Cell-to-cell communication in multicellular organisms is established through the transmission of various kinds of chemical substances such as proteins. It is well known that gene expression triggered by a chemical substance in individuals has stable spatial patterns despite the individual differences in concentration patterns of the chemical substance. This fact reveals an important property of multicellular organisms called “robustness”, which allows the organisms to generate their forms while maintaining proportion. Robustness has been conventionally accounted for by the stability of solutions of dynamical equations that represent a specific interaction network of chemical substances. However, any biological system is composed of autonomous elements. In general, an autonomous element does not merely accept information on the chemical substance from the environment; instead, it accepts the information based on its own criteria for reaction. Therefore, this phenomenon needs to be considered from the viewpoint of cells. Such a viewpoint is expected to allow the consideration of the autonomy of cells in multicellular organisms. This study aims to explain theoretically the robust patterning of gene expression from the viewpoint of cells. For this purpose, we introduced a new operator for transforming a state variable of a chemical substance from an external coordinate system to an internal coordinate system of each cell, which describes the observation of the chemical substance by cells. We then applied this operator to the simplest reaction–diffusion model of the chemical substance to investigate observation effects by cells. Our mathematical analysis of this extended model indicates that the robust patterning of gene expression against individual differences in concentration pattern of the chemical substance can be explained from the viewpoint of cells if there is a regulation field that compensates for the difference between cells seen in the observation results. This result provides a new insight into the investigation of the mechanism of robust patterning in biological systems composed of individual elements.
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