 Methodology
 Open Access
Empiric mathematical model for predicting the content of alphaacids in hop (Humulus lupulus L.) cv. Aurora
 Siniša Srečec^{1}Email author,
 Barbara Čeh^{2},
 Tanja Savić Ciler^{1} and
 Alenka Ferlež Rus^{2}
 Received: 5 November 2012
 Accepted: 6 February 2013
 Published: 19 February 2013
Abstract
The aim of this research is to find a simple mathematical model due to sum of effective temperatures and rainfalls from second germination after spring pruning till the technological maturity of hop cones, in order to achieve reliable prognosis of alphaacids content in hop cv. Aurora. After mathematical analyses of experimental data by Eurequa Formulize 0.96 Beta software 17 equations were offered, and after substituting the values of dependent and independent variables in all equations only one equation was chosen with p = 0.034 (p<0.05). This equation is not reliable in extremely drought year if crop evapotranspiration ET_{0} in July is higher than 4.5, primarily because of negative influence on formation and development of hop glandular trichomes. Considering achieved results it is possible to suggest following general equation for alphaacids accumulation in hop: y = [(k _{1} w) − k _{2} − (k _{3} w ^{2})/x] ÷ (−10) ↔ ET _{0 July } ≤ 4.5. Where y is alpha acids content in dry matter (%), x = sum of effective temperatures and w = sum of rainfals, both from second germination after spring pruning till technological maturity of hop cones. Coefficients k _{ 1 }, k _{ 2 } and k _{ 3 } are determined for cultivar Aurora (53.8, 453 and 1.33, respectively).
Keywords
 Hop
 Humulus lupulus L
 Alphaacids
 Accumulation of alphaacids
 Empiric mathematical model
 Eurequa software
Introduction

– A – content of alphaacids in dry matter (%)

– T – sum of effective temperatures from second germination to hop harvest (°C)

– I – total hours of sun shining from second germination to hop harvest

– F3 – second germination after pruning

– F9 – technological maturity of hop cones (harvest time)

– ET _{ 0(VII)} – average daily reference crop evapotranspiration in July (mm day^{1})
This equation, based on significant analytical data, unfortunately does not allow reliable prognosis of alphaacids accumulation in technological maturity of hop cones.
However, nowadays because of very high hop supply, prognosis of alphaacids accumulation become very important in order to estimate the commodities. Thus, the aim of this research is to find a simple mathematical model due to sum of effective temperatures and rainfall from second germination after spring pruning till the technological maturity of hop cones, in order to achieve reliable prognosis of alphaacids accumulation in hop cv. Aurora.
Materials and methods
Research was carried out on hop cv. Aurora planted in hop garden in Croatia, near the village of Gregurovec (close to Križevci), during the six vegetation years (2001 – 2006, Srečec et al.2008). The soil type of examined hop garden is an eutric pseudogley or eutric podzoluvisol. Content of physiological active phosphorus and potash, analysed by the ALmethod, was medium. Average content of P_{2}O_{5} and K_{2}O during the all six experimental years was 19.1 and 15.4 mg per 100 grams of soil, respectively. Content of humus in soil was very low, only 1.48%. Fertilization was provided on the basis of plant uptake for phosphorus and potash and in three splits of nitrogen (50 + 70 + 50 kg/ha N on 20^{th} May, 10^{th} June and 5^{th} July) in all 6 experimental years. Meteorological data in Croatia were collected in Agro Meteorology Station in Križevci, placed five kilometres far away from the hop garden.
Hop cones were sampled from the same plants each year in the phenological phase of technological maturity.
The empiric mathematical equation achieved in Croatia was checked in year of 2012 at the same cultivar grown in completely different agroecological conditions in Žalec, Slovenia. P_{2}O_{5} and K_{2}O in the hop garden was 46.0 and 20.1 mg per 100 grams of soil, respectively. Content of humus in soil was 2.9%. The soil is middle heavy, young alluvial soil, poorly developed on sandygravelly deposits of two rivers.
Meteorological data in Slovenia for year 2012 were collected in Agro Meteorology Station at the Slovenian Institute of Hop Research and Brewing placed near by the hop garden.

Dmin T  daily minimal temperature,

Dmax T  daily maximal temperature and
5°C is minimal temperature required for beginning of hop vegetation.
The content of alphaacids was determined by the method of lead conductance value of hops, powder and pellets prescribed by Analytica  EBC 7.4 (Anon. 1998) and dry matter content by the method of moisture content of hops and hop products prescribed by Analytica  EBC 7.2 (Anon. 1998). The boarder of repeatability (r_{95}) for method of lead conductance value is 0.2 and the boarder for reproducibility (R_{95}) is1.
During the examined vegetation years in Croatia, samples of hop cones were handpicked and analysed from the same 35 plants, randomly chosen and marked in the first research year of 2001 (Srečec et al. 2008) in the same hop garden, which means five control plots with seven plants per each plot within the same hop garden. On the other hand, samples of hop cones in Slovenia in year of 2012 were also handpicked, but sampled randomly from different plants in the same hop garden.
Analyses of achieved analytical results were provided by software Eureqa Formulize 0.96 Beta (Nutonian, Inc.). Eureqa Formulize is a scientific data mining software package that searches for mathematical patterns hidden in data. Formulize’s user interface is organized as a set of seven tabs that correspond to the normal workflow through the program, and user guide is organized around those tabs (see: http://www.nutonian.com). In mathematical analyses of experimental data sum of effective temperatures and total rainfall were treated as independent variables (marked as x and w variables) and content of alphaacids (%) as dependent variable (y variable).
Results and discussion
Where:
y – alphaacids content in dry matter (%)
x – sum of effective temperatures (°C) from second germination after the spring pruning till technological maturity
w – sum of total raifalls (mm) for the same period
Reliability of mathematical model for accumulation of alphaacids in hop cultivar Aurora in location of Gregurovec, Croatia during the six vegetation periods (20012006*)
Crop year  Sum of effective temperatures (°C)  Sum of total rainfalls (mm)  ET _{ 0 }in July  Calculated content of alphaacids (%) in dry matter  Analysed content of alphaacids (%) in dry matter (mean)  Difference (alpha calc. – alpha analysed) 

2001  1698.4  393.7  4.26  10.9  11.6  0.7 
2002  1932.1  425.8  4.20  11.3  11.1  0.2 
2003  1994,4  175.2  5.5  2.5  6.7   4.2 (n/r) 
2004  1856.7  398.5  4.28  10.2  10.0  0.2 
2005  1920.8  403.3  4.38  10.1  9.7  0.5 
2006  1872.1  382.0  4.42  9.2  9.3  0.1 
Reliability of mathematical model for accumulation of alphaacids in hop cultivar Aurora in location of Žalec, Slovenia in 2012
Crop year  Sum of effective temperatures (°C)  Sum of total rainfalls (mm)  ET _{ 0 }in July  Calculated content of alphaacids (%) in dry matter (mean)  Detected content of alphaacids (%) in dry matter (mean)  Difference (alpha calc. – alpha detect.) 

2012  1766.2  400.4  4.2  10.8  10.2  0.6 
It is obvious that the differences between calculated and detected content of alphaacids varied from 0.1 to 0.7, which is within boarders of repeatability (r_{95} = 0.2) and reproducibility (R_{95} = 1) for EBC 7.4 method (Anon 1998). However, this equation is not reliable in extremely drought year, like the year of 2003 was. That confirms results of Srečec et al. (2008), who found a negative correlation, determined by Spearman’s rank correlation, during the phonological phase of hop cones formation, between average daily reference crop evapotranspiration (ET_{0}) in July and yield of hop cones, r_{ s } = − 0.75 (p < 0.05), as well as between ET_{0} and yield of alphaacids in the same period, r_{ s } = − 0.88 (p < 0.05). This is also possible to explain with results of Pavlovič et al. (2012), according to them, rainfall quantity from June 18 to July 22 shows the highest correlation with alphaacid contents and impact of rainfall begins to decline after July 29. However, in time after July 29, the formation of glandular trichomes starts and the positive Spearman’s rank correlations were found between the average number of glandular trichomes and the content of alphaacids (r_{ s } = 0.90; p<0.05) and also between the average volume of glandular trichomes and content of alphaacids (r_{ s } = 0.97; p<0.05) (Srečec et al. 2011).
In case of Aurora hop cultivar coefficients k _{ 1 }, k _{ 2 } and k _{ 3 } are determined, which have to be determined for the other hop cultivars.
Conclusion
Achieved results confirms the results of Srečec et al. (2008) and using the Eureqa Formulize 0.96 Beta software allows reliable mathematical analyses but only if linear and multiple correlations of experimental data are previously provided. However, these results as well as results of previous authors show that weather conditions, during the hop vegetation have a stronger influence on accumulation of alphaacids in technological maturity of hop cones than soil conditions.
Declarations
Authors’ Affiliations
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