Modeling Vascular Networks with Applications

Instructor: Van Savage Spring 2015 Quarter

4/6/2015 Meeting time: Monday and Wednesday,

2:00pm-3:50 pm

Total volume calculated

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Vnet = πrN2lNn

N n−(N−k )β−2(N−k )γ−(N−k )k= 0

N

∑

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netV = NN NV1− (nγ 2β )−(N +1)

1− (nγ 2β )−1

What is scaling between number of terminal units and network volume?

Terminal units are also special because that is where exchange of resources usually occurs

Total volume calculated

By self similarity, as we will see we expect power laws,

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βk ≡k+1rkr

= n−a and

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γ k ≡k+1lkl

= n−b

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netV = NN NV1− n(2a+b−1)(N +1)

1− n2a+b−1= NV

NN(2a+b ) − n1−2a−bNN1− n1−2a−b

What more can we say about b and g?

Simple pipe model for plants

area preserving

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nπ k+12r = π k2r ⇒β ≡ k+1rkr

= −1/ 2n First noticed by Da Vinci

Shinozaki and West et al. Science (1997)

space filling Nk+1c k+13l = Nkc k3l ⇒ γ ≡k+1lkl= −1/3n to collect and distribute

resources

Plant size changes network size

Terminal units are invariant and set overall scale

Aorta

Capillaries

level 0 Level 0

Level 1

level N

Scaling of network volume with number of terminal units is now

completely determined

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Vnet = NVNN

4 / 3 − n−1/ 3NN1− n−1/ 3

~ NN4 / 3

NV1− n−1/ 3

∝ NN 4 / 3

West et al. Science (1997) €

NN ∝Vnet3 / 4

We will return to this ¾ again and again as an example

Outline

I.  General review of power laws (continued) 1. identifying power laws

2. some examples 3. self similarity 4. universality classes and critical points II. Dimensional analysis III. Biological Allometry and Scaling

II. Review of power laws

1. Types of Power Laws

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y = Axb

 1.  Physical laws--planetary orbits, parabolic motion of thrown objects, classical forces, etc. (these are really idealized notions and do not exist in real world) 2. Scaling relations--relate two fundamental parameters in a system like lifespan to body mass in biology (Physical Laws are special/strong case) 3. Statistical distributions

Identifying Power Laws

•  Need big range on x- and y-axes to determine power laws because this minimizes effects of noise and errors

•  Can give good measure of b, the exponent •  r2 is property of data and measures how much

variance in y is explained by variance in x. It is NOT really a measure of goodness of fit!

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ln y = bln x + lnaLinear plot: slope=b and intercept=ln a

Identifying Power Laws •  Maximum likelihood methods are good for

identifying power laws if used in correct way •  Whether to curve fit in linear or logarithmic

space depends on distribution of errors because regressions make assumptions about these: homoscedascity->variance in y is independent of value of x

•  Body size (for population, variance in body size or heart rate varies linearly with x)--logarithmic space

2. Examples of Power Laws

Parabolic Motion (Type 1)

Rates related to vascular networks (Type 2)

Savage, et al., Func. Eco., 2004

Word  usage  (Type  3)  

Word  Usage  (Type  3)  

Web  Sites   (Type 2)

3. Self Similarity and Fractals

Imagine taking a picture of smaller piece and magnifying it, and then it looks like original part.

•  Once a useful process is found in nature, it tends to be used over and over again--physics constrains possible processes and evolution tends to maintain it because most mutations are harmful

 

Other examples of self similarity

1.  Branching process that is self similar and repeated at every scale 2.  Changes our concepts of measuring distances by ruler and thus our concept of area, volumes, and dimensions 3. Watch “Hunting the Hidden Dimension”: http://www.pbs.org/wgbh/nova/fractals It is a nova program this year that talk about the development of fractals over the last half century and importance for many applications, including vascular networks.

What does a picture like this have to do with vascular networks?

Look  familiar?    

How  do  you  think  this  one  was  generated?    

One  movie  just  for  fun    

hCp://www.ericbigas.com/fractals/Hlm/Tropic_Isle_Level_Mandel.mpg  

Power Laws Self Similarity

-‐>                                                                                

4. Fixed Points and Universality

Dynamical systems flow relative to fixed points

What is functional form as fixed point is approached? This describes region and dynamics that are relevant for many scientific questions.

Non-power-law functions often behave as power laws near critical points

•  Other functions commonly occur in nature: ex, sin(x), cosh(x), (x), Ai(x)

•  These functions can generally be expressed in Taylor or power series near critical points (phase transitions, etc.). When x is close to x*, difference is small, and first term dominates.

p-exponent of leading-order term Any functions with the same first term in their series

expansion behave the same near critical points, which is of great physical interest, even if they behave very differently elsewhere. Source of universality classes.

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f (x) = (x − x*)k

k!f (k )(x − x*)

k= p

N

∑ ~ C(x − x*) p

!Jυ

ln(1+x)

x  

sin(x)

x  

Near x=0, both of these functions scale like x!

 III. Dimensional Analysis and Power Laws

Dimensional Analysis •  Often used in physics •  For reasons given thus far, many processes

should scale as a power law. •  Given some quantity, f, that we want to

determine, we need to intuit what other variables on which it must depend, {x1,x2,…,xn}.

•  Assume f depends on each of these variables as a power law.

•  Use consistency of units to obtain set of equations that uniquely determine exponents.

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f (x1,x2,...,xn ) = x1p1 x2

p2 ...xnpn

Example 1: Pythagorean Theorem

•  Hypotenuse, c, and smallest angle, θ, uniquely determine right triangles.

•  Area=f(c, θ), DA implies Area=c2g(θ).  

θ

φ

θ φ

c a

b

Area of whole triangle=sum of area of smaller triangles

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a2g(θ) + b2g(θ) = c 2g(θ)⇒ a2 + b2 = c 2