BACKGROUND
non-random mating, including size-assortative mating  <cit> , is regarded as a powerful evolutionary force  <cit> , that may cause speciation and morphological evolution.
size-assortative mating may be caused by a combination of various factors, including mate preferences, mate availability and constraints on mating  <cit> .
many studies focused on the effect of mate preferences on size-assortative mating  <cit> .
the effects of mating constraints, on size-assortative mating were less frequently studied.
although physical constraints on intromission, constraints on sending courtship signals, and loading constraints were suggested  to induce true size-assortative mating in some species, the potential of constraints as an important mechanism that causes true size-assortative mating and evolution of sexual size dimorphism  has not been emphasized.
size-assortative mating can shape evolution of ssd, and water striders served as model study subjects in this area.
fairbairn  <cit>  suggested that prolonged pairing causes increased ssd through the size-assortative mating: females of species with longer mating durations show stronger preferences for smaller males in order to decrease their "loading costs" .
however, fairbairn  <cit>  suggested that the loading constraints mechanism contributes only to some extent to ssd in a. remigis, and she proposed that ssd in gerridae may be affected by a number of factors including mate choice, fecundity selection on females, and viability selection on both sexes.
this illustrates that in research on water striders, like in the studies of many sexually size-dimorphic species, the potential power of a single mechanism to explain the population level ssd has not been emphasized, and that the "differential equilibrium model" explanations assuming many counter-balancing factors are common  <cit> .
in this study we examine how a specific manner of grasping the female by mating males combined with correlations between body length and leg morphology results in a simple mechanical constraint on coercive mate-grasping.
we ask whether this constraint may cause size-assortative mating in natural populations of water striders, gerridae.
mating initiation in water striders, gerridae, often comprises a forceful attempt by a male to mount the female.
females struggle and attempt to throw the male off because, beyond certain frequency of mating they do not benefit from repeated frequent mating  <cit>  and carrying an extra load  may even lead to fitness costs for females  <cit> .
water strider males grasp tightly the female's thorax with their forelegs in order to overcome the female's resistance.
males are very persistent in opposing female attempts of throwing them off because frequency of mating is positively associated with male fitness  <cit> .
after the end of sperm transfer the male typically remains on the female in a "guarding" position.
among water strider species, guarding may last from several minutes to several days, sometimes several weeks .
because female water striders often lay a batch of eggs every day, by guarding for a long time males assure their paternity of the eggs laid by the female during the long guarding.
female resistance is usually lower after the initial moments of mating, and the male's success in overcoming the female resistance at the initiation of mating  should have an important effect on male fitness, especially in species where guarding lasts for many hours or days .
the male is initially attached to the female's body by pressing his genitalia against the female's abdomen tip, and later during mating by inserting and engaging his genitalia with the female genitalia, while his forelegs grasp the bottom of female midcoxa .
in some species, like gerris gracilicornis, the insertion into female genitalia does not occur until the female opens the genitalia  <cit> , which happens three to five minutes after mating initiation.
in this position, the male water strider has to overcome female attempts to throw off the male.
female reluctance to mate  is expressed as struggling-jumping, somersaulting, pushing and rubbing off a male that attempts to remain on the female by forcefully grasping female's thorax with his forelegs  <cit> .
we hypothesized that due to correlations between total body length, length of body sections relevant for the grasping model, and leg morphology  the strength/stability of the forelegs' grasp depends on the male's relative body length .
if this is true, and if we accept body length as a general indicator of body size, then the resulting size-assortative mating pattern may be predicted from mechanical model of the male grasping behavior .
we also examine whether the size-assortative mating pattern that is predicted from the mechanical model may be sufficient to explain the observed ssd in natural populations of gerris gracilicornis, gerridae.
mechanical model of mate-grasping in gerris gracilicornis
model description
we applied standard methods used in robotics for calculating grasping forces of mechanical devices.
we used a  <dig> d model for calculating maximum pulling force by a male as a function of the size ratio between the male and the female of the mating pair, given the existing correlations between body length and foreleg morphology.
the technical description of the model is presented in additional file  <dig>  here we only present those fragments of the description that are less technical and illustrate the main ideas and predictions.
for our analysis we employ the concept of the force polytope  <cit> , which graphically represents the maximum realizable pulling or pushing tip force given limits on the maximum input joint forces.
the male ftg  and female mcg  constitute the main axes of the mechanism, and the planar model was constructed based on the premise that the bending forelegs of the male  give rise to pulling forces at the anterior end of the mcg.
.
the foreleg joints  of a male water strider were modeled as revolute joints, so that joint muscle forces can be cast as joint torques.
to further simplify matters, we assumed that the attachment of the genitalia  was strong enough for the male to hold the female water strider, and considered only the foreleg joints.
we also consider only the pressing force exerted at the tip of the foreleg , ignoring other points of possible attachment.
based on the mean value measured from photos of mating pairs , we present the predictions for the angle between the female mcg  and male ftg  fixed at  <dig> degrees, but using a range of angles from  <dig> to  <dig> resulted in similar conclusions .
also, we fixed the tip of the male's foreleg  at the bottom of the female midcoxa.
the female midcoxa is the most commonly used grasping site by males, probably due to its convex shape.
if the mounting male applied his foreleg to the smooth area of the female thorax, the stability of connection would most likely be weaker because the male legs would slip more easily.
we assumed that the joint torque limit tmax is the same for joints in the male forelegs .
we incorporated in the model the empirical relationships between ftg and the forefemur length , between the ftg and foretibia length , and between ftg and the width of the forefemur .
increase of grasping force was assumed to be positively correlated with the width of the forefemur.
some further assumptions were made to calculate the maximum pulling force as a function of the sexual size ratio of a pair .
the maximum force vector can be expressed by the length of the arrowed vector  situated inside the force polytope  along the given force line.
the shape of the force polytope  change when the direction of the pulling force changes.
therefore it is important to set the direction of the pulling forces.
the direction of the pulling force chosen for final calculations was based on our observations of interacting animals.
we observed that during initial interactions the body of a male  may occasionally shift forward, so that the male abdomen's tip slides forward over the female genitalia.
it implies that the direction of pulling force may be slightly tilted forward relative to the direction perpendicular to the female axis.
it is consistent with the impression received from direct behavioral observations: a male appears to use the components of the pulling force in order to press his genitalia against female's genitalia.
thus, we chose to calculate the force along the direction slightly tilted forward relative to the line perpendicular to the female axis.
we describe the consequences of varying the direction of this force on the predictions and on our conclusions in the discussion and in the additional file  <dig>  under the assumptions described above and according to the additional detailed description in the additional file  <dig>  the maximal pulling force can be computed.
model predictions
maximum pulling force was found when the female mcg/male ftg ratio was close to  <dig> .
it varied only to a small degree  when we varied the angle between the body axes  within the natural range of values .
hence the prediction appears to be robust regardless of variation in some of the parameters used in the model.
if maximum pulling force substantially increases the male's ability to overcome the females' attempts to dislodge the male, then males paired with females at the mcg/ftg ratio of  <dig> will be the most successful in mating.
due to correlations between mcg and female body length as well as between ftg and male body length , this corresponds to the ratio f/m ttl of about  <dig>  .
we assumed that the line of given force was tilted forward relative to the perpendicular line to the female axis.
if the line of force was perpendicular to the female body axis, the maximum achievable pulling force was obtained at the ratio mcg/ftg =  <dig>  .
all the calculations indicate that the optimal mcg/ftg ratio is near the value of  <dig>
RESULTS
male success in the initiation of mating - the test of mechanical model
the two types of variables, body length and the length of the body sections crucial for the mechanical model  are closely correlated in males -ftg: rma regression slope  =  <dig>  ±  <dig> , r <dig> =  <dig> ; allometric : slope =  <dig>  ±  <dig> , r <dig> =  <dig> ) and in females -mcg: rma regression slope =  <dig>  ±  <dig> , r <dig> =  <dig> , allometric : slope =  <dig>  ±  <dig> , r <dig> =  <dig> ).
therefore, the effect of the mcg/ftg ratio, which is directly relevant for the mate grasping mechanism, on male mating initiation success is statistically associated with the effect of f/m ttl ratio, which is indirectly relevant in the mate grasping model, on male mating initiation success.
in order to test predictions from the mechanical model, we setup an experiment to focus on the effect of f/m ttl ratio, and the correlated mcg/ftg ratio, on the success of males in interactions during mating attempts.
after determining that there was no significant difference  =  <dig> , p =  <dig> ) in the level of female resistance  among different f/m ttl ratios , as well as between female of different sizes  =  <dig> , p =  <dig> ), we asked whether the relationships between the f/m ttl ratio, as well as the mcg/ftg ratio, and male success in initiation of mating  agree with the relationships predicted from the mechanical model .
the highest male mating initiation success was for mcg/ftg values within the range  <dig> - <dig>   - the range that matches the predictions from the mechanical model .
in terms of the f/m ttl ratio, the highest success was for males that are about  <dig> - <dig>  of female size .
there was a significant effect of the f/m ttl ratio on the male mating success .
this effect remained significant after trial was included as a blocking variable in addition to the f/m ttl ratio category as an independent variable, and the presence or absence of mating success as a dependent variable in the generalized linear model .
the distribution of mcg/ftg ratio in successful pairs was significantly different from the distribution in unsuccessful pairs .
these distributions did not differ in their means as there was no statistically significant difference between successful and unsuccessful mating attempts in the mean mcg/ftg ratio and f/m ttl ratio =- <dig> , p =  <dig> ; f/m ttl, t=- <dig> , p = <dig> ; table 1).
the mean values for mcg/ftg  were similar to the ratio predicted in the mechanical model for the maximum grasping force by males , and they were close to the ratio between the population means of mcg and ftg .
the two distributions  differed because the variability, expressed as coefficient of variation, in the female/male ratios  was smaller in successful than in the non-successful mating pairs .
this indicates that mostly the pairs with the ratios close to the mean ratio showed successful male mating attempts.
the shapes of the modeled distribution of mating initiation success as a function of mcg/ftg ratio  and the respective distribution observed in the laboratory  were similar with respect to the slight skeweness: the left-most mcg/ftg ratios  have higher male mating initiation success than the right-most mcg/ftg ratios.
this further suggests that the laboratory results are consistent with the model.
additional analyses of the experiment are in table a <dig>
note - analyses were done using t-test and a z-test for difference between two coefficients of variation  <cit> .
preference by males
males g. gracilicornis initiate matings by forcefully mounting the females.
mating preferences of  <dig> males were measured by their frequency of mating attempts with females of different sizes in situations where two small females and two large unmated females were equally available to males .
in  <dig> mating attempts of small males and  <dig> mating attempts of large males, neither small nor large males preferred any particular size of a female at the moment of attempting to mate .
among all  <dig> mating, males' mating attempts did not depend on f/m ttl categories .
it appeared that experimental males mounted on the female who was seen at first in the male's proximity
mating in the natural population in the light of laboratory experiments
there was a significant relationship between female mcg length and male ftg length  as well as between female ttl and male ttl  among mating pairs in the field population.
these two results are two alternative  ways of illustrating size-assortative mating because the female mcg as well as the male ftg are significantly and strongly correlated with the female and respectively male body lengths.
can the the male mating initiation success in the laboratory be successfully used to explain mating pattern in the natural population?
to answer this question, we compared the field-based regression coefficient between female mcg and male ftg with the distribution of  <dig> hypothetical regression coefficients derived  for the same field population of  <dig> individuals  by simulation of mating.
we assumed that males attempt to mate with random females and that the male mating success directly imitates the laboratory-based relationship between mcg/ftg ratio and male mating initiation success .
the value of the observed rma regression coefficient  was located within the 95% confidence interval  of the sample of  <dig> computed coefficients .
this indicates that the size assortative mating in the field conforms to predictions based on the laboratory data.
distribution of mcg, ftg and body length in the natural population
in the natural population the mean female mcg length matched the mean male ftg length , and was close to the ratio associated with the largest pulling force in the mechanical model , as well as with the highest success in mating initiation by males in the laboratory .
because of the correlations between body length and mcg in females or ftg in males, the mean ratio of total body length  was larger than  <dig> , indicating female biased ssd in g. gracilicornis.
mean male body size  n = 144) was  <dig> % smaller than the mean female body size  n = 104), and there was no overlap in body length distribution between males and females: the smallest female was larger than the largest male.
natural selection on the male ftg and the body length in the field
can the relationship between female/male ratio  and the male mating success cause a noticeable natural selection on the male size in the studied population of specific frequency distributions of male and female body lengths ?
because in typical water striders male fitness, unlike female fitness, is positively correlated with the mating frequency  <cit> , the field data can be used to reasonably estimate selection coefficients from the information of male's, but not the female's, mating status at the time of sampling.
we have estimated selection coefficients on male ttl and, separately, on male ftg in the natural population of  <dig> males and  <dig> females sampled.
the linear  selection gradient as well as the variance  selection gradient estimated from the  <dig> mated and  <dig> unmated males were not significant .
for consistency we also calculated analogical selection gradients for the females .
however, because female fitness in water striders is typically not strongly related to the mating frequency  <cit> , and because our study design focuses on estimating male mating success and selection on males, we further focused on males only.
in simulation  <dig> female size distribution was shifted towards larger values.
in simulation  <dig> the variation of male size distribution was increased.
p-values in the table were calculated by the least squares analysis.
but p-values were also determined by logistic regression and the resulting statistically significant  selection coefficients are given in bold.
we asked whether the laboratory-based relationship between mcg/ftg ratio  and male mating initiation success  creates a potential for natural selection on male size given the female size is shifted away from distribution detected in the field.
we simulated  <dig> matings among the  <dig> males of the natural population with the  <dig> females whose body sizes were shifted towards larger body by  <dig>  mm.
the resulting selection coefficients indicate significant directional, but not variance, selection on male body size and on the ftg .
we also asked whether the laboratory-based relationship between mcg/ftg ratio and male mating success  may cause stabilizing selection on male size under size distributions slightly different from the natural population.
we calculated selection coefficients in a simulation where male size distribution was changed by expanding the variation .
the resulting selection coefficients indicate significant stabilizing, but not directional, selection on male body size and on the ftg .
thus, when size distributions differed slightly from the distributions present in the natural population the selection coefficients on male ftg and on the correlated male body length became significant.
discussion
"mechanical constraints" hypothesis and population level assortative mating pattern
we focused on a species, where successful mating initiation by a male relies on an efficient grasp of a female body and on successful overcoming of initial female resistance.
we determined that mechanical constraints, resulting from relationships between body size and foreleg morphology, may affect mating initiation success of male water striders according to the mechanical model of mating interactions.
the predicted optimal ratio of male/female morphological characters crucial for a good grasp  was associated with the highest success in coercive mating initiation by males in laboratory experiments.
we also considered an additional contribution to the observed laboratory results from a hypothetical effect of potential mismatch of male and female genitalia discussed in some insects  <cit> .
we scored a mating initiation attempt as successful  when the male abdomen tip was firmly pressed " against the female abdomen's tip.
however, in this species it is not equivalent with intromission.
this is the first stage leading to successful intromission, which happen only after an extended period of stable situation when a the female shows relatively little active resistance  while the male produces vibratory signals and waits for the female to extend her genitalia for intromission  <cit> .
because the genitalia mismatch mechanism concerns mostly the intromission process, when genitalia engage, we think that the it is unlikely to explain our laboratory relationship between the relative male size  and the mating initiation success in our species .
finally, during various experiments on g. gracilicornis  <cit> , we did not observed clear indications that genitalia mismatching may cause any significant portion of mating being unsuccessful in terms of intromission.
hence, we propose that the mate-grasping mechanics is a feasible explanation of the close match between our laboratory measurements of male mating initiation success and the model predictions.
furthermore, using computer simulation we showed that if the laboratory-based relationship between mcg/ftg ratio and male mating initiation success is assumed then the population level mating simulation model correctly predicts regression coefficient between female mcg and male ftg of mated pairs in a natural population.
we also determined that the optimal female/male size ratio predicted in the mechanical model was similar to the mean degree of ssd detected in the natural population.
finally, although we did not detect current significant selection on male size in the field population , the selection coefficients calculated for hypothetical situations with size frequency distributions of females or males slightly modified from the natural situation showed that the mechanical constraint mechanism could create significant selection pressures that modify the male body size distribution given the size distribution of females, leading to the currently observed frequency distributions for which males of different sizes have similar fitness.
hence, it appears that fitness of a given size class of males depends on the frequency distribution of both male and female body lengths.
because the success of a male depends on his relative size  the fitness of males from specific size class is clearly dependent on the frequency of female sizes in the population.
the male fitness is also affected by the frequency of other male sizes in the population because the more male competitors of the size similar to the focal male's phenotype the less chances the male has of encountering a female of the "matching" size before a competitor mates with her.
hence, this hypothetical selection on the ftg  mediated by the mate-grasping mechanical model is frequency dependent in a manner more complex than a simple classical frequency-dependence.
the core feature is that fitness of a male size phenotype depends on the size distributions of males and females in the population.
in summary, the results are consistent with the idea that the observed mean ssd, and the lack of current evidence of selection on male size may be an outcome of the past natural selection on male body size operating through the size-assortative mating according to the mechanical model.
this indicates that true size-assortative mating and ssd - the phenomena typically explained by complex interactions among various biological factors  - may, in certain conditions, be greatly affected by a simple physical mechanism that has not been previously considered.
if this is true then the results illustrate how population level phenomenon, the size-assortative mating, can be directly predicted from a simple mechanical model of interacting individuals.
however, despite this very suggestive match between field data and the predictions based on our simulation model, and the arguably low probability of obtaining such a match by pure chance, we cannot entirely exclude the possibility that other processes might have accidentally produced the size-assortative mating pattern.
further evaluation of the mechanical model against alternative mechanisms in the natural population should reveal whether our proposition of the importance of the mechanical mate-grasping model is upheld.
with the exception of two studies that showed only weak true size-assortative mating  <cit> , or no assortative mating  <cit> , majority of research on water striders detected directional size-selective mating or apparent size-assortative mating, such like large male advantage .
we hypothesize that in certain conditions a single mechanism may dominate the other counter-balancing selective factors, and that this may explain the true size-assortative mating success of g. gracilicornis, and maybe other species where the mechanical model has not been applied yet.
if this is true, then the mechanics of mating grasp cannot be ignored in evolutionary studies of the true size-assortative mating and ssd in sexually dimorphic species where mate grasping behavior is crucial for male fitness, and where it has similar mechanics to the behavior described here.
insects  <cit> , crustaceans  <cit>  and amphibians  <cit>  provide numerous examples of such grasping behaviors.
we think that we can reject two other potential causes of the true size-assortative mating pattern in g. gracilicornis: size-dependent mate choice or size-related differences in availability of mates.
first, because both sexes did not prefer to mate with a partner of specific size, we reject the mate choice hypothesis.
second, in contrast to some other water strider species  <cit>  there was no significant female size difference in g. gracilicornis among habitats or emergence periods .
therefore the "availability of mates" is unlikely to explain the pattern of size-assortative mating detected in our field observations.
the real mechanics between mating water striders undoubtedly is more complicated than our simplified model, and consequences of some of the assumptions are discussed in additional file  <dig>  however, in spite of the simplifications, the close agreement between the model predictions and the behavior of water striders may indicate that even this simple two-dimensional model properly captures the core mechanism involved in the constraints on mating initiation in male water striders.
mechanical constraints and the evolution of ssd
if there is a positive correlation between success in mating initiation and male fitness, then we propose that the mating mechanics may not only be the main mechanism sufficient to explain the true size-assortative mating pattern in a population, but it may also contribute to the natural selection that shapes male size distribution given the local female size distribution in g. gracilicornis, and possibly in other water strider species.
the significant directional selection on male size in response to the simulated  shift in female size distribution , and the significant stabilizing selection on male size in response to the simulated increase in variation of the male size distribution relative to the female size distribution  illustrate the feasible natural selection forces that can adjust the male size distribution to the female size distribution in accordance with the mechanical constraint model.
natural systems under such a frequency dependent selection are often expected to be in an equilibrium state  <cit> .
this may be the reason for the lack of significant selection coefficients on male size in the field data.
if evolution of male size in g. gracilicornis was shaped by the frequency-dependent selection for adaptation to efficiently grasp females during mating, then we expect that not only mcg/ftg ratio within mating pairs is very close to the optimal  <dig>  but that also the population level mcg/ftg ratio will be close to  <dig>  as expected, mcg/ftg ratio for mating pairs as well as the average population level mcg/ftg ratio was close to the optimal mcg/ftg ratio of  <dig> predicted from the model.
given these results, and assuming the existing correlations between body length and foreleg morphology, we hypothesize that ssd in g. gracilicornis may be maintained by the natural selection on male size through the "mechanical constraints" process.
we hypothesize that the mechanical constraints process creates selection towards population level male size distribution that is tuned to the female body size distribution .
similarity between field data and the optimal size ratio predicted from simple mechanical model is consistent with the hypothesis that the male size distribution in g. gracilicornis may be under selection to evolutionarily follow changes in female size , such that the average relative size ratio is kept close to the level optimal for male grasping success.
any ecological factor that contributes to the association between success in mate grasping and male's fitness will increase the likelihood that the mechanics of mating becomes the main natural selection mechanism responsible for ssd.
for example, we propose that evolutionary trend towards extremely increased duration of the post-copulatory guarding by males, which is a male adaptation to sperm competition  <cit> , observed in some species among gerridae , may contribute to the importance of mating mechanics in the evolution of ssd .
if a male guards a female for a very long time after copulation then the successful initiation of mating may lead to substantial increase in male fitness  <cit>  either because the males secures fatherhood of most of the eggs laid by the female during this guarding duration, or because long guarding shortens the time between termination of guarding and oviposition by the female, and this shortened time decreases the risk of female being inseminated by another male before oviposition.
hence, in species with long post-copulatory guarding durations, a good grasp at initiation should be more strongly associated with an increase of the male fitness , than in species with short-guarding duration .
additionally, if the population of long-guarding species is male-biased then a single male that lost mating attempt faces low probability of finding another single female for mating, because the females are monopolized for extensive time periods .
finally, we propose that the same grasping force that is important at mating initiation stage may also help a male to overcome female resistance  against longer duration of post-copulatory guarding.
longer guarding increases male's fitness because it increases the chances that female will lay eggs during the time of post-copulatory mate guarding .
thus, we predict that even in species with no distinct ssd, the evolution of long post-copulatory guarding may trigger the evolution of ssd according to the "mechanical constraints" selection on male body size .
obviously, the mechanical constraints on mating are not the only factor that may affect the evolutionary changes of male size in insects.
the following factors of selection on body size have been identified or discussed in gerridae: higher longevity of smaller individuals of either sex  <cit> , large female fecundity advantage  <cit> , small male advantages , developmental constraints  <cit> , small size advantage for migration by flight  <cit> , "loading constraint" selection for smaller males , size selective mating  <cit> , and genetic correlation that restricts rapid evolution of dimorphism  <cit> .
five of these mechanisms may concern direct selection on the male body size: size selective mating, small male advantage due to longevity, due to lower maintenance, or due to loading costs to females.
two of them directly concern selection on male size relative to the female size: loading constraint and size selective mating.
loading constraint hypothesis predicts that females should prefer smaller males who create lower load and therefore lower costs to the mating male-carrying female.
we believe that the "loading constraint hypothesis" is an unlikely major explanation of the observed ssd because we do not have clear indications of female mate preferences for smaller males, or male preferences for larger females.
for the same reason, and because the complex interactions between preferences for wing morph and body size present in gerridae with wing-size polymorphism  <cit>  are absent in g. gracilicornis , we think that mate preference hypothesis is not a likely explanation of our results either.
our results are consistent with the idea that mechanics of mate grasping may be the major factor maintaining the ssd in g. gracilicornis, rather than the alternative explanation, that the differential equilibrium model combining all the various counter-balancing effects  <cit>  coincidentally produced sexual size ratio consistent with the predictions of the mechanical model.
CONCLUSIONS
in summary, by applying robotics to animal behavior, we demonstrated how, in a situation when mate grasping increases male fitness, one simple physical mechanism may be sufficient to account for processes at two levels of biological organization: individual  and population .
similar mechanical formulas, yielding different numerical predictions, can be tested in many insects, where the males ride on females in a mating/guarding grasp  <cit> , in amphibians, where males grasps females during reproduction  <cit>  or even in crustaceans, where larger males grasp smaller females for a precopulatory guarding  <cit>  and energetic considerations due to carrying the female by the grasping male should be added to the models.

 non-random mating, including size-assortative mating, is regarded as a powerful evolutionary force, that may cause speciation and morphological evolution. size-assortative mating may be caused by a combination of various factors, including mate preferences, mate availability and constraints on mating. many studies focused on the effect of mate preferences on size-assortative mating. the effects of mating constraints, on size-assortative mating were less frequently studied. although physical constraints on intromission, constraints on sending courtship signals, and loading constraints were suggested to induce true size-assortative mating in some species, the potential of constraints as an important mechanism that causes true size-assortative mating and evolution of sexual size dimorphism has not been emphasized. size-assortative mating can shape evolution of ssd, and water striders served as model study subjects in this area. fairbairn suggested that prolonged pairing causes increased ssd through the size-assortative mating: females of species with longer mating durations show stronger preferences for smaller males in order to decrease their "loading costs". however, fairbairn suggested that the loading constraints mechanism contributes only to some extent to ssd in a. remigis, and she proposed that ssd in gerridae may be affected by a number of factors including mate choice, fecundity selection on females, and viability selection on both sexes. this illustrates that in research on water striders, like in the studies of many sexually size-dimorphic species, the potential power of a single mechanism to explain the population level ssd has not been emphasized, and that the "differential equilibrium model" explanations assuming many counter-balancing factors are common. in this study we examine how a specific manner of grasping the female by mating males combined with correlations between body length and leg morphology results in a simple mechanical constraint on coercive mate-grasping. we ask whether this constraint may cause size-assortative mating in natural populations of water striders, gerridae. mating initiation in water striders, gerridae, often comprises a forceful attempt by a male to mount the female. females struggle and attempt to throw the male off because, beyond certain frequency of mating they do not benefit from repeated frequent mating and carrying an extra load may even lead to fitness costs for females. water strider males grasp tightly the female's thorax with their forelegs in order to overcome the female's resistance. males are very persistent in opposing female attempts of throwing them off because frequency of mating is positively associated with male fitness. after the end of sperm transfer the male typically remains on the female in a "guarding" position. among water strider species, guarding may last from several minutes to several days, sometimes several weeks. because female water striders often lay a batch of eggs every day, by guarding for a long time males assure their paternity of the eggs laid by the female during the long guarding. female resistance is usually lower after the initial moments of mating, and the male's success in overcoming the female resistance at the initiation of mating should have an important effect on male fitness, especially in species where guarding lasts for many hours or days. the male is initially attached to the female's body by pressing his genitalia against the female's abdomen tip, and later during mating by inserting and engaging his genitalia with the female genitalia, while his forelegs grasp the bottom of female midcoxa. in some species, like gerris gracilicornis, the insertion into female genitalia does not occur until the female opens the genitalia, which happens three to five minutes after mating initiation. in this position, the male water strider has to overcome female attempts to throw off the male. female reluctance to mate is expressed as struggling-jumping, somersaulting, pushing and rubbing off a male that attempts to remain on the female by forcefully grasping female's thorax with his forelegs. we hypothesized that due to correlations between total body length, length of body sections relevant for the grasping model, and leg morphology the strength/stability of the forelegs' grasp depends on the male's relative body length. if this is true, and if we accept body length as a general indicator of body size, then the resulting size-assortative mating pattern may be predicted from mechanical model of the male grasping behavior. we also examine whether the size-assortative mating pattern that is predicted from the mechanical model may be sufficient to explain the observed ssd in natural populations of gerris gracilicornis, gerridae. mechanical model of mate-grasping in gerris gracilicornis model description we applied standard methods used in robotics for calculating grasping forces of mechanical devices. we used a d model for calculating maximum pulling force by a male as a function of the size ratio between the male and the female of the mating pair, given the existing correlations between body length and foreleg morphology. the technical description of the model is presented in additional file here we only present those fragments of the description that are less technical and illustrate the main ideas and predictions. for our analysis we employ the concept of the force polytope, which graphically represents the maximum realizable pulling or pushing tip force given limits on the maximum input joint forces. the male ftg and female mcg constitute the main axes of the mechanism, and the planar model was constructed based on the premise that the bending forelegs of the male give rise to pulling forces at the anterior end of the mcg.. the foreleg joints of a male water strider were modeled as revolute joints, so that joint muscle forces can be cast as joint torques. to further simplify matters, we assumed that the attachment of the genitalia was strong enough for the male to hold the female water strider, and considered only the foreleg joints. we also consider only the pressing force exerted at the tip of the foreleg, ignoring other points of possible attachment. based on the mean value measured from photos of mating pairs, we present the predictions for the angle between the female mcg and male ftg fixed at degrees, but using a range of angles from to resulted in similar conclusions. also, we fixed the tip of the male's foreleg at the bottom of the female midcoxa. the female midcoxa is the most commonly used grasping site by males, probably due to its convex shape. if the mounting male applied his foreleg to the smooth area of the female thorax, the stability of connection would most likely be weaker because the male legs would slip more easily. we assumed that the joint torque limit tmax is the same for joints in the male forelegs. we incorporated in the model the empirical relationships between ftg and the forefemur length, between the ftg and foretibia length, and between ftg and the width of the forefemur. increase of grasping force was assumed to be positively correlated with the width of the forefemur. some further assumptions were made to calculate the maximum pulling force as a function of the sexual size ratio of a pair. the maximum force vector can be expressed by the length of the arrowed vector situated inside the force polytope along the given force line. the shape of the force polytope change when the direction of the pulling force changes. therefore it is important to set the direction of the pulling forces. the direction of the pulling force chosen for final calculations was based on our observations of interacting animals. we observed that during initial interactions the body of a male may occasionally shift forward, so that the male abdomen's tip slides forward over the female genitalia. it implies that the direction of pulling force may be slightly tilted forward relative to the direction perpendicular to the female axis. it is consistent with the impression received from direct behavioral observations: a male appears to use the components of the pulling force in order to press his genitalia against female's genitalia. thus, we chose to calculate the force along the direction slightly tilted forward relative to the line perpendicular to the female axis. we describe the consequences of varying the direction of this force on the predictions and on our conclusions in the discussion and in the additional file under the assumptions described above and according to the additional detailed description in the additional file the maximal pulling force can be computed. model predictions maximum pulling force was found when the female mcg/male ftg ratio was close to. it varied only to a small degree when we varied the angle between the body axes within the natural range of values. hence the prediction appears to be robust regardless of variation in some of the parameters used in the model. if maximum pulling force substantially increases the male's ability to overcome the females' attempts to dislodge the male, then males paired with females at the mcg/ftg ratio of will be the most successful in mating. due to correlations between mcg and female body length as well as between ftg and male body length, this corresponds to the ratio f/m ttl of about. we assumed that the line of given force was tilted forward relative to the perpendicular line to the female axis. if the line of force was perpendicular to the female body axis, the maximum achievable pulling force was obtained at the ratio mcg/ftg =. all the calculations indicate that the optimal mcg/ftg ratio is near the value of  male success in the initiation of mating - the test of mechanical model the two types of variables, body length and the length of the body sections crucial for the mechanical model are closely correlated in males -ftg: rma regression slope = ±, r =; allometric: slope = ±, r = ) and in females -mcg: rma regression slope = ±, r =, allometric: slope = ±, r = ). therefore, the effect of the mcg/ftg ratio, which is directly relevant for the mate grasping mechanism, on male mating initiation success is statistically associated with the effect of f/m ttl ratio, which is indirectly relevant in the mate grasping model, on male mating initiation success. in order to test predictions from the mechanical model, we setup an experiment to focus on the effect of f/m ttl ratio, and the correlated mcg/ftg ratio, on the success of males in interactions during mating attempts. after determining that there was no significant difference =, p = ) in the level of female resistance among different f/m ttl ratios, as well as between female of different sizes =, p = ), we asked whether the relationships between the f/m ttl ratio, as well as the mcg/ftg ratio, and male success in initiation of mating agree with the relationships predicted from the mechanical model. the highest male mating initiation success was for mcg/ftg values within the range - - the range that matches the predictions from the mechanical model. in terms of the f/m ttl ratio, the highest success was for males that are about - of female size. there was a significant effect of the f/m ttl ratio on the male mating success. this effect remained significant after trial was included as a blocking variable in addition to the f/m ttl ratio category as an independent variable, and the presence or absence of mating success as a dependent variable in the generalized linear model. the distribution of mcg/ftg ratio in successful pairs was significantly different from the distribution in unsuccessful pairs. these distributions did not differ in their means as there was no statistically significant difference between successful and unsuccessful mating attempts in the mean mcg/ftg ratio and f/m ttl ratio =-, p =; f/m ttl, t=-, p =; table 1). the mean values for mcg/ftg were similar to the ratio predicted in the mechanical model for the maximum grasping force by males, and they were close to the ratio between the population means of mcg and ftg. the two distributions differed because the variability, expressed as coefficient of variation, in the female/male ratios was smaller in successful than in the non-successful mating pairs. this indicates that mostly the pairs with the ratios close to the mean ratio showed successful male mating attempts. the shapes of the modeled distribution of mating initiation success as a function of mcg/ftg ratio and the respective distribution observed in the laboratory were similar with respect to the slight skeweness: the left-most mcg/ftg ratios have higher male mating initiation success than the right-most mcg/ftg ratios. this further suggests that the laboratory results are consistent with the model. additional analyses of the experiment are in table a note - analyses were done using t-test and a z-test for difference between two coefficients of variation. preference by males males g. gracilicornis initiate matings by forcefully mounting the females. mating preferences of males were measured by their frequency of mating attempts with females of different sizes in situations where two small females and two large unmated females were equally available to males. in mating attempts of small males and mating attempts of large males, neither small nor large males preferred any particular size of a female at the moment of attempting to mate. among all mating, males' mating attempts did not depend on f/m ttl categories. it appeared that experimental males mounted on the female who was seen at first in the male's proximity mating in the natural population in the light of laboratory experiments there was a significant relationship between female mcg length and male ftg length as well as between female ttl and male ttl among mating pairs in the field population. these two results are two alternative ways of illustrating size-assortative mating because the female mcg as well as the male ftg are significantly and strongly correlated with the female and respectively male body lengths. can the the male mating initiation success in the laboratory be successfully used to explain mating pattern in the natural population? to answer this question, we compared the field-based regression coefficient between female mcg and male ftg with the distribution of hypothetical regression coefficients derived for the same field population of individuals by simulation of mating. we assumed that males attempt to mate with random females and that the male mating success directly imitates the laboratory-based relationship between mcg/ftg ratio and male mating initiation success. the value of the observed rma regression coefficient was located within the 95% confidence interval of the sample of computed coefficients. this indicates that the size assortative mating in the field conforms to predictions based on the laboratory data. distribution of mcg, ftg and body length in the natural population in the natural population the mean female mcg length matched the mean male ftg length, and was close to the ratio associated with the largest pulling force in the mechanical model, as well as with the highest success in mating initiation by males in the laboratory. because of the correlations between body length and mcg in females or ftg in males, the mean ratio of total body length was larger than, indicating female biased ssd in g. gracilicornis. mean male body size n = 144) was % smaller than the mean female body size n = 104), and there was no overlap in body length distribution between males and females: the smallest female was larger than the largest male. natural selection on the male ftg and the body length in the field can the relationship between female/male ratio and the male mating success cause a noticeable natural selection on the male size in the studied population of specific frequency distributions of male and female body lengths? because in typical water striders male fitness, unlike female fitness, is positively correlated with the mating frequency, the field data can be used to reasonably estimate selection coefficients from the information of male's, but not the female's, mating status at the time of sampling. we have estimated selection coefficients on male ttl and, separately, on male ftg in the natural population of males and females sampled. the linear selection gradient as well as the variance selection gradient estimated from the mated and unmated males were not significant. for consistency we also calculated analogical selection gradients for the females. however, because female fitness in water striders is typically not strongly related to the mating frequency, and because our study design focuses on estimating male mating success and selection on males, we further focused on males only. in simulation female size distribution was shifted towards larger values. in simulation the variation of male size distribution was increased. p-values in the table were calculated by the least squares analysis. but p-values were also determined by logistic regression and the resulting statistically significant selection coefficients are given in bold. we asked whether the laboratory-based relationship between mcg/ftg ratio and male mating initiation success creates a potential for natural selection on male size given the female size is shifted away from distribution detected in the field. we simulated matings among the males of the natural population with the females whose body sizes were shifted towards larger body by mm. the resulting selection coefficients indicate significant directional, but not variance, selection on male body size and on the ftg. we also asked whether the laboratory-based relationship between mcg/ftg ratio and male mating success may cause stabilizing selection on male size under size distributions slightly different from the natural population. we calculated selection coefficients in a simulation where male size distribution was changed by expanding the variation. the resulting selection coefficients indicate significant stabilizing, but not directional, selection on male body size and on the ftg. thus, when size distributions differed slightly from the distributions present in the natural population the selection coefficients on male ftg and on the correlated male body length became significant. discussion "mechanical constraints" hypothesis and population level assortative mating pattern we focused on a species, where successful mating initiation by a male relies on an efficient grasp of a female body and on successful overcoming of initial female resistance. we determined that mechanical constraints, resulting from relationships between body size and foreleg morphology, may affect mating initiation success of male water striders according to the mechanical model of mating interactions. the predicted optimal ratio of male/female morphological characters crucial for a good grasp was associated with the highest success in coercive mating initiation by males in laboratory experiments. we also considered an additional contribution to the observed laboratory results from a hypothetical effect of potential mismatch of male and female genitalia discussed in some insects. we scored a mating initiation attempt as successful when the male abdomen tip was firmly pressed " against the female abdomen's tip. however, in this species it is not equivalent with intromission. this is the first stage leading to successful intromission, which happen only after an extended period of stable situation when a the female shows relatively little active resistance while the male produces vibratory signals and waits for the female to extend her genitalia for intromission. because the genitalia mismatch mechanism concerns mostly the intromission process, when genitalia engage, we think that the it is unlikely to explain our laboratory relationship between the relative male size and the mating initiation success in our species. finally, during various experiments on g. gracilicornis, we did not observed clear indications that genitalia mismatching may cause any significant portion of mating being unsuccessful in terms of intromission. hence, we propose that the mate-grasping mechanics is a feasible explanation of the close match between our laboratory measurements of male mating initiation success and the model predictions. furthermore, using computer simulation we showed that if the laboratory-based relationship between mcg/ftg ratio and male mating initiation success is assumed then the population level mating simulation model correctly predicts regression coefficient between female mcg and male ftg of mated pairs in a natural population. we also determined that the optimal female/male size ratio predicted in the mechanical model was similar to the mean degree of ssd detected in the natural population. finally, although we did not detect current significant selection on male size in the field population, the selection coefficients calculated for hypothetical situations with size frequency distributions of females or males slightly modified from the natural situation showed that the mechanical constraint mechanism could create significant selection pressures that modify the male body size distribution given the size distribution of females, leading to the currently observed frequency distributions for which males of different sizes have similar fitness. hence, it appears that fitness of a given size class of males depends on the frequency distribution of both male and female body lengths. because the success of a male depends on his relative size the fitness of males from specific size class is clearly dependent on the frequency of female sizes in the population. the male fitness is also affected by the frequency of other male sizes in the population because the more male competitors of the size similar to the focal male's phenotype the less chances the male has of encountering a female of the "matching" size before a competitor mates with her. hence, this hypothetical selection on the ftg mediated by the mate-grasping mechanical model is frequency dependent in a manner more complex than a simple classical frequency-dependence. the core feature is that fitness of a male size phenotype depends on the size distributions of males and females in the population. in summary, the results are consistent with the idea that the observed mean ssd, and the lack of current evidence of selection on male size may be an outcome of the past natural selection on male body size operating through the size-assortative mating according to the mechanical model. this indicates that true size-assortative mating and ssd - the phenomena typically explained by complex interactions among various biological factors - may, in certain conditions, be greatly affected by a simple physical mechanism that has not been previously considered. if this is true then the results illustrate how population level phenomenon, the size-assortative mating, can be directly predicted from a simple mechanical model of interacting individuals. however, despite this very suggestive match between field data and the predictions based on our simulation model, and the arguably low probability of obtaining such a match by pure chance, we cannot entirely exclude the possibility that other processes might have accidentally produced the size-assortative mating pattern. further evaluation of the mechanical model against alternative mechanisms in the natural population should reveal whether our proposition of the importance of the mechanical mate-grasping model is upheld. with the exception of two studies that showed only weak true size-assortative mating, or no assortative mating, majority of research on water striders detected directional size-selective mating or apparent size-assortative mating, such like large male advantage. we hypothesize that in certain conditions a single mechanism may dominate the other counter-balancing selective factors, and that this may explain the true size-assortative mating success of g. gracilicornis, and maybe other species where the mechanical model has not been applied yet. if this is true, then the mechanics of mating grasp cannot be ignored in evolutionary studies of the true size-assortative mating and ssd in sexually dimorphic species where mate grasping behavior is crucial for male fitness, and where it has similar mechanics to the behavior described here. insects, crustaceans and amphibians provide numerous examples of such grasping behaviors. we think that we can reject two other potential causes of the true size-assortative mating pattern in g. gracilicornis: size-dependent mate choice or size-related differences in availability of mates. first, because both sexes did not prefer to mate with a partner of specific size, we reject the mate choice hypothesis. second, in contrast to some other water strider species there was no significant female size difference in g. gracilicornis among habitats or emergence periods. therefore the "availability of mates" is unlikely to explain the pattern of size-assortative mating detected in our field observations. the real mechanics between mating water striders undoubtedly is more complicated than our simplified model, and consequences of some of the assumptions are discussed in additional file however, in spite of the simplifications, the close agreement between the model predictions and the behavior of water striders may indicate that even this simple two-dimensional model properly captures the core mechanism involved in the constraints on mating initiation in male water striders. mechanical constraints and the evolution of ssd if there is a positive correlation between success in mating initiation and male fitness, then we propose that the mating mechanics may not only be the main mechanism sufficient to explain the true size-assortative mating pattern in a population, but it may also contribute to the natural selection that shapes male size distribution given the local female size distribution in g. gracilicornis, and possibly in other water strider species. the significant directional selection on male size in response to the simulated shift in female size distribution, and the significant stabilizing selection on male size in response to the simulated increase in variation of the male size distribution relative to the female size distribution illustrate the feasible natural selection forces that can adjust the male size distribution to the female size distribution in accordance with the mechanical constraint model. natural systems under such a frequency dependent selection are often expected to be in an equilibrium state. this may be the reason for the lack of significant selection coefficients on male size in the field data. if evolution of male size in g. gracilicornis was shaped by the frequency-dependent selection for adaptation to efficiently grasp females during mating, then we expect that not only mcg/ftg ratio within mating pairs is very close to the optimal but that also the population level mcg/ftg ratio will be close to as expected, mcg/ftg ratio for mating pairs as well as the average population level mcg/ftg ratio was close to the optimal mcg/ftg ratio of predicted from the model. given these results, and assuming the existing correlations between body length and foreleg morphology, we hypothesize that ssd in g. gracilicornis may be maintained by the natural selection on male size through the "mechanical constraints" process. we hypothesize that the mechanical constraints process creates selection towards population level male size distribution that is tuned to the female body size distribution. similarity between field data and the optimal size ratio predicted from simple mechanical model is consistent with the hypothesis that the male size distribution in g. gracilicornis may be under selection to evolutionarily follow changes in female size, such that the average relative size ratio is kept close to the level optimal for male grasping success. any ecological factor that contributes to the association between success in mate grasping and male's fitness will increase the likelihood that the mechanics of mating becomes the main natural selection mechanism responsible for ssd. for example, we propose that evolutionary trend towards extremely increased duration of the post-copulatory guarding by males, which is a male adaptation to sperm competition, observed in some species among gerridae, may contribute to the importance of mating mechanics in the evolution of ssd. if a male guards a female for a very long time after copulation then the successful initiation of mating may lead to substantial increase in male fitness either because the males secures fatherhood of most of the eggs laid by the female during this guarding duration, or because long guarding shortens the time between termination of guarding and oviposition by the female, and this shortened time decreases the risk of female being inseminated by another male before oviposition. hence, in species with long post-copulatory guarding durations, a good grasp at initiation should be more strongly associated with an increase of the male fitness, than in species with short-guarding duration. additionally, if the population of long-guarding species is male-biased then a single male that lost mating attempt faces low probability of finding another single female for mating, because the females are monopolized for extensive time periods. finally, we propose that the same grasping force that is important at mating initiation stage may also help a male to overcome female resistance against longer duration of post-copulatory guarding. longer guarding increases male's fitness because it increases the chances that female will lay eggs during the time of post-copulatory mate guarding. thus, we predict that even in species with no distinct ssd, the evolution of long post-copulatory guarding may trigger the evolution of ssd according to the "mechanical constraints" selection on male body size. obviously, the mechanical constraints on mating are not the only factor that may affect the evolutionary changes of male size in insects. the following factors of selection on body size have been identified or discussed in gerridae: higher longevity of smaller individuals of either sex, large female fecundity advantage, small male advantages, developmental constraints, small size advantage for migration by flight, "loading constraint" selection for smaller males, size selective mating, and genetic correlation that restricts rapid evolution of dimorphism. five of these mechanisms may concern direct selection on the male body size: size selective mating, small male advantage due to longevity, due to lower maintenance, or due to loading costs to females. two of them directly concern selection on male size relative to the female size: loading constraint and size selective mating. loading constraint hypothesis predicts that females should prefer smaller males who create lower load and therefore lower costs to the mating male-carrying female. we believe that the "loading constraint hypothesis" is an unlikely major explanation of the observed ssd because we do not have clear indications of female mate preferences for smaller males, or male preferences for larger females. for the same reason, and because the complex interactions between preferences for wing morph and body size present in gerridae with wing-size polymorphism are absent in g. gracilicornis, we think that mate preference hypothesis is not a likely explanation of our results either. our results are consistent with the idea that mechanics of mate grasping may be the major factor maintaining the ssd in g. gracilicornis, rather than the alternative explanation, that the differential equilibrium model combining all the various counter-balancing effects coincidentally produced sexual size ratio consistent with the predictions of the mechanical model.  in summary, by applying robotics to animal behavior, we demonstrated how, in a situation when mate grasping increases male fitness, one simple physical mechanism may be sufficient to account for processes at two levels of biological organization: individual and population. similar mechanical formulas, yielding different numerical predictions, can be tested in many insects, where the males ride on females in a mating/guarding grasp, in amphibians, where males grasps females during reproduction or even in crustaceans, where larger males grasp smaller females for a precopulatory guarding and energetic considerations due to carrying the female by the grasping male should be added to the models. 
here, we explore the consequences of a simple, and previously ignored, fact that in a grasping posture the position of the male's grasping appendages relative to the female's body is often a function of body size difference between the sexes. through field sampling and simulation modeling of a natural population we determine that the simple mechanical model, which ignores most of the other hypothetical counter-balancing selection pressures on body size, is sufficient to account for size-assortative mating pattern as well as species-specific sexual dimorphism in body size of g. gracilicornis. a major challenge in evolutionary biology is to understand the typically complex interactions between diverse counter-balancing factors of darwinian selection for size assortative mating and sexual size dimorphism. the results indicate how a simple and previously overlooked physical mechanism common in many taxa is sufficient to account for, or importantly contribute to, size-assortative mating and its consequences for the evolution of sexual size dimorphism. mate-grasping by males, using specialized adaptive morphologies of their forelegs, midlegs or even antennae wrapped around female body at specific locations, is a general mating strategy of many animals, but the contribution of the mechanics of this wide-spread behavior to the evolution of mating behavior and sexual size dimorphism has been largely ignored.
 a major challenge in evolutionary biology is to understand the typically complex interactions between diverse counter-balancing factors of darwinian selection for size assortative mating and sexual size dimorphism. it appears that rarely a simple mechanism could provide a major explanation of these phenomena. mechanics of behaviors can predict animal morphology, such like adaptations to locomotion in animals from various of taxa, but its potential to predict size-assortative mating and its evolutionary consequences has been less explored. mate-grasping by males, using specialized adaptive morphologies of their forelegs, midlegs or even antennae wrapped around female body at specific locations, is a general mating strategy of many animals, but the contribution of the mechanics of this wide-spread behavior to the evolution of mating behavior and sexual size dimorphism has been largely ignored.  here, we explore the consequences of a simple, and previously ignored, fact that in a grasping posture the position of the male's grasping appendages relative to the female's body is often a function of body size difference between the sexes. using an approach taken from robot mechanics we model coercive grasping of females by water strider gerris gracilicornis males during mating initiation struggles. we determine that the male optimal size, which gives the males the highest grasping force, properly predicts the experimentally measured highest mating success. through field sampling and simulation modeling of a natural population we determine that the simple mechanical model, which ignores most of the other hypothetical counter-balancing selection pressures on body size, is sufficient to account for size-assortative mating pattern as well as species-specific sexual dimorphism in body size of g. gracilicornis.  the results indicate how a simple and previously overlooked physical mechanism common in many taxa is sufficient to account for, or importantly contribute to, size-assortative mating and its consequences for the evolution of sexual size dimorphism. 