10408
J. Phys. Chem. 1995,99, 10408-10411
Kinetics of Precipitation of Monodisperse Short Chain Polypeptides? Hiroshi Maeda* and Akio Nakaishi Department of Chemistry, Faculty of Science, Kyushu University, Fukuoka 812, Japan Received: October 24, 1994; In Final Form: March 13, 1995@ Kinetics of precipitation of monodisperse samples (degrees of polymerization x = 11, 12, and 13) of uncharged
poly[S-(carboxymethyl)-L-cysteine] was measured at 25 f 1 “C in salt-free solutions at pH 2.20 f 0.02 as functions of molar affinity 4 (=RT ln(Ct/S)) of the reaction, with C, and S denoting the initial concentration and the solubility. The induction time tp was shorter for longer chains. Dependence of tp on 4 was given common to three chains as follows: log tp = -(2.9 f 0.2)4 a,where a slightly depended on the chain length. A scheme is proposed to account for the result, consisting of the fast pre-equilibrium for trimer formation followed by the rate-limiting step of the growth of the two-dimensional aggregates through the addition of the trimers. The rate of the addition is proportional to exp (uIRT), where cr represents the surface tension of the trimer. According to the model, the chain length dependence of a is given by the rate of detachment of a single chain from the surface of aggregates, which was shown to be very weakly dependent on the chain length in ther present case.
+
Short chain homopolypeptides are incapable of forming a secondary structure through intramolecular interactions. They, instead, tend to associate in water particularly in an uncharged state through hydrogen bonding between peptide groups. This type of association leads to a two-dimensional aggregate @sheet), adjacent chains usually being in antiparallel arrangement. However, P-sheets consisting of short chains are subject to further association through side chain interactions to form threedimensional aggregates which eventually tend to precipitate out from the solution. The few studies on soluble P-sheets consisting of short chains in aqueous solutions are on polydisperse samples’,*or on chains with a solubilizing moiety such as polyethylene glyc01.~~~ Little is known about the kinetics of precipitation of short chain polypeptides. When the P-sheet formation of monodisperse samples of poly[S-(carboxymethyl)L-cysteine] (poly[Cys(CH2COOH)]) was examined in aqueous solutions, well-defined solubilities were ob~erved.~ It tumed out that crystallization took place and that soluble P-sheets did not occur in solutions even in equilibrium with precipitates. In the present study, we will report the kinetics of precipitation of monodisperse samples of poly[Cys(CH2COOH)] with the degrees of polymerization x ranging from 11 to 13. The present system has some interesting characteristics from the viewpoint of kinetics of crystallization. The first of them concerns the chain nature of the molecule. Generally, a polymer has many different conformations in solution but these are reduced to one most stable conformation in the crystal. Another concems the anisotropic nature of the crystal growth in the case of polypeptides. Along one of the two directions perpendicular to the molecular axis, the hydrogen bond between peptide groups is operating as depicted schematically in Figure 1, while along the other direction side chain-side chain interactions develop. In the present polypeptides, the hydrogen bond between carboxyl groups makes the major contribution. Along the remaining direction, Le., the direction along the molecular chain, interactions between the two molecular ends are weak. There are two possible types of crystal growth in this direction. If the aggregation of the kinetic units (not necessarily single chains) occurs in the “in register” manner, two-dimensional aggregates (sheets) will be formed at first. The surfaces of these sheets are covered with nonpolar methyl and ethyl groups. These Dedicated to Professor Jtirgen Engel to celebrate his 60th birthday. @Abstractpublished in Advance ACS Abstmcts, June 1, 1995. +
0022-3654/95/2099-10408$09.00/0
sheets aggregate further through hydrophobic interaction leading to the formation of three-dimensional aggregates. If, on the other hand, the aggregation of kinetic units takes palce in a staggered way, we can expect the growth will take place in three dimensions more or less isotropically. As to the ,&sheet formation of single polypeptide chains, we have reported the kinetics of chain folding of a random coil poly[Cys(CH2COOH)] into the unassociated intramolecular B-sheet6s7
Experimental Section Monodisperse samples of poly[S-(carboxymethyl)- cysteine] with the degrees of polymerization x ranging from 11 to 13 were prepared as described previou~ly.~Amino and carboxyl ends were blocked with acetyl and ethylamide groups, respectively. The induction time tp was defined at 25 f 1 “C as the time when the solution began to be turbid after its pH was brought to 2.20 f 0.02 from neutral pH. The mixing time, the time required for attaining the uniform pH, was about 30 s or less. Turbidity of the solutions was detected by eye. At pH 2.2, all carboxyl groups are expected to be protonated since pKo = 3.2.8
Results Induction Time. Induction times tp of three samples at different residue concentrations Cpare shown in Figure 2. When plotted against log C,, log tp decreases linearly, but deviations are seen for those t p values shorter than about 1 min. In this time range, tp was not determined accurately. However, in the first approximation, three straight lines with a slope -3 are drawn in Figure 2 for monodisperse samples. Results on a polydisperse sample containing several different kinds of chains (x ranging from 8 to 12) are also shown in Figure 2. For this sample at a residue concentration higher than about 20 mM, log t p shows some deviation from the linearity with C,,although tp values for this sample were long enough to be accurately determined. This is ascribed to the presence of soluble aggregates which have been found in the case of polydisperse samples.’.* Polypeptides in the soluble aggregates take the P-conformation, as in the crystals. When the concentration is normalized with the solubility S, dependence of tp on the reduced concentration becomes identical 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 25, 1995 10409
Monodisperse Short Chain Polypeptides
IO’
.-c
E
%
10
1
Figure 2. Dependence of the induction time tp on the residue concentration Cp for different samples. Monodisperse samples: x = 13 (triangles), x = 12 (filled circles), and x = 11 (open circles). A polydisperse sample consisting of chains with x ranging from 8 to 12 (filled squares). Straight lines are drawn with a slope of -3.
for three samples, as shown in Figure 3. The dependence can be well described with eq 1 in terms of the affinity of the reaction. log(t4min) = -(2.9 f 0.2) log(CJs)
+a
(1)
In eq 1, Ct denotes the chain concentration, and the solubilities S are also expressed in molarity. Values of S were (6.2 f 0.2) x (2.4 f 0.1) x and (1.34 & 0.02) x M for the samples of x = 13, 12, and 11, re~pectively.~ The solubilities in the residue molarity are obtained by multiplying S by x. The intercept a at C, = S could be approximately regarded as common to the three samples, although closer inspection reveals it is chain length dependent: values of intercept a are 2.84, 2.68, and 2.52 for x = 11, 12, and 13, respectively. Spacings between two adjacent lines are constant, A d & = -0.16. The present results will be summarized as follows. (1) Soluble aggregates of monodisperse samples were present in negligible amounts, contrary to the case of polydisperse samples.’,2 This indicates that the chains in the aggregates of monodisperse samples are “in register” alignment. (2) Induction time is shorter for longer chains. This strongly suggests the presence of a fast pre-equilibrium for the formation of some stable intermediate(s). (3) Different rates for different chain lengths can be described with a common equation (1) when the concentrations are scaled by respective solubilities. (4) The concentration dependence of the rate -d log tdd log Ct is approximately 3.
C t IS
Figure 3. Induction time tp as a function of the reduced concentrations CJS.
Characteristics of the Kinetics of the Present System. In many association reactions where both monomer (A,) and aggregates (An: n-mer) have similar kinetic characteristics, the addition of monomers to aggregates is the main pathway of growth since the monomer is predominant over other aggregates with respect to number concentration? In the case of linear chain molecules, on the contrary, it is unlikely that this process is the main pathway since it would take a long time for a chain molecule to settle onto the surface in a correct manner, i.e. “register” manner in the present case. It is expected that the rate of the addition of the monomer is much smaller than those of the combination of two aggregates because of the flexible chain nature of the monomer against the fixed conformation of chains in aggregates An (n 1 2). Hence we assume that the main pathway of aggregation is the combination reactions and that the contribution from the addition of the monomer is neglected hereafter. As to the backward reaction, detachment of single chains from the surface of aggregates is much more likely than splitting of an aggregate into two small aggregates. Among many possible combination reactions, we assume that the growth takes place by the addition of a few kinds of species (“basic units”) to the aggregates of various sizes. In conformity with result (2), a fast pre-equilibrium is suggested between monomers and a stable intermediate (i-mer). The basic units are related to the stable intermediate, i-mer, and we assume the simplest case that the i-mer itself serves as the basic unit. Important intermediates may distribute over several different sizes, but we approximate them with a single species, i-mer.
Maeda and Nakaishi
10410 J. Phys. Chem., Vol. 99, No. 25, 1995
Due to the assumptions given above, the main reactions of the present system can be simplified to the following sets of reactions. K,
iA, = Ai Ai
k, + A, 7 Am+i(m 2 i)
k- 1
(3)
Fast Pre-equilibrium. Chemical potentials of the chains in the precipitates (crystals), the i-mer, and the monomer are designated as pppt, pi, and P I , respectively. For the latter two their values at the standard states are denoted as pi* and PI*. From the assumed fast pre-equilibrium of reaction 2, we have eq 5.
PI =Pi
(5)
The formation constant of the i-mer, K,, of reaction 2 is defined as eq 6 and is related to the solubility S by eq 7.
K, = C, ICl‘
(6)
+ RTi In S
(7)
= iPppt+ (J,
pi* + R T l n S = p p p t
(13)
When eq 1 is compared with eq 13, we have found that i = 3. The observed scaling of the concentration in terms of the solubility S requires the term k- to be independent of the chain length x or weakly dependent on x. The dissociation process takes place only through the detachment of the monomer from the surface of aggregates, as shown in reaction 4. If we take this into account, the backward rate of reaction 3, k-, can be approximated as k-lli. Here we neglect the time required for i dissociated chains to form an i-mer since we assume the fast pre-equilibrium for the i-mer formation. It is to be noted that the affinity A of reaction 3 is given as eq 14.
+
A = (si RTln Ci
(14)
When eqs 12 and 14 are introduced into eq 10, we obtain the following relation between the rate and the affinity of reaction 3. (15)
Dependence of k-1 on the Chain Length x. From eq 13, the chain length dependence of the intercept a of eq 1 is given as follows.
(9)
A d A x = A(ln k-)lAx = A(ln k-,)lAx
The Rate Limiting Step. The aggregation of the basic units leads to the formation of nuclei. The standard theory of nucleation based on the isotropic growth of the aggregates failed to explain the observed results (3) and (4), however, when applied to the aggregation of the basic units, as is briefly discussed later. Hence, the nucleation in the present case is expected to take place in two steps. In the first step, the aggregation of the basic units proceeds in the two-dimensional way: in the plane perpendicular to the molecular axis. The thickness of the sheets corresponds to the end-to-end distance of the chain in the ,!?-conformation, which is proportional to x and about 4 nm. The surfaces of the two-dimensional sheets (stacked ,!?-sheets) are covered with nonpolar methyl and ethyl groups. The hydrophobic interaction among these sheets leads to the formation of the three-dimensional nucleus. As a simple model, the precipitation is considered to take place when the surface area reaches a certain critical value. Then, the rate of nucleation, J , is proportional to the rate of growth of the sheets, which is given as follows in terms of a constant independent of x , since the critical surface area is independent of x . J = constant k+ C,
+ i ln(CjS) + constant
(8)
Solubility, S, is related to the true phase equilibrium
(10)
From eqs 6 , 7, and 10,
+ +
-
J = constant k- exp(AIR7‘)
The “surface tension” of the i-mer, u,, is defined as PI*
(12)
The total concentration Ct is generally approximated with the free monomer concentration CI,since the amount of precipitates is negligible at the moment tp. The rate of nucleation J has been considered to be inversely proportional to the induction time tp.9,10 Taking these into account, we have finally
- In tp In J = In k-
+
A,+[ -Am A, (m 2 i) (4) The rate constants k+, k-, and k-1 are assumed to be common to aggregates of different sizes.
-RT In K, = p,* - ipl* = a,
k+ = k- exp(ajRT)
+ +
In J = ln(constant k+C,) = In k , ln(K,C,’) constant = In k+ - (J, /RT i ln(C,/S) constant (1 1) The equilibrium constant of reaction 3 is given as k+lk-, which is related to the standard free energy change of the reaction (-ai) as follows.
(16)
Here we examine the dependence of k-1 on x . Under the condition that the hydrogen bond between peptide groups is favored, a bond in the middle of a sequence of the bonds cannot be broken until its neighbor is broken. On the long time scale, however, each bond is characterized with the same equilibrium constant. The rate k-1 is mainly determined by this cooperative nature of a sequence of the hydrogen bonds rather than less cooperative side chain-side chain interactions in the twodimensional sheets. We consider the kinetics of breaking a sequence of the hydrogen bonds in the text below. Chains that detach from the surface of aggregates are connected to the surface through n, hydrogen bonds, n, being 12, 13, and 14 for x = 11, 12, and 13, as shown in Figure 1 schematically for the case of x = 11. The rate of detachment consists of two contributions: opening one of the two ends or both followed by the propagation or “unzipping” toward the other end. The rate of the initiation step is independent of the chain length, but the unzipping step is rate-limiting and hence it should be examined here. The present problem is identical to the unraveling model introduced by Spodheim and Neumann in the kinetics of triple-helix formation of polynucleotides.’ The time tl (mean first transit time) required to break all n, bonds starting from one end is given by Jackson and Silberberg as follows.”
’
Here p and q denote the probabilities of forming and breaking one bond, respectively, and z is the average time constant for this elementary process. According to eq 17, tl depends on n,. In the present problem, the “unraveling” can be initiated at both ends of an array of n, bonds. The problem will be better
Monodisperse Short Chain Polypeptides
J. Phys. Chem., Vol. 99, No. 25, 1995 10411
described as evaluating the average time for the two vortices starting from both ends to meet each other somewhere in the sequence of n, bonds. For simplicity, however, we examine the dependence of the rate on nx according to eq 17. We roughly evaluate the probabilities p and q based on the equilibrium data as follows. We can write the following relation in terms of fhb, the fraction of the hydrogen bond contribution to the total stability.
The value of [(RT In S)/x] has been given5 as -3.8 k 0.1 kJ mol-' independent of x in the examined range of x. On the other hand, the free energy difference (per residue) of long chain uncharged poly[Cys(CH2COOH)] between the random coil state and the unaggregated intramolecular P-sheet in solution has been evaluated8 as -1.7 f 0.4 kJ mol-'. The latter value can be taken as representing the maximum contribution from the peptide hydrogen bond: (&,)ma = 1.7/3.8 = 0.45. Then, we have ln(p/q) = 1.5(Xft,/nx) l.5(fhb)max 0.68. We then evaluate p / q = 2.0 and p = 0.67 and q = 0.33. Expected variation of the intercept of the plot log tp vs log Ct amounts to
-
N
A a = A log k- = A log t , = A[2(p - q)nx]/2.303
n* = (2q/34)3
Discussion There seems to be few existing theories capable of explaining the present results (1)-(4). According to the classical nucleation theory of varying nucleus size and isotropic growth, the rate of nucleation J is given as follows.9
+ In k- + (1/2) ln($/n*) + constant (20)
where k- and n* represent, respectively, the dissociation rate of a unit from the aggregate (assumed to be independent of the aggregate size) and the size of the critical nucleus. When the
(21)
The major contribution among the terms of the rhs of eq 20 comes from the first term. Hence, In J varies linearly with n*#, which becomes #-2 according to eq 21. This dependence on # cannot explain the obtained results in the present study. Next we apply the above theory to the addition reaction of the basic unit. To avoid confusion, the species under consideration are denoted as B,: Bl = Ai. Equation 20 is rewritten as follows when approximated with the first term on the rhs.
In terms of the equilibrium concentration of the monomeric basic unit (CBI),
-
Finally, we have In J hP2#-*. We see, thus, that the standard theory of nucleation cannot explain the observed results even when it is combined with the concept of the basic unit. Oosawa has proposed a theory of protein crystallization in terms of the nucleus of a fixed size determined by the chemical nature of the protein.I2 According to Oosawa's theory, the time to required for a given fraction of monomers to be crystallized is given as follows under certain approximations in terms of the affinity # and the size of the nucleus i. N
(19)
Hence we have AalAn, = 2(p - q)/2.303 = 0.30. The estimated value 0.30 is larger than the experimentally found value (0.1-0.2). There are at least two factors leading to smaller calculated A a values. A smaller value offib yields p / q values smaller than 2.0. The unraveling process from both ends provides a weaker dependence of tl on n, than that given by eq 17. If we take these into account, the above estimation is consistent with the observed result. Conclusions. We propose the following scheme for the precipitation of monodisperse polypeptides observed in the present study. It consists of the fast pre-equilibrium for the trimer formation and the rate-determining step of the growth of the two-dimensional sheets through the addition of the trimers. Three-dimensional aggregates are formed when the sheets reach a certain critical surface area. For a trimer two possibilities exist: a planar three-strand P-sheet and a species with a two-stranded P-sheet and single strand on top of it. Probably both species may serve as the basic unit. Any size of aggregates of the P-sheet or extended chains could work as the basic unit for aggregation. It is reasonable and interesting that the experimental results suggest a trimer rather than a dimer as the basic unit for the growth of aggregates. As the size of the basic unit increases, its stability increases, but its population decreases due to a kinetic situation. Dimers are more readily formed but less stable. Tetramers are more stable than trimers, but their formation from disordered chains takes longer. Trimers are the best compromise for both stability and population requirements.
In J = -n*4/(2kT)
surface tension of the n-mer is denoted as q ~ n ~n*' ~is, given as follows.
log to = -(i/2) log4
+ constant
(25)
However, the time to in eq 25 differs from the induction time tp in the present study. The Oosawa theory refers to the rate of crystallization, while in the present study the induction time is studied. Experimental studies should be extended to confirm the validity of the reaction scheme proposed in the present study.
Acknowledgment. We thank Dr. Olsson (Gambro AB) for discussions and sending us a copy of ref 10. We thank Drs. F. Oosawa and T. Ooi for discussions. This work was partly supported by a Grant-in-Aid (No. 02403004) from the Ministry of Education, Science and Culture, Japan, and by the Nippon Oil & Fats Co. Ltd. References and Notes (1) Saito, K.; Maeda, H.; Ikeda, S. Biophys. Chem. 1982, 16, 67. (2) Maeda, H.; Kadono, K.; Ikeda, S. Macromolecules 1982, 15,822. ( 3 ) Mutter, M.; Mutter, H.; Uhmann, R.; Bayer, E. Biopolymers 1976, 15, 97. (4) Toniolo, C.; Bonora, G. M.; Salardi, S.; Mutter, M. Macromolecules 1979, 12, 620. ( 5 ) Nakaishi, A.; Maeda, H.; Tomiyama, T.; Ikeda, S.; Kobayashi, Y.; Kyogoku, Y. J . Phys. Chem. 1988, 92, 6161. (6) Fukada, K.; Maeda, H.; Ikeda, S. Macromolecules 1989,22, 640. (7) Fukada, K.; Maeda, H. J . Phys. Chem. 1990, 94, 3843. (8) Kimura, M.; Maeda, H.; Ikeda, S. Biophys. Chem. 1988, 30, 185. (9) Nielsen, A. E. Kinetics of Precipitation; Pergamon Press: London, 1964. (10) Garside, J. Nucleation. In Biological Mineralization and Demineralization; Nancollas, G. H., Ed.; Springer-Verlag: Berlin, 1982. (11) Spodheim, M.; Neumann, E. Biopolymers 1977, 16, 289. (12) Oosawa, F.; Asakura, S. Thermodynamics of the Polymerization of Protein; Academic Press: London, 1975; Chapter 4. Oosawa, F. Seitui No Kagaku 1986, 37, 526. JP9428202
