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How fast do living organisms move: Maximum speeds from bacteria

to elephants and whales

Nicole Meyer-Vernet

a)

LESIA, Observatoire de Paris, CNRS, UPMC, Universit



e Paris Diderot, 92195 Cedex Meudon, France

Jean-Pierre Rospars

b)

INRA, UMR 1392, Institut d’Ecologie et des Sciences de l’Environnement de Paris, 78000 Versailles, France

(Received 30 October 2014; accepted 29 March 2015)

Despite their variety and complexity, living organisms obey simple scaling laws due to the

universality of the laws of physics. In the present paper, we study the scaling between maximum

speed and size, from bacteria to the largest mammals. While the preferred speed has been widely

studied in the framework of Newtonian mechanics, the maximum speed has rarely attracted the

interest of physicists, despite its remarkable scaling property; it is roughly proportional to length

throughout nearly the whole range of running and swimming organisms. We propose a simple

order-of-magnitude interpretation of this ubiquitous relationship, based on physical properties

shared by life forms of very different body structure and varying by more than 20 orders of

magnitude in body mass.

V

C

2015 American Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4917310]

I. INTRODUCTION

A fundamental property of many living organisms is au-

tonomous locomotion. On Earth, which is a rocky planet of

surface temperature T ’ 300 K with liquid water and an

atmosphere, animals can run, swim, and/or fly. Each individ-

ual has a preferred speed of locomotion which is determined

by its size and by dynamical constraints,

1,2

depending on the

means of locomotion and the ambient conditions. Newtonian

mechanics tells us that a walker of size L approximated as an

inverted pendulum in the Earth’s gravitational field g, moves

at angular frequency ðg=LÞ

1=2

; hence its preferred speed V is

of order of magnitude the step length L times the fre-

quency: V ðLgÞ

1=2

=2p. More elaborate arguments have

been used to propose scalings for various means of locomo-

tion, depending on such factors as the gravity and the density

of the surrounding medium.

3–7

However, when living organisms are driven by circum-

stances to move as fast as possible, they may increase their

speed above the preferred speed. As human beings, we can

increase our speed by one order of magnitude by running.

The fastest human sprint speed on record is 12.2 m/s for

Usain Bolt (size L ¼ 1:96 m and mass M ¼ 86 kg in 2009),

8

whereas the recent 50 m swimming record of Florent

Manaudou (L ¼ 1:99 m, M ¼ 99 kg) corresponds to an aver-

age speed of 2.5 m/s. For comparing organisms of widely

different sizes, it is more appropriate to express the speed in

terms of the body length. The above values yield 6.1 and 1.2

lengths per second, respectively. Near the lower extreme of

the size range, the 2.5-lm-long bacterium Bacillus subtilis

can swim at 15 lm/s, or 6 lengths/s, strikingly close to Usain

Bolt’s running performance. A 4-mm-long ant runs at

60 mm/s (15 lengths/s), whereas a 2.1-m ostrich runs at

23 m/s (11 lengths/s).

4

These examples reflect a ubiquitous

property of living organisms: the maximum speed of running

and swimming lies between 1 and 100 lengths per second, in

an overall mass range covering nearly 20 orders of magni-

tude, as first noted by Bonner in a classic book.

9

Indeed, while maximum speeds V

max

and lengths L vary

by nearly 7 orders of magnitude (and mass by three times

more), maximum relative speeds V

max

=L remain constant at

ten per second with an accuracy of a factor of ten,

4,9–11

even

though different scalings hold over narrower mass

ranges.

12,13

This large-scale pattern, which holds for organ-

isms as diverse in overall structure, means of locomotion,

and surrounding medium as eight-legged mites and two-

legged ostriches, from small bacteria to large mammals, has

apparently not been appreciated by the physics community

and requires a basic explanation. A solution was proposed by

Dusenbery,

11

but being based on viscous drag it only applies

to the small range of organisms swimming at low Reynolds

numbers.

In the present paper, we illustrate the approximate linear

relation between maximum speed and body size for a large

number of running and swimming species from micro-

organisms to the largest mammals (Sec. II), and we propose a

simple order-of-magnitude interpretation, based on three ba-

sic properties of living beings that constrain their performan-

ces: their density q, the applied force per unit cross-sectional

area r, and the maximum rate of energy consumption per unit

mass b

M

. These three quantities are known to be roughly in-

dependent on size over virtually the full mass range of mov-

ing species, as summarized in Sec. III. We shall briefly

discuss in Sec. IV two exceptions to the large-scale relation

V

max

=L  10 s

1

: very large organisms, whose maximum

speed tends to level off,

4,12

and flyers.

The huge mass range studied enables us to estimate orders

of magnitude by neglecting specific details that are important

in scaling studies over narrower mass ranges. In this sense,

the present paper can be viewed as an exercise at the bound-

ary between comparative zoology and order-of-magnitude

physics inspired by Victor Weisskopf’s physics courses.

14,15

As usual, the symbol  means that two quantities are equal

to within one order of magnitude or so, whereas ’ means

equal within a factor of two or so. Unless otherwise stated,

units are SI.

II. EMPIRICAL RESULTS

Figure 1 shows the measured maximum relative speed

V

max

=L as a function of mass for running and swimming,

from micro-organisms to the largest mammals. The mass

719 Am. J. Phys. 83 (8), August 2015 http://aapt.org/ajp

V

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2015 American Association of Physics Teachers 719

range goes from mites to the African bush elephant for run-

ning and from micrometer-sized bacteria to whales for swim-

ming. Almost all the data lie in the range 1 < V

max

=L < 100.

This range is remarkably narrow compared to the 10

20

-fold

variation in body mass and confirms the striking constancy

of the maximum relative speed first noted by Bonner.

9

The

human world records for running and swimming are plo tted

as asterisks (red and blue, respectively). Both lie in the lower

range of animal running and swimming relative speeds,

respectively, illustrating the low rank of human beings in the

animal world for sprinting and swimming. Nevertheless,

these records still lie within an order of magnitude of the

scaling V

max

=L ¼ 10 s

1

.

Figure 1 also suggests that the maximum speed tends to

level-off for large masses,

4

a question that we shall discuss

in Sec. IV. We have not plotted flying speeds, which follow

a different scaling law (see Sec. IV).

4

III. ESTIMATION OF MAXIMUM SPEED

In order to propose a basic interpretation of the observed

scaling, let us consider the three universal properties of

living species which constrain their maximum speed of loco-

motion: mass density q, applied force per unit cross-

sectional area r, and maximum power per unit mass b

M

(maximum metabolic rate).

A. Three ubiquitous properties of living species

First, the mass density of organisms is roughly that of

liquid water, on which life on Earth is based

q ’ 10

3

kg m

3

: (1)

Second, the applied force per unit cross-sectional area of

tissue

6,17

is of order of magnitude

r  2  10

5

Nm

2

; (2)

from micro-organisms to the largest animals.

18

This is an

example of the rule dating back to Galileo that the strength

of an object is proportional to its cross-section. Here, Eq. (2)

is not the resistance to fracture, the so-called tensile strength,

but the average active tension applied by organisms for their

locomotion. This tension has a similar value for all organ-

isms because it is based on biological molecular motors of

similar basic properties. Biological motors are molecules

converting chemical energy into mechanical energy via a

conformational change in their molecular structure.

19

This

3-dimensional structure is held together by non-covalent

bonds, with the typical free energy

W

0

 10 k

B

T; (3)

which prevents their destruction by thermal agitation, and

their typical size is

20

a

0

 e

2

=4p

0

W

0

 6nm; (4)

despite the complexity of electrostatic interactions within

large molecules.

21

Basically, a molecular motor uses an

energy W

0

for moving by one “step” via a change in 3-D

structure, so that the “step” length is a

0

. The elementary

force is thus

F

0

 W

0

=a

0

 7 pN (5)

over an equivalent cross-section area whose order of magni-

tude is a

2

0

, so that the force per unit cross-section area is

r  F

0

=a

2

0

 W

0

=a

3

0

: (6)

Substituting Eqs. (3) and (4) into Eq. (6) yields Eq. (2).

This order of magnitude holds for muscles of animals,

which are made of filaments containing hundreds of elemen-

tary motors (myosin), as well as for the moving appendages

of micro-organisms.

18,22,23

Third, consider the power available. Transport of heat and

nutrients takes place across surfaces, which are expected to

scale as the square of size, and thus to vary with body mass

as M

2=3

; therefore, the energy consumption rate of living

beings (the so-called “metabolic rate”) per unit mass is

expected to scale as M

2=3

=M ¼ M

1=3

. Reality is more com-

plicated because body shape and structure change with size,

so that different scalings are observed

24

with an exponent

closer to 1=4 than to 1 =3. After decades-long controver-

sies,

25,26

it has been shown, albeit rarely appreciated in the

physics community, that the basal metabolic rate per unit

mass remains roughly constant across life forms.

27,28

More

precisely, for the vast majority of organisms it remains

within a 30-fold range,

29

which is remarkably narrow com-

pared with the 10

20

-fold body mass range concerned. Since

Fig. 1. Maximum relative speed versus body mass for 202 running species (157 mammals plotted in magenta and 45 non-mammals plotted in green), 127

swimming species and 91 micro-organisms (plotted in blue). The sources of the data are given in Ref. 16. The solid line is the maximum relative speed

[Eq. (13)] estimated in Sec. III. The human world records are plotted as asterisks (upper for running and lower for swimming). Some examples of organisms of

various masses are sketched in black (drawings by Franc¸ois Meyer).

720 Am. J. Phys., Vol. 83, No. 8, August 2015 N. Meyer-Vernet and J.-P. Rospars 720

we wish to estimate the maximum speed, the relevant prop-

erty is not the basal metabolic rate but rather the maximum

metabolic rate. The order of magnitude of this parameter has

been shown to be roughly constant, too, when scaled to the

mass, with the value

b

M

 2  10

3

Wkg

1

(7)

per unit of working tissue.

27,30,31

B. Maximum relative speed

If the maximum relative speed V

max

=L only depends on

the parameters q, r, and b

M

, dimensional analysis can be

used to deduce its scaling. In terms of the three dimensions

½M , ½L; ½T, the density scales as

q /½M ½L

3

: (8)

Since r is a force (/½M½L½T

2

) per unit cross-section

(/½L

2

), it sca les as

r /½M ½L

1

½T

2

; (9)

and since b

M

is a power (/½M ½L

2

½T

3

) per unit mass, it

scales as

b

M

/½L

2

½T

3

: (10)

Therefore, since V

max

=L /½T

1

, we deduce

V

max

=L / b

M

q=r: (11)

In order to make a quantitative estimate, let us go a step

further than dimensional analysis. First, consider running and

swimming of animals beyond the micro-organism range. At

zero order, both means of locomotion can be considered as a

cyclic process (of frequency f) in which an organism of length

L moves by one “step” of length L during each cycle, by

contracting muscles. Consider an organism of cross-section S

and length L:

•

its mass is M  qSL,

•

moving by one step of length L by applying the force

rS requires the energy per unit mass w  rSL=M  r=q,

•

since f steps per second consume the energy fw per unit

mass, which must be smaller than b

M

, the maximum step

rate is f

max

 b

M

=w  b

M

q=r.

The maximum speed equals the step length L times the

maximum step rate f

max

, whence

V

max

=L  f

max

 b

M

q=r: (12)

Substituting Eqs. (1), (2) , and (7) into Eq. (12) yields

V

max

=L  10 s

1

; (13)

which is the large-scale relation mentioned in the

Introduction.

Consider now micro-organisms. They move by rotating or

undulating flagella, cilia, or pili, which are operated by mo-

lecular motors as are the muscles of larger organisms, even

though the number of motors is much smaller for micro-

organisms. In this case, it is more enligh tening to consider

the microscopic level. During one period of rotation or undu-

lation,

23,32

a micro-organism of length L moves along a dis-

tance L using energy W

0

[given in Eq. (3)] per molecular

motor. With f cycles per second, the power spent is fW

0

.

For a motor of size a

0

given in Eq. (4) and mass qa

3

0

, the

power cannot exceed the maximum metabolic rate b

M

qa

3

0

.

This yields f ⱗ b

M

qa

3

0

=W

0

, whence

V

max

=L  b

M

qa

3

0

=W

0

: (14)

With n motors, both the numerator and the denominator of

Eq. (14) are multiplied by n, which does not change the

result. Since from Eq. (6) r  W

0

=a

3

0

, Eq. (14) is equivalent

to Eq. (12).

Hence, both micro-organisms and larger animals should

have a similar maximum relative speed for running and

swimming, given by Eqs. (12)–(13), in agreement with the

data plotted in Fig. 1.

IV. CONCLUDING REMARKS

There are two exceptions to the scaling derived above: fly-

ing species and ver y large organisms.

Flying is outside the scope of our simplified model

because in that case the muscles essentially govern wing

flapping, and this frequency does not yield the total relative

speed of the organism. In addition, air drag represents the

dominant constraint at large flying speeds.

4

Consider now large running and swimming organisms, for

which V

max

=L tends to decrease (Fig. 1), even though the

data do not lie below one order of magnitude of the scaling

(13) except for the largest animal. Several effects become

important at high speeds, such as friction and excess heat

production. However, Fig. 1 suggests a similar trend for run-

ning and swimming, which points to a more fundamental li-

mitation, independent of the surrounding medium.

Let us consider an organism of cross-section S and length

L, as in Sec. III B, and approximate the locomotion as a peri-

odic motion of legs (for running) or tail (for swimming) of

length L. The maximum frequency is constrained not only

by the power available, as considered in Sec. III B, but also

by the maximum angular acceleration that muscles can pro-

vide. With the torqu e C ⱗ rSL and moment of inertia

I  ML

2

 qSL

3

, the angular acceleration d

2

h=dt

2

 C=I is

constrained by

d

2

h=dt

2

ⱗ r=ðqL

2

Þ: (15)

Integrating Eq. (15) twice yields the order of magnitude of

the time for the append age to be accelerated up to a fixed

angle h:

t ⲏ Lðqh= rÞ

1=2

: (16)

Setting h  1 in Eq. (16) yields the frequency f 

1=t ⱗ ðr=qÞ

1=2

=L and therefore the upper limit of the maxi-

mum speed

V

max

ⱗ ðr=qÞ

1=2

: (17)

Hence, the value of V

max

=L in Eq. (12) can only hold for

L ⱗ ðr=qÞ

1=2

=ðb

M

q=rÞ¼ðr=qÞ

3=2

=b

M

: (18)

721 Am. J. Phys., Vol. 83, No. 8, August 2015 N. Meyer-Vernet and J.-P. Rospars 721

Substituting Eqs. (1), (2), and (7) into Eq. (18) yields L ⱗ 1: 4

m. This limitation prevents larger organisms from following

Eq. (12), which suggests that the maximum speed should

increase linearly with body length onl y up to (approxi-

mately) meter-sized organisms, in agreement with Fig. 1.

Consider the blue whale (blue point at M ’ 1:5  10

5

kg),

which lies below one order of magnitude of the scaling

(12)–(13); with its length L ’ 26 m, Eq. (17) yields

V

max

=L ⱗ 0:5s

1

, a limit close to the observed value plotted

in Fig. 1.

Finally, one should be reminded that in the spirit of this

paper, Eqs. (12) and (17) are order-of-magnitude results.

Because of the huge diversity of organisms and sizes, we

have ignored the specific methods of locomotion, using dras-

tic approximations for the applied forces, cross-sections and

distances involved, as well as approximating by unity the ef-

ficiency of energy conversion and the proportion of active

tissue. The numerous correction factors tend to cancel out in

the final order-of-magnitude result.

In conclusion, we explain the ubiquity of the maximum

relative speed at about ten lengths per second for running or

swimming, from bacteria to large mammals, by the ubiquity

of the density, the applied force (per unit cross-sectional

area), and the maximum metabolic rate (per mass of active

tissue). The maximu m absolute speed is limited by the maxi-

mum acceleration that mu scles can provide, which may

explain why animals larger than the ostrich do not move

faster.

ACKNOWLEDGMENTS

The authors thank three anonymous reviewers for helpful

comments and suggestions. The authors are grateful to

Franc¸ois Meyer, who did the original drawings superimposed

on Fig. 1.

a)

Electronic mail: nicole.meyer@obspm.fr

b)

Electronic mail: jean-pierre.rospars@versailles.inra.fr

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