The control of organ size presents a fundamental open problem in biology. A declining growth rate is observed in all studied higher animals, and the growth limiting mechanism may therefore be evolutionary conserved. Most studies of organ growth control have been carried out in Drosophila imaginal discs. We have previously shown that the area growth rate in the Drosophila eye primordium declines inversely proportional to the increase in its area, which is consistent with a dilution mechanism for growth control. Here, we show that a dilution mechanism cannot explain growth control in the Drosophila wing disc. We computationally evaluate a range of alternative candidate mechanisms and show that the experimental data can be best explained by a biphasic growth law. However, also logistic growth and an exponentially declining growth rate fit the data very well. The three growth laws correspond to fundamentally different growth mechanisms that we discuss. Since, as we show, a fit to the available experimental growth kinetics is insufficient to define the underlying mechanism of growth control, future experimental studies must focus on the molecular mechanisms to define the mechanism of growth control.
Patterning and growth are linked during early development and have to be tightly controlled to result in a functional tissue or organ. During the development of the Drosophila eye, this linkage is particularly clear: the growth of the eye primordium mainly results from proliferating cells ahead of the morphogenetic furrow (MF), a moving signaling wave that sweeps across the tissue from the posterior to the anterior side, that induces proliferating cells anterior to it to differentiate and become cell cycle quiescent in its wake. Therefore, final eye disc size depends on the proliferation rate of undifferentiated cells and on the speed with which the MF sweeps across the eye disc. We developed a spatio-temporal model of the growing eye disc based on the regulatory interactions controlled by the signals Decapentaplegic (Dpp), Hedgehog (Hh) and the transcription factor Homothorax (Hth) and explored how the signaling patterns affect the movement of the MF and impact on eye disc growth. We used published and new quantitative data to parameterize the model. In particular, two crucial parameter values, the degradation rate of Hth and the diffusion coefficient of Hh, were measured. The model is able to reproduce the linear movement of the MF and the termination of growth of the primordium. We further show that the model can explain several mutant phenotypes, but fails to reproduce the previously observed scaling of the Dpp gradient in the anterior compartment.
Patterning and growth of a tissue are linked during early development and have to be tightly controlled. During the development of the Drosophila eye, this linkage is particularly clear: A moving signaling wave sweeps across the tissue that will eventually develop into the eye of the fly. This wave is responsible for the transition from cells undergoing cell divisions in front of the wave into differentiated, specialized cells that are not dividing anymore and that eventually develop into the many individual eye units of the compound eye. Therefore, the final size of this tissue depends on how fast cells in front of the wave are growing and dividing and on the speed with which the signaling wave sweeps across the tissue. We developed a computational model based on regulatory interactions that have been experimentally determined in order to explore how the signaling patterns affect the movement of the signaling wave and impact on tissue growth. The model captures the movement of the signaling wave at a constant speed and the growth termination of the developing tissue. We further show that the model can explain the abnormal size of the eye that can be observed in several genetically modified fly strains.
Summary: MATLAB is popular in biological research for creating and simulating models that use ordinary differential equations (ODEs). However, sharing or using these models outside of MATLAB is often problematic. A community standard such as Systems Biology Markup Language (SBML) can serve as a neutral exchange format, but translating models from MATLAB to SBML can be challenging—especially for legacy models not written with translation in mind. We developed MOCCASIN (Model ODE Converter for Creating Automated SBML INteroperability) to help. MOCCASIN can convert ODE-based MATLAB models of biochemical reaction networks into the SBML format.
Availability and implementation: MOCCASIN is available under the terms of the LGPL 2.1 license (http://www.gnu.org/licenses/lgpl-2.1.html). Source code, binaries and test cases can be freely obtained from https://github.com/sbmlteam/moccasin.
Supplementary information: More information is available at https://github.com/sbmlteam/moccasin.
Quantitative data from the Drosophila wing imaginal disc reveals that the amplitude of the Decapentaplegic (Dpp) morphogen gradient increases continuously. It is an open question how cells can determine their relative position within a domain based on a continuously increasing gradient. Here we show that pre-steady state diffusion-based dispersal of morphogens results in a zone within the growing domain where the concentration remains constant over the patterning period. The position of the zone that is predicted based on quantitative data for the Dpp morphogen corresponds to where the Dpp-dependent gene expression boundaries of spalt (sal) and daughters against dpp (dad) emerge. The model also suggests that genes that are scaling and are expressed at lateral positions are either under the control of a different read-out mechanism or under the control of a different morphogen. The patterning mechanism explains the extraordinary robustness that is observed for variations in Dpp production, and offers an explanation for the dual role of Dpp in controlling patterning and growth. Pre-steady-state dynamics are pervasive in morphogen-controlled systems, thus making this a probable general mechanism for the scaled read-out of morphogen gradients in growing developmental systems.
BMP signaling plays a crucial role in the establishment of the dorso-ventral body axis in bilaterally symmetric animals. However, the topologies of the bone morphogenetic protein (BMP) signaling networks vary drastically in different animal groups, raising questions about the evolutionary constraints and evolvability of BMP signaling systems. Using loss-of-function analysis and mathematical modeling, we show that two signaling centers expressing different BMPs and BMP antagonists maintain the secondary axis of the sea anemone Nematostella. We demonstrate that BMP signaling is required for asymmetric Hox gene expression and mesentery formation. Computational analysis reveals that network parameters related to BMP4 and Chordin are constrained both in Nematostella and Xenopus, while those describing the BMP signaling modulators can vary significantly. Notably, only chordin, but not bmp4 expression needs to be spatially restricted for robust signaling gradient formation. Our data provide an explanation of the evolvability of BMP signaling systems in axis formation throughout Eumetazoa.
•A gradient of BMP signaling maintains the directive axis in Nematostella•BMP signaling regulates Nematostella Hox genes and patterns the endoderm•BMP network topology is similar in Nematostella and Xenopus•Constraint analysis reveals the evolvability of BMP network components
Similar to Bilateria, anthozoans possess a secondary directive body axis. Genikhovich et al. examine the bone morphogenetic protein (BMP) signaling network required to compartmentalize the directive axis and regulate staggered Hox gene expression in a sea anemone. Mathematical modeling reveals the constraints guiding the evolution of the axis-forming BMP networks in animals.
Tight spatio-temporal signaling of cytoskeletal and adhesion dynamics is required for localized membrane protrusion that drives directed cell migration. Different ensembles of proteins are therefore likely to get recruited and phosphorylated in membrane protrusions in response to specific cues.
Here, we use an assay that allows to biochemically purify extending protrusions of cells migrating in response to three prototypical receptors: integrins, recepor tyrosine kinases and G-coupled protein receptors. Using quantitative proteomics and phospho-proteomics approaches, we provide evidence for the existence of cue-specific, spatially distinct protein networks in the different cell migration modes.
The integrated analysis of the large-scale experimental data with protein information from databases allows us to understand some emergent properties of spatial regulation of signaling during cell migration. This provides the cell migration community with a large-scale view of the distribution of proteins and phospho-proteins regulating directed cell migration.
Fibroblast; Directional cell migration; Signaling; Proteomics; Phosphorylation
Mathematical modeling of biological networks can help to integrate a large body of information into a consistent framework, which can then be used to arrive at novel mechanistic insight and predictions. We have previously developed a detailed, mechanistic model for the regulation of σ F during sporulation in Bacillus subtilis. The model was based on a wide range of quantitative data, and once fitted to the data, the model made predictions that could be confirmed in experiments. However, the analysis was based on a single optimal parameter set. We wondered whether the predictions of the model would be stable for all optimal parameter sets. To that end we conducted a global parameter screen within the physiological parameter ranges. The screening approach allowed us to identify sensitive and sloppy parameters, and highlighted further required datasets during the optimization. Eventually, all parameter sets that reproduced all available data predicted the physiological situation correctly.
Many organs of higher organisms are heavily branched structures and arise by an apparently similar process of branching morphogenesis. Yet the regulatory components and local interactions that have been identified differ greatly in these organs. It is an open question whether the regulatory processes work according to a common principle and how far physical and geometrical constraints determine the branching process. Here, we review the known regulatory factors and physical constraints in lung, kidney, pancreas, prostate, mammary gland and salivary gland branching morphogenesis, and describe the models that have been formulated to analyse their impacts.
branching; computational modelling; in silico organogenesis
The development of ovarian follicles hinges on the timely exposure to the appropriate combination of hormones. Follicle stimulating hormone (FSH) and luteinizing hormone (LH) are both produced in the pituitary gland and are transported via the blood circulation to the thecal layer surrounding the follicle. From there both hormones are transported into the follicle by diffusion. FSH-receptors are expressed mainly in the granulosa while LH-receptors are expressed in a gradient with highest expression in the theca. How this spatial organization is achieved is not known. Equally it is not understood whether LH and FSH trigger distinct signalling programs or whether the distinct spatial localization of their G-protein coupled receptors is sufficient to convey their distinct biological function.
We have developed a data-based computational model of the spatio-temporal signalling processes within the follicle and (i) predict that FSH and LH form a gradient inside the follicle, (ii) show that the spatial distribution of FSH- and LH-receptors can arise from the well known regulatory interactions, and (iii) find that the differential activity of FSH and LH may well result from the distinct spatial localisation of their receptors, even when both receptors respond with the same intracellular signalling cascade to their ligand.
The model integrates the large amount of published data into a consistent framework that can now be used to better understand how observed defects translate into failed follicle maturation.
Ovarian follicle development; PDE model; Computational biology; Bovine
Turing models have been proposed to explain the emergence of digits during limb development. However, so far the molecular components that would give rise to Turing patterns are elusive. We have recently shown that a particular type of receptor-ligand interaction can give rise to Schnakenberg-type Turing patterns, which reproduce patterning during lung and kidney branching morphogenesis. Recent knockout experiments have identified Smad4 as a key protein in digit patterning. We show here that the BMP-receptor interaction meets the conditions for a Schnakenberg-type Turing pattern, and that the resulting model reproduces available wildtype and mutant data on the expression patterns of BMP, its receptor, and Fgfs in the apical ectodermal ridge (AER) when solved on a realistic 2D domain that we extracted from limb bud images of E11.5 mouse embryos. We propose that receptor-ligand-based mechanisms serve as a molecular basis for the emergence of Turing patterns in many developing tissues.
Proper lung functioning requires not only a correct structure of the conducting airway tree, but also the simultaneous development of smooth muscles and vasculature. Lung branching morphogenesis is strongly stereotyped and involves the recursive use of only three modes of branching. We have previously shown that the experimentally described interactions between Fibroblast growth factor (FGF)10, Sonic hedgehog (SHH) and Patched (Ptc) can give rise to a Turing mechanism that not only reproduces the experimentally observed wildtype branching pattern but also, in part counterintuitive, patterns in mutant mice. Here we show that, even though many proteins affect smooth muscle formation and the expression of Vegfa, an inducer of blood vessel formation, it is sufficient to add FGF9 to the FGF10/SHH/Ptc module to successfully predict simultaneously the emergence of smooth muscles in the clefts between growing lung buds, and Vegfa expression in the distal sub-epithelial mesenchyme. Our model reproduces the phenotype of both wildtype and relevant mutant mice, as well as the results of most culture conditions described in the literature.
Lung development; Computational biology; Smooth muscles; Vasculogenesis; Branching morphogenesis
Many organs of higher organisms, such as the vascular system, lung, kidney, pancreas, liver and glands, are heavily branched structures. The branching process during lung development has been studied in great detail and is remarkably stereotyped. The branched tree is generated by the sequential, non-random use of three geometrically simple modes of branching (domain branching, planar and orthogonal bifurcation). While many regulatory components and local interactions have been defined an integrated understanding of the regulatory network that controls the branching process is lacking. We have developed a deterministic, spatio-temporal differential-equation based model of the core signaling network that governs lung branching morphogenesis. The model focuses on the two key signaling factors that have been identified in experiments, fibroblast growth factor (FGF10) and sonic hedgehog (SHH) as well as the SHH receptor patched (Ptc). We show that the reported biochemical interactions give rise to a Schnakenberg-type Turing patterning mechanisms that allows us to reproduce experimental observations in wildtype and mutant mice. The kinetic parameters as well as the domain shape are based on experimental data where available. The developed model is robust to small absolute and large relative changes in the parameter values. At the same time there is a strong regulatory potential in that the switching between branching modes can be achieved by targeted changes in the parameter values. We note that the sequence of different branching events may also be the result of different growth speeds: fast growth triggers lateral branching while slow growth favours bifurcations in our model. We conclude that the FGF10-SHH-Ptc1 module is sufficient to generate pattern that correspond to the observed branching modes.
Most organs of higher organisms, such as the vascular system, lung, kidney, pancreas, liver and glands, are heavily branched structures. The branching process during lung development has been studied in great detail and is remarkably stereotyped. The branched tree is generated by the sequential, non-random use of three geometrically simple modes of branching. While the branching sequence is identical in mice of identical genetic background it differs between mouse strains. This suggests that the positioning of branch points and the type of branching sensitively depends on information encoded in the genome. Encoding every branching point independently in the genome would require a large number of genes, and it is more likely that a recursive, self-organized process exists that determines the patterning. While many regulatory molecules have been identified an integrated understanding of the regulatory network (program) is missing. Based on available experimental data we have developed a model for lung branching. The model correctly predicts branching phenotypes in mutants and suggests that also the growth speed of the lung tip can affect the positioning and type of the next branching event.
Biological functionality arises from the complex interactions of simple components. Emerging behaviour is difficult to recognize with verbal models alone, and mathematical approaches are important. Even few interacting components can give rise to a wide range of different responses, that is, sustained, transient, oscillatory, switch-like responses, depending on the values of the model parameters. A quantitative comparison of model predictions and experiments is therefore important to distinguish between competing hypotheses and to judge whether a certain regulatory behaviour is at all possible and plausible given the observed type and strengths of interactions and the speed of reactions. Here I will review a detailed model for the transcription factor σF, a regulator of cell differentiation during sporulation in Bacillus subtilis. I will focus in particular on the type of conclusions that can be drawn from detailed, carefully validated models of biological signaling networks. For most systems, such detailed experimental information is currently not available, but accumulating biochemical data through technical advances are likely to enable the detailed modelling of an increasing number of pathways. A major challenge will be the linking of such detailed models and their integration into a multiscale framework to enable their analysis in a larger biological context.
Integrin signaling regulates cell migration and plays a pivotal role in developmental processes and cancer metastasis. Integrin signaling has been studied extensively and much data is available on pathway components and interactions. Yet the data is fragmented and an integrated model is missing. We use a rule-based modeling approach to integrate available data and test biological hypotheses regarding the role of talin, Dok1 and PIPKI in integrin activation. The detailed biochemical characterization of integrin signaling provides us with measured values for most of the kinetics parameters. However, measurements are not fully accurate and the cellular concentrations of signaling proteins are largely unknown and expected to vary substantially across different cellular conditions. By sampling model behaviors over the physiologically realistic parameter range we find that the model exhibits only two different qualitative behaviors and these depend mainly on the relative protein concentrations, which offers a powerful point of control to the cell. Our study highlights the necessity to characterize model behavior not for a single parameter optimum, but to identify parameter sets that characterize different signaling modes.
The family of TGF-β ligands is large and its members are involved in many different signaling processes. These signaling processes strongly differ in type with TGF-β ligands eliciting both sustained or transient responses. Members of the TGF-β family can also act as morphogen and cellular responses would then be expected to provide a direct read-out of the extracellular ligand concentration. A number of different models have been proposed to reconcile these different behaviours. We were interested to define the set of minimal modifications that are required to change the type of signal processing in the TGF-β signaling network.
To define the key aspects for signaling plasticity we focused on the core of the TGF-β signaling network. With the help of a parameter screen we identified ranges of kinetic parameters and protein concentrations that give rise to transient, sustained, or oscillatory responses to constant stimuli, as well as those parameter ranges that enable a proportional response to time-varying ligand concentrations (as expected in the read-out of morphogens). A combination of a strong negative feedback and fast shuttling to the nucleus biases signaling to a transient rather than a sustained response, while oscillations were obtained if ligand binding to the receptor is weak and the turn-over of the I-Smad is fast. A proportional read-out required inefficient receptor activation in addition to a low affinity of receptor-ligand binding. We find that targeted modification of single parameters suffices to alter the response type. The intensity of a constant signal (i.e. the ligand concentration), on the other hand, affected only the strength but not the type of the response.
The architecture of the TGF-β pathway enables the observed signaling plasticity. The observed range of signaling outputs to TGF-β ligand in different cell types and under different conditions can be explained with differences in cellular protein concentrations and with changes in effective rate constants due to cross-talk with other signaling pathways. It will be interesting to uncover the exact cellular differences as well as the details of the cross-talks in future work.
The distribution of most genes is not random, and functionally linked genes are often found in clusters. Several theories have been put forward to explain the emergence and persistence of operons in bacteria. Careful analysis of genomic data favours the co-regulation model, where gene organization into operons is driven by the benefits of coordinated gene expression and regulation. Direct evidence that coexpression increases the individual's fitness enough to ensure operon formation and maintenance is, however, still lacking. Here, a previously described quantitative model of the network that controls the transcription factor σF during sporulation in Bacillus subtilis is employed to quantify the benefits arising from both organization of the sporulation genes into the spoIIA operon and from translational coupling. The analysis shows that operon organization, together with translational coupling, is important because of the inherent stochastic nature of gene expression, which skews the ratios between protein concentrations in the absence of co-regulation. The predicted impact of different forms of gene regulation on fitness and survival agrees quantitatively with published sporulation efficiencies.
coupled gene expression; noise; operon; signalling networks; sporulation