# Back to the Future: Revisiting the Application of an Enzyme Kinetic Equation to Maize Development Nearly Four Decades Later

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{t}) as a function of absolute temperature (T):

_{A}= entropy of activation of the reaction, (19.49 cal/mol/°K), ΔH

_{A}= enthalpy of activation of the reaction, (9170 cal/mol), R = real gas constant, (1.9858775 cal/deg/mol), ΔS

_{L}= entropy for the low-temperature denaturation, −95.70, ΔH

_{L}= enthalpy for the low-temperature denaturation, (27,794 cal/mol), ΔS

_{H}= entropy for the high-temperature denaturation, (211.8 cal/mol/°K), ΔH

_{H}= enthalpy for the high-temperature denaturation, (65,203 cal/mol).

_{t}) is a unitless value that varies from 0.0 at zero-development to 1.0 at the optimum temperature. This was the first time both high- and low-temperature denaturation were taken into account in one equation. These values for the parameters were the result of Sharpe and DeMichele fitting the equation to Lehenbauer’s [10] constant temperature maize shoot elongation data. Tonnang et al. [2] included this enzyme kinetic equation in their comparisons with maize development in Africa.

_{b}and T

_{m}. To obtain the decay at high temperatures, a square root was included to allow a high slope when the temperature values approached T

_{m}. By combining the products of different powers of temperature, an inflection point occurs, yielding the Briere_1 model as

_{mean}– T

_{base}if T

_{mean}is not above T

_{optimum},

_{optimum}− T

_{base}) × (1.0 − (T

_{mean}− T

_{optimum})/(T

_{high}− T

_{optimum}) if T

_{mean}is above T

_{optimum},

_{mean}is the mean daily temperature, T

_{base}is the base temperature (8 °C), T

_{optimum}is the optimum temperature, and T

_{high}is the high temperature where the development is zero, with all temperatures in °C.

## 2. Materials and Methods

_{i}and Ŷ

_{i}are the observed and predicted values, respectively.

_{j}_AICc, were also used to determine the level of support for each model [22]. The AICc with the minimum value, denoted by AICc_min, refers to the best model. The Δ

_{j}_AICc for the jth model is the difference between the AICc

_{j}and AICc_min, shown as follows:

_{j}_AICc is

Δ_{j}_AICc | Level of support of Model j |

0–2 | There is substantial evidence that supports the model j. |

4–7 | The Model j has considerably less support. |

>10 | The Model j has essentially no support. |

_{j}, were used as a measure of the strength of evidence for each model; the Akaike weights count the probability that model j is the best among the set of models [22]. They are the ratio of each model‘s AICc differences relative to the sum of the AICc differences for all models, shown as follows:

_{j}_AICc, and w

_{j}for the Briere model were obtained and derived from the values reported in the original paper [2].

## 3. Results and Discussion

_{j}_AICc over all maize varieties was 0.05, which was lower than the heat stress and Poikilotherm equations. This indicates that the heat stress equations fitted separately for each variety were not superior to the heat stress equation with the same set of parameters for all varieties. Thus, separate fitting of such an equation may not be necessary for a large number of varieties.

_{j}_AICc over all maize varieties was 26, which means that the Briere_1 equation was unlikely to be the best model. It is apparent that the heat stress equation was closer to the measured values for the highest temperature data point, for the lower temperature data point, and for the data point with the greatest development rate.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Poikilotherm equation with original parameters from Lehenbauer’s maize coleoptile elongation rate data, applied to the first variety (VE 201). RMSE indicates the root-mean-square error for the maize variety’s development rate.

**Figure 3.**Heat stress regressions (forced through a base temp. of 8 °C) fit to the first variety (VE 201) and then applied to all the varieties.

**Figure 4.**Regression fit to the data for each variety separately. These results are for the equation without the highest temperature data point.

**Table 1.**Estimated parameters of Briere—1 and Briere—2 models for eight maize varieties (Tonnang et al. [2]).

Vriety | Tb (°C) | Tm (°C) | a | |
---|---|---|---|---|

VE 201 | 8.98 | 40.11 | 0.0003 | |

VE 203 | 10.14 | 10.14 | 0.0003 | |

VE 206 | 9.51 | 38.49 | 0.0003 | |

VE 208 | 8.57 | 38.587 | 0.0003 | |

VE 210 | 9.70 | 39.12 | 0.0003 | |

VE 212 | 9.03 | 37.24 | 0.0003 | |

VE 218 | 8.11 | 40.06 | 0.0029 | |

Variety | μ | Tb (°C) | Tm (°C) | a |

VE 220 | 1.03 | 10.03 | 40.43 | 0.00004 |

**Table 2.**Results with root-mean-square error (RMSE) for the published equations of Tonnang et al. [2] (Briere), with the Poikilotherm (Pk) equation [11] fit as described herein, for the heat stress equation fitted for just the first variety (Htstr), and for the heat stress fitted separately for each variety (Indiv fitted); also shown are Akaike’s Information Criterion (AIC), AICc, AICc differences (Δ

_{j}AICc), and the derived Akaike weights (ω

_{j}) computed from the data. The Tbase is the x intercept for each equation that was fitted for each variety, as discussed herein. The last variety (VE 220) was not included in these analyses as discussed in this paper. K indicates the number of parameters that were used to calculate the maize development rate. The AICc-selected best model is shown here in bold.

Variety | Model | K | RMSE | AIC | AICc | Δ_{j}AICc | ω_{j} |
---|---|---|---|---|---|---|---|

VE 201 | Htstr | 2 | 0.003 | −94.779 | −92.379 | 0.000 | 0.720 |

Indiv fitted | 3 | 0.003 | −93.671 | −87.671 | 4.708 | 0.068 | |

PK | 2 | 0.003 | −92.327 | −89.927 | 2.451 | 0.211 | |

Brierea ^{a} | 3 | 0.001 | −74.339 | −68.339 | 24.039 | 0.000 | |

VE 203 | Htstr | 2 | 0.003 | −91.961 | −89.561 | 0.000 | 0.738 |

Indiv fitted | 3 | 0.003 | −91.320 | −85.320 | 4.241 | 0.088 | |

PK | 2 | 0.004 | −89.073 | −86.673 | 2.888 | 0.174 | |

Brierea ^{a} | 3 | 0.002 | −68.789 | −62.789 | 26.772 | 0.000 | |

VE 206 | Htstr | 2 | 0.003 | −93.860 | −91.460 | 0.000 | 0.885 |

Indiv fitted | 3 | 0.003 | −90.808 | −84.808 | 6.652 | 0.032 | |

PK | 2 | 0.004 | −89.120 | −86.720 | 4.740 | 0.083 | |

Brierea ^{a} | 3 | 0.002 | −71.788 | −65.788 | 25.672 | 0.000 | |

VE 208 | Htstr | 2 | 0.003 | −93.433 | −91.033 | 0.000 | 0.773 |

Indiv fitted | 3 | 0.003 | −91.182 | −85.182 | 5.851 | 0.041 | |

PK | 2 | 0.000 | −90.579 | −88.179 | 2.854 | 0.186 | |

Brierea ^{a} | 3 | 0.002 | −70.744 | −64.744 | 26.289 | 0.000 | |

VE 210 | Htstr | 2 | 0.003 | −92.947 | −89.947 | 0.000 | 0.819 |

Indiv fitted | 3 | 0.003 | −91.572 | −83.572 | 6.375 | 0.034 | |

PK | 2 | 0.004 | −89.507 | −86.507 | 3.439 | 0.147 | |

Brierea ^{a} | 3 | 0.002 | −70.381 | −64.381 | 25.566 | 0.000 | |

VE 212 | Htstr | 2 | 0.003 | −94.621 | −92.221 | 0.000 | 0.938 |

Indiv fitted | 3 | 0.003 | −91.415 | −85.415 | 6.805 | 0.031 | |

PK | 2 | 0.004 | −87.810 | −85.410 | 6.810 | 0.031 | |

Brierea ^{a} | 3 | 0.002 | −72.364 | −66.364 | 25.856 | 0.000 | |

VE 218 | Htstr | 2 | 0.003 | −93.067 | −90.667 | 0.000 | 0.689 |

Indiv fitted | 3 | 0.004 | −89.953 | −83.953 | 6.714 | 0.024 | |

PK | 2 | 0.003 | −91.313 | −88.913 | 1.754 | 0.287 | |

Brierea ^{a} | 3 | 0.002 | −70.996 | −64.996 | 25.671 | 0.000 |

^{a}K, RMSE, and AIC values for Briere equation were obtained from Tonnang et al. [2], and values of AICc, Δ

_{j}AICc, and ω

_{j}were computed based on the published values.

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**MDPI and ACS Style**

Kiniry, J.R.; Kim, S.; Tonnang, H.E.Z. Back to the Future: Revisiting the Application of an Enzyme Kinetic Equation to Maize Development Nearly Four Decades Later. *Agronomy* **2019**, *9*, 566.
https://doi.org/10.3390/agronomy9090566

**AMA Style**

Kiniry JR, Kim S, Tonnang HEZ. Back to the Future: Revisiting the Application of an Enzyme Kinetic Equation to Maize Development Nearly Four Decades Later. *Agronomy*. 2019; 9(9):566.
https://doi.org/10.3390/agronomy9090566

**Chicago/Turabian Style**

Kiniry, James R., Sumin Kim, and Henri E. Z. Tonnang. 2019. "Back to the Future: Revisiting the Application of an Enzyme Kinetic Equation to Maize Development Nearly Four Decades Later" *Agronomy* 9, no. 9: 566.
https://doi.org/10.3390/agronomy9090566