We cast the metabolism of interacting cells within a statistical mechanics framework considering both the actual phenotypic capacities of each cell and its interaction with its neighbors. Reaction fluxes will be the components of high-dimensional spin vectors, whose values will be constrained by the stochiometry and the energy requirements of the metabolism. Within this picture, finding the phenotypic states of the population turns out to be equivalent to searching for the equilibrium states of a disordered spin model. We provide a general solution of this problem for arbitrary metabolic networks and interactions. We apply this solution to a simplified model of metabolism and to a complex metabolic network, the central core of Escherichia coli, and demonstrate that the combination of selective pressure and interactions defines a complex phenotypic space. We also present numerical results for cells fixed in a grid. These results reproduce the qualitative picture discussed for the mean-field model. Cells may specialize in producing or consuming metabolites complementing each other, and this is described by an equilibrium phase space with multiple minima, like in a spin-glass model.

中文翻译：

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相互作用的代谢网络的统计力学。

考虑到每个细胞的实际表型能力及其与邻居的相互作用，我们在统计力学框架内进行相互作用细胞的代谢。反应通量将是高维自旋向量的组成部分，其值将受化学计量和代谢能量需求的约束。在这张照片中，发现总体的表型状态等同于寻找无序自旋模型的平衡状态。我们为任意代谢网络和相互作用提供了该问题的一般解决方案。我们将此解决方案应用于简化的代谢模型和复杂的代谢网络（大肠杆菌的核心），并证明选择性压力和相互作用的结合定义了一个复杂的表型空间。我们还提供了固定在网格中的单元格的数值结果。这些结果再现了针对均场模型讨论的定性图。细胞可能专长于产生或消耗彼此互补的代谢产物，这通过具有多个极小值的平衡相空间来描述，就像在旋转玻璃模型中一样。