page

Self-similar dynamics of morphogen gradients

We discovered a class of self-similar solutions in nonlinear models describing the formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differention in developing tissues. These models account for diffusion and self-induced degration of locally produced chemical signals. When production starts, the signal concentration is equal to zero throughout the system. We found that in the limit of infinitely large signal production strength the solution of this problem is given by the product of the steady state concentration profile and a function of the diffusion similarity variable. We derived a nonlinear boundary value problem satisfied by this function and used a variational approach to prove that this problem has a unique solution in a natural setting. Using the asymptotic behavior of the solutions established by the analysis, we constructed these solutions numerically by the shooting method. Finally, we demonstrated that the obtained solutions may be easily approximated by simple analytical expressions, thus providing an accurate global characterization of the dynamics in an important class of non-linear models of morphogen gradient formation. Our results illustrate the power of analytical approaches to studying nonlinear models of biophysical processes.Comment: 17 pages, 5 figure

Quantifying the Gurken morphogen gradient in Drosophila oogenesis

Quantitative information about the distribution of morphogens is crucial for understanding their effects on cell-fate determination, yet it is difficult to obtain through direct measurements. We have developed a parameter estimation approach for quantifying the spatial distribution of Gurken, a TGFα-like EGFR ligand that acts as a morphogen in Drosophila oogenesis. Modeling of Gurken/EGFR system shows that the shape of the Gurken gradient is controlled by a single dimensionless parameter, the Thiele modulus, which reflects the relative importance of ligand diffusion and degradation. By combining the model with genetic alterations of EGFR levels, we have estimated the value of the Thiele modulus in the wild-type egg chamber. This provides a direct characterization of the shape of the Gurken gradient and demonstrates how parameter estimation techniques can be used to quantify morphogen gradients in development

Gene Regulation by MAPK Substrate Competition

SummaryDeveloping tissues are patterned by coordinated activities of signaling systems, which can be integrated by a regulatory region of a gene that binds multiple transcription factors or by a transcription factor that is modified by multiple enzymes. Based on a combination of genetic and imaging experiments in the early Drosophila embryo, we describe a signal integration mechanism that cannot be reduced to a single gene regulatory element or a single transcription factor. This mechanism relies on an enzymatic network formed by mitogen-activated protein kinase (MAPK) and its substrates. Specifically, anteriorly localized MAPK substrates, such as Bicoid, antagonize MAPK-dependent downregulation of Capicua, a repressor that is involved in gene regulation along the dorsoventral axis of the embryo. MAPK substrate competition provides a basis for ternary interaction of the anterior, dorsoventral, and terminal patterning systems. A mathematical model of this interaction can explain gene expression patterns with both anteroposterior and dorsoventral polarities

RASopathies: unraveling mechanisms with animal models

RASopathies are developmental disorders caused by germline mutations in the Ras-MAPK pathway, and are characterized by a broad spectrum of functional and morphological abnormalities. The high incidence of these disorders (∼1/1000 births) motivates the development of systematic approaches for their efficient diagnosis and potential treatment. Recent advances in genome sequencing have greatly facilitated the genotyping and discovery of mutations in affected individuals, but establishing the causal relationships between molecules and disease phenotypes is non-trivial and presents both technical and conceptual challenges. Here, we discuss how these challenges could be addressed using genetically modified model organisms that have been instrumental in delineating the Ras-MAPK pathway and its roles during development. Focusing on studies in mice, zebrafish and Drosophila, we provide an up-to-date review of animal models of RASopathies at the molecular and functional level. We also discuss how increasingly sophisticated techniques of genetic engineering can be used to rigorously connect changes in specific components of the Ras-MAPK pathway with observed functional and morphological phenotypes. Establishing these connections is essential for advancing our understanding of RASopathies and for devising rational strategies for their management and treatment

Temporal ordering and registration of images in studies of developmental dynamics

Abstract Dynamics of developmental progress is commonly reconstructed from imaging snapshots of chemical or mechanical processes in fixed embryos. As a first step in these reconstructions, snapshots must be spatially registered and ordered in time. Currently, image registration and ordering is often done manually, requiring a significant amount of expertise with a specific system. However, as the sizes of imaging data sets grow, these tasks become increasingly difficult, especially when the images are noisy and the examined developmental changes are subtle. To address these challenges, we present an automated approach to simultaneously register and temporally order imaging data sets. The approach is based on vector diffusion maps, a manifold learning technique that does not require a priori knowledge of image features or a parametric model of the developmental dynamics. We illustrate this approach by registering and ordering data from imaging studies of pattern formation and morphogenesis in three different model systems. We also provide software to aid in the application of our methodology to other experimental data sets

Pattern formation by a moving morphogen source

Abstract During Drosophila melanogaster oogenesis, the follicular epithelium that envelops the germline cyst gives rise to an elaborate eggshell, which houses the future embryo and mediates its interaction with the environment. A prominent feature of the eggshell is a pair of dorsal appendages, which are needed for embryo respiration. Morphogenesis of this structure depends on broad, a zinc-finger transcription factor, regulated by the EGFR pathway. While much has been learned about the mechanisms of broad regulation by EGFR, current understanding of processes that shape the spatial pattern of broad expression is incomplete. We propose that this pattern is defined by two different phases of EGFR activation: an early, posterior-to-anterior gradient of EGFR signaling sets the posterior boundary of broad expression, while the anterior boundary is set by a later phase of EGFR signaling, distributed in a dorsoventral gradient. This model can explain the wild-type pattern of broad in D. melanogaster, predicts how this pattern responds to genetic perturbations, and provides insight into the mechanisms driving diversification of eggshell patterning. The proposed model of the broad expression pattern can be used as a starting point for the quantitative analysis of a large number of gene expression patterns in Drosophila oogenesis