Scaling functional patterns of skeletal and cardiac muscles: New non-linear
elasticity approach

.

Valery B. Kokshenev

.

Departamento de Fisica, Instituto de Ciencias Exatas, Universidade Federal de
Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, Minas Gerais,
Brazil. Email<EMAIL_ADDRESS>
.

Submitted to the Physica D 15 August 2009

.

Abstract: Responding mechanically to environmental requests, muscles show a
surprisingly large variety of functions. The studies of in vivo cycling
muscles qualified skeletal muscles into four principal locomotor patterns:
motor, brake, strut, and spring. While much effort of has been done in
searching for muscle design patterns, no fundamental concepts underlying
empirically established patterns were revealed. In this interdisciplinary
study, continuum mechanics is applied to the problem of muscle structure in
relation to function. The ability of a powering muscle, treated as a
homogenous solid organ, tuned to efficient locomotion via the natural
frequency is illuminated through the non-linear elastic muscle moduli
controlled by contraction velocity. The exploration of the elastic force
patterns known in solid state physics incorporated in activated skeletal and
cardiac muscles via the mechanical similarity principle yields analytical
rationalization for locomotor muscle patterns. Besides the explanation of the
origin of muscle allometric exponents observed for muscles in legs of running
animals and wings of flying birds, the striated muscles are patterned through
primary and secondary activities expected to be useful in designing of
artificial muscles and modeling living and extinct animals.

.

PACS: 89.75.Kd, 89.75.Da, 87.10.Pq, 87.19.Ff

Key words: Dynamic patterns, Scaling laws, Non-linear elasticity, Muscles.

##  I Introduction

The mechanical role of muscles varies widely with their architecture and
activation conditions. Striated (skeletal and cardiac) muscles are diverse in
their contractive interspecific and intraspecific functional properties
observed among and within animal species, nevertheless, ”the smaller muscles
and muscles of smaller animals are quicker”. This generic feature of skeletal
muscles was established by Hill [1]. More recently, the physiological
adaptation of skeletal muscles resulting in beneficial changes in muscle
function has been recognized by a number of investigators. It was learned that
long-fibre muscles commonly contract at over larger length ranges and
relatively higher velocities producing the greatest muscle forces the lowest
relative energetic costs [2]. Muscles having shorter fibres expose smaller
length change, but their cost of force generation is relatively less, e.g.
[3]. Searching for determinants of evolution of shape, size, and force output
of cardiac and skeletal muscle, a little is known about the regulation of
directional processes of mass distribution [4,5]. Although skeletal muscles
grow in length as the bones grow, most studies only involve force increasing
with cross-sectional area. Following the idea that the muscle force production
function is a critical evolutionary determinant [5], I develop a physical
study of muscle form adaptation to a certain primary activity with growth of
size preserving muscle shape.

When designing architecture of the striated muscle built from repeating units
(fibres and sarcomeres) at least three distinct muscle activities should be
distinguished [5]: (i) the _concentric_ contraction defined as the production
of active tension while the muscle is shortening and performing positive work,
(ii) the _eccentric_ contraction defined as the contraction during lengthening
performing negative work in a controlled fashion, and (iii) the _isometric_
contraction when the muscle force output is produced without changing of
length and performing net work. The corresponding _mechanical work patterns_
called by Russel et al. [5] as ”concentric work” and ”eccentric work” (that
might be extended by ”isometric work”) were carefully studied via in vivo
measurements of length-force cycling of individual skeletal muscles in active
animals, such as (i) the pectoralis in flying birds, (ii) leg extensors in
running cockroaches, and (iii) gastrocnemius in the level running turkey. The
corresponding  _muscle locomotor patterns_ were called as (i) _motor_ , (ii)
_brake_ , and (iii) _strut_ functions [6].

The seminal research by Hill [1] on dynamics of electrically stimulated
_isolated_ muscles was restricted to a single isotonic shortening. The studies
of the corresponding motor function resulted in famous force-inverse-velocity
master curve remaining the major dynamic constraint of all real (slow-fibre,
fast-fibre, and superfast) muscles [7] and computationally modeled muscles,
e.g. [8]. Besides, other two fundamental rules of muscle dynamics were noted
by Hill [1]. Examining hovering humming and sparrow birds, he recognized that
the ”frequencies of wings are roughly in inverse proportion to the cube roots
of the weights, i.e. to linear size”. Moreover, because the linear
proportionality between the stroke period and body length was found equally in
electrically stimulated isolated muscles, the intrinsic _frequency-length_
feature constrained by scaling rule $T_{m}^{-1}\varpropto L_{m}^{-1}$ beyond
the nervous control is likely more universal than previously appreciated.
Second _velocity-length_ Hill’s constraint states that ”the intrinsic speed of
muscle has to vary inversely to length”, i.e. $V_{m}\varpropto L_{m}^{-1}$.
Both Hill’s scaling rules remain a challenge to viscoelastic models of
transient-state mechanics and other existing theories of muscle contraction
[9].

The earliest theories of muscle motor function supposed muscle to be an
elastic body which, when stimulated, was converted in an active state
containing elastic energy causing the muscle to shorten. Such _elastic-energy
theories_ failed to explain mechanisms of the force production in terms of
viscoelastic characteristics. To a certain extent, poor experimental
approaches providing often conflicting clues to muscle structure in relation
to function may explain a little progress in understanding of contractile
properties of a muscle [9,4]. Moreover, physiological muscle properties
accounted for theories of muscle contractions developed at both molecular and
macroscopic scales are primarily focused on the reproduction of force-velocity
curve [9]. Besides, the existing phenomenological frameworks such as Hill-type
muscle models only mimic the proper mechanical characteristics of muscles by
means of passive viscoelastic springs attached to muscle contractive element
in series [10,11,3] or in parallel [12] and recruited when muscle is
activated. By ignoring the proper muscle function of force production and
force transmission throughout the muscle organ, these models are able to
explain no one of Hill’s principal constraints in muscle dynamics. On the
other hand, there exist experimental evidences of the adaptive ability of
skeletal muscle to exchange _elastic_ strain energy during force production
[2]. In line with this concept, it has recently communicated on a possibility
of the modeling of the adaptive muscle elasticity by elastic force patterns
[13].

In the present paper, I develop an integrative theoretical framework to the
problem of forces, structure, and contractive non-linear dynamics in striated
muscles. Instead of Hill-type modeling of in vitro motor function, e.g. [3],
brake function, e.g. [12, 2], and strut function, e.g. [14], or study of
muscle design by means of simulation of phenomenological force-length and/or
force-velocity constraints [8], the powerful method of continuum mechanics
generally providing macroscopic characterization and modeling of soft tissues,
e.g. [15, 16], is employed. By further exploration of the elastic force
patterns, I propose a self-consistent depiction of the three velocity-distinct
characteristic points well distinguished in all in vivo force-length loops of
the naturally activated muscles. Unlike the earliest elastic theories based on
minimization of energy, I develop the physical concept of similarity between
the force output and reaction active elastic forces that permits to avoid the
details of muscle activation process. The theory is validated by a comparison
to phenomenological scaling rules including both mentioned Hill’s dynamic
constraints and therefore may be hopefully helpful in designing artificial
muscles [15] and modeling living and extinct organisms [17].

##  II Theory

###  II.1 Theoretical Background

####  II.1.1 McMahon’s scaling models

The engineering models by McMahon [18, 19] develop previous Hill’s approach to
the problem of scaling of parameters of animal performance to _body weight_
$W=Mg$. Using Hill’s geometric similarity models [1,19] equally applied to
animal body, long bone, or individual muscle, each one was approximated by a
cylinder of longitudinal _length_ $L$ and _cross-sectional area_ $A$ (or
diameter $D\backsim\sqrt{A}$). Then, the assumption on the _weight-invariance_
of for the _tissue density_ , namely

$\rho_{tiss}=\frac{M}{AL}\varpropto W^{0}\text{,}$ (1)

was adopted. In mammalian long-bone allometry, this invariant was verified and
observed with a high precision [20]. Mechanical models of bending bones and
shortening muscles were introduced by McMahon via the weight-invariant
_elastic modulus_ $E_{tiss}$, _stress_ $\sigma_{tiss}$, and _strain_
$\varepsilon_{tiss}$, namely

$E_{tiss}=\frac{\sigma_{tiss}}{\varepsilon_{tiss}}\varpropto W^{0}\text{, with
}\sigma_{tiss}=\frac{\Delta F}{A}\text{ and }\varepsilon_{tiss}=\frac{\Delta
L}{L}\text{.}$ (2)

Here $\Delta L$ ($=L-L_{0}$) is the _length change_ accompanied by the _force
change_ $\Delta F$ ($=F-F_{0}$) counted off from the _resting length_ $L_{0}$.

While searching for functional mechanical patterns of biological systems
determined by _maximal_ forces using Eqs. (1) and (2), the maximal
stress/strain scaling relations

$\sigma_{geom}^{(\max)}\varpropto W^{1/3}\text{,
}\sigma_{elast}^{(\max)}\varpropto W^{1/4}\text{, and
}\sigma_{stat}^{(\max)}\varpropto W^{1/5}\text{,}$ (3)

could be readily derived from McMahon’s _geometric_(isometric), _elastic_
(buckling stress) and _static_ (bending elastic stress) _similarity models_
distinguished through McMahon’s scaling relations

$L_{geom}\backsim D\text{, }L_{elast}\varpropto D^{2/3}\text{, and
}L_{stat}\varpropto D^{1/2}\text{.}$ (4)

Instead, the _maximum_ stress and strain

$\sigma_{tiss}^{(\max)}\text{ }\varpropto\varepsilon_{tiss}^{(\max)}\varpropto
W^{0}\text{,}$ (5)

were postulated (see Table 4 in [19]) extending groundlessness his exact
result for the _mean_ stress $\sigma_{elast}^{(mean)}\varpropto W^{0}$,
obtained within the static stress similarity model (see Fig. 1 in [19]). The
improved self-consistent maximal stresses shown in Eqs. (3) follow
straightforwardly from McMahon’s cross-sectional areas

$A_{geom}^{(isom)}\varpropto W^{2/3}\text{, }A_{elast}^{(buck)}\varpropto
W^{3/4}\text{, and }A_{static}^{(bend)}\varpropto W^{4/5}$ (6)

applied to Eq. (2), along with McMahon’s idea on the dominating gravitational
forces in bones, muscles, and bodies, i.e. $\Delta F\backsim gM_{b}\backsim
gM_{m}\backsim W$. As shown in [20], the structure of long bones is driven by
peak muscle forces, but not by gravity.

####  II.1.2 Muscle shape and structure

After Alexander [21], the _physiologic_ cross-sectional area $A_{0m}$ (PCSA)
of the isolated skeletal _muscle_ $m$__ of _mass_ $M_{m}$ composed of $N$
bundles of masses $m_{i}$ was commonly estimated, e.g. [22], with the help of
the cylinder-geometry relation $A_{i}=m_{i}/\rho_{musc}L_{i}$, where $L_{i}$
is directly measured muscle fibre length. The spindle-like shape of the muscle
as whole organ was therefore determined by the muscle PCSA, namely __

$\text{
}A_{0m}={\displaystyle\sum\limits_{i=1}^{N}}A_{i}=\frac{M_{m}}{\rho_{musc}L_{0m}}\text{,
and
}\frac{1}{L_{0m}}=\frac{1}{M_{m}}{\displaystyle\sum\limits_{i=1}^{N}}\frac{m_{i}}{L_{i}}\text{,
with }M_{m}={\displaystyle\sum\limits_{i=1}^{N}}m_{i}\text{,}$ (7)

resulted in the sum of areas of muscle and the muscle length $L_{0m}$ of the
parallel-linked contractible subunits described statistically by the length-
unversed sum weighed by masses. Such a coarse-grained characterization of the
_muscle structure_ generally ignores the arrangement of muscle fibres relative
to generated force axis, distinguished by _pinnate angles_.

In scaling models, the evolution of the muscle structures across different-
sized animals of _body mass_ $M$ is observed statistically via _allometric
exponents_ $a_{m}$, $l_{m}$, and $\alpha_{m}$ determined by common rules
[21,23,25]:

$A_{0m}\varpropto M_{m}^{a_{m}}\text{, }L_{0m}\varpropto M_{m}^{l_{m}}\text{,
and }M_{m}\varpropto M^{1+\alpha_{m}},$ (8)

where the _muscle mass index_ $\alpha_{m}$ plays the same role as Prangel’s
index $\beta$ in bones, as noted in [26].

When the muscle-density invariance employed implicitly in Eq. (7) and
specified in Eq. (1) is applied to different skeletal muscles, the muscle
shape approximated by cylinder geometry is also preserved. Consequently, the
_muscle functional volume_

$A_{m}L_{m}=A_{0m}L_{0m}=\frac{M_{m}}{\rho_{0m}}\text{, with
}\rho_{0m}=\rho_{musc}\varpropto M_{m}^{0}\varpropto M^{0}\text{,}$ (9)

holding in all muscle work loops plays the role of the mechanical muscle
invariant. This condition is ensured by functional change
$\Delta\rho_{musc}/\rho_{musc}$ not exceeding $5\%$ [24]. Hence, the function-
independent _muscle-shape constraint_ [13]

$a_{m}+l_{m}=1+\alpha_{m}$ (10)

straightforwardly follows from Eqs. (8) and (9). Likewise the case of hindlimb
mammalian bones of the mean structure $a_{b}^{(\exp)}=2d_{b}^{(\exp)}=0.752$,
$l_{b}^{(\exp)}=0.298$, and $\beta^{(\exp)}=0.04$ [20,26], Eq. (10) is also
empirically observable in muscle allometry (see analysis in Table 5 below).

###  II.2 General Muscle Characterization

####  II.2.1 Maximal force and stress

In in vivo work loops, the muscle locomotor patterns can be generally
specified regardless of details of activation-deactivation conditions. In Fig.
1, the linear-slope characteristics $L_{1m}$ can be introduced in the force-
length cycling by the domains: $L_{2m}<L_{1m}<L_{3m}\approx L_{0m}$, for the
motor function, $L_{2m}>L_{1m}>L_{3m}$, for the brake function, and by
$L_{2m}\gtrsim L_{1m}\gtrsim L_{3m}\approx L_{0m}$, for the strut function
showing nearly isometric muscle contractions.

.

Place Fig. 1

.

Moreover, such a qualitative general characterization of the activated
individual muscle $m$ of _resting length_ $L_{0m}$ can be rationalized on the
basis of common _two-point_ force-length characterization, namely

$F_{musc}^{(\exp)}(L_{2m})=F_{musc}^{(\max)}=F_{2m}\text{ and
}F_{musc}^{(\exp)}(L_{1m})=F_{1m}\text{,}$ (11)

__ introduced by the maximum force __ $F_{2m}^{(\max)}$ and the _optimum_
muscle length [28, 29] $L_{1m}$ . The _instant dynamic length_
$L_{m}=L_{1m}\pm\Delta L_{1m}$ is counted off from the characteristic point
$L_{1m}$ via the optimum length change $\Delta L_{1m}$ (11) shown in Fig. 1
for all functions.

First, the linearization of the in vivo muscle force-length curve allows one
to determine _trial_ peak stress and corresponding strain by

$\sigma_{musc}^{(\max)}=\frac{F_{musc}^{(\max)}}{A_{2m}}\text{ and
}\varepsilon_{musc}^{(\max)}=\frac{\Delta L_{1m}^{(\max)}}{L_{2m}}\text{, with
}\Delta L_{1m}^{(\max)}=|L_{2m}-L_{1m}|\text{. }$ (12)

The corresponding force change $\Delta F_{musc}^{(\max)}$ observed near the
_optimum force_ $F_{1m}^{(\max)}$ (11) provides

$F_{musc}^{(\max)}=F_{musc}^{(\exp)}(L_{1m})+\Delta
F_{musc}^{(\max)}=F_{1m}+K_{musc}^{(\max)}\Delta L_{1m}^{(\max)}\text{ }$ (13)

that determinates _effective_ _muscle stiffness_ and _effective modulus_ ,
respectively

$\displaystyle K_{musc}^{(\max)}$ $\displaystyle\equiv
K_{2m}=\left|\frac{dF_{musc}}{dL_{m}}\right|_{F_{1m}^{(\max)}}\thickapprox\frac{\Delta
F_{musc}^{(\max)}}{\Delta L_{1m}^{(\max)}}=\frac{\Delta
F_{musc}^{(\max)}}{F_{musc}^{(\max)}}E_{musc}^{(\max)}\frac{A_{2m}}{L_{2m}}\text{,
}$ $\displaystyle\text{and }E_{musc}^{(\max)}$ $\displaystyle\equiv
E_{2m}=\frac{\sigma_{musc}^{(\max)}}{\varepsilon_{musc}^{(\max)}}\text{,}$
(14)

following from Eqs. (12) and (13).

####  II.2.2 Active stiffness and resonant muscle mechanics

Secondly, treating the maximum-force crossover state (11) as the generic
transient-neutral __ state [26] the _resonant_ _frequency_
$1/T_{musc}^{(\max)}=T_{2m}^{-1}$ related to point $2$ in Fig. 1 associated
with maximum efficiency of muscle cycling, e.g. [29], can also be introduced
as _natural frequency_ [19], namely

$T_{2m}^{-1}\thicksim
2\pi\sqrt{\frac{K_{2m}}{M_{m}}}\thicksim\sqrt{\frac{E_{musc}^{(\max)}}{\rho_{0m}}}\left(\frac{\Delta
F_{musc}^{(\max)}}{F_{musc}^{(\max)}}\right)^{1/2}\frac{1}{L_{2m}}\text{,}$
(15)

and analyzed by Eqs. (9) and (14).

One can see that Eq. (15) yields first Hill’s general constraint discussed in
Introduction. However, the following three conditions are required: (i) the
preservation of dynamic functional volume __(9), (ii) the weight-invariance of
the elastic modulus $E_{musc}^{(\max)}$ (2), and (iii) the existence of force
similarity between the exerted force $F_{musc}^{(\max)}$ and its change
$\Delta F_{musc}^{(\max)}$ (13). Therefore, the muscle _force-similarity
principle_ , namely

$F_{musc}\cong\Delta F_{musc}\cong F_{prod}\cong F_{elast}\cong\Delta
F_{elast}\text{,}$ (16)

implying a coexistence of all forces in biomechanically equivalent states [26]
must be adopted. Here the _active elastic force_ $\Delta F_{elast}$ (shown
schematically as $F_{act}$ in Fig. 1D) is also included. The total state-
transient elastic force $F_{elast}$ is the superposition of common _passive
elastic force_ $F_{pass}$ provoked by external loads and _active elastic
force_ $\Delta F_{elast}$ caused by the _production_ force $F_{prod}$. The
correspondence sign $\cong$ indicates that though the involved physical
characteristics belong to the same mechanical state, they may differ in both
physical and numerical parameters stipulating this state.

Given that the peak _active muscle stress_ $\sigma_{m}$ always exceeds the
corresponding passive stress, e.g. [14], in further I focus on the fully
activated _transient states_ described by

$\sigma_{m}=\frac{\Delta F_{elast}}{A_{m}}=E_{m}\frac{\Delta
L_{m}}{L_{m}}\text{. }$ (17)

Unlike Eqs. (2) and (12), $\sigma_{m}$ is the true intrinsic _elastic stress_
in a certain (not specified) dynamic state. This reveals the maximum-amplitude
_elastic force_ of the fully activated muscle

$\Delta F_{elast}\equiv\Delta F_{m}=K_{m}\Delta L_{m}=E_{m}A_{m}\frac{\Delta
L_{m}}{L_{m}}$ (18)

and in turn provides the corresponding _active_ _muscle stiffness_

$K_{m}=E_{m}\frac{A_{m}}{L_{m}}\text{.}$ (19)

The underlying mechanical _sarcomere elastic stiffness_ $K_{s}$ is related via
the muscle-volume average, namely

$K_{m}=\frac{1}{A_{m}L_{m}}\int K_{s}(r_{m})\text{ }d^{3}r_{m}\text{,}$ (20)

originated from end-to-end intercellular overlapping [31, 12].

The _muscle energy change_

$\Delta U_{m}\backsim K_{m}\Delta L_{m}^{2}\cong E_{m}A_{m}\frac{\Delta
L_{m}^{2}}{L_{m}}$ (21)

stored or released during active-period contraction provides the mechanical
_cost of energy_

$CU_{m}=\frac{\Delta U_{m}}{\Delta L_{m}}\cong E_{m}A_{m}\frac{\Delta
L_{m}}{L_{m}}\text{.}$ (22)

These relations demonstrate how the observable mechanical characteristics can
be linked to the underlying muscle elastic forces using the force-similarity
principle (16). In turn, the _contraction velocity_

$V_{m}=\overline{V_{m}(t)}\equiv\frac{1}{\Delta t_{m}}\int_{0}^{\Delta
t_{m}}\left[\frac{dL_{m}(t)}{dt}\right]dt\backsim\left[\frac{dL_{m}(t)}{dt}\right]_{t=\Delta
t_{m}}\cong\frac{L_{m}}{T_{m}}$ (23)

is defined by the instant velocity $V_{m}(t)$ averaged over _activation time_
$\Delta t_{m}$.

####  II.2.3 Fast and slow activated muscles

According to the most general classification of diverse muscles, three types
are conventionally distinguished: red (slow fibre) muscles, white (fast fibre)
muscles, and intermediate type, mixed fibre muscles. Although collective
mechanisms of muscle contractions are poor understood, e.g. [32], physically,
the two limiting situations of dynamic accommodation of local forces generated
by cross bridge attachments can be generally rationalized. As schematically
drawn in Fig. 1D, in an activated muscle, the dynamic process of equilibration
between the production intrinsic forces and external loads (not shown) is
followed by the spatiotemporal relaxation of elastic forces. For the simplest
case of _slow muscles_ , the dynamic equilibration occurs via the slow channel
of relaxation, assumably common for both active, $F_{prod}^{(slow)}$, and
passive elastic forces. Since passive forces in solids are short of range
[33], both the forces are proportional to muscle _surface_. In contrast, it is
plausible to adopt that in _fast muscles_ the fast-twitch fibres transmit the
locally generated forces in all directions, i.e. along and across fibres,
resulting in the overall maximum force output $F_{prod}^{(fast)}$ to be linear
with dynamic muscle _volume_. Basing on such a general physical picture, a
function-independent and regime- independent characterization of the force
production function, namely

$F_{prod}^{(fast)}\varpropto A_{rm}L_{rm}\text{ and
}F_{prod}^{(slow)}\varpropto A_{rm}\text{, with }r=1,2\text{, and }3\text{,}$
(24)

is proposed via the force-size scaling rules for all three distinct states
shown in Fig. 1 and hereafter distinguished by symbol $r$.

The widely adopted by biologists linear-displacement regime is discussed in
Eq. (2) via $\Delta L\varpropto L$ resulted in the weight-independent strain
(5). The corresponding optimum-velocity regime $r=1$, attributed to the
instant length-independent elastic strains,
$\varepsilon_{m}^{(opt)}=|L_{m}-L_{3m}|/L_{m}\varpropto L_{m}^{0}$ (2) with
$L_{m}$ lying between $L_{1m}$ and $L_{3m}\approx L_{0m}$, is now clarified by
the scaling equations

$E_{1m}^{(fast)}=E_{fast}^{(opt)}\varpropto L_{m}^{1}\text{ and
}E_{1m}^{(slow)}=E_{slow}^{(opt)}\varpropto L_{m}^{0}$ (25)

characteristic of fast and slow muscles. Such a muscle description follows
from the similarity (16) between the active elastic force $\Delta
F_{1m}=\Delta F_{elast}^{(opt)}=E_{1m}A_{1m}\varepsilon_{1m}$ (18) and
corresponding production force (24). The _optimum_ force-velocity muscle
mechanics is rationalized below in Table 1 and then tested by empirical data.

Similarly, the bilinear-displacement regime $r=2$ introduced by the dynamic
length change $\Delta L_{2m}=|L_{m}-L_{1m}|\varpropto L_{m}^{2}$, with $L_{m}$
lying between $L_{2m}$ and $L_{1m}$, and the _maximum_ active elastic force
$\Delta F_{2m}=\Delta
F_{elast}^{(\max)}=E_{2m}A_{2m}\varepsilon_{2m}^{(\max)}$ (18) results in the
maximal elastic moduli

$E_{fast}^{(\max)}=E_{2m}^{(fast)}\varpropto L_{m}^{0}\text{ and
}E_{slow}^{(\max)}=E_{2m}^{(slow)}\varpropto L_{m}^{-1}\text{,}$ (26)

adjusted with the muscle production function (24) via the force similarity
principle (16). Finally, the high-velocity trilinear regime $r=3$ is suggested
by the moderate-force and moderate-elastic muscle determined by

$E_{fast}^{(\operatorname{mod})}=E_{3m}^{(fast)}\varpropto L_{m}^{-1}\text{
and }E_{slow}^{(\operatorname{mod})}=E_{3m}^{(slow)}\varpropto
L_{m}^{-2}\text{. }$ (27)

This condition specifies point $3$ in Fig. 1, along with the underlying cubic-
power muscle displacements $\Delta L_{3m}\varpropto L_{m}^{3}$ scaled by
dynamic $L_{m}$ lying above or below the characteristic length $L_{3m}$ in any
muscle acting as motor, brake or strut (see Fig. 1).

###  II.3 Muscle Functions

Likewise the naturally curved mammalian long bones biomechanically adapted to
the maximum longitudinally bending [20, 26], the muscle _motor function_ is
assigned to locomotor muscles showing concentric positive work exerted by
elastic bending forces. Given that the _elastic force patterns_ coincide for
bending and torsion [26], both kinds of unpinnate and uni-pinnate skeletal
muscles, having respectively close to zero and non-zero fixed pinnate angles,
may be expected to be structured by the same motor function. The specific-
function mechanical characterization is described in Appendix B and results
are summarized in Table 2.

##  III Results

###  III.1 Assumptions and predictions

The following assumptions are made regarding elastic striated muscles:

1\. The muscles are considered at macroscopic scale as individual homogeneous
organs. Within the continuum mechanics, the coarse-grained approach ignores
the details of heterogeneous microstructure and pinnate angles.

2\. When activated under different boundary loaded conditions, the muscles do
not undergo changes in shape and whole volume. The emerging elastic fields
follow patterns established for long solid cylinders.

3\. The mechanical similarity adopted between the extrinsic forces exerted by
the muscle and intrinsic elastic reaction forces, as well as the dynamic
similarity adopted for contraction velocities and frequencies are observable
in all biomechanically equivalent states.

4\. The natural ability of the non-linear elastic tuning of fast and slow
muscles to distinct locomotor states can be characterized by the elastic
moduli sensitive to evolving dynamic variable associated here with the regime-
characteristic muscle length.

.

The function-independent mechanical characterization of muscles is provided in
Table 1.

.

Place Table 1.

.

The specific case of muscle structure accommodation in the bilinear regime is
described in Table 2.

.

Place Table 2

.

The rules of mass distribution across and along the muscle axis provided in
Table 2 in terms of the muscle-structure scaling exponents [$a_{2m}$,
$l_{2m}$] are characteristic for slow, fast, and mixed muscles producing
maximum force. In Table 3, these scaling rules are compared with the finding
for the optimal-force state [$a_{1m}$, $l_{1m}$] and moderate-force state
[$a_{3m}$, $l_{3m}$].

.

Place Table 3

.

The dynamic characteristics of distinct-velocity contractions are predicted in
Table 4.

.

Place Table 4

.

The consequences of the theoretical scaling framework are:

1\. The peak forces generated in all regimes scales as muscle volume or
surface in fast or slow muscles, respectively.

2\. A general, function-independent mechanical description of the striated
muscle activated in the liner-displacement regime is predicted for each type
of muscles (Table 1).

3\. The muscle-type independent locomotor functions and related mechanical and
dynamic characteristics of the striated muscle activated in the bilinear
regime are predicted (Table 2).

4\. The muscle-type independent varied dynamic structures are predicted for
all muscle regimes and functions (Table 3).

5\. The function-independent dynamic scaling characteristics are obtained in
Table 4 for all type of muscles.

In what follows, all theoretical findings are tested by the available from the
literature data.

##  IV Discussion

”What determines the shape, size, and force output of cardiac and skeletal
muscle?” (Louis Sullivan quoted in [5]). The provided coarse-grained study of
conservative striated muscles suggests that the size-dependent peak elastic
forces determine fiber-type-independent patterns of the functionally adapted
structures preserving muscle shape. The size-dependent peak force output is
determined by the muscle volume and area for white and red muscles, regardless
of muscle structure and function.

###  IV.1 Function against structure

####  IV.1.1 General muscle characterization

Being composed of bundles of muscle fibres including all other contractible
components (neural, vascular, and collagenous reticulum), the striated muscle
is thought of as a heterogeneous _continuum medium_ transmitting the produced
tension internally and externally, e.g. [34]. Primarily, I address the problem
of mechanical design of striated muscle to a general, function-independent
characterization of the individual muscle organ loaded by tension, reaction,
and gravity through tendons, ligaments, and bones. My non-energetic approach
is physically grounded by the existence of linear force-displacement regions
(shown by the solid arrows in Figs. 1A, 1B, and 1C) in all in vivo work loops
regardless of dynamic details of approaching to the maximum exerted force
$F_{musc}^{(\max)}$. Hence, the mechanical characterization of the maximum-
force activated muscle arises from the muscle __ stiffness $K_{m}^{(\max)}$
(14) underlaid by sarcomere stiffness $K_{s}^{(\max)}$ (20). Consequently, all
forces involved in muscle contraction following by active and passive elastic
strains allow common mechanical description (shown in Fig. 1D) not depending
on their biochemical, inertial, or reaction origin.

The analytical justification of Hill’s first frequency-length constraint
arises from the analysis of Eq. (15) that requires eventually the usage of the
similarity between all intrinsic muscle forces, Eq. (16). The constraint
$T_{m}^{-1}\varpropto L_{m}^{-1}$ and other mechanical characteristics for
slow muscles accumulated in Table can be applied to _steady-speed_ regimes of
locomotion modes where all forces are generally equilibrated and controlled by
slow-fibre muscles [35]. In the case of non-steady transient locomotion when
fast-twitch fibres and nervous control are additionally requested [35], Hill’s
first constraint transforms [by Eqs. (15) and (25)] into a new one,
$T_{m}^{-1}\varpropto L_{m}^{-1/2}\varpropto 1/V_{fast}^{(opt)}$ (Tables 1 and
4), well known for animals running with maximal _optimum_ speed [36,37]
$V_{run}^{(\max)}\backsim
V_{fast}^{(opt)}\varpropto\sqrt{L}\varpropto\sqrt{L_{m}}$. We have therefore
demonstrated how the suggested _dynamic similarity_ establishes a link between
the body-propulsion speed and locomotor-muscle contraction velocity, also
described by Rome et al. [38]. Being united with the muscle-force similarity,
both constraints yield _mechanical similarity_ , the key principle explored in
this research.

####  IV.1.2 Maximum force output against structure and velocity

In muscle physiology, the functional effect of muscle conceptual architecture
simply states that muscle force output is proportional to PCSA. The proposed
study of adaptation of the muscle structure via the force production function
seems to be in qualitative agreement with this statement, because in all cases
exposed in Eq. (24) the muscle force output is _proportional_ to $A_{m}$. Such
a simplified treatment of the fast-muscle mechanics (formally substituted by
that for slow muscles) arrived at the widely adopted opinion that the peak
muscle stress $F_{prod}^{(slow)}/A_{m}$ , specifying the case of slow muscles
in linear dynamic regimes with $\sigma_{m}^{(slow)}\varpropto L_{m}^{0}$
(Table 1), is generic for any muscle, as already discussed in Eq. (5).
Although the proposal on scaling of the maximum production force (and active
stress) with muscle size (24) is a challenge for further research, the
provided fairly general physical grounds are supported by empirical
observations by Marden and Allen [39]. They established statistically that the
maximum force output in all biological (and human-made) motors falls into two
fundamental scaling laws: (i) in fast-cycling motors, presented by flying
insects, bats and birds, swimming fishes, and running animals it scales as
(_motor mass)_ 1 and (ii) in slow-cycling motors, such as myosin molecules,
muscle cells, and some (unspecified) ”whole muscles” the force at output
scales as (_motor mass_)2/3. The ”motor mass” was associated with muscle (and
fuel) mass. That fact that the authors observed muscle motors from sarcomere
to whole muscle organ passing through the single-fibre level of muscle
organization, makes a basis for the discussed below _micro-macro scale
correspondence_.

The proposed treatment of the in vivo force-length curves is provided for
three distinct force-velocity characteristic points (shown in Fig. 1)
correlated by the inequalities

$F_{2m}>F_{1m}>F_{3m}\text{ and }V_{2m}<V_{1m}<V_{3m}\text{.}$ (28)

These three generic function-independent states are associated with the linear
($r=1$), bilinear ($r=2$), and trilinear ($r=3$) muscle dynamics determined
via the muscle elastic moduli $E_{rm}$ in Eqs. (25), (26), and (27),
respectively. The mechanical characterization of slow and fast striated
muscles is therefore provided in terms of the maximum ($\Delta F_{2m}$),
optimum ($\Delta F_{1m}$) and moderate ($\Delta F_{3m}$) active elastic forces
developed at the measurable maximum ($V_{3m}$), optimum ($V_{1m}$), and
moderate ($V_{2m}$) contraction velocities (Table 4). The stabilization of the
dynamic regimes is expected at the natural frequencies, which also are scaled
in Table 4 to the dynamic length $L_{rm}$.

####  IV.1.3 Muscle functions against size and shape

Searching for answer on ”what features make a muscular system well-adapted to
a specific function?” [28], it has been shown preliminary [13] that such
features are related to natural conditions of the stabilization or tuning to
the moderate-velocity regime $r=2$ via the mean dynamic length of the fast-
twitch fibers adapted by the best way to one of the patterns of muscle
locomotor functions. In this study such features specify the role of slow-
twitch fibers.

The elastic-force patterns underlying concentric, eccentric, isometric, and
cardiac contractions are suggested in Eqs. (35), (39), (42), and (44),
respectively. The solutions to the muscle-force and muscle-shape constraints
are accumulated in Table 2 as patterned functions well distinguished by the
muscle _structure parameter_ ($\eta_{m}=d\ln A_{m}/d\ln L_{m}$) established
for the motor ($\eta_{1}=4$), brake ($\eta_{2}=3$), strut ($\eta_{3}=\infty$)
skeletal muscles, an extended by the pump ($\eta_{5}=1$) cardiac muscle and
one spring ($\eta_{4}=2$) striated muscle. These structurally adapted muscles
are thought of as to be suited to efficient work during powering when,
respectively, shortening ($m=1$), lengthening ($m=2$), or remaining in the
nearly isometric dynamic state ($m=3$), high-pressure-resistant state ($m=4$),
and likely energy-saving state ($m=5$). The found new pump function is in
accord with the observation by Russel et al. [5] that ”the heart chamber,
unlike skeletal muscles, can extend in both longitudinal and transverse
directions, and cardiac cells can grow in length and width”, that implies
$\eta_{5}<$ $\eta_{1},\eta_{2}$, or $\eta_{3}$. Given that only a few patterns
exist in elastic theory of solids [26], it is not striking that the spring,
brake, and motor functions resembles McMahon’s ”geometric”, ”elastic”, and
”static” stress similarities discussed in Eqs. (3) and (4).

In Table 3, conceivable stable dynamic structures corresponding to muscle
activity in different dynamic regimes are analyzed. As in the case of Table 2,
the solutions of dynamic constraints follow from the similarity between force
output (16) and elastic-force patterns. The resulting _dynamic_ states are
discussed in terms of the scaling exponents for the muscle _dynamic structure_
[$A_{rm}$, $L_{rm}$] preserving muscle shape and volume (9). Other related
observable mechanical characteristics are exemplified in Tables 1 and 2. The
major outcome of the analysis in Tables 2 and 3 is that both slow-twitch and
fast-twitch fibres belonging to the same muscle $m$ should manifest concerted
behavior coordinated by the dynamic active elastic forces.

Another significant feature of the analysis in Table 3 is a striking
prediction of the mechanical functions which are expected to be shown by a
given striated muscle $m$ of certain specialization (_primary functions_
indicated by regime $r=2$, see proof below) when its cycling dynamics is
switched to regimes $r=1$ and $3$ by tuning to the corresponding natural
frequencies $T_{rm}^{-1}$. In case of regime $r=1$, both types of arbitrary
slow muscle tuned to $T_{1slow}^{-1}$ and fast muscle tuned to
$T_{1fast}^{-1}$ (Table 4) are expected to show maximum workloop efficiency
when acting as controlled spring. In the efficient nonlinear regime $r=3$ the
slow and fast struts ($m=3$ in Table 2) will not show another function, but
any type of brakes ($m=2$ in Table 2) will work as motors, whereas motor are
expected to expose a new function, say, $m=6$ [determined by
$\eta_{6}=(6/7)/(1/7)=6$] that is closer to the brake activity ($\eta_{2}=3$)
than the strut ($\eta_{3}=\infty$). The cardiac muscles seem to display a
crucial dynamic state, say $m=0$ with $\eta_{0}=0$, which flatters the heart.
Such predicted _secondary functions_ and unusual ($m=0$ and $6$) muscles
adapted to new functions is a challenge deserving further study by
experimentalists.

###  IV.2 Direct observation of muscle specialization

”If a muscle is specialized for a particular mechanical role how this is
reflected in it architecture?” [40]. The stated problem is approached here by
the comparative analysis between the muscle allometric exponents and those
predicted for particular efficient activities describing the trends of biomass
accommodation via PCSA and along a muscle.

####  IV.2.1 Isolated muscles in hindlimb of mammals and birds

In Table 5, the _morphometric data_ on the allometric exponents for the mean
cross-sectional area $A_{0m}^{(\exp)}$ and length $L_{0m}^{(\exp)}$ of four
skeletal muscles in the mammalian hindlimb for $35$ quadrupedal species of
body-mass domain exceeding four orders in magnitude are studied.

.

Place Table 5

.

First, let us verify the cylinder-shape similarity of skeletal muscles
described by Eq. (9). The muscle mass index $\alpha_{0m}$ estimated in Eq.
(10) via experimental data $a_{0m}^{(\exp)}$ and $l_{0m}^{(\exp)}$ is compared
in Table 5 with the measured indexes $\alpha_{0m}^{(\exp)}$.

.

Place Fig. 2

.

In Figs. 2 and 3, the method of determination of the primary mechanical
function is illustrated: the adapted muscle structure is indicated by the
appropriate theoretical point located most closely to the datapoint.

.

Place Fig. 3

.

The found reliable estimates $\alpha_{0m}^{(est)}$ were used then in the
muscle-function analysis in Figs. 2 and 3. The established small indices
$\alpha_{0m}$ generally validate the muscle biomechanics by proving a high-
precision observation of locomotory muscle patterns via muscle morphometry and
_functional physiology_. This implies that the effect of biomechanical
adaptation of muscle design to active elastic forces predominates over effects
of biological adaptation assigned to small $\alpha_{0m}^{(\exp)}$.

Secondly, the analysis in Figs. 2 and 3 indicates strong correlations between
the morphometrically characterized structure of skeletal muscle and one of the
primary locomotor functions described in Table 2. The primary functions
indicated in Table 5 are found with a high degree of certainty. Indeed, as
illustrated in Fig. 2, the deviations of distances measured along the dashed
line, corresponding to a given muscle, between the datapoint and distant
challengers for the primary function, from the smallest distance indicating
the primary candidate, always exceed the experimental uncertainty.

Thirdly, the found muscle mechanical specifications do not conflict with the
_physiological categorization_ established for joint extensors and flexors,
which muscle structures are shown to be adapted to the brake and motor
functions via activation of eccentric and concentric elastic forces. The found
structure parameter $\eta_{plant}\thickapprox 18$ indicates the foot support
activity for plantaris as the primary function (Table 5) that is in accord
with in vivo workloop presented in Fig. 1C. As shown in Table 3, the struts
are most conservative muscles no changing their support function in non-linear
regimes. In contrast, the gastrocnemius in mammals manifests their motor,
strut, and brake functions in, respectively, uphill, level, and incline
running of animals. Through the motor adapted structure with
$\eta_{gast}\thickapprox\eta_{1}=4$, the analysis in Fig. 3 establishes the
motor activity for gastrocnemius as the primary function naturally selected
for the significant mechanical task of uphill running exploring the bilinear
muscle dynamics. The effective trilinear gastrocnemius-displacement dynamics
is most close to the brake-like activity $\eta_{6}=6$, attributed to the
secondary function of the motor experimentally observed in gastrocnemius of
incline running turkey [27] and hopping tammar wallabies [24].

In Fig. 4, the overall muscle peak stress data measured in limb muscles of
animals in strenuous activity, reviewed by Biewener [25], are re-examined and
re-analyzed accounting for the primary functions of hindlimb muscles
established in Table 5.

.

Place Fig. 4

.

The uphill-motor specialization of gastrocnemius is independently supported by
the compressive-stress analysis made in Fig. 4 for fast running, jumping, and
hopping mammals. The stress scaling exponents ($s_{m}$) predicted for the
motor ($s_{1}=1/5$), strut ($s_{3}=0$), and control ($s_{4}=0$) functions are
shown to be distinguishable in work-specific mammalian muscles described in
Table 2. Hence, although the overall-function data by Biewener [25] indeed
expose almost weight-independent muscle stress, earlier postulated by McMahon
in Eq. (5) and only in part justified here by the slow-fibre muscles (Table 1)
and strut muscles (Table 2), the analyses in Fig. 4 demonstrates how the
function-specific muscle stress may serve as a new tool for the direct
observation of muscle specialization ignored in all previous overall-function
analyses.

I have also investigated an interesting question: whether the primary function
established for a certain leg muscle in mammals specialized to fast running
coincides with that for the same muscle in birds? The pioneering data on
individual leg muscles in $8$ running birds, ranging in size from $0.1$ _kg_
quail to $40$ _kg_ ostrich, are analyzed in Table 6 and Fig. 5.

.

Place Table 6

.

Place Fig. 5

.

In running and non-running birds (Fig. 5), the _gastrocnemius_ is employed as
the brake and spring, in contrast to the motor function in mammals (Table 5).
This is in accord with Bennett [23], who noted that ”the full force-generated
capacity of gastrocnemius is only used occasionally, such as during take-off,
when a bird attempts to throw itself into the air”. This explains our indirect
observation: the primary function of the gastrocnemius in running specialists
is attributed to the foot flexor in mammals and ankle extensor in birds (Table
6). In _non-running_ birds, the legs are designed to control the ground
locomotion (Fig. 5), whereas the wings may share motor and brake functions
(Table 3), in accord with the review by Dickinson et al. [6].

####  IV.2.2 Micro-macro scale correspondence

There are many striking examples when skeletal muscles expose adaptation to a
specific function, e.g. [43, 3]. The striated muscles anatomically suited to
concentric or eccentric work [2] are structurally distinct having,
respectively, long thin cells or short wide cells [5]. This observation
suggests the _microscopic level_ of muscle-cell adaptation introduced here by

$A_{cell}^{(conc)}>A_{cell}^{(ecent)}\text{ and
}L_{cell}^{(ecent)}>L_{cell}^{(conc)}$ (29)

for the _cellular_ cross-sectional area $A_{cell}$ ($\equiv A_{s}$) and length
$L_{cell}$ ($\equiv L_{s}$). Adopting these function specific trends, one may
expect to observe the cell-structure parameters $\eta_{s}=4$ and $3$ for
sarcomeres accommodated to efficient shortening or stretching of muscle as a
whole.

A general question arises whether allometric coefficients of proportionality
omitted above in all structure-function power-law (scaling) relations are also
attributed to active elastic strains accompanying maximum force production?
Or, alternatively, other microscopically justified mechanisms, c.f. [44], or
additional parameters (such as pinnate angle) may result in different general
macroscopic consequences? Given the highly conservative nature of contractive
units of _skeletal_ muscles and their well pronounced organization [25], the
_specific-function trends_ of the muscle cross-sectional area

$A_{strut}^{(\text{{isom}})}>A_{motor}^{(conc)}>A_{brake}^{(eccen)}>A_{contr}^{(sprin)}\text{
}$ (30)

and muscle-fibre length

$L_{contr}^{(sprin)}>L_{brake}^{(eccen)}>L_{motor}^{(conc)}>L_{strut}^{(\text{{isom}})}$
(31)

are generally expected from Table 2. The suggested trends become observable
via the primary functions established in Table 5 for gastrocnemius ($m=1$),
DDF ($m=1$), CDE ($m=2$), and plantaris ($m=3$), when the regression data [22]
on passive-muscle structure [$A_{0m}^{(\exp)}(M)$, $L_{0m}^{(\exp)}(M)$] are
taken additionally into consideration:
$A_{plant}^{(\exp)}>A_{gast}^{(\exp)}\gtrsim
A_{DDF}^{(\exp)}>A_{CDE}^{(\exp)}$ and
$L_{CDE}^{(\exp)}>L_{gast}^{(\exp)}\gtrsim
L_{DDF}^{(\exp)}>L_{plant}^{(\exp)}$, starting with $M>1$ $kg$.

Similarly, the trend for active stiffness

$K_{strut}^{(\max)}>K_{motor}^{(\max)}>K_{brake}^{(\max)}\text{ and,
generally, }K_{fast}^{(\max)}>K_{slow}^{(\max)}\text{ }$ (32)

straightforwardly follows from Table 2. Given that the _optimum velocity_ for
fast fibres $V_{1m}\varpropto L_{m}^{1/2}$ (Table 1), Eq. (31) provides

$V_{brake}^{(opt)}>V_{motor}^{(opt)}>V_{strut}^{(\text{{opt}})}$ (33)

Moreover, a crude estimate for the _cost energy_

$CU_{motor}^{(\max)}>CU_{strut}^{(\max)}>CU_{brake}^{(\max)}$ (34)

follows from $CU_{fast}^{(\max)}\varpropto M_{m}$ (22) and the experimental
data by Pollock and Shadwick [22],
$M_{1}^{(\exp)}>M_{3}^{(\exp)}>M_{2}^{(\exp)}$, considered at the same body
mass $M$. The finding (34) is in accord with the experimental observation
[44]: muscles contracting nearly isometrically (strut function) generate force
more economically than muscles involved in concentric work (via motor
function).

####  IV.2.3 Muscle dynamics of mammalian legs and dragonfly wings

Given that _mammalian leg extensors_ are active mostly during lengthening [2],
the brake primary function ($m=2$ in Table 2) could be assigned to leg muscles
specified by effective length $L_{leg}\varpropto M^{1/4}$ ($\alpha_{leg}=0$ is
adopted). In accord with Hill’s second constraint, underlaid by the proper
frequency $T_{3m}^{-1}\varpropto L_{m}^{-2}$ (Table 4), the theory predicts
$V_{leg}^{(\max)}\varpropto L_{leg}^{-1}\varpropto M^{-1/4}$ that results in
$1/T_{leg}^{(\max)}\varpropto L_{leg}^{-2}\varpropto M^{-1/16}$. Similarly,
for the wing-motor muscles in _flying birds_ ($m=1$ in Table 2) one should
expect $V_{wing}^{(\max)}\varpropto L_{wing}^{-1}\varpropto M^{-1/5}$, for
contraction velocity, and $1/T_{wing}^{(\max)}\varpropto M^{-1/25}$, for the
frequency or, alternatively, $1/T_{wing}^{(opt)}\varpropto M^{-1/5}$, in the
optimum-velocity regime (see Table 4). Hence, analytically revealed Hill’s
constraint becomes observable via the empirical regression data by Medler
[43]: on the maximum-amplitude contraction velocities for the locomotor
muscles in leg of terrestrial animals, $V_{leg}^{(\exp)}\varpropto M^{-0.25}$,
and that for wings in flying birds, bats, and insects,
$V_{wing}^{(\exp)}\varpropto M^{-0.20}$. Moreover, the experimental data by
Schilder and Marden [45] of the wingbeat frequency
$1/T_{wing}^{(\exp)}\varpropto M_{m}^{-0.20}$ scaled by mass $M_{m}$ (and
length $L_{0m}$) of the basalar muscle in dragonflies indicate that the motor-
type muscles ($L_{0m}\varpropto M_{m}^{1/5}$, see analysis in Fig. 6) were
studied self-consistently in the optimum, steady-velocity motion regime.

.

Place Fig. 6

.

In the same optimum-velocity regime (Table 1), the maximum-amplitude _static
force_ $F_{stat}^{(\exp)}\cong\Delta F_{1m}^{(slow)}\varpropto M_{m}^{2/3}$
and net  _lever-system force_ $F_{ind}^{(\exp)}\cong\Delta
F_{1m}^{(fast)}\varpropto M_{m}$ reported by Schilder and Marden [45] may be
associated with the slow and fast activated fibres in the basalar muscles
tuned elastically to the linear regime through the dynamic PCSA
$A_{1m}^{(dyn)}\varpropto M_{m}^{2/3}$ and length $L_{1m}^{(dyn)}\varpropto
M_{m}^{1/3}$. The observed dynamic force output $F_{dyn}^{(\exp)}\varpropto
M_{m}^{0.83}$ can be therefore suggested as the mixed-fibre force
$F_{dyn}^{(pred)}\cong\Delta F_{1m}^{(mix)}\varpropto M_{m}^{5/6}$ (Table 1),
i.e. as a compromise of the forces $F_{stat}^{(\exp)}$ and $F_{ind}^{(\exp)}$.
These estimates challenge further analysis of the reported dynamic forces.

##  V Conclusion

A theoretical framework for mechanical characterization of the three transient
activated states of the striated muscles passing in force-length cycles
through the three distinct dynamic regimes is proposed. The explicit
analytical description of muscle locomotor functions and related mechanical
characteristics is provided on the basis of two concepts: (i) the preservation
of spindle-type shape in skeletal muscles and egg-type shape in cardiac
muscles related to the preservation of dynamic muscle volume and (ii) the
mechanical similarity between action and reaction forces emerging in
biomechanically equivalent states. Exploring known patterns of elastic forces
in continuum mechanics, the macroscopic study of the force production and its
functional and structural accommodation in the loaded muscle organ as a whole
provides the following major points.

1\. It is demonstrated how the dynamic (frequency-velocity) constraints for
muscle contractions, first observed by Hill in hovering birds and then
revealed in locomotor muscles of running animals and flying birds, bats, and
insects, can be derived from the generic principle of mechanical (force and
velocity) similarity.

2\. It is shown how relations in classical mechanics of solids can be explored
in soft tissues. The study is grounded by the active-force muscle stiffness
reliably derived in all muscle work loops nearby and below the maximum-
amplitude exerted forces. The muscle stiffness, underlaid by sarcomere
stiffness, is shown to be dependent on muscle geometry and dynamic functional
variable, underlaid by elastic moduli, which encompass all contractive
elements acting as an elastic continuum medium.

3\. The theoretical prediction that the fast and slow muscles should generate
maximum forces linear, respectively, with the muscle volume and cross-
sectional area, regardless of muscle function and structure, is in part
validated by the direct empirical observation of maximal forces exerted by
animals and by the provided indirect observation of the adapted (primary)
muscle functions in legs of mammals and birds.

4\. The macroscopic structures of locomotor skeletal muscles observable
directly by muscle allometry are found to be adapted to the maximum-force
state, following moderate-velocity dynamic regime, instead of the expected
optimum velocity regime. Such a bilinear-displacement muscle dynamics
involving both fast-twitch and slow-twitch powering muscle fibres sheds light
on the origin of allometric power laws and muscle specialization. The adapted
structures are examined via available empirical data: the legs are brakes in
mammals and springs in non-running birds, whereas the wings are motor-brake
engines in flying species. Suggested pump function for the cardiac muscles
needs further experimental tests.

5\. The provided study of the muscle specialization in mammalian hindlimb
indicates that the properly tuned force production function is a dominated
factor in the accommodation of muscle structure. This finding also indicates
the predomination role of mechanical effects over biological adaptive
mechanisms assigned to the relatively small muscle-mass index. As the result,
a new investigation tool for indirect statistical observation of the
biomechanical adaptation of individual locomotor muscles is proposed through
the regression analysis of in vivo muscle stresses in synergists scaled across
different-sized animals.

6\. The assumption on that the muscle tuning muscle ability of animals can be
modeled by active elastic forces via non-linear muscle elastic moduli is
validated by the observation of the theoretical predictions for muscle
dynamics of legs and wings in running and flying specialists. Predictions are
made for the experimental modelling the primary and secondary function by
tuning the cycling muscle to the corresponding natural frequency and
controlling its efficiency.

7\. The conservative character of architecture and related mechanical
characteristics of striated muscles suggests general trends following from
mechanical and shape constraints. The trends dictated by primary functions
explain, in particular, why the muscles having larger fibre and sarcomere
lengths and suited to efficient eccentric work, tend toward higher optimum
contraction velocities, but show lower maximum stiffness and mechanical energy
cost.

8\. As an intriguing outcome of the analysis of maximal contraction muscle
velocities and frequencies, the maximum-speed steady locomotion is revealed to
be _controlled_ by non-linear elasticity of slow-fibre muscles generating
moderated force. This finding deserves further evaluation in finite muscle
element analysis studying top speeds of living and extant animals.

.

Acknowledgments

I thank Andrew A. Biewener and James H. Marden for careful reading of the
draft of this paper and helpful critical comments. Rudolf J. Schilder and
James H. Marden are appreciated for giving datapoints in Fig. 6. The financial
support by CNPq is also acknowledged.

.

Appendix A. List of abbreviations

.

PCSA - physiologic cross-sectional area

.

Mathematical signs and symbols

$=$ \- common equality sign

$\equiv$ \- identity sign implying ”by definition”

$\approx$ \- approximate equality sign

$\sim$ \- proportionality relation symbol omitting only numerical coefficients

$\cong$ \- here used as similarity sign supporting only physical dimension
units

$\propto$ \- here used as scaling rule symbol not supporting dimension units

.

Physical and geometrical notations

$\alpha_{m}$\- muscle-mass allometric index

$\varepsilon_{m}^{(opt)}$\- muscle strain in the optimum dynamic regime

$\eta_{m}$\- muscle geometry parameter

$\rho_{tiss}$\- tissue density

$\sigma_{tiss}^{(\max)}$\- peak tissue stress

$\Delta L$ \- length change

$\Delta F$ \- force change

$\Delta t_{m}$\- activation timing of muscle $m$

$A_{rm}$\- cross-sectional area of muscle $m$ in passive ($r=0$) and active
($r\neq 0$) states

$a$\- scaling exponent for cross-sectional area

$D$ \- diameter of ideal cylinder

$E_{rm}$ \- active-muscle elastic modulus establishing the dynamic regime
$r\neq 0$

$e$\- strain scaling exponent

$\Delta F_{elast}^{(\max)}=\Delta F_{m}^{(\max)}$\- maximum active elastic
force

$F_{prod}^{(fast)}$\- production force by fast muscle

$F_{motor}^{(conc)}$\- elastic force adapted to concentric work in motor
muscle

$F_{musc}^{(\max)}$\- maximum force exerted by muscle

$K_{m}$\- active muscle stiffness

$K_{s}$\- sarcomere/cellular stiffness

$L$\- length of an ideal cylinder

$L_{m}$ \- variable muscle length in non-specified dynamics

$L_{rm}$ \- dynamic muscle length in the regime $r$

$l$ \- length exponent

$m$ \- muscle in unspecified function

$M$ \- body mass of animals

$M_{m}$ \- muscle mass

$r$ \- numerical parameter indicating transient dynamic states via optimum-
velocity ($r=1$), moderate-velocity ($r=2$), and high-velocity ($r=3$) dynamic
regimes, distinct of passive muscle state ($r=0$).

$T_{rm}$\- period of cycling in the adapted regime $r$

$V_{rm}$\- muscle contraction velocity in the dynamic regime $r$

$W$ \- body weight

.

Appendix B. Scaling Muscle Functions

.

The _motor function_ is associated with the active force $F_{prod}^{(\max)}$
generated during muscle shortening at moderate contraction velocity at the
turning points $2$ in Figs. 1A, 1B, and 1C. In _fast-fibre_ muscles, the
corresponding _concentric force_

$F_{motor}^{(conc)}=F_{elast}^{(\max)}=\Delta F_{2m}^{(conc)}\sim
E_{2m}^{(fast)}A_{2m}^{3/2}L_{2m}^{-1}\text{ }\cong F_{prod}^{(fast)}$ (35)

is described by the known universal pattern of the maximal elastic forces [33]
equally applied to pure bending, pure torsion, or complex bending-torsion
loads subjected to long cylinder of length $L_{2m}$ and cross-sectional area
[26] $A_{2m}$. The exploration of Eq. (35) though Eqs. (8), (16), (24), and
(26) results in the fast-muscle-force constraint
$3a_{m}/2-l_{m}=1+\alpha_{m}$. It is remarkable that the case of slow-fibre
muscle, namely

$F_{motor}^{(conc)}=F_{elast}^{(\max)}=\Delta F_{2m}^{(conc)}\sim
E_{2m}^{(slow)}A_{2m}^{3/2}L_{2m}^{-1}\text{ }\cong F_{prod}^{(slow)}$ (36)

results in the slow-muscle-force constraint $3a_{m}/2-2l_{m}=a_{m}$, which is
exactly the same as fast muscle, in view of function-independent Eq. (10).
Therefore, any muscle tuned to the motor locomotor function should expose its
_dynamic structure_ scaled by

$a_{motor}^{(conc)}=\frac{4}{5}(1+\alpha_{motor})\text{ ,
}l_{motor}^{(conc)}=\frac{1}{5}(1+\alpha_{motor})\text{, }$ (37)

regardless of the fibre type content. This finding follows from both the
muscle force constraints solved with the help of the function-independent
muscle-shape constraint (10). Moreover, as shown in [26], the principal
component of the compressive stress $\sigma_{m}^{(conc)}$ specifying Eq. (17)
may be caused by the peak transverse-tensile _strains_

$\varepsilon_{motor}^{(conc)}=\frac{\Delta D_{m}^{(\max)}}{L_{m}}\varpropto
M_{m}^{e_{m}}\text{, with
}e_{m}=e_{motor}^{(conc)}=\frac{a_{m}}{2}-l_{m}\text{,}$ (38)

where $\Delta D_{m}^{(\max)}\thicksim D_{m}\backsim A_{m}^{1/2}$ is transverse
muscle deformation.

Likewise, the maximum elastic _eccentric force_

$F_{brake}^{(eccen)}=\Delta F_{2m}^{(eccen)}\sim
E_{2m}^{(eccen)}A_{2m}^{2}L_{2m}^{-2}\cong F_{prod}^{(fast)}\text{,}$ (39)

associated with the _brake muscle function_ (Fig. 1B) provides the maximum
elastic stress

$\sigma_{brake}^{(\max)}=\frac{F_{brake}^{(eccen)}}{A_{m}}\varpropto
M_{m}^{s_{m}}\text{, with }s_{m}=s_{brake}^{(eccen)}=a_{m}-2l_{m}\text{,}$
(40)

following from Eqs. (17) and (39). The unique solution to both fast-muscle-
force constraint, $2a_{m}-2l_{m}=1+\alpha_{m}$, and slow-muscle-force
constraint, $2a_{m}-3l_{m}=a_{m}$, is

$a_{brake}^{(eccen)}=\frac{3}{4}(1+\alpha_{brake})\text{ ,
}l_{brake}^{(eccen)}=\text{
}s_{brake}^{(eccen)}=\frac{1}{4}(1+\alpha_{brake})\text{.}$ (41)

The _strut muscle function_ treated as antagonistic to both motor and brake
functions drives nearly isometric contractions characteristic of small, but
non-zero length change ($\Delta L_{m}\ll L_{m}$) achieved near peak forces
(see Fig. 1C). This suggests the nearly _isometric force_

$F_{strut}^{(isom)}=E_{2m}^{(isom)}\varepsilon_{2m}^{(isom)}A_{2m}\cong
F_{prod}^{(fast)}\text{, with }\varepsilon_{2m}^{(isom)}=\Delta
L_{2m}^{(isom)}/L_{2m}\text{,}$ (42)

in fast muscles. Again, one solves the _muscle strut_ constraints
$2a_{m}+l_{m}=1+\alpha_{m}$ and $2a_{m}+2l_{m}=a_{m}$ resulting in

$a_{strut}^{(isom)}=1+\alpha_{strut}\text{ and
}l_{strut}^{(isom)}=s_{strut}^{(isom)}=0\text{, with }\Delta
L_{2m}^{(isom)}\varpropto L_{2m}^{2}\text{,}$ (43)

for any type of muscles.

A new antagonist (to strut muscle) tuned to the _cardiac_ type contractions
via active elastic force

$F_{pump}^{(card)}=\Delta F_{2m}^{(card)}\thicksim
E_{2m}^{(card)}L_{2m}^{2}\cong F_{prod}^{(fast)}$ (44)

is associated with, say, _pump function_ providing the fast-muscle-force
constraint $2l_{m}=1+\alpha_{m}$. This yields

$a_{pump}^{(card)}=l_{pump}^{(card)}=\text{
}s_{pump}^{(card)}=e_{pump}^{(card)}=\frac{1}{2}(1+\alpha_{pump})\text{,}$
(45)

equally applied to slow-fibre muscles resulting in the slow-force constraint
$l_{m}=a_{m}$.

To complete the intrinsic-force description, the spring-type _control
function_ associated with the optimum-regime elastic force

$F_{contr}^{(sprin)}=F_{elast}^{(opt)}\varpropto
E_{1m}^{(slow)}M_{m}^{2/3}\varpropto
E_{1m}^{(slow)}A_{m}^{2/3}L_{m}^{2/3}\cong F_{prod}^{(slow)}$ (46)

in slow-fiber muscles results in

$a_{cont}^{(sprin)}=\frac{2}{3}(1+\alpha_{cont})\text{,
}l_{cont}^{(sprin)}=\frac{1}{3}(1+\alpha_{cont})\text{, with
}s_{cont}^{(sprin)}=e_{cont}^{(sprin)}=0\text{,}$ (47)

that follows from the slow-force and fast-force constraints
$2(a_{m}+l_{m})/3=a_{m}$ and $(2a_{m}+5l_{m})/3=1+\alpha_{m}$ and therefore is
valid for any type of muscle tuned to velocity-optimum regime. All obtained
specific-function mechanical characteristics are summarized in Table 2.

.

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Figure Captions

.

Fig. 1. The qualitative analysis of the in vivo muscle force-length data. The
muscle _motor function_ is presented by gastrocnemius powering during
shortening in uphill running turkey (_inset A_ , adapted from [27]). The
lateral gastrocnemius and plantaris act as brake (_inset B_) and strut (_inset
C_) in hopping tammar wallabies [24]. The solid (and dashed) _arrows_ indicate
rasing (and decreasing) of the exerted force near its maximum magnitude
$F_{\max}$. The regions of the linear force-length domain are displayed by the
force change $\Delta F_{1m}$ and length change $\Delta L_{1m}$, both estimated
from $L_{1m}$, and the starting datapoint $F_{1m}$ of the force enhancement
achieved at the optimum contraction velocity $V_{opt}$. Similar to physical
pendulum, the resting length $L_{0m}$ is expected to be passed at near maximum
velocities $V_{\max}$ and lower forces $F_{3m}$. The origin of intrinsic
muscle forces (_inset D_): in both cases of the powering shortening (_motor_)
and lengthening (_brake_) muscles the resulted force $F_{\max}$ is a
superposition of the production force output $F_{prod}$ and reaction passive
$F_{pass}$ and active $F_{act}$ _elastic_ forces [see also text below Eq.
(16)].

.

Fig. 2. The indirect observation of the primary activity of mammalian
plantaris. The _solid symbol_ is the datapoint [22] presented in Table 5 and
the bars indicate experimental error. The _open symbols_ are theoretical
estimates for stable dynamic structures established for the motor, brake,
strut, or control functions described in Table 2, with
$\alpha_{m}=\alpha_{0m}^{(est)}$ taken from Table 5.

.

Fig. 3. The observation of the primary mechanical function in some isolated
individual muscles in mammals. The analysis and notations correspond to those
in Fig. 2. The experimental (and theoretical) data for gastrocnemius, DDF
(deep digital flexor), and CDE (common digital extensor) are shown,
respectively, by the closed (and open) inverted triangles, regular triangles,
and circles. All the data are taken from Table 5.

.

Fig. 4. The qualitative study of the in vivo data on the peak stress in
individual leg muscles of animals in strenuous activity. The symbols employed
above in Figs. 2 and 3 are extended by the open circles (triceps) for the data
on peak muscle stress taken from Table 1 in [25], with the exclusion of the
slow-mode data on cantering goat and trotting cat. The data [41] on the
activated isometric stress in isolated white rabbit tibialis are added. The
_dashed line_ shows the brake-functional stress indicated by the stress
scaling exponent $s=1/4$. The _solid lines_ are drawn by $115\cdot M^{1/5}$,
for the motor function, and by $215$ _kPa_ , for the strut and spring
functions. All coefficients are adjusted by eye.

.

Fig. 5. The analysis of the primary mechanical functions for leg muscles in
running and non-running birds. The measured (and estimated) data taken from
Table 6 (and Table 2) for gastrocnemius, femorotibialis, and digital flexors
are shown by the closed (and open) inverted triangles, circles, and regular
triangles, respectively. The semi-open triangles are the data by Bennett [23]
for non-running birds.

.

Fig. 6. The qualitative scaling of the basalar structure to muscle mass in
male dragonflies (Odonata and Anisoptera, listed in Fig. 5 in [45]. The
datapoints for muscle length $L_{0m}^{(\exp)}$ is a courtesy by the authors.
The estimated muscle cross-sectional area $A_{0m}^{(est)}$ is obtained on the
basis of Eq. (1) taken with $\rho_{musc}^{(\exp)}=$ $1060$ $\emph{kg/m}^{3}$
[46]. The _solid lines_ are $L_{motor}=0.052\cdot M_{m}^{1/5}$ and
$A_{motor}=0.018\cdot M_{m}^{4/5}$. The _dashed lines_ indicated by the
scaling exponents are drawn according to muscle specialization shown in Table
2. All pre-exponential coefficients are adjusted by eye.

Tables

.

Optimum muscle characteristics, (equations) | Fast fibres | Slow fibres | Mixed fibres
---|---|---|---
Optimum length change, $\Delta L_{1m}$, (25) | $L_{m}$ | $L_{m}$ | $L_{m}$
Production/active-elastic force, $\Delta F_{1m}$, (16), (18), (25) | $A_{m}L_{m}$ | $A_{m}$ | $A_{m}L_{m}^{1/2}$
Optimum stiffness, $K_{1m}=E_{1m}A_{1m}/L_{1m}$, (19) | $A_{m}$ | $A_{m}L_{m}^{-1}$ | $A_{m}L_{m}^{-1/2}$
Optimum elastic stress, $\sigma_{1m}=\Delta F_{1m}/A_{1m}$, (17) | $L_{m}$ | $L_{m}^{0}$ | $L_{m}^{1/2}$
Contraction frequency, $T_{1m}^{-1}\thicksim\sqrt{E_{1m}/\rho_{0m}}/L_{1m}$, (15) | $L_{m}^{-1/2}$ | $L_{m}^{-1}$ | $L_{m}^{-3/4}$
Optimum velocity, $V_{1m}=V_{musc}^{(opt)}$, (23) | $L_{m}^{1/2}$ | $L_{m}^{0}$ | $L_{m}^{1/4}$
Optimum power, $P_{1m}=F_{1m}V_{1m}$ | $A_{m}L_{m}^{3/2}$ | $A_{m}$ | $A_{m}L_{m}^{3/4}$

Table 1. General mechanical characteristics of the striated muscles tuned to
linear-displacement dynamic regime scaled to dynamic fiber length
$L_{m}=L_{1m}$. The _mixed-fibre_ scaling dynamic exponents (shown in the last
column) are modeled by the common means for the fast-muscle and slow-muscle
exponents (established in the second and third columns), i.e.
$F_{mix}\backsim\sqrt{F_{fast}F_{slow}}$; $A_{m}$ and $L_{m}$ are attributed
to the stabilized _dynamic_ muscle geometry constrained by muscle volume (9).

.

| Locomotor pattern, regime
---
(equation)
| Motor, r=2
---
(37)
| Brake, r=2
---
(41)
| Strut, r=2
---
(43)
| Control, r=1
---
(47)
| Pump, r=2
---
(45)
| Force pattern, muscle
---
(equation)
|  $F_{{\small motor}}^{{\small(conc)}}$, m=1
---
(35)
|  $F_{{\small brake}}^{{\small(eccen)}}$, m=2
---
(39)
|  $F_{{\small strut}}^{{\small(isom)}}$, m=3
---
(42)
|  $F_{{\small contr}}^{{\small(sprin)}}$, m=4
---
(46)
|  $F_{{\small pump}}^{{\small(card)}}$, m=5
---
(44)
Maximum force output, (24) | $1+\alpha_{1}$ | $1+\alpha_{2}$ | $1+\alpha_{3}$ | $\frac{2}{3}(1+\alpha_{4})$ | $1+\alpha_{5}$
Muscle fibre length, (8) | $\frac{1}{5}(1+\alpha_{1})$ | $\frac{1}{4}(1+\alpha_{2})$ | $0$ | $\frac{1}{3}(1+\alpha_{4})$ | $\frac{1}{2}(1+\alpha_{5})$
Cross-sectional area, (8) | $\frac{4}{5}(1+\alpha_{1})$ | $\frac{3}{4}(1+\alpha_{2})$ | $1+\alpha_{3}$ | $\frac{2}{3}(1+\alpha_{4})$ | $\frac{1}{2}(1+\alpha_{5})$
Structure parameter, $\eta_{m}$=$a_{m}l_{m}^{-1}$ | $4$ | $3$ | $\ \infty$ | $2$ | $1$
Length change*, (13) | $\frac{2}{5}(1+\alpha_{1})$ | $\frac{1}{2}(1+\alpha_{2})$ | $0$ | $\frac{1}{3}(1+\alpha_{4})$ | $1+\alpha_{5}$
Maximum stress/strain*, (12) | $\frac{1}{5}(1+\alpha_{1})$ | $\frac{1}{4}(1+\alpha_{2})$ | $0$ | $0$ | $\frac{1}{2}(1+\alpha_{5})$
Maximum stiffness*, (19) | $\frac{3}{5}(1+\alpha_{1})$ | $\frac{1}{2}(1+\alpha_{2})$ | $1+\alpha_{3}$ | $\frac{1}{3}(1+\alpha_{4})$ | $0$
Natural frequency*, (15) | $-\frac{1}{5}(1+\alpha_{1})$ | $-\frac{1}{4}(1+\alpha_{2})$ | $0$ | $-\frac{1}{3}(1+\alpha_{4})$ | $-\frac{1}{2}(1+\alpha_{5})$
Energy change*, (21) | $\frac{7}{5}(1+\alpha_{1})$ | $\frac{3}{2}(1+\alpha_{2})$ | $1+\alpha_{3}$ | $1+\alpha_{4}$ | $2(1+\alpha_{5})$
Moderate velocity* (23) | $0$ | $0$ | $0$ | $0$ | $0$

Table 2. The locomotor functions and their mechanical characteristics scaled
to dynamic structures. The all-type powering individual muscles $m=1,2,3,$ and
$5$ are tuned to the maximum-force bilinear dynamic regime $r=2$ [described in
Eq. (26)] and muscles $m=4$ act in the linear regime $r=1$ [Eq. (25)]. *The
data shown for fast muscles. The allometric exponents are related to animal’s
body mass via Eq. (8).

.

Dyn. regimes | Force | Funct. | Structure | Funct. | Structure | Funct. | Structure | Funct. | Structure
---|---|---|---|---|---|---|---|---|---
$r=1,2,3$ |  ${\small F}_{{\small prod}}^{{\small\max}}$ ${\small\varpropto}$ | ${\small F}_{{\small motor}}^{{\small conc}}{\small\varpropto}$ |  | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small F}_{{\small brake}}^{{\small eccen}}{\small\varpropto}$ |  | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small F}_{{\small strut}}^{{\small isom}}{\small\varpropto}$ |  | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small F}_{{\small plun}}^{{\small card}}{\small\varpropto}$ |  | ${\small a}_{m}$ | ${\small l}_{m}$
---|---
${\small E}_{1m}^{(slow)}{\small\varpropto L}_{m}^{{\small 0}}$ | ${\small A}_{m}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-}1}$ |  | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-2}$ |  | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}$ |  | ${\small nc}$ | ${\small nc}$
---|---
${\small L}_{m}^{2}$ |  | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small E}_{1m}^{(fast)}{\small\varpropto L}_{m}$ | ${\small A}_{{\small m}}{\small L}_{{\small m}}$ | ${\small A}_{m}^{\frac{3}{2}}$ |  | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-1}$ |  | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small A}_{m}{\small L}_{m}$ |  | ${\small nc}$ | ${\small nc}$
---|---
${\small L}_{m}^{3}$ |  | $\frac{{\small 2}}{3}$ | $\frac{{\small 1}}{3}$
---|---
${\small E}_{2m}^{(slow)}{\small\varpropto L}_{{\small m}}^{{\small-1}}$ | ${\small A}_{m}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-2}}$ |  | $\frac{\mathbf{4}}{\mathbf{5}}$ | $\frac{\mathbf{1}}{\mathbf{5}}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-3}$ |  | $\frac{\mathbf{3}}{\mathbf{4}}$ | $\frac{\mathbf{1}}{\mathbf{4}}$
---|---
${\small L}_{m}^{-1}{\small A}_{m}$ |  | $\mathbf{1}$ | $\mathbf{0}$
---|---
${\small L}_{m}^{1}$ |  | $\frac{\mathbf{1}}{\mathbf{2}}$ | $\frac{\mathbf{1}}{\mathbf{2}}$
---|---
${\small E}_{2m}^{(fast)}{\small\varpropto}L_{m}^{0}$ | ${\small A}_{{\small m}}{\small L}_{{\small m}}$ | ${\small A}_{{\small 0}}^{\frac{3}{2}}{\small L}_{{\small 0}}^{{\small-1}}$ |  | $\frac{\mathbf{4}}{\mathbf{5}}$ | $\frac{\mathbf{1}}{\mathbf{5}}$
---|---
${\small A}_{0}^{2}{\small L}_{0}^{-2}$ |  | $\frac{\mathbf{3}}{\mathbf{4}}$ | $\frac{\mathbf{1}}{\mathbf{4}}$
---|---
${\small A}_{0}$ |  | $\mathbf{1}$ | $\mathbf{0}$
---|---
${\small L}_{0}^{2}$ |  | $\frac{\mathbf{1}}{\mathbf{2}}$ | $\frac{\mathbf{1}}{\mathbf{2}}$
---|---
${\small E}_{3m}^{(slow)}{\small\varpropto L}_{m}^{{\small-2}}$ | ${\small A}_{m}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-3}}$ |  | $\frac{{\small 6}}{7}$ | $\frac{{\small 1}}{7}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-4}$ |  | $\frac{{\small 4}}{5}$ | $\frac{{\small 1}}{5}$
---|---
${\small L}_{m}^{-2}{\small A}_{m}$ |  | ${\small 1}$ | ${\small 0}$
---|---
${\small L}_{m}^{0}$ |  | ${\small 0}$ | ${\small 1}$
---|---
${\small E}_{3m}^{(fast)}{\small\varpropto L}_{m}^{{\small-1}}$ | ${\small A}_{{\small m}}{\small L}_{{\small m}}$ | ${\small A}_{m}^{\frac{3}{2}}{\small L}_{{\small m}}^{{\small-2}}$ |  | $\frac{{\small 6}}{7}$ | $\frac{{\small 1}}{7}$
---|---
${\small A}_{m}^{2}{\small L}_{m}^{-3}$ |  | $\frac{{\small 4}}{5}$ | $\frac{{\small 1}}{5}$
---|---
${\small A}_{m}{\small L}_{m}^{-1}$ |  | ${\small 1}$ | ${\small 0}$
---|---
${\small L}_{m}^{1}$ |  | ${\small 0}$ | ${\small 1}$
---|---

Table 3. Locomotor functions predicted by dynamic structured for slow and fast
striated muscles tuned to distinct dynamic regimes. The primary functions
($r=2$) are shown by bold type. The analysis of functional muscle structures
made in terms of elastic-force patterns: the active-muscle optimum-velocity
($r=1$), moderate-velocity ($r=2$), and high-velocity ($r=3$) dynamic regimes
are described in the first column via the muscle elastic moduli $E_{rm}$ [Eqs.
(25), (26), and (27)] and specified by slow and fast force output [Eq. (24)],
shown in the second column. The third and next odd columns show the elastic
force functional scaling in concentric, eccentric, isometric, and pump
contractions. The corresponding solutions to scaling equations underlaid by
the force similarity principle (16) are shown for simplicity with
$\alpha_{rm}=0$, in the forth and next even columns. _Notation:_ $nc$
indicates non-conclusive solution.

.

Dynamic regimes | Optimum, | $r=1$ | Moderate, | $r=2$ | Maximum, | $r=3$
---|---|---|---|---|---|---
Muscle type | slow | fast | slow | fast | slow | fast
Natural frequency, Eq. (15) | $L_{m}^{-1}$ | $L_{m}^{-1/2}$ | $L_{m}^{-3/2}$ | $L_{m}^{-1}$ | $L_{m}^{-2}$ | $L_{m}^{-3/2}$
Contraction velocity, Eq. (23) | $L_{m}^{0}$ | $L_{m}^{1/2}$ | $L_{m}^{-1/2}$ | $L_{m}^{0}$ | $L_{m}^{-1}$ | $L_{m}^{-1/2}$

Table 4. Dynamic characterization of the red (slow) and white (fast) striated
muscles in the optimum-, moderate-, and maximum-velocity dynamic regimes
$r=1,2$, and $3$ described in Table 3.

.

Individual mammalian muscles | $a_{0m}^{(\exp)}$ | $l_{0m}^{(\exp)}$ | ${\small\alpha}_{0m}^{(\exp)}$ | ${\small\eta}_{{\small 0}m}$ | ${\small\alpha}_{0m}^{(est)}$ | $\mathit{a}_{m}$ | $\mathit{l}_{m}$ | Prim. functions
---|---|---|---|---|---|---|---|---
Gastrocnemius (and soleus) | ${\small 0.77\pm}.{\small 02}$ | ${\small 0.21\pm.02}$ | -${\small 0.03}$ | ${\small 3.7}$ | -${\small 0.02}$ | ${\small 0.78}$ | ${\small 0.20}$ | motor, ${\small m=1}$
Deep digital flexor (DDF)∗) | ${\small 0.85\pm.03}$ | ${\small 0.18\pm.02}$ | ${\small\ 0.03}$ | ${\small 4.7}$ | ${\small\ 0.03}$ | ${\small 0.82}$ | ${\small 0.21}$ | motor, ${\small m=1}$
Comm. digit. extensor (CDE) | ${\small 0.69\pm.04}$ | ${\small 0.24\pm.02}$ | -${\small 0.07}$ | ${\small 2.9}$ | -${\small 0.07}$ | ${\small 0.70}$ | ${\small 0.23}$ | brake, ${\small m=2}$
Plantaris (SDF) | ${\small 0.91\pm.04}$ | ${\small 0.05\pm.04}$ | -${\small 0.03}$ | ${\small 18}$ | -${\small 0.04}$ | ${\small 0.96}$ | ${\small 0.00}$ | strut, ${\small m=3}$
Ankle-joint muscle group | ${\small 0.81\pm.03}$ | ${\small 0.17\pm.03}$ | -${\small 0.03}$ | ${\small 4.8}$ | -${\small 0.03}$ | ${\small 0.78}$ | ${\small 0.19}$ | motor, ${\small g=1}$

Table 5. The analysis of the allometric data by Pollock and Shadwick [22]
provided on the basis of Eq. (10) and Table 2. The shown statistical error is
approximated by the symmetrized $95\%$ confidence interval. The methodology of
the analysis is illustrated in Fig. 2. The primary functions found in Figs. 2
and 3 are described following Table 2, with $\alpha_{m}=\alpha_{0m}^{(est)}$.
The overall muscle group ($g=1$) is determined as the standard mean over all
muscles. ∗)DDF includes individual flexor hallucis and flexor digitorum
longus; SDF means superficial digital flexor.

.

Running birds | $\ a_{0m}^{(\exp)}$ | $\alpha_{0m}^{(\exp)}$ | $l_{0m}^{(est)}$ | $a_{0m}^{(\exp)}/l_{0m}^{(est)}$ | $a_{2m}$ | $l_{2m}$ | Primary function/force
---|---|---|---|---|---|---|---
Gastrocnemius | $0.81\pm 0.14$ | $\ 0.14$ | $0.33$ | $2.5$ | $0.85$ | $0.29$ | brake/eccentric
Digital flexors (DF) | $0.76\pm 0.22$ | $-0.03$ | $0.21$ | $3.6$ | $0.78$ | $0.19$ | motor/concentric
Femorotibialis | $0.80\pm 0.12$ | $-0.02$ | $0.18$ | $4.4$ | $0.78$ | $0.20$ | motor/concentric
Overall group | $0.79\pm 0.16$ | $\ 0.03$ | $0.24$ | $3.3$ | $0.77$ | $0.26$ | brake/eccentric

Table 6. The analysis of the allometric data by Maloiy et al. [42]. The shown
large error is due to relatively wide confidence limits. The mean exponents
$l_{0m}^{(est)}$ are estimated via Eq. (10). The overall muscle group is
determined as the standard mean over all muscles. The indicated primary
functions and active elastic forces are described by the evaluated dynamic-
structure exponents $a_{2m}$ and $l_{2m}$ found as most close to the
experimental resting-volume data on $a_{0m}^{(\exp)}$ and $l_{0m}^{(\exp)}$
and therefore assigned to regime $r=2$ (Table 2).