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Cooperativity and Contact Order in Protein Folding

The effects of cooperativity are studied within Go-Lennard-Jones models of proteins by making the contact interactions dependent on the proximity to the native conformation. The kinetic universality classes are found to remain the same as in the absence of cooperativity. For a fixed native geometry, small changes in the effective contact map may affect the folding times in a chance way and to the extent that is comparable to the shift in the folding times due to cooperativity. The contact order controlls folding scenarios: the average times necessary to bring pairs of amino acids into their near native separations depend on the sequential distances within the pairs. This dependence is largely monotonic, regardless of the cooperativity, and the dominant trend could be described by a single parameter like the average contact order. However, it is the deviations from the trend which are usually found to set the net folding times.

There are many indications that properties of a protein, such as the folding time, t f old , depend on the topology of its native state [1][2][3]. Furthermore, the number of possible native folds is known to be limited [4]. These two facts provide support for geometry based modelling of proteins such as the tube-like [5] and Go-like approaches [6]. There are two geometrical parameters that have been hypothesized as setting a scale for t f old : N -the number of amino acids (or the backbone length of a protein) and CO -the relative contact order parameter [2]. The latter is defined as an average sequential distance between pairs of amino acids that interact, or make a contact, and normalized by N. A compilation of t f old that were measured at room temperature [2,3] suggested a correlation with CO but no dependence on N. On the other hand, theoretical studies yielded t f old growing with N [7,8] and not depending on CO [8]. Is it the theories or the interpretation of the experimental data that are wrong?
Jewett, Pande and Plaxco [9] have recently suggested that the theories do not properly account for cooperativity effects, i.e. for the fact that the strength of effective amino acid interactions should depend on a conformation. One of the reasons for such a dependence is that changes in the conformation may lead to variations in the degree of exposure to water molecules. Specifically, Jewett et al. have considered a 27-mer lattice Go model in which the contact energy, ǫ ′ , is related to the native contact energy, ǫ, through where Q is the fraction of established native bonds and s is a control parameter which introduces a conformation dependence for s > 1. The result of their simulations is that t f old correlates with CO at the 57% correlation level for s=3 as compared to 5% for s=1 and to 80% reported in the experimental data. A related study of a similar model by Kaya and Chan [10] indicates an even higher degree of correlation induced by this kind of cooperativity.
What would this prescription for cooperativity yield in off-lattice models of actual proteins? In this Paper, we consider 14 α-type (no native β-sheets) and 16 β-type (no native helices) short proteins which were studied in Ref. [8]. The model is coarse-grained, i.e. it involves only the C α atoms, it is Go-like, i.e. the potential is chosen so that the ground state agrees with the experimentally determined native structure and it also favors the native sense of chirality. We describe the contact interactions by the Lennard-Jones potential which is scaled by the energy parameter ǫ ′ as in eq. (1). This parameter is identical for all native contacts. We take s=3 and adopt an adiabatic way to smooth out any sudden changes in Q during the time evolution.
We find that the scaling of t f old with N and the kind of the dependence on CO do not change on switching from s = 1 to s = 3, but the folding times become, on average, about 2.25 longer since the couplings provide a weaker pull at the begining of the folding process. In particular, we confirm existence of the structure related kinetic universality classes [8]. Even though the single CO parameter itself does not correlate with t f old , the contact order, in a more general sense, is important for the folding scenarios. One can characterize the folding scenarios by plotting the average first times, t c , needed to establish specific native contacts as a function of the corresponding sequence distance. We find that the folding scenarios are governed by the distance itself in a fairly monotonic way. However, the control is incomplete and often it is the out-of-trend deviations that set the time to form the last contact, i.e that set t f old . Thus the local structures, like helices, do tend to form first (in agreement with numerous experimental findings) but the way the non-local structures form is not necessarily in agreement with the contact order. Furthermore, we find that the effects of cooperativity may be less important than those of the precise determination of the contacts considered native in the Go model. Removing some contacts from the native set or declaring some reasonable non-native contacts to be effectively native may affect t f old more significantly than the tinkering with the strength of the couplings. These kinds of adjustments in the contact map physically correspond to considering sets of sequences which are different and yet folding to nearly the same conformations, i.e. belonging to the same native fold.
An important feature of our studies is that for each model protein we determine the dependence of t f old on the temperature, T , and take t f old at the optimal temperature, T min , as the characteristic duration of folding that is relevant for scaling studies. When one considers measurement or calculation at a fixed value of T (such as the room temperature) then this T can be in any distance away from the minimum of the typically U-shaped dependence of t f old on T . Thus such, essentially arbitrary, choices of T may significantly affect comparatory analysis of proteins. The arbitrariness is removed if the kinetics are observed at T min . The experimental studies do not involve optimization of this kind.
When constructing the model, we follow references [11]. Briefly, the amino acids are represented by particles of mass m located at the positions of the C α atoms. They are tethered by a strong harmonic potential with a minimum at the peptide bond length. The native structure of a protein is taken from the Protein Data Bank [12] and the interactions between the amino acids are divided into native and non-native. The distinction is based on the atomic representation of the amino acids in the native state. We check for overlaps between the atoms by associating spherical volume to them. The assigned radii are 1.24 times the van der Walls values and the multiplication factor accounts for the softness of the potential [13]. The overlapping amino acids (i and j) are considered to be making contacts.
The resulting C α − C α contact separation ranges between 4.3 and 12.8Å. These pairs are endowed with the Lennard-Jones potential in which the length parameter σ ij is chosen pair by pair so that the native C α -C α distance agrees with the minimum of the potential. The non-native contacts are purely repulsive (in the basic model) and truncated at the distance of 2 1/6 σ where σ = 5Å. During the time evolution, a native contact is considered to be established when the distance between the amino acids involved is less than 1.5 σ ij . The thermal fluctuations away from the native state are accounted for through the Langevin noise with the damping constant γ of 2 m/τ , where τ is mσ 2 /ǫ. This leads to negligible inertial effects [8] but a more realistic account of the water environment requires γ to be at least an order of magnitude larger. However, t f old at T min has been found to be linear in γ [11] so the physical time scales can be accessed by a simple rescaling. Figure 1 illustrates the way the model with the cooperativity effect is defined by showing the time evolution of ρ. The adiabatic way to incorporate variations in ρ is as follows.
The integration of the equations of motion is based on the discretization of τ into 200 segments. With each time advancement by 1 200 τ , Q that enters eq.1 becomes updated through Q new = 0.99Q old + 0.01Q current to eliminate rapid jumps in Q current . (Q current , the instantaneous value, is meant as Q in Figure 1). The resulting ρ is 1 3 in the fully unfolded state as then Q is zero. In the native state, Q = ρ = 1. It is seen that at about 1/3 through of the folding evolution, the variations in ρ start mirroring those in Q. The inset of Figure   1 shows that the initial reduction in ρ, compared to the native value, results in a longer t f old . However, the results for t f old at s=3 correlate strongly with those at s=1. It should be noted that the cooperativity effect is expected to enhance the thermodynamic stability as it makes non-native local energy minima less stable relative to the native native state.
On the other hand, we have found that the values of T min become lower (in the record cases by 0.12). These two effects combined suggest that the energy landscape gets sculpted in a way that enhances the folding funnel. Figure 2 shows that cooperativity does not affect the scaling curves. The largest value of N considered here is 154 and the smallest -35. In this range, it is hard to distinguish between the power law and the exponential dependencies, even though the correlation levels for the α proteins favor the former slightly. However, there continues to be a support for existence of kinetic universality classes that depend on the type of the secondary structure.
When the power law fits are used, the exponent for the α-proteins is about 1.7 and for the β-proteins -about 3.2 (in the mixed case it is about 2.5). The scaling trends seen in Figure   2 become disturbed, but still identifiable, when calculations are performed not at T min but at a fixed T .
Cooperativity does not affect the dependence on CO either, as is shown in Figure 3. It is seen that a given value of CO may correspond to a big span of the values of t f old and there is no trend that can be demonstrated. Significant variations in t f old can be obtained by staying with one native geometry and adjusting the list of contacts that are considered native in the dynamics. We focused on three proteins: 1rpo, 1csp, and 1efn with the calculated numbers of the contacts of 194 (N=61), 169 (N=67), and 150 (N=57), respectively. We identified all non-native contacts with the spatial C α -C α range of less than 12Å and considered systems with 5,10,20,30, and 40 such contacts, chosen randomly, as providing additional active contacts. In addition, we considered systems in which 7 long range native contacts were removed from the native list. We studied variations of t f old with CO within the three families of systems, each consisting of 7 members. The family of 1rpo shows a growing trend with CO. This trend gets disturbed in the case of 1efn. On the other hand, the variations around 1csp are chaotic. It should be noted that the variations within the families are not significant in the plots on the N-dependence, even in the case of 1rpo. It should also be pointed out that our data contain two pairs of proteins with identical values of N (2abd and 1imq in the α case and 1tit and 1ten in the β case) and in these pairs t f old is in fact longer for the protein with the bigger CO.
In the Author's opinion, the experimental evidence, at the fixed T , for the trends in CO is not definitive. The inset of Figure 4 presents these data [3] separately for the αand βproteins (the data in the original paper are not plotted split into the three structural classes: α, β, and α−β ). If one focuses just on the β proteins then it emerges that four very different folding times correspond to almost the same CO. A similar point is demonstrated in the inset of Figure 5 which presents the β-protein entries in the data compiled by Galzitskaya et al. [14]. The β-proteins form the crucial test case of the approach since they involve long range contacts. In the case of α-proteins, CO is more a measure of the helical content, h, in the protein than a measure of the sequence range which is short. The lower right panel of Figure 6 shows that h, if non-zero, is in fact anticorrelated with CO to a fair degree (the correlation coefficient of 0.74).
The folding scenarios, however, do depend on the contact order. Not on its average value but on the full set of values. This is illustrated in Figures 4 and 5 for the 1rpo (α-type) and 1csp (β-type) proteins. The figures show t c as a function of the sequence length |j − i|. In the case of the helical 1rpo, the dependence is monotonic and as such it could be represented by a single parameter, e.g. the relative (or average) contact order. Note that there is no qualitative difference between the s=3 and s = 1 cases and the same goes to 1csp (s = 1 not shown in Figure 5). Thus cooperativity does not introduce any new features in the folding scenarios other than general shifts. Note that when 30 additional contacts that are consistent with the native topology are introduced in 1rpo (s=3) then the shifts in the data points are of the size that is comparable to the very introduction of the cooperativity. Thus cooperativity acts as if it was affecting the number of effective contacts. Figure 5 shows that the β-protein 1csp has a structured form of the plot of t c vs. |j − i|.
This kind of branched form is found in many other proteins both of the β-type, like in 1tit, and of the α-type, like in 1f63, 1ycc, or 256b. In 1csp, there are many contacts of the same |j −i| which are established at different times. More importantly, the last to form are not the longest ranged contacts corresponding to (i, j) = (1,62) and (1,64) but the medium ranged (9,41), (8,43) and then (6,44), (6,45). Adding the 20 contacts to 1csp shifts the pattern downward but does not affect it in any fundamental manner. Note that the formation time of the shortest ranged contacts is not affected by the additional contacts. The acceleration of folding begins in the second branch of contacts around |j − i| of 11.
We have checked that the folding scenarios are not sensitive to the details of the Gomodeling, such as the choice of the contact potential (the 10-12 potential instead of the Lennard-Jones) or to the addition of terms that depend on angular parameters, such as the dihedral angles. Furthermore, we have considered other variants of the cooperativity effects.
One of them is to replace Q by the ratio of the full potential energy of the system to its native value. The results are qualitatively similar to those reported here but the range of variations of the ρ parameter during the simulations is reduced, making it less effective, compared to the contact based definition.
We conclude that incorporation of elements of cooperativity in the couplings does not affect the folding process in the off-lattice Go models in any qualitative manner other than making the folding times longer despite the better sculpted folding funnel. Notice that the parameter Q does not generally couple to the contact order. Thus it is puzzling why its incorporation into the cooperativity effects appears to enhance the CO-dependence in the N=27 lattice models. [9,10] Perhaps the lattice geometry itself imposes some sort of coupling that emerges when considering systems of a fixed value of N.
Since the off-lattice Go models, with or without the cooperativity, do not yield a correlation of the folding times with CO it is interesting to ask whether some new quality arises if CO is replaced by the helical content parameter in the case of α and α − β proteins. Figure   6 shows that this is not so. The experimental data points compiled by Galzitskaya et al. [14] on two-state proteins containing helices (the lower left panel) do anti-correlate with h (the correlation coefficient is 0.68), though not as strongly as they correlate with CO (not shown; the correlation coefficient is 0.86). On the other hand, our model calculations (the top two panels) remain uncorrelated both with h and with CO even though some tendency to grow with h might be identified in the α − β case. We should reemphasize that the experimental data do not pertain to the characteristic folding time that could be uniquely associated with a protein. The characteristic time must be obtained through an optimization process, i.e. it must be measured at T min . A room temperature based measurement need not coincide with the conditions at T min and can yield almost any value of t f old , depending on the the precise choice of the control parameters such as T and pH.
We have considered a well controlled model of proteins and demonstrated lack of dependence of the folding times on the relative contact order parameter, irrespective of whether the cooperativity is taken into account or not. We have also demonstrated that, for a fixed geometry, or a fixed native fold, one can get very different folding times depending on the sequence, i.e. depending on what precisely constitutes the set of active contacts. These findings do not mean that geometry of the native state is irrelevant. Rather, they mean that the single relative contact order parameter may be inappropriate to characterize it. It is the full native contact map that has a kinetic meaning.
MC appreciates fruitful discussions with T. X. Hoang and his help. The correspondence with K. W. Plaxco and especially his gift of reference [9] before publication are also appreciated. A. Sienkiewicz helped polish the manuscript. This work was funded by KBN (grant 2 P03B 032 25).
[14] O. V. Galzitskaya, S. O. Garbuzynskiy, D. N. Ivankov, and A. V. Finkelstein, Proteins 51, 162-166 (2003).  In the left-hand panels, the N scale is logarithmic and the corresponding power law exponents are indicated. In, the right-hand panels, the N scale is linear and the corresponding correlation lengths are indicated. The PDB codes of the α-proteins studied are 1ce4, 1bba, 2pdd, 1bw6, 1rpo, 1hp8, 1ail, 2abd, 1imq, 1lmb, 1ycc, 1hrc, 256b, 1f63. The codes of the β-proteins are: 1cbh, 1ixa, 1ed7, 1bq9, 1efn, 2cdx, 1csp, 2ait, 1bdo, 1tit, 1ten, 1wit, 1who, 6pcy, 1ksr, 4fgf. The correlation levels of the power law (exponential) fits are 0.978 and 0.956 (0.960 and 0.971) for the α and β proteins respectively. Fig. 3. The dependence of t f old on the relative contact order for the α-(triangles) and βproteins (circles). The dotted line separates the values of CO that were found for the αfrom those found for the β-proteins. The filled symbols correspond to three selected proteins 1rpo (circles), 1csp (squares), and 1efn (stars) in which extra contacts were added or some contacts were subtracted. The families of such systems are connected by lines. The results corresponding to the true native contact maps are shown by the larger symbols. Within each family, t f old at an individually determined optimal temperature is displayed. If a fixed temperature is used instead (the one which is optimal for the true native contact map) the plots would look similar in character.  shows compilation of the experimental results, based on the data from [3]. The relative contact order CO P is calculated somewhat differently than CO in that it involves nonhydrogen atoms in a distance less than a cutoff value of 6Å as discussed further in ref. [8]. CO involves only the C α atoms but existence of a contact is based on the atomic overlap. The inset shows the two-state protein data compiled by Galzitskaya et al. [14] and plotted vs. the relative contact order parameter, CO G , as calculated by them -usually it coincides with CO P . Fig. 6 The dependence on the helical content parameter h defined as the ratio of the number of amino acids that belong to α-helices to the total number of amino acids in a protein. The top two panels shows results of the model calculations for s=1 as obtained in Ref. [8] for the α − β and α proteins respectively. The proteins considered are those listed in this reference. The bottom panel on the right shows the dependence of CO on h for the same proteins. The bottom panel on the left plots the experimental data points [14] for a different set of proteins (there is a substantial overlap with the set considered in the simulations).