INTRODUCTION

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Functional protein sequences exhibit the ability to fold spontaneously into a unique native state (Creighton, 1993). A natural step in order to understand this crucial property is to compare good and bad folding sequences in simple models where conformational space can be properly explored. Most such studies have been directed toward identifying physical characteristics of good folders, and in this important area some progress has been made (Sli et al., 1994; Bryngelson et al., 1995; Klimov and Thirumalai, 1998; Nymeyer et al., 1998). In this paper we address the question of how good folders differ from random sequences in purely statistical terms. A related but different topic is how sequences that share the same (unique) native state are distributed in sequence space. This question and its evolutionary implications have recently attracted considerable attention (Li et al., 1996; Bornberg-Bauer, 1997; Govindarajan and Goldstein, 1997a,b; Bastolla et al., 1999; Broglia et al., 1999; Bornberg-Bauer and Chan, 1999; Tiana et al., 2000).

In a recent study of a hydrophobic/polar off-lattice model, it was found that good folders tend to show negative hydrophobicity correlations along the chains (Irbäck et al., 1997). The analogous calculations gave, moreover, qualitatively similar results for a major class of real proteins, corresponding to typical total hydrophobicities (Irbäck et al., 1996). On the other hand, the opposite behavior, positive hydrophobicity correlations, has been reported for a class of designed model sequences that display certain protein-like features (Khokhlov and Khalatur, 1998, 1999). These designed sequences are, for instance, not meant to have unique native states, so the different results do not represent a contradiction. However, it shows that sequence correlations in proteins is a delicate issue that requires a careful analysis.

The main goal of this paper is to test the robustness of the conclusion that good folding model sequences as well as functional proteins show negative hydrophobicity correlations. To this end we perform new calculations for both model and real sequences. The model we study is the minimal HP model on the square lattice (Lau and Dill, 1989; Dill et al., 1995). This choice makes it possible for us to improve significantly on the statistics in the previous study (Irbäck et al., 1997), which was based on an off-lattice model. The real sequences studied are single-domain enzymes taken from the CATH protein structure classification database (Orengo et al., 1997), which we hope displays statistical properties representative of functional (globular) folding units. With this restriction on protein type, it turns out that the previous, somewhat artificial, restriction on total hydrophobicity (Irbäck et al., 1996) can be lifted.

	METHODS

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Sequences

Let us first define the sequences studied. The real sequences studied are the 173 nonhomologous single domain enzymes found in the October 1998 release of the CATH database (Orengo et al., 1997). These sequences are transformed into binary hydrophobicity strings, by taking the six amino acids Leu, Ile, Val, Phe, Met, and Trp as hydrophobic (_i = 1) and the others as hydrophilic (_i = 1). This choice is somewhat arbitrary. Therefore, we also tried a 20-valued hydrophobicity scale, which did not affect any of the conclusions below. In CATH, the most general level of classification is denoted "class" and describes the relative content of  helices and  sheets. Below, the class dependence of our results is checked by separate calculations for each of the three major classes: mainly , mainly , and . A fourth class, low secondary structure content, exists but it is not considered separately, as only 3 of the 173 sequences belong to it. In our calculations we also divide the sequences into extracellular and intracellular ones. Following Martin et al. (1998), we take the presence of a disulphide bridge as an indicator of extracellular location. The number of enzymes in the different subsets studied can be found in Table 3 below.

The model we use is the minimal two-dimensional HP model (Lau and Dill, 1989), whose behavior is known in quite some detail (Dill et al., 1995). It contains only two types of amino acids, H (hydrophobic, _i = 1) and P (polar, _i = 1), and the chain conformation is represented as a self-avoiding walk on a lattice. The formation of a hydrophobic core is favored by defining the energy as minus the number of HH pairs that are nearest neighbors on the lattice but not along the chain. On the square lattice, it turns out that this simple choice of energy function is sufficient in order to get a significant number of sequences with unique ground states (Chan and Dill, 1994; Irbäck and Sandelin, 1998); complete enumeration of all possible sequences and structures shows that the fraction of such sequences is roughly 2% for N  18. Throughout this paper we consider all HP sequences that have unique ground states as good folding sequences. Also central is that the sequences are able to fold fast into their native states, a requirement that we ignore. This is a reasonable simplification because the sequences are short and because almost all have the same energy gap between ground state and next lowest level.

Sequence correlations

Our statistical analysis of hydrophobicity strings can be divided into two parts. The first part deals with the distribution of hydrophobicity along the chains; how does a "good" sequence with length N and total hydrophobicity

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differ from a typical sequence with the same N and M? This question can be addressed by monitoring variables such as the number of hydrophobic and hydrophilic clumps along the chain (White and Jacobs, 1990), Fourier amplitudes (Irbäck et al., 1996), or random walk (Brownian bridge) representations (Pande et al., 1994). In this paper we work with block variables, a widely used technique that has proven useful in studies of DNA sequences (Peng et al., 1992) as well as proteins (Irbäck et al., 1996).

In addition to the distribution of hydrophobicity along the chains, we also study the distribution of the total hydrophobicity M. This analysis relies entirely on comparisons between observed sequences, which makes it statistically more difficult, especially for the real sequences with varying N.

The blocking method

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The distribution of total hydrophobicity

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Individual structures

As mentioned in the Introduction, several recent model studies have addressed the question of how sequences that fold to the same native state are related. In particular, using an HP-like model with compact structures only, Li et al. (1996) found that structure-preserving mutations tend to be largely independent for highly designable structures. To see whether this behavior is consistent with our analysis, we perform two measurements for different fixed structures, too.

Consider a given structure r, and let {h_i^(r)} be the corresponding hydrophobicity profile (h_i^(r) is the probability that _i = 1). The first quantity we calculate is

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where ^(r) is defined as  in Eq. 5 but for fixed structure. ^(r) measures the average _i, _j correlation for fixed structure (see Eq. 6). The second quantity is the entropy

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for a system of independent _i with hydrophobicity profile {h_i^(r)}. If the _i values are approximately independent, then e^S provides an order-of-magnitude estimate of the actual number of sequences, N_r. If this is not the case, then e^S overestimates N_r.

	RESULTS

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In this section we present the results of our analyses of the mean-square block fluctuations ^(s) and the distribution of total hydrophobicity, M, for model and real sequences. We end the section with some comments on our model results and related studies of similar models.

The blocking method

Model sequences

View larger version (10K): [in this window] [in a new window]   FIGURE 1   The mean-square block fluctuation (s) against block size s for good N = 18 sequences in the HP model. Shown are results both for the full sequences (+) and for the subsequences consisting of the central 14 amino acids (×). The straight line represents random sequences; see Eq. 4.

Enzymes

View larger version (12K): [in this window] [in a new window]   FIGURE 2   (a) Hydrophobicity profile h() for the enzymes. The horizontal line indicates the mean   0.29. (b) k(4) as a function of  for the enzymes. The horizontal line represents random sequences.

View larger version (11K): [in this window] [in a new window]   FIGURE 3   The mean-square block fluctuation (s) against block size s for different groups of enzymes. (a) All sequences (data points connected by dashed line) and intracellular (IC)/extracellular (EC) sequences. (b) Division of the sequences into three structural classes: mainly , mainly , and . The straight lines represent random sequences.

The distribution of total hydrophobicity

Model sequences

View this table: [in this window] [in a new window]   TABLE 2   h = (1 + MN/N)/2 and the normalized variance  of M for good HP sequences for different N

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Enzymes

View larger version (13K): [in this window] [in a new window]   FIGURE 4   Total hydrophobicity M against N for the enzymes. The data points are averages over intervals of length 30 in N. The straight line is a least-square fit.

Comments

Our study of HP sequences has been focused on structure-independent properties. The question of how sequences that share the same (unique) native structure are related has recently been examined using similar models (Li et al., 1996; Bornberg-Bauer, 1997; Bornberg-Bauer and Chan, 1999). From these studies, a simple picture seems to emerge for structures that are highly designable. For high-N_r structures (N_r is the number of sequences that fold to the structure r), it has been found that the sequences tend to form a single cluster connected by one-point mutations, called a "neutral net" (Bornberg-Bauer, 1997), and that structure-preserving mutations tend to be largely independent (Li et al., 1996). The latter property was observed in a model with compact structures only. We checked that it holds in the present model too, which is illustrated in Fig. 5. From this figure it can be seen that the quantities e^S/N_r and |^(r)|, as defined in Methods, indeed tend to be small for high N_r. Also indicated in this figure is whether or not the sequences form a neutral net, results first obtained by Bornberg-Bauer (1997).

View larger version (15K): [in this window] [in a new window]   FIGURE 5   (a) eS/Nr, Nr and (b) (r), Nr scatter plots for the 1475 designable N = 18 structures in the HP model. The shape of the plot symbol indicates whether the sequences form a neutral net (+) or not ().

The fact that structure-preserving mutations are largely independent for high N_r does not contradict our previous results. To verify this, we calculated  from the known hydrophobicity profiles {h_i^(r)} under the assumption that the _i values are independent for each structure. The value obtained this way, ₂, can be found in Table 2 above, and is indeed a relatively good approximation to the observed .

Admittedly, the model used in this study is crude. In particular, Buchler and Goldstein (1999, 2000) have recently argued, based on a study of compact lattice chains, that the use of a two-letter alphabet leads to designability artifacts, which disappear with increasing alphabet size. Let us stress, therefore, that the analyses discussed in this paper can be tested on real proteins in a direct manner. Let us also comment on the stability of our results. First, we note that the dependence on chain length N is weak. This was explicitly shown for , and is true for ^(s) too, although our discussion focused on one system size in this case. Second, we note that our results are in nice agreement with those obtained earlier using a simple hydrophobic/polar off-lattice model (Irbäck et al., 1997). To further explore the model dependence of our results, we also did calculations for a "solvation-like" two-letter model discussed by Ejtehadi et al. (1998a,b) and by Buchler and Goldstein (1999, 2000). This model differs from the HP model in that the interaction strength is additive [(H, H) = 2, (H, P) =  and (P, P) = 0], which means that the total energy can be expressed as a simple sum of monomer contributions. Buchler and Goldstein argued that HP-like models, unlike pair-contact models with larger alphabets, tend to have solvation-like designability properties. It is therefore interesting to note that when analyzing sequences with unique ground states in the solvation-like model defined above, we obtained results qualitatively different from those for the HP model. More precisely, it turns out that the block fluctuations are significantly larger, close to random, for the solvation-like model.

	Summary and Discussion

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Hydrophobicity plays a key role in the formation of protein structures, which makes it of utmost interest to understand the statistical distribution of hydrophobicity along the chains. In this paper we have analyzed hydrophobic/polar sequences in the two-dimensional HP lattice model. Whenever statistically feasible, the analogous calculations were performed for a set of real enzymes, too. Our main findings are as follows.

1.

Both model sequences and enzymes show mean-square block fluctuations (s) that are smaller than for random sequences. In particular, this implies that the enzymes display the same behavior that had been found previously for general proteins with typical total hydrophobicities (Irbäck et al., 1996). The present analysis was performed without any restriction on total hydrophobicity.

2.

The average total hydrophobicity M varies approximately linearly with chain length N over the range of N studied, both for model sequences and enzymes. This implies, contrary to what one naively might expect, that the fraction of hydrophobic amino acids does not grow with increasing N. The fluctuations in M are difficult to study for the enzymes, due to statistical uncertainties. For the model sequences it turns out that the normalized variance  is significantly smaller than for random sequences.

We also divided the enzymes into different groups according to their structural content, and to whether they reside in an intra- or extracellular environment. The fluctuations in total hydrophobicity appeared to depend on group. However, whether this dependence is significant or not is difficult to say, due to statistical uncertainties. The mean-square block fluctuations are statistically much easier to measure, and show only a weak dependence on group. The conclusion that ^(s) is suppressed is, in particular, the same for all the different groups.

A full explanation of the suppression of ^(s) is probably hard to give. Let us note, however, that long hydrophobic or hydrophilic stretches in the amino acid sequence are likely to lead to degenerate structures, and the suppression of sequences containing such stretches should indeed tend to make ^(s) smaller.

The nonrandomness of the block fluctuations provides an indirect confirmation of the important role played by hydrophobicity in the formation of protein structures. Furthermore, it is tempting to take the similarity with the model results as an indication that the ability to form a stable structure represents a significant selective advantage in the evolution of proteins. It would be interesting to check that the behavior remains the same in more realistic models.