Fractal dimension based geographical clustering of COVID-19 time series data.

Understanding the local dynamics of COVID-19 transmission calls for an approach that characterizes the incidence curve in a small geographical unit. Given that incidence curves exhibit considerable day-to-day variation, the fractal structure of the time series dynamics is investigated for the Flanders and Brussels Regions of Belgium. For each statistical sector, the smallest administrative geographical entity in Belgium, fractal dimensions of COVID-19 incidence rates, based on rolling time spans of 7, 14, and 21 days were estimated using four different estimators: box-count, Hall-Wood, variogram, and madogram. We found varying patterns of fractal dimensions across time and location. The fractal dimension is further summarized by its mean, variance, and autocorrelation over time. These summary statistics are then used to cluster regions with different incidence rate patterns using k-means clustering. Fractal dimension analysis of COVID-19 incidence thus offers important insight into the past, current, and arguably future evolution of an infectious disease outbreak.

Random Coordinate Descent with Spinor-matrices and Geometric Filters for Efficient Loop Closure.

Protein loop closure constitutes a critical step in loop and protein modeling whereby geometrically feasible loops must be found between two given anchor residues. Here, a new analytic/iterative algorithm denoted random coordinate descent (RCD) to perform protein loop closure is described. The algorithm solves loop closure through minimization as in cyclic coordinate descent but selects bonds for optimization randomly, updates loop conformations by spinor-matrices, performs loop closure in both chain directions, and uses a set of geometric filters to yield efficient conformational sampling. Geometric filters allow one to detect clashes and constrain dihedral angles on the fly. The RCD algorithm is at least comparable to state of the art loop closure algorithms due to an excellent balance between efficiency and intrinsic sampling capability. Furthermore, its efficiency allows one to improve conformational sampling by increasing the sampling number without much penalty. Overall, RCD turns out to be accurate, fast, robust, and applicable over a wide range of loop lengths. Because of the versatility of RCD, it is a solid alternative for integration with current loop modeling strategies.

Spinor product computations for protein conformations.

Spinor operators in geometric algebra (GA) can efficiently describe conformational changes of proteins by ordered products that act on individual bonds and represent their net rotations. Backward propagation through the protein backbone yields all rotational spinor axes in advance allowing the efficient computation of atomic coordinates from internal coordinates. The introduced mathematical framework enables to efficiently manipulate and generate protein conformations to any arbitrary degree. Moreover, several new formulations in the context of rigid body motions are added. Emphasis is placed on the intimate relationship between spinors and quaternions, which can be recovered from within the GA approach. The spinor methodology is implemented and tested versus the state of the art algorithms for both protein construction and coordinate updating. Spinor calculations have a smaller computational cost and turn out to be slightly faster than current alternatives.

FTIR 2D correlation spectroscopy of α1 and α2 fractions of an alkali-pretreated gelatin.

An alkali-pretreated gelatin (pI~4.9) was fractionated by means of alcohol coacervation and semi-preparative gel chromatography. The thermal responses of the isolated α fractions, the coacervate and the total gelatin were investigated by 2D-correlation FTIR spectroscopy in the amide I band region (1600-1700 cm⁻¹). The gelation temperature was the same for all examined samples (24.5°C) while the melting temperature of the α2 fraction was lower (30°C) than that of the other samples (32.5°C). The 2D COS plots indicate that on cooling (gelation) the core sequence of conformational changes is the same for all samples. On heating, however, the α2 fraction deviates from the α1-containing samples and shows an earlier disappearance of the triple helix signal in the event sequence. The lower melting temperature (less thermostable gelatin gel) of the α2 fraction thus results from a different conformational cascade of the α2 chains upon melting. In all samples the initial conformational changes take place in the ß-turns, providing further evidence for the models proposed previously.

Application of geometric algebra for the description of polymer conformations.

In this paper a Clifford algebra-based method is applied to calculate polymer chain conformations. The approach enables the calculation of the position of an atom in space with the knowledge of the bond length (l), valence angle (theta), and rotation angle (phi) of each of the preceding bonds in the chain. Hence, the set of geometrical parameters {l(i),theta(i),phi(i)} yields all the position coordinates p(i) of the main chain atoms. Moreover, the method allows the calculation of side chain conformations and the computation of rotations of chain segments. With these features it is, in principle, possible to generate conformations of any type of chemical structure. This method is proposed as an alternative for the classical approach by matrix algebra. It is more straightforward and its final symbolic representation considerably simpler than that of matrix algebra. Approaches for realistic modeling by means of incorporation of energetic considerations can be combined with it. This article, however, is entirely focused at showing the suitable mathematical framework on which further developments and applications can be built.