Measurement and Modeling of Respiration Rate of Guava (CV. Baruipur) for Modified Atmosphere Packaging

Abstract

Several experiments were conducted at different storage temperatures for generating respiration data using close system method for respiration. A respiration rate model, based on enzyme kinetics and the Arrhenius equation was proposed for predicting the respiration rates of Guava as a function of O₂ and CO₂ concentrations and storage temperature. Temperature was found to influence the model parameters. In this model, the dependence of respiration rate on O₂ and CO₂ was found to follow the uncompetitive inhibition. The enzyme kinetic model parameters, calculated from the respiration rate at different O₂ and CO₂ concentration were used to fit the Arrhenius equation against different storage temperature. The activation energy and respiration pre-exponential factor were used to predict the model parameters of enzyme kinetics at any storage temperature between 0–30°C. The developed models were tested for its validity at 12°C and it was found to be in good agreement (the mean relative deviation moduli between the predicted and experimental respiration rates were found to be 8.95% and 8.02% for O₂ consumption and CO₂ evolution, respectively) with the experimentally estimated respiration rates.

Keywords:

• Guava
• Respiration rate
• Enzyme kinetics
• Regression coefficient
• Modeling

INTRODUCTION

Guava (Psidium guajava L.) is an important tropical fruits and is native to the tropical America stretching from Mexico to Peru.[1,2] Guava claims superiority over other fruits by virtue of its commercial and nutritional values. It is considered as common man's fruits and is rightly called as the apple of the tropics.[1] Although the guava plant was domesticated more than 2000 years ago, it was not until 1526 when the first commercial cultivation of guava was reported in the Caribbean islands. Later it was spread by explorers into the Philippines and India. From there it quickly spread to most of the tropical and subtropical regions of the world.[3] It has been adopted in India so well that it appears to be an Indian fruit.[2] The world production of Guava is estimated to be 5.0 MT. India leads the world in Guava production with annual production of 1.80 MT with the production area of 0.15 Mha.[4] Guava is the fourth most widely grown fruit crop in India. The popular varieties of guava grown in India are Allahabad Safeda, Lucknoe-49, Baraipur, Nagpur Seedless, Dharwar, etc.[4] In India, Bihar is the leading state in guava production with 0.30 MT followed by Andhra Pradesh, and Uttar Pradesh. The other states in India, where guava is widely grown are Gujarat, Karnataka, Punjab, and Tamil Nadu.

Guava is an important fruit crop of India and has gained considerable importance because of its high nutritive value, availability at moderate price, a pleasant aroma, good flavor, delicious taste, and remunerative nature of crops. Guava is one of the most common fruit liked by both the rich and the poor.[5] Guava fruits consist of about 20% peel, 50% fleshy portion, and 30% seed core. The physicochemical characteristics of fruits change significantly with maturity.[6,7] Guava fruit usually contains 74–87% water, 13–26% dry matter, 0.8–1.5% protein, 0.4–0.7% fat, 2–7.2% crude fiber, 2.45% acid, 4.45% reducing sugars, 5.23% non-reducing sugars, 9.73 °brix, 0.5–1.0% ash and 260 mg vitamin C/100 g fruit. The compositions differ with the cultivar, stage of maturity and season.[1,5,8] Ripe guava fruit is an excellent source of ascorbic acid (vitamin C), dietary fiber and pectin. Guava is also a good source of vitamin A, phosphorus, calcium and iron as well as thiamine, niacin, riboflavin and carotene.[3] The total pectin content of Guava fruit is in the range of 0.5–1.8%. Reportedly, consumption of guava fruit reduces serum cholesterol levels, triglycerides and hypertension while increasing the level of good cholesterol (high density lipoprotein).[9] Both ripe and green fruits, the root, leaves and bark are used throughout the guava growing region in local medicine for treating gastroenteritis, diarrhea and dysentery.[3,10,11] The fruit, exuding a strong, sweet, musky odor when ripe, may be round, ovate, or pear-shaped, 4–10 cm long, with 4 or 5 protruding floral remnants (sepals) at the apex; and thin, light-yellow skin, frequently blushed with pink. Next to the skin is a layer of somewhat granular flesh, 3–12.5-mm thick, white, yellowish, light- or dark-pink, or near-red, juicy, acid, subacid, or sweet and flavorful. The central pulp, concolorous or slightly darker in tone, is juicy and normally filled with very hard, yellowish seeds, 3-mm long, though some rare types have soft, chewable seeds. The average diameter of fruit varies from 4–8 cm and the weight may range from 50–500 g. Actual seed counts have ranged from 112 to 535 but some guavas are seedless or very limited in numbers.[9,10,12]

The fruit is relished when mature or ripe and freshly plucked from the tree. Guava fruits are used for both, fresh consumption and processing. It is eaten as fresh fruit when ripe, used in sauce and chutney, or cooked as a vegetable when green. It is also processed into a variety of products such as jam, jelly, cheese, toffee, juice, squash, wine, dried fruits, canned fruits, as well as flavoring for other foods.[1,10,12] It excels most other fruit trees in productivity, hardiness, adaptability and vitamin-C content. Besides its high nutritive value, it bears heavy crop every year and gives good economic returns involving very little input.[12] This has prompted several farmers to take up guava cultivation on a commercial scale.

The main objective of storage of perishable commodities such as guava is to prolong their availability on the market throughout year. Certain post harvest constraints like short shelf life, chilling sensitivity, and susceptibility to diseases limit its long duration storage and transportation. Many researchers have studied the influence of altered atmosphere on the commodity storage life. Two main types of altered atmosphere are modified and controlled atmosphere (MA and CA). The use of these methods provide some control of fruit and vegetable ripening, retardation of senescence, or browning in cut produce[13,14] and ultimately prolongs the shelf life. Depending on the specific requirements, Modified Atmosphere (MA) and Control Atmosphere (CA) can prolong market availability of the produce up to the next harvest.

Storage operations can be classified with respect to construction as O₂ control systems, CO₂ control systems, temperature and relative humidity control systems, ethanol vapor generators, and ethylene absorbent.[15,16] In CA storage, the atmosphere is modified and its composition is precisely adjusted according to the specific requirements throughout the storage period.[17–19] Modified atmosphere packaging (MAP) of fresh produce relies on the modification of the atmosphere inside the package, achieved by the natural interplay between the respiration of the product and the transfer of the gases through the packaging. This atmosphere can potentially reduce respiration rates, ethylene sensitivity and production, decay and physiological changes, namely oxidation.[20,21] MAP is used when the composition of the atmosphere is not closely controlled, whereas CA monitors and controls the gaseous composition more precisely. Hence, the need was realized on the post harvest management of guava fruit in medium/hi-tech way like CA and MA storage. However, accurate measurement of respiration rates and modeling is pivotal to the design of MA/CA for agricultural commodity.[17,20,22]

Respiration is a metabolic process that provides the energy for plant biochemical processes. Various substrates used in important synthetic metabolic pathways in the plant are formed during respiration.[23] Aerobic respiration consists of oxidative breakdown of organic reserves to simpler molecules, including carbon dioxide (CO₂) and water, with release of energy. The organic substrates broken down in this process may include carbohydrates, lipids, and organic acids. The process consumes oxygen (O₂) in a series of enzymatic reactions. Due to these metabolic processes in fruits, which continue even after harvest, the shelf life of the product is reduced. However, the rates of these metabolic processes are generally directly proportional to the storage temperature, thus increasing the shelf life to a certain level at lower temperatures.

Measurement of respiration rate of produce at a particular storage temperature and gas compositions is time consuming and needs special equipments for gas analysis. Various mathematical models have been developed to correlate the respiration rate with different storage parameters such as gas composition, i.e., O₂ and CO₂ and temperature. Yang and Chinnan[24] used a quadratic function to correlate the respiration rates of tomato with O₂ and CO₂ concentrations and storage time. The effect of temperature on respiration rate was not considered in their models. The models adequately predicted the respiration rate of tomatoes stored in a controlled atmosphere. Lee et al.[25] suggested an enzyme kinetics theory for modeling of respiration rate. Cameron et al.[26] used an empirical approach to measure the respiration rate as a function of O₂ concentration by observing the rate of O₂ depletion in a closed system. Talasila et al.[27] developed a non-linear empirical model for predicting the respiration rate of strawberry as a function of temperature, O₂ and CO₂ concentration. Most of the model have not incorporated one or other dependent factors such as O₂, CO₂, temperature and time, and hence, are not flexible enough to predict the respiration rate at various storage conditions.

The recent approach for modeling respiration rates by using Michaelis-Menten type equation is based on enzyme kinetics.[25] This provides a simple description of respiration, based on the assumption that diffusion and solubility of O₂ and CO₂ in plant tissue regulates reactions catalyzed by enzymes. For agricultural commodity, both O₂ and CO₂ concentrations have an influence on quality and shelf life. At high CO₂ concentrations, quality loss could be reduced for products such as broccoli,[28] pear[29] and mushrooms.[30] High carbon dioxide concentrations can reduce the oxygen consumption rate of a number of fruits and vegetables. The role of CO₂ in respiration is suggested to be mediated via inhibition mechanism of the Michaelis-Menen equations. This can be modeled by four types of inhibition in an enzyme kinetics model: (1) the competitive type; (2) the uncompetitive type; (3) a combination of competitive and uncompetitive types; and (4) the non-competitive type. All these models describe the inhibitory effects of CO₂ on respiration rate.[31] These inhibition models were tested for describing the CO₂ influence on O₂ consumption using experimental data supplemented with data from literature.[31] They found the clear influence of high CO₂ on the gas exchange rates of some produce and estimated the O₂ consumption rate using the inhibition models. This supports the use of Michaelis-Menten kinetics for modeling O₂ consumption and CO₂ evolution rate. Peppelenbos and Leven[31] suggested the simultaneous existence of both competitive and uncompetitive type of inhibition. However, for reasons of simplicity they preferred the non-competitive type of inhibition.

McLaughlin and O'Beirne[32] described the effect of CO₂ on dry coleslaw mix respiration rate in three ways: competitive inhibition, uncompetitive inhibition and non-competitive inhibition. The uncompetitive inhibition model fitted best with the experimental results. But the model was found to be inflexible to include temperature as the independent parameter. However, the Arrhenius equation describes the dependence of the respiration rate on temperature.[17,33–35] have predicted the respiration rate of some commodities using principles of enzyme kinetics with uncompetitive inhibition. However, due to the specificity of the model parameters, the same need to be identified and quantified for each fruit and vegetable product. Hence, the present work was undertaken to develop and verify a combined respiration rate model based on the principle of enzyme kinetics, for dependence of O₂ and CO₂ and based on Arrhenius equation for dependence of storage temperature.

THEORETICAL CONSIDERATIONS

Enzyme Kinetics

The principles of enzyme kinetics have been suggested as it is being applicable to the respiration rate of fresh produce and Michaelies-Menten equation has been fitted to respiration rates.[25] Considering that CO₂ acts as a respiration inhibitor, the effect of CO₂ on the product respiration can be described by the un-competitive inhibition.[31,32,36] The uncompetitive type of inhibition occurs where the inhibitor (CO₂) does not bind with the enzyme, but binds reversely with the enzyme-substrate complex, reducing the rate of product formation. In this case the increase of substrate concentration (O₂ concentration) at high CO₂ concentrations has almost no influence on the O₂ consumption rate, i.e., increase in the substrate concentration cannot reverse it.[37] Thus the maximum respiration rate is not much influenced at high CO₂ concentration. At high levels of CO₂ concentration (17–18%), however the respiration mechanism changes form aerobic to anaerobic pathway.[17] Hence principles of enzyme kinetics with uncompetitive inhibition were used to develop a model[25,31,33,38–41] for predicting the respiration rate of products. Equations (1) and (2) express the uncompetitive inhibition mechanisms for respiration process in terms of O₂ consumption and CO₂ evolution rate respectively. The model has three parameters viz., , and for both O₂ consumption and CO₂ evolution. The model parameters are determined using the experimental respiration data.

(1)

(2)

where is the respiration rate, ml [O₂] kg⁻¹h⁻¹; is the respiration rate, ml [CO₂] kg⁻¹h⁻¹, , and are the gas concentrations for O₂ and CO₂ respectively in percentage; and are the maximum respiration rate for O₂ consumption; and CO₂ evolution respectively, and are the Michaelis-Menten constant for O₂ consumption and CO₂ evolution,% O₂, respectively; and and are the inhibition constants for O₂ consumption, CO₂ evolution,% CO₂, respectively.

The parameters of Eqs. (1) and (2) may be estimated by linearization of the equation and subsequent linear regression analysis[25,32,34,35,38–43] or directly by non-linear regression analysis.[31,44–48] However linearizing the equations is equivalent to changing the weight given to the data in the estimation procedure and thus should be avoided.[20,31]

Arrhenius Equations

Temperature is the most important environmental factor in the post harvest life of fresh produce because of its dramatic effect on rates of biological reaction including respiration. Within the physiological range (0 to 30°C), the respiration rate of fresh fruits and vegetables generally increases two to three folds for every 10°C rise in temperature.[49] The Arrhenius relationship describes the dependence of the biological reactions, including respiration of fresh produce, on temperature. The activated complex theory for chemical rates is the basis for the Arrhenius equation, which relates reaction rates to the absolute temperature.[50] Temperature has also major effect on the respiration rate, which has not been considered in the equation of the uncompetitive inhibition of enzyme kinetics (Eqs. 10 and 11). As per the studies conducted by Bhande et al.,[34] Das,[35] Hong et al.,[51] and Lakakul et al.[52] on banana, sapota fruit, fresh-cut green onion, and sapota slices respectively, the model parameter of Michaelis-Menten type enzyme kinetics was found to vary with the storage temperature. They used Arrhenius plot to describe the relationship between model parameters and storage temperature. The dependence of model parameters of uncompetitive inhibition kinetics on storage temperature can be expressed by Eq. (3).[52]

(3)

where, R_m is the model parameter of enzyme kinetics; R_p is the respiration pre-exponential factor; E_a is the activation energy, kJ/g mole; T_abs is the storage temperature; and K and R is the Universal gas constant (8.314 kJ/kg mole K). The Eq. (3) can be expressed in a linearized form[53] as shown in Eq. (4):

(4)

The model parameters obtained previously would be used to fit in Eq. (4) at different storage temperatures to estimates the values of E_a and R_p.

MATERIALS AND METHODS

Fruit Material

Fresh mature Guava fruit (Psidium guajava L.) of local popular variety Baraipur local were obtained from the orchard. Fruits were washed to remove adhering dirt, and used for the study. Care was taken to ensure that the fruits were of uniform size and weight.

Physicochemical Properties of Fruits

The physicochemical properties were determined in the laboratory to specify the maturity level of fruits prior to the respiration study. Initial fruit color and firmness was determined using Hunter Lab Colorimeter and Texture analyzer respectively.[17,54] The weight of guava fruits was measured using a precision balance (Essae Teraoka, Japan) having an accuracy of 0.01 g. True volume, ascorbic acid, total soluble solids and titrable acidity were determined as suggested by Ranganna[55] and Gungor and Sengul.[56] All the physical and chemical properties were determined for 10 representative samples and the average values are presented in Table 1.

Table 1 Physicochemical properties of guava fruits cv. Baruipur

Parameter	Value*
Physical properties
Fruit weight, g	250 ± 25
Fruit volume, ml	228 ± 12
True density, kg/m^3	1097 ± 43
Specific gravity	1.1 ± 0.15
Physical dimension
Major diameter, mm	72 ± 5
Intermediate diameter, mm	66 ± 2
Minor diameter, mm	65 ± 5
Sphericity	0.94 ± 0.02
Firmness, g	9527 ± 1011
Puncture strength, g	759 ± 142
Hunter Color parameters:
L*	58 ± 5
a*	−18 ± 3
b*	39 ± 3
Hue angle	114 ± 3
Chroma	43 ± 3
Chemical properties
Titrable acidity,% malic acid	0.67 ± 0.06
Ascorbic acid, mg/100 ml of juice	284 ± 8
Total soluble solids, °Brix	8.5 ± 1.2
*Values presented are average of 10 representative guava fruits and ± indicates the standard deviation.

Closed System Respirometer for Gas Exchange Measurement

Measurement of O₂ consumption by the fruit tissue as a function of O₂ and CO₂ concentration in a flow through system is technically difficult, since it requires highly accurate analytical equipment.[26] Hence, the respiration rate data for guava fruits was experimentally generated for different temperatures using the closed system method[31,33–35,41,42] as described in Fig. 1. Closed systems can provide a convenient way of characterizing respiration of fresh produce in a single set of experiment.[20,38,41,51,57] In this process, the changes in O₂ and CO₂ concentrations resulting from respiration within a sealed container could be measured directly. Air-tight respirometer chambers of size 0.125 m × 0.175 m × 0.23 m made up of acrylic (Perspex) sheet was used. A Ni-Cr thermocouple was inserted into one side of the respirometer and used to measure the storage temperature. Fruits were kept in respirometer from the open topside and were closed with the lid while inserting neoprene gasket in between. Lid was closed with nuts and bolts provided on the respirometer to make it airtight. The system was kept in a humidity control chamber (Digitek System, Kolkata, India) which was maintained at the desired temperature with a tolerance limit of ±0.2°C and relative humidity of 92 ± 2%. Gas composition of the respirometer was analyzed at regular intervals depending on the storage temperature of guava. Typically, the interval chosen were 2 h for the temperatures at 30 and 25°C, 4 h at 20 and 15°C, 8 h at 10, 5 and 0°C, respectively. Gas analysis was done until the CO₂ concentration reached 18% and until aerobic respiration persisted.[38] Changes in the concentration of O₂ and CO₂ over a certain period of time were measured and used to estimate respiration rates. After an interval of time, gas samples of the container were analyzed for O₂ and CO₂ concentration. Weight of fruits taken during the experiment and its corresponding free volume is shown in Table 2. Free volume of the respirometer was the total volume of the respirometer minus volume occupied by its content. The free volume of the respiration chamber () was determined by nitrogen injection method.[58] For this a predetermined quantity (Q_N) of N₂, which could cause a measurable change in N₂ level of container's atmosphere was injected. The increase in N₂ level was determined by analyzing the gas sample on gas chromatograph. By incorporating these values in Eq. (5), the free volume of the respiration container was calculated:

Table 2 Free volume of respirometer and weight of guava taken for generating the respiration data

Storage temperature °C	Weight of fruits (W), kg	Free volume of respirometer (), ml
0	1.311	4892.47
5	1.397	4800.28
10	1.308	4895.82
15	1.377	4820.36
20	1.384	4814.65
25	1.325	4878.37
30	1.314	4888.52

(5)

Figure 1 Schematic sketch for the closed system method for generation of respiration data.

where, V_f is the free volume of the respiration chamber in cm³; Q_N is the volume of N₂ injected into the respiration chamber in cm³; and N_i and N_f are the initial and final N₂ concentration in%.

Gas Analysis

Storage gas sample of about 1 ml was taken periodically with an airtight syringe and was analyzed quantitatively for O₂ and CO₂ concentrations using gas chromatograph (GC model 100, Knaur, Germany). A column packed with Spherocarb (18–100 mesh, 1.83-m long) was used for separation of CO₂, O₂ and nitrogen. Hydrogen was used as carrier gas at a flow rate of 25 ml/min. Oven temperature was kept at 50°C whereas injector and detector were set at 125 and 150°C, respectively, with the detector voltage of 5 volt. The sampling intervals were different for different storage temperatures as mentioned in the previous section. Each time three replications were taken for the gas analysis and the average gas composition was recorded. The respiration study was replicated in triplicate for accuracy of measurement and analysis.

Measurement of Respiration rate

Respiration rates can be measured by observing the concentration of O₂ consumption or CO₂ evolution per unit time per unit weight of the produce. Various external factors such as O₂ concentration, CO₂ concentration, temperature and time affect respiration.[17,25,38,58,59] The experimental respiration rate was calculated from the concentration difference, weight of fruit and free volume of the chamber. The respiration rates in terms of O₂ consumption and CO₂ evolution at a given temperature are calculated using the following Eqs. (6) and (7) as given by Kays (1991).[59]

(6)

(7)

where: R_O2 is the respiration rate, ml [O₂] kg⁻¹h⁻¹, R_CO2 is the respiration rate, ml [CO₂] kg⁻¹h⁻¹, and are the gas concentrations for O₂ and CO₂, respectively, t is the storage time in h, Δt is the time difference between two gas measurements, is the free volume of the respiration chamber in ml and W is the weight of the fruit in kg.

Verification of the Model

Since the respiration rates were generated at temperature between 0°C to 30°C with a step of 5°C, it is necessary to verify whether the developed models are capable of predicting the respiration rates at any temperature within this domain of experimental temperatures. The experimental respiration data for O₂ and CO₂ were obtained by the closed system respirometer kept at 12°C. Free volume of respirometer and weight of guava taken for verification of the model were 4852.37 ml and 1.348 kg, respectively. The experimental respiration rates for guava at different combinations of O₂ and CO₂ at 12°C temperature were determined using Eqs. (6) and (7), respectively.

Statistical Analysis

The standard deviations of the physicochemical properties were estimated using standard statistical formulae.[60] The values obtained are reported in Table 1. The SYSTAT 8.0 software was used to estimate the model parameters of enzyme kinetics model along with their coefficients of determination. The one way ANOVA was carried out to ascertain the effect of temperature on model parameters and effect of temperature on regression coefficients using SYSTAT 8.0 with completely randomized block design. The mean relative percentage deviation modulus is widely adopted throughout the literature to evaluate the goodness of fit of mathematical expression. Moduli below 10% are indicative of reasonably good fit, 10–20% fairly good fit and 20–30% not satisfactory fit for all practical purposes. The goodness of fit between predicted and experimental respiration rates was obtained by calculating the mean relative percentage deviation modulus.[32] In general, lower the modulus better is the agreement between experimental and predicted values.

RESULTS AND DISCUSSION

Assessment of Maturity Level

The average unit weight and volume were found to be 250.38 g and 228.15 ml respectively (Table 1). The L*, a* and b* color values were measured to be 57.83, −17.58 and 39.41, respectively. The negative value of ‘a’ indicates the greenness of guava fruits at maturity. The flesh firmness, puncture strength, and TSS were 9527.40 g, 759.34 g, and 8.5, respectively indicating that the fruits were hard and matured. The results obtained were found to be in agreement with those reported by Adsule and Kadam,[1] Palaniswamy and Shanmugavelu,[8] Morton,[9] and Wilson.[12]

Gas Exchange Rate

Figure 2 shows the periodical changes in O₂ and CO₂ concentrations inside the closed respirometer containing guava at different storage temperatures. The O₂ concentration of the respirometer decreased in proportion to increase in CO₂ concentration with the storage period, the rate being higher at the higher storage temperature.

Modeling and Analysis

A regression function is often used to fit the data of gas concentration versus time, and the respiration rate at given time is determined from the first derivative of the regression function.[26,38,40] Figure 2 shows the actual gas composition in terms of O₂ and CO₂ concentrations inside the closed respirometer at different storage temperature. By using the generated experimental respiration data, a nonlinear regression analysis (Origin 6.1) was done to fit the curve of O₂ concentration and CO₂ concentration at different storage periods. The resultant regression equations for O₂ consumption and CO₂ evolution are shown in Eqs. (8) and (9) to determine the values of the coefficients ‘a’ and ‘b’:

(8)

(9)

Figure 2 Experimental data for O₂ consumption and CO₂ evolution at different temperatures; ▴ is Y_O2, in%; ▪ is Z_CO2, in%.

where a and b are the regression coefficients; a is unit less and b is in h; t is storage period in h, and are the gas concentrations for O₂ and CO₂, respectively in decimal. The first derivative of the regression functions were used to determine the rate of change of gas concentration as outlined in Eqs. (10) and (11):

(10)

(11)

The respiration rate at any given time (at each sampling time) was then calculated by substituting the values of dY_O2 /dt and dZ_CO2 /dt obtained from Eqs. (10) and (11) in Eqs. (12) and (13), respectively.

(12)

(13)

The respiration rate was found to have decreased with time due to decreasing O₂ concentrations and increasing CO₂ concentrations. The regression coefficients a, and b of Eqs. (8) and (9), and correlation coefficients (r²) at different storage temperatures are shown in Table 3. The above regression functions described the experimental data very well (r² ≥ 0. 997) for guava. From the values of the regression coefficients a and b in the Table 3, it can be inferred that both the parameters were influenced by the storage temperature and that coefficient ‘b’ was more influenced by temperature than coefficient ‘a’. The analysis of variance of temperature with regression coefficients a and b is given in Table 4. It can be inferred that the temperature has a significant effect (1% level of significance) on regression coefficients. For any unknown temperature, the values of a and b can be determined by linear interpolation of the known values. The corresponding values of a and b when substituted in Eqs. (10) and (11), allow to estimate dY_O2 /dt and dZ_CO2 /dt, respectively. The respiration rates for the given temperature can then be determined using Eqs. (12) and (13).

Table 3 Values of regression coefficients for regression model

		Regression coefficients
Storage temperature of guava, °C	Respiration expression in terms of	A	b	r^2
0	O2 consumption	2.595	341.579	0.999
	CO2 evolution	2.622	368.065	0.998
5	O2 consumption	2.615	290.991	0.998
	CO2 evolution	2.432	332.881	0.997
10	O2 consumption	2.965	204.228	0.998
	CO2 evolution	3.071	211.820	0.999
15	O2 consumption	2.678	174.790	0.998
	CO2 evolution	2.716	185.263	0.998
20	O2 consumption	3.366	79.790	0.999
	CO2 evolution	3.483	82.354	0.998
25	O2 consumption	2.555	80.342	0.997
	CO2 evolution	2.468	88.221	0.998
30	O2 consumption	2.875	51.486	0.999
	CO2 evolution	2.482	53.928	0.998

Table 4 Analysis of variance of effect of temperature on regression coefficients

Factor	DF	MSS	F value	MSS	F value	MSS	F value	MSS	F value
Replication	2	0.006	0.496	0.001	0.120	0.086	0.175	540.599	1.105
Temperature	6	0.240	18.844**	0.449	39.978**	37582.787	7677.572**	40752.578**	83.315
Error	12	0.013	—	0.011	—	4.895	—	489.139	—
**Significant at 1% level.

The verification of the regression model for respiration rate was done at 12°C storage temperature. The values of the regression coefficient a and b at 12°C were estimated by the linear interpolation of the values of the same coefficients at 10 and 15°C as shown in Table 3. The above values of a and b are substituted in Eqs. (8) and (9) to obtain the concentration of O₂ and CO₂ at each sampling time. Then the values of dY_O2/dt and dZ_CO2/dt were obtained from Eqs. (10) and (11). The respiration rate at each sampling time was then calculated by substituting the values of dY_O2 /dt and dZ_CO2 /dt in Eqs. (12) and (13), respectively. The experimental respiration rates were also obtained using Eqs. (6) and (7). The goodness of fit was obtained by calculating the mean relative percentage deviation modulus.

The mean relative percentage deviation modulus was found to be 1.68 and 2.49% for O₂ consumption and CO₂ evolution, respectively. This suggests that the respiration rates predicted by the regression model were in good agreement with the experimental respiration rates data at different storage periods. Similarly the respiration rate at any storage temperature can be obtained by employing regression model, Eqs. (12) and (13).

Estimation of Model Parameters

The model parameters of Eqs. (1) and (2) were determined by fitting the experimental data at each temperature using non-linear regression analysis (SYSTAT 8.0, SPSS). According to Eqs. (1) and (2), respiration rate was considered as the dependent variable whereas the percent O₂ concentration () and percent CO₂ concentration were considered as the independent variables for the regression analysis. The model parameters were calculated from the constants of non-linear regression analysis, which are given in Table 5 along with their corresponding coefficients of determination (R²) values. With the help of these model parameters, Eqs. (1) and (2) could now predict the respiration rates with any combination of O₂ and CO₂ concentration. The values in Table 5 show that the model parameters were indeed dependent on the storage temperature. The coefficients of determination was found to be higher or equal to 0.975 indicating that the relationship between respiration rate and O₂ and CO₂ concentrations fitted well with the uncompetitive inhibition enzyme kinetics model.

Table 5 Model parameters for uncompetitive inhibition enzyme kinetics for different storage temperatures

Storage Temperature, °C	Respiration expressed in terms of	Vm, ml/kg h	Km,% O2	Ki,% CO2	R^2
0	O2	15.62	9.37	11.27	0.992
	CO2	12.65	4.89	9.29	0.995
5	O2	18.54	10.24	9.46	0.985
	CO2	15.38	6.10	8.16	0.980
10	O2	21.78	11.33	8.22	0.990
	CO2	18.82	7.43	7.23	0.985
15	O2	28.43	12.76	7.15	0.985
	CO2	25.16	8.56	6.35	0.975
20	O2	37.25	13.92	6.42	0.991
	CO2	34.83	9.38	5.67	0.990
25	O2	48.37	14.74	5.80	0.975
	CO2	44.25	10.15	5.10	0.980
30	O2	63.59	15.62	5.17	0.992
	CO2	60.47	10.86	4.53	0.990

The analysis of variance of temperature with model parameters of enzyme kinetics based model is given in Table 6. It can be inferred that the temperature has a significant effect (1% level of significance) on model parameters , , , , , and of enzyme kinetics.

Table 6 Analysis of variance of effect of temperature on model parameters of Enzyme kinetics model

Factor	DF	MSS	F value	MSS	F value	MSS	F value	MSS	F value	MSS	F value	MSS	F value
Replication	2	0.14	0.12	0.14	0.12	0.14	0.12	0.07	0.31	0.09	0.35	0.01	0.02
Temperature	6	923.51	808.07**	908.82	795.22**	16.53	14.47**	13.86	64.65**	14.10	51.90**	8.83	106.08**
Error	12	1.14	—	1.14	—	1.14	—	0.21	—	0.27	—	0.08	—
**Significant at 1% level.

Arrhenius Equation

Since the model parameters such as , , and for O₂ and CO₂ concentrations were found to vary with temperature and hence, Arrhenius equation was used to correlate the model parameters at different storage temperatures. As per Eq. (4), the model parameters were plotted at different storage temperatures; by plotting the log values of the model parameters against the inverse of corresponding temperature in absolute units. The resulting linear plot is shown as Fig. 3. The slope and the Y-axis intercept of Eq. (4) for model parameters , , and are shown in Table 7. The activation energy was calculated from the slope of the straight line and the pre-exponential factor was calculated from the Y-axis intercept. Table 8 shows the activation energy and pre-exponential factor for different model parameters of enzyme kinetics. Activation energy was found to be in negative side for and valued at −17.54 and −16.40 kJ/g-mol, respectively. The negative values of activation energies for could be attributed to the inhibitory effect of CO₂ concentration on respiration rate. By using these constants (values of activation energy and respiration pre exponential factor), the model parameters of enzyme kinetics at any temperatures can be predicted by using Eq. (3) and then, the respiration rate at the given temperature can be estimated by employing Eqs. (1) and (2), in terms of O₂ consumption and CO₂ evolution, respectively.

Table 7 Slope (−E_a/R) and Y axis intercept (ln R_p) of Arrhenius relation for different model parameters of enzyme kinetics

	Vm		Km		Ki
	O2	CO2	O2	CO2	O2	CO2
Slope	−3927.0	−4375.6	−1461.9	−2168.1	2110.6	1973.2
Y axis intercept	17.044	18.479	7.597	9.603	−5.336	−4.998
r^2	0.984	0.988	0.990	0.960	0.995	0.990

Table 8 Activation energy and pre-exponential factor of Arrhenius type equation for different model parameters of uncompetitive inhibition

	Vm		Km		Ki
Parameters for Arrhenius equation	O2	CO2	O2	CO2	O2	CO2
Ea (kJ/g-mole)	32.64	36.37	12.15	18.02	−17.54	−16.40
Rp	2.5 × 10^7	1.0 × 10^8	1.9 × 10^3	1.4 × 10^4	5.0 × 10^−3	7.0 × 10^−3

Figure 3 Arrhenius relation for different model parameters of enzyme kinetics; ▴ is ; ▪ ; ♦ is ; о is ; Δ is ; □ is .

Verification of Respiration Rate Model

The respiration rate predicted by the enzyme kinetics model was verified with the experimental respiration rate at 12°C storage temperature. By using activation energy and pre-exponential factor from Table 8, the model parameters at 12°C for both O₂ consumption and CO₂ evolution were calculated and are shown in Table 9. Then the respiration rates of guava in terms of O₂ consumption and CO₂ evolution were estimated by using Eqs. (1)and (2). The experimental respiration rates of guava were obtained by the closed system method. The predicted and actual respiration rates of guava are shown in Fig. 4. The mean relative deviation moduli between predicted and experimental respiration rates of guava at 12°C were found to be 8.95% and 8.02% for O₂ consumption and CO₂ evolution respectively. This indicates that the predicted respiration rates for guava were in good agreement with the experimental respiration rates.

Table 9 Model parameters of uncompetitive inhibition enzyme kinetics at 12°C storage temperature

Respiration expression in terms of	Maximum respiration rate (Vm), ml kg^−1h^−1	Michaelis-Menten constant (Km), % O2	Inhibition constant (Ki), % CO2
O2 consumption	26.18	11.79	7.92
CO2 evolution	22.78	7.35	6.85

Figure 4 Experimentally estimated and predicted respiration rates for guava at 12°C ‐‐◊‐‐ is for experimental respiration rate; ‐‐□‐‐ is for respiration rate predicted by regression model; ‐‐Δ‐‐ is for respiration rate predicted by enzyme kinetics model.

CONCLUSIONS

The respiration rates were found to decrease with storage time due to diminishing concentrations of O₂ concentration and increase in CO₂ concentrations in the respirometer. In the enzyme kinetics model, the dependence of respiration rate on O₂ and CO₂ was found to follow the uncompetitive inhibition. Because of the inflexibility of the model to incorporate the temperature terms, a new model was proposed in which the model parameters were correlated with storage temperature as per the Arrhenius equations. The activation energy and pre-exponential factors of the Arrhenius equations were used for predicting the model parameters at any temperature between 0–30°C. The applicability of the model for the prediction of respiration rate was verified at 12°C storage temperature. There was a good agreement between the predicted and experimental respiration rates for O₂ consumption and CO₂ evolution.

Nomenclature
	=	Regression coefficient
	=	Regression coefficient, h
E	=	Mean relative deviation modulus,%
Ea	=	Activation energy, kJ g−1mol−1
	=	Michaelis-Menten constant for O2 consumption,% O2
	=	Michaelis-Menten constant for CO2 evolution,% O2
	=	Inhibition constants for O2 consumption,% CO2
	=	Inhibition constants for CO2 evolution,% CO2
	=	Number of respiration data points
Ni	=	Initial N2 concentration of the respirometer,%.
Nf	=	Final N2 concentration of the respirometer,%.
QN	=	Volume of N2 injected into the respiration chamber, cm3
R	=	Universal gas constant, 8.314 kJ kg−1mol−1K−1
	=	Respiration rate, ml [CO2] kg−1h−1
Rexp	=	Experimental respiration rate, ml kg−1 h−1
Rm	=	Model parameter of enzyme kinetic
Rpre	=	Predicted respiration rate, ml kg−1 h−1
	=	Respiration rate, ml [O2] kg−1h−1
Rp	=	Respiration pre-exponential factor
T	=	Storage temperature, °C
Tabs	=	Storage temperature, K
	=	Storage time, h
	=	Time difference between two gas measurements
	=	Free volume of the respiration chamber, ml
	=	Maximum respiration rate for CO2 evolution, ml/kg-h
	=	Maximum respiration rate for O2 consumption, ml/kg-h
	=	Weight of guava fruit, kg
	=	Oxygen concentration, decimal
	=	Carbon dioxide concentration, decimal