Funnel sculpting for in silico assembly of secondary structure elements of proteins

Abstract

We present a method for designing a funnel-shaped free-energy surface that reproducibly assembles secondary structure elements of proteins into their native conformations from a random extended configuration. Assuming a priori knowledge of secondary structure, our method can design a funnel-shaped surface for folding of α, β, and αβ structures individually. We design energy surfaces that fold up to five unrelated sequences with the same energy parameters. We develop a measure of the foldability of an energy landscape in silico and present an alternative way to view energy landscapes.

In 1973, Anfinsen (1) showed that the native state of a protein corresponds to a minimum of free energy. Using his discovery as a starting point, scientists have tried to derive the free energy surface of proteins and to predict the folded protein structure.

One of the dominant prediction methods used today is the decoy-then-scoring paradigm (45). First, a large number of decoys are generated to sample the conformation space. Usually considerable effort is exerted to explore and enrich the near-native fraction of the decoy set. After the sampling, a discrimination function picks out the best structure(s) from the ensemble. In this approach, the sampling of conformation space and the scoring of structures are separate. This paradigm is suitable for today's rapid increase of computational power: sampling is trivially parallelizable and, consequently, progress can result from simple increase in the number of operations spent on the task. Dovetailed into the decoy-generate-then-score paradigm are several different methods of extracting energy parameters for proteins. We mention a few examples: Z-score minimization (2–4) requires that the native state energy lie as low as possible relative to some reference state, usually represented by an ensemble of compact structures. Linear optimization (5, 6) maximizes a variety of conditions subject to the logical constraint that the energy of the native state (suitably parametrized as a linear sum) is simply lower than all alternate conformations. Potentials of mean force (7) assume that some relevant variables, e.g., residue-residue distances, are Boltzmann-distributed with the native state being the overwhelmingly more populated one. Finally, some approaches have trial-and-error components, where researchers notice and introduce features that make their predictions more native-like (8, 9).

The price extracted by the generate-decoy-then-score separation is the necessity of elaborate and computationally intensive sampling techniques to uncover the low energy states. Several clever schemes (see ref. 9 and references therein) use sequence information by either finding a close homologue or using short segment homology in fragment insertion. So far, researchers have not been able to consistently fold proteins using the generate-then-score paradigm (10, 11).

Folding Funnels

Scientists are not alone in having trouble folding proteins with current methods. A polypeptide can assume such a vast number of conformations that, as Levinthal noted (12), if proteins tried all possible conformations, they would not be able to fold up either. Yet they do. The apparent paradox is resolved by noticing that Anfinsen (1) proved not only that the native is at the minimum of free energy but also that proteins fold reliably into this minimum. A solution, pioneered by Bryngelson and Wolynes (13) and Leopold et al. (14), is that the free energy surface is a funnel that guides the chain toward its native state. Much theoretical and experimental framework has been laid down in this area [Shakhnovich and Gutin (15), Karplus and Shakhnovich (16), Boczko and Brooks (17), Dill et al. (18), Karplus et al. (19), Onuchic et al. (20), Wolynes et al. (21), Tsai et al. (22), reviewed in Dill and Chan (23), Karplus (24), Lazaridis and Karplus (25), and Dill (26)]. The majority of applications of the folding funnel concept to protein structure prediction have concentrated on the principle of minimal frustration brought over from spin-glass theory (13). The implication of minimal frustration is that the energy gap between the native state and the molten-globule ensemble is as large as possible. Good results in assembly of helical proteins using this technique have been reported by Hardin et al., Koretke et al., and Nanias et al. (refs. 27–29 and references therein). The Discussion contains further consideration of these methods and a comparison with our technique.

Inspired by the widely accepted existence of a funnel, in vivo, we set out to unite sampling and energy functions and to design funnel-shaped free-energy surfaces that fold proteins in silico. We imagine the computer model of a protein as a stochastic minimizer, able to overcome relatively small obstacles in the energy landscape (including intermediate states) and tumble down to the global minimum. We are interested in landscapes that can fold† a protein with the available computational resources, and we isolate the distinguishing properties of such free-energy surfaces.

This Work

We present a method of constructing a landscape that can guide a stochastic minimizer to the native state from an arbitrary starting point. The procedure works individually on a structural variety of proteins: α, β, and αβ. It also finds energy parameters that work simultaneously on five α-helical proteins at once with no degradation in performance. This finding heralds a future simultaneous optimization of the energy parameters on a larger training set. The procedure is stably convergent in training: after achieving its goal, subsequent optimization rounds do not destroy the funnel-like shape of the energy surface.

This is an appropriate time to state several disclaimers. The method presented in this article is only a proof of concept; we have not solved the protein folding problem. First, we assume knowledge of secondary structure. Second, our training set of five structures is far smaller than the 100 or more required (30, 31) to extract a true folding energy expression. Self-avoidance, a preference for forming disulfide bonds, and a rudimentary compactness/hydrophobicity are the only training features that carry over to a test sequence. (We do not enforce a priori self-avoidance into the force fields. The optimization procedure learns to eliminate clashes, in itself a satisfying result.) Third, at this point, the procedure has limited applicability (it seems to work best on helical proteins).

Some of the limitations are built in on purpose: our minimizer uses very simple moves, and this is one of the reasons we are reluctant to roll up helices from extended chains. Others are not: we fix secondary structure as we feel we have not understood or modeled hydrogen bonding adequately.

As a concept, however, our proposal has significant merit. Our scheme launches the protein downhill over and over again, correcting only the parts of the landscape responsible for impeding its progress, until virtually every trajectory converges to near the native state. Funnel sculpting systematically and algorithmically incorporates incorrect predictions into subsequent iterations to incrementally improve the performance of folding. The funnel-like nature of the energy surface also ensures that degradation of performance is gradual: those mistakes that have not been eliminated at some training stage become progressively more near-native and do not land wildly far away from the desired result.

Model

Representation of the Protein. We choose a simplified “lollipop” model of protein structure for the same reasons that prompted Levitt (32) to invent it: it offers computational simplification while retaining biologically relevant resolution. Our model contains the backbone heavy atoms, the virtual side chain interaction centers, and the polar hydrogen atoms involved in backbone hydrogen bonding. The model is shown in Fig. 5, which is published as supporting information on the PNAS web site, www.pnas.org. The virtual side-chain (R) is a point 3.0 Å from the C_α along the C_α—C_β vector. (C_α and C_β refer to the α and β carbons of the polypeptide chain.) We position the GLY centroid at the C_α atom. This representation has a resolution of ≈3 Å. Our degrees of freedom are the dihedral angles ϕ and ψ. More details can be found in refs. 31 and 32.

Energy Function. Because our previous work (32) contains a full description of the energy function, we provide only a brief summary here. The free energy consists of three types of interactions:

[1]

The first term in Eq. 1 is residue-specific and depends on the number of the given residue's neighbors (a neighbor is any other side chain within 10 Å). This energy component is modeled after the hydrophobic interaction. Formally, the energy is

[2]

where N is the length of the protein, A indexes the amino acid type, T_k(x) is the Chebyshev polynomial of order k, and n is the number of neighbors within a 10-Å radius from the amino acid i. C^A k are the energy parameters to be determined (120 in all).

The second component of the free energy is a pairwise residue-residue-specific potential that depends on the distance between side chain centroids (see Fig. 5). The expression is

[3]

where, once again, T_k(x) is the Chebyshev polynomial of order k and A and B indexes the amino acid type (e.g., CYS). It is the small r component of this interaction that will provide the clash avoidance when the potential is fully formed. We keep the precision of this energy contribution to four Chebyshev terms, giving a total of 840 C^AB k energy parameters to be determined.

Finally, we include a hydrogen bonding energy that counts the number of hydrogen bonds formed, as well as whether the bonds form networks (e.g., helices and, sheets).

[4]

The first term in Eq. 4 is a step function in angle and distance from the donor and acceptor (33): it simply records that a hydrogen bond has been formed. The bond is considered formed if the distance between H and O is <2.8 Å, and the corresponding bond angle is >90° (the angle for a perfect hydrogen bond is 180°). The second term in Eq. 4 is a correction that depends on the bonds' environment. We define the environment by the presence of another hydrogen bond in nearest and next-nearest neighbors (along the chain) on both sides of the bond in question. A total of five parameters, E_a, quantify the interaction.

The values of the coefficients (E_a, C^A k, and C^AB k) and, therefore, the shape of the potential need to be determined by the optimization procedure. We take the Chebyshev expansion to order 4 in pairwise and order 6 in burial components. Thus we have a total of 5 + 6 × 20 + 4 × 210 = 965 parameters. The expansion needs a relatively small number of coefficients to represent an arbitrary shaped curve. Ref. 32 discusses this and other benefits of energy expressions 2 and 3 in more detail. The particular choice of energy functions is not crucial to our method.

Energy Optimization and Iterative Refinement

Overall Strategy. We start the optimization by setting all 965 parameters to zero; this gives a flat (E = 0) free energy surface. We then proceed with cycles of minimization and energy optimization until proteins fold reliably, starting from a random conformation structure to within 4, Å crmsd² (crmsd, coordinate Cα rms deviation) of the native structure (the resolution of our model). Here reliable folding means that the lowest energy structure of the 10 runs of the cycle is within 4 Å to the native state. In our experiments, most of the time every single optimized trajectory produced a near-native fold (4 Å or less).

Fig. 1 is a graphical illustration of the process. In the first round (E = 0), we seed the procedure with 100 random structures for each protein. At subsequent rounds, 10 conformations are produced by 10 minimization runs (described in Search Algorithm). These conformations settle in local minima, which either are lower than the native or are surrounded by barriers that make escape difficult. These 10 new decoys join the ensemble of existing decoys. We then design a new energy landscape that attempts to arrange the decoy ensemble in a funnel-like fashion. The design algorithm (see Optimization of Energy Parameters) optimizes the energy parameters to locally smooth out the rugged features that prevented a complete descent to native, while preserving the desirable funnel-like nature of the preceding landscape. The whole process is reminiscent of panel beating and high spot sanding in auto coachwork; hence our title “funnel sculpting.”

Fig. 1.

An illustration of the funnel-sculpting strategy. We start with a flat energy landscape (E = 0) and proceed with rounds of alternating minimization and landscape smoothing until all random starting points “roll” into the native-like neighborhood.

An important property of the overall sculpting process is that the smoothing is tightly connected with the capabilities of the search algorithm. If, for example, the search can escape a local minimum on its own, then it will not leave a decoy for the next round marking the spot to be flattened. Thus funnel sculpting, round after round, teaches the energy function to fold proteins with the available search strategy.‡ Another important feature of the method is the fact that it is stably convergent. The procedure finds the energy parameters that fold the protein, and consequent rounds of optimization do not destroy the foldability of the landscape (see, for instance, Fig. 2). This stability permits us to keep just 10 folds in each round and not worry that proteins of different size will require very different numbers of decoys to map out their conformational space.

Fig. 2.

Gradual optimization of the folding landscape for two proteins, 1erp and 1aca. The sculpting for two sequences is shown side by side. The graphs (Top) plot the improvements in rmsd for successive rounds of sculpting. The black lines mark individual members of the folding ensembles for each round, and the red line traces the rmsd of the lowest-energy decoy for that particular round. The series of graphs (Middle) show 1D snapshots of the energy landscapes at different stages of sculpting. The shapes (Bottom) are the projections of the energy landscapes into two dimensions. The projections were made by using the techniques of the Appendix, which is published as supporting information on the PNAS web site. For both proteins, the sculpting produces smoother and more correct energy funnels, and in each round the trajectories terminate closer and closer to the native state. The 3D plots (Bottom) can be seen and manipulated at www.stanford.edu/~bfain/landscape (Java-enabled browser required).

This mapping out of the conformational space is similar in spirit to the probabilistic roadmaps used in robotics for motion planning (34) and recently applied to binding site landscapes (35).

Search Algorithm. To find the minima of our energy surface, we fold the structures using either Adaptive Simulated Annealing (ASA, www.ingber.com/#ASA) or our own Monte Carlo (36) minimization routine. Our degrees of freedom are the dihedral angles of the loop residues; the secondary structure of proteins is fixed at the ideal angles. Consequently, we are not working on protein folding in the conventional sense but rather on the easier problem of tertiary structure assembly. Instead of annealing the temperature, we periodically readjust it to be of the order of the roughness of the surrounding landscape (e.g., Reanneal Cost <–1 in ASA). On the other hand, the generating functions for the parameter moves are slowly frozen. Each minimization run consists of 40,000 steps, and the acceptance ratio is quite low (10^–3 to 10^–4). We describe the search as “simulated oozing,” with the algorithm ambling slowly downhill but able to some extent to jump over local barriers and out of local minima. At the end of each run, we apply 1,000 steps of simplex minimization (37).

Optimization of Energy Parameters

At each stage of optimization, we have an ensemble of decoys plus the native conformation. We want to smooth the false minima and energy barriers that prevented the search algorithm (described in Search Algorithm) from tumbling into the native state, yet we also wish to preserve the partially correct funnel-like landscape constructed in the previous round. We achieve this goal by changing the 965 energy parameters to maximize the funnel smoothness coefficient, ℱ, defined below. ℱ is optimized by using initial Monte Carlo maximization (36) for 3,000 iterations, followed by refinement with Brent's minimization (38) for 1,000 iterations.

We shall denote the distance between a pair of conformations i and j as d(i, j). The distance to the native conformation will be denoted by d(i, 0) or just d_i. The metric d can be cmrsd (39), the recently developed scaled gauss metric (40), distance (d)rmsd, % native contacts, or any combination of these. In the current work, we use crmsd as our metric.§

We now define the funnel smoothness coefficient, ℱ. The coefficient is inspired by the nonparametric correlation coefficient called Kendall's τ, and the following definition is similar to one found in chapter 14 of Numerical Recipes (41). Consider a set of N backbone conformations (N–1 decoys + Native). We call a pair (i, j) concordant or con if either d_i < d_j and E_i < E_j,orif d_i > d_j and E_i > E_j. Conversely a pair is tagged discordant or dis if either d_i < d_j and E_i > E_j,orif d_i > d_j and E_i < E_j. In adding up the concordant or discordant conformations, we count only those (i, j) pairs for which the connection matrix C_i,j defined in Eq. 5 is not 0.

The connection matrix C_i,j is set to 1 for those pairs that are closer to each other than some cutoff D_max, and the absolute value of the difference in their distances to the native is greater than some parameter D_tie. C_i,j is also set to 1 if either i or j is the native conformation. We set D_tie = ln(1 + d(i, j))) and D_max is taken to be of the order of the radius of gyration of the native structure. Otherwise, if i and j are either too close or too far away from each other, C_i, j is set to 0. Formally:

[5]

Eq. 5 is illustrated in Fig. 6, which is published as supporting information on the PNAS web site.

The funnel smoothness coefficient is then the following combination of these counts:

[6]

Paraphrasing sentences into equations, the coefficient is:

[7]

The density correction ρ_i is introduced to prevent overemphasis of highly populated regions (e.g., near-native if the method is working). We found it sufficient to set it proportional to the number of neighbors of conformation i to the first power (this correction depends on the effective dimension of the conformation space of each protein). The constant of proportionality is adjusted to keep ℱ between –1 and 1.

[8]

Comments

First, let us explore the meaning of Eq. 7. Recall that we are designing a funnel-like energy surface that will guide a stochastic search algorithm (e.g., Monte Carlo minimization) toward the native state. The function ℱ takes its values between –1 and 1. If we ignore for the moment the connection matrix (i.e., we compare all pairs), we see that corresponds to a funnel monotonically sloping toward the native. The actual value of the slope along the funnel can vary due to the nonparametric nature of the correlation imposed by ℱ, but the closer a conformation is to native, the lower is its energy. [ produces a random energy surface, a spin-glass, and corresponds to a monotonic surface with (negatively) correlated d and E, a “hill” with the native at its peak.]

We now move on to the details of Eq. 5. There is no physical, conceptual, or pragmatic reason for distant points on the funnel to correlate. Basic physics of the problem allows two conformations that are both, say, 10 Å away from native to have different and, furthermore, unrelated energies if they are also far away from each other. Because our minimizer needs only local gradients to progress, we need to ensure that only those exist. The limit D_max in Eq. 5 discards comparisons of conformations on opposite sides of the funnel. The introduction of D_tie permits local variations of the energy surface that our search algorithm can jump over. We do not want to over-smooth the energy surface and, in fact, we do not need to so long as our stochastic search algorithm can overcome local obstacles. The introduction of D_tie relaxes the optimization requirements and enables the optimization procedure to concurrently accommodate a larger set of proteins.

The second line of Eq. 5 connects every conformation to the native state and, consequently, allows us to make the native the global minimum in energy if we so choose. In our experiments, we set Å to be of the scale of local moves of the search algorithm and to increase at larger distances. D_tie was set to approach 0 at the native state [D_tie = ln(1 + d(i, j)) Å] to achieve more precise refinement near the ground state.

Results

Individual Sculpting. In all our folding studies, we held the native secondary structure fixed and varied the loop dihedral angles. We first applied the sculpting procedure individually to each of the seven small and three medium-size proteins: 1pga, 1fc2, 1erp, 1vii, 1cbn, 1mbh, 4pti, 1ctf, 1mba, 2hp8, and 1aca. The left half of Table 1 lists a summary of the properties of these peptides. The proteins are structurally diverse in that they belong to different SCOP folds (42) and have no detectable sequence similarity.

Table 1. The 11 proteins used to test individual landscape sculpting.

Protein Data Bank ID code	Structural classification of protein	Number res.	α	β	Number of degrees of freedom	rmsd range, Å best-average-worst	rmsd chosen (lowest energy)	Number runs to folding below 4 Å
1vii	a.14.1.1	36	3(21)	0	30	2.33-2.54-2.73	2.33	5
1erp	a.10.1.1	38	3(26)	0	24	1.93-3.10-7.07	1.93	5
1fc2	a.8.1.1	43	2(23)	0	40	2.25-2.82-4.29	2.25	6
1cbn	g.13.1.1	48	2(18)	2(4)	52	2.23-3.32-4.14	3.20	8
1mbh	a.4.1.3	52	3(30)	0	44	2.42-4.01-7.52	2.42	8
1pga	d.15.7.1	56	1(15)	4(24)	34	2.40-3.10-4.80	2.90	3
4pti	g.8.1.1	58	2(13)	2(23)	44	2.21-3.75-6.23	2.95	10
1ctf	d.45.1.1	68	3(36)	2(18)	28	2.34-3.88-4.54	2.34	12
2hp8	a.17.1.1	68	3(43)	0	50	2.93-3.78-4.58	2.93	18
1aca	a.11.1.1	86	4(60)	0	52	3.60-4.94-5.87	3.60	20
1mba	a.1.1.2	147	7(114)	0	66	3.78-5.48-8.21	4.32	34

The left side of the table describes the proteins: the length, the CATH (www.biochem.ucl.ac.uk/bsm/cath) designation, the α and β content, and the number of degrees of freedom. The right side reports the folding performance of the potentials obtained in the optimization procedure: the best folding results and the number of optimization rounds required to start folding structures. In all cases, after folding was achieved, the subsequent potentials all performed well (see Fig. 2 for illustration).

In each case, the training procedure quickly derived the set of energy parameters that fold each protein. The right half of Table 1 summarizes how well the trained energy function folds each of the proteins individually. In all cases, it took the sculpting 10 rounds or less to yield a set of parameters that folds the proteins to within 4 Å of their respective native conformations. As expected, longer proteins required longer training: their conformation space is larger, requiring more decoys to map it out and sculpt it. In Fig. 2, we graphically illustrate the training procedure for two proteins, 1erp and 1aca. In Fig. 2 Top, we plot thecrmsd of thefolding runs for each training round, as well as the crmsd of the lowest-energy decoy produced in that round. Even if decoys from another round score lower, they are not displayed. As the training progresses, all 10 folding trajectories (Folding Results, black bars in Fig. 2), as well as the lowest-energy conformations (red bars), move closer and closer to the native state. Once good folding is achieved, it is not ruined by subsequent optimization. Such stable convergence is highly desirable: when we optimize several proteins simultaneously (e.g., training set), smaller proteins have to wait for larger proteins to optimize. We introduced the density correction (Eq. 7) for this exact purpose. Fig. 2 Middle and Bottom shows 1D and 2D (using methods in the Appendix) snapshots of the energy landscapes in various stages of optimization. One can clearly see the elimination of false minima by sculpting.

Although β proteins did fold to within 4 Å of their native state, the resulting conformations did not form good β-sheets. This happens for two reasons: our simple sampling technique has trouble aligning strands, and, frankly, we do not model hydrogen bonding very well. Although these issues are not fundamental, they do obscure the development of funnel sculpting. Consequently, we have postponed including β proteins in concurrent optimization until we solve these problems.

Concurrent Sculpting. The results presented in the previous section are very exciting. In a larger perspective, however, we have folded something with a priori knowledge of the answer. Although we did not include the desired distances into the force-field, it is possible that the native structure had somehow wormed its way into the energy parameters. Judging from the number of parameters in our model (965) and previous experience (32), we anticipate that prediction of protein structure will require simultaneous folding of a training set of 50–100 sequences. To prove that our model was extensible to a larger decoy set, we set out to sculpt an energy surface that folds five proteins at once. Satisfyingly, the simultaneous sculpting was a success without any loss of performance from the individual organizations. The numerical results of the optimization are presented in Table 2, and a graph of the lowest energy rmsds is in Fig. 3. The best energy parameters fold all five proteins to within 4 Å of native, with smaller proteins landing considerably closer. The major difference in going from individual to simultaneous training is that the overall procedure converges more slowly for the smaller proteins. In concurrent sculpting, the presence of other proteins forces each sequence to wander in a larger portion of its conformational space that, in turn, requires more decoys to map out the landscapes. Fig. 4 shows how funnels change during concurrent optimization.

Table 2. Results of concurrent funnel sculpting for five proteins.

Protein Data Bank ID code	rmsd range, Å best—average—worst	rmsd chosen (lowest energy)
1erp	2.67-3.30-4.64	3.10
1fca	3.18-3.90-4.59	3.90
1vii	2.65-3.16-3.62	3.23
2hp8	3.59-4.83-7.73	4.01
1aca	3.95-5.50-7.50	3.95

Several runs were good (see Fig. 2); we present the results of sculpting round no. 38 (of 50) runs.

Fig. 3.

The rmsd for five proteins sculpted simultaneously. There are several energy parameters (e.g., nos. 36, 38, and 48) that fold all five sequences to within 4 Å from the native conformation. The convergence for 1erp is much slower than in Fig. 2.

Fig. 4.

The sculpting of the funnels for 1erp and 1aca during concurrent optimization. The landscapes start out flat (not shown here), become generally funnel-like with many false minima, and then finally assume the desired shape. The landscapes shown are for rounds 3 and 48, respectively.

Discussion

Comparison with Other Methods. Several methods (3, 27–29, 44) have the same goal of determining the free energy of protein folding and then assembling tertiary structure from secondary structure elements (the list is by no means exhaustive).

Linear optimization, which requires only that the native state is lower than all alternate conformations, is logically faultless. In practice, it usually works up to a point, and then it does not (6). It is difficult to say which inequalities made the solution infeasible and which are to be kept. It is also hard to determine whether the failure was caused by the insufficient detail of the energy field or the incorrect placement of the minima and alternatives. Furthermore, to weed out among the great many landscapes where the native is lowest, one has to specify an additional criterion (objective function) to select a set of parameters within the feasible polytope.

Many works embrace the convincing theoretical arguments advanced by Wolynes et al. (21) that the native state is characterized by a large energy gap. Although this may be true, we disagree with the implication that the gap should be made as large as possible. This insistence (31, 32) can make searching for the deep minimum prohibitively difficult. In lattice models, the built-in coarse-graining of conformation space ensures that a large energy gap translates (somewhat) into foldability. But in the realistic approximation of the present form of the potential in this work, insisting on the largest possible Z-score of the native creates a “golf-course” landscape of a random surface with a deep and narrow crater in the middle. It may very well be that in vivo landscapes do not look exactly like those in Figs. 2 and 4 but more like those suggested by the principle of minimal frustration. We are more interested in discovering which properties of the free-energy landscape can help us fold proteins in silico with available computational resources.

There are more specific differences between this work and, e.g., the excellent work of Hardin et al. (27). In energy-gap optimization, the reference “molten globule” state asserts itself only through a state average, which leaves the landscape vulnerable to false (or true) multiple minima. In contrast, we take into account the complete landscape or at least those portions that our stochastic minimizer happens to visit. However, we very much like the authors' successive updates of the molten globule state with misfolded structures.

There are also some parallels in our learning technique to the algorithms of the Scheraga group (see refs. 29 and 44 and similar articles). We appreciate the iterative update of the reference ensemble, motivated by the desire to map out conformational space. We also concur with their practice of taking updates from successive minimizations. The difficulty experienced by Nanias et al. (29) in locating the minimum as proteins get more complex, as well as the fact that their results are a family of low-lying states with some near-native members, may be due to their use of energy-gap considerations.

Finally, in contrast with Hardin et al. (27), we do not exploit knowledge of existing protein structures either in the Hamiltonian (associative memory) or in sampling (fragment insertion). The use of such information is definitely worthwhile. We kept it out of this work to isolate the development of funnel sculpting as a concept.

Pros and Cons. At this point, our work has several limitations. First, inefficient sampling forces us to fix native secondary structure. This can be overcome with fragment insertion methods (9). Second, our hydrogen bonds form poor cooperative networks, which means that we must improve the way we model hydrogen bonding. The final and most important barrier is the question whether the method has predictive power. The solution to this problem obviously is to try a larger set and attempt prediction.

At the same time, funnel sculpting has significant merit. Our method gradually forms a landscape that consistently folds several sequences to near-native conformations. It also shows no decrease in resolution when applied to larger sets, pointing the way toward a future simultaneous optimization in a training scenario. The most important feature of funnel-sculpted landscapes is that they allow proteins to fold in silico in a very short time (of the order of minutes in our experiments) by guiding the minimizer toward the native state. Our method can embrace energy functions and search methods of varying degrees of complexity and adopt itself accordingly. We start only with the knowledge of the native conformation and the desire to fold to it, leaving out even van der Waals' interactions and other clash-reducing considerations; our initial energies are completely flat! (Avoidance of clashes is invariably an a priori feature of knowledge-based potentials.)

Funnel sculpting is not limited to ab initio protein structure prediction but can also be easily adopted to various force fields and objectives. It is suitable for determining effective energies for protein and domain docking, drug design, and homology modeling.

The canonical paradigm of physics is to ensure that the ground (and excited) states of the Hamiltonian agree with those found in nature. We extend it by requiring in addition that the ground states be reachable by the available search methods.