# Comparative Analysis of Biologically Relevant Response Curves in Gene Expression Experiments: Heteromorphy, Heterochrony, and Heterometry

## Abstract

**:**

## 1. Introduction

## 2. Identifying Biologically Relevant Response Curves that Fit Well

_{j}denote the j

^{th}observed response and x

_{j}denote the j

^{th}observed time. We fit the model to responses {y

_{1}, y

_{3}, y

_{5}, y

_{7}, y

_{9}, y

_{11}, y

_{13}} at times {x

_{1}, x

_{3}, x

_{5}, x

_{7}, x

_{9}, x

_{11}, x

_{13}}. We call {(x

_{1}, y

_{1}), (x

_{3}, y

_{3}), (x

_{5}, y

_{5}), (x

_{7}, y

_{7}), (x

_{9}, y

_{9}), (x

_{11}, y

_{11})} the fitted points. We evaluate the model at {(x

_{2}, y

_{2}), (x

_{4}, y

_{4}), (x

_{6}, y

_{6}), (x

_{8}, y

_{8}), (x

_{10}, y

_{10}), (x

_{12}, y

_{12})}, which we call the evaluation points Let {f(x

_{2}), f(x

_{4}), f(x

_{6}), f(x

_{8}), f(x

_{10}), f(x

_{12}), f(x

_{14})}denote the predicted responses of a particular model corresponding to the evaluation points.

_{j in J}{y

_{j}− f(x

_{j})}

^{2}/7]

^{1/2}/[max

_{j in J}{f(x

_{j})} − min

_{j in J}{f(x

_{j})}]

_{A}. In the first example Curve A is impulseU. If RPE

_{A}> 10%, report no curve; otherwise proceed to Step 2.

_{B}denote the set of response curves with fewer parameters than Curve A. In the first example CurveSet

_{B}= {flat, lineU, sigmoidU}. Let CurveSubset

_{B}denote a subset of response curves in CurveSet

_{B}such that RPE ≤ RPE

_{A}+ 5%. In the first example CurveSubset

_{B}is {lineU, sigmoidU}. If CurveSubset

_{B}is the empty set, we identify Curve B as the curve with the most parameters in CurveSet

_{B}(sigmoidU in the second example) but select Curve A as the reported curve. If CurveSubset

_{B}is not empty we identify Curve B as the curve in CurveSubset

_{B}with the fewest parameters (lineU in the first example) and select Curve B as the reported curve.

_{A}≤ 10%. The curve reported for each gene in the pair is either Curve A or Curve B, whichever was selected via the curve selection algorithm.

_{j in J}{y

_{j}− f(x

_{j})}

^{2}/7] + 2 × (number of parameters). Although this is a non-standard use of AIC because it applies to evaluation points instead of fitted points, it is still instructive. Figure 1 plots points for genes with good fitting models in both species of frogs. The points labeled Curve A (Curve B) selected correspond to reporting Curve A (Curve B) in the curve selection algorithm. Most Curve A selected points, which require RPE

_{A}− RPE

_{B}> 5%, correspond to AIC

_{A}− AIC

_{B}> 0 (the upper right quadrant). Most Curve B selected points, which require RPE

_{A}− RPE

_{B}≤ 5%, correspond to AIC

_{A}− AIC

_{B}≤ 0 (the lower left quadrant).

**Figure 1.**Comparison of a change in relative prediction error (RPE) with a change in Akaike Information Criterion (AIC) among response curve pairs. The red points corresponding to Curve A require RPE

_{A}− RPE

_{B}≤ 5% (so are above the horizontal 5% line) The green points corresponding to Curve B require RPE

_{A}− RPE

_{B}≤ 5% (so are below the horizontal 5% line). A value of AIC

_{A}− AIC

_{B}≤ 0 (so on the left of vertical line) would indicate selection of Curve B.

## 3. Fitting Algorithms

#### 3.1. Flat

_{FLA}(x) = α

_{FLA}.

#### 3.2. Linear

_{LIN}(x) = α

_{LIN}+ β

_{LIN}·x, for β

_{LIN}≠ 0. Letting b

_{LIN}denote the estimate of β

_{LIN}, we designated the linear model as lineD if b

_{LIN}< 0 and lineU if b

_{LIN}> 0, where D stands for downward and U stands for upward.

#### 3.3. Sigmoid

_{SIG}and γ

_{SIG}specify levels of the steady states. The parameter δ

_{SIG}is the horizontal point corresponding to the maximum slope, β

_{SIG}, between the steady states. The sign of b

_{LIN}is not always a reliable guide to the trend of the sigmoid curve, which we determine by simply comparing the first and last points on the sigmoid curve. We designated the downward and upward trending sigmoid curves as sigmoidD and sigmoidU, respectively.

#### 3.4. Hockey Stick

#### 3.5. Transition

#### 3.6. Impulse

_{dbs}= a

_{sig}, β

_{dbs}= (a

_{sig}+ g

_{sig})/2, γ

_{dbs}= g

_{sig}, δ

_{dbs}= δ

_{sig}, ϵ

_{dbs}= 0, and ø

_{dbs}= 0, where a

_{sig}and g

_{sig}are the estimates of α

_{sig}and γ

_{sig}, respectively.

_{dbs}and γ

_{dbs}correspond to levels of the flat sections. For example with b

_{LIN}> 0 and ϵ

_{dbs}> 0, f

_{DBS}(x) is approximately (1/β

_{dbs}) × α

_{dbs}× β

_{dbs}= α

_{dbs}for small values of x and approximately (1/β

_{dbs}) × β

_{dbs}× γ

_{dbs}= γ

_{dbs}for large values of x. The parameter β

_{dbs}determines the level of the impulse. The parameter ϵ

_{dbs}, which appears in each sigmoid factor, determines the slope of the peak or trough. Mathematically, we identified the impulse curve as a double sigmoid curve in which the minimum or maximum did not occur at the endpoints. We designated an impulse curve with a trough and peak as impulseD and impulseU, respectively.

#### 3.7. Step

#### 3.8. Impulse+

_{dbs}in Equation (3) by an additional parameter λ

_{dbs}. We used the parameter estimates from the impulse curve as starting values with λ

_{dbs}= 0. We identified the impulse+ curve as a generalized double sigmoid curve in which the minimum or maximum was not at the endpoints. We designated an impulse+ curve with a trough and peak as impulse+D and impulse+U, respectively.

#### 3.9. Step+

## 4. Measuring Heterometry and Heterochrony

#### 4.1. Heterometry

X.laevis | X.tropicalis | Total | Heterochrony only | Heterometry Only | Heterochrony and heterometry |
---|---|---|---|---|---|

sigmoidU | sigmoidU | 694 | 18 | 347 | 70 |

lineU | sigmoidU | 146 | 0 | 0 | 0 |

sigmoidU | hockeyU | 73 | 0 | 0 | 0 |

sigmoidD | sigmoidD | 48 | 3 | 22 | 0 |

lineU | lineU | 47 | 0 | 39 | 0 |

hockeyU | hockeyU | 30 | 0 | 18 | 1 |

sigmoidU | lineU | 20 | 0 | 0 | 0 |

lineU | hockeyU | 14 | 0 | 0 | 0 |

hockeyU | sigmoidU | 13 | 0 | 0 | 0 |

sigmoidU | impulseD | 9 | 0 | 0 | 0 |

impulseD | sigmoidU | 8 | 0 | 0 | 0 |

sigmoidD | lineD | 8 | 0 | 0 | 0 |

sigmoidD | hockeyD | 5 | 0 | 0 | 0 |

#### 4.2. Heterochrony

## 5. Results

**Figure 2.**Example of sigmoid curves for one gene pair. X.laevis and X.tropicalis are two species of frogs. The red points denoted “fitted” were used for model fitting. The black points denoted “evaluation” were used for model evaluation and computation of RPE. The curves with blue and green labels and lines are the reported curves (Curve A or Curve B, whichever was selected). The curves with orange labels and lines are included for comparison (Curve A or Curve B, whichever was not selected). HC and HM are measures of heterochrony and heterometry, respectively.

**Table 2.**Counts for response curve pairs with downward trends for both X.laevis and X.tropicalis. The total number is 68.

X.tropicalis | |||||||||
---|---|---|---|---|---|---|---|---|---|

X.laevis | flat | lineD | tranD | hocD | sigD | impD | stepD | imp+D | step+D |

Flat | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

lineD | 0 | 3 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

tranD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

hocD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

sigD | 0 | 8 | 0 | 5 | 48 | 1 | 0 | 0 | 0 |

impD | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |

stepD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

impD+ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

stepD+ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 3.**Counts for response curve pairs with downward trends for X.laevis and upward trends for X.tropicalis. The total number is 14.

X.tropicalis | |||||||||
---|---|---|---|---|---|---|---|---|---|

X.laevis | flat | lineU | tranU | hocU | sigU | impU | stepU | imp+U | step+U |

flat | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

lineD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

tranD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

hocD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

sigD | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |

impD | 0 | 4 | 0 | 0 | 8 | 0 | 0 | 0 | 0 |

stepD | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

impD+ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

stepD+ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 4.**Counts for response curve pairs with upward trends for X.laevis and downward trends curves for X.tropicalis. The total number is 16.

X.tropicalis | |||||||||
---|---|---|---|---|---|---|---|---|---|

X.laevis | flat | lineD | tranD | hocD | sigD | impD | stepD | imp+D | step+D |

flat | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

lineU | 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 | 0 |

tranU | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

hocU | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

sigU | 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 | 0 |

impU | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

stepU | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

imp+U | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

step+U | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table 5.**Counts for response curve pairs that are upward trends for both X.laevis and X.tropicalis. The total number is 1,052.

X.tropicalis | |||||||||
---|---|---|---|---|---|---|---|---|---|

X.laevis | flat | lineU | tranU | hocU | sigU | impU | stepU | imp+U | step+U |

flat | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

lineU | 0 | 47 | 0 | 14 | 146 | 0 | 1 | 0 | 0 |

tranU | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

hocU | 2 | 4 | 0 | 30 | 13 | 1 | 0 | 0 | 0 |

sigU | 0 | 20 | 0 | 73 | 694 | 3 | 3 | 0 | 0 |

impU | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

stepU | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

imp+U | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

step+U | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

## 6. Discussion

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Baker, S.G.
Comparative Analysis of Biologically Relevant Response Curves in Gene Expression Experiments: Heteromorphy, Heterochrony, and Heterometry. *Microarrays* **2014**, *3*, 39-51.
https://doi.org/10.3390/microarrays3010039

**AMA Style**

Baker SG.
Comparative Analysis of Biologically Relevant Response Curves in Gene Expression Experiments: Heteromorphy, Heterochrony, and Heterometry. *Microarrays*. 2014; 3(1):39-51.
https://doi.org/10.3390/microarrays3010039

**Chicago/Turabian Style**

Baker, Stuart G.
2014. "Comparative Analysis of Biologically Relevant Response Curves in Gene Expression Experiments: Heteromorphy, Heterochrony, and Heterometry" *Microarrays* 3, no. 1: 39-51.
https://doi.org/10.3390/microarrays3010039