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The measurement of the similarity of RNA secondary structures, and in general of contact structures, of a fixed length has several specific applications. For instance, it is used in the analysis of the ensemble of suboptimal secondary structures generated by a given algorithm on a given RNA sequence, and in the comparison of the secondary structures predicted by different algorithms on a given RNA molecule. It is also a useful tool in the quantitative study of sequence-structure maps. A way to measure this similarity is by means of metrics. In this paper we introduce a new class of metrics $d_{m}$, $m\geq 3$, on the set of all contact structures of a fixed length, based on their representation by means of edge ideals in a polynomial ring. These metrics can be expressed in terms of Hilbert functions of monomial ideals, which allows the use of several public domain computer algebra systems to compute them. We study some abstract properties of these metrics, and we obtain explicit descriptions of them for $m=3,4$ on arbitrary contact structures and for $m=5,6$ on RNA secondary structures., Comment: Presented at the Biomolecular Mathematics Special Session of the First Joint AMS-RSME Meeting (Sevilla, june 2003)

A computer model was developed for estimation of the kinetics of microbial inactivation by pulsed electric field. The model is based on the electroporation theory of individual membrane damage, where spherical cell geometry and distribution of cell sizes are assumed. The variation of microbial cell sizes was assumed to follow a statistical probability distribution of the Gaussian type. Surviving kinetics was approximated by Weibull equation. The dependencies of two Weibull parameters (shape \textit{n} and time $\tau $, respectively) versus electric field intensity E and width of cell diameters distribution was studied., Comment: 4 pages, 4 figures, RevTex4

We present a connectionist architecture that can learn a model of the relations between perceptions and actions and use this model for behavior planning. State representations are learned with a growing self-organizing layer which is directly coupled to a perception and a motor layer. Knowledge about possible state transitions is encoded in the lateral connectivity. Motor signals modulate this lateral connectivity and a dynamic field on the layer organizes a planning process. All mechanisms are local and adaptation is based on Hebbian ideas. The model is continuous in the action, perception, and time domain., Comment: 9 pages, see http://www.marc-toussaint.net/

We present a theory to investigate electro-kinetic behavior, namely, electrorotation and dielectrophoresis under alternating current (AC) applied fields for a pair of touching inhomogeneous colloidal particles and biological cells. These inhomogeneous particles are treated as graded ones with physically motivated model dielectric and conductivity profiles. The mutual polarization interaction between the particles yields a change in their respective dipole moments, and hence in the AC electrokinetic spectra. The multipolar interactions between polarized particles are accurately captured by the multiple images method. In the point-dipole limit, our theory reproduces the known results. We find that the multipolar interactions as well as the spatial fluctuations inside the particles can affect the AC electrokinetic spectra significantly., Comment: Revised version with minor changes: References added and discussion extended

This paper develops a two gene, single fitness peak model for determining the equilibrium distribution of genotypes in a unicellular population which is capable of genetic damage repair. The first gene, denoted by $ \sigma_{via} $, yields a viable organism with first order growth rate constant $ k > 1 $ if it is equal to some target ``master'' sequence $ \sigma_{via, 0} $. The second gene, denoted by $ \sigma_{rep} $, yields an organism capable of genetic repair if it is equal to some target ``master'' sequence $ \sigma_{rep, 0} $. This model is analytically solvable in the limit of infinite sequence length, and gives an equilibrium distribution which depends on $ \mu \equiv L\eps $, the product of sequence length and per base pair replication error probability, and $ \eps_r $, the probability of repair failure per base pair. The equilibrium distribution is shown to exist in one of three possible ``phases.'' In the first phase, the population is localized about the viability and repairing master sequences. As $ \eps_r $ exceeds the fraction of deleterious mutations, the population undergoes a ``repair'' catastrophe, in which the equilibrium distribution is still localized about the viability master sequence, but is spread ergodically over the sequence subspace defined by the repair gene. Below the repair catastrophe, the distribution undergoes the error catastrophe when $ \mu $ exceeds $ \ln k/\eps_r $, while above the repair catastrophe, the distribution undergoes the error catastrophe when $ \mu $ exceeds $ \ln k/f_{del} $, where $ f_{del} $ denotes the fraction of deleterious mutations., Comment: 14 pages, 3 figures. Submitted to Physical Review E

We examine the dependence of the physical quantities of the rotatory molecular motor, such as the rotation velocity and the proton translocation rate, on the chemical reaction rate using the model based only on diffusion process. A peculiar behavior of proton translocation is found and the energy transduction efficiency of the motor protein is enhanced by this behavior. We give a natural explanation that this behavior is universal when certain inequalities between chemical reaction rates hold. That may give a clue to examine whether the motion of the molecular motor is dominated by diffusion process or not., Comment: 12 pages, 8 figures

We introduce a new interacting particle system intended to model an example of ecological succession involving two species: the bracken and the european beech. The objective is to exhibit phase transitions by proving that there exist three possible evolutions of the system to distinct ecological balances. More precisely, the whole population may die out, the european beech may conquer the bracken, and coexistence of both species may occur., Comment: 16 pages with 8 figures

Mechanical coupling between a cell and substrate relies on focal adhesions, clusters of adhesion proteins linking stress fibers (bundles of actin proteins) inside the cell with surrounding tissue. Focal adhesions have been demonstrated to both measure and regulate the mechanical traction along the stress fibers. We present a quantitative model for focal adhesion mechano-sensing and stress regulation based on stress amplification at the critical point of a condensation transition of the adhesion proteins., Comment: 4 pages, 2 figures

In this paper we model several simple biochemical operations on RNA molecules that modify their secondary structure by means of a suitable variation of Gro\ss e-Rhode's Algebra Transformation Systems., Comment: 10 pages

We present a generic model for DNA at the base-pair level. We use a variant of the Gay-Berne potential to represent the stacking energy between neighboring base-pairs. The sugar-phosphate backbones are taken into account by semi-rigid harmonic springs with a non-zero spring length. The competition of these two interactions and the introduction of a simple geometrical constraint leads to a stacked right-handed B-DNA-like conformation. The mapping of the presented model to the Marko-Siggia and the Stack-of-Plates model enables us to optimize the free model parameters so as to reproduce the experimentally known observables such as persistence lengths, mean and mean squared base-pair step parameters. For the optimized model parameters we measured the critical force where the transition from B- to S-DNA occurs to be approximately $140{pN}$. We observe an overstretched S-DNA conformation with highly inclined bases that partially preserves the stacking of successive base-pairs., Comment: 15 pages, 25 figures. submitted to PRE

Understanding of the evolutionary origins of protein structures represents a key component of the understanding of molecular evolution as a whole. Here we seek to elucidate how the features of an underlying protein structural "space" might impact protein structural evolution. We approach this question using lattice polymers as a completely characterized model of this space. We develop a measure of structural comparison of lattice structures that is analgous to the one used to understand structural similarities between real proteins. We use this measure of structural relatedness to create a graph of lattice structures and compare this graph (in which nodes are lattice structures and edges are defined using structural similarity) to the graph obtained for real protein structures. We find that the graph obtained from all compact lattice structures exhibits a distribution of structural neighbors per node consistent with a random graph. We also find that subgraphs of 3500 nodes chosen either at random or according to physical constraints also represent random graphs. We develop a divergent evolution model based on the lattice space which produces graphs that, within certain parameter regimes, recapitulate the scale-free behavior observed in similar graphs of real protein structures., Comment: 27 pages, 7 figures

This paper focuses on the outline of some computational methods for the approximate solution of the integral equations for the neuronal firing probability density and an algorithm for the generation of sample-paths in order to construct histograms estimating the firing densities. Our results originate from the study of non-Markov stationary Gaussian neuronal models with the aim to determine the neuron's firing probability density function. A parallel algorithm has been implemented in order to simulate large numbers of sample paths of Gaussian processes characterized by damped oscillatory covariances in the presence of time dependent boundaries. The analysis based on the simulation procedure provides an alternative research tool when closed-form results or analytic evaluation of the neuronal firing densities are not available., Comment: 10 pages, 3 figures, to be published in Scientiae Mathematicae Japonicae

Recent work has focused attention on statistical inference for the population distribution of the number of sexual partners based on survey data. The characteristics of these distributions are of interest as components of mathematical models for the transmission dynamics of sexually-transmitted diseases (STDs). Such information can be used both to calibrate theoretical models, to make predictions for real populations, and as a tool for guiding public health policy. Our previous work on this subject has developed likelihood-based statistical methods for inference that allow for low-dimensional, semi-parametric models. Inference has been based on several proposed stochastic process models for the formation of sexual partnership networks. We have also developed model selection criteria to choose between competing models, and assessed the fit of different models to three populations: Uganda, Sweden, and the USA. Throughout this work, we have emphasized the correct assessment of the uncertainty of the estimates based on the data analyzed. We have also widened the question of interest to the limitations of inferences from such data, and the utility of degree-based epidemiological models more generally. In this paper we address further statistical issues that are important in this area, and a number of confusions that have arisen in interpreting our work. In particular, we consider the use of cumulative lifetime partner distributions, heaping and other issues raised by Liljeros et al. in a recent working paper., Comment: 22 pages, 5 figures in linked working paper

We introduce new quantitative approaches to study sleep-stage transitions with the goal of addressing the two following questions: (i) Can the new approaches provide more information on the structure of sleep-stage transitions? (ii) How does sleep fragmentation in patients with sleep apnea affect the structure of sleep-stage transitions? Our new results show that the distribution of sleep and wake duration have different functional forms, indicating fundamental differences in the dynamics between sleep and wake control. The difference remains even in the fragmented sleep of sleep apnea. The fragmentation of sleep in sleep apnea results in a shorter wake duration and interrupts the structure of sleep-stage transitions of sleep apnea subjects, causing the loss of certain particular transition paths., Comment: 17 pages, 5 figures

A feed-forward neural net with adaptable synaptic weights and fixed, zero or non-zero threshold potentials is studied, in the presence of a global feedback signal that can only have two values, depending on whether the output of the network in reaction to its input is right or wrong. It is found, on the basis of four biologically motivated assumptions, that only two forms of learning are possible, Hebbian and Anti-Hebbian learning. Hebbian learning should take place when the output is right, while there should be Anti-Hebbian learning when the output is wrong. For the Anti-Hebbian part of the learning rule a particular choice is made, which guarantees an adequate average neuronal activity without the need of introducing, by hand, control mechanisms like extremal dynamics. A network with realistic, i.e., non-zero threshold potentials is shown to perform its task of realizing the desired input-output relations best if it is sufficiently diluted, i.e. if only a relatively low fraction of all possible synaptic connections is realized.

The time evolution of an exactly solvable layered feedforward neural network with three-state neurons and optimizing the mutual information is studied for arbitrary synaptic noise (temperature). Detailed stationary temperature-capacity and capacity-activity phase diagrams are obtained. The model exhibits pattern retrieval, pattern-fluctuation retrieval and spin-glass phases. It is found that there is an improved performance in the form of both a larger critical capacity and information content compared with three-state Ising-type layered network models. Flow diagrams reveal that saddle-point solutions associated with fluctuation overlaps slow down considerably the flow of the network states towards the stable fixed-points., Comment: 17 pages Latex including 6 eps-figures

We introduce a phase field approach for diffusion inside and outside a closed cell with damping and with source terms at the interface. The method is compared to exact solutions (where possible) and the more traditional finite element method. It is shown to be very accurate, easy to implement and computationally inexpensive. We apply our method to a recently introduced model for chemotaxis by Rappel et al. [Biophys. J. 83, 1361 (2002)]., Comment: 4 pages, 4 figures

We study the persistent recoverable prevalence and the extinction of computer viruses via e-mails on a growing scale-free network with new users, which structure is estimated form real data. The typical phenomenon is simulated in a realistic model with the probabilistic execution and detection of viruses. Moreover, the conditions of extinction by random and targeted immunizations for hubs are derived through bifurcation analysis for simpler models by using a mean-field approximation without the connectivity correlations. We can qualitatively understand the mechanisms of the spread in linearly growing scale-free networks., Comment: 9 pages, 9 figures, 1 table. Update version after helpful referee comments

Benign Paroxysmal Positional Vertigo (BPPV) is a mechanical disorder of the vestibular system in which calcite particles called otoconia interfere with the mechanical functioning of the fluid-filled semicircular canals normally used to sense rotation. Using hydrodynamic models, we examine the two mechanisms proposed by the medical community for BPPV: cupulolithiasis, in which otoconia attach directly to the cupula (a sensory membrane), and canalithiasis, in which otoconia settle through the canals and exert a fluid pressure across the cupula. We utilize known hydrodynamic calculations and make reasonable geometric and physical approximations to derive an expression for the transcupular pressure $\Delta P_c$ exerted by a settling solid particle in canalithiasis. By tracking settling otoconia in a two-dimensional model geometry, the cupular volume displacement and associated eye response (nystagmus) can be calculated quantitatively. Several important features emerge: 1) A pressure amplification occurs as otoconia enter a narrowing duct; 2) An average-sized otoconium requires approximately five seconds to settle through the wide ampulla, where $\Delta P_c$ is not amplified, which suggests a mechanism for the observed latency of BPPV; and 3) An average-sized otoconium beginning below the center of the cupula can cause a volumetric cupular displacement on the order of 30 pL, with nystagmus of order $2^\circ$/s, which is approximately the threshold for sensation. Larger cupular volume displacement and nystagmus could result from larger and/or multiple otoconia., Comment: 15 pages, 5 Figures updated, to be published in J. Biomechanics

It is known that storage capacity per synapse increases by synaptic pruning in the case of a correlation-type associative memory model. However, the storage capacity of the entire network then decreases. To overcome this difficulty, we propose decreasing the connecting rate while keeping the total number of synapses constant by introducing delayed synapses. In this paper, a discrete synchronous-type model with both delayed synapses and their prunings is discussed as a concrete example of the proposal. First, we explain the Yanai-Kim theory by employing the statistical neurodynamics. This theory involves macrodynamical equations for the dynamics of a network with serial delay elements. Next, considering the translational symmetry of the explained equations, we re-derive macroscopic steady state equations of the model by using the discrete Fourier transformation. The storage capacities are analyzed quantitatively. Furthermore, two types of synaptic prunings are treated analytically: random pruning and systematic pruning. As a result, it becomes clear that in both prunings, the storage capacity increases as the length of delay increases and the connecting rate of the synapses decreases when the total number of synapses is constant. Moreover, an interesting fact becomes clear: the storage capacity asymptotically approaches $2/\pi$ due to random pruning. In contrast, the storage capacity diverges in proportion to the logarithm of the length of delay by systematic pruning and the proportion constant is $4/\pi$. These results theoretically support the significance of pruning following an overgrowth of synapses in the brain and strongly suggest that the brain prefers to store dynamic attractors such as sequences and limit cycles rather than equilibrium states., Comment: 27 pages, 14 figures

A perturbative method is developed for calculating the effects of recurrent synaptic interactions between neurons embedded in a network. A series expansion is constructed that converges for networks with noisy membrane potential and weak synaptic connectivity. The terms of the series can be interpreted as loops of interactions between neurons, so the technique is called a loop-expansion. A diagrammatic method is introduced that allows for construction of analytic expressions for the parameter dependencies of the spike probability function and correlation functions. An analytic expression is obtained to predict the effect of the surrounding network on a neuron during an intracellular current injection. The analytic results are compared with simulations to test the range of their validity and significant effects of the the recurrent connections in network are accurately predicted by the loop-expansion., Comment: 13 pages, 8 figures (v2, new appendix, minor revisions)

Experimental observations suggest that proteins follow different pathways under different environmental conditions. We perform molecular dynamics simulations of a model of the SH3 domain over a broad range of temperatures, and identify distinct pathways in the folding transition. We determine the kinetic partition temperature --the temperature for which the SH3 domain undergoes a rapid folding transition with minimal kinetic barriers-- and observe that below this temperature the model protein may undergo a folding transition via multiple folding pathways. The folding kinetics is characterized by slow and fast pathways and the presence of only one or two intermediates. Our findings suggest the hypothesis that the SH3 domain, a protein for which only two-state folding kinetics was observed in previous experiments, may exhibit intermediates states under extreme experimental conditions, such as very low temperatures. A very recent report (Viguera et al., Proc. Natl. Acad. Sci. USA, 100:5730--5735, 2003) of an intermediate in the folding transition of the Bergerac mutant of the alpha-spectrin SH3 domain protein supports this hypothesis., Comment: 16 pages, 4 figures To be published in the "Journal of Molecular Biology"

An In Silico model to relate the properties of proteins to the structure, sequence, function and evolutionary history of proteins is shown. The derived ideal sequences for amino acid residues in proteins can then be considered as attractors for structures of actual protein families and their functions. Hopfield networks1 can then act as GCMs (General Content Addressable Memories) when they are trained to recall unique pre-determined states when presented with information associated with that state. The tertiary structure of proteins is determined by structurally enforced interactions between residues and they help to elucidate common motifs exhibited in proteins. The acceptance (or rejection) of substitution on the protein backbone conforms to a nonlinear response and can lead to emergent behavior., Comment: 19 pages, 14 figures

We compute explicitly several abstract metrics for RNA secondary structures defined by Reidys and Stadler., Comment: 6 pages

Several controversial issues concerning the packing of linear DNA in bacteriophages and globules are discussed. Exact relations for the osmotic pressure, capsid pressure and loading force are derived in terms of the hole size inside phages under the assumption that the DNA globule has a uniform density. A new electrostatic model is introduced for computing the osmotic pressure of rodlike polyelectrolytes at very high concentrations. At intermediate packing, a reptation model is considered for DNA diffusing within a toroidal globule. Under tight packing conditions, a model of Coulomb sliding friction is proposed. A general discussion is given of our current understanding of the statics and dynamics of confined DNA in the context of to the following experiments: characterization of the liquid crystalline phases, X-ray scattering by phages, osmotic stress measurements, cyclization within globules and single-molecule determination ofthe loading forces., Comment: 32 pages, 5 figures, submitted to Phil.Trans.Roy.Soc.A ; special issue entitled "Mechanics of DNA", editor J.M.T.Thompson

The hair cells of the vertebrate inner ear convert mechanical stimuli to electrical signals. Two adaptation mechanisms are known to modify the ionic current flowing through the transduction channels of the hair bundles: a rapid process involves calcium ions binding to the channels; and a slower adaptation is associated with the movement of myosin motors. We present a mathematical model of the hair cell which demonstrates that the combination of these two mechanisms can produce `self-tuned critical oscillations', i.e. maintain the hair bundle at the threshold of an oscillatory instability. The characteristic frequency depends on the geometry of the bundle and on the calcium dynamics, but is independent of channel kinetics. Poised on the verge of vibrating, the hair bundle acts as an active amplifier. However, if the hair cell is sufficiently perturbed, other dynamical regimes can occur. These include slow relaxation oscillations which resemble the hair bundle motion observed in some experimental preparations., Comment: 13 pages, 6 figures,REVTeX 4, To appear in Biophysical Journal

We introduce a spherical Hopfield-type neural network involving neurons and patterns that are continuous variables. We study both the thermodynamics and dynamics of this model. In order to have a retrieval phase a quartic term is added to the Hamiltonian. The thermodynamics of the model is exactly solvable and the results are replica symmetric. A Langevin dynamics leads to a closed set of equations for the order parameters and effective correlation and response function typical for neural networks. The stationary limit corresponds to the thermodynamic results. Numerical calculations illustrate our findings., Comment: 9 pages Latex including 3 eps figures, Addition of an author in the HTML-abstract unintentionally forgotten, no changes to the manuscript

We incorporate the generic hierarchical architecture of foodwebs into a "{\it unified}" model that describes both "micro" and "macro" evolutions within a single theoretical framework. This model describes the "micro" -evolution in detail by accounting for the birth, ageing and natural death of individual organisms as well as prey-predator interactions on a hierarchical dynamic food web. It also provides a natural description of random mutations and speciation/orgination of species as well as their extinctions. The distribution of lifetimes of species follows an approximate power law only over a limited regime., Comment: 6 pages, 8 embedded EPS figures, REVTEX

In this paper, we consider a series of events observed at spaced time intervals and present a method of representation of the series. To explain an idea, by dealing with a set of gene expression data, which could be obtained from developmental biology, the procedures are sketched with comments in some details. We mean representation by choosing a proper function, which fits well with observed data of a series, and turning its characteristics into numbers, which extract the intrinsic properties of fluctuating data. With help of a machine learning techniques, this method will give a classification of developmental biological data as well as any varying data during a certain period and the classification can be applied for diagnosis of a disease., Comment: 10 pages

We present a study of information flow that takes into account the observation that an item relevant to one person is more likely to be of interest to individuals in the same social circle than those outside of it. This is due to the fact that the similarity of node attributes in social networks decreases as a function of the graph distance. An epidemic model on a scale-free network with this property has a finite threshold, implying that the spread of information is limited. We tested our predictions by measuring the spread of messages in an organization and also by numerical experiments that take into consideration the organizational distance among individuals.

We describe a simple model for the melting and optical properties of a DNA/gold nanoparticle aggregate. The optical properties at fixed wavelength change dramatically at the melting transition, which is found to be higher and narrower in temperature for larger particles, and much sharper than that of an isolated DNA link. All these features are in agreement with available experiments. The aggregate is modeled as a cluster of gold nanoparticles on a periodic lattice connected by DNA bonds, and the extinction coefficient is computed using the discrete dipole approximation. Melting takes place as an increasing number of these bonds break with increasing temperature. The melting temperature corresponds approximately to the bond percolation threshold., Comment: 5 pages, 4 figure. To be published in Phys. Rev. B

A mathematical description of the refractoriness period in neuronal diffusion modeling is given and its moments are explicitly obtained in a form that is suitable for quantitative evaluations. Then, for the Wiener, Ornstein-Uhlenbeck and Feller neuronal models, an analysis of the features exhibited by the mean and variance of the first passage time and of refractoriness period is performed., Comment: 12 pages, 1 figure

A jump-diffusion process is proposed to describe the displacements performed by single myosin heads along actin filaments during the rising phases. The process consists of the superposition of a Wiener and a jump process, with jumps originated by sequences of Poisson-distributed energy-supplying pulses. In a previous paper, the amplitude of the jumps was described by a mixture of two Gaussian distributions. To embody the effects of ATP hydrolysis, we now refine such a model by assuming that the jumps' amplitude is described by a mixture of three Gaussian distributions. This model has been inspired by the experimental data of T. Yanagida and his co-workers concerning observations at single molecule processes level., Comment: 9 pages, 4 figures

With reference to the experimental observations by T. Yanagida and his co-workers on actomyosin interaction, a Brownian motor of fluctuating ratchet kind is designed with the aim to describe the interaction between a Myosin II head and a neighboring actin filament. Our motor combines the dynamics of the myosin head with a chemical external system related to the ATP cycle, whose role is to provide the energy supply necessary to bias the motion. Analytical expressions for the duration of the ATP cycle, for the Gibbs free energy and for the net displacement of the myosin head are obtained. Finally, by exploiting a method due to Sekimoto (1997, J. Phys. Soc. Jpn., 66, 1234), a formula is worked out for the amount of energy consumed during the ATP cycle., Comment: 15 pages. 1 figure

We investigate the dynamics of a particle moving randomly along a disordered hetero-polymer subjected to rapid conformational changes which induce superdiffusive motion in chemical coordinates. We study the antagonistic interplay between the enhanced diffusion and the quenched disorder. The dispersion speed exhibits universal behavior independent of the folding statistics. On the other hand it is strongly affected by the structure of the disordered potential. The results may serve as a reference point for a number of translocation phenomena observed in biological cells, such as protein dynamics on DNA strands., Comment: 4 pages, 4 figures

We study a simple replicator model with non-symmetric and nonlocal interactions. Hierarchical structures with prey-predator relations are self-organized from a homogeneous state, induced by the dynamical instability of nonlinear interactions., Comment: 7 pages, 2 figures

The formation of local denaturation zones (bubbles) in double-stranded DNA is an important example for conformational changes of biological macromolecules. We study the dynamics of bubble formation in terms of a Fokker-Planck equation for the probability density to find a bubble of size n base pairs at time t, on the basis of the free energy in the Poland-Scheraga model. Characteristic bubble closing and opening times can be determined from the corresponding first passage time problem, and are sensitive to the specific parameters entering the model. A multistate unzipping model with constant rates recently applied to DNA breathing dynamics [G. Altan-Bonnet et al, Phys. Rev. Lett. 90, 138101 (2003)] emerges as a limiting case., Comment: 9 pages, 2 figures

The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.

This work describes a novel image analysis approach to characterize the uniformity of objects in agglomerates by using the propagation of normal wavefronts. The problem of width uniformity is discussed and its importance for the characterization of composite structures normally found in physics and biology highlighted. The methodology involves identifying each cluster (i.e. connected component) of interest, which can correspond to objects or voids, and estimating the respective medial axes by using a recently proposed wavefront propagation approach, which is briefly reviewed. The distance values along such axes are identified and their mean and standard deviation values obtained. As illustrated with respect to synthetic and real objects (in vitro cultures of neuronal cells), the combined use of these two features provide a powerful description of the uniformity of the separation between the objects, presenting potential for several applications in material sciences and biology., Comment: 14 pages, 23 figures, 1 table, 1 reference

The ensemble of RNA secondary structures of uniform sequences is studied analytically. We calculate the partition function for very long sequences and discuss how the cross-over length, beyond which asymptotic scaling laws apply, depends on thermodynamic parameters. For realistic choices of parameters this length can be much longer than natural RNA molecules. This has to be taken into account when applying asymptotic theory to interpret experiments or numerical results., Comment: 10 pages, 13 figures, published in Phys. Rev. E

We present a temperature-independent Monte Carlo method for the determination of the density of states of lattice proteins that combines the fast ground-state search strategy of the nPERM chain growth and multicanonical reweighting for sampling the complete energy space. Since the density of states contains all energetic information of a statistical system, we can directly calculate the mean energy, specific heat, Gibbs free energy, and entropy for all temperatures. We apply this method to HP lattice proteins and for the examples of sequences considered, we identify the transitions between native, globule, and random coil states. Since no special properties of heteropolymers are involved in this algorithm, the method applies to polymer models as well., Comment: 4 pages, RevTeX, 5 Postscript figures, Author Information under http://www.physik.uni-leipzig.de/CQT

This paper develops an analytically tractable model for determining the equilibrium distribution of mismatch repair deficient strains in unicellular populations. The approach is based on the single fitness peak (SFP) model, which has been used in Eigen's quasispecies equations in order to understand various aspects of evolutionary dynamics. As with the quasispecies model, our model for mutator-nonmutator equilibrium undergoes a phase transition in the limit of infinite sequence length. This "repair catastrophe" occurs at a critical repair error probability of $ \epsilon_r = L_{via}/L $, where $ L_{via} $ denotes the length of the genome controlling viability, while $ L $ denotes the overall length of the genome. The repair catastrophe therefore occurs when the repair error probability exceeds the fraction of deleterious mutations. Our model also gives a quantitative estimate for the equilibrium fraction of mutators in {\it Escherichia coli}., Comment: 4 pages, 2 figures (included as separate PS files)

The distribution functions of the codon usage probabilities, computed over all the available GenBank data, for 40 eukaryotic biological species and 5 chloroplasts, do not follow a Zipf law, but are best fitted by the sum of a constant, an exponential and a linear function in the rank of usage. For mitochondriae the analysis is not conclusive. A quantum-mechanics-inspired model is proposed to describe the observed behaviour. These functions are characterized by parameters that strongly depend on the total GC content of the coding regions of biological species. It is predicted that the codon usage is the same in all exonic genes with the same GC content. The Shannon entropy for codons, also strongly depending on the exonic GC content, is computed., Comment: Latex 25 pages, 21 figures

This talk will review a little over a decade's research on applying certain stochastic models to biological sequence analysis. The models themselves have a longer history, going back over 30 years, although many novel variants have arisen since that time. The function of the models in biological sequence analysis is to summarize the information concerning what is known as a motif or a domain in bioinformatics, and to provide a tool for discovering instances of that motif or domain in a separate sequence segment. We will introduce the motif models in stages, beginning from very simple, non-stochastic versions, progressively becoming more complex, until we reach modern profile HMMs for motifs. A second example will come from gene finding using sequence data from one or two species, where generalized HMMs or generalized pair HMMs have proved to be very effective.

The influx of calcium ions into the dendritic spines through the N-metyl-D-aspartate (NMDA) channels is believed to be the primary trigger for various forms of synaptic plasticity. In this paper, the authors calculate analytically the mean values of the calcium transients elicited by a spiking neuron undergoing a simple model of ionic currents and back-propagating action potentials. The relative variability of these transients, due to the stochastic nature of synaptic transmission, is further considered using a simple Markov model of NMDA receptos. One finds that both the mean value and the variability depend on the timing between pre- and postsynaptic action-potentials. These results could have implications on the expected form of synaptic-plasticity curve and can form a basis for a unified theory of spike time-dependent, and rate based plasticity., Comment: 14 pages, 10 figures. A few changes in section IV and addition of a new figure

We investigate the stability of negative image equilibria in mean synaptic weight dynamics governed by spike-timing dependent plasticity (STDP). The neural architecture of the model is based on the electrosensory lateral line lobe (ELL) of mormyrid electric fish, which forms a negative image of the reafferent signal from the fish's own electric discharge to optimize detection of external electric fields. We derive a necessary and sufficient condition for stability, for arbitrary postsynaptic potential functions and arbitrary learning rules. We then apply the general result to several examples of biological interest., Comment: 13 pages, revtex4; uses packages: graphicx, subfigure; 9 figures, 16 subfigures

Qualitative attributes of the region between order and disorder are examined to explore models of genetic and protein networks. Results show how the connectivity of vertices and the strength of their connections are related and how their stability is related to their geometry. It has been possible to relate the interaction energies and chemical potentials of general lattices with the coordination numbers of such lattice models. The results seem to agree with some of those obtained by means of other methods, e.g., those of Barabasi, Strognatz, Dorogovtsev, and Kauffman (BSDK). This can yield new perspectives to the cause, treatment and remedies of disease other than the present mode of drug discovery prevalent: the docking of a single ligand on to a target molecule. In order to utilize such results more efficiently, the pathway of biological processes need be elucidated. A method is available for determining biochemical-reaction or metabolic pathways through its systematic synthesis. It is based on a rigorous graph-theoretic method for identifying pathways of catalytic reactions. It synthesizes networks of metabolic pathways using a highly exacting combinatorial method. It generates not only all feasible, independent reaction networks but also those combinations of independent pathways. This method can determine the mechanisms of complex chemical reactions and is applicable to biochemical reactions. It is important to combine this result with the mechanisms existent in gene regulatory networks. Training Genetic Regulatory Networks for feasible biochemical reaction networks or pathways and incorporating such knowledge into DNA would be a superb technique for vanquishing complex diseases., Comment: 13 pages, 7 figures

It has been argued that the sum of usage probabilities for codons, belonging to quartets, that have as third nucleotide C or A, is independent of the biological species for vertebrates. The comparison between the theoretical correlation matrix derived from these sum rules and the experimentally computed matrix for 26 species shows a satisfactory agreement. The Shannon entropy, weakly depending on the biological species, gives further support. Suppression of codons containing the dinucleotides CG or AU is put in evidence.

Localization properties of residue fluctuations in globular proteins are studied theoretically by using the Gaussian network model. Participation ratio for each residue fluctuation mode is calculated. It is found that the relationship between participation ratio and frequency is similar for all globular proteins, indicating a universal behavior in spite of their different size, shape, and architecture., Comment: 4 pages, 3 figures. To appear in Phys. Rev. E

Physical mechanisms underlying the empirical correlation between relative contact order (CO) and folding rate among naturally-occurring small single-domain proteins are investigated by evaluating postulated interaction schemes for a set of three-dimensional 27mer lattice protein models with 97 different CO values. Many-body interactions are constructed such that contact energies become more favorable when short chain segments sequentially adjacent to the contacting residues adopt native-like conformations. At a given interaction strength, this scheme leads to folding rates that are logarithmically well correlated with CO (correlation coefficient $r=0.914$) and span more than 2.5 orders of magnitude, whereas folding rates of the corresponding G\=o models with additive contact energies have much less logarithmic correlation with CO and span only approximately one order of magnitude. The present protein chain models also exhibit calorimetric cooperativity and linear chevron plots similar to that observed experimentally for proteins with apparent simple two-state folding/unfolding kinetics. Thus, our findings suggest that CO-dependent folding rates of real proteins may arise partly from a significant positive coupling between nonlocal contact favorabilities and local conformational preferences., Comment: 23 pages, 4 postscript figures (will appear on Proteins)