Annals of Communications in Mathematics

AbstractThis work lies in the whole biomathematics framework which has as general goal the solving of biological problems with mathematical tools. The main objective of the present paper is to predict the transmembrane helices of proteins using wavelet denoising techniques. As a case of matter, we particularly highlight the interest in solving the problem of localizing these helices. Indeed, these helices play a vital role in the human body, notably in photosynthesis, respiration, neuronal signaling, immune response, absorption of nutrition, and have an important link with drugs as receptors coupled to many proteins. However, due to technical constraints, the crystallization of these helices remains very complex, which limits the exploration of their structure [27]. To overcome these difficul- ties, different prediction tools have been developed, initially based on hydrophobicity. In this paper, we serve the Farey wavelet as a last alternative mother wavelet constructed in [4] to develop a mathematical model suitable for protein series description. We precisely apply a new type of wavelets constructed recently in [4] to localize the and/or predict the position of the transmembrane proteins alpha-helices in a coronavirus strain. The results are compared to existing works for performance, accuracy, and efficiency.

AbstractThe wavelet analysis of a function passes through its so-called wavelet transform. Such a transform is mathematically defined as a convolution product of the analyzed function with another analyzing function known as the mother wavelet by involving the scale and the translation parameters. This means that the mother wavelet construction is the starting and major point in the wavelet analysis. Besides, the choice of the mother wavelet remains a major problem in wavelet applications such as statistical series, time series, signal, and image processing. This needs more candidates of mother wavelets to be constructed. The main aim of the present paper is to construct a new mother wavelet by exploiting the well-known Farey map. We showed indeed that such a map may be a mother wavelet owing properties such as admissibility, moments, 2-scale relation, and reconstruction rule already necessary in the wavelet analysis of functions. By a suitable choice of translation-dilation parameters on the original Farey map, we succeeded to prove the main properties of a Farey wavelet analysis. The constructed mother looks to be suitable for many complicated applications such as hyperbolic PDEs.