Thomas Mejer Hansen

Correlation between mechanical strength of messenger RNA pseudoknots and ribosomal frameshifting

Programmed ribosomal frameshifting is often used by viral pathogens including HIV. Slippery sequences present in some mRNAs cause the ribosome to shift reading frame. The resulting protein is thus encoded by one reading frame upstream from the slippery sequence and by another reading frame downstream from the slippery sequence. Although the mechanism is not well understood, frameshifting is known to be stimulated by an mRNA structure such as a pseudoknot. Here, we show that the efficiency of frameshifting relates to the mechanical strength of the pseudoknot. Two pseudoknots derived from the Infectious Bronchitis Virus were used, differing by one base pair in the first stem. In Escherichia coli, these two pseudoknots caused frameshifting frequencies that differed by a factor of two. We used optical tweezers to unfold the pseudoknots. The pseudoknot giving rise to the highest degree of frameshifting required a nearly 2-fold larger unfolding force than the other. The observed energy difference cannot be accounted for by any existing model. We propose that the degree of ribosomal frameshifting is related to the mechanical strength of RNA pseudoknots. Our observations support the “9 Å model” that predicts some physical barrier is needed to force the ribosome into the −1 frame. Also, our findings support the recent observation made by cryoelectron microscopy that mechanical interaction between a ribosome and a pseudoknot causes a deformation of the A-site tRNA. The result has implications for the understanding of genetic regulation, reading frame maintenance, tRNA movement, and unwinding of mRNA secondary structures by ribosomes.

Linear inverse Gaussian theory and geostatistics

Inverse problems in geophysics require the introduction of complex a priori information and are solved using computationally expensive Monte Carlo techniques (where large portions of the model space are explored). The geostatistical method allows for fast integration of complex a priori information in the form of covariance functions and training images. We combine geostatistical methods and inverse problem theory to generate realizations of the posterior probability density function of any Gaussian linear inverse problem, honoring a priori information in the form of a covariance function describing the spatial connectivity of the model space parameters. This is achieved using sequential Gaussian simulation, a well-known, noniterative geostatisticalmethod for generating samples of a Gaussian random field with a given covariance function. This work is a contribution to both linear inverse problem theory and geostatistics. Our main result is an efficient method to generate realizations, actual solutions rat...

Inverse problems with non-trivial priors: efficient solution through sequential Gibbs sampling

Markov chain Monte Carlo methods such as the Gibbs sampler and the Metropolis algorithm can be used to sample solutions to non-linear inverse problems. In principle, these methods allow incorporation of prior information of arbitrary complexity. If an analytical closed form description of the prior is available, which is the case when the prior can be described by a multidimensional Gaussian distribution, such prior information can easily be considered. In reality, prior information is often more complex than can be described by the Gaussian model, and no closed form expression of the prior can be given. We propose an algorithm, called sequential Gibbs sampling, allowing the Metropolis algorithm to efficiently incorporate complex priors into the solution of an inverse problem, also for the case where no closed form description of the prior exists. First, we lay out the theoretical background for applying the sequential Gibbs sampler and illustrate how it works. Through two case studies, we demonstrate the application of the method to a linear image restoration problem and to a non-linear cross-borehole inversion problem. We demonstrate how prior information can reduce the complexity of an inverse problem and that a prior with little information leads to a hard inverse problem, practically unsolvable except when the number of model parameters is very small. Considering more complex and realistic prior information thus not only makes realizations from the posterior look more realistic but it can also reduce the computation time for the inversion dramatically. The method works for any statistical model for which sequential simulation can be used to generate realizations. This applies to most algorithms developed in the geostatistical community.

VISIM: Sequential simulation for linear inverse problems

Linear inverse Gaussian problems are traditionally solved using least squares-based inversion. The center of the posterior Gaussian probability distribution is often chosen as the solution to such problems, while the solution is in fact the posterior Gaussian probability distribution itself. We present an algorithm, based on direct sequential simulation, which can be used to efficiently draw samples of the posterior probability distribution for linear inverse problems. There is no Gaussian restriction on the distribution in the model parameter space, as inherent in traditional least squares-based algorithms. As data for linear inverse problems can be seen as weighed linear averages over some volume, block kriging can be used to perform both estimation (i.e. finding the center of the posterior Gaussian pdf) and simulation (drawing samples of the posterior Gaussian pdf). We present the kriging system which we use to implement a flexible GSLIB-based algorithm for solving linear inverse problems. We show how we implement such a simulation program conditioned to linear average data. The program is called VISIM as an acronym for Volume average Integration SIMulation. An effort has been made to make the program efficient, even for larger scale problems, and the computational efficiency and accuracy of the code is investigated. Using a synthetic cross-borehole tomography case study, we show how the program can be used to generate realizations of the a posteriori distributions (i.e. solutions) from a linear tomography problem. Both Gaussian and non-Gaussian a priori model parameter distributions are considered.

A Frequency Matching Method: Solving Inverse Problems by Use of Geologically Realistic Prior Information

The frequency matching method defines a closed form expression for a complex prior that quantifies the higher order statistics of a proposed solution model to an inverse problem. While existing solution methods to inverse problems are capable of sampling the solution space while taking into account arbitrarily complex a priori information defined by sample algorithms, it is not possible to directly compute the maximum a posteriori model, as the prior probability of a solution model cannot be expressed. We demonstrate how the frequency matching method enables us to compute the maximum a posteriori solution model to an inverse problem by using a priori information based on multiple point statistics learned from training images. We demonstrate the applicability of the suggested method on a synthetic tomographic crosshole inverse problem.

SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information

From a probabilistic point-of-view, the solution to an inverse problem can be seen as a combination of independent states of information quantified by probability density functions. Typically, these states of information are provided by a set of observed data and some a priori information on the solution. The combined states of information (i.e. the solution to the inverse problem) is a probability density function typically referred to as the a posteriori probability density function. We present a generic toolbox for Matlab and Gnu Octave called SIPPI that implements a number of methods for solving such probabilistically formulated inverse problems by sampling the a posteriori probability density function. In order to describe the a priori probability density function, we consider both simple Gaussian models and more complex (and realistic) a priori models based on higher order statistics. These a priori models can be used with both linear and non-linear inverse problems. For linear inverse Gaussian problems we make use of least-squares and kriging-based methods to describe the a posteriori probability density function directly. For general non-linear (i.e. non-Gaussian) inverse problems, we make use of the extended Metropolis algorithm to sample the a posteriori probability density function. Together with the extended Metropolis algorithm, we use sequential Gibbs sampling that allow computationally efficient sampling of complex a priori models. The toolbox can be applied to any inverse problem as long as a way of solving the forward problem is provided. Here we demonstrate the methods and algorithms available in SIPPI. An application of SIPPI, to a tomographic cross borehole inverse problems, is presented in a second part of this paper.

SIPPI: A Matlab toolbox for sampling the solution to inverse problems with complex prior information: Part 2—Application to crosshole GPR tomography

Abstract We present an application of the SIPPI Matlab toolbox, to obtain a sample from the a posteriori probability density function for the classical tomographic inversion problem. We consider a number of different forward models, linear and non-linear, such as ray based forward models that rely on the high frequency approximation of the wave-equation and ‘fat’ ray based forward models relying on finite frequency theory. In order to sample the a posteriori probability density function we make use of both least squares based inversion, for linear Gaussian inverse problems, and the extended Metropolis sampler, for non-linear non-Gaussian inverse problems. To illustrate the applicability of the SIPPI toolbox to a tomographic field data set we use a cross-borehole traveltime data set from Arrenaes, Denmark. Both the computer code and the data are released in the public domain using open source and open data licenses. The code has been developed to facilitate inversion of 2D and 3D travel time tomographic data using a wide range of possible a priori models and choices of forward models.

Kriging interpolation in seismic attribute space applied to the South Arne Field, North Sea

Seismic attributes can be used to guide interpolation in-between and extrapolation away from well log locations using for example linear regression, neural networks, and kriging. Kriging-based estimation methods (and most other types of interpolation/extrapolation techniques) are intimately linked to distances in physical space: If two observations are located close to one another, the implicit assumption is that they are highly correlated. This may, however, not be a correct assumption as the two locations can be situated in very different geological settings. An alternative approach to the traditional kriging implementation is suggested that frees the interpolation from the restriction of the physical space. The method is a fundamentally different application of the original kriging formulation where a model of spatialvariability is replaced by a model of variability in an attribute space. To the extent that subsurface geology can be described by a set of seismic attributes, we present an automated mult...

Geostatistical inference using crosshole ground-penetrating radar

High-resolution tomographic images obtained from crosshole geophysical measurements have the potential to provide valu- able information about the geostatistical properties of unsaturat- ed-zonehydrologic-statevariablessuchasmoisturecontent.Un- der drained or quasi-steady-state conditions, the moisture con- tentwillreflectthevariationofthephysicalpropertiesofthesub- surface, which determine the flow patterns in the unsaturated zone.Deterministicleast-squaresinversionofcrossholeground- penetrating-radarGPRtraveltimesresultinsmooth,minimum- variance estimates of the subsurface radar wave velocity struc- ture,whichmaydiminishtheutilityoftheseimagesforgeostatis- tical inference. We have used a linearized stochastic inversion technique to infer the geostatistical properties of the subsurface radar wave velocity distribution using crosshole GPR travel- times directly. Expanding on a previous study, we have deter- minedthatitispossibletoobtainestimatesofglobalvarianceand mean velocity values of the subsurface as well as the correlation lengths describing the subsurface velocity structures. Accurate estimation of the global variance is crucial if stochastic realiza- tions of the subsurface are used to evaluate the uncertainty of the inversion estimate. We have explored the full potential of the geostatisticalinferencemethodusingseveralsyntheticmodelsof varying correlation structures and have tested the influence of different assumptions concerning the choice of covariance func- tionanddatanoiselevel.Inaddition,wehavetestedthemethod- ology on traveltime data collected at a field site in Denmark. There, inferred correlation structures indicate that structural dif- ferences exist between two areas located approximately 10 m apart,anobservationconfirmedbyaGPRreflectionprofile.Fur- thermore, the inferred values of the subsurface global variance and the mean velocity have been corroborated with moisture- content measurements, obtained gravimetrically from samples collectedatthefieldsite.

Attribute-guided well-log interpolation applied to low-frequency impedance estimation

Several approaches exist to use trends in 3D seismic data, in the form of seismic attributes, to interpolate sparsely sampled well-log measurements between well locations. Kriging and neural networks are two such approaches. We have applied a method that finds a relation between seismic attributes (such as two-way times, interval velocities, reflector roughness) and rock properties (in this case, acoustic impedance) from information at well locations. The relation is designed for optimum prediction of acoustic impedances away from well sites, and this is accomplished through a combination of cross validation and the Tikhonov-regularized least-squares method. The method is fast, works well even for highly underdetermined problems, and has general applicability. We apply it to two case studies in which we estimate 3D cubes of low-frequency impedance, which is essential for producing good porosity models. We show that the method is superior to traditional least squares: Numerous blind tests show that estimated low-frequency impedance away from well locations can be determined with an accuracy very close to estimations obtained at well locations.