Realistic boundary conditions for stochastic simulations of reaction-diffusion processes

Many cellular and subcellular biological processes can be described in terms of diffusing and chemically reacting species (e.g. enzymes). Such reaction-diffusion processes can be mathematically modelled using either deterministic partial-differential equations or stochastic simulation algorithms. The latter provide a more detailed and precise picture, and several stochastic simulation algorithms have been proposed in recent years. Such models typically give the same description of the reaction-diffusion processes far from the boundary of the simulated domain, but the behaviour close to a reactive boundary (e.g. a membrane with receptors) is unfortunately model-dependent. In this paper, we study four different approaches to stochastic modelling of reaction-diffusion problems and show the correct choice of the boundary condition for each model. The reactive boundary is treated as partially reflective, which means that some molecules hitting the boundary are adsorbed (e.g. bound to the receptor) and some molecules are reflected. The probability that the molecule is adsorbed rather than reflected depends on the reactivity of the boundary (e.g. on the rate constant of the adsorbing chemical reaction and on the number of available receptors), and on the stochastic model used. This dependence is derived for each model.Comment: 24 pages, submitted to Physical Biolog

R-Smad Competition Controls Activin Receptor Output in Drosophila

Animals use TGF-β superfamily signal transduction pathways during development and tissue maintenance. The superfamily has traditionally been divided into TGF-β/Activin and BMP branches based on relationships between ligands, receptors, and R-Smads. Several previous reports have shown that, in cell culture systems, “BMP-specific” Smads can be phosphorylated in response to TGF-β/Activin pathway activation. Using Drosophila cell culture as well as in vivo assays, we find that Baboon, the Drosophila TGF-β/Activin-specific Type I receptor, can phosphorylate Mad, the BMP-specific R-Smad, in addition to its normal substrate, dSmad2. The Baboon-Mad activation appears direct because it occurs in the absence of canonical BMP Type I receptors. Wing phenotypes generated by Baboon gain-of-function require Mad, and are partially suppressed by over-expression of dSmad2. In the larval wing disc, activated Baboon cell-autonomously causes C-terminal Mad phosphorylation, but only when endogenous dSmad2 protein is depleted. The Baboon-Mad relationship is thus controlled by dSmad2 levels. Elevated P-Mad is seen in several tissues of dSmad2 protein-null mutant larvae, and these levels are normalized in dSmad2; baboon double mutants, indicating that the cross-talk reaction and Smad competition occur with endogenous levels of signaling components in vivo. In addition, we find that high levels of Activin signaling cause substantial turnover in dSmad2 protein, providing a potential cross-pathway signal-switching mechanism. We propose that the dual activity of TGF-β/Activin receptors is an ancient feature, and we discuss several ways this activity can modulate TGF-β signaling output

BMP signaling components in embryonic transcriptomes of the hover fly Episyrphus balteatus (Syrphidae)

<p>Abstract</p> <p>Background</p> <p>In animals, signaling of Bone Morphogenetic Proteins (BMPs) is essential for dorsoventral (DV) patterning of the embryo, but how BMP signaling evolved with changes in embryonic DV differentiation is largely unclear. Based on the extensive knowledge of BMP signaling in <it>Drosophila melanogaster</it>, the morphological diversity of extraembryonic tissues in different fly species provides a comparative system to address this question. The closest relatives of <it>D. melanogaster </it>with clearly distinct DV differentiation are hover flies (Diptera: Syrphidae). The syrphid <it>Episyrphus balteatus </it>is a commercial bio-agent against aphids and has been established as a model organism for developmental studies and chemical ecology. The dorsal blastoderm of <it>E. balteatus </it>gives rise to two extraembryonic tissues (serosa and amnion), whereas in <it>D. melanogaster</it>, the dorsal blastoderm differentiates into a single extraembryonic epithelium (amnioserosa). Recent studies indicate that several BMP signaling components of <it>D. melanogaster</it>, including the BMP ligand Screw (Scw) and other extracellular regulators, evolved in the dipteran lineage through gene duplication and functional divergence. These findings raise the question of whether the complement of BMP signaling components changed with the origin of the amnioserosa.</p> <p>Results</p> <p>To search for BMP signaling components in <it>E. balteatus</it>, we generated and analyzed transcriptomes of freshly laid eggs (0-30 minutes) and late blastoderm to early germband extension stages (3-6 hours) using Roche/454 sequencing. We identified putative <it>E. balteatus </it>orthologues of 43% of all annotated <it>D. melanogaster </it>genes, including the genes of all BMP ligands and other BMP signaling components.</p> <p>Conclusion</p> <p>The diversification of several BMP signaling components in the dipteran linage of <it>D. melanogaster </it>preceded the origin of the amnioserosa.</p> <p>[Transcriptome sequence data from this study have been deposited at the NCBI Sequence Read Archive (SRP005289); individually assembled sequences have been deposited at GenBank (<ext-link ext-link-id="JN006969" ext-link-type="gen">JN006969</ext-link>-<ext-link ext-link-id="JN006986" ext-link-type="gen">JN006986</ext-link>).]</p

Binding between Crossveinless-2 and Chordin Von Willebrand Factor Type C Domains Promotes BMP Signaling by Blocking Chordin Activity

BACKGROUND: Crossveinless-2 (CV2) is an extracellular BMP modulator protein of the Chordin family, which can either enhance or inhibit BMP activity. CV2 binds to BMP2 via subdomain 1 of the first of its five N-terminal von Willebrand factor type C domains (VWC1). Previous studies showed that this BMP binding is required for the anti-, but not for the pro-BMP effect of CV2. More recently, it was shown that CV2 can also bind to the BMP inhibitor Chordin. However, it remained unclear which domains mediate this binding, and whether it accounts for an anti- or pro-BMP effect. PRINCIPAL FINDINGS: Here we report that a composite interface of CV2 consisting of subdomain 2 of VWC1 and of VWC2-4, which are dispensable for BMP binding, binds to the VWC2 domain of Chordin. Functional data obtained in zebrafish embryos indicate that this binding of Chordin is required for CV2's pro-BMP effect, which actually is an anti-Chordin effect and, at least to a large extent, independent of Tolloid-mediated Chordin degradation. We further demonstrate that CV2 mutant versions that per se are incapable of BMP binding can attenuate the Chordin/BMP interaction. CONCLUSIONS: We have physically dissected the anti- and pro-BMP effects of CV2. Its anti-BMP effect is obtained by binding to BMP via subdomain1 of the VWC1 domain, a binding that occurs in competition with Chordin. In contrast, its pro-BMP effect is achieved by direct binding to Chordin via subdomain 2 of VWC1 and VWC2-4. This binding seems to induce conformational changes within the Chordin protein that weaken Chordin's affinity to BMP. We propose that in ternary Chordin-CV2-BMP complexes, both BMP and Chordin are directly associated with CV2, whereas Chordin is pushed away from BMP, ensuring that BMPs can be more easily delivered to their receptors

Mathematics and biology: a Kantian view on the history of pattern formation theory

Driesch’s statement, made around 1900, that the physics and chemistry of his day were unable to explain self-regulation during embryogenesis was correct and could be extended until the year 1972. The emergence of theories of self-organisation required progress in several areas including chemistry, physics, computing and cybernetics. Two parallel lines of development can be distinguished which both culminated in the early 1970s. Firstly, physicochemical theories of self-organisation arose from theoretical (Lotka 1910–1920) and experimental work (Bray 1920; Belousov 1951) on chemical oscillations. However, this research area gained broader acceptance only after thermodynamics was extended to systems far from equilibrium (1922–1967) and the mechanism of the prime example for a chemical oscillator, the Belousov–Zhabotinski reaction, was deciphered in the early 1970s. Secondly, biological theories of self-organisation were rooted in the intellectual environment of artificial intelligence and cybernetics. Turing wrote his The chemical basis of morphogenesis (1952) after working on the construction of one of the first electronic computers. Likewise, Gierer and Meinhardt’s theory of local activation and lateral inhibition (1972) was influenced by ideas from cybernetics. The Gierer–Meinhardt theory provided an explanation for the first time of both spontaneous formation of spatial order and of self-regulation that proved to be extremely successful in elucidating a wide range of patterning processes. With the advent of developmental genetics in the 1980s, detailed molecular and functional data became available for complex developmental processes, allowing a new generation of data-driven theoretical approaches. Three examples of such approaches will be discussed. The successes and limitations of mathematical pattern formation theory throughout its history suggest a picture of the organism, which has structural similarity to views of the organic world held by the philosopher Immanuel Kant at the end of the eighteenth century

Patterning of the Dorsal-Ventral Axis in Echinoderms: Insights into the Evolution of the BMP-Chordin Signaling Network

Formation of the dorsal-ventral axis of the sea urchin embryo relies on cell interactions initiated by the TGF beta Nodal. Intriguingly, although nodal expression is restricted to the ventral side of the embryo, Nodal function is required for specification of both the ventral and the dorsal territories and is able to restore both ventral and dorsal regions in nodal morpholino injected embryos. The molecular basis for the long-range organizing activity of Nodal is not understood. In this paper, we provide evidence that the long-range organizing activity of Nodal is assured by a relay molecule synthesized in the ventral ectoderm, then translocated to the opposite side of the embryo. We identified this relay molecule as BMP2/4 based on the following arguments. First, blocking BMP2/4 function eliminated the long-range organizing activity of an activated Nodal receptor in an axis rescue assay. Second, we demonstrate that BMP2/4 and the corresponding type I receptor Alk3/6 functions are both essential for specification of the dorsal region of the embryo. Third, using anti-phospho-Smad1/5/8 immunostaining, we show that, despite its ventral transcription, the BMP2/4 ligand triggers receptor mediated signaling exclusively on the dorsal side of the embryo, one of the most extreme cases of BMP translocation described so far. We further report that the pattern of pSmad1/5/8 is graded along the dorsal-ventral axis and that two BMP2/4 target genes are expressed in nested patterns centered on the region with highest levels of pSmad1/5/8, strongly suggesting that BMP2/4 is acting as a morphogen. We also describe the very unusual ventral co-expression of chordin and bmp2/4 downstream of Nodal and demonstrate that Chordin is largely responsible for the spatial restriction of BMP2/4 signaling to the dorsal side. Thus, unlike in most organisms, in the sea urchin, a single ventral signaling centre is responsible for induction of ventral and dorsal cell fates. Finally, we show that Chordin may not be required for long-range diffusion of BMP2/4, describe a striking dorsal-ventral asymmetry in the expression of Glypican 5, a heparin sulphated proteoglycan that regulates BMP mobility, and show that this asymmetry depends on BMP2/4 signaling. Our study provides new insights into the mechanisms by which positional information is established along the dorsal-ventral axis of the sea urchin embryo, and more generally on how a BMP morphogen gradient is established in a multicellular embryo. From an evolutionary point of view, it highlights that although the genes used for dorsal-ventral patterning are highly conserved in bilateria, there are considerable variations, even among deuterostomes, in the manner these genes are used to shape a BMP morphogen gradient

The Drosophila DPP signal is produced by cleavage of its proprotein at evolutionary diversified furin-recognition sites

Maturation of bone morphogenetic proteins (BMPs) requires cleavage of their precursor proteins by furin-type proprotein convertases. Here, we find that cleavage sites of the BMP2/4/decapentaplegic (DPP) subfamily have been evolutionary diversified and can be categorized into 4 different types. Cnidaria BMP2/4/DPP is considered to be a prototype containing only 1 furin site. Bilateria BMP2/4/DPP acquired an additional cleavage site with either the combination of minimal–optimal or optimal–optimal furin sites. DPPs belonging to Diptera, such as Drosophila and mosquito, and Lepidoptera of silkworm contain a third cleavage site between the 2 optimal furin sites. We studied how the 3 furin sites (FSI–III) of Drosophila DPP coordinate maturation of ligands and contribute to signals in vivo. Combining mutational analysis of furin-recognition sites and RNAi experiments, we found that the Drosophila DPP precursor is initially cleaved at an upstream furin-recognition site (FSII), with consequent cleavages at 2 furin sites (FSI and FSIII). Both Dfurin1 and Dfurin2 are involved in the processing of DPP proproteins. Biochemical and genetic analyses using cleavage mutants of DPP suggest the first cleavage at FSII to be critical and sufficient for long-range DPP signaling. Our data suggest that the Drosophila DPP precursor is cleaved in a different manner from vertebrate BMP4 even though they are functional orthologs. This indicates that the furin-cleavage sites in BMP2/4/DPP precursors are tolerant to mutations acquired through evolution and have adapted to different systems in diversified species

Fat facets deubiquitylation of Medea/Smad4 modulates interpretation of a Dpp morphogen gradient.

The ability of secreted Transforming Growth Factor beta (TGF beta) proteins to act as morphogens dictates that their influence be strictly regulated. Here, we report that maternally contributed fat facets (faf; a homolog of USP9X/FAM) is essential for proper interpretation of the zygotic Decapentaplegic (Dpp) morphogen gradient that patterns the embryonic dorsal-ventral axis. The data suggest that the loss of faf reduces the activity of Medea (a homolog of Smad4) below the minimum necessary for adequate Dpp signaling and that this is likely due to excessive ubiquitylation on a specific lysine. This study supports the hypothesis that the control of cellular responsiveness to TGF beta signals at the level of Smad4 ubiquitylation is a conserved mechanism required for proper implementation of a morphogen gradient