Artificial intelligence to solve the X-ray crystallography phase problem: a case study report.

The determination of three dimensional structures of macromolecules is one of the actual challenge in biology with the ultimate objective of understanding their function. So far, X-ray crystallography is the most popular method to solve structure, but this technique relies on the generation of diffracting crystals. Once a correct data set has been obtained, the calculation of electron density maps requires to solve the so-called phase problem using different approaches. The most frequently used technique is molecular replacement, which relies on the availability of the structure of a protein sharing strong structural similarity with the studied protein. Its success rate is directly correlated with the quality of the models used for the molecular replacement trials. The availability of models as accurate as possible is then definitely critical. Very recently, a breakthrough step has been made in the field of protein structure prediction thanks to the use of machine learning approaches as implemented in the AlphaFold or RoseTTAFold structure prediction programs. Here, we describe how these recent improvements helped us to solve the crystal structure of a protein involved in the nonsense-mediated mRNA decay pathway (NMD), an mRNA quality control pathway dedicated to the elimination of eukaryotic mRNAs harboring premature stop codons.

The Effect of the Range of a Modulating Phase Mask on the Retrieval of a Complex Object from Intensity Measurements

In many fields of science, it is often impossible to preserve the information about the phase of the electromagnetic field, and only the information about the magnitude is available. This is known as the phase problem. Various algorithms have been proposed to recover the information about phase from intensity measurements. Nowadays, iterative algorithms of phase retrieval have become popular. Many of these algorithms are based on modulating the object under study with several masks and retrieving the missing information about the phase of an object by applying mathematical optimization methods. Several of these algorithms are able to retrieve not only the phase but also the magnitude of the object under study. In this study, we investigate the effect of the range of modulation of a mask on the accuracy of the retrieved magnitude and phase map. We conclude that there is a sharp boundary of the range of modulation separating the successfully retrieved magnitude and phase maps from those retrieved unsuccessfully. A decrease in the range of modulation affects the accuracy of the retrieved magnitude and phase map differently.

Infinitely many homoclinic solutions for double phase problem on integers

We consider a discrete double phase problem on integers with an unbounded potential and reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our results extend some recent findings expressed in the literature.

Magnetic Resonance Pore Imaging, a new tool for porous media research

<p>Porous media are highly prevalent in nature and span a wide range of systems including biological tissues, chemical catalysts or rocks in oil reservoirs. Imaging of the structure of the constituent pores is therefore highly desirable for life sciences and technological applications. This thesis presents the new development and application of a nuclear magnetic resonance (NMR) technique to acquire high resolution images of closed pores. The technique is a further development of diffusive-diffraction Pulsed Gradient Spin Echo (PGSE) NMR, which has been shown to image the pore auto-correlation function averaged over all pores. Until recently it was conventional wisdom that diffusive-diffraction PGSE NMR can only measure the magnitude of the form factor, due to its similarity to diffraction techniques such as x-ray and neutron scattering. In diffraction applications the loss of phase information is commonly referred to as the “phase problem”, which prevents the reconstruction of images of the pore space by inverse Fourier transform. My work is based on a recently suggested modification of the diffusive-diffraction PGSE NMR method, which creates a hybrid between Magnetic Resonance Imaging (MRI) and PGSE NMR. Therefore, we call this approach Magnetic Resonance Pore Imaging (MRPI). We provide experimental confirmation that MRPI does indeed measure the diffractive signal including its phase and thus the “phase problem” is lifted. We suggest a two-dimensional version of MRPI and obtain two-dimensional average pore images of cylindrical and triangular pores with an unprecedented resolution as compared to state of the art MRI. Utilizing a laser machined phantom sample we present images of microscopic pores with triangular shape even in the presence of wall relaxation effects. We therefore show that MRPI is able to reconstruct the pore shape without any prior knowledge or assumption about the porous system under study. Furthermore, we demonstrate that the MRPI approach integrates seamlessly with known MRI concepts. For instance we introduce “MRPI mapping” which acquires the MRPI signal for each pixel in an MRI image. This enables one to resolve pore sizes and shapes spatially, thus expanding the application of MRPI to samples with heterogeneous distributions of pores.</p>

Magnetic Resonance Pore Imaging, a new tool for porous media research

<p>Porous media are highly prevalent in nature and span a wide range of systems including biological tissues, chemical catalysts or rocks in oil reservoirs. Imaging of the structure of the constituent pores is therefore highly desirable for life sciences and technological applications. This thesis presents the new development and application of a nuclear magnetic resonance (NMR) technique to acquire high resolution images of closed pores. The technique is a further development of diffusive-diffraction Pulsed Gradient Spin Echo (PGSE) NMR, which has been shown to image the pore auto-correlation function averaged over all pores. Until recently it was conventional wisdom that diffusive-diffraction PGSE NMR can only measure the magnitude of the form factor, due to its similarity to diffraction techniques such as x-ray and neutron scattering. In diffraction applications the loss of phase information is commonly referred to as the “phase problem”, which prevents the reconstruction of images of the pore space by inverse Fourier transform. My work is based on a recently suggested modification of the diffusive-diffraction PGSE NMR method, which creates a hybrid between Magnetic Resonance Imaging (MRI) and PGSE NMR. Therefore, we call this approach Magnetic Resonance Pore Imaging (MRPI). We provide experimental confirmation that MRPI does indeed measure the diffractive signal including its phase and thus the “phase problem” is lifted. We suggest a two-dimensional version of MRPI and obtain two-dimensional average pore images of cylindrical and triangular pores with an unprecedented resolution as compared to state of the art MRI. Utilizing a laser machined phantom sample we present images of microscopic pores with triangular shape even in the presence of wall relaxation effects. We therefore show that MRPI is able to reconstruct the pore shape without any prior knowledge or assumption about the porous system under study. Furthermore, we demonstrate that the MRPI approach integrates seamlessly with known MRI concepts. For instance we introduce “MRPI mapping” which acquires the MRPI signal for each pixel in an MRI image. This enables one to resolve pore sizes and shapes spatially, thus expanding the application of MRPI to samples with heterogeneous distributions of pores.</p>

Coexistence of structural and magnetic phases in van der Waals magnet CrI3

CrI3 has raised as an important system to the emergent field of two-dimensional van der Waals magnetic materials. However, it is still unclear why CrI3 which has a ferromagnetic rhombohedral structure in bulk, changed to anti-ferromagnetic monoclinic at thin layers. Here we show that this behaviour is due to the coexistence of both monoclinic and rhombohedral crystal phases followed by three magnetic transitions at TC1 = 61 K, TC2 = 50 K and TC3 = 25 K. Each transition corresponds to a certain fraction of the magnetically ordered volume as well as monoclinic and rhombohedral proportion. The different phases are continuously accessed as a function of the temperature over a broad range of magnitudes. Our findings suggest that the challenge of understanding the magnetic properties of thin layers CrI3 is in general a coexisting structural-phase problem mediated by the volume-wise competition between magnetic phases already present in bulk.

Singular Finsler Double Phase Problems with Nonlinear Boundary Condition

In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.