Metabolic Scaling Theory

Assymetric Scaling 

Brummer et al. 2017 PloS Bio A general model for metabolic scaling in self-similary asymetric networks. 

Why are metabolic rates important?

Brose et al. 2006 - Allometric sacling enhances stability in complex food webs, Ecol. Letters

Harte 2011 Maximum Entropy and Ecology

Michaletz et al. 2014 Convergence of terrestrial plants.

Herman et al. 2011

Vascular Morphology

Tekin et al. PloS Comp Bio 2016

Hosoi et al. 2013

Leave's have loops in them which make them hard to model.

History of vascular models

~1920s C. Murray optimizes resistance to constant flow vs maintenance

~1950's Wormersley develops model for pulsing flow in elastic tube

~1970's M. Zamir improves Murray's approach elaborates on maintenance

Other side of model

~1960's B. Mandelbrot formalizes fractals (self-similarity with scale)

~15th Century L. DaVinci notes cross sectional preservation of tree branching

~1980's - early 1990's Observations of fractal scaling reportedi n many fields of science.

1990s WBE merge pulsing constant flow with fractals match 3/4 - assume symmetry

2200's to present WBE framework extends ecosystems cities medicine (syymetric assumption)

Today

motivate asymmetric branching

Discuss theoretical branching

Make some predictions

Why model asymmetric branching?

histogram of mouse lung vs pinyon branching

Lambda_length = lenght_sm_child | length_lrg_child

Lambda_radius = r_sm_cihld | r_lrg_child

Symmetric WBE Model (1997)

• Resource distribution network -> scaling laws /allometries

• symmetric within generations

• space filling fractals

• minimization of energy loss

• terminaal units are invariant

• all resource transfer occurs at terminal units

Space-filling Fractal Trees (Peano Curve)

Space-filling a d-dimensional space (euclidean) L^D parent = l^D child1 + l^D child2

for 3-d space filling l^3 = l^3 child1 l^3 + child 2

l^2 = l^2 

l^1.43 = l^1.43 + l^1.43 

Maximizing Efficiency

network tip

constant flow regime | minimize resistance constrianed by: volume, space-filling, and mass

hagen-poiseuille

resistance

Z approx 1/r^4

Netowrk trunk

pulsing flow regime | impedance matching (no pulse reflections)

resistance

Z_j approx 1/r^2

Model Predictions

Aorta | Capillaries

Pulsatile flow regime | constant laminar flow regime

positive asymmetry = child branches are longer than wide

r^2 = r^2 child1 + r^2 child2 | r^3 = r^3 child1 + r^3 child2

l^3 parent = l^3 child2 +l^3 child2 | l^3 parent = l^3 child2 +l^3 child2

negative asymmetry = child branches shorter and wider

Asymmetric Coordinates

symmetric | positive asymmetric | negative asymmertric

scale factors | radius | length

beta = 1/2(r_c1+r_c2)/r_p | gama = 1/2(l_c1+l_c2)/l_p

delta beta = 1/2(r_c1-r_c2)/r_p | delta gama 1/2(l_c1+lc_2)/l_p

average and difference

Metabolic scaling exponent. vs asyymmetry

B_tot = N_capillary * B_cap

V_total = big equation

N_cap ~= (v_tot/v_cap)

theta = (3/4s term)

Cool figure - Metabolic scaling exp vs Asymmetry (Pulsing flow network)

Length difference Scale Factor (delta gamma) vs Radial difference scale factor (delta beta)

the center of the graph overs around 0.75

Metabolic scaling exponent theta

Impose a fixed cut-off size for the asymmetric force

Digital trees of finite size, pulsing flow.

Total number of branch ends (length difference vs radial difference)

Total Netowrk resistance to flow normalized by symmetric value - network resistance decreases as asymmetry arizes

Digital trees of finitce size, constant flow.

Total number of branch ends (center around 0 by 0 with two centroids >0.05) branch number of ~ 4500

Total network resistance to flow logarithmic scale (3.25)

Slight amounts of asymmetric branching will give larger number of branches. 

Trees become self pruning as laminar flow becomes more commong.

Comparison study to vascularture of animals and plants

MRI of mouse lung and human head and torso cardiovascular system

Brummer etal. in prep.

In plants 3 species of gymnosperms and angiosperms, 50cm long terminal clippings, whole tree destructively measured (Bentley et al), Roots destructively measured

Convergence in vascular branching

both plants and animals have vascular branching (broken lineages ~1.6 billion years ago)

Indistinguishable distributions -> convergence in branching (same distributions)

Distingishuable distributions -> no convergence in branching (different distributions)

Comparing mammals versus plants grouped together

radial scale factors | counts of beta vs delta beta

length scale factors | counts of lambda vs delta lambda

mammals = consistent with area increasing scaling - speeds slow down 

plants = constant with eulerian (beam) buckling

Comparing human head and torso vs mouse lung vs plant

mammals and plants are distinguishable 

within mammals - mouse and human head and torso are indistinguishable

within plants - aged individuals distinguishable, roots and young individuals indistinguishable.

Transitions in branchign patterns

relative gneration (c) = 1 - Transition generation / total generations

yes transition = huma, balsa, pinyon

no transition = mouse lung, ponderosa, roots*, tips* | *not enough generations

transitions in branching pattenrs - huma head - transtiions near tips, shift from negative to positve or negative to symmetric

Symmetric WBE

Asymmetric WBE | 3/4s term 

Asymmetric WBE with transition

estimated Metabolic Scaling Exponent

Recalling Kleibers law

Conclusions

Asymmetric branching allows for greater variation in vascular form

can distinguish between species

Identifying common morphological patterns of these transitions

Improving vascular-based estimates of metabolic scaling

Looking forward

Expand these vascular datasets

Adapt theory for further branching structures

• greater than two child branches

• looping structures

Check vascular level estimates of metabolism against actual measurements