Maximum Entropy Regularization Research Articles

Regularization methods for the extraction of depth profiles from simulated ARXPS data derived from overlayer/substrate models

Starting from posited input depth profiles of silicon oxide on silicon, 100 sets of noisy simulated ARXPS data were created for each oxide layer thickness of 3, 6, 9, 12, 15, 18, 21, 24 and 27 Å. Oxygen depth profiles were then recovered from the noisy simulated data using regularized inversion methods, including maximum entropy and Tikhonov regularization. Three regularization parameters were used: one determined by the S-curve method, one determined by the L-curve method and a third corresponding to the closest correspondence between the input and extracted profiles. The various regularization schemes evaluated were ranked with respect to their ability to reproduce the input profile.

Deriving the radial-velocity variations induced by stellar activity from high-precision photometry

Stellar activity induces apparent radial velocity (RV) variations in late-type main-sequence stars that may hamper the detection of low-mass planets and the measurement of their mass. We use simultaneous measurements of the active planet host star HD189733 with high-precision optical photometry by the MOST satellite and high-resolution spectra by SOPHIE. We apply on this unique dataset a spot model to predict the activity-induced RV variations and compare them with the observed ones. The model is based on the rotational modulation of the stellar flux. A maximum entropy regularization is applied to find a unique and stable solution for the distribution of the active regions versus stellar longitude. The RV variations are synthesized considering the effects on the line profiles of the brightness perturbations due to dark spots and bright faculae and the reduction of the convective blueshifts in the active regions. The synthesized RV time series shows a remarkably good agreement with the observed one although variations on timescales shorter than 4-5 days cannot be reproduced by our model. Persistent active longitudes are revealed by the spot modelling. They rotate with slightly different periods yielding a minimum relative amplitude of the differential rotation of \Delta \Omega / \Omega = 0.23 \pm 0.10. The method proves capable of reducing the power of the activity-induced RV variations by a factor from 2 to 10 at the rotation frequency and its harmonics up to the third. Thanks to the high-precision space-borne photometry delivered by CoRoT, Kepler, or later PLATO, it is possible to map the longitudinal distribution of active regions in late-type stars and apply the method presented in this paper to reduce remarkably the impact of stellar activity on their RV jitter allowing us to confirm the detection of low-mass planets or refine the measurement of their mass.

The boundary structure in the analysis of reversibly interacting systems by sedimentation velocity

Sedimentation velocity (SV) experiments of heterogeneous interacting systems exhibit characteristic boundary structures that can usually be very easily recognized and quantified. For slowly interacting systems, the boundaries represent concentrations of macromolecular species sedimenting at different rates, and they can be interpreted directly with population models based solely on the mass action law. For fast reactions, migration and chemical reactions are coupled, and different, but equally easily discernable boundary structures appear. However, these features have not been commonly utilized for data analysis, for the lack of an intuitive and computationally simple model. The recently introduced effective particle theory (EPT) provides a suitable framework. Here, we review the motivation and theoretical basis of EPT, and explore practical aspects for its application. We introduce an EPT-based design tool for SV experiments of heterogeneous interactions in the software SEDPHAT. As a practical tool for the first step of data analysis, we describe how the boundary resolution of the sedimentation coefficient distribution c( s) can be further improved with a Bayesian adjustment of maximum entropy regularization to the case of heterogeneous interactions between molecules that have been previously studied separately. This can facilitate extracting the characteristic boundary features by integration of c( s). In a second step, these are assembled into isotherms as a function of total loading concentrations and fitted with EPT. Methods for addressing concentration errors in isotherms are discussed. Finally, in an experimental model system of alpha-chymotrypsin interacting with soybean trypsin inhibitor, we show that EPT provides an excellent description of the experimental sedimentation boundary structure of fast interacting systems.

Imaging Protein Statistical Substate Occupancy in a Spectrum-Function Phase Space

Hemeprotein ligand rebinding studies reveal varying IR absorbance and rebinding function across a cryogenic ensemble. Since IR-active vibrations and rebinding barriers couple to structural coordinates, spectral and functional heterogeneity arise from conformational heterogeneity. Modeling rebinding data as a spectrally resolved superposition of first order rate processes and employing maximum entropy regularization, protein heterogeneity is imaged as ensemble occupancy of a spectrum-function phase space. Results from myoglobin rebinding carbonmonoxide are discussed.

Comparison of regularization methods for the inversion of ARXPS data

Starting from a posited input depth profile of oxygen in a hydrocarbon polymer surface, 100 sets of noisy simulated ARXPS data were created at each of 5 noise levels, 0.3%, 1%, 3%, 6% and 9%. Oxygen depth profiles were then recovered from the noisy simulated data using nine regularized methods, including Maximum Entropy and Tikhonov regularization. The various regularization schemes evaluated were ranked with respect to their ability to reproduce the input profile, as a function of the level of added noise.

Imaging Protein Statistical Substate Occupancy in a Spectrum-Function Phase Space

Hemeprotein ligand rebinding studies reveal varying IR absorbance and rebinding functions across a cryogenic ensemble. Since IR-active vibrations and rebinding barriers couple to structural coordinates, spectral and functional heterogeneity arise from conformational heterogeneity. Modeling rebinding data as a spectrally resolved superposition of first-order rate processes and employing maximum entropy regularization, protein heterogeneity is imaged as an ensemble occupancy of a spectrum-function phase space. Results from myoglobin rebinding carbon monoxide are discussed.

Maximum Entropy Regularization Method of Ionospheric Occultation

With the increasing of applications of Global Positioning System (GPS), the research on the factors affecting the radio signals is becoming more and more important. One of the most significant effects on ionosphere monitoring is the ionospheric radio occultation (IRO), and the IRO data is now serving as one of the most important monitoring sources for the ionospheric measurements. Meanwhile the inversion of the occultation data is a popular topic. The traditional Abel inversion process through compensated total electron content (TEC) is a linear inversion method, thus it would transfer the measurement errors directly to the inversion results. In order to improve the occultation results, we introduce in this paper two nonlinear methods, namely the regularization method and the maximum entropy regularization method. Through designed simulative experiments, we verify and compare these three methods, and conclude that the maximum entropy regularization method can reduce significantly the influence of measurement errors.

Maximum entropy regularization method for electrical impedance tomography combined with a normalized sensitivity map

Electrical impedance tomography (EIT) aims to estimate the electrical properties at the interior of an object from current–voltage measurements on its boundary. To overcome ill-posedness, regularization techniques such as Tikhonov regularization as well as some iterative methods were developed. In difference imaging between two different conductivity distributions, a conductivity change can be seen relatively non-negative to the medium with lower conductivity through some safeguard techniques. Therefore, the concept of maximum entropy from information theory and statistic mechanics can be used for this purpose. Furthermore, because the sensing field is “soft-field” and non-uniform, the same anomaly may produce different reconstruction signatures depending on its location within the image plane. Therefore, in this paper, maximum entropy based on general Tikhonov regularization, combined with normalized sensitivity map, is proposed to solve the inverse problem of EIT. Image reconstruction was carried out by maximum entropy regularization (MER) with a normalized sensitivity map and compared with the results from conjugate gradient method (CG), Tikhonov regularization, and CG with a normalized sensitivity map accordingly. Simulation and experiment results indicate that reconstructed images with higher quality can be obtained by MER with a normalized sensitivity map.

Heterodimerization of the Sialidase NEU1 with the Chaperone Protective Protein/Cathepsin A Prevents Its Premature Oligomerization

Lysosomal neuraminidase-1 (NEU1) forms a multienzyme complex with beta-galactosidase and protective protein/cathepsin A (PPCA). Because of its association with PPCA, which acts as a molecular chaperone, NEU1 is transported to the lysosomal compartment, catalytically activated, and stabilized. However, the mode(s) of association between these two proteins both en route to the lysosome and in the multienzyme complex has remained elusive. Here, we have analyzed the hydrodynamic properties of PPCA, NEU1, and a complex of the two proteins and identified multiple binding sites on both proteins. One of these sites on NEU1 that is involved in binding to PPCA can also bind to other NEU1 molecules, albeit with lower affinity. Therefore, in the absence of PPCA, as in the lysosomal storage disease galactosialidosis, NEU1 self-associates into chain-like oligomers. Binding of PPCA can reverse self-association of NEU1 by causing the disassembly of NEU1-oligomers and the formation of a PPCA-NEU1 heterodimeric complex. The identification of binding sites between the two proteins allowed us to create innovative structural models of the NEU1 oligomer and the PPCA-NEU1 heterodimeric complex. The proposed mechanism of interaction between NEU1 and its accessory protein PPCA provides a rationale for the secondary deficiency of NEU1 in galactosialidosis.

Fluorescence Detection of a Lipid-induced Tetrameric Intermediate in Amyloid Fibril Formation by Apolipoprotein C-II

The misfolding and self-assembly of proteins into amyloid fibrils that occurs in several debilitating and age-related diseases is affected by common components of amyloid deposits, notably lipids and lipid complexes. We have examined the effect of the short-chain phospholipids, dihexanoylphosphatidylcholine (DHPC) and dihexanoylphosphatidylserine (DHPS), on amyloid fibril formation by human apolipoprotein C-II (apoC-II). Micellar DHPC and DHPS strongly inhibited apoC-II fibril formation, whereas submicellar levels of these lipids accelerated apoC-II fibril formation to a similar degree. These results indicate that the net negative charge on DHPS, compared with the neutrally charged DHPC, is not critical for either the inhibition or activation process. We also investigated the mechanism for the submicellar, lipid-induced activation of fibril formation. Emission data for fluorescently labeled apoC-II indicated that DHPC and DHPS stimulate the early formation and accumulation of oligomeric species. Sedimentation velocity and equilibrium experiments using a new fluorescence detection system identified a discrete lipid-induced tetramer formed at low apoC-II concentrations in the absence of significant fibril formation. Seeding experiments showed that this tetramer was on the fibril-forming pathway. Fluorescence resonance energy transfer experiments established that this tetramer forms rapidly and is stabilized by submicellar, but not micellar, concentrations of DHPC and DHPS. Several recent studies show that oligomeric intermediates in amyloid fibril formation are toxic. Our results indicate that lipids promote on-pathway intermediates of apoC-II fibril assembly and that the accumulation of a discrete tetrameric intermediate depends on the molecular state of the lipid.

A Bayesian Approach for Quantifying Trace Amounts of Antibody Aggregates by Sedimentation Velocity Analytical Ultracentrifugation

Sedimentation velocity analytical ultracentrifugation (SV-AUC) has become an important tool for the characterization of the purity of protein therapeutics. The work presented here addresses a need for methods orthogonal to size-exclusion chromatography for ensuring the reliable quantitation of immunogenic oligomers, for example, in antibody preparations. Currently the most commonly used approach for SV-AUC analysis is the diffusion-deconvoluted sedimentation coefficient distribution c(s) method, previously developed by us as a general purpose technique and implemented in the software SEDFIT. In both practical and theoretical studies, different groups have reported a sensitivity of c(s) for trace oligomeric fractions well below the 1% level. In the present work we present a variant of c(s) designed for the purpose of trace detection, with customized Bayesian regularization. The original c(s) method relies on maximum entropy regularization providing the most parsimonious distribution consistent with the data. In the present paper, we use computer simulations of an antibody system as example to demonstrate that the standard maximum entropy regularization, due to its design, leads to a theoretical lower limit for the detection of oligomeric traces and a consistent underestimate of the trace populations by approximately 0.1% (dependent on the level of regularization). This can be overcome with a recently developed Bayesian extension of c(s) (Brown et al., Biomacromolecules, 8:2011-2024, 2007), utilizing the known regions of sedimentation coefficients for the monomer and oligomers of interest as prior expectation for the peak positions in the distribution. We show that this leads to more clearly identifiable and consistent peaks and lower theoretical limits of quantization by approximately an order of magnitude for some experimental conditions. Implications for the experimental design of SV-AUC and practical detection limits are discussed.

Investigation of a two-point maximum entropy regularization method for signal enhancement applied to magnetoencephalography data

A two-point maximum entropy method (TPMEM) was investigated for post-acquisition signal recovery in magnetoencephalography (MEG) data, as a potential replacement of a low-pass (LP) filtering technique currently in use. We first applied TPMEM and the LP filter for signal recovery of synthetically noise corrupted MEG “phantom” data sets in which the true underlying signal was known. Results were quantified with the use of visual plots, percent error histograms, and the statistical parameters root mean squared error and Pearson’s correlation coefficient. Synthetically noise corrupted data from a simulated magnetic dipole was used to quantify the improvements gained in using TPMEM over LP filters in reconstructing known dipole parameters such as position, orientation, and magnitude. Finally, we applied TPMEM and LP filters to a sample MEG patient data set. Our results show that TPMEM has improved noise-reduction and signal recovery capabilities than those of the LP filter, and furthermore data processed with TPMEM shows less error in the reconstructed dipole parameters. We propose that TPMEM can be used for MEG signal processing, resulting in improved MEG source characterization.

Maximum entropy regularization of the geomagnetic core field inverse problem

SUMMARY The maximum entropy technique is an accepted method of image reconstruction when the image is made up of pixels of unknown positive intensity (e.g. a grey-scale image). The problem of reconstructing the magnetic field at the core‐mantle boundary from surface data is a problem where the target image, the value of the radial field Br, can be of either sign. We adopt a known extension of the usual maximum entropy method that can be applied to images consisting of pixels of unconstrained sign. We find that we are able to construct images which have high dynamic ranges, but which still have very simple structure. In the spherical harmonic domain they have smoothly decreasing power spectra. It is also noteworthy that these models have far less complex null flux curve topology (lines on which the radial field vanishes) than do models which are quadratically regularized. Problems such as the one addressed are ubiquitous in geophysics, and it is suggested that the applications of the method could be much more widespread than is currently the case.

Maximum entropy regularization of time-dependent geomagnetic field models

SUMMARY We incorporate a maximum entropy image reconstruction technique into the process of modelling the time-dependent geomagnetic field at the core‐mantle boundary (CMB). In order to deal with unconstrained small lengthscales in the process of inverting the data, some core field models are regularized using a priori quadratic norms in both space and time. This artificial damping leads to the underestimation of power at large wavenumbers, and to a loss of contrast in the reconstructed picture of the field at the CMB. The entropy norm, recently introduced to regularize magnetic field maps, provides models with better contrast, and involves a minimum of a priori information about the field structure. However, this technique was developed to build only snapshots of the magnetic field. Previously described in the spatial domain, we show here how to implement this technique in the spherical harmonic domain, and we extend it to the time-dependent problem where both spatial and temporal regularizations are required. We apply our method to model the field over the interval 1840‐1990 from a compilation of historical observations. Applying the maximum entropy method in space—for a fit to the data similar to that obtained with a quadratic regularization—effectively reorganizes the magnetic field lines in order to have a map with better contrast. This is associated with a less rapidly decaying spectrum at large wavenumbers. Applying the maximum entropy method in time permits us to model sharper temporal changes, associated with larger spatial gradients in the secular variation, without producing spurious fluctuations on short timescales. This method avoids the smearing back in time of field features that are not constrained by the data. Perspectives concerning future applications of the method are also discussed.

MEG signal enhancement using a two-point maximum entropy regularization method

A two-point maximum entropy regularization method (TPMEM) was investigated for signal enhancement of magnetoencephalography (MEG) data, as a potential replacement for current low-pass (LP) filtering techniques. TPMEM was found to have superior noise reduction and signal recovery capabilities over the LP filter, and MEG data processed with TPMEM yields less error in reconstructed dipole position and orientation.

Using prior knowledge in the determination of macromolecular size-distributions by analytical ultracentrifugation.

Analytical ultracentrifugation has reemerged as a widely used tool for the study of ensembles of biological macromolecules to understand, for example, their size-distribution and interactions in free solution. Such information can be obtained from the mathematical analysis of the concentration and signal gradients across the solution column and their evolution in time generated as a result of the gravitational force. In sedimentation velocity analytical ultracentrifugation, this analysis is frequently conducted using high resolution, diffusion-deconvoluted sedimentation coefficient distributions. They are based on Fredholm integral equations, which are ill-posed unless stabilized by regularization. In many fields, maximum entropy and Tikhonov-Phillips regularization are well-established and powerful approaches that calculate the most parsimonious distribution consistent with the data and prior knowledge, in accordance with Occam's razor. In the implementations available in analytical ultracentrifugation, to date, the basic assumption implied is that all sedimentation coefficients are equally likely and that the information retrieved should be condensed to the least amount possible. Frequently, however, more detailed distributions would be warranted by specific detailed prior knowledge on the macromolecular ensemble under study, such as the expectation of the sample to be monodisperse or paucidisperse or the expectation for the migration to establish a bimodal sedimentation pattern based on Gilbert-Jenkins' theory for the migration of chemically reacting systems. So far, such prior knowledge has remained largely unused in the calculation of the sedimentation coefficient or molecular weight distributions or was only applied as constraints. In the present paper, we examine how prior expectations can be built directly into the computational data analysis, conservatively in a way that honors the complete information of the experimental data, whether or not consistent with the prior expectation. Consistent with analogous results in other fields, we find that the use of available prior knowledge can have a dramatic effect on the resulting molecular weight, sedimentation coefficient, and size-and-shape distributions and can significantly increase both their sensitivity and their resolution. Further, the use of multiple alternative prior information allows us to probe the range of possible interpretations consistent with the data.

Bayesian reconstruction of the cosmological large-scale structure: methodology, inverse algorithms and numerical optimization

We address the inverse problem of cosmic large-scale structure reconstruction from a Bayesian perspective. For a linear data model, a number of known and novel reconstruction schemes, which differ in terms of the underlying signal prior, data likelihood and numerical inverse extraregularization schemes are derived and classified. The Bayesian methodology presented in this paper tries to unify and extend the following methods: Wiener filtering, Tikhonov regularization, ridge regression, maximum entropy and inverse regularization techniques. The inverse techniques considered here are the asymptotic regularization, the Jacobi, Steepest Descent, Newton–Raphson, Landweber–Fridman and both linear and non-linear Krylov methods based on Fletcher–Reeves, Polak–Ribiere and Hestenes–Stiefel conjugate gradients. The structures of the up-to-date highest performing algorithms are presented, based on an operator scheme, which permits one to exploit the power of fast Fourier transforms. Using such an implementation of the generalized Wiener filter in the novel argo software package, the different numerical schemes are benchmarked with one-, two- and three-dimensional problems including structured white and Poissonian noise, data windowing and blurring effects. A novel numerical Krylov scheme is shown to be superior in terms of performance and fidelity. These fast inverse methods ultimately will enable the application of sampling techniques to explore complex joint posterior distributions. We outline how the space of the dark matter density field, the peculiar velocity field and the power spectrum can jointly be investigated by a Gibbs-sampling process. Such a method can be applied for the redshift distortions correction of the observed galaxies and for time-reversal reconstructions of the initial density field.

Determination of Radical Re-encounter Probability Distributions from Magnetic Field Effects on Reaction Yields

Measurements are reported of the effects of 0-23 mT applied magnetic fields on the spin-selective recombination of Py*- and DMA*+ radicals formed in the photochemical reaction of pyrene and N,N-dimethylaniline. Singlet <--> triplet interconversion in [Py*- DMA*+] radical pairs is probed by investigating combinations of fully protonated and fully deuterated reaction partners. Qualitatively, the experimental B1/2 values for the four isotopomeric radical pairs agree with predictions based on the Weller equation using known hyperfine coupling constants. The amplitude of the "low field effect" (LFE) correlates well with the ratio of effective hyperfine couplings, aDMA/aPy. An efficient method is introduced for calculating the spin evolution of [Py*- DMA*+] radical pairs containing a total of 18 spin-1/2 and spin-1 magnetic nuclei. Quantitative analysis of the magnetic field effects to obtain the radical re-encounter probability distribution f (t )-a highly ill-posed and underdetermined problem-is achieved by means of Tikhonov and maximum entropy regularization methods. The resulting f (t ) functions are very similar for the four isotopomeric radical pairs and have significant amplitude between 2 and 10 ns after the creation of the geminate radical pair. This interval reflects the time scale of re-encounters that are crucial for generating the magnetic field effect. Computer simulations of generalized radical pairs containing six spin-1/2 nuclei show that Weller's equation holds approximately only when the radical pair recombination rate is comparable to the two effective hyperfine couplings and that a substantial LFE requires, but is not guaranteed by, the condition that the two effective hyperfine couplings differ by more than a factor of 5. In contrast, for very slow recombination, essentially any radical pair should show a significant LFE.

Zinc Induces Dimerization of the Class II Major Histocompatibility Complex Molecule That Leads to Cooperative Binding to a Superantigen

Dimerization of class II major histocompatibility complex (MHC) plays an important role in the MHC biological function. Mycoplasma arthritidis-derived mitogen (MAM) is a superantigen that can activate large fractions of T cells bearing specific T cell receptor Vbeta elements. Here we have used structural, sedimentation, and surface plasmon resonance detection approaches to investigate the molecular interactions between MAM and the class II MHC molecule HLA-DR1 in the context of a hemagglutinin peptide-(306-318) (HA). Our results revealed that zinc ion can efficiently induce the dimerization of the HLA-DR1/HA complex. Because the crystal structure of the MAM/HLA-DR1/hemagglutinin complex in the presence of EDTA is nearly identical to the structure of the complex crystallized in the presence of zinc ion, Zn(2+) is evidently not directly involved in the binding between MAM and HLA-DR1. Sedimentation and surface plasmon resonance studies further revealed that MAM binds the HLA-DR1/HA complex with high affinity in a 1:1 stoichiometry, in the absence of Zn(2+). However, in the presence of Zn(2+), a dimerized MAM/HLA-DR1/HA complex can arise through the Zn(2+)-induced DR1 dimer. In the presence of Zn(2+), cooperative binding of MAM to the DR1 dimer was also observed.

Design and commissioning of a new diagnostic for routine determination of Zeff from visible bremsstrahlung measurements on TEXTOR

On TEXTOR a new diagnostic was commissioned for the determination of the profile of the ion effective charge Zeff from bremsstrahlung emissivity measurements in the visible. The line-integrated bremsstrahlung emissivity is measured with high spatial resolution along 28 fiber optic channels, in one poloidal cross section of the tokamak. Twenty-four channels are coupled to a cooled fast charge-coupled device (CCD) camera and four to photomultiplier tubes for higher sampling rate. The design of the diagnostic is described, including a technique for coping with CCD image smearing, without the need for the calculation of the point spread function of the system. An in-vessel calibration was performed by means of an integrating sphere. A convenient alternative way is outlined for determining the relative calibration from plasma emissivity measurements only. Bremsstrahlung emissivity profiles can be reconstructed by Abel inversion of the line integrals measured with the camera. Overestimation of edge Zeff values due to nonbremsstrahlung contributions to the continuum radiation can be minimized via Tikhonov or maximum entropy regularization. The propagation of such edge continuum contributions towards the center of the reconstructed profile is illustrated in the case of ergodization of the edge plasma.