Temperature evolution of Trp-cage folding pathways: An analysis by dividing the probability flux field into stream tubes

Abstract

Owing to its small size and very fast folding rate, the Trp-cage miniprotein has become a benchmark system to study protein folding. Two folding pathways were found to be characteristic of this protein: pathway I, in which the hydrophobic collapse precedes the formation of α-helix, and pathway II, in which the events occur in the reverse order. At the same time, the relative contribution of these pathways at different temperatures as well as the nature of transition from one pathway to the other remain unclear. To gain insight into this issue, we employ a recently proposed hydrodynamic description of protein folding, in which the process of folding is considered as a motion of a “folding fluid” (Chekmarev et al., Phys. Rev. Lett. 100(1), 018107 2008). Using molecular dynamics simulations, we determine the field of probability fluxes of transitions in a space of collective variables and divide it into stream tubes. Each tube contains a definite fraction of the total folding flow and can be associated with a certain pathway. Specifically, three temperatures were considered, T = 285K, T = 315K, and T = 325K. We have found that as the temperature increases, the contribution of pathway I, which is approximately 90% of the total folding flow at T = 285K, decreases to approximately 10% at T = 325K, i.e., pathway II becomes dominant. At T = 315K, both pathways contribute approximately equally. All these temperatures are found below the calculated melting point, which suggests that the Trp-cage folding mechanism is determined by kinetic factors rather than thermodynamics.

Electronic supplementary material

The online version of this article (10.1007/s10867-017-9470-7) contains supplementary material, which is available to authorized users.

Introduction

Trp-cage, a designed 20-residue miniprotein [1], has become a benchmark system to study protein folding due to its small size and very short time of folding (4.1μ s at temperature T ≈ 300K [2]). It consists of an α-helix, a 3₁₀-helix, and a polyproline II (PPII) helix, which form a hydrophobic core (“cage”) with the tryptophan buried in the center. The Trp-cage folding thermodynamics and kinetics have been studied in many details both by experiment [1–14] and simulation [15–41]. In the experimental studies, various techniques were employed—nuclear magnetic resonance and circular dichroism [1, 5, 6, 8, 9, 12–14], laser temperature-jump [2, 4, 7], and fluorescence correlation and time-resolved vibrational spectroscopies [3, 10, 11]. Mostly, the general character of folding kinetics was studied. In some works [2, 6, 13, 14], the folding kinetics were found to be two-state, whereas in some others [3, 4, 8, 9, 11, 12] not two-state. Several works indicated the presence of well-defined on-pathway intermediates [3, 4, 7, 8, 11, 12, 14] in contrast to the suggestion that the folding is downhill [9, 10]. Also, it was found that either the hydrophobic collapse precedes the formation of α-helix [4, 5] or the α-helix is formed first [7, 11, 13]. The simulations were performed using a “direct” molecular dynamics (MD) [15, 18, 19, 24, 28, 30, 36–38, 40] and replica-exchange molecular dynamics (REMD) [16, 17, 20, 21, 23, 25–27, 33, 39], both in explicit [17, 20–24, 27, 29, 30, 32–34, 36–39] and implicit [15, 16, 18, 25, 28, 35, 40] solvents. To analyze the simulation results, various techniques were employed such as the transition path [20] and interface [33] sampling methods, the transition path theory [25, 29–31], the p-fold technique [25, 30], and a Markov state model analysis [29] of the ultralong MD trajectory [24]. The performed experimental [4, 5, 7, 11, 13] and simulation [18, 20–22, 25, 29, 30, 32–34, 37, 40] studies revealed that Trp-cage can fold through different pathways. Most characteristic among them are two pathways: in one pathway (I), the collapse of the hydrophobic core precedes the formation of the α-helix, and in the other (II), the events occur in the reverse order. At the same time, the contribution of these pathways to the process of folding as well as the nature of transition from one pathway to the other remain unclear. Some MD studies of Trp-cage folding [18, 20, 30, 40] (and also of its mutants [39]) suggest that pathway I is dominant (or even exclusive [35] as in experiment [4, 5]), whereas the others [19, 21, 29] indicate that pathway II prevails, similar to what was observed experimentally [7, 11, 13]. Characteristically, pathway I was observed in the simulations at room and nearby temperatures [18, 20, 30, 40], and pathway II appeared in the simulations at melting and higher temperatures [21, 29]. It should be noted here that the melting temperature in simulations is typically obtained much higher than the experimental value T _m ≈ 315K [1, 2], both in the simulations with an explicit solvent (440K [17], and 455K [21]) and with an implicit solvent (400K [16], 450K [25] and 468K [29]). In several recent simulations with modified force fields, reasonable values of the temperature were obtained (321K [23], 324K [26], and 329K [35]), but folding pathways were not analyzed.

The obtained results [18, 20, 21, 25, 29, 30, 40] suggest that the transition from pathway I to pathway II takes place somewhere between room and melting (or higher) temperatures. However, systematic studies of folding pathways in the whole temperature range are rare [21, 25]. In one work [21], the simulations were performed using an explicit solvent model, and the temperature was varied from 280K to 553.9K, with T _m ≈ 455K. As the authors indicated, pathway II was observed almost exclusively [21]. In the other work [25], an implicit solvent model was used, and the simulations were performed for the temperature range of 270–566K, with the folding temperature being around 450K. It was found that at 363K, i.e., below the melting temperature, pathway I dominates, and at 465K, i.e., slightly above the melting temperature, the contribution of pathway II becomes comparable to the contribution of pathway I. Therefore, the situation with folding pathways remains ambiguous. First, there is no agreement between these studies, and, second, if the transition from pathway I to pathway II occurs as the temperature increases [25], which seems plausible according to the all other works, it is not clear at which temperature it takes place, because folding pathways at intermediate temperatures were not considered there. If this temperature is close to the melting temperature, the transition from pathway I to pathway II can be associated with thermodynamic factors, but if it is far below the melting temperature, then kinetic factors are likely responsible for this transition. In other words, the nature of the transition between pathways I and II remains unclear.

In the present paper, we employ a conceptually different approach [42], which is based on the analogy between the process of protein folding and fluid motion [43]. More specifically, using the simulated folding trajectories, the probability fluxes of transitions between protein states in a reduced conformational space are calculated. These fluxes are then considered as the fluxes of a “folding fluid”, whose density is proportional to the probability for the protein to be in the current state. Given the space distribution of the fluxes, the flow streamlines are determined that separate different folding pathways.

We consider the process of the “first-passage folding”, which is of particular interest because it mimics physiological conditions [44]. In this case, folding trajectories are initiated in unfolded states of the protein and terminated upon reaching the native state. The process of folding can then be viewed as a steady flow of folding fluid from an unfolded state to the native state. Accordingly, the field of probability fluxes can be divided into stream tubes separated by streamlines. Since each stream tube contains a certain fraction of the total folding flow, this allows us to estimate the contribution of different folding pathways. To have meaningful statistics on folding pathways at a reasonable computational cost (100 folding trajectories were run for each temperature), the MD simulations were performed with an implicit solvent model; for this, the CHARMM program [45] was employed. We perform a systematic study of Trp-cage folding in a temperature range of 285–325K, where pathway I (90% of folding trajectories at T = 285K) transforms to pathway II (90% of folding trajectories at T = 325K). At T = 315K, both pathways contribute approximately equally. This temperature range is found to be below the calculated melting point (≈ 420K), which suggests that the Trp-cage folding mechanism is determined by kinetic factors rather than thermodynamics.

The paper is organized as follows. Section 2 describes the system we study and gives a brief survey of the methods we used to simulate and analyze the process of Trp-cage folding (for more details, see [40]). Section 3 presents the results and their discussion, and Section 4 contains concluding remarks.

System and methods

Trp-cage is a 20-residue miniprotein (1L2Y.pdb), which has the sequence NLYIQ WLKDG GPSSG RPPPS in the TC5b variant [1]. It consists of a N-terminal α-helix (residues 2-8), a 3₁₀-helix (residues 11-14), and a C-terminal polyproline II helix (residue 17-19), which form a hydrophobic core with the Trp6 buried in the center (Fig. 1). The simulation were performed using the CHARMM program (version 35b3) [45]. More specifically, all heavy atoms and the hydrogen atoms bound to nitrogen or oxygen atoms were considered explicitly; PARAM19 force field [46] and a default cutoff of 7.5Å for the nonbonding interactions were used. To describe the main effects of the aqueous solvent, a meanfield approximation based on the solvent-accessible surface-area (SASA) [47] was employed. To control the temperature, the Berendsen thermostat with a coupling constant of 5 ps was used. The SHAKE algorithm [48] was applied to fix the length of the covalent bonds involving hydrogen atoms, which allowed the integration time step of 2 fs. The atomic coordinates (“frames”) were saved every 20 ps.

Fig. 1.

(Color online) The native structure of the Trp-cage miniprotein (1L2Y.pdb) in a ribbon representation. The Trp6 residue is shown in blue sticks

To see the applicability of the present approach to the simulation of Trp-cage folding, we have performed three different tests. First, we started the MD trajectory in the native state of Trp-cage (the first NMR structure in the 1L2Y.pdb file) and run it at T = 300K for 400ns, i.e., for the time that is one order of magnitude larger than the mean first-passage time (MFPT) equal to 36ns [40] (Supporting Information, Fig. S1). It was found that the protein periodically visited the native-like states (with the C _α RMSD less than 2.2Å) and spent there approximately one-fifth of the simulation time. It follows that the native state of Trp-cage in the framework of the SASA implicit solvent model [47] is marginally stable, which is in line with the notion of the native state from the viewpoint of protein functioning [49–52]. Another substantial test of the reliability of the model was to calculate the variation of the fraction of native contacts with temperature. As has been shown [53], the reliable models are characterized by the fraction of native contacts at room temperatures not less than ≈ 0.7, in agreement with experiment. We assumed a native contact to be formed if the distance between the C _α-atoms which are not neighbors in the sequence is less than 6.5Å, and this contact is present in most (≈ 87%) of the NMR structures [1]; this resulted in 35 native contacts. The calculations showed (Supporting Information, Fig. S2) that the dependence of the fraction of native contacts on temperature was similar to whose obtained for Trp-cage previously [16, 17, 21, 25, 26, 29, 35] and had the fraction of native contacts ≈ 0.74 at T = 300K. Finally, we performed limited simulations of the first-passage Trp-cage folding in explicit solvent using the CHARMM22/CMAP force field and the TIP3P model for water molecules [45] (11 folding trajectories were run) and compared the free energy surfaces (FESs) depending on the C _α RMSD and radius of gyration for the SASA and explicit solvent (Supporting Information, Fig. S3a and b, respectively). Both surfaces were found to be similar, with the basins for native-like states being in close proximity to the native state in agreement with previous studies [17, 20]; in the present case they lay between approximately 2Å to 3Å in the C _α RMSD from the native state. According to these tests, the results for Trp-cage folding obtained with the SASA implicit solvent model [47] are expected to be reasonable. Regarding the applicability of the present approach, it is also worthy to mention one known shortcoming of implicit solvent models. Such models typically overestimate the folding rates because the friction of protein atoms against the solvent is absent. However, the relative rates of formation of secondary structural elements are obtained comparable to the values observed experimentally - α-helices and β-hairpins fold approximately 100 times faster [54] than those in experiment [55]. Therefore, the characteristic times obtained in the simulations with an implicit solvent can be corrected by multiplying them by ∼ 100 times; in particular, for Trp-cage at T = 300K, this leads to a good agreement of the calculated MFPT (36ns [40]) with the experimental time (4.1μ s [2]).

The simulations were performed at three temperatures, specifically at T = 285K, T = 315K, and T = 325K. In each case, 100 folding trajectories were generated. The trajectories were initiated in unfolded states of the protein and terminated upon reaching the native state. The unfolded states were prepared using the standard CHARMM protocol [45]; i.e., an extended conformation of the protein was first minimized and then heated and equilibrated at the temperature of interest (for 5 × 10³ time steps).

Current protein conformations were characterized by the distances between the C _α-atoms that formed native contacts. The space of the distances was then transformed to a space of orthogonal collective variables using a principal component analysis (PCA) [56]. Figure 2 shows the corresponding spectra of the eigenvalues for T = 285K, T = 315K, and T = 325K. For each temperature, the first three modes are well separated from the others, with the second and third modes being close to each other and considerably smaller than the first mode. Therefore, to determine a two-dimensional space of collective variables g = (g ₁, g ₂), the variable g ₁ was chosen as the first eigenvector, and the variable g ₂ was determined as a sum of the second and third eigenvectors weighted according to their eigenvalues. Since the PCA variables are linear combinations of the original variables, g ₁ and g ₂ are measured in the same units as the distances, i.e., in angstroms. One advantage of the PCA variables over the other frequently used collective variables, e.g., the fraction of native contacts and radius of gyration [17, 20], is that the PCA variables are independent.

Fig. 2.

Eigenvalue spectra: T = 285K (triangles-up), T = 315K (circles), and T = 325K (triangles-down)

Using the simulated folding trajectories, there were constructed the FESs, kinetics networks, fields of the probability fluxes and streamlines of the folding flows.

The local free energy was calculated as F(g ₁, g ₂) = −k _B T lnp(g ₁, g ₂), where p(g ₁, g ₂) is the probability for the system to be at the (g ₁, g ₂) point, and k _B is the Boltzmann constant. Since the thermodynamic relations are of limited application in the case of first-passage folding, the correspond FESs should be considered as quasi-equilibrium ones, or, more practically, as the surfaces that characterize the residence probability.

To construct kinetic networks, the representative points of the protein states in the g space were initially clustered using the MCLUST method [57], in which the collection of points is approximated by a set of multidimensional (in our case 2D) Gaussian functions. Since some initial clusters fell into the same basins on the FES, all initial clusters were grouped into “consolidated clusters”, taking into account the proximity of the initial clusters in the (g ₁, g ₂) space and their kinetic connectivity [40]. To determine cluster connectivity, free energy disconnectivity graphs [58] were used, which separated the initial clusters into consolidated clusters such that the free energy barriers between the consolidated clusters were considerably higher than the barriers between the initial clusters within the consolidated clusters (Supporting Information, Figs. S4, S5, and S6). We note that in few cases, the proximity of initial clusters in the (g ₁, g ₂) space and their kinetic connectivity came into conflict, so that one initial cluster, which should be related to a certain consolidated cluster due to connectivity was found relatively distant in the (g ₁, g ₂) space from the latter, which introduced some uncertainty into the distribution of the initial clusters among the consolidated ones (cf. the positions of the initial clusters in Tables SI, SII, and SIII of the Supporting Information with the corresponding disconnectivity graphs of Figs. S4–S6). However, since this happened only for clusters with low weight (∼ 1%), the overall picture did not change essentially. Both initial and consolidated clusters were also provided with characteristic protein conformations, which were chosen as ones that had most frequent secondary structure strings encoded with the DSSP alphabet [59] in these clusters; the program WORDOM [60] was used to perform the analysis. The corresponding secondary structure states were determined with the Avogadro program [61].

The probability fluxes were calculated according to the hydrodynamic description of protein folding [42]. The g ₁-component of the flux j ( g ) was calculated as

$$\begin{array}{lll}{j}_{g_1}\left(\mathbf{\text{g}}\right)=\left[\sum _{{\mathbf{\text{g}}}^{\prime },{\mathbf{\text{g}}}^{\mathrm{\prime \prime }}(\mathbf{\text{g}}\subset {\mathbf{\text{g}}}^{\ast })}^{g_1^{\mathrm{\prime \prime }}-g_1^{\prime }>0}n\right({\mathbf{\text{g}}}^{\mathrm{\prime \prime }},{\mathbf{\text{g}}}^{\prime })& & \\ -\sum _{{\mathbf{\text{g}}}^{\prime },{\mathbf{\text{g}}}^{\mathrm{\prime \prime }}(\mathbf{\text{g}}\subset {\mathbf{\text{g}}}^{\ast })}^{g_1^{\mathrm{\prime \prime }}-g_1^{\prime }<0}n({\mathbf{\text{g}}}^{\mathrm{\prime \prime }},{\mathbf{\text{g}}}^{\prime })]/(M{\overline{t}}_{\mathrm{f}}\mathrm{\Delta }{g}_2)& & \end{array}$$ 1

where M is the total number of simulated trajectories, ${\overline{t}}_{\mathrm{f}}$ is the MFPT, n(g″, g′) is the number of transitions from state g′ to g″, and g ⊂ g ^∗ is a symbolic designation of the condition that the transitions included in the sum have the straight line connecting points g′ to g″, which crosses the line g ₁ = const within the segment of the length of Δg ₂ centered at the point g. The g ₂-component of j(g) is determined in a similar way, except that one selects the transitions crossing the line g ₂ = const within the current segment Δg ₁. The calculations were performed on a grid with discretization Δg ₁ = Δg ₂ = 1Å. Given j(g), a stream function of the flow [43] was calculated

$$\mathrm{\pi }(g_1,g_2)={\int }_0^{g_2}{j}_{g_1}(g_1,g_2^{\prime })d{g}_2^{\prime }$$ 2

The equality π(g ₁, g ₂) = const determines a streamline of the flow, which is tangent to the flux vectors j(g). Correspondingly, two streamlines π(g ₁, g ₂) = C ₁ and π(g ₁, g ₂) = C ₂, which satisfy the condition C ₂ > C ₁, create a stream tube that contains the (C ₂ − C ₁)/G fraction of the total folding flow G.

We note that the flow field can also be divided into stream tubes by grouping the folding pathways on the basis of the similarity between the structures along two neighboring pathways [25, 29]. An essential difference between the present approach and the latter studies is that in those studies the pathways were based on the folding fluxes which advanced p-fold (the committor probability) values. At the same time, the local fluxes of transitions are not necessarily directed to the folded state of the protein [42, 44, 62, 63], in particular, they can form flow vortices [40, 42, 64] (see also Figs. 5c, d, 6c, d and 7c, d below).

Fig. 5.

(Color online) T = 285K: (a) consolidated clusters of conformations (red circles) superimposed on the FES; (b) kinetic network; (c) streamlines of the folding flows superimposed on the FES; (d) streamlines of the folding flow superimposed on the vorticity distribution of the folding flows. The figures at or within the circles in panels (a) to (d) denote cluster numbers according to Tables 1 and 2, and the numbers at the streamlines in panels (c) and (d) indicate the fractions of the total flow restricted by these streamlines. The cluster sizes (areas of the circles) in panels (a) and (b) are proportional to the cluster weights in Table 1 (except for cluster 1, which is too small), although the absolute sizes of the clusters in these panels are taken different for illustration purpose

Fig. 6.

(Color online) T = 315K: All notations are similar to those in the caption to Fig. 5, except that the cluster characteristics are given in Tables 3 and 4

Fig. 7.

(Color online) T = 325K: All notations are similar to those in the caption to Fig. 5, except that the cluster characteristics are given in Tables 5 and 6

Results and discussion

We started with a temperature slightly below room temperature (T = 285K) and increased it until the scenario of Trp-cage folding would be completely changed. As a result, three temperature were studied, specifically, T = 285K, T = 315K, and T = 325K. To relate these temperatures to the melting temperature, the temperature-dependent heat capacity was calculated as C _V = (〈E ²〉 − 〈E〉²)/k _B T ² [65], where E is the protein energy, and angle brackets denote the ensemble averages. For each temperature, an “equilibrium” (7μ s) MD trajectory was run, during which the protein experienced many folding/unfolding events. The resulting heat capacity curve (Fig. 3) indicates that the melting is gradual with no pronounced pre-melting effects, which is in agreement with the calculation of the temperature-dependent fraction of native contacts, both in previous works [16, 17, 21, 25, 29] and in our calculations (Supporting Information, Fig. S2). According to Fig. 3, the melting temperature is ≈ 420K, i.e., it is much higher than the experimental value T _m ≈ 315K [1, 2], similar to what was often found in previous studies [16, 17, 21, 25, 29]. We note that the midpoint of the temperature-dependent fraction of native contacts gives the same value of melting temperature (Supporting Information, Fig. S2). The primary conclusion we derive from this consideration is that all three selected temperatures (T = 285K, T = 315K and T = 325K) lie below the melting point.

Fig. 3.

Melting curve. The heat capacity, C _V, is measured in kcal/(mol ⋅ K)

Figure 4 shows the distributions of the first-passage times for the given temperatures, in the form of survival probabilities. The distributions are approximately exponential, which indicates two-state kinetics, such as in Refs. [2, 6, 13–15, 17, 23, 25, 29, 40]. The corresponding values of the MFPT are ≈ 44ns (T = 285K), ≈ 37ns (T = 315K) and ≈ 45ns(T = 325K). With the above-mentioned correction in the rates of formation of secondary structural elements (Section 2), i.e., that the simulations with implicit solvent overestimate the rates by approximately two orders of magnitude, the presents MFPTs are in good agreement with the experimental time of ≈ 4.1μ s [2].

Fig. 4.

Cumulative distributions (labels) and best-fits (lines) of the first-passage times. T = 285K (triangles-up, solid line), T = 315K (circles, dashed line) and T = 325K (triangles-down, dash-dotted line).

T = 285K

Figure 5 presents the results for T = 285K. Figure 5a shows the FES as well as the positions and relative sizes of the consolidated clusters of conformations (indicated by red circles). The numeration and characteristics of the consolidated clusters are given in Table 1 (the corresponding characteristics of the initial clusters and the disconnectivity graph, which was used to group these clusters in consolidated clusters, are presented in the Supporting Information, Table SI and Fig. S4, respectively). The areas of the circles representing the clusters are proportional to the cluster weights. In Table 1, there are also given the RMSDs of the conformations in the clusters, which characterize cluster diffusivity in the (g ₁, g ₂) space. The clusters are located in or close to the basins on the FES, with a small exception for cluster 5, for which the corresponding basin is very shallow and, to be seen, requires a considerably higher resolution of the FES. As follows from Table 2, which presents the numbers of inter- and intracluster transitions, the protein dwells for a long time in clusters 2 to 7, and for a short time in cluster 1. Below we will call such clusters as a “residence cluster” (R-cluster) and “transition cluster” (T-cluster), respectively.

Table 1.

Characteristics of consolidated clusters, T = 285K

Cluster	g 1	g 2	Weight	RMSD
1	12.8	9.4	0.1	5.8
2	30.0	13.7	5.3	1.7
3	28.0	18.6	4.6	1.6
4	29.1	25.3	20.5	2.6
5	35.5	19.7	10.8	2.4
6	41.6	19.0	27.2	1.3
7	39.0	26.9	31.5	1.7

Table 2.

Numbers of transitions between consolidated clusters, T = 285K (column - from, and line - to)

Cluster	Extended	1	2	3	4	5	6	7	Native
Extended	0	0	0	0	0	0	0	0	0
1	100	114	10	12	11	0	0	0	0
2	0	45	15039	1270	17	651	4	0	0
3	0	44	1278	10991	1497	827	0	0	0
4	0	44	19	1528	58909	2668	15	2557	0
5	0	0	671	834	2669	25726	2539	2122	0
6	0	0	9	1	16	2515	81229	3312	0
7	0	0	0	1	2621	2174	3295	92659	0
Native	0	0	0	0	0	0	0	100	0

Figure 5b depicts the corresponding kinetic network. The arrows connecting the clusters represent the transitions between the clusters, with the widths of the lines being proportional to the net number of transitions. At each cluster, a protein conformation characteristic of this cluster is shown. To make the kinetic network more transparent, only the arrows for ten and larger number of net transitions are shown. In particular, they do not include the transitions that were observed only in a few percentage of folding trajectories, which are negligible with respect to the other transitions that occurred many times per trajectory (Table 2). The network indicates that there are two parallel folding pathways: one (A) is presented by the sequence of clusters U → 1 → 3 → 4 (1 → 4) → 7 → N, and the other (B) by the sequence U → 1 → 2 → 5 → 7 → N, where U and N stand for the unfolded and native states, respectively. According to the characteristic protein conformations in these clusters, both pathways can be related to pathway I, in which the collapse of the hydrophobic core precedes the formation of α-helix. The α-helix appears in cluster 6, which presents a deviation of the protein from the general pathway (cluster 7) in the attempt to form the 3₁₀-helix. The dominance of pathway I is well supported by the folding streamlines projected on the FES (Fig. 5c). To separate pathways I and II, there was chosen a streamline that leaves clusters 1, 3, 4, 5, and 7 on one side of the FES, and clusters 2 and 6 on the other side (it is shown in red color). This streamline corresponds to the fraction of the total flow equal to 0.1, which indicates that ≈ 90% of the folding flow follow pathway I. We note that the selection of the streamline to separate pathways I and II is not precise (with the uncertainty ≈ 0.05 fraction of the total flow), mostly because of a finite size of the clusters (see the RMSDs in Table 1). This value of the fraction of the total flow for pathway I is in good agreement with the results of previous studies for T = 300K [18, 20, 30, 40]. Figure 5d also shows the projection of the streamlines on the distribution of the flow vorticity $\omega (g_1,g_2)=\partial j_{g_2}/\partial g_1-\partial j_{g_1}/\partial g_2$, where ω < 0 and ω > 0 correspond to clockwise and anticlockwise motion, respectively. It is seen that the flow field is filled with a number of alternate clockwise and anticlockwise vortices of small size. One general feature of the overall vorticity distribution is that the vorticity is generally clockwise in the lower part of the flow field and anticlockwise in the upper part, because the folding flow weakens toward the periphery of the flow field [66]; it is most characteristic of the initial stage of folding, where the flow is well directed to the native state. One specific element of the present vorticity field is the clockwise vortex in the center of the triangle formed by clusters 7, 6 and 5, which corresponds to the deviation of the protein from the general pathway in the kinetic network of Fig. 5b. In general, the behavior of the protein at the present temperature is very similar to that at T = 300K [40]: two-state folding kinetics, a very close value of the MFPT, a similar FES landscape and disposition of the clusters on the FES, a similar kinetic network and alignment of streamlines, and, eventually, the same percentage of contribution of pathway I to the total folding flow (≈ 90%).

T = 315K

As the temperature increases, the overall picture of the Trp-cage folding changes (Fig. 6a-d, Tables 3 and 4, and Table SII and Fig. S5 of the Supporting Information). Only two basins that are close to the native state are retained, to which R-clusters 4 and 5 in Fig. 6a belong, similar to R-clusters 6 and 7 in Fig. 5a, respectively. Also, instead of three R-clusters (2, 3, and 4) at T = 285K, clusters 2 and 3 appear, which are positioned similar to clusters 2 and 4 at T = 285K, but they are rather T-clusters than R-clusters (Table 4). The kinetic network, presented in Fig. 6b, suggests that at the given temperature, the pathway presented by the sequence of clusters U → 1 → 3 → 2 → 5 → N becomes significant, in which the α-helix is formed first (cluster 3), and then the collapse of the hydrophobic core occurs through the transitions 3 → 2 → 5 → N. Thus, as the temperature increases from T = 285K to T = 315K, the transition from pathway I to pathway II takes place. The streamlines projected on the FES (Fig. 6c) suggest that in consistency with the kinetic network, pathways I and II can be separated by the streamline corresponding to the 0.5 fraction of the total flow (the red colored curve), although not exactly because cluster 2 is diffusive (Table 3). It is significant that the transition from cluster 3 to 2 is not performed by a “direct hop”, as might be thought from the kinetic network (Fig. 6b), but involves a complex forward-backward motion in conformational space.

Table 3.

Characteristics of consolidated clusters, T = 315K

Cluster	g 1	g 2	Weight	RMSD
1	25.5	12.1	0.1	10.3
2	56.2	24.3	25.2	3.4
3	56.5	16.6	6.8	2.6
4	65.9	18.0	29.3	2.2
5	64.9	26.5	38.6	2.1

Table 4.

Numbers of transitions between consolidated clusters, T = 315K (column - from, and line - to)

Cluster	Extended	1	2	3	4	5	Native
Extended	0	0	0	0	0	0	0
1	100	90	43	34	10	2	0
2	0	82	39172	1509	1815	10534	0
3	0	79	1465	8707	3876	180	0
4	0	20	1821	3896	52700	3447	0
5	0	8	10611	161	3483	67271	0
Native	0	0	0	0	0	100	0

T = 325K

A further increase of the temperature makes pathway II dominant (Fig. 7a-d, Tables 5 and 6, and Table SIII and Fig. S6 of the Supporting Information). More specifically, as can be seen from Fig. 7b, the protein mostly follows the sequence of consolidated clusters U → 1 → 3 → 4 → N, in which the formation of the α-helix (cluster 3) precedes the hydrophobic collapse (transition from cluster 2 to 4). The projection of the streamlines on the FES suggests the pathways I and II to be separated by the streamline corresponding to the fraction of the total flow equal to 0.9 (the red colored curve in Fig. 7c). Thus, approximately 90% of the folding flow follow pathway II, instead of ≈ 10% at T = 285K, making pathway II dominant. This is in line with Refs. [21, 29], where the Trp-cage folding was studied at elevated temperatures. The change of the order in which the hydrophobic collapse and formation of α-helix occur with increasing temperature may have an entropic origin. As the temperature becomes higher, the hydrogen bonds responsible for the formation of α-helix weaken while the strength of hydrophobic interaction becomes stronger [67, 68]. Accordingly, the fraction of conformational space associated with the interhelix contacts increases, and the fraction of space associated with the formation of tertiary contacts decreases. Therefore at higher temperatures, taking into account that the protein was initially equilibrated at the given temperature (Section 2), statistically it is more probable that the folding trajectory will start with the formation of α-helix.

Table 5.

Characteristics of consolidated clusters, T = 325K

Cluster	g 1	g 2	Weight	RMSD
1	33.4	12.7	0.2	9.3
2	58.0	18.9	24.6	3.8
3	66.0	13.3	35.1	3.0
4	66.9	21.6	40.1	2.4

Table 6.

Numbers of transitions between consolidated clusters, T = 325K (column - from, and line - to)

Cluster	Extended	1	2	3	4	Native
Extended	0	0	0	0	0	0
1	100	181	240	87	14	0
2	0	269	44076	8716	13379	0
3	0	159	8699	80571	5071	0
4	0	13	13423	5126	89444	0
Native	0	0	2	0	98	0

Figure 7d also reveals that two opposite directed vortices are formed at the basins corresponding to clusters 3 and 4: a relatively small anticlockwise vortex (positive vorticity) at the basin for cluster 3, and a large clockwise vortex (negative vorticity) at the basin at cluster 4. Figure 8 shows sequences of characteristic protein conformations in these vortices that are based on the transitions between the initial (unconsolidated) clusters (Table SIII and Fig. S6 of the Supporting Information). The present vortices can be considered as intermediates, particularly because they are related to catchment basins on the FES. According to the kinetic network and the fact that the folding time distribution is essentially single-exponential (Fig. 4), these vortices can be treated as on-pathway intermediates.

Fig. 8.

(Color online) Sequences of protein conformations corresponding to two flow vortices in Fig. 7d: (a) at cluster 3 (positive vorticity, anticlockwise motion), and (b) at cluster 4 (negative vorticity, clockwise motion). The numbers beside the conformations denote the initial cluster numbers (Table SIII of the Supporting Information)

Conclusions

In the present paper, we have studied how the scenario of Trp-cage miniprotein folding changes with temperature. The case of the “first-passage folding” was under consideration, when the folding trajectories were initiated in unfolded states of the protein and terminated upon reaching the native state. Three temperatures have been considered, specifically, T = 285K, T = 315K and T = 325K. To have a statistically significant ensemble of folding trajectories at a reasonable computational cost (100 folding trajectories were simulated at each temperature), a meanfield approximation was used to describe the main effects of the aqueous solvent (the SASA model [47]). The MD simulations were performed with the CHARMM program [45]. To determine folding pathways, we employed the hydrodynamic description of protein folding [42]. Calculating the probability fluxes of transitions between protein states, the streamlines of folding flows were constructed, which divided the flow field into stream tubes. Each stream tube contained a definite fraction of the total folding flow and was associated with a certain pathway (I or II) on the basis of the kinetic network, which described the transitions between the clusters of characteristic protein conformations. As a result, we were able to determine the fractions of pathways I and II and see how they change with temperature.

We have found that as the temperature increases, the pathway I, in which the hydrophobic collapse precedes the formation of the α-helix, gradually transforms into pathway II, in which the events occur in the reverse order, i.e., the α-helix is initially formed, which is followed by the collapse of the hydrophobic core. Such a transition from pathway I to II at elevated temperature is in general agreement with previous experimental [4, 5, 7, 11, 13] and simulation [18–21, 29, 30, 40] studies in that the pathway I was found to be dominant at lower temperatures [4, 5, 18, 20, 30, 40], and pathway II was mostly observed at higher temperatures [7, 11, 13, 19, 21, 29], although the transition from pathway I to II itself was not examined. Specifically, we have found that at T = 285K, approximately 90% of the total flow follow pathway I, similar to what was previously found at T = 300K [18, 20, 30, 40]. At T = 315K, the fraction of the flow through pathway I decreases to ≈ 50%, i.e., pathways I and II contribute approximately equally. Finally, at T = 325K, the pathway II becomes dominant (≈ 90%), which is in line with the studies of Trp-cage folding at elevated temperatures [21, 29]. Such folding pathways are in agreement with the melting curves and folding kinetics in that the process of Trp-cage folding is a cooperative process [69, 70]. This is supported by the calculation of the times of formation of the interhelix and tertiary contacts, which are rather close to each other at all temperatures (Supporting Information, Fig. S7a–c). We have also found that the folding flow field is filled with small- and large-scale vortices, which reflect the events of partial folding and unfolding of the protein. Due to these vortices, the streamlines are not well directed toward the native state but can even turn in the opposite direction. Accordingly, the stream tubes representing the pathways vary in width not monotonically but may experience very abrupt changes.

According to the obtained results, T = 315K can be considered as a characteristic temperature at which the mechanism of Trp-cage folding changes. The calculated melting curves show that this temperature is far below the calculated melting temperature (≈ 420K), and no pronounced pre-melting effects are observed at this temperature. This suggests that the mechanism of Trp-cage folding, and the transition between pathways I and II in particular, is determined by kinetic factors rather than thermodynamics. Although the approach we used to simulate Trp-cage folding does not provide the correct value of the melting temperature, in all other aspects the results are consistent among themselves and are mostly in agreement with the previous results. We thus infer that the division of the probability flux field into stream tubes on the basis of hydrodynamic description of folding dynamics [42] can be successfully used to discriminate folding pathways.

Electronic supplementary material

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Compliance with Ethical Standards

This work was performed under a grant from the Russian Science Foundation (No. 14-14-00325). The authors declare that they have no conflicts of interest.