Biology Reference
In-Depth Information
1.7 Scaling: A Fundamental Concept in Systems Biology
Size is a crucial biological property. As the size and complexity of a biological
system increases, the relationship among its different components and processes
must be adjusted over a wide a range of scales so that the organism can continue to
function (Brown et al. 2000 ). Otherwise stated, the organism must remain self-
similar. Self-similarity is a main attribute of fractals—a concept introduced by
Mandelbrot ( 1977 )—therefore the relationships among variables from different
processes can be described by a fractal dimension or a power function. Geometri-
cally, fractals can be regarded as structures exhibiting scaling in space: this is
because their mass as a function of size, or their density as a function of distance,
behave as a power law. If a variable changes according to a power law when the
parameter on which it depends is growing linearly, we say it scales , and the
corresponding exponent is called scaling exponent, b :
Y o M b
Y
(1.1)
¼
where Y can be a dependent variable, e.g., metabolic rate; M is some independent
variable, e.g., body mass, while Y o is a normalization constant (Brown et al. 2000 ).
If b
1, the relationship represented by ( 1.1 ) is called isometric, whereas when
¼
b
1 is called allometric—a term coined by Julian Huxley ( 1932 ). An important
allometric relationship in biology is the existing between metabolic rate and body
mass, first demonstrated by Kleiber ( 1932 ). Instead of the expected b
2/3
according to the surface law (i.e., surface to volume area), Kleiber showed that
b
¼
(i.e., 0.75 instead of 0.67), meaning that the amount of calories dissipated by
a warm-blooded animal each day scales to the
¼ ¾
of its mass (Whitfield 2006 ).
Scaling not only applies to spatial organization but to temporal organization as
well. The dynamics of a biological system—visualized through time series of its
variables (e.g., membrane potential, metabolites concentration)—exhibits fractal
characteristics. In this case, short-term fluctuations are intrinsically related to the
long-term trends through statistical fractals. On these bases, we can say that scaling
reflects the interaction between the multiple levels of organization exhibited by
cells and organisms, thus linking the spatial and temporal aspects of their organiza-
tion. The discovery of chaotic dynamics by Lorenz ( 1963 ) and criticality in phase
transitions by Wilson ( 1983 ) enabled the realization that scaling is a common
fundamental and foundational concept of chaos and criticality as it is with fractals.
¾
1.8 Networks
The concept of networks is basic for understanding biological organization. Specifi-
cally, networks enable address of the problems of collective behavior and large-scale
response to stimuli and perturbations exhibited by biological systems (Alon 2007 ;
Barabasi and Oltvai 2004 ). Scaling and topological and dynamical organization of
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