A mathematical framework for understanding the spontaneous emergence of complexity applicable to growing multicellular systems
Fig 2
Maximum positional information pattern formation in the two-gene system.
(a). Possible intracellular and intercellular interactions in the two-gene system. The number of possible genetic networks is 6561. The black arrow represents the possible interaction relationship, like an activation, an inhibition or an absence of interaction. Solid and dotted lines represent intracellular and intercellular interaction relationships, respectively. (b). The distribution diagram of the number of genetic networks corresponding to different maximum positional information under four different initial cell states. (c). The exclusive genetic network (T1) in the two-gene system that regulates maximum positional information pattern formation in a stable state. (d). The expression pattern of two genes throughout time and space. This pattern formation regulated by the genetic network T1 in (c). At step 14, system reaches 2.0 bit positional information. (e). Mechanisms for increasing positional information from 1.0 bit to 2.0 bit. When cells 4 to 7 divide, cells 8 to 15 inherit cell states from their mothers. At this time, the potential information in the concatenated matrix ([S,C∙S]) is 2.0 bit. With the time moving forward, updated cell states according to the genetic network T1 in (c), cell 8, cell 11, cell 12 and cell 15 alter their states. At step 14, positional information increased from 1.0 bit to 2.0 bit. (f). Transition linkages across four cell states under the regulation of genetic network T1. The number and color of the circles next to the line indicate the number and state of the cell’s neighbors, and an arrow points from the initial state of the cell to the final state of the cell. The similar cell state transition that took place in (d) is shown by circles with a gray background. (g). The relationship between maximum positional information and the proportion of times a regulatory motif occurs. The proportion of self-inhibition and direct-lateral-activation motif occurrence are negatively correlated with the maximum positional information. While the proportion of direct-lateral-inhibition motif occurrence is positively connected with the maximum positional information. (h). According to the maximum positional information achievable in the development system directed by genetic networks, networks with the same maximum positional information are grouped together. The first column represents the maximum positional information value for each genetic network group. The second column indicates the number of genetic networks in each group. The third column represents the ratio of direct lateral inhibition motif present in the group. The fourth column represents the ratio of indirect lateral inhibition motif present in the group. The fifth column represents the proportion of networks in the group that do not exhibit lateral inhibition motif.