A Statistical Model for Helices with Applications

Abstract

Motivated by a cutting edge problem related to the shape of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> -helices in proteins, we formulate a parametric statistical model, which incorporates the cylindrical nature of the helix. Our focus is to detect a "kink," which is a drastic change in the axial direction of the helix. We propose a statistical model for the straight <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> -helix and derive the maximum likelihood estimation procedure. The cylinder is an accepted geometric model for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> -helices, but our statistical formulation, for the first time, quantifies the uncertainty in atom positions around the cylinder. We propose a change point technique "Kink-Detector" to detect a kink location along the helix. Unlike classical change point problems, the change in direction of a helix depends on a simultaneous shift of multiple data points rather than a single data point, and is less straightforward. Our biological building block is crowdsourced data on straight and kinked helices; which has set a gold standard. We use this data to identify salient features to construct Kink-detector, test its performance and gain some insights. We find the performance of Kink-detector comparable to its computational competitor called "Kink-Finder." We highlight that identification of kinks by visual assessment can have limitations and Kink-detector may help in such cases. Further, an analysis of crowdsourced curved <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi></mml:math> -helices finds that Kink-detector is also effective in detecting moderate changes in axial directions.

Statistical Shape Methodology for the Analysis of Helices

“…Dickerson and Geis (1969) give a value of 1.75 radians = 100.3 • . Recent confirmation can be obtained from the detailed analysis of the maximum likelihood estimation of β carried out for 129 straight helices from crowdsourcing data in the web supplement (Web Figure 5( Mardia et al (2018); for this data it was found that the mean estimate of β is 99.1 • with a standard error of 1.2 • .…”

“…Alfahad et al In more detail, the paper is organized as follows. Section 2 introduces the statistical helix model (similar in spirit to Mardia et al (2018), but more geometrically explicit). Section 3 gives the details behind the optimized least squares algorithm.…”

“…This procedure is investigated on simulated data in Section 7. In Section 8 it is applied to several protein examples and compared to Kink-Detector (Mardia et al, 2018).…”

“…a point where the helix axis locally changes direction (Wilman et al, 2014b;Mardia, 2014). Various statistical methods in the literature have been developed to analyze kinks as local change points, such as Helanal by Bansal et al (2000), Kinkfinder by Wilman (2014a), and Kink-Detector by Mardia et al (2018). Kinkfinder uses all the atoms on the helix, whereas the Helanal and Kink-Detector use just the C α atoms.…”

“…These methods estimate the kink position by looking for a local change point in the direction of the helix axis. For further review of the topic, see Mardia (2013) and Mardia et al (2018). Blundell et al (1983) and Barlow and Thornton (1988) initiated the work on helix structure based on curvature; they used main classifications:…”