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Protein Science (2003), 12:2057-2062.
Copyright © 2003 The Protein Society

Contact order revisited: Influence of protein size on the folding rate

Dmitry N. Ivankov1, Sergiy O. Garbuzynskiy1, Eric Alm2, Kevin W. Plaxco3, David Baker2 and Alexei V. Finkelstein1

1 Institute of Protein Research, Russian Academy of Sciences, 142290 Pushchino, Moscow Region, Russia
2 Department of Biochemistry and Howard Hughes Medical Institute, University of Washington, Seattle, Washington 98195, USA
3 Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, USA

Reprint requests to: Alexei V. Finkelstein, Institute of Protein Research, Russian Academy of Sciences, 142290 Pushchino, Moscow Region, Russia; e-mail: afinkel{at}vega.protres.ru; fax: 7095-924-0493.

(RECEIVED January 16, 2003; FINAL REVISION May 23, 2003; ACCEPTED May 28, 2003)

Article and publication are at http://www.proteinscience.org/cgi/doi/10.1110/ps.0302503.


   Abstract
TOP
 Abstract
Introduction
 Results and Discussion
 References
 
Guided by the recent success of empirical model predicting the folding rates of small two-state folding proteins from the relative contact order (CO) of their native structures, by a theoretical model of protein folding that predicts that logarithm of the folding rate decreases with the protein chain length L as L2/3, and by the finding that the folding rates of multistate folding proteins strongly correlate with their sizes and have very bad correlation with CO, we reexamined the dependence of folding rate on CO and L in attempt to find a structural parameter that determines folding rates for the totality of proteins. We show that the Abs_CO = CO x L, is able to predict rather accurately folding rates for both two-state and multistate folding proteins, as well as short peptides, and that this Abs_CO scales with the protein chain length as L0.70 ± 0.07 for the totality of studied single-domain proteins and peptides.

Keywords: Protein folding kinetics; two-state kinetics; multistate kinetics; contact order; protein size; protein topology; rate of folding


   Introduction
TOP
 Abstract
 Introduction
Results and Discussion
 References
 
Many proteins fold and unfold by a simple two-state transition lacking observable intermediates at any solvent conditions (Jackson 1998). Many other proteins exhibit a more complicated multistate transition; namely, they have observable folding intermediates under physiological conditions. However, the boundary between these two groups of proteins is not as well defined.

It is known that some proteins can be switched from two-state to multistate folding, and vice versa, by point mutations or even by changing conditions such as the salt concentration or temperature (Jackson 1998). In addition, multistate folding is observed only far from the point of thermodynamic equilibrium between the native and denatured states, whereas, close to this point, all proteins fold without any observable intermediates (Privalov 1979; Jackson 1998; Finkelstein and Ptitsyn 2002).

Small two-state folding proteins have attracted particular attention of experimentalists and theorists. It was demonstrated that the logarithms of in-water folding rates of these proteins correlate with their gross topological parameter called relative contact order (CO; Plaxco et al. 1998b). The latter is defined as

(1)
where N is the number of contacts (within 6 Å) between nonhydrogen atoms in the protein, L is the length of the protein in amino acid residues, and {Delta}Lij is the number of residues separating the interacting pair of nonhydrogen atoms (adjacent residues are assumed to be separated by one residue, etc.).

CO is a renormalization of the perhaps more intuitive measure, absolute contact order (Abs_CO),


(2)
which, however, was found to be less correlated than CO with folding rates of the two-state folders (Plaxco et al. 1998b; Grantcharova et al. 2001).

The CO was invented to compare differences in topology (rather than in size) between proteins of different length. This parameter is small for proteins stabilized mainly by local interactions and is large when residues in a protein interact frequently with partners far away in the protein sequence. The latter should lead to slower folding (Plaxco et al. 1998b; Fersht 2000). Indeed, negative correlation between the CO and the logarithm of folding rates was found to be very strong, ~ -0.8 (Plaxco et al. 1998b; Fersht 2000) for two-state folding proteins (which also holds for all two-state folding proteins studied to date; Fig. 1Go, circles).



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  		Figure 1. Natural logarithm of observed folding rate in water, ln(kf), versus relative contact order (CO) for various proteins and peptides: proteins having two-state folding kinetics at all the denaturant concentrations (circles ), proteins having multistate folding kinetics in water (and at low denaturant concentrations; triangles ), and short peptides (crosses ). The figure includes peptides and proteins listed in Table 1GoGo; CO is computed after Equation 1Go from the PDB coordinates (Bernstein et al. 1977). If several folding rates are observed for some protein (see Table 1GoGo), ln(kf) is the mean value of their natural logarithms. The dashed line represents the best linear fit for two-state folders only (the negative correlation coefficient is as significant as -0.75; the fitted dependence is y = 16.94 - 0.76x); the dotted line represents the best linear fit for multistate folders only (the correlation coefficient is +0.26; namely, it has the opposite sign compared with that for the two-state folders; y = -1.55 + 0.26x); the solid line represents the best linear fit for the totality of all peptides and proteins presented (the correlation coefficient is insignificant, +0.10 only; y = 2.37 + 0.10x).

 
However, examining a whole set of proteins studied to date (Table 1GoGo), we see that CO, although it still gives good results for two-state folding proteins, fails to predict the folding rates of short peptides and large multistate folding proteins (Fig. 1Go). It seems the reason is that CO takes into account topology only and pays no explicit attention to the protein size.


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  		Table 1. List of proteins and polypeptides a
 

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  		Table 1. List of proteins and polypeptides a
 
A number of basic correlations between protein size and folding rate have been suggested (Thirumalai 1995; Gutin et al. 1996; Finkelstein and Badretdinov 1997a,b). All of them stress that, as might be expected, folding rate decreases monotonically with protein size, but all indicate different scaling laws for this decrease. It should be noted that some recent simulations of folding of off-lattice protein models with simplified potentials (Koga and Takada 2001) indicate that the logarithms of protein folding rate decrease with the chain length as L0.61 ± 0.18, which is in accordance with both Finkelstein and Badretdinov’s (1997a, b) and Thirumalai’s (1995) theories.

It has been shown, however, that the protein size by itself determines folding rates of only multistate folding proteins and fails to predict those for two-state folders (Galzitskaya et al. 2003): For multistate folders, the negative correlation between LP (L being the number of residues in the chain and P a free parameter) and the logarithm of folding rates is as high as -0.80 in the broad range of power P from zero to one, whereas for two-state folders any correlation between folding rate and size is virtually absent.

This study is aimed to develop a general parameter for predicting the protein folding rates of two-state folding proteins, multistate folding proteins, and small peptides. This general estimate, if found, would be useful for two reasons: (1) Attribution of proteins to two-state or multistate folders is somewhat arbitrary, at least for proteins which can be switched from the two-state to the multistate behavior by point mutations or changing solvent conditions, and (2) it is useful to estimate the folding rate of a protein when one does not know a priori if it is two-state or multistate folding protein.


   Results and Discussion
TOP
 Abstract
 Introduction
 Results and Discussion
References
 
The simplest way to obtain such a parameter is to take into account both the protein topology and its size, that is, to combine a length-based theory with empirical topology effect (Plaxco et al. 1998b). Here we describe such a combination.

Specifically, a theory of Finkelstein and Badretdinov’s (1997a, b) predicted that in a vicinity thermodynamic midtransition, folding rates of all single-domain proteins should decrease with their lengths, L as exp[-(0.5 ÷ 1.5) L2/3], and where the size-independent coefficient C = 0.5 ÷ 1.5 depends on the topology of the protein: C is close to 0.5 when a protein is stabilized mainly by local interactions, so that semifolded protein does not contain closed loops protruding from the folding nucleus, and C is close to 1.5 when a protein has many long-range contacts, so that many closed loops protrude from the nucleus. Later it was shown (Galzitskaya et al. 2001) that the range kf = exp(0.5L2/3) x 10ns ÷ exp(1.5L2/3) x 10ns is valid for all the studied peptides and single-domain proteins of a great variety of lengths, topologies, and folding behaviors.

Although Finkelstein and Badretdinov did not give an algorithm to compute their coefficient, C, from protein structure, it is clear that a physical sense of C is similar to those of the CO of Plaxco et al. Both are small for proteins with local contacts (i.e., {alpha}-helical proteins), and both are large for proteins with predominantly long-range contacts, which cannot avoid having many loops in a semifolded state. Therefore, the values of C and CO should correlate.

The simplest combination of CO and L, which seems to follow from theories of Plaxco et al. and Finkelstein and Badretdinov, may look like CO x L2/3. However, because we observe that CO is not a chain length–independent parameter (as the value C of Finkelstein and Badretdinov should be) but anticorrelates with the chain length, L (Fig. 2Go), for totality of proteins and peptides, we summarize CO and L in a general parameter, the "size-modified contact order" (SMCO), as

(3)



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  		Figure 2. Logarithm of relative contact order versus logarithm of chain length. See legend to Figure 1Go for specification of the symbols and other details. The dashed line represents the best linear fit for two-state folders only (the correlation coefficient is 0.02; y = 2.68 + 0.01x); the dotted line represents the best linear fit for multistate folders only (the correlation coefficient is -0.54; y = 4.41 - 0.40x); the solid line represents the best linear fit for the totality of all peptides and proteins (the correlation coefficient is -0.50; y = 3.95 - 0.30x). The linear regression coefficients 0.01, -0.40, and -0.30 are determined with errors ±0.16, ±0.13, and ±0.07, respectively.

 
One can see that P = 0 corresponds to SMCO = CO, whereas P = 1 corresponds to SMCO = Abs_CO.

The correlation of SMCO and ln(kf), depending on the power P value, is presented in the inset in Figure 3Go. One can see that although any P > 0.7 results in approximately the same correlation for the totality of proteins and peptides, the best correlation is achieved at P {approx} 1, that is, when SMCO {approx} Abs_CO. The correlation of Abs_CO and ln(kf) is presented in Figure 3Go.



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  		Figure 3. Logarithm of observed folding rate in water ln(kf) versus Abs_CO = CO x L. See legend to Figure 1Go for specification of the symbols and other details. The dashed line represents the best linear fit for two-state folders only (the fitted dependence is y = 9.44 - 0.36x; the correlation coefficient is -0.51); the dotted line represents the best linear fit for multistate folders only (the fitted dependence is y = 8.56 - 0.44x; the correlation coefficient is -0.78); the solid line represents the best linear fit for the totality of all peptides and proteins presented (the fitted dependence is y = 11.15 - 0.54x; the correlation coefficient is -0.74). (Inset) Correlation coefficients between the logarithm of experimental folding rate in water and the value of CO x LP depending on the value of power P. Error bars, standard errors in correlation coefficients. The curve "2-STATE" concerns the two-state folding proteins only, and the curve "ALL" concerns the totality of all studied peptides and proteins; the curve for the multistate folding proteins is not shown because it is close to the curve "ALL" up to the standard error in correlation coefficients.

 
It should be mentioned, however, that for the two-state folders, the best ln(kf)–to–SMCO correlation is achieved when P = 0 ÷ 0.5 rather than 1 (Fig. 3Go, inset).

However, this difference between the scaling laws observed for two-state folders and the other proteins correlates, to a certain extent, with the finding (Fig. 2Go) that CO is independent on the chain length for the two-state folders, whereas it decreases with the chain length, L, in proportion to L-0.4 for multistate folders, and for the totality of proteins and peptides, CO decreases with their chain length, L, in proportion to L-0.30 ± 0.07 on the average.

It is noteworthy that CO scales namely as L-0.30 ± 0.07 for the totality of proteins and peptides (Fig. 2Go, dashed line). This means that the value Abs_CO = CO x L (which has the highest correlation with ln[kf] for the totality of proteins and peptides; Fig. 3Go, inset) scales with the chain length as L0.70 ± 0.07. This is in a very good concordance with a general scaling law L2/3 predicted by Finkelstein and Badretdinov 1997a,b; although the Thirumalai’s [1995] scaling law L0.5 has only a little worse correlation with experiment, and thus, cannot be ruled out; Fig. 3Go, inset), and agrees with an empirical scaling L0.61 ± 0.18 resulting from simplified off-lattice folding simulations of Koga and Takada (2001).


   Acknowledgments
 
We are grateful to Blake Gillespie and Oxana Galzitskaya for discussions and some computations, and to David Thirumalai for discussions and his results on correlation of ln(kf) with CO x L1/2. This work was supported in part by the Russian Foundation for Basic Research, by an International Research Scholar’s Award to A.V.F. from the Howard Hughes Medical Institute, and by the Institute of Theoretical Physics (Santa Barbara University, ITP work no. NSF-ITP-01-173).

The publication costs of this article were defrayed in part by payment of page charges. This article must therefore be hereby marked "advertisement" in accordance with 18 USC section 1734 solely to indicate this fact.


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T. R. Sosnick
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Protein Sci., August 1, 2008; 17(8): 1308 - 1318.
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Z. Ouyang and J. Liang
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Protein Sci., July 1, 2008; 17(7): 1256 - 1263.
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C.-P. Chng and A. Kitao
Thermal Unfolding Simulations of Bacterial Flagellin: Insight into its Refolding Before Assembly
Biophys. J., May 15, 2008; 94(10): 3858 - 3871.
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S. Dalal, D. Canet, S. E. Kaiser, C. M. Dobson, and L. Regan
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Protein Eng. Des. Sel., March 1, 2008; 21(3): 197 - 206.
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B. Barua, J. C. Lin, V. D. Williams, P. Kummler, J. W. Neidigh, and N. H. Andersen
The Trp-cage: optimizing the stability of a globular miniprotein
Protein Eng. Des. Sel., March 1, 2008; 21(3): 171 - 185.
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A. Y. Istomin, D. J. Jacobs, and D. R. Livesay
On the role of structural class of a protein with two-state folding kinetics in determining correlations between its size, topology, and folding rate
Protein Sci., November 1, 2007; 16(11): 2564 - 2569.
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X. Qi and J. J. Portman
Excluded volume, local structural cooperativity, and the polymer physics of protein folding rates
PNAS, June 26, 2007; 104(26): 10841 - 10846.
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E. Capriotti and R. Casadio
K-Fold: a tool for the prediction of the protein folding kinetic order and rate
Bioinformatics, February 1, 2007; 23(3): 385 - 386.
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T. Kimura, J. C. Lee, H. B. Gray, and J. R. Winkler
Site-specific collapse dynamics guide the formation of the cytochrome c' four-helix bundle
PNAS, January 2, 2007; 104(1): 117 - 122.
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S. R. K. Ainavarapu, J. Brujic, H. H. Huang, A. P. Wiita, H. Lu, L. Li, K. A. Walther, M. Carrion-Vazquez, H. Li, and J. M. Fernandez
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M. M. Gromiha, A. M. Thangakani, and S. Selvaraj
FOLD-RATE: prediction of protein folding rates from amino acid sequence.
Nucleic Acids Res., July 1, 2006; 34(Web Server issue): W70 - W74.
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P. Das, C. J. Wilson, G. Fossati, P. Wittung-Stafshede, K. S. Matthews, and C. Clementi
Characterization of the folding landscape of monomeric lactose repressor: Quantitative comparison of theory and experiment
PNAS, October 11, 2005; 102(41): 14569 - 14574.
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U. Bastolla and L. Demetrius
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Protein Eng. Des. Sel., September 1, 2005; 18(9): 405 - 415.
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D. Kihara
The effect of long-range interactions on the secondary structure formation of proteins
Protein Sci., August 1, 2005; 14(8): 1955 - 1963.
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S. Wallin and H. S. Chan
A critical assessment of the topomer search model of protein folding using a continuum explicit-chain model with extensive conformational sampling
Protein Sci., June 1, 2005; 14(6): 1643 - 1660.
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J. Faraone-Mennella, H. B. Gray, and J. R. Winkler
Early events in the folding of four-helix-bundle heme proteins
PNAS, May 3, 2005; 102(18): 6315 - 6319.
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K. L. Maxwell, D. Wildes, A. Zarrine-Afsar, M. A. De Los Rios, A. G. Brown, C. T. Friel, L. Hedberg, J.-C. Horng, D. Bona, E. J. Miller, et al.
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W. Y. Yang and M. Gruebele
Folding {lambda}-Repressor at Its Speed Limit
Biophys. J., July 1, 2004; 87(1): 596 - 608.
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D. N. Ivankov and A. V. Finkelstein
Prediction of protein folding rates from the amino acid sequence-predicted secondary structure
PNAS, June 15, 2004; 101(24): 8942 - 8944.
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L. Hedberg and M. Oliveberg
Scattered Hammond plots reveal second level of site-specific information in protein folding: {phi}' ({beta}{ddagger})
PNAS, May 18, 2004; 101(20): 7606 - 7611.
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G. Bulaj, R. E. Koehn, and D. P. Goldenberg
Alteration of the disulfide-coupled folding pathway of BPTI by circular permutation
Protein Sci., May 1, 2004; 13(5): 1182 - 1196.
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N. Bascos, J. Guidry, and P. Wittung-Stafshede
Monomer topology defines folding speed of heptamer
Protein Sci., May 1, 2004; 13(5): 1317 - 1321.
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