Ontogenetic scaling of burrowing forces in the earthworm Lumbricus terrestris Kim Quillin J Exp Biol 203, (2000)

Anatomy of the earthworm

FAFA FRFR FRFR Earthworm Locomotion

Open burrow setup Dead-end burrow setup Apparatus for measuring burrowing forces

Apparatus controls (1) Burrow diameter(2) Soil Properties

Measured force traces

Scaling Laws L L3L3 L2L2 leg length segment length height surface area cross-sectional area muscle force volume mass weight mb1mb1 m b 1/3 m b 2/3

Scaling Laws Isometric Scaling if y ∝ L, measure b = 1/3 if y ∝ L 2, measure b = 2/3 if y ∝ L 3, measure b = 1 Allometric Scaling if the condition for isometry is not met (i.e. any other value for b) y = some Parameter y ∝ m b b what is b?

Scaling Laws y = some Parameter y ∝ m b b what is b? Body Mass (m b ) Parameter (y) log m b log y [log y] = b [log m b ] + C y = am b b

Scaling Laws y = some Parameter y ∝ m b b what is b? b is the slope of the log-log plot log m b log y b = 2/3 b = 1 b = 1/3

Scaling Laws y = Burrowing Force y ∝ m b b what is b? log m b log Force b = 2/3 b = 1/3 Laplace’s Law: σCσC Δp r t = σLσL 1 2 = σCσC F ∝ m b 0.67 F ∝ muscle cross sectional area b = 2/3 If muscle properties constant during development: If wall thickness (t), max stess (σ C ) constant: Δp = ∝ σC tσC t r 1 r ∝ Force Area b = 1/3 F ∝ L ∝ m b 0.33 If wall thickness scales with length: Δp = ∝ σC tσC t r ∝ Force Area C F ∝ L 2 ∝ m b 0.67 b = 2/3

Actual: b ≈ Radial burrow-enlarging forces >> radial anchoring forces - Axial and radial enlarging forces about same magnitude Scaling of burrowing force Cmb0mb0 L2L2 L2L2 ∝∝ H 0 : b = 0.67

ab H 0 : b = H 0 : b = 0 Scaling of burrowing force L -1 m b -1/3 Force Weight L2L2 L3L3 ∝∝∝ Cmb0mb0 Force Area L2L2 L2L2 ∝∝∝ Pressure Force Area* = *Area of plane of the force transducer ⊥ to the force

Scaling of burrowing force 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 Small worms can push 500 times body weight, large worms can only push 10 times body weight Burrowing force does not scale isometrically:

1) Muscle Area F ∝ CSA of muscle CSA ∝ m b 0.67 H0:H0: CSA ∝ m b b HA:HA: b < 0.67 Actual: b > 0.67

2) Muscle Stress Cmb0mb0 Force Area L2L2 L2L2 ∝∝∝ σ muscle Force Area* = *Area of muscle cross- section H 0 : σ muscle constant across all body sizes H A : σ muscle less in large worms untested

3) Mechanical Advantage MA Length diameter ∝ MA A B = b a = L = 102 m b 0.34 d 15 = 5.3 m b 0.34 d 50 = 4.2 m b 0.32 Quillin (1998) 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 H A : Muscles of larger worms working at higher strains→produce less force untested

5) Soil Properties untested

Scaling of burrowing force 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 Small worms can push 500 times body weight, large worms can only push 10 times body weight Burrowing force does not scale isometrically:

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 MA Length diameter ∝ MA A B = b a = L = 102 m b 0.34 d 15 = 5.3 m b 0.34 d 50 = 4.2 m b 0.32 Quillin (1998) Length and diameter both scale isometrically → no expected change in MA