From: <complex-science@necsi.org> (Stanley Salthe)
Sender: <y3list1@necsi.org> (Yaneer Bar-Yam)
To: complex-science
Date: Wed, 06 Aug 2008 00:02:06 -0400
Message-ID: <redirect-22507454@necsi.org>
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In-Reply-To: <list-22236611@necsi.org>
References: <list-22236611@necsi.org>
Subject: Re: The 'Brookhavenator': Self-organizing systems with critical
 properties
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Sung -- Aha!  Thanks!  This distribution that was found seems to me 
in a general way to serve the same 'philosophical interests' as if it 
had been power law.  Thus, we can suppose that there is a broad 
spectrum of activity magnitudes for any kind of delimitable entity at 
any scale.  I don't think that power law itself is necessary for this 
understanding, and again, it might actually be an artefact of 
observation.  The main point is that scientifically observed average 
activities (in rate constants, etc.) really are only 'averages'.

STAN


>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).
>>>
>>>
>>>--------------------------------------------------
>>>For information about this discussion group visit
>>>http://necsi.org/discuss/discuss.html
>>
>
>--------------------------------------------------
>For information about this discussion group visit
>http://necsi.org/discuss/discuss.html