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Kinetics of Protein Aggregation with Formation of Unreactive Intermediates V. P. Zhdanov*,†,‡ and B. Kasemo† Department of Applied Physics, Chalmers University of Technology, S-412 96 Go¨ teborg, Sweden, and Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Received October 30, 2003. In Final Form: February 6, 2004 Irreversible protein aggregation resulting in formation and deposition of insoluble fibrils or amorphous precipitates is usually assumed to occur via sequential attachment of monomers to soluble intermediates. We complement this scheme by slow conversion of the intermediates to a relatively stable form so that they do not react with monomers but can be trapped by precipitates. For reasonable values of parameters, our model predicts that the aggregation kinetics order may be between 2.0 and 2.5. In particular, the model can be used to explain the reaction order, 2.17 ( 0.09, observed for aggregation of recombinant human granulocyte colony stimulating factor.
Irreversible protein aggregation (IPA) resulting in formation and deposition of insoluble fibrils or amorphous precipitates may cause severe neurodegenerative disorders including Alzheimer’s and Parkinson’s diseases.1,2 For this and other reasons, IPA has attracted attention during the past decade.2,3 Due to the complexity of the protein behavior, the understanding of the physics behind IAP is however very incomplete (one of the problems here is that the concentration of transient species mediating IPA is low and can hardly be measured). The available experimental data3 indicate that the IPA kinetics can formally be divided into at least three groups. The apparently simplest kinetics, observed4 for example for human interferon-γ, exhibit first-order behavior suggesting that the rate-determining step is unimolecular. There are also kinetics [e.g., for recombinant human granulocyte colony stimulating factor5 (rhGCSF)] with an order close to 2, indicating that the rate-limiting step is bimolecular. The third group (e.g., R-synuclein6) shows a distinct lag phase (attributed usually to nucleation) followed by rapid native protein loss. Interpretation of the IPA kinetics is often based on various versions of the Lumry-Eyring model.7-9 In particular, the simplest scheme corresponding to the second group of IPA kinetics is represented as9 * Corresponding author. Fax: 46-31-7723134. E-mail: zhdanov@ fy.chalmers.se or
[email protected]. † Chalmers University of Technology. ‡ Russian Academy of Sciences. (1) Koo, E. H.; Lansbury, P. T.; Kelly, J. W. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 9989-9990. (2) Rochet, J.-C.; Lansbury, P. T. Curr. Opin. Struct. Biol. 2000, 10, 60-68. (3) Chi, E. Y.; Krishnan, S.; Randolph, T. W.; Carpenter, J. F. Pharm. Res. 2003, 20, 1325-1336. (4) Kendrick, B. S.; Cleland, J. L.; Lam, X.; Nguyen, T.; Randolph, T. W.; Manning, M. C.; Carpenter, J. F. J. Pharm. Sci. 1998, 87, 10691076. (5) Krishnan, S.; Chi, E. Y.; Webb, J. N.; Chang, B. S.; Shan, D. X.; Goldenberg, M.; Manning, M. C.; Randolph, T. W.; Carpenter, J. F. Biochemistry 2002, 41, 6422-6431. (6) Wood, S. J.; Wypych, J.; Steavenson, S.; Louis, J. C.; Citron, M.; Biere, A. L. J. Biol. Chem. 1999, 274, 19509-19512. (7) Zale, S. E.; Klibanov, A. M. Biotechnol. Bioeng. 1983, 25, 22212230. (8) Sanchez-Ruiz, J. M. Biophys. J. 1992, 61, 921-935. (9) Roberts, C. J. J. Phys. Chem. B 2003, 107, 1194-1207.
NhR
(1)
R + R f A(2)
(2)
A(2) + R f A(3) ... (n-1)
A
+RfA
(3) (n)
where N and R are the monomeric protein in the native and reactive states, Ai (2 e i e n - 1) are soluble aggregates, and A(n) is the protein complex having appreciable reactivity with respect to formation of insoluble fibrils or amorphous precipitates. The kinetic equations corresponding to this scheme can easily be solved9 assuming that step 1 is close to equilibrium (i.e., [R] ) p[N], where p , 1 is the equilibrium constant), steps 3 are rapid and accordingly the formation and conversion of the intermediates (with 2 e i e n - 1) is close to steady state, and step 2 is rate-limiting. In this case, the IPA rate is given by
d[N]/dt ) -nk1p2[N]2
(4)
where k1 is the rate constant of step 2. The scheme above predicts second-order IPA kinetics. It can be complemented9 by steps A(i) + A(j) f A(i+j) and/or by introducing solubility-related restrictions on [A(i)]. With these modifications, the reaction order may become lower than 2 or, more specifically, decrease to 1. As already mentioned, the first-order IPA kinetics have been observed. On the other hand, the experimental studies of the rhGCSF aggregation kinetics by Krishnan and coworkers5 indicate that the reaction order, 2.17 ( 0.09, is somewhat higher than 2. To interpret their results, Krishnan et al. used the Lumry-Eyring model and theory of diffusion-limited reactions but did not explain why the observed reaction order is above 2. Driven partly by this puzzle and partly by an analogy between IPA and protein adsorption (see below), we complement schemes 1-3 by slow steps,
A(i) f B(i)
10.1021/la030400a CCC: $27.50 © 2004 American Chemical Society Published on Web 02/28/2004
(5)
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Langmuir, Vol. 20, No. 7, 2004
Letters
resulting in formation of relatively stable agglomerates, Bi (2 e i e n - 1), which do not react with R but can eventually be attached to insoluble fibrils or amorphous precipitates. To motivate inclusion of steps 5 into the model, it is appropriate to notice that steps 2 and 3 are physically similar to protein adsorption. In both cases, the kinetics are driven by the interplay between the hydrophilic and hydrophobic interactions. Energetically, location of hydrophilic fragments is usually preferable on the waterprotein interface. Just after adsorption, the protein conformation does not however correspond to an energy minimum in the adsorbed state, and accordingly there is a tendency to denaturation. Denaturation may be rapid, but the transition to a new native state (i.e., to an ordered structure, formed upon adsorption after denaturation and refolding, with a different secondary and/or tertiary structure compared to the bulk native state) is usually much slower.10 During steps 2 and 3, the energy of the agglomerate formed just after R attachment is, in analogy with the surface case, also expected to be above minimum and accordingly there should be a tendency toward conformational changes. As long as the energy is appreciably above the energy minimum, an aggregate is expected to be more reactive with respect to attachment of additional monomers compared to the situation when the energy is close to minimum. Introducing steps 5, we in fact use the simplest two-state (A and B) approximation corresponding to this concept. [Physically the above scenarios are based on the multiple state dynamics of proteins; they continuously fluctuate between nearly isoenergetic states (e30kBT) and in some of these states expose binding sites (e.g., hydrophobic patches) to contact a surface or another protein.] According to our scheme, the kinetic equations for the agglomerate concentrations, [A(2)] and [A(i)] (with 3 e i e n - 1), are as follows:
d[A(2)]/dt ) k1[R]2 - (k2[R] + κ2)[A(2)] (i)
(i-i)
d[A ]/dt ) ki-1[R][A
P2 )
Pi )
κi
κ2
i-1
∏Fj k p[N] + κ j)2 i
(i)
] - (ki[R] + κi)[A ]
(7)
d[N]/dt ) -ηk1p2[N]2
(8)
where n
iPi ∑ i)2
(9)
is the average number of monomers consumed per one elementary reaction event 2, and Pi are the probabilities that such an event results in the formation of Bi if 2 e i e n - 1 or A(n) if i ) n. Physically, eqs 8 and 9 simply represent the balance of monomer consumption. To use these equations, one should derive explicit expressions for the probabilities Pi. This step is elementary, because eqs 6 and 7 are linear. (10) Zhdanov, V. P.; Kasemo, B. Proteins 2001, 42, 481-494.
(10)
k2p[N] + κ2 (2 e i e n - 1)
(11)
i
n-1
Pn )
Fj ∏ j)2
(12)
where Fj ≡ kjp[N]/(kjp[N] + κj). Equations 8 and 9 are simple, but expressions 10-12 are somewhat cumbersome. To illustrate what the model can predict, it is instructive to discuss briefly the dependence of the rate constants kj and κj on j and analyze explicitly a simple generic example. The attachment rate constant kj is expected to slightly increase with increasing j due to an increase of the aggregate size. This effect seems however to be of minor importance, and one can put kj ) k, where k is the average value of kj (this approximation has already been used in the literature9). In contrast, the reconfiguration rate constant κj is expected to rapidly decrease with increasing j due to an increase of sterical constraints for reconfiguration. In the latter case, it makes sense to allow step 5 only for the first m intermediates. Specifically, we use κj ) κ for 2 e j e 1 + m (κ is the average value of κj for this j interval) and κj ) 0 for j > 1 + m. In addition, taking into account that reconfiguration is slow, we should have kp[N] . κ. Employing for example m ) 10 and n ) 100, we can conclude that steps 5 (with 2 e j e 2 + m) will slightly reduce Pn, but the formation of A(n) will still play the key role in IPA. This means that we may keep only the main term in eq 9, because the other terms (with 2 e j e 2 + m) are negligible due to Pj , Pn and m , n, that is,
(6)
where ki and κi are the rate constants of steps 3 and 5. To solve eqs 6 and 7, we use the conventional approach already outlined above. In particular, step 1 is assumed to be close to equilibrium, that is, [R] ) p[N]. In addition, we employ the steady-state approximation for formation and conversion of A(i). With these approximations, the IPA kinetics are described as
η)
Specifically, we have
η = nPn ) n
(
kp[N] kp[N] + κ
)
m
(13)
Substituting this expression into eq 8, we get
(
)
kp[N] m 2 d[N] ) -nk1p2 [N] dt kp[N] + κ
(14)
To derive eq 14, we have assumed that the formation of relatively stable unreactive agglomerates (step 5) occurs only for 2 e j e 1 + m. In principle, the formation of such agglomerates for small j (=2) may thermodynamically be unfavorable and accordingly the j window may be shifted to higher j-values. In this case, eq 14 will obviously hold as well. Thus, the limits of the applicability of eq 14 are somewhat wider than it might appear. The reaction order is defined as the slope of the dependence of the logarithm of the reaction rate, ln(d[N]/dt), on ln [N]. Taking into account that the first three factors, n, k1, and p2, in the right-hand part of eq 14 do not contribute to the slope, the reaction order can in our case be obtained by constructing the dependence of ln([N]2{kp[N]/(kp[N] + κ)}m) on ln [N]. To apply this expression, we assume that [N] is normalized so that the minimum value is 1 and then increase [N] from 1 to 20 (this is a typical range of the changes of [N] in experiments5). With this normalization, the ratio κ/kp is dimensionless. Using for example κ/kp ) 0.06 and m ) 10, we obtain the average reaction order of 2.17 ( 0.03 (Figure 1). This value exactly corresponds to the rhGCSF ag-
Letters
Figure 1. Function f ) ln([N]2{kp[N]/(kp[N] + κ)}m) on ln [N] vs ln([N]) for κ/kp ) 0.06 and m ) 10. The slope of this curve is 2.17 ( 0.03.
gregation kinetics.5 Scrutinizing Figure 1, one can find that the reaction order slightly decreases with increasing
Langmuir, Vol. 20, No. 7, 2004 2545
concentration (quantitatively, this effect is comparable with the error bar of the experiment5). The perfect agreement between our calculations and the experiment5 has been obtained due to ad hoc choice of the model parameters to fit the experimental data. Changing the parameters, one can get the reaction order in the range from 2.0 to about 2.5. These values specify the type of the kinetics which are predicted and/or can be described by our model. In summary, we have proposed a kinetic model of IPA, including the formation of relatively stable agglomerates, which do not react with monomers but can eventually be attached to insoluble fibrils or amorphous precipitates. For reasonable values of the parameters, our model predicts that the reaction order may be between 2.0 and 2.5. In particular, the model makes it possible to describe the reaction order observed for rhGCSF. The latter of course does not guarantee that the model is really applicable to this case (because we have no experimental data on the intermediates). However, our results extend the conceptual basis for interpretation of the IPA kinetics. LA030400A
