Understanding Complex Systems Symposium

Abstracts

Multi-Resolution Modeling of Protein-DNA Systems

We present a multi-resolution approach to modeling protein-DNA systems containing long DNA loops. The approach combines a coarse-grained model of the DNA loops, based on the classical theory of elasticity, with all-atom model of proteins and protein-DNA interfaces. The coarse-grained part of the model uses boundary conditions obtained from the all-atom part; molecular dynamics simulations of the all-atom part employ forces obtained from the coarse-grained model. An application of the multi-scale modeling to the complex of bacterial proteins lac repressor, catabolite gene activator (CAP), and a 106 bp DNA loop is demonstrated.

Division of labor in ants is studied as one example of a general biological problem: how does the global behavior of a system arise from the behavior of individual agents, and how is the behavior of these agents programmed? Division of labor in ants will be described as a repertoire of tasks, a set of workers, and a mapping of workers to tasks, with the challenge being to understand this mapping. Our working hypothesis is based on variation in response thresholds: we assume that there are stimuli specific to each task, and that workers vary in their responsiveness to each stimulus. This simple model can explain two striking features of the division of labor: specialization of workers on restricted sets of tasks, and behavioral flexibility in response to changes in task needs or in the set of workers.

Experiments are being done with leaf-cutting ants (Atta ) performing a single task, undertaking, for which the stimulus (a dead ant) is known and can be systematically varied. Results thus far show that 1) about 30% of the workers in a colony perform undertaking, 2) if these workers are removed they will be replaced in part, 3) the colony response becomes more rapid and reliable with repeated stimulation. Preliminary results also suggest that individual responsiveness is not conditioned by experience, and that varying the stimulus quality of the dead ant does not affect the colony response.

The potential implications of these results will be discussed with respect to the response threshold model and in the broader context of division of labor and colony design in ant colonies.

We use numerical modeling to study the combined effects of concurrent physical transport (by advection, diffusion, and dispersion), chemical reaction, and biological populations on groundwater flows in the subsurface of the Earth's crust. Natural groundwater flows host significant microbial populations, and these populations are distributed in zones according to metabolism. Some zones, for example, contain bacteria that use dioxygen to oxidize organic compounds, other zones contain sulfate reducers or methanogens. In examining this problem, we first examine the rate of microbial metabolism, which we show depends on both kinetic factors (how strongly do substrate species bind to enzymes?) and thermodynamic concerns (how much energy is available in the environment?). We then show that a proper description of metabolic rate coupled to laws of physical transport and chemical reaction leads to a system of equations that describes self-organization of the groundwater flow.

Foreshocks, subevents, and aftershocks are pragmatic, intuitive notions to characterize earthquake sequences in time. In such a scheme, a critical factor is the true size of an earthquake, the seismic moment tensor, which must be determined for each earthquake from observed displacement fields. The seismic moment tensor also contains the orientation of earthquake rupture. In an earthquake sequence, individual ruptures should lie in the same seismogenic fault system. However, the notion of spatial configuration is often overlooked.
The devastating Chi-Chi earthquake sequence of September 20, 1999 in central Taiwan occurred within several dense seismic arrays which yield a vast data set of unprecedented resolution. In addition, information on subsurface geology is available from exploration work for hydrocarbon, well-logging, and geodesy. Using this combined data set, we show that this earthquake sequence took place over several distinct seismogenic zones of different fault geometry with a wide range in pressure and temperature. The complex sequence of events in time seem to settle into conventional patterns when the role of spatially distinct seismogenic zones are taken into account. At the same time, close proximity of events in both space and time suggests complex interaction between seismogenic zones. This poses a major challenge in that none of the seismogenic zones are predicted by commonly accepted mechanical models such as the critical taper.

A bulk 1cm x 1cm x 1cm sample of water containing polystyrene nanospheres, 20nm in diameter at a 7 percent volume fraction is optically transparent, and can be represented as a homogeneous thermodynamic liquid phase. This homogeneous thermodynamic phase can be viewed as a simple fluid containing 20nm diameter indistinguishable spherical molecules. If the size and volume fraction of the nanosperes in the water is varied, it is observed in experiment that the transparency and homogeneity of the bulk medium is lost, if the nanospheres exceed 70nm in diameter at greater than 1 percent volume fraction. In this circumstance a complex fluid , containing small distinguishabe classical objects emerges. These considerations make it possible to draw a line of demarkation between small distinguishable classical objects and large indistinguishable spherical molecules. A photon that passes through a homogeneous or inhomogeneous medium eluscidate Gibbs', Einstein's, and Dirac's works.

If everthing in a turbulent systems is in constant flux, how is it that humans are able to distinguish different kinds of turbulence? Hopf's answer was that dynamics drives a given spatially extended system through a repertoire of unstable patterns; as we watch a given ``turbulent'' system evolve, every so often we catch a glimpse of a familiar pattern. For any finite spatial resolution, the system follows approximately for a finite time a pattern belonging to a finite alphabet of admissible patterns, and the long term dynamics can be thought of as a walk through the space of such patterns, just as chaotic dynamics with a low dimensional attractor can be thought of as a succession of nearly periodic (but unstable) motions.

Hopf's proposal is in its spirit very different from most ideas that animate current turbulence research. It is not the Kolmogorov's 1941 homogeneous turbulence with no coherent structures fixing the length scale, here all the action is in specific coherent structures. It is emphatically not universal; spatiotemporally periodic solutions are specific to the particular set of equations and boundary conditions. And it is not probabilistic; everything is fixed by the deterministic dynamics with no probabilistic assumptions on the velocity distributions or external stochastic forcing.

I will describe a modest implementation of Hopf's program on a 1-dimensional spatially extended Kuramoto-Sivashinsky system, a PDE that describes interfacial instabilities such as unstable flame fronts.

Magnets, earthquake faults, the stockmarket and many other systems respond to slowly changing external conditions with discrete, impulsive events that span a huge range of sizes (Barkhausen noise or avalanches in the case of magnets, and earthquakes in the case of the earth). We model Barhausen noise in disordered magnets as a representative of these systems and compute predictions for the universal aspects of the behavior on long length scales as a function of disorder and history, using ideas from phase transitions and disordered systems theory. Similar ideas can also be applied to the interpretation of the Gutenberg-Richter scaling law in the statistics of earthquakes.

For a layered, electrostatic self-assembly, we present a simple, quantitative model to explain the observed layer-dependent ionization of a weak polyacid as layers of polyelectrolytes are sequentially deposited. The model provides an understanding, and expected general features, of the layer-dependent fractional ionization of an embedded polyacid by including the effects of electrostatics and of chemical response of the solution within the limits set by buffer-capacity.

The oscillation spectrum of vortex rings has a long history, dating back to W. Thomson (Lord Kelvin) and J. J. Thomson, and is known in the long wavelength limit as the Kelvin mode spectrum. In this talk, a derivation of the oscillation spectrum of vortex rings in incompressible, inviscid fluids, within a geometrical cutoff procedure for the vortex core, is presented. A new feature, as compared to the Kelvin mode spectrum, is that the spectrum bends down after the initial increase of energy with wavevector, and approaches the zero of energy for wavelengths which are about six times the core diameter. This result should be of relevance for the description of (superfluid) turbulence, in particular the dissipation at the smallest scales of the energy cascade, as well as for the related vortex reconnection and nucleation events.

Rimmed-pool carbonate terraces in terrestrial hot springs at Yellowstone National Park and subterranean cold springs in Illinois Caverns are strikingly similar with respect to their crystalline morphology and depositional patterns. Yet the two depositional settings are dramatically different with respect to all fundamental environmental parameters, including water temperature, water chemistry, carbonate mineralogy, microbial ecology, and local climatic conditions. These architecturally similar yet ecologically different deposits provide a unique natural setting in which to discriminate between abiotic and biotic carbonate mineral precipitates, which will permit more accurate interpretation of the geological record of microbial life on earth and potentially other planets.

We are in the process of identifying the fundamental processes operating in these hot spring and cave environments to address the following:

1. What physical, chemical, and microbiological environmental conditions are present during the precipitation of rimmed-pool carbonates in hot springs and caves?
2. Can quantitative analysis of the emergent scale-invariant patterns of rimmed-pool carbonate terraces from these extremely different environments identify unifying physical controls?
3. Once identified, can these inorganic physical controls on carbonate precipitation be used to discriminate between biotic and abiotic carbonate deposition in these and other settings?

One important aspect of complex systems is multiplicity of persistent internal configurations that can be realized for the same set of externally controlled conditions. A change in the externally controlled conditions results in processes where the system moves from one persistent configuration to another in an irreversible manner resulting in hysteresis. From a purely mathematical point of view hysteresis can be viewed as a branching phenomenon. The simplest form of this branching occurs when the external control can be described by a scalar input variable, such as magnetic field amplitude, in magnetic systems. In this case, transitions from one branch to another take place when the input variable reverses its direction of variation. Remarkably, many complex systems of different physical origins display certain universal behavior when it comes to choosing hysteresis branches. Somehow these systems choose to follow only those branches that can form closed minor hysteresis loops upon the very first return of the input to its former reversal point. Moreover, after the minor loop closure, the system continues to respond to its input as if the minor loop did not occur at all. This property has been called return-point memory. Return-point memory is observed many magnetic [1], magnetostrictive, and superconducting [2] materials. It has been found to hold in a simplified microscopic hysteresis model called the Random Field Ising model [3]. Virtual closure of minor loops (which is a partial indicator of return-point memory) has also been observed in numerical experiments with the Sherrington-Kirpatrick model [4] at zero temperature. In this talk a review of phenomenological models of hysteresis with return-point memory will be provided. Our recent numerical experiments attempting to investigate the origins of return-point memory in complex systems, such as micro-magnetic systems consisting of many similar non-linear interacting elements, will be described. In addition to the investigation of different hysteretic systems with return memory, reduced mathematical representations of such systems and their application to problems of finding the minimum energy configuration will be discussed.

REFERENCES
[1]I.D. Mayergoyz, Mathematical models of hysteresis, Springer-Verlag, New York, 1990
[2]G. Friedman, L.Liu, J. Kouvel, J. Appl. Phys., Vol. 75 No. 10, pp 5683-5687, 1994
[3]J.P. Sethna, K. Dahmen, S. Kartha, J.A. Krumhansl, B.W. Roberts, J.D. Shore, Phys.Rev.Lett., Vol. 70, No. 21, pp. 3347-3350, 1993
[4]F. Pazmandi, G. Zarand, G.T. Zimanyi, Phys. Rev. Lett. Vol. 83, No.5, pp. 1034-1037, 1999

Modeling and simulation testbed development: Simulation is a critical methodology in multi-agent systems research, and a number of sharable testbeds such as SWARM, RePast, MASS, MADKIT, etc. exist. Widely available testbeds have significant deficiencies in the areas of useful representations for large-grain agents, scalability, visualization, flexibility, integration, and theories of environment modeling. We've developed the core of a large-grain agent testbed designed to remedy these issues, which has been used for simulations of over 5000 agents, 10,000 tasks, and 10M messages. The testbed is multi-threaded for concurrency, and we're about to start scalability testing on NCSA's SGI Origin multiprocessor.

When f(z) is analytic and has a fixed point, then fⁿ(z) can be defined in all generality for n where n is a complex number, instead of being limited to the whole numbers. This is achieved by showing that the combinatoric structure, the total partitions of m (sequence A000311 in the On-Line Encyclopedia of Integer Sequences), is isomorphic to the m^th derivative of fⁿ(z). When evaluated at a fixed point, each tree associated with the total partitions of m represents a different nested summation of the Lyponouv exponent. This leads to the idea that most recursion in physics could be implemented by a single mathematical mechanism and that supersymmetry could be extended to include recursion. This would explain the mystery of why supersymmetry is so useful in the study of chaotic systems, assuming that the systems are chaotic due to being generated recursively.

Recent measurements on the power fluctuations in a closed turbulent cell reveal scaling properties that are very close to the magnetisation fluctuations in a finite 2D XY model. The same type of scaling is observed in a simplified statistical model of species abundance in an ecosystem. I show how these properties follow from hyperscaling, and present a simple physical model that accounts for the main experimental features of the turbulent flow.

This work was performed in collaboration with Vivek Aji and supported by the National Science Foundation through grant NSF-DMR-99-70690.

Several lines of evidence combine to suggest that the observed variability in the frequency of occurrence of El Niño and La Niña is likely to be the product of global change. Global change is, in turn driven significantly by anthropogenic causes, solar variability, and volcanism, all of which may be legitimately considered as external to ENSO, regardless of how much of the ordinary climate system is regarded associated with ENSO. This attribution of variability to outside sources may change the way we view ENSO, and its perceived role as a classical example of a complex system. For a measure of amends, I offer a couple of other geophysical systems, which may function as classical non-linear systems.

The ensemble of particles in accelerators forms a weakly nonlinear system, and high-order Taylor transfer maps can represent the system well. The maps allow to address various questions, and sometimes the verification is required. For example, the long-term stability in repetitive systems is an important question. The normal form method performs coordinate transformations to make the phase space motion highly regular. The deviation from the regular motion givesa measure for the stability, turning the problem into a verified global optimization problem of a complicated multi-dimensional function. The method of Taylor models combines high-order computational differentiation and the interval method for verification. The bulk of the functional dependency is kept in high-order Taylor polynomials, so the interval blow-up problem becomes negligible, allowing to give practical answers to the verified global optimization problem. The method also avoids typical technical difficulties in verified ODE integrations, allowing the verification of transfer map computation, and among other things,making the stability analysis fully rigorous.

Extended condensed matter systems driven through disorder exhibit a rich collective dynamics, including history dependence of the response, memory effects, and novel nonequilibrium phases. Examples include the motion of domain walls in disordered magnets, flux flow in type-II superconductors, charge density wave transport in anisotropic conductors and sliding friction at various scales. These systems can be divided in two broad classes. If the disorder is weak, transport can be described in terms of the distortion and motion of an elastic structure, not unlike a rubber sheet, which exhibits a continuous depinning transition with universal critical behavior. If the disorder is strong, the elastic structure breaks up and the flow occurs along channels, with strong spatial and temporal inhomogeneities. The depinning transition becomes history dependent, as seen in many experimental systems. This talk will describe a model that exhibits a crossover from continuous elastic depinning to hysteretic depinning to viscous fluid flow. Hystereric depinning is accompanied by stick-slip type instability, not unlike what observed in vortex lattice motion in type-II superconductors.

A mystery of central importance to cell biology is the question of exactly how the millimeter to meter long chromosomes in cells are organized (folded?). I will explain how and why understanding of this problem will require input from statistical mechanics. I will show how micromechanical techniques developed over the past decade - the interpretation of which demands the use of statistical mechanics - have greatly improved our understanding of the basic physical properties of large DNA molecules. I'll then discuss what our group has learned about the folding up of chromosomes during cell division, using a combination of micromechanical and biochemical dissection techniques. In particular I will show that a completely compacted chromosome is best thought of as a rather loose polyelectrolyte gel, with slowly fluctuating internal degrees of freedom, and therefore with a folding scheme that ultimately must be described statistically.

In the emergence of self-organized structures in complex systems, time-scales play a critical role in determining if and when which structures will eventually win the evolutionary fitness battle. If one views the history of the universe from that perspective one also observes that accelerated temporal time-scales can be associated with the emergence of systems of increasing complexity. The invariant pattern that appears to describe that phenomenon in that a large enough number of subsystems has to be able to communicate efficiently and fast enough with all members of the complex system. In 1994 we pointed out that a network with a small world structure satisfies this criteria. We predicted the conditions (in terms connectivity time-scales and network size) when the computers and users of the Internet can be expected to self-organize into global information structures that we referred to as “Global Brains”.

In this presentation we want to give an update of developments towards a Global Brain both in applications, technical, and ethical discussions. Links to original sources can be found in the archives of our electronic newsletter Complexity Digest www.comdig.org.

We present a model for self-adjusting dynamical systems which treats the control parameters of the system as slowly varying, rather than constant. The dynamics of these variables is governed by a low-pass filtered feedback from the dynamical variables. We apply this model to the logistic map and examine the behavior of the control parameter. We find that the parameter leaves the chaotic regime. We observe a high probability of finding the parameter at the boundary between periodicity and chaos, also known as the edge of chaos. In addition, we find that the parameter may occasionally re-enter the chaotic regime. We study the duration of these chaotic outbreaks and find that the duration can be either an exponential or power law scaling.

Finding the lowest energy state of a system with many degrees of freedom and heterogeneous interactions can sometimes be accomplished numerically in relatively little time. Numerical calculations, based on mapping physical problems to graph theory problems from computer science, can now address subtle questions, such as the nature of the thermodynamic limit. I will review some of these algorithms and applications and describe recent work on phase transitions in the dynamic behavior of these algorithms.

We show that suspensions of ultrasmall silicon nanoparticles form exotic membrane-like systems in bulk. We demonstrate self-assembly of flexible web-like membranes in Si nanoparticle suspensions in water. The suspension is a mixture of higly luminescent and nonluminescent particles of sizes in the range 1-4 nm. Optical imaging shows that the membrane is a Si network of spherical aggregates (~ 40 micrometer in diameter), interconnected by fibers (~50 micrometer in length and 1 micrometer in diameter), and decorated with crystallites. Fluorescence spectroscopy shows that the membrane is nonfluorescent, while the crystallites are luminescent.

This work is being performed in collaboration with Prof. Hassan Aref and is supported by the Center for Simulation of Advanced Rockets at UIUC.

Though adobe is composed of just soil and water, under the right conditions structures made of adobe brick can stand for a thousand years. The defining feature of adobe is in its distribution of particle sizes. Though the smallest particle and the largest particle differ in radius by several orders of magnitude, their respective numbers are such that each size of particle contributes the same volume to the whole. We are using numerical simulations to show that when adobe is subject to environmental pressures the resulting forces can propagate along many different length scales. We hypothesize that this results in a material with an unusual ability to adapt to its environment.

Oscillatory behavior is common in biology. Many of these behaviors (circadian rhythms, heartbeat) are due to complex, multi-scale interactions. Groups of enzymes give rise to oscillations in the rate of glycolysis. The only single-enzyme oscillator known is that based on peroxidase enzymes. Oscillatory consumption of NADH and oxygen has been observed for peroxidases from several plants, fungi, and mammals. As a result, the reaction has been closely studied as a sufficiently simple complex system that detailed modeling is feasible.

We present our experiments and modeling of the horseradish peroxidase oscillator. Discrepancies between experimental data and model led us to identify weaknesses in our understanding of nicotinamide adenine dinucleotide chemistry as a major limitation. We subsequently corrected a long-standing error in the mechanism, revealing a selective biochemical reaction previously unsuspected. Thus, the pursuit of a precise model for a dynamical system revealed previously unsuspected reactivity. The potential biological implications are indicated.

We do a numerical investigation of randomly generated, competing trees on regular, square lattices of varying size and find several scaling laws. We find that the number of trees which sprout scales as N_t=a*L where L is the linear dimension of the grid, while the number of leaves scales as N_l=b*L². Both N_l and N_t have small variance. The number of trees of given size s scales as N_s=c*L*s^-5/4. We discuss the model’s application to a system of self-assembling agglomerates of metal particles in castor oil and are led to consider the future study of dissipation as an order parameter in the proposed model.

What kind of crystals do classical electrons form when constrained to live on a "complex" geometry? This problem can be successfully tackled by mapping the problem to a theory where the relevant degrees of freedom are the topological disclination defects. The accuracy of this new theory is discussed in detail for the case of electrons on a sphere (The Thomson problem) and an elastic spring model on a torus. Finally, This new approach allows for a solution of the Thomson problem in the limit of a very large number of particles.

Self-adjusting, or adaptive, systems have gathered much recent interest. We present a model for self-adjusting systems which treats the control parameters of the system as slowly varying, rather than constant. The dynamics of these parameters is governed by a low-pass filtered feedback from the dynamical variables of the system. We apply this model to a symmetrical and asymmetrical Chua oscillator in both simulations and experiments and examine the behavior of the control parameter. We find that the parameter leaves the chaotic regime. We observe that the probability of finding the system parameter at the boundary between periodicity and chaos increases steadily as the number of parameter adjustments increases. We therefore find that this system exhibits adaptation to the edge of chaos.

Jerky response to a smooth applied force is found in a wide variety of systems ranging from earthquakes to the magnetization of disordered materials. Recent efforts to understand the similarities between such avalanche-like response in seemingly disparate systems has lead to a model that suggest an underlying disorder induced phase transition to be the reason for the apparent universality observed in many systems. The zero-temperature non-equilibrium random field Ising model has been studied in depth in the adiabatic (infinitely slow field sweep rate) regime. Here we introduce the adiabatic critical point and discuss how the behavior changes upon increasing the sweep rate.

This work is concerned a novel class of biomolecular self-assemblies, where new condensed phases of various biopolymers are formed through their interactions with oppositely charged ions of varying complexity, from point-like multivalent ions to charged amphiphilic molecules. Intuitively, two like-charged macromolecules in aqueous solution are expected to repel one another, which is essentially the prediction of prevailing mean-field theories. In the presence of oppositely charged multivalent ions, however, many biopolymers actually attract one another and condense into compact, ordered states. We have examined the global phase behavior of a few model charged biopolymers: DNA, cytoskeletal F-actin, Fd and M13 viruses. These simple systems can exhibit a rich and complex range of behavior. For example, we unambiguously demonstrate the existence of two distinct condensed phases in F-actin. At low multivalent ionic strengths, a homogeneous liquid of uncondensed filaments is observed. At high multivalent concentrations, the filaments condense into uniaxial bundles, in the form of close-packed parallel arrays of individual filaments. At intermediate multivalent concentrations, however, we find a new phase of liquid crystalline matter, in the form of a multi-axial network. In contrast, cationic amphiphilic molecules can condense F-actin into hierarchically organized tubules with no direct analog in simple membrane systems. Using high resolution small angle x-ray scattering, confocal microscopy and electron microscopies, we will present a systematic structural investigation of these condensed biopolymer phases, and the resultant implications for our understanding of polyelectrolyte physics.

The existence of non-exponential dephasing in molecular systems has been verified by a substantial number of observations. Yet approximate theories predicting exponential decay dynamics are often assumed to be valid. An understanding of some factors that result in deviations from these standard treatments has been arrived at through numerical results obtained using a local random matrix model. An approximate realization of this model is the vibrational Hamiltonian of thiophosgene. A state space view of quantum evolution allows for a simple, geometric interpretation of the dynamics in this case.