Scaling in Biology and in Society

Definition of scaling: How do attributes of a system change as the system’s size increases?

http://static-www.icr.org/i/wide/mouse_elephant_wide.jpg

http://static-www.icr.org/i/wide/mouse_elephant_wide.jpg

How do attributes of organisms change as their body mass increases?

http://us.123rf.com/400wm/400/400/aprilphoto/aprilphoto0811/aprilphoto081100005/3858120-small-city-in-italy-is-located-in-a-valley-among-hills-the-top-view.jpg

http://cybraryman.com/images/newyor1.jpg

http://us.123rf.com/400wm/400/400/aprilphoto/aprilphoto0811/aprilphoto081100005/3858120-small-city-in-italy-is-located-in-a-valley-among-hills-the-top-view.jpg

http://cybraryman.com/images/newyor1.jpg

How do attributes of cities change as their population increases?

Proportionality

Proportionality

y ∝ x means y = c x, where c is a constant.

Proportionality

y ∝ x means y = c x, where c is a constant. For example

y = 2 x

Proportionality

y ∝ x means y = c x, where c is a constant. For example

y = 2 x

or

y = (− ⅓) x

Proportionality

y ∝ x means y = c x, where c is a constant. For example

y = 2 x

or

y = (− ⅓) x

Proportionality is a linear relationship

Scaling Example

Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom.

2 × Room Length

x

Room Length

x

Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.

2 × Room Length

x

Room Length

x

Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.

2 × Room Length

x

Room Length

x

Bed Length

Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.

2 × Room Length

x

Room Length

x

Bed Length

2 × Bed Length

Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.

2 × Room Length

x

Room Length

x

Bed Length

2 × Bed Length

bed length ∝room length

Scaling Example Suppose you move from a small (square) bedroom to a larger (square) bedroom. The length of the bed you can fit in the room scales linearly with the length of the room.

2 × Room Length

x

Room Length

x

Bed Length

2 × Bed Length

bed length ∝room length

bed length

room length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

2 × Room Length

x

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

The area of a (square) rug you can fit in the room scales quadratically with the length of the room.

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom. rug area ∝(room length)2

Room Length

Scaling Example

x

2 × Room Length

The area of a (square) rug you can fit in the room scales quadratically with the length of the room.

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom. rug area ∝(room length)2

Room Length

Scaling Example

x

2 × Room Length

The area of a (square) rug you can fit in the room scales quadratically with the length of the room.

rug area

room length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

The volume of laundry you can pile to the ceiling scales cubically with the length of the room.

(Assume the length, width, and height of the room are approximately equal.)

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

The volume of laundry you can pile to the ceiling scales cubically with the length of the room.

(Assume the length, width, and height of the room are approximately equal.)

volume of laundry ∝(room length)3

laundry volume

room length

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

The volume of laundry you can pile to the ceiling scales cubically with the length of the room.

(Assume the length, width, and height of the room are approximately equal.)

volume of laundry ∝(room length)3

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

The volume of laundry you can pile to the ceiling scales cubically with the length of the room.

(Assume the length, width, and height of the room are approximately equal.)

volume of laundry ∝(room length)3

More generally, many attributes have power-law scaling: attribute ∝(size)α , where α is a constant

x

Suppose you move from a small (square) bedroom to a larger (square) bedroom.

Room Length

Scaling Example

x

2 × Room Length

The volume of laundry you can pile to the ceiling scales cubically with the length of the room.

(Assume the length, width, and height of the room are approximately equal.)

volume of laundry ∝(room length)3

More generally, many attributes have power-law scaling: attribute ∝(size)α , where α is a constant

That is,

attribute =c(size) α , where α is a constant or y = c xα

Power-Law Scaling

Power-Law Scaling

Two equivalent mathematical expressions for a power law:

Power-Law Scaling

Two equivalent mathematical expressions for a power law:

y = cxα

log y =α log x + logc

Power-Law Scaling

Two equivalent mathematical expressions for a power law:

y = cxα

log y =α log x + logc

y

x

Power-Law Scaling

Two equivalent mathematical expressions for a power law:

36

Straight line on a log-log plot

y = cxα

log y =α log x + logc

y

x

Power-Law Scaling

Two equivalent mathematical expressions for a power law:

37

Straight line on a log-log plot

y = cxα

log y =α log x + logc

y

x

log y

log x

Power-Law Scaling

Two equivalent mathematical expressions for a power law:

38

Straight line on a log-log plot

y = cxα

log y =α log x + logc

y

x

log y

log x

Slope is α

Examples of power law scaling in nature

Gutenberg-Richter law of earthquake magnitudes

Metabolic scaling in animals

K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Important? Cambridge, 1984

L. Bettencourt and G. West, A Unified Theory of Urban Living, Nature, 467, 912–913, 2010

Scaling crime, income, etc. with city population

Metabolic Scaling in Biology

Metabolic Scaling in Biology

•  Metabolic rate: Amount of energy expended by an organism per unit time.

Metabolic Scaling in Biology

•  Metabolic rate: Amount of energy expended by an organism per unit time.

–  Can be measured as the amount of heat emitted by the organism per unit time.

Metabolic Scaling in Biology

•  Metabolic rate: Amount of energy expended by an organism per unit time.

–  Can be measured as the amount of heat emitted by the organism per unit time.

It has been known for a long time that metabolic rate is a function of body mass, but how, exactly, does metabolic rate scale with body mass?

Metabolic Scaling in Biology

•  Metabolic rate: Amount of energy expended by an organism per unit time.

–  Can be measured as the amount of heat emitted by the organism per unit time.

It has been known for a long time that metabolic rate is a function of body mass, but how, exactly, does metabolic rate scale with body mass?

Theories of metabolic scaling

Theories of metabolic scaling

•  Early on, some assumptions were made:

Theories of metabolic scaling

•  Early on, some assumptions were made –  Body is made of cells, in which metabolic reactions take place.

Theories of metabolic scaling

•  Early on, some assumptions were made –  Body is made of cells, in which metabolic reactions take place. –  Can “approximate” body mass by a sphere of cells with radius r.

Theories of metabolic scaling

•  Early on, some assumptions were made: –  Body is made of cells, in which metabolic reactions take place. –  Can “approximate” body mass by a sphere of cells with radius r.

r

Mouse

Radius ∝ r

Mouse

Radius ∝ r

Surface Area ∝ r2

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal

Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal

Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal

125,000 times the heat of a mouse radiating over an area only 2500 times the surface area of a mouse

Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

BIG Problem: Mass is proportional to volume of animal but heat can radiate only from surface of animal

125,000 times the heat of a mouse radiating over an area only 2500 times the surface area of a mouse

Hypothesis 1: metabolic rate ∝ body mass (where body mass ∝ volume)

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Note that Surface Area ∝ (Volume)2/3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Note that Surface Area ∝ (Volume)2/3

because Surface Area ∝ r2 = (r3)2/3

∝ (Volume)2/3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Note that Surface Area ∝ (Volume)2/3

because Surface Area ∝ r2 = (r3)2/3

∝ (Volume)2/3

Hypothesis 2: metabolic rate ∝ (body mass)2/3

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Note that Surface Area ∝ (Volume)2/3

because Surface Area ∝ r2 = (r3)2/3

∝ (Volume)2/3

Hypothesis 2: metabolic rate ∝ (body mass)2/3 (“Surface Hypothesis”)

Mouse

Radius ∝ r

Surface Area ∝ r2

Volume ∝ r3

Hamster

Radius ∝ 2 r

Surface Area ∝ 4 r2

Volume ∝ 8 r3

Hippo

Radius ∝ 50 r

Surface Area ∝ 2500 r2

Volume ∝ 125,000 r3

Note that Surface Area ∝ (Volume)2/3

because Surface Area ∝ r2 = (r3)2/3

∝ (Volume)2/3

Hypothesis 2: metabolic rate ∝ (body mass)2/3 (“Surface Hypothesis”)

This was believed for many years!

Actual data:

Actual data: metabolic rate ∝ (body mass)3/4

Actual data: metabolic rate ∝ (body mass)3/4

Hypothesis 3 (“Kleiber’s law): metabolic rate ∝ mass3/4

Actual data: metabolic rate ∝ (body mass)3/4

Hypothesis 3 (“Kleiber’s law): metabolic rate ∝ mass3/4

For sixty years, no explanation

y=x2/3

y=x3/4

y=x3/4

y=x2/3

More “efficient”, in sense that metabolic rate (and thus rate of distribution of nutrients to cells) is larger than surface area would predict.

Other Observed Biological Scaling Laws

Heart rate ∝ body mass-1/4

Blood circulation time ∝ body mass1/4

Life span ∝ body mass1/4

Growth rate ∝ body mass-1/4

Heights of trees ∝ tree mass1/4

Sap circulation time in trees ∝ tree mass1/4

Other Observed Biological Scaling Laws

Heart rate ∝ body mass-1/4

Blood circulation time ∝ body mass1/4

Life span ∝ body mass1/4

Growth rate ∝ body mass-1/4

Heights of trees ∝ tree mass1/4

Sap circulation time in trees ∝ tree mass1/4

No*onof“quarterpowerscaling”

West, Brown, and Enquist’s Theory (1990s)

West, Brown, and Enquist’s Theory (1990s)

General idea: “metabolic scaling rates (and other biological rates) are limited not by surface area but by rates at which energy and materials can be distributed between surfaces where they are exchanged and the tissues where they are used.”

West, Brown, and Enquist’s Theory (1990s)

General idea: “metabolic scaling rates (and other biological rates) are limited not by surface area but by rates at which energy and materials can be distributed between surfaces where they are exchanged and the tissues where they are used.”

How are energy and materials distributed?

Distribution systems

West, Brown, and Enquist’s Theory (1990s)

West, Brown, and Enquist’s Theory (1990s)

•  Assumptions about distribution network:

West, Brown, and Enquist’s Theory (1990s)

•  Assumptions about distribution network: –  branches to reach all parts of three-dimensional organism (i.e., needs to be as “space-filling” as possible)

West, Brown, and Enquist’s Theory (1990s)

•  Assumptions about distribution network: –  branches to reach all parts of three-dimensional organism (i.e., needs to be as “space-filling” as possible) –  has terminal units (e.g., capillaries) that do not vary with size

among organisms

West, Brown, and Enquist’s Theory (1990s)

•  Assumptions about distribution network: –  branches to reach all parts of three-dimensional organism (i.e., needs to be as “space-filling” as possible) –  has terminal units (e.g., capillaries) that do not vary with size

among organisms

–  evolved to minimize total energy required to distribute resources

Because distribution network has fractal branching structure,

Euclidean geometry is the wrong way to view scaling; one should

use fractal geometry instead!

Because distribution network has fractal branching structure,

Euclidean geometry is the wrong way to view scaling; one should

use fractal geometry instead!

With detailed mathematical model using three assumptions, they derive

metabolic rate ∝ body mass3/4

West, Brown, and Enquist’s interpretation of their model

•  Metabolic rate scales with body mass like surface area scales with volume...

West, Brown, and Enquist’s interpretation of their model

•  Metabolic rate scales with body mass like surface area scales with volume... but in four dimensions.

West, Brown, and Enquist’s interpretation of their model

•  Metabolic rate scales with body mass like surface area scales with volume... but in four dimensions.

•  “Although living things occupy a three-dimensional space, their internal physiology and anatomy operate as if they were four-dimensional. . . Fractal geometry has literally given life an added dimension.”

h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/

Metabolicrate∝ volume2/3(ormass2/3)

h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/

Metabolicrate∝ volume2/3(ormass2/3)

Youare3-dimensional

h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/

Metabolicrate∝ volume2/3(ormass2/3)

Youare3-dimensional

Metabolicrate∝ volume3/4(ormass3/4)

h=p://www.happehtheory.com/TheDailyInsight/2008/11/29/the-daily-insight-11-29-08-the-big-hand-view-of-the-human-body/

Metabolicrate∝ volume2/3(ormass2/3)

Youare3-dimensional

Youare4-dimensional

Metabolicrate∝ volume3/4(ormass3/4)

Critiques of their model

Critiques of their model

Critiques of their model

Critiques of their model

Bottom line: Model is interesting and elegant, but both the explanation and the underlying data are controversial.

Critiques of their model

Bottom line: Model is interesting and elegant, but both the explanation and the underlying data are controversial. Also, note that there have been many updated versions of their model since their original paper.

L. Bettencourt and G. West, A Unified Theory of Urban Living, Nature, 467, 912–913, 2010

Do fractal distribution networks explain scaling in cities?

Wrapping Up

Large networks of simple interacting elements,

which, following simple rules, produce emergent,

collective, complex behavior.

What are Complex Systems?

Core Disciplines of the Sciences of Complexity

Dynamics: The study of continually changing structure and behavior of

systems

Information: The study of representation, symbols, and communication

Computation: The study of how systems process information and act on the

results

Evolution / Learning: The study of how systems adapt to constantly

changing environments

Goals of this course:

•  To give you a sense of how these topics are integrated in the study of complex systems

•  To give you a sense of how idealized models can be used to study these topics

What did we cover?

Let’s review...

Dynamics and Chaos

•  Provides a “vocabulary” for describing how complex systems change over time –  Fixed points, periodic attractors, chaos, sensitive dependence on initial

conditions

•  Shows how complex behavior can arise from iteration of simple rules

•  Characterizes complexity in terms of dynamics

•  Shows contrast between intrinsic unpredictability and “universal” properties

Fractals

•  Provides geometry of real-world patterns

•  Shows how complex patterns can arise from iteration of simple rules

•  Characterizes complexity in terms of fractal dimension

Information Theory

•  Makes analogy between information and physical entropy

•  Characterizes complexity in terms of information content

Genetic Algorithms

•  Provides idealized models of evolution and adaptation

•  Demonstrates how complex behavior/shape can emerge from simple rules (of evolution)

Cellular Automata

•  Idealized models of complex systems

•  Shows how complex patterns can emerge from iterating simple rules

•  Characterizes complexity in terms of “class” of patterns

Models of Self-Organization

•  Idealized models of self-organizing behavior

•  Attempt to find common principles in terms of dynamics, information, computation, and adaptation

Firefly synchronization Flocking / Schooling Ant Foraging

Ant Task Allocation Immune System Cellular Metabolism, …

Models of Cooperation

•  Idealized model of how self-organized cooperation can emerge in social systems

•  Demonstrates how idealized models can be used to study complex phenomena

Prisoner’s dilemma El Farol Problem

Networks

•  Vocabulary for describing structure and dynamics of real-world networks –  small-world, scale-free, degree distribution, clustering,

path-length

•  Shows how real-world network structure can be captured by simple models (e.g., preferential attachment)

Scaling

•  Gives clues to underlying structure and dynamics of complex systems (e.g., fractal distribution networks)

Goals of the Science of Complexity

•  Cross-disciplinary insights into complex systems

•  General theory?

√

?

Can we develop a general theory of complex systems?

That is, a mathematical language that unifies dynamics,

information processing, and evolution in complex systems ?

I.e., a “calculus of complexity” ?

Isaac Newton, 1643–1727

infinitesimal

limit

derivative

integral

“He was hampered by the chaos of language

—words still vaguely defined and words not

quite existing. . . . Newton believed he could

marshal a complete science of motion, if only

he could find the appropriate lexicon. . . .”

― James Gleick, Isaac Newton

emergence

self-organization

network

adaptation

Complex Systems, c. 2013

attractor criticality

information computation

bifurcation

nonlinearity

equilibrium

entropy fractal chaos

.

.

.

.

.

.renormalization randomness

scaling

power law

“I do not give a fig for the simplicity on this side of

complexity, but I would give my life for the simplicity on

the other side of complexity.”

― O. W. Holmes (attr.)