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作物学报 ›› 2019, Vol. 45 ›› Issue (1): 144-152.doi: 10.3724/SP.J.1006.2019.81014

• 研究简报 • 上一篇    下一篇

小麦旗叶衰老过程不同数学模型拟合比较及衰老特征分析

吕国锋1,2(),范金平2,张伯桥2,高德荣2,王慧2,刘业宇2,吴素兰2,程凯2,王秀娥1,*()   

  1. 1 南京农业大学, 江苏南京210095
    2 江苏里下河地区农业科学研究所 / 国家小麦改良中心扬州分中心, 江苏扬州225007
  • 收稿日期:2018-02-03 接受日期:2018-10-08 出版日期:2018-11-07 网络出版日期:2018-11-07
  • 通讯作者: 王秀娥
  • 基金资助:
    本研究由江苏省重点研发计划项目(BE2017340);国家重点研发计划项目资助(2017YFD0100800)

Comparison of different mathematical models describing flag leaf senescence process of wheat and characteristics of leaf senescence process

Guo-Feng LYU1,2(),Jin-Ping FAN2,Bo-Qiao ZHANG2,De-Rong GAO2,Hui WANG2,Ye-Yu LIU2,Su-Lan WU2,Kai CHENG2,Xiu-E WANG1,*()   

  1. 1 Nanjing Agricultural University, Nanjing 210095, Jiangsu, China
    2 Institute of Agricultural Sciences for Lixiahe Region of Jiangsu Province / Yangzhou Sub-center of National Wheat Improvement Center, Yangzhou 225007, Jiangsu, China
  • Received:2018-02-03 Accepted:2018-10-08 Published:2018-11-07 Published online:2018-11-07
  • Contact: Xiu-E WANG
  • Supported by:
    This study was supported by the Key Research and Development Programs of Jiangsu Province(BE2017340);the National Key Research and Development Programs of China(2017YFD0100800)

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摘要：

为了准确了解小麦叶片的衰老特征, 筛选适合描述小麦叶片衰老过程的数学模型, 2011年和2012年分别以91个和105个小麦品种(系)为材料, 用“S”型曲线中的Logistic、Gompertz和Richards模型拟合了试验品种叶片的衰老过程, 解析了其叶片衰老特征。结果表明, 试验品种可分为延绿、中等延绿、中等早衰和早衰4种类型。其旗叶衰老过程可分为衰老起始期、快速衰老期和衰老结束期3个阶段, 3个阶段旗叶的衰老速度表现为“慢-快-慢”, 不同延绿类型品种开花后旗叶的绿色叶面积百分比下降主要在衰老过程的中后期。3种模型对不同延绿类型品种旗叶衰老过程均可以拟合, Gompertz和Richards模型拟合度接近, 高于Logistic模型。Gompertz模型的拟合度以早衰>中等早衰>中等延绿>延绿类型。不同延绿类型品种旗叶衰老曲线特征参数达到最大衰老速度时间(TMRS)、平均衰老速度(ARS)和绿色叶面积持续期(GLAD)存在显著差异, TMRS和GLAD以延绿>中等延绿>中等早衰>早衰, ARS以早衰>中等早衰>中等延绿>延绿。Gompertz模型对小麦叶片衰老过程的拟合度优于Logistic模型。叶片衰老过程特征参数可以用于品种延绿性差异评价。

关键词: 小麦, 叶片衰老模型, 非线性拟合, 延绿

Abstract:

To clarity leaf senescence patterns and characteristics of wheat, we used 91 and 105 varieties or advanced lines in 2011 and 2012 respectively, to fit the leaf senescence process by Logistic, Gompertz, and Richards models. The varieties tested were classified into stay green (SG), moderately stay green (MSG), moderately non-stay green (MNSG) and non-stay green (NSG) types. Development of the flag leaf senescence was divided into initial senescence stage, rapid senescence stage and terminal senescence stage in all variety types. The senescence rates of the three stages showed a slow-fast-slow changing pattern. The decrease of green leaf area (% GLA) after anthesis occurred mainly in the middle and late senescence stages. Logistic, Gompertz and Richards models were able to simulate the leaf senescence process of all variety types and the fitness of Richards and Gompertz models was better than that of Logistic model. The fitting goodness of Gompertz model ranked as NSG type > MNSG type > MSG type > SG type. There was a significant difference in time of maximum rate of senescence, green leaf area duration and average rate of senescence derived from Gompertz equation in all varieties. The average rate of senescence showed a trend of NSG type > MNSG type > MSG type > SG type, while the remaining two parameters showed a trend of SG type > MSG type > MNSG type > NSG type. Our results indicate that Gompertz model is more suitable than Logistic model to describe the leaf senescence process of wheat and the characteristic parameters for leaf senescence process can be used to evaluate stay green difference between wheat varieties.

Key words: wheat, leaf senescence model, curve fitting, stay green

表1

Gompertz、Logistic和Richards模型拟合方程的特征参数"

模型
Model		 数学式
Equation		 最大衰老速度时间
TMRS		 最大衰老速度时的%
GLA ymax		 最大衰老速度
MRS		 绿色叶面积持续期
GLAD		 平均衰老速度
ARS
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ $c$ $\frac{a}{\text{e}}$ $\frac{ab}{e}$ $c-\frac{\ln (-\ln 0.01)}{b}$ $\frac{-99a}{\text{GLAD}}$
Logistic $y=\frac{a}{1+{{\text{e}}^{-b(x-c)}}}$ $c$ $\frac{a}{2}$ $\frac{ab}{4}$ $c-\frac{\ln 99}{b}$ $\frac{-99a}{\text{GLAD}}$
Richards $y=\frac{a}{{{[1+{{\text{e}}^{-}}^{b(x-c)}]}^{\frac{1}{d}}}}$ $c-\frac{\lg d}{b}$ $\frac{a}{{{(d+1)}^{\frac{1}{d}}}}$ $\frac{ab}{{{(d+1)}^{(\frac{1}{d}+1)}}}$ $c-\frac{\ln ({{100}^{d}}-1)}{b}$ $\frac{-99a}{\text{GLAD}}$

表2

不同延绿类型品种旗叶的%GLA衰减的过程"

类型
Type		 品种数
Variety number		 DAA10 DAA15 DAA20 DAA25 DAA30
2011 2012 2011 2012 2011 2012 2011 2012 2011 2012 2011 2012
延绿 SG 10 22 99.2 a 96.6 a 98.7 a 91.7 a 95.6 a 82.3 a 89.0 a 56.5 a 53.0 a 11.4 a
中等延绿 MSG 21 32 98.0 ab 96.3 a 95.7 b 90.0 a 91.5 ab 75.1 b 74.5 b 37.8 b 9.8 b 3.7 b
中等早衰 MNSG 28 40 97.5 b 94.6 b 95.0 b 85.8 b 89.7 b 64.4 c 45.9 c 13.2 c 0.8 c 0.7 c
早衰 NSG 32 11 98.3 ab 92.7 c 95.2 b 81.5 c 85.6 c 26.9 d 11.6 d 0.5 d 0 c 0 c

表3

Logistic、Gompertz和Richards模型对不同延绿类型品种拟合结果"

类型
Type		 品种数
No. of varieties		 模型
Model		 方程
Equation		 方程系数Coefficients 模型适合性
Fitness of model
a b c d R2 RMSE
2011
延绿
Stay green		 10 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 98.40 -3.67 1.35 0.9536 4.149
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 98.53 -3.15 1.48 0.9577 3.961
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 99.20 -1.97 6.30 1.57×10-4 0.9426 4.662
中等延绿
Moderately stay green		 21 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 95.59 -4.93 0.93 0.9646 6.436
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.71 -2.95 1.09 0.9710 5.798
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 97.05 -2.62 7.40 6.85×10-8 0.9713 5.858
中等早衰
Moderately non-stay green		 28 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 95.43 -5.39 0.72 0.9846 4.760
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.91 -3.12 0.86 0.9893 3.976
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 97.10 -3.07 5.76 2.98×10-7 0.9887 4.098
早衰
Non-stay green		 32 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 97.39 -7.58 0.33 0.9948 3.167
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 97.47 -5.07 0.50 0.9949 3.124
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 97.36 -3.86 3.54 8.40×10-6 0.9880 4.832
2012
延绿
Stay green		 22 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 94.31 -2.84 0.78 0.9516 7.150
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.29 -1.95 0.98 0.9609 6.424
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 96.29 -2.04 6.01 3.62×10-5 0.9596 6.564
中等延绿
Moderately stay green		 32 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 94.81 -2.79 0.52 0.9735 5.808
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 98.08 -1.78 0.73 0.9797 5.075
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 98.07 -1.78 3.52 7.01×10-3 0.9795 5.118
类型
Type		 品种数
No. of varieties		 模型
Model		 方程
Equation		 方程系数Coefficients 模型适合性
Fitness of model
a b c d R2 RMSE
中等早衰
Moderately non-stay green		 40 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 93.31 -3.42 0.20 0.9809 5.367
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 96.30 -2.12 0.39 0.9840 4.917
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 95.96 -2.16 4.57 1.17×10-4 0.9836 4.992
早衰
Non-stay green		 11 Logistic $y=a/1+{{\text{e}}^{-b(x-c)}}$ 93.46 -4.30 -0.23 0.9916 3.783
Gompertz $y=a{{\text{e}}^{-{{\text{e}}^{-b(x-c)}}}}$ 94.44 -3.17 -0.08 0.9917 3.768
Richards $y=a/{{[1+{{\text{e}}^{-b(x-c)}}]}^{\frac{1}{d}}}$ 94.03 -3.65 0.02 0.50 0.9898 4.216

表4

Gompertz模型对试验品种旗叶衰老过程的拟合度"

年份
Year		 品种数
Number of varieties		 R2均值
Mean of R2		 R2变幅
Range of R2		 RMSE均值
Mean of RMSE		 RMSE变幅
Range of RMSE
2011 91 0.997 0.948-1.000 2.062 0.007-10.640
2012 105 0.996 0.964-1.000 2.546 0-8.750

表5

不同延绿类型品种旗叶3个衰老阶段的参数"

类型
Type		 衰老起始期Initial senescence stage 快速衰老期Rapid senescence stage 衰老结束期Terminal senescence stage
T1 (d) P1 (%) V1 (% d-1) T2 (d) P2 (%) V2 (% d-1) T3 (d) P3 (%) V3 (% d-1)
2011
延绿 SG 28.9 a -31.4 a -1.1 a 5.9 a -60.3 a -11.4 a 1.7 a -6.2 a -4.0 a
中等延绿 MSG 25.8 b -30.8 a -1.2 b 4.5 ab -59.1 a -15.1 a 1.3 ab -6.1 a -5.3 a
中等早衰 MNSG 23.6 c -30.8 a -1.3 c 4.2 b -59.1 a -14.8 a 1.2 b -6.1 a -5.2 a
早衰 NSG 21.9 d -31.0 a -1.4 d 3.4 b -59.6 a -19.5 b 1.0 b -6.2 a -6.9 b
2012
延绿 SG 23.7 a -30.6 a -1.3 a 7.1 ab -58.8 a -9.0 a 2.1 ab -6.0 a -3.1 a
中等延绿 MSG 21.3 b -31.3 a -1.5 b 7.7 a -60.1 a -8.6 a 2.2 a -6.2 a -3.0 a
中等早衰 MNSG 19.7 c -30.5 a -1.6 b 6.3 bc -58.6 a -10.2 a 1.8 bc -6.0 a -3.6 a
早衰 NSG 17.3 d -30.0 a -1.7 c 4.4 c -57.5 a -13.6 b 1.3 c -5.9 a -4.7 b

图1

Gompertz方程对4种延绿类型品种旗叶衰老过程的拟合曲线 g1~g4分别代表延绿、中等延绿、中等早衰和早衰类型Gompertz拟合的曲线; gla_g1 vs t1、gla_g2 vs t2、gla_g3 vs t3和gla_g4 vs t4分别代表这4种延绿类型品种不同日龄下的%GLA。"

表6

不同延绿类型品种的旗叶衰老过程参数"

类型
Type		 TMRS MRS GLAD ARS
2011 2012 2011 2012 2011 2012 2011 2012
延绿SG 32.14 a 26.71 a -16.64 a -13.09 a 37.60 a 32.08 a -2.67 a -3.00 a
中等延绿MSG 27.92 b 24.55 b -24.20 b -11.74 a 31.04 b 31.10 a -3.14 b -3.15 a
中等衰老MNSG 25.27 c 22.63 c -22.53 b -13.90 a 28.57 c 27.65 b -3.38 c -3.42 b
早衰NSG 23.34 d 19.94 d -29.44 c -15.29 b 25.90 d 24.67 c -3.78 d -3.85 c
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