Department of Mathematics, Western Washington University, Bellingham, WA, 98229, USA.
Institute of Fundamental Technological Research Polish Academy of Sciences, 02-106, Warsaw, Poland.
Math Biosci. 2023 Nov;365:109054. doi: 10.1016/j.mbs.2023.109054. Epub 2023 Aug 5.
Previous work showed that Gal-1A and Gal-8, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction-diffusion-adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle-node bifurcation from a monostable state. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1A and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.
先前的研究表明，Gal-1A 和 Gal-8 这两种属于半乳糖结合凝集素家族的蛋白是胚胎鸡肢骨骼元素形态发生的最早决定因素，而且，它们在胚胎肢芽中的实验确定的相互作用可以通过反应扩散粘附（2GL：两个凝集素加上配体）模型来解释。在这里，我们使用基于常微分方程的方法来分析 2GL 网络的固有切换模式，并独立于形态发生机制的扩散和粘附臂来表征网络行为。我们确定了两种状态：两种凝集素的浓度分别可以忽略不计和非常高。这个双稳态开关样系统是通过从单稳态状态的鞍结分岔产生的。对于质量作用产生项的情况，我们为系统提供了一个显式的李雅普诺夫函数，表明它没有周期解。因此，我们的模型预测，凝集素网络可能存在于空间或时间上分离的低表达和高表达状态，而没有任何中间状态。我们在使用鸡肢间充质细胞高密度培养进行的实验中检验了这些预测，并观察到软骨前原纤维中的细胞表达 Gal-1A 的速度比细胞外的速度高得多，而细胞外的 Gal-1A 则可以忽略不计。因此，基于 Gal-1A 和 -8 的模式形成网络足以将间充质细胞群体分为具有不同发育（软骨形成与非软骨形成）命运的两个离散细胞状态。当将其纳入具有粘附和扩散功能的框架中时，该系统可以生成具有空间图案的肢骨骼。
