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

Q (r) = B rb Self-Similarity Scaling 11/30/2018 Self-Similarity Scaling 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)

Q (r) = B rbf ( ) where f(1+x) = f(x) 11/30/2018 Self-Similarity Scaling Q (ar) = k Q(r) Q (r) = B rb f(Log[r]/Log[a]) 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 ( ) log ar log a log ar + log r log a 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)

Scaling Relationships 11/30/2018 Scaling Relationships most common form: Power Law Q (r) = B rb Q (r) Log Q (r) the measuremnt Logarithm of measurement Log r r resolution used to make the measurement Logarithm of the resolution used to make the measurement

Scaling Relationships 11/30/2018 Scaling Relationships less common, more general form: log r log a Q (r) = B rb f ( ) Q (r) Log Q (r) the measuremnt Logarithm of measurement r Log r resolution used to make the measurement Logarithm of the resolution used to make the measurement

How Long is the Coastline of Britain? 11/30/2018 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 Log10 (Total Length in Km) GERMAN LAND-FRONTIER, 1900 WEST COAST OF BRITIAN 3.0 LAND-FRONTIER OF PORTUGAL 1.0 1.5 2.0 2.5 3.0 3.5 LOG10 (Length of Line Segments in Km)

Scaling of Membrane Area 11/30/2018 Scaling of Membrane Area Paumgartner, Losa, and Weibel 1981 J. Microscopy 121:51 - 63

imi inner mitochondrial membrane omi outer mitochondrial membrane 11/30/2018 imi inner mitochondrial membrane omi outer mitochondrial membrane er endoplasmic reticulum 30 [1 - D] [1 - D] 10 -0.54 imi -0.09 omi BOUNDARY LENGTH DENSITY IN M-1 -0.72 er 1 4 6 8 10 20 40 60 x10-6 RESOLUTION SCALE M-1

teff in msec Scaling of Ion Channel Kinetics keff in Hz 11/30/2018 Scaling of Ion Channel Kinetics Liebovitch et al. 1987 Math. Biosci. 84:37-68 70 pS Channel, on cell, Corneal Endothelium 1000 100 keff in Hz effective kinetic rate constant 10 1 1 10 100 1000 teff in msec effective time scale

Two Interpretations of the Fractal Scalings 11/30/2018 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

Two Interpretations of the Fractal Scalings 11/30/2018 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

Biological Examples of Scaling Relationships 11/30/2018 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

Biological Examples of Scaling Relationships 11/30/2018 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

Scaling Resolution 1 cm Perimeter = 8 cm Perimeter = 12 cm Resolution 11/30/2018 Scaling Resolution 1 cm Perimeter = 8 cm Perimeter = 12 cm Resolution 0.5 cm

Scaling one measurement: not so interesting 11/30/2018 Scaling one measurement: not so interesting scaling relationship: much more interesting one value slope the measuremnt Logarithm of the measuremnt Logarithm of Logarithm of the resolution used to make the measurement Logarithm of the resolution used to make the measurement