Hi Stan,

On the fact that power laws have not been shown in molecular biology,
this may just be a matter of not being able to visualize the
activities of molecules at the individual level.

If we could watch
one molecule in a reaction, I think it plausible that we would see a
power law distribution of the magnitude of is activity distributed
around the log normal mean of those activity magnitudes.

Such experiments have been done, for example in Ref. [1] in my prvious
email (see below). They measured the rate constants of a single molecule
of cholesterol oxidase through many turnovers of the same molecule. What
was surprising was that the rate constants are not fixed (as many had
thought) but distributed over a wide range covering two orders of
magnitudes. When they groups the rate constants and plotted their
frequency, the distribution was not a power law but waht I found to be
mixed, i.e., the product of a power law and an exponential law.

To me this does not necessarily mean that self-organized critical (SOC)
behaviors cannot occur in molecular systems but rather that molecular SOC
behaviors may obey laws other than a power law. In other words, the power
law behaviors may not be the sole and universal criterion of SOC, just as
the laws of Newtonian mechanics are not the universal mathematical
experssions of all mechanical processes in nature.

It just has
never been practically important to inquire into such things before.

STAN

With all the best.


Sung

(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).


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