Ulrich Behn

Idiotypic networks: toward a renaissance?

Summary:  Idiotypic networks, after being a dominating paradigm for more than a decade, have fallen out of fashion in parallel with the rapid success of molecular immunobiology. Today signs of a possible renaissance in idiotypic network studies are visible. For system biologists and also for physicists, the network idea remains attractive. Herein, a short account of the historical development of the paradigm is given. The necessary technical and conceptual ingredients for a theoretical description of idiotypic networks are briefly reviewed, and previous approaches are discussed. We also describe a minimalistic model developed in our group that allows for understanding the random evolution toward a highly non‐trivial complex architecture. In the network, a connected large cluster of idiotype clones and many disconnected ones coexist, thus resembling the notion of central and peripheral parts proposed in the ‘second‐generation’ version of the paradigm. The connected cluster consists of groups of idiotypic clones with clearly distinct statistical properties. The simplicity of the model allows for calculating the size of the groups and the number of inter‐ and intragroup links, which define the architecture. Aspects of idiotypic interactions in experimental medicine are discussed, along with the challenges to theory and experimentation.

Fundamental scaling laws of on-off intermittency in a stochastically driven dissipative pattern forming system

Noise-driven electroconvection in sandwich cells of nematic liquid crystals exhibits on-off intermittent behavior at the onset of the instability. We study laser scattering of convection rolls to characterize the wavelengths and trajectories of the stochastic amplitudes of the intermittent structures. The pattern wavelengths and statistics of these trajectories are in quantitative agreement with simulations of the linearized electrohydrodynamic equations. The fundamental tau(-3/2) distribution law for the durations tau of laminar phases as well as the power law of the amplitude distribution of intermittent bursts are confirmed in the experiments. Power spectral densities of the experimental and numerically simulated trajectories are discussed.

Electrohydrodynamic instabilities in nematic liquid crystals driven by a dichotomous stochastic voltage

Abstract The stability analysis of the one-dimensional model of Dubois-Violette et al. shows the possibility of a discontinuous behaviour of the threshold toward the Williams domain for large amplitude and finite correlation time of the stochastic voltage.

Patterns in randomly evolving networks: Idiotypic networks

We present a model for the evolution of networks of occupied sites on undirected regular graphs. At every iteration step in a parallel update, I randomly chosen empty sites are occupied and occupied sites having occupied neighbor degree outside of a given interval (t(l),t(u)) are set empty. Depending on the influx I and the values of both lower threshold and upper threshold of the occupied neighbor degree, different kinds of behavior can be observed. In certain regimes stable long-living patterns appear. We distinguish two types of patterns: static patterns arising on graphs with low connectivity and dynamic patterns found on high connectivity graphs. Increasing I patterns become unstable and transitions between almost stable patterns, interrupted by disordered phases, occur. For still larger I the lifetime of occupied sites becomes very small and network structures are dominated by randomness. We develop methods to analyze the nature and dynamics of these network patterns, give a statistical description of defects and fluctuations around them, and elucidate the transitions between different patterns. Results and methods presented can be applied to a variety of problems in different fields and a broad class of graphs. Aiming chiefly at the modeling of functional networks of interacting antibodies and B cells of the immune system (idiotypic networks), we focus on a class of graphs constructed by bit chains. The biological relevance of the patterns and possible operational modes of idiotypic networks are discussed.

Electrohydrodynamic Convection in Nematics

The purpose of this review is to present a status report on the electrohydrodynamic convection in nematic liquid crystals. The considerable progress achieved in the past two years is emphasized.

Orbits and phase transitions in the multifractal spectrum

We consider the one-dimensional classical Ising model in a symmetric dichotomous random field. The problem is reduced to a random iterated function system (RIFS) for an effective field. The Dq-spectrum of the invariant measure of this effective field exhibits a sharp drop of all Dq with q<0 at some critical strength of the random field. We introduce the concept of orbits, which naturally group the points of the support of the invariant measure. We then show that the pointwise dimension at all points of an orbit has the same value and calculate it for a class of periodic orbits and their so-called offshoots as well as for generic orbits in the non-overlapping case. The sharp drop in the Dq-spectrum is analytically explained by a drastic change of the scaling properties of the measure near the points of a certain periodic orbit at a critical strength of the random field, which is explicitly given. A similar drastic change near the points of a special family of periodic orbits explains a second, hitherto unnoticed transition in the Dq-spectrum. As it turns out, a decisive role in this mechanism is played by a specific offshoot. We furthermore give rigorous upper and/or lower bounds on all Dq in a wide parameter range. In most cases the numerically obtained Dq coincide with either the upper or the lower bound. The results in this paper are relevant for the understanding of RIFSs in the case of moderate overlap, in which periodic orbits with weak singularity can play a decisive role.

Critical behavior of nonequilibrium phase transitions to magnetically ordered states

We describe nonequilibrium phase transitions in arrays of dynamical systems with cubic nonlinearity driven by multiplicative Gaussian white noise. Depending on the sign of the spatial coupling we observe transitions to ferromagnetic or antiferromagnetic ordered states. We discuss the phase diagram, the order of the transitions, and the critical behavior. For global coupling we show analytically that the critical exponent of the magnetization exhibits a transition from the value 1/2 to a nonuniversal behavior depending on the ratio of noise strength to the magnitude of the spatial coupling.

Architecture of idiotypic networks: Percolation and scaling Behavior

We investigate a model where idiotypes (characterizing B lymphocytes and antibodies of an immune system) and anti-idiotypes are represented by complementary bit strings of a given length d allowing for a number of mismatches (matching rules). In this model, the vertices of the hypercube in dimension d represent the potential repertoire of idiotypes. A random set of (with probability p) occupied vertices corresponds to the expressed repertoire of idiotypes at a given moment. Vertices of this set linked by the above matching rules build random clusters. We give a structural and statistical characterization of these clusters, or in other words of the architecture of the idiotypic network. Increasing the probability p one finds at a critical p a percolation transition where for the first time a large connected graph occurs with probability 1. Increasing p further, there is a second transition above which the repertoire is complete in the sense that any newly introduced idiotype finds a complementary anti-idiotype. We introduce structural characteristics such as the mass distribution and the fragmentation rate for random clusters, and determine the scaling behavior of the cluster size distribution near the percolation transition, including finite size corrections. We find that slightly above the percolation transition the large connected cluster (the central part of the idiotypic network) consists typically of one highly connected part and a number of weakly connected constituents and coexists with a number of small, isolated clusters. This is in accordance with the picture of a central and a peripheral part of the idiotypic network and gives some support to idealized architectures of the central part used in recent dynamical mean field models.

First-passage and first-exit times of a Bessel-like stochastic process

We study a stochastic process X(t) which is a particular case of the Rayleigh process and whose square is the Bessel process, with various applications in physics, chemistry, biology, economics, finance, and other fields. The stochastic differential equation is dX(t)=(nD/X(t))dt+√(2D)dW(t), where W(t) is the Wiener process. The drift term can arise from a logarithmic potential or from taking X(t) as the norm of a multidimensional random walk. Due to the singularity of the drift term for X(t)=0, different natures of boundary at the origin arise depending on the real parameter n: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behavior is observed in the case of a regular boundary.

Network Description of the Immune System: Dormant B Cells Stabilize Cycles

The role of dormant B cells and cycles is analyzed in the context of a Lotka-Volterra network. It is shown that dormant B cells stabilize a cycle and that in this way both cooperate to preserve the internal image (memory) of an antigen. The network is embedded in a hierarchical scheme which allows adaptation, learning, and innovation by biased and random mutation.