The possible states for the spatial formation of the selected group by analyzing the time trajectories of individuals’ motion. We investigate two different cases with and without distribution of MS023MedChemExpress MS023 chemoattractant in the environment. Considering the case without chemoattractant distribution in the environment (Fig. 5a), the analysis identifies four states for the formation of the group motion. When the transition probability of one state has higher level, it indicates that the amount of energy needed for the group of ONO-4059 web bacteria to create that formation is relatively lower. Based on our analysis, the first state has the lowest energy in the landscape compared to others. This state with the lower energy formation has the highest transition probability in the transition matrix, which means it is the most probable state for the group formation over time. We can order other states relatively to this state based on energy landscape from lowest energy to the highest. We categorize them into two groups of stable and transitioning states. When the group has higher probability to stay in the same state over time, we consider it as a stable state, which can be recognized as local equilibrium state forScientific RepoRts | 6:27602 | DOI: 10.1038/srepFrom energy landscape to estimation of missing information, self-organization and complexity of three different natural groups. Bacteria. Cellular collective groups can create complex patternswww.nature.com/scientificreports/Figure 5. Transition probability matrix and complexity analysis for different natural collective motions. (a) Transition probabilities among the possible states for a collective group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment without chemoattractant gradient. (b) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment without chemoattractant gradient. This plot shows the level of change in missing information when the collective motion leaves each identified state to evolve to a new state (Note S1 in Supplementary Documents explains this in more details). It also demonstrates the relative emergence and relative self-organization and relative complexity of the swarm when evolving from any of the identified state to the first and most probable state. (c) Transition probabilities between the possible states for a group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment with chemoattractant gradient. (d) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment with chemoattractant gradient. (e) Transition probabilities between the possible states for a group ofScientific RepoRts | 6:27602 | DOI: 10.1038/srepwww.nature.com/scientificreports/9 pigeons in free flight. (f) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 9 pigeons in free flight. (g) Transition probabilities between the possible states for a group of 8 pigeons in home flight. (h) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 8 pigeons in home flight. (i) Transition probabilities between the.The possible states for the spatial formation of the selected group by analyzing the time trajectories of individuals’ motion. We investigate two different cases with and without distribution of chemoattractant in the environment. Considering the case without chemoattractant distribution in the environment (Fig. 5a), the analysis identifies four states for the formation of the group motion. When the transition probability of one state has higher level, it indicates that the amount of energy needed for the group of bacteria to create that formation is relatively lower. Based on our analysis, the first state has the lowest energy in the landscape compared to others. This state with the lower energy formation has the highest transition probability in the transition matrix, which means it is the most probable state for the group formation over time. We can order other states relatively to this state based on energy landscape from lowest energy to the highest. We categorize them into two groups of stable and transitioning states. When the group has higher probability to stay in the same state over time, we consider it as a stable state, which can be recognized as local equilibrium state forScientific RepoRts | 6:27602 | DOI: 10.1038/srepFrom energy landscape to estimation of missing information, self-organization and complexity of three different natural groups. Bacteria. Cellular collective groups can create complex patternswww.nature.com/scientificreports/Figure 5. Transition probability matrix and complexity analysis for different natural collective motions. (a) Transition probabilities among the possible states for a collective group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment without chemoattractant gradient. (b) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment without chemoattractant gradient. This plot shows the level of change in missing information when the collective motion leaves each identified state to evolve to a new state (Note S1 in Supplementary Documents explains this in more details). It also demonstrates the relative emergence and relative self-organization and relative complexity of the swarm when evolving from any of the identified state to the first and most probable state. (c) Transition probabilities between the possible states for a group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment with chemoattractant gradient. (d) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 9 bacteria selected from a population density of 108 bacteria/cm3 moving in an environment with chemoattractant gradient. (e) Transition probabilities between the possible states for a group ofScientific RepoRts | 6:27602 | DOI: 10.1038/srepwww.nature.com/scientificreports/9 pigeons in free flight. (f) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 9 pigeons in free flight. (g) Transition probabilities between the possible states for a group of 8 pigeons in home flight. (h) Complexity analysis for different states compared to the first identified state with lowest energy level for a group of 8 pigeons in home flight. (i) Transition probabilities between the.