Cellular Translocation of Biomolecules using Ratcheting

The translocation of biopolymers through nanopores is a crucial process in cellular biology [1]. Such processes have been modelled by means of a Brownian ratchet [2].
In such models, diffusion through the nanopore occures by thermal fluctuations. In addition, target sites at the translocated biomolecule prevent movement of the biomolecule from the lumen back to the cytosol. Such target sites are often assumed to be fixed positions of the translocated biomolecule [2].

However, as described by [3], the translocation of proteins into the Endoplasmatic Reticulum (ER) is based on the random binding of an ATPase to the protein inside the ER lumen. Thus the movement of the protein into the ER depends on the process of binding and unbinding of ATPases.

Brownian ratchet models have been used to describe polymerization of actin filaments against a barrier. While these processes have been thoroughly studied [2,4], the understanding of ratchet models for translocation with random target sites is not complete. The most important question is: What is the average velocity of a biomolecule depending on binding and unbinding rates of target molecules?

Andrej Depperschmidt and Peter Pfaffelhuber

Literature:

[1] T. Ambjornsson, M. A. Lomholt and R. Metzler, Directed motion emerging from two coupled random processes: translocation of a chain through a membrane nanopore driven by binding proteins. Journal of Physics: Condensed Matter, 2005, 17, S3945-S3964.

[2] C. S. Peskin, G. M. Odell and G. F. Oster, Cellular motions and thermal fluctuations: the Brownian ratchet. Biophys J, 1993, 65, 316-324.

[3] W. Liebermeister, T. A. Rapoport and R. Heinrich, Ratcheting in post-translational protein translocation: A mathematical model. J Mol Biol, 2001, 305, 643-656.

[4] H. Qian, A stochastic analysis of a Brownian ratchet model for actin-based motility. Mech Chem Biosyst, 2004, 1, 267-278.

Aging and the loss of telomere sequences

Eucaryotic DNA is organized linearly and is replicated when a cell proliferates. The replication mechanism is not perfect in that telomeres are shortened by each round of DNA replication. This mechanism has been suggested to play a role in cellular aging: if the length of the telomere falls below the Hayflick constant the cell loses its ability to replicate, a phenomenon known as replicative senescence.
Several quantitative models have been suggested to describe replicative senescence. An important process is the evolution of the frequency of proliferating cells. The approach of [1] who use a complete binary tree has been extended by [2] by using branching processes. This process can describe the gradual increase of senesced cells in experiments. This approach has been complemented by [3] by explicitely taking telomerase activity into account and by [4] who model the loss of telomere lengths stochastically.
Already in the simplest formulation several questions about cellular senescence remain unanswered. Most interesting is the stochastic process of the gradual decrease of proliferating cells which can describe experimental data. The average decrease is already well studied in most cases. Moreover, a detailed analysis of all effects is possible through the connection of branching processes to the theory of random binary search trees [5]. A linking model of these processes has been described by [6].

Katharina Surovcik and Peter Pfaffelhuber

Literature:

[1] M. Z. Levy, R. C. Allsopp, A. B. Futcher, C. W. Greider and C. B. Harley, Telomere end-replication problem and cell aging. J Mol Biol, 1992, 225, 951-960.

[2] O. Arino, M. Kimmel and G. F. Webb, Mathematical modeling of the loss of telomere sequences. J Theor Biol, 1995, 177, 45-57.

[3] N. Arkus, A mathematical model of cellular apoptosis and senescence through the dynamics of telomere loss. J Theor Biol, 2005, 235, 13-32.

[4] T. Antal, K. B. Blagoev, S. A. Trugman and S. Redner, Aging and immortality in a cell proliferation model. J Theor Biol, 2007, 248, 411-417.

[5] H. Mahmoud. Evolution of Random Search Trees. Wiley, 1992.

[6] D. Aldous and P. Shields, A diffusion limit for a class of randomly-growing binary trees. Probability Theory and Related Fields, 1988, 79, 509-542.

Spatial Phenomena in Systems Biology

Pattern formation and wave propagation are central in the mathematical
description of biological systems. They arise in fields like chemical
reaction kinetics (as intra- as well as inter-cellular dynamical
phenomena), population models, bacterial patterns, wound healing, tumor
growth, and the spread of infectious diseases. [1]

While such spatial phenomena have been mainly modeled in the deterministic
limit, there is growing interest in a stochastic description of chemical
reaction networks [2] to account for kinetic phenomena concerned with
rather small numbers of chemical species like enzymes within a cell.

Our research therefore aims for a closer (mathematically exact) look at
possible stochastic phenomena in the spatial organization of chemical
reaction networks and their impact on perturbation phenomena in pattern
and wave formation.

Heinz Weisshaupt and Peter Pfaffelhuber

Literature:

[1] J. D. Murray, Mathematical Biology I + II, Springer Verlag

[2] K. Ball, T.G. Kurtz, L. Popovic, G. Rempala, Asymptotic Analysis of Multiscale Approximations to Reaction Networks. The Annals of Applied Probability, 2006, Vol. 16, No. 4, 1925–1961.

Autoinducer signalling in rhizobia

Intercellular communication by means of small signal molecules synchronizes gene expression and coordinates functions among bacteria. This population density-dependent regulation is known as quorum sensing. Quorum sensing is frequently mediated by N-acylhomoserine lactone (AHL) autoinducers. We investigate the molecular mechanism and regulation of quorum sensing in aiming at establishing a predictive mathematical model of autoinducer signalling in this organism. To this end, we describe the dynamical system of the reactions responsible for the autoinducing behaviour within the bacterial cell and extend it to include coupling via a common environment. In a further step, we investigate the spatial organisation of the communicating cells.

Katharina Surovcik and Peter Pfaffelhuber,
together with the group of Anke Becker