# Research

## Research

One of the main unifying concepts of biology is the theory of self-organization. The key idea is that individuals in an ensemble act upon information from their local environment. Nevertheless, even though only such local (neighbor) interactions between the elements constituting the system exist, large-scale patterns can appear. Typical examples of such processes are dense crowds of people and swarms of fish. Certain aspects of excitations propagating in an excitable medium (e.g. fire fronts in a forest) can be best understood using the theory of self-organization. It is one of the major challenges of modern bioinformatics to formulate observations on the large-scale correlation structure of DNA sequences in the language of self-organizing systems.

In a more formal sense, self-organization corresponds to a phase transition towards higher spatial order. The local interactions determine properties of the emerging patterns. It is one of the central tasks of data analysis research to, conversely, extract properties of the local interactions (the "rules" of self-organization) from the patterns experimentally observed in a biological system.

Our research addresses the question of large-scale pattern formation, in particular the fascinating special case of long-range correlations, from a variety of theoretical points of view.

(1) Correlations in DNA sequences

key reference: M. Dehnert, W.E. Helm and M.-Th. Hütt, "A discrete autoregressive process as a model for short-range correlations in DNA sequences", Physica A, 327 (2003) 535-553.[pdf]

We attempt to address the basic question of, how "complex" a DNA sequence is, when thought of as a string of successive symbols rather than an entity embedded in some biological organism. Our approach is to study the long- and short-range correlations in such symbol sequences with methods from information theory. In addition, we use concepts from the theory of stochastic processes to parameterize certain forms of correlations. An example of such a parameterization is given by so-called "discrete autoregressive processes".

A discrete autoregressive process of the order p, DAR(p), is a formal algorithm for recursively generating a symbol sequence.

When estimating the parameters of a DAR(30) process from a given DNA sequence (the human chromosome 22), we are able to reproduce the observed correlation structure (as given by the mutual information function I(k) for 0<k<=30) with good accuracy.

(2) Networks of nonlinear oscillators: data analysis

key reference: M.-Th. Hütt, R. Neff, H. Busch and F. Kaiser, "A method for detecting the signature of spatiotemporal stochastic resonance", Phys. Rev. E 66 (2002) 026117.[pdf]

The following Figure summarizes - in a very naive form - the antagonistic roles of simulation and data analysis.

The development of new analysis techniques for experimental data can be complemented by studying model systems. This is particularly true for the experimental investigation of nonlinear systems. The idea is to generate sample data using models and then put similar restrictions on these sample data as in the case of an actual experiment. Examples for typical restrictions are (1) only one of the dynamical variable is measured, (2) the sampling rate is reduced or (3) the values of internal parameters for different time series are unknown.

With the help of such sample data one can test, how well the analysis tools are capable of handling real-life data. In many cases one can improve the analysis techniques significantly on the basis of such tests.

In a series of previous studies we have formulated spatiotemporal filters translating neighborhood constellations into quantitative estimates of a certain system property (see [1] for definitions and first tests). We have used these tools to study the phenomenon of spatiotemporal stochastic resonance [2], to quantify synchronization properties of specific biological patterns [3] and to develop an algorithm for evaluating independently the contributions of measurement noise and internal noise to a spatiotemporal data set [4].

One particular example of such a spatiotemporal filter is the fluctuation number Omega, that is summarized in the following Figure.

The key idea here for quantification of noise in a spatiotemporal data set is to use the relative movement of neighbors of a particular cell as a means of separating directed and undirected (eventually stochastic) change of the state of a cell. If the discretization of the spatiotemporal data set in space (due to the finite cell size) and time (due to the finite number of images) is small enough, directed and stochastic changes will have very different scales in time and space.

We tested this method on data from mathematical model systems displaying spatiotemporal stochastic resonance. The term spatiotemporal stochastic resonance denotes a phenomenon when patterns in a spatially extended system are most pronounced at intermediate intensity of the noise present in the system. One of the main difficulties when attempting to find this phenomenon in natural systems has been that in most cases noise intensity is not immediately accessible by experiment. Our technique allows to reconstruct noise intensity from the spatiotemporal data set itself. The following Figure shows typical snapshots from such model data (taken from [2]), namely from a system developed by Jung and Mayer-Kress [5], the system, indeed, for which the phenomenon has first been described.

It is clearly seen that at intermediate noise intensity (in particular snapshot #2) the spiral wave propagating through the system is most clearly visible. In the lower part of this Figure some measure of spatial order, the homogeneity, is plotted as a function of noise intensity. The homogeneity of a pattern is given by the average number of equal nearest neighbors (see [1] for details). The characteristic feature of a spatiotemporal stochastic resonance is the pronounced peak at intermediate noise intensity. The following Figure (part B) shows our attempt to reconstruct the noise intensity from the data alone. There, the fluctuation number is shown as a function of the real noise intensity used to generate the data. The monotonous relation between fluctuation number and noise intensity allows one to reconstruct the characteristic peak from the data alone, by plotting homogeneity and fluctuation number in a correlation diagram, as shown in the lowest part of this Figure. These tools form a basis for identifying the phenomenon of spatiotemporal stochastic resonance and other noise-induced phenomena in spatiotemporal data sets from biological systems.

(3) Networks of nonlinear oscillators: biological variability

key reference: M.-Th. Hütt, H. Busch and F. Kaiser, "The effect of biological variability on spatiotemporal patterns: model simulations for a network of biochemical oscillators", Nova Acta Leopoldina, in press.

Noise has an important effect on spatiotemporal patterns in biological systems. In contrast to noise, biological variability (or disorder) is a static system property. Nevertheless it can have dynamical implications, as the magnitude and the statistical properties of the biological variability influence the capabilities of the elements to synchronize or form patterns.

By variability we understand a parameter distribution within an extended system with the parameter values being constant in time.

We study such influences in a chain of coupled nonlinear oscillators, each of which can be thought of as a simple form of oscillating biochemical reaction.

With our numerical investigation we find that under certain conditions an increase in variability can induce spatial waves and complex spatiotemporal patterns.

In particular, it is seen that the mutual information quantifying the complexity of the spatiotemporal patterns can depend resonantly on variability.

(4) Networks of nonlinear oscillators: connectivity

key reference: M.-Th. Hütt and U. Lüttge, "An analysis scheme for biological network dynamics", J. Theor. Biol., submitted.

Our hypothesis is that with a combination of heterogeneity and flutuation one can extract the connectivity of the underlying network from the observed time series of the individual units. The following Figure shows typical realizations of graphs obtained via the small-world prescription outlined in [6] (see also [7] for details on the graph-theoretical properties). When some dynamical units (e.g. some model of biochemical oscillations) are placed on the positions of the vertices, one arrives at a network of dynamical elements with synchronization properties depending on the topology of the underlying graph.

By simulating the time development of this system one obtains spatiotemporal patterns that, to a certain extend, reflect properties of the underlying network. These patterns are studied with two-oscillator variants of the spatiotemporal filters introduced above. The resulting curves in the heterogeneity-fluctuation plane are also included in the Figure below.

It is seen that the curves arrange themselves in this plane according to the link density (or connectivity) of the system. With this method of analyzing the spatiotemporal dynamics of a regulatory network it is thus possible to determine, which of the data comes from a more densely connected system. In particular, one can find out, under what circumstances a system is most densely connected.

References

**[1] M.-Th. Hütt and R. Neff, "Quantification of spatiotemporal phenomena by means of cellular automata techniques", Physica A 289 (2001) 498.**

**[2] M.-Th. Hütt, R. Neff, H. Busch and F. Kaiser, "A method for detecting the signature of spatiotemporal stochastic resonance", Phys. Rev. E 66 (2002) 026117.**

**[3] U. Rascher, M.-T. Hütt, K. Siebke, B. Osmond, F. Beck and U. Lüttge, "Spatio-temporal variation of metabolism in a plant circadian rhythm: the biological clock as an assembly of coupled individual oscillators", Proc. Natl. Acad. Sc. USA, 98 (2001) 11801.**

**[4] H. Busch and M.-Th. Hütt, "Scale-dependence of spatiotemporal filters inspired by cellular automata", Int. J. Bif. Chaos, in press.**

**[5] P. Jung and G. Mayer-Kress, "Spatio-Temporal Stochastic Resonance in Excitable Media", Phys. Rev. Lett. 74 (1995) 2130. **

**[6] D. Watts and S. Strogatz, "Collective dynamics of 'small world' networks", Nature 393 (1998) 440. **

**[7] S. Strogatz, "Exploring complex networks", Nature 410 (2001) 268.**