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Edited by: Naomi Nakayama, University of Edinburgh, UK

Reviewed by: David Smyth, Monash University, Australia; Vincent Mirabet, École Normale Supérieure de Lyon, France

*Correspondence: Koichi Fujimoto, Department of Biological Sciences, Graduate School of Science, Osaka University, 1-1 Machikaneyama-cho, Toyonaka, Osaka 560-0043, Japan e-mail:

This article was submitted to Plant Evolution and Development, a section of the journal Frontiers in Plant Science.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Stochasticity ubiquitously inevitably appears at all levels from molecular traits to multicellular, morphological traits. Intrinsic stochasticity in biochemical reactions underlies the typical intercellular distributions of chemical concentrations, e.g., morphogen gradients, which can give rise to stochastic morphogenesis. While the universal statistics and mechanisms underlying the stochasticity at the biochemical level have been widely analyzed, those at the morphological level have not. Such morphological stochasticity is found in foral organ numbers. Although the floral organ number is a hallmark of floral species, it can distribute stochastically even within an individual plant. The probability distribution of the floral organ number within a population is usually asymmetric, i.e., it is more likely to increase rather than decrease from the modal value, or vice versa. We combined field observations, statistical analysis, and mathematical modeling to study the developmental basis of the variation in floral organ numbers among 50 species mainly from Ranunculaceae and several other families from core eudicots. We compared six hypothetical mechanisms and found that a modified error function reproduced much of the asymmetric variation found in eudicot floral organ numbers. The error function is derived from mathematical modeling of floral organ positioning, and its parameters represent measurable distances in the floral bud morphologies. The model predicts two developmental sources of the organ-number distributions: stochastic shifts in the expression boundaries of homeotic genes and a semi-concentric (whorled-type) organ arrangement. Other models species- or organ-specifically reproduced different types of distributions that reflect different developmental processes. The organ-number variation could be an indicator of stochasticity in organ fate determination and organ positioning.

Biological systems ubiquitously and inevitably exhibit stochasticity in traits from the molecular level to the multicellular and morphological level. The stochasticity in the numbers of protein molecules within single cells has been extensively analyzed in species ranging from bacteria to mammals (McAdams and Arkin,

Here, we focus on the discrete stochastic variation appearing in floral organ numbers. Although the floral organ number is a hallmark of eudicot species, it can distribute stochastically, even within an individual plant or a continuous population of a single species (Figure

We performed a review and statistical comparison of six hypothetical mechanisms for the stochastic determination of floral organ numbers in eudicots. We combined field observations, statistical analysis, and mathematical modeling to study the developmental basis of variation in floral organ numbers. The statistical selection of the best model to describe the observed variation in floral organ numbers clarified that a distribution based on a modified error function (the modified ERF) widely reproduced the asymmetric variation found in nature. The error function is derived from mathematical modeling of the floral organ positioning, and its parameters represent measurable distances on the floral bud morphologies. Moreover, the model predicts several mechanisms for the observed distributions (e.g., stochastic shifts in the expression boundaries of genes). The modified ERF model requires a semi-concentric organ arrangement (i.e., the whorled-type arrangement) to give an asymmetric distribution, whereas it does not require such an arrangement to give a symmetric distribution. Other models, i.e., the Gaussian, the Poisson, and the log-normal distributions, reproduced different types of variations species- or organ-specifically that reflect different developmental processes. The organ-number variation could be an indicator of stochasticity in organ fate determination and organ positioning during floral development.

Populations of flowers were studied in natural and cultivated environments. The sampling of each floral population was limited both temporally (1–8 days) and spatially (diameter up to 100 m), because seasonal (Weldon,

The fitting of the measured probability distribution to six statistics was determined using the non-linear least-square (NLS) method, where the probability of each organ number was a single data point. Because the organ number in each population does not distribute to a very large number of states (e.g., five states in Figure

One of the most popular and statistically rigorous criteria for selecting the best-fit model is the Akaike-Information Criterion (AIC), which is represented by the parameter number of the model

The AIC can be used to autonomously select the best-fit statistical distribution, which gives the minimum value of the AIC. When the number of states

which must satisfy

Using the NLS algorithm, we fit the probability distribution of floral organ numbers to four continuous distributions (the standard Gaussian, the log-normal, the gamma, and the beta) and two discrete distributions (the Poisson and the modified ERF). We chose the four continuous models because the standard Gaussian distribution is the most basic distribution in statistics, the log-normal (Furusawa et al.,

The probability density function of the standard Gaussian distribution is given by

This function exhibits a bell-shaped curve that is symmetric to the mean μ with standard deviation σ (Figure

The probability density function of the log-normal distribution is given by

This function represents a Gaussian distribution when

The probability density function of the gamma distribution, which is also skewed to larger values of

which has two parameters, the shape

The probability density function of the beta distribution is given by

where B(α, β) is the beta function. The probability density function is not only skewed to either larger or smaller values of _{max}_{min}_{max}_{min}_{max}_{min}

where

Suppose that each flower has a special number

The probability of the Poisson distribution is given by

where the parameter λ corresponds to the mean (Figure

Some of the stochasticity in floral organ numbers is induced by so-called homeotic transformations, i.e., the variations in the determination of floral organ identities (Goethe,

where _{r}, and σ_{r} denote the radial distance from the meristem center and the average and standard deviation of the distance within the population, respectively (Figure

_{r} and standard deviation σ_{r} (Equation 12). _{r} to the last (innermost) primordium of the first whorl and the first (outermost) primordium of the second whorl, respectively, (Equation

In addition to the boundary variation, we assume that the organs take on the semi-whorled arrangement that is widely observed in the Ranunculaceae (Ronse De Craene, _{r} is located between the _{d}_{d}_{d} = σ/

where erf is the error function

The first line of Equation 13 (

We analytically calculated the integrals using the NORM.DIST function in Microsoft Excel, which is the cumulative distribution function of the Gaussian distribution (Figures _{d}_{d}_{d}_{d}_{d}_{d}_{d}_{d}_{d}_{d}_{d}_{d} (Figure

We measured the frequencies of different floral organ numbers in populations of eudicots. By normalizing the frequency of each organ number by the population size (

There are two types of the asymmetric distribution: positively and negatively skewed distributions. In a positively skewed distribution, as in the tepals of

To find the best model for each pattern of floral organ-number variation and to elucidate whether there is any common law that unifies the patterns, we performed non-linear least-square fitting of each data set containing more than five states (histogram in Figure

An.fl | Mt. Ibuki | 2519 | ^{***} |
^{*} |
^{**} |
32.06 | ^{***} |
^{***} |

An.fl | Mt. Ibuki | 2445 | ^{***} |
^{***} |
18.94 | NA | ^{***} |
^{**} |

An.fl | Mt. Ibuki | 1033 | ^{***} |
^{**} |
^{**} |
NA | ^{***} |
^{***} |

An.fl | Mt. Kongo | 1717 | ^{***} |
31.51 | ^{**} |
NA | ^{***} |
^{***} |

An.fl | Mt. Kongo | 671 | ^{***} |
^{***} |
^{***} |
23.04 | ^{***} |
^{***} |

An.fl | Mt. Kongo | 1528 | ^{***} |
42.60 | ^{*} |
NA | ^{***} |
^{***} |

An.fl | Mt. Kongo | 1384 | ^{***} |
12.12 | ^{***} |
41.55 | ^{***} |
^{***} |

An.fl | Mt. Kongo (sum) | 5702 | ^{***} |
17.70 | ^{**} |
47.57 | ^{***} |
^{***} |

An.fl | Sasayama City | 355 | ^{***} |
15.13 | 13.35 | NA | ^{***} |
^{**} |

An.fl | Hanno City Ohno, |
1624 | ^{***} |
^{***} |
^{***} |
17.70 | ^{***} |
^{***} |

Er.pi | Maibara City | 47 | ^{***} |
20.38 | 20.38 | NA | 12.67 | ^{**} |

Er.pi | Maibara City | 165 | ^{***} |
10.29 | 10.34 | 17.58 | 6.74 | 9.67 |

Er.pi | Maibara City | 153 | ^{***} |
26.73 | 26.73 | 48.16 | 23.58 | ^{**} |

Er.pi | Maibara City | 252 | ^{***} |
11.99 | 15.59 | 22.05 | ^{**} |
^{***} |

Er.pi | Maibara City | 98 | ^{***} |
6.85 | 7.12 | ^{***} |
^{***} |
^{**} |

Er.pi | Sasayama City | 184 | ^{***} |
3.08 | 2.58 | 0.00 | 7.02 | 2.62 |

Er.pi | Sasayama City | 127 | ^{***} |
10.36 | 10.38 | 39.37 | ^{***} |
9.63 |

Er.pi | Sasayama City | 239 | ^{***} |
18.54 | 18.84 | 20.58 | 20.74 | ^{***} |

The modified ERF model was the best fit for the outer floral organs in half (50/99) of the Ranunculaceae data sets [

The Poisson distribution was the second dominant model (best fit for 24/99 data sets) for the Ranunculaceae perianth organs. Especially for the petal numbers in genus

The ERF model was the best model for many other data sets in which the modal number has an extraordinarily high probability (i.e., the modal number is very stable), and the organ number varies on both sides of the mode. On the semi-log plot of

The Poisson distribution of floral organ numbers was originally proposed for interspecific hybrids of

In Oleaceae flowers measured by Roy (

The best-fit models differed not only among the species, genera, and families described above but also among the organ types. In order to demonstrate that, we performed ΔAICc-based model selection for the distributions of three organ types in the same species (i.e., the petals, ovules, and seeds of

The organ-specific model selection can even occur independently of floral development. The number distribution of the seeds, which develop from ovules only after pollination, differed from that of the ovules: in

Although the ray floret is not a floral organ but itself a flower, it has similarity with a perianth organ in developmental and morphological aspects: they develop from the meristem and surround the compact inner organs. Historically, the Asteraceae ray florets were a main target of research on organ-number variation along with the Ranunculaceae floral organs (de Vries,

The modified ERF distribution requires three properties during floral development: (1) The concentric expression of homeotic genes, (2) A Gaussian distribution of the gene expression boundary, and (3) A semi-whorled arrangement. Here we discuss the biological bases of these three assumptions.

The identity of floral organs in

Little is known about the stochastic variation in MADS-box and _{gene}

Many Ranunculaceae species such as

We suggested a development-based model for the modified ERF distribution. The easiest way to confirm such a model based on homeotic stochasticity (Figure

In some of the data sets for which the modified ERF distribution produced the lowest ΔAICc value, the right tail was rather closer to the log-normal distribution (Figure

We hardly discussed multimodal distributions, which have attracted some researchers who suggest that there is a relation between peak organ numbers and the Fibonacci series, especially among Asteraceae heads (Ludwig,

The parameters after the convergence of the non-linear least-square method generally depend on the initial set of fitting parameters. Although we set the initial conditions intuitively to avoid divergent parameter fitting, there might be other initial conditions resulting better parameter sets (i.e., lower AICc). The fitting parameters were occasionally sensitive to initial conditions for the log-normal, the gamma, and the beta distributions when

The variation in the floral organ numbers of various eudicot species was fit to six statistical models. Statistical model selection revealed that the selection of the best model was reproducible for each species and organ. The modified ERF model, which we first proposed by assuming a semi-concentric arrangement of organ primordia following helical initiation and stochasticity in the concentric determination of organ fate during floral development, was widely selected for the perianth organs of eudicots, and even for the stamens and ray florets of some core eudicots. The standard Gaussian and log-normal distributions were selected, respectively, for the Oleaceae petals, which show the simultaneous initiation of the primordia, and the Papaveraceae ovules, which have a totally different developmental process compared with the perianth organs. We showed that the different distributions of morphological traits reflect different developmental processes. The modeling of developmental process of these organs and the statistical analyses of other species and organs will shed more light on the developmental and evolutionary sources of morphological variation.

Conceived and designed the experiments: Miho S. Kitazawa, Koichi Fujimoto. Performed the data collection and the statistical analysis: Miho S. Kitazawa, Koichi Fujimoto. Analyzed the data: Miho S. Kitazawa, Koichi Fujimoto. Contributed materials/analysis tools: Miho S. Kitazawa, Koichi Fujimoto. Wrote the paper: Miho S. Kitazawa, Koichi Fujimoto.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Miho S. Kitazawa is JSPS Research Fellow (24.1243). Koichi Fujimoto is supported by the Osaka University Life Science Young Independent Researcher Support Program through the Special Coordination Program to Disseminate Tenure Tracking System, and Grant-in-Aid for Scientific Research on Innovative Areas (26113511), MEXT. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

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