Accumulation of Biomass and Mineral Elements with Calendar Time by Corn: Application of the Expanded Growth Model

Transcription

1 Accumulation of Biomass and Mineral Elements with Calendar Time by Corn: Alication of the Exanded Growth Model Allen R. Overman, Richard V. Scholtz III* Agricultural & Biological Engineering Deartment, University of Florida, Gainesville, Florida, United States of America Abstract The exanded growth model is develoed to describe accumulation of lant biomass (Mg ha 21 ) and mineral elements (kg ha 21 ) in with calendar time (wk). Accumulation of lant biomass with calendar time occurs as a result of hotosynthesis for green land-based lants. A corresonding accumulation of mineral elements such as nitrogen, hoshorus, and otassium occurs from the soil through lant roots. In this analysis, the exanded growth model is tested against high quality, ublished data on corn (Zea mays L.) growth. Data from a field study in South Carolina was used to evaluate the alication of the model, where the lanting time of Aril 2 in the field study maximized the cature of solar energy for biomass roduction. The growth model redicts a simle linear relationshi between biomass yield and the growth quantifier, which is confirmed with the data. The growth quantifier incororates the unit rocesses of distribution of solar energy which drives biomass accumulation by hotosynthesis, artitioning of biomass between light-gathering and structural comonents of the lants, and an aging function. A hyerbolic relationshi between lant nutrient utake and biomass yield is assumed, and is confirmed for the mineral elements nitrogen (N), hoshorus (P), and otassium (K). It is concluded that the rate limiting rocess in the system is biomass accumulation by hotosynthesis and that nutrient accumulation occurs in virtual equilibrium with biomass accumulation. Citation: Overman AR, Scholtz RV III (2011) Accumulation of Biomass and Mineral Elements with Calendar Time by Corn: Alication of the Exanded Growth Model. PLoS ONE 6(12): e doi: /journal.one Editor: Randall P. Niedz, United States Deartment of Agriculture, United States of America Received July 25, 2011; Acceted November 9, 2011; Published December 14, 2011 Coyright: ß 2011 Overman, Scholtz. This is an oen-access article distributed under the terms of the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original author and source are credited. Funding: This analysis was funded by the Florida Agricultural Exeriment Station. The funders had no role in study design, data collection and analysis, decision to ublish, or rearation of the manuscrit. Cometing Interests: The authors have declared that no cometing interests exist. * rscholtz@ufl.edu Introduction In a recent ublication the authors discussed a model of yield resonse of corn to lant oulation and absortion of solar energy within the lant canoy [1]. Data from three field studies formed the emirical foundation for the mathematical model. The simle exonential model contained two arameters: one for uer limit on yield at high lant oulation and an exonential resonse coefficient. The model described the data very well and exhibited similarities among the three studies. In a textbook the authors have discussed various asects of cro growth and yield [2], including a mathematical model of cro growth with calendar time. The exanded growth model incororates the three basic rocesses of an energy driving function, artitioning of biomass between light-gathering and structural comonents of the lants, and an aging function. This model is used in the resent analysis. A simlified theory of biomass roduction by hotosynthesis has been ublished by the authors [3]. The theory incororates basic rinciles from mathematics and hysics and uses data from the literature for a warm-season erennial grass as an emirical base. The strategy follows the rocedure of emergence as described by Robert Laughlin [4] which means that develoment of the theory is guided by measurement and observation. This aroach examines behavior of a large assemblage of matter, in contrast to the classical reductionist aroach which breaks a system into its smallest arts and then describes interactions among the arts. Data from a field study at Florence, SC, USA are used to evaluate alication of the model [5] for corn (Zea mays L.). Many field studies have been conducted on the growth of corn, noted as examles in references [6 11]. Key measurements of accumulation of lant biomass as well as the mineral elements nitrogen, hoshorus, and otassium were established in the 1948 study by Sayre [6]. Effect of alied N and P fertilization on biomass accumulation was evaluated by Bar-osef and Kafkafi [7]. Interactions of lant oulation and alied N on biomass accumulation were measured by Rhoads and Stanley [8]. Deendence of yield on measured evaotransiration for three different soils was reorted by Tolk and Howell [9]. An exonential relationshi was clearly demonstrated. The resent article is not intended as a general literature review of either mathematical models or field studies on cro growth. It describes concets and rocedures for the exanded growth model on high quality data from a field study with growth of corn. Methods The first ste is to define relevant quantities (variables and model arameters): t is calendar time (referenced to Jan. 1), wk; is biomass yield (dry matter), Mg ha 21 ; N u is lant nutrient utake (N, P, or K), kg ha 21 ; N c = N u / is lant nutrient concentration (N, P, or K), gkg 21. A common reference time is used to facilitate comarison among various studies. The second ste is to utilize a mathematical PLoS ONE 1 December 2011 Volume 6 Issue 12 e28515

2 model which relates biomass accumulation to calendar time. For this urose we adot the exanded growth model discussed in Section 3.5 of Overman and Scholtz [2], which can be written as ~A Q where A is a yield factor, Mg ha 21 ;andq is a dimensionless growth quantifier, defined by Q~ ð1{kx i ffiffiffi ex 2 scxi Þ½erf x{erf x i Š{ k ffiffiffi ð1þ ex {x 2 {ex {x 2 : i ð2þ in which the dimensionless time variable x is defined by x~ t{m ffiffi 2 sc ffiffiffi z ð3þ 2 s 2 with the arameters in Eq. (3) defined as m time to the mean of the solar energy distribution (referenced to Jan. 1 for the northern ffiffi hemishere), wk; 2 s the time sread of the solar energy distribution, wk; k the artition coefficient between light-gathering and structural comonents of the lants, and c an aging coefficient for the lant secies, wk 21. It follows that x i corresonds to the time of initiation of significant lant growth t i, wk. These arameters are discussed in more detail in the next section on alication to the corn study at Florence, SC,USA.The errorfunction,erfx, in Eq. (2) is defined by erf x~ 2 ffiffiffi ð x 0 ex {u 2 du ð4þ with u as the variable of integration for the Gaussian distribution ex {u 2.Valuesoftheerfx can be obtained from a handbook of mathematical functions (see Table 7.1 [12]). Examination of data of couling between lant nutrient accumulation N u and lant biomass leads to the hyerbolic hase relation N u ~ N um K y z with N um as otential maximum lant nutrient accumulation at high and K y is the value of at which N u = N um /2. This subject is exlored in Table 1. Accumulation of biomass and mineral elements by corn at Florence, SC, USA. ð5þ more detail in the next section of alication to data from Florence, SC, USA. Results Data for this analysis are adated from a field study by D.L. Karlen and associates [5] with Pioneer 3382 corn on Norfolk loamy fine sand (fine-loamy, siliceous, thermic Tyic Paleudult) at the USDA-ARS Coastal Plains Soil, Water, and Plant Research Center at Florence, SC, USA. Data for 1982 and lant oulation density of 7 lants m 22 are used here. Planting date was Aril 2 (t = 15.0 wk). Fertilizer alication was N-P-K = kg ha 21. Data are given in Table 1 for each samling time for calendar time t, biomass yield, and lant nutrient utake and lant nutrient concentration for nitrogen, hoshorus, and otassium. Analysis of data from various ffiffi studies has led to arameter estimates: m~26:0 wk, 2 s~8:00 wk, k~5, c~ 0:20 wk {1. Now by varying the time of initiation, x i, it can be shown that maximum utilization of solar energy is obtained for x i = 0. This choice of the arameters leads to x~ t{m ffiffi 2 sc ffiffi z ~ t{26:0 2 s 2 8:00 z (8:00)(0:20) ~ t{19:6 2 8:00 It follows from Eq. (6) that x i = 0 corresonds to t i = 19.6 wk. Aarently a time interval of 4.6 weeks is required for germination of seeds and develoment of corn lants to reach otimum cature of solar energy and significant lant growth. Note that the effect of the aging function is to shift reference time from 26.0 wk to 19.6 wk. The growth quantifier equation becomes Q~ ð1{kx i Þ½erf x{erf x i Š{ k ffiffiffi ffiffi ex 2 scxi ~ ½erf x{0š{2:821 ex {x 2 {1 ð6þ ex {x 2 {ex {x 2 : i Values adated from the exeriment are listed in Table 2. Linear regression of biomass yield on the growth quantifier leads to ð7þ ^ ~0:290z7:274 Q r~0:9973 ð8þ Table 2. Correlation of biomass accumulation () with the growth quantifier (Q) for corn at Florence, SC, USA. t x erf x ex (2x 2 ) Q T N u N c P u P c K u K c wk Mg ha 21 kg ha 21 gkg 21 kg ha 21 gkg 21 kg ha 21 gkg lanting (Aril 2) doi: /journal.one t001 wk Mg ha doi: /journal.one t002 PLoS ONE 2 December 2011 Volume 6 Issue 12 e28515

3 Figure 1. Correlation of biomass yield () with growth quantifier (Q). Biomass yield data for corn at USDA research center at Florence, SC, USA [5]. Line is drawn from Eq. (8). doi: /journal.one g001 where ^ signifies an estimator of biomass yield. The quality of the correlation is confirmed in Figure 1. Equation (8) is in agreement with Eq. (1), the simle linear model. The next ste is to Figure 3. Correlation of lant biomass to hoshorus utake ratio (A) and of lant hoshorus utake (B) with biomass yield. Cro data for corn at USDA research center at Florence, SC, USA [5]. Line is drawn from Eq. (12). Curve is drawn from Eq. (13). doi: /journal.one g003 exlore the couling between accumulation of lant nutrients and lant biomass. Equation (5) can be rearranged to the linear form N u ~ K y N um z 1 N um ð9þ Data for nitrogen, hoshorus, and otassium are now used to test the validity of Eq. (9). The value of /N u corresonding to each value of is calculated from Table 1. Linear regression then leads to Nitrogen: N u ~0:0251z0:00367 r~0:9967 ð10þ ^N u ~ 273 6:84z ð11þ Phoshorus: Figure 2. Correlation of lant biomass to nitrogen utake ratio (A) and of lant nitrogen utake (B) with biomass yield. Cro data for corn at USDA research center at Florence, SC, USA [5]. Line is drawn from Eq. (10). Curve is drawn from Eq. (11). doi: /journal.one g002 P u ~0:269z0:0172 r~0:9965 ð12þ ^P u ~ 58:1 15:6z ð13þ PLoS ONE 3 December 2011 Volume 6 Issue 12 e28515

4 Figure 4. Correlation of lant biomass to otassium utake ratio (A) and of lant otassium utake (B) with biomass yield. Cro data for corn at USDA research center at Florence, SC, USA [5]. Line is drawn from Eq. (14). Curve is drawn from Eq. (15). doi: /journal.one g004 Potassium: K u ~0:00635z0:00316 r~0:9894 ð14þ ^K u ~ 317 ð15þ 2:01z Results are shown grahically in Figures 2, 3, and 4 for nitrogen, hoshorus, and otassium, resectively. The high correlations confirm the validity of the hase relations for this study. Discussion The next ste is to rovide simulation of biomass () and lant nitrogen (N u ) with calendar time (t). Accumulation of the growth quantifier (Q) with calendar time follows from Eq. (7). Couling of biomass yield with growth quantifier follows from Eq. (8). Couling of lant nitrogen with biomass yield follows from Eq. (11). Couling of lant nitrogen concentration is then defined by N c = N u /. Simulation curves are shown in Figure 5 along with values from the exeriment. Close agreement between the estimated and measured values should be noted. The decline in lant nitrogen concentration with calendar time is exlained by the shift from dominance of light-gathering (leaf) fraction with higher lant N in young lants toward dominance of structural (stalk) fraction with lower lant N in older lants. In cases where Figure 5. Accumulation of biomass yield (A), lant nitrogen utake (B), and lant nitrogen concentration (C) with calendar time. Cro data for corn at USDA research center at Florence, SC, USA [5]. Curves are drawn from Eqs. (7), (8), and (11). (A). doi: /journal.one g005 correlations are much lower than obtained in this analysis could signify either large scatter in the data and/or that the linear relationshi is not valid. The high efficiency of nitrogen utilization by the lants in this study may be noted. Potential nitrogen utake from Eq. (11) is 273 kg ha 21 for alied nitrogen of 268 kg ha 21, for an efficiency ratio of 273/268 = Numerous factors influence the value of the yield factor A. These include lant oulation, level of alied nutrients, water availability (such as rainfall or irrigation), and frequency of harvest (for erennial grasses). Some of these interactions have been detailed in Overman and Scholtz [1&2]. We can now interret the meaning of this analysis. Both and N u are accumulating with calendar time and therefore reresent rate rocesses. The rate limiting rocess in the system is biomass accumulation by hotosynthesis. Phase relations (Eqs. (11), (13), and (15)) imly that accumulation of the mineral elements (N, P, and K) occur in virtual equilibrium with biomass accumulation. This conclusion is suorted by the simlified theory of biomass roduction [3]. An excellent discussion of hotosynthesis has been PLoS ONE 4 December 2011 Volume 6 Issue 12 e28515

5 resented by Morton [13], with emhasis on what has been learned and what remains as oen questions. The exanded growth model is derived via classical methods. By making simlifying assumtions, an analytical solution can be found from linear differential equations that are based on key fundamental rocesses. Thus, the exanded growth model eliminates the need to use comuter algorithms to solve the inherent comlexity, and along with the nutrient accumulation hase relationshi has been shown to closely agree with the data resented by Karlen et al. [5]. It is suggested that this rocedure should be tested for other cases, including other cro secies and exerimental conditions, and other mineral elements (such as Ca, References 1. Overman AR, Scholtz RV (2011) Model of ield Resonse of Corn to Plant Poulation and Absortion of Solar Energy. PLoS One 6(1): e doi: /journal.one Overman AR, Scholtz RV (2002) Mathematical Models of Cro Growth and ield. New ork: Taylor and Francis Overman AR, Scholtz RV (2010) A Memoir on A Simlified Theory of Biomass Production by Photosynthesis. University of Florida. Gainesville FL Laughlin RB (2005) A Different Universe: Reinventing Physics from the Bottom Down. Cambridge MA: Basic Books Karlen DL, Sadler EJ, Cam CR (1987) Dry matter, nitrogen, hoshorus, and otassium accumulation rates by corn on Norfolk loamy sand. Agronomy J 79: Sayre JD (1948) Mineral accumulation in corn. Plant Physiology 23: Bar-osef B, Kafkafi U (1972) Rates of growth and nutrient utake of irrigated corn as affected by N and P fertilization. Soil Sci Soc Amer Proc 36: Mg, etc.). Some data suggest that the rocedure utilizing nutrient accumulation hase relationshi does aly for Ca and Mg [11]. The authors lan to examine the yields of light-gathering and structural comonents and the effect on nutrient accumulation using the exanded growth model in a future ublication. Author Contributions Conceived and designed the exeriments: AO. Analyzed the data: AO RS. Contributed reagents/materials/analysis tools: AO RS. Wrote the aer: AO. Reviewed and reared the manuscrit for ublication, this includes grahs and tables: RS. 8. Rhoads FM, Stanley RL (1979) Effect of oulation and fertility on nutrient utake and yield comonents of irrigated corn. Soil Cro Sci Soc Fla Proc 38: Mutti LSM (1984) Dynamics of water and nitrogen stresses on corn growth, yield, and nutrient utake. PhD Dissertation. University of Florida, Gainesville, FL Tolk JA, Howell TA, Evett SR (1998) Evaotransiration and yield of corn grown on three high lains soils. Agronomy J 90: Overman AR, Scholtz RV (1999) Model for accumulation of dry matter and lant nutrients by corn. Commun Soil Sci Plant Anal 30: Abramowitz M, Stegun IA (1965) Handbook of Mathematical Functions. New ork: Dover Publications Morton O (2007) Eating the Sun: How Plants Power the Planet. London: HarerCollins Publishers PLoS ONE 5 December 2011 Volume 6 Issue 12 e28515