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ScienceWeek
SCIENCEWEEK E-BOOK
Complex Systems
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Section 1
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1. Introduction.
2. Chaos and Complexity
3. Biological vs. Engineering Complexity
4. Complex Networks
5. Complex Fluids
6. Complexity in Chemistry
7. Proteins and Complexity
8. Consciousness and Complexity
Notices and Subscription Information
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Section 2
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1. INTRODUCTION.
PHYSICS AND COMPLEXITY
"For the vast majority of scientists physics is a marvelous
algorithm explaining natural phenomena in terms of the building
blocks of the universe and their interactions. Planetary motion;
the structure of genetic material, of molecules, atoms or nuclei;
the diffraction pattern of a crystalline body; superconductivity;
the explanation of the compressibility, elasticity, surface
tension or thermal conductivity of a material, are only a few
among the innumerable examples illustrating the immense success
of this view, which presided over the most impressive
breakthroughs that have so far marked the development of modern
science since Newton.
"Implicit in the classical view, according to which physical
phenomena are reducible to a few fundamental interactions, is the
idea that under well-defined conditions a system governed by a
given set of laws will follow a unique course, and that a slight
change in the causes will likewise produce a slight change in the
effects. But, since the 1960s, an increasing amount of
experimental data challenging this idea has become available, and
this imposes a new attitude concerning the description of nature.
Such ordinary systems as a layer of fluid or a mixture of
chemical products can generate, under appropriate conditions, a
multitude of self-organization phenomena on a macroscopic scale -
- a scale orders of magnitude larger than the range of
fundamental interactions -- in the form of spatial patterns or
temporal rhythms. States of matter capable of evolving (states
for which order, complexity, regulation, information and other
concepts usually absent from the vocabulary of the physicist
become the natural mode of description) are, all of a sudden,
emerging in the laboratory. These states suggest that the gap
between 'simple' and 'complex', and between 'disorder' and
'order', is much narrower than previously thought. They also
provide the natural archetypes for understanding a large body of
phenomena in branches which traditionally were outside the realm
of physics, such as turbulence, the circulation of the atmosphere
and the oceans, plate tectonics, glaciations, and other forces
that shape our natural environment: or, even, the emergence of
replicating systems capable of storing and generating
information, embryonic development, the electrical activity of
brain, or the behaviour of populations in an ecosystem or in an
economic environment."
Gregoire Nicolis: in: Paul Davies (ed.): The New Physics.
Cambridge University Press 1989, p.316.
GENES AND COMPLEXITY
"The [human] genome project is the ultimate celebration of the
gene. Until recently we believed that the complete biological
history of a human being was encoded in the 3 billion letters of
the helical DNA. To be sure, the mapping of the human genome
revolutionized biological research. But it also showed us what a
small fraction of the vast world is really known to us and how
much more is left to be explored.
"In 1996 the decoding of the yeast genome gave the scientific
community a shock: It contained as many as 6,300 genes. Only
about a quarter of these were expected and could be assigned
vague functions. To be on the safe side, and boosted by humans'
perceived importance as the pinnacle of evolution, biologists
estimated that the human genome would have at least 100,000
genes. This number was believed to be sufficient to account for
the high complexity of Homo sapiens. Then came February 2001 and
the publication of the human genome. It turned out that we have
less than a third of the anticipated genes only about 30,000.
Therefore, a mere one-third increase in genes must explain the
difference between us and the unsophisticated Caenorhabditis
elegans worm -- quite a provocative idea when we consider that
the 20,000 genes of C. elegans need to encode only three hundred
neurons, whereas our extra 10,000 genes have to account for the
billion nerve cells present in our brain.
"In short, it is now clear that the number of genes is not
proportional to our perceived complexity. Then what does
complexity mean? Networks point to the answer. Framed in terms of
networks, our question becomes: How many different potentially
distinct behaviors can a genetic network display with the same
number of genes? In principle, two cells that are identical
except that a specific gene is on in the first cell and off in
the second could behave differently. Assuming that each gene can
be turned on or off independently, a cell with N genes could
display V distinct states. If we adopt as a measure of complexity
the potential number of distinct behaviors displayed by a typical
cell, the difference between the worm and humans is staggering:
Humans could be viewed as 10^(3000) times more complex than our
wormy relatives!
"Whereas the twentieth century was seen as the century of
physics, the twenty-first is often predicted to be the century of
biology. A decade ago it would have been tempting to call it the
century of the gene. Few people would dare say that any longer
about the century we have just entered. It will most likely be a
century of complexity..."
Albert-Laszlo Barabasi: Linked: The New Science of Networks.
Perseus Publishing 2002, p.196.
SUPRAMOLECULES AND COMPLEXITY
"Supramolecular chemistry is paving the way towards comprehending
chemistry as an information science in which molecules are
instructed by judicious structural design to interact in specific
ways -- with each other, with light, with their environment, with
natural systems. Supramolecular chemistry is developing from its
origins in recognition processes, through assembly and
organization, to a paradigm in which information is stored,
retrieved, transferred and processed at the molecular and
supramolecular level. There is no question that molecular science
can by itself make these things possible: biology shows us that.
DNA is a digital molecular data bank, whose genetic information
is read out during the transcription of messenger RNA. This
information is used to make a programmed molecular or
supramolecular material: a protein. Feedback mechanisms act to
modulate and regulate this process of information retrieval, so
that the expression of certain genes takes place only at the
correct stages of the cell cycle, or in the correct tissues.
There are editing processes, parallel processing pathways and
redundancies to make the system robust against errors or
breakdowns. The system is self-regulating, self-regenerating and
self-replicating. These are complex chemical systems! It is worth
considering to what extent artificial supramolecular systems of
this kind can be designed from first principles and to what
extent their creation might instead have to involve a process of
self-selection in the face of competitive pressures.
"'Complex' is not the same as 'complicated'. The beauty of
evolution lies not in the very complicated structures of its
products but in the simplicity of its principles. Similarly,
while the creation of complex, functional supramolecular systems
will involve considerable synthetic sophistication, we hope that
these systems will be characterized by elegance and economy of
design.
"In chemical and biological systems, complexity commonly implies
a degree of organizational hierarchy, defined by several length
scales. Because each level of hierarchy generally contains
features that do not and even cannot exist at the level below,
supramolecular science cannot be a reductionist discipline but
rather must be an integrative one, connecting one level to the
others by integrating species and interactions to describe the
increasing complexity of behavior. Much of the appeal lies also
in an exploration of the mesoscales, described by Wolfgang
Ostwald in 1915 as 'the world of neglected dimensions', where
molecular properties have been left behind but bulk properties
not yet attained. This is perhaps the scale at which complexity
is manifested most profoundly: before the homogeneity of the
bulk, after the discreteness of the molecule. We can see the
tension between these two extremes emerging in a number of fields
whose links with supramolecular chemistry are evident: colloid
and cluster science, nanotribology (molecular-scale friction),
quantum electronics. Perhaps most enticingly of all, somewhere in
this middle ground we cross the no-man's land between inanimate
matter and life -- and the bridge is provided by chemistry and
its ever more complex entities."
J-M. Lehn and P. Ball: in: Nina Hall (ed.): The New Chemistry.
Cambridge University Press 2000, p.349.
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2. CHAOS AND COMPLEXITY
ON CHAOTIC SYSTEMS
Andreas Albrecht (University of California Davis, US) discusses
chaotic systems. Chaotic behavior is well understood from a
classical perspective, and is typically discussed in the context
of a mathematical "phase space" in which there are dimensions for
both position (x) and momentum (p). A particle at a given instant
can be specified as a point in classical phase space, and the
time development of the particle describes a curve or trajectory
in phase space. In chaotic systems, particles that start out in
virtually identical states (i.e., at very close points in phase
space) rapidly evolve into completely different states (i.e.,
distant parts of phase space). Because nothing is ever measured
with absolute precision, one can never realistically talk about
"points" in phase space. Instead, every point (x,p) in phase
space is typically assigned a probability P(x,p). For a well-
specified particle, this probability peaks sharply at a localized
point in phase space. For an ordinary classical object, such as a
single billiard ball, a phase-space probability distribution that
starts out sharply peaked will remain peaked over time; a small
uncertainty in the starting point results in a similar small
degree of ignorance at a later time. Chaotic systems are
dramatically different. A sharply peaked initial distribution
gets torn apart by the chaotic evolution, as neighboring phase-
space trajectories rapidly head off in different directions. A
small amount of ignorance at the beginning rapidly translates
into huge uncertainties later on, as the distribution becomes
highly delocalized.
Nature 2001 412:687
Related Background:
THERMODYNAMICS, CHAOS, AND COMPLEXITY
Ilya Prigogine (Free University Brussels, BE), who received the
Nobel Prize in Chemistry in 1977 for his work in nonequilibrium
thermodynamics, was among the first theoreticians to deal with
the applications of the second law of thermodynamics to complex
systems. The second law of thermodynamics effectively holds that
physical systems tend to slide spontaneously and irreversibly
toward a state of disorder (an increase of entropy). There is no
explanation in classical thermodynamics, however, of how complex
systems can arise spontaneously from less ordered states and
maintain themselves in apparent defiance of the tendency toward
entropy. Prigogine has proposed that as long as systems receive
energy and matter from an external source, nonlinear systems
("dissipative structures") can pass through periods of
instability and then self-organization, resulting in more complex
systems whose characteristics cannot be predicted except as
statistical probabilities. The work of Prigogine has been
influential in a wide variety of fields, ranging from physical
chemistry to biology, and this work has been fundamental in the
new disciplines of chaos theory and complexity theory.
What is called "complexity theory" is a theory that proposes that
certain systems manifest behaviors that are completely
inexplicable by any conventional analysis of the constituent
parts of the system. These behaviors, commonly called "emergent
behaviors", apparently occur in many complex systems involving
living organisms. One example is the idea that human
consciousness is an emergent property of a complex network of
neurons in the brain. The major problem of complexity theory is
how to model such emergent behavior: how to devise mathematical
laws that allow emergent behavior to be explained and predicted.
This effort to establish a solid theoretical foundation for the
description of complex systems has attracted mathematicians,
physicists, biologists, economists, and social scientists.
In the research context, complexity and "chaotic behavior" are
not synonymous. If one focuses attention on the time evolution of
an emergent behavior, e.g., daily changes in temperature, that
behavior may well be completely deterministic yet
indistinguishable from a random process: the behavior is chaotic.
However, although chaos is often associated with complex systems,
not all complex systems manifest chaotic behavior. From the
standpoint of systems theory, it is the interactions of
components that create emergent patterns that are important, and
not any chaotic behavior these may patterns may display.
Massimo Pigliucci (University of Tennessee Knoxville, US)
presents a review of current ideas in chaos and complexity
theory, the author making the following points:
1) The author points out that in common non-scientific usage the
term "chaos" is a synonym for randomness, for completely non-
deterministic and irregular phenomena. In mathematical theory,
however, the term "chaos" refers to a deterministic (i.e., non-
random) phenomenon characterized by special properties that make
the predictability of outcomes very difficult: chaotic behavior
is such that although it does not occur randomly, it has the
appearance of a series of random occurrences.
2) Chaotic dynamics are usually (but not always) a property of
nonlinear systems (i.e., systems whose behavior can be described
by sets of nonlinear equations). However, the converse is not
true: not all nonlinear dynamics generate chaotic behavior.
Typically, a given system of equations can produce both non-
chaotic and chaotic outcomes, depending on the range of values
assumed by the parameters of the equations. In many systems, one
can increase the value of a key parameter and obtain a
progression of outcomes from a steady equilibrium state to
regular oscillations with two equilibria, to more complex cycles
with multiple equilibria, to finally producing the chaotic
condition.
3) Another phenomenon typically associated with chaos is the so-
called "butterfly effect": chaos is analogous to a situation in
which the flapping of a butterfly's wings in Brazil ends up
starting a cascade of events that results in a tornado in Texas.
The term for this is "sensitivity to initial conditions": a small
perturbation of a system can cause a series of effects that
eventually lead to macroscopic consequences later in the time
sequence. Had that perturbation been of a different nature, an
entirely different series of events would have occurred. a more
formal way to describe the butterfly effect is to state that the
predictability of the system decreases exponentially with time:
our predictions of where the system will be are relatively good
for the immediate future, but lose accuracy for slightly longer
intervals of time, and are soon completely useless.
4) In general, a chaotic system is one whose mathematical
function is characterized by at least one of the following: a)
The system has sensitive dependence on initial conditions on its
domain; and/or b) the system has a positive *Lyapunov exponent at
each point in its domain that is not eventually periodic. A
"Lyapunov exponent" is a convenient measure of how fast the
trajectories of the system diverge in *phase space: if the
exponent is negative, the system actually converges at an
equilibrium point; if the exponent is near zero, the system
behaves with periodic regularity; if the exponent is positive,
the system is either chaotic or (for very large positive
exponents) random.
5) Chaos theory is a component of a larger but more vague
theoretical framework called "complexity theory". Essentially,
complexity theory is an attempt to study systems that satisfy two
conditions: a) the system is made of many interacting parts; b)
the interactions result in emergent properties that are not
immediately reducible to a simple sum of the properties of the
individual components. In general, complexity theory uses
nonlinear dynamical modeling to account for the behavior of
orderly complex systems. The dynamics manifested by a given
system depend fundamentally on two parameters: the number of
parts (N) that compose the system, and the average number of
connections (K) among the parts within the system. So-called "NK"
systems then fall into 3 types, depending on the relationship
between N and K:
... ... a) K very small compared to N: Number of connections very
small compared to the total number of parts: Each part behaves
essentially independently of other parts, and the properties of
the system are the properties of the individual parts. Such
systems tend to be static or reach simple dynamic equilibria, and
are sometimes called "sub-critical".
... ... b) K increasing compared to N: The dynamics becomes more
complex and emergent properties appear: Local changes propagate
to distant parts of the system as a consequence of connectivity,
but this propagation usually does not cause global change, since
the ratio of K to N is still relatively small. Such systems are
called "edge of chaos" systems, or "critical systems".
... ... c) K approaches N: Most components of the system are
connected to almost every other component: This creates the
determinate but unstable "supercritical" systems described by
chaos theory.
In terms of Lyapunov exponents: a) subcritical NK systems have a
negative Lyapunov exponent; b) critical NK systems have a
Lyapunov exponent near zero; c) chaotic NK systems are
characterized by a positive Lyapunov exponent. Most classical
mathematics, physics, and biology deal with subcritical systems;
chaos theory and fractal geometry deal with supercritical
systems; complexity theory focuses on critical systems and the
transition between system types. Alleged examples of critical
systems (i.e., systems on the "edge of chaos") include the
evolution of natural populations, the developmental biology of
plants and animals, the stock market, the global economy, and the
dynamics of galaxy clusters.
Skeptic 2000 vol.8 No.3
Notes:
... ... *Lyapunov exponent: See related background material
below.
... ... *phase space: See related background material below.
Related Background:
THEORETICAL PHYSICS: CHAOS AND NONLINEAR DYNAMICS
In general, a nonlinear dynamical system is a system described by
time-dependent differential equations such that the rates of
change of one or more dependent variables of the system depend in
a nonlinear fashion on the variables themselves. Certain
nonlinear dynamical systems, some of which are of great
scientific interest, exhibit "chaotic dynamics". In this context,
the term "chaos" refers to unpredictable behavior arising in a
system that obeys deterministic laws but exhibits
unpredictability. The essential idea is that in certain systems
small perturbations may produce a cascade of larger
perturbations, so that eventually the behavior of such systems
cannot be predicted from prior states no matter if the systems
appear simple and obey deterministic laws. Examples of chaotic
nonlinear dynamical systems are the weather and populations of
organisms, and instances of chaotic dynamics have now been
documented in most scientific disciplines. Because the
differential equations for many nonlinear systems are often
intractable (i.e., no explicit quantitative solutions are
possible), a focus of theoretical research on nonlinear systems
has been on analysis of the qualitative behavior of such systems,
in particular on analysis of the "phase space" and "trajectories"
in the phase spaces of such systems. The idea is essentially as
follows: If the state of a system depends upon N variables, the
instantaneous state of the system can be viewed as a point (phase
point) in an N-dimensional space (phase space; system
hyperspace), and as the state of the system changes, its phase
point can be viewed as describing a trajectory in its phase
space. Qualitative analysis of the possible families of solutions
of nonlinear differential equations can provide information about
such phase space trajectories, and there are certain real systems
for which qualitative analysis of the phase space trajectories of
the system has revealed significant properties of the system
otherwise difficult to delineate.
J.P. Gollub and M.C. Cross (2 installations, US) present a
commentary on recent research on chaotic nonlinear dynamics, the
authors making the following points:
1) The techniques of nonlinear dynamics are well-developed, but
the impact of this field has been largely confined to phenomena
in which there are only a few important time-dependent
quantities. Unfortunately, this excludes a vast range of
important problems in which the behavior of one point in space
can be quite different (though statistically similar) to that at
another location. A particular example is convective behavior.
2) The traditional approach to studying nonlinear dynamical
behavior is to plot the dynamical variables of the system as a
multidimensional phase space graph indicating how the behavior
changes over time. For example, a simplified model of the Solar
System consisting of the Sun and 9 planets would require a phase
space with as many as 60 dimensions (3 position and 3 momentum
coordinates for each body). In the case of a convecting fluid, a
complete description of the flow pattern requires knowledge of
the velocity and temperature at a very large number of locations,
so the number of dimensions of the phase plot are enormous (from
thousands to millions, depending on the desired spatial
resolution). As a result, the methods of nonlinear dynamics are
cumbersome and progress has been slow, even though many
interesting examples of spatiotemporal chaos have been explored
both experimentally and numerically.
3) Recent research (D.A. Egolf et al: Nature 404:733 2000)
involving numerical studies of an accepted model of thermal
convection indicates that the origin of unpredictable motion in
chaotic thermal convective systems, at least in one particular
form of spatiotemporal chaos, lies in what occurs in small
regions of space and over short time-scales. These local changes
in the organization of the flow affect the surrounding regions in
such a way that the entire future evolution is affected. The
authors state: "This is something akin to Ed Lorenz's famous
remark [E.N. Lorenz: J. Atmos. Sci. 20:130 1963] that the
localized flapping of a butterfly's wings might change the
weather dramatically over the entire world a few weeks later."
Although such sensitivity to localized fluctuations has never
been confirmed as the source of the unpredictability of the
weather, it is apparently the origin of chaotic dynamics in
thermal convection.
4) The authors conclude: "The methods used by Egolf et al should
apply to many other forms of chaos in spatially extended systems
(physical, chemical, and biological) for which reliable model
equations are available, so that the key processes leading to the
complex dynamics can be identified. Applications to areas as
diverse as cardiology and atmospheric dynamics might be expected
eventually. Moreover, it is not unreasonable to imagine that
insight into the processes leading to unpredictability will also
lead to progress in modifying or controlling the dynamics of
these systems."
Nature 2000 404:710
Related Background:
EXPERIMENTAL EVIDENCE FOR MICROSCOPIC CHAOS
In the study of physical systems, the term "chaotic behavior" has
a specific meaning: the behavior of a system is said to be
"chaotic" if its final state is so sensitive to the system's
precise initial conditions that the behavior of the system is in
effect unpredictable and cannot be distinguished from a random
process, even though the behavior of the system is strictly
determinate in a mathematical sense. In other words, a
deterministic system characterized by extremely sensitive
instabilities, despite the system being determinate, can exhibit
behavior that is unpredictable, and the system is then called
"chaotic". During the past several decades, the analysis of such
chaotic systems has intrigued both physicists and mathematicians.
In general, in the study of physical systems, the term "phase
space" refers to a multidimensional space, each point of which
(phase point) completely represents the state of the system. For
example, in the study of dynamical systems, each phase point in
the phase space completely represents the values of all the
generalized coordinates and corresponding momenta. As the phase
point of a system moves in the phase space (e.g., changes with
time), the phase point follows a trajectory in the phase space,
and this trajectory is called the "phase point trajectory". In
the mathematical analysis of a particular phase space and its
phase point trajectories, "*Lyapunov exponents" are coefficients
that describe the rates at which nearby phase point trajectories
converge or diverge, and the Lyapunov exponents can be shown to
provide estimates of how long the behavior of a dynamical system
is predictable before chaotic behavior sets in. Chaotic behavior
of a system is characterized by the existence of positive
Lyapunov exponents.
Gaspard et al present the results of an experimental study of
"microscopic chaos". The authors point out that many macroscopic
dynamical phenomena, for example in hydrodynamics and oscillatory
chemical reactions, have been observed to display erratic or
random time evolution, despite the deterministic character of
their dynamics -- a phenomenon known as "macroscopic chaos". On
the other hand, it has been long supposed that the existence of
chaotic behavior in the microscopic motions of atoms and
molecules in fluids or solids is responsible for their
equilibrium and non-equilibrium properties. But, the authors
state, this hypothesis of microscopic chaos has never been
verified experimentally. The authors now report direct
experimental evidence for microscopic chaos in fluid systems, the
study involving the *observation of brownian motion of a
colloidal particle suspended in water. The authors report finding
a positive lower bound on the sum of positive Lyapunov exponents
of the system composed of the brownian particle and the
surrounding fluid. They suggest their results and quantitative
analysis provide strong experimental evidence for microscopic
chaos. They conclude: "On the assumption that the system is
deterministic, and given our knowledge of the molecular structure
of the fluid, this evidence supports, in particular, the
hypothesis that large systems -- which may be treated by
statistical mechanics -- are typically chaotic. The result also
supports the role of dynamical instability in non-equilibrium
fluids."
Nature 1998 394:865
Notes:
... ... *Lyapunov: A.M. Lyapunov (1857-1918) developed a general
theory of dynamic stability applicable to both linear and
nonlinear systems. His work was largely buried and forgotten
until it was exhumed nearly 30 years after his death.
... ... *observation of brownian motion: The experiment here
involved a colloidal particle of 2.5 microns diameter moving in
suspension in deionized water at 22 degrees Celsius, with
recorded observations of 145,612 positions over a total time
interval of approximately 2430 seconds, the observations
involving a microscope and video camera, the smallest resolution
stated as 25 nanometers. Particles of this size undergo
sedimentation, which may confound the results with non-Brownian
effects, but the authors report studies of non-sedimenting
smaller particles substantiate their observations, the larger
particle simply allowing tracking observations for a longer time.
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3. BIOLOGICAL VS. ENGINEERING COMPLEXITY
Although natural selection does not guarantee that organisms will
increase in complexity as they evolve, it is apparent that the
complexity of certain lineages, such as our own, has increased
during evolution. But despite our intuitive notion of biological
complexity -- in terms of morphological or behavioral complexity,
or the variety of cell types in an organism -- the term itself is
notoriously difficult to define. Is the number of genes in the
genome of an organism an appropriate measure of biological
complexity? It has been assumed that eukaryotes have more genes
than bacteria, that animals have more genes than plants, and that
vertebrates have more genes than invertebrates -- which fits with
the traditional idea of a "scala naturae". But recent completed
genome sequences indicate this is not necessarily the case:
surprisingly, it turns out that the nematode worm C. elegans has
18,424 genes in its genome, the fruit fly. Drosophila has 13,601
genes, the plant Arabidopsis approximately 25,498, and humans
approximately 35,000 genes. This suggests there must be other and
more sensible genomic measures of complexity than the mere number
of genes.
Transcription factors are DNA binding proteins that switch target
genes on and off. For all transcription factor families, their
members increase in number in the order yeast, nematode, fruit
fly, human. The diversity of cell types in these organisms also
increases in that order. This makes sense, given that maintaining
the differential state of increasingly diverse cell types
requires the presence of more and more molecular switches.
Claverie (2001) has suggested that we define biological
complexity in terms of the number of "transcriptome" states that
the genome of an organism can achieve, with a transcriptome
defined as the complete set of RNA transcripts. (Szathmary et al:
Science 2001 292:1315)
M.E. Csete and J.C. Doyle (University of Michigan, US) discuss
biological complexity, the authors making the following points:
1) The theory and practice of complex engineering systems have
progressed so radically that they often embody Arthur C. Clarke's
dictum, "Any sufficiently advanced technology is
indistinguishable from magic." Systems-level approaches in
biology have a long history (1, 2) but are just now receiving
renewed mainstream attention (3-5), whereas systems-level design
has consistently been at the core of modern engineering,
motivating its most sophisticated theories in controls,
information, and computation. The hidden nature of complexity
("magic") and discipline fragmentation within engineering have
been barriers to a dialog with biology. A key starting point in
developing a conceptual and theoretical bridge to biology is
robustness, the preservation of particular characteristics
despite uncertainty in components or the environment.
2) Biologists and biophysicists new to studying complex networks
often express surprise at a biological network's apparent
robustness. They find that "perfect adaptation" and homeostatic
regulation are robust properties of networks, despite
"exploratory mechanisms" that can seem gratuitously uncertain.
Some even conclude that these mechanisms and their resulting
features seem absent in engineering. However, ironically, it is
in the nature of their robustness and complexity that biology and
advanced engineering are most alike. Good design in both cases
(e.g., cells and bodies, cars and airplanes) means that users are
largely unaware of hidden complexities, except through system
failures. Furthermore, the robustness and fragility features of
complex systems are both shared and necessary. Although the need
for universal principles of complexity and corresponding
mathematical tools is widely recognized, sharp differences arise
as to what is fundamental about complexity and what mathematics
is needed.
3) The differences between biology and technology (and between
organisms) are obvious, particularly at the molecular and device
level. Nevertheless, convergent evolution, a well-established
concept in both engineering and evolutionary biology, yields
remarkable similarities at higher levels of organization.
Recently, engineering systems have begun to have almost
biological levels of complexity. For example, a Boeing 777 is
fully "fly-by-wire" with 150,000 different subsystem modules,
organized via elaborate protocols into complex control systems
and networks, including roughly 1000 computers that can automate
all vehicle functions. In terms of cost and complexity, the 777
is essentially a vast control system and computer network that
just happens to fly. The consequence of good design is that its
regulatory complexity is hidden from passengers (except when they
use entertainment systems). The internal activity level is
staggering, however (e.g., the data rate recorded on the internal
state during final production testing is nearly equivalent to one
human genome every minute). Commercial aircraft are not the only
systems undergoing such explosions in complexity as a result of
advanced controls and embedded networking; virtually all
technologies are evolving similarly. The authors suggest that
this technological evolution of complexity is convergent with
that of biology.
References (abridged):
1. L. von Bertalanffy, Modern Theories of Development: An
Introduction to Theoretical Biology (Oxford Univ. Press, New
York, 1933).
2. M. A. Savageau, Biochemical Systems Theory (Addison-Wesley,
Reading, MA, 1976).
3. M. A. Savageau, Genetics 149, 1665 (1998)
4. C. V. Rao and A. P. Arkin, Annu. Rev. Biomed. Eng. 3, 391
(2001)
5. M. Dickinson, et al., Science 288, 100 (2000)
Science 2002 295:1664
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4. COMPLEX NETWORKS
ON THE DIVERSITY OF COMPLEX NETWORKS
Steven H. Strogatz (Cornell University, US) discusses complex
networks, the author making the following points:
1) Networks and the dangers of networks are currently a focus of
attention. On 10 August 1996, a fault in two power lines in
Oregon led, through a cascading series of failures, to blackouts
in 11 US states and two Canadian provinces, leaving approximately
7 million customers without power for up to 16 hours(1). The Love
Bug worm, the worst computer attack to date, spread over the
Internet on 4 May 2000 and inflicted billions of dollars of
damage worldwide. At the other extreme, networks have been the
theme of various parlor games(2,3).
2) Empirical studies have shed light on the topology of food
webs(4,5), electrical power grids, cellular and metabolic
network, the World-Wide Web, the Internet backbone, the neural
network of the nematode worm Caenorhabditis elegans, telephone
call graphs, co-authorship and citation networks of scientists,
and the quintessential 'old-boy' network, the overlapping boards
of directors of the largest companies in the United States. These
databases are now easily accessible, courtesy of the Internet.
Moreover, the availability of powerful computers has made it
feasible to probe their structure; until recently, computations
involving million-node networks would have been impossible
without specialized facilities.
3) Why is network anatomy so important to characterize? Because
structure always affects function. For instance, the topology of
social networks affects the spread of information and disease,
and the topology of the power grid affects the robustness and
stability of power transmission. From this perspective, the
current interest in networks is part of a broader movement
towards research on complex systems. In the words of E. O.
Wilson, "The greatest challenge today, not just in cell biology
and ecology but in all of science, is the accurate and complete
description of complex systems. Scientists have broken down many
kinds of systems. They think they know most of the elements and
forces. The next task is to reassemble them, at least in
mathematical models that capture the key properties of the entire
ensembles."
4) In summary: The study of networks pervades all of science,
from neurobiology to statistical physics. The most basic issues
are structural: how does one characterize the wiring diagram of a
food web or the Internet or the metabolic network of the
bacterium Escherichia coli? Are there any unifying principles
underlying their topology? From the perspective of nonlinear
dynamics, we would also like to understand how an enormous
network of interacting dynamical systems -- be they neurons,
power stations or lasers -- will behave collectively, given their
individual dynamics and coupling architecture. Researchers are
only now beginning to unravel the structure and dynamics of
complex networks.
References (abridged):
1. Western Systems Coordinating Council (WSCC). Disturbance
Report for the Power System Outage that Occurred on the Western
Interconnection on August 10th, 1996 at 1548 PAST
http://www.wscc.com (October 1996).
2. Anonymous. Media: Six degrees from Hollywood. Newsweek 11
October 1999, 6 (1999).
3. Kirby, D. & Sahre, P. Six degrees of Monica. New York Times 21
February 1998, op. ed. page (1998).
4. Cohen, J. E., Briand, F. & Newman, C. M. Community Food Webs:
Data and Theory (Springer, Berlin, 1990).
5. Williams, R. J. & Martinez, N. D. Simple rules yield complex
food webs. Nature 404, 180-183 (2000).
Nature 2002 410:268
Related Background:
STATISTICAL MECHANICS OF COMPLEX NETWORKS
R. Albert and A-L. Barabasi (University of Notre Dame, US)
discuss complex networks, the authors making the following
points:
1) Complex weblike structures comprise a wide variety of systems
of high technological and intellectual importance. For example,
the biological cell is best described as a complex network of
chemicals connected by chemical reactions; the Internet is a
complex network of routers and computers linked by various
physical or wireless links; fads and ideas spread on the social
network, whose nodes are human beings and whose edges represent
various social relationships; the World Wide Web is an enormous
virtual network of Web pages connected by hyperlinks. These
systems represent just a few of the many examples that have
recently prompted the scientific community to investigate the
mechanism that determine the topology of complex networks, and
the desire to understand such interwoven systems has encountered
significant challenges.
2) Physics, a major beneficiary of reductionism, has developed an
arsenal of successful tools for predicting the behavior of a
system as a whole from the properties of its constituents. For
example, we now understand how magnetism emerges from the
collective behavior of millions of spins, or how quantum
particles lead to such spectacular phenomena as Bose-Einstein
condensation or superfluidity. The success of these modeling
efforts is based on the simplicity of the interactions between
the elements: there is no ambiguity as to what interacts with
what, and the interaction strength is uniquely determined by the
physical distance.
3) We are at a loss, however, to describe systems for which
physical distance is irrelevant or for which there is ambiguity
as to whether two components interact. While for many complex
systems with nontrivial network topology such ambiguity is
naturally present, in the past few years it has been increasingly
recognized that the tools of statistical mechanics offer an ideal
framework for describing these interwoven systems as well. These
developments have introduced new and challenging problems for
statistical physics and unexpected links to major topics in
condensed matter physics, ranging from percolation to Bose-
Einstein condensation. Concerning complex networks in general,
while traditionally such systems have been modeled in random
graphs, it has become increasingly recognized that the topology
and evolution of real networks that deviate from random graphs
are governed by robust organizing principles that need
quantitative formulation.
Rev. Mod. Phys. 2002 74:47
References (abridged):
1. Aiello, W., F. Chung, and L. Lu, 2000, Proceedings of the 32nd
ACM Symposium on the Theory of Computing (ACM, New York), p.171.
2. ben Avraham, D., and S. Havlin, 2000, Diffusion and Reactions
in Fractals and Disordered Systems (Cambridge University Press,
Cambridge/New York).
3. Bollobas, B., 1985, Random Graphs (Academic, London). Bunde,
A., and S. Havlin, 1996, Eds., Fractals and Disordered Systems
(Springer, Berlin).
4. Crisanti, A., G. Paladin, and A. Vulpiani, 1993, Products of
Random Matrices in Statistical Physics (Springer, Berlin).
Related Background:
ON THE INTERNET AS A COMPLEX SYSTEM
W. Willinger et al (ATT Labs-Research, US) discuss the Internet,
the authors making the following points:
1) Today's Internet is a prime example of a large-scale, highly
engineered, yet highly complex system. It is characterized by an
enormous degree of heterogeneity any way one looks and continues
to undergo significant changes over time. In terms of size, by
mid-2001, the Internet consisted of approximately 120 million
hosts, or endpoints, and more than 100,000 distinct networks,
totaling millions of routers and links connecting the hosts to
the routers and the routers to one another. These links differ
widely in speed (from slow modem connections to high-speed
"backbone links) as well as in technology (e.g., wired, wireless,
satellite communications).
2) At the largest scale, the Internet is divided into autonomous
systems, each of which is a collection of routers and links under
a single administrative domain. The global Internet currently
consists of several thousand separate autonomous systems
interlinked to give users the illusion of a single seamlessly
connected network capable of providing a universal data-delivery
service. The foundation of the ubiquitous connectivity is a
datagram (packet) delivery mechanism termed the Internet Protocol
(IP).
3) Given all of the efforts devoted to understanding today's
Internet, it is surprising how often networking researchers
observe "emergent phenomena" -- measurement-driven discoveries
that come as a complete surprise, cannot be explained or
predicted within the framework of the traditionally considered
mathematical models -- phenomena which depend crucially on the
large-scale nature of the Internet, and which are almost always
absent when considering small-scale IP networks.
Proc. Nat. Acad. Sci. 2002 99:2573
Related Background:
ERROR AND ATTACK TOLERANCE OF COMPLEX NETWORKS
As the Internet and the World Wide Web continue to tie the
disparate parts of Earth together in a vast multi-functional
communications network, the question of the vulnerability of this
network to attack becomes more and more important.
R. Albert et al (University of Notre Dame, US) report an analysis
of complex networks, the authors making the following points:
1) Many complex systems display a surprising degree of tolerance
against errors. For example, relatively simple organisms grow,
persist, and reproduce despite drastic pharmaceutical or
environmental interventions, an error tolerance attributed to the
robustness of the underlying metabolic network. Complex
communications networks also display a surprising degree of
robustness: although key components regularly malfunction, local
failures only rarely lead to the loss of the global information-
carrying ability of the network. The stability of these and other
complex systems is often attributed to the redundant wiring of
the functional web defined the components of the system.
2) The authors report their analysis demonstrates that error
tolerance is not shared by all redundant systems: it is displayed
only by a class of inhomogeneous wired networks, called "*scale-
free networks", which include the World Wide Web, the Internet,
social networks, and biological cells. Such networks display an
unexpected degree of robustness, the ability of their nodes to
communicate being unaffected even by unrealistically high failure
rates. Error tolerance, however, comes at a high price in that
these networks are extremely vulnerable to attacks (i.e., to the
selection and removal of a few nodes that play a vital role in
maintaining the network's connectivity). The authors suggest such
error tolerance and attack vulnerability are generic properties
of communications networks.
3) The authors conclude: "Although it is generally thought that
attacks on networks with distributed resource management are less
successful, our results indicate otherwise. The topological
weaknesses of the current communication networks, rooted in their
inhomogeneous connectivity distribution, seriously reduce their
attack survivability. This could be exploited by those seeking to
damage these systems."
Nature 2000 406:378
Notes:
... ... *scale-free networks: The authors suggest that existing
empirical and theoretical results indicate that complex networks
can be divided into two major classes based on their connectivity
distribution P(k), giving the probability that a node in the
network is connected to k other nodes. The first class of
networks is characterized by a P(k) that peaks at an average [k]
and decays exponentially for large k. Examples are certain random
graph models and "small-world" models known to mathematicians
[e.g., work of Erdos and Renyi (1960), Watts and Strogatz
(1998)], both leading to a fairly homogeneous network in which
each node has approximately the same number of links. In
contrast, results on the World Wide Web, the Internet, and other
large networks indicate that many systems belong to a class of
inhomogeneous networks, called "scale-free networks", for which
P(k) decays as a power-law free of a characteristic scale.
Whereas the probability that a node has a very large number of
connections is practically prohibited in exponential networks,
highly connected nodes are statistically significant in scale-
free networks.
Related Background Brief:
MOLECULAR INTERACTION MAP OF THE MAMMALIAN CELL CYCLE CONTROL AND
DNA REPAIR SYSTEMS. Eventually, to understand the integrated
function of the cell cycle regulatory network, we must organize
the known interactions in the form of a diagram, map, and/or
database. The author presents a diagram convention capable of
unambiguous representation of networks containing multiprotein
complexes, protein modifications, and enzymes that are substrates
of other enzymes. To facilitate linkage to a database, each
molecular species is symbolically represented only once in each
diagram. Molecular species can be located on the map by means of
indexed grid coordinates. Each interaction is referenced to an
annotation list where pertinent information and references can be
found. Parts of the network are grouped into functional
subsystems. The map shows how multiprotein complexes could
assemble and function at gene promoter sites and at sites of DNA
damage. It also portrays the richness of connections between the
p53-Mdm2 subsystem and other parts of the network. K.W. Kohn: Mol
Biol Cell 1999 10:2703.
Related Background Brief:
THE LARGE-SCALE ORGANIZATION OF METABOLIC NETWORKS. In a cell or
microorganism, the processes that generate mass, energy,
information transfer and cell-fate specification are seamlessly
integrated through a complex network of cellular constituents and
reactions. However, despite the key role of these networks in
sustaining cellular functions, their large-scale structure is
essentially unknown. The authors present a systematic comparative
mathematical analysis of the metabolic networks of 43 organisms
representing all three domains of life. The authors demonstrate
that, despite significant variation in their individual
constituents and pathways, these metabolic networks have the same
topological scaling properties and show striking similarities to
the inherent organization of complex non-biological systems. The
authors suggest this may indicate that metabolic organization is
not only identical for all living organisms, but also complies
with the design principles of robust and error-tolerant scale-
free networks, and may represent a common blueprint for the
large-scale organization of interactions among all cellular
constituents. H. Jeong et al: Nature 2000 407:651.
Related Background Brief:
COLLECTIVE DYNAMICS OF "SMALL-WORLD" NETWORKS. Networks of
coupled dynamical systems have been used to model biological
oscillators, Josephson junction arrays,, excitable media, neural
networks, spatial games, genetic control networks and many other
self-organizing systems. Ordinarily, the connection topology is
assumed to be either completely regular or completely random. But
many biological, technological and social networks lie somewhere
between these two extremes. The authors explore simple models of
networks that can be tuned through this middle ground: regular
networks "rewired" to introduce increasing amounts of disorder.
The authors find that these systems can be highly clustered, like
regular lattices, yet have small characteristic path lengths,
like random graphs. The authors call them "small-world" networks,
by analogy with the small-world phenomenon, (popularly known as
six degrees of separation). The neural network of the worm
Caenorhabditis elegans, the power grid of the western United
States, and the collaboration graph of film actors are shown to
be small-world networks. Models of dynamical systems with small-
world coupling display enhanced signal-propagation speed,
computational power, and synchronizability. In particular,
infectious diseases spread more easily in small-world networks
than in regular lattices. D.J. Watts and S.H. Strogatz: Nature
1998 393:440.
Related Background Brief:
CLASSES OF SMALL-WORLD NETWORKS. The authors report a study of
the statistical properties of a variety of diverse real-world
networks. The authors present evidence of the occurrence of three
classes of small-world networks: (a) scale-free networks,
characterized by a vertex connectivity distribution that decays
as a power law; (b) broad-scale networks, characterized by a
connectivity distribution that has a power law regime followed by
a sharp cutoff; and (c) single-scale networks, characterized by a
connectivity distribution with a fast decaying tail. The authors
note for the classes of broad-scale and single-scale networks
that there are constraints limiting the addition of new links.
The authors suggest their results indicate that the nature of
such constraints may be the controlling factor for the emergence
of different classes of networks. L.A. Amaral et al: Proc. Nat.
Acad. Sci. 2000 97:11149.
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5. COMPLEX FLUIDS
ON THE FLOW BEHAVIOR OF COMPLEX FLUIDS
D. Bonn et al (Ecole Normal Superieure Paris, FR) discuss complex
fluids, the authors making the following points:
1) The flow behavior of "complex fluids" such as, for instance,
colloidal suspensions is of both practical and fundamental
interest [1]. The large length scales present in these systems,
when compared to molecular dimensions, can lead to interactions
between the flow field and the organization of the complex
fluids. A structural change can affect the viscosity of the fluid
and thus, in turn, modify the flow field. This leads to a non-
Newtonian viscosity: in general, the resistance to flow decreases
with increasing flow velocity [1,2]. Unfortunately, because of
the existence of long-range hydrodynamic interactions in these
systems, it has turned out to be nearly impossible to predict the
non-Newtonian viscosity on the basis of the structure and/or
interactions in these systems [2].
2) It is for this reason that a completely different approach has
recently been tried to predict non-Newtonian behavior. Instead of
taking all the hydrodynamic interactions into account, one starts
[3-5] from a model of a glassy system that has slow degrees of
freedom: certain states are said to be jammed. This jamming is a
common property of a large number of complex fluids such as
foams, gels, and granular systems, which in general hardly flow
if a small stress is exerted on them. For a glassy system, the
slow modes are affected by an external forcing, which is
associated with a flow [3-5]. These models have the advantage
that both the linear (viscoelastic) response and the nonlinear
behavior under flow, i.e., the non-Newtonian viscosity, can be
calculated explicitly. The second advantage is that this opens,
for the first time, the possibility to relate the macroscopic
rheological behavior to the microscopic dynamics [3-5].
3) The authors report a study of both the nonlinear rheological
behavior and the microscopic dynamics for a typical "soft glassy
material" (the colloidal glass of Laponite), to see whether these
ideas are applicable to a real system. The detailed predictions
that result from the different models and simulations are the
following [3-5]: (i) Without an external forcing, the systems
evolve spontaneously: they are said to age, meaning that the
relaxation time of the slow mode increases in time. (ii) Under an
external drive, the system can reach a steady state: the aging
stops, and the relaxation time is constant. (iii) Upon increasing
the forcing, the relaxation time in steady state decreases, (iv)
Both in the presence of an external drive and during the aging,
the viscosity is given by the (distribution of) relaxation
time(s) of the "slow mode" of the glassy system, (v) The
viscosity decreases strongly with the shear rate (the velocity
gradient) applied to the system.
4) In summary: The authors report a study the nonlinear
rheological behavior and the microscopic particle dynamics for a
colloidal glass, to see whether recently developed models for
driven glassy systems can be applied to predict the rheology.
Qualitatively, all the findings predicted by the models can be
retrieved in the present system system. Notably, the viscosity
decreases strongly with the shear rate. Since it is difficult to
predict non-Newtonian viscosities of colloidal systems due to
long-ranged hydrodynamic interactions, the authors suggests this
demonstrates the promise of this approach for predicting flow
behavior. In addition, the measurements allow one to relate the
microscopic diffusion dynamics to the macroscopic viscosity of
the system.
References (abridged):
1. R.G. Larson, The Structure and Rheology of Complex Fluids
(Oxford University Press, New York, 1999)
2. H.A. Barnes, J. F. Hutton, and K. Walters, An Introduction to
Rheology (Elsevier, Amsterdam, 1989).
3. R. Yamamoto and A. Onuki, Europhys. Lett. 40, 61 (1997); Phys.
Rev. ESS, 3515 (1998).
4. L.F. Cugliandolo, J. Kurchan, P. Le Doussal, and L. Peliti,
Phys. Rev. Lett. 78, 350-353 (1997).
5. P. Sollich, F. Lequeux, P. Hebraud, and M. Gates, Phys. Rev.
Lett. 78, 2020 (1997); P. Sollich, Phys. Rev. E 58, 738 (1998);
S.M. Fielding, P. Sollich, and M.E. Cates, J. Rheol 44, 323
(2000)
Phys. Rev. Lett. 2002 89:015701
Related Background:
ON SIMPLE ORDERING IN COMPLEX FLUIDS
A colloid is a system of particles 1 to 1000 nanometers in
diameter dispersed in another phase. A colloidal crystal is a
periodic array of suspended colloidal particles, the array
arising spontaneously in a "monodisperse" colloidal system under
proper conditions. A monodisperse colloidal system is simply a
colloidal system in which the suspended particles have identical
size, shape, and interaction.
A.P. Gast and W.B. Russel (2 installations, US) review current
research on ordering in complex fluids and 2-dimensional
crystals, the authors making the following points:
1) One important feature of colloidal suspensions is that the
sub-micron particles are subject to constant Brownian motion from
the thermal fluctuations in the surrounding solvent. Thus, to
some degree, the particles can be considered as effective
molecules and treated according to the theories of statistical
mechanics. Because the solvents often contain -- in addition to
the colloidal particles -- dissolved ions, polymer molecules,
surface-active molecules, and other small solutes, colloids are
referred to as a general class of complex fluids.
2) Perhaps one of the most important answers to the dreams of
physicists has been the development of colloidal particles that
interact by means of hard-sphere repulsions... Entropy is usually
thought to bring about disorder. But in a system of hard spheres,
particles gain entropy by arranging themselves equidistantly from
one another to maximize the space in their vicinity, and thus
they are compelled to order.
3) The addition of a "soft" (i.e., long-range) repulsion to
colloidal particles can keep them sufficiently separated that the
*van der Waals attraction is negligible, rendering the suspension
stable against aggregation under a variety of conditions. Most
commonly, aqueous suspensions are stabilized by the *screened
electrostatic repulsion between charges imparted by the *Debye
length, which scales inversely with the square root of the *ionic
strength of the suspension.
4) The disorder-order transition for particles having purely
repulsive interactions persists when attractive interactions are
added... In aqueous systems, adding electrolytes to screen the
electrostatic repulsion between charged particles can induce
aggregation by means of van der Waals attraction.
5) The complex structure of proteins can often be deciphered
using the power of crystallography -- but only if the proteins
can be crystallized. Some proteins form 2-dimensional arrays when
attached to a lipid monolayer floating on top of an aqueous
solution, and the monolayer of proteins can then be transferred
to an electron microscope grid for imaging and study by electron
diffraction... Although proteins remain complex in their detailed
structure and interactions, they provide ample opportunity to
study the general phenomenon of crystallization at both colloidal
and molecular scales.
Physics Today December 1998
Notes:
... ... *van der Waals attraction: (also spelled Van der Waals)
Considering molecules that have permanent dipoles, and molecules
that can have dipoles induced by the electric fields of other
molecules, there are three possible mechanisms recognized in the
formation of the van der Waals bonds: 1) the orientation effect,
in which molecules rearrange themselves in their mutual
electrical fields, the rearrangements involving reorientations of
whole molecules; 2) the static induction effect, in which
molecules that are static monopoles (ions) or dipoles may induce
a static rearrangement of the electron distribution of other
molecules; 3) the dynamic induction effect, or "dispersion"
effect, in which any molecule, polar or nonpolar, may induce in
other molecules transient electron distribution rearrangements
that are time-variant. All these mechanism involve interaction
energies, and they are "bonds" in the sense that they all involve
energetic couplings between molecules.
... ... *screened: In general, screening is a reduction of the
effective electric field at a point, the reduction due to the
space charge of ambient charged particles of sign opposite to the
source of the field.
... ... *Debye length: (Debye shielding length, Debye-Hueckel
screening radius) A characteristic distance in a system of
particles beyond which the electric field of a charged particle
is shielded by particles having charges of the opposite sign.
... ... *ionic strength: A measure of the average electrostatic
interactions among ions in an electrolyte. Quantitatively defined
as one-half the sum of the terms obtained by multiplying the
molality of each ion by its valence squared.
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6. COMPLEXITY IN CHEMISTRY
The term "complexity" is fashionable in science these days, the
interest presumed to indicate a movement away from reductionism,
away from the idea that the behavior of a system is best
understood in terms of how the components of the system behave
and interact. A focus on "complexity", however, is not perforce
anti-reductionist. Indeed, in practice, with real systems, the
behavior of a system is often not predictable from knowledge of
the behavior of its components, but most often this is simply
because that knowledge is incomplete, and not because of any
_principle_ barring prediction of the behavior of the system from
knowledge of its parts. Even systems exhibiting *chaotic
fluctuations are not necessarily non-reductionist, since such
systems are mathematically deterministic. In any case, faced with
an apparent unpredictability of a system given available
information about its parts, one looks for predictive global
methods to understand the system, methods that do not depend upon
a detailed knowledge of the behavior of the components of the
system. Thermodynamics is exactly such a global method of great
utility in chemistry and physics, and since thermodynamics is a
method of analysis that goes back to its originator Carnot in
1824, one can safely say that the idea of special methods to deal
with "complexity" is quite old. In our time, at least for ideal
systems, we can derive the equations of thermodynamics from
statistical mechanics, i.e., derive the global equations from
equations for the behavior of components. But Nicolas Sadi Carnot
(1796-1832) never heard of statistical mechanics, which was
introduced by Boltzmann (1844-1906) in 1871; Carnot founded
thermodynamics as a predictive global method to deal with an
important "complex" system of his time -- the steam engine.
G.M. Whitesides and R.F. Ismagilov (Harvard University, US)
present a review of current ideas in chemistry concerning
"complexity", the authors making the following points:
1) Chemistry has its own understandings of the term "complexity".
In one characterization, a complex system is one whose evolution
is very sensitive to initial conditions or to small
perturbations, one in which the number of independent interacting
components is large, or one in which there are multiple pathways
by which the system can evolve. Analytical descriptions of such
systems typically require nonlinear differential equations. A
second characterization is more informal; that is, the system is
"complicated" by some subjective judgment and is not amenable to
exact description, analytical or otherwise.
2) Faced with the impossibility of handling many real systems
exactly, chemists have evolved a series of approaches to the
treatment of complex systems. These treatments include reasoning
by analogy, averaging, linearization, drastic approximation, pure
empiricism, and detailed analytical solution. The emphasis in
thinking about complicated systems has been to find methods that
are predictive, even if they are non-analytical. "Complexity" per
se, the study of nonlinear processes with high sensitivity to
conditions, has not been the focus of major effort.
3) Chemistry has relied heavily on the ability of ensemble
properties that are obtained through thermodynamics and
statistical mechanics to make it unnecessary to consider the
behavior of individual molecules. However, single-molecule
chemistry is now making it possible to inquire about individual
molecular behaviors, and the behavior of macromolecules is a
promising area of research because of the existence of many
possible molecular conformations, each with different properties.
4) At the core of chemical interest in complexity are the two
fundamental problems concerning life: a) how collections of
molecules give rise to the varieties of behaviors that
characterize cells and organisms; and 2) how individual molecules
might have originally assembled into collections that had the
characteristics of life (energy dissipation, self-replication,
and adaptation). Whether the understanding of complexity at the
molecular level will reveal important elements of the structure
of life is unclear.
5) One of the opportunities in fundamental chemical research is
to learn from biology and to use what is learned to design non-
biological systems that dissipate energy, replicate, and adapt.
Whether such systems would model life is not critical; they would
unquestionably be interesting and probably important.
Science 1999 284:89
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7. PROTEINS AND COMPLEXITY
COMPLEXITY, PROTEINS, AND THE ENERGY LANDSCAPE
Hans Frauenfelder (Los Alamos National Laboratory, US) discusses
proteins and complexity, the author making the following points:
1) What is complexity? Which systems are complex? What are the
crucial concepts in complex systems? A system can be called
complex if it can assume a large number of states or
conformations and if it can carry information. One often hears
even biologists talk about "astronomically large numbers."
Astronomically large numbers are actually very small compared
with biological numbers. They are of the order of 10^(200) or log
n(subastro) of approximately 200. Consider now DNA. It is built
from four different units (bases) and may contain 10^(9) bases.
The number of conceivable DNAs is therefore log n(subbio) of
approximately 10^(8), which is much greater than log n(subastro).
The number of possible proteins is of the order of log n(subprot)
>> 200. Even the number of states that an individual protein can
assume is very large. Biological systems clearly also carry
information. Hence proteins, and in general biological systems,
are complex.
2) Proteins are built from 20 different amino acids (1, 2).
Directed by the DNA, of the order of a few hundred of these
building blocks are linked together into a linear polypeptide
chain. The order in which the different amino acids are inserted
determines structure, function, and dynamics. In the proper
solvent, the chain folds into a compact structure that is often
globular and that has linear dimensions of a few nanometers.
Proteins perform essentially all functions in biological systems.
The textbook picture of a protein is clear: The folded structure
is unique; each atom is in its proper place. The pictures
obtained by x-ray diffraction techniques appear to support this -
- at first sight -- appealing situation. Such proteins would be
aptly characterized by Schroedinger's words, "aperiodic crystals"
(3). Reality, however, is different. Proteins are dynamic and not
static systems, and they must perform motions to execute their
functions. Motions are possible only if a given protein can
assume a large number of somewhat different conformations, for
instance with open and closed channels. Actually, the motions
involve the atoms not just of the protein itself but also of the
hydration shell, a layer of water surrounding the protein. The
structure and dynamics of the protein and the hydration shell can
be characterized by the energy or conformation landscape.
3) The energy landscape is a construct in 3N dimensions, where N
is the number of atoms in the protein and the hydration shell.
The energy landscape contains valleys and saddle points between
valleys. We call each valley a conformational substate. A
substate describes the structure of the entire protein, because
it characterizes the positions of all atoms. Transitions between
substates correspond to protein motions. Unfortunately, it is
difficult to visualize the landscape, because it lives in a
hyperspace. One- or two-dimensional cross sections can give a
misleading impression. One difference between such a
representation and the complete landscape is the path between two
substates. In the low-dimensional cross section, it may appear
that the protein has to overcome many saddles, whereas in reality
only one or two steps may be necessary. One goal of the physics
approach to proteins is the exploration of the energy landscape.
In no protein is the entire landscape known. This state is not
surprising if one contemplates how many years it took to
determine the energy levels of complex nuclei or atom systems
that are far simpler than proteins.
References (abridged):
1. Stryer, L. (1995) Biochemistry (Freeman, New York).
2. Fersht, A. (1999) Structure and Mechanism in Protein Science
(Freeman, New York).
3. Schroedinger, E. (1944) What Is Life? (Cambridge Univ. Press,
Cambridge, U.K.).
Proc. Nat. Acad. Sci. 2002 99:2479
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8. CONSCIOUSNESS AND COMPLEXITY.
G. Tononi and G.M. Edelman (Neurosciences Institute, US) discuss
consciousness and complexity, the authors making the following
points:
1) What is the neural substrate of conscious experience? While
William James concluded that it was the entire brain (1), recent
approaches have attempted to narrow the focus: are there neurons
endowed with a special location or intrinsic property that are
necessary and sufficient for conscious experience? Does primary
visual cortex contribute to conscious experience? Are brain areas
that project directly to prefrontal cortex more relevant than
those that do not (2)? Although heuristically useful, these
approaches leave a fundamental problem unresolved: How could the
possession of some particular anatomical location or biochemical
feature render some neurons so privileged that their activity
gives rise to subjective experience? Conferring this property on
neurons seems to constitute a category error, in the sense of
ascribing to things properties they cannot have (3).
2) The authors pursue a different approach. Instead of arguing
whether a particular brain area or group of neurons contributes
to consciousness or not, the strategy of the authors is to
characterize the kinds of neural processes that might account for
key properties of conscious experience. The authors emphasize two
properties: conscious experience is integrated (each conscious
scene is unified) and at the same time it is highly
differentiated (within a short time, one can experience any of a
huge number of different conscious states).
3) The authors first consider neurobiological data indicating
that neural processes associated with conscious experience are
highly integrated and highly differentiated. The authors then
provide tools for measuring integration (called functional
clustering) and differentiation (called neural complexity) that
are applicable to actual neural processes. This leads to the
formulation of operational criteria for determining whether the
activity of a group of neurons contributes to conscious
experience. These criteria are incorporated into the dynamic core
hypothesis, a testable proposal concerning the neural substrate
of conscious experience (4).
4) In summary: Conventional approaches to understanding
consciousness are generally concerned with the contribution of
specific brain areas or groups of neurons. By contrast, the
authors consider what kinds of neural processes can account for
key properties of conscious experience. Applying measures of
neural integration and complexity, together with an analysis of
extensive neurological data, leads to a testable proposal -- the
dynamic core hypothesis -- about the properties of the neural
substrate of consciousness.
References (abridged):
1. W. James, The Principles of Psychology (Holt, New York, 1890).
2. F. Crick and C. Koch, Cold Spring Harbor Symp. Quant. Biol.
55, 953 (1990) [Medline] ; Nature 375, 121 (1995); S. Zeki and A.
Bartels, Proc. R. Soc. London Ser. B 265, 1583 (1998).
3. G. Ryle, The Concept of Mind (Hutchinson, London, 1949).
4. G. M. Edelman, The Remembered Present (Basic Books, New York,
1989); ___ and G. Tononi, Consciousness: How Matter Becomes
Imagination (Basic Books, New York, in press); see also G. Tononi
and G. M. Edelman, in Consciousness, H. Jasper et al., Eds.
(Plenum, New York, 1998). pp. 245-280.
Science 1998 282:1846
Related Background Brief:
THE ASYNCHRONY OF CONSCIOUSNESS. The authors present below a
simple hypothesis concerning what they believe is a
characteristic of visual consciousness. The hypothesis is derived
from facts about the visual brain revealed in the past quarter of
a century, but it relies most especially on psychophysical
evidence which shows that different attributes of the visual
scene are consciously perceived at different times. This temporal
asynchrony in visual perception reveals, the authors believe, a
plurality of visual consciousnesses that are asynchronous with
respect to each other, reflecting the modular organization of the
visual brain. The authors further hypothesize that when two
attributes (e.g. color and motion) are presented simultaneously,
the activity of cells in a given processing system is sufficient
to create a conscious experience of the corresponding attribute
(e.g. colour), without the necessity for interaction with the
activities of cells in other processing systems (e.g. motion).
Thus, any binding of the activity of cells in different systems
should be more properly thought of as a binding of the conscious
experiences generated in each system. S. Zeki and A. Bartels:
Proc R Soc Lond B Biol Sci 1998 265:1583.
Related Background Brief:
DUAL-TASK INTERFERENCE IN SIMPLE TASKS: DATA AND THEORY. People
often have trouble performing 2 relatively simple tasks
concurrently. The causes of this interference and its
implications for the nature of attentional limitations have been
controversial for 40 years, but recent experimental findings are
beginning to provide some answers. Studies of the psychological
refractory period effect indicate a stubborn bottleneck
encompassing the process of choosing actions and probably memory
retrieval generally, together with certain other cognitive
operations. Other limitations associated with task preparation,
sensory-perceptual processes, and timing can generate additional
and distinct forms of interference. These conclusions challenge
widely accepted ideas about attentional resources and probe
reaction time methodologies. They also suggest new ways of
thinking about continuous dual-task performance, effects of
extraneous stimulation (e.g., stop signals), and automaticity.
Implications for higher mental processes are discussed. H.
Pashler: Psychol Bull 1994 116:220.
Related Background Brief:
TIME-LOCKED MULTIREGIONAL RETROACTIVATION: A SYSTEMS-LEVEL
PROPOSAL FOR THE NEURAL SUBSTRATES OF RECALL AND RECOGNITION. The
author outlines a theoretical framework for the understanding of
the neural basis of memory and consciousness, at systems level.
It proposes an architecture constituted by: (1) neuron ensembles
located in multiple and separate regions of primary and first-
order sensory association cortices ("early cortices") and motor
cortices; they contain representations of feature fragments
inscribed as patterns of activity originally engaged by
perceptuomotor interactions; (2) neuron ensembles located
downstream from the former throughout single modality cortices
(local convergence zones); they inscribe amodal records of the
combinatorial arrangement of feature fragments that occurred
synchronously during the experience of entities or events in
sector; (3) neuron ensembles located downstream from the former
throughout higher-order association cortices (non-local
convergence zones), which inscribe amodal records of the
synchronous combinatorial arrangements of local convergence zones
during the experience of entities and events in sector; (4) feed-
forward and feedback projections interlocking reciprocally the
neuron ensembles in (1) with those in (2) according to a many-to-
one (feed-forward) and one-to-many (feedback) principle. The
author proposes that (a) recall of entities and events occurs
when the neuron ensembles in (1) are activated in time-locked
fashion; (b) the synchronous activations are directed from
convergence zones in (2) and (3); and (c) the process of
reactivation is triggered from firing in convergence zones and
mediated by feedback projections. This proposal rejects a single
anatomical site for the integration of memory and motor processes
and a single store for the meaning of entities of events. Meaning
is reached by time-locked multiregional retroactivation of
widespread fragment records, and only the latter records can
become contents of consciousness. A.R. Damasio: Cognition 1989
33:25.
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