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For training, functions immediately glaze the protective Oath, which, among dark discoveries, is that weeks ' pose no scientist ' to their elements. Here we show that jumping striders tune their leg rotation speed to reach the maximum jumping speed that water surface allows. We find that the leg stroke speeds of water strider species with different leg morphologies correspond to mathematically calculated morphology-specific optima that maximize vertical takeoff velocity by fully exploiting the capillary force of water.

These results improve the understanding of correlated evolution between morphology and leg movements in small jumping insects, and provide a theoretical basis to develop biomimetic technology in semi-aquatic environments. It is widely known that the superhydrophobic hairy legs of water striders enable them to float only on tarsi 1 , 2 , and that the striders transport the momentum via vortices and capillary waves to propel themselves across the water surface 1 , 3 , 4 , 5.

But what are the mechanical characteristics of jumping off the water surface?

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Vertical and near-vertical jumps are performed in natural habitats in a series of frequent jumps that are triggered by attacks of predators, such as fish and backswimmers predatory insects of the genus Notonecta , from under the water surface 6 , 7. While it was suggested that water striders push the water surface with their legs to generate the upward capillary force on jumping 3 , 8 , 9 , 10 , the mechanics of this swift mode of locomotion have not been fully understood.

Recently, a robotic water strider was built that uses the basic principle of momentum transfer observed in one of the larger species of water striders, Aquarius paludum A simple mechanical model of interactions between A. Upon the basis of the initial kinematic calculations 11 and the insights from high speed videos of jumping, we develop here a theoretical model of insect leg movements that enables the predictions of conditions for optimal jump performance in water striders.

We then empirically verify the optimal predictions using individuals from five water strider species of different body sizes and leg morphologies. We show that, despite having different morphological constraints on leg dimensions, species tune their leg rotation speed to optimize the takeoff velocity from the water surface. From high-speed videos see Methods for details of three species of water striders Gerris latiabdominis, G. At rest, a water strider Fig. Once the insect initiates a jump by pushing the surface with the middle and hind legs downwards, dimples are made Fig.

The dimple depth h Fig. Error bars indicate standard deviation between right and left legs. Error bars indicate standard deviation among four legs. Body velocity profile in g is the same data as that of water strider 2 in Koh et al. In the pushing stage, the tarsus and tibia of each leg Fig. Thus, the average wetted length of the legs l w is assumed to be almost constant and equal to the average of sum of tarsi and tibiae lengths of four legs l t. Owing to the decreasing wetted length l w that interacts with the water surface, the increase rate in upward velocity of the insect body in the closing stage is lower than that in the pushing stage as shown in Fig.

Based on these observations, we build a model to calculate the vertical force that produces the vertical jump as a function of the depth of the dimple formed by the legs. The force can be used to estimate the takeoff velocity, which allows us to seek the optimal stroke condition that the water strider should perform to maximize jumping speed.

Fast jumping is important for the water striders to escape from the predators, such as backswimmers or fish, attacking from under the water surface. Therefore, evolution of abilities to maximize takeoff speed is expected under natural selection. These ratios suggest that the capillary force dominates over the other forces. Hence, the seemingly complex phenomenon of jumping on the water surface can be simplified as a surface tension-dominant interaction of a long thin flexible cylinder with the water surface.

Assuming that all the legs involved in the propulsion move synchronously and leave the surface at the same time Supplementary Fig. The force F increases monotonically with the depth of dimple h until the surface penetration occurs. Then, the temporal change of the dimple depth h t is given by.

The solid lines correspond to the model assuming the sinusoidal model of leg rotation and the circles correspond to the average of measurement of four legs from the same movie used in Fig. The error bars indicate standard deviation among the four legs. We also calculated the time derivative of the empirically measured from video average angle of legs with respect to the horizontal plane through the body centre Supplementary Fig. In addition, as the legs close in the closing stage their inclination angles from the water surface increase leading to an apparent decrease of flexibility factor C We observed several cases in which a leg quickly sank under the water surface after the distal end of the leg pierced the meniscus during the stroke at the average depth of 3.

The jumps that do not involve meniscus breaking can be theoretically categorized into two types: pre-takeoff closing and post-takeoff closing jumps, depending on when the legs are fully closed. Here, t c indicates the instant of the end of leg closing movements. Only two of these modes of jumping, the post-takeoff closing and meniscus breaking jumps surface breaking by one or two legs were observed in water striders as described in Fig.

This classification is important for model predictions Fig. The pre-takeoff closing jump was not observed in the experiments.

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  7. M represents the ratio of body mass to the maximal mass of water that the legs can displace by pushing against the water surface. Water striders observed in this study have M near 0. The arrows in a show that the longer leg indicated by red arrows should move slower, for example, than the leg indicated black arrows to get maximum takeoff speed. White dashed lines indicate the boundary of meniscus breaking jump. The phase diagram includes empirical results from the jump characteristics of females filled symbols and males unfilled symbols of G.

    When we solved equation 3 and used the parameters extracted directly from the videos of jumps of the actual individual water striders, the model reasonably predicted the observed maximum dimple depth Supplementary Note 5 and Supplementary Fig.

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    These calculations indicated that the model correctly approximates the physical processes involved in jumping. Using the model, we derived the theoretical predictions about the optimal vertical jumping behaviour assuming that predator-mediated natural selection maximizes the vertical takeoff speed. Fast vertical takeoff is important for survival because it quickly removes the insect from the vicinity of the approaching predators such as fish and backswimmers, which attack upwards from under the water surface 6 , 7.

    We compared the theoretical predictions with empirical data from slow motion videos of five species of water striders with different body size and leg morphology: G.

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    But, if an insect of the same M had short legs, then its optimal leg rotation would be fast black arrows in Fig. The insects should be careful to not go over the optimal leg rotation speed because it may result in piercing of the water surface and a sudden decrease in jump performance notice sharp transition from yellow to blue especially for large values of L ; Fig. Moreover, Fig.

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    The red lines in Fig. We found that the jumping of water striders occurs always near the condition for the maximal takeoff velocity as shown in Fig. Most jumps occurred just a little below the critical line of meniscus breaking Fig. The reasonable match between the model predictions and the empirical findings for the maximal takeoff speeds suggests that water striders have the ability to adjust their behaviour angular leg movements to reach the optimal conditions for the fastest jump away from danger.

    The morphological traits appearing in our model, that is, body mass m , tibia plus tarsus length l t , whole leg length l l and leg radius r , are not likely to have been optimally designed for only a single particular function, such as jump escape. Our model enables the pursuit of a variety of similar questions in the future because, in principle, it may be used to predict the effect of leg angular movements as well as morphological features on jump performance.

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