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

2. 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)

3. 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)

4. 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

5. 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

6. 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:
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)

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

8. 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

9. teff in msec Scaling of Ion Channel Kinetics keff in Hz
11/30/2018
Scaling of Ion Channel Kinetics
Liebovitch et al 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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