We consider a simplified model for gene regulation, where gene expression is regulated by transcription factors (TFs), which are single proteins or protein complexes. Proteins are in turn synthesised from expressed genes, creating a feedback loop of regulation. This leads to a directed bipartite network in which a link from a gene to a TF exists if the gene codes for a protein contributing to the TF, and a link from a TF to a gene exists if the TF regulates the expression of the gene. Both genes and TFs are modelled as binary variables, which indicate, respectively, whether a gene is expressed or not, and a TF is synthesised or not. We consider the scenario where for a TF to be synthesised, all of its contributing genes must be expressed. This results in an ``AND'' gate logic for the dynamics of TFs. By adapting percolation theory to directed bipartite graphs, evolving according to the AND logic dynamics, we are able to determine the necessary conditions, in the network parameter space, under which bipartite networks can support a multiplicity of stable gene expression patterns, under noisy conditions, as required in stable cell types. In particular, the analysis reveals the possibility of a bi-stability region, where the extensive percolating cluster is or is not resilient to perturbations. This is remarkably different from the transition observed in standard percolation theory. Finally, we consider perturbations involving single node removal that mimic gene knockout experiments. Results reveal the strong dependence of the gene knockout cascade on the logic implemented in the underlying network dynamics, highlighting in particular that avalanche sizes cannot be easily related to gene-gene interaction networks.

中文翻译：

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基因调控网络上的渗透

我们考虑基因调控的简化模型，其中基因表达受转录因子 (TF) 的调控，转录因子是单一蛋白质或蛋白质复合物。蛋白质依次由表达的基因合成，形成调节的反馈回路。这导致了一个有向二分网络，其中如果基因编码对 TF 有贡献的蛋白质，则存在从基因到 TF 的链接，如果 TF 调节基因的表达，则存在从 TF 到基因的链接。基因和 TF 都被建模为二元变量，分别表示基因是否表达，以及是否合成了 TF。我们考虑这样一种情况，即要合成 TF，必须表达其所有贡献基因。这导致 TF 动态的“与”门逻辑。通过将渗流理论应用于有向二分图，根据 AND 逻辑动力学进化，我们能够确定网络参数空间中的必要条件，在这些条件下，二分网络可以在嘈杂的条件下支持多种稳定的基因表达模式，根据稳定细胞类型的需要。特别是，该分析揭示了双稳态区域的可能性，其中广泛的渗透簇对扰动具有或不具有弹性。这与标准渗透理论中观察到的转变有显着不同。最后，我们考虑涉及模拟基因敲除实验的单节点去除的扰动。结果揭示了基因敲除级联对底层网络动态中实现的逻辑的强烈依赖性，