Comparison of Different Reweighting Approaches for the Calculation of Conformational Variability of Macromolecules from Molecular Simulations.

Conformational variability and heterogeneity are crucial determinants of the function of biological macromolecules. The possibility of accessing this information experimentally suffers from severe under-determination of the problem, since there are a few experimental observables to be accounted for by a (potentially) infinite number of available conformational states. Several computational methods have been proposed over the years in order to circumvent this theoretically insurmountable obstacle. A large share of these strategies is based on reweighting an initial conformational ensemble which arises from, for example, molecular simulations of different qualities and levels of theory. In this work, we compare the outcome of three reweighting approaches based on radically different views of the conformational heterogeneity problem, namely Maximum Entropy, Maximum Parsimony and Maximum Occurrence, and we do so using the same experimental data. In this comparison we find both expected as well as unexpected similarities.

A critical assessment of methods to recover information from averaged data.

Conformational heterogeneity is key to the function of many biomacromolecules, but only a few groups have tried to characterize it until recently. Now, thanks to the increased throughput of experimental data and the increased computational power, the problem of the characterization of protein structural variability has become more and more popular. Several groups have devoted their efforts in trying to create quantitative, reliable and accurate protocols for extracting such information from averaged data. We analyze here different approaches, discussing strengths and weaknesses of each. All approaches can roughly be clustered into two groups: those satisfying the maximum entropy principle and those recovering ensembles composed of a restricted number of molecular conformations. In the first case, the solution focuses on the features that are common to all the infinite solutions satisfying the experimental data; in the second case, the reconstructed ensemble shows the conformational regions where a large probability can be placed. The upper limits for conformational probabilities (MaxOcc) can also be calculated. We also give an overview of the mainstream experimental observables, with considerations on the assumptions underlying their usage.

Information content of long-range NMR data for the characterization of conformational heterogeneity.

Long-range NMR data, namely residual dipolar couplings (RDCs) from external alignment and paramagnetic data, are becoming increasingly popular for the characterization of conformational heterogeneity of multidomain biomacromolecules and protein complexes. The question addressed here is how much information is contained in these averaged data. We have analyzed and compared the information content of conformationally averaged RDCs caused by steric alignment and of both RDCs and pseudocontact shifts caused by paramagnetic alignment, and found that, despite the substantial differences, they contain a similar amount of information. Furthermore, using several synthetic tests we find that both sets of data are equally good towards recovering the major state(s) in conformational distributions.

On the choice of retrieval variables in the inversion of remotely sensed atmospheric measurements.

In this paper we introduce new variables that can be used to retrieve the atmospheric continuum emission in the inversion of remote sensing measurements. This modification tackles the so-called sloppy model problem. We test this approach on an extensive set of real measurements from the Michelson Interferometer for Passive Atmospheric Sounding. The newly introduced variables permit to achieve a more stable inversion and a smaller value of the minimum of the cost function.

Maximum occurrence analysis of protein conformations for different distributions of paramagnetic metal ions within flexible two-domain proteins.

Multidomain proteins are composed of rigid domains connected by (flexible) linkers. Therefore, the domains may experience a large degree of reciprocal reorientation. Pseudocontact shifts and residual dipolar couplings arising from one or more paramagnetic metals successively placed in a single metal binding site in the protein can be used as restraints to assess the degree of mobility of the different domains. They can be used to determine the maximum occurrence (MO) of each possible protein conformation, i.e. the maximum weight that such conformations can have independently of the real structural ensemble, in agreement with the provided restraints. In the case of two-domain proteins, the metal ions can be placed all in the same domain, or distributed between the two domains. It has been demonstrated that the quantity of independent information for the characterization of the system is larger when all metals are bound in the same domain. At the same time, it has been shown that there are practical advantages in placing the metals in different domains. Here, it is shown that distributing the metals between the domains provides a tool for defining a coefficient of compatibility among the restraints obtained from different metals, without a significant decrease of the capability of the MO values to discriminate among conformations with different weights.

Narrowing the conformational space sampled by two-domain proteins with paramagnetic probes in both domains.

Calmodulin is a two-domain protein which in solution can adopt a variety of conformations upon reorientation of its domains. The maximum occurrence (MO) of a set of calmodulin conformations that are representative of the overall conformational space possibly sampled by the protein, has been calculated from the paramagnetism-based restraints. These restraints were measured after inclusion of a lanthanide binding tag in the C-terminal domain to supplement the data obtained by substitution of three paramagnetic lanthanide ions to the calcium ion in the second calcium binding loop of the N-terminal domain. The analysis shows that the availability of paramagnetic restraints arising from metal ions placed on both domains, reduces the MO of the conformations to different extents, thereby helping to identify those conformations that can be mostly sampled by the protein.

Iterative approach to self-adapting and altitude-dependent regularization for atmospheric profile retrievals.

In this paper we present the IVS (Iterative Variable Strength) method, an altitude-dependent, self-adapting Tikhonov regularization scheme for atmospheric profile retrievals. The method is based on a similar scheme we proposed in 2009. The new method does not need any specifically tuned minimization routine, hence it is more robust and faster. We test the self-consistency of the method using simulated observations of the Michelson Interferometer for Passive Atmospheric Sounding (MIPAS). We then compare the new method with both our previous scheme and the scalar method currently implemented in the MIPAS on-line processor, using both synthetic and real atmospheric limb measurements. The IVS method shows very good performances.

Paramagnetism-based NMR restraints provide maximum allowed probabilities for the different conformations of partially independent protein domains.

An innovative analytical/computational approach is presented to provide maximum allowed probabilities (MAPs) of conformations in protein domains not rigidly connected. The approach is applied to calmodulin and to its adduct with alpha-synuclein. Calmodulin is a protein constituted by two rigid domains, each of them composed by two calcium-binding EF-hand motifs, which in solution are largely free to move with respect to one another. We used the N60D mutant of calmodulin, which had been engineered to selectively bind a paramagnetic lanthanide ion to only one of its four calcium binding sites, specifically in the second EF-hand motif of the N-terminal domain. In this way, pseudocontact shifts (pcs's) and self-orientation residual dipolar couplings (rdc's) measured on the C-terminal domain provide information on its relative mobility with respect to the domain hosting the paramagnetic center. Available NMR data for terbium(III) and thulium(III) calmodulin were supplemented with additional data for dysprosium(III), analogous data were generated for the alpha-synuclein adduct, and the conformations with the largest MAPs were obtained for both systems. The MAP analysis for calmodulin provides further information on the variety of conformations experienced by the system. Such variety is somewhat reduced in the calmodulin-alpha-synuclein adduct, which however still retains high flexibility. The flexibility of the calmodulin-alpha-synuclein adduct is an unexpected result of this research.

Efficiency of paramagnetism-based constraints to determine the spatial arrangement of alpha-helical secondary structure elements.

A computational approach has been developed to assess the power of paramagnetism-based backbone constraints with respect to the determination of the tertiary structure, once the secondary structure elements are known. This is part of the general assessment of paramagnetism-based constraints which are known to be relevant when used in conjunction with all classical constraints. The paramagnetism-based constraints here investigated are the pseudocontact shifts, the residual dipolar couplings due to self-orientation of the metalloprotein in high magnetic fields, and the cross correlation between dipolar relaxation and Curie relaxation. The relative constraints are generated by back-calculation from a known structure. The elements of secondary structure are supposed to be obtained from chemical shift index. The problem of the reciprocal orientation of the helices is addressed. It is shown that the correct fold can be obtained depending on the length of the alpha-helical stretches with respect to the length of the non helical segments connecting the alpha-helices. For example, the correct fold is straightforwardly obtained for the four-helix bundle protein cytochrome b562, while the double EF-hand motif of calbindin D9k is hardly obtained without ambiguity. In cases like calbindin D9k, the availability of datasets from different metal ions is helpful, whereas less important is the location of the metal ion with respect to the secondary structure elements.