Identification of approximate symmetries in biological development

Virtually all forms of life, from single-cell eukaryotes to complex, highly differentiated multicellular organisms, exhibit a property referred to as symmetry. However, precise measures of symmetry are often difficult to formulate and apply in a meaningful way to biological systems, where symmetries and asymmetries can be dynamic and transient, or be visually apparent but not reliably quantifiable using standard measures from mathematics and physics. Here, we present and illustrate a novel measure that draws on concepts from information theory to quantify the degree of symmetry, enabling the identification of approximate symmetries that may be present in a pattern or a biological image. We apply the measure to rotation, reflection and translation symmetries in patterns produced by a Turing model, as well as natural objects (algae, flowers and leaves). This method of symmetry quantification is unbiased and rigorous, and requires minimal manual processing compared to alternative measures. The proposed method is therefore a useful tool for comparison and identification of symmetries in biological systems, with potential future applications to symmetries that arise during development, as observed in vivo or as produced by mathematical models. This article is part of the theme issue ‘Recent progress and open frontiers in Turing’s theory of morphogenesis’.

Shannon (Information) Measures of Symmetry for 1D and 2D Shapes and Patterns

Informational (Shannon) measures of symmetry are introduced and analyzed for the patterns built of 1D and 2D shapes. The informational measure of symmetry Hsym (G) characterizes the an averaged uncertainty in the presence of symmetry elements from the group G in a given pattern; whereas the Shannon-like measure of symmetry Ωsym (G) quantifies averaged uncertainty of appearance of shapes possessing in total n elements of symmetry belonging to group G in a given pattern. Hsym(G1)=Ωsym(G1)=0 for the patterns built of irregular, non-symmetric shapes. Both of informational measures of symmetry are intensive parameters of the pattern and do not depend on the number of shapes, their size and area of the pattern. They are also insensitive to the long-range order inherent for the pattern. Informational measures of symmetry of fractal patterns are addressed. The mixed patterns including curves and shapes are considered. Time evolution of the Shannon measures of symmetry is treated. The close-packed and dispersed 2D patterns are analyzed.

Estimating Proprioceptive Position Sense of Upper Limbs Using Joint Distance and Joint Angle Symmetries

Background Proprioceptive position sense (PPS) plays a critical role in movement and coordination of the upper limbs. It is commonly affected in neurological disorders such as unilateral spastic cerebral palsy (USCP) and stroke. However, UL PPS is often not considered during assessment or treatment in motor rehabilitation programs, in part due to limitations of current methods to quantify PPS. Traditional methods assess the similarity or symmetry of angles of a single joint and may not be sufficiently sensitive nor accurate. Emerging methods such as the use of robotic technology typically are limited to 2D and are also not standardizable as they are not easily available to clinics and research labs. This study aims to introduce a standardized and accessible way to quantify multi-joint PPS using 3D kinematic measures. Methods We developed novel multi-joint angle and distance-based measures of symmetry to assess upper limb PPS. Healthy adults (N = 17) and children with USCP (N=9) participated in this study. Kinematics were extracted during upper limb 3D pose matching tasks using a VICON Nexus system. Brunner-Munzel permutation tests were used for statistical comparisons between groups. Pearson’s correlation coefficient was used to investigate the relationship between the different symmetry measures and pose matching tasks. Results We introduced novel angle and distance-based measures of symmetry to estimate PPS. Healthy adults scored higher on both these symmetry measures compared to children with USCP. Angle and distance-based symmetries did correlate with each other. Exploratory factor analysis indicated that both angle symmetry and joint distance symmetry were highly loaded by the latent factor that best explained variance in the data. These results suggest that the two measures of pose symmetry are not redundant. It was also observed that distance symmetry was less sensitive to the different poses while angle symmetry varied across poses. Conclusions Symmetry measures derived from kinematics during upper limb pose matching tasks can estimate multi-joint PPS. Angle and distance-based symmetries together provide information about an individual’s ability to sense joint position. This framework will allow clinicians and researchers to measure PPS of upper limbs only using kinematic acquisition devices.

Andreas’ Device: the Labyrinth and Hidden Symmetries

This article is in two parts: a series of drawings depicting walls, and a series of lyrical or imaginative commentaries on walls. The drawings show walls in multifarious forms, more or less abstract, more or less figurative, constrained by geometrical regularity. The format of presentation elicits the perception of rhythm in the lateral survey of the drawings, which in turn suggests the operation of forms comparable to those encountered in the perusal of poetic language organized over many pages in serial verse. In particular, comprising a visual series, the drawings may be discussed in terms of poetic devices such as echo, verse-line, and meter, and their repeating elements. Besides verbal art, the drawings also function as a schematic representation of walled edifices that embody the ratios and measures of symmetry in physical architecture (a cathedral interior, a city block, a Daedalian labyrinth, for example). Mindful of both analogies – with poetic language and physical architecture – the commentaries attempt reflection on the experience of symmetry in the drawings.

The Levenshtein distance as a measure of mirror symmetry and homogeneity for binary digital patterns

<p><strong>Abstract.</strong> The complexity of a digital pattern, image, map, or sequence of symbols is a salient feature that finds numerous applications in a variety of domains of knowledge [1], [7], [10], [11]. Two features of patterns that form inherent components of pattern complexity, are mirror (reflection) symmetry and homogeneity [8], [9]. In the raster graphics representation mode, a pattern consists of a two-dimensional array (matrix) of elements (pixels, symbols). It is assumed here that the elements are binary-valued (black-white). With such a representation it is common to compute properties of 2-dimensional patterns, such as complexity, mirror-symmetry, and homogeneity, along the 1-dimensional rows, columns, and diagonals of the array [4]. In addition, within each row, mirror symmetries may be analysed either globally or locally [3]. A pattern that does not exhibit <i>global</i> mirror symmetry may still possess an abundant number of <i>local</i> mirror symmetries. Local symmetries permit graded measures of symmetry rather than all-or-nothing decisions. One powerful type of local symmetry is the <i>sub-symmetry</i>, a <i>contiguous</i> subset of elements of the pattern that is palindromic (has mirror symmetry). It has been shown empirically that the total number of sub-symmetries present in a pattern may serve as an excellent predictor of the perception of both <i>visual</i> pattern complexity [5], and <i>auditory</i> pattern complexity [6]. The present research project explores how two well-known measures of the distance between binary patterns and their inversions, correlate with sub-symmetries, as well as other measures of symmetry and homogeneity.</p>