## Research

Work of the group lies on the interface between statistical mechanics and quantitative biology. At the centre is the relationship between fluctuations and noise in biological systems, the corresponding statistical ensembles, and biological function. This connection emerges at very different levels and timescales, from stochastic modeling of gene expression to the population dynamics of regulatory DNA.### Stochastic dynamics of gene expression

Gene transcription takes place out of equilibrium, with the rate of expression of a
gene constantly changing due to the presence or absence of specific
proteins called transcription factors. Random fluctuations of
molecular concentrations can have large effects on cells, such as
transitions between metastable states. In a collaboration with
experimentalists of the
Veening lab (Groningen), we have pinpointed the fluctuation causing
the transition to the induced state of the paradigmatic lactose-uptake pathway in
E. coli and developed a statistical theory of this
transition [1]. Past work was on
out-of-equilibrium properties of gene expression [2], the determination of
regulatory interactions between genes from gene expression time
series [3], and fluctuations in the dynamics of non-coding RNA [4].
| The rate at which the lac-pathway switches to the induced state (log-scale) against the inverse of the inducer concentration. Our model predicts a simple relationship that is borne out very well experimentally (orange triangles), see [1]. |

### Statistical mechanics, inverse problems, and inference

We study pure and applied problems at the interface between
statistical mechanics and machine learning. An example is the
inverse Ising problem: how to reconstruct couplings and fields of an
Ising model given observables such as spin-spin correlations. We
contributed a novel method based on the Bethe approximation, which is
exact on tree-like lattices [1], and extended mean-field approaches to
both ferromagnetic and glassy low-temperature phases of Ising models [2]. Applications lie in
the reconstruction of expression levels of the constituents of
complex tissues from mixtures (in collaboration with Roman
Müller and Thomas Benzing at the
Nephrology
Lab, Cologne)
and, more recently, cancer genomics
(in collaboration with Roman Thomas at the Department of Translational Medicine, Cologne).
| Reconstruction error at different temperatures for model with a glassy low-T phase (SK-model) against inverse temperature and for different numbers of thermodynamic states. See [1] for details. |

### Statistical mechanics of bio-molecular networks

We have developed an equilibrium statistical
mechanics of networks with local connectivity correlations
[1], which links a network Hamiltonian with the corresponding
statistical observables. This statistical approach forms the
theoretical basis for identifying functional units in biological
networks. Examples are repeated patterns in networks, called network
motifs, which can be identified using Bayesian inference on models of
correlations within networks [2]. The evolutionary dynamics of
networks of related species can be traced by graph alignment, which is
based on a stochastic evolution model for biological interactions and
for DNA sequences [3,4]. We have used this approach to analyze
coexpression networks of human and mouse, and the protein interaction
networks of different herpesviruses. A software package called GraphAlignment
is available for download.
| Alignment of the protein interaction networks of two different Herpes viruses [4]. Animation by Jörn Meier. |

### Evolution of regulatory interactions

Transcription factors function by binding to specific binding sites on
DNA, thereby affecting the expression rate of nearby genes.
These binding sites encode regulatory interactions between
genes at the level of the DNA sequence. Over evolutionary time
scales binding sites can grow weaker or stronger, or disappear
completely from the sequence. Higher organisms, which largely share
the same set of genes, owe much of their diversity to the evolutionary
changes of binding sites.
A stochastic model for binding site evolution, including point mutations, genetic drift and natural selection, shows how quickly a new binding site can be generated as a response to selective pressure on a population [1]. The resulting equilibrium distribution of binding strengths agrees well with the empirical binding-site statistics found in bacterial genomes.
At a genome-wide level, many different regulatory networks can produce
a given set of expression patterns.
A simple model of gene
regulation allows to investigate the ability of regulatory networks to
reproduce given expression levels [2]. We find an exponentially large
space of regulatory networks compatible with a given set of expression
levels, giving rise to an extensive entropy of networks.
| A population of binding sites, driven by mutations and selection for binding a transcription factor. |

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