Leucine

ABSTRACT A lattice-based model of a protein and the Monte Carlo simulation method are used to calculate the entropy loss of dimerization of the GCN4 leucine zipper. In the representation used, a protein is a sequence of interaction centers arranged on a cubic lattice, with effective interaction potentials that are both of physical and statistical nature. The Monte Carlo simulation method is then used to sample the partition functions of both the monomer and dimer forms as a function of temperature. A method is described to estimate the entropy loss upon dimerization, a quantity that enters the free energy difference between monomer and dimer, and the corresponding dimerization reaction constant. As expected, but contrary to previous numerical studies, we find that the entropy loss of dimerization is a strong function of energy (or temperature), except in the limit of large energies in which the motion of the two dimer chains becomes largely uncorrelated. At the monomer-dimer transition temperature we find that the entropy loss of dimerization is approximately five times smaller than the value that would result from ideal gas statistics, a result that is qualitatively consistent with a recent experimental determination of the entropy loss of dimerization of a synthetic peptide that also forms a two-stranded alpha-helical coiled coil.

INTRODUCTION

Reduced models of a protein have been shown to provide a possible route for the estimation of the free energy of dimerization of relatively short coiled coils (Vieth et al., 1995, 1996; Mohanty et al., 1999). A key step in the calculation of the free energy of dimerization concerns the entropy loss upon bringing two monomer chains together to form the dimer. A practical method for the numerical estimation of this entropy loss by Monte Carlo simulation is discussed in this paper.

The focus of our work is on the calculation of free energies of dimerization, and in particular a re-analysis of prior computational research on the folding thermodynamics of the GCN4 leucine zipper (Vieth et al., 1996; Mohanty et al., 1999). Leucine zippers belong to the class of structural motifs that are known as coiled coils. Generically, they comprise right-handed alpha-helices wrapped around each other with a small left-handed super-helical twist (Crick, 1953). While leucine zippers can exist in both monomeric or dimeric form, GCN4 forms a dimer in both the crystalline phase (O'Shea et al., 1991), and in solution (d'Avignon et al., 1998).

Numerical calculations of the free energy difference between the monomeric and dimeric forms of GCN4 have already been given by Vieth et al. (1996) and Mohanty et al. (1999). In the former case, the free energy of the monomer was estimated by transfer matrix methods, whereas the entropy change of dimerization was estimated by Monte Carlo methods. To obtain the latter, two monomers were placed in a parallel configuration and in registry with one another. As we discuss further below, the actual configuration space volume available to the dimer was substantially underestimated due to the restriction that the alignment of the two monomers introduced. This numerical analysis of the dimerization equilibrium was later extended by Mohanty et al. (1999) by using the Entropy Sampling Monte Carlo method of Hao and Scheraga (1994), although they used the same values of the entropy loss of dimerization that have been given by Vieth et al. (1996). Several thermodynamic quantities were then computed, including the dimer fraction as a function of temperature and monomer concentration, the helical content of the monomer and dimer, and an analysis of the existence of possible folding intermediates was given.

In our present work, we use the Replica Exchange Monte Carlo method (Swendsen and Wang, 1986; Geyer, 1992; Hukushima and Nemoto, 1996) to obtain the canonical distribution of both monomer and dimer forms separately. According to this method, one considers r independent Monte Carlo simulations at a set of constant temperatures T^sub 1^, T^sub 2^, . . . , T^sub r^. At fixed intervals, two configurations at different temperatures ("replicas") are chosen at random, and their respective temperatures exchanged with probability that preserves detailed balance as given by the canonical probability distribution. By exchanging configurations that have been equilibrated at different temperatures, this method is more efficient in sampling complex configuration spaces than the standard Monte Carlo method. Reweighting methods as first introduced by Ferrenberg and Swendsen (1989a, b) are then used to calculate the free energy as a function of temperature. Energy histograms collected at the set of temperatures Ti are combined ("reweighted") to minimize the statistical error in the resulting density of states. As a byproduct of the calculation, the method yields the thermodynamic free energy F^sub i^ = F(T^sub i^) at the set of temperatures T^sub i^. Because the simulations for the monomer and dimer are carried out separately, we finally describe how to place the free energies of the monomer and dimer on the same relative scale so that the free energy difference between them and the corresponding dimerization constant can be computed.

The calculation of the entropy loss of dimerization that we present in this paper is free from any assumptions made in previous treatments. Given the reduced model of the protein and the related set of interaction potentials, there are no further restrictions in the region of configuration space sampled in the simulations. Furthermore, we show that the entropy loss is a strong function of energy (or temperature) so that approximate treatments that factorize contributions to the partition function arising from internal degrees of freedom of the chains and contributions from relative chain translational and rotational degrees of freedom are not correct in general. We find, for example, that the volume available to the second chain of the dimer relative to the first chain had been previously underestimated by about two orders of magnitude (Vieth at al., 1996), a factor that corresponds to overestimating the entropy loss of dimerization by ~4.5 k^sub B^, where k^sub B^ is Boltzmann's constant. Finally, and although we do not take the solvent explicitly into account in our calculations, we compare the values of the entropy loss of dimerization that we obtain with experimental values obtained for the case of a short synthetic peptide that forms a two-stranded, a-helical coiled coil (Yu et al., 2001). In this latter work, it was found that the standard entropy change of dimerization is ~- 5 k^sub B^, or a tenth of the value predicted by ideal gas statistics (== -50 k ^sub B^, at standard temperature and concentration). Our calculations show a similar reduction, although our estimate of the entropy loss at the transition temperature is only 20-25% of the ideal gas value, or a factor of two larger than the corresponding experimental ratio. We finally note that the method presented in this paper to compute the entropy loss of dimerization is not restricted to our particular protein model, but that it can be straightforwardly extended to any number of computational schemes that use a more refined description of the peptide chain.

THE PROTEIN MODEL

Our analysis is based on a reduced model in which a protein is represented by a sequence of virtual bonds connecting effective particles (Kolinski and Skolnick, 1994, 1996). Each particle represents one side chain, but is assumed to be located at the center of mass of the side chain in question plus the corresponding backbone alpha-carbon. The effective particles are embedded in a regular cubic lattice of fixed spacing that allows for a fairly accurate representation of the backbone of known protein structures. This geometric part of the model has been checked against all structures contained in the protein data bank (Bernstein et al., 1977). The observed root-mean-squared deviation between the lattice representation of any protein and its resolved structure is typically below 0.8 Angstrom (Kolinski et al., 1999). The actual resolution of the model is of course lower, typically of the order of 2 Angstrom for small proteins.

Effective interactions (force fields) are introduced among the particles that include generic (sequence-independent) and sequence-specific contributions. The potentials associated with the generic type of interactions are defined to introduce a bias toward reasonable secondary structures. One such potential is introduced to account for the fact that proteins exhibit a characteristic bimodal distribution of neighbor residue distances, especially between the ith and (i + 4)th residues. Configurations corresponding to the larger distances in the distribution are associated with proteins that exhibit either beta-type or expanded coils, whereas shorter distances correspond to helices and turns (Kolinski et al., 1999). A second generic interaction further introduces a bias toward certain packing structures such as helices and beta-type states. The first potential produces the required stiffness of the polypeptide chain, whereas the second provides for cooperative packing.

Sequence-specific interactions are of three types and include short-range interactions, long-range, pairwise interactions, and many body interactions. The short-range pairwise potentials are of statistical origin and are fitted to non-homologous reference structures that do not include any coiled coils. This is accomplished by considering the known distances between pairs of amino acids that are separated by one through four bonds along the chain. Chirality is also introduced by considering an interaction between the ith and (i + 3)rd and (i + 6)th bonds to produce the correct pitch. Long-range interactions are also of statistical origin and are assumed to depend not only on relative distances between the effective particles, but also on the relative orientation between the corresponding side chains (for example, residues of opposite charges are attractive when the corresponding bonds are parallel to each other, whereas the interaction is weak or repulsive when they are anti-parallel).

Finally, although multi-body interactions are implicitly included in the pairwise potentials determined from inter-residue distances and bond angles, two additional terms are added to model hydrophobic interactions and the known probability of a residue to have a given number of parallel and anti-parallel contacts. The hydrophobic potential is estimated from the surface exposure of a given side chain, i.e., of all possible contacts of a side chain, those that are not effectively occupied by contacts with neighboring chains. The second multi-body potential introduces a bias toward known propensities of various amino acids to pack their side chains in a parallel or anti-parallel orientation. Recent applications of the methodology include the improvement of threading based structure prediction (Kolinski et al., 1999; Kihara et al., 2001), and direct ab initio folding studies (Kolinski et al., 2000). Further details can be found in Appendix A.

This research was supported by National Science Foundation Grant 9986019. J.V. is also supported by National Institutes of General Medical Sciences Grant GM64150-01.

REFERENCES

Bernstein, F. C., T. F. Koetzle, G. J. B. Williams, E. F. Meyer, M. D. Brice, J. R. Rodgers, 0. Kennard, T. Shimanouchi, and M. Tasumi. 1977. The protein data bank: a computer-based archival file for macromolecular structures. J. Mol. Biol. 112:535.

Crick, F. H. C. 1953. The packing of a-helices: simple coiled-coils. Acta Crystallogr. 6:689-697.

d'Avignon, D. A., G. L. Bretthorst, M. E. Holtzer, and A. Holtzer. 1998. Site-specific thermodynamics and kinetics of a coiled-coil transition by spin inversion transfer NMR. Biophys. J. 74:3190-3197.

Ferrenberg, A. M., and R. H. Swendsen. 1989a. Optimized Monte Carlo data analysis. Computers in Physics. September/October:101-104. Ferrenberg, A. M., and R. H. Swendsen. 1989b. Optimized Monte Carlo

data analysis. Phys. Rev. Lett. 63:1195-1198.

Geyer, C. J. 1992. Practical Markov chain Monte Carlo. Stat. Sci. 7:437-483.

Hao, M.-H., and H. A. Scheraga. 1994. Monte Carlo simulation of a first order transition for protein folding. J. Chem. Phys. 98:4940-4948.

Harbury, P. B., T. Zhang, P. S. Kim, and T. Alter. 1993. A switch between two-, three-, and four-stranded coiled coils in GCN4 leucine zipper mutants, Science. 262:1401-1407.

Hill, T. L. 1986. An Introduction to Statistical Thermodynamics. Dover, New York.

Holtzer, M. E., G. L. Bretthorst, D. A. d'Avignon, R. H. Angeletti, L. Mints, and A. Holtzer. 2001. Temperature dependence of the folding and unfolding kinetics of the GCN4 leucine zipper via C-13(a)-NMR. Biophys. J. 80:939-951.

Hukushima, K., and K. Nemoto. 1996. Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn. 65:1604-1608. Kenar, K. T., B. Garcia Moreno, and E. Freire. 1995. A calorimetric

characterization of the salt dependence of the stability of the GCN4 leucine zipper. Protein Sci. 4:1934-1938.

Kihara, D., A. Kolinski, and J. Skolnick. 2001. TOUCHSTONE: An ab initio protein structure prediction method that uses threading-based tertiary restraints. Proc. Natl. Acad. Sci. 98:10125-10130.

Kolinski, A., P. Rotkiewicz, B. Ilkowski, and J. Skolnick. 1999. A method for the improvement of threading-based protein models. Proteins: Struct., Funct., Genet. 37:592-610.

Kolinski, A., P. Rotkiewicz, B. Ilkowski, and J. Skolnick. 2000. Protein folding: flexible lattice models. Prog. Theor. Phys. SuppL 138:292-300. Kolinski, A., and J. Skolnick. 1994. Monte Carlo simulations of protein

folding. 1. Lattice model and interaction scheme. Proteins: Struct., Funct., Genet. 18:338-352.

Kolinski, A., and J. Skolnick. 1996. Lattice models of protein folding, dynamics and thermodynamics. R. G. Landes, Austin.

Kolinski, A., and J. Skolnick. 1998. Assembly of protein structure from sparse experimental data: an efficient Monte Carlo model. Proteins: Struct., Funct., Genet. 32:475-494.

Ma, S.-K. 1985. Statistical Mechanics. World Scientific, Singapore. Mayer, J. E., and M. G. Mayer. 1963. Statistical Mechanics. John Wiley and Sons, New York.

Mohanty, D., A. Kolinski, and J. Skolnick. 1999. De novo simulations of the folding thermodynamics of the GCN4 leucine zipper. Biophys. 177:54-69. O'Shea, E. K., J. D. Klemm, P. S. Kim, and T. Alber. 1991. X-ray structure

of GCN4 leucine zipper, a two stranded, parallel coiled coil. Science. 254:539-544.

Skolnick, J., M. Vieth, A. Kolinski, and C. L. Brooks 111. 1995. De novo simulations of the folding of GCN4 and its mutants. In Modeling of Biomolecular Structures and Mechanisms. A. Pullman, J. Jortner, and B. Pullman, editors. Kluwer Academic, The Netherlands.

Swendsen, R. H., and J.-S. Wang. 1986. Replica Monte Carlo simulation of spin glasses. Phys. Rev. Lett. 57:2607-2610.

Vieth, M., A. Kolinski, C. L. Brooks III, and J. Skolnick. 1995. Prediction of quaternary structure of coiled coils. Application to mutants of the GCN4 leucine zipper. J. Mol. Biol. 251:448-467.

Vieth, M., A. Kolinski, and J. Skolnick. 1996. Method for predicting the state of association of discretized protein models. Application to leucine zippers. Biochemistry. 35:955-967.

Yu, B. Y., P. L. Privalov, and R. S. Hodges. 2001. Contribution of translational and rotational motions to molecular association in aqueous solution. Biophys. J. 81:1632-1642.

Jorge Vinals,* Andrzej Kolinski,*^ and Jeffrey Skolnick*

*Laboratory of Computational Genomics, Donald Danforth Plant Science Center, St. Louis, Missouri 63132 USA, and ^Department of Chemistry, University of Warsaw, 02-093 Warsaw, Poland

Submitted April 1, 2002, and accepted for publication July 3, 2002.

Address reprint requests to Jorge Vinals, Florida State University, Tallahassee, FL 32306-4120. Tel.: 850-644-1010; E-mail: jvinals@ danforthcenter.org.

Copyright Biophysical Society Nov 2002
Provided by ProQuest Information and Learning Company. All rights Reserved