Graph Kernels

We present a unified framework to study graph kernels, special cases of which include the random walk (Gärtner et al., 2003; Borgwardt et al., 2005) and marginalized (Kashima et al., 2003, 2004; Mahé et al., 2004) graph kernels. Through reduction to a Sylvester equation we improve the time complexity of kernel computation between unlabeled graphs with n vertices from O(n^6) to O(n^3). We find a spectral decomposition approach even more efficient when computing entire kernel matrices. For labeled graphs we develop conjugate gradient and fixed-point methods that take O(dn^3) time per iteration, where d is the size of the label set. By extending the necessary linear algebra to Reproducing Kernel Hilbert Spaces (RKHS) we obtain the same result for d-dimensional edge kernels, and O(n^4) in the infinite-dimensional case; on sparse graphs these algorithms only take O(n^2) time per iteration in all cases. Experiments on graphs from bioinformatics and other application domains show that these techniques can speed up computation of the kernel by an order of magnitude or more. We also show that certain rational kernels (Cortes et al., 2002, 2003, 2004) when specialized to graphs reduce to our random walk graph kernel. Finally, we relate our framework to R-convolution kernels (Haussler, 1999) and provide a kernel that is close to the optimal assignment kernel of Fröhlich et al. (2006) yet provably positive semi-definite

Classes of fast and specific search mechanisms for proteins on DNA

Problems of search and recognition appear over different scales in biological systems. In this review we focus on the challenges posed by interactions between proteins, in particular transcription factors, and DNA and possible mechanisms which allow for a fast and selective target location. Initially we argue that DNA-binding proteins can be classified, broadly, into three distinct classes which we illustrate using experimental data. Each class calls for a different search process and we discuss the possible application of different search mechanisms proposed over the years to each class. The main thrust of this review is a new mechanism which is based on barrier discrimination. We introduce the model and analyze in detail its consequences. It is shown that this mechanism applies to all classes of transcription factors and can lead to a fast and specific search. Moreover, it is shown that the mechanism has interesting transient features which allow for stability at the target despite rapid binding and unbinding of the transcription factor from the target.Comment: 65 pages, 23 figure

Dynamics of Alpha-Helix Formation in the CSAW Model

We study the folding dynamics of polyalanine (Ala$_{20}$), a protein fragment with 20 residues whose native state is a single alpha helix. We use the CSAW model (conditioned self-avoiding walk), which treats the protein molecule as a chain in Brownian motion, with interactions that include hydrophobic forces and internal hydrogen bonding. We find that large scale structures form before small scale structures, and obtain the relevant relaxation times. We find that helix nucleation occurs at two separate points on the protein chain. The evolution of small and large scale structures involve different mechanisms. While the former can be describe by rate equations governing the growth of helical content, the latter is akin to the relaxation of an elastic solid.Comment: 18 pages, 10 figure

Modeling transcription factor binding events to DNA using a random walker/jumper representation on a 1D/2D lattice with different affinity sites

Surviving in a diverse environment requires corresponding organism responses. At the cellular level, such adjustment relies on the transcription factors (TFs) which must rapidly find their target sequences amidst a vast amount of non-relevant sequences on DNA molecules. Whether these transcription factors locate their target sites through a 1D or 3D pathway is still a matter of speculation. It has been suggested that the optimum search time is when the protein equally shares its search time between 1D and 3D diffusions. In this paper, we study the above problem using a Monte Carlo simulation by considering a very simple physical model. A 1D strip, representing a DNA, with a number of low affinity sites, corresponding to non-target sites, and high affinity sites, corresponding to target sites, is considered and later extended to a 2D strip. We study the 1D and 3D exploration pathways, and combinations of the two modes by considering three different types of molecules: a walker that randomly walks along the strip with no dissociation; a jumper that represents dissociation and then re-association of a TF with the strip at later time at a distant site; and a hopper that is similar to the jumper but it dissociates and then re-associates at a faster rate than the jumper. We analyze the final probability distribution of molecules for each case and find that TFs can locate their targets fast enough even if they spend 15% of their search time diffusing freely in the solution. This indeed agrees with recent experimental results obtained by Elf et al. 2007 and is in contrast with theoretical expectation.Comment: 24 pages, 9 figure

Base sequence dependent sliding of proteins on DNA

The possibility that the sliding motion of proteins on DNA is influenced by the base sequence through a base pair reading interaction, is considered. Referring to the case of the T7 RNA-polymerase, we show that the protein should follow a noise-influenced sequence-dependent motion which deviate from the standard random walk usually assumed. The general validity and the implications of the results are discussed.Comment: 12 pages, 3 figure

Intermittent search strategies

This review examines intermittent target search strategies, which combine phases of slow motion, allowing the searcher to detect the target, and phases of fast motion during which targets cannot be detected. We first show that intermittent search strategies are actually widely observed at various scales. At the macroscopic scale, this is for example the case of animals looking for food ; at the microscopic scale, intermittent transport patterns are involved in reaction pathway of DNA binding proteins as well as in intracellular transport. Second, we introduce generic stochastic models, which show that intermittent strategies are efficient strategies, which enable to minimize the search time. This suggests that the intrinsic efficiency of intermittent search strategies could justify their frequent observation in nature. Last, beyond these modeling aspects, we propose that intermittent strategies could be used also in a broader context to design and accelerate search processes.Comment: 72 pages, review articl