Download Supplementary Materials The model used in this article incorporates

Supplementary Materials
The model used in this article incorporates the mass distribution elements
proposed by Gü nther and colleagues [13] into a phenomenological Hill-type muscle
model. The symbols and parameters used by the model are listed in Table S1. To
distribute the muscle’s mass throughout its tissue, the total muscle mass M is
divided among N in-series point masses of equal mass m. Between each point mass
is a force-producing contractile segment that contains an activation-dependent
contractile element (CE) and a parallel elastic element (PEE). The number of
segments is equivalent to one less than the number of point masses. Initially, each
point mass is distributed evenly along the x-axis of this 1-dimensional model, such
that the initial segment length is equal to the initial muscle length divided by the
number of segments. The position of the first point mass is fixed throughout all
simulations, and the last point mass is attached to an external load Fe that was
varied for each simulation.
(a) Forces in each contractile segment
The normalized force of each contractile element 𝐹̂CE is the product of its
normalized activation state 𝑞̂, normalized active force-strain 𝐹̂a (𝜀s ), and normalized
force-strain-rate 𝐹̂ (𝜀̇s ) characteristics, where:
𝐹̂CE = 𝑞̂𝐹̂a (𝜀s )𝐹̂ (𝜀̇s )
(S1)
𝑟
𝐹̂a (𝜀s ) = 𝑒
(𝜀 +1)𝑤 −1
−| s
|
𝑠
(S2)
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𝑤 is the skewness, 𝑟 is the roundness, 𝑠 is the width of the active force-length
relationship, and 𝜀s is the segment strain [S1]. Additionally:
𝐹̂ (𝜀̇s ) =
𝜀̇
(1− s )
𝜀̇ 0
,
𝜀̇
(1+ ṡ )
for
𝜀̇s < 0
(S3a)
for
𝜀̇s ≥ 0
(S3b)
𝑘𝜀0
𝐹̂ (𝜀̇s ) = 1.5 − 0.5
𝜀̇
(1+ s )
𝜀̇ 0
7.56𝜀̇ s
(1−
)
𝑘𝜀̇ 0
,
where 𝜀̇s is the strain-rate of the segment, 𝜀̇0 is the maximum unloaded strain-rate of
the segment, and 𝑘 is the curvature of the force-strain-rate relationship [S2]. Both 𝜀̇0
and 𝑘 vary depending on the fibre-type composition of the muscle.
The normalized force of each parallel elastic element 𝐹̂PEE is due only to its
normalized passive force-strain 𝐹̂𝑝 (𝜀s ) behaviour, where:
𝐹̂PEE = 𝐹̂𝑝 (𝜀s ) = 𝑒 (𝑐1 +𝑐2(𝜀s +1))
(S4)
both c1 and c2 are constants derived from experimental data [S3]. The values of the
constants used in equations S3-S4 can be found in Table S1.
The force of each contractile segment 𝐹s is the sum of the normalized forces
𝐹̂CE and 𝐹̂PEE , scaled by the maximum isometric force of the muscle 𝐹0 :
𝐹s = 𝐹0 (𝐹̂CE + 𝐹̂PEE ).
(S5)
(b) Equations of motion for the point masses
In the current model, each point mass has position xn, and its acceleration is
dependent on the forces applied to it on both its left and right sides:
𝑥𝑛 = 0 ,
𝑑2
𝐹s,𝑛 − 𝐹s,𝑛−1 = 𝑚 ∙ 𝑑𝑡 2 (𝑥𝑛 ) ,
for
𝑛=1
(S6a)
for
1<𝑛<𝑁
(S6b)
2
𝑑2
𝐹e − 𝐹s,𝑛 = 𝑚 ∙ 𝑑𝑡 2 (𝑥𝑛 ) ,
for
𝑛=𝑁
(S6c)
where 𝑛 is the point mass number, 𝑁 is the total number of point masses, 𝑡 is time,
and Fe is the external force that is applied to the distal end of the muscle on the
right-hand point mass. The equations for whole system were determined by
substituting equations S1-S5 into equations S6a-S6c, and the set of equations was
solved for all point masses in the system.
(c) Model scaling
The base model had dimensions to represent the plantaris muscle in the rat
(optimal muscle length L0 of 34 mm, cross-sectional area 29 mm2, density of 1060 kg
m-3, and thus a mass of 1.05 g, maximum isometric stress 200 k Pa; [14]). This model
was geometrically scaled to larger sizes, with the linear dimensions being scaled by
a factor of 25, the forces by a factor of 252, and masses by a factor of 253. The largest
muscle had a mass of 16.33 kg, which reasonably approximates the mass of an
elephant leg muscle [15].
(d) Activation state
The normalized activation constant 𝑞̂, which varied between 0.2 and 1.0,
represents the relative activation of the contractile elements within the muscle. This
activation constant thus affects the balance between the contractile forces and the
inertial forces within the muscle. The contractile forces scale with the square of the
linear dimension, whereas the inertial forces scale with the cube of the linear
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dimensions. Thus, the effect of the inertial forces would be expected to dominate at
large muscle sizes, as well as at low activation levels.
(e) Fibre-type properties
Two distinct fibre-type variants were examined in the present model. When
the model is designated as slow or fast, it is assumed that the muscle being modelled
is composed of a homogeneous population of contractile elements. The maximum
unloaded strain-rate of the contractile elements 𝜀̇0 and the curvature of the forcestrain-rate relationship 𝑘 for the slow and fast muscle were designated as -5 and -10
s-1, and 0.18 and 0.29, respectively [S4]. Both 𝜀̇0 and 𝑘 are held constant throughout
each simulation.
(f) Comparison to experimental data
One of the purposes of this study was to test previous experimental findings
[12] that the activation-dependent differences in the maximum shortening velocity
of the rat plantaris could be explained by inertial effects, even when they resulted
from contractions with different fibre-types active. In this current study each model
was assumed to be of a single, homogeneous fibre-type: but the fibre-type
differences influence only the active forces, and not the passive forces from each
contractile segment. Thus this model can simulate situations [12] when a single
fibre-type is activated, provided the intrinsic properties of that fibre-type are
represented in the active force-strain-rate properties described in equations S3aS3b.
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The rat plantaris consists of many muscle fibres that act in parallel, and is
thus typical of mammalian muscle. Traditional Hill-type muscle models consider
whole muscles to act as if they were scaled-up muscle fibres [6-9]. We have recently
demonstrated that Hill-type models that have fast and slow contractile elements
acting in parallel can result in improved force predictions over traditional models
[S4-S5]. However, even in such formulations the length and thus the velocity of the
fast and slow contractile elements that are in parallel are assumed to be the same.
The current model represents the muscle as a scaled-up muscle fibre that is
subdivided into contractile segments and point masses that are arranged in series.
Each contractile segment can be considered to represent the action of a set of
myofilaments or even muscle fibres acting in parallel, provided that they all have the
same length that is set by the spacing between the adjacent point masses. Thus this
model can represent whole muscles, even when they contain parallel fibres within
the context of this simplified Hill-type model representation. However,
inhomogeneity in fibre strains that may exist between parallel elements would
require a more detailed modelling approach, such as using the finite-element
method [e.g. S6-S10], however such models have not accounted for the inertial mass
distribution to date.
(g) Model output measurements
The forces acting on and thus the accelerations of the point masses vary
along the muscle. During contraction the right-hand masses (Fig. 1) experience
greater accelerations and faster speeds than masses on the left, and this is
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highlighted in Fig. S1. The overall muscle length L was the sum of all the segment
lengths (Table S1), and the maximum velocity of the whole muscle was determined
for each contraction. This velocity was expressed in terms of a strain-rate for the
whole muscle 𝜀̇m to aid in the comparison between different muscle scales. The
maximum strain-rate for the muscle was determined for each muscle size, activation
level and fibre-type and this maximum strain-rate was calculated for: (I) the first set
of simulations that started with the segments at their optimal lengths to give 𝜀̇max,I ;
and (II) for the second set of simulations that started with stretched segments and
that achieved maximum velocities when the whole muscle was at its optimal length
to give 𝜀̇max,II .
(h) Neglected forces and model assumptions
In order to isolate the effects of the inertial mass on the contractile
performance of whole muscles possible internal forces within the muscle, other than
those described in equations S1-S6c, have been neglected. Similar to previous
approaches [9], this model has treated the whole muscle as a scaled-up muscle fibre
and as such ignored the effects of inhomogeneity in tissue strains and material
properties that may occur in parallel across the muscle, and also the effect of muscle
architecture. Thus forces that may occur between parallel fibres such as drag or
friction due to movement of fibres relative to surrounding tissue, changes in fibre
pennation angle, heterogeneous fibre-type populations and compartmentalization of
activities are not considered. Series elasticity, such as that arising from the tendon,
and the effects of time-varying activation are also ignored. Nonetheless, this simple
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representation of the muscle as a series of point masses separating the contractile
segments is sufficient to capture the salient features of sub-maximal activation of
different fibre-types in a whole muscle [12], and predicts a substantial effect of
muscle size on the contraction speed of the muscle.
Supplementary Material References
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the locomotor function of rat ankle extensor muscles. J. Exp. Biol. 213, 318 –
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analysis tuned for determining the timing and level of activation in different
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Table S1. Model parameters.
Parameter Definition
Description
N
Total number of point masses
n
Point mass number
M
Muscle mass
m
=
𝑀
𝑁
Mass of each point mass
x
Position
t
Time
ls
Segment length
l0
Optimal segment length
𝜀s
𝜀̇s
=
𝑙s − 𝑙0
𝑙0
Segment strain
=
𝑑
(𝜀 )
𝑑𝑡 s
Strain-rate of segment
𝜀̇0
Maximum unloaded strain-rate of contractile element
L
Muscle length
L0
𝜀̇m
= (𝑁 − 1)𝑙0
=
𝑑 𝐿 − 𝐿0
(
)
𝑑𝑡
𝐿0
Optimal muscle length
Strain-rate of muscle
𝜀̇max,I
Maximum strain-rate of muscle for simulations I
𝜀̇max,II
Maximum strain-rate of muscle for simulations II
𝐹̂CE
Normalized, active force of contractile element within
segment
𝐹̂PEE
Normalized, passive force from parallel elastic element
within segment
Fs,n
Force of segment n
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F0
Maximum isometric force of segment
Fe
External force applied to point mass n=N
𝑞̂
Normalized activation (0-1)
w
= 0.6
Skewness of active force-strain relationship
r
= 2.3
Roundness of active force-strain relationship
s
= 0.3
Width of active force-strain relationship
k
Curvature of force-strain rate relationship
c1
= -13.816
Constant of passive force-strain relationship
c2
= 9.210
Constant of passive force-strain relationship
Figure S1. Displacement (A), velocity (B) and acceleration (C) of point masses as a
function of time within a single model for N=16. The simulation shown is for a fast
muscle with total mass of 1.05 g contracting at 20% activation against an isotonic
external force of 0.01 Fe. The motion of the last point mass n=N is displayed as a
dashed line.
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