Download Ontogenetic scaling of burrowing forces in the earthworm Lumbricus

Ontogenetic scaling
of burrowing forces in
the earthworm
Lumbricus terrestris
Kim Quillin
J Exp Biol 203, 2757-2770 (2000)
Anatomy of the earthworm
Earthworm Locomotion
FR
FA
FR
Apparatus for measuring burrowing forces
Open burrow setup
Dead-end burrow
setup
Apparatus controls
(1) Burrow diameter
(2) Soil Properties
Measured
force
traces
Scaling Laws
L
L2
L3
leg length
segment length
height
surface area
cross-sectional area
muscle force
volume
mass
weight
mb1/3
mb2/3
mb1
Scaling Laws
y = some Parameter
y ∝ mbb
what is b?
Isometric Scaling
if y ∝ L, measure b = 1/3
if y ∝ L2, measure b = 2/3
if y ∝ L3, measure b = 1
Allometric Scaling
if the condition for isometry is not met
(i.e. any other value for b)
Scaling Laws
y = some Parameter
y ∝ mbb
Parameter (y)
what is b?
y = ambb
log y
Body Mass (mb)
[log y] = b [log mb] + C
log mb
Scaling Laws
y = some Parameter
y ∝ mbb
what is b?
b is the slope of the log-log plot
b=1
log y
b = 2/3
b = 1/3
log mb
Scaling Laws
y = Burrowing Force
what is b?
b = 2/3
log
Force
y ∝ mbb
b = 1/3
log mb
If wall thickness (t), max stess (σC) constant:
σC t
1
Δp = r ∝ r
If muscle properties
constant during
development:
F ∝ muscle cross
sectional area
F ∝ mb0.67
b = 2/3
F ∝ L ∝ mb0.33
Force
∝ Area
b = 1/3
Laplace’s Law:
σC
σL
=
=
Δp r
σC t
Δp = r ∝ C
t
1
2
If wall thickness scales with length:
σC
F ∝ L2 ∝ mb0.67
Force
∝ Area
b = 2/3
Scaling of
burrowing force
H0: b = 0.67
Actual: b ≈ 0.5
- Radial burrow-enlarging forces >> radial anchoring
forces
- Axial and radial enlarging forces about same magnitude
Scaling of
burrowing force
Force
Weight
∝
L2
L3
∝
L-1
a
∝
mb-1/3
b
H 0:
b = -0.33
H 0:
b=0
Pressure
Force
Area
=
∝
Force
Area*
L2
L2
∝
C
∝
m b0
*Area of plane of the force transducer ⊥ to the force
Scaling of burrowing force
Burrowing force does not scale isometrically:
Small worms can push 500 times body weight,
large worms can only push 10 times body weight
Hypotheses for cause of relatively weak large worms:
1) Muscle area might not increase isometrically with body size
2) Muscle stress might not be constant across body sizes
3) Mechanical advantage of segments might change with body
size
4) Burrowing kinematics different for small & large worms
5) Soil deformation resistance might depend on scale of
deformation
1) Muscle Area
H 0:
F ∝ CSA of
muscle
CSA ∝ mb0.67
HA:
CSA ∝ mbb
b < 0.67
Actual:
b > 0.67
2) Muscle Stress
σmuscle
=
Force
Area*
*Area of muscle crosssection
Force
Area
∝
L2
L2
∝
C
∝
mb0
H0: σmuscle constant across all body sizes
HA: σmuscle less in large worms
untested
3) Mechanical Advantage
Quillin (1998)
MA
=
A
B
=
L = 102 mb0.34
d15 = 5.3 mb0.34
d50 = 4.2 mb0.32
b
a
MA
Length
∝ diameter
Length and diameter both scale
isometrically → no expected change in
MA
4) Burrowing Kinematics
larger worms→fewer strides per second
larger worms→more elongated
during crawling
HA: Muscles of larger worms working at higher
strains→produce less force
untested
5) Soil Properties
untested
Scaling of burrowing force
Burrowing force does not scale isometrically:
Small worms can push 500 times body weight,
large worms can only push 10 times body weight
Hypotheses for cause of relatively weak large worms:
1) Muscle area might not increase isometrically with body size
2) Muscle stress might not be constant across body sizes
TEST: σmuscle ↓ during development
3) Mechanical advantage of segments might change with body size
4) Burrowing kinematics different for small & large worms
TEST: larger earthworm muscles working at larger strains
5) Soil deformation resistance might depend on scale of deformation
Maximum forces in earthworms
compared with other animals
point here is that lever-like
systems can’t scale with
BOTH geometric and static
stress similarity -- but
hydrostatic skeletons can;
worms grow isometrically, so
these and dynamic stresses
scale.
3) Mechanical Advantage
Quillin (1998)
L = 102 mb0.34
d15 = 5.3 mb0.34
d50 = 4.2 mb0.32
MA
=
A
B
=
b
a
MA
Length
∝ diameter
Length and diameter both scale
isometrically → no expected change in
MA