Biological Journal of The Linnean Sociely (1985),26: 345-358
The ideal and the feasible:
physical constraints on evolution
R. McN. ALEXANDER
Departmenl of Pure and Applied ,zbology, University of Lceds, Leeds LS2 93T
ilccepledfor publicalion Ju!y 1985
Thc laws of physics and the properties of the physical environment impose constraints o n evolution.
Structures and processes that may be imagined cannot in some cases be evolved, because they are
physically impossible. This paper explores the consequences of the particulate nature of matter and
of light; of the wave nature of light and sound; of the laws of diffusion and heat exchange; of the
mechanical properties of materials; of limits to aerodynamic and hydrodynamic performance; and
of the behaviour of electririty.
KEY WORDS:-Physics
-
biophysics
Introduction . . . . . .
Particles . . . . . . .
Waves.
. . . . . . .
Diffusion and heat exchange . .
Merhaniral properties . . . .
Aerodynamics and hydrodynamics .
Electricity . . . . . . .
Conclusion
. . . . . .
Summary . . . . . . .
Acknowledgements
. . . .
References. . . . . . .
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INTRODUCTION
An ideal animal might have a skeleton made of some unbreakable material
which had infinite strength but no mass. Its muscles would be capable of
responding instantaneously to stimuli, exerting unlimited force and contracting
(if required) at infinite speed. It could see infinitesimally small objects and could
distinguish the finest gradations of illumination from utter darkness to infinite
brightness. It would possess a psychic sense which informed it infallibly of the
positions and intentions of all other animals. Its metabolic processes would
proceed with lOOo/, efficiency. Similarly, an ideal plant might intercept all the
sun’s radiation and utilize it all in photosynthesis, wasting none of its energy.
Alternatively, the ideal animal and plant might both create all the energy they
needed, instead of depending on food or photosynthesis. The distinction between
animals and plants would be destroyed.
Invited paper read at the Population Genetirs Group meeting at Manchester, January 1985.
0024-4066/85/120345+ 14 SOS.OO/O
345
0 1985 The Linnean Society of London
346
R. McN. ALEXANDER
Such ideal organisms are not feasible in the real world, which has
characteristics that cannot be altered by evolution. Organisms are subject to the
laws of physics. They may synthesize new materials, but the ranges of physical
properties that can be produced are strictly limited. They cannot alter the
properties of the air or water in which they live. The pressure and composition
of the atmosphere, and the intensity and composition of solar radiation, are
beyond their control. Such physical considerations impose constraints on
evolution. Structures and processes that might be imagined cannot evolve,
because they are physically impossible.
This paper examines some of the constraints and enquires how closely they
are approached by known organisms.
PARTICLES
Some of the physical constraints on evolution arise because matter consists of
molecules which cannot be subdivided without a change of chemical properties,
and because light consists of indivisible photons.
Molecular size sets a lower limit to the size of organisms. Viruses can be
extremely small because they do not carry the full equipment for life, but
depend on the metabolic apparatus of the host cell. The smallest organisms
known to be capable of independent life on a non-living medium seem to be
mycoplasms of diameter 300 nm (Pirie, 1973). All cells need enclosing
membranes, and those of mycoplasms are about 8 nm thick. It has been
suggested that about 45 different enzymes may be necessary for life, and it has
been estimated that one molecule of each, plus one each of the ribosomes needed
to make them, would fill a 60 nm cube (Pirie, 1973). Imagine a spherical
organism of diameter 100 nm. If its cell membrane were 8 nm thick it would
occupy 40% of the cell volume. A single set of enzyme molecules and ribosomes
would occupy a further 40% of the volume, leaving only 20% for other cell
components. It seems most unlikely that so small an organism could exist: the
minimum feasible diameter presumably lies somewhere between this and the
observed minimum (for mycoplasms) of 300 nm. However, this argument
depends on the assumptions that the cell membrane cannot be made thinner,
and that 45 enzymes are needed. Both assumptions are questionable.
The molecular nature of matter also sets limits to the sensitivities of olfaction
and of hearing. In the case of olfaction the limit occurs because one molecule is
the smallest detectable quantity of an odour. It has been shown that action
potentials can be elicited in the receptor cells on silk moth (Bornbyx) antennae by
one or at most two molecules of the sex pheromone bombycol (Kaissling &
Priesner, 1970). In the case of hearing, a limit occurs because very faint sounds
would be masked by the random Brownian movement of molecules (Harris,
1968).
The roots of plants are believed to sense the direction of gravity by means of
dense particles that tend to sink in the cytoplasm. It has been argued that for
this to work, the particles must have diameters of at least several micrometres.
Otherwise their rates of sinking would be masked by Brownian movement
(Hoppe, Lohmann, Mark1 & Ziegler, 1983).
Light arrives at eyes as discrete photons. It has been shown both for
PHYSICAL LIMITS TO EVOLUTION
347
vertebrates and for insects that single photons evoke detectable electrical events
in the receptors (see Land, 1980), so we can think of vision as a process of
counting arriving photons. Consider how bright patches in a field of view can be
distinguished from dimmer ones. A receptor of area A looking at a particular
patch receives on average nA photons per unit time. The brightness of the patch
can be assessed by counting the photons arriving in some time interval At. The
mean number arriving in intervals of this length is nA At and the standard
deviation (if photons are emitted at random times) is (nA At)+. If fine
discriminations of brightness are to be made, the standard deviation must be
small compared to the mean, which implies that large numbers of photons must
be counted. Land (1980) showed, for example, that discrimination of 10%
differences of brightness with 95% reliability required counting at least 768
photons per receptor. The number of photons available for counting can be
increased by increasing the time interval Al, but this makes the eye less able to
observe rapid changes. Alternatively, i t can be increased by making the
receptors larger (by increasing A ) or by pooling the counts from adjacent
receptors, but this makes the eye less able to resolve fine detail. Thus,
discriminations of brightness and of temporal and spatial detail have competing
requirements. An improvement in one may be obtainable only be sacrificing one
of the others, or by evolving a larger eye. The problems are particularly severe
for animals such as the dragonfly figomma, which pursue moving prey in dim
light. a g o m m a hunts at dusk and has envolved much larger ommatidia than
dragonflies that hunt at midday (Snyder, 1977).
WAVES
Other constraints apply to evolution, because light and sound have wave
properties.
The radiation emitted by the sun spans a wide spectrum of wavelengths but
most of the energy is associated with wavelengths of the order of 0.5 pm.
Accordingly, such wavelengths are used for photosynthesis and for vision. The
chemical processes involved have to be driven by the quanta of energy
associated with photons of these wavelengths.
Diffraction effects limit visual acuity, making it impossible to distinguish
objects of which the images on the retina are less than about 1F apart (see, for
instance, Land, 1980). Here 1 is the wavelength and F is the F-number, the
ratio of the focal length to the aperture. T o distinguish images spaced 1F apart,
the retinal receptors should be 1F/2 apart (giving a receptor to receive each
image, and one between them). This suggests that an eye with an F-number of
2, receiving light of wavelength 0.5 pm, should have retinal receptors 0.5 pm
apart. Smaller F-numbers are possible (for example, 1.3 in teleost fishes; Land,
1980) but i t is impossible to makes lenses with very small F-numbers unless
materials of high refractive index are available. This suggests that eyes should
have the lowest possible F-numbers and correspondingly closely packed retinal
receptors. However, waveguide effects make receptors of diameter less than
1 pm unsatisfactory (Land, 1980). This is not a fundamental limit, but depends
on the refractive indices of the receptors and the surrounding fluid. If receptors
cannot have diameters less than 1 pm, no gain in acuity is obtained by reducing
348
R. McN. ALEXANDER
the F-number below 4. Also, images to be distinguished must be at least 2 pm
apart and the minimum angle between objects that can be resolved is 2lf, where
f is the focal length of the eye in micrometres. I n practice, eyes seem always to
have receptors more than 1 pm apart, and correspondingly poorer acuity.
The sounds that animals can produce and detect depend on their sizes.
Typically, the wavelengths are similar in magnitude to the linear dimensions of
the body. For example, the musical range of human voices (bass to soprano) is
approximately from 80 to 800 Hz, corresponding to wavelengths from 4 to
0-4m. Several reasons combine to restrict wavelengths in this way. The
principles involved are explained by Alexander (1983) and in textbooks of
acoustics. First, the behaviour of fluid-filled resonators, such as the oral and
nasal cavities of man and the vocal sacs of frogs, depends on their sizes. The
wavelength, corresponding to the resonant frequency, tends to be somewhat
larger than (but similar in magnitude to) the linear dimensions of the resonator.
Loudspeakers, and sound-emitting organs of animals, are inefficient if their
diameters are too small. They work best if the diameter is at least one-sixth of
the emitted wavelength. The direction from which sound comes can be judged
most easily, by an animal with ears on its head, if the diameter of the head is not
too small compared to the wavelength. This is true, whether direction is judged
by comparing phases or times of arrival at two ears, or by using the sound
shadow of the head.
For these reasons, animals tend to be adapted to produce and to hear sounds
of which the wavelengths are not too large, compared to the linear dimensions
of the body. It has even been shown that toads use the pitch of rivals’ croaking
to assess fighting ability: a deep-voiced toad is large and therefore dangerous
(Davies & Halliday, 1978). However, various animals have become adapted for
efficient production of sound of long wavelength (compared to the body) by
enlargement of the sound-producing organs. I n bladder cicadas (Cystosorna,
Simmons & Young, 1978) the whole abdomen vibrates and serves as a
loudspeaker. In mole crickets (Cryllotalpa, Bennet-Clark, 1970) the burrow serves
as an extension of the sound-producing organs.
Fish with swim-bladders present exceptions to the general rule. A gas-filled
bubble, submerged in water, resonates at the frequency corresponding to a
wavelength (in water) of 200 times its diameter. Swim-bladders resonate in
similar fashion and increase the sensitivity of their possessors to sounds near the
resonant frequency. This is especially effective in species that have evolved a
connection between the swim-bladder and the ear (Alexander, 1983). Also,
some species use the swim-bladder as a resonating sound producer: examples
include cod (Gadus) and some catfishes (Blaxter & Tytler, 1978). Thus, many
fish are most sensitive to, and some produce, sounds of long wavelength
compared to the dimensions of the body. The size of the swim-bladder is
generally set by the requirements of buoyancy, and in turn determines the
resonant frequency.
Sounds used for echo-location must not have excessively long wavelengths,
compared to the dimensions of the objects that are to be detected. Otherwise,
the objects would scatter very little of the sound, and so would produce very
little echo. Insect-earing bats generally produce sounds of frequencies
30-100 kHz, wavelengths 3-10 mm. Most of the insects they eat have bodies of
at least 1 mm diameter.
PHYSICAL LIMITS T O EVOLUTION
349
DIFFUSION AND HEAT EXCHANGE
Processes of diffusion are very important in organisms. Oxygen must diffuse to
tissues that are respiring and carbon dioxide to tissues that are
photosynthesizing. Waste products may be lost by diffusion.
Gases diffuse from regions where their partial pressures are high to regions
where they are low. The rate of diffusion, whether through a liquid or through a
mixture of gases, is proportional to the gradient of partial pressure and to the
diffusion constant of the gas in the particular medium. Differences in partial
pressure, available to drive diffusion, may be limited by environmental
conditions. For example, the partial pressure of oxygen in the atmosphere is 0.21
atmospheres so the partial pressure difference, driving oxygen into a respiring
tissue, cannot exceed 0.21 atmospheres. Also, the diffusion constant has a
particular value for a particular gas diffusing through a particular medium.
Large animals have circulatory systems that carry oxygen from the surface of
the body (or from lungs or gills) to the tissues, but many small animals rely on
diffusion. The larger they are, the larger the distance over which diffusion must
occur, and the smaller the gradient of partial pressure (for a given partial
pressure difference). If the animal is too large, diffusion cannot supply oxygen
fast enough to satisfy its needs. It has been argued in this way that cylindrical
turbellarian worms could not have diameters greater than 1.5 mm, and that
typical (flattened) ones could not be more than 1 mm thick (Alexander, 1979). I
know of no free-living flatworms which transgress these rules, but the rules do
not apply to parasitic flukes that depend on anaerobic metabolism. When
diffusion occurs through air, longer diffusion distances are possible because the
diffusion constant is then higher. Even the very active flight muscles of insects
can be supplied by diffusion along radial tracheae from an axial (ventilated)
trachea. It has been calculated that the maximum diameter for a wing muscle so
supplied is about 3.5 mm (Weis-Fogh, 1964; Alexander, 1979), but the
calculation depends on the fraction of the muscle volume occupied by tracheae,
as well as on the metabolic rate of the muscle. Only a few of the very largest
insects have larger wing muscles than this, and diffusion in them is probably
supplemented by pumping. The limits stated in this paragraph depend on the
metabolic rates required. Thicker flatworms and wing muscles could be supplied
by diffusion if their metabolic rates were lower.
Diffusion constants have ‘particularly clear limiting effects when two gases
diffuse along the same pathway. Thus, oxygen diffuses into developing birds’
eggs through the pores in their shells, and water vapour diffuses out by the same
route. A given rate of respiration of the embryo implies a rate of water loss
which is inevitable unless the humidity of the nest can be increased (Rahn, Ar &
Paganelli, 1979). Similarly, carbon dioxide diffuses into leaves through the
stomata and water vapour diffuses out by the same route. Any given rate of
photosynthesis implies a minimum rate of water loss. However, losses can be
reduced in the case by the ‘trick’ of crassulacean acid metabolism, whereby
carbon dioxide is taken up at night (when the saturation deficit of the air is low)
for use in photosynthesis when daylight returns. This trick is used by cactus and
some other desert plants (Walker, 1966).
Heat travels from regions of higher temperature to regions of lower
temperature, by conduction, convection and radiation. Rates of gain and loss of
heat can be altered by moving to a different environment, or by a change of
350
R. McN. ALEXANDER
shape which alters surface area. Heat can be lost by allowing water to
evaporate, as in the sweating of mammals and transpiration of plants. However,
if the environment and surface area are fixed and water is to be conserved, the
processes of heat exchange allow only limited control. Rates of convection can
be modified to some extent by changes of shape, without changes of area. The
deeply lobed shape of the leaves on the sunny side of oak trees promotes
convective cooling by reducing the mean thickness of the boundary layer at any
given wind speed (Vogel, 1970). Rates of uptake of solar radiation can be
modified by changes of colour. Thus, black beetles absorb 60% of incident solar
radiation but some desert-living beetles with white elytra absorb only 20%, and
so reduce the danger that the animal will overheat (Edney, 1971). However,
evolution can do virtually nothing to control exchange of long-wave radiation
with the environment. Plants and terrestrial animals of all colours behave
essentially as black bodies to long-wave radiation (Monteith, 1973; Cena &
Clark, 1978). Metallic surfaces do not behave as black bodies but have lower
absorbtivities and emissivities, and have been used for fire-fighting clothes and
for the ‘space blankets’ carried by mountaineers, but they have not been evolved
by organisms.
There is limited scope for evolution to modify rates of conduction of heat, in
and out of animals. Air, trapped in fur or feathers, seems to be the best heatinsulating material available to them. The thermal conductivities of animal
coats are always rather higher than the conductivity of still air, because heat is
transmitted through them by radiation and convection as well as by conduction
(Cena & Clark, 1978). Animals of given dimensions can conveniently bear only
a limited thickness of fur or feathers. (A mouse would have trouble walking if it
had fur as thick as a sheep’s fleece.) Even if there were no practical limit to the
thickness of the coat, its insulating effect would still be limited. This is because
successive layers of insulation, added one outside the next, have successively
greater areas and therefore present less and less resistance to heat flow. Think of
an animal as a sphere of radius r, with added insulation of thermal conductivity
k. It produces heat by metabolism at a rate M. No matter how thick the coat is,
the temperature difference between the skin and the outer surface of the coat
cannot exceed Ml4nkr. If the coat thickness equals the radius of the enclosed
body, the temperature difference is half this much (Cena & Clark, 1978). For
related animals of different sizes, M can be expected to be proportional to
(body mass) O a 7 and r to (body mass) 0.3 (Schmidt-Nielsen, 1984). Thus, small
animals cannot maintain large temperature differences between themselves and
their environments. This is probably why no birds or mammals have masses less
than about 2 g.
Heat insulation may also set an upper limit to mammal size: excessively large
mammals would overheat. The blood of animals must be separated from the
environment by a protective layer of skin, which has limited thermal
conductivity. No materials of high thermal conductivity (like metals) seem to be
available for use in skin construction. If a mammal producing heat at a rate M
has skin of area A, thickness s and thermal conductivity k’, the temperature
difference across the skin is Ms/Ak’. As already noted, M can be expected to be
proportional to (body mass) O e 7 5 . For geometrically similar animals, A is
proportional to (body mas^)^'^'. It seems reasonable to assume that s must
increase with body mass. (An elephant needs thicker skin to protect it than a
PHYSICAL LIMITS T O EVOLUTION
35 I
mouse does). If so, the temperature difference across fur-free skin must increase
as body mass increases, and excessively larger mammals would be liable to
overheat. A mammal larger than an elephant might overheat in a tropical
climate, but whales do not overheat because they live in permanently cold
water.
Organisms cannot function as effective heat engines, converting heat to other
forms of energy. This follows indirectly from the second law of thermodynamics,
which says that entropy (disorder) tends to increase. One consequence of the
law is that a heat engine in which the absolute temperature of a substance falls
from temperature TI to Tz cannot have an efficiency greater than (TI- T z ) / T l
(Mitton, 1961). Steam engines can convert a useful fraction of heat to work,
because different parts of the engine have substantially different temperatures,
but only small differences in absolute temperature are possible within an
organism, so only trivial fractions of heat could be converted. Thus, a runner
travelling downhill uses his muscles as brakes, to degrade potential energy to
heat, but he cannot use the heat energy obtained in this way to propel himself
up the next hill.
M K H A N I CAL PROPERTIES
Animals and plants have evolved a wide variety of structural materials, most
of which are either fibres (such as collagen and silk) or composites (such as
wood, bone and mollusc shell). Composite materials are mixtures of several
components which retain their identity: collagen and hydroxyapatite in bone,
resin and glass in fibreglass.
Natural materials seem much more subtly designed than man-made ones.
The subtlety may be at the molecular level: man-made polymers are
monotonous chains of (usually) identical monomer units but organisms produce
proteins in which about 20 amino-acid species are connected in precisely
controlled patterns. Alternatively, the subtlety may concern the arrangement of
molecules or groups of molecules: man-made composites may have fibres
arranged in chosen directions but they cannot match the complexity and
precision of fibre arrangement found in the helicoidal cuticle of insects or the
Haversian bone of mammals (see Wainwright, Biggs, Currey & Gosline, 1976).
Despite this capability for producing varied materials, there are limits to the
ranges of possible properties. The properties depend on the arrangement and
interconnection of molecules, more than on their chemical nature. Metals,
ceramics and glasses consist of closely packed molecules, tightly bonded
together. They therefore tend to be relatively strong and stiff, with yield
strengths of the order of 1 GPa and Young’s moduli of the order of 100 GPa
(Ashby & Jones, 1980). They stretch very little before yielding. Rubbers and
other amorphous, cross-linked polymers consist of long, flexible molecules with
relatively few interconnections, and are consequently relatively weak and
deformable. Both their yield strengths and their Young’s moduli are commonly
of the order of 0.01 GPa. Many of them stretch to double their initial length
before yeilding. Fibrous polymers consist of long molecules which are loosely
connected in some places, but closely packed in crystalline arrays in others.
Their properties depend on the degree of crystallinity, overlapping those of
ceramics at one extreme and rubbers at the other. Metals are both strong (they
352
R. McN. ALEXANDER
withstand large static forces) and tough (they require a lot of energy to break
them, and are therefore resistant to impacts) but organisms seem incapable of
producing them. Ceramics such as hydroxyapatite crystals can be produced by
organisms and are strong but brittle (the opposite of tough). The useful
combination of strength and toughness can however be obtained by making a
composite of a ceramic and another material, as in bone or mollusc shell. Wood
and insect cuticle are composites in which the stiff phase is a highly crystalline
polymer (cellulose or chitin) rather than a ceramic, Most rigid skeletons are
composite materials.
Thus, the properties required of a structural material tend to dictate its
nature (ceramic, fibre, composite etc.) but not its chemical composition. Also,
there are limits to the ranges of properties available. In particular, it seems to be
impossible to produce tough materials with yield strengths higher than about
2 GPa (Ashby & Jones, 1980; Vincent & Currey, 1980). Silks, cellulose fibres
and insect apodemes (Bennet-Clark, 1975) attain strengths of the order of
1 GPa. It is rather surprising that collagen is so widely used as a structural
material although its strength is only about 0.1 GPa.
The fibres and composite materials produced by animals resemble fibres and
composites used in engineering. Muscle, however, is quite unlike any man-made
material in being able to use chemical energy to do mechanical work.
Vertebrate striated muscle seems to be rather uniform in strength, exerting
maximum isometric stresses of about 0.3 MPa (see Weis-Fogh & Alexander,
1977). Some other muscles are much stronger: locust leg muscles exert 0.8 MPa
(Bennet-Clark, 1975) and mollusc muscles exert up to 1.4 MPa (Ruegg, 1968).
There is no apparent fundamental limit to muscle strength. The stronger
muscles have longer thick filaments, so that more cross-bridges can attach to
each filament. The stresses in the filaments, when muscles contract, are modest.
For example, a vertebrate striated muscle exerting 0.3 MPa has stresses of the
order of 10 MPa in its thin filaments.
The fastest known muscles seem to be a mouse toe muscle and a rat eye
muscle which are capable of shortening at up to 25 fibre lengths per second
(Close & Luff, 1974). Weis-Fogh & Alexander (1977) calculated from their
properties that the maximum power output obtainable from anaerobic muscles
might be about 500 W/kg (this is mean power output over a series of
contractions and relaxations), but they gave no reasons why higher power might
not be obtainable from as-yet-unknown muscles. The power output of aerobic
muscles may be limited by the mitochondria, which house the enzymes of the
Krebs cycle. Pennycuick & Rezende (1984) estimated that various bird and
insect muscles produce mechanical power per unit mass of mitochondria
amounting to about 900 W/kg. This is consistent with the finding of Mathieu el
al. (1981) that the maximum aerobic power consumptions of running mammals
are proportional to the masses of mitochondria in their leg muscles. There may
be a fixed maximum rate at which mitochondria can function, but we know no
physical reason for this limit.
Geometrically similar animals have weights proportional to the cubes of their
linear dimensions, but have cross-sectional areas (and therefore strengths)
proportional only to the squares. Therefore, it is more difficult for large animals
than for small ones to support their weight. Terrestrial animals have to support
their weight but aquatic animals are supported largely by buoyancy. T h e
PHYSICAL LIMITS TO EVOLUTION
353
largest dinosaur of which a reasonably complete skeleton has been found
(Brachiosaurus) is estimated to have had a mass of about 80 tonnes (Colbert,
1962) and was thus less massive than the largest whales. I t is tempting to suggest
that dinosaur size was iimited by the strengths of bone and muscle. However,
the dimensions of dinosaur skeletons seem consistent with fairly active life,
without requiring excessive stresses (Alexander, 1985). Also, inspection of the
skeletons suggests that the bones and muscles could have grown thicker without
making the animal too unwieldy. It seems difficult to identify any principal of
structural engineering that would prevent the evolution of a terrestrial animal
larger than the largest dinosaurs.
Denny, Daniel & Koehl (1985) argued that wave-swept organisms experience
inertia forces proportional to their volumes, but have strengths and adhesive
forces proportional only to areas. Thus, there should be an upper limit to size on
any particular shore. They showed that intertidal organisms tend to be small,
compared to subtidal organisms, but (as in the similar case of large terrestrial
animals) it would be difficult to argue that known species approach any
absolute physical limit.
There is a limit to the performance of suction pumps, which might be
expected to limit the height to which sap can be raised in trees. Ordinary
suction pumps cannot raise water more than about 10 m (corresponding to a
pressure drop of almost 1 atmosphere) because bubbles of vapour tend to
develop in liquid water when zero pressure is approached. Existing bubbles
grow, but surface tension makes it difficult to start new bubbles. For this reason,
fine tubes of bubble-free water can withstand strongly negative pressures,
without forming bubbles. Experiments by Scholander, Hammel, Bradstreet &
Hemmingsen (1965) seem to show that pressures of - 15 atmospheres are
developed in the xylem of tall trees, and as low as -80 atmospheres in desert
shrubs. The tallest trees are about 100 m tall, requiring a minimum pressure
drop of 10 atmospheres to raise water from the ground, and are a long way from
any limit to height that might be imposed by the tensile strength of water,
which is at least 20 MPa (-200 atmospheres, Hoppe et al., 1983).
AERODYNAMICS AND HYDRODYNAMICS
This section is concerned with animals swimming in water or flying through
air, and with stationary animals in currents of water and air. We will be
concerned with organisms of different sizes, and with two different fluids. It will
therefore be necessary to refer to Reynolds numbers, quantities that take
account of size, speed and fluid properties and indicate the circumstances in
which patterns of flow around bodies of different sizes can be expected to be
similar. The Reynolds number is lu/v where 1 is a characteristic linear dimension
(usually the length) of the body, u is the speed of the fluid relative to the body
m2/s for water,
and v is the kinematic viscosity of the fluid ( 1 . 0 ~
1.5 x
m2/sfor air at 20°C). Small bodies moving slowly have low Reynolds
numbers but large bodies moving fast have high ones. Thus, the Reynolds
numbers of swimming spermatozoa, water beetles and dolphins, tend to be of
lo4 and lo’, respectively.
the order of
When a body moves through a fluid, the fluid exerts a drag force on it,
backwards in the direction of movement. Similarly, drag acts on a stationary
354
R. McN. ALEXANDER
body in a moving fluid. If the Reynolds number is 100 or more (which it will
generally be, if we are dealing with macroscopic rather than microscopic
organisms) the drag is conveniently represented by the expression +pu2ACD.
Here p is the density of the fluid, u is the speed, A is (usually) the frontal area of
the body and C, is a quantity called the drag coefficient, which varies somewhat
with Reynolds number but generally changes only a little over a wide range of
Reynolds numbers. The drag coefficient is about 0.5 for spheres (over a wide
range of Reynolds numbers) but varies from less than 0.1 for streamlined bodies
to 1.4 for parachute shapes and 2.3 for long gutter-like shapes with their
concave faces facing the flow (Vogel, 1981; Alexander, 1983). Thus, changes of
shape can be used to alter drag, but only within these limits. For some
organisms (for example, thistledown) a high drag coefficient is advantageous
and parachute-like shapes have evolved. For others (for example, swimming
penguins) drag is disadvantageous. Bilo & Nachtigall (1980) measured drag
coefficients of 0.07 for penguins, which is reasonably close to the value of 0.04
obtainable for man-made streamlined shapes at the same Reynolds number.
An abrupt change in the pattern of flow occurs as the Reynolds number
increases past lo6. The boundary layer becomes turbulent with the result that
the drag coefficient drops abruptly (for unstreamlined bodies) or rises (for
streamlined ones). There is some scope for altering the Reynolds number at
which the transition occurs, and several mechanisms have been suggested that
might prevent the boundary layer from becoming turbulent in dolphins.
However, it is doubtful whether the proposed mechanisms work (Vogel, 1981).
At low Reynolds numbers, drag coefficients for streamlined bodies approach
those for unstreamlined ones. Streamlining becomes ineffectual.
Drag acts backwards along the direction of motion but a component of force
called lift may act at right-angles to the direction of motion. Aeroplanes are
supported by lift acting on their wings. Aerofoils and hydrofoils are structures
designed to obtain lift, and their merits can be assessed by comparing lift
coefficents, which are defined in similar fashion to drag coefficients (see
Alexander, 1983). The lift coefficient depends on the angle at which the aerofoil
is set, relative to the direction of motion, but cannot be increased beyond some
maximum value. Well-designed man-made aerofoils give lift coefficients up to
about 1.5 at Reynolds numbers around lo6, and rather less at lower Reynolds
numbers. Coefficients up to 3.9 are obtainable from multi-slotted aerofoils,
made from several narrow aerofoils set parallel to each other (Landolt &
Bornstein, 1955). There is no reason to expect animal aerofoils to be capable of
exceeding these limiting values. However, it should be remembered that manmade aerofoils generally work in steady conditions, in which the relative
velocity of air and aerofoil makes no sudden changes. The limits apply only to
such conditions. The wings of birds, bats and insects often work in highly
unsteady fashion and calculations of lift coefficients may lead to ‘impossible’
values such as 5.3 for a hovering flycatcher (Ficedula, Norberg, 1975). This does
not indicate any superiority of animal aerofoils over man-made ones, but merely
that lift can be produced in unsteady movement by mechanisms that do not
apply in steady conditions. In steady conditions, as in wind-tunnel tests and in
gliding, animal aerofoils give lift coefficients in the expected range (Vogel,
1981).
Geometrically similar aircraft have weights proportional to the cubes of their
PHYSICAL LIMI’I’S ‘1’0EVOLUTION
355
linear dimensions, but wing areas proportional only to the squares of linear
dimensions. For this reason, large aircraft have to fly faster than small ones, to
produce the lift needed to keep themselves airborne. Similarly, optimum flying
speeds, both for powered flight and for gliding, are generally larger for larger
aircraft. A designer of gliders may aim to make the sinking speed (the rate of
loss of height) as small as possible. Large gliders fly best at higher speeds than
small models but the minimum obtainable sinking speed, for gliders of a wide
range of sizes, seems to lie between 0.5 and 1.0 m/s. This seems to be a practical
limit, but it is not easily explained in terms of basic physical principles. It has
been shown that various gliding animals (birds, insects and a bat) have
minimum sinking speeds only a little higher than well-designed man-made
gliders and models (data summarized by Vogel, 1981; and Alexander, 1982).
Seeds of sycamore ( h e r ) and of some other trees are aerofoils which spin as
they fall, producing lift which slows their fall and increases the chance that they
will land well away from the parent tree. The rate of fall can be calculated by
considering how the seed affects the momentum of the air through which i t
moves. The minimum rate of fall of an ideal seed proves to be 1.5 ( W / p A ) * ,
where W is the weight of the seed, A is the area of the circle swept out by its tip
(as seen from above) and p is the density of the air. Rates of fall of real seeds are
little greater than this minimum (Norberg, 1973).
ELEC‘I‘RICI’I‘Y
The most important electrical events in cells are the action potentials that
occur in neurones, in muscle fibres and in the electric organs of some fishes. The
potential differences involved are small. Even in the strong electric organs of the
electric eel (Eleclrophorus electricus) the potential differences across individual cell
membranes are no more than about 150 mV (Aidley, 1978). Action potentials
are produced by allowing sodium ions to diffuse into the cell and approach
equilibrium, so the membrane potential may ap roach (but cannot pass) the
Nernst potential for sodium. This is 58 log,, (a,rga/ a p ) m V , where a t a and ap
represent the activities of sodium ions inside and outside the cell. Thus a
150 mV requires a ratio ap/ap of at least 390. A
membrane potential of
substantially larger membrane potential, of 1 V, would require a ratio of
1.8 x lo1’. The activity of sodium ions in the blood of animals is of the same
order of magnitude as in seawater (0.5 mol I - , ) . There are -6 x l o z 3ions in a
mole. The volumes of cells are typically of the order of lo4 pm3(10- I 1). For
u t a / a y a to reach 1.8 x lo”, each cell would have to contain only a tiny fraction
of an ion. Thus membrane potentials must be restricted to fractions of volts, and
electric eels can build up large potential differences only by having numerous
cells in series.
It is generally advantageous for axons to conduct action potentials as fast as
possible, to minimize delays in the transmission of information. It can be shown
that the speed of conduction depends on axon diameter (thicker axons conduct
faster) and on the electrical properties of the axon and its outer membrane or
myelin sheath (Aidley, 1978; Stein, 1980) but it is not clear why the membranes
have not evolved even more favourable electrical properties. T h e electrical
behaviour of a particular axon is described by its space constant (which gives a
scale of length for electrical disturbances in the axon) and its time constant
+
356
R. McN. ALEXANDER
(which gives a scale of time). The space constant is proportional to R* (where R
is the electrical resistance of unit area of membrane) and the time constant to R,
so speed of conduction should be proportional to R*/R = R-*. I t is not clear why
R cannot be reduced by putting more pores in the lipid membrane, so that
speed can be increased for axons of any given diameter.
The rate at which an axon can transmit information is limited by the
precision with which the time intervals between action potentials can be
registered (Griffith, 1971). If the intervals can be measured to the nearest
millisecond, and if frequencies up to 1000 Hz are possible, information can be
transmitted at up to 1000 bits per second. Frequencies u p to about 1000 Hz are
observed, but it is not clear whether any fundamental physical principle forbids
higher frequencies and correspondingly higher rates of transmission of
informa tion.
CONCLUSION
The scope of this paper is limited by ignorance, both of biology and of
physics. There are apparent constraints that I cannot explain, and there must
be many others that I have not perceived. In many cases (for example, in the
discussion of gliding performance) the fundamental limits that determine the
practical ones are not fully understood.
The scope of the paper is also limited rather arbitrarily: it could be argued
quite plausibly that the topic, the physical constraints on evolution, should
embrace the whole of comparative physiology.
SUMMARY
The laws of physics and the physical properties of the environment and of
available materials impose constraints on evolution.
The particulate nature of matter and of light impose limits on the sensitivity
of olfaction and hearing and on the ability of eyes to discriminate brightness,
and set a minimum size for gravity sensors.
The wavelengths of light impose limits on the resolving power of eyes, and the
principles of acoustics tend to restrict the wavelengths of sound used by animals.
The diffusion constant of oxygen in tissue limits the dimensions of animals
without circulatory systems. The thermal conductivities of available materials
set limits to the ranges of size of homoiothermic animals. Animals and plants
behave as black bodies to long-wave radiation, irrespective of colour. Organisms
cannot convert heat to other forms of energy.
Despite the variety of possible structural materials, only restricted ranges of
mechanical properties are possible. The sizes of terrestrial animals and of trees
are subject in principle to limits, due to the mechanical properties of skeletal
materials and of water, but do not seem to approach these limits.
The forces on bodies moving through air or water depend on their size, speed
and shape, but the effects of changes of shape alone are strictly limited. There
seems to be a practical limit to the performance of gliders, and there is a more
fundamental limit to the performance of auto-rotating seeds.
Membrane potentials are restricted to fractions of a volt, by the requirements
of the Nernst equation.
PHYSICAL LIMITS TO EVOLUTION
357
ACKNOWLEDGEMENTS
I am grateful to the Population Genetics Group for stimulating me to write
on this topic, by inviting me to speak on it. I have had useful discussions with
Dr R. F. Ker.
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