More about FRACTALS AND SCALING

1. More about FRACTALS AND SCALING Larry Liebovitch, Ph.D. Florida Atlantic University 2004

2. Scaling Self-Similarity Q (ar) = k Q(r) Q (r) = B rb Self-Similarity can be satisfied by the power law scaling: Q (r) = B rb Proof: using the scaling relationship to evaluate Q(a) and Q(ar), Q (r) = B rb Q (ar) = B ab rb if k = ab then Q (ar) = k Q (r)

3. log ar log a log ar + log r log a Self-Similarity Scaling Q (r) = B rb f(Log[r]/Log[a]) Q (ar) = k Q(r) Self-Similarity can be satisfied by the more complex scaling: log r log a Q (r) = B rbf ( ) where f(1+x) = f(x) Proof: using the scaling relationship to evaluate Q(a) and Q(ar), log r log a Q (r) = B rbf ( ) Q (ar) = B ab rbf ( ) = Babrbf ( ) log ar log a log r log a = B ab rbf ( ) = Babrbf ( ) 1+ if k = ab then Q (ar) = k Q (r)

4. Scaling Relationships most common form: Power Law Q (r) = B rb Q (r) Log Q (r) Logarithm of the measuremnt measurement Log r r resolution used to make the measurement Logarithm of the resolution used to make the measurement

5. log r log a Scaling Relationships less common, more general form: Q (r) = B rb f ( ) Log Q (r) Q (r) Logarithm of the measuremnt measurement r Log r resolution used to make the measurement Logarithm of the resolution used to make the measurement

6. How Long is the Coastline of Britain? Richardson 1961 The problem of contiguity: An Appendix to Statistics of Deadly Quarrels General Systems Yearbook 6:139-187 AUSTRIALIAN COAST 4.0 CIRCLE SOUTH AFRICAN COAST 3.5 GERMAN LAND-FRONTIER, 1900 Log10 (Total Length in Km) WEST COAST OF BRITIAN 3.0 LAND-FRONTIER OF PORTUGAL 3.5 1.0 2.0 3.0 1.5 2.5 LOG10 (Length of Line Segments in Km)

7. Scaling of Membrane Area Paumgartner, Losa, and Weibel 1981 J. Microscopy 121:51 - 63

8. imi inner mitochondrial membrane omi outer mitochondrial membrane er endoplasmic reticulum 30 [1 - D] [1 - D] -0.54 10 imi -0.09 omi BOUNDARY LENGTH DENSITY IN M-1 -0.72 er 1 x10-6 40 60 4 6 20 8 10 RESOLUTION SCALE M-1

9. Scaling of Ion Channel Kinetics Liebovitch et al. 1987 Math. Biosci. 84:37-68 70 pS Channel, on cell, Corneal Endothelium 1000 100 keffin Hz effective kinetic rate constant 10 1 1 1000 10 100 teffin msec effective time scale

10. Two Interpretations of the Fractal Scalings Structural The scaling relationship reflects the distribution of the activation energy barriers between the open and closed sets of conformational substates. closed open Energy

11. Two Interpretations of the Fractal Scalings Dynamical The scaling relationship reflects the time dependence of the activation energy barrier between the open and closed states. t3 t2 t1 closed time Energy open

12. Biological Examples of Scaling Relationships spatial • area of endoplasmic reticulum • membrane • area of inner mitochondrial membrane • area of outer mitochondrial membrane • diameter of airways in the lung • size of spaces between endothelial • cells in the lung • surface area of proteins

13. Biological Examples of Scaling Relationships temporal • kinetics of ion channels • reaction rates of chemical • reactions limited by diffusion • washout kinetics of substances • in the blood

14. Scaling Resolution 1 cm Perimeter = 8 cm Perimeter = 12 cm Resolution 0.5 cm

15. Scaling scaling relationship: much more interesting one measurement: not so interesting one value slope Logarithm of the measuremnt Logarithm of the measuremnt Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement