# Body Size and Scaling

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06-Feb-2016Category

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Body Size and ScalingSize Matters in Physiology

Living Organisms Come in a Huge Range of Sizes!Pleuropneumonia-like organisms (Mycoplasma) 0.1 pg (10-13 g)Rotifers0.01 g (10-8 g)Blue Whale (Balaenoptera musculus)10,000 kg (108 g)Giant Redwood Trees (Sequoia spp.) = even bigger!Living organisms range 1021+ in size Animals range 1016 in size

Size Profoundly Influences PhysiologyGravityCirculationMovement and LocomotionSurface Area/Volume RatioRespirationDigestionWater BalanceThermoregulation

ScalingChanges in body proportion often accompany changes in body sizeboth ontogenic and phylogenice.g. changes in proportions from human fetus to adult

Allometry1. The study of differential growth 2. The study of biological scaling

Allometric EquationY = aXb (a power function)

Y = body part being measured in relationship to the size of the organismX = measure of size used for basis of comparison usually a measure of whole body sizea = initial growth index size of Y when X = 1b = scaling exponent proportional change in Y per unit X

The Scaling Exponent (b) Defines the Type of Scaling RelationshipIf b = 1, there is no differential growththe relative size of Y to X is the same at all values of Xisometry (geometric similarity)

The Scaling Exponent (b) Defines the Type of Scaling RelationshipIf b < 1, Y increases at a slower rate than Xas X increases, Y becomes relatively smallernegative allometryHead Length (cm)

The Scaling Exponent (b) Defines the Type of Scaling RelationshipIf b > 1, Y increases at a faster rate than Xas X increases, Y becomes relatively largerpositive allometry

AllometryAllometric Data Can Also Be Expressed as Linear Functions of Log-Transformed Data

Y = aXblog Y = log a + b log X

the slope of the line (b) indicates the type of scaling relationship

Types of Scaling RelationshipsIf b = 1, isometry (geometric similarity)If b < 1, negative allometryIf b > 1, positive allometry

The Catch: Above is true only when we compare like dimensions (e.g. length to length, mass to mass).

Isometry for Different DimensionsExample: Head Length vs. Body LengthLinear dimension (m1) vs. linear dimension (m1)Isometry: m1/m1, b = 1/1 = 1.0Example: Head Length vs. Body MassLinear Dimension (m1) vs. Cubic Dimension (m3)Isometry: m1/m3, b = 1/3 = 0.33Example: Surface Area vs. Body MassSquare Dimension (m2) vs. Cubic Dimension (m3)Isometry: m2/m3, b = 2/3 = 0.67

Differential Scaling is Common

1 cm1 cm10 cm10 cm10 cmLimb cross-sectional area = 1 cm X 1 cm = 1 cm2Volume = 10 cm X 10 cm X 10 cm = 1000 cm3Estimated mass = 1000 gWeight loading on limb = 1000g / 1 cm2 Hypothetical one-legged cuboid animalExample: Support of Weight by the Limbs

10 cm10 cm100 cm100 cm100 cmLimb cross-sectional area = 10 cm X 10 cm = 100 cm2Volume = 100 cm X 100 cm X 100 cm = 1,000,000 cm3Estimated mass = 1,000,000 gWeight loading on limb = 1,000,000g / 100 cm2 = 10,000 g/cm2Increase all linear dimensions tenfoldExample: Support of Weight by the Limbs

33.3 cm33.3 cm100 cm100 cm100 cmLimb cross-sectional area = 33.3 cm X 33.3 cm = 1000 cm2Volume = 100 cm X 100 cm X 100 cm = 1,000,000 cm3Estimated mass = 1,000,000 gWeight loading on limb = 1,000,000g / 1000 cm2 = 1000 g/cm2Need a proportionately thicker limb bone to support the increased massExample: Support of Weight by the Limbs

Scaling of Skeleton MassExpect b = 1 for isometryHowever, because of increased weight loading, may expect b > 1Typical scaling: b = 1.12skeleton becomes relatively more massive with increased body sizecompensates for increased weight loading

Scaling of Skeleton MassScaling of 1.12 does not fully compensate for weight loading with increased mass:Length Mass1/3Skeletal Mass Cross-Sectional Area * LengthCross-Sectional Area Body MassSkeletal Mass Body Mass * LengthSkeletal Mass Body Mass * Body Mass1/3Skeletal Mass4/3Skeletal Mass1.33 expected for isometry of weight loading

Scaling of Skeleton MassWhy not scale at 1.33?Mass of skeleton contributes to mass of animalincreased skeletal mass limits movement and ability of skeleton to absorb physical shocksthe thicker the skeleton, the greater the chance of fracture

Scaling of Skeleton MassHow can animals increase in size without becoming all skeleton?Animals accept lower safety factorgenerally, skeleton is about 10x as strong as needed to support the animals weightDecreased locomotor performanceAlter morphology to reduce stress on the skeletonAlter chemical composition of skeletone.g. human limb bones - relatively more slender in adults

How Does Differential Scaling Arise?Differences in size among animals are due primarily to differences in cell numberDuring embryonic development, cells differentiate and give rise to germinal centerseach part of an organism arises from one or more germinal centers

How Does Differential Scaling Arise?Rate at which a part grows depends on number of germinal centers and rate of cell divisionFor Y = aXba number of germinal centers contributing to a particular body regionb ratio of the frequencies of cell division between Y and X