Biotechnoi. Bog. 1004, 10, 389-397
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Acid Hydrolysis of K-Carrageenan in a Batch Reactor: Stochastic Simulation of Change of Molecular Weight Distribution with Time S. K.Singh Department of Pharmaceutical Analysis, R&D Therapeutics, Pharmacia AB, 5-112 87 Stockholm, Sweden
B.C. Shen,S. T.Chou, and L. T.Fan* Department of Chemical Engineering, Durland Hall, Kansas State University, Manhattan, Kansas 66506
The degradation of K-carrageenan by acid hydrolysis in a LiCl/HCl (pH 2) buffer solution has been studied at various temperatures in a batch reactor. The reduction in the weight-average molecular weight and the variation in the molecular weight distribution (MWD) have been followed over time by an SEC-MALLS-RI chromatography system. While a deterministic rate equation can be fit to the average molecular weight-time data, this is not the case for the MWD-time data. Since depolymerization by hydrolysis usually proceeds via nonspecific (random) bond cleavage, a stochastic approach can be appropriate for the resultant MWD-time data. In the present work, the master equation has been derived from the stochastic population balance for this batch hydrolysis system characterized by a set of discrete states, each representing a particular molecular weight range. The governing equations for the means, variances, and covariances of the random variables, each representing the number of carrageenan molecules in an individual state, have been extracted from the master equation. The equations for the means have been applied to the MWD-reaction time data, and the unknown reaction rate parameters have been estimated. These parameters can be used to generate the MWD a t any hydrolysis reaction time from any initial MWD. The applicability, limitations, advantages, and sources of error of the derived model have been discussed.
Introduction Carrageenans are water-soluble linear polysaccharides of cell wall extracted from certain members of the class of red seaweeds (Rhodophyceae). They are composed of alternating CY(1+3)- and 8(1+4)-linked D-gdaCtOSe residues. Three primary forms, K - , A-, and r-carrageenans, are identified on the basis of modification of the disaccharide repeating unit resulting from the Occurrence of ester sulfate or anhydride formation in the 4-linked residue. The K-carrageenan comprises alternating CY-(1-3)-D-gaht05e4-sulfate and 8-(1-.4)-3,6-anhydro-~-galactose (seeFigure 1). The K- and r-carrageenans form thermoreversible gels in solution, while A-carrageenans and other minor forms yield highly viscous solutions that do not gel. The gelling behavior is independent of the nature and concentration of the cations present in the solution [see, for example, Clark and Ross-Murphy (198711. Significant interactions also occur with proteins, e.g., those in milk, and with hydrocolloids, such as starch, locust bean gum, and guar gum (Descampset al.,1986). Because of these properties, carrageenans are extensively employed in the food industry for a variety of functions, such as viscosity/gel or texture enhancers and stabilizers (Christensen, 1964). The same properties are exploited by the pharmaceutical/cosmetic industry through the incorporation of carrageenans as ingredients in various products, such as lotions, creams, toothpastes, and cough preparations. Due to their widespread consumption, carrageenans have been extensively evalulated toxicologically. The major problem that has been identified is attributable to
* Correspondence should be addressed to this author. 8756-7938/94/3010-0389$04.50/0
1
r
-0,so.
~
o
OH
~
OH
o
1
=
n
Figure 1. Idealized structure of K-carrageenan. low molecular weight (
c
0.2 0
0.0 2
a
6
4
Molecular Weight (g/mole
10
X l 0')
Figure 5. Fit of the model to the experimentally obtained cumulative molecular weight distributionsat 45 O C , yielding 1.78 min-' for a (reaction time (min),exp data, model): ,.,3 -; 13, V, - -; 20, V,- -; 30, cl, ---; 40, W,
e-;
51, A,- -; 65, A,- *
-.
tive MWD data, such as those presented in Figure 3, at every chosen reaction time. In other words, each of the experimental cumulative MWDs has been lumped into an MWD among the 10 states. These recalculated distributions based on the pooled data are shown in Figures 4,5, and 6 for 35, 45, and 55 OC, respectively, while the corresponding experimental weight-average molecular weights are given in Figure 7.
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394
,
45
1 .o
-x
I
o 0.8
35
40
it
0.6 25
30
0.4
2o
t
0.2 /
0.0 0
d,
, 4
I
2
I
,
I
6
8
10
M o i e s u l o r W e i g h t (g/mole
I
1
10
0
SO
20
40
50
70
60
80
R e a c t i o n Time (min.)
12.0, V,--; 20.0, V, --; 30.0,0,---;40.0,. ,45.5, A, - * - ; 50.0, A, - -; 60.0, 0 , *e*;
Simulations have been carried out by numerically solving eq 15 for 1 = 10 states for all three reaction temperatures. Due to the error involved in the initial heating and dissolution step described previously, the cumulative MWD measured from the fiist aliquot for each temperature has been chosen as the initial condition for every simulation. The simulation proceeds up to a reaction time of approximately 70 min. To simplify the computation, the hydrolysis intensities, aij, have been assumed to be identical: aii = a
l5 10
X10')
Figure 6. Fit of the model to the experimentally obtained cumulativemolecular weight distributionsat 55 O C , yielding 4.29 min-1 for a (reaction time (min), exp data, model): 4.0, 0 , -;
-.
I
(22)
This assumption implies that all of the 41-3) and 8(1-4) bonds are equally susceptible to hydrolysis. The values of a have been recovered through least-squres fitting of the calculated weight-average molecular weights, Mwcd,obtained by substituting the numerical resujts of eq 15into eq 16,to those measured experimentally, MwexP, i.e., by minimizing Q:
Figure 7. Comparison of the experimentallyobtained weightaverage molecular weight as a function of the reaction time with the stochastic and empirical fist-order models: 0,exp data, -, stochastic model; ---,empirical first-order model (eq 26). Table 1. Parameter Estimates for the Acid Hydrolysis of K-Carrsrteenan at Various Temwratures ~
hydrolysis overall average temperature intensity hydrolysis intensity a (min-1) a* (min-1) T ("(2)
reaction
35 45 55
0.59 1.78 4.29
0.0032 0.0095 0.0229
pseudo-reaction rate kM (min-I) 0.0039 0.0117 0.0206
are plotted in F i p e 7 as a function of the reaction time. The calculated M, values are generally in excellent accord with the experimental data. The same data have also been empirically fit to the first-order rate equation,
Mw(t)= Afw(0)exp(-k,t)
(24)
The resultant values of the pseudo-rate constant, k ~ are , listed in Table 1. To compare k~ with the transition intensities, we first calculate ai, signifying the intensity of the event that a K-carrageenan molecule in state i will split into any two smaller segments. Substitution of eq 22 into eq 5 yields
zaij [i/Zl
where the summation is over the entire reaction time at each temperature. In performing the least-squares fitting, numerical computations were started by assigning an initial value to a,with the measured initial molecular weight distribution as the initial conditions. The simulation was continued until a certain reaction time was reached. The calculgted molecular weight distribution was then converted to Mw* through eq 16. This was repeated over the entire span of reaction times for which experimental data were available. Subsequently, the objective function, Q, was calculated from eq 23. The computations were terminated when the minimum of Q was reached. The hydrolysis intensities, a, obtained at the three temperatures are given in Table 1. The corresponding simulated cumulative MWDs are plotted in Figures 4-6 for these reaction temperatures. The overall agreement between the experimental data and the model appears to be excellent, except for the low molecular weights at long reaction times. This is probably attributable to, among other factors, the experimentalerror involved in measuring the MWDs, as will be discussed later. The weight-average molecular weights, Mw, during the course of hydrolysis at 35,45, and 55 OC have also been calculated from the model, specifically from eq 16. These
aiE
= [i/2]a i = 2,3, ..., 10
(25)
I-=]
Each state in this study has a molecular weight range of about 106g/mol, corresponding to about (106/192)galactose residues. In other words, if the number of galactose residues in K-carrageenan molecules is in the range between (i- 1)X 105/192andi X 106/192,the molecule is considered to be in state i. A bond cleavage in any of these molecules is considered as a breakage of state i. The maximum or possible number of such bond cleavagesis essentially ( l e / 192). Then, the average cleavage intensity of the K-CUrageenan molecules with the molecular weights in the weight range of state i, ai*, can be estimated as ai*=
W21a . ai =I = 2,3, ..., 10 105/192 105/192
(26)
Let the overall average hydrolysis intensity of all the K-carrageenan molecules, a*,be defined as
(10- 1)
(27)
Bbtechnoi. Pro$., 1994, Vol. 10, No. 4
( i / ~ ) x1 o
so5
- (K-') ~
Figure 8. Arrhenius plot of the hydrolysis intensities.
Then, substitution of eq 26 into this expression gives rise to a*=-=25a/9
106/192
(5.3333 x lo-%
(28)
The calculated values of a* are given in Table 1. The agreement between a* and k~ seems to be fairly good in light of the fact that a* represents a fundamental parameter of the hydrolysis process, while k~ is essentially a regression parameter; see eq 24. Note that for ordinary first-order reactions of the type
lead to errors in the calculated values of W,(t) for the first and last states (m = 1, 10); the errors in the initial distribution, W,(O), are especially critical. Furthermore, the inaccuracies in the high molecular weight end of the distribution can lead to disproportionately large errors in the calculated weight-average molecular weight. The errors in the experimental data, specifically in the MWDs available at the outset, which serve as the initial conditions for solving eq 15,will be reflected in the simulated MWDs also. Moreover, the accuracy of the simulated MWDs depends on the value of 1 chosen to represent the molecular weight range. The larger the value of 1, the more accurate the representation of the MWD obtained. The increase in accuracy, however, must be weighed against the increase in computational effort required, since eq 15 must be separately solved for 1 states. Features of the Model. A merit of the model, and perhaps its drawback, is its detailed nature. The number of states needs to be reduced, as done in this work, to solve the set of governing equations within a reasonable time; this has resulted in a loss of accuracy. This situation, however, can be remedied if some estimate of the variation in the MWD is available. One way of accomplishing this is by measuring MWDs through repeated experiments at the same temperature and for the same reaction times. The main parameter recovered from the model is the reaction intensity (rate), CY+ The model has been derived wherein differing reaction intensities for different bonds can be easily incorporated. For the purpose of simulation in the present work, the assumption aij= a = constant
k
A-B+C
(29)
the concentration of A decays according to the exponential expression, with the reaciton rate constant k as the parameter. For such reactions, it can be shown that the stochastic reaction intensity a is equal to the deterministic rate constant k [see, for example, Chou et al. (1988)l. The validity of the hydrolysis intensity, a,for a complex depolymerization reaction can also be seen from a semilogarithmic plot of a against the inverse of the absolute reaction temperature; the resultant linear plot (Figure 8) indicates that the Arrhenius relationship is obeyed. The estimated activation energy for bond cleavage for the acid hydrolysis of K-carrageenan, obtained from the slope of this plot, is 84 400 J/g.mol. This value is somewhat lower than those obtained by other researchers, which range between 104 700 and 161 170J/g.mol; however, the values in the literature have been obtained exclusively from a consideration of Mwversus t data, as reviewed by Singh and Jacobsson (1994). Variation in reported activation energies can be attributed to differences in hydrolysis conditions, including ionic strength, nature of cations present, and presence of dissolved oxygen. Furthermore, the various methods employed in the literature to estimate the molecular weight (end-group titration, capillary viscometry, sedimentation equilibrium, and slit viscometry) measured slightly different quantities and involvevarying degrees of error. Sources of Error. The MWD data obtained from the present experimental setup have the largest inaccuracies near the edges of the distribution. This is attributable to the mode of detection adopted;the light-scattering detector is not as sensitive to the low scattering intensities from the low molecular weight fractions, while the RI (concentration) detector is less sensitive to the high molecular weight fractions due to their low concentrations. These
has been made. The available experimental data do not warrant the use of two reaction intensities for the types of bonds in K-carrageenan. In a situation where the reaction intensities of the two bonds are different, the simulation could lead to error. If one bond hydrolyzes very slowly, the overall rate of hydrolysis can be considered to be halved, as the effective monomer weight is doubled. In the case of widely differing but still significant hydrolytic rates, one would have to keep track of the type of bond being hydrolyzed at any particular instant during the simulation. Experiments are currently being carried out by NMR and FT Raman spectroscopy to examine t h susceptibility of the two K-carrageenan bonds to hydrogis. Other situations where differing reaction intensities for different bonds can be of significance are bond-specific enzymatic hydrolysis and hydrolysis of a copolymer with differing comonomers. Various fractions of carrageenan contain the same a(1-3)- and B(l+li)-linkages as K-carrageenan, although the disaccharide residues are different. It is likely, therefore, that the mechanism and kinetics of hydrolysis are the same. In this case, the parameters obtained here can be employed again for h- and L-carrageenans. The present model is generally applicable to polymer degradation by bond cleavage. No other chemical species has been considered; i.e., any other reacting species would have to be present in excess for eq 1to be valid. In order to adopt this model, the initial intensities must be recovered from a fit of the model to the data obtained by performing a set of experiments where MWDs are measured during hydrolysis, as was done in the present work. If the reaction intensities are known at three or more temperatures, we can resort to the Arrhenius relationship to extrapolate to other temperatures, as long as the mechanism of hydrolysis remains unchanged. The effect of processing temperature and time can then be
Biotechnol. Rog., 1994, Vol. 10, No. 4
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simulated for any starting MWD. This would make possible the estimation of low molecular weight fractions under particular conditions. Moreover, this would give rise to the estimation of other properties that depend on Mwand MWD, e.g., viscosity of solution and gel strength. In t h e present work, the reaction intensities have been obtained at a pH of 2 and in the presence of Li+ ions; i t is likely that H+ ions take part in the hydrolysis reaction. The reaction intensities within the range pH 2-3 can be considered to be approximately equal. Cations causing K-carrageenan to gel, e.g., K+, Rb+, Cs+, or Ca2+,would reduce the intensity, however. T h e superiority of the present model lies in ita ability t o model t h e MWD as well as Mw.A first-order fit can be made only to t h e Mw data, and as discussed in t h e Introduction, this is insufficient for detailed analysis.
Concluding Remarks The acid hydrolysis of K-carrageenan in a LiCl/HCl (pH 2) buffer solution has been studied at three temperatures in a batch reactor. The resultant molecular weight distributions and average molecular weights are compared with those obtained through the stochastic analysis. The parameters recovered from the comparison have been shown to conform to the Arrhenius equation. T h e sources of error, limitations, advantages, and applicability of the model have been considered. The conditions to which t h e parameters obtained in this work can be extrapolated have been discussed.
Notation D E[] kM
I M m
MR Mt a v
N n N m
P t Wm a 6 6
one-step operator mean or expectation pseudo-rate constant number of states in a batch average molecular weight of the K-carrageenan molecules in state m weight range of the K-carrageenan molecules inside the reactor total weight of the K-carrageenan molecules inside the reactor weight-average molecular weight random vect representing the distribution of the K-c ageenan molecules realization vector of random vector N random variable representing the number of molecules in state m probability of the K-carrageenan molecules to have a specific population distribution time weight fraction of the K-carrageenanmolecules in state m hydrolysis intensity function increment of the molecular weight Kronecker delta vector
x
Subscripts
i, j , k , m,q , s states of the K-carrageenan molecules 1
highest state of K-carrageenan molecules
Supplementary Material Available: Appendix A, derivation of eqs 8 and 11;Appendix B, derivation of eq 12;Appendix C, derivation of eq 17;and Appendix D, derivation of eqs 19 and 20 (10pages). Ordering information is given on any current masthead page.
Literature Cited Andersen, A. G. Carrageenan in Toothpaste. Manufacturing Chemist 1988 (September), 33-38. Badui, S.; Desai, N.; Hansen, P. M. T. Heat Degradation of Carrageenan in a Milk Salt System. J. Agric. Food Chem. 1978,26,675-679. Bradley, T. D.; Mitchell, J. R. The Determination of the Kinetics of Polysaccharide Thermal Degradation Using High Temperature Viscosity Measurements. Carbohydr. Polym. 1988, 9,257-267. Casella, G.; Berger, R. L. Statistical Inference; Pacific Grove, CA, 1990a;p 58. Casella, G.; Berger, R. L. Statistical Inference; Pacific Grove, CAI 1990b;p 162. Chiang, C. L. AnIntroduction to StochasticProcesses and Their Applications;Robert E. Krieger Pub.: Huntington, NY, 1980; pp 477-498. Chou, S. T.;Fan, L. T.;Nassar, R. Modeling of Complex Chemical Reactions in a Continuous-Flow Reactor: A Markov Chain Approach. Chem. Eng. Sci. 1988,43,2807-2815. Christensen, 0. Carrageenan, A Useful Food-Additive. Food Manuf. 1964,39,49-52. Clark, A. H.; Ross-Murphy, S. B. Structural and Mechanical Properties of Biopolymer Gels. Adu. Polym. Sci. 1987,83, 57-192. Desai, N.; Hansen, P. M. T. Heat Stability of Carrageenan. In Gums and Stabilizers for the Food Industry, 3; Phillips, G. O., Wedlock, D. J., Williams, P. A., Eds.; Elsevier Applied Science Publishers: Barking, Essex, U.K., 1986,pp 341-349. Descamps, 0.; Langevin, P.; Combs, D. H. Physical Effect of Starch/Carrageenan Interactions in Water and Milk. Food Technol. 1986 (April), 81-88. EkstrBm, L.-G. Molecular-Weight-Distributionand the Behavior of Kappa-Carrageenan on Hydrolysis. Carbohydr.Res. 1985, 135,283-289. Fox, R. 0.; Fan, L. T. Stochastic Modeling of Chemical Process Systems, Part 1. Introduction. Chem. Eng. Ed. 1990a,24, 56-60. Fox, R. 0.; Fan, L. T. Stochastic Modeling of Chemical Process Systems,Part 2. The Master Equation. Chem.Eng. Ed. 199Ob, 24,88-92. Fox, R. 0.;Fan, L. T. Stochastic Modeling of Chemical Process Systems, Part 3. Application. Chem.Eng. Ed. 1990c,24,164167. Jackson, C.; Nilsson, L. M.; Wyatt, P. J. Characterization of Biopolymers Using a Multi-Angle Light Scattering Detector with Size Exclusion Chromatography. J. Appl. Polym. Sei. 1989,43,99-106. Lecacheux, D.; Panaras, R.; Brigand, G.; Martin, G. Molecular Weight Distribution of Carrageenans by Size Exclusion Chromatography and Low Angle Laser Light Scattering. Carbohydr. Polym. 1985,5,423-440. Marcus, A. J.; Marcus, S. N.; Marcus, R.; Watt, J. Rapid Production of Ulcerative Disease of the Colon in the NewlyWeaned Guinea-pigs by Degraded Carrageenan. J. Pharm. Pharmacol. 1989,41,423-426. Masson, C. R. The Degradation of Carrageenan, I, Kinetics in Aqueous Solution at pH 7. Can. J. Chem. 1955,33,597-603. Masson, C. R.; Santry, D.; Caines, G. W. The Degradation of Carrageenan, 11,Influence of Further Variables. Can.J.Chem. 1955,33,1088-1096. Morris, E. R.; Rees, D. A.; Robinson, G. Cation Specific Aggregation of Carrageenan Helices. Domain Model of Polymer Gel Structure. J. Mol. Biol. 1980,138, 349-362. Oppenheim, I.; Shuler, K. E.; Weiss, G. H. Stochastic Processes in Chemical Physics: The Master Equation; MIT Press: Cambridge, MA, 1977;pp 1-45. Singh, S. K.; Jacobsson, S. P. Kinetics of Acid Hydrolysis of Kappa-carrageenanas Determined by Molecular Weight (SECMALLS-RI), Gel Breaking Strength, and Viscosity Measurements. Carbohydr. Polym. 1994,23,89-103. Smidsr~rd,0.; Grasdalen, H. Some Physical Properties of Carrageenan in Solution and Gel State. Carbohydr. Polym. 1982,2,270-272.
Blotechnol. hog.., 1994, Vol. 10, No. 4
Tanford, C. Physical Chemistry of Macromolecules; John Wiley and Sons: New York, 1961; pp 611-618. van Kampen, N. G. Stochastic Processes in Physics and Chemistry;North-Holland (Elsevier): Amsterdam, 1981a;pp 101-138. van Kampen, N. G. Stochastic Processes in Physics and Chemistry; North-Holland (Elsevier): Amsterdam, 1981b; p 145.
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von Kampen, N. G. Stochastic Processes in Physics and Chemistry;North-Holland (Elsevier): Amsterdam, 1981c;pp 180-208. Wyatt TechnologyCorporationASTRA SoftwareManwl;Wyatt Technology Corp.: Santa Barbara, CA, 1991. Accepted ~ ~19, 1994.0 ~ i l @
Abstract published in Advance ACS Abstracts, June 15,1994.
