Power  Laws  in  Biology  

Stephanie  Forrest  Introduc9on  to  Scien9fic  Modeling  

CS  365,  2012  

On  Growth  and  Form  •  D’Arcy  Thompson  “On  Growth  and  Form”  (1917)  

–  Structuralism  vs.  Survival-­‐of-­‐the-­‐FiUest  –  Structuralism:  Physical  laws  govern  the  form  of  species,  in  addi9on  to  

evolu9on  –  AUempted  to  account  for  differences  in  the  forms  of  related  animals  

could  be  described  by  means  of  rela9vely  simple  mathema9cal  transforma9ons  

–  Style  was  descrip9ve  

Transforma9on  of  Argyropelecus  into  Sternoptyx  diaphana  by  applying  a  20°  shear  mapping  

Allometry:  The  study  of  rela9onships  between  body  size  and  shape  

Metabolic  Scaling  Theory  A  general  theory  for  the  origin  of  allometric  scaling  

laws  in  biology  (1997)  

In biology, bigger networks are slower when they are centralized"

Kleiber’s  Law    

Hemmingson 1960

3/4  

Observed Metabolic Scaling: ""B ∝ M3/4"

"B is the rate of energy (oxygen) use"B: the master biological rate governs"

"ecological interactions""food webs & ecosystem dynamics""growth and reproduction"

"Mass Specific Scaling"Other biological rates ∝ M-1/4"

Biological times ∝ M1/4"

“There is a unity of the single system of energy, ecology, and economics…

let us here seek common sense overview

which comes from overall energetics.”    

H.  T.  Odum  (1973)  

Organisms  have  evolved  networks  to  distribute  energy  efficiently  

Social  insects  use  networks  to  acquire  energy  and  communicate  

Global  Shipping  Routes  Halpern  et  al  Science  2008  

Human  engineered  networks  span  the  globe    

Metabolic  Scaling  Theory  

"Bigger organisms require bigger networks"•  Pipe lengths (L) are longer"•  Cross sectional areas (A) are larger"•  # of capillaries increases slower than pipe volume"

"N = cV3/4"

!Metabolism: B = cM3/4"""""Increasing volume (mass) 100 times increases delivery rate 30 times!""Diminishing returns: Network size grows faster than network delivery rate"""

Elements  of  the  Theory  Metabolic  Ecology:  A  Scaling  Approach  (2012)  

•  All  cells  need  nutrients  and  oxygen  –  Delivered  by  an  internal  space-­‐filling,  hierarchical  (fractal)  distribu9on  

network  –  This  assump9on  has  been  adjusted  in  later  versions  of  the  theory  

•  The  final  branch  of  the  network  (capillary)  is  constant  size,  independent  of  organism  size  (invariant  terminal  units)  –  This  assump9on  has  been  adjusted  in  later  versions  of  the  theory  

•  The  energy  required  to  distribute  resources  is  minimized  (network  design  is  op9mized)  

•  Network  is  area  preserving  at  every  level  of  hierarchy  

Network  scaling  concepts  

•  Network  volume  (Vnet)  increases  faster  than  number  of  capillaries  (Nc)        [1]  Vnet  ∝  Nc

4/3    

Diminishing  returns:  Network  volume  grows  faster  than  delivery  rate  A  network  100  9mes  bigger  delivers  only  30  9mes  more  blood  per  unit  9me    •  Each  capillary  is  the  same:  B  ∝  Nc    

 [2]  Vnet  ∝  B4/3    

 •  Biological  constraint,  blood  volume  is  a  constant  %  of  mass:  Vnet  ∝  M        

 [3]  B  ∝  M3/4    Controversy,  but  accepted  that  centralized  distribu9on  networks  generate    

Vnet  ∝  NcD+1/D  

       

Network scaling accurately predicts rates and times  

•  Physiology"•  Individual Growth"•  Population growth"•  Reproduction"•  Disease spread"•  Lifespan"•  Photosynthesis & carbon flux "•  …"

Biomass  Produc9on:                                  P∝M 3/4

Physiological  Rates:      B∝M −1/4

-1/4!

Scaling  9mes  associated  with  disease  

Time to first symptoms! Time to death!

1/4!

Metabolic  scaling  determines  growth  rates  

€

Emdmdt

= B0m3 / 4 − Bmm

Rate  energy  is  available  for  growth  =    incoming  metabolic  rate  -­‐  maintenance  metabolic  rate  

West et al 2001 Moses et al 2008

(m/M

)1/4  

T=(at/4M)1/4-­‐ln(1-­‐(m0/M)1/4)  

1-­‐e-­‐T  

Metabolic  scaling  during  growth:  B∝M 3/4

Life9me  reproduc9ve  effort:  %  of  mother’s  mass  invested  in  offspring  

Predicted  LRE  =  1/δ  =  4/3    (in  theory)      δ  =  0.7  in  prac>ce  

A female lizard lays 1.4 times her own body mass in eggs (hatchling mass)"A female mammal raises to weaning offspring totaling 1.4 times her body mass"  

What  about  computa9onal  networks?  

•  Modern  microprocessors  contain  ~1  billion  transistors  

•  operate  at  power  densi9es  (W/m2)    approaching  a  nuclear  blast  

•  Wire-­‐scaling  drives  increased  power  on  single  core  chips  

Predic9ng  &  Minimizing  Power  Requirements  of  Microprocessors    

Scaling  in  compu9ng  (Moses  and  Forrest)  

Summed area of transistors on a chip as a function of die area (observed b = 0.53, predicted b = 1/2, data from www.icknowledge.com/history/history.html  

Internet backbone bandwidth as a function of the processing power of Internet hosts (observed b = 0.66, predicted b = 2/3, data from www.isc.org/index.pl?/ops/ds/host-count-history.php and www.zakon.org/robert/ internet/timeline/

Challenges    for  a  MST  Theory  of  Computer  Chips  

•  Distributed  networks:  Computer  networks  are  not  centralized  like  the  vascular  system  

•  Density  dependence    – MST  assumes  cells  are  constant  size  –  Transistor  densi9es  have  increased  exponen9ally  over  9me  from  thousands  to  millions  of  transistors  per  sq.  mm.  

•  The  last  mile:  network  delivers  resource  to  a  service  unit  and  then  it  is  delivered  to  des9na9on  using  other  methods    –  E.g.,  the  isochronic  region  in  chips  

•  Superlinear  scaling:  The  wire  scaling  problem  

Clock  trees:  Scale  like  circulatory  networks  

Hierarchical  Space  filling  Fractal  branching  Varia9on  in  length  of  terminal  wires  Aclock  ∝  NAchip

1/2  

 

Decentralized  vs.  Centralized  Networks  

Computer  Chips  

0.5 1 1.5 2 2.50.5

1

1.5

2

2.5

log10

Predicted Power From Decentralized Modello

g10 O

bse

rve

d P

ow

er

€

P ∝λV 2Ntr1/ 2Achip

1/ 2 fSlope = 0.95!

-4 -3 -2 -1 0 1 2 3 4!1.5

!1

!0.5

0

0.5

1

1.5

2

2.5

log

ob

serv

ed

po

we

r

log predicted power

€

P ∝λV 2NtrAchip1/ 2 f

Slope = 0.5!

Each  Wire  length  depends  on  radius:  N  *  (A1/2)  

Each  Wire  length  depends  on  distance  between  nearest  components:  N  *  (ρ-­‐1/2)  

Decentralized  vs.  Centralized  Networks  

Centralized Network Prediction " " Decentralized Network Prediction"

Road  Networks  

Hierarchical  Fractal  Networks  

Aorta: k = 0 Capillaries: k = K

Area  preserving  branching    

   

€

rkrk+1

= b1/ 2

Space-­‐filling  branching  

€

lklk+1

= b1/ 3

 Metabolic  rate  B~  Nc  Nc  =  bK  

€

Nc = bK

lK = lcrK = rc

€

N0 =1r0 ~ rcb

K / 2

l0 ~ lcbK / 3

€

Vnet = πr02l0 b− i / 3

i= 0

K

∑

Vnet  ∝  Nc4/3  

b  is  the  branching  ra9o  

LRE  =(offspring/year)*(offspring  mass  at  independence)*(adult  lifespan)  /                            (adult  mass)  

dm/dt:  Simple  growth  &  produc9on  model  δ:  metabolic  exponent    R0:  Fitness=total  #  of  offspring  per  life9me      α:  age  at  1st  reproduc9on      S:  prob  of  living  to  age  α      E:  adult  lifespan      Maximize  R0:  set  deriva9ve  wrt  α  =  0    Z:  Instantaneous  mortality  rate    Z=  1/E    

Calcula9ng  Op9mal  Life9me  Reproduc9ve  Effort  

__  __    __    __          __  

≈  4/3  Alterna9ve    growth  model