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Ant experimentsAnt coloniesSeven colonies of the garden ant L. niger collected from the Soka University and a nearby park were used in this study (Extended Data Table S1). They were placed in plastic cases (35 × 25 × 6 cm). Water was provided ad libitum. They were fed a sucrose solution, and were starved for 2–5 days before the start of the experiment. The colonies were queen-less colonies with 200–700 workers. Aqueous sucrose solution was used as a food resource (bait) in the experiments. The laboratory room where the experiments were performed and the ant colonies were kept was maintained at a temperature of 25–27 °C and a humidity of 60–70%. Artificial lights were also installed in this room.ApparatusWe used an apparatus, called “the main apparatus,” with two paths from the nest to the feeding site (length: 30 cm, width: 2 cm, height: 12 cm for the outward path and 15 cm for the return path) (Fig. 1). This apparatus could separate the outward path (bridge) from the inward path (bridge). Here, the outward path refers to that taken by ants from the nest to the feeding site, whereas the inward path refers to the path taken by ants from the feeding site to the nest.Figure 1The main apparatus used in the three experiments (the main experiment and the comparison experiments 1 and 2). Nests are connected to the experimental apparatus by a slope. In the main experiment, on the outward path, there is ant traffic from the nest to the feeding site on a pheromone trail, and on the inward path, there is ant traffic from the feeding site to the nest on a pheromone trail. In the comparison experiment 1, only a pheromone trail is present on both the outward and inward paths. In the comparison experiment 2, no pheromone trail or ant traffic is present on both the outward and inward paths.Full size imageTwo important features of this apparatus were as follows: firstly, it allowed ants to only enter the outward path from the nest. A rat-guard structure at the end of the inward path prevented the ants on the outward path from entering the inward path (Extended Data Fig. S1A). Secondly, we installed a vertical structure at the end of the outward path (height: 4 cm). After climbing the vertical structure, ants were not allowed to return to the outward path (Extended Data Fig. S1B). Moreover, we installed partitions on the feeding site, which also prevented ants from returning to the outward path after reaching the feeding site (Extended Data Fig. S1C). After entering the feeding site, ants had to pass through a narrow gap (width: 0.5 cm) created by the partition. No visual cues were offered as the apparatus was surrounded on all four sides by plastic walls.In this experiment, we made another apparatus for a single ant (target ant), which would be joining the ant trail on the main bridges (Extended Data Fig. S2). This apparatus, called “the confluence device,” was a detachable device that could be connected at right angles to the outward and inward bridges of the main apparatus. To connect this device to the outward bridge, we made the confluence path of this device under the inward bridge of the main apparatus, since the outward bridge was lower than the inward bridge. Thus, we made a slope on the outward confluence path connected to the outward bridge of the main apparatus. Further, because placing the ants directly on the sidewalk sometimes caused them to fall off the sidewalk owing to panic, we constructed a free space and a wall (height: 5 cm) in the middle of the confluence device on which the ants were placed calmly. Owing to this modification, we could let each target ant calm down and then access the main bridge whenever they wanted to. The apparatus used in this experiment was made of white plastic plates.Pheromone trail with ant trafficThis main experiment was limited to once a day for each colony. A sucrose solution was dripped into the feeding site. Target ants, which were walking on a plastic case as foragers, had been moved from their nests to another case immediately before a trail of (nontarget) ants was formed. Thus, dozens of ants were moved in advance to the case to be used as target ants. Subsequently, a trail of (nontarget) ants was formed from the nest to the main apparatus. Considering that it took some time for the ants that had finished foraging and returned to the nest to recruit their mates, the ants were left for approximately 40 min to an hour until a permanent ant trail was formed. It was difficult to form an ant trail immediately after the start of the experiment since no foraging pheromones could be produced in the first foraging trip on the outward path and since experienced foraging ants may make foraging pheromones on the outward path2,21,22. The target ants were allowed to enter bridges of the main apparatus after the establishment of a permanent ant trail. At that time, trails of individual target ants were started. Target ants were allowed to join at right angles to the path on the apparatus, one by one from the confluence device. Individual target ants were allowed to enter the main apparatus at four different points: (1) Left-Left (LL), located at the left side of the center of the outward path. The outward path was on the left side, whereas the inward path was on the right side for the experimenter when seen from the nest. (2) Left–Right (LR), located at the right side of the center of the outward path. (3) and (4) Right-Left (RL) and Right-Right (RR), located at the left and right sides of the center of the inward path, respectively (Fig. 2). We had set these four points to check if target ants tended to turn their body to a certain direction when entering the main bridges, regardless of the movement direction of the other ants. A video camera (Panasonic, AVCHD 30fps) was used to record the migration of ants to the feeding site or nest. Videos were taken from above, and target ants were used only once.Figure 2Four joining points (LL, LR, RL, and RR) and the confluence device (joining device). The confluence (joining) device was connected at right angles to the center of the outward and inward bridges of the main apparatus. Here, the LR version is shown as an example.Full size imageThe goal lines were set at 15 cm from the center of the main paths. We checked the side (nest side or feeding site side) from which a target ant passed the goal line.Pheromone trail with no ant trafficThis comparison experiment 1 was limited to once a day for each colony. Dozens of ants were moved in advance to another case to be used as target ants in a similar manner to the main experiment. The (nontarget) ants were left for about 40 min to an hour until a permanent ant trail was formed. Subsequently, we removed all the ants from the device. Then, target ants were allowed to enter on the side path one by one. In this case, we left the bait in place to control this experiment under the same condition as the main experiment. As the pheromone trail was created on the outward path as well as on the inward path, it was the only decision-making factor for the ants to join at the main path (outward/inward paths). We checked the side (nest side or feeding site side) from which a target ant passed the goal line in a similar manner to the main experiment.No pheromone trail or ant trafficThis comparison experiment 2 was limited to once a day for each colony. Dozens of ants were moved in advance to another case to be used as target ants in a similar manner to the main experiment. This experiment was conducted to investigate ant behavior under the following two conditions: (1) no ant trails and (2) no pheromones trails. The bait was in place in the same manner. We checked the side (nest side or feeding site side) from which a target ant passed the goal line in a similar manner to the main experiment. After each trial (the target ant passed the goal line), we wiped the apparatus with ethanol solution before the next target ant was allowed to enter the main paths.AnalysisThe goal lines were set at 15 cm from the center of the main paths. We checked which goal side the target ants reached the goal line on each trial. A reverse run referred to the goal to the nest on the outward path and the goal to the feeding site on the inward path. A normal run referred to the goal to the feeding site on the outward path and the goal to the nest on the inward path.In some cases of the main experiment, foraging (nontarget) ants that could not reach the feeding site on their outward path or could not return to the nest on their inward path would be against the ant flows. On the outward path, we considered that the ants conducted a “reverse flow” if the position of their heads was on the nest side compared with the position of their stomach. If not, we defined that the ants conducted a “normal flow” (Extended Data Fig. S3). On the inward path, we defined that the ants conducted a “reverse flow” if the position of their head was on the feeding site side compared with the position of their stomach. If not, we defined that the ants conducted a “normal flow” (Extended Data Fig. S3). We focused on the target ants that came in contact with ants with normal flow. Therefore, if an ant with reverse flow was located within 10 cm of the target ant, that trial was excluded from the analysis.Furthermore, we also evaluated if target ants coming in contact with foraging (nontarget) ants immediately after entering the trail would affect the goal choice. Therefore, we conducted an analysis focusing on the contact using the data from the main experiment. We examined whether or not the target ant made contact with other foraging ants until it passed a point 2 cm from the center of the path. As already mentioned, if the target ant came in contact with another ant moving against the normal flow of the ant trail, this contact was excluded from the counts. Moreover, we also excluded cases in which the body of target ants was on a point 2 cm from the center of the path by visual evaluation. Thus, we examined the goal choice of target ants by focusing on whether or not they came in contact with other ants immediately after joining the main bridges.We also conducted a preliminary experiment using a single path apparatus to investigate bi-directional trail behaviour. Please see the Extended Data File S1.Model descriptionThe models were coded using the C programming language. The model description follows the Overview, Design concepts, and Details protocol23,24.PurposeThe purpose of the model was to examine the mechanistic understanding of our findings. We adopted an action of target agents obtained from our ant experiments and compared it with another action of target agents on a trail that was contrary to the fact. To be more precise, target agents were allowed to obey an alignment rule in which they tended to move in the same direction with other agents. We named the former model as the reverse-rule model and the latter model as the alignment-rule model. By doing so, we could find the significance of our findings from ant experiments.Entities, state variables, and scalesWe developed two different models (reverse-rule model and alignment-rule model) that included two types of entities: agents and cells. The agent has the state variable Navigational state, which has two values: Navigational state = {wandering, foraging}. The cell has the state variable Pheromone; this value represents the amount of pheromones in each cell. We used a 2D lattice field and set a straight bridge with 61 cells × 5 cell sizes. We also set goal lines at x-coordinate =  − 30 and 30. If the agents reached coordinates satisfying their x-coordinate =  − 30 or 30, they were removed from the system. If the agents reached y-axis boundaries, their movement direction was restricted. Each trial continued until the target agent reached one of the two goal lines. However, trials were forcibly finished if the target agent never reached any goal line by t = 500-time steps. In total, we conducted 1000 trials.Process overview and schedulingAt the beginning of each trial, an artificial target ant (Navigational state = wandering) was introduced at the center of an artificial simulation field. Foraging agents (Navigational state = foraging) were randomly distributed on the simulation field in advance.Agents on the simulation field selected one direction from two directions (+ x and − x) on each time step and updated their positions. Briefly, an agent at coordinate (x, y) selected one direction from two directions (+ x and − x) and updated its position with one of the three coordinates—(x − 1, y), (x − 1, y + 1), or (x − 1, y − 1)—if it selected the − x direction, or—(x + 1, y), (x + 1, y + 1), or (x + 1, y − 1)—if it selected the + x direction by scanning pheromones on these three coordinates. For example, if an agent at coordinate (0, 2) decided to move in + x direction at one time, the position of this agent was replaced with one of (1, 3), (1, 2) and (1, 1) from (0, 2) by scanning pheromones on these three coordinates. The target agent selected the − x/ + x direction with equal probability on each time step until it met the foragers. In contrast, foraging agents tended to decide to move in the − x direction on each time step with a high probability and therefore they tended to select the − x direction for position updating. Foraging agents deposited pheromones before leaving the current cell (see submodel entitled “Position updating” and submodel entitled “Pheromone updating”). In contrast, the target agents did not deposit pheromones.Using above submodels, artificial ants sometimes met other agents. If the target agent (Navigational state = wandering) met the foragers (Navigational state = foraging), the target agent tended to select one direction from two directions (+ x and − x) on each time step thereafter with a high probability, which was dependent on which direction the met foragers came from. More strictly, in the reverse-rule model, the target agent tended to move in an opposite direction from the foragers if it met the foragers coming from the opposite direction. On the contrary, the target agent in the alignment-rule model tended to move in the same direction with foragers if it met the foragers moving in the same direction (see submodel entitled “The interaction between the target agent and foragers”). For example, in the reverse-rule model, if the target agent at coordinate (x, y), whose previous coordinate was (x − 1, y), met the forager coming from the opposite direction, whose previous coordinate was (x + 1, y), the target agent decided to move in + x direction on each time step thereafter with a high probability until similar events occurred.Design conceptThe mean goal time was the emergent property of the model. Sensing was important as the agents scanned the pheromone concentrations. Stochasticity was used to determine in which direction the agent moved and to select one cell using the pheromone concentrations.InitializationWe set a single agent (target agent) on the coordinate (0, 2) and its Navigational state was set to wandering (Extended Data Fig. S5A). We also set N foraging agents on the bridge whose Navigational state was set to foraging. Therefore, N + 1 agents were on the test field at the beginning of each trial. A target agent was the agent k = 0, whereas foraging agents were agents k = 1, 2, …, N. These foragers were randomly distributed on the bridge. Thus, x(k) (in) {n |− 30 ≤ n ≤ 30, n is an integer} and y(k) (in) {n | 0 ≤ n ≤ 4, n is an integer} for k  > 0.Foraging agents were set to move in the -x direction (Direction(k) for k  > 0 = − x). On the other hand, the target agent randomly chose one direction from two directions at the beginning of each trial (Direction(0) was set to + x or − x with equal probability). Herein, Direction(k) can be − x or + x, which implies bias in the movement direction. The parameter prob(k) indicates the probability of moving in Direction(k). The target agent selected the − x/+ x direction with equal probability on each time step until it met the foragers. Therefore, the parameter prob was set to 0.50 for the target ant (prob(0) = 0.50), whereas prob was set to 0.80 for foraging agents (prob(k) = 0.80 for k  > 0). The amount of pheromones on each cell was set to 1 at the beginning of each trial (pheromone(x, y) = 1) and the pheromone evaporation rate q was set to 0.99.The model descriptions are explained using submodels. A Submodel: the interaction between the target agent and foragers causes differences between two rules (the reverse-rule model and the alignment-rule model).SubmodelsSubmodel: the interaction between the target agent and foragersThe parameters Direction(0) and prob(0) were replaced with new ones whenever the following events occurred.In the reverse-rule model, for any agent k (k  > 0),Herein, (xt(k), yt(k)) indicates the x–y-coordinate for the agent k at time t. Furthermore, (xt(0), yt(0)) = (xt(k), yt(k)) means that the target agent and the agent k occupy the same cell at time t while (xt(0) − xt−1(0)) × (xt(k) − xt−1(k)) =  − 1 indicates that the target agent meets the agent k came from the opposite direction. The target agent replaces Direction(0) with an opposite direction from the forager k (see Extended data Fig. S5B).In the alignment-rule model, for any agent k (k  > 0),(xt(0) − xt−1(0)) × (xt(k) − xt−1(k)) = 1 indicates that the target agent meets the agent k came from the same direction. The target agent replaces Direction(0) with a same direction with the forager k (See Extended Data Fig. S5B).In the reverse-rule model, these events are driven from the experimental observations of real ants. Target ants appear to move against the trail and seem to move straight by contacting those other nestmates that come from the opposite direction. Also, target ants seem to select the reverse goal even if physical contact with ant nestmates does not occur immediately after entering the bridge. So, regarding parameter replacements, we did not consider the position at which the target agent met another agent. Note that foraging agents did not change these parameters until the end of each trial. Further, Direction(0) can be replaced with − x from + x and vice versa whenever the target agent meets foragers that come from the opposite direction.In the alignment-rule model, the target agent tends to move in the same direction with other agents. This is contrary to the experimental observations of real ants.Submodel: position updatingFor all k agents (k = 0–N), the movement direction and position updates are shown as follows (Extended Data Fig. S5C);Here, rnt(k) indicates a random number. Thus, rnt(k) (in) [0.00, 1.00].Prob(0) for the target agent is initially set to 0.50. Therefore, the target agent selects one direction from the two (− x and + x) on each time step randomly before the condition described in submodels—the interaction between the target agent and foragers is satisfied. On the other hand, foraging agents select − x direction with a high probability (= Prob(k)) on each time step. After selecting one direction from two (− x and + x), agents scan three cells in the direction of movement. Using pheromone concentrations on those three cells, they update their positions.If agents reach coordinates satisfying their y-coordinate = 4 or 0, those agents update their position by selecting not three but two coordinates since they are located on the edges of the bridge.Submodel: pheromone updatingForaging agents (k  > 0) deposited pheromones on the current cell when leaving that cell.Then, pheromones are evaporated using the evaporation rate q.For each time iteration, these submodels operated in the following order.STEP 1: The interaction between the target agent and foragers.STEP 2: Position updating.STEP 3: Pheromone updating.AnalysisTo check the accuracy of our model, we counted which goal side the target agent entered the goal line from using the reverse-rule model by setting N = 9. If the target agent passed the goal line at x-coordinate =  − 30 (30), we considered that it reached the normal (reverse) goal. Note that trials in which the target agent never reached any goal lines by t = 500 were excluded from this analysis. Furthermore, to investigate the adaptability of the reverse run mechanism, we examined the time until the target agent reached the goal lines using the reverse-rule model and the alignment-rule model. Herein, we set two different conditions with respect to the number of foraging agents (N = 4 and 9). More

Study subjects and measurement of skin physiological parametersWe analyzed skin microbiome and mycobiome from cheeks and foreheads of healthy younger (19–28 years old, Y-group) and older (60–63 years old, O-group) Korean women who were free from cutaneous disorders (Table 1 and Supplementary Table S1). All 61 subjects had been living in Seoul, Korea, for more than 3 years with normal skin conditions. We preferentially selected those who had sebum secretion greater than 30 arbitrary units and moisture greater than 50 arbitrary units in both groups. Among the measurements of moisture content, pH, sebum content, and transepidermal water loss (TEWL), only sebum and TEWL decreased significantly in the O-group compared to the Y-group in the cheeks (P = 2.25e−06, Wilcoxon rank-sum test; P = 0.019, Welch two-sample t test) and forehead (P = 1.33e−06, Wilcoxon rank-sum test; P = 0.003, Welch two-sample t test). Whereas no significant differences were found in the average values for moisture (cheeks: Y-group, 59.9; O-group, 56.6; forehead: Y-group, 61.1; O-group, 58.7) and pH (cheeks: Y-group, 6.0; O-group, 5.8; forehead: Y-group, 6.0; O-group, 5.6) between the two age groups.Table 1 Characteristics of subjects for aged related skin microbiome and mycobiome study.Full size tableComparisons in cheek and forehead microbiome and mycobiome between the two age groupsWe analyzed bacterial communities from 27 Y-group samples (cheeks, n = 13; forehead, n = 14) and 24 O-group samples (cheeks, n = 12; forehead, n = 12) and fungal communities from 28 Y-group samples (cheeks, n = 15; forehead, n = 13) and 32 O-group samples (cheeks, n = 16; forehead, n = 16), except for samples that were eliminated from the Illumina Mi-Seq sequencing due to low sequence reads (bacteria,  3. 0) (Fig. 4). Pathways belonging to the metabolism category were dominant in each age group. In the cheek of the Y-group, pathways involved in energy metabolism by bacteria, such as glycolysis/gluconeogenesis, citrate cycle, pentose phosphate pathway, fructose and mannose metabolism, galactose metabolism, d-alanine metabolism, and thiamine metabolism, were predominant, whereas in the cheek of the O-group, degradation-related pathways, such as fatty acid degradation, synthesis and degradation of ketone bodies, benzoate degradation, and chloroalkane and chloroalkene degradation, were predominant. In the forehead of the Y-group, glycolysis/gluconeogenesis, pentose phosphate pathway, fructose and mannose metabolism, galactose metabolism, d-glutamine and d-glutamate metabolism, d-alanine metabolism, and thiamine metabolism pathway were significantly more abundant, whereas in the forehead of the O-group, fatty acid degradation, synthesis and degradation of ketone bodies, valine/leucine and isoleucine degradation, and limonene/pinene degradation pathway were significantly more abundant.Figure 4Heat map for significantly different predicted functional pathways on (a) cheeks and (b) foreheads of Korean women by age based on LEfSe analysis (LDA score  > 3.0).Full size imageThe metabolism pathway for biotin, a water-soluble vitamin that is effective for skin health and essential for keratin production15, was more prevalent in the cheek and forehead of the Y-group. Interestingly, the metabolism pathway for lipoic acid, which is known to possess beneficial effects against skin aging and is used widely in cosmetic and dermatological products16,17, was significantly higher in the foreheads of the Y-group. We tracked the specific ASVs possessing these pathways, in both biotin metabolism and lipoic acid metabolism, Cutibacterium sp. (ASV2136 and ASV2130) and Staphylococcus sp. (ASV3008) were predicted to have the top three relative abundances in KOs. The relative abundances in biotin metabolism and lipoic acid metabolism of Cutibacterium sp. (ASV2136) were 24.9% and 26.1%, respectively. The relative abundances for each pathway for Staphylococcus sp. (ASV3008) were 10.2% and 18.7%, and for Cutibacterium sp. (ASV2130), they were 9.3% and 10.0%, respectively. We confirmed these two pathways in the genome of skin bacteria, C. acnes (Supplementary Fig. S2). These additional analyses support the reliability of the function in the skin environment of Cutibacterium. Interestingly, from the LEfSe result, Cutibacterium sp. (ASV2136) had a significantly higher abundance in the cheek and forehead microbiome of the Y-group. The pathway of biosynthesis of lipopolysaccharide, also known as bacterial endotoxins, showed higher abundance in the cheek and forehead microbiome of the O-group. The ASVs that contribute to inferring the LPS biosynthesis pathway were identified as Paraburkholderia sp. (ASV5030) and B. vesicularis (ASV4155). Also, pathways related to antibiotic biosynthesis (biosynthesis of vancomycin group antibiotics) and bacterial motility (bacterial chemotaxis and flagellar assembly; both belonging to the cellular processes category) were prominent in the cheek and forehead of the O-group. PICRUSt2 analysis implied that, regardless of skin site differences, the potential functions of the microbial community that compose the skin microbiome were similar according to age.Network analysis on cheek and forehead microbiome and mycobiomeWe performed SParse InversE Covariance estimation for Ecological Association Inference (SPIEC-EASI) analysis to evaluate the overall network of the skin microbes. The results of network density (D) on 81 cheek and 87 forehead ASVs, calculated using the ratio of the number of edges, showed higher network density in the skin microbiome of the Y-group (D = 0.015 and D = 0.001, in cheek and forehead, respectively) than the O-group (D = 0.007 and D = 0.007, respectively) (Fig. 5). To examine network correlation between bacteria and fungi, network density for Bacteria–Fungi (DBF) was calculated by the actual number of edges and a potential number of edges in a correlation ([bacterial nodes × fungal nodes]/2). We confirmed higher network density in the cheek of the Y-group (DBF = 0.008) than the O-group (DBF = 0) and edges of the major bacterial and fungal taxa, such as Staphylococcus sp. (ASV3008)—M. sympodialis (ASV500) and Roseomonas sp. (ASV4088)—M. restricta (ASV482), were observed in the cheek of the Y-group. In the forehead, edges of Methylobacterium sp. (ASV4314)—M. globosa (ASV454), Methylobacterium sp. (ASV4314)—Zygosaccharomyces rouxii (ASV208), and Venionella sp. (ASV3575)—M. sympodialis (ASV500) were observed in the Y-group, and edges of Cutibacterium sp. (ASV2107)—M. globosa (ASV461), Staphylococcus sp. (ASV3024)—M. arunalokei (ASV446), and Methylobacterium (ASV4314)—M. dermatis (ASV448) were observed in the O-group (DBF = 0.004). We found a network between bacteria and fungi with different kingdom levels in the skin microbiome, and especially, we confirmed that different genus or species level microbe was involved in the microbial network according to skin location and Y-, O-group.Figure 5Network analysis of the ASVs on (a) cheeks and (b) forehead of Korean women. Each node represents the ASV and the size of the node is based on relative abundance of each ASV. Color markings indicate the major taxa except for unidentified bacteria or fungi. Shapes represent the level of kingdom, Bacteria (bold) and Fungi (dotted line). The ASVs were selected for bacterial ASVs found in more than half of all samples on the cheeks and forehead, respectively, and for the fungal ASVs with a relative abundance of more than 0.1% in each of the cheeks and forehead samples. The D value is network density calculated using the ratio of the number of edges.Full size image More

Analytical targetsTwo species of undulation motion rays with different pectoral fin shapes bred in KAIYUKAN were analyzed: sharpnose stingray Dasyatis acutirostra and pitted stingray Dasyatis matsubarai (Fig. 1a,b). Blender 2.7925 was used to construct stingray models from pictures26,27 as accurately as possible; Blender is a free and open-source 3D creation suite used to make realistic characters for movies, etc. Detailed information on how to construct models using Blender is provided in our previous paper28. To focus on the effects of pectoral fin movements, we did not consider the body’s shape as in the previous studies12,29. The height and disk width (WD) of all models were set to 0.01 m and 0.44 m, respectively, considering the previous studies30,31. The disk length of each model was determined from WD, referring to the aspect ratio of the rays’ photographs26,27; the disk length (LD) of D. acutirostra and D. matsubarai are 0.348 m and 0.344 m, respectively.Figure 1Analytical targets and description of motion. (a) Analytical model of D. acutirostra. (b) Analytical model of D. matsubarai. (c) Description of motion, (d) the relationship between any two points on the surface before and after the deformation.Full size imageMotionThe motion was given to satisfy the following equations:$$z = left{ {begin{array}{*{20}l} {A;sin left( {omega left( {t – kTleft( {frac{{angl{text{e}}left( {x_{i} ,y_{i} } right) – 10^{{text{o}}} }}{{All;angl{text{e}}}} – theta } right)} right)h_{1} h_{2} } right.} hfill & {left( {10^{{text{o}}} le angl{text{e}}left( {x_{i} ,y_{i} } right) le 170^{{text{o}}} } right)} hfill \ {A;sin left( {omega left( {t – kTleft( {frac{{350^{{text{o}}} – left( {angl{text{e}}left( {x_{i} ,y_{i} } right) – 10^{{text{o}}} } right)}}{{All;angl{text{e}}}}} right)} right)h_{1} h_{2} } right.} hfill & {left( {190^{{text{o}}} le angl{text{e}}left( {x_{i} ,y_{i} } right) le 350^{{text{o}}} } right)} hfill \ end{array} } right.$$
(1)
$$begin{array}{c}{h}_{1}=a{r}_{i}^{3}+b{r}_{i}^{2}+c{r}_{i}end{array}$$
(2)
$$h_{2} = left{ {begin{array}{*{20}l} {dleft( {angleleft( {x_{i} ,y_{i} } right) – 10^{ circ } } right)^{2} + eleft( {angleleft( {x_{i} ,y_{i} } right) – 10^{ circ } } right)} hfill & {left( {10^{ circ } le angleleft( {x_{i} ,y_{i} } right) le 170^{ circ } } right)} hfill \ {dleft( {350^{ circ } – left( {angleleft( {x_{i} ,y_{i} } right) – 10^{ circ } } right)} right)^{2} + eleft( {350^{ circ } – left( {angleleft( {x_{i} ,y_{i} } right) – 10^{ circ } } right)} right)} hfill & {left( {190^{ circ } le angleleft( {x_{i} ,y_{i} } right) le 350^{ circ } } right)} hfill \ end{array} } right.$$
(3)
$$begin{array}{c}{left({r}_{i}-{r}_{i-1}right)}^{2}+{left({z}_{i}-{z}_{i-1}right)}^{2}={left({r}_{i}^{mathrm{^{prime}}}-{r}_{i-1}^{mathrm{^{prime}}}right)}^{2}+{left({z}_{i}^{mathrm{^{prime}}}-{z}_{i-1}^{mathrm{^{prime}}}right)}^{2}end{array}$$
(4)
$$begin{array}{c}angleleft({x}_{i},{y}_{i}right)= angleleft({x}_{i}^{^{prime}},{y}_{i}^{^{prime}}right).end{array}$$
(5)
Equation (1) represents the amount of movement of the model surface in the z-axis direction, where (A) is the amplitude of the pectoral fin tip, (omega) is the angular velocity, (t) is time, (k) is the wavenumber, (T) is the period, angle(({x}_{i},{y}_{i})) is the angle made by the line connecting the center of rotation and any point (({x}_{i},{y}_{i})) on the model surface with the x-axis, Allangle is the range where the motion is given (160°), and (theta) is the phase difference between the movements of the right and left pectoral fins (Fig. 1c). ({h}_{1}) is the weighting from the center to the radial direction: it is necessary to set the amplitude at the ray’s center to zero and increase the amplitude toward the pectoral fin tip (a = 119.786, b = -7.957, c = 0.498). ({h}_{2}) is the weighting in the circumferential direction: it is necessary to increase the amplitude from the anterior to the tip of the pectoral fin and decrease the amplitude from the tip of the pectoral fin to the posterior (d = − 1.563 × 10–4, e = 0.025). Equation (4) is the condition in which the distance between two neighboring points in the same radial direction is equal before and after the movement (Fig. 1d). (r) is the distance between the center of rotation and any point (({x}_{i},{y}_{i})), defined as (sqrt{{x}_{i}^{2}+{y}_{i}^{2}}). Equation (5) defines angle(({x}_{i},{y}_{i})) as being constant before and after the move (Fig. 1c). The variables after the move are marked with ‘. Variables used in the analysis are A = 0.089 m, T = 0.499, k = 1.270, and ω = 12.599 rad/s. Videos of the created motion from the front and the side are shown in the “Supplement” (Supplement Movies 3, 4).Analytical conditionsAnalysis cases were conducted with eight conditions: two types of pectoral fin shape (Fig. 1a,b) and four types of phase difference (0 (T), 0.25 (T), 0.5 (T), and 0.75 (T)). These conditions were set for investigating the effects of phase differences between left and right pectoral fin movements on swimming and how these effects vary with pectoral fin shape.Numerical methodsA CFD simulation of the ray models in the water flow was performed using OPENFOAM, an open-source finite volume method CFD toolbox32, to calculate the forces acting on the rays in each axial direction and the moment around each axis. The governing equations were the continuity equation and the three-dimensional incompressible Reynolds-averaged Navier–Stokes equation, expressed by:$$begin{array}{*{20}c} {nabla cdot u = 0} \ end{array}$$
(6)
$$begin{array}{*{20}c} {frac{partial u}{{partial {text{t}}}} + nabla cdot left( {uu} right) = – nabla p + nabla cdot left( {vnabla u} right) + nabla cdot left[ {nu left{ {left( {nabla u} right)^{T} – frac{1}{3}nabla cdot uI} right}} right], } \ end{array}$$
(7)
where (u) is the velocity vector, t is the time, p is the static pressure divided by the reference density, (nu) is the kinematic viscosity, and I is the unit tensor. The Reynolds number was defined regarding the previous studies3 as:$$begin{array}{c}{R}_{e}=frac{U{L}_{D}}{nu },end{array}$$
(8)
where (U) (/ms) is the given flow speed3 (1.5 × LD/ms), ({L}_{D}) (m) is the length of the ray models, and (nu) is the kinematic viscosity of water at 20 °C (1.0 × 10–6 m2/s). The Reynolds number in this study is 1.8 × 105; considering this, we used the k–ω shear stress turbulence model33,34. The k–ω shear stress turbulence model is a type of Reynolds-averaged Navier–Stokes equation (RANS) turbulence model that is widely used to calculate for the fish swimming flow35,36,37. The overset grid method was used in this study; it is a generic implementation of overset meshes. For both static and dynamic cases, cell-to-cell mapping between multiple, disconnected mesh regions is employed to generate a composite domain38,39. This method permits complex mesh motions and interactions without the penalties associated with deforming meshes. The process is described in detail by Noack40. The calculation volume was 5.4 WD in length, 5.4 WD in height, and 5.4 WD in width (Fig. 2a,b). A hexahedral volume mesh was created using the snappyHexMesh of OPENFOAM. The fluid region was divided into two parts: the overset region and the background region (Fig. 2a,b). The overset region moves and transforms to match the motion of the ray and was made with fine meshes around the analysis target and coarse meshes in the outlying areas; a one-layer boundary layer mesh was created around the analysis target. The overset region shape is an ellipsoid (Fig. 2a,b). The minimum mesh volume is 7.3 × 10–10 (m3), and the maximum mesh volume is 2.6 × 10–2 (m3). The total number of meshes was 9.0 × 105. At the outlet boundary, the average static relative pressure was set to 0 Pa. The surfaces of the fish model were formed into non-slip surfaces.Figure 2Meshes for CFD simulation and differences in force between different meshes. (a) Meshes at the coronal plane of the whole fluid region. (b) Frontal cross-section of the fluid region at the green line in (a). The red region is the overset region. (c,d) Comparison of the instantaneous drag coefficient and the moment coefficient around the y-axis of D. matsubarai between the coarse, fine, and dense mesh.Full size imageThe drag coefficient ({C}_{D}left(tright)), the lateral force coefficient ({C}_{l}left(tright)), the lift coefficient ({C}_{L}left(tright)), the moment coefficient around the x-axis ({C}_{mx}left(tright)), the moment coefficient around the y-axis ({C}_{my}left(tright)) and the moment coefficient around the z-axis ({C}_{mz}left(tright)) were calculated as:$$begin{array}{c}{C}_{D}left(tright)=frac{Dleft(tright)}{frac{1}{2}rho {U}^{2}{L}_{D}{W}_{D}}end{array}$$
(9)
$$begin{array}{c}{C}_{l}left(tright)=frac{lleft(tright)}{frac{1}{2}rho {U}^{2}{L}_{D}{W}_{D}}end{array}$$
(10)
$$begin{array}{c}{C}_{L}left(tright)=frac{Lleft(tright)}{frac{1}{2}rho {U}^{2}{L}_{D}{W}_{D}}end{array}$$
(11)
$$begin{array}{c}{C}_{mx}left(tright)=frac{{M}_{psi }left(tright)}{frac{1}{2}rho {U}^{2}{L}_{D}^{2}{W}_{D}}end{array}$$
(12)
$$begin{array}{c}{C}_{my}left(tright)=frac{{M}_{phi }left(tright)}{frac{1}{2}rho {U}^{2}{L}_{D}^{2}{W}_{D}}end{array}$$
(13)
$$begin{array}{c}{c}_{mz}left(tright)=frac{{M}_{theta }left(tright)}{frac{1}{2}rho {U}^{2}{L}_{D}^{2}{W}_{D}},end{array}$$
(14)
where (Dleft(tright)) is the calculated drag, (lleft(tright)) is the calculated lateral force, (Lleft(tright)) is the calculated lift, ({M}_{psi }left(tright)) is the calculated moment around the x-axis, ({M}_{phi }left(tright)) is the calculated moment around the y-axis, ({M}_{theta }left(tright)) is the calculated moment around the z-axis, and (rho) (kg/m3) is the density of water at 20 °C (998 kg/m3). As shown in a previous study41. the propulsive efficiency (eta) is defined as the ratio of output power ({P}_{o}) to input power ({P}_{e}) which can be written as:$$begin{array}{c}{P}_{o}left(tright)=frac{1}{T}{int }_{0}^{T}Dleft(tright)Udtend{array}$$
(15)
$$begin{array}{c}{P}_{e}left(tright)=frac{1}{T}{int }_{0}^{T}left[Dleft(tright)dot{x}left(tright)+lleft(tright)dot{y}left(tright)+Lleft(tright)dot{z}left(tright)right]dtend{array}$$
(16)
$$begin{array}{c}eta =frac{{P}_{o}}{{P}_{e}}.end{array}$$
(17)
An in-house program calculated the forces acting on rays in each axial direction and the moment around each axis. The numerical method’s validity and reliability were verified by comparing previous experimental and numerical analytical studies of heaving and pitching on airfoil naca001341. A high degree of similarity to previous studies was confirmed; the mean difference in the propulsive efficiency from the previous study of analysis was 5%, and the difference from the previous study of the experiment was 9%. Detailed information such as mesh, length, and velocity, of this analysis method’s verification is provided in the “Supplement”.A grid sensitivity study was conducted using three meshes: coarse, fine, and dense. The coarse mesh has 8.1 × 105 elements, the fine mesh has 9.0 × 105 elements, and the dense mesh has 9.9 × 105 elements. The analysis was conducted using a condition with no phase difference of D. matsubarai. As shown in Fig. 2c,d, the drag coefficient and the moment coefficient around the y-axis are almost the same when the mesh is fine and when the mesh is dense. The mean drag and propulsive efficiency error of fine and coarse meshes are 2.7% and 3.5%, respectively. The fine mesh was used in all simulation cases considering accuracy. We used the same meshes for all cases. More