# Fluctuating Asymmetry: Methods, Theory, and Applications

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## Abstract

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## 1. Introduction

## 2. Symmetries

#### 2.1. Types of Symmetry

#### 2.2. Symmetry Groups

_{∞}-symmetry (see Figure 7d). It differs from rotational and dihedral symmetries because the number of rotational and reflective axes of symmetry is infinite. An object displays translational symmetry if each of its segments is equal to the other segments of the object located at equal translational distances [1].

#### 2.3. Types of Objects

## 3. Measuring Deviations from Perfect Symmetry

#### 3.1. Measures of Dispersion

_{i}= l

_{i}– r

_{i}, where d

_{i}is the left-right asymmetry of individual i, l

_{i}is the value of the trait on the left side, and r

_{i}is the value of the same trait on the right side, then a measure of fluctuating asymmetry is Var(d

_{i}), or its square root (standard deviation). Trait values (l, r) can be either continuous or discrete.

_{i}– μ|, where x

_{i}is an individual observation and μ is the population mean. Assuming μ = 0, which is routine in studies of fluctuating asymmetry, the mean absolute value of d is E|d

_{i}|. It is less sensitive to outliers than the variance [57], and |d

_{i}| lends itself to analysis of variance [58,59,49]. If d

_{i}is normally distributed, then |d

_{i}| has a half-normal distribution. One can normalize |d

_{i}| with a power transform: (|d

_{i}|+ 0.00005)

^{0.33}[54]. The value of the mean absolute deviation is approximately 0.8 of the standard deviation [60].

_{i}), where x

_{i}is the length of arm i. Because all starfish have five arms, this would be a population variance, rather than a sample variance (if all five arms were measured). Used in this way, Var(x

_{i}) is analogous to d

_{i}, and is a measure of individual asymmetry. The mean of Var(x

_{i}) (or the standard deviation) would be the actual measure of fluctuating asymmetry for a sample of starfish. In addition to the variance, the mean absolute deviation can be applied to other kinds of symmetry. For the starfish mentioned earlier, individual asymmetry would be E|x

_{i}– μ|, where x

_{i}is the length of arm i and μ is the mean of all five arms. Similar approaches can be taken with translational symmetry. For reasons discussed previously, we prefer the mean absolute deviation.

#### 3.2. Landmark Methods for Shape Asymmetry

#### 3.3. Continuous Symmetry Measures (CSM)

_{3}symmetry group, namely, the rotational symmetry group of order 3. For a set of orbits, we calculate the CSM independently for every orbit, with the appropriate symmetry operation applied about the center of mass of all points rather than about the origin. This algorithm can be implemented for all symmetry groups in any dimension. Proof and details can be found in [65,68,26]. An example of a hexagon and its CSM value and symmetry transform for various symmetry groups is given in Figure 16.

#### 3.3.1. CSM in Physics and Chemistry

#### 3.3.2. Measuring Asymmetry in the Absence of Landmarks

#### 3.3.3. Analysis of Asymmetry Using Anchor Points

_{k}matches point Q

_{k}). When a secondary vein is ‘missing’, we insert a new zero-length vein, marked as two endpoints located at the bifurcation point. Figure 17C shows the paired points, where P

_{1}and P

_{2}are the endpoints of the new vein inserted as a match for Q

_{1}and Q

_{2}, respectively. One can estimate the optimal pairing of points with a dynamic programming algorithm [23,24].

^{rd}and 4

^{th}order veins [81]. These structures are inherently hierarchical, where every sub-structure is itself a bifurcating structure, as can be seen in Figure 19.

^{rd}and 4

^{th}order veins than when using only the 2

^{nd}order veins [81].

#### 3.3.4. CSM-Developmental Stability Measurement

## 4. Measurement Error

#### 4.1. Mixed-Model ANOVA

^{2}

_{S x I}should appear in the expected mean square for individuals. The model used by Palmer and Strobeck [48], Samuels’ Model 2, includes σ

^{2}

_{S x I}. Her Model 1, however, excludes it. Although Samuels [84] favors Model 1, the best model for studies of fluctuating asymmetry is still unclear. According to Samuels, Model 1 is more appropriate for nonrandomized observational designs like those represented by fluctuating asymmetry. Indeed, her example of such a design is density of neurons in right and left hemispheres of the brain. Nevertheless, Model 2 is more appropriate when the goal of a mixed-model ANOVA is estimating components of variance. This is because all of the random terms in Model 2 are uncorrelated [84].

_{I}= ½Var(l + r)

_{S x I}= ½Var(l − r).

_{I}− MS

_{S x I}). It is the variation among individuals; the stochastic component is the variation within individuals.

_{o}: μ

_{l}= μ

_{r}). The two means are tested by comparing variances (the mean squares). For random effects, however, one does not (cannot) compare levels. For a random effect, the levels of the effect are a random sample from a large population. It is more meaningful to estimate variance components.

_{0}: σ

^{2}

_{S x I}= 0, under Model 2), it does not make much sense to do so. For metric traits, no two individuals are identical, no two sides in an otherwise symmetrical individual are ever identical, and no two measurements of the same object are ever identical. Variation is pervasive. One can always find a difference, no matter how small. Thus, we know beforehand that the null hypothesis for a random effect is not true; it cannot be true. The only reason one might consider testing the null hypothesis that fluctuating asymmetry is not significantly greater than zero would be during trait selection. Traits having small amounts of asymmetry may not be worth the effort.

^{2}

_{m,}σ

^{2}

_{S x I}, σ

^{2}

_{I}), and test only for differences between sides. The variance component for individuals (σ

^{2}

_{I}) is an estimate of the shape and size variation among individuals. The variance component for the interaction effect (σ

^{2}

_{S x I}) is an estimate of the non-directional asymmetry (fluctuating asymmetry and antisymmetry). The variance component for replicates (σ

^{2}

_{m}) is an estimate of measurement error. Provided that the sides effect is insignificant, and that there is no obvious antisymmetry, one can interpret σ

^{2}

_{I}and σ

^{2}

_{S x I}as factorial and stochastic variances. Their sum is the total phenotypic variance, minus measurement error.

_{o}: μ

_{l}≠ μ

_{r}), the variance component estimate for the interaction (i.e., fluctuating asymmetry) may be biased, depending upon how directional asymmetry is manifested [52]. This is because the ANOVA’s additivity assumption may be violated. Nothing more can be done with the mixed-model ANOVA. Alternate methods of partitioning out directional asymmetry have been proposed by Graham et al. [52] and Van Dongen et al. [53]. The mixed-model ANOVA should be viewed then as the beginning, not the end of an asymmetry analysis.

^{2}

_{m}) is equal to or greater than fluctuating asymmetry (σ

^{2}

_{S x I}) does not mean fluctuating asymmetry is insignificant. Finally, it is always best to express measurement error (MS

_{error}or σ

^{2}

_{m}) as a percentage of either the total phenotypic variation (MS

_{I}) or the asymmetry variance (σ

^{2}

_{S x I}).

#### 4.2. Fuzzy Analysis for Quantifying Measurement Error

_{n}-symmetry) is used to evaluate the most probable symmetric shape under the maximum likelihood criterion. This variant considers the fuzzy-input points as n measurements, ${Q}_{\begin{array}{l}i\\ \end{array}}$~$({P}_{i};{A}_{i})$, where P

_{i}is the mean and A

_{i}is the covariance of the point-location distribution. The folding-unfolding method is applied to these n measurements. (See Figure 23, which shows the method applied to a rotationally symmetrical object of order n. The approach for mirror symmetry, or any other symmetry, is similar.)

## 5. Error Models and Size Scaling

#### 5.1. Error Models and Probability Distributions

_{t}at time t is a random proportion of its size X

_{t}

_{-1}at a previous time t – 1, such that X

_{t}= X

_{t}

_{-1}+ ε

_{t}X

_{t}

_{-1}, then X

_{t}is asymptotically lognormally distributed [96,97]. Thus, multiplicative error generates a lognormal distribution (but see [98]). Examples of such active growth include leaves, petals and stems of plants, and bones and soft tissues of animals (Figure 25A).

_{t}at time t is independent of its size X

_{t}

_{-1}at a previous time t – 1, such that X

_{t}= X

_{t}

_{-1}+ ε

_{t}, then X

_{t}is asymptotically normally distributed. Additive error generates a normal distribution. Examples of such inert growth include nails, scales, bristles, feathers, teeth, and exoskeletons. Measurement error is additive as well.

#### 5.2. Transformations for Size Scaling

^{2}– ((σ

^{2}/n)/ σ

^{2})100 with each replicate averaged.

#### 5.3. Recommended Approaches

_{i}| by the average of that trait for the population. An alternative for highly leptokurtic data is to total the ranks of the asymmetry values.

_{i}(location and variance), one should also report the third and fourth moments of the distribution (i.e., skew and kurtosis). Moreover, for the sake of future meta-analyses, one should report both signed and unsigned asymmetries. For a full discussion of sampling and statistical considerations, see Palmer and Strobeck [49].

## 6. Developmental Homeostasis, Canalization, and Developmental Stability

## 7. Random Developmental Variation

#### 7.1. Nature, Nurture, and Noise

^{2}

_{p}) into genetic (σ

^{2}

_{g}) and environmental (σ

^{2}

_{e}) components, fluctuating asymmetry is part of the environmental component [118]. But the stochastic component is often as large as, or larger than, the genetic and true environmental components. Consequently, several authors have argued that it should stand on an equal footing [119,120,117,121]. Kozhara [89,90], as mentioned previously, decomposes σ

^{2}

_{p}into factorial (σ

^{2}

_{f}) and stochastic (σ

^{2}

_{s}) components. The factorial component is the variation among individuals, and the stochastic component is the variation within individuals. For symmetrical traits, σ

^{2}

_{p}= ½Var(l + r), and σ

^{2}

_{f}= ½Var(l + r) − ½Var(l − r). The factorial component combines genetic and environmental variation.

#### 7.2. Deterministic Chaos

#### 7.3. Population and Individual Fluctuating Asymmetry

## 8. Origins of Developmental Homeostasis

#### 8.1. Adaptation, Coadaptation, and Heterozygosity

^{2}) for fluctuating asymmetry are small, but not zero [158,167]. Nevertheless, estimates of heritability require large sample sizes, and most are not large enough to distinguish between heritabilities h

^{2}of 0.1 and 0.5 [168,169]. In contrast, epistatic interactions account for much of the genetic variation in fluctuating asymmetry within laboratory populations of Mus musculus [170,158].

#### 8.2. Distributed Robustness

^{-}

^{γ}, distribution. Most nodes have few links, but a few hubs may have hundreds or thousands of links. The hubs connect the less connected nodes to the system. These systems are typically scale-free [179] and hierachical [186].

_{i}– μ| among different populations or genotypes.) If the response to a specific, local perturbation to a node or link is proportional to the connectedness of the node, then fluctuating asymmetry should also be distributed as a negative power-law. Clearly, this does not occur, because a small amount of noise is necessary for normal development and gene regulation [122,123]. We have never seen a continuous trait that did not have some asymmetry. Thus, one would expect a skewed distribution with fat tails. The Double Pareto lognormal distribution is such a distribution; it is a mixture of lognormal and Pareto distributions. Babbitt and colleagues [188,189,190] have fit the double Pareto lognormal distribution to asymmetry data (though the data appear to contain artifactual antisymmetries). Nevertheless, the distribution may still be useful in testing the hypothesis that distributed robustness mediates developmental instability.

#### 8.3. Environmental Stress

## 9. Applications

#### 9.1. Stress

_{1}and F

_{2}hybrids (Salicaceae: Angiospermae)). Although they sampled only one leaf from each plant, they sampled leaves from 2359 plants. Genetic stress was more important than environmental stress. F

_{2}hybrids had the greatest leaf asymmetry, but water stress, pathogen attack, and competition had no effect on leaf asymmetry. Fluctuating asymmetry also decreased with plant size.

#### 9.2. Fitness

_{i}– μ|, where the x

_{i}s are the individual observations on left and right sides and μ is the mean. Consequently, the mean deviation is Σ|x

_{i}– μ|/n, and with n = 2, it may be reduced to |x

_{l}– x

_{r}|/2.

#### 9.3. Developmental Integration

#### 9.4. Diverse Fluctuating Asymmetries

#### 9.4.1. Fluctuating Rotational, Dihedral, and Radial Asymmetries

_{i}), for example, is analogous to |l – r|, and is a measure of individual dihedral and rotational asymmetry. The mean of Var(x

_{i}) (or the standard deviation) is the actual measure of fluctuating asymmetry for a sample. Unfortunately, this method completely ignores shape and rotation. Therefore, it is usually a poor measure of fluctuating rotational, dihedral, and radial asymmetries. Occasionally, as in the lengths of ray flowers in the inflorescence of a sunflower, it may be the best approach.

#### 9.4.2. Fluctuating Translational Asymmetry

_{y∙x}of leaf length on leaf order as a measure of developmental instability. The standard error of the estimate, however, varies with the mean $\overline{y}$ of the dependent variable. Zar [260] prefers S

_{y∙x}/$\overline{y}$ as a measure of variation around the regression.

_{y∙x}/$\overline{y}$ is used to estimate fluctuating translational asymmetry. S

_{y∙x}/$\overline{y}$ is estimated on a per individual basis, and the mean of S

_{y∙x}/$\overline{y}$ across the sample is the estimate of fluctuating translational asymmetry.

^{a}e

^{-bN}, where L is internode length, N is internode order starting from the base of the shoot, e is the base of natural logarithms, and k, a, and b are fitted constants [263]. The kN

^{a}reflects the allometric relationship between internode length and node order, and e

^{-bN}reflects apical inhibition. Stress presumably decreases the accuracy of the curve fitting. The standard error of the estimate (S

_{y∙x}/$\overline{y}$), the statistical noise in the allometric relation, is a measure of individual asymmetry.

#### 9.4.3. Fluctuating Helical Asymmetry

^{θ cot Φ},

_{e}r on log

_{e}a + θ cot Φ for each individual snail and used the standard error of the estimate, divided by the mean of the dependent variable, (S

_{y∙x}/$\overline{y}$) as an estimate of individual asymmetry. Populations exposed to ammonia emissions and pesticides showed higher levels of asymmetry. Individuals in the population having the greatest helical asymmetry also showed erosion of their periostracum, which was not evident in the other two populations.

#### 9.4.4. Fractal Dimension

_{y∙x}/$\overline{y}$) be used as an index of developmental instability. There are, however, many methodological problems associated with such estimates [281]. For example, if techniques devised to estimate monofractals are blindly applied to multifractals, then D will be biased and S

_{y∙x}/$\overline{y}$ will be inflated, because multifractals are more complex than monofractals [282,271]. More research is needed.

#### 9.5. Fluctuating Asymmetry and Developmental Instability of Diverse Taxonomic Groups

#### 9.5.1. Viruses

#### 9.5.2. Archaea and Eubacteria

#### 9.5.3. Fungi

#### 9.5.4. Plants

#### 9.5.5. Animals

#### 9.6. Fossil Material

#### 9.7. Molecular Robustness

#### 9.8. Specific Disciplines

#### 9.8.1. Ecotoxicology

#### 9.8.2. Conservation Biology

#### 9.8.3. Anthropology and Evolutionary Psychology

#### 9.8.4. Medicine and Public Health

#### 9.8.5. Agriculture and Aquaculture

## 10. Fluctuating Asymmetry: Problems and Solutions

#### 10.1. Inconsistencies among Studies of Fluctuating Asymmetry

#### 10.2. The “Evolution Canyon” Microsites

#### 10.3. Ongoing Research on Fluctuating Asymmetry at the “Evolution Canyon” Microsites

## 11. Conclusions

## Acknowledgements

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**Figure 1.**Spherical and circular symmetries. A. Volvox (© www.micrographia.com). B. Cap of a mushroom. C. Centric diatom. (© www.micrographia.com).

**Figure 2.**Dihedral and rotational symmetries. A. Starfish (Bill Davin, Department of Biology, Berry College). B. Aster inflorescence. C. Periwinkle flower.

**Figure 3.**Bilateral symmetry. A. Skull of a turtle. B. Leaf of Quercus calliprinos. C. Flower of Clitoria. D. Sand dollar.

**Figure 4.**Translational symmetry. A. Filament of a cyanobacterium, Nostoc (Ashwani K. Rai, Department of Botany, Banaras Hindu University). B. Internodes of a horsetail, Equisetum arvense (Eric Guinther, public domain). C. Metameres of the polychaete Glycera (NOAA Photo Library).

**Figure 5.**Helical symmetry. A. The snail Cepaea nemoralis. B. DNA double helix (Richard Wheeler [Zephyris], http://en.wikipedia.org/wiki/DNA). C. Tobacco mosaic virus (Agricultural Research Center, USDA, at http://emu.arsusda.gov/typesof/pages/tmv2.html).

**Figure 6.**Fractal symmetry. A. Human skull suture. B. Staghorn coral (Bill Davin, Department of Biology, Berry College).

**Figure 7.**Example of symmetries: a) rotational symmetry (C

_{8}-symmetry), b) mirror symmetry, c) dihedral symmetry (D

_{8}-symmetry), d) radial symmetry (C

_{∞}-symmetry) (from [26]).

**Figure 8.**Three types of objects for quantifying metric asymmetry. A. Structures having consistent topology and number of landmarks (Drosophila melanogaster wings). B. Structures having consistent topology, but varying number of corresponding landmarks among specimens (the secondary veins of Daucus carota leaves). C. Variable structures having no consistent topology, no quantitative consistency, and sometimes no matching points (lobes and sinuses on the leaves of Quercus alba).

**Figure 13.**Rotate the wings to achieve an optimal fit of the landmarks [adapted from 21].

**Figure 14.**Asymmetry of eight landmarks. The diagram shows the original configuration (solid black) and its reflected and relabeled copy (dotted black), after least-squares Procrustes superimposition. The Procrustes average (dashed-dotted red) is perfectly symmetrical. (Adapted from [39]).

**Figure 15.**The folding-unfolding algorithm applied to a single orbit (from [26]). CSM with respect to rotational symmetry of order 3 is calculated. The points ${\widehat{P}}_{0},$ ${\widehat{P}}_{1},$ ${\widehat{P}}_{2}$ are associated with the rotations ${R}_{3}^{0},{R}_{3}^{1},{R}_{3}^{2}$, respectively: a) Original 3 points ${\left\{{P}_{i}\right\}}_{i=0}^{2}$. b) Fold – apply the inverse symmetry element to each point (i.e., rotate ${P}_{i}$ by ${R}_{3}^{-i}$). Points ${\left\{{P}_{i}\right\}}_{i=0}^{2}$ are obtained. c) Average – points ${\left\{{P}_{i}\right\}}_{i=0}^{2}$ are averaged, obtaining ${\widehat{P}}_{0}=\frac{1}{3}{\displaystyle {\sum}_{i=0}^{2}\overline{P}.}$ d) Unfold – apply the symmetry elements to average point ${\widehat{P}}_{0}$ (i.e., rotate by ${R}_{3}^{i}$), obtaining points ${\left\{{P}_{i}\right\}}_{i=0}^{2}$). CSM is calculated: CSM = $\frac{1}{3}({\Vert {{P}_{0}}^{\prime}-\widehat{P}\Vert}^{2}+{\Vert {{P}_{1}}^{\prime}-\widehat{P}\Vert}^{2}+{\Vert {{P}_{2}}^{\prime}-{\widehat{P}}_{2}\Vert}^{2})$.

**Figure 16.**Hexagon and its CSM value and symmetry transform for various symmetry groups (from [26]).

**Figure 17.**A bifurcating structure consisting of a main vein and secondary veins branching off the main vein. A. The bifurcating structure is a set of points interconnected by segments. B. Points P

_{k}and Q

_{k}on paired veins have been matched. C. A new vein is inserted as points P

_{1}and P

_{2}(matching Q

_{1}and Q

_{2}), the endpoints of the new vein. Points P

_{1}, P

_{2}, and Q

_{2}coincide.

**Figure 18.**Computing the CSM using the folding-unfolding method applied to two paired secondary veins, each represented by 2 points. A. Original bifurcating structure with matching points marked. B. Fold – Vein Q is reflected across the symmetry axis, obtaining the folded points ${\tilde{\mathrm{Q}}}_{1}$ and ${\tilde{\mathrm{Q}}}_{2}$. C. Average – The folded points are averaged with their matching points, obtaining the points ${\widehat{\mathrm{P}}}_{1}$ and ${\widehat{\mathrm{P}}}_{2}$. D. Unfold – The average points are reflected back across the symmetry axis, obtaining the unfolded points ${\widehat{\mathrm{Q}}}_{1}$ and ${\widehat{\mathrm{Q}}}_{2}$. The CSM value is the average distance squared between the original points and the corresponding unfolded points: $\frac{1}{4}\left[{\left({P}_{1}-{\widehat{P}}_{1}\right)}^{2}+{\left({P}_{2}-{\widehat{P}}_{2}\right)}^{2}+{\left({Q}_{1}-{\widehat{Q}}_{1}\right)}^{2}+{\left({Q}_{2}-{\widehat{Q}}_{2}\right)}^{2}\right]$.

**Figure 20.**Symmetry of bifurcating structures. a) Global Symmetry. b) Local Symmetry. c) Complete Symmetry.

**Figure 23.**Folding a set of n measurements: a) A configuration of 6 measurement points: Q

_{0}.....Q

_{5}. b) Each measurement Q

_{i}is rotated by 2π/6 radians about the centroid of the expected point location (marked as +), obtaining measurements Q

_{0}.....Q

_{5}(from [26]).

**Figure 24.**Probability distribution of CSM values depends on measurement error. a-d) Measurement error of the points increases from left to right. e) Probability density and symmetry values (${C}_{6}$-symmetry) for configurations a-d (from [26]).

**Figure 25.**Frequency distributions of lobe lengths of 300 fig leaves, Ficus carica (from [82]). A. Lognormal distribution of lobe length. B. Normal distribution of the same data after logarithmic transformation.

**Figure 26.**Size-scaling of asymmetry (from [82]). l and r are lognormally distributed and correlated. A. Size-scaling of asymmetry l – r plotted against size (l + r)/2. B. Lack of size-scaling of asymmetry log l – log r plotted against size (log l + log r)/2.

**Figure 27.**Positive and negative size scaling of leaf asymmetry in Olea europaea from the “African” slope of “Evolution Canyon” I. A. Untransformed data showing positive size scaling. B. Log transformation generates negative size scaling. C. Power transformation (λ = 0.6) removes all size scaling. R’ and L’ are transformed variates.

**Figure 28.**The four “Evolution Canyons” in Israel (EC I – EC IV). Note the interslope divergence in vegetation, even in EC III in the Negev Desert [151].

**Table 1.**Variance components σ

^{2}in a mixed-model ANOVA (after [55]). N is the number of individuals and R is the number of replicate measurements per individual. <σ

^{2}

_{S}> is not a true variance component, since sides are fixed. σ

^{2}

_{m}is a random component due to measurement error.

Source | df | MS | Expected mean squares | Interpretation |
---|---|---|---|---|

Sides | 1 | MS _{S} | σ ^{2}_{m} + R (σ ^{2}_{S x I} + N <σ ^{2}_{s} >) | Directional asymmetry |

Individuals | N – 1 | MS _{I} | σ ^{2}_{m} + R (σ ^{2}_{S x I} + 2σ ^{2}_{I} ) | Size/shape variation |

Sides x Individuals | N – 1 | MS _{S x I} | σ ^{2}_{m} + R σ ^{2}_{S x I} | FA and antisymmetry |

Replicates (S x I) | N(R – 1) | MS_{error} | σ ^{2}_{m} | Measurement error |

Transformation | Equation | References |
---|---|---|

Division by the mean | d or |d| divided by the trait mean (l + r)/2 | [48] |

Log transform | log l – log r | [99,52,82,49] |

Power transform | [(l ^{λ} – 1)/ λ] – [(r ^{λ} – 1)/ λ] for λ ≠ 0 | |

log l – log r for λ = 0 | ||

Half-normal transform | (|d + 0.00005)^{0.33} | [54] |

**Table 3.**Natural environmental stress and fluctuating asymmetry: a small sampling of studies and stressors. All studies are of fluctuating bilateral asymmetry.

Stressor | Taxa | Traits | Result^{1} | Reference |
---|---|---|---|---|

Water limitation/ | Quercus ilex, wet site | leaf | ↑FA | [222] |

drought | Quercus ilex, dry site | leaf | 0 | [222] |

Phaseolus vulgaris | leaf | ↑FA | [223] | |

Salix sericea | leaf | 0 | [224] | |

Salix eriocephala | leaf | 0 | [224] | |

Flooding | Betula pubescens | leaf | 0 | [218] |

High salinity | Glycine max | leaf | 0 | [216] |

UV-B radiation | Dimorphotheca sinuate | leaf | ↑FA | [225] |

Heat shock | Bicyclus anynana | eyespots | 0 | [226] |

Scathophaga stercoraria | tibia and wing | ↑FA | [227] | |

High elevation/cold | Betula pubescens | leaf | ↑FA | [228] |

Food limitation | Sternus vulgaris | primary feathers | ↑FA | [229] |

Scathophaga stercoraria | tibia and wing | 0 | [227] | |

Coenagrion puella | femurs, wings | 0^{2} | [230] | |

Cyrtodiopsis dalmanni | eye stalks, wings | 0 | [200] | |

Nutrient limitation | Acer platanoides | leaf | 0 | [205] |

Betula pendula | leaf | 0 | [205] | |

Nitrogen enrichment | Betula pubescens | leaf | ↑FA | [213] |

Lythrum salicaria | leaf | ↑FA | [214] | |

Penthorum sedoides | leaf | 0 | [214] | |

Competition | Salix sericea | leaf | 0 | [224] |

Salix eriocephala | leaf | 0 | [224] | |

Insect attack | Betula pubescens | leaf | 0 | [228] |

Grazing/browsing | Betula pubescens | leaf | ↑FA | [218] |

Infection | Salix sericea | leaf | 0 | [224] |

Salix eriocephala | leaf | 0 | [224] | |

Natural disaster | Peromyscus leucopus | femur length | ↑FA | [231] |

P. maniculatus | femur length | ↓FA | [231] |

^{1}↑FA indicates a significant increase in fluctuating asymmetry. ↓FA indicates a significant decrease in fluctuating asymmetry. 0 indicates no significant change.

^{2}Food limitation facilitated the effect of pesticides, increasing fluctuating asymmetry before the molt.

Stressor | Species | Traits | Result^{1} | Reference |
---|---|---|---|---|

Inbreeding | Scathophaga stercoraria | tibia and wing | 0 | [227] |

Hybridization | Dalechampia scandens | leaf | 0 | [234] |

Piriqueta caroliniana | leaf | ↑FA | [235] | |

Salix spp. | leaf | ↑FA | [224] | |

Enneacanthus spp. | meristic | ↑FA | [154] | |

Directional selection | Drosophila melanogaster | wing shape | 0 | [217] |

Heterozygosity | Alectoris chukar | toe length | 0 | [236] |

Molecular chaperones | Drosophila melanogaster | bristles, wings | 0 | [171] |

(Hsp90, Hsp83) | Drosophila melanogaster | wing shape | 0 and ↑FA^{2} | [173] |

Arabidopsis thaliana | hypocotyl | ↑V_{p} ^{3} | [174] | |

DNA damage | Dimorphotheca sinuate | leaf | ↑FA | [225] |

Transposons | Drosophila melanogaster | bristles | 0 | [237] |

^{1}↑FA indicates a significant increase in fluctuating asymmetry. ↓FA indicates a significant decrease in fluctuating asymmetry. 0 indicates no significant change.

^{2}Increase in fluctuating asymmetry only for Hsp83 allele introgressed into a control line. No effect of Hps90 inhibition.

^{3}Developmental instability was estimated as phenotypic variation (V

_{p}) of hypocotyl length among clone mates. Because the plants were inbred, and the environment was standardized, V

_{p}is measuring developmental noise.

Stressor | Taxa | Traits | Result^{1} | Reference |
---|---|---|---|---|

Heavy metals | Drosophila melanogaster | bristles | ↑FA | [301] |

Drosophila melanogaster | bristles | 0 | [302] | |

Sorex araneus | various | ↑FA | [303] | |

Paper mill effluent | Gambusia holbrooki | various | ↑FA | [304] |

Urban air pollution | Platanus | leaf | ↑FA | [305] |

SO^{2} emmissions | Betula spp. | leaf | ↑FA^{2} | [45] |

Pinus sylvestris | leaf | ↑FA | [293,306] | |

Acidification | Rana arvalis | skeleton | ↑FA | [307] |

Pesticides | Lucilia cuprina | bristle, wing | ↑FA^{3} | [299] |

Elevated CO^{2}Dioxin | Quercus spp.Mus musculus | leafmandible | ↓FA0^{4} | [300][308] |

teeth | ↑FA | [309] | ||

Magnetic fields | Drosophila melanogaster | bristles | 0 | [310] |

wing veins | ↑↓FA^{5} | [310] | ||

phenodeviants | ↑↓FA^{5} | [310] | ||

Military training | Rhus copallinum | leaf | ↓FA^{6} | [311] |

Ipomoea pandurata | leaf | ↓FA^{6} | [311] | |

Cnidoscolus stimulosus | leaf | ↑FA | [312] |

^{1}↑FA indicates a significant increase in fluctuating asymmetry. ↓FA indicates a significant decrease in fluctuating asymmetry. 0 indicates no significant change.

^{2}Treatment with lime alleviates the stress.

^{3}Fluctuating asymmetry returned to normal after several generations in the presence of the pesticide.

^{4}While FA did not increase, mandible size was smaller and its shape changed.

^{5}Wing asymmetry was greater at 80 μT than at 1.5 μT, but wing asymmetry was less at 1.5 μT than in the controls. Abdominal phenodeviants paralleled the wing vein asymmetry.

^{6}Decreased asymmetry is a likely consequence of greater light availability in disturbed sites.

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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**MDPI and ACS Style**

Graham, J.H.; Raz, S.; Hel-Or, H.; Nevo, E.
Fluctuating Asymmetry: Methods, Theory, and Applications. *Symmetry* **2010**, *2*, 466-540.
https://doi.org/10.3390/sym2020466

**AMA Style**

Graham JH, Raz S, Hel-Or H, Nevo E.
Fluctuating Asymmetry: Methods, Theory, and Applications. *Symmetry*. 2010; 2(2):466-540.
https://doi.org/10.3390/sym2020466

**Chicago/Turabian Style**

Graham, John H., Shmuel Raz, Hagit Hel-Or, and Eviatar Nevo.
2010. "Fluctuating Asymmetry: Methods, Theory, and Applications" *Symmetry* 2, no. 2: 466-540.
https://doi.org/10.3390/sym2020466