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ARCHIVES OF BIOCHEMISTRY AND BIOPHYSICS Vol. 196, No. 1, August, pp. 23-321979

Potentiometric Titration Behavior of Polylysine and Copolymer of Lysine with Alanine Prepared by Thermal Polycondensation

ETSUO KOKUFUTA,’ TAKAICHI TERADA, MINORU TAMURA, SHINNICHIRO SUZUKI, AND KAORU HARADAZ

Department of Chemistry, The University of Tsukuba, Sakura-mura, Niihari-gun, Ibaraki, 30041, Japan

Received July 7, 1978; revised January 12, 1979

The dissociation behavior of polyamino acids prepared by thermal polycondensations of free L-lysine and of a mixture of free Llysine and L-alanine were studied by means ofpotentio- metric titrations at various ionic strengths. The dependence of apparent dissociation constant against the degree of dissociation and the Henderson-Hasselbalch plots were investigated. These results indicated that thermally prepared polylysine and copoly(lys, ala) behave as a branched random-chain polymer. On the other hand, the intrinsic dissociation constants (p&) of amino groups attached to lysyl residues and the species of the amino groups were evaluated by the analyses of titrational data. It was found that the a-amino groups, which were characterized by pK, values of ‘7.1-7.4, contain about 93-96% of the titratable amino groups in the polymers. Furthermore, the e-amino groups, which were characterized by pK, values of 10.4-10.7, contain about 4-7%. The pK, values of a- and e-amino groups mentioned above were similar to those of proteins such as ribonuclease and bovine plasma albumin. The results obtained here are discussed in connection with possible peptide chain propagation by the thermal polycondensation of lysine.

Polyamino acids can easily be prepared by the thermal polycondensation of a-amino acids (1). These are homopolymers of glycine (Z), lysine (3, 4), or aspartic acid (5, 6) and several copolymers of acidic and basic amino acids (‘7) or proteinoid-containing amino acid residues common to protein (8, 9) which have many properties of the proteinoid (10, 11) and the other polyamino acids have been reported (12). However, most of the previous studies have dealt almost exclusively with the chemical and biological properties. Few studies have been made on the physico- chemical properties of polyamino acids prepared by thermal polycondensation.

In the previous studies (13, 14), the potentiometric titration behavior of polyaspartic acid (PAA) and copoly(glu, ala) pre-

1 Present address: Institute of Applied Biochem- istry, The University of Tsukuba, Sakura-mura, Niihari-gun, Ibaraki, 300-31, Japan.

* To whom correspondence should be addressed. 3 Abbreviations used: PAA, polyaspartic acid; PL,

polylysine; DNP, dinitrophenyl; Ew, equivalent weight; GC, glycol chitosan; PEl, polyethyleneimine.

pared by thermal polycondensation was investigated. It was reported that evalua- tion of the intrinsic dissociation constants (p&J of the carboxyl groups in the polypeptides gave information about the linkages of aspartyl and glutamyl residues. PAA contains (Y- and p-linked aspartyl residues (7:3), and the two carboxyl groups are characterized by pK, values of 4.35 and 3.25, respectively (13). On the other hand, the glutamyl residue in copoly(glu, ala) is mostly or entirely y-linked and the carboxyl group is characterized by a pK, value of 3.92 (14).

It is also interesting to investigate the potentiometric titration behavior of thermally prepared polyamino acids consisting of lysyl residues in order to obtain the information about the dissociational property of the amino group of the lysyl residue and about the linkage of the lysyl residue. In the present study, polylysine (PL) and copoly- (lys, ala) were used for the potentiometric titration. From the analyses of the titrational data, it was found that the PL and copoly(lys, ala) behave as a branched random-chain polymer, in contrast to Leuchs

23 0003-9861/79/090023-10$02.00/O Copyright 0 1979 by Academic Press, Inc. All rights of reproduction in any form reserved

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KOKUFUTA ET AL.

polyamino acids such as poly+lysine (15). On the other hand, the lysyl residue in PL and copoly(lys, ala) is found to be mostly e-linked because about 93-96% of titratable amino groups are a-amino groups. The a-amino group was characterized by pK, values of 7.1-7.4, while the pK, values of the e-amino group were 10.4-10.7. These pK, values were similar to those of amino groups in proteins such as ribonuclease (16) and bovine plasma albumin (17).

MATERIALS AND METHODS

Material. Free L-lysine was obtained by the dehy- drochlorination of L-lysine.HCl using a Dowex 1 col- umn and was recrystallized by a method described in the literature (18). The thermal polycondensation of free L-lysine (0.04 mol) was carried out at 200°C for 3 h under a nitrogen atmosphere. On the other hand, the copoly(lys, ala) was also prepared by thermal polycondensation of a mixture of free L-lysine (0.02 mol) and L-alanine (0.02 mol) at 190°C for 8.5 h under a nitrogen atmosphere. The reaction product was dissolved in 0.4 N HCl (100 ml), and the solution was dialyzed against distilled water for 5 days. After dialysis, a slightly yellowish product was obtained by lyophilization. The samples used in this study were dried at 70°C over phosphorous pentaoxide under reduced pressure until the weight reached a constant value.

The PL and copoly(lys, ala) were hydrolyzed with 6 N hydrochloric acid for 24 h at llo”C, and the resulting amino acid mixtures were analyzed by an amino acid analyzer (Yanagimoto Model LCdS). The amino acid composition of PL is: lysine, 99.2 mol%; glycine, 0.8 mol%, with total recovery of amino acid 54%. That of copoly(lys, ala) is: lysine, 52.6 mol%; alanine, 43.7 mol%; glycine, 3.7 mol%, with total recovery of amino acid 73%.

The infrared spectra of PL and copoly(lys, ala) showed moderate absorptions at 1660 and 1550 cm-‘, which are assigned to amide I and amide II, respectively. The number average molecular weights of PL and copoly(lys, ala) were 1.8 x lo5 and 9.8 x 106,4 respectively, as estimated by the osmometric method.

Analyses of the hydrolyzates of dinitrophenylated PL and copoly(lys, ala). In order to estimate the per- centage of (Y- and e-amino groups and to confirm the existence of branching points, the hydrolysate of the dinitrophenylated polymer was analyzed. The PL and copoly(lys, ala) were converted to dinitrophenyl (DNP)

4 Measurements by the osmometric method yield high molecular weights which might not be reliable because of the nature of the method employed. However, the molecular weight of PL is sufficiently high for the study of potentiometric titration.

derivatives by a modification (4) of the method of Sanger and Thompson (19). The DNP derivatives were hydrolyzed with 6 N HCl in an evacuated and sealed tube at 110°C for 24 h. The time of hydrolysis is limited to about 24 h because a-DNP-lysine decomposes at longer times (4). The a- and PDNP-lysines and free lysine in the hydrolyzate were extracted with water, and cu,p- di-DNP-lysine was extracted with ether. The cyy and l -DNP-lysines in the aqueous phase were separated by paper chromatography, and the amounts were determined by calorimetry after extraction. The amount of free lysine was also determined by the amino acid analysis of the extracted aqueous solution. The a,edi-DNP- lysine in the ether phase was separated by paper chromatography and then determined by a spectrophotom- eter after extraction. Authentic cY-DNP-lysine was synthesized by a method described in the literature (20), and the other DNP derivatives were obtained from Mann Research Laboratories.

Potentiometric titration. About 25 mg of PL or about 35 mg of copoly(lys, ala) was dissolved in aqueous solution (35 ml) containing 1.27 x 10m4 mop of NaOH and various concentrations of NaCl in order to adjust the ionic strength of the titration system. The sample solution was titrated at 25 + O.Ol”C in a nitrogen atmosphere with 0.101 N HCl using a Toadenpa automatic titration apparatus (Model HMS-10 A pH stat equipped with a Model XTR recorder). Each system was titrated at least twice, and only differences between duplicates of0.04 pH units or less were tolerated.

RESULTS AND DISCUSSION

Titration Curve

The polymers prepared as described above have the ammonium ion as an ionizable group. The polymers were dissolved in a solvent containing sodium hydroxide to convert the ammonium ion to the amino group and then back-titrated with hydrochloric acid. Typical examples of titration curves are shown in Fig. 1. It is observed that the titration curves have two inflection points. The first inflection point is related to the neutralization point of excess of sodium hydroxide added to the sample solution.

The equivalent weight (Ew) was obtained by dividing the weight of polyamino acid by

5 The amount of NaOH was determined from pre- liminary experiments in which the potentiometric titrations were carried out for the sample solutions (35 ml) containing 10-3-10-5 mol of NaOH and 20-40 mg of polyamino acid. The amount of alkali used here is sufficient to convert the ammonium ion to an amino group in the polyamino acid.

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TITRATION BEHAVIOR OF POLYLYSINE AND LYSINE COPOLYMER 25

0.103N HCI volume Vim/)

0 8 12

10

l---i-"""

B

10

II_;_\: 0.5 I. 0 1. 5 0.103 N HCI volume V (ml)

FIG. 1. Potentiometric titration curves of PL (A) and copoly(lys, ala) (B) at an ionic strength of 0.1. The degree of neutralization (a,,) of amino group with hydrochloric acid was calculated as follows: (Y. = (V - V,)/(V, - VI), where VI represents titrant volume (V) at first inflection point, and VP is V at the second inflection point.

the total mole number of ionizable groups which was calculated from the titrant volume between the two inflection points. The Ew value is independent of the ionic strength within experimental error (~8%). The average values of Ew are as follows: 303 for PL and 467 for copoly(lys, ala). These results obtained by the titration curves showed clearly that the thermal copolymeri- zation of neutral amino acid decreases the density of ionizable groups in the resulting polyamino acid. On the other hand, the Ew value of PL does not agree with the value (164.5) calculated by the assumption that PL is a homopolymer having the repeating unit of -NH-CH(C,H,NH, HCl)-CO-. This disagreement can be explained by assuming that the PL contains some residues

which lost titratable amino groups during the thermal polycondensation.

The Ew value can also be calculated from the results of amino acid analysis by dividing the weight of polyamino acid by the total number of lysyl residue. The Ew values obtained are as follows: 313 for PL and 461 for copoly(lys, ala). These values agreed ap- proximately with those estimated by the potentiometric titrations; however, the recovery of amino acids was not quantitative after acid hydrolysis. The agreement of Ew values obtained by potentiometric titration and by amino acid analysis suggests that the titratable amino groups are in the lysyl residues of PL and copoly(lys, ala).

Dependence of Apparent Dissociation Constant on the Degree of Dissociation

In order to evaluate the dissociational behavior of PL and copoly(lys, ala), the dependence of the apparent dissociation constant (pK,) on the degree of dissociation (1~~) of protons from ammonium groups in the polyamino acid was investigated at various ionic strengths. The values of pKa and ffd are calculated by the following equations:

pKa = pH + log [(I - @)/ad] [II and

(Yd = (1 - a,> + [cc,,,- - c,+)lc,l, L21

where a, represents the degree of neutralization of the base with acid (see Fig. l), CH+ and COH- are the molarity of protons and hydroxyl ions, and C, is the molar concentration of the amino groups of PL or copoly(lys, ala) in a titration system. The curves of pK, vs ffd obtained at various ionic strengths are shown in Fig. 2. It was found that the curves of PL and copoly(lys, ala) show monotonous changes, whereas that for poly-L-lysine prepared by Leuchs method showed a characteristic pattern due to the helix-coil transition (15). From the pattern of the curves of pKa vs (Yd for PL and copoly(lys, ala), it was suggested that these polyamino acids behave as a random coil in the titration process. This was supported by the facts that the specific rotations of PL and copoly(lys, ala) were almost zero.

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26 KOKUFUTA ET AL.

0.5 ‘d

FIG. 2. Dependence of apparent dissociation constant (p&J on degree of dissociation (q,) for PL (A) and copoly(lys, ala) (B) at various ionic strengths (0, 0.1; n, 0.5; 0, 1.0).

On the other hand, the magnitude of the change in pKn on ffd is not weakened by the addition of a neutral salt or by the copoly- merization of alanine. This indicates that the reduction of the interaction between protons and polyion is not induced by the electrostatic screening of the polyion charge with counterions and/or by the decrease in the density of ionizable groups with the copoly- merization of neutral amino acid. These be- haviors for PL and copoly(lys, ala) are in contrast to those of linear chain polymeric bases such as glycol chitosan (GC) (21, 22), and are similar to those of branched chain polymeric bases such as polyethyleneimine

(PEI) (22-24). It is known that the interaction between protons and GC ions decreases rapidly because of the electrostatic screening of the polyion charge with counterions (21, 22). However, the interaction between protons and PEI ion is not weakened by electrostatic screening of the polyion charge, since the protonated amino groups form hydrogen bonds with the nearest neighboring amino groups in the branched chain (22-24). The presence of the branching points in PL and copoly(lys, ala) was qualitatively estimated by the analyses of the hydrolysates of dinitrophenylated PL and copoly(lys, ala), and free lysine was iso- lated from the hydrolyzates (see Table I). Therefore, the dependence of pKa on (Yd for PL and copoly(lys, ala) is explained by the assumption that the dissociation of protons from the polyion could be affected by hydrogen bond formation between protonated amino groups and the nearest neighboring amino groups due to the branching of the peptide chain.

This finding is also supported by the results obtained from the analysis of titrational data by the Henderson-Hasselbalch equation:

pH = pK - ?‘l log [( 1 - a&&j], ]31

where pK represents the average dissociation constant, and n is the empirical constant which denotes the magnitude of the interaction between protons and polyion as the deviation of n from unity. As is shown in Fig. 3, the Henderson-Hasselbalch plots {the PlOtS Of pH VS lOg[(l - (Y,j)/(Yd]} at various ionic strengths represent a linear relationship in the (Yd region from 0.2 to 0.8. The values of n obtained from these plots are listed in Table II, together with those of GC (21, 22) and PEI (22, 23). The n value of GC decreases with increasing ionic strength and reaches unity at an ionic strength of 0.2, indicating that the interaction between protons and GC ion is weakened by the electrostatic screening of the polyion charge with counterions. In contrast to GC, the n value of PE I becomes larger with increasing ionic strength because of hydrogen bond formation between the protonated amino groups and the nearest neighboring amino groups due to the branched chain. The

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TABLE I

PROPORTIONOFVARIOUSDNP-LYSINESANDOFFREE LYSINE INTHE HYDROLYSATES OF DINITROPHENYLATED PL AND COPOLY(LYS, ALA)

Polymer cr-DNP-lys l -DNP-lys Di-DNP-lys Free lys

PL 43” 20 3 34

(44)b (11) (11) (33)

Copoly(lys, ala) 34 11 1 54

a Values given are percentages. b The values in parentheses were reported by Heinrich et al. (4), using a PL sample prepared by thermal

polycondensation at 195°C for 0.25-4 h. -

variations of n values with ionic strength for PL and copoly(lys, ala) resemble that of PEI and are different from that of GC. Hence, the results obtained by the Hender- son-Hasselbalch equation also indicate the branched chain structure of PL and copoly- (lys, ala) prepared by thermal polycondensation.

Intrinsic Dissociation Constant

The dissociation behavior of PL and copoly(lys, ala) were also evaluated by the intrinsic dissociation constant (pK,) of the amino groups in the polyamino groups. The pK, value was calculated from the titrational data by the following equation (25):

PKCI = (PK,)a+ - 0.434(e2&DkT) [4]

or pK, = pK - 0.434(e2d3DkT) [51

Here, the term 0.434(e2d3DkT) (e, electronic charge; K, Debye-Hiickel parameter; D, the dielectric constant of the solvent; k, Boltzman constant; T, absolute tempera- ture) represents the correction term related to the free energy consumed for building up the ionic atmosphere. The (pK,),, value was estimated by the graphical extrapolation of the curve of pKa vs ad (see Fig. 2). The pK value was also estimated by the Henderson-Hasselbalch plot at (Ye = 0.5 (see Fig. 3). The results are shown in Table III, together with those of poly-L-lysine (15), ribonuclease (16), bovine plasma albumin (17), GC (ZZ), and PEI (22). It is found that the pK, values for PL and copoly(lys, ala) are clearly different from that of poly-L-

lysine prepared by Leuchs method and are close to those of a-amino groups in proteins such as ribonuclease and bovine plasma albumin. These results indicate that the lysyl residues in the thermally prepared PL and copoly(lys, ala) carry mostly a-amino groups. On the other hand, from Table III, it is also found that the pK, values which were corrected by the term related to the free energy consumed for building up the ionic atmosphere, are more or less depend- ent on ionic strength and that the dependence of pK, on ionic strength for PL is stronger than that for copoly(lys, ala). Furthermore, some disagreement between

TABLE II

EFFECTOF IONIC STRENGTH (~)oN THE ~VALUEOBTAINEDBYHENDERSON-

HASSELBALCH PLOTS

Polymer P n Reference

0.1 1.26 PL 0.5 1.40

1.0 1.49 Present study

0.1 1.48 Copoly(lys, ala) 0.5 1.53

1.0 1.59

0 1.83

PEI 0.05 2.05 0.1 2.10 C%W 0.2 2.15

0 1.29

GC 0.05 1.13 0.1 1.11 (21,22) 0.2 1.00

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28 KOKUFUTA ET AL.

8

$

7

-05 0 0. 5

9

8

h

7

.

6 8111'11~~ -05 0

I-ad log- *d

1

FIG. 3. Henderson-Hasselbalch plots for PL (A) and copoly(lys, ala) (B) at various ionic strengths (0, 0.1; A, 0.5; 0, 1.0).

the pK, values calculated by the Eqs. [4] and [5] was observed. As was reported in previous papers (13, 14), the pK, values obtained from pK,,, by the same analytical method, agreed with those obtained from pK. These pKo values were independent of ionic strength in the range above 0.5. Therefore, the disagreement observed here could be inter-

preted by the interaction between the nearest neighboring amino groups as was discussed in the previous section.

Ratio of (Y- and E-Amino Groups Attached to the Lysyl Residues

In order to obtain detailed information on the species of amino groups in the thermally prepared PL and copoly(lys, ala), the titrational data were analyzed in the same manner as described previously (14). The molar concentrations of two species of amino groups (C, for e-ammonium ion and C, for a-ammonium ion) can be expressed by the following equations (see Appendix):

Y=C1+CJ [W

y = (K, + C,+>U - ah>C, C

[6bl H+

x= K, + CH+

K2 + CH+ ’ WI

if the correction for the activity coefficients is negligible. Here, K, and K2 represent the dissociation constants of protons from E- and a-ammonium groups, o& is the sum of the degree of dissociation for each ammonium groups, and C, is the total amino group concentration. Equation [6] is considered to be the formal resemblance of Speakman’s equation (26) which is used to determine the dissociation constants of weakly dibasic acid such as adipic acid. The values of C, and C, can be estimated by means of the interception on the Y-axis and the slope of a straight line which was obtained by plotting Y against X, if the values of K, and K, are known. In the present stage of the study on the thermal polyamino acids, however, little is known about the K values of (Y- and e-amino groups in PL and copolymer of lysine with other amino acid. In this study, some values of K, and K, in the pK range from 1 to 13 were selected for the estimation of C, and C, from the titrational data by Eq. [6], and the reasonable values of K, and K2 were determined at conditions where the plots of Y vs X were best fit to a linear relationship. This fitness was evaluated by means of correlation coefficient (r) obtained by the

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TABLE III

INTRINSIC DISSOCIATION CONSTANTS OF AMINO GROUPS FOR THERMALLY PREPARED PL AND COPOLY(LYS, ALA) AND FOR VARIOUS POLYPEPTIDES

AT 25°C AS FUNCTION OF IONIC STRENGTH

Polymer 0.434eQ

Ir. (P&l),-+, PK 3DkT (p&Y (p&Y Reference

PL

Copoly(lys, ala)

Poly-L-lysine

Ribonuclease

Bovine plasma albumin

0.1 7.40 0.5 7.68 1.0 7.93

0.1 7.75 0.5 7.89 1.0 7.98

0.1 10.22 0.4 10.39

10.53 0.7 1.2 10.56

(Y-NH, (l)c e-NH, (10) a-NH2 (l)c e-NH, (66)c

7.21 0.11 7.54 0.24 7.76 0.34

7.41 0.11 7.68 0.24 7.79 0.34

- 0.11 - 0.22 - 0.28 - 0.37

7.29 7.20 7.44 7.30 7.59 7.42

7.64 7.30 7.65 7.44 7.64 7.45

10.11 - 10.17 - 10.25 - 10.19 -

(pKo)d = 7.8 (pKJ = 10.2 (pK,) = 7.8 (pK,) = 9.8

Present study

(15)

(16)

(17)

PEI

GC

Primary, secondary, and tertiary amino groups

(pK,)” = 8.66

(pK,)’ = 6.55

(22)

(22)

o Calculated from (pK,),,, by Eq. [4]. * Calculated from pK by Eq. [5]. c Number of amino groups obtained from amino acid analysis. d Average value obtained from the titration curves at the ionic strength range of 0.03-0.2. e Obtained by the extrapolation of linear plots of pK, vs electrophoretic mobility to zero mobility.

following equation:

& ; (Xi - X’)(Yi - P) N i=l

Y= 7 m UXUY

where B and Y are average values of Xi and Yi, respectively, ox and (TV are the standard deviations of Xi and Yi, respectively, and N is the data number. The calculation of X, Y, and r at various K, and Kz, and the determination of K,, Kt, Cl, and Cz at r maximum were performed on an electronic computer (TOSPAC 5600) using the FORTRAN program.

The typical examples of the plots of Y vs X are shown in Fig. 4 (represented by open circles). The results of pK1, pK2, and r measurements obtained by the analytical method mentioned above are also shown in Table IV. It was found that the content

of a-amino group is about 93-96% of the titratable amino groups in PL and copoly- (lys, ala). Furthermore, the values of K, and K2 which were chosen for the determination of C, and Cz to give a maximum r, increased with the increasing of ionic strength. However, the values corrected by the term of 0.434 (eW3DkT) in Eq. [4] or [5], which are related to the free energy consumed for building up the ionic atmosphere, are almost constant in the ionic strength range above 0.5 and are similar in the value to the intrinsic dissociation constants of (Y- and e-amino groups of proteins such as ribonuclease and bovine plasma albumin (see Table III). These sug- gest that the corrected pK values are intrinsic dissociation constants of (Y- and e-amino groups in the PL and copoly- (lys, ala).

On the other hand, the ratio of (Y- and

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30 KOKUFUTA ET AL.

e-amino groups can be estimated by the contents of (Y- and l -DNP-lysine in the hydrolysates of dinitrophenylated PL and copoly(lys, ala) (see Table I). The percentages of (Y- and e-amino groups are: CY, 67%, and E, 33%, for PL; and a, 74%, and E, 26%, for copoly(lys, ala). These results do not agree with those obtained by potentiometric titration. In order to investigate this disagreement, the results estimated using the DNP derivatives are compared with those of potentiometric ti-

trations. In Fig. 4, the relationship of Eq. [6a], which agrees with the results obtained by the content of DNP derivatives, is expressed by the broken dashed lines. The black circles are plots of Eq. [6a] at the values of C, and C, which most closely match those expressed by the dashed lines. Under the conditions used, the values of pK and r at ionic strength 0.1 are: pK, (c-amino group), 7.6; pK, (a- amino group), 7.0; and r, 0.799 for PL; pK, (e-amino group), 8.4; pK, (a-amino group), 7.0; and r 0.963, for copoly(lys, ala). From these results, it was found that pK, values were particularly lower than those of e-amino groups in general polypeptides and proteins. Therefore, the disagreement between the chemical method and potentiometric titration could be considered due to the incomplete dinitrophenylation of the lysine polymers and/or the incomplete hydrolysis of the dinitrophenylated polymers.

The fact that the lysyl residue in the thermal polymers is composed mainly or entirely of e-linked residue, might indicate the following propagation of the lysyl residue in the polycondensation process. Lysine was first converted to a seven- membered a-amino lactam. The amino group of the other amino acid reacted with the lactam to form E-lysyl residue by a transamination reaction, and the transamination processes were repeated to form polymers containing E-lysyl residues.

X

FIG. 4. The plots of Y against X for PL (A) and copoly(lys, ala) (B) at ionic strength of 0.1. Open circles are the plots at maximal correlation coefficient, i.e., the case in which the values of K, and K, in Eqs. [6b] and [6c] were selected to obtain the best fit to a linear relationship. Dashed lines present the data of Eq. [6a] expressed by the values of C, and C, obtained from the results shown in Table I: Y = 0.00083 + 0.00165X for PL and Y = 0.00054 + 0.00157X for copoly(lys, ala). Black circles indicate the case in which the values of K, and K, were selected so that the plots would be the best fit to the dashed lines expressing the above-mentioned relationships.

FI fi + ~NH(CH~)~~HC~..INHFHC~

NH R

-SO Ctt~~i~

P’2k s

APPENDIX

We consider a system consisting of a mixture of two species of ammonium ions (R,-NH3+ and R,-NH,+) whose dissociation constants overlap each other over a wide pH range. Assuming that the activity

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TABLE IV

THE VALUESOF K, AND K2 CHOSENTO GIVE MAXIMUM~AT VARIOUS IONIC STRENGTHS(~) ANDTHE VALUESOFC,ANDC~AT~,,,~

Polymer p PK, PKZ rmax c, X Iv(M) c, X 103(M) c&c, + c,) c, X lo3 (M)

PL 0.1 10.5(10.4) 7.2C7.1) 0.997 0.10 2.33 0.96 2.48 0.5 10.8(10.6) 7.5(7.3) 0.991 0.18 2.34 0.93 2.57 1.0 lO.g(lO.6) 7.6(7.3) 0.990 0.15 2.38 0.94 2.49

Copoly- 0.1 10.7GO.6) 7.4(7.3) 0.992 0.15 2.02 0.93 2.11 (lys, ala) 0.5 lO.g(lO.7) 7.6(7.4) 0.991 0.11 2.02 0.95 2.09

1.0 ll.O(lO.7) 7.7C7.4) 0.991 0.13 1.98 0.94 2.15

a The K value was expressed as the negative logarithmic form by the relation of pK = -1ogK. The values in parentheses are pK values corrected by the term of 0.434(e2K/3DkT) in Eq. [4] or [5]. Total concentration ( Ct) of amino groups in the titration system was obtained by potentiometric titration curve.

coefficients are unity, the dissociation constants of ammonium ions may be defined:

K 1

= [H+l[R1-NHzl [RI-NW1

IA-11

K 2

= W+lFL-N&l [R,-NH,+] ’ LA-21

where brackets signify the concentration of the species. The total concentrations (C, for R,-NH: and C, for R,-NH:) of each ammonium ion in the system are given as:

Cl = [RI-NH,+] + [RI-NH,] [A-3]

C, = [R,-NH,+] + [R,-NH,] [A-4]

When a strong monobasic acid, which may be completely dissociated, has been added to give a molar concentration (M), the requirement of “the electroneutrality” is defined:

[H+] + [RI-NHs+] + [Rz-NHs+l

= [OH-] + [X-l - M [A-5]

where [X-l is total concentration of the counterions for the two ammonium ions. From the combination of Eq. [A-l] with Eq. [A-3] and that of Eq. [A-2] with Eq. [A-4], we obtain:

[&-NH;] = C1’H+l [H+l + K,

[A-61

[&-NH;] = Cz[H+l [H+l + G

IA-71

Introducing the terms of [R,-NH:] and [R,-NH,+] into Eq. [A-5], the following expression can be obtained:

([X-l - M + [OH-] - [H+])(K, + [H+])

[H+l

= c + K, + W+l 1

& + [H+l CP [A-f31

Eq. [A-8] takes the same form as Eq. [6] from the previous section when the left- hand side of Eq. [A-8] is rewritten using the relationship between the sum of the degree of dissociation (a:) for each ammonium ion and the total concentration of ammonium ions (C,):

a5 = [%-NH,1 + &-N&l Cl + c2

= C, - WG-NH,+1 + WrNWl) Ct

= 1 _ C-1 -M + W-1 - [H+l Cl

,A-9l

ACKNOWLEDGMENT

This work was partially supported by a scientific Research Grant from the Ministry of Education, Japan.

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