Can the complexity of molecular networks account for the bewildering diversity, e.g., of higher animals and their rapid adaptability to different environments? Interactions between genes are themselves encoded in the genome, and this proves crucial for their dynamics. Specific proteins (called transcription factors) bind to their corresponding target sites on the DNA, thereby enhancing or reducing the activity of the nearest gene. Gene activity has fitness consequences, which in turn adapts the sequences of the target sites on evolutionary time scales through mutations and Darwinian selection. We have analyzed exemplaric cases of this evolution and arrive at a dynamical picture of the genome with relatively static genes ("hardware") and adaptively evolving interactions between them ("software") [1]. We derive consequences for the architecture of genomes in their regulatory regions that compare favorably with genomic data and lead to quantitative predictions of promoter evolution [2].

Time-dependent selection causes the adaptive evolution of new phenotypes, and this dynamics can be traced in genomic data. We have developed a new Bayesian inference method for adaptations, which is based on a model for evolution under time-dependent selection and is more sensitive than the standard population-genetic tests [1,2]. We find evidence Analyzing substitutions and polymorphisms in Drosophila by this method shows that selection itself is strongly time-dependent, resembling more a seascape than a static fitness landscape. Our results suggest that selection acts not only as a constraint but as a major driving force of genomic change [1]. The joint statistics of genetic drift and selection fluctuations resembles the physics of systems with quenched disorder [3].

On a coarse-grained level, the statistics of bio-molecular networks can be studied at the level of the network topology (i.e., the ensemble of its nodes and links). Are there topological observables that quantify the deviation of a biological network from a simple random graph? We have established the statistical mechanics of graphs with neighbor connectivity correlations [1]. We have constucted an empirically grounded evolution model for protein networks, which is based on gene duplications and functional mutations of existing proteins and leads to correlated graphs whose structure is in good agreement with the experimental data in yeast [2].

A major application of statistical mechanics to biological networks is in graph alignment. We have established algorithms that find and quantify mutually similar subgraphs within one network or the amount of similarity between different networks, pointing at putative biologically functional units. The underlying statistical theory is grounded on a stochastic evolution model for network topologies. Initial applications have been the identification of functional motifs in the transcription network of E. coli [3] and the analysis of conservation and innovation in gene co-expression networks between human and mouse [4].

We study a minimal model for genome evolution whose elementary processes are single site mutation, duplication and deletion of sequence regions and insertion of random segments. These processes are found to generate long-range correlations in the composition of letters as long as the sequence length is growing, i.e., the combined rates of duplications and insertions are higher than the deletion rate. For constant sequence length, on the other hand, all initial correlations decay exponentially. These results are obtained analytically and by simulations. They are compared with the long-range correlations observed in genomic DNA, and the implications for genome evolution are discussed [1]. A more comprehensive analyisis of the underlying nonequilibrium stochastic process is contained in [2].

We study secondary structures of random RNA molecules by means of a renormalized field theory based on an expansion in the sequence disorder. We show that there is a continuous phase transition from a molten phase at higher temperatures to a low-temperature glass phase . Primary freezing occurs above the critical temperature: local islands of stable folds form within the molten phase, the size of these islands defining the correlation length of the transition. Our results include critical exponents at the transition and in the glass phase.

Sequences are similar due to their common evolutionary ancestry, and they differ due to mutations. The recognition and analysis of sequence similarities is one of the most widely used mathematical tools in modern molecular biology. It is crucial to the understanding of genome data. Using concepts and methods from theoretical physics, we have developed a statistical theory of sequence similarity recognition. A recent outcome has been a new algorithm for the detection of distant evolutionary relationships.