Fig 1.
One-dimensional division-decision growing model and its maximum information pattern in the single-gene system.
(a). A diagram illustrating our one-dimensional division-decision growing model. In our model, cells with the identical genome divide and differentiate from a single cell state to produce a multicellular system with several cell states. Distinct colors are used to indicate different cell states. (b). Possible intracellular and intercellular interactions in the single-gene system. There are nine possible types of genetic networks. 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. (c). A distribution diagram of the maximum positional information in the single-gene system. (d). The exclusive genetic network which controls maximum positional information pattern formation in the single-gene system. (e). The pattern of gene expression varies over time in a single-gene system regulated by the genetic network O1. At step 8, the system obtains 1.0 bit positional information. The time-course of positional information is calculated and shown in the right of the pattern. The binarized gene A expression level determines the cell state. Green circles represent gene A expressed while yellow circles represent gene A unexpressed. (f). Explanation of increased positional information in the single-gene system. The maximum positional information 1.0 bit is created by the new space network resulted from cell division. Cell 4, cell 5, cell 6, and cell 7 inherit cell states from their mothers when cell 2 and cell 3 divide. The expression state matrix (S) and the cell-contacting state matrix (CS) comprised the concatenated matrix ([S,C∙S]), which changed after cell division. The new concatenated matrix contains 1.0 bit potential information which could potentially distinguish different two cell states. With the time moving forward, updated cell states according to the genetic network O1 in (d), cell 5 and cell 6 acquires new cell states. At same time, positional information in 4-cell stage increases to 1.0 bit. (g). A diagram illustrating the mechanics of information growth in division-decision systems. The cell contacting matrix (C), together with the current cell expression states (S), form the potential information source manifested in the concatenated matrix ([S,C∙S]). Appropriate genetic networks convert the potential information contained in the concatenated matrix ([S,C∙S]) into the updated expression states (S), resulting in an increase in the positional information. Cell division may change the cell contacting matrix (C), which therefore increases the potential information.
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.
Fig 3.
Basin size and stability of multicellular stable states.
(a). The basin size of 4-cell stable states and the distribution diagram of the corresponding number of stable states under the regulation of genetic network T1. The red dotted box and red arrow denote the position of the 4-cell stable state formed by the growing system regulated by the genetic network T1. (b). In logarithmic coordinates, the basin size of 8-cell stable states and the distribution diagram of the corresponding number of stable states under the regulation of genetic network T1. The red arrow denotes the position of the 8-cell stable state formed by the growing system regulated by the genetic network T1. (c). Basin size and positional information corresponding to distinct 4-cell stable states under the regulation of genetic network T1. (d). In logarithmic coordinates, basin size and positional information corresponding to distinct 8-cell stable states under the regulation of genetic network T1. (e). Under the regulation of genetic network T1, multicellular stable states corresponding to various basin sizes were displayed in each generation in the schematic diagram. The areas of the smaller, colored circles represent the basin sizes of the distinct multicellular stable states, while the large gray circle represents the cumulative basin size of all stable states for each generation. And the circle’s color corresponds to the positional information of the multicellular stable state. Red lines indicate stable states that appeared in the one-dimensional division-decision growing system under the regulation of genetic network T1. (f). To make the cell state update with a certain probability of mistake, noise is introduced to the growing system. The noise that occurred during cell state update is depicted on the left panel. And the relationship between the probability of cell state update mistakes and pattern accuracy at various cell cycle lengths was illustrated in the right panel. (g). The left panel illustrates a representative instance of asynchronous cell division, while the right panel delineates the relationship between the standard deviation (σ) of cell cycle lengths and the accuracy of patterning under conditions of asynchronous division.
Fig 4.
The spatiotemporal expression patterns of genes in the continuous model and the noise resistance of particular patterns.
(a). The AND-gate regulatory function of the genetic network O1 in the continuous model and the spatiotemporal expression pattern of gene A simulated by this function. (b). The OR-gate regulatory function of the genetic network O1 in the continuous model and the spatiotemporal expression pattern of gene A simulated by this function. (c). The OR-gate regulatory function of the genetic network O2 in the continuous model and the spatiotemporal expression pattern of gene A simulated by this function. (d). Resistance to noise (λ) of patterns generated in generation 3 under various regulatory settings in single-gene systems. (e). The spatiotemporal expression patterns regulated by the genetic network T1 in the continuous model, and corresponding equations which regulate this pattern formation. (f). Resistance to noise (λ) of patterns generated in generation 4 under various regulatory settings in two-gene systems.
Fig 5.
Examples of positional information increase during C.elegans embryonic development.
(a). Positional information and potential information are estimated at each step of the development of the C.elegans embryo, from the fertilized egg to the 200-cell stage. The cell state is defined by the binarized gene expression profiles. Different states are marked by different colors, and named by the lower case of the first cell entering this state. (b). ABa cell transitions from the old cell state “ab” to the new cell state “aba” because it is not in contact with P2 cell during the 4-cell stage. (c). At the 12-cell stage, cell ABarp inherited the cell state "aba" and changed to "abarp" upon contact with C cell. Although the cell ABpra is also in contact with the C cell, its inherited cell state is "ab", so its cell state does not change.