Force per cross-sectional area from molecules to muscles: a general property of biological motors

We propose to formally extend the notion of specific tension, i.e. force per cross-sectional area—classically used for muscles, to quantify forces in molecular motors exerting various biological functions. In doing so, we review and compare the maximum tensions exerted by about 265 biological motors operated by about 150 species of different taxonomic groups. The motors considered range from single molecules and motile appendages of microorganisms to whole muscles of large animals. We show that specific tensions exerted by molecular and non-molecular motors follow similar statistical distributions, with in particular, similar medians and (logarithmic) means. Over the 1019 mass (M) range of the cell or body from which the motors are extracted, their specific tensions vary as Mα with α not significantly different from zero. The typical specific tension found in most motors is about 200 kPa, which generalizes to individual molecular motors and microorganisms a classical property of macroscopic muscles. We propose a basic order-of-magnitude interpretation of this result.

JPR, 0000-0003-0797-5153 We propose to formally extend the notion of specific tension, i.e. force per cross-sectional area-classically used for muscles, to quantify forces in molecular motors exerting various biological functions. In doing so, we review and compare the maximum tensions exerted by about 265 biological motors operated by about 150 species of different taxonomic groups. The motors considered range from single molecules and motile appendages of microorganisms to whole muscles of large animals. We show that specific tensions exerted by molecular and nonmolecular motors follow similar statistical distributions, with in particular, similar medians and (logarithmic) means. Over the 10 19 mass (M) range of the cell or body from which the motors are extracted, their specific tensions vary as M α with α not significantly different from zero. The typical specific tension found in most motors is about 200 kPa, which generalizes to individual molecular motors and microorganisms a classical property of macroscopic muscles. We propose a basic order-ofmagnitude interpretation of this result.

Background
Living organisms use biological motors for various functions, which range from internal transport of ions and molecules in cells to motion of microorganisms and animals, the latter being driven by muscles. The forces developed by muscles are generally expressed as force per cross-sectional area, called specific tension or stress. It has been known for a long time that the vertebrate striated muscles can exert maximum tensions at constant length (isometric tension) of about 200- approximation independent of the muscle and the body mass [1]. This rule was extended to arthropod muscles with values in the range 300-700 kPa [2], although in some mollusc muscles stresses up to 1400 kPa were reported [3]. Later, a review of the literature based on muscles of 72 species of different taxonomic groups, including mammals, birds, reptiles, amphibians, molluscs, insects and crustaceans [4] concluded that there was no significant relationship between body mass and isometric tension, although isometric tension was found to be significantly higher in molluscs, crustaceans and amphibians than in other groups.
Despite their diversity, all these motors are based on protein machines generating forces. Macroscopic muscles are based on the myosin motor, whereas microorganisms and cells use other types of molecular motors. For comparing motors of so many different sizes, the convenient parameter is not the force F, which varies from several 10 −12 N for the myosin globular motor of cross-sectional area A ∼ 40 nm 2 to approximately 500 N for a large muscle of cross section approximately 20 cm 2 , but, as we intend to show, the specific tension F/A (all symbols and abbreviations are defined in table 1). In muscles, the approximate conservation of F/A between animals is an extension of a rule dating back to Galileo, that the strength of a structure is proportional to its cross section. Now, it turns out from the above numbers that the tension of the myosin molecular motor is of the same order of magnitude as the tension of macroscopic muscles (all references to tension here and elsewhere refer to specific tension unless otherwise noted). We will show that this property is not a coincidence but stems from the basic arrangement of cross-bridges in striated muscles. Furthermore, because biological molecular motors are based on protein machines that convert chemical energy into mechanical energy in similar ways (with the possible exception of pili and jump muscles), their tensions are expected to be of the same order of magnitude as that of myosin. Therefore, we propose to extend to molecular motors the concept of tension of macroscopic muscles and to compare their applied forces per unit cross-sectional area. That the forces per unit cross-sectional area may be similar for molecular motors and muscles agrees with results by Marden & Allen [18] and Marden [19], who show in a class of motors that maximum force output scales as the two-thirds power of motor's mass, close to the motor's cross-sectional area.
In order to make a meaningful comparison, we need to consider a representative set of muscle tensions, as well as the tension of the myosin motor and those of various other molecular motors. So, we analysed 329 published values of maximum forces or tension for approximately 265 diverse biological motors. These motors include single molecules, molecular assemblies, muscle cells and whole muscles with various functional demands. They come from free-living cells and multicellular organisms of diverse phyla spanning more than 18 orders of magnitude in mass from 10 −16 to 10 3 kg. Our primary interest was for motors involved in whole body motion, whereas the other motors were kept for comparison.
The three main questions we addressed on this basis are as follows. Can the notion of specific tension of muscles (force per cross-sectional area) be formally extended to propulsion of organelles and to individual molecular motors? How does this tension compare with that in muscles, and can the results be understood in terms of the basic structures of both molecular motors and muscle fibres? How does tension in motors devoted to cell or body motion compare with tension in other motors?

Motor forces
The main variable of interest in this paper is the force generated by molecules, molecular assemblies, muscle fibres and muscles. Our dataset includes 13 motor types aggregated in five motor classes depending on the nature of the generated force.    (ii) forces produced by large molecular assemblies (denoted M2): F 0 F 1 -ATPase, bacterial flagella, pili, spasmonemes and myofibrils. These motors can be also classified as non-locomotory (ATPase) and locomotory (the others) or as rotary (ATPase, bacterial flagella) and linear (the others); (iii) forces produced by single muscle fibres (i. e. muscle cells) or bundles of a few muscle fibres (both denoted FI), frequently demembranated (skinned), while maximally stimulated and clamped at constant length (isometric contraction), with electrical or chemical stimulations; (iv) maximum force produced by dissected large bundles of fibres or isolated whole muscles stimulated isometrically with electrical stimulation of the nerve or the muscle (denoted MU); and (v) forces measured in behaving animals engaged in a wide range of activities including running, jumping, swimming and biting (denoted MV).
Single molecules (M1) and molecular assemblies (M2) are collectively called here 'molecular motors'. The other motors, muscle fibres (FI) and whole muscles (MU and MV) are called 'non-molecular motors'.

Identification of study reports
Values of forces generated by molecular and non-molecular motors were taken from 173 articles published in peer-reviewed journals for a wide variety of cells and animals. We sought a sample that is representative of the widest range of sizes and design varieties for as many species as possible (approx. 150 species were found) representing several different taxonomic groups, including bacteria, protozoa, algae, fungi, echinoderms, insects, crustaceans, molluscs, fishes, amphibian, reptiles, birds and mammals.
For molecular motors, we searched for articles providing the main variables of interest (either force for linear motors or torque and lever arm for rotary motors) for the 10 types listed above. Other types were not considered. For example, of the 14 classes of kinesin, only the most studied kinesin I was included and in the myosin superfamily which consists of at least 18 classes of motor proteins involved in a large variety of physiological processes, only class II myosin (conventional) responsible for muscle contraction was included; the other classes involved in phagocytosis, cell motility and vesicle transport were excluded. For each type, potentially relevant papers were searched using the Google Scholar database using as keywords the motor type plus 'force', 'torque' or 'pN'.
For non-molecular motors, we proceeded in two steps. First, relevant papers were identified from previous review papers [1,2,4,18]; all their cited references were included, except the rare cases for which the full text was not available or the paper could not be feasibly translated into English. Second, other potentially relevant papers were searched without restriction on language or date in the Google Scholar database using keywords ('specific tension', 'muscle stress', 'fibre', 'fiber', 'N/m 2 ', 'N m −2 ', 'N/cm 2 ', 'N cm −2 ', 'N mm −2 ', 'pascal', 'kPa', 'physiological cross-sectional area', 'PCSA', 'CSA', etc.). Bibliographic searches were discontinued in April 2015.
The papers in this preliminary list were screened based on their title and abstract to exclude those unrelated to biological motors, then collected. The useful information was extracted from each of them (see below) with independent checks by the two authors for most of them. Papers without original measurements were excluded. Data published more than once by the same author(s) or reproduced by other authors were identified and only the paper with the original measurement was kept in the reference list. Measurements not fulfilling our criteria (stall force of single molecular motor, maximum isometric tension of non-molecular motors) were not considered. No relevant papers were excluded.

Motor tensions
For all motors, the measured forces F were normalized per cross-sectional area A (tension f = F/A expressed in Newton per square-metre or equivalently kilopascal).
For molecular motors the tensions were calculated from the published values (measured force or for rotary motors, torque and lever arm, tables 2 and 3) with the area A calculated from the volume V of the motor (with the order-of-magnitude approximation A = V 2/3 , table 2), except for a few elongated shapes (pilus and spasmoneme) for which we estimated A from the diameter of the molecular assembly. For myosin, A was estimated from the head of the molecule.
For non-molecular motors the tensions (f = F/A) were always given in the articles cited.  [21], Carter et al. [22] .     All tensions were expressed in kilopascal. In papers giving several values or minimum and maximum, their mean was calculated. Values from different papers were never pooled. In tables 3 (molecular motors) and 4 (non-molecular motors) tensions given by different authors in different conditions for the same motor are listed separately (329 values). If the same motor of the same species, studied by different authors or the same authors in different conditions, are counted only once, the number of different motors is approximately 265 (the uncertainty arises from a few measurements in table 4 which were made on a mixture of distinct fibres or several muscles together).

Other motor classifications
The data were also analysed with respect to the structure of motors, their function and the taxonomic position of the organisms.
For comparing structures, the original 13 types, from molecules to muscles, were aggregated in five classes (M1, M2, FI, MU, MV) or two classes (molecular M1 + M2 and non-molecular) as defined above. In some figures and table 5, MF, for which the cross-section was indicated in the articles cited, was shown separately from the other M2 motors.
The functional groups were defined by the contribution of the motor to the overall movement of their parent organism, the four basic categories being swimming (Swim), flying (Fly), moving with respect to a solid surface (terrestrial Terr) and no direct contribution to locomotion (non-loc). Examples of non-loc motors are RNA polymerase, cytoplasmic dynein, kinesin, F 0 /F 1 -ATPase and various muscular motors (heart, diaphragm, wing closer, gill pump, claw closer, larynx, eye).
For taxonomic comparisons, groups 5 with number of f values less than 5 (protozoa, algae, fungi, echinoderms, arachnids) were excluded.

Body mass
Finally, the tensions were analysed with respect to the mass M of the 'body' that the motor contributes to move. For molecular motors this is the mass of the cell from which the motor was extracted. When not reported, cell masses were estimated from other sources or calculated from the cell size. In nonmolecular motors, tensions were analysed with respect to the mass M of the corresponding animal. When not reported, body masses were also estimated from other sources. Note that as a consequence of these choices a different mass was used for a myosin molecule (molecular motor) and a muscle fibre (non-molecular motor) from the same organism. The organisms considered range in mass from the bacterium Escherichia coli (1.3 × 10 −15 kg) to the muscular fibre (5 × 10 −8 kg) for the cells, and from the mite Archegozetes longisetosus (10 −7 kg) to the elephant (2500 kg) for the multicellular organisms.
For both f and M, means of a series of equivalent measurements by the same author(s) were preferred when available. When only minimum and maximum values were given, we took their mean.

Statistics
Statistical distributions were compared with the Kolmogorov-Smirnov test [194]. Multiple distributions were compared with the one-way analysis of variance (ANOVA) and corresponding multiple comparison of means using Tukey-Kramer adjustment. Slopes of least-square regressions of log 10 (f ) versus log 10 (M) were compared with 0 using the F test. Details of statistical analyses are given as the electronic supplementary material, tables S1-S6 for ANOVA and multiple comparison of means and tables S7-S12 for regressions. All tests were performed with the MATLAB STATISTICAL TOOLBOX (The Mathworks, Natick, USA).

Results
The data have been analysed in terms of the maximum force per cross-sectional area f. We consider separately motors made of single molecules (denoted M1) and molecular assemblies (M2, MF) that we collectively call 'molecular motors', whereas the other motors, muscle fibres (FI) and whole muscles (MU for dissected muscles or MV for behaving animals) are called 'non-molecular motors'. We have also analysed the data in terms of the mass M of the 'body' that the motor contributes to move and to whether the motor contributes to the overall movement of the parent organism.

3.3.
There is no large-scale variation with cell or body mass We also looked for 'local' trends based on the different categories defined previously. For motor types, some slight positive and negative slopes of the regression lines f versus M were found (electronic supplementary material, tables S8 and S9). For taxonomic groups (electronic supplementary material, tables S10 and S11) and motor functions (electronic supplementary material, table S12), either the slope is not significantly different from zero (according to the F-test at level 1%), or the slope is smaller or equal to 0.02 in absolute value.

Discussion
We discuss in order the choice of specific tension for normalizing forces developed by widely different motors, the similarity of specific tension in molecular and non-molecular motors, the factors explaining the variability of tension, especially in muscles, and the relationship between tension invariance and force-mass scaling.

Specific tension as a size-independent measure of force
In order to compare forces developed by biological motors as different as molecules and muscles, whose spatial scale varies by nearly 7 orders of magnitude and whose applied force varies by nearly 14 orders of magnitude, it is useful to express them in relative values. Because most non-molecular motor forces F (FI, MU, MV) are expressed as specific tension (F/A) in the literature, it is natural to try to express molecular motors similarly.
As F/A is not available for molecular motors, in order to avoid bias, we defined the cross-section A in the most basic way, i.e. from the volume V as A = V 2/3 , which holds for a cube and still holds in order of magnitude for shapes of moderate elongation. This is in line with results of Marden & Allen [18] who found F proportional to motor mass m 2/3 for a class of molecular motors, and to the fact that these forces depend on chemical bonds (mainly hydrogen bonds), whose number acting in parallel is expected to depend on the cross section. For defining the cross-section, we were extremely careful to select the acting part of the motor (ignoring the 'passive' tails) so that the shape was of moderate elongation. For example to estimate the volume of the myosin motor, we only considered the heads and ignored the tail which does not contribute to the actin-myosin interaction. We will return to this topic in the last subsection 'Scaling with motor's mass' and suggest below an order-of-magnitude interpretation.

Invariance of specific tension in molecular and non-molecular motors
The main characteristics found here for the values of tension f in both molecular (M1, M2, MF) and nonmolecular motors (FI, MU, MV) are (table 5): (i) their almost equal median tensions (approx. 170 kPa), (ii) their similar ranges of variation (60 < f < 350 kPa for 90% of motors), and (iii) the approximately five times higher tensions exerted by pili (600 < f < 2000 kPa). These three characteristics can be understood from basic physical considerations.

Molecular motors
Molecular motors are proteins that produce mechanical energy by changing their three-dimensional conformation. They move in steps whose length is of the order of magnitude of their size a 0 , which is typically a 0 ∼ 6 nm [195,196]. The steps are mainly powered by ATP with free energy W 0 12kT 0.5 × 10 −19 J/molecule at T = 300 K [197]. Therefore, the elementary force F 0 developed by motor proteins is of order of magnitude F 0 ∼ W 0 /a 0 ∼ 8 pN and the corresponding force per unit crosssectional area f is f ∼ F 0 /a 0 2 W 0 /a 0 3 ∼ 200 kPa. This is close to the average value found for molecular motors (M1, M2 and MF, table 5). This order-of-magnitude estimate is based on a perfect transduction of chemical into mechanical energy. Taking into account the actual efficiency would not change this order of magnitude since molecular motors are known to have a high efficiency-often exceeding 50% (e.g. [198,199]), in particular, 80-95% for kinesin [197] and up to 100% for F1-ATPase [8]. Molecular motors, like other proteins, owe their properties to a three-dimensional structure mainly held by H-bonds and other weak forces [200,201]. In order to act near (but not at) thermal equilibrium and not to break the motor protein, the elementary motor force should not exceed kT divided by the distance over which H-bonds operate, i.e. the size of the water molecule, a H 2 O 0.3 nm. This yields the minimum size, a 0 > a H 2 O × (W 0 /kT) 4 nm, and maximum tension, f W 0 /a 0 3 < 800 kPa, of molecular motors. This order of magnitude estimate is similar to the maximum tension observed in molecular motors (table 5) with the notable exception of pili. Pili, which are virtually universal in prokaryotes [202], have exceptional mechanical properties of stretching and adhesion, and some of them can withstand extreme forces, with an important role played by covalent bonds (e.g. [203]) so that the above order-of-magnitude estimate, based on weak forces, does not apply to them. In order to compare pili with other structures, we have only considered steady-state unwinding forces (e.g. [60]). Even then, pili can still reach extreme specific tensions, with a median four times higher than that of other motors.

Non-molecular motors
The most striking result of this paper is that the formally defined tension of molecular motors turns out to be similar to the value f 200 kPa typical of muscle fibres. A hint to this uniformity stems from the basic arrangement of myosin motors in striated muscles (reviewed in e.g. [13,204]). Most of the space within muscle fibres is occupied by protein thick filaments along which groups of myosin globular motors (heads) are protruding with an axial spacing e = 14.6 nm. These motors are cyclically attaching to (and detaching from) adjacent thin filaments of actin to form the cross-bridges, and enable thin and thick filaments to slide past each other. Along each half thick filament (of total length 2l 1.6 µm, neglecting for this order-of-magnitude estimate a bare zone of smaller length free of motors) about 150 myosin molecules exert forces that add in parallel and only about one-third of the cross-bridges are attached during isometric contraction [47,205]. Therefore, the number of active individual myosin motors along each half thick filament is N 50. (Note that since l/e 50, this might imply that only one motor per group of three can attach simultaneously, a likely consequence of steric constraints brought about by the three-dimensional structure enabling transitory conformational changes.) With N motors acting in parallel each exerting a force F myosin the total force per thick filament is NF myosin . Each thick filament and its associated lattice of thin filaments occupies an equivalent cross-section s d 2 , where d 40 nm is the lateral spacing of thick filaments, so the total tension in the structure is f fibre NF myosin /s which acts (in series) along the length of the fibre. Tables 3 and 4 show that the myosin motor, of equivalent cross-sectional area A 36 nm 2 , exerts a mean force F myosin f myosin A 7 pN. Substituting the values of F myosin , N and s in the above formula yields the tension in the structure f fibre 240 kPa.
This rough estimate enables us to understand why the tension of muscles ( f fibre ) is of the same order of magnitude as the tension of the myosin motor f myosin 190 kPa. Indeed, the tensions of muscle fibres and of myosin motors are in the ratio f fibre /f myosin NA/s, and the myosin motors are arranged so that the number N of them acting simultaneously in parallel is approximately equal to the ratio s/A of the equivalent cross-sectional area of each thick-thin filament structure to that of an individual myosin motor head, which is not surprising because of steric constraints.

Origins of variability of specific tension in various motors
Overall, tensions in most molecular and non-molecular motors are distributed around their means according to similar Gaussian functions with coefficients of variation s.d./mean 0.5. This variability may arise from methodological, experimental and biological factors.

Methodological and experimental factors
The cross-section A of molecular motors was estimated from their mass m using the formulae A = V 2/3 and V = m/ρ with protein density ρ 10 −3 pg nm −3 . This is admittedly rough, since the longer dimension of the motors considered can differ from the cross-diameter by nearly a factor of 2. The resulting error may not be negligible compared with the observed variability of specific tension in molecular motors, in which more than 80% of f values are within one-third of the median and twice the median (see Q10, Q90 and median in table 5, second line).
Although we did not have to estimate the cross-section for muscles, their tensions show the same variability on f as molecular motors (Q10 is one-third the median and Q90 twice the median, see table 5, third line). Their cross-sectional area has sometimes been corrected for the area occupied by mitochondria (dragonfly, [116]), sometimes not (beetle, [115]) and never for the sarcoplasmic reticulum (e.g. [206]). The pennation angle has not always been taken into account. Temperature during the experiments has been noted and is usually close to the working temperature of the muscle. Although data are not fully homogeneous, the similarity of the distributions of specific tensions measured in vivo and in vitro suggests that uncorrected factors do not introduce important bias. In principle, corrections for these factors should lead to less variable data. However, no corrections have been attempted for two reasons. First, the information needed is not always provided, so corrections cannot be done systematically. Second, these corrections would probably have no incidence on the qualitative conclusions, and might even be less convincing than unmodified data.
Isometric tension in single skeletal muscle fibres (FI) is approximately 35% smaller than in whole muscles (MU or MV) (figure 3a). This difference probably results from the experimental conditions, most measurements of single fibres being performed after chemical or mechanical skinning. It produces swelling of the fibres and reduces the specific tension. Median tension is about the same for whole muscles when measured in vitro (MU, 200 kPa) and in vivo (MV, 227 kPa) ( figure 3a,b). This indicates that the tension for muscles in behaving animals is close to the maximum they can develop in in vitro conditions. It must also be realized that detailed physiologically and ecologically relevant comparisons between similar motors in different taxonomic groups are hindered by their unequal levels of investigation; for example, muscles MU have been studied in 29 vertebrate species, but only 13 invertebrate species (table 4).

Biological factors
Further sources of variability are probably biological. At the molecular level, variability stems from differences within and across families of single motor proteins (M1). At the supramolecular level, notably in propulsion organelles and muscles, elementary molecular forces are expressed via an organization that introduces further variations and specific adaptations to the diversity of mechanical problems they had to solve. More factors being involved, the values of their tension is a priori less easy to predict, explaining the variability observed. Nonetheless, as shown in figure 3a, after removal of pili, the variability of specific tension between the different types of molecular motors studied is larger in motors M1 and M2 than in myofibrils. The structural and functional homogeneity of myofibrils contrasts with the heterogeneity of the other molecular motors.
Neglecting experimental errors and pili being set aside, tensions of non-molecular motors (FI, MU, MV) vary approximately in the same range as tensions of molecular motors (M1, M2 and MF) with the same statistical distribution ( figure 1c,d). So, notwithstanding their myosin-based molecular homogeneity, the diversity in geometry and adaptation of muscular motors leads to variations in tension equivalent to those resulting from the diversity of molecules and their arrangements in molecular motors. It is remarkable that so many different mechanisms lead to the same final distributions of force per cross-sectional area at the microscopic and macroscopic levels.

Variability of tensions in whole muscles
The variability of tension in muscles has been the subject of thorough research. An important adaptive factor is sarcomere length. As predicted by the sliding filament model of muscle contraction, long filaments and long overlap between thick and thin filaments should occur in fibres with long sarcomeres. As in long overlap zones more actin-myosin cross-bridges should be formed, the maximum tension which a fibre can produce should be correlated with sarcomere length [207,208]. The resting sarcomere length exhibits little variation in insect and vertebrate muscles (2-4 µm), but much greater variations in crustacean muscles (7-17 µm). Overall, tension scales isometrically with the resting sarcomere length [157]. In particular, the claw closer muscles of cancer crabs exhibit both the longest sarcomere lengths and extreme mean crushing forces (525-1030 kPa; table 4 and figure 3c). This is a special adaptation of shell-crushing non-locomotory motors which is not found in locomotors (figure 3d).
Many other factors have been invoked to explain the variations in muscle tension, such as the density of the myosin filaments, the non-uniformity of sarcomere length along the fibres, the diameter of myofibrillar bundles, the actin : myosin filament ratios and the cross-bridge duty factors. For example, the slightly higher tension than in other groups found in amphibians and molluscs (except crustaceans; figure 3c) may be explained by their higher proportion of fast oxidative fibres and their higher relative myofibrillar volume [4,206]. However, these various factors apparently play a minor role in arthropod and vertebrate muscles as more than 80% of the variation in muscle tension in a series of muscles from these groups can be explained by the resting sarcomere length ( [157] and references therein).
Two characteristics other than tension contribute to muscle performance: speed of contraction (and relaxation) and endurance. They influence tension because high tension requires that most of the crosssectional area of a fibre be myofibrils, whereas high endurance requires a large mitochondrial volume and short twitch duration requires an extended sarcoplasmic reticulum. Therefore, trade-offs are inherent in the functional design of muscles so that a muscle cannot be simultaneously strong, enduring and rapid. This is the reason why rapid muscles are weak (either enduring, e.g. katytid singing muscles, or not, e.g. lobster sound-producing muscles with their hypertrophied SR) [208]. However, special adaptations in the oscillatory (asynchronous) flight muscles of insects result in high contraction frequencies without a large volume of SR, which leaves room for more mitochondria, but their strength is nevertheless limited by the endurance requirements of flight [208]. They are built optimally for maximum output of energy in their narrow contraction range, whereas most vertebrate sarcomeres are optimized for optimal mechanical conversion of chemical energy across a wider contraction range [209]. These different adaptions contribute to the variability observed. Overall, the similarity of muscle tensions is essentially owing to the similarity of fibre structure and thick filament length across muscles and species, in contrast with the variability of muscle speeds which are affected by the variability of thin filament lengths (e.g. [210]).
It is remarkable that tension is smaller in flight locomotors (median 79 kPa) than in terrestrial locomotors (median 187 kPa) and in swim locomotors (183 kPa), although only the difference for terrestrial locomotors is significant according to ANOVA at level 5% (figure 3e). Despite the high power needed for flight, the high frequencies required may impose a large concentration of mitochondria and, at least in birds, of sarcoplasmic reticulum at the expense of myofibrils. Solving this issue will need further investigation.

Absence of large-scale trend with cell's or body's mass
Given the constancy in both central value (mean or median) and dispersion (s.d. or interquartile range) of f in molecular and non-molecular motors, it is not surprising that the regressions in a log-log plot of f against M, the mass of the cell (for subcellular motors) or body (for cellular and supracellular motors) from which the motor is extracted, give no evidence of overall trend ( figure 4a,b). Other variables for the mass might be used, but their implementation is difficult because they are often ill-defined or unknown. This is the reason why we chose for the horizontal axis a proxy of the mass that the motor movesthe mass of the next higher hierarchical level, i.e. the cell's mass for subcellular motors (M1, M2, MF) and the animal's mass for cellular and supracellular motors (FI, MU, MS). This definition is simple, unambiguous, known in almost all cases and discriminant with a range extending over 18 orders of magnitude. If we had chosen the motor's mass m for the horizontal axis, the range would have been still wider since the minimum mass would be 10 −22 kg (kinesin) and the maximum mass > 1 kg (muscle), so that as the overall range of f would remain the same, the slope of the regression line would become still closer to zero. The absence of global trends does not preclude the existence of 'local' trends, i.e. regression lines with slope significantly different from zero, for specific classes of motors extending on a narrower mass range. Several examples of such significant trends were found (see the electronic supplementary material, tables S8-S12) but their slopes are small and difficult to interpret. These small-scale relationships are outside the scope of this paper which focuses on a large-scale study. The wide range of size, mass and area considered allows one to transcend the possible variations specific to certain categories.

Scaling with motor's mass
A different approach based on force F and motor mass m strengthens this conclusion. Indeed, Marden & Allen [18] studied the scaling of forces with motor's mass for two classes of animal-and humanmade motors and found that one of them, 'Group 1' motors, producing translational motion, scale allometrically with motor mass m, as F 10 3 m 2/3 (with F in Newtons and m in kilograms). We show below that this scaling, expressed in terms of specific tension f, is in good agreement with the typical specific tension found in the present paper (approx. 200 kPa). Consider first the order-ofmagnitude approximation of cubes of section A. With the mass density ρ 10 3 kg m −3 , the motor mass is m ρA 3/2 , so that the scaling above F 10 3 (ρA 3/2 ) 2/3 yields the tension f = F/A 10 3 ρ 2/3 100 kPa. This is a minimum value since replacing the cubic approximation by an elongated shape, with a ratio length/width r, with width d A 1/2 , would yield m rρA 3/2 , whence f 100 r 2/3 kPa. Thus, the massforce scaling for Group 1 motors found by Marden & Allen [18] implies the constancy of their specific tension with a constant value consistent with that found here.
The above argument might also explain why three 'molecular motors' corresponding in part to our 'M2 motors' (bacterial flagellum, mammalian flagellum and spasmoneme) are shifted to the right of the fitted line (see red circles in fig. 1 of [18]). Indeed, the mass m considered is the mass of the whole organelle, whose length far exceeds the square root of the section (i.e. r 1). This implies that m is much larger than ρA 3/2 , so that a constant value of f yields a smaller value of F/m 2/3 . However, for the other group of motors (Group 2) defined by Marden & Allen [18], the biological motor forces are generally deduced from the motion of the whole organism against gravity, which implies various joints and lever arms connecting the motor to the organism. It is, therefore, difficult to compare these data with those considered in this paper, which are directly measured at the level of the muscle (or of the fibre or the molecular motor).

Concluding remarks
The main result of this paper is that, despite their diversity, molecular and macroscopic biological motors do exert similar forces per unit cross-sectional area, which enables us to unify biological motors of different sizes and varied functions, from the motion of animals and microorganisms to cargo transport in cells or DNA transcription. The similarity of tensions of macroscopic muscles and fibres is not surprising as it stems from the similarity of fibres' basic architecture. In turn, the similarity of the tensions of molecular motors is owing to the basic physical properties of protein machines, and we have given an order-of-magnitude estimate of this tension from basic physics. Finally, we have shown that the tension in muscle fibres is similar to that of the myosin motor in particular because of the arrangement of these motors in the fibres, owing to steric constraints.
The approximate constancy of the maximum force per unit area f found in this paper from molecules to muscles implies general scaling laws for the motion of organisms [211] and raises the question of relating these laws to basic biological and physical constraints. Moreover, it calls for an explanation of why human-engineered motors, which are not based on ATP hydrolysis and hydrogen bond forces, show very similar specific tension to biological motors [18,19].
Data accessibility. All supporting data are made available in tables 2-5 and the electronic supplementary material, tables S1-S12.