Use of Nonlinear Regression for Analyzing Î²-Lactoglobulin

Chapter 4

Use of Nonlinear Regression for Analyzing β-Lactoglobulin Denaturation Kinetics in Skim Milk 1

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Downloaded by EAST CAROLINA UNIV on January 4, 2018 | http://pubs.acs.org Publication Date: November 19, 1996 | doi: 10.1021/bk-1996-0650.ch004

D. J.Oldfield ,Harjinder Singh , M. W.Taylor ,and K. N. Pearce 1

Department of Food Technology, Massey University, Palmerston North, New Zealand New Zealand Dairy Research Institute, Private Bag 11029, Palmerston North, New Zealand 2

Five different methods were used to calculate the kinetic parameters for the denaturation ofβ-lactoglobulin(genetic variant A). The activation energy (E ) and pre-exponential term (ln(k )) determined by nonlinear regression (NLR) had smaller confidence intervals and more degrees of freedom than those obtained by the commonly used two-step method, which uses linear regression. Of the methods investigated, NLR was the preferred method for analysing denaturation kinetics. a

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Heat-induced denaturation of whey proteins in skim milk is important in determining the functional properties of the final milk product. A number of researchers have used kinetics to quantify whey protein denaturation in milk (7-5). The rate of protein loss as a function of temperature, time and concentration is typically described by the general rate equation 1 and the Arrhenius equation 2.

(1)

dt

where C = t

K = n

1

protein concentration (g kg" ), time (s), rate constant ((g kg" ) s" ), and reaction order. !

(ln)

1

0097-6156/96/0650-0050$15.00/0 © 19% American Chemical Society

Parris et al.; Macromolecular Interactions in Food Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

4. OLDFIELD ET AL.

0-LactoglobuIin Denaturation Kinetics in Slant Milk (2)

R T

k - k e n

n

0

where ! (1

n)

1

pre-exponential term ((g kg" ) " s*), activation energy (kJ mol" ), universal gas constant (8.314 x 10" kJ mol" K" ), and temperature (K).


1

R = T =

3

1

1

Whilst equations 1 and 2 are used to quantify denaturation reactions they are not necessarily correct mechanistic descriptions of what actually occurs. Though they often fit the observations well. The procedure most commonly used to determine kinetic parameters has been described as a two-step method (6) and uses linear regression. The first step is to determine the concentration dependence of the rate at a fixed temperature. The temperature dependence of the rate constant is then found using the Arrhenius equation 2. The advantages of the two-step method are the ease of use and the small number of calculations required. Any non-uniform distribution of the variances of the raw data and the rates are not taken into account, and logarithmic transformation may produce trends in the residuals (7, 8). It has been suggested that better estimates of the kinetic parameters can be obtained by nonlinear regression (NLR) of concentration, temperature and time data (9). With advances in computing technology, methods such as NLR, which require more computational power, can be within easy reach of a researcher, namely through statistical software packages or one's own written software. In this study, the two-step (linear regression) method was compared with NLR, using P-lactoglobulin genetic variant A (p-lg A) denaturation in milk as an example. The precisions of the kinetic parameters, E and k were calculated for each method so that comparisons could be made. a

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Materials and Methods Heat Treatment of Milk. Raw whole milk was obtained from the No. 1 dairy farm, Massey University, Palmerston North, New Zealand. The milk was separated at 40°C using a hermetic milk separator (Alfa-Laval, Sweden), and the resultant skim milk was stored at 5°C. The skim milk was then processed on a pilot-scale UHT plant (Type D, Alfa-Laval, Sweden). For each run, the skim milk was heated by direct steam injection (DSI) to the required temperature; DSI allowed for a rapid step change in the temperature. A range of temperatures (100-130°C) and a range of holding times (3-160 s) were investigated. After the holding tube, a flash vessel was used to reduce the milk temperature to approximately 65°C, effectively ending any further denaturation. The milk was collected and further cooled to 20°C in an ice bucket.



52

MACROMOLECULAR INTERACTIONS IN FOOD TECHNOLOGY

Protein Analysis of Milk. Milk samples were sealed in disposable plastic centrifuge tubes (13.5 ml, part No. 344322, Beckman, Palo Alto, CA) and placed in a Type 80 Ti rotor (Beckman, Palo Alto). Ultracentrifugation was carried out in a Beckman L8-80M centrifuge at 50,000 rpm (average 175,000 g) for 1 h at 20°C. After centrifugation, the top of the tube was cut open and all the supernatant was carefully removed. The supernatants were then analyzed for native p-lg A by polyacrylamide gel electrophoresis (native-PAGE) under non-dissociating conditions (10). The gels were scanned using a laser densitometer (Molecular Dynamics, Sunnyvale, CA), and the integrated intensities of the P-lg A protein bands were calculated by a software program, ImageQuant (Molecular Dynamics). A concentration standard of p-lg A (Sigma No. L-7880, lot 13H7020, Sigma Corp., St. Louis, MO) was run on the gel to convert the band intensities to concentration units of g kg" . 1

Statistical Analysis of Kinetic Data. The concentration/time data were analyzed by a variety of methods using the statistical software package SPSS (version 4.0.1, SPSS Inc., Chicago, IL). Five different methods of data analysis based on the general rate equation 1 and the Arrhenius equation 2 were used (77). The concentration/time data of native P-lg A, determined by native-PAGE, were used to test the different methods. As a break in the Arrhenius plot is observed at 90°C (7, 5), a temperature region of 100-130°C was chosen to avoid this break. The data used were unweighted so that all the data points were treated equally by the program. (i) Two-step Method (Linear Regression). The first step was to determine the rate constant from the integrated general rate equation. The resulting integrated equations are given in equations 3 and 4 (7). When n + 1,

(3)

- 1 • (n-1) k C„ t n

where C, = Co =

1

concentration of undenatured protein at t = t (g kg" ), and concentration of undenatured protein at t = 0 (g kg" ). 1

When n = 1 (first order reaction),

\ In


(4)

4.

OLDFIELD ET AL.

where kj =

0-Lactoglobulin Denaturation Kinetics in Skim 53 Milk

1

rate constant for first order reaction (s ).

The left-hand side of the equation was plotted against time and k was calculated from the slope. The second step was to plot \n(k ) against 1/T so that the Arrhenius equation could be used to calculate E and k from the slope and intercept, respectively. Calculation of k by equation 2 involved finding the intercept at 1/T = 0 or when T-°°. As this point is a long way from where the data points lie on the Arrhenius plot, a small change in E causes a large change in k (7). Linear regression of equation 2 assumed that the two parameters could be determined independently. n

n

a

0

0


a

0

(ii) Two-Step Method with Adjusted Intercept (Linear Regression). The Arrhenius equation was modified so that a reference temperature was used in the exponential term (equation 5). This creates a new pre-exponential term, k which is related to k by equation 6 (7). np

0

k -e ref

where k = T =

-I R

1

(5)

r

1 0 0

1

reference pre-exponential term ((g kg" ) " * s" ), and reference temperature (K).

ref

ref

k = k ,e n

0

(6)

*

ref

By placing the intercept in the middle of the data set,fc becomesindependent of the slope (2s ), and the estimated error in the pre-exponential term will be reduced (equation 5). The reference temperature chosen was 115°C, being in the middle of the temperature range 100-130°C. Thus the kinetic parameters were renamed k and T r

Use of Nonlinear Regression for Analyzing Î²-Lactoglobulin

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