Noise induced dissipation in Lebesgue-measure preserving maps on d−d-dimensional torus

We consider dissipative systems resulting from the Gaussian and $alpha$-stable noise perturbations of measure-preserving maps on the $d$ dimensional torus. We study the dissipation time scale and its physical implications as the noise level \vep vanishes. We show that nonergodic maps give rise to an O(1/\vep) dissipation time whereas ergodic toral automorphisms, including cat maps and their $d$-dimensional generalizations, have an O(\ln{(1/\vep)}) dissipation time with a constant related to the minimal, {\em dimensionally averaged entropy} among the automorphism's irreducible blocks. Our approach reduces the calculation of the dissipation time to a nonlinear, arithmetic optimization problem which is solved asymptotically by means of some fundamental theorems in theories of convexity, Diophantine approximation and arithmetic progression. We show that the same asymptotic can be reproduced by degenerate noises as well as mere coarse-graining. We also discuss the implication of the dissipation time in kinematic dynamo.Comment: The research is supported in part by the grant from U.S. National Science Foundation, DMS-9971322 and Lech Wolowsk

Infinite Order Differential Operators in Spaces of Entire Functions

We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane. We obtain integral representations of some particular cases of these operators and apply these results to obtain explicit solutions to some Cauchy problems for diffusion equations with nonconstant drift term

Pade approximants of random Stieltjes series

We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Pade approximants of the function S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional disordered system, and the results of Marklof et al. (2005), we obtain explicit formulae (in terms of modified Bessel functions) for the almost-sure rate of convergence of these approximants, and for the almost-sure distribution of their poles.Comment: To appear in Proc Roy So

Relaxation Time of Quantized Toral Maps

We introduce the notion of the relaxation time for noisy quantum maps on the 2d-dimensional torus - a generalization of previously studied dissipation time. We show that relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit ($\hbar$ -> 0) together with the limit of small noise strength (\ep -> 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime $\hbar1) in which classical and quantum relaxation times share the same asymptotics: in this regime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical correspondence for noisy maps on the torus, up to times logarithmic in$\hbar^{-1}$. On the other hand, we show that in the ``quantum regime''$\ep$<<$\hbar$ << 1, quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps), we obtain the exact asymptotics of the quantum relaxation time and precise the regime of correspondence between quantum and classical relaxations.Comment: LaTeX, 27 pages, former term dissipation time replaced by relaxation time, new introduction and reference

High-quality, high-throughput measurement of protein-DNA binding using HiTS-FLIP

In order to understand in more depth and on a genome wide scale the behavior of transcription factors (TFs), novel quantitative experiments with high-throughput are needed. Recently, HiTS-FLIP (High-Throughput Sequencing-Fluorescent Ligand Interaction Profiling) was invented by the Burge lab at the MIT (Nutiu et al. (2011)). Based on an Illumina GA-IIx machine for next-generation sequencing, HiTS-FLIP allows to measure the affinity of fluorescent labeled proteins to millions of DNA clusters at equilibrium in an unbiased and untargeted way examining the entire sequence space by Determination of dissociation constants (Kds) for all 12-mer DNA motifs. During my PhD I helped to improve the experimental design of this method to allow measuring the protein-DNA binding events at equilibrium omitting any washing step by utilizing the TIRF (Total Internal Reflection Fluorescence) based optics of the GA-IIx. In addition, I developed the first versions of XML based controlling software that automates the measurement procedure. Meeting the needs for processing the vast amount of data produced by each run, I developed a sophisticated, high performance software pipeline that locates DNA clusters, normalizes and extracts the fluorescent signals. Moreover, cluster contained k-mer motifs are ranked and their DNA binding affinities are quantified with high accuracy. My approach of applying phase-correlation to estimate the relative translative Offset between the observed tile images and the template images omits resequencing and thus allows to reuse the flow cell for several HiTS-FLIP experiments, which greatly reduces cost and time. Instead of using information from the sequencing images like Nutiu et al. (2011) for normalizing the cluster intensities which introduces a nucleotide specific bias, I estimate the cluster related normalization factors directly from the protein Images which captures the non-even illumination bias more accurately and leads to an improved correction for each tile image. My analysis of the ranking algorithm by Nutiu et al. (2011) has revealed that it is unable to rank all measured k-mers. Discarding all the clusters related to previously ranked k-mers has the side effect of eliminating any clusters on which k-mers could be ranked that share submotifs with previously ranked k-mers. This shortcoming affects even strong binding k-mers with only one mutation away from the top ranked k-mer. My findings show that omitting the cluster deletion step in the ranking process overcomes this limitation and allows to rank the full spectrum of all possible k-mers. In addition, the performance of the ranking algorithm is drastically reduced by my insight from a quadratic to a linear run time. The experimental improvements combined with the sophisticated processing of the data has led to a very high accuracy of the HiTS-FLIP dissociation constants (Kds) comparable to the Kds measured by the very sensitive HiP-FA assay (Jung et al. (2015)). However, experimentally HiTS-FLIP is a very challenging assay. In total, eight HiTS-FLIP experiments were performed but only one showed saturation, the others exhibited Protein aggregation occurring at the amplified DNA clusters. This biochemical issue could not be remedied. As example TF for studying the details of HiTS-FLIP, GCN4 was chosen which is a dimeric, basic leucine zipper TF and which acts as the master regulator of the amino acid starvation Response in Saccharomyces cerevisiae (Natarajan et al. (2001)). The fluorescent dye was mOrange. The HiTS-FLIP Kds for the TF GCN4 were validated by the HiP-FA assay and a Pearson correlation coefficient of R=0.99 and a relative error of delta=30.91% was achieved. Thus, a unique and comprehensive data set of utmost quantitative precision was obtained that allowed to study the complex binding behavior of GCN4 in a new way. My Downstream analyses reveal that the known 7-mer consensus motif of GCN4, which is TGACTCA, is modulated by its 2-mer neighboring flanking regions spanning an affinity range over two orders of magnitude from a Kd=1.56 nM to Kd=552.51 nM. These results suggest that the common 9-mer PWM (Position Weight Matrix) for GCN4 is insufficient to describe the binding behavior of GCN4. Rather, an additional left and right flanking nucleotide is required to extend the 9-mer to an 11-mer. My analyses regarding mutations and related delta delta G values suggest long-range interdependencies between nucleotides of the two dimeric half-sites of GCN4. Consequently, models assuming positional independence, such as a PWM, are insufficient to explain these interdependencies. Instead, the full spectrum of affinity values for all k-mers of appropriate size should be measured and applied in further analyses as proposed by Nutiu et al. (2011). Another discovery were new binding motifs of GCN4, which can only be detected with a method like HiTS-FLIP that examines the entire sequence space and allows for unbiased, de-novo motif discovery. All These new motifs contain GTGT as a submotif and the data collected suggests that GCN4 binds as monomer to these new motifs. Therefore, it might be even possible to detect different binding modes with HiTS-FLIP. My results emphasize the binding complexity of GCN4 and demonstrate the advantage of HiTS-FLIP for investigating the complexity of regulative processes

Noise Induced Dissipation in Discrete-Time Classical and Quantum Dynamical Systems

We introduce a new characteristics of chaoticity of classical and quantum dynamical systems by defining the notion of the dissipation time which enables us to test how the system responds to the noise and in particular to measure the speed at which an initially closed, conservative system converges to the equilibrium when subjected to noisy (stochastic) perturbations. We prove fast dissipation result for classical Anosov systems and general exponentially mixing maps. Slow dissipation result is proved for regular systems including non-weakly mixing maps. In quantum setting we study simultaneous semiclassical and small noise asymptotics of the dissipation time of quantized toral symplectomorphisms (generalized cat maps) and derive sharp bounds for semiclassical regime in which quantum-classical correspondence of dissipation times holds.Comment: PhD Dissertation, University of California at Davis, June 2004, LaTex, 195 page