The Structure, Function, and Evolution of Biological Systems. Instructor: Van Savage Winter 2015 Quarter 3/3/2015

Transcription

1 The Structure, Function, and Evolution of Biological Systems Instructor: Van Savage Winter 2015 Quarter 3/3/2015

2 Review of power laws

3 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

4 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!

5 Examples of Power Laws

6 Parabolic Motion (Type 1)

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

8 Word usage (Type 3)

9 Word Usage (Type 3)

10 Web Sites (Type 2)

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 Parabolic motion (error in distance measures is independent of y or x)--linear space Body size (for population, variance in body size or heart rate varies linearly with x)--logarithmic space

12 Self Similarity and Fractals

13 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

14 Other examples of self similarity

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16 Hun8ng the Hidden Dimension special on Nova: hap:// hidden- dimension.html Benoit Mandelbrot

17 How do you think this one was generated?

18 One movie just for fun hap:// Tropic_Isle_Level_Mandel.mpg

19 Power Laws Self Similarity Equation form for self similarity: f (λx) = λ p f (x) - > <- Differentiate with respect to λ 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) Free to choose =1 x df (x) dx f (x) = ax p = pf (x) df f = p dx x

20 Fixed Points and Universality

21 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.

22 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 (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! " $ # d k f dx % (x x*) k ' ~ 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.

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

24 Dimensional Analysis and Power Laws

25 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

26 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 θ φ c 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

27 Example 2: Nuclear Blast US government wanted to keep energy yield of nuclear blasts a secret. Pictures of nuclear blast were released in Life magazine with time stamp Using DA, G. I. Taylor determined energy of blast and government was upset because they thought there had been a leak of information

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29 Radius, R, of blast depends on time since explosion, t, energy of explosion, E, and density of medium, ρ, that explosion expands into [R]=m, [t]=s, [E]=kg*m 2 /s 2, ρ=kg/m 3 R=t p E q ρ k 1 = 2q 3k 0 = p 2q 0 = q + k q=1/5, k=-1/5, p=2/5 m s kg R (E / ρ) 1/5 t 2/5 E R5 ρ unknown constant coefficient can be determined from y-intercept of regression of log-log plot of time series t 2

30 Pitfalls of Dimensional Analysis Miss constant factors Miss dimensionless ratios But, can get far with a good bit of ignorance!!!

31 Summary Self-similarity and fractalsè Power Laws Behavior near critical point è Power Laws But, Power Lawsè near critical points Dimensional Analysis assumes power law form and this is partially justified by necessity of matching units

32 If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. -John von Neumann

33 Theories are approximations that hope to impart deeper understanding We all know that art [theory] is not truth. Art [theory] is a lie that makes us realize truth, at least the truth that is given us to understand. The artist [theorist] must know the manner whereby to convince others of the truthfulness of his lies. --Pablo Picasso

34 A little philosophy of science Many general patterns are power laws Can often explain these without knowledge of all the details of the system Art of science is knowing system well enough to have intuition about which details are important

35 Single prediction models are not enough Much better to predict value of exponent and not just that it is a power law To really believe a theory we need multiple pieces of evidence (possibly n multiple power 1/3 laws) and need to be able to predict many of these. Understanding dynamics and some further details allows one to predict deviations from power law, and that is a very strong test and leads to very precise results

36 Barabasi et al.

37 Power laws in statistical distributions and scaling of growth of networks

38 Erdos-Renyi random graphs

39 Examples of distributions of real data Actors Internet Power grid

40 Growth models of networks 1. Start with m 0 unconnected nodes 2. Network grows one vertex at a time (gene duplication, species invasion, etc) 3. Add 1 node at a time and form m new connections between this node and existing nodes. (Why?) 4. Connections are formed with probability k i / k j where k i is the connectivity of node i. (Preferential attachment/rich get richer/proportional model) This is a type of self similarity! Power laws are to be expected.

41 Scaling of connectivity Total number of nodes at time t is m 0 +t Total number of edges is mt= k j /2 Connectivity at next time step is on average (edges can t be lost): k i (t +1) = k i (t) + m[k i (t) / k j ] Number of new edges at next 8me step Probability of that connec8on going to node i k i (t) t

42 Scaling of connectivity

43 Scaling of probability density " P(k i > k) = $ m # k (" * $ % # ' * &* * ) mt m 0 + t k % + '- &- - -, Average connec8vity of any node at 8me t Connec8vity at 8me that node i came into existence P(k i < k) =1 P(k i > k) =1 Connec8vity for comparison and probability m 2 t k 2 (m 0 + t) Probability density dp dk k 3

44 Scaling of probability density

45 Scaling of probability density

46 Scaling of probability density Adding edges without adding new nodes does not reach equilibrium

47 Metabolic networks

48 E coli metabolic substrate network

49 Scaling of probability density in metabolic networks Archaea E coli C elegans Average across 43 organisms

50 Distributions of pathway lengths

51 Scaling of connectivity

52 Scaling of lengths with node removal

53 Return to motifs

54 Similar frequencies of subgraphs in real networks

55 Universal scaling exponents in metabolic networks γ is degree exponent and α is clustering exponent. Exponents characterize local and global network organiza8on

56 Statistically significant subgraphs (i.e., motifs) for power law networks EDGES NODES

57 Scaling of probability density of triangles Dis8nct components T is number of selected subgraphs passing by a node

58 Scaling of probability density of penta-graphs Giant component T is number of selected subgraphs passing by a node

59 Phase transitions in hex-graphs: Before transition are motifs Aggrega8on/clustering decreases with number of edges in subgraph. Normalized by size of largest connected component

60 Free Network software 1. Network workbench (NWB) 2. Fanmod 3. Gephi 4. Prism.m Quick demos. Can use these in your research!

61 All humans are caught in an inescapable network of mutuality. - Mar8n Luther King, Jr.

62 Can display same network in different ways

63 Can look at different network or levels for given system

64 Can look at different 8mes

65 Can look at different 8mes

66 What have we learned? 1. Evolu8onary theory selec8on, drig, muta8ons, epistasis, neutrality, coalescence 2. Network theory mo8fs, subgraph structure, branching hierarchical, op8mal, clustering, growth, preferen8al aaachment (genes, proteins, food webs, disease) 3. Kolmogorov/Diffusion/Fokker Planck how to combine direc8onal and non- direc8onal processes, cell migra8on, cancer, gene expression, species abundance 4. Scaling power laws, self similarity, asympto8c expansions 5. How to read a modeling paper and work through the math and logic without just reading figures and punch lines 6. How to relate assump8on to figures and fundamental equa8ons and predic8ons 7. Fixed points, ODEs, sums, delta func8on, gamma func8on

67 THANKS to all of you! Good luck with your projects!