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Motivation: Computational studies of the energetics of protein association are important for revealing the underlying fundamental principles and for designing better tools to model protein complexes. The interaction cutoff contribution to the ruggedness of protein–protein energy landscape is studied in terms of relative energy fluctuations for 1/rn potentials based on a simplistic model of a protein complex. This artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase.
Results: The critical values of the cutoff were calculated for each of 11 popular power-type potentials with n=0÷9, 12 and for two thresholds of 5% and 10%. The artificial ruggedness decreases to tolerable thresholds for cutoffs larger than the critical ones. The results showed that for both thresholds the critical cutoff is a non-monotonic function of the potential power n. The functions reach the maximum at n=3÷4 and then decrease with the increase of the potential power. The difference between two cutoffs for 5% and 10% artificial ruggedness becomes negligible for potentials decreasing faster than 1/r12. The analytical results obtained for the simple model of protein complexes agree with the analysis of artificial ruggedness in a dataset of 62 protein–protein complexes, with different parameterizations of soft Lennard–Jones potential and two types of protein representations: all-atom and coarse-grained. The results suggest that cutoffs larger than the critical ones can be recommended for protein–protein potentials.
Protein binding can be explained in terms of the funnel-based concept initially developed to describe protein folding (Bryngelson et al., 1995; Camacho and Vajda, 2001; Camacho et al., 1999; Dill, 1999; Elcock et al., 2001; Hunjan, 2008; Tovchigrechko and Vakser, 2001; Tsai et al., 1999; Vakser, 1996; Wang and Verkhivker, 2003; Wang et al., 2000; Wolynes, 2005). The concept suggests that unbound proteins are guided by the slope of the rugged energy landscape funnel into the bound state. The nature of the ruggedness and related effects is a subject of active research (Ferreiro et al., 2007; O'Toole and Vakser, 2008; Ruvinsky and Vakser, 2008a; Sutto et al., 2007). Highly frustrated interactions are observed on the protein surface near the binding site (Ferreiro et al., 2007). Mechanical unfolding experiments to measure the scale of the landscape ruggedness of proteins and RNAs have been suggested (Hyeon and Thirumalai, 2003) and performed (Nevo et al., 2005).
The amplitude of the protein–protein energy landscape ruggedness has a component associated with the range of the energy potentials (Ruvinsky and Vakser, 2008a). The range of non-bonded inter-atomic interactions and related truncation methods are known to play an important role in protein folding (Abkevich et al., 1995; Buchete et al., 2004; Doyle et al., 1997; Faisca et al., 2005; Gō and Taketomi, 1978; Govindarajan and Goldstein, 1995; Gromiha and Selvaraj, 1999), protein–protein docking (Ruvinsky and Vakser, 2008b; Vakser, 1996) and all-atom molecular dynamics and Monte-Carlo simulations of macromolecules and liquids ((Brooks et al., 1985; Gilson, 1995; Haluk and McCammon, 2000; Harvey, 1989; Izvekov et al., 2008; Loncharich and Brooks, 1989; Norberg and Nilsson, 2000; Sagui and Darden, 1999). The cutoff is one of only two parameters used in coarse-grained normal mode analysis and elastic networks of proteins and their assemblies (Bahar et al., 1997; Hinsen, 2008; Tirion, 1996). The choice of the cutoff affects the functional form and performance of knowledge-based potentials in small molecule docking (Ruvinsky and Kozintsev, 2005a). The importance of long-range interactions for protein stability (Grimsley et al., 1999), protein folding (Klein-Seetharaman et al., 2002) and RNA binding (Lafuente et al., 2002) has been revealed experimentally.
The interactions are usually truncated at specific cutoff distances to reduce a number of interacting pairs of atoms or atomic groups in order to make feasible large-scale macromolecular calculations. Despite the considerable progress achieved in methodology and computer power, the cutoff-related artifacts are still a bottleneck in macromolecular modeling. In comparison with other modeling approaches, the protein docking community has been less focused on the problem. Our attention to the cutoff problem is motivated by observations that the choice of larger cutoffs results in the ruggedness depression (Ruvinsky and Vakser, 2008a) and thus in smooth protein–protein energy landscapes (Vakser, 1996; Vakser et al., 1999), which according with the principal of minimal frustration ((Bryngelson et al., 1995; Bryngelson and Wolynes, 1989; Wolynes, 2005) better approximate the actual binding landscape. Similar effects of the energy landscape smoothing due to the cutoff extension have been found in studies of liquids and atomic clusters (Braier et al., 1990; Doye and Wales, 1996; Miller et al., 1999; Stillinger and Stillinger, 1990; Wawak et al., 1992; Whitfield and Straub, 2002), helix dimers (Pappu et al., 1999; Vakser, 1996; Vakser and Jiang, 2000) and protein complexes (Tovchigrechko et al., 2002; Vakser, 1996; Vakser et al., 1999).
It has been recently shown (Ruvinsky and Vakser, 2008a) that short cutoffs perturb protein–protein energy landscape and thus lead to false minima, changed positions and altered shape of true conformation-based minima. Such changes of the landscape impede the search for the global minimum in protein docking (Vakser, 1996) and introduce errors in calculations of binding free energy (Alsallaq and Zhou, 2007; Minh et al., 2005; Ruvinsky, 2007; Ruvinsky and Kozintsev, 2005b). The false minima cause the artificial ruggedness of the energy landscape. The fine structure of the funnel or conformational substates (Frauenfelder et al., 2001) can be blurred due to the artificial ruggedness. The amplitude of the artificial ruggedness decreases with the increase of the cutoff (Ruvinsky and Vakser, 2008a). Thus, it is important to know the cutoffs for different potentials that form a minimally frustrated funnel-like landscape, while allowing extensive calculations. The low boundary of the optimal range, called further the critical cutoff, corresponds to a tolerable frustration of the energy landscape.
In this article, we focus on determination of critical cutoffs for 11 power-type potentials at two thresholds of 5% and 10% of the artificial ruggedness. For cutoffs longer than the critical ones, the artificial ruggedness drops below these thresholds. We analyze dependence of the critical cutoffs on the potential power n and discuss practical implications of the results for protein docking and protein folding.
A simplistic model of a sandwich-like protein complex (Ruvinsky and Vakser, 2008a; see also Lukatsky et al., 2006) can be used to describe the interaction energy between an atom A in Proteins 2 and Protein 1 atoms located at the distance r (Fig. 1) as
where δr is the thickness of the spherical layer in Protein 1, ρ1 is Protein 1 atom density, x is a distance from the atom A to the surface of Protein 1 and θo=arccos(x/r). For simplicity, we assume that potential ε(r) does not depend on atom type. The total energy of the atom A is
where R is the interaction cutoff. In Equation (2) we assumed that the interface, restricted by polar and azimuth angles, is flat (Fig. 1). Thus, the energy of interaction between two proteins can be written as
where S(x) is the area formed by Protein 2 atoms located at distance x from the interface, and a is the minimal distance between the two proteins. It is reasonable to assume that both proteins have equal densities ρ = ρ1 = ρ2 and S(x) is a weak function of the distance x. Thus, we can rewrite Equation (3) as
where S is the average interface area. If we slightly enlarge the distance between two proteins, the energy of the new conformation will be
where δa is the relative shift of proteins, Sc is the new interface. The artificial ruggedness of the landscape manifests itself as the deviation of the relative energy change δE(R)/E(R)=(Ec(R)−E(R))/E(R) from its value at the asymptotically large cutoff RL
Intermolecular energy can be written as a sum of pairwise inter-atomic interactions described by model potentials and direct electrostatic potentials (Kaplan, 2006). As a rule, these potentials (e.g. Lennard–Jones potentials and their modifications, multipole–multipole potentials) have a form of an expansion over r−n. The simple form of the power potentials allows analytical evaluation of Equation (6) for all integer n. Further, assuming ε (r) ~ r−n, Sc ≈ S, a = 2.8 Å, δa=0.5 Å and RL = 30 Å, we compute two critical cutoffs for each of 11 potentials with n=0 ÷ 9, 12 under the condition that the artificial ruggedness drops below the threshold of 10% or 5% for cutoffs longer than the critical ones.
The results of the calculations of the asymptotic behavior of the relative energy change at large cutoffs, of the artificial ruggedness, and the critical cutoffs for different power-type potentials are summarized in Table 1. The relative energy change δE(R)/E(R) asymptotically approaches zero for n ≤ 4, and approaches a constant −(n − 4)δa/a for n > 4. The artificial ruggedness is a decreasing function of the cutoff for each of 11 potentials. The results show that both critical cutoffs depend non-monotonically on the potential power n (Fig. 2). They increase up to the maximum at n = 3÷4 and then decrease with the power increase. The non-monotone character is readily explained by the interplay of the density-related term r2 − rx and the energy ε(r)~ r−n in the double integral of Equations (4) and (5). The integral is dominated by the density-related term for slow-decreasing potentials (n < 3) and by the energy term for fast decreasing potentials (n > 4). The estimates of the critical cutoff for n = 6 and 12 are in a good agreement with our previously published results based on use of a soft Lennard–Jones potential on a set of 66 protein complexes (Ruvinsky and Vakser, 2008a). The difference between two cutoffs, which correspond to artificial ruggedness of 10% and 5%, decreases for n > 6, and becomes negligible for potentials decreasing faster than 1/r12.
The cutoff effect was analyzed on a dataset of 62 protein–protein complexes selected from a docking benchmark set (Gao et al., 2007). Our protein docking program GRAMM (http://vakser.bioinformatics.ku.edu/main/resources.php) was used to generate 5000 matches for each pair of proteins. The closest near-native match for each complex was selected for analysis. The average root mean square deviation (RMSD) of the near-native match relative to the native one was 1.7 Å. The average RMSD of the interface was 1.0 Å. Both values of the average RMSD play the role of the shift δ a in the analysis above. In addition, one-bead coarse-grained models were built for the native and near-native conformations of each complex. Within the coarse-graining procedure, we replaced all atoms of each amino acid with a bead at the amino acid center of mass. A soft Lennard–Jones potential
implemented in the GRAMM-X docking server (Tovchigrechko and Vakser, 2005) was used to estimate the average ruggedness ∑i=162 |νarti(R)|/62 (i is the number of the protein–protein complex in the set) of the energy landscapes formed by all atom-to-atom or bead-to-bead interactions. The results for the entire dataset, obtained for different parameterizations of the potential, are shown in Figures 3 and and4.4. The artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase. The artificial ruggedness drops below the threshold of 10% or 5% for cutoffs that are close to the critical cutoffs of potentials 1/r6 and 1/r12 (Table 1). The amplitude of the ruggedness is larger for the all-atom representation than for the bead representation for cutoffs <6 Å. In the case of all-atom representation, the ruggedness decreases as α or A increase. This can be explained by the decrease of the ratio of the potential wall ε(0) and the depth of the energy well εmin for greater values of α or A.
Since protein folding and protein binding are similar processes in terms of the landscape characteristics, including the funnel concept, we may expect that our results have implications to protein folding. Systematic attempts have been undertaken to design pair potentials for protein folding (Tobi and Elber, 2000; Tobi et al., 2000; Vendruscolo and Domany, 1998; Vendruscolo et al., 1999). Using machine learning algorithms, the authors of these studies clearly showed that a set of contact potentials with cutoffs of 8.5 Å or 9 Å, which guarantees the native structure energies lower than those of the decoys, does not exist. Then, using different resolutions of the potential functions, the same learning algorithm, and the 9 Å cutoff, the flexible functional forms of potentials were optimized. Based on the performance of the potentials, it was noted that it is impossible to find a pair potential with the flexible form that recognizes all native folds (Tobi and Elber, 2000; Tobi et al., 2000). Developing contact potentials with the cutoff of 7.5 Å for predicting stability changes in proteins upon mutations, Khatun et al. (2004) note that, ‘it is impossible to reach experimental accuracy and derive fully transferable contact parameters using the contact models of potentials’. The choice of the cutoff may partly explain these results and thus encourage new attempts to parameterize potentials for longer ranges. Indeed, the 9 Å range is less than the critical cutoffs of power potentials for n ≤ 6 and the artificial ruggedness threshold of 5%, or for n ≤ 8 and the artificial ruggedness threshold of 10% (Table 1). For example, the artificial ruggedness of the energy landscape described by contact or Coulomb potentials cutoff at 8 − 9 Å is 17–19%. Since substantially frustrated landscapes are not adequate approximations of actual energy profiles due to the principle of minimal frustration, (Bryngelson et al., 1995; Dill, 1999; Miller and Dill, 1997; Tsai et al., 1999; Wolynes, 2005), the above studies had limited chances to detect the actual parameters of the interactions. Our results suggest that using longer cutoffs with such algorithms may improve the potentials.
Studies of ruggedness of protein–protein energy landscape are important for understanding the connection between protein structure, function and dynamics. We have analyzed energy fluctuations and the artificial contribution to the ruggedness of the protein–protein energy landscape by limited range interactions described by 1/rn potentials. The results show that the undesirable artificial ruggedness exists for short cutoffs and gradually disappears with the cutoff increase. We calculated the critical values of the cutoff for each of 11 popular power−type potentials with n=0÷9, 12 and for two thresholds of 5% and 10%. We showed that for both thresholds, the critical cutoff is a non-monotonic function of the potential power n. These functions reach the maximum at n = 3÷4 and then decrease with the increase of the potential power. The difference between the cutoffs for 5% and 10% artificial ruggedness becomes negligible for potentials decreasing faster than 1/r12. The analytical results were validated on the dataset of 62 protein–protein complexes, with different parameterizations of the soft Lennard–Jones potential and two types of protein representations: all-atom and coarse-grained. The results suggest that the cutoffs larger than the critical ones can be recommended for protein–protein potentials.
Funding: National Institutes of Health (R01 GM074255).
Conflict of Interest: none declared.