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WIREs Dev Biol
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The benefits differential equations bring to limb development

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Abstract Systems biology is a large field, offering a number of advantages to a variety of biological disciplines. In limb development, differential‐equation based models can provide insightful hypotheses about the gene/protein interactions and tissue differentiation events that form the core of limb development research. Differential equations are like any other communicative tool, with misuse and limitations that can come along with their advantages. Every theory should be critically analyzed to best ascertain whether they reflect the reality in biology as well they claim. Differential equation‐based models have consistent features which researchers have drawn upon to aid in more realistic descriptions and hypotheses. Nine features are described that highlight these trade‐offs. The advantages range from more detailed descriptions of gene interactions and their consequence and the capacity to model robustness to the incorporation of tissue size and shape. The drawbacks come with the added complication that additional genes and signaling pathways that require additional terms within the mathematical model. They also come in the translation between the mathematical terms of the model, values and matrices, to the real world of genes, proteins, and tissues that constitute limb development. A critical analysis is necessary to ensure that these models effectively expand the understanding of the origins of a diversity of limb anatomy, from evolution to teratology. This article is categorized under: Vertebrate Organogenesis > Musculoskeletal and Vascular Gene Expression and Transcriptional Hierarchies > Regulatory Mechanisms Establishment of Spatial and Temporal Patterns > Repeating Patterns and Lateral Inhibition
(a) The BGSF model with its interactions and the outputs (Benazet et al., ). (b) The BSW model with the interactions, the limb tissue models and Hoxd13 and FGF static gradients, finally, the digital pattern output (Raspopovic et al., ). Skeletal representations are caricatures of wild‐type and relevant mutant forelimbs found in Bandyopadhyay et al. () and Benazet et al. ()
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Tissue size and shape as a model input. Caricature example of modifying size and shape with a large tissue resembling a wild‐type limb tissue (a), reduced tissue (b) and tissue reduced even more (c) all running the same reaction parameters. Purple represents the digital condensations. Results from the author's implementation of a model from Sheth et al. () in MATLAB
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The trait of contexts is presented using two models of limb development. The inputs of the BGSF model are the wild‐type genes, while its outputs, the size and shape of the limb bud and the proximal‐distal polarized tissue from the FGF gradient are two important, though not the only, inputs into the BSW model (Benazet et al., ; Raspopovic et al., ). This chain of contexts can then progress with the primary outputs of the BSW model being its initiated digits
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(a) The modeled versus observed results of the BSW model, first, showing the five digits in the wild‐type simulation and the observations, second, the three digits in the BMP/WNT double inhibited limb (figures are after: Raspopovic et al., ). (b–e) The different roles for the Hox‐13 paralog genes and FGF factors in two models using DEs of limb development (Raspopovic et al., ; Sheth et al., ). The outputs above are from a two‐gene model using DEs that is a personal implementation in MATLAB of the model found in Sheth et al. with some modifications (2012). The four combinations, (b–e), show the various uses of the gradients. The first model shows the model without the gradients. (b) The Hox gradient restricts the model's activity to one part of the spatial model, (c,e), and the FGF gradient aligns the stripes (d,e)
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(a) The principal BSW model and four others capable of producing similar patterns described in the supplement to Raspopovic et al. () as discussed in feature 1. The principal model is at the top of the figure with the subsequent models labeled T3–T6 (T2 is not included). The variable interactions are highlighted in light purple. (b) The contradicting interactions that balance the BGSF model discussed in feature 2 (Benazet et al., ). The pathway through FGFs and Shh, activate BMP4 by inhibiting Grem, while the direct activating interaction with Grem inhibits BMP4. (c) The BSW tissues are represented by the activity of three genes in the BSW model as discussed in feature 3 (after Raspopovic et al., ). The interdigit is represented by BMP and WNT, while digital tissue is represented by Sox9
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Establishment of Spatial and Temporal Patterns > Repeating Patterns and Lateral Inhibition
Gene Expression and Transcriptional Hierarchies > Regulatory Mechanisms
Vertebrate Organogenesis > Musculoskeletal and Vascular