From: <complex-science@necsi.org>
Sender: <y3list1@necsi.org> (Yaneer Bar-Yam)
To: complex-science
Date: Fri, 04 Jul 2008 14:58:19 -0400
Message-ID: <redirect-22042368@necsi.org>
Subject: The 'Brookhavenator': Self-organizing systmes with critical 
     properties
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(Yaneer, if it is not too late, can you replace my previous post with this
one?  There were 3 minor typos.  Thanks.)

I just finished reading Per Bak's "How Nature Works: The Science of
Self-Organized Criticality" (Springer, 1996).

Self-organized criticality (SOC) refers to the property of a complex
system that is slowly driven to a critical state where small perturbations
lead to "avalanches" of all sizes affecting the whole system. SOC was
discovered in 1987 by Bak, Tang and Wiesenfeld (BTW) and has been applied
since then to 1) geophysics, 2) cosmology, 3) evolutionary biology, 4)
brain physiology, 5) quantum gravity, 6) solar physics, 7) plasma physics,
8) neurobiology, 9) sociology, 10) economics, and others.

BTW visualized SOC as a "sandpile" to which grains are slowly added from
above to cause avalanches or slides. If the number of avalanches, N, of a
given size, s, is plotted against the size of avalanches in a double
logarithmic plot, one obtains an excellent straight line, indicating that
a power law is involved:

                  N(s) = As^-k          . . . . . . . . . . (1)

where A and k are constants. Power laws similar to Eq. (1) were found in
all of the 10 fields listed above.

Clearly SOC is a major discovery of the 20th century science.  It may be
considered to be the first mathematical theory of complex systems and
hence be referred to as the 'first law of complex systems', comparable to
the first and second laws of thermodynamic systems.

Since the tradition is well established in physical chemistry that all
self-organizing systems are named after the city where the research is
carried out, followed by the suffix "-ator", we may refer to all
self-organizing systems with critical properties like sandpiles as the
"Brookhavenator", although the Brookhaven National Laboratory (BNL) where
SOC was discovered is located in Upton, N.Y., some tens of miles Northeast
of Brookhaven.  It is interesting to note that, though BNL
produced 6 Nobel laureates in particle physics, it is through SOC that its
name may be 'immortalized' in the form of the Brookhavenator.

Also interesting is the fact that SOC has never been reported in molecular
biology, to the best of my knowledge.  One possible reason for this
conspicuous lack of any report of SOC in molecular biology so far may be
that the usual criterion for SOC, namely, the power law, e.g., Eq. (1),
may be too simple to capture the mechanisms of the SOC occurring in
molecular dynamical systems.

I have been analyzing the so-called 'dynamic heterogeneity' of rate
constants observed in single-molecule cholesterol oxidase reported in 1998
by Lu, Sun and Xie [1]. These authors found that a single molecule of
cholesterol oxidase catalyzes the oxidation of cholesterol to cholesterone
with rate constants that are not uniform but vary widely by a factor of
about 25. And the distribution of the rate constants are not Gaussian but
obeys the distribution that is similar to the black-body radiation formula
of M. Planck [2], which is of a mixed form, being a product of a power law
and an exponential law:

                p(w) = (aw^-5)e^(-b/w)   . . . . . . . . . . . (2)

where a and b are constants, w is the waiting time (which is inversely
related to rate constants), and p(w) is the probability of the occurrence
of a given waiting time, w.  A waiting time is the time the enzyme waits
until the next catalytic event. The mixed distribution law, Eq. (2), can
be derived using the sandpile as an analogy, where i) the sandpile is
replaced with an enzyme, ii) sand grains with conformons (packets of
energy and information), and iii) avalanches with catalytic events.

So, if my analysis of SOC briefly described here turns out to be correct,
there may be two kinds of SOC--the one obeying the power law, e.g., Eq.
(1), and the other obeying what may be referred to as "the mixed law",
e.g., Eq. (2).

With all the best.

Sung

___________________________________________
Sungchul Ji, Ph.D.
Department of Pharmacology and Toxicology
Rutgers University
Piscataway, N.J,. 08855


References:
   [1]  Lu, H. P., Xun, L., and Xie, X. S. (1998).  Single-Molecule
Enzymatic Dynamics.  Science 282:1877-1882.
   [2] Ji, S. (2009).  Molecular Theory of the Living Cell: Conceptual
Foundations, Molecular Mechanisms and Applications.  Springer, N.Y. (to
appear).