Targeted Conformational Search with Map-Restrained Self-Guided Langevin Dynamics: Application to Flexible Fitting of Electron Microscopy Images

Abstract

We present a map-restrained self-guided Langevin dynamics (MapSGLD) simulation method for efficient targeted conformational search. The targeted conformational search represents simulations under restraints defined by experimental observations and/or by user specified structural requirements. Through map-restraints, this method provides an efficient way to maintain substructures and to set structure targets during conformational searching. With an enhanced conformational searching ability of self-guided Langevin dynamics, this approach is suitable for simulating large-scale conformational changes, such as the formation of macromolecular assemblies and transitions between different conformational states. Using several examples, we illustrate the application of this method in flexible fitting of atomic structures into density maps from cryo-electron microscopy.

Introductions

Conformational search is essential in computational biology for identifying conformations of interest. Due to the large number of degrees of freedom in biomolecular systems, conformational space is often huge. This makes a thorough search prohibitively expensive at best, and impossible in practice for most applications. de novo protein folding that relies solely on force fields has achieved success for some small proteins with massive computing power (Lindorff-Larsen et al., 2011). However, due to the limit in computing resource, macromolecular systems are often overly simplified in simulation studies, such as approximation in solvent representation and arbitrarily assigned charge states of ionizable residues, which often causes inconsistencies between theoretical simulations and experimental observations.

These difficulties and inconsistencies can be partially overcome by introducing experimental constraints or restraints such as secondary structures from circular dichroism (CD)(Johnson, 1988), atomic distances from nuclear magnetic resonance (NMR)(Duggan et al., 2001), residue distances from fluorescence resonance energy transfer (FRET)(Yu et al., 2013) or residue contacts from mutagenesis studies(Warshel and Sussman, 1986). We use the term “targeted conformational search (TCS)” to describe simulations with structural restraints. The structural restraints can be either obtained from experiments such as X-ray crystallography, NMR spectroscopy, and FRET measurements, or derived from existing structural information such as secondary structure predictions and homologous protein structures.

The advance of cryo-electron microscopy (EM) is beginning to open a new window to the analysis of large biomolecular assemblies under biologically relevant conditions. Even though EM images are low in resolution, they have been used to produce complex structures based on individual protein structures from X-ray or NMR methods, often through rigid fitting (Antzutkin et al., 2002; Milne et al., 2006; Milne et al., 2002; Roseman, 2000; Spahn et al., 2000; Wriggers and Birmanns, 2001; Wriggers et al., 1999; Wu et al., 2003).

For systems with flexible components, rigid fitting is difficult to apply. Proteins often adopt different conformations in different states, such as in bound and unbound states. In addition, proteins have certain conformational flexibility and can adapt to different environmental conditions. To accommodate the conformational change, a process called flexible fitting is used to change structures from X-ray or NMR to match electron microscopy images.

A series of methods have been developed to perform flexible fitting. For example, Tama et al. proposed a method that uses a linear combination of low-frequency normal modes from elastic network description to deform the structure to conform to the low-resolution electron density map(Tama et al., 2004). DiMaio et al. (DiMaio et al., 2009) presented a method based on Rosetta structure refinement (Bradley et al., 2005). It uses a local measure of the fit to guide structure refinement and has been shown to achieve near-atomic resolution in some instances starting from density maps at 4-6 Å resolution. Trabuco et al. described a molecular dynamics flexible fitting (MDFF) method(Trabuco et al., 2008; Trabuco et al., 2009). This method uses a grid potential and calculates forces by interpolation. Orzechowski and Tama presented a method based on molecular dynamics simulation(Orzechowski and Tama, 2008) that used a correlation based potential function to induce molecules to fit the map. Grubisic and colleagues presented a coarse-grained approach using a Go-model to represent biological molecules, and used a biased molecular dynamics search to allow conformational transitions(Grubisic et al., 2010). Zheng proposed a coarse-grained pseudo-energy minimization method(Zheng, 2011), which uses two-bead-per-residue to reduce the number of degrees of freedom and to speed up the calculation.

All these methods have contributed to the development of flexible fitting and have their unique attributes and advantages. However, one major problem faced by these methods is that the conformation can end up being trapped in states of local minima. It is desirable to have a method for large-scale conformational searching that has a strong ability to overcome these local energy barriers. Along this direction, Vashisth et al combined MDFF with an enhanced conformational sampling method, temperature-accelerated molecular dynamics (TAMD)(Vashisth et al., 2012). Their comparison simulations of adenylate kinase in explicit solvent showed very limited enhancement (the time to reach the final conformation reduced from ∼0.7ns for MDFF to ∼0.4ns for TAMDFF). In addition, TAMD needs prior knowledge of a molecular system to define collective variables. Self-guided Langevin dynamics (SGLD) is a method developed to dramatically accelerate conformational searching (Wu and Brooks, 2003; Wu et al., 2012). This method is unique in the way that it selectively enhances and suppresses molecular motions based on their frequency to accelerate conformational searching without modifying energy surfaces or raising temperatures. It has been applied to studies of many long time scale events such as protein folding and signal transduction (Damjanovic et al., 2008; Lee and Chang, 2010; Olson and Lee, 2013; Pendse et al., 2010). Recent progress in the understanding of SGLD conformational distribution (Wu and Brooks, 2011a) makes SGLD especially suitable for quantitative studies of molecular systems, for example, in determining free energy differences (König et al., 2012).

This work presents a method that uses maps to define structural targets and uses SGLD to achieve efficient conformational searching. We call this method the map-restrained SGLD method (MapSGLD). The restraint maps are included in simulations as movable objects called rigid domains, which interact with simulation systems to achieve the restraining effect and can move with their restraining atoms. With the advantages of the map-restraints and the searching ability of SGLD, this method has been applied previously in the study of macromolecular assemblies based on EM images (Elegheert et al., 2011; Jayasinghe et al., 2012). In this work, we focus on the description of the method and the demonstration of its application in deriving protein structures with flexible components.

Methods

The map-restraint potential

A map represents a distribution over a spatial region. Typically, a map is described by quantities on lattice grid points:

$$M(N_x,N_y,N_z)=\underset{i,j,k}{\overset{N_x,N_y,N_z}{\cup }}\rho (x_i,y_j,z_k)$$ (1)

Here, the map grids are defined by grid pints, (x_i, y_j, z_k). The numbers of grids in the x, y, and z directions are N_x, N_y, and N_z, respectively. The symbol, $\underset{i,j,k}{\overset{N_x,N_y,N_z}{\cup }}$, represents a union of all entities with i ∈ N_x, j ∈ N_y, and k∈ N_z. The x, y, and z directions can have any angle between them. For convenience, in this work, we assume them to be orthogonal to each other. The distribution property, ρ(x_i,y_j,z_k), is the electron density for electron microscopy maps.

To avoid arbitrary units and background levels of map densities, it is convenient to work with a normalized map where:

$$\widehat{\rho }(x_i,y_j,z_k)=\frac{\rho (x_i,y_j,z_k)-\overline{\rho }}{\delta \rho }$$ (2)

Here, ρ̄ is the map average and δρ is the map standard deviation. They are calculated with the following equations:

$$\overline{\rho }=\frac{1}{N_x{N}_y{N}_z}\sum _{i,j,k}^{N_x,N_y,N_z}\rho (x_i,y_j,z_k)$$ (3)

$$\delta \rho =\sqrt{\frac{1}{N_x{N}_y{N}_z}\sum _{i,j,k}^{N_x,N_Y,N_z}{(\rho (x_i,y_j,z_k)-\overline{\rho })}^2}$$ (4)

As can be seen from eqs. (2)-(4), a normalized map has ρ̂̄ = 0 and δρ ^ = 1.

If we set a map as the target of a group of N atoms, at the target conformation, these atoms should produce this map and these atoms must all sit at high-density positions. Because atomic masses correlate to their numbers of electrons, we simply define a map-restraint potential by the products between the atomic mass, m_a, and the normalized density at the atom position, ρ̂(x_a, y_a, z_a):

$$E_{\text{map}}=-c_{\text{map}}\sum _a^N{m}_a\widehat{\rho }(x_a,y_a,z_a)$$ (5)

The restraint constant, c_map, sets the strength of the map-restraint. The units of m_a and c_map are g/mol and kcal/g, respectively. Eq.(5) produces an energy landscape in the shape of the density distribution, −c_mapρ̂(x_a,y_a,z_a), for every restrained atom, a. It induces atoms to move to positions of lower energy, or of higher density. Obviously, eq.(5) represents a simplified correlation between atom masses and the map density distribution.

This map-restraint potential captures the low-resolution characteristics of molecular systems and is not designed to reproduce atomic structures by itself. Instead, when combined with an all-atom force field, which contains bonded interactions (bond lengths, bond angles, dihedral angles) and nonbonded interactions (van der Waals, electrostatic interactions, solvation), the map-restraint potential can help to stabilize conformations that match the restraint map. It is the combination of a force field and the map-restraint that drives a system to the target conformation. This map-restraint potential has an order of O(N) and is very efficient to calculate as compared to other pair wise non-bonded interactions.

Eq.(5) needs densities at the positions of the atoms, but a map has densities only at discrete grid points. To obtain the density at an atom's position, (x_a, y_a, z_a), one can interpolate from its neighboring grid points.

$$\widehat{\rho }(x_a,y_a,z_a)=\sum _{i,j,k}^{N_x,N_y,N_z}s(x_a-x_i,y_a-y_j,z_a-z_k)\widehat{\rho }(x_i,y_j,z_k)$$ (6)

Here, s(x_a− x_i, y_a− y_j, z_a− z_k) is an interpolation function. Many options are available for interpolation. Considering computing efficiency, this work uses b-spline to do the interpolation:

$$s(x_a-x_i,y_a-y_j,z_a-z_k)=b_m(x_a-x_i)b_m(y_a-y_j)b_m(z_a-z_k)$$ (7)

Here, b_m, represents a b-spline function of the mth order, which can be calculated with the following recursive equations:

$$b_0(x_a-x_i)=\{\begin{array}{cc}1& x_i\le x_a<x_{i+1}\\ 0& \text{otherwise}\end{array}$$ (8a)

$$b_m(x_a-x_i)=\frac{x_a-x_i}{x_{m+i}-x_i}{b}_{m-1}(x_a-x_i)+\frac{x_{m+i+1}-x_a}{x_{m+i+1}-x_{i+1}}{b}_{m-1}(x_a-x_{i+1})$$ (8b)

b-spline is not only convenient for interpolation, but also efficient for derivative calculation, which is crucial for efficient molecular dynamics simulation.

$$\frac{\partial b_m(x_a-x_i)}{\partial x_a}=b_{m-1}(x_a-x_i)-b_{m-1}(x_a-x_{i+1})$$ (9)

Based on eq.(9), the restraint force, ${\mathbf{f}}_a^{(\text{map})}=(f_{\mathit{\text{ax}}}^{(\text{mab})},f_{\mathit{\text{ay}}}^{(\text{map})},f_{\mathit{\text{az}}}^{(\text{map})})$, on atom a can be calculated by the following equation:

$$\begin{array}{l}{f}_{\mathit{\text{ax}}}^{\left(\text{map}\right)}=-\frac{\partial }{\partial x_a}{E}_{\text{map}}=c_{\text{map}}{m}_a\frac{\partial }{\partial x_a}\widehat{\rho }(x_a,y_a,z_a)\\ =c_{\text{map}}{m}_a\sum _{i,j,k}^{N_x,N_y,N_z}\widehat{\rho }(x_i,y_j,z_k)b_m(y_a-y_j)b_m(z_a-z_k)\frac{\partial }{\partial x_a}{b}_m(x_a-x_i)\end{array}$$ (10)

The restraint map is often obtained from electron microscopy experiments. However, it can also be created from molecular structures by spreading atom masses over the grid points:

$$\rho (x_i,y_j,z_k)=\sum _a^{N}s(x_a-x_i,y_a-y_j,z_a-z_k)m_a$$ (11)

The resolution of the created map can be conveniently controlled with the grid intervals and the order of b-spline,

$$R_x\approx \frac{m\Delta x}{2},R_y\approx \frac{m\Delta y}{2},R_z\approx \frac{m\Delta z}{2},$$

which correspond to resolutions in the x, y, and z directions, respectively. A map created with above resolution-specified grid intervals can be converted to a map of other grid intervals by interpolating.

The map-restraints work in a cooperative way so that atoms interacting through the force field contribute together to match the map density distribution. The map-restraints have the following characteristics that are helpful for a targeted conformational search:

1. They have soft energy surfaces that make large-scale conformational transition feasible.

2. They are atom identity blind, so the restraint energy calculation is of O(N).

3. They allow flexibility for the restrained systems and tolerate noise.

4. They can be extended to represent other properties such as partial charges, desolvation energies and van der Waals interactions (Wu and Brooks, 2007).

Self-Guided Langevin Dynamics

The self-guided Langevin dynamics(SGLD) simulation method (Wu and Brooks, 2003; Wu and Brooks, 2011a; Wu and Brooks, 2011b; Wu et al., 2012) was developed for conducting an efficient search of conformational space. Here, we briefly describe the concept of this method.

For any particle, a, the equation of the self-guided motion has the following general form:

$${\dot{\mathbf{p}}}_a={\mathbf{f}}_a^{(\text{ff})}+{\mathbf{f}}_a^{(\text{map})}+{\mathbf{g}}_a-{\gamma }_a{\mathbf{p}}_a+{\mathbf{R}}_a$$ (12)

Where p˙_a is the time derivative of momentum. ${\mathbf{f}}_a^{(\text{ff})}$ is the interaction force due to the force field. ${\mathbf{f}}_a^{(\text{map})}$, is the force from the restraint map. R_a represents a random force, which is related to the mass, m_a. the collision frequency, γ_a, and the simulation temperature, T, by the following equation:

$$<{\mathbf{R}}_a(0){\mathbf{R}}_a(t)>=2m_a\mathit{\text{kT}}{\gamma }_a\delta (t)$$ (13)

Eq. (12) contains a guiding force, g_a, which is calculated based on the momentum, p_a, and the low frequency momentum, p̃ _a:

$${\mathbf{g}}_a(t)=\lambda {\gamma }_a({\stackrel{\sim }{\mathbf{p}}}_a(t)-\xi {\mathbf{p}}_a(t))$$ (14)

Here, λ is the guiding factor, which defines the strength of the guiding force. When λ = 0, eq. (12) reduces to the equation of motion of Langevin dynamics. The parameter, ξ, is an energy conservation factor to eliminate any net energy input from the guiding force and is calculated at each dynamic step from the following equation:

$$\xi =\frac{\sum _i{\gamma }_i{\stackrel{\sim }{\mathbf{p}}}_i.{\dot{\mathbf{r}}}_i}{\sum _i{\gamma }_i{\mathbf{p}}_i.{\dot{\mathbf{r}}}_i}$$ (15)

The low frequency momentum, p̃_a, is calculated as a local average in the following progressive way:

$${\stackrel{\sim }{\mathbf{p}}}_a(t)=(1-\frac{\delta t}{t_L}){\stackrel{\sim }{p}}_a(t-\delta t)+\frac{\delta t}{t_L}{\mathbf{p}}_a(t)$$ (16)

Here, t_L, is the local averaging time and δt is the simulation time step. From the low frequency momentum, we define the low frequency temperature, T̃, as:

$$\stackrel{\sim }{T}=\frac{1}{N_{\text{DF}}k}\langle \sum _i\frac{{\stackrel{\sim }{\mathbf{p}}}_i^2}{m_i}\rangle$$ (17)

To quantitatively describe the conformational search ability of an SGLD simulation, we define the self-guiding temperature, T_SG, as:

$$T_{\text{SG}}=\frac{\stackrel{\sim }{T}(T-{\stackrel{\sim }{T}}_0)}{{\stackrel{\sim }{T}}_0(T-\stackrel{\sim }{T})}T$$ (18)

where T̃₀ is the reference low frequency temperature, which corresponds to the low frequency temperature when the guiding factor, λ, is zero. The self-guiding temperature, T_SG, provides a rough measurement of the conformational searching ability in the unit of temperature. An SGLD simulation with a self-guiding temperature of T_SG has conformational search ability comparable to that of a high temperature simulation at T=T_SG. In SGLD simulations, λ, or set a target self-guiding temperature, $T_{\text{SG}}^0$, and let the guiding factor, λ, be automatically adjusted so that T_SG approaches $T_{\text{SG}}^0$.

Langevin motion of restraint maps

Besides flexible fitting of EM images, there are many other applications for targeted conformational search. A typical case for macromolecular assemblies is to search conformations without unfolding of individual proteins or domains. This can be achieved by using the map-restraints to maintain the folded structures and allowing these maps to move with the individual proteins or domains they are restraining.

Because a map contains a large amount of data, it is inconvenient to move the map itself. For example, a map rotation will result in its grid points moving off lattice, which requires interpolations to calculate the properties on the new grid points. To avoid repeated interpolations and error accumulations, we first define the idea of using “rigid domains” to represent maps (Fig. 1). A rigid domain contains only the identity of the map object it represents, and the position and orientation vectors related to the map object. Because the position and orientation vectors are continuous quantities, rigid domains can be manipulated conveniently and accurately. Each rigid domain has a unique map identity and many rigid domains can refer to the same map.

Fig.1.

Use of rigid domains to represent restraint maps in MapSGLD simulations. (a) initial map; (b) a rigid domain representing the position and orientation of a map; (c) translation of a rigid domain; (d) rotation of a rigid domain; (e) projection of a map from a rigid domain.

For each rigid domain, its translation vector, T, and rotational matrix, U, have the following form:

$$\mathbf{T}=\left(\begin{array}{c}{t}_x\\ t_y\\ t_z\end{array}\right)$$ (19)

$$\mathbf{U}=\left(\begin{array}{ccc}\begin{array}{c}{u}_{11}\\ u_{21}\\ u_{31}\end{array}& \begin{array}{c}{u}_{12}\\ u_{22}\\ u_{32}\end{array}& \begin{array}{c}{u}_{13}\\ u_{23}\\ u_{33}\end{array}\end{array}\right)$$ (20)

The operations, translations and rotations, are performed through these vectors

$${\mathbf{T}}^{(i+1)}={\mathbf{T}}^{(i)}+\Delta {\mathbf{T}}^{(i+1)}=\left(\begin{array}{c}{t}_x^{(i)}\\ t_y^{(i)}\\ t_z^{(i)}\end{array}\right)+\left(\begin{array}{c}\Delta t_x^{(i+1)}\\ \Delta t_y^{(i+1)}\\ \Delta t_z^{(i+1)}\end{array}\right)$$ (21)

$${\mathbf{U}}^{(i+1)}={\Omega }^{(i+1)}\times {\mathbf{U}}^{\left(i\right)}=\left(\begin{array}{ccc}{\omega }_{11}^{(i+1)}& {\omega }_{12}^{(i+1)}& {\omega }_{13}^{(i+1)}\\ {\omega }_{21}^{(i+1)}& {\omega }_{22}^{(i+1)}& {\omega }_{23}^{(i+1)}\\ {\omega }_{31}^{(i+1)}& {\omega }_{32}^{(i+1)}& {\omega }_{33}^{(i+1)}\end{array}\right)\times \left(\begin{array}{ccc}{u}_{11}^{\left(i\right)}& u_{12}^{\left(i\right)}& u_{13}^{\left(i\right)}\\ u_{21}^{\left(i\right)}& u_{22}^{\left(i\right)}& u_{23}^{\left(i\right)}\\ u_{31}^{\left(i\right)}& u_{32}^{\left(i\right)}& u_{33}^{\left(i\right)}\end{array}\right)$$ (22)

and many operations can be accumulated:

$${\mathbf{T}}^{(n)}={\mathbf{T}}^{(n-1)}+\Delta {\mathbf{T}}^{(n)}={\mathbf{T}}^{(n-2)}+\Delta {\mathbf{T}}^{(n)}+\Delta {\mathbf{T}}^{(n-1)}={\mathbf{T}}^{(0)}+\sum _{i=1}^n\Delta {\mathbf{T}}^{(i)}$$ (23)

$${\mathbf{U}}^{\left(n\right)}={\Omega }^{\left(n\right)}\times {\mathbf{U}}^{(n-1)}={\Omega }^{\left(n\right)}\times {\Omega }^{(n-1)}\times {\mathbf{U}}^{(n-2)}=\prod _{i=1}^n{\Omega }^{\left(i\right)}\times {\mathbf{U}}^{\left(0\right)}$$ (24)

At any moment, the instantaneous map that a rigid domain represents can be projected by applying the translation and rotation to the original map:

$$\mathbf{X}=\mathbf{T}+\mathbf{U}\times {\mathbf{X}}^{\left(\mathit{\text{ref}}\right)}$$ (25)

We use the Langevin position equation of motion(Allen and Tildesley, 1987) to describe the movement of rigid domains. For translation, we have:

$${\gamma }_{\text{map}}{\mathbf{p}}_{\text{map}}={\mathbf{f}}_{\text{map}}=-\sum _a^N{\mathbf{f}}_a^{\left(\text{map}\right)}$$ (26)

Here, τ_map is the map collision frequency used to control the speed of the map and f_map is the total force on this map, which is opposite to the sum of map-restraint forces, ${\mathbf{f}}_a^{(\text{map})}$, on all restrained atoms. Similarly, for rotation, we have:

$${\gamma }_{\text{map}}{\mathrm{L}}_{\text{map}}={\tau }_{\text{map}}=-\sum _a^N{\mathrm{r}}_a\times {\mathrm{f}}_a^{\left(\text{map}\right)}$$ (27)

Here, τ_map is the torque on the map. After each time step, δt, the map changes its position:

$$\Delta T=\frac{{\mathbf{f}}_{\text{map}}\delta t}{{\gamma }_{\text{map}}{m}_{\text{map}}}$$ (28)

and rotates around the torque axis:

$$\Delta \mathbf{\alpha }=\frac{{\mathbf{\tau }}_{\text{map}}\delta t}{{\gamma }_{\text{map}}{I}_{\text{map}}}$$ (29)

The step rotation matrix is:

$$\mathbf{\Omega }=\left(\begin{array}{ccc}cos\Delta \alpha +{\widehat{\tau }}_x^2(1-cos\Delta \alpha )& {\widehat{\tau }}_x{\widehat{\tau }}_y(1-cos\Delta \alpha )-{\widehat{\tau }}_{s}sin\Delta \alpha & {\widehat{\tau }}_x{\widehat{\tau }}_z(1-cos\Delta \alpha )+{\widehat{\tau }}_{y}sin\Delta \alpha \\ {\widehat{\tau }}_z{\widehat{\tau }}_x(1-cos\Delta \alpha )+{\widehat{\tau }}_{z}sin\Delta \alpha & cos\Delta \alpha +{\widehat{\tau }}_y^2(1-cos\Delta \alpha )& {\widehat{\tau }}_y{\widehat{\tau }}_z(1-cos\Delta \alpha )-{\widehat{\tau }}_{x}sin\Delta \alpha \\ {\widehat{\tau }}_z{\widehat{\tau }}_x(1-cos\Delta \alpha )+{\widehat{\tau }}_{y}sin\Delta \alpha & {\widehat{\tau }}_z{\widehat{\tau }}_y(1-cos\Delta \alpha )+{\widehat{\tau }}_{x}sin\Delta \alpha & cos\Delta \alpha +{\widehat{\tau }}_z^2(1-cos\Delta \alpha )\end{array}\right)$$ (30)

Where $m_{\text{map}}=\sum _a^N{m}_a$ and $I_{\text{map}}=\sum _a^N{m}_a{({\mathbf{r}}_a-{\mathbf{r}}_{\text{COM}})}^2$ are the mass and moment of inertia of the map, calculated using the atoms restrained by the map. Here, we assume an isotropic distribution of atomic masses to simplify the rotation of rigid domains. The movement of a rigid domain is carried out by updating its position, T, and rotation matrix, U, according to eqs. (23) and (24), respectively.

Simulation details

The MapSGLD method has been implemented into AMBER(Case et al., 2005; Case et al., 2012) and CHARMM(Brooks et al., 1983; Brooks et al., 2009). The simulation results presented here were obtained with the SANDER module of AMBER 12. Users can use the test cases in AMBER 12 (test/emap/Run.emap) and in CHARMM c38 (test/c38test/mapsgld.inp) as examples to use the method.

Several parameters control a MapSGLD simulation. The map-restraint constant, c_map, defines the restraint strength. When a restraint map is generated from a structure, the map resolution, R_map, controls the broadness of the restraint potential. For simulations allowing map movement, the map collision frequency, γ_map, can be defined to control how easy the restraint map moves. The collision frequency, γ, determines the coupling with the thermal bath and acts as a “friction” factor to the movement of the simulated system. The local averaging time, t_L, defines the frequency range of the enhanced motion. The guiding factor, λ, or the guiding temperature, T_SG, determines how much the low frequency motion to be enhanced. In the example presented here, we set c_map=0.05 kcal/g, R_map=3Å, γ_map=1/ps, γ=1/ps, t_L=1.0ps, λ=1, unless noted otherwise.

In MapSGLD simulations, systems are set up exactly the same way as in a molecular dynamics (MD) simulation. Users can choose to use explicit solvent or implicit solvation models. Simulations can be performed in bulk solvent, vacuum, or interfacial region. In the examples presented here, the AMBER force field FF99SB(Wickstrom et al., 2009) was used. For a small protein, the B1 domain of streptococcal protein G, the generalized Born model (Mongan et al., 2007) was used to describe solvation effect. For examples with chaperonin assemblies, simulations were performed in vacuum and the isotropic periodic sum (3D IPS) method(Wu and Brooks, 2005; Wu and Brooks, 2008; Wu and Brooks, 2009; Wu and Brooks, 2012) was used for non-bonded energy calculations. The cutoff distance was set to 9 Å. The electron microscopy maps for the chaperonin examples were downloaded from the EM databank(Lawson et al., 2011). All simulations were run in parallel on the LOBOS cluster of the Laboratory of Computational Biology, NHLBI, NIH.

Results and Discussions

As a general method for targeted conformational search, MapSGLD uses map-estraints to set conformation targets and uses SGLD to perform efficient conformational searching. This section explains the behavior and application of this method. We first use a small protein to examine the effects of the MapSGLD simulation parameters. Next, we demonstrate the application of this method in flexible fitting and in the simulation of state transitions based on published EM maps.

1. Effects of MAPSGLD parameters

In protein folding studies, it has been observed that secondary structure elements fold first, followed by their arrangement to form tertiary structures. To study how the secondary structure elements assemble the tertiary structures, it requires to maintain these secondary structures during simulations. Similarly, for protein assemblies, it is desired to simulate how individual proteins assemble to form the complex structure without unfolding. These simulations are typical examples of targeted conformational search. Here, we use a small protein, the B1 domain of streptococcal protein G, abbreviated here as GB1, to illustrate the application of MapSGLD in this type of application and to examine the effect of simulation parameters.

GB1 has 56 residues with one α-helix and one β-sheet. The β-sheet is made of two β-hairpins. Fig.2 shows the NMR structure (PDB: 1gb1) and a partially unfolded conformation to illustrate the helix and β-hairpins. Three restraint maps for the three secondary structure motifs were generated from the NMR structure: residues 1 to 19 for the N-terminal β-hairpin, residues 22 to 37 for the helix, and residues 42 to 56 for the C-terminal β-hairpin. In the following MapSGLD simulations, we will examine how well the map-restraints maintain the structures, and how fast these structure elements move to search the conformation space.

Fig.2.

The NMR structure and a partially unfolded conformation of the B1 domain of streptococcal protein G (GB1). The protein has three secondary structure motifs: one α-helix (yellow) and two β-hairpins (red for the N-terminal one and green for the C-terminal one). In the folded state, the two β-hairpins form a β-sheet and in the partial unfolded state these motifs separate from each other. The restraint maps are generated from the folded structure motifs at a resolution of 3 Å.

1.1 Strength of map-restraints

First, we examine how well the map-restraint potential maintains a structure. The folded and partially unfolded conformations of GB1 were restrained with maps generated from their three secondary structure elements. These substructure maps were not movable so we could examine how well the map can maintain the overall structure. Fig.3 shows the root-mean-square deviations (rmsd) of the simulated conformations for the folded (bottom panel) and the partially unfolded (top panel) structures. All root-mean-square-deviation (rmsd) values in this work were calculated with the backbone atoms, N, Cα, and C. Because of strong intra protein interactions, the folded structure has little tendency to deviate from the folded state. As can be seen from the bottom panel of Fig.3, at c_map =0 (no map-restraint), the rmsd is around 1.1 Å. As c_map increases, the rmsd decreases and at c_map=0.1 kcal/g, the rmsd is around 0.25 Å. Clearly, the map-restraint can maintain the folded structure accurately.

Fig.3.

Conformation changes of GB1 restrained with maps of different restraint strengths. The rmsds are calculated using the backbone atoms against the initial conformations.

For the partially unfolded structure, intra protein interactions are much weaker and the conformation has a stronger tendency to change. From the top panel of Fig.3 we can see that with no map-restraint (c_map=0), the conformation quickly reached a rmsd above 6 Å. At c_map>0.002, the protein can maintain the structure around its initial conformation, and at c_map=0.01 kcal/g the rmsd is maintained around 0.4Å. Again, the map-restraint can hold the partially unfolded structure accurately. Typically, we use c_map =0.01∼0.1 kcal/g in our MapSGLD simulations. A strong map-restraint, e.g., c_map =0.1 kcal/g, is sufficient to maintain a structure, while a weak map-restraint, e.g., c_map =0.01 kcal/g, would allow large-scale conformational changes such as during state transitions.

1.2 Resolution of restraint maps

Besides the restraint constant, the map resolution also affects the restraining effect. For experimental maps, the resolution is not an option to choose. But for maps generated from atomic structures, which we often use as map restraints to main substructures, we can select a resolution to control the broadness of the restraint potential. A lower resolution (a large R_map value) results in a broader distribution of electron density and a looser restraint potential, which allows larger rmsd values and more structural flexibility.

Fig. 4 shows the rmsd profiles during the simulations restrained with fixed maps of different resolutions. The folded system remains folded but rmsd values increase with R_map (Fig. 4, bottom panel). The partially unfolded system shows much larger rmsd(top panel of Fig.4). Under these conditions, a resolution of R_map =5 Å can keep rmsd values <1 Å. However, a resolution of R_map =10 Å or R_map =20 Å allows the unfolded state to deviate by ∼ 2 Å or ∼ 4 Å, respectively. These results confirm that increasing R_map allows the system to have more flexibility and using R_map =3∼5 Å allows the structure to remain folded.

Fig.4.

Conformation changes of GB1 restrained with maps of different resolutions. The rmsds are calculated using the backbone atoms against the initial conformations.

1.3 Friction of the restraint maps

When searching the conformational space with movable map-restraints, the moving speed of rigid domains depends on the force, torque, and the map collision frequency. The lower panel of Fig.5 shows the conformational changes at different map friction factors. When the map collision frequency, γ_map, increases from 1/ps to 100/ps, the rmsd change over the 1000 ps simulation decreases from 8∼10Å to <1 Å. Typically, we choose γ_map=1/ps to allow the restraint map to easily follow the movement of its restraining structure.

Fig.5.

Conformation changes of the partially unfolded GB1 during MapSGLD simulations with different map frictions (lower panel) and different guiding temperatures (upper panel).

1.4 The self-guiding effect

The map motion can be accelerated with the SGLD guiding force by choosing a large guiding factor, λ, or a higher guiding temperature, T_SG. The upper panel of Fig. 5 shows the conformational changes of the partially unfolded GB1 at different guiding temperatures, ranging from 300K to 1000K. As can be seen, higher guiding temperatures result in faster motion of the restrained domains. Typically, we use T_SG=500 K to gain sufficient acceleration in conformational search.

1.5 Folding simulations with MapSGLD

An important application of targeted conformational search is to fold proteins to conformations defined by a target map. For example, in homology modeling, proteins with high sequence similarity are assumed to have similar 3D structures. Therefore, it is reasonable to use the structure of a homologous protein to generate a target map and search for folded structures through MapSGLD simulations. Here, we show the folding simulations of GB1 from the partially unfolded conformation.

Multiple map-restraints were applied in this targeted conformational search. First, the three secondary structure elements were maintained with movable maps generated from the initial conformation with a resolution of 3 Å and a restraint constant of 0.1 kcal/g. Second, the target map of the whole protein was generated from the NMR structure (PDB: 1gb1) with a resolution of 5 Å. As explained before, the resolution defines how broad the map-restraint potential is. A lower resolution allows conformational change easier. The target map is not movable and the restraint constant for the target map is 0.01 kcal/g. Fig.6 shows two simulation results, one with T_SG =300 K and one with T_SG =500K. When T_SG =300 K=T, the conformational search was not enhanced and the SGLD simulation was reduced to a regular Langevin dynamics (LD) simulation. In this case, the simulation failed to reach the folded conformation in up to 100 ns. While in the case of T_SG =500K, accelerated conformational motion prompted the protein to reach the folded state in 9.5 ns. In other words, the map-restraint itself is not enough to bring the protein to the folded state in 100 ns, but SGLD can significantly accelerate the search for the target conformation.

Fig.6.

Map-induced folding of GB1 through MapSGLD simulations. The secondary structures are restrained with movable maps generated from the initial conformation. The whole system is restrained with a fixed map generated from the pdb structure: 1gb1. The rmsd is calculated against the NMR structure using the backbone atoms. Some conformations during the simulations are shown to illustrate conformational changes.

2. Flexible fitting to EM maps

A major application of MapSGLD is the flexible fitting of macromolecules into EM maps. MapSGLD searches conformational space according to the energy landscape defined by all-atom force fields and the map-restraint potentials to obtain high-resolution structures that match the EM maps. Here, we show the flexible fitting of GroEL to demonstrate the application of this method.

We downloaded the EM maps of a GroEL chaperonin from the EM databank (Lawson et al., 2011). The EM maps of the GroEL at seven conformational states were obtained by Clare et al.(Clare et al., 2012) and are named EMD-1997, EMD-1998, EMD-1999, EMD-2000, EMD-2001, EMD-2002, and EMD-2003. Their resolutions are between 7 and 10 Å. The PDB structure, 1OEL, was used as the starting conformation for these MapSGLD simulations.

The map, EMD-1997, is assumed to be in the same state as the x-ray structure 1OEL, so we can use 1OEL as a reference to demonstrate the effects of simulation parameters on the flexible fitting. Using EMD-1997 as the restraint map, five simulations with c_map=0, 0.01, 0.02, 0.05, and 0.1 kcal/g were performed and the results are shown in Fig.7. From the superimposed conformations in Fig.7, we can see a slightly improved alignment in secondary structure elements after the simulation. To examine the conformational change between the x-ray structure and the flexible fitting result, we show the rmsd as functions of vertical (top), horizontal (middle), and radial (bottom) distances in the right panel of Fig.7. As the restraint constant, c_map, increases, the rmsd decreases in all measurements. The assembly is hollow inside and the horizontal and radial distances cross the inner surface first and reach the outer surface last. From the bottom panel of Fig.7 we can see a peak at the inner surface, indicating that the inner surface is the most flexible region, which may be related to the function of the GroEL folding chamber. From the middle panel of Fig.7 we can see peaks at the inner and outer surface region. The inner surface peak is higher than the outer one, indicating that the inner surface is more flexible than the outer surface. The top panel of Fig.7 shows the rmsd as a function of the distance from the x-y plane crossing the center. The farther from the center, the more flexible the assembly becomes. The apical domains at the opening region have the largest distance to the x-y plane and are the most flexible. Without the map-restraint (c_map=0), we see significant conformational changes at the top and bottom region, which indicate that this simulation condition cannot maintain the initial structure. The inability to maintain the structure is mainly because this simulation condition was overly simplified. For example, solvent was not included and the charge states of ionizable residues were assigned without consider the pH environment. As c_map increases, the rmsd decreases, indicating the map-restraint can help to overcome the effect of these simplifications and to maintain the structure to match the EM map.

Fig. 7.

Flexible fitting of GroEL with MapSGLD. The left panel shows the map of EMD-1997 and the conformations before (red) and after (green) the flexible fitting with c_map =0.05 kcal/g. The initial GroEL structure is the PDB structure 1OEL. The right panel shows rmsds of the backbone atoms as functions of distances in radial, horizontal, and vertical directions. Curves in different colors are simulation results with different restraint constants (c_map, kcal/g), as labeled in the top panel.

Fig.8 shows the simulation profiles during the MapSGLD simulations. The top panel shows the rmsd against the x-ray structure. At c_map=0, no map-restraint is on the system, and the conformation quickly deviates from its starting conformation. When c_map increases, the conformation deviation decreases and reaches a limit of about 1.5 Å. This limit represents a combination of the structural difference between the EM structure and the x-ray structure, the structure distortion due to the noises in the map, and structural fluctuation due to thermal motion.

Fig.8.

Simulation profiles during the MapSGLD flexible fitting of the GroEL into EMD-1997. Top: The rmsds of the backbone atoms against the initial conformation (PDB:1OEL). Middle: the fitting scores defined by eq.(31). Bottom: molecular energies excluding map-restraints.

The agreement between a structure and a restraint map can be measured by the fitting score defined by the following equation:

$$S_{\text{map}}=\frac{\sum _a^N{m}_a\widehat{\rho }(x_a,y_a,z_a)}{\sum _a^N{m}_a}$$ (31)

This fitting score measures the average density at atom positions. As can be seen from the middle panel of Fig. 8, with c_map =0.01 or 0.02 kcal/g, the fitting scores decreases during the simulations, indicating that these restraint strengths are not enough to maintain the conformation. With c_map =0.05 or 0.1 kcal/g, the fitting scores increase during the simulations, indicating that these restraint strengths can improve the fitting. Therefore, we suggest using c_map =0.05 kcal/g or larger for flexible fitting. As can be seen from eq.(5), the map-restraint energy is proportional to system size. The same is true for other extensive properties, such as kinetic energy and potential energy. Therefore, the map-restraint constant, c_map, is an intensive property and independent of system sizes. The suggested value, c_map =0.05 kcal/g, should apply to all systems, even though optimal values may vary, depending on the force field and simulation conditions.

The bottom panel of Fig. 8 shows the molecular energies during these simulations. We can see that as c_map increases, the molecular energy increases. This is because the map-restraint allows deviation from the minimum of the force field. A stronger map-restraint results in a larger deviation from the force field minimum state. However, the map-restraint helps to maintain the native structure. When c_map =0 kcal/g, there is no map-restraint on the system and the overly simplified simulation conditions, such as lack of solvent, prompt the conformation to drift away from the native structure. As can be seen from the bottom panel of Fig.8, the simulation with c_map =0 kcal/g leads to conformations with higher energies than those reached with c_map =0.01 or 0.02 kcal/g.

EM maps often have significant noise that may cause distortions due to over fitting, such as stretched bonds, bended bond angles, and misfolded secondary structures. These distortions can be suppressed by imposing a map-restraint generated from an undistorted structure. With MapSGLD, we can restrain individual domains or sub domains with maps generated from their x-ray structures to minimize distortions caused by over fitting. Because map-restraints are soft and identity blind, using maps to maintain domain structures does not prevent small conformational changes such as in side chains and in the terminus regions. For the GroEL assembly, each monomer contains three domains: the apical domain (residues 190-373), the intermediate domain(residues 136-189, 374-408) and the equatorial domain(residues 1-135, 409-524). For each domain, we generated a movable restraint map to maintain its structure. The GroEL assembly has 14 monomers, and therefore has 42 domains. We generated 42 movable domain maps from the x-ray structure to maintain all domain structures. These restraint maps were created with a resolution of 3 Å and were applied with a restraint constant of 0.1 kcal/g. In addition to these movable domain maps, the experimental EM maps were applied as fixed restraint maps with a restraint constant of 0.05 kcal/g to induce the assembly to match these maps. By fitting the x-ray structure to the seven EM maps, we obtained atomic structures of these seven states as shown in Fig.9. We label the results for EMD-1997, EMD-1998, …, EMD-2003 as (a) - (g), respectively. During the simulations, the rmsd reached equilibrium value in less than 100 ps. The final rmsds from the initial conformation are 0.68, 5.41, 5.15, 6.60, 7.76, 7.46, 11.3 Å, for the fitting results, (a) - (g), respectively.

Fig. 9.

Comparison of the GroEL structures before (blue) and after (rainbow) flexible fitting with MapSGLD simulations. Results (a)∼(g) correspond to the results with maps EMD-1997∼EMD-2003 from the EM databank. The initial GroEL structure is 1OEL from PDB. The left column shows the top views superimposed with the EM maps. The middle and right columns are the top and side views without the EM maps.

The result for EMD-1997, structure (a) in Fig.9, has an rmsd of 0.68 Å, while the result without the domain map-restraints shown in Fig.8 has an rmsd of 1.56Å. This reduction in rmsd is a result of the domain map-restraints limiting the conformational space and suppressing the effect of noise.

Fig.10 shows the rmsd values of residues along the polypeptide chains in the final structures. Residues in the equatorial domain have relatively small rmsd values, while those in the apical domain have the largest rmsd values. Detailed analysis of the conformational differences is beyond the scope of this work. Please refer to the original EM study (Clare et al., 2012) for a more detailed interpretation of the significance of these conformational changes for GroEL function.

Fig. 10.

Residue deviations in the seven states. Results (a)∼(g) correspond to the results with maps EMD-1997∼EMD-2003 from EM databank. The rmsd is calculated using the backbone atoms against the initial conformation (PDB:1OEL).

3. Dynamics of a group II chaperonin

One advantage of using SGLD for the targeted conformational search is to promote large-scale conformational changes necessary for protein functions. We chose the open-close transitions of a group II chaperonin to demonstrate the application of this method.

The EM maps of the open and close states of a group II chaperonin, mn-cpn, are available in the EM databank: EMD-5138 (close state) and EMD-5140 (open state)(Zhang et al., 2010). Zhang and colleagues have modeled the structures of the closed state (PDB:3J03)(Zhang et al., 2011) and the open state (PDB:3IYF)(Zhang et al., 2010). We used these model structures as starting conformations and use the maps of the opposite states as restraints to perform MapSGLD simulations.

For the chaperonin assembly, each monomer contains three domains, the apical domain (residues 205-334), the intermediate domain(residues 136-204, 335-371) and the equatorial domain(residues 1-135, 372-491). These domains were maintained with movable map-restraints generated from the initial conformation with a resolution of 3 Å and a map-restraint constant of 0.1 kcal/g. The EM maps were applied to the system as fixed map-restraints with a restraint constant of 0.05 kcal/g.

Fig. 11 shows the conformations during the opening and closing of the folding chamber. We can see the apparent difference in the starting conformation from the EM maps and the agreement of the final conformations with their restraint maps.

Fig. 11.

Opening (top) and closing (bottom) of the chaperonin folding chamber during the MapSGLD simulations. The last column compares the final conformations with their corresponding pdb entries (in grey). The opening simulation started from PDB structure 3J03 and was restrained with EMD-5140. The closing simulation started from PDB structure 3IYF and was restrained with EMD-5138.

Fig. 12 shows the energies and rmsd during these simulations. Comparing the rmsd shown in the middle panel of Fig. 12, we can see that the closing simulation reached within 1.51 Å from the closed structure in 20 ps and remained there with little change afterward. The opening simulation reached within 2.01 Å from the opened structure in 20 ps and remained there with little change afterward.

Fig. 12.

The opening and closing simulation profiles during the MapSGLD simulations. Bottom panel: the energy profile during the MapSGLD simulations. The energies are molecular potential energies and do not include the map-restraint energies. Middle panel: the rmsd profiles during the MapSGLD simulations. The rmsds are calculated using the backbone atoms against the PDB structures, 3J03(opened) and 3IYF(closed). Top panel: the rmsd profiles during the MD simulations continuing from the final conformation of the MapSGLD simulations.

Examining the energy profiles (bottom panel of Fig.12), we can see that in order for the closed assembly to open up, it first went through an energy barrier. In about 15ps, it reached a peak and began an energy decrease throughout the rest of the simulation. For the closing simulation, there was no energy barrier. The efficient overcoming of the energy barrier during the opening process demonstrates the benefit of the SGLD simulation method.

With the map-restraint potentials, MapSGLD simulations search conformations matching the EM maps. It is because of these map-restraint potentials that the targeted conformation reaches a minimum free-energy state. Without these map-restraint potentials, experimental structures may not be the global free energy minimum states at this simplified simulation conditions. There are many reasons for this, such as inaccuracies in the force field, overly simplified set up of the simulated system, or inadequate description of the effects of the solvent. To illustrate this point, we performed conventional molecular dynamics simulations from the final conformations of the MapSGLD simulations and the rmsd profiles are shown in the top panel of Fig.12. We can see that in the MD simulations, the systems slowly drifted away from the corresponding states. Again, this result shows that the simplification in simulation conditions can cause conformational deviations from experimental observations and the map-restraint can help overcome the effect of the simplification in simulation conditions.

Conclusions

A map is a data type with abundant structural information and plays an increasingly important role in the study of macromolecular systems. The map-restrained SGLD simulation method presented in this work provides an efficient way to utilize maps for targeted conformational search. We have demonstrated that map-restraints can accurately maintain structures and conveniently set conformational searching targets.

One important feature of this method is that the restraint maps are objects that not only interact with simulation systems, but also move with the restraining structures. Through SGLD, this method is capable of large scale conformational searching.

Using a small protein, we examined the effects of simulation parameters. We demonstrated that the map-restraints can maintain a targeted structure accurately. The map-restraints are identity-blind and are non-prohibitive for conformational transitions. Movement of map objects can be controlled by map frictional coefficients, as well as by the guiding factor or guiding temperature of SGLD simulations.

With EM maps downloaded from the EM databank, we showed examples of flexible fitting for a GroEL assembly to obtain conformations in a series of states. Through the EM maps of a chaperonin in open and close states, we simulated the opening and closing procedures of the chaperonin. Using EM maps as complementary to atomic force fields, this method provides a useful way for the study of macromolecule assemblies.