Although the expansion in the RED required a complete set of basis functions in principle, it is usually not necessary to include too many basis functions in practice. Usually, it is easy to choose the initial configurations so that its distribution P_init+ (q) is already local equilibrium, thus only slow processes need to be reweighted by the RED. Therefore Ω (q) in the all-atomic q space can be replaced by Ω (x) = P_eq(x)/[P_init+ (x)] in a coarse-grained x space, where x represents the slow (coarse-grained) degrees of freedom, P(x) is the corresponding distribution function in the x space. For example, in proteins, if we disregard the first few nanoseconds of relaxation simulation and reset the zero point of time, it is safe to suppose only some slow degrees of freedom, such as torsion angles of backbone, positions of side-chain atomic groups, be x, thus only functions in the x space should be applied as basis functions. In many cases, systems might reach equilibrium within the τ time (length of individual trajectory) except for only a few collective variables, thus we even apply the RED in a low-dimensional space. In addition, we usually suppose there are many potential-energy basins (or super-basins) separated by high (free) energy barriers in the conformational space; each (super-)basin corresponds to a metastable state wherein a τ -length trajectory is sufficient to reach local equilibrium. Thus, Ω (x) looks like a step-like function, which is almost constant in each metastable state, variations only happening at boundaries between states. We approximately have

where Θ _α (x) is character function of the α metastable state, which is unity if x is inside the state, zero otherwise. The set of character functions of states {Θ _α (x)} provides a natural set of basis functions to expand Ω (x), thus the required number of basis functions is equal to the number of states. Since the boundaries of these states are unknown a priori, we usually use far more basis functions to linearly combine the step-like Ω (x) function. We may use some usual collective variables with clear physical meaning as basis functions. In biological macromolecules, dihedral angles and their analytical functions, pair distances between some key atomic groups, potential energy or its parts, solved area of proteins, radius of gyration, native contact number, number of hydrogen, etc., can be applied as basis functions. The variance– covariance matrix g^{μ ν} will ensure the consistent consideration of different classes of basis functions, and provide the measure of their relative importance by the orthonormalization process. In addition, the number of independent basis functions is also limited by the finite size of init+ , thus more selected basis functions will be discarded after considering the matrix g_{μ ν} , then do not significantly contribute to final results. Another possible scheme about the selection of basis functions is to use local functions in the conformational space. For example, for a set of sampled configurations, {x_i}, i = 1, … , M, one may group them into some subsets by clustering algorithms based on their geometric similarity, and use the character functions of clusters as basis functions. Thus the number of basis functions is the number of clusters, n_cl = M/m. Here M and m are the total number of sampled conformations and the (average) number of conformations in each cluster. The idea is actually already applied in the Markov State Model (MSM) to divide the whole conformational space into metastable states.^{[24, 30– 34]} In this paper, we use physically meaningful collective variables as basis functions, the application of the sample-based discrete cluster functions will appear elsewhere.

The completeness of selected basis functions can be directly test-based on sampled configurations in the RED. For a set of basis functions {A^μ (x)}, we can estimate the difference between two samples (or the corresponding distributions), such as the initial distribution P_init+ (x) and the equilibrium distribution P_eq(x),

The definition gives a measurement about the similarity of samples, and can be generally applied to compare two samples. For example, we had used the formula to optimize coarse-grained models of the all-atomic water.^[37] For assessing the completeness of basis functions in the expansion of P_eq(x)/[P_init+ (x)], a reasonable method is to check the convergence of while adding more and more basis functions.

To calculate the sample for equilibrium distribution P_eq(x), thus the trajectory weights {w_i}, are necessary. We propose two methods to bypass this difficulty. First, before reweighting, P_eq(x) is replaced by and we roughly select basis functions. Second, after reweighting, can be calculated with the resulting trajectory weights. This procedure can be used to test the former selection. If the rigorously calculated has reached its saturation region with previously roughly selected basis functions, the results should be satisfiable. However, this posterior examination is restricted to the cases that the trajectory weights can be uniquely determined. In practice, the distribution , has the same local equilibrium characteristics in each metastable region as P_init+ and P_eq, thus is similarly a step function, which is also constant in each meta-stable region, only the constant values are different from those of Ω _{init+ , eq}. Thus we believe the a priori test is already enough, and the following results are all calculated in this way.

Since the statistical errors exist due to the finite sample sizes, S² would not strictly stop growing with the number of basis functions, it just increases slowly compared to the initial growing phase with increasing basis functions. Similarly, when a large error occurs, S² may abnormally increase within the saturation region. We design a scheme to get rid of the statistically unreliable basis functions. We orthonormalize the functions {A^μ (x)}, leading to the explicitly identifiable value and error contribution from the orthonormalized basis functions to S². In the present paper, we implement the Gram– Schmidt orthonormalization method to one-by-one derive out the set of orthonormalized basis functions {Â ^μ (x)}, thus To estimate the error of S², we approximately estimate the error of s_μ , and combine them by

Here, Err( ) denotes the statistical error of the quantity in brackets. For each orthonormalized basis function, Â ^μ , if Err(s_μ ) is larger than | s_μ | , i.e., the relative error of s_μ is larger than 1.0, this basis function will not be selected in expansion. One-by-one selection will give out the final basis function set and the estimated error of the completeness measure S².

The scheme introduced here provides us with the knowledge about how many basis functions are enough in the RED. After orthonormalization, the contribution of the basis functions to S² clearly reflect their relative importance in trajectory reweighting. The basis functions that contribute more are more important for portraying the difference between P_init+ (x) and P_eq(x), and the ones that contribute quite little can be linearly expanded by other existing basis functions. Furthermore, various kinds of basis functions can be safely added to investigate the related phenomena. For illustration, we will show that the basis functions of dihedral angles and those of distances between heavy atoms can be consistently treated in calculation, and the effect of a solvent on the conformation of biological macromolecules can be studied by additional basis functions characterising the solute– solvent relationship.

There is only one free parameter in the RED, for collecting the init+ samples in Eq.  (5). Large may bring systematical errors, and small will reduce the sample number in init+ , leading to a larger statistical uncertainty. is usually applied in the RED with satisfactory results obtained. We also point out that, for the purpose of discovering the states in conformational space, the results are not sensitive to

The variable t̃ actually reflects the timescale of the process that we are interested in. If it were set to the single trajectory length τ , then the trajectory weights solved by Eq.  (6) will be equal to each other. In this case, we are focusing on the processes with a timescale much longer than τ , thus all the trajectories are still in the relaxation to local equilibrium, and should have the same weight. Usually, if a system can be divided into several metastable states under the focused timescale, t̃ should be selected comparable to or smaller than the smallest life time of these states. Given that the kinetic states of the system are frequently unknown, t̃ should be selected relatively short. In practice, it is possible to use shorter and shorter t̃ and check the convergence of the obtained free energy surface. Thus, a proper t̃ may be found as a compromise between correctness and statistical reliability (small fluctuation of trajectory weights). Another practical consideration is that a few percent of τ would be the safe choice for t̃ . This is because the transition events with timescales prominently smaller than τ have reached equilibrium in a single trajectory (thus do not need reweighting), we only need to focus on the ones with time-scales similar to or slightly smaller than τ . We usually choose t̃ /τ in the interval of [0.01, 0.1], and the results are satisfactory. In this paper, we will explicitly show that the variable within the range is not sensitive to our results.

In TDQW system, the eigenvalues of the matrix H in Eq.  (6) are plotted in Fig.  2(a). As expected, there are four zero eigenvalues at temperature 0.3, implying the four disconnected states. At higher temperatures, the second to fourth eigenvalues deviate from zero to larger values due to the increasing fraction of the interstate transition trajectories. More illustrative perspective can be obtained from Figs.  2(b), 2(c), and 2(d), where the trajectories are mapped into the three-dimensional space by projecting the trajectory-mapped vectors, {Gⁱ} defined in Eq.  (7), to the second, third, and fourth eigenvectors of H, denoted as where u_α is the α -th eigenvector of H. At temperature 0.3, the trajectories agglomerate into four disconnected clusters that locate respectively at the four vertexes of a tetrahedron. Inside each cluster are the trajectories isolated in the same potential well. At temperature 0.85, part of the trajectories can climb over the potential barriers to the other states. As a result, the points corresponding to the transition trajectories are found to gather along the straight lines that shoot out of the vertexes, and the ones corresponding to the non-transition trajectories are still locating on the vertexes. In Fig.  2(c), each vertex only connects to two neighboring vertexes, correctly reflecting the topology of TDQW system. At temperature 1.5, it is relatively easier for the trajectories to transfer between different states. The tetrahedron structure is still preserved, and its interior is filled by multiple-transition trajectories which transit among more than two states.

Fig.  2.

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	Fig.  2. The eigenvalues and projection maps for TDQW potential under different temperatures. (a) The first ten eigenvalues of H matrix calculated at different temperatures. For the three temperatures of 0.3 (b), 0.85 (c), and 1.50 (d), the projection of {Gi} to the second, third, and fourth eigenvectors of H are also plotted.

In MHAT system, the 70 smallest eigenvalues of H are shown in Fig.  3(a). Different from TDQW, the eigenvalues of H gradually increase from zero at the low temperature T = 0.05, and lots of small eigenvalues exist. The number of the eigenvalues that obviously deviate from 1.0 is always more than the number of physical states in MHAT system, till the temperature is high enough that there is only one zero-eigenvalue apparently different from 1.0. This behavior simply reflects the entropic nature of the outer potential well. At low temperature, a single trajectory cannot reach local equilibrium in the state, consequently a few small eigenvalues exist, as there are many sub-states in the well. At higher temperatures, the trajectories that start from the outer potential well usually climb over the potential barrier before reaching local equilibrium within the state, thus the inter-trajectory difference will not be extinguished, until the two states in MHAT kinetically become one state at high enough temperature. The projections of {Gⁱ} onto the eigenvectors of H are shown in Figs.  3(b), (3c), and 3(d). At both temperatures 0.05 and 0.30, the trajectories are divided into two groups. In one group, the trajectories dispersively locate on the same plane which is perpendicular to the axis. These trajectories are found to be isolated in the outer potential well. In the other group, the trajectories concentrate to nearly one point. They are found to be isolated in the inner potential well. These two groups of trajectories are clearly differentiated by the projections along the second eigenvector of H, i.e., the one indicating the most important contribution to ergodicity broken. At T = 0.05, each trajectory only explores a very limited region in the outer state, leading to the distribution on the perimeter of a circle. At T = 0.30, since each trajectory can explore a larger area, the circle has been filled with points. At T = 0.85, the trajectories are found to locate in a cone. The points in the bottom surface and the vertex of the cone correspond to the non-transition trajectories inside the outer and inner potential wells, respectively, and the other points are transition trajectories between the two states. Once again, the trajectory-projected space clearly provides the topologies of this system.

Fig.  3.

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	Fig.  3. The eigenvalues and projection maps for MHAT potential under different temperatures. (a) The first seventy eigenvalues of H matrix calculated at different temperatures. For the three temperatures of 0.05 (b), 0.30 (c), and 0.85 (d), the projection of {Gi} to the second, third, and fourth eigenvectors of H are also plotted.

In the two 2D models, triangle functions in the 2D space are applied as basis functions, and the details are presented in Appendix A. We did the S² analysis mentioned in Subsection  2.2 at T = 0.85. For both systems, S² initially grows fast with increasing number of basis functions, then the growth slows down and shows a saturation behavior of S² as expected [see Figs.  4(a) and 4(b)]. Once S² has approximately reached the platform region, more basis functions will not change the calculation results anymore (data not shown). Thus, for TDQW and MHAT systems, about 24 basis functions are sufficient.

Fig.  4.

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Fig.  4. The S2 curve, effect of t̃ , and illustration of statistics improvement. The behavior of S2 with increasing number of basis functions are shown in panel  (a) (TDQW) and panel  (b) (MHAT). (c) The difference between the reconstructed equilibrium distribution and the theoretical distribution are shown for MHAT potential at T = 0.85. The horizontal axis denotes the method for reproducing the equilibrium distribution. ‘ eq’ means the results without reweighting (i.e., wi = 1), numbers mean the selected ‘ SI1’ and ‘ SI2’ label the two methods for statistics improvement, respectively. The trajectory weights calculated with equal to 0.01 are plotted as the inset. In panels (a), (b), and (c), the statistical errors are also shown as error bars. (d) The first 40 eigenvalues of H for MHAT potential at T = 0.05. The ones calculated with normal data set (stars), multiple trajectory data set (one-fifth initial configurations and five trajectories for each initial configuration) with the standard RED (circles), multiple trajectory data set with the SI2 (squares) are shown.

We also study the effect of free parameter t̃ in the RED with the simulation data of MHAT at T = 0.85. In this case, since there is only one zero eigenvalue of H, the whole free energy surface can be reconstructed. We uniformly divide the whole conformational space into small cells, k = 1, … , N_p (N_p is selected as 1600 here, i.e., 40 bins along each dimension), and the distribution probability in each cell is reconstructed with different We define the function as

to measure the difference between the reconstructed distribution and the theoretical distribution. Here is the distribution in the k-th cell which is reproduced by the RED using is the theoretical distribution in the cell. As shown in Fig.  4(c), indeed decreases with shorter in calculation. For in the [0.01, 0.1] interval, the values are almost the same, and are prominently smaller than the value of the non-weighted distribution of the simulation data. In the non-weighted (all w_i is unity) distribution, the major deviation from theoretical distribution resides in the inner potential well of MHAT, where the distribution is too high for the non-weighted data. In the reweighting process, the trajectories that start from the inner potential well are usually specified lower weights [see Fig.  4(c), inset], thus the over occupation in the inner state can be offset. The other variation in trajectory weights further adjust the intrastate distribution for both the inner and the outer states.

We test the statistics improvement methods, SI1 and SI2. In here and below, we always randomly select one-fifth of the initial configurations, then five trajectories are spawned from each configuration to test the improvement. At T = 0.85, we apply both SI1 and SI2 to reconstruct the energy surface of MHAT. From Fig.  4(c), it can be seen that both methods correctly reproduce the energy surface with even smaller deviation from the theoretical one, as compared to the standard RED. At T = 0.05, the eigenvalues of the H matrix are calculated with SI2 method. As shown in Fig.  4(d), no matter whether one trajectory or multiple trajectories are generated from one initial configuration, the eigenvalues calculated by the standard RED are almost the same. However, analyzing by the SI2 method significantly alters the structure of eigenvalues. By generating more trajectories from single initial configuration, and combining these trajectories together in analysis, the statistics of the single trajectory are prominently improved, and the difference between trajectories within the outer state are reduced to a large extent. As a result, some of the eigenvalues increase, in particular for the ones deviating but close to 1.0. In other words, the SI2 method has depressed the statistical noise in the spectral analysis of H, leaving the physically relevant states recognized.

With thorough investigation, the two dihedral angles ϕ and ψ are the major coordinates to characterize the conformational motion of the alanine dipeptide molecule. As in the previous study, ^[35] we adopted the first-to-fourth order twodimensional trigonometrical functions of ϕ and ψ as the basis functions. The 10 smallest eigenvalues of H are shown in Fig.  6(a). The smallest zero eigenvalue is intrinsic due to the construction of H. The second eigenvalue corresponds to the kinetic separation between the region with ϕ > 0 (containing and α _L) and the region with ϕ < 0 (containing and α _R). Consequently, by projecting the trajectory-mapped vectors {Gⁱ} to the second eigenvector of H, i.e., through , the trajectories can be clearly classified into the abovementioned two regions. The third eigenvalue of H corresponds to the partially kinetic connection between the and α _R states. Only a fraction of the trajectories in can climb over the free energy barriers to the α _R state, and vice versa. Thus, we may use the to classify the trajectories that are confined in the super-state involving and α _R into three groups. Since the two potential wells are well defined with their internal relaxation time smaller than the transition timescale, we can manipulate the trajectories to predict the transition times and locate the transition state ensemble.^[35] The fourth smallest eigenvalue of H is also smaller than 1.0. Although it no longer indicates a new state in the conformational space and should be taken as the reflection of statistical difference between trajectories, we recently find that, by observing two trajectories that transiently walk into the α _L region can be picked out. One of the two trajectories initially located in the α _L region, then quickly moved into , another just occasionally visited α _L. Actually, at the temperature of 300  K, the α _L region seems not very stable with the applied force field. The other eigenvalues of H are very close to 1.0 without observable meaningful information.

Fig.  6.

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Fig.  6. RED analysis with the basis functions of dihedral angles and inter-atomic distances. Shown are the results for solvated alanine dipeptide, which is simulated at T = 300  K with the modified force field. (a) The eigenvalues of H matrix calculated with different sets of basis functions, including the full set of inter-atomic distances (squares), 12 inter-atomic distances (upward triangles), first-to-fourth order trigonometrical functions of ϕ and ψ (circles), and first-to-second order trigonometrical functions of ϕ and ψ (stars). (b) The behavior of S2 is shown either with the inter-atomic distances (distance-angle) at first, or with the trigonometrical functions of ϕ and ψ at first (angle-distance). The values calculated with truncated trajectories (80 percent length) versus the ones calculated with full trajectories are shown. The results are obtained either with the first-to-second order trigonometrical functions of ϕ and ψ (c), or with the inter-atomic distances (d). The arrows label the intersection points between the dashed and dotted lines.

We also test the effects of distances between atoms in the backbone of dipeptide by using them as basis functions. We compare two aspects, i.e., the eigenvalues of H and the manipulations on for extracting the kinetic information between and α _R. There are 10 non-hydrogen heavy atoms in alanine dipeptide. Except for the directly bonded atoms, 36 pairs of atoms exist, corresponding to 36 interatomic distances. Together with the two distances for the possible intramolecular hydrogen bonds (see Table  1), we select the 38 distances as basis functions. The resulting eigenvalues of H are shown in Fig.  6(a). While the three smallest eigenvalues are almost the same as the results obtained with the functions of dihedral angles, the fourth eigenvalue is prominently lifted to a value closer to 1.0. However, the discrimination of the two α _L-visited trajectories is still possible. We plot the values calculated with the trajectories truncated to 80-percent length (i.e., the 0.2τ ending segment of the trajectories are discarded in analysis) versus the ones calculated with full trajectories^[35] in Fig.  6(d). For comparison, the same figure generated with the first-to-second order two-dimensional trigonometrical functions of torsion angles is shown in Fig.  6(c). The two figures have quite similar structure. The points corresponding to the non-transition trajectories in α _R and are plotted as upward and downward (black) triangles, respectively. The points corresponding to the transition trajectories that start from α _R and are plotted as (red) squares and (blue) circles, respectively. The points along the inclined dashed lines correspond to the trajectories with only one time transition, and the points along the horizontal dotted lines correspond to the trajectories that do not happen to transition within the initial 0.8τ . Thus the intersection between the dotted and dashed lines corresponds to a single-transition trajectory with its transition happening exactly at 0.8τ . These points can be used to predict the times the transitions happen in the single-transition trajectories, consequently the transition state ensemble can be constructed. All the other points correspond to the trajectories with an even number of transitions which all happen within 0.8 τ , or with early transitions occurring within t̃ , or the multiple transition trajectories with transition happening both within 0.8 τ and after 0.8 τ . More details about detecting transition time thus transition conformation ensemble from the truncated-trajectory plots are presented in Ref.  [35]. The similarity between Fig.  6(c) and Fig.  6(d) suggests both the torsion angles and the inter-atomic distances can be applied as basis functions.

Table 1.

Table 1.

Table 1. Basis function list. Group Number Function Internal freedom of 40 Two-dimensional trigonometric functions of ϕ and ψ alanine dipeptide 8 N1– O1, N2– O2, N1H– O1, N2H– O2 and their squaresa 2 Interactionb between peptide and water Interaction between 8 Interaction between O1, O2, N1, N2, and water solvent and solute 2 Water number around peptide (3.3  Å , 5.5  Å ) 1 Solvent accessible surface area 20 Water number around O1, O2, N1, N2 (3.3  Å , 3.7  Å , 4.1  Å , 4.5  Å , 4.9  Å ) Solvent structure 10 Integration of the OO radial distribution function, g(r), of the bulk waterc, i.e., (r0 is selected to be 2.9  Å , 3.3  Å , 3.7  Å , 4.1  Å , 4.5  Å , 4.9  Å , 5.3  Å , 5.7  Å , 6.1  Å , and 6.5  Å ). aThe four atoms O1, O2, N1, and N2 are labelled in Fig.  5, left panel. N1H (N2H) is the hydrogen atom bonded to N1 (N2). O1– N1, and O2– N2 may form intra-molecule hydrogen bonds, or may be connected by water bridge.[38]; bIn here and below, interaction means the electrostatic energy part and the Van de Waals energy part; cThese functions reflect the packing of bulk water.

Table 1. Basis function list.

We also apply the RED to study the low-temperature (T = 150  K) properties of the dipeptide. We note that, at this temperature, the physical phenomena are quite complicated; the force field and the simulation procedure that we adopt may not reflect the physical reality. Here, we are only meant to illustrate the application of the RED in the extreme condition, instead of providing physical insight into the system.

The eigenvalues of the H matrix are calculated with the 91 basis functions listed in Table  1, and are plotted in Fig.  7(b). The solvent-related quantities do affect the eigenvalues of H now, leading to even more small eigenvalues. Correspondingly, the S², as shown in Fig.  7(d), almost always increases with the number of basis functions. At such a low temperature, the free energy surface becomes quite rough, and the diffusive dynamics^[39] in the conformational space slows down seriously. As a result, the simulation trajectories may only explore a small region in the conformational space. Some degrees of freedom, like the solvent motion around the solute molecule and the conformational motion perpendicular to the ϕ – ψ plane, are no longer equilibrated in a single trajectory. Thus, the inter-trajectory difference becomes larger, more states are detected automatically, and more basis functions are required to expand the rugged distribution of the samples, leading to the abnormal behavior of the eigenvalues of H and S².

Manipulating the eigenvectors of H can provide further information of the simulation data. The projections of {Gⁱ} to the second and the third eigenvectors of H pick out two completely separated states, and the super-state containing α _R, and [Fig.  8(a), at the current low temperature, the ϕ and ψ angles may not be enough to characterize the conformational motion, thus the notations, α _R, and , are used mainly for the purpose of illustration]. With the aid of the histogram of [see Fig.  8(a), inset], we can roughly classify the trajectories inside the super-state into three groups, i.e., the trajectories in the leftmost two bins, in the rightmost two bins and the remaining ones. The conformational distribution of trajectories in different groups are shown in Fig.  8(c). The two groups of the leftmost or rightmost bins both possess single peak distribution, either in region or α _R region, just like the non-transition trajectories identified in the simulations at higher temperature. In contrast, the trajectories in the remaining group have a significant occupation in the intermediate region between and α _R. Obviously, these trajectories are trapped at the intermediate region due to slow dynamics, and are different from the transition trajectories identified with the similar procedure at 300  K. We plot the calculated with 80 percent trajectory length with the ones with full trajectory length in Fig.  8(c). The behaviors presented in Fig.  6 no longer exist. In terms of kinetics, the transition between the two regions, α _R and , can neither be approximated as Markovian process nor described with the first-order transition process any more. Consequently, with the current simulation data, it is only possible to reconstruct the intrastate equilibrium distribution in the local regions of conformational space, or to study the more localized kinetic behavior.

Fig.  8.

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Fig.  8. Results for the low-temperature simulation of alanine dipeptide. (a) The projection of trajectories into the space of and . The inset is the histogram of values. The different states, into which the trajectories are roughly classified, are labeled. (b) The distributions along the ψ angle in the groups are shown in panel  (a). Shown are the distributions for the trajectories in the leftmost two bins (downward triangles), the rightmost two bins (upward triangles) and the other trajectories (circles). (c) The values calculated with truncated trajectories (80 percent length) versus the ones calculated with full trajectories. (d) The first 70 eigenvalues of H calculated with normal data set (blue stars), multiple trajectory data set (one-fifth initial configurations and five trajectories for each initial configuration) with the standard RED (red circles), multiple trajectory data set with SI2 method (green squares) are shown.

Due to the numerous local minimums at the low temperature, the α _R and regions are just like two entropicdominated states. We also test the statistics improvement method in this system. With one-fifth initial conformations and five trajectories from each initial configuration, the inter-trajectory difference, thus the statistical noise, is indeed suppressed as shown in Fig.  8(d).

As a summary, in the dipeptide system at T = 300  K with modified potential, we find the redundant basis functions (including functions of both the dihedral angles and the inter-atomic distances) are consistently treated in the RED, and the inclusion of the solvent-related basis functions gives out the expected results. At pretty low temperature of 150  K, the roughness of the free energy surface is reflected in the results of RED. The conformational space can still be roughly divided into parts, however, the kinetics becomes too complicated to be unambiguously characterized.