On February 12, Roger Critchlow posted a reference to "sperm pelotons",
which inspired me to read the Nature article and to think a bit about how
principles of peloton interactions could be applied to sperm aggregations.
I've outlined some thoughts below.
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DRAFT
Applications of a peloton model to sperm aggregration dynamics
An analysis of article: Fisher, H., Hoekstra, H. (2010) Competition drives
cooperation among closely related sperm of deer mice. Nature. Vol. 463, 11
Feb 801-803
Hugh Trenchard
Abstract
The Nature article by Fisher and Hoekstra suggests that a mechanism exists
among the sperm of certain species of mice to identify genetic relatives.
The identification mechanism itself is not apparent and, based upon
observations of analogous processes in bicycle pelotons, an alternative
hypothesis is suggested. There are similarities between bicycle pelotons
and sperm aggregations: they are both competitive dynamical systems, and
there are energy savings mechanisms by which agents couple and facilitate
self-organized aggregate formations. A model for the division of a peloton
at critical output levels is shown and suggested as analogous to certain,
but not all, sperm aggregations, and a model for the relative energy
consumption of coupled and non-coupled aggregates is shown, which suggests
how sub-aggregates may form that are composed of agents within a narrowed
fitness range, and also why the strongest individual agents may not always
reach the target objective first. This suggests that no mechanism is
required for the identification of genetic relatives, but that sorting
occurs according to a self-organized metabolic process whereby sperm with
close fitness levels will aggregate. Sorting among sperm is hypothesized to
occur at a critical output threshold, and is more likely to occur among
promiscuous species than monogamous species because sperm velocity of
monogamous species may not be high enough to reach the critical sorting
threshold. Genetically related sperm are more likely to have closer average
fitness levels, and so will naturally sort into groups composed of
predominantly related sperm. Thus proposed is an alternative framework by
which to analyze the data.
_______________________
Introduction
Fisher and Hoekstra (2010) provide evidence that supports the
hypothesis that sperm identify related sperm, aggregate and cooperate with
them and, through increased velocity when travelling in aggregations,
provide an advantage to genetically related sperm in advancing one of their
kind to impregnate the egg. The authors report a species of mouse whose
sperm exhibits "the ability to recognize sperm based on genetic relatedness
and preferentially cooperate with the most closely related sperm." The
question was raised: "how do sperm identify their brothers?" (FRIAM, 2010).
The question reveals a problem in Fisher's and Hoekstra's analysis, and a
clear mechanism for this identification process does not appear to be
suggested in their article.
Observations of peloton dynamics allow an alternative explanation to the
cooperative aggregates that Fisher and Hoekstra (2010) have observed. Here
presented, instead, is the hypothesis that any aggregation among genetically
related sperm is coincidental to what is better explained by aggregates that
form due to coupling among groups of sperm as a result of an energy savings
effect that occurs when sperm travel closely together, an effect that is
similar to drafting in a bicycle peloton. This is a self-organized process
and, as such, no mechanism is required for sperm to identify genetically
related sperm to adjust their positions to be near each other. This process
includes a sorting of individual sperm into groups with proportionately high
numbers of sperm whose swimming fitness is closest to their own.
Genetically related sperm are more likely to have similar swimming fitness
levels than are unrelated sperm. Hence grouping is based upon swimming
fitness and not genetic relatedness, which also partially explains why
aggregates are not entirely homogenous according to relatedness: genetically
unrelated sperm with fitness levels near others, who may be related, will
group with them.
For simplicity, here this self-organized energetic process is referred to as
drafting, although for sperm the energy savings mechanism is a hydrodynamic
one (Lauga and Powers, 2009; Woolley et al, 2009). Similarly, the
interactive dynamic between sperm that allows for this energy savings to
occur is referred to as coupling. Coupling of this nature has been
described as a synchronization of flagellar motion and optimal positioning
of sperm-heads for friction reduction and increased sperm velocities when
travelling in coupled formations as opposed to individually (Woolley, et al,
2009). Woolley et al (2009) describe the mechanism for coupling in bull
sperm as follows:
The subject of the present study, the flagellar synchronisations, resulted
from chance contacts between individual spermatozoa. These events will be
called 'conjunctions'. In a few instances, the two spermatozoa separated
again after a period of conjunction and they resumed the swimming speeds and
beat frequencies that they had shown before the conjunction.
Woolley et al. (2009) go on to show distinct increases in mutual speeds
when coupled (i.e. in conjunction states). Their article does not, however,
appear to discuss overall average savings in energy as sperm accelerate and
decelerate while alternating between conjunctive and separated states
between different sets of coupled sperm, nor do they appear to discuss the
durations of conjunctions/separations, which would provide clues as to
relative differences in inherent fitness and whether sperm aggregations form
with sperm whose range of fitness is relatively narrow. Here it is
hypothesized that this is, however, what does in fact occur: this narrowed
fitness range among sperm sub-aggregates due to sorting is more likely to be
the mechanism underlying the genetically related sperm aggregations in the
Fisher and Hoekstra (2010) findings.
The Woolley et al (2009) article suggests that, similarly to cyclists who
save energy by coupling in a peloton, it is unnecessary for sperm to be of
equal physical fitness to travel at the same (mutually increased) speed: to
travel at equal velocity while being of unequal fitness is facilitated by a
coupled energy savings mechanism. Similarly, Riedel et al (2005), and Lauga
and Powers (2009), for example, appear to support the notion that there is
in fact some form of energy savings occurring among sperm aggregates.
The peloton sorting model
In bicycle pelotons, drafting allows riders within a range of output
capacities to sustain the same speed: weaker riders drafting can maintain
the same speed as stronger riders ahead according to the equation
PDR = (Wa-Wb/Wa) / D*100
¡ Wa is maximum sustainable power (watts) of cyclist A at any given moment
¡ Wb is maximum sustainable power of cyclist B at any given moment
¡ D/100 is the percent energy savings at the speed travelled
This is referred to as the Peloton Divergence Ratio (PDR) (Trenchard, 2009;
2005). Cyclists save energy by drafting at approximately 1% per mile an hour
(Hagberg and McCole, 1990). So, if for example, cyclists are traveling at
25 mph, they save approximately 25% energy by drafting riders ahead.
Extending the illustration, if stronger cyclist A has a maximum sustainable
output at 400w at 25mph, and cyclist B has a maximum sustainable output of
300w, cyclist B could not sustain the same speed as cyclist A if they were
travelling individually and without drafting. Thus where cyclist B has only
75% the output capacity of cyclist A, PDR = (400-300/400) /D*100; PDR=1. As
long as PDR is <1, cyclists can maintain the same speed. If PDR>1, cyclists
will not be able to maintain the same speed and will diverge, or decouple.
So, at a speed of 25mph, all cyclists within a range of 25% output capacity
can travel at the same speed.
Here it is suggested that PDR applies to certain types of sperm
aggregations, though not necessarily to all types, as there are several
different types of sperm morphology (Immler, et al. 2007), and their
respective energy savings mechanisms therefore cannot be assumed to be the
same. In fact, PDR does not appear to apply to the Woolley et al (2009)
bull sperm observations, although it may to the Riedel (2005) observations,
the Moore et al. (2002) and Immler (2007) observations of "train"
aggregations, which in pelotons occur during a distinctive phase of energy
output when all riders are riding at or near maximum sustainable speeds, or
when riders are at or near PDR=1. The phase is unstable and small
increases in speed or disturbances in rider positioning can put riders at
PDR>1 and precedes peloton separations and the formation of sub-pelotons.
When cyclists in a peloton approach PDR=1, a sorting process occurs whereby
sub-pelotons form that are composed of cyclists within a smaller range of
inherent fitness levels; i.e. each cyclist in the group has an inherent
fitness level (max sustainable output) that is closer to the average of the
sub-group than it is to the larger aggregate. When peloton divisions occur
at points of instability (PDR >1) and cyclists in a competitive situation
exert maximal efforts to remain among the composition of the group ahead,
but are unable to do so, it is self-evident that the average fitness of the
group behind is less than that ahead, and that each of the groups contain
cyclists of closer average fitness than when among the undivided aggregate.
The range of fitness within each sub-group is also effectively narrowed
further by the drafting process, as evidenced by the fact that sub-pelotons
in a mass-start bicycle race finish a race with nearly identical finishing
times (eg. see data in Trenchard, H., Mayer-Kress, G., 2005). This would not
be self-evident or a reasonable conclusion if the groups were not all
proceeding at maximum sustainable outputs, but had divided for
non-competitive reasons.
This conclusion, however, does not preclude the possibility that some
cyclists with fitness levels which could sustain them in faster groups do
end up in slower groups (i.e. fitness levels that are substantially above
the average of the group), and so there may be a small proportion of
cyclists with fitness levels that overlap the ranges of different groups, as
would there be among sperm sub-aggregates.
The sorting process and formation of aggregates with close average fitness
is well illustrated by imagining a peloton composed of 75 cyclists with a
broad range of abilities: 25 cyclists are professional level and can sustain
speeds of 50k an hour on the flat without drafting, 25 cyclists are medium
amateur level and can sustain speeds of 30km on the flat without drafting;
25cyclists are kids who can sustain speeds of 15km per hour on the flat
without drafting. If they all start together, the peloton is 75 strong up
to approximately 20km/h (because the kids can draft, they can go faster than
they could without drafting); at 21 km/h, the peloton sorts into two groups:
25 kids, and 50 medium and pro cyclists. The group of 50 accelerates, and
when they travel at approximately 36km/h the peloton divides again. It
divides at 36km/h and not 30km/h because the medium-level riders can draft
up to speeds approximately 20 percent faster than they could achieve on
their own without drafting. When speeds of 36km/h are sustained, eventually
all the medium-level cyclists will be separated from the professional
cyclists, and most, if not all, will end up together in a group. At this
point the original peloton has divided into three groups containing riders
with fitness levels near to the average of the group. In an actual
competition and peloton that is composed of all professional riders, the
sorting process is more subtle because average fitness of all the cyclists
is very close from the outset, but the effect is fundamentally the same.
Applying the peloton model to sperm aggregations and the Fisher and Hoekstra
findings
Here it is hypothesized that a similar sorting dynamic occurs in sperm
aggregations and may provide a clue as to the composition of the
sub-aggregates and the proportional representation of conspecific and
heterospecific sperm in any given aggregate, as identified by Fisher and
Hoekstra (2010). Thus if the sperm of two males, say heterospecific in the
first example the authors provide, is mixed into an initial single
aggregate, the aggregate will begin to divide according to PDR as the sperm
accelerate. Sorting occurs as weaker sperm end up being "dropped" into
trailing sub-aggregates, as in the peloton illustration above. Thus, if a
set of sperm from an individual conspecific male are, as a group, fitter
than those of the heterospecific competitor, there will be a self-organized
tendency for sperm of close physical fitness to group together. Some
individual sperm from other groups, however, will be capable of sustaining
the speed of fitter sperm if they group with fitter sperm, as long as they
are at PDR<1 The proposition is thus that genetically related sperm are
naturally closer in physical fitness and therefore will tend to aggregate
together through self-organized coupling dynamics, as presented here.
The composition of sperm aggregates is thus determined by individual sperm
fitness levels and the energy savings due to drafting at the velocity
travelled. Divergences in the aggregates occur at critical individual
output/speed levels. This is particularly so in the case of the promiscuous
species, P. maniculatus, and here it is assumed that sperm in a competitive
situation naturally swim at or near maximum sustainable speeds. This is a
reasonable assumption in a competitive situation, in which all sperm are
seeking to reach the egg first.
However, as indicated in the Fisher and Hoekstra article, in the case of P.
polionatus, the monogamous species, sperm may not travel at or near maximum
sustainable speeds, which suggests that the same degree of sorting does not
occur as among their faster swimming P. maniculatus counterparts. This
provides an explanation why P. polionatus sperm tend to mix
indiscriminately, as the authors describe (see Table 1); i.e. the sperm have
adapted to swimming at less than maximum speeds as an intrinsic
characteristic of monogamous species, and the sorting of sperm into groups
with nearly equal fitness does not occur because the critical output
threshold is not reached for this to happen.
Table 1 summarizes findings presented in the Fisher and Hoekstra article and
provides an alternative peloton model explanation. Fisher and Hoekstra show
results for three experiments involving different combinations of mouse
species sperm mixes. Table 1 summarizes both the results of the Fisher and
Hoekstra study, and the alternative peloton model explanation:
Test
Result
Peloton model explanation
1
Sperm from one heterospecific (P. polionotus) male and one conspecific
(P. maniculatus) male are mixed in vivo assay
"found that overall groups were composed of significantly more
conspecific sperm than expected at random"
Sperm from each of the conspecific males exhibit closer physiological
fitness than as between heterospecifics; i.e. conspecific males have average
fitness close to each other, as do rival heterospecifics to each other .
Some sperm for each sets, however, exhibit close physiological fitness
levels, and these represent the proportion of the aggregates that are not
from conspecific males. There are also percentages of each of
heterospecifics and conspecifics whose fitnesses are such that they are
capable of travelling with groups, but which are "trapped" in the slower
travelling groups.
2
Sperm from two unrelated male P. maniculatus, a promiscuous species,
are mixed
"sperm group significantly more often with sperm of the same male than
expected at random"
The explanation above applies to the related conspecific maniculatus
males
3
Sperm from two unrelated conspecific males of P. polionatus, a
monogamous species, are mixed
"aggregations form indiscriminately in assays"
The speed at which these sperm aggregations travel is relevant. It may
be that the sperm of monogamous species travel slower (due to decreased
competition) than the critical speed at which self-organized sorting occurs.
4
Sperm from related P. maniculatus was mixed
"found a greater proportion of sperm from the same male grouped
together than was expected at random."
The explanation in cases 1 and 2 above applies to the related
conspecific maniculatus males
Table 1
In cases 1,2 and 4, the sorting process described in the foregoing provides
a reasonable alternative explanation to the formation of sperm aggregrations
with close average fitness, and the proposition that it is likely that sperm
of one male have closer average fitness than another male, whether it is
related or not. Some proportion of the two sets of sperm will mix, but at
the critical threshold when sorting occurs, sperm with nearest average
fitness will aggregate.
In case 3, the lower vitality of monogamous sperm, as the authors' finding
indicate, and the smaller testes of P. polionatus suggests that sperm
swimming speeds are slower and/or do not proceed at near maximal output
levels. The slower sperm speeds of monogamous species is supported by other
finding (Nascimento, 2008; Fitzpatrick et al. 2009), although it is not
clear whether sperm are simply slower with a lesser maximum output capacity,
or whether they are in fact capable of swimming faster, but simply do not.
If the peloton model holds, then the inference is that the sperm of
monogamous species are capable of swimming faster but do not do so and, as a
result, are less likely to reach the critical output threshold by which they
will sort into sub-aggregates that contain sperm of near equal fitness
levels. These relative speeds and output levels should be investigated and
confirmed.
Conclusion
The analysis here presents an alternative hypothesis for the findings
presented in the Fisher and Hoekstra article. Based upon analogous behavior
observed in bicycle-pelotons, it provides an analytical and experimental
framework by which existing data could be re-analyzed or further experiments
conducted to test for the observations and principles outlined.
References
FRIAM email listserv Feb 12, 2010, R. Critchlow. Vertical axis windmills and
sperm pelotons.
Hagberg, J. and McCole, S. 1990. The effect of drafting and aerodynamic
equipment on energy expenditure during cycling. Cycling Science 2:20.
Immler, S., Harry D.M. Moore, H., William G. Breed, W., Birkhead, R. (2007).
By Hook or by Crook? Morphometry, Competition and Cooperation in Rodent
Sperm, PLoS ONE. 2(1): e170.
Published online 2007 January 24.
Lauga, E., Powers, T. The hydrodynamics of swimming microorganisms. Rep.
Prog. Phys. 72 (2009) 096601
Moore, H., Dvorakova, K., Jenkins, N. Breed., W. (2002). Exceptional sperm
cooperation in the wood mouse. Nature 418, 174-177
Nascimento JM, et al.(2008) The use of optical tweezers to study sperm
competition and motility in primates. J R Soc Inter 5:297-302. Published
online 2007 July 24. doi: 10.1098/rsif.2007.1118.
Fitzpatrick, J., Montgomerie, R., Desjardins, J., Kelly, S., Kolm, N.,
Balshine, S. Female promiscuity promotes the evolution of faster sperm in
cichlid fishes PNAS January 27, 2009 vol. 106 no. 4 1128-1132
Riedel, et al. 2005. A Self-Organized Vortex Array of Hydrodynamically
Entrained Sperm Cells
Science 8 July 2005: 300-303
Trenchard, H., Mayer-Kress, G. (2005) Self-organized oscillator coupling
and synchronization in bicycle pelotons during mass-start bicycle racing.
Intl Conference on Control and Synchronization of Dynamical Systems. Oct 4-7
Leon, Gto. Mexico.
Trenchard, H. (2009). Self-organized coupling dynamics and phase
transitions in bicycle pelotons. AAAI Fall Symposium, Arlington VA.
Technical Report Series FS-09-03.
Woolley, D., Crockett, R., Grook. W., Revell, S. (2009). A study of
synchronisation between the flagella of bull spermatozoa, with related
Observations. The Journal of Experimental Biology 212, 2215-2223
Yang, Y., Elgeti, J, Gompper, G. (2008) Cooperation of sperm in two
dimensions: synchronization, attraction, aggregation through dynamic
interactions. Phys Rev. E. 061903
Appendix A
Further development of the peloton/sperm aggregation model
Note that PDR is a useful model if an energy savings mechanism exists
whereby one of two coupled sperm benefits from the energy savings mechanism,
while the other does not. This may be the mechanism in the mouse species
described in the Moore (2002) and Immler (2007) articles, as indicated by
the "train" formation, which is similar to "single paceline" peloton
formations when cyclists are aligned near or at PDR=1 to each other
(Trenchard, 2009; 2005).
There do appear, however, to be other energy savings mechanisms in sperm
aggregates, such as for example the conjunction and synchronized dynamic of
bull sperm (Woolley et al, 2009). In a sperm conjunction (Woolley et al,
2009), the PDR equation does not strictly apply and must be adjusted because
the stronger sperm also appears to benefit from the coupled formation, which
does not occur to any significant degree between coupled cyclists; i.e. in a
peloton the front riding cyclist does not receive any reduction in output
from the rider behind, while the rider behind benefits substantially by
drafting; for bull sperm, it appears that both coupled sperm benefit by
increased velocity.
This leads to a further hypothesis that the stronger sperm increases speed
with some reduction in metabolic cost, while the weaker sperm increases
speed with little or no change in metabolic cost, although they both travel
faster than they would individually. Thus it is the stronger sperm that
benefits by energy savings, while the weaker sperm benefits by increased
speed, but with no savings in energy. There is an implication that the
stronger sperm will always be able to advance to the front of the sperm
aggregation faster than weaker sperm. However, this is not necessarily so,
as it depends on the relative durations of coupling and separations. That
is to say, a weaker sperm could advance farther and faster than a stronger
sperm if it spends sufficiently more time coupled than a stronger sperm
which may spend relatively more time isolated. Thus the faster sperm are
not necessarily those that are stronger, but those whose proportion of total
coupling time exceeds the percent differences in their relative fitness
levels.
The following model is descriptive for coupled organisms that may alternate
durations of time spent coupled with time spent non-coupled, and which
mutually benefit from coupling because it accounts for the proportions of
total time spent both saving energy in coupled positions and not saving
energy in non-coupled positions. It is a simplified model because there are
other factors that affect total output than time spent in energy-saving
positions (coupled) and positions where there is no energy savings.
However, it provides insight into the energetic dynamics of coupled agents
of varying degrees of fitness and why they do not necessarily achieve
positions based on inherent fitness.
TO = Wa-(Wa*%E) * T) / Wb-(Wb*%E) * T)
· Where TO is ratio of total output of two agents in coupled
positions (not necessarily with each other) with identical objective (e.g.
to win a race or impregnate an egg); here, sperm or cyclist; in the case of
sperm, millijoules; for cyclists, calories
· T is total time spent in coupled positions and travelling at
mutually faster velocities than achievable in isolation
· %E is percent energy savings in coupled formation
· Wa is the maximum sustainable power output; picoNewtons for sperm,
watts for cyclists of stronger agent A (cyclist or sperm) at a given moment
, assuming that in a competitive situation, agents are travelling as fast as
their metabolisms will allow.
· Wb is the maximum sustainable power output at a given moment of
weaker agent B, again assuming that in a competitive situation, agents
travel as fast as metabolisms will allow.
For example (quantities and units for illustration only): if stronger sperm
A has max sustainable output of 50pN, and B has max sustainable output of
45pN, and A saves an average of 10% output when coupled and spends a total
of 10 minutes coupled, total output for A is 50-5*10, or 450mj, while the
weaker sperm saves no energy when coupled, but spends 11 minutes in coupled
positions, we have 45-0*11, 495; and 450/495 = 0.91. Thus where this ratio
is <1, the weaker sperm potentially can be ahead of the stronger sperm over
the duration of the coupling interactions. Thus this ratio indicates why it
is not necessarily the case that stronger sperm will impregnate the egg.
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