Modeling Vascular Networks with Applications. Instructor: Van Savage Spring 2015 Quarter 4/6/2015 Meeting time: Monday and Wednesday, 2:00pm-3:50 pm

Transcription

1 Modeling Vascular Networks with Applications Instructor: Van Savage Spring 2015 Quarter 4/6/2015 Meeting time: Monday and Wednesday, 2:00pm-3:50 pm

2 Total volume calculated V net = πr N 2 l N n N N k= 0 n (N k ) β 2(N k ) γ (N k) V net = N N V 1 (nγ 2 N 1 (nγ (N +1) β ) 2 β ) 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

3 Total volume calculated What more can we say about b and g? By self similarity, as we will see we expect power laws, r β k k +1 rk = n a l and γ k k +1 lk = n b V net = N N V 1 n(2a +b 1)(N +1) N = V 1 n 2a +b 1 N N (2a +b ) N n 1 2a b N N 1 n 1 2a b

4 Simple pipe model for plants 2 2 area preserving nπ r k +1 = π r k β r k +1 rk = 1/ 2 n First noticed by Da Vinci 3 3 space filling N k+1 cl k+1 = N k cl k γ k+1 l l k = 1/3 n to collect and distribute resources Shinozaki and West et al. Science (1997)

5 Terminal units are invariant and set overall scale Capillaries Aorta Level 0 level 0 Level 1 Plant size changes network size level N

6 Scaling of network volume with number of terminal units is now completely determined V net = V N N N 4 / 3 n 1/ 3 N N 1 n 1/ 3 4 / 3 V N ~ N N 1 n 1/ 3 N N 4 / 3 N N V 3 / 4 net We will return to this ¾ again and again as an example West et al. Science (1997)

7 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

8 II. Review of power laws

9 1. Types of Power Laws y = Ax b 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

10 Identifying Power Laws ln y = bln x + lna Linear plot: slope=b and intercept=ln a 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 r 2 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!

11 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

12 2. Examples of Power Laws

13 Parabolic Motion (Type 1)

14 Rates related to vascular networks (Type 2) Savage, et al., Func. Eco., 2004

15 Word usage (Type 3)

16 Word Usage (Type 3)

17 Web Sites (Type 2)

18 3. Self Similarity and Fractals

19 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

20 Other examples of self similarity

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

22 Look familiar?

23 How do you think this one was generated?

24 One movie just for fun hcp:// Tropic_Isle_Level_Mandel.mpg

25 Power Laws Self Similarity Equation form for self similarity: f (λx) = λ p f (x) - > <- Differentiate with respect to l Chain Rule f (x) = ax p f (λx) = a(λx) p = λ p ax p = λ p f (x) pλ p 1 f (x) = df (λx) dλ = d(λx) dλ df (λx) d(λx) = x df (λx) d(λx) λ Free to choose =1 x df (x) dx f (x) = ax p = pf (x) df f = p dx x

26 4. Fixed Points and Universality

27 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.

28 Non-power-law functions often behave as power laws near critical points Other functions commonly occur in nature: e x, sin(x), cosh(x),! J υ(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. f (x) = N k= p (x x*) k k! f (k) (x x*) ~ C(x x*) p 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.

29 ln(1+x) x sin(x) x Near x=0, both of these functions scale like x!

30 III. Dimensional Analysis and Power Laws

31 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, {x 1,x 2,,x n }. 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. f (x 1,x 2,...,x n ) = x 1 p 1 x 2 p 2...x n p n

32 Example 1: Pythagorean Theorem Hypotenuse, c, and smallest angle, θ, uniquely determine right triangles. Area=f(c, θ), DA implies Area=c 2 g(θ). a φ c θ φ θ b Area of whole triangle=sum of area of smaller triangles a 2 g(θ) + b 2 g(θ) = c 2 g(θ) a 2 + b 2 = c 2