Denavit-Hartenberg paramaters for chain description

In this section we introduce the parametric representation of a linear chain used to derive the equations at the core of our algorithm. We consider a linear chain composed of N + 1 atoms linked serially in which each of the N bonds can be labeled with numbers from 1 to N. We describe the chain by using the Denavit-Hartenberg (DH) notation [17]. According to DH a local reference system 𝒪_i can be built on each bond composing the chain: the ẑ_i axis lies on the bond while x̂_i is oriented as ẑ_i−1 × ẑ_i. The ŷ_i axis is given, as usual, by the right-hand rule. The origin o_i of each reference frame is always located along ẑ_i: in case ẑ_i is co-planar to ẑ_i−1, it lies on the first atom defining the bond; otherwise it lies on the common normal to ẑ_i and ẑ_i−1.

A vector ${\overrightarrow{a}}^j$ in the reference frame 𝒪_j can be expressed relative to 𝒪_j−1 as (1) where ${\mathbf{\text{R}}}_j^{j-1}$ is a 3 × 3 orthogonal matrix that expresses the orientation of 𝒪_j relative to 𝒪_j−1 and ${\overrightarrow{S}}_j^{j-1}$ is a vector describing the position of the origin o_j with respect to 𝒪_j−1. The matrix ${\mathbf{\text{R}}}_j^{j-1}$ and the vector ${\overrightarrow{S}}_j^{j-1}$ can be completely described by using four parameters named link offset, link twist, link length and joint angle. These are defined by the following rules:

• the link offset d_i is the distance along x̂_i from o_i to the intersection of the x̂_i and the ẑ_i−1 axes (i.e. the minimum distance between ẑ_i−1 and ẑ_i axis);
• the link twist α_i is the angle between ẑ_i−1 and ẑ_i measured about x̂_i;
• the link length r_i is the distance along ẑ_i−1 from o_i−1 to the intersection of the x̂_i and the ẑ_i−1 axes and
• the joint angle θ_i is the angle between x̂_i−1 and x̂_i measured about ẑ_i−1.

Fig. 1 shows how the DH parameters are defined for the general disconnected case (i.e. link offset d_i > 0). This is actually a typical case, when the structure of a protein backbone chain with all its heavy atoms is considered, since the ω torsional angle around the peptide bond is a hard degree of freedom with a well defined typical value. If ω is then kept strictly fixed, the DH convention allows to “spare” that degree of freedom, defining a disconnected chain as shown in Fig. 1. In the simplest case, when all bonds included in the DH description are connected with each other (i.e. all link offsets d_i = 0), the DH variables have a well defined physical meaning. Link lengths are bond lengths, link twists are supplementary of bond angles, and joint angles are torsional angles. The DH formalism is in this case equivalent to the one routinely used by software programs that reconstruct biomolecular structures subject to experimental restraints [18, 19] and that employ efficient internal dynamics algorithms that update only the values of torsional angles [20].

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Fig 1. General graphical representation of a chain according to the Denavit-Hartenberg convention, as discussed in the text.

Thick lines represent the physical bonds and spheres the atoms. α, θ, r and d are the DH parameters describing the chain and o is the origin of each local frame . The structure of a portion of a protein backbone chain with all its heavy atoms is shown superimposed.

https://doi.org/10.1371/journal.pone.0118342.g001

By using the DH definitions the matrix ${\mathbf{\text{R}}}_j^{j-1}$ and the vector ${\overrightarrow{S}}_j^{j-1}$ can be explicitly expressed as (2a) (2b) With a more compact notation we rewrite Eq. (1) with the following (3) where ${\mathsf{\text{T}}}_j^{j-1}$ is a 4 × 4 matrix given by (4) and a is the vector $\mathbf{a}\text{=}{\text{(}\overrightarrow{a},1\text{)}}^T$. With this notation it is easier to relate any 𝒪_j with any other 𝒪_i (j > i); indeed the following equation holds: (5)

It should be now clear that the chain can be described by the whole set {r_i, α_i, d_i, θ_i}_{i = 1,⋯,N}. For simplicity we denote {r_i}_{i = 1,⋯,N} with r, {α_i}_{i = 1,⋯,N} with α, {d_i}_{i = 1,⋯,N} with d and {θ_i}_{i = 1,⋯,N} with θ; the chain configuration is then given by the set {r, α, d, θ}.

Performing the concerted local move

It is possible to deform an initial configuration {r₀, α₀, d₀, θ₀} by changing at least one of the parameters describing it.

Consider n DH parameters ξ̃_μ, with μ = 1, ⋯, n and the n_b bonds to which the n parameters are related. The number of bonds to be considered is always smaller or equal to the number of DH parameters because in principle two or more parameters could be related to the same bond. As already stated we consider the general case in which these bonds can be non-consecutive. There are two particularly interesting bonds among the n_b: the first and the last i.e. the one with the lowest label and the one with the highest one. These two bonds delimit the region of the chain we are interested in modifying with an opportune change of $\tilde{\xi }$, leaving the atoms outside this region unmodified (see Fig. 2). Such a change can be highly non-trivial and could not always be obtained: we will see later a condition that ensures us that this change can be performed.

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Fig 2. Schematic representation of the portion of a linear chain involved in a local modification.

Bonds (1) and (n_b + 1) are colored in blue while all the others are in red. The degrees of freedom which are varied (ξ₁, …, ξ_n) are arbitrarily distributed inside the region. When they are concertedly changed to new values $({\xi }_1^{\prime },\dots ,{\xi }_n^{\prime })$, the new pink configuration is obtained while all the bonds outside the region (in black) remain fixed in space.

https://doi.org/10.1371/journal.pone.0118342.g002

For convenience, we re-label the bonds we are interested in with numbers from 1 to n_b + 1, where the latter is the first bond that remains fixed subsequent to the moved portion of the chain. The condition that needs to be imposed in order to ensure the locality of the change {r₀, α₀, d₀, θ₀} → {r, α, d, θ} is (6) that is requiring that the local reference frame 𝒪_nb+1 does not move with respect to the first one. In order to explicit the variables $\tilde{\xi }$ inside the relation in Eq. (6) we define, with abuse of notation, (7) in such a way that Eq. (6) can be rewritten as (8) Given the form of ${\mathsf{\text{T}}}_i^j$ (described in Eq. (4)), the 16 equations that are implicit in Eq. (8) can be reduced to 6 equations in the n variables $\tilde{\xi }$. Three equations are needed in order to set the translational part of ${\mathsf{\text{T}}}_i^j$ and other three for the rotational part. We choose, for instance, (9) or, in a more compact form, $\mathbf{\text{f}}(\tilde{\xi })=\mathbf{\text{0}}$. If n is greater than 6 and if the system of equation described in Eq. (9) is non degenerate then the solutions lie on a manifold with dimension n − 6. If, on the contrary, the system is degenerate the solutions lie on a manifold with dimension greater than n − 6.

Before to proceed it is important to notice that since the degrees of freedom $\tilde{\xi }$ can describe both spatial or angular quantities the space defined by these variables is in general dimensionally non-homogeneous. For this reason we introduce a set of n scalar multipliers λ_i chosen in such a way that every ξ_i = λ_iξ̃_i is dimensionless. The space defined by the rescaled variables ξ_i is now homogeneous and an appropriate metric can be defined in it by means of the usual scalar product. In the case that a set of homogeneous variables are chosen as system variables (e.g. all dihedral angles) the introduction of the scalar multipliers is unnecessary but nonetheless it turns out to be useful (see Section “Tuning of fluctuations by rescaling variables”).

The problem of finding a new configuration ξ starting from an existing ξ₀ in such a way that Eq. (8) is satisfied can now be visualized as the problem of moving on an (n − 6)-dimensional manifold embedded in an n-dimensional space. The most intuitive way to perform this non-trivial task is that of generating an intermediate configuration ξ^′ that may not lie on the manifold but that it is not far from it. This first step is called by other authors pre-rotation [15, 21], and we will adapt to this nomenclature. Starting from ξ^′ it is then possible to compute a true solution with numerical methods, e.g. root finding algorithms, or analytical ones [12].

In order to simplify the description of the pre-rotation step we introduce two new quantities n_m and n_v that are respectively the dimension of the tangent space M to the manifold at ξ₀ and the dimension of the orthogonal space V to the manifold at ξ₀. Obviously n_m + n_v = n and n_v = 6 if the system Eq. (9) is non-degenerate. In general n_v is the number of linearly independent functions in (9). The procedure we use to find a basis of the tangent space to the manifold takes advantage of the implicit function theorem in order to compute the derivatives $\frac{\partial {\xi }_i}{\partial {\xi }_j}$. Indeed we consider n_m among the n variables as independent and we denote them with the subscript x. The other n_v are labeled with a subscript y and will be written as a function of ξ_x. With this notation we can write a set of n_m n-dimensional vectors that span the tangent space as (10) where the derivative $\left(\frac{\partial \xi }{\partial {\xi }_{\mathbf{x},\mathbf{i}}}\right)$ can be performed by computing separately the contribute of the dependent variables ξ_y and that of the independent ones ξ_x. The former (in the form of an n_v-dimensional vector $\frac{\partial {\xi }_{\mathbf{y}}}{\partial {\xi }_{\mathbf{x},\mathbf{i}}}$) can be easily computed by applying the implicit function theorem (11) while the latter is given by the n_m relations $\frac{{\partial \xi }_{\mathbf{x},\mathbf{j}}}{\partial {\xi }_{\mathbf{x},\mathbf{i}}}={\delta }_{i,j}$. In the cases in which the matrix $\left(\frac{\partial \mathbf{f}({\tilde{\xi }}_0)}{\partial {\xi }_{\mathbf{y}}}\right)$ is not invertible it is sufficient to choose a different set of ξ_x as independent variables. Vectors e_x,i can be orthonormalized to compute a basis {ê_x,i}_i for the tangent space. The intermediate configuration ξ^′ can finally be computed by simply summing an arbitrary linear combination of ê_x,i to the initial configuration ξ₀.

The intermediate configuration is an open configuration in which the position and orientation of the last reference frame do not correspond to the target ones. We therefore adjust the coordinates on the orthogonal space V by using a root-finding algorithm to obtain the final configuration. Fig. 3 depicts an example move in case the manifold is one-dimensional (the full example is addressed in the Section “Workout example”). The solution manifold (blue) and the initial configuration are represented. The pre-rotation step corresponds to the first update (thick red arrow) while the second step along the dotted line allows to converge back to the manifold by means of a root-finding algorithm. This is in general possible only for small enough pre-rotation steps. The brown arrows show the case when a too big pre-rotation step does not allow the procedure to converge back to the manifold.

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Fig 3. Graphical sketch of the updating of the conformations in the n-dimensional space of the degrees of freedom which are changed.

The blue line represents the manifold of the solutions of Eq. (23), as discussed in the Section “Workout example” within “Material and Methods”. The manifold is plotted within the (θ, ρ) plane, with the independent variable being ξ_x = ρ and the dependent variable ξ_y = θ. M and V are respectively the tangent and the orthogonal space to the manifold in the starting conformation ξ₀. The degrees of freedom are first changed along M (red continuous arrow) and a new conformation satisfying Eq. (9) is reached by moving along V (dashed red arrow). If the pre-rotation step along M is too large (brown continuous arrow), the post-rotation closure step along V (dashed brown arrow) fails to fall back to the manifold.

https://doi.org/10.1371/journal.pone.0118342.g003

A basis for V can be efficiently computed starting from the knowledge of a basis for M and by using ad-hoc algorithms (e.g. QR-decomposition algorithm). With this strategy the chain closure step usually takes few iterations of the Broyden’s root finding algorithm. Notice that it is in principle possible to use a root-finding algorithm that takes advantage of the easy-to-compute gradient $\frac{\partial \mathbf{f}(\xi )}{\partial \xi }$ to boost the search for the solution. However our tests have shown that the time saved by the root-finding algorithm usually does not compensate for the time necessary for computing the gradient.

The full algorithm can be summarized in the following steps:

1. Compute a basis for the tangent space M at ξ₀, as well as a basis for the orthogonal space V;
2. Choose an arbitrary direction $\widehat{\eta }$ in M and an arbitrary step length ds;
3. Generate an intermediate configuration ${\xi }^{\prime }={\xi }_0+ds\cdot \widehat{\eta }$;
4. Use a root-finding algorithm in order to converge to a solution of Eq. (9) by moving on V;

Detailed balance

In this section we will denote the three-dimensional configuration of an N-atom chain at time t with the 3N-dimensional vector $R^t=\{r_1^t,r_2^t,\cdots r_N^t\}$, where each $r_k^t={(x_k^t,y_k^t,z_k^t)}^T$ represents the three-dimensional position of atom k at time t.

In order to demonstrate that the scheme we proposed for changing the backbone configuration satisfies the detailed balance condition (12) we will show separately that (1) with an appropriate change of variables, it is possible to uniformly sample the whole configurational space of the system without changes to the algorithm and (2) the probability of moving from R^t to R^t+Δt can be chosen in such a way that Π(R^t → R^t+Δt) = Π(R^t+Δt → R^t).

We start by showing that point (1) is true. If the DH’s variables α, θ, r are used for describing the chain, the volume element in the configurational space (13) can be rewritten as (14) where $J_k^t=\text{sin}({\alpha }_k^t){\left(r_k^t\right)}^2$. Since the determinant $J^s={\prod }_{k=1}^N{J}_k^s$ of the change of variables R_t → {α, θ, r} is not constant, a uniform sampling of the space of the DH’s variables does not result in a uniform sampling of the configurational space. In this case uniformity can be achieved by accepting the configuration change {α, θ, r}₁ → {α, θ, r}₂ with probability $p=\text{min}(1,\frac{J^{t+\Delta t}}{J^t})$. Such essential choice compromises efficiency both by increasing the number of calculations per time step and by reducing the probability of obtaining a new configuration. With our approach the problem is overcome without performing any additional time-consuming calculation. Indeed it is easy to notice that the algorithm proposed in the previous section does not rely on the particular form of ξ or $\mathbf{\text{f}}(\tilde{\xi })$ but rather on the possibility of computing $\mathbf{f}(\tilde{\xi })$ and its derivatives. For this reason every differentiable, invertible function of $\tilde{\xi }$, whose inverse is differentiable, can be used as degree of freedom of the system without jeopardizing the efficiency of the full scheme. If, for instance, $\psi =g(\tilde{\xi })$ is used as degree of freedom, the following trivial relations holds: (15a) (15b) If DH’s parameters can be interpreted as bond lengths, bond angles and torsional angles (θ is always a torsional angle, α is a bond angle for bonds connected to the previous one, r is a bond length for bonds connected to both the previous and the subsequent one) it is sufficient to use the variables ${\tilde{\xi }}_{\alpha }=cos(\alpha )$ and ${\tilde{\xi }}_{\mathbf{r}}=\frac{r^3}{3}$ as new degrees of freedom in order to guarantee the determinant of the Jacobian of the change of variables to be constant: this implies that the new configuration can always be accepted. This result relies only on the fact that the Jacobian does not depend on ξ; we derived it independently from the values of the rescaling factors.

Notice that even if the inverse of the cosine function is not differentiable in 0 and π, these points are at the boundary of the domain in which the bond angle α is defined. Hence the analysis is valid in the open domain (0, π) but not in its closure. The same is true for $\frac{r^3}{3}$, whose inverse is not differentiable in r = 0. Of course these drawbacks are not relevant for most applications, since α = 0, π and r = 0 are far from physical values. Therefore a uniform sampling of the space of the new variables $\{{\tilde{\xi }}_{\alpha },{\tilde{\xi }}_{\mathbf{r}},\theta \}$ is sufficient to ensure a uniform sampling of the configurational space.

In order to ensure the detailed balance condition to be valid it is now necessary to check whether the step size ds that has been used in the forward update ξ₁ → ξ₂ is the same that would be necessary for the reverse transformation ξ₂ → ξ₁. Indeed the step size is typically chosen from a fixed probability distribution and therefore if the backward step size ds^′ is different from the forward one, it is necessary to take into account the different probability of proposing such a step. The backward step can be easily computed as the norm of the projection of ξ₂ − ξ₁ onto the tangent space to the manifold at ξ₂. If the step size is chosen from a normal distribution with zero average and variance σ² it is sufficient to accept the update with probability (16) to ensure the equivalence between forward and backward probability. In many practical cases $\frac{d{s}^{\prime }}{ds}$ is close to one and, as a consequence, the probability of rejecting a Monte Carlo move is negligible. This calculation can be done for whatever choice of the factors λ_i. The detailed balance can therefore be satisfied independently from the choice of each λ_i.

Tuning fluctuations by rescaling variables

The role of rescaling factors λ_i previously introduced can be further exploited. As already briefly discussed, their introduction has been necessary to map the original non-homogeneous space of the variables $\tilde{\xi }$ in the homogeneous one of the variables $\xi =\lambda \cdot \tilde{\xi }$ where a metric can be defined. But on top of that they can be used in order to tune the fluctuations of the corresponding degrees of freedom.

Given a starting configuration ${\tilde{\xi }}_0$ suppose that some of the ${\tilde{\xi }}_{0,i}$ are hard degrees of freedom (${\tilde{\xi }}_{0,H}$) while the others are soft (${\tilde{\xi }}_{0,S}$). We can use two different values of λ_i, depending on the class of the corresponding degrees of freedom, thus mapping (17) where λ_H > λ_S. As described in previous sections the algorithm is applied to the initial conformation in the homogeneous, deformed space. For simplicity we consider in the following discussion the particular case in which the soft degrees of freedom correspond to the independent variables, in such a way that the new configuration ξ can be written as a function of the variation vectors Δξ_S and Δξ_H = ∇_ξS ξ_H ⋅ Δξ_S as (18) where ∇_ξS ξ_H is the matrix of the derivatives of the hard degrees of freedom with respect to the soft ones.

It is possible to express the norm of the variation vector for independent variables $\Delta {\tilde{\xi }}_S$ as a function of the size of the actual move performed in our algorithm, the step size ds in the tangent space: $ds={({\lambda }_S^2\mid \mid \Delta {\tilde{\xi }}_S\mid {\mid }^2+{\lambda }_H^2\mid \mid \Delta {\tilde{\xi }}_H\mid {\mid }^2)}^{\frac{1}{2}}$. The previous equation allows to define the function $g_{{\lambda }_s,{\lambda }_H}({\tilde{\xi }}_S,{\tilde{\xi }}_H)$ through $\mid \mid \Delta {\tilde{\xi }}_S\mid \mid =\frac{g_{{\lambda }_s,{\lambda }_H}({\tilde{\xi }}_S,{\tilde{\xi }}_H)}{{\lambda }_S}ds$. The function $g_{{\lambda }_s,{\lambda }_H}({\tilde{\xi }}_S,{\tilde{\xi }}_H)$ describes the reduction of the variation of the soft independent variables, for a fixed step size ds, due to the orientation of the tangent space with respect to the space of independent variables. It depends, in general, both on the position of the initial point in the manifold and on the chosen rescaling factors.

Each point of the deformed manifold can be remapped to the original manifold with the inverse transformation $\tilde{\xi }=\frac{1}{{\lambda }_i}\xi$. Therefore the new configuration found in Eq. (18) corresponds to a final configuration in the original space equal to (19) where ${\widehat{\Delta \tilde{\xi }}}_S$ is a normalized vector.

The previous equation holds also in the λ_S = λ_H = 1 case, when no rescaling occurs. The rescaling factor K_S for the variation of the independent variables ${\tilde{\xi }}_S$ as a consequence of the introduction of λ_S, λ_H ≠ 1 is then easily computed: (20) while the rescaling factor K_H for the variation of the hard degrees of freedom ${\widetilde{\mathbf{\text{ξ}}}}_H$ reads (21) This is the central equation of this section, as it shows to what extent the variation vector for hard degrees of freedom is modified differently than for soft degrees of freedom, due to the rescaling of the original variables. Besides the global tuning factor $\frac{{\lambda }_S}{{\lambda }_H}$ a second factor $\frac{\mid \mid {\nabla }_{{\xi }_S}{\xi }_H\cdot {\widehat{\Delta \tilde{\xi }}}_S\mid \mid }{\mid \mid {\nabla }_{{\tilde{\xi }}_S}{\tilde{\xi }}_H\cdot {\widehat{\Delta \tilde{\xi }}}_S\mid \mid }$ appears, related to the local geometrical properties of the considered manifold.

For one-dimensional manifolds, it is easy to see that ${\nabla }_{{\xi }_S}{\xi }_H=\frac{{\lambda }_H}{{\lambda }_S}{\nabla }_{{\tilde{\xi }}_S}{\tilde{\xi }}_H$, so that $K_S({\tilde{\xi }}_S,{\tilde{\xi }}_H)=K_H({\tilde{\xi }}_S,{\tilde{\xi }}_H)$; the variations of both the hard and the soft degrees of freedom are rescaled in the same way, as if a new effective step size $ds({\tilde{\xi }}_S,{\tilde{\xi }}_H)=K_S({\tilde{\xi }}_S,{\tilde{\xi }}_H)ds$ were used.

For manifold of higher dimension, if ${\nabla }_{{\xi }_S}{\xi }_H\ne \frac{{\lambda }_H}{{\lambda }_S}{\nabla }_{{\tilde{\xi }}_S}{\tilde{\xi }}_H$ then $K_S({\tilde{\xi }}_S,{\tilde{\xi }}_H)\ne K_H({\tilde{\xi }}_S,{\tilde{\xi }}_H)$ and the λ factors can effectively be used in order to tune the amplitude of the fluctuations of hard degrees of freedom with respect to the soft ones. It is obviously not easy to quantify ‘a priori’ the local geometric factor $\frac{\mid \mid {\nabla }_{{\xi }_S}{\xi }_H\cdot {\widehat{\Delta \tilde{\xi }}}_S\mid \mid }{\mid \mid {\nabla }_{{\tilde{\xi }}_S}{\tilde{\xi }}_H\cdot {\widehat{\Delta \tilde{\xi }}}_S\mid \mid }$.

In the most general case, with no restriction on the soft degrees of freedom being either independent or dependent variables, we expect to find qualitatively similar results both for high-dimensional and one-dimensional manifolds.

Workout example

In order to better explain each step of the algorithm we provide a simple example that can be solved exactly. Beyond the context of concerted local movements in chain molecules, the simplest possible case for our algorithm corresponds to the motion along a one-dimensional manifold defined by a single constraint within a two-dimensional space. In this spirit, we consider a physical system in which a single particle is constrained to move within the (x, y) plane on the right branch (x ≥ 1) of a hyperbola of equation (22) We introduce new polar coordinates $(\tilde{\rho }=\sqrt{x^2+y^2},\tilde{\theta }=\mathrm{\text{atan}}\left(\frac{y}{x}\right))$ that, in this example, will have the same role the DH parameters have for the description of a chain molecule. Notice that θ̃ is defined in the open interval $(-\frac{\pi }{4},\frac{\pi }{4})$. We also introduce two scalar quantities λ_ρ and λ_θ defined in such a way that ρ = λ_ρ ρ̃ and θ = λ_θ θ̃ are both dimensionless quantities. Constraints are defined by the analogous of Eq. (9) (23) that, as a function of the rescaled variables becomes (24) The tangent space M and the orthogonal space V in the initial configuration (ρ₀, θ₀) can be easily computed starting from the expression of the derivatives of f₁(ρ, θ) (25) by means of the implicit function theorem. Indeed, if the initial configuration is such that $\frac{\partial f_1(\rho ,\theta )}{\partial \theta }\ne 0$ (i.e.θ ≠ 0) it is possible to choose ρ as independent variable and to compute the derivative $\frac{\partial \theta }{\partial \rho }=\frac{{\lambda }_{\theta }}{\rho }\mathrm{\text{cotan}}\left(2\frac{\theta }{{\lambda }_{\theta }}\right)$. The tangent space M is therefore generated by the vector $\eta (\rho ,\theta )={(1,\frac{\partial \theta }{\partial \rho })}^T$ and any intermediate configuration can be selected as $({\rho }^{\prime },{\theta }^{\prime })=({\rho }_0,{\theta }_0)+ds\cdot \widehat{\eta }({\rho }_0,{\theta }_0)$, where ds is an arbitrary step-size and $\hat{\eta }$ is normalized. Once η is computed, the QR algorithm is used in order to generate a basis for the orthogonal space V. In this simple example V is one-dimensional and the generating vector can be computed explicitly as ${\eta }^{\perp }(\rho ,\theta )={(-\frac{\partial \theta }{\partial \rho },1)}^T$. A root finding algorithm is finally used in order to find a solution to the equation f₁(ρ, θ) = 0 with $(\rho ,\theta )=({\rho }^{\prime },{\theta }^{\prime })+k\cdot {\widehat{\eta }}^{\perp }$ and with varying k. In case θ = 0, it is not possible to use ρ as the independent variable, and θ has to be chosen instead. Notice that in order to ensure numerical stability it is safer to use θ as independent variable in a suitable interval centered in θ = 0. Also, the existence of a solution is in general ensured only for small enough step sizes, as depicted in Fig. 3.

We now consider within this example the effect of introducing the rescaling factors λ_ρ and λ_θ, as discussed in the previous section. We assume θ̃ ≠ 0, so that ρ can be used as the independent variable. In this case we see that (26) and therefore any rescaling induced for dρ is applied to dθ as well, as expected for a one-dimensional manifold. The relation between the step size ds tangent to the manifold and the variation dρ̃ of the independent unrescaled variable is $ds=d\tilde{\rho }{[{\lambda }_{\rho }^2+{\lambda }_{\theta }^2{\left(\frac{\partial \tilde{\theta }}{\partial \tilde{\rho }}\right)}^2]}^{\frac{1}{2}}$, that allows to recover the function $g_{{\lambda }_{\rho },{\lambda }_{\theta }}(\tilde{\rho },\tilde{\theta })=\frac{{\lambda }_{\rho }}{ds/d\tilde{\rho }}={[1+\frac{{\lambda }_{\theta }^2}{{\lambda }_{\rho }^2}\frac{{\mathrm{\text{cotan}}}^2\left(2\tilde{\theta }\right)}{{\tilde{\rho }}^2}]}^{-\frac{1}{2}}$. We finally obtain the rescaling function from Eq. (20) as

Detailed balance

In order to verify that the detailed balance is satisfied we performed a Monte Carlo simulation on a 63-residue long protein fragment (pdb code 1CTF) by using our algorithm. We allowed only the Ramachandran’s ϕ and ψ angles to vary during the simulation; these modifications were obtained by randomly selecting either a Pivot move around a randomly selected bond or our locally concerted move on a set of randomly selected angles. The introduction of the Pivot move was necessary to move the last bond of the chain and thus ensure simulation ergodicity. Fig. 4 shows the distribution of the values of different torsional angles of a protein chain that has been simulated using the proposed schema. As expected when detailed balance holds, the distribution is flat, within natural stochastic fluctuations.

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Fig 4. Distribution of the values of torsional angles of a 67 residues long protein (1CTF) during a 3.5 ⋅ 10⁵ time-steps Monte-Carlo simulation.

Each of the stacked barchart is relative to a different torsional angle. At each simulation step the structure is deformed by applying our algorithm to a randomly chosen portion of the chain (with probability 0.7) or by a pivot move (with probability 0.3). In the former case the step size ds is chosen from a normal distribution with mean 0 rad and variance 0.08 rad². In the latter case a randomly chosen ϕ or ψ angle is perturbed by adding to it a quantity that is chosen from a normal distribution with mean 0 rad and variance 0.4 rad².

https://doi.org/10.1371/journal.pone.0118342.g004

Fluctuation tuning by rigidity rescaling

We also studied how the dynamics of the exploration of the solution manifold is affected by the rigidity factors λ_i used to rescale the original variables (see Materials and Methods). This has been done by comparing the distribution of the fluctuations Δθ of two different torsional angles upon changing the rigidity of one of the two. The data have been obtained by performing short simulations with fixed step-size ds = 0.01 along a randomly chosen direction onto the tangent space to the manifold. First a nine degrees of freedom fragment, corresponding to a three-dimensional manifold, was explored for different values of λ₁. Fig. 5 shows the distribution of the fluctuations of the corresponding torsional angle θ₁ and of another torsional angle (θ₃) used as negative control. A rescaling of λ₁ by a factor of 3 does not globally affect the distribution of Δθ₃, but rescales the corresponding distribution of Δθ₁ by a factor roughly 3. This shows that the main role in Eq. (19) is played by the global rescaling factor 1/λ, wheres the role of local manifold geometry is in practice less relevant, at least in the considered example. On the other hand, the effect of local manifold geometry may explain some reshaping that can be appreciated in the plotted distributions apart from the global rescaling.

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Fig 5. Distribution of variability of θ₁ and θ₃ angles for different values of λ₁ for a 9 degrees of freedom chain.

The distribution of θ₃ is largely not affected by the change of λ₁ while the distribution of θ₁ is rescaled according to Eq. (19).

https://doi.org/10.1371/journal.pone.0118342.g005

Similarly, Fig. 6 shows the distribution of the fluctuations of the angles θ₁ and θ₃ obtained while exploring the solution manifold of a seven degrees of freedom fragment with a fixed step-size ds for different values of λ₁. In this case the solution manifold is one-dimensional and therefore the fluctuations of θ₁ and θ₃ are always related. Indeed, after the rigidity λ₁ is increased, both distributions are globally rescaled by the same quantity. The dynamics on the deformed manifold corresponds, in this case, to a dynamics on the original manifold with a rescaled step size. For the one-dimensional example as well, some reshaping can be seen in the plotted distributions apart from the global rescaling. Again, this observation may be explained as an effect of local manifold geometry, consistently with Eq. (19).

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Fig 6. Distribution of variability of θ₁ and θ₃ angles for different values of λ₁ for a 7 degrees of freedom chain.

In this case it is not possible to change a single dof without altering the others: as a consequence a rescaling of λ₁ affects the distributions of θ₁ as well as of the other angles (shown: θ₃). In this case the effect of changing the rigidity of one or more parameters is the same as rescaling the step size used during the simulation.

https://doi.org/10.1371/journal.pone.0118342.g006

Exploring the conformational space

The first application we describe is the exploration of the whole conformational space of a small portion of a polypeptide, i.e. finding all configurations of that fragment that are compatible with the locality constraints. The problem can be resolved by finding a method to compute all the solutions of Eq. (9). This is not a simple task when the number of degrees of freedom is large but, on the contrary, if the number of variables is 7 the exploration of the manifold is simple. In this case the manifold has dimension 1 and so it is sufficient to move on the manifold always along the same direction. This can be done by choosing at each step k an initial direction ${\hat{\eta }}_k$ in such a way that ${\hat{\eta }}_{k-1}\cdot {\hat{\eta }}_k>0$. In practice, we choose ${\hat{\eta }}_k$ to be parallel to the projection of ${\hat{\eta }}_{k-1}$ onto the tangent space to the manifold in the actual configuration.

Fig. 7 shows a three-dimensional projection of a one-dimensional manifold obtained by changing 7 consecutive ϕ and ψ Ramachandran’s angles of a portion of a protein. Some configurations of the polypeptide that have been generated during the exploration are also plotted. There are no other configurations of the selected region that are compatible with the constraints imposed by the locality requirement and that can be generated with a continuous modification of the original configuration.

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Fig 7. Three-dimensional projection of the solution manifold.

The whole manifold lies in a seven-dimensional space. Some configurations are highlighted: the starting configuration is emphasized with a red circle while two other possible solutions are emphasized in green. The red part of the structure is the portion which has been modified with our moves.

https://doi.org/10.1371/journal.pone.0118342.g007

Higher dimensional manifolds can be explored as well but, in these cases, there is not a general strategy that allows an efficient exploration of the whole space. This exploration can be achieved with a Monte-Carlo simulation or with ad hoc procedure as in the next example. As a proof of concept, with the only purpose of showing the viability of the method in higher dimensions, we consider a small cyclic molecule (cyclooctatetraene) and we assume that all its 8 torsional angles are soft degrees of freedom that can be modified. Fig. 8 shows four conformations of cyclooctatetraene obtained with our procedure and Fig. 9 shows the whole bidimensional solution manifold.

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Fig 8. Four conformations of the ciclic molecule cyclooctatetraene obtained while exploring its whole conformational space by changing the 8 torsional angles degrees of freedom.

Bond length and bond angles are kept fixed. This is a toy model to illustrate the efficiency of the method: the molecule alternates double and single bonds and therefore half of the torsional angles are constrained and only four are completely free. Images of molecular structures have been generated with PyMol [22].

https://doi.org/10.1371/journal.pone.0118342.g008

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Fig 9. A three-dimensional projection of the solution manifold for our model of cyclooctotetraene molecule (B) and a schematics of how the manifold has been explored (A).

First a set of seven angles are chosen and the relative one-dimensional manifold is visited (red line). Then the one-dimensional manifold relative to a different choice of degrees of freedom is explored using each of the generated structure as a starting point (green dots).

https://doi.org/10.1371/journal.pone.0118342.g009

We highlight the efficiency of the method. The full exploration of cyclooctatetraene configurational space has been performed in about 5 minutes on a single core 2.0 Ghz computer, collecting more than 10⁵ different structures with a sample-step of 10⁻² radians.

Protein backbone mobility

In this section we describe how it is possible to estimate the mobility of a portion of protein backbone (local backbone mobility) by using a simple schema based on the algorithm proposed in this paper. The hypothesis we use is that the local mobility is proportional to the number of configurations that can be explored locally without modifying the rest of the chain, the local backbone volume. In principle, the number of configurations taken into account in this counting could be reduced by eliminating those conformations that exhibit steric clashes. Also, it would be possible to introduce a pair-wise potential in order to consider interaction effects. Here we limit the study of the mobility to non-interacting chains.

Consider the 3N-dimensional configurational space describing the position of each atom composing the system. In analogy with the usual definition of entropy, we define the local backbone entropy as the logarithm of the local backbone volume, that is the volume of the solution manifold measured in the 3N-dimensional configurational space. This volume can be computed by integrating the Gramian of the transformation {α, θ, r} → R^t on the solution manifold in the DH’s variable space. The Gramian function specifies how the (n − 6)-dimensional volume element of the manifold in the space of DH parameters is mapped into the 3N-dimensional configurational space. If s are the coordinates on the manifold, determined within an orthonormal system of basis vectors in the tangent space, the Jacobian of the transformation can be written as ∇_s R^t, and the Gramian is computed as (27) (see [23, 24]).

Values of entropy can be assigned to different protein fragments, i.e. to different subsets of degrees of freedom, by considering the corresponding manifolds defined by their concerted variations. We first estimate the entropy S(i) associated to a single residue i as the sum of the entropies computed for all the different fragments comprising that residue. For simplicity, we restrict our calculation to fragments that comprise seven consecutive ϕ and ψ Ramachandran’s angles. We then call the map i ↦ A × exp(S(i)) the mobility profile of the structure. The factor A is a proportionality constant which has the dimension of an Angstrom squared.

In Fig. 10, we compare the mobility profile with the corresponding experimental data, i.e. the variance of the positions of the α carbon atoms in different NMR models of the same structure. Note that the predicted mobility profile is matched to the experimental one by fitting A, so that only the ratio between the mobilities of different regions of the same protein structure can be considered as a real prediction of our model.

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Fig 10. Comparison between the experimental mobility and the theoretical mobility profile of different structures.

Data in (A) are relative to an all-α protein (1YGM) while (B) shows the mobility of mixed β and α protein (2ITH). The experimental profile has been computed as the variance in the position of α-carbons in different NMR models of the same protein. Red lines represent theoretical calculations, black lines experimental values. Boxes below the horizontal axis locate the position of secondary structures along the chain: red for α-helices and yellow for β-sheet. The matching between the theorethical and experimental profiles has been obtained by optimizing the proportionality constant A with a least square fit. In (A) the fit has been computed over the whole length of the chain, while in (B) the fit concerned only the α regions.

https://doi.org/10.1371/journal.pone.0118342.g010

Remarkably, the mobility of 1YGM, an all-α structure, is described with a good degree of accuracy by the theoretical estimations. This fact, confirmed by similar analysis on other all-α structures (data not shown) suggests that the local geometrical constraints of the protein backbone taken into account by our method are enough to predict the relative mobility of helical and non helical regions. The surprising conclusion is that the presence of both steric and energetic effects reduces the available phase space, and hence the mobility, by the same amount for both regions. On the contrary, β-sheet mobility is not captured at all, probably because we do not consider in our analysis the non-local inter-strand interactions that are crucial for their stability.

Structure refinement

The ability of exploring completely the conformation space of a limited fragment of a protein can be exploited for reconstructing or refining a small portion of a polypeptide structure. Let us suppose to have a putative fragment of a protein which has been roughly reconstructed by experimental methods: the hard degrees of freedom (bond angles and bond lengths) of this portion have correct values, whereas some of the torsional angles may not all be consistent with the allowed values of the Ramachandran’s plot. Given such initial configuration Γ₀, our approach allows to modify exhaustively the soft degrees of freedom and to check if it is feasible to obtain a solution which is compatible with the standard Ramachandran’s plot.

To test this possibility, we first partition the Ramachandran’s plot in a grid of squares with size 2^° × 2^°. By analyzing the databank Top500 formed by a non-redundant, specially refined set of 500 high resolution X-ray crystallographic structures of globular proteins [25], we consider as good, those bins which have a fractional occupancy higher than 0.04. We investigate if the condition of having good Ramachandran’s angles in a small fragment of a protein is a condition sufficient to reconstruct/refine the protein backbone. Therefore we randomly explored the conformational space of different portions of a protein. For each portion we store only the acceptable configurations, i.e. those configurations with good Ramachandran’s angles.

Different 5 residues long fragments were analyzed, from α, β, or coil structures (that are neither in α nor in β). Fig.s 11 and 12 show all the acceptable configurations that have been generated during the exploration of an α-helix and a β-strand portion, respectively. Each subplot shows the values sampled by a different pair of Ramachandran’s angles within the considered fragment. From Fig. 11, we can notice that all acceptable solutions generated from an α-helix structure lie in a very narrow region of the Ramachandran’s plot. The exploration of different α-helix regions confirms this finding and shows that usually about the 90% of the configurations that are compatible with the imposed constraints are acceptable, and hence shown in Fig. 11.

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Fig 11. All acceptable Ramachandran angles obtained during the exploration of the solution manifold of a 5-residue-long α fragment.

All points form a unique cluster, meaning that there is only one acceptable conformation of the helix that is compatible with the initial configuration. Thus, having fixed the first and the last bond, there is only one possible way to reconstruct the missing helix.

https://doi.org/10.1371/journal.pone.0118342.g011

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Fig 12. All acceptable Ramachandran angles obtained during the exploration of the solution manifold of a 5-residue-long β fragment.

Points form different clusters, meaning that there are different acceptable conformations of the strand that are compatible with the initial configuration. It is not possible to safely reconstruct or refine a β fragment by only requiring the final configuration to have acceptable Ramachandran’s angles.

https://doi.org/10.1371/journal.pone.0118342.g012

Going back to the original problem of refining a rough initial estimation of a protein portion, our results imply that, for an α-helix, it is easy to reconstruct a solution that is essentially unique, within very small changes of Ramachandran’s angles. Results are different in the case of β-structure; in such situation the algorithm works smoothly to find solutions but most of them are not in good regions of the Ramachandran’s plot (more than 90%) and those satisfying such constraints, the ones shown in Fig. 12, are more widely distributed in the good region. Therefore, a reconstruction starting by a β-like structure is feasible within our approach, but produces a broader range of possible solutions with respect to α-helical fragments. For both the α and the β fragments, in order to evaluate how exhaustive is our exploration strategy, the stored configurations were analyzed with a sophisticated cluster algorithm [26], checking that the conformational clusters found in each half of the search trajectories do not change.

Repeating the same analysis for coil fragments, it is found that only a very low percentage (usually below 1%) of configurations have good Ramachandran’s angles, whereas the acceptable configurations span all possible allowed values (data not shown). Under such circumstances, an exhaustive search of all possible acceptable solutions can be quite time consuming.