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A kinetic model for whey protein denaturation at different moisture contents and temperatures
Accepted Manuscript A kinetic model for whey protein denaturation at different moisture contents and temperatures James C. Atuonwu, Joydeep Ray, Andrew G.F. Stapley PII:
S0958-6946(17)30150-4
DOI:
10.1016/j.idairyj.2017.07.002
Reference:
INDA 4197
To appear in:
International Dairy Journal
Received Date: 23 May 2017 Revised Date:
17 July 2017
Accepted Date: 17 July 2017
Please cite this article as: Atuonwu, J.C., Ray, J., Stapley, A.G.F., A kinetic model for whey protein denaturation at different moisture contents and temperatures, International Dairy Journal (2017), doi: 10.1016/j.idairyj.2017.07.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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ACCEPTED MANUSCRIPT A kinetic model for whey protein denaturation at different moisture
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contents and temperatures
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5 6 7 James C. Atuonwu, Joydeep Ray, Andrew G.F. Stapley*
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Department of Chemical Engineering, Loughborough University, Loughborough,
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Leicestershire, LE11 3TU, UK.
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*Corresponding author. Tel.: +44 1509 222 525
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E-mail address: [email protected] (A. G. F. Stapley)
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_____________________________________________________________________
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ABSTRACT
26 The denaturation of whey protein samples that had previously undergone heat-
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treatment for different times at different temperatures and moisture contents was
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analysed by differential scanning calorimetry (DSC), using the DSC enthalpy as a
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measure of residual undenatured protein. Data were fitted to first order irreversible or
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reversible kinetic expressions, and the resulting rate constants were found to increase
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with both temperature and moisture content. The whole data set was then fitted as a
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function of time, temperature and moisture content, with rate constants varying
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according to either Arrhenius or Williams-Landel-Ferry (WLF) kinetics and with
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selected fit parameters made empirical functions of moisture content. The best fits
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were obtained using reversible WLF kinetics, which could be further slightly simplified
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without loss of accuracy. The model provides a platform for single- and multi-objective
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drying trajectory optimisation with respect to protein denaturation in dairy products.
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1.
Introduction
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Proteins are highly desirable components in food products because of their high nutritional value and contributions to sensory attributes such as colour, flavour and
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texture. Whey proteins, in particular, play a huge role due to the widespread
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consumption of dairy produce. Constituting about 20% of the proteins in milk, they
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have the highest biological value among all known proteins (Anandharamakrishnan,
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Reilly, & Stapley, 2007; Wijayanti, Bansal, & Deeth, 2014). They are used as additives
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in formulated products such as bakery products, processed meats, beverages, pasta,
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ice cream, confectionery, infant foods, spreads, dips and as encapsulating agents in
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pharmaceuticals.
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For convenience and to extend their shelf life, liquid dairy products are often subjected to dehydration processes such as evaporation and spray drying. Spray
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drying, in particular, is the generally-accepted method for producing dry powder. The
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dehydration process involves thermal treatment, during which whey proteins can lose
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their normal secondary, tertiary or quaternary structures. Denaturation, as this process
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is called, transforms the protein from its native folded, biologically-active state to an
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unfolded, biologically-inactive state. Physically, this can manifest in aggregation,
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coagulation, reduced solubility for powders and loss of some functional properties such
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as gelling and emulsification (Haque, Aldred, Chen, Barrow, & Adhikari, 2013a,b). In
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addition, the denaturation of whey proteins is accompanied by the release of small
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sulphur-containing compounds such as methanethiol and hydrogen sulphide that
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produce cooked flavours in heated milk (Al-Attabi, D’Arcy, & Deeth, 2009). The
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aggregation that results from denaturation can potentially lead to processing downtime
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due to heat exchanger fouling and nozzle blocking in spray dryers. Consequently,
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whey protein denaturation is a major issue in dairy processing.
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The study of denaturation during spray drying is complicated by the variation of moisture content and temperature with both position in the droplet/particle and time.
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Understanding the mechanism of whey protein denaturation with the ability to predict
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its extent over a wide range of process and product conditions is important for deriving
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dehydration strategies to minimise this phenomenon. For this, reliable kinetic models
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are required. Studies on the thermal denaturation of whey proteins abound in the
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literature (e.g., Dissanayake, Ramchandran, Donkor, & Vasiljevic, 2013; Lamb, Payne,
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Xiong, & Castillo, 2013; Oldfield, Singh, & Taylor, 2005; Wolz & Kulozik, 2015),
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especially in fully hydrated systems (Wolz & Kulozik, 2015). However, most of these
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studies have not examined how denaturation behaviour varies with moisture content.
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An exception is the work of Haque et al. (2013a,b), who investigated the denaturation
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of whey protein isolate solutions in suspended droplets whilst drying. They assumed a
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model equation used by Meerdink and van’t Riet (1995) for enzyme inactivation to
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describe the variation of kinetics with temperature and moisture content. This was
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inserted into a drying model which predicted the moisture content and temperature
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history of the droplets from which the kinetic parameters for the denaturation were
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inferred via inverse model fits. However, this experimental study focussed mainly on
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high moisture content levels (40% and above), over limited temperatures (65 and 80
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°C), and with results sampled at 60 s intervals, wh ich exceeds the times for most
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commercial spray drying operations. Moreover, since the experiments were in a
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convective drying environment, continuous temperature and moisture content
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variations occurred as a result of which, the denaturation kinetics at constant
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temperature and moisture levels were not resolved (two experiments were performed
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at constant temperature and moisture content but these appear to have been at high
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moisture contents only). Therefore, although it is well known that protein damage
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occurs at temperatures above 60 °C in fully wet sys tems, little understanding exists on
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the ability of proteins to withstand heat when in a partially dried state. Quantifying the simultaneous impact of moisture levels and temperatures on
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protein denaturation kinetics will better enable the development of optimum
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dehydration profiles with respect to protein denaturation. Such an approach has been
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undertaken with assessing lysine losses in dairy powders, whereby powders are
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equilibrated at different water activities (using salt solutions) and then subject to
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thermal treatment in hermetically sealed cans for different time-temperature
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combinations (Schmitz, Gianfrancesco, Kulozik, & Foerst, 2011; Schmitz-Schug,
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Kulozik, & Foerst, 2014).
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This work presents a similar experimental method using thermal treatment in
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combination with partial hydration followed by differential scanning calorimetry (DSC)
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analysis to generate experimental data that quantify the combined effects of time,
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temperature and moisture content on whey protein denaturation. From this, a set of
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four mathematical models have been derived in a three-step process to explicitly
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describe the denaturation kinetics. The models are based on a combination of either
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first-order reversible or irreversible kinetics, with rate constant temperature
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dependencies expressed using either William-Landel-Ferry (WLF)-type or Arrhenius
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equations. Semi-mechanistic monotonic functions of moisture content are proposed for
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some of the parameters, so a global fit of the whole data set can then be made as
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functions of time, temperature and moisture content. The four mathematical models
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are then compared and possibilities for optimising drying processes considered.
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2.
Materials and methods
2.1.
Experimental
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Samples (10–20 mg) of as-received low-heat whey protein isolate (WPI) powder (Impact Whey isolate, MyProtein, Northwich, UK) were loaded into high-pressure
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stainless steel DSC pans (TA Instruments, New Castle, DE, USA), and weighed. The
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samples used in the experiment were obtained from products manufactured using a
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combination of membrane filtration and low-temperature drying to avoid protein
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denaturation prior to the experiment. The composition (w/w) of the WPI powder stated
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by the manufacturer was 90% protein, 2.5% lactose, 0.3% fat, 0.5% salt and 6.7%
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other (including moisture). Samples were equilibrated with different saturated salt
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solutions to various intermediate moisture levels over a few days. The salts used were
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magnesium chloride (Acros Organics, Fair Lawn, NJ, USA), sodium chloride (Fisher
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Scientific, Fair Lawn, NJ, USA) and potassium chloride (Hopkin and Williams Limited,
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Harrogate, UK). Saturated solutions of these salts gave water activities and equilibrium
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moisture levels (kg kg-1) as follows: magnesium chloride, 0.33 and 0.08, respectively;
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sodium chloride, 0.75 and 0.18, respectively; potassium chloride, 0.86 and 0.23,
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respectively. The relationship between the water activity generated by the saturated
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salt solutions and the equilibrium moisture level is based on the sorption isotherms for
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WPI powder presented by Foster, Bronlund, and Paterson (2005). The equilibration
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vessels were 1000 mL Kilner jars to which 100 mL of RO (reverse osmosis) water was
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added, and an excess of the desired salt such that crystals were always present. A
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total of 10 loaded DSC pans were placed on a Petri dish, which was then placed on a
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raised platform within the jar which was located above the level of the salt solution.
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The jars were placed in a water bath at 25 °C. The pans were weighed at daily
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intervals until no significant further change in mass was observed (a second Petri dish
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was used to cover the pans to minimise any changes in moisture content whilst
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samples were outside of the Kilner jar). Other dry basis moisture content (kg kg-1)
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powder at ambient temperature for 24 h), 0.05 (the initial moisture content of the as-
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received powder) and 2.5 (full hydration). Samples were then sealed and heated in
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either a stirred water bath (for temperatures < 100 °C) or a silicone oil bath (> 100 °C),
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for different times. The pans were then plunged into a beaker of cold water, removed,
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dried on the outside, and then opened and water added to the samples using a
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micropipette (to a ratio 1:2.5, w/v) to ensure full hydration as required for DSC analysis
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(Anandharamakrishnan et al., 2007). All mass measurements were performed with
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Sartorius microbalances (Sartorius AG, UK) of accuracy 0.1 mg.
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The pans were then resealed and analysed by DSC (Q10, TA Instruments),
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together with an identical but empty reference pan. In the DSC analysis, the sample
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and reference were first equilibrated at 20 °C, and then heated at a linear rate of 5 °C
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min-1 to 100 °C. The experiments were performed in tripl icate. Protein denaturation
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was observed in the peaks produced in the heat flow-temperature diagram. The area
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under the peak is produced by the denaturation in the DSC of any undenatured
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proteins left in the sample after the previous heat treatment in the water/oil bath. It is
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thus a measure of the amount of undenatured proteins left after heat treatment.
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Enthalpies were obtained assuming a linear baseline, and were compared with
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those of a hydrated sample of as-received powder, which was used as a reference.
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The latter yielded the highest enthalpy which was taken to represent zero denaturation
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(or 100% native protein concentration). The fractional decrease in enthalpy from this
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value was then taken to be the fraction of denaturation that occurred in the sample
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during the previous heat treatment.
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2.2.
Mathematical modelling
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The existing literature on whey protein denaturation kinetic models (e.g., Dissanayake et al., 2013; Haque et al., 2013a,b; Oldfield, Singh, Taylor, & Pearce,
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1998; Wolz & Kulozik, 2015) can be categorised into 1st, 2nd and higher-order models
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with Arrhenius rate constant temperature dependencies. The majority of these models
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are concerned with whey protein components either individually or in milk, whey and
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model systems without specific reference to moisture content effects. The overall
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denaturation process has been described as a two-step process consisting of a
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conditionally-reversible unfolding step, and a subsequent irreversible aggregation step
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(Chen, Chen, Nguang, & Anema, 1998; Parris, Purcell, & Ptashkin, 1991; Tolkach &
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Kulozik, 2007).
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In this work, two rate constant temperature dependencies, the Williams-LandelFerry (WLF)-type and the Arrhenius equations were tested in conjunction with both
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reversible and irreversible first order kinetic models. Both 2nd order and “nth” order
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kinetic models were also tested, but these generally gave poorer fits to data and were
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therefore discarded. A full set of nomenclature for all models is provided in Table 1.
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The moisture content dependence was then built in by making some of the WLF or Arrhenius parameters functions of moisture content. The overall modelling strategy
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was performed in three stages.
In the first stage, concentration versus time data were fitted to first-order
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reversible and irreversible kinetics, which have the following rate laws, respectively:
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dc = −k f c + k r (c0 − c ) dt
(1)
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dc = −k i c dt
(2)
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where c represents the concentration (or fraction) of undenatured proteins, c0 the initial
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value of c, such that (c0 - c) represents the concentration of denatured proteins. kf and
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reversible kinetic model, while ki is the rate constant for the irreversible kinetic model.
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Upon integration (since rate constants do not vary with time) these yield,
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c=
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c =c
0
(− (k f
+ k r ) t )))
−k) it
(3) (4)
K eq =
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(For the reversible scheme an equilibrium constant can be defined as:
kf kr
(5)
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exp
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c (k r + k f ( kf + kr
exp
respectively: 0
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Values for ki, kf, and kr (and hence Keq) were obtained for each combination of
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moisture content and temperature by fitting concentration versus time data using a
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least squares method. This allowed an initial assessment of how these parameters
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vary with temperature and moisture content.
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In stage two the temperature variation was explicitly modelled. At each moisture
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content, the rate constants (kf and ki) were assumed to vary with temperature
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according to either WLF-type (equations 6, 7) or Arrhenius (equations 8, 9) equations:
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kf = k
0
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ki = k
0
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f
i
(7)
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E Af − RT
(8)
exp
k i = k iTd
exp
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T −T i di C +T −T i di
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−C
(6)
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k f = k fTd
1 2
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T −T f df C +T −T f df
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−C
E Ai − RT
(9)
The term “activation energy” is routinely used in conjunction with the Arrhenius equation. It is used here purely as a parameter that models the temperature 9
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barrier (as protein unfolding is generally regarded to be driven instead by differences in
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thermodynamic stability between folded and unfolded states). It was found that
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attempting to model the reverse rate constant (kr) using similar equations to those
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above produced sporadic results in a full simulation. Instead, a more reliable method
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was to model the equilibrium constant, and use this together with the forward rate
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constant to deduce the reverse rate constant (linked via equation 5). The temperature
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variation of the equilibrium constant was modelled using the van’t Hoff equation for
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both the Arrhenius and WLF-type dependencies respectively as:
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K eq = K eqref (
) WLF
K eq = K eqref (
) Arr
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) WLF
1
1
E Aeq − ( Arr) R
− T T( ref
1
E Aeq − ( WLF) R
1
exp
) Arr
(10)
(11)
The full parameter list for each of the resulting four models is shown in Table 2.
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exp
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These models (in combination with equation 3 or 4 as appropriate) were fitted to
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concentration data using the least squares method. Fits were initially made to data
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pertaining to specific moisture contents, so that individual fits for each parameter in
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Table 2 were produced for each moisture content. However, trends in the resulting fit
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parameters were unclear. This was attributed to over-parameterisation (redundancy),
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which made the fits susceptible to the effects of noise. To combat this, based on
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observations made in stage one of the fitting process, some parameters (Tref, Td, kTd
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and Keqref,) were made constant across all moisture contents, whilst the remaining
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parameters were allowed to assume different values for different moisture contents.
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This concluded stage two of the fitting process.
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The resulting fit parameters were then fed to the final third stage, where the moisture varying parameters from stage two were made explicit functions of moisture 10
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content (as also performed by Schmitz-Schug et al., 2014). The functional forms were
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empirically derived from the moisture content variation of the fit parameters obtained in
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stage two, as follows: For the reversible WLF model,
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max 1
1
exp
Cf
Cf =
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E Aeq ( WLF ) = E Aeq ( WLF ) ∞ + (E Aeq ( WLF ) − − E Aeq ( WLF ) ∞ )( −
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where X is the dry basis moisture content (kg kg-1) and W is the solids fraction defined
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as:
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W =
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For the reversible Arrhenius model,
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k 0f max k 0f = exp 1 + Bf exp(− βf X )
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E Aeq ( Arr ) = E Aeq ( Arr ) ∞ + (E Aeq ( Arr ) − − E Aeq ( Arr ) ∞ )( −
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For the irreversible WLF model,
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Ci =
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For the irreversible Arrhenius model,
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k
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E Ai =
+ Af
(− α f X ) exp 1
(− ε WLFW ))
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1
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(− α i X )
0
(− β i X )
) Arr
exp
exp
max
+ Gi
1
1
AC C
exp
0
k(i + Bi E Ai
(− ε Arr W ))
(12) (13)
(14)
(15)
(16)
(17)
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exp
+ Ai
exp 1
1
max 1
1
Ci
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+X
1
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i
=
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1
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(− γ i X )
(18)
(19)
The model performances were judged using the R2 and adjusted R2 values
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where Np is the number of parameters in each model, and N, the total number of data
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points:
11
(
X t)−c
2
N R 2 Adj = 1 − 1 − R 2 N − N p − 1
(T
−
exp
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∑ c
, ,
R = −
, ,
X t ) − c fit (T X t ))
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2
, ,
, ,
exp exp
1
2
∑ (c (T
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)
(21)
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3.
Results and discussion
3.1.
Experimental results and first stage of the kinetic analysis
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(20)
(T X t )
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Fig. 1 shows example DSC thermograms for samples after 2 min of isothermal
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hold at (a) different temperatures (at X = 0) and (b) different moisture contents (at T =
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90 °C). It can be seen that increasing either the s ample moisture or temperature
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results in a reduction of the residual denaturation peak, indicating that more
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denaturation occurred during the previous isothermal hold. The same trends were
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observed across all samples. For wet samples, it is very well established that
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increasing the temperature produces a more rapid denaturation, but this can now be
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seen across the whole moisture content range. It is also seen that the presence of
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moisture promotes denaturation. This agrees with findings reported by Haque et al.
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(2013a). The presence of more moisture might also improve heat transfer within the
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pans, but as this is considered only to affect temperatures within the first minute of
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heat treatment (or less) it is unlikely to explain the large differences in behaviour
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observed.
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The first stage of the kinetic analysis was to plot, for each moisture content, the
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first-order reversible and irreversible rate constants (equations 3, 4 and 5) and
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equilibrium constants against temperature and inverse temperature, as shown in Fig. 12
ACCEPTED MANUSCRIPT 2. As expected, for each moisture content, the rate constant increases with
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temperature. It can also be seen that an increase in moisture content brings about an
283
increase in the rate constant for the same temperature. From the left-hand plots of Fig.
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2, it is generally observed that there is a near-linear relationship between the log of the
285
forward and irreversible rate constants and temperature, with the gradients rising as
286
moisture contents increase. Hence, as moisture content drops, progressively higher
287
temperatures are needed to achieve the same effect. The exception to this rule are the
288
rate constants for the X = 2.5 data that appear to level off at ~80–90 °C. A similar
289
“sharp bend” was also observed by Tolkach and Kulozik (2007) at a similar
290
temperature in their study of β-lactoglobulin at high hydration. Tolkach and Kulozik
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(2007) proposed that the change in slope was due to a change in limiting mechanism.
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Below 90 °C the kinetics are limited by the initial unfolding step, and above 90 °C they
293
are limited by the subsequent aggregation step. The rationale was that DSC
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endotherms of denaturation of hydrated samples tend to finish at around 87 °C, and as
295
these are sensitive to the unfolding rather than aggregation step, this suggests that this
296
step is complete by 90 °C. Another possible explana tion is a thermal lag effect, in that
297
heat transfer into the pans [or, in the case of Tolkach & Kulozik (2007), steel tubes]
298
becomes a rate limiting factor when the experimental time scales become short (at
299
high reaction rates). This is possible as in the experiments reported here, the effect
300
was only seen when rate constants exceeded 3 min-1 (log10 k = 0.5) which only
301
occurred with the fully hydrated samples.
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At lower temperatures, the rate constant lines appear to converge such that
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there is a temperature at which the rate constant becomes independent of moisture
304
content (and is also very low). This can be viewed as a base or reference temperature
305
Td, and the rate of denaturation at any temperature T, is essentially dependent on the
306
difference between this temperature and Td, with slight nonlinearities. This and the 13
ACCEPTED MANUSCRIPT near-linear relationship of the plots in Fig 2, tends to favour a WLF-type form of
308
equation (rather than an Arrhenius type), but with the glass transition temperature Tg in
309
the normal WLF equation replaced by Td (see equations 5 and 6). However, a
310
fundamental difference between the standard WLF equation and the one used here is
311
that Td (unlike Tg) is independent of moisture content. The link with moisture content
312
(which explains differences in rate constants as T > Td) can be established through the
313
C-parameters. From the left-hand plots of Fig. 2, Td can be projected to be between 60
314
and 70 °C for the reversible kinetics and 50 and 70 °C for the irreversible kinetics (as
315
will be confirmed by fitting results presented in the next section). The presence of
316
these apparent convergence points was the rationale behind the use of the modified
317
WLF equation, and also the reason why the parameters Tdf, Tdi, kfTd and kfTi were
318
assumed constant in the second stage of the kinetic analysis.
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The equilibrium constant Keq (top-right of Fig. 2) also tends to increase with moisture content and temperature. The Keq lines appear to converge at a point
321
between 50 and 70 °C. Consequently, Tref(WLF), Tref(Arr), Keqref(WLF) and Keqref(Arr) were
322
also assumed constant in the second stage of the kinetic analysis. The complete set of fitting results from the second stage of the modelling
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process is shown in Table 3. Some parameters, notably C1f and C1i from the WLF
325
models, and k0f, k0i, EAeq(Arr) and EAi from the Arrhenius models showed evidence (in
326
Table 3) of a sigmoidal variation with moisture content X, as the values appear to
327
reach a limiting value as X is increased. This would make sense physically as once the
328
proteins have become fully hydrated the effect of additional water should not make any
329
further difference. Sigmoidal functions of X were therefore chosen to represent these
330
quantities in the global model. This is different to the functions used by Schmitz-Schug
331
et al. (2014) for lysine loss, as that exhibited an optimum intermediate moisture content
332
where reaction rates were highest.
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3.2.
Final parameter fitting results: models and data comparison
335 336
The globally fitted values of the model parameters from the final (third) fitting stage are presented in Table 4. Good agreement is observed between the
338
experimental data and the four models in most cases, with the WLF-type models
339
performing better than the Arrhenius-based models as seen in the table of R2 and
340
adjusted-R2 values (Table 5). Of these, the reversible-WLF model performs the best,
341
both in terms of R2 and adjusted-R2 suggesting that the superior performance is due to
342
its better predictive power, rather than just the higher number of parameters.
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Figs. 3–6 show examples of the denaturation kinetic data (undenatured protein concentration against time) compared with the different model fits at different
345
temperatures and moisture contents. The slower kinetics associated with lower
346
moisture contents can be seen by the higher temperatures used in Fig. 4 (X = 2.5)
347
compared with Fig. 3 (X = 0.23), to achieve similar denaturation. For certain
348
combinations of moisture content and temperature (e.g., X = 2.5 and T = 65 °C; X =
349
0.23 and T = 80 °C; X = 0, 0.05, 0.08 and T = 100 °C – see Figs. 3–5), it can be seen
350
from the data that denaturation is incomplete even at long times, which justifies the use
351
of the reversible equilibrium model to fit the data. In reality, it is unlikely that denatured
352
proteins are able to convert back to undenatured proteins, but the value of the
353
reversible equation is its ability to predict a non-zero concentration after an infinite
354
time, i.e., ceq = krc0/(kf+kr) (from equation 3) which could be useful in extrapolating
355
results to low-temperature systems. The reversible models are also observed to have
356
excellent performance in fitting the early kinetics (the first 2 minutes) of denaturation;
357
approaching an R2 of about 0.99 for this range. This is particularly important for short
358
residence-time dehydration processes such as spray drying.
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ACCEPTED MANUSCRIPT In terms of the model parameter values and equation behaviours, the following
359
are observed. First, the base denaturation temperature (Td) used by the reversible
361
WLF model is 65 °C as against 58 °C for the irrever sible WLF model. This is possibly
362
due to the existence of the reverse kinetic kr=kf/Keq term, which tends to limit
363
denaturation in the reversible model, leading to the requirement of a higher
364
temperature (compared to the irreversible model) to initiate the denaturation process.
365
Also, the moisture-independent reference temperatures in the equations for the
366
equilibrium constants Tref(WLF) and Tref(Arr) are 54 and 58 °C, respectively. These results,
367
obtained from the global fitting agree with our earlier postulations from the preliminary
368
kinetic analysis of Fig. 2.
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The C2f value in the reversible WLF model is very high and dwarfs the (T - Td)
369
term, suggesting that the denominator term in the WLF exponent (C2f + T- Td) is
371
unnecessary and can dispensed with. This reduces the number of fit parameters by
372
one. The resulting equation is thus of the form:
373
kf = kfTd 10
374
where
375
β=
df
)
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(C1f max / C2f ) = C1f β max = C2f 1 + Af exp (− α f X ) 1 + Af exp (− α f X )
(22)
(23)
The fitted value of βmax is 0.181 K-1 (the same as the ratio C1fmax/C2f from the
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(
β T −T
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377
reversible WLF model) with the other parameter values as given in Table 4 from the
378
reversible WLF model. The functional form of equation 22 also exactly mirrors the
379
linear relationships observed in Fig. 2.
380
In all the models, the model parameters are explicit algebraic functions of
381
moisture content. This ensures easy computation of protein denaturation within usually
382
complex drying models described by partial differential equations. The models thus
16
ACCEPTED MANUSCRIPT 383
provide a platform for drying trajectory optimisation with respect to protein denaturation
384
in dairy products.
385 386
3.3.
Possibilities for drying trajectory optimisation
388
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387 Usually, in the optimisation of drying processes, many variables are considered simultaneously. These may include energy consumption or quality indicators such as
390
protein stability. Hence, it is important to know if there exists a Pareto space within
391
which the same performance can be achieved to enhance multi-objective optimisation
392
possibilities. Lines of equal native protein concentration derived from the reversible
393
WLF model are shown in Fig. 7 (only three lines 10%, 50% and 90% native protein
394
concentrations are shown). These indicate that for each processing time, there is a
395
family of temperature-moisture content combinations that achieve equal denaturation
396
(raising the moisture content lowers the temperature required). The Pareto space is
397
seen to exist within a strip (from 0% to 100% protein concentration), and of course,
398
optimisation will usually aim towards the top part of the strip. The width of the space is
399
observed to increase with processing time. For low moisture content values, the
400
operating line of equal denaturation is sensitive to both moisture content and
401
temperature changes. However, as moisture content rises, it becomes independent of
402
moisture content but still strongly sensitive to temperature changes. The operating
403
window of process conditions for which the same denaturation can occur is thus,
404
predicted by the models. The trend is the same for the other models. Hence, the
405
models serve as useful inputs for multi-criteria optimisation in cases where conflicts
406
exist between protein levels (as a quality indicator) and some other processing
407
objective, as is often the case in practice.
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ACCEPTED MANUSCRIPT In general, as seen from the surface plots of protein denaturation (Fig. 8), the
408
denaturation kinetics are very fast for very high temperature-moisture content
410
combinations. As time progresses, more lower temperature-moisture content
411
combinations, “push upwards”, towards achieving complete (100%) denaturation, with
412
increased steepness lying on the high-temperature side. Hence, the models can be
413
used in fixing the end-time of the dehydration process based on desired objectives.
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409
414 4.
Conclusions
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415 416
DSC analysis of heat treated whey protein samples clearly shows that the rate
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of protein denaturation is an increasing function of moisture content as well as
419
temperature. At lower temperatures, the data showed that denaturation is unlikely to
420
go to completion, even after very long times. This was reflected in the model fits which
421
showed the reversible models having a better performance.
422
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Better model fits were found with the WLF-type temperature dependency formulation than the commonly used Arrhenius formulation. The best combination of
424
complexity and accuracy was the simplification of the reversible WLF model in which
425
the denaturation rate varies exponentially with the difference between the processing
426
temperature and a moisture-independent reference temperature. The model provides
427
reaction rate constants as explicit functions of temperature and moisture content and
428
this permits easy computation of protein denaturation within usually complex drying
429
models. They thus provide a platform for drying trajectory optimisation with respect to
430
protein denaturation in dairy products. The models also establish operating windows of
431
process conditions for which the same denaturation occurs. They are thus useful
432
inputs for multi-criteria optimisation in cases where conflicts exist between protein
433
levels (as a quality indicator) and some other processing objective as is often the case
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ACCEPTED MANUSCRIPT in practice. The models can be used to predict denaturation in variable moisture and
435
temperature systems, and incorporated into spray drying and other dehydration
436
models. However a recommendation for future work is to assess whether the presence
437
of other components (such as lactose or salt) have any effect on denaturation kinetics
438
over and above changes to water activity.
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Acknowledgement
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The research leading to these results has received funding from the European
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Research Council under the European Union’s Seventh Framework Programme
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(FP7/2007-2013)/ ERC grant agreement no. 613732 – ENTHALPY – www.enthalpy-
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fp7.eu
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Al-Attabi, Z., D’Arcy, B. R., & Deeth, H. C. (2009). Volatile sulphur compounds in UHT milk. Critical Reviews in Food Science and Nutrition, 49, 28–47.
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References
Anandharamakrishnan, C., Rielly, C. D., & Stapley, A. G. F. (2007). Effects of process variables on the denaturation of whey proteins during spray drying. Drying
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Technology, 25, 799–807.
Chen, X. D., Chen, Z. D., Nguang, S. K., & Anema, S. (1998). Exploring the reaction kinetics of whey protein denaturation/aggregation by assuming the denaturation
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step is reversible. Biochemical Engineering Journal, 2, 63–69.
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Dissanayake, M., Ramchandran, L., Donkor, O. N., & Vasiljevic, T. (2013).
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Denaturation of whey proteins as a function of heat, pH and protein
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concentration. International Dairy Journal, 31, 93–99. 19
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Foster, K. D., Bronlund, J. E., & Paterson, A. H. J. (2005). The prediction of moisture sorption isotherms for dairy powders. International Dairy Journal, 15, 411–418. Haque, M.A., Aldred, P., Chen, J., Barrow, C.J., & Adhikari, B. (2013a). Comparative study of denaturation of whey protein isolate (WPI) in convective air drying and
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isothermal heat treatment processes. Food Chemistry, 141, 702–711.
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Haque, M. A., Putranto, A., Aldred, P., Chen, J., & Adhikari, B. (2013b). Drying and
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denaturation kinetics of whey protein isolate (WPI) during convective air drying
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process. Drying Technology, 31, 1532–1544.
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Lamb, A., Payne, F., Xiong, Y. L., & Castillo, M. (2013). Optical backscatter method for determining thermal denaturation of β-lactoglobulin and other whey proteins in
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milk. Journal of Dairy Science, 96, 1356–1365.
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Meerdink, G., & van’t Riet, K. (1995). Prediction of product quality during spray drying. Food and Bioproducts Processing, 73, 165–170.
Oldfield, D. J., Singh, H., & Taylor, M. W. (2005). Kinetics of heat-induced whey protein
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denaturation and aggregation in skim milks with adjusted whey protein
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concentration. Journal of Dairy Research, 72, 369–378. Oldfield, D. J., Singh, H., Taylor, M. W., & Pearce, K. N. (1998). Kinetics of
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denaturation and aggregation of whey proteins in skim milk, heated in an ultra-
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high temperature (UHT) pilot plant. International Dairy Journal, 8, 311–318.
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Parris, N., Purcell, J. M., & Ptashkin, S. M. (1991). Thermal denaturation of whey proteins in skim milk. Journal of Agricultural and Food Chemistry, 39, 2167– 2170.
Schmitz, I., Gianfrancesco, A., Kulozik, U., & Foerst, P. (2011). Kinetics of lysine loss
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in an infant formula model system at conditions applicable to spray drying.
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Drying Technology, 29, 1876–1883.
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Schmitz-Schug, I., Kulozik, U., & Foerst, P. (2014). Reaction kinetics of lysine loss in a
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model dairy formulation as related to the physical state. Food and
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BioprocessTechnology, 7, 877–886. Tolkach, A., & Kulozik, U. (2007). Reaction kinetic pathway of reversible and
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irreversible thermal denaturation of β-lactoglobulin. Lait, 87, 301–315.
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Wijayanti, H. B., Bansal, N., & Deeth, H. C. (2014). Stability of whey proteins during
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thermal processing: A review. Comprehensive Reviews in Food Science and
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Food Safety, 13, 1235–1251.
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protein concentrations. International Dairy Journal, 49, 95–101.
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Wolz, M., & Kulozik, U. (2015). Thermal denaturation kinetics of whey proteins at high
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Figure legends
2 Fig. 1. DSC thermograms showing the dependence of denaturation peaks on (a)
4
previous hold temperature and (b) sample moisture content, after a hold time of 2
5
minutes.
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3
6
Fig. 2. Logarithms (to the base 10) of the kinetic rate and equilibrium constants plotted
8
versus temperature or inverse temperature, for each moisture content (X): , X = 2.5;
9
, X = 0.23; , X = 0.18; , X = 0.08; , X = 0.05; , X = 0).
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10 11
Fig. 3. Evolution of undenatured protein concentration versus time at different
12
temperatures and for a dry basis moisture content (X) of 2.5 kg kg-1. Plots show
13
experimental data () and model fits: rWLF (
14
reversible Arrhenius; iWLF (
), reversible WLF; rARR (
), irreversible Arrhenius).
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), irreversible WLF; iARR (
),
15
Fig. 4. Evolution of undenatured protein concentration versus time at different
17
temperatures and for a dry basis moisture content (X) of 0.23 kg kg-1. Plots show
18
experimental data () and model fits: rWLF (
19
reversible Arrhenius; iWLF (
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16
), reversible WLF; rARR (
), irreversible WLF; iARR (
),
), irreversible Arrhenius).
21
Fig. 5. Evolution of undenatured protein concentration versus time at different dry
22
basis moisture contents (X) and at a temperature of 100 °C. Plots show experi mental
23
data () and model fits: rWLF (
24
iWLF (
), reversible WLF; rARR (
), irreversible WLF; iARR (
), reversible Arrhenius;
), irreversible Arrhenius).
25
1
ACCEPTED MANUSCRIPT 26
Fig. 6. Evolution of undenatured protein concentration versus time at different dry
27
basis moisture contents (X) and temperatures (T). Plots show experimental data ()
28
and model fits: rWLF (
29
(
), reversible WLF; rARR (
), irreversible WLF; iARR (
), reversible Arrhenius; iWLF
), irreversible Arrhenius).
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30 31
Fig. 7. Lines of equal (percent) undenatured protein concentration at different times
32
(12 s, 30 s, 1 min and 2 min)
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33
Fig. 8. Surface plots of percentage denaturation against temperature and moisture
35
combinations at different times.
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2
ACCEPTED MANUSCRIPT Table 1 Nomenclature.
Initial, at temperature = ∞ at X=∞ Activation Arrhenius Minimum denaturation Experimental data Equilibrium Forward Fitted value Irreversible Maximum Reverse Reference William-Landel-Ferry
%
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Subscripts 0 ∞ A Arr d exp eq f fit i max r ref WLF
Unit
°C J mol-1 min-1
SC
Meaning Model parameters Fraction undenatured protein WLF-type parameter 1 WLF-type parameter 2 Energy (specific) Rate constant Total number of data points Total number of model parameters X-dependent parameter equation forms Parameter values of q Gas constant Coefficient of determination Time Temperature Solids fraction Moisture content (dry basis) Model steepness factors
J mol-1 K-1
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Symbol A, B, G c C1 C2 E k N Np Q Q R R2 t T W X α, β, γ, ε
min ⁰C, K kg kg-1 kg kg-1 kg kg-1
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Table 2
WLF Reversible
kfTd
C1f
Arrhenius Reversible
k0f
EAf
WLF Irreversible
kfTd
C1i
Arrhenius Irreversible
k0i
EAi
C2f
C2i
Tdf
Tdf
Keqref(WLF)
EAeq(WLF)
Tref(WLF)
Keqref(Arr)
EAeq(Arr)
Tref(Arr)
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Parameters
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Model
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Fit parameters used in the second stage of model development. a
ACCEPTED MANUSCRIPT Table 3 Fitted parameter values from second stage fits for different moisture contents (X).
-1
-1
X = 0.05
X = 0.08
X = 0.18
X = 0.23
X = 2.5
-79.39 76,910 0.0129
-94.90 76,910 0.0128
-111.3 94,890 0.0129
-196.8 118,500 0.0129 0.0316 5533.4 65.1 54.1
-218.1 197,100 0.0128
-1000 394,200 0.0129
2.715E7 73,530
5.320E7 73,530
7.7995E7 3.352E8 79,050 113,700 62,640 0.0197 58.0
5.400E8 153,500
4.985E9 361,800
-13.11
-14.58
-16.73
-27.27
-70.80
7.907E9 83,320
7.898E11 96,080
1.505E13 104,000
7.616E19 142,900
1.095E33 222,700
min K °C °C
Arrhenius reversible -1 k0f min -1 -1 EAeq(Arr) J K mol -1 -1 EAf J K mol Keqref(Arr) °C Tref(Arr) °C WLF irreversible C1i -1 kfTd min C2i K Tdf °C
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Arrhenius irreversible -1 k0i min -1 -1 EAi J K mol
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-1
J K mol
X=0
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WLF reversible C1f EAeq(WLF) Keqref(WLF) kfTd C2f Tdf Tref(WLF)
Units
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Parameter
-22.56 70,960 341.6 57.8
4.584E17 130,300
ACCEPTED MANUSCRIPT Table 4 Final fitted parameter values for the WLF and Arrhenius models. WLF
ARR
Parameter
Value
Unit
Parameter
Value
Unit
kfTd
0.0316
min
k0fmax
22.33
min
C1fmax
-1000
Bf
0.3208
Af
11.85
βf
4.647
αf
5.406
kg kg
EAf
62640
C2f
5533
K
KEQref(Arr)
0.0197
Tdf
65
°C
EAQref(Arr)∞
4.901E+05
J K mol
KEQref(WLF)
0.0122
EAQref(Arr) -1
-3.384E+06
J K mol
EAQref(WLF)∞
530 000
J K mol
EAQref(WLF) -1
-5.664E+09
J K mol
Reversible
-1
kg kg -1
-1
-1
-1
J K mol
-1
-1
-1
εArr
0.1174
kg kg
-1
-1
Tref(Arr)
54.6
°C
k0i(Arr)
76.08
min
Bi
2.338
εWLF
8.14E-05
kg kg
Tref(WLF)
54
°C
kiTd
7.18E-04
min
C1imax
-70.81
Ai
4.652
αi
4.523
C2i
341.6
Tdi
58
Irreversible
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-1
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-1
-1
-1
-1
-1
βi
5.487
kg kg
kg kg
EAimax
223 000
J K mol
K
Gi
1.672
°C
γi
4.772
-1
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-1
-1
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-1
-1
-1
-1
kg kg
ACCEPTED MANUSCRIPT Table 5 R2 and Adjusted-R2 values for the different models. Model type
Number of
R
2
Adjusted-R
2
11
0.9128
0.9067
Irreversible WLF
6
0.9006
0.897
Reversible Arrhenius
9
0.8953
0.8887
Irreversible Arrhenius
6
0.8555
0.8493
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Reversible WLF
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2 3 4
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Figure 1.
8
1
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2 log (kf (1/min))
X=2.5 X=0.23
3
X=0.18 X=0.05 X=0.08 X=0
2 log(Keq)
1 0
1 0
-1 90 110 130 o Temperature ( C)
150
-1 50 1 log(ki(1/min))
log(ki(1/min)
1 0
-1
-1
-2
-2
9
70
90 110 130 o Temperature ( C)
10 11 12
14 15
2.4E-3 2.6E-3 2.8E-3 1/Temperature (1/K)
3E-3
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Figure 2.
-3 2.2E-3
150
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150
90 110 130 o Temperature ( C)
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16
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70
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50
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ACCEPTED MANUSCRIPT 18
40 20 0 0
Concentration (%)
100
Data iWLF rWLF iARR rARR 10 20 30 Time (mins)
40 20 0 0 100
40
60
40
40
20
20
5 10 Time (mins)
20 21
15
0 0
60
80oC
1
2 3 Time (mins)
4
5
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20 40 Time (mins)
80
60
19
23 24
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26
28
60
75oC
80
0 0
27
80
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60
70oC
SC
80
100 Concentration (%)
65oC
M AN U
Concentration (%)
100
Figure 3.
3
ACCEPTED MANUSCRIPT 80
60
40
40 iWLF iARR 10 20 Time (mins)
Concentration (%)
30
0 0
100oC
80
80
60
60
40
40
20
20
0 0 29
10 20 Time (mins)
100
100
1
2 3 Time (mins)
30 31
4
5
0 0
30
110oC
1
2 3 Time (mins)
4
5
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32 33 34
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35 36
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39
20
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Data rWLF rARR
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20
90oC
80
60
0 0
38
100
80oC
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Concentration (%)
100
Figure 4.
4
ACCEPTED MANUSCRIPT 40 X=0.18
60
iWLF iARR
100
60
40
40
20
20
0 0
5 10 Time (mins)
15
60 40 20
X=0.05
41
10 20 Time (mins)
42
44 45
60 40 20
5 10 Time (mins)
30
0 0
15
10 20 Time (mins)
X=0
30
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51
80
X=0.08
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50
Concentration (%)
100
80
0 0
0 0
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Concentration (%)
100
49
80
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80
Data rWLF rARR
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Concentration (%)
100
Figure 5.
5
60
40
20
20
Concentration (%)
1
2 3 Time (mins)
0 0
40
40
20
20 2 3 Time (mins)
54
5
0 0
2
1
2 3 Time (mins)
4
5
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55
4
1 1.5 Time (mins)
X=0 T=150oC
80 60
1
0.5
100
60
53
56 57
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58 59
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60
63
5
X=0 T=140oC
80
52
62
4
100
0 0
61
60
40
0 0
X=0.05 T=150oC
80
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80
100
Data iWLF rWLF iARR rARR
X=0.05 T=140oC
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100
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Concentration (%)
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Figure 6.
6
ACCEPTED MANUSCRIPT 64 10 9050
130
110
110
90
2.5
50 0 50
130
130
110
110
70
65
9090
5010 90
50 10
0.5 1 1.5 2 kg-1) Moisture content (kg (kg/kg)
66 67
2.5
50 0
5010
5010
90
2.5
t=2 min
10 50
10 50 90
10 50 90
0.5 1 1.5 2 kg-1) Moisture content(kg (kg/kg)
10 50 90
2.5 2.5
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90
0.5 1 1.5 2 kg-1) Moisture content(kg (kg/kg)
150 10
t=1 min
5010
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70
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0.5 1 1.5 2 kg-1) Moisture content (kg (kg/kg)
5010
50 0
69 70
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71 72
AC C
73
75
5010
90
90
10 50
90
90 5010
70
90
t=0.5 min
90
90
150
74
150 5010
130
50 0
Temperature (°C)
t=0.2 min
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150
Figure 7.
7
(kg kg-1)
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Figure 8.
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S0958-6946(17)30150-4
DOI:
10.1016/j.idairyj.2017.07.002
Reference:
INDA 4197
To appear in:
International Dairy Journal
Received Date: 23 May 2017 Revised Date:
17 July 2017
Accepted Date: 17 July 2017
Please cite this article as: Atuonwu, J.C., Ray, J., Stapley, A.G.F., A kinetic model for whey protein denaturation at different moisture contents and temperatures, International Dairy Journal (2017), doi: 10.1016/j.idairyj.2017.07.002. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
ACCEPTED MANUSCRIPT A kinetic model for whey protein denaturation at different moisture
2
contents and temperatures
3
5 6 7 James C. Atuonwu, Joydeep Ray, Andrew G.F. Stapley*
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9
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Department of Chemical Engineering, Loughborough University, Loughborough,
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Leicestershire, LE11 3TU, UK.
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*Corresponding author. Tel.: +44 1509 222 525
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E-mail address: [email protected] (A. G. F. Stapley)
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ACCEPTED MANUSCRIPT 24
_____________________________________________________________________
25
ABSTRACT
26 The denaturation of whey protein samples that had previously undergone heat-
28
treatment for different times at different temperatures and moisture contents was
29
analysed by differential scanning calorimetry (DSC), using the DSC enthalpy as a
30
measure of residual undenatured protein. Data were fitted to first order irreversible or
31
reversible kinetic expressions, and the resulting rate constants were found to increase
32
with both temperature and moisture content. The whole data set was then fitted as a
33
function of time, temperature and moisture content, with rate constants varying
34
according to either Arrhenius or Williams-Landel-Ferry (WLF) kinetics and with
35
selected fit parameters made empirical functions of moisture content. The best fits
36
were obtained using reversible WLF kinetics, which could be further slightly simplified
37
without loss of accuracy. The model provides a platform for single- and multi-objective
38
drying trajectory optimisation with respect to protein denaturation in dairy products.
39
_____________________________________________________________________
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1.
Introduction
41 42
Proteins are highly desirable components in food products because of their high nutritional value and contributions to sensory attributes such as colour, flavour and
44
texture. Whey proteins, in particular, play a huge role due to the widespread
45
consumption of dairy produce. Constituting about 20% of the proteins in milk, they
46
have the highest biological value among all known proteins (Anandharamakrishnan,
47
Reilly, & Stapley, 2007; Wijayanti, Bansal, & Deeth, 2014). They are used as additives
48
in formulated products such as bakery products, processed meats, beverages, pasta,
49
ice cream, confectionery, infant foods, spreads, dips and as encapsulating agents in
50
pharmaceuticals.
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For convenience and to extend their shelf life, liquid dairy products are often subjected to dehydration processes such as evaporation and spray drying. Spray
53
drying, in particular, is the generally-accepted method for producing dry powder. The
54
dehydration process involves thermal treatment, during which whey proteins can lose
55
their normal secondary, tertiary or quaternary structures. Denaturation, as this process
56
is called, transforms the protein from its native folded, biologically-active state to an
57
unfolded, biologically-inactive state. Physically, this can manifest in aggregation,
58
coagulation, reduced solubility for powders and loss of some functional properties such
59
as gelling and emulsification (Haque, Aldred, Chen, Barrow, & Adhikari, 2013a,b). In
60
addition, the denaturation of whey proteins is accompanied by the release of small
61
sulphur-containing compounds such as methanethiol and hydrogen sulphide that
62
produce cooked flavours in heated milk (Al-Attabi, D’Arcy, & Deeth, 2009). The
63
aggregation that results from denaturation can potentially lead to processing downtime
64
due to heat exchanger fouling and nozzle blocking in spray dryers. Consequently,
65
whey protein denaturation is a major issue in dairy processing.
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The study of denaturation during spray drying is complicated by the variation of moisture content and temperature with both position in the droplet/particle and time.
68
Understanding the mechanism of whey protein denaturation with the ability to predict
69
its extent over a wide range of process and product conditions is important for deriving
70
dehydration strategies to minimise this phenomenon. For this, reliable kinetic models
71
are required. Studies on the thermal denaturation of whey proteins abound in the
72
literature (e.g., Dissanayake, Ramchandran, Donkor, & Vasiljevic, 2013; Lamb, Payne,
73
Xiong, & Castillo, 2013; Oldfield, Singh, & Taylor, 2005; Wolz & Kulozik, 2015),
74
especially in fully hydrated systems (Wolz & Kulozik, 2015). However, most of these
75
studies have not examined how denaturation behaviour varies with moisture content.
76
An exception is the work of Haque et al. (2013a,b), who investigated the denaturation
77
of whey protein isolate solutions in suspended droplets whilst drying. They assumed a
78
model equation used by Meerdink and van’t Riet (1995) for enzyme inactivation to
79
describe the variation of kinetics with temperature and moisture content. This was
80
inserted into a drying model which predicted the moisture content and temperature
81
history of the droplets from which the kinetic parameters for the denaturation were
82
inferred via inverse model fits. However, this experimental study focussed mainly on
83
high moisture content levels (40% and above), over limited temperatures (65 and 80
84
°C), and with results sampled at 60 s intervals, wh ich exceeds the times for most
85
commercial spray drying operations. Moreover, since the experiments were in a
86
convective drying environment, continuous temperature and moisture content
87
variations occurred as a result of which, the denaturation kinetics at constant
88
temperature and moisture levels were not resolved (two experiments were performed
89
at constant temperature and moisture content but these appear to have been at high
90
moisture contents only). Therefore, although it is well known that protein damage
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occurs at temperatures above 60 °C in fully wet sys tems, little understanding exists on
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the ability of proteins to withstand heat when in a partially dried state. Quantifying the simultaneous impact of moisture levels and temperatures on
93
protein denaturation kinetics will better enable the development of optimum
95
dehydration profiles with respect to protein denaturation. Such an approach has been
96
undertaken with assessing lysine losses in dairy powders, whereby powders are
97
equilibrated at different water activities (using salt solutions) and then subject to
98
thermal treatment in hermetically sealed cans for different time-temperature
99
combinations (Schmitz, Gianfrancesco, Kulozik, & Foerst, 2011; Schmitz-Schug,
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Kulozik, & Foerst, 2014).
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This work presents a similar experimental method using thermal treatment in
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combination with partial hydration followed by differential scanning calorimetry (DSC)
103
analysis to generate experimental data that quantify the combined effects of time,
104
temperature and moisture content on whey protein denaturation. From this, a set of
105
four mathematical models have been derived in a three-step process to explicitly
106
describe the denaturation kinetics. The models are based on a combination of either
107
first-order reversible or irreversible kinetics, with rate constant temperature
108
dependencies expressed using either William-Landel-Ferry (WLF)-type or Arrhenius
109
equations. Semi-mechanistic monotonic functions of moisture content are proposed for
110
some of the parameters, so a global fit of the whole data set can then be made as
111
functions of time, temperature and moisture content. The four mathematical models
112
are then compared and possibilities for optimising drying processes considered.
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2.
Materials and methods
2.1.
Experimental
115 116
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Samples (10–20 mg) of as-received low-heat whey protein isolate (WPI) powder (Impact Whey isolate, MyProtein, Northwich, UK) were loaded into high-pressure
120
stainless steel DSC pans (TA Instruments, New Castle, DE, USA), and weighed. The
121
samples used in the experiment were obtained from products manufactured using a
122
combination of membrane filtration and low-temperature drying to avoid protein
123
denaturation prior to the experiment. The composition (w/w) of the WPI powder stated
124
by the manufacturer was 90% protein, 2.5% lactose, 0.3% fat, 0.5% salt and 6.7%
125
other (including moisture). Samples were equilibrated with different saturated salt
126
solutions to various intermediate moisture levels over a few days. The salts used were
127
magnesium chloride (Acros Organics, Fair Lawn, NJ, USA), sodium chloride (Fisher
128
Scientific, Fair Lawn, NJ, USA) and potassium chloride (Hopkin and Williams Limited,
129
Harrogate, UK). Saturated solutions of these salts gave water activities and equilibrium
130
moisture levels (kg kg-1) as follows: magnesium chloride, 0.33 and 0.08, respectively;
131
sodium chloride, 0.75 and 0.18, respectively; potassium chloride, 0.86 and 0.23,
132
respectively. The relationship between the water activity generated by the saturated
133
salt solutions and the equilibrium moisture level is based on the sorption isotherms for
134
WPI powder presented by Foster, Bronlund, and Paterson (2005). The equilibration
135
vessels were 1000 mL Kilner jars to which 100 mL of RO (reverse osmosis) water was
136
added, and an excess of the desired salt such that crystals were always present. A
137
total of 10 loaded DSC pans were placed on a Petri dish, which was then placed on a
138
raised platform within the jar which was located above the level of the salt solution.
139
The jars were placed in a water bath at 25 °C. The pans were weighed at daily
140
intervals until no significant further change in mass was observed (a second Petri dish
141
was used to cover the pans to minimise any changes in moisture content whilst
142
samples were outside of the Kilner jar). Other dry basis moisture content (kg kg-1)
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powder at ambient temperature for 24 h), 0.05 (the initial moisture content of the as-
145
received powder) and 2.5 (full hydration). Samples were then sealed and heated in
146
either a stirred water bath (for temperatures < 100 °C) or a silicone oil bath (> 100 °C),
147
for different times. The pans were then plunged into a beaker of cold water, removed,
148
dried on the outside, and then opened and water added to the samples using a
149
micropipette (to a ratio 1:2.5, w/v) to ensure full hydration as required for DSC analysis
150
(Anandharamakrishnan et al., 2007). All mass measurements were performed with
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Sartorius microbalances (Sartorius AG, UK) of accuracy 0.1 mg.
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The pans were then resealed and analysed by DSC (Q10, TA Instruments),
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together with an identical but empty reference pan. In the DSC analysis, the sample
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and reference were first equilibrated at 20 °C, and then heated at a linear rate of 5 °C
155
min-1 to 100 °C. The experiments were performed in tripl icate. Protein denaturation
156
was observed in the peaks produced in the heat flow-temperature diagram. The area
157
under the peak is produced by the denaturation in the DSC of any undenatured
158
proteins left in the sample after the previous heat treatment in the water/oil bath. It is
159
thus a measure of the amount of undenatured proteins left after heat treatment.
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Enthalpies were obtained assuming a linear baseline, and were compared with
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those of a hydrated sample of as-received powder, which was used as a reference.
162
The latter yielded the highest enthalpy which was taken to represent zero denaturation
163
(or 100% native protein concentration). The fractional decrease in enthalpy from this
164
value was then taken to be the fraction of denaturation that occurred in the sample
165
during the previous heat treatment.
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2.2.
Mathematical modelling
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The existing literature on whey protein denaturation kinetic models (e.g., Dissanayake et al., 2013; Haque et al., 2013a,b; Oldfield, Singh, Taylor, & Pearce,
171
1998; Wolz & Kulozik, 2015) can be categorised into 1st, 2nd and higher-order models
172
with Arrhenius rate constant temperature dependencies. The majority of these models
173
are concerned with whey protein components either individually or in milk, whey and
174
model systems without specific reference to moisture content effects. The overall
175
denaturation process has been described as a two-step process consisting of a
176
conditionally-reversible unfolding step, and a subsequent irreversible aggregation step
177
(Chen, Chen, Nguang, & Anema, 1998; Parris, Purcell, & Ptashkin, 1991; Tolkach &
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Kulozik, 2007).
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In this work, two rate constant temperature dependencies, the Williams-LandelFerry (WLF)-type and the Arrhenius equations were tested in conjunction with both
181
reversible and irreversible first order kinetic models. Both 2nd order and “nth” order
182
kinetic models were also tested, but these generally gave poorer fits to data and were
183
therefore discarded. A full set of nomenclature for all models is provided in Table 1.
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The moisture content dependence was then built in by making some of the WLF or Arrhenius parameters functions of moisture content. The overall modelling strategy
186
was performed in three stages.
In the first stage, concentration versus time data were fitted to first-order
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reversible and irreversible kinetics, which have the following rate laws, respectively:
189
dc = −k f c + k r (c0 − c ) dt
(1)
190
dc = −k i c dt
(2)
191
where c represents the concentration (or fraction) of undenatured proteins, c0 the initial
192
value of c, such that (c0 - c) represents the concentration of denatured proteins. kf and
8
ACCEPTED MANUSCRIPT kr are the rate constants of the forward and reverse reactions respectively for the
194
reversible kinetic model, while ki is the rate constant for the irreversible kinetic model.
195
Upon integration (since rate constants do not vary with time) these yield,
197
c=
198
c =c
0
(− (k f
+ k r ) t )))
−k) it
(3) (4)
K eq =
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(For the reversible scheme an equilibrium constant can be defined as:
kf kr
(5)
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exp
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c (k r + k f ( kf + kr
exp
respectively: 0
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Values for ki, kf, and kr (and hence Keq) were obtained for each combination of
201 202
moisture content and temperature by fitting concentration versus time data using a
203
least squares method. This allowed an initial assessment of how these parameters
204
vary with temperature and moisture content.
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In stage two the temperature variation was explicitly modelled. At each moisture
205 206
content, the rate constants (kf and ki) were assumed to vary with temperature
207
according to either WLF-type (equations 6, 7) or Arrhenius (equations 8, 9) equations:
210
kf = k
0
211
ki = k
0
212 213
f
i
(7)
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E Af − RT
(8)
exp
k i = k iTd
exp
209
T −T i di C +T −T i di
10
−C
(6)
1 2
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k f = k fTd
1 2
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T −T f df C +T −T f df
10
−C
E Ai − RT
(9)
The term “activation energy” is routinely used in conjunction with the Arrhenius equation. It is used here purely as a parameter that models the temperature 9
ACCEPTED MANUSCRIPT dependence of the rate constant, rather than quantifying an actual activation energy
215
barrier (as protein unfolding is generally regarded to be driven instead by differences in
216
thermodynamic stability between folded and unfolded states). It was found that
217
attempting to model the reverse rate constant (kr) using similar equations to those
218
above produced sporadic results in a full simulation. Instead, a more reliable method
219
was to model the equilibrium constant, and use this together with the forward rate
220
constant to deduce the reverse rate constant (linked via equation 5). The temperature
221
variation of the equilibrium constant was modelled using the van’t Hoff equation for
222
both the Arrhenius and WLF-type dependencies respectively as:
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K eq = K eqref (
) WLF
K eq = K eqref (
) Arr
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1
E Aeq − ( Arr) R
− T T( ref
1
E Aeq − ( WLF) R
1
exp
) Arr
(10)
(11)
The full parameter list for each of the resulting four models is shown in Table 2.
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exp
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These models (in combination with equation 3 or 4 as appropriate) were fitted to
227
concentration data using the least squares method. Fits were initially made to data
228
pertaining to specific moisture contents, so that individual fits for each parameter in
229
Table 2 were produced for each moisture content. However, trends in the resulting fit
230
parameters were unclear. This was attributed to over-parameterisation (redundancy),
231
which made the fits susceptible to the effects of noise. To combat this, based on
232
observations made in stage one of the fitting process, some parameters (Tref, Td, kTd
233
and Keqref,) were made constant across all moisture contents, whilst the remaining
234
parameters were allowed to assume different values for different moisture contents.
235
This concluded stage two of the fitting process.
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The resulting fit parameters were then fed to the final third stage, where the moisture varying parameters from stage two were made explicit functions of moisture 10
ACCEPTED MANUSCRIPT 238
content (as also performed by Schmitz-Schug et al., 2014). The functional forms were
239
empirically derived from the moisture content variation of the fit parameters obtained in
240
stage two, as follows: For the reversible WLF model,
241
max 1
1
exp
Cf
Cf =
243
E Aeq ( WLF ) = E Aeq ( WLF ) ∞ + (E Aeq ( WLF ) − − E Aeq ( WLF ) ∞ )( −
244
where X is the dry basis moisture content (kg kg-1) and W is the solids fraction defined
245
as:
246
W =
247
For the reversible Arrhenius model,
248
k 0f max k 0f = exp 1 + Bf exp(− βf X )
249
E Aeq ( Arr ) = E Aeq ( Arr ) ∞ + (E Aeq ( Arr ) − − E Aeq ( Arr ) ∞ )( −
250
For the irreversible WLF model,
251
Ci =
252
For the irreversible Arrhenius model,
253
k
254
E Ai =
+ Af
(− α f X ) exp 1
(− ε WLFW ))
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1
1
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(− α i X )
0
(− β i X )
) Arr
exp
exp
max
+ Gi
1
1
AC C
exp
0
k(i + Bi E Ai
(− ε Arr W ))
(12) (13)
(14)
(15)
(16)
(17)
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+ Ai
exp 1
1
max 1
1
Ci
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i
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(− γ i X )
(18)
(19)
The model performances were judged using the R2 and adjusted R2 values
256
where Np is the number of parameters in each model, and N, the total number of data
257
points:
11
(
X t)−c
2
N R 2 Adj = 1 − 1 − R 2 N − N p − 1
(T
−
exp
259
∑ c
, ,
R = −
, ,
X t ) − c fit (T X t ))
258
2
, ,
, ,
exp exp
1
2
∑ (c (T
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(21)
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3.
Results and discussion
3.1.
Experimental results and first stage of the kinetic analysis
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262 263
(20)
(T X t )
264
Fig. 1 shows example DSC thermograms for samples after 2 min of isothermal
266
hold at (a) different temperatures (at X = 0) and (b) different moisture contents (at T =
267
90 °C). It can be seen that increasing either the s ample moisture or temperature
268
results in a reduction of the residual denaturation peak, indicating that more
269
denaturation occurred during the previous isothermal hold. The same trends were
270
observed across all samples. For wet samples, it is very well established that
271
increasing the temperature produces a more rapid denaturation, but this can now be
272
seen across the whole moisture content range. It is also seen that the presence of
273
moisture promotes denaturation. This agrees with findings reported by Haque et al.
274
(2013a). The presence of more moisture might also improve heat transfer within the
275
pans, but as this is considered only to affect temperatures within the first minute of
276
heat treatment (or less) it is unlikely to explain the large differences in behaviour
277
observed.
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The first stage of the kinetic analysis was to plot, for each moisture content, the
279
first-order reversible and irreversible rate constants (equations 3, 4 and 5) and
280
equilibrium constants against temperature and inverse temperature, as shown in Fig. 12
ACCEPTED MANUSCRIPT 2. As expected, for each moisture content, the rate constant increases with
282
temperature. It can also be seen that an increase in moisture content brings about an
283
increase in the rate constant for the same temperature. From the left-hand plots of Fig.
284
2, it is generally observed that there is a near-linear relationship between the log of the
285
forward and irreversible rate constants and temperature, with the gradients rising as
286
moisture contents increase. Hence, as moisture content drops, progressively higher
287
temperatures are needed to achieve the same effect. The exception to this rule are the
288
rate constants for the X = 2.5 data that appear to level off at ~80–90 °C. A similar
289
“sharp bend” was also observed by Tolkach and Kulozik (2007) at a similar
290
temperature in their study of β-lactoglobulin at high hydration. Tolkach and Kulozik
291
(2007) proposed that the change in slope was due to a change in limiting mechanism.
292
Below 90 °C the kinetics are limited by the initial unfolding step, and above 90 °C they
293
are limited by the subsequent aggregation step. The rationale was that DSC
294
endotherms of denaturation of hydrated samples tend to finish at around 87 °C, and as
295
these are sensitive to the unfolding rather than aggregation step, this suggests that this
296
step is complete by 90 °C. Another possible explana tion is a thermal lag effect, in that
297
heat transfer into the pans [or, in the case of Tolkach & Kulozik (2007), steel tubes]
298
becomes a rate limiting factor when the experimental time scales become short (at
299
high reaction rates). This is possible as in the experiments reported here, the effect
300
was only seen when rate constants exceeded 3 min-1 (log10 k = 0.5) which only
301
occurred with the fully hydrated samples.
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At lower temperatures, the rate constant lines appear to converge such that
303
there is a temperature at which the rate constant becomes independent of moisture
304
content (and is also very low). This can be viewed as a base or reference temperature
305
Td, and the rate of denaturation at any temperature T, is essentially dependent on the
306
difference between this temperature and Td, with slight nonlinearities. This and the 13
ACCEPTED MANUSCRIPT near-linear relationship of the plots in Fig 2, tends to favour a WLF-type form of
308
equation (rather than an Arrhenius type), but with the glass transition temperature Tg in
309
the normal WLF equation replaced by Td (see equations 5 and 6). However, a
310
fundamental difference between the standard WLF equation and the one used here is
311
that Td (unlike Tg) is independent of moisture content. The link with moisture content
312
(which explains differences in rate constants as T > Td) can be established through the
313
C-parameters. From the left-hand plots of Fig. 2, Td can be projected to be between 60
314
and 70 °C for the reversible kinetics and 50 and 70 °C for the irreversible kinetics (as
315
will be confirmed by fitting results presented in the next section). The presence of
316
these apparent convergence points was the rationale behind the use of the modified
317
WLF equation, and also the reason why the parameters Tdf, Tdi, kfTd and kfTi were
318
assumed constant in the second stage of the kinetic analysis.
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The equilibrium constant Keq (top-right of Fig. 2) also tends to increase with moisture content and temperature. The Keq lines appear to converge at a point
321
between 50 and 70 °C. Consequently, Tref(WLF), Tref(Arr), Keqref(WLF) and Keqref(Arr) were
322
also assumed constant in the second stage of the kinetic analysis. The complete set of fitting results from the second stage of the modelling
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process is shown in Table 3. Some parameters, notably C1f and C1i from the WLF
325
models, and k0f, k0i, EAeq(Arr) and EAi from the Arrhenius models showed evidence (in
326
Table 3) of a sigmoidal variation with moisture content X, as the values appear to
327
reach a limiting value as X is increased. This would make sense physically as once the
328
proteins have become fully hydrated the effect of additional water should not make any
329
further difference. Sigmoidal functions of X were therefore chosen to represent these
330
quantities in the global model. This is different to the functions used by Schmitz-Schug
331
et al. (2014) for lysine loss, as that exhibited an optimum intermediate moisture content
332
where reaction rates were highest.
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3.2.
Final parameter fitting results: models and data comparison
335 336
The globally fitted values of the model parameters from the final (third) fitting stage are presented in Table 4. Good agreement is observed between the
338
experimental data and the four models in most cases, with the WLF-type models
339
performing better than the Arrhenius-based models as seen in the table of R2 and
340
adjusted-R2 values (Table 5). Of these, the reversible-WLF model performs the best,
341
both in terms of R2 and adjusted-R2 suggesting that the superior performance is due to
342
its better predictive power, rather than just the higher number of parameters.
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Figs. 3–6 show examples of the denaturation kinetic data (undenatured protein concentration against time) compared with the different model fits at different
345
temperatures and moisture contents. The slower kinetics associated with lower
346
moisture contents can be seen by the higher temperatures used in Fig. 4 (X = 2.5)
347
compared with Fig. 3 (X = 0.23), to achieve similar denaturation. For certain
348
combinations of moisture content and temperature (e.g., X = 2.5 and T = 65 °C; X =
349
0.23 and T = 80 °C; X = 0, 0.05, 0.08 and T = 100 °C – see Figs. 3–5), it can be seen
350
from the data that denaturation is incomplete even at long times, which justifies the use
351
of the reversible equilibrium model to fit the data. In reality, it is unlikely that denatured
352
proteins are able to convert back to undenatured proteins, but the value of the
353
reversible equation is its ability to predict a non-zero concentration after an infinite
354
time, i.e., ceq = krc0/(kf+kr) (from equation 3) which could be useful in extrapolating
355
results to low-temperature systems. The reversible models are also observed to have
356
excellent performance in fitting the early kinetics (the first 2 minutes) of denaturation;
357
approaching an R2 of about 0.99 for this range. This is particularly important for short
358
residence-time dehydration processes such as spray drying.
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ACCEPTED MANUSCRIPT In terms of the model parameter values and equation behaviours, the following
359
are observed. First, the base denaturation temperature (Td) used by the reversible
361
WLF model is 65 °C as against 58 °C for the irrever sible WLF model. This is possibly
362
due to the existence of the reverse kinetic kr=kf/Keq term, which tends to limit
363
denaturation in the reversible model, leading to the requirement of a higher
364
temperature (compared to the irreversible model) to initiate the denaturation process.
365
Also, the moisture-independent reference temperatures in the equations for the
366
equilibrium constants Tref(WLF) and Tref(Arr) are 54 and 58 °C, respectively. These results,
367
obtained from the global fitting agree with our earlier postulations from the preliminary
368
kinetic analysis of Fig. 2.
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The C2f value in the reversible WLF model is very high and dwarfs the (T - Td)
369
term, suggesting that the denominator term in the WLF exponent (C2f + T- Td) is
371
unnecessary and can dispensed with. This reduces the number of fit parameters by
372
one. The resulting equation is thus of the form:
373
kf = kfTd 10
374
where
375
β=
df
)
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(C1f max / C2f ) = C1f β max = C2f 1 + Af exp (− α f X ) 1 + Af exp (− α f X )
(22)
(23)
The fitted value of βmax is 0.181 K-1 (the same as the ratio C1fmax/C2f from the
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(
β T −T
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370
377
reversible WLF model) with the other parameter values as given in Table 4 from the
378
reversible WLF model. The functional form of equation 22 also exactly mirrors the
379
linear relationships observed in Fig. 2.
380
In all the models, the model parameters are explicit algebraic functions of
381
moisture content. This ensures easy computation of protein denaturation within usually
382
complex drying models described by partial differential equations. The models thus
16
ACCEPTED MANUSCRIPT 383
provide a platform for drying trajectory optimisation with respect to protein denaturation
384
in dairy products.
385 386
3.3.
Possibilities for drying trajectory optimisation
388
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387 Usually, in the optimisation of drying processes, many variables are considered simultaneously. These may include energy consumption or quality indicators such as
390
protein stability. Hence, it is important to know if there exists a Pareto space within
391
which the same performance can be achieved to enhance multi-objective optimisation
392
possibilities. Lines of equal native protein concentration derived from the reversible
393
WLF model are shown in Fig. 7 (only three lines 10%, 50% and 90% native protein
394
concentrations are shown). These indicate that for each processing time, there is a
395
family of temperature-moisture content combinations that achieve equal denaturation
396
(raising the moisture content lowers the temperature required). The Pareto space is
397
seen to exist within a strip (from 0% to 100% protein concentration), and of course,
398
optimisation will usually aim towards the top part of the strip. The width of the space is
399
observed to increase with processing time. For low moisture content values, the
400
operating line of equal denaturation is sensitive to both moisture content and
401
temperature changes. However, as moisture content rises, it becomes independent of
402
moisture content but still strongly sensitive to temperature changes. The operating
403
window of process conditions for which the same denaturation can occur is thus,
404
predicted by the models. The trend is the same for the other models. Hence, the
405
models serve as useful inputs for multi-criteria optimisation in cases where conflicts
406
exist between protein levels (as a quality indicator) and some other processing
407
objective, as is often the case in practice.
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ACCEPTED MANUSCRIPT In general, as seen from the surface plots of protein denaturation (Fig. 8), the
408
denaturation kinetics are very fast for very high temperature-moisture content
410
combinations. As time progresses, more lower temperature-moisture content
411
combinations, “push upwards”, towards achieving complete (100%) denaturation, with
412
increased steepness lying on the high-temperature side. Hence, the models can be
413
used in fixing the end-time of the dehydration process based on desired objectives.
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414 4.
Conclusions
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415 416
DSC analysis of heat treated whey protein samples clearly shows that the rate
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of protein denaturation is an increasing function of moisture content as well as
419
temperature. At lower temperatures, the data showed that denaturation is unlikely to
420
go to completion, even after very long times. This was reflected in the model fits which
421
showed the reversible models having a better performance.
422
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Better model fits were found with the WLF-type temperature dependency formulation than the commonly used Arrhenius formulation. The best combination of
424
complexity and accuracy was the simplification of the reversible WLF model in which
425
the denaturation rate varies exponentially with the difference between the processing
426
temperature and a moisture-independent reference temperature. The model provides
427
reaction rate constants as explicit functions of temperature and moisture content and
428
this permits easy computation of protein denaturation within usually complex drying
429
models. They thus provide a platform for drying trajectory optimisation with respect to
430
protein denaturation in dairy products. The models also establish operating windows of
431
process conditions for which the same denaturation occurs. They are thus useful
432
inputs for multi-criteria optimisation in cases where conflicts exist between protein
433
levels (as a quality indicator) and some other processing objective as is often the case
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ACCEPTED MANUSCRIPT in practice. The models can be used to predict denaturation in variable moisture and
435
temperature systems, and incorporated into spray drying and other dehydration
436
models. However a recommendation for future work is to assess whether the presence
437
of other components (such as lactose or salt) have any effect on denaturation kinetics
438
over and above changes to water activity.
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439 440
Acknowledgement
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The research leading to these results has received funding from the European
442
Research Council under the European Union’s Seventh Framework Programme
444
(FP7/2007-2013)/ ERC grant agreement no. 613732 – ENTHALPY – www.enthalpy-
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fp7.eu
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Al-Attabi, Z., D’Arcy, B. R., & Deeth, H. C. (2009). Volatile sulphur compounds in UHT milk. Critical Reviews in Food Science and Nutrition, 49, 28–47.
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References
Anandharamakrishnan, C., Rielly, C. D., & Stapley, A. G. F. (2007). Effects of process variables on the denaturation of whey proteins during spray drying. Drying
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Chen, X. D., Chen, Z. D., Nguang, S. K., & Anema, S. (1998). Exploring the reaction kinetics of whey protein denaturation/aggregation by assuming the denaturation
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Dissanayake, M., Ramchandran, L., Donkor, O. N., & Vasiljevic, T. (2013).
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Denaturation of whey proteins as a function of heat, pH and protein
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Foster, K. D., Bronlund, J. E., & Paterson, A. H. J. (2005). The prediction of moisture sorption isotherms for dairy powders. International Dairy Journal, 15, 411–418. Haque, M.A., Aldred, P., Chen, J., Barrow, C.J., & Adhikari, B. (2013a). Comparative study of denaturation of whey protein isolate (WPI) in convective air drying and
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isothermal heat treatment processes. Food Chemistry, 141, 702–711.
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Haque, M. A., Putranto, A., Aldred, P., Chen, J., & Adhikari, B. (2013b). Drying and
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denaturation kinetics of whey protein isolate (WPI) during convective air drying
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process. Drying Technology, 31, 1532–1544.
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Lamb, A., Payne, F., Xiong, Y. L., & Castillo, M. (2013). Optical backscatter method for determining thermal denaturation of β-lactoglobulin and other whey proteins in
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milk. Journal of Dairy Science, 96, 1356–1365.
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Meerdink, G., & van’t Riet, K. (1995). Prediction of product quality during spray drying. Food and Bioproducts Processing, 73, 165–170.
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denaturation and aggregation in skim milks with adjusted whey protein
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concentration. Journal of Dairy Research, 72, 369–378. Oldfield, D. J., Singh, H., Taylor, M. W., & Pearce, K. N. (1998). Kinetics of
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denaturation and aggregation of whey proteins in skim milk, heated in an ultra-
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high temperature (UHT) pilot plant. International Dairy Journal, 8, 311–318.
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Parris, N., Purcell, J. M., & Ptashkin, S. M. (1991). Thermal denaturation of whey proteins in skim milk. Journal of Agricultural and Food Chemistry, 39, 2167– 2170.
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in an infant formula model system at conditions applicable to spray drying.
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Drying Technology, 29, 1876–1883.
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Schmitz-Schug, I., Kulozik, U., & Foerst, P. (2014). Reaction kinetics of lysine loss in a
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model dairy formulation as related to the physical state. Food and
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BioprocessTechnology, 7, 877–886. Tolkach, A., & Kulozik, U. (2007). Reaction kinetic pathway of reversible and
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irreversible thermal denaturation of β-lactoglobulin. Lait, 87, 301–315.
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Wijayanti, H. B., Bansal, N., & Deeth, H. C. (2014). Stability of whey proteins during
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thermal processing: A review. Comprehensive Reviews in Food Science and
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Food Safety, 13, 1235–1251.
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protein concentrations. International Dairy Journal, 49, 95–101.
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Wolz, M., & Kulozik, U. (2015). Thermal denaturation kinetics of whey proteins at high
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Figure legends
2 Fig. 1. DSC thermograms showing the dependence of denaturation peaks on (a)
4
previous hold temperature and (b) sample moisture content, after a hold time of 2
5
minutes.
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6
Fig. 2. Logarithms (to the base 10) of the kinetic rate and equilibrium constants plotted
8
versus temperature or inverse temperature, for each moisture content (X): , X = 2.5;
9
, X = 0.23; , X = 0.18; , X = 0.08; , X = 0.05; , X = 0).
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Fig. 3. Evolution of undenatured protein concentration versus time at different
12
temperatures and for a dry basis moisture content (X) of 2.5 kg kg-1. Plots show
13
experimental data () and model fits: rWLF (
14
reversible Arrhenius; iWLF (
), reversible WLF; rARR (
), irreversible Arrhenius).
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), irreversible WLF; iARR (
),
15
Fig. 4. Evolution of undenatured protein concentration versus time at different
17
temperatures and for a dry basis moisture content (X) of 0.23 kg kg-1. Plots show
18
experimental data () and model fits: rWLF (
19
reversible Arrhenius; iWLF (
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16
), reversible WLF; rARR (
), irreversible WLF; iARR (
),
), irreversible Arrhenius).
21
Fig. 5. Evolution of undenatured protein concentration versus time at different dry
22
basis moisture contents (X) and at a temperature of 100 °C. Plots show experi mental
23
data () and model fits: rWLF (
24
iWLF (
), reversible WLF; rARR (
), irreversible WLF; iARR (
), reversible Arrhenius;
), irreversible Arrhenius).
25
1
ACCEPTED MANUSCRIPT 26
Fig. 6. Evolution of undenatured protein concentration versus time at different dry
27
basis moisture contents (X) and temperatures (T). Plots show experimental data ()
28
and model fits: rWLF (
29
(
), reversible WLF; rARR (
), irreversible WLF; iARR (
), reversible Arrhenius; iWLF
), irreversible Arrhenius).
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Fig. 7. Lines of equal (percent) undenatured protein concentration at different times
32
(12 s, 30 s, 1 min and 2 min)
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33
Fig. 8. Surface plots of percentage denaturation against temperature and moisture
35
combinations at different times.
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2
ACCEPTED MANUSCRIPT Table 1 Nomenclature.
Initial, at temperature = ∞ at X=∞ Activation Arrhenius Minimum denaturation Experimental data Equilibrium Forward Fitted value Irreversible Maximum Reverse Reference William-Landel-Ferry
%
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Subscripts 0 ∞ A Arr d exp eq f fit i max r ref WLF
Unit
°C J mol-1 min-1
SC
Meaning Model parameters Fraction undenatured protein WLF-type parameter 1 WLF-type parameter 2 Energy (specific) Rate constant Total number of data points Total number of model parameters X-dependent parameter equation forms Parameter values of q Gas constant Coefficient of determination Time Temperature Solids fraction Moisture content (dry basis) Model steepness factors
J mol-1 K-1
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Symbol A, B, G c C1 C2 E k N Np Q Q R R2 t T W X α, β, γ, ε
min ⁰C, K kg kg-1 kg kg-1 kg kg-1
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Table 2
WLF Reversible
kfTd
C1f
Arrhenius Reversible
k0f
EAf
WLF Irreversible
kfTd
C1i
Arrhenius Irreversible
k0i
EAi
C2f
C2i
Tdf
Tdf
Keqref(WLF)
EAeq(WLF)
Tref(WLF)
Keqref(Arr)
EAeq(Arr)
Tref(Arr)
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Parameters
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Model
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Fit parameters used in the second stage of model development. a
ACCEPTED MANUSCRIPT Table 3 Fitted parameter values from second stage fits for different moisture contents (X).
-1
-1
X = 0.05
X = 0.08
X = 0.18
X = 0.23
X = 2.5
-79.39 76,910 0.0129
-94.90 76,910 0.0128
-111.3 94,890 0.0129
-196.8 118,500 0.0129 0.0316 5533.4 65.1 54.1
-218.1 197,100 0.0128
-1000 394,200 0.0129
2.715E7 73,530
5.320E7 73,530
7.7995E7 3.352E8 79,050 113,700 62,640 0.0197 58.0
5.400E8 153,500
4.985E9 361,800
-13.11
-14.58
-16.73
-27.27
-70.80
7.907E9 83,320
7.898E11 96,080
1.505E13 104,000
7.616E19 142,900
1.095E33 222,700
min K °C °C
Arrhenius reversible -1 k0f min -1 -1 EAeq(Arr) J K mol -1 -1 EAf J K mol Keqref(Arr) °C Tref(Arr) °C WLF irreversible C1i -1 kfTd min C2i K Tdf °C
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Arrhenius irreversible -1 k0i min -1 -1 EAi J K mol
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-1
J K mol
X=0
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WLF reversible C1f EAeq(WLF) Keqref(WLF) kfTd C2f Tdf Tref(WLF)
Units
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Parameter
-22.56 70,960 341.6 57.8
4.584E17 130,300
ACCEPTED MANUSCRIPT Table 4 Final fitted parameter values for the WLF and Arrhenius models. WLF
ARR
Parameter
Value
Unit
Parameter
Value
Unit
kfTd
0.0316
min
k0fmax
22.33
min
C1fmax
-1000
Bf
0.3208
Af
11.85
βf
4.647
αf
5.406
kg kg
EAf
62640
C2f
5533
K
KEQref(Arr)
0.0197
Tdf
65
°C
EAQref(Arr)∞
4.901E+05
J K mol
KEQref(WLF)
0.0122
EAQref(Arr) -1
-3.384E+06
J K mol
EAQref(WLF)∞
530 000
J K mol
EAQref(WLF) -1
-5.664E+09
J K mol
Reversible
-1
kg kg -1
-1
-1
-1
J K mol
-1
-1
-1
εArr
0.1174
kg kg
-1
-1
Tref(Arr)
54.6
°C
k0i(Arr)
76.08
min
Bi
2.338
εWLF
8.14E-05
kg kg
Tref(WLF)
54
°C
kiTd
7.18E-04
min
C1imax
-70.81
Ai
4.652
αi
4.523
C2i
341.6
Tdi
58
Irreversible
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-1
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-1
-1
-1
-1
-1
βi
5.487
kg kg
kg kg
EAimax
223 000
J K mol
K
Gi
1.672
°C
γi
4.772
-1
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-1
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-1
-1
-1
-1
kg kg
ACCEPTED MANUSCRIPT Table 5 R2 and Adjusted-R2 values for the different models. Model type
Number of
R
2
Adjusted-R
2
11
0.9128
0.9067
Irreversible WLF
6
0.9006
0.897
Reversible Arrhenius
9
0.8953
0.8887
Irreversible Arrhenius
6
0.8555
0.8493
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Reversible WLF
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2 3 4
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5 6 7
Figure 1.
8
1
ACCEPTED MANUSCRIPT
2 log (kf (1/min))
X=2.5 X=0.23
3
X=0.18 X=0.05 X=0.08 X=0
2 log(Keq)
1 0
1 0
-1 90 110 130 o Temperature ( C)
150
-1 50 1 log(ki(1/min))
log(ki(1/min)
1 0
-1
-1
-2
-2
9
70
90 110 130 o Temperature ( C)
10 11 12
14 15
2.4E-3 2.6E-3 2.8E-3 1/Temperature (1/K)
3E-3
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Figure 2.
-3 2.2E-3
150
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150
90 110 130 o Temperature ( C)
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16
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70
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ACCEPTED MANUSCRIPT 18
40 20 0 0
Concentration (%)
100
Data iWLF rWLF iARR rARR 10 20 30 Time (mins)
40 20 0 0 100
40
60
40
40
20
20
5 10 Time (mins)
20 21
15
0 0
60
80oC
1
2 3 Time (mins)
4
5
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80
60
19
23 24
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28
60
75oC
80
0 0
27
80
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60
70oC
SC
80
100 Concentration (%)
65oC
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100
Figure 3.
3
ACCEPTED MANUSCRIPT 80
60
40
40 iWLF iARR 10 20 Time (mins)
Concentration (%)
30
0 0
100oC
80
80
60
60
40
40
20
20
0 0 29
10 20 Time (mins)
100
100
1
2 3 Time (mins)
30 31
4
5
0 0
30
110oC
1
2 3 Time (mins)
4
5
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32 33 34
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35 36
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37
39
20
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Data rWLF rARR
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20
90oC
80
60
0 0
38
100
80oC
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100
Figure 4.
4
ACCEPTED MANUSCRIPT 40 X=0.18
60
iWLF iARR
100
60
40
40
20
20
0 0
5 10 Time (mins)
15
60 40 20
X=0.05
41
10 20 Time (mins)
42
44 45
60 40 20
5 10 Time (mins)
30
0 0
15
10 20 Time (mins)
X=0
30
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X=0.08
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Concentration (%)
100
80
0 0
0 0
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100
49
80
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Data rWLF rARR
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Concentration (%)
100
Figure 5.
5
60
40
20
20
Concentration (%)
1
2 3 Time (mins)
0 0
40
40
20
20 2 3 Time (mins)
54
5
0 0
2
1
2 3 Time (mins)
4
5
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4
1 1.5 Time (mins)
X=0 T=150oC
80 60
1
0.5
100
60
53
56 57
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58 59
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60
63
5
X=0 T=140oC
80
52
62
4
100
0 0
61
60
40
0 0
X=0.05 T=150oC
80
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80
100
Data iWLF rWLF iARR rARR
X=0.05 T=140oC
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100
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Figure 6.
6
ACCEPTED MANUSCRIPT 64 10 9050
130
110
110
90
2.5
50 0 50
130
130
110
110
70
65
9090
5010 90
50 10
0.5 1 1.5 2 kg-1) Moisture content (kg (kg/kg)
66 67
2.5
50 0
5010
5010
90
2.5
t=2 min
10 50
10 50 90
10 50 90
0.5 1 1.5 2 kg-1) Moisture content(kg (kg/kg)
10 50 90
2.5 2.5
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0.5 1 1.5 2 kg-1) Moisture content(kg (kg/kg)
150 10
t=1 min
5010
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70
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0.5 1 1.5 2 kg-1) Moisture content (kg (kg/kg)
5010
50 0
69 70
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71 72
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73
75
5010
90
90
10 50
90
90 5010
70
90
t=0.5 min
90
90
150
74
150 5010
130
50 0
Temperature (°C)
t=0.2 min
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150
Figure 7.
7
(kg kg-1)
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76
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Figure 8.
87
8