Metastability and small eigenvalues in Markov chains

In this letter we announce rigorous results that elucidate the relation between metastable states and low-lying eigenvalues in Markov chains in a much more general setting and with considerable greater precision as was so far available. This includes a sharp uncertainty principle relating all low-lying eigenvalues to mean times of metastable transitions, a relation between the support of eigenfunctions and the attractor of a metastable state, and sharp estimates on the convergence of probability distribution of the metastable transition times to the exponential distribution.Comment: 5pp, AMSTe

Infinite-dimensional diffusions as limits of random walks on partitions

The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.Comment: AMSTex, 33 pages. Version 2: minor changes, typos corrected, to appear in Prob. Theor. Rel. Field

Direct observation of nuclear reorganization driven by ultrafast spin transitions

One of the most basic molecular photophysical processes is that of spin transitions and intersystem crossing between excited states surfaces. The change in spin states affects the spatial distribution of electron density through the spin orbit coupling interaction. The subsequent nuclear reorganization reports on the full extent of the spin induced change in electron distribution, which can be treated similarly to intramolecular charge transfer with effective reaction coordinates depicting the spin transition. Here, single-crystal [FeII(bpy)3] (PF6)2, a prototypical system for spin crossover (SCO) dynamics, is studied using ultrafast electron diffraction in the single-photon excitation regime. The photoinduced SCO dynamics are resolved, revealing two distinct processes with a (450 ± 20)-fs fast component and a (2.4 ± 0.4)-ps slow component. Using principal component analysis, we uncover the key structural modes, ultrafast Fe–N bond elongations coupled with ligand motions, that define the effective reaction coordinate to fully capture the relevant molecular reorganization

Cavitation induced by explosion in a model of ideal fluid

We discuss the problem of an explosion in the cubic-quintic superfluid model, in relation to some experimental observations. We show numerically that an explosion in such a model might induce a cavitation bubble for large enough energy. This gives a consistent view for rebound bubbles in superfluid and we indentify the loss of energy between the successive rebounds as radiated waves. We compute self-similar solution of the explosion for the early stage, when no bubbles have been nucleated. The solution also gives the wave number of the excitations emitted through the shock wave.Comment: 21 pages,13 figures, other comment

Differences between Ca2+ and Mg2+ in DNA binding and release by the SfiI restriction endonuclease: implications for DNA looping

Many enzymes acting on DNA require Mg2+ ions not only for catalysis but also to bind DNA. Binding studies often employ Ca2+ as a substitute for Mg2+, to promote DNA binding whilst disallowing catalysis. The SfiI endonuclease requires divalent metal ions to bind DNA but, in contrast to many systems where Ca2+ mimics Mg2+, Ca2+ causes SfiI to bind DNA almost irreversibly. Equilibrium binding by wild-type SfiI cannot be conducted with Mg2+ present as the DNA is cleaved so, to study the effect of Mg2+ on DNA binding, two catalytically-inactive mutants were constructed. The mutants bound DNA in the presence of either Ca2+ or Mg2+ but, unlike wild-type SfiI with Ca2+, the binding was reversible. With both mutants, dissociation was slow with Ca2+ but was in one case much faster with Mg2+. Hence, Ca2+ can affect DNA binding differently from Mg2+. Moreover, SfiI is an archetypal system for DNA looping; on DNA with two recognition sites, it binds to both sites and loops out the intervening DNA. While the dynamics of looping cannot be measured with wild-type SfiI and Ca2+, it becomes accessible with the mutant and Mg2+

DNA looping by FokI: the impact of synapse geometry on loop topology at varied site orientations

Most restriction endonucleases, including FokI, interact with two copies of their recognition sequence before cutting DNA. On DNA with two sites they act in cis looping out the intervening DNA. While many restriction enzymes operate symmetrically at palindromic sites, FokI acts asymmetrically at a non-palindromic site. The directionality of its sequence means that two FokI sites can be bridged in either parallel or anti-parallel alignments. Here we show by biochemical and single-molecule biophysical methods that FokI aligns two recognition sites on separate DNA molecules in parallel and that the parallel arrangement holds for sites in the same DNA regardless of whether they are in inverted or repeated orientations. The parallel arrangement dictates the topology of the loop trapped between sites in cis: the loop from inverted sites has a simple 180° bend, while that with repeated sites has a convoluted 360° turn. The ability of FokI to act at asymmetric sites thus enabled us to identify the synapse geometry for sites in trans and in cis, which in turn revealed the relationship between synapse geometry and loop topology

Dissecting protein-induced DNA looping dynamics in real time

Many proteins that interact with DNA perform or enhance their specific functions by binding simultaneously to multiple target sites, thereby inducing a loop in the DNA. The dynamics and energies involved in this loop formation influence the reaction mechanism. Tethered particle motion has proven a powerful technique to study in real time protein-induced DNA looping dynamics while minimally perturbing the DNA–protein interactions. In addition, it permits many single-molecule experiments to be performed in parallel. Using as a model system the tetrameric Type II restriction enzyme SfiI, that binds two copies of its recognition site, we show here that we can determine the DNA–protein association and dissociation steps as well as the actual process of protein-induced loop capture and release on a single DNA molecule. The result of these experiments is a quantitative reaction scheme for DNA looping by SfiI that is rigorously compared to detailed biochemical studies of SfiI looping dynamics. We also present novel methods for data analysis and compare and discuss these with existing methods. The general applicability of the introduced techniques will further enhance tethered particle motion as a tool to follow DNA–protein dynamics in real time

Concerted action at eight phosphodiester bonds by the BcgI restriction endonuclease

The BcgI endonuclease exemplifies a subset of restriction enzymes, the Type IIB class, which make two double-strand breaks (DSBs) at each copy of their recognition sequence, one either side of the site, to excise the sequence from the remainder of the DNA. In this study, we show that BcgI is essentially inactive when bound to a single site and that to cleave a DNA with one copy of its recognition sequence, it has to act in trans, bridging two separate DNA molecules. We also show that BcgI makes the two DSBs at an individual site in a highly concerted manner. Intermediates cut on one side of the site do not accumulate during the course of the reaction: instead, the DNA is converted straight to the final products cut on both sides. On DNA with two sites, BcgI bridges the sites in cis and then generally proceeds to cut both strands on both sides of both sites without leaving the DNA. The BcgI restriction enzyme can thus excise two DNA segments together, by cleaving eight phosphodiester bonds within a single-DNA binding event

Multivariate curve resolution of time course microarray data

BACKGROUND: Modeling of gene expression data from time course experiments often involves the use of linear models such as those obtained from principal component analysis (PCA), independent component analysis (ICA), or other methods. Such methods do not generally yield factors with a clear biological interpretation. Moreover, implicit assumptions about the measurement errors often limit the application of these methods to log-transformed data, destroying linear structure in the untransformed expression data. RESULTS: In this work, a method for the linear decomposition of gene expression data by multivariate curve resolution (MCR) is introduced. The MCR method is based on an alternating least-squares (ALS) algorithm implemented with a weighted least squares approach. The new method, MCR-WALS, extracts a small number of basis functions from untransformed microarray data using only non-negativity constraints. Measurement error information can be incorporated into the modeling process and missing data can be imputed. The utility of the method is demonstrated through its application to yeast cell cycle data. CONCLUSION: Profiles extracted by MCR-WALS exhibit a strong correlation with cell cycle-associated genes, but also suggest new insights into the regulation of those genes. The unique features of the MCR-WALS algorithm are its freedom from assumptions about the underlying linear model other than the non-negativity of gene expression, its ability to analyze non-log-transformed data, and its use of measurement error information to obtain a weighted model and accommodate missing measurements

Heartbeat of the Sun from Principal Component Analysis and prediction of solar activity on a millenium timescale

yesWe derive two principal components (PCs) of temporal magnetic field variations over the solar cycles 21–24 from full disk magnetograms covering about 39% of data variance, with σ = 0.67. These PCs are attributed to two main magnetic waves travelling from the opposite hemispheres with close frequencies and increasing phase shift. Using symbolic regeression analysis we also derive mathematical formulae for these waves and calculate their summary curve which we show is linked to solar activity index. Extrapolation of the PCs backward for 800 years reveals the two 350-year grand cycles superimposed on 22 year-cycles with the features showing a remarkable resemblance to sunspot activity reported in the past including the Maunder and Dalton minimum. The summary curve calculated for the next millennium predicts further three grand cycles with the closest grand minimum occurring in the forthcoming cycles 26–27 with the two magnetic field waves separating into the opposite hemispheres leading to strongly reduced solar activity. These grand cycle variations are probed by α − Ω dynamo model with meridional circulation. Dynamo waves are found generated with close frequencies whose interaction leads to beating effects responsible for the grand cycles (350–400 years) superimposed on a standard 22 year cycle. This approach opens a new era in investigation and confident prediction of solar activity on a millenium timescale