Identification of approximate symmetries in biological development

Abstract

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'.

Keywords: approximate symmetries; asymmetry measure; information theory and entropy; morphogenesis.

Figure 1.

Transformation information (TI) as a measure for analysing rotation symmetries of the pentagonal Turing pattern in [20]. (a) Turing pentagonal pattern on a disk. (b) TI measure as a function of the angle of rotation. A maximum of $-\mathrm{TI}$ is marked with a downward triangle, and a minimum is marked with an upward triangle. (c) TI as a function of the angle of rotation, with the same maximum (downward triangle) and minimum (upward triangle) as in (b) indicated with lines in polar coordinates. (d) The difference in pixel values between the original image (a) and the transformed image corresponding to the local maximum of $-\mathrm{TI}$ marked with a downward triangle in (b,c). (e) The difference in pixel values between the original image (a) and the transformed image corresponding to the local minimum of $-\mathrm{TI}$ marked with an upward triangle in (b,c). Regions where the original image is higher intensity are indicated in red, while regions where the transformed image is higher are indicated in blue. (Online version in colour.)

Figure 2.

(a) An image of a bay leaf (Laurus nobilis). (b) The original image with its vertical reflection superimposed. (c) The difference in greyscale pixel intensity of the reflected and original image. (d) The optimal axis of bilateral symmetry as determined by minimizing TI is shown with a solid green line. (e) TI bilateral symmetry about axes between the dashed lines in panel (a). The optimal axis is marked by a solid vertical line. (f) The optimal axis of bilateral symmetry as determined by minimizing SI is shown with a solid orange line. (g) The difference in greyscale pixel intensity of the reflected and original image for the optimal symmetry axis indicated in (d). Red (blue) indicates higher pixel intensity for the original (transformed) image. (h) SI bilateral symmetry about axes between the dashed lines in panel (f). The optimal axis is marked by a solid vertical line. (i) The difference in area above and below the optimal axis indicated in (f). (Online version in colour.)

Figure 3.

The top row shows an image of the desmid Micrasteria undergoing cellular regeneration in the centre panel (b). The difference in pixel intensity between its reflection about the solid black line and the original image is shown for an axis through (a) the midpoint of the organism, and (c) the axis defining its upper and lower structure. The bottom row shows (d) the processed image on which TI is computed along with the optimal axes associated with vertical and horizontal reflections in solid orange and solid green, respectively. Reflection TI as a function of the location of the axis relative to the centre of the image is shown for reflections about (e) the vertical axis and (f) the horizontal axis. In each case, the optimal location that minimizes TI is marked by the solid line. (Online version in colour.)

Figure 4.

(a) An image of a Pachypodium flower. (b) The processed image on which the TI is being computed. (c) Manual identification of the floral edges and distances from the centre to each petal edge. (d) A peak in the magnitude $\parallel \mathrm{d}TI/\mathrm{d}\theta \parallel$ (high values in yellow) for rotations about a fixed location indicates the inferred centre point of the flower shown in (a)–(c). (e) The centre point found in (b) based on rotational TI (orange asterix) provides a much better approximation to the manually identified centre (black target) than the centre found by equalizing areas above/below and left/right of the black circled up arrow mark. (f) The vertical position of the optimal centre as found by reflection TI (dashed green) is well above the centre points found by other methods. However, a smaller peak in reflection TI, indicated by a dotted line, does appear near the manually identified centre. (Online version in colour.)

Figure 5.

(a) Translation information as a function of axis angle for rotations about the manually determined centrepoint of the flower shown in figure 4a–c. The top ranked symmetries among these are (i) rotation by ${72}^{\circ }$, (ii) rotation by $-{73}^{\circ }$. (b) The measure in [36] (denoted by ZI), which calculates the sum of the squared distances between nearest petal tips of the original and rotated image. (c) Translation information as a function of axis angle for reflections about the manually determined centrepoint of the same flower. The top ranked symmetry among these is (iii) reflection about axis with angle ${95}^{\circ }$. (d) Corresponding difference in pixel intensities between the rotation (i) in (a) and the original image. (e) Corresponding difference in pixel intensities between the rotation (ii) in (a) and the original image. (f) Corresponding differences in pixel intensities between the reflection (iii) in (c) and the original image. (Online version in colour.)

Figure 6.

Patterns obtained from the Turing model (2.2), (2.3) with the same random initial condition and an advection term of the form $c\delta (\mathrm{\partial }u/\mathrm{\partial }y)$ added to equation (2.2). Simulations are carried out on a $10\times 10$-unit periodic domain, with advection rate increasing from top to bottom: $c=0,3,6$. In each row, the left panel (a) shows the spatial distribution of $u+v$ at $t=100$ with yellow indicating high values. The centre panel (b) shows translation TI for the periodically extended state as a function of the magnitude of the vertical and horizontal shifts. The right panel (c) shows the Fourier spectrum of the deviation from average of $u+v$ for comparison. (Online version in colour.)

Figure 7.

Transformation Information associated with rotation by angle $\theta$ and rescaling by a constant factor for a transverse section through a catnip stem bearing pairs of opposite leaves (Nepeta cataria). Panel (a) shows the location of local minima (with low values in blue) in TI of the image of leaf development shown in panel (b). Panels (c,d) show the original image superimposed, with panel (c) rotated by ${88}^{\circ }$ and rescaled by a factor of 0.57, and panel (d) rotated by ${180}^{\circ }$ and rescaled by a factor of 0.35. (Online version in colour.)