Beyond the French Flag Model: Exploiting Spatial and Gene Regulatory Interactions for Positional Information

A crucial step in the early development of multicellular organisms involves the establishment of spatial patterns of gene expression which later direct proliferating cells to take on different cell fates. These patterns enable the cells to infer their global position within a tissue or an organism by reading out local gene expression levels. The patterning system is thus said to encode positional information, a concept that was formalized recently in the framework of information theory. Here we introduce a toy model of patterning in one spatial dimension, which can be seen as an extension of Wolpert’s paradigmatic “French Flag” model, to patterning by several interacting, spatially coupled genes subject to intrinsic and extrinsic noise. Our model, a variant of an Ising spin system, allows us to systematically explore expression patterns that optimally encode positional information. We find that optimal patterning systems use positional cues, as in the French Flag model, together with gene-gene interactions to generate combinatorial codes for position which we call “Counter” patterns. Counter patterns can also be stabilized against noise and variations in system size or morphogen dosage by longer-range spatial interactions of the type invoked in the Turing model. The simple setup proposed here qualitatively captures many of the experimentally observed properties of biological patterning systems and allows them to be studied in a single, theoretically consistent framework.

多细胞生物早期发育的关键环节之一，是建立基因表达的空间模式——这类模式后续将指导增殖细胞获得不同的细胞命运。这些模式使细胞能够通过读取局部基因表达水平，推断自身在组织或生物体内的全局位置。因此，这类模式形成系统被认为可编码位置信息，这一概念最近在信息论框架下得到了形式化定义。本研究提出了一个一维空间模式形成的玩具（toy）模型，该模型可视为沃尔珀特（Wolpert）经典“法国国旗”模型的拓展，适用于受内源性与外源性噪声影响的多个相互作用、空间耦合基因的模式形成过程。我们的模型作为伊辛（Ising）自旋系统的变体，能够系统性地探索最优编码位置信息的基因表达模式。研究发现，最优模式形成系统会如法国国旗模型中那样利用位置线索，并结合基因间相互作用，生成用于编码位置的组合编码，我们将其命名为“Counter”模式。“Counter”模式还可通过图灵（Turing）模型中引入的长程空间相互作用，抵御系统尺寸或形态发生因子剂量的噪声与变化。本文提出的这一简单框架，能够在定性层面捕捉生物模式形成系统的诸多实验观测特性，并可在单一的理论自洽框架下对其开展研究。