Simple Kinetic Models of Potato Sloughing

Abstract

Potato cooking was analysed on the basis of kinetic theory of potato softening. The application of the theory makes it possible to connect cooking time, i.e., time of cooking prior to the start of disintegration of the potato, with some terms of the kinetic theory, mainly kinetic coefficient and critical concentration of the intracellular bonds. Simple models were developed for description of the disintegration part in terms of the cooking part parameters. Two years of data from three different varieties, in one case in six different cultivation modes, shows that deep discrepancy exists between parameters for the non-salad varieties, which correlated with each other, and the parameters obtained for the salad variety Nicola. Nicola differed from the other analysed varieties mainly much lower cooking sensitivity to starch content. This difference was the main source of the observed loss of correlation among the different cooking parameters.

Keywords:

• Potato
• Cooking
• Texture
• Sloughing
• Disintegration
• Density
• Starch
• Cooking time
• Kinetics

INTRODUCTION

The phenomenon of “sloughing”, the flaking and disintegration of the outer layers of potato tubers cooked in water, is considered to be one of the principal characteristics of potato texture.[1] Potatoes are classified into different cooking types that are part of the potato texture profile. Potato texture results from many factors, including starch content, cell size, cell-wall composition, starch swelling pressure and breakdown of the cell-wall middle lamella during cooking.[2,3],[4] Among these factors, tuber density and the degree of disintegration after cooking are often correlated, particularly within one cultivar. However, contradictory results have also been reported regarding the relationship between specific gravity and sloughing.[1]

Different modifications of sloughing tests based on measurements of the degree of cell separation were used.[4,5,6][7,8] Direct methods for assessment of sloughing are mostly based on CPM (cooked potato mass) tests. This group of tests used to be referred to as CPW (cooked potato weight) tests. During the CPM test,[9] potato flakes are cooked in a stirred water bath then decanted and sluiced down on a sieve with 2 mm mesh. The mass of the remaining cooked potato tissue on the sieve is recorded and the characteristic time CT₁₀₀ (the maximum cooking time up to which the initial 100 g potato mass remains on the sieve) is determined by interpolation as a measure for the resistance to sloughing.

The CPM test was modified into the CPEM (cooked potato effective mass) sloughing method in our previous paper.[10] The CPEM method involves cooking the potato flakes on the sieve in a stirred water bath and determining their effective mass periodically during cooking. The effective mass is the difference between the real mass of the cooked specimen and the mass of the actual cooking medium of the same volume. The method was set up in such a way that temperature and density variations in both the tested sample and the cooking medium during a test did not affect results. The CPEM method enables us to determine cooking properties of small samples consisting of only a few tubers. Two main parts are distinguished (Fig. 1): the cooking part and the breaking part, from which two main parameters were derived. The cooking part represents the part of test in which no sample disintegration is observed. In contrast to the cooking part, the sample disintegration is characteristic for the breaking part. The cooking time (CT) corresponds to the transition from the cooking to the breaking part. In the breaking part, the initial part was fitted linearly with time and the absolute value of the obtained slope was called the slope of the breaking part (SBP).

Figure 1 Scheme of the CPEM cooking curve: CT (cooking time) defined as intersection of the IEM (initial effective mass) and the linear approximation of the breaking part by excluding its last part, SBP (slope of the breaking part) derived as the slope of the approximation with opposite sign.

The CPEM sloughing method was applied to cooking properties assessment of potato tubers with different density, i.e., with different content of starch for three potato varieties (Nicola, Agria, and Saturna) in the other previous paper.[11] It was shown that both the basic cooking parameters: CT and SBP are described in frames of the individual variety as linear functions of tuber density.

Cooking softening can be described in terms either by the chemical kinetics[12] or in terms of changes in the tuber texture.[13] In the first case, the simplest kinetic equation of the first order can be applied:

(1a)

with the solution:

(1b)

where k (min⁻¹) is kinetic coefficient; c (in units e.g. m⁻³) denotes actual bond concentration at time t; and c₀ the initial bond concentration in the raw sample. Changes in bond concentrations can be expressed by relative values c/c₀ . The equation (1a) was modified for starch gelatinization and vegetable cooking by other authors[14,15] into the following form:

(2a)

with solution:

(2b)

Parameter c_u in Eqs. (2a) and (2b) represents the asymptotic bond concentration at infinite time; it could be understood as the remained bond concentration after thermal processing. In this paper the results obtained previously in the CPEM potato tests are analysed using a simple theory based on the cooking kinetics.

RESULTS AND DISCUSSION

Cooking Part

The Cooking Part of a CPEM curve is accompanied by molecular bond destruction in the whole volume of the cooked tuber tissue. This process can be simply described by Eq. (1) with kinetic coefficient k dependent on the cooking temperature. The higher the kinetic coefficient indicates the quicker the cooking process. Texture changes during cooking can be described by the same formulas as (1a) and (1b), in which the bond concentration is replaced by some important texture parameter, mostly modulus of elasticity.[16,17,18]

The cooking part ends at cooking time CT when the cell wall bond concentration reaches the critical value c_{CT .} This state can be also expressed by relative critical bond concentration c_cr = c_CT/c_{0 .} Using this definition, CT is expressed from Eq. (1b) as:

(3)

The critical relative bond concentration represents the portion of the initial bond concentration c₀ that was not destroyed in the cooking part. Eq. (3) is the equation connecting three important cooking parameters: relative critical bond concentration c_cr, cooking time CT and kinetic coefficient k (relatively independent on potato density[16]). The cooking time can be determined by the CPEM method (e.g., Fig. 1) and k can be estimated from texture-time plots obtained during potato cooking[13,16]. Figure 2 shows that experimental CT values decrease approximately linearly with increasing potato density within all the tested varieties. The differences between the individual plots are given by the coefficients a_CT and b in the linear regression:

(4a)

Figure 2 Cooking time CT plotted against density. Coefficients in Eq. (4b) see Table 1. Nicola (R² = 0.683), Saturna (R² = 0.746), Agria1 (R² = 0.916), data from growing year 2005 in Hejlová and Blahovec.[23]

Table 1 Quantities measured and calculated (CAL) in the simple kinetic model

Variety quantity	Agria	Nicola	Saturna
aCT0 * (min)	19.82	23.89	21.17
b* (min.m^3.kg^−1)	0.1605	0.1077	0.1556
k (min^−1)	0.178**	0.081***	0.130^+
CAL kb (m^3.kg^−1)	0.286	0.0087	0.0202
CAL ccr (−)ρ = 1070 kg.m^−3	0.170	0.253	0.236
CAL ccr (−)ρ = 1085 kg.m^−3	0.260	0.289	0.320
CAL ccr (−)ρ = 1100 kg.m^−3	0.400	0.3289	0.433
CAL x/c0 (m^3/kg) ρ = 1070 kg.m^−3	0.00485	0.00221	0.00477
CAL x/c0 (m^3/kg) ρ = 1085 kg.m^−3	0.00744	0.00252	0.00647
CAL x/c0 (m^3/kg) ρ = 1100 kg.m^−3	0.01142	0.00287	0.00876
Notes: * data in[23] ** unpublished data, *** data in[16],^+ data in[16] obtained for Panda.

The CT sensitivity to density is expressed mainly by the parameter b that takes part also in the modified relation between CT and density ρ[11]:

(4b)

where a_CT0 = a_CT − bρ ₀ and b are characteristic constants and ρ ₀ is a hypothetical tuber density corresponding to zero starch content that was approximated under von Scheele's experimental equations to 1005.5 kg m⁻³.[19,20,11] Figure 2 shows that CT values in salad variety Nicola are higher than in other two varieties with higher sensitivity to sloughing. Parameter b in Eqs. (4a,b) indicates CT depressing role of starch as the main source of potato density differences. This fact can be explained by different sensitivity of intercellular hemi-cellulose to wall tensions caused by intracellular starch expansion during cooking. It means that varieties more sensitive to sloughing are more sensitive to cells' expansions that destabilise cooked tissue at higher relative concentration of intercellular bonds than in case of the salad varieties.

The relative critical bond concentrations were calculated from our experimental data (Fig. 2) using Eq. (3). The kinetic coefficients were approximated from our previous experiments in which the elastic module was measured during potato cooking—see Table 1. Results of the calculations for three potato densities are also given in Table 1. They confirm that in the salad variety Nicola the intermolecular bonds are less sensitive to cell wall tensions caused by starch swelling: at higher tuber densities we obtained lower relative critical bond concentrations in Nicola than in the other two varieties. The role of the partial parameters a_CT0 and b in critical bond concentration can be assessed by putting Eq. (4b) into Eq. (3):

(5a)

that can be rewritten using Taylor's set at ρ = ρ ₀ :

(5b)

The curvilinear character of the c_cr dependence on density has a special polynomial character, in which the product kb plays an important role. The higher the value of kb, the steeper and more curved the c_cr -ρ dependence. The examples of the calculated c_cr are given in Table 1. The kb value in Nicola is less than half of the corresponding values obtained in the other two varieties. Parameter b termed as CT-sensitivity to starch represents a relatively stable variety characteristic[11],[[21]. On the other hand, the initial value of the c_cr (in relation to the density ρ ₀ ) is equal to the first term in Eq. (5b): exp(-ka_CT0) that is determined by product of k and a_CT0 .

The expression for CT-starch sensitivity b can be derived if k is independent on ρ as follows:

(6)

where x is a simplified expression for the derivation dc_CT /dρ. For known b the ratio x/c₀ was expressed as:

(6a)

Eq. (6a) shows that x/c₀ is proportional to the relative critical bond concentration with the proportionality constant kb described previously. Using Eq. (5a), Eq. (6a) can be rewritten into the following form:

(6b)

The importance of the product kb for calculating the dependence of critical bond concentration on density dc_CT /dρ is clearly expressed in Eq. (6b). Numerical results for x/c₀ are given in Table 1. This quantity increases with increasing density and is much lower for Nicola than for the other two varieties. The formulas (6), (6a), and (6b) show that CT-starch sensitivity b increases with increasing slope of critical bond concentration dc_CT /dρ. The higher the dc_CT /dρ indicates the higher the CT-starch sensitivity b. The role of the initial bond concentration c₀ in a raw tuber is given by formulas (6), (6a), and (6b), expressing changes caused by different cultivation & growing conditions. These conditions influence mainly the parameter a_CT0 with a very limited influence on CT-starch sensitivity b[21] so that the c₀ can enter into Eqs. (5b) and (6b) only by means of the member exp(-ka_CT0 ). Cooking part theory for a more sophisticated reversible kinetic model is described in the Appendix. The reversible kinetic model is used for discussion of the experimental values in the Applications.

Breaking Part

The initial slope of the breaking part, SBP (see Fig. 1), can be expressed as the initial rate of depression of the effective potato mass at the time of cooking in a CPEM test t = CT in the following manner:

(7)

where the second term follows easily from Eq. (1b):

(7a)

The first term in Eq. (7) expresses change of the effective potato mass inside the sieve basket due to changes of the cell wall bond concentration close to its critical value c_CT . It is determined by getting the potato fragments through the sieve meshes. It could be estimated on the base of its anticipated asymptotic values[22] :

(8a)

(8b)

As one of suitable functions with the required asymptotic properties was analysed the generalized power function:

(9)

with the parameter n > 0. The SBP then can be expressed in the following form[22]:

(10)

For constant K, k, and n Π(0, 1), the SBP behaves as a decreasing (exponential) function of the cooking time CT. The density derivative of Eq. (10) then can be expressed using Eq. (4a) as:

(11)

indicating that the derivative is not constant in dependence on density via cooking time CT (see Eqs. 4a,b), so that the SBP cannot be a linear function of density in fact.

Notes to the Materials and CPEM Method

This part contains the most important information on previous experiments (materials, method, and main results in CPEM tests) and applications to the model. The potatoes were harvested in October 2005 and kept in cold store (at temperature 6°C and 95% humidity) until testing in November and December 2005 (Agria) and in January and February 2006 (Nicola and Saturna). Potato sample for a CPEM test was prepared from three or four tubers of approximately the same size and mean density (determined by weighing the tubers in air and in tap water by considering the water temperature). Rectangular slices 10 × 10 × 1.5 mm were cut mainly from the inner parenchyma, immediately before testing washed in a sieve (15 seconds) and dried (3–5 minutes) leaving on a textile tissue at room temperature. The sample, 100 g of washed and dried potato flakes, was then tested by the CPEM method.[10]

Main CPEM Results

Both CPEM parameters are analysed as functions of potato tissue density. The dependence of cooking time CT on density is described by linear regression equations in individual potato groups creating the linear model of cooking stage [see (4a,b) and Fig. 2]. Similarly, the dependence of SBP on density is characterised in linear model of disintegration stage (Fig. 3).

Figure 3 SBP plotted against density. Nicola (SBP = 0.00395ρ − 3.958, R² = 0.435), Saturna (SBP = 0.01018ρ −10.575, R² = 0.400), Agria1 (SBP = 0.01972 ρ −20.885, R² = 0.443), data from growing year 2005 in Hejlová and Blahovec.[23]

In the transformed regression equation (4b), ρ ₀ is an approximation of a fictive tuber density without starch derived from Scheele´s empirical formulas measuring the relationship among potato tuber density, dry matter content, and starch content.[19,20] Hence, the intercept a_CT0 represents the fictive cooking time required for sloughing of potato tissue with zero starch content and is associated with the cell wall properties. The regression coefficient b was termed as CT- sensitivity to starch content.[11] It represents a new cooking parameter typical for variety and relatively independent on cultivation and growing conditions.[11,21]

Data Application and Discussion

The SBP estimation was written as product of two components in Eq. (10): The model component ^{ke–kCT (1–n)} is derived from the cooking part and its kinetic model and and a suitable n-value[22]. The other component Kc₀ represents an additional mass factor, which can be estimated from Eq. (10). Statistical analysis of our experimental data in [22] indicates optimum n-values in the range 0.2<n<0.4, for which the Kc₀ values are close to 10 g, i.e. 10% of the initial mass of the sample in a CPEM test, and weakly correlated with tissue density (Table 2). Then the main source of SBP dependence on density (Fig. 3) laid in the dependence of its model component SBP/Kc₀ on density in individual tested groups (see Fig. 4 and[22]). Hence the parameter n and the factor Kc₀ could express some inner disintegration relation associated with experiment conditions and relatively independent on the tested variety.[22]

Table 2 Product Kc ₀ calculated from Eq. (10) using experimental CT and SBP data and k-values from Table 1, parameter n = 0.2

		Kc0
Variety	Number of tests	MV (g)	SD (g)	VC (%)
Agria	100	10.91	2.34	21.47
Nicola	13	11.21	1.49	13.33
Saturna	12	9.40	0.86	9.12

Figure 4 Model component SBP/(Kc₀) calculated using Eq. (10) and plotted against density. Nicola (SBP/(Kc₀) = 0.00043ρ − 0.4328, R² = 0.709), Saturna (SBP/(Kc₀) = 0.00421ρ − 4.5324, R² = 0.768), Agria1 (SBP/(Kc₀) = 0.00336ρ − 3.5875, R² = 0.963), CT data from Fig. 2 and Hejlová and Blahovec.[22]

The parameter b in Eq. (4) is a relatively well-defined variety parameter nearly independent on cultivation conditions. The correlation matrix in Table 3, based on two years data for 8 different potato variants,[23] confirms this clearly. It seems that this fact has roots in the loss of correlation between a_CT0 and b, which is displayed in Fig. 5. It illustrates the real state well: there exists a relatively tight increasing relationship between the parameters b and a_CT0 for all the tests on varieties Agria and Saturna, but the results on variety Nicola, even if different in two years, differ in both years significantly from results obtained for Agria and Saturna. It seems that cooking of the salad varieties is controlled by another mechanism than cooking of the other varieties. This difference represents the main source of low correlations in Table 3. The CT-starch sensitivity b is given by Eq. (6); it can be rewritten into the following form:

(12)

expressing the dependence either on the kinetic coefficient k or on the critical bond concentration in dependence on starch content. The salad varieties are the varieties with low level parameters b so that the product kb = d(lnc_CT )/dρ, expressing the increasing dependence of critical bond concentration c_CT on tuber density, is low too.

Figure 5 Parameter b plotted against parameter a_CT0 from the Eq. (4b). Two years data of potato cooking tests[23]. The plots for Agria and Saturna are approximated by the common linear equation: b = 0.0114 a_CT0 − 0.0798, R² = 0.875.

Table 3 Correlation matrix of parameters describing dependence of sloughing on tuber density

aCT0	0.273	−0.512
	b	−0.153
		RC
Note: two years results in three varieties,[23] RC (m^3.min^−1) denotes regression coefficient in relation SBP versus density.

The application of the more sophisticated partly reversed kinetics of potato cooking developed in the appendix is limited by a lack of suitable data describing the further parameter k₂ - see Eqs. (A1) and (A2). This model could be preferred for c_cr approaching to c_u/c₀ , i.e. as is clear from Eq. (A2a) for c_cr being a little bigger than ratio k₂/k. The potatoes are well cooked at relative bond concentrations 0.1[16] so that partly reversed kinetics can lead to important differences in comparison to the simple model at k₂/k values higher than 0.1. Using of k₂/k = 0.1 and density 1090 kg m⁻³ in Eq. (A3) leads to c_cr values 0.196 for Agria and 0.171 for Nicola. At lower densities the obtained c_cr values decrease consequently (see Table 1). At these conditions the partly reversible model can give results significantly different from the simple one. Similarly as in Eq. (6) the CT-starch sensitivity b can be expressed in a partly reversible model in the following:

(13a)

and the ratio x/c₀ consequently as:

(13b)

decreasing with increasing k₂ approximately linearly as plotted in Fig. 6. This decrease is much greater for Agria than for Nicola. For lower potato density the lower was x/c₀ (see also Table 1). When the expression in the brackets of Eq. (13b) is less than zero the ratio x/c₀ is negative indicating that c_cr is lower than the asymptotic concentration c_u in expression (A2a), i.e. the critical state cannot be reached in any final cooking time. The variety plots in Fig. 6 show that this asymptotic state is reached for both Agria and Nicola at approximately the same value of k₂/k ∼ 0.13-0.15 (at density 1090 kg m⁻³). The corresponding relative critical concentrations are ∼0.14 and ∼0.13 for Agria and Nicola, respectively. Knowing the special cooking properties of the salad varieties like Nicola the partly reversible kinetic mechanism of softening could be expected in this case.

Figure 6 Derivation x/c₀ calculated from Eqs. (14b) and (A3) plotted against ratio k₂/k. Data for Agria and Nicola at constant density 1090 kg.m⁻³ from Table 1.

Our analysis based on the potato cooking kinetics showed that the variety susceptibility to sloughing is simply expressed via CT-starch sensitivity b that has roots in i/ dependence of critical bond concentration on density, ii/ initial critical bond concentration, and possibly iii) some details of more complicated kinetics of softening that are not yet fully clear.

CONCLUSIONS

Potato cooking is controlled by sensitivity of the intercellular bonds to the stresses produced by starch swelling. The simple kinetic model shows that cooking time is predetermined by two main parameters: kinetic coefficient of the tissue softening on one hand and the critical bond concentration in the intercellular space at the stage of starting disintegration on the other. Potatoes, in which the process of softening is slower, i.e. kinetic coefficient is of a lower value, and/or in which disintegration starts at lower bond concentrations in the intracellular spaces, the susceptibility to sloughing is lower than in the more susceptible varieties. Some role is played also by the initial bond concentration as a source of cultivation induced differences. The important role in potato sloughing can be also played by details of more complicated but still not fully known kinetics of softening. Knowing the main parameters of cooking, i.e. kinetic coefficient of the potato softening and the relative critical concentration of the intracellular bonds, the main parameters of disintegration can be estimated using the model proposed. There are some principal differences between most of potato varieties and the salad varieties consisting in much lower cooking sensitivity of salad varieties to the starch content. This difference is a source of reduction of the correlations among different cooking parameters that are usually observed with the other varieties than with the salad ones.

NOMENCLATURE

aCT (min)	=	constant in the linear regression relation between CT and density
aCT0 (min)	=	hypothetical CT value for potato tissue with zero starch content
b (min.m3.kg−1)	=	slope in the linear regression relation between CT and density, CT-sensitivity to density
c (m−3)	=	cell wall bond concentration
ccr = cCT/c0 (−)	=	critical relative cell wall bond concentration
cCT (m−3)	=	critical cell wall bond concentration (corresponding to the cooking time CT)
cu (m−3)	=	asymptotic cell wall bond concentration in partly reversible model
c0 (m−3)	=	initial cell wall bond concentration in the raw tuber
CT (min)	=	“cooking time”, time of cooking in CPEM test just before indication of the disintegration process
CV	=	coefficient of variations
CPEM	=	“cooked potato effective mass”, modified CPM test[10],[11]
CPM	=	“cooked potato mass”, official test of potato sloughing[9]
CPW	=	“cooked potato weight”, term commonly used for CPM tests[9]
k (min−1)	=	kinetic coefficient, in partly reversible model k = k1 + k2
K (g.m3)	=	parameter of Eq. (9), defined by Eq. (8b)
k1 (min−1)	=	main kinetic coefficient in the partly reversible model
k2 (min−1)	=	kinetic coefficient in partly reversible model that is responsible for reversibility
MV	=	mean value
m (g)	=	effective sample mass
RC (m3.min−1)	=	slope in the linear regression relation between SBP and density
R2 (−)	=	coefficient of determination
SBP (g.min−1)	=	“slope of the breaking part”, initial slope (absolute value of regression coefficient) of the breaking part in a CPEM test
SD	=	standard deviation
x = dcCT /dρ (kg−1)	=	density derivative of cell wall bond concentration at cooking time CT
ρ (kg.m−3)	=	density
ρ 0 = 1005.5 (kg.m−3)	=	hypothetical potato density corresponding to zero starch content (estimation based on the Von Scheele's empirical formulas)

ACKNOWLEDGMENT

The research was supported in part by the Research Intention MSM 6046070905 (Czech Republic).

APPENDIX: TWO MEMBER KINETIC THEORY OF COOKING PART

Instead of Eq. (1a) the kinetics of softening could be also described by the following two member linear kinetic equation:

(A1)

This equation is an example of reversible kinetic equations[12] describing a superposition of the original (k₁ ) and the reversible (k₂ ) processes. Even if in our case the process of cooking could be understood as rather irreversible one, the reverse part in Eq. (A1), i.e. k₂(c₀ – c) can be used to description of the stress induced process activated during cooking new and new intercellular bonds that did not originally participate in the product cohesion. The starch swelling stresses seem to play some important role in processes of this kind.

The solution of Eq. (A1) can be written easily in the following form:

(A2)

that is reduced to Eq. (1b) when k₂ = 0 and is similar to Eq. (2b). Eq. (A2) has asymptotic solution:

(A2a)

for infinite cooking time. The initial part of Eq. (A2) is quasi-exponential in time with quasi-simple kinetic coefficient ∼ (k₁ + k₂ ) = k. For higher time values Eq. (A2) more and more differs from Eq. (1b). Differences of this type were also observed in the previous precise study of potato cooking[16]. Eq. (A2) can be understood as a modification of Eq. (2b), that was successfully used for description of thermally induced texture softening of different agricultural products[14],[15].

The relative critical bond concentration c_cr defined for simple kinetic equation in Eq. (3) is then expressed in the following form:

(A3)

that could be rewritten similarly as in Eq. (5b):

(A4)

The parameter k = k₁ + k₂ plays in Eq. (A4) a similar role as the kinetic coefficient k in Eq. (5b). The relative critical bond concentration contains two parts in Eq. (A4): the first one independent on cooking parameters and proportional to kinetic parameter k₂ and the second one that is sensitive to potato density and determined mainly by the product kb similarly as in Eq. (5b). When k₂ = 0 then Eq. (A4) has the same form as Eq. (5b].