General Trends of Dihedral Conformational Transitions in a Globular Protein

Abstract

Dihedral conformational transitions are analyzed systematically in a model globular protein, cytochrome P450cam, to examine their structural and chemical dependences through combined conventional molecular dynamics (cMD), accelerated molecular dynamics (aMD) and Adaptive Biasing Force (ABF) simulations. The aMD simulations are performed at two acceleration levels, using dihedral and dual boost, respectively. In comparison with cMD, aMD samples protein dihedral transitions ~2 times faster on average using dihedral boost, and ~3.5 times faster using dual boost. In the protein backbone, significantly higher dihedral transition rates are observed in the Bend, Coil and Turn flexible regions, followed by the β bridge and β sheet, and then the helices. Moreover, protein sidechains of greater length exhibit higher transition rates on average in the aMD-enhanced sampling. Sidechains of the same length (particularly N_χ = 2) exhibit decreasing transition rates with residues when going from hydrophobic to polar, then charged and aromatic chemical types. The reduction of dihedral transition rates is found to be correlated with increasing energy barriers as identified through ABF free energy calculations. These general trends of dihedral conformational transitions provide important insights into the hierarchical dynamics and complex free energy landscapes of functional proteins.

I. Introduction

Proteins often undergo structural changes during their function, such as, for example, flipping of residue sidechains in ligand binding^1,2, conformational changes of flexible loops during substrate/drug binding^3,4, the opening and closing of ion channels⁵ and rearrangements of structural domains during protein-protein binding and cellular signaling processes^6–8. Since dihedral angles are key degrees of freedom determining protein conformation, structural changes in the proteins are usually accompanied by dihedral transitions. Notably, conformational changes of dihedrals have been investigated in previous studies on both the protein backbone^9,10 and sidechains during protein-ligand binding and protein-protein associations^11–13. However, dihedral transitions are known to take place on a wide range of timescales (from picoseconds to milliseconds and longer) and energy barriers (~0.3–12 kcal/mol)^9,10,14–17. Such highly heterogeneous dynamics of protein dihedral transitions presents a challenge for computational modeling and statistical analysis. It is thus important to characterize the physicochemical dependences of dihedral transitions in order to understand the structural dynamics and free energy landscapes of functional proteins. This will provide useful information for the development of improved protein simulation methods and structure-based drug design techniques, e.g. induced fit docking¹⁸.

In this study, cytochrome P450cam, a globular protein with 414 amino acid residues, is used as a model system for statistical analysis of the protein dihedral transitions. Cytochrome P450 is a large family of hemoprotein monooxygenases that catalyze a wide variety of biochemical reactions involved in carcinogenesis, drug metabolism, lipid and steroid biosynthesis and degradation of toxins in higher organisms¹⁹. The chemical reactions catalyzed include hydroxylation, sulfoxidation, epoxidation, dehalogenation, deformylation, dealkylation and C-C coupling ²⁰. A P450 enzyme from Pseudomonas putida, Cytochrome P450cam catalyzes the regio- and stereo-specific hydroxylation of camphor and has long served as a model for studying the dynamics of P450s and proteins in general ²¹.

Molecular dynamics (MD), a widely used computational technique for simulating the structural dynamics of proteins, provides a detailed picture of protein dynamics at an atomistic level, including the dihedral transitions. However, with current computing power conventional MD (cMD) simulations are limited typically to hundreds of nanoseconds to microsecond timescales, which are much shorter than the timescales of many protein conformational changes (which can reach milliseconds or even longer) ¹⁷. This often leads to insufficient sampling and un-converged simulations of proteins ²².

Accelerated molecular dynamics (aMD) has been developed to enhance the conformational sampling of proteins. AMD often works by adding a non-negative boost potential, ΔV to the energy surface, effectively decreasing energy barriers and thus accelerating transitions between the low-energy states^23,24. AMD has been successfully applied to a number of biological systems^25–30 and significant enhanced sampling is achieved in both globular and membrane proteins^8,31,32, including opening and closing of the substrate access channel in cytochrome P450cam³³ . In principle, aMD simulation frames can be reweighted by the Boltzmann factors of the corresponding boost potential (i.e., e^ΔV/k_BT) to obtain the canonical ensemble and recover the original free energy profiles of simulated systems. This has been, in particular, demonstrated on the alanine dipeptide biomolecular model system in a number of previous studies^23,34,35 . However, the exponential reweighting of aMD simulations is known to suffer from large energetic noise in practical calculations^36–39 because the Boltzmann reweighting factors are often dominated by a very few frames with high boost potential. By comparing a list of available reweighting methods, we showed in a previous study that when the boost potential follows near Gaussian distribution, the cumulant expansion to the 2^nd order (also referred to “Gaussian approximation”) provides significantly improved reweighting compared with the exponential average and Maclaurin series expansion methods as used earlier³⁵. Such improved reweighting has been applied to aMD simulations of several fast-folding protein, including the chignolin, Trp-cage, villin and WW-domain³⁰. The protein folding free energy profiles obtained from reweighting hundreds-of-nanosecond aMD simulations were found to agree quantitatively with those of ~1000 times longer Anton cMD simulations. However, with increasing system size, the aMD boost potential exhibits high fluctuations with wider distributions. Accurate reweighting of aMD simulations through cumulant expansion to the 2^nd order is still limited to proteins with ≤ ~35 residues³⁰, but not larger systems³⁸. While this precludes obtaining canonical ensemble from aMD simulations of the cytochrome P450cam, here we aim to characterize the aMD acceleration relative to cMD by analyzing the protein dihedral conformation transitions and examine the dependences of dihedral transitions on the structural and chemical properties of the protein residues.

Using both cMD and aMD simulations on cytochrome P450cam, we investigate the transition rates of protein backbone and sidechain dihedrals (ϕ, ψ, χ₁, χ₂, χ₃, χ₄) as illustrated in Fig. 1. The protein dihedrals are categorized into those in the backbone with different secondary structures, and those in the sidechains of different length. Moreover, the protein sidechains are divided into different chemical types as in a previous study⁴⁰, including the charged (ARG, LYS, seven doubly-protonated HIS (17, 21, 80, 176, 308, 337 and 355) and all ASP and GLU except the protonated ASP297 and GLU366^41,42), polar (SER, THR, ASN, GLN, CYS, TYR, protonated ASP297 and GLU366, and all HIS except the seven doubly-protonated), hydrophobic (ALA, LEU, ILE, VAL, MET and PRO) and aromatic (PHE and TRP) residues.

Fig. 1.

(A) Schematic representation of the X-ray structure of cytochrome P450cam (PDB: 3L61). The protein is shown in ribbons (blue) and the heme group in sticks. Residues 1 to 9 in the N-terminus, which is missing in the X-ray structure, is added as random coil and the other missing residues 90 to 97 and 104 in the B/C loop are added using coordinates from the camphor-bound X-ray structure (PDB: 3L63) after aligning the two protein structures. The added residues are highlighted in yellow. (B) Illustration of protein backbone and sidechain dihedrals (ϕ, ψ, χ₁, χ₂, χ₃, χ₄) in a lysine residue.

Finally, to understand the observed trend of dihedral transition rates in different chemical types of protein residues, adaptive biasing force (ABF) calculations have been performed to determine the dihedral free energy profiles. ABF is an enhanced sampling method that calculates the thermal average force along selected reaction coordinates (also called collective variables or order parameters) and then applies an adaptive force in the reverse direction to facilitate barrier-crossing events in the system^43,44. In principle, uniform sampling of the system along selected reaction coordinates is achieved upon convergence of the ABF simulations. ABF has been shown to greatly improve the efficiency of free energy calculations in a wide variety of biomolecular systems, including solvation of small molecules⁴⁵, dihedral conformational transitions in the alanine dipeptide⁴⁴, reversible unfolding of deca-alanine⁴⁵ and large-scale conformational changes in proteins^29,46. Here, selected dihedrals in cytochrome P450cam are set as input reaction coordinates for ABF simulation to calculate the dihedral free energy profiles and estimate their transition energy barriers.

In the following, we analyze how much acceleration is obtained in the dihedral transitions from cMD to aMD, the dependence of dihedral transition rates on the protein secondary structures in the backbone, and effects on transition rates of the sidechain length and chemical type. The correlations between dihedral transition rates and energy barriers are examined through ABF free energy calculations as well. Although protein dihedral transitions are highly dynamically heterogeneous, these statistical analyses will provide important insights into the hierarchical structural dynamics and free energy landscapes of proteins¹⁴.

Methods

System Setup

Simulations of cytochrome P450cam were carried out using the 3L61 X-ray crystal structure that was determined at 1.5 Å resolution⁴⁷. Protein residues were set to the standard CHARMM protonation states at neutral pH, with the following exceptions, following the results of pK_a calculations of Ref. 48: ASP297 and GLU366 are protonated, HIS62 is protonated on the δ-nitrogen, and seven other histidines (17, 21, 80, 176, 308, 337 and 355) are doubly protonated. The coordination of the heme and the system setup follow the scheme described previously^41,42,49. The CHARMM22 force field^50,51 was used for the protein using the TIP3P model ⁵² for water molecules. Standard CHARMM force field parameters⁵³ were used for the heme group (toppar_all22_prot_heme.str) including an explicit Fe-S bond to CYS357. The partial charge of the SG atom in CYS357 was adjusted to −0.07e to account for the resting state of P450cam in both apo and camphor-bound forms⁴⁸. CHARMM parameters for camphor were obtained from Ref. 54. Periodic boundary conditions were applied on the simulation system.

Conventional Molecular Dynamics (cMD) Simulations

Conventional Molecular Dynamics (cMD) simulations of cytochrome P450cam were performed using NAMD2.8⁵⁵. A cutoff distance of 12 Å was used for the van der Waals and short-range electrostatic interactions and the long-range electrostatic interactions were computed with the Particle-Mesh Ewald summation method⁵⁶ using a grid point density of 1/Å. The SHAKE⁵⁷ algorithm was applied to all hydrogen-containing bonds. A 2 fs integration time-step was used for the MD runs and a multiple-time-stepping algorithm⁵⁵ was employed with bonded and short-range nonbonded interactions computed every time-step and long-range electrostatic interactions every two time-steps.

The system was initially energy minimized for 1000 steps fixing the location of atoms given in the crystal structures and another 1000 steps with all atoms free to move using the conjugate gradient algorithm. The system was then gradually heated to 300 K at a rate of 60 K/ps and equilibrated at 300 K for 5 ns in constant volume (i.e, NVT ensemble) with the crystallographically-identified atoms fixed. The system was further equilibrated for 5 ns by using NPT (300 K and 1atm with Nosé-Hoover Langevin piston pressure control) ensemble with all atoms free to move and a 100 ns NPT production run was obtained for analysis.

Accelerated Molecular Dynamics (aMD) Simulations

Accelerated molecular dynamics (aMD) enhances the conformational sampling of functional biomolecules, often by adding a non-negative boost potential to the potential energy surface when the system potential is lower than a reference energy^23,24,37:

$$V^{*}(r)=V(r),V(r)\ge E,$$

$$V^{*}(r)=V(r)+\mathrm{\Delta }V(r),V(r)<E,$$ (1)

where V(r) is the original potential, E is the reference energy, and V*(r) is the modified potential. The boost potential, ΔV(r) is given by:

$$\mathrm{\Delta }V(r)=\frac{{(E-V(r))}^2}{\alpha +E-V(r)},$$ (2)

where α is the acceleration factor. As the acceleration factor α decreases, the potential energy surface is flattened and biomolecular transitions between the low-energy states are increased.

Restarting from the final structure of the 100 ns cMD simulation, aMD simulations were performed on P450cam using NAMD2.8³⁴ at two acceleration levels, i.e., dihedral²³ and “dual-boost”²⁴. In dihedral aMD, a boost potential is applied to all dihedral angles in the system with input parameters (E_dihed, α_dihed). In dual-boost aMD, a total boost potential is applied to all atoms in the system in addition to the dihedral boost, i.e., (E_dihed, α_dihed; E_total, α_total). The input parameters are calculated as the following:

$$E_{\mathit{\text{dihed}}}=V_{\mathit{\text{dihed}}\_\mathit{\text{avg}}}+4N_{\mathit{\text{res}}},{\alpha }_{\mathit{\text{dihed}}}=4N_{\mathit{\text{res}}}/5,$$

$$E_{\mathit{\text{total}}}=V_{\mathit{\text{total}}\_\mathit{\text{avg}}}+0.2N_{\mathit{\text{atoms}}},{\alpha }_{\mathit{\text{total}}}=0.2N_{\mathit{\text{atoms}}},$$ (3)

where N_res is number of protein residues, N_atoms is the total number of atoms, and V_{dihed_avg} and V_{total_avg} are the average dihedral and total potential energies calculated from the 100 ns cMD simulation, respectively. Specifically, the E_dihed, α_dihed, E_total and α_total were calculated as 3813 kcal/mol, 333.6 kcal/mol, −116417 kcal/mol and 9958 kcal/mol, respectively (see Appendix S1 for parameter settings used in the aMD simulation of P450cam).

Adaptive Biasing Force Calculations

Adaptive biasing force (ABF) calculations of dihedrals in cytochrome P450cam were performed using the Collective Variables Module^45,58 implemented in NAMD2.8⁵⁵. Similar to a pervious study that analyzed free energy profiles of all methyl-containing residues in one ABF simulation¹⁵, a total of 178 χ₁ sidechain dihedrals in the protein residues with N_χ=2 were selected as input reaction coordinates for the ABF simulation of P450cam. This includes 98, 28, 29 and 23 sidechain dihedrals in the hydrophobic (LEU, ILE and PRO), polar (ASN, TYR and HIS), charged (ASP and HSP) and aromatic (PHE and TRP) residues. A threshold of 300 force samples was collected prior to applying the adaptive biasing mean force. The potential of mean force (PMF) profiles were calculated with a bin size of 1° in the range of [−180°, 180°] for all dihedrals except [−60°, 60°] for χ₁ in the PRO residues (see Appendix S2 for parameter settings of the ABF calculations, particularly for χ₁ in the example ILE5 and PRO13 residues). The convergence of the PMF profiles was monitored by calculating their root-mean squared deviation (RMSD) over time:

$${\mathit{\text{RMSD}}}_{\mathit{\text{PMF}}}=\sqrt{\frac{1}{N}\sum _{\chi }[\mathit{\text{PMF}}(\chi ,t+\mathrm{\Delta }t)-\mathit{\text{PMF}}(\chi ,t)]},$$ (4)

where N is the total number of bins (120 for χ₁ in the PRO residues and 360 for the others), t is the simulation time and Δt is the time interval at which PMF differences are computed. Here, Δt was set to 5 ns for ABF calculations of P450cam. The PMF profiles were considered to be converged when the RMSD decreases to ≤0.05 kcal/mol. Convergence was achieved for most χ₁ dihedrals within 300 ns simulation time (Fig. S2), except for those in 11 residues (HSP17, ASP25, TRP55, HIS62, TYR78, ASN149, PHE163, ASP297, PHE307, PHE350, PHE381) that were excluded from further analysis. The sampled probability versus angle and angle versus time obtained from the 300 ns ABF simulation of P450cam were plotted in Fig. S3 for PHE381 and another four representative amino acid residues in the protein, including the ILE5 (hydrophobic), ASN8 (polar), ASP25 (charged) and PHE81 (aromatic). Notably, the ASN8 residue exhibits nearly uniform sampling of the entire dihedral range of χ₁. Fairly even sampling was also achieved for different regions of χ₁ in the ILE5, ASP25 and PHE81 residues. In comparison, the PHE381 residue did not reach such even sampling of χ₁ in the ABF simulation, which largely contributes to un-convergence of the PMF calculation. However, RMSDs of the PMF profiles calculated for the 11 protein residues keep decreasing through the 300 ns ABF simulation (Fig. S2) and convergence is thus expected in a potentially longer simulation.

Simulation Analysis

Root-mean square deviation (RMSD) of the protein backbone relative to the starting structure was calculated for the cMD, aMD and dual-boost aMD simulations of P450cam (Fig. S1). During the 100 ns cMD simulation, the RMSD appears to converge quickly to ~2 Å with no significant structural change in the protein. In comparison, greater RMSD values (larger protein conformational changes) were found in the aMD enhanced sampling simulations. Nevertheless, the RMSD levels off to ~4 Å and ~6 Å during the second half of the dihedral and dual-boost aMD simulations, respectively, although certain fluctuations were observed in the dual-boost aMD simulation.

The g_chi tool in the GROMACS package⁵⁹ was applied to analyze conformational transitions of all dihedral angles in the protein. Statistical analysis was then performed on the dihedral transition rates for the entire protein, as well as the protein backbone of different secondary structures (the Bend, Coil, Turn, β bridge, β sheet, 3₁₀ Helix and α Helix) and residue sidechains that were categorized according to their length (N_x = 1, 2, 3 and 4) and chemical type (the hydrophobic, polar, charged and aromatic). For the ABF free energy profiles of the χ₁ sidechain dihedrals in the protein residues with N_x=2, Matlab scripts were implemented to extract the energy minimum and maximum values and calculate the dihedral transition energy barriers.

II. Results

Significantly Increased Dihedral Transitions from cMD to aMD

The total number of dihedral transitions per ns is plotted as a function of time in Fig. 2B during 100 ns cMD, aMD and dual-boost aMD simulations of cytochrome P450cam. Taking protein dihedrals in both the backbone and sidechains together, i.e., (ϕ, ψ, χ₁, χ₂, χ₃, χ₄), the average dihedral transition rate is ~20 transitions/ns in the cMD simulation. In comparison, the transition rates of the protein dihedrals are found to be significantly higher in the aMD simulations. By adding a boost potential to the protein dihedral energy (see Methods for details), the average transition rate of protein dihedrals is increased to ~40 transitions/ns. With the second boost potential applied to the system total potential energy, the average transition rate of protein dihedrals is further increased to ~70 transitions/ns in the dual-boost aMD simulation. Therefore, dihedral aMD samples dihedral transitions ~2 times faster on average than cMD, and ~3.5 times faster using the dual-boost aMD.

Fig. 2.

(A) Schematic illustration of accelerated molecular dynamics, which works by adding a non-negative boost potential to enhance conformational sampling when the system potential V(r) is lower than a reference energy E. α is an acceleration factor that can be adjusted to smoothen the energy surface and thus accelerate protein conformational transitions between different low-energy states. (B) The total number of dihedral transitions per ns in P450cam is plotted as a function of time in 100 ns cMD, aMD and dual-boost aMD (aMD_DB) simulations. Thick lines depict running average over 1 ns (100 simulation frames).

Dependence of Backbone Dihedral Transitions on Protein Secondary Structures

Fig. 3A shows the different types of protein secondary structures calculated from the average structure of P450cam as obtained from the 100 ns cMD simulation. Residues 90–97, which are missing in the starting X-ray structure and were added as a helix using the atomic coordinates of the camphor-bound P450cam, exhibit random coil conformation in the simulation-derived average structure. This is consistent with previous crystallographic and computational studies that suggest that the B/C loop containing the above residues is highly flexible in the absence of substrate binding and adopts random coil^41,47,49,60. The newly-added N-terminus also remains in the coil conformation. With this, 16.2%, 22.5%, 8.7%, 1.7%, 8.5%, 1.4% and 41.1% of the total 414 protein residues are identified in the Bend, Coil, Turn, β bridge, β sheet, 3₁₀ helix and α helix secondary structures, respectively (Fig. 3A).

Fig. 3.

(A) Different types of protein secondary structures in the average structure of P450cam as calculated from the 100 ns cMD simulation. In a total of 414 protein residues, 16.2%, 22.5%, 8.7%, 1.7%, 8.5%, 1.4% and 41.1% of the residues are identified in the Bend (cyan), Coil (red), Turn (blue), β bridge (brown), β sheet (yellow), 3₁₀ helix (pink) and α helix (purple). (B) The average number of transitions per ns of the backbone dihedrals (ϕ, ψ) are plotted for each type of the protein secondary structures in P450cam as obtained from the 100 ns cMD, aMD and dual-boost aMD (aMD_DB) simulations. Error bars indicate the standard deviation of ϕ and ψ transition rates.

The average transition rates of the backbone dihedrals (ϕ, ψ) obtained from the MD simulations are plotted in Fig. 3B for each type of secondary structure. A low rate of dihedral transitions is observed for the cMD simulation, for which only the ϕ dihedrals in Bend configurations undergo more than 1 transition/ns, while the others are slower, especially in the helices (Table 1). This is consistent with severely limited conformational sampling in 100ns-timescale cMD simulations of proteins.

Table 1.

The transition rates of backbone dihedrals (ϕ, ψ) in different secondary structures of cytochrome P450cam. The table has been sorted by the average transition rate of ϕ and ψ in the dihedral aMD simulation (bold column). Units are transitions/ns

Secondary Structure	cMD			aMD			aMD_DB
	ϕ	ψ	ravg	ϕ	ψ	ravg	ϕ	ψ	ravg
Bend	1.06	0.60	0.83	9.24	4.71	6.98	13.67	8.94	11.31
Coil	0.78	0.32	0.55	7.41	3.37	5.39	13.23	6.00	9.62
Turn	0.21	0.27	0.24	5.90	3.68	4.79	9.45	7.25	8.35
β Bridge	0.20	0.12	0.16	3.32	2.79	3.06	6.09	5.77	5.93
β Sheet	0.42	0.17	0.29	4.07	1.66	2.87	8.74	3.76	6.25
310 Helix	0.00	0.13	0.07	0.54	1.68	1.11	1.94	2.27	2.11
α Helix	0.01	0.03	0.02	0.43	0.61	0.52	1.03	1.34	1.18

In the aMD simulations, with the boost potential applied, the backbone dihedrals exhibit significantly increased transition rates, as shown in Fig. 3B. The transition rate of dihedrals in the Bend, Coil, and Turn flexible regions is increased to ~5–7 transitions/ns in dihedral aMD, and ~8–11 transitions/ns in the dual-boost aMD (Table 1). This leads to greatly enhanced sampling of this aspect of the protein conformations. Furthermore, a clear trend is identified in the dihedral transition rates of different protein secondary structures: significantly higher rates of dihedral transitions are found in the Bend, Coil and Turn flexible regions of the protein backbone, followed by the β bridge and β sheet, and then the helices.

Dependence of Sidechain Dihedral Transitions on Residue Length and Chemical Type

Next, we focus on conformational transitions of the dihedrals in protein sidechains. 9.2% of the total 414 residues in P450cam have four sidechain dihedrals, 15.9% three, 43.0% two, 18.1% one and 13.8% zero sidechain dihedrals. For each class of the protein residues, with different numbers of sidechain dihedrals, the average transition rates of the sidechain dihedrals (χ₁, χ₂, χ₃, χ₄) obtained from the 100 ns cMD, aMD and dual-boost aMD simulations are plotted in Fig. 4A.

Fig. 4.

(A) The average number of transitions per ns of the sidechain dihedrals (χ₁, χ₂, χ₃, χ₄) are plotted for protein residues with different number of sidechain dihedrals (N_χ) as obtained from the 100 ns cMD, aMD and dual-boost aMD (aMD_DB) simulations. (B) The average number of transitions per ns of χ₁ are plotted for different chemical types of the protein residues with N_χ=2 (the hydrophobic, polar, charged and aromatic) as obtained from the 100 ns cMD, aMD and dual-boost aMD (aMD_DB) simulations. Error bars indicate the standard deviation of sidechain dihedral transition rates.

In the 100 ns cMD simulation of P450cam, residues with four sidechain dihedrals exhibit a lower transition rate (0.94 transitions/ns on average) than those with three or two sidechain dihedrals (>2 transitions/ns), due largely to insufficient conformational sampling of the ARG and LYS charged residues (Table 2). In the aMD simulations, sidechains of greater length display a higher transition rate on average. Notably, sidechains with 2–4 dihedrals exhibit transition rates greater than 12 transitions/ns in the dihedral aMD simulation and >20 transitions/ns in the dual-boost aMD simulation (Table 2 and Fig. 4A).

Table 2.

The transition rates of sidechain dihedrals (χ₁, χ₂, χ₃, χ₄) in protein residues with different number of sidechain dihedrals (N_χ). Units are transitions/ns.

Nχ	cMD					aMD					aMD_DB
	χ1	χ2	χ3	χ4	ravg	χ1	χ2	χ3	χ4	ravg	χ1	χ2	χ3	χ4	ravg
4	0.17	1.56	1.06	0.99	0.94	6.18	14.07	20.47	13.45	13.54	10.77	23.35	30.01	22.46	21.65
3	0.60	0.97	5.23		2.27	9.94	8.64	20.81		13.13	15.93	15.74	35.44		22.37
2	1.27	3.61			2.44	6.48	18.58			12.53	11.06	29.66			20.36
1	0.50				0.50	5.54				5.54	11.22				11.22

In addition to the sidechain length, we also analyzed the dependence of dihedral transition rates on the chemical type of residue, distinguishing between charged, polar, hydrophobic and aromatic residues as listed in Table S2. Whereas only subsets of chemical types are found in sidechains with one, three and four dihedrals (N_χ = 1, 3 and 4), sidechains with two dihedrals consist of all four chemical types of residues (Table 3). Another trend can be identified in dihedral transition rates of these residue sidechains. Generally, sidechains of the same length (particularly N_χ = 2) exhibit decreasing transition rates going from hydrophobic to polar, charged and aromatic residues. Compared with the hydrophobic residues, the polar residues exhibit ~15 and ~23 fewer transitions/ns on average in the dihedral and dual-boost aMD simulation, respectively. While smaller differences appear among the polar, charged and aromatic sidechains, the trend is clearly observed in all cMD, dihedral aMD and dual-boost aMD simulations (Fig. 4B). This is consistent with a previous finding that hydrophobic sidechains appear to be less “resilient”, exhibiting anharmonic dynamics first on increasing the temperature from T = 0 K, followed by the hydrophilic (polar and charged) residues^40,61. Particularly, the methylene (-CH₂-) groups in the PRO sidechains and other hydrophobic residues undergo faster dihedral transitions than those in the ARG and LYS sidechains.

Table 3.

The transition rates of sidechain dihedrals (χ₁, χ₂, χ₃, χ₄) in protein residues with different number of sidechain dihedrals (N_χ) and chemical type, including the hydrophobic, polar, charged and aromatic. Units are transitions/ns.

Nχ	Type	cMD					aMD					aMD_DB
		χ1	χ2	χ3	χ4	ravg	χ1	χ2	χ3	χ4	ravg	χ1	χ2	χ3	χ4	ravg
4	Charged	0.17	1.56	1.06	0.99	0.94	6.18	14.07	20.47	13.45	13.54	10.77	23.35	30.01	22.46	21.65
3	Hydrophobic	0.74	3.40	9.80		4.65	13.33	23.48	60.86		32.56	20.66	36.25	95.88		50.93
	Polar	0.54	0.61	5.67		2.28	10.69	8.25	19.25		12.73	17.28	15.79	31.51		21.53
	Charged	0.60	0.49	3.54		1.54	8.39	4.41	9.76		7.52	13.56	9.49	19.87		14.31
2	Hydrophobic	2.19	5.66			3.92	9.93	30.45			20.19	16.58	47.08			31.83
	Polar	0.12	1.57			0.85	3.41	6.40			4.90	5.47	12.52			9.00
	Charged	0.14	1.51			0.83	2.03	4.77			3.40	4.60	10.36			7.48
	Aromatic	0.15	0.03			0.09	1.12	0.25			0.68	2.55	0.67			1.61
1	Hydrophobic	0.23				0.23	2.97				2.97	6.52				6.52
	Polar	0.62				0.62	6.73				6.73	13.37				13.37

In addition, we analyzed the transition rates of sidechain dihedrals for amino acid residues found in different locations (particularly the surface and core) of the globular protein. Residues with any heavy atom found within 2.4 Å of solvent (water or ion) are defined to be roughly on the protein surface and the others are assigned to the protein core. The surface and core account for 63.0% and 37.0% of the protein residues, respectively. As shown in Table 4, although higher dihedral transition rates are found in the protein surface for sidechains with one and four dihedrals (N_χ = 1 and 4), sidechains with two dihedrals (N_χ = 2) exhibit similar dihedral transition rates regardless of the location. Furthermore, sidechains with three dihedrals (N_χ = 3) undergo even slower dihedral transitions on the protein surface than in the protein core (Table 4). Therefore, no direct correlation was identified between transition rates of the sidechain dihedrals and their location in P450cam.

Table 4.

The transition rates of sidechain dihedrals (χ₁, χ₂, χ₃, χ₄) in protein residues with different number of sidechain dihedrals (N_χ) and location, including surface and core of the protein. Units are transitions/ns.

Nχ	Type	cMD					aMD					aMD_DB
		χ1	χ2	χ3	χ4	ravg	χ1	χ2	χ3	χ4	ravg	χ1	χ2	χ3	χ4	ravg
4	Surface	0.21	1.29	1.41	1.29	1.05	6.57	15.80	20.81	17.56	15.18	11.60	26.30	32.03	26.57	24.13
	Core	0.13	1.83	0.70	0.69	0.84	5.78	12.34	20.13	9.33	11.90	9.94	20.39	27.99	18.35	19.17
3	Surface	0.60	0.77	4.85		2.07	9.43	7.08	15.65		10.72	15.40	13.72	28.41		19.17
	Core	0.63	2.10	7.38		3.37	12.79	17.39	49.71		26.63	18.93	27.10	74.84		40.29
2	Surface	1.38	4.12			2.75	6.97	18.33			12.65	11.93	28.85			20.39
	Core	1.11	2.93			2.02	5.83	18.91			12.37	9.90	30.76			20.33
1	Surface	0.55				0.55	6.58				6.58	13.43				13.43
	Core	0.42				0.42	3.77				3.77	7.44				7.44

Finally, we calculated the average transition rates of sidechain dihedrals for protein residues that belong to different backbone secondary structures. Results showed that for sidechains of various lengths (N_χ = 1, 2, 3 and 4), higher dihedral transition rates are not found in the protein backbone and residue sidechains simultaneously (see Table S3). In contrast, in the case of protein sidechains with N_χ = 2, residues of the most flexible backbone secondary structures (Bend, Coil and Turn) exhibit lower sidechain dihedral transition rates than those of the β bridge and 3₁₀ helix structures. This suggests no direct correlation between the flexibility of protein in the backbone and sidechains. Rather, the sidechain length and chemical type appear to be the determining factors for the sidechain dihedral transitions as identified above.

Correlations of Dihedral Transition Rates with Free Energy Barriers

To understand the observed trend of decreasing transition rates in protein residues going from hydrophobic to the polar, charged and aromatic, we first examined the dihedral force field parameters retrieved from CHARMM22⁵¹ as used for the present simulations. Table S4 lists the force constants of sidechain dihedrals in the protein residues as categorized by the number of sidechain dihedrals (N_χ) and chemical type (hydrophobic, polar, charged and aromatic). Overall, the force constants are closely similar to each other for dihedrals in sidechains of different lengths and chemical types. No correlation appears between the dihedral force constants and the decreasing trend of transition rates going from hydrophobic to polar, charged and aromatic residues. This instead implies that the chemical microenvironment of protein residues (e.g., electrostatic and van der Waals interactions) plays an important role in the dihedral transitions.

Next, we performed 300 ns ABF simulation of cytochrome P450cam to determine the dihedral free energy profiles. Particularly, a total of 178 sidechain dihedrals corresponding to χ₁ in all protein residues with N_χ=2 were selected for ABF free energy calculations (see Methods). This includes 98, 28, 29 and 23 sidechain dihedrals in the hydrophobic (LEU, ILE and PRO), polar (ASN, TYR and HIS), charged (ASP and HSP) and aromatic (PHE and TRP) residues. As shown in Fig. 5, all PMF profiles exhibit three energy minima separated by free energy barriers at ~−60°, ~60° and ~±180° that correspond to the gauche⁻, gauche⁺ and trans conformations, except the PRO sidechains with two energy minima found at ~−30° and ~30° corresponding to the down and up puckering conformations. For each energy barrier between two low energy wells, there are two possible barrier-crossing events in the forward and backward directions. Thus, six free energy barriers can be obtained for transitions of each χ₁ sidechain dihedral in non-PRO residues and two barriers for χ₁ in PRO residues. A summary of the resulting free energy barriers, E_a of the χ₁ sidechain dihedral in residues with N_χ=2 is provided in Table S5 for each chemical type of the protein residues, including the hydrophobic, polar, charged and aromatic. The energy barriers are distributed between 0.3–11.5 kcal/mol, with the lowest found in PRO residues for the puckering transitions and the highest in the charged and aromatic residues for the gauche-trans conformational transitions (Table S5 and Fig. 5). The average energy barriers are calculated as 4.32±1.72, 5.28±0.42, 5.70±0.49 and 5.31±0.49 kcal/mol for χ₁ in the hydrophobic, polar, charged and aromatic sidechains with N_χ=2, respectively (Table 5). The increasing energy barriers appear correlated with the decreasing transition rates going from the hydrophobic to the polar, charged and aromatic sidechains, except for the aromatic residues.

Fig. 5.

The potential of mean force (PMF) profiles of the χ₁ sidechain dihedrals are plotted for each chemical type of the protein residues with N_χ=2 as obtained from adaptive biasing force (ABF) calculations: (A) hydrophobic, (B) polar, (C) charged and (D) aromatic. The PMF profiles exhibit three energy minima separated by free energy barriers at ~−60°, ~60° and ~±180° that correspond to the gauche⁻, gauche⁺ and trans conformations, except that the PRO sidechains (hydrophobic) have two energy minima found at ~−30° and ~30° corresponding to the down and up puckering conformations.

Table 5.

Free energy barriers of χ₁ in the hydrophobic, polar, charged and aromatic protein residues with N_χ=2. N_res is the number of protein residues belonging to each chemical type.

Type (Nχ=2)	Residues	Nres	Ea (kcal/mol)
Hydrophobic	LEU ILE PRO	98	4.32±1.72
Polar	ASN TYR HIS	28	5.28±0.42
Charged	ASP HSP	29	5.70±0.49
Aromatic	PHE TRP	23	5.31±0.49

Given the free energy barriers, we also constructed histograms using a bin size of 2 kcal/mol to examine the probability distributions of all χ₁ sidechain dihedral barriers and those belonging to each of the four chemical types of protein residue as shown in Fig. 6. Overall, the relative probability distribution of the free energy barriers exhibits an intense peak at ~5 kcal/mol (Fig. 6A). In comparison, the hydrophobic residues exhibit higher probability of low energy barriers (≤ 4 kcal/mol) than the others; with the polar residues exhibiting significantly higher probability in the middle energy barrier range of 4–8 kcal/mol; while the charged and aromatic residues dominate the highest energy barriers between 8–12 kcal/mol. This trend is also reflected in the accumulated probability plot in Fig. 6B. With the increased energy barriers, the hydrophobic residues exhibit a sharper increase in the accumulated probability to 100%, followed by the polar, charged and finally the aromatic residues. Therefore, whereas the dihedral transitions exhibit great dynamic heterogeneity among each chemical group of the protein residues, a general trend of increased energy barriers emerges for the dihedral transitions going from the hydrophobic to the polar, charged and aromatic residues, being correlated with their decreasing transition rates. Other than the sidechain length, the chemical microenvironment induced by the electrostatic and van der Waals interactions of the protein residues appears to be another key factor in determining the kinetics and free energy landscapes of the dihedral conformational transitions.

Fig. 6.

(A) Relative and (B) accumulated probability distributions of the free energy barriers (E_a) obtained for χ₁ sidechain dihedrals in the protein residues with N_χ=2 are plotted for all the residues and different chemical types, including the hydrophobic, polar, charged and aromatic.

III. Discussion

This study combined extensive conventional MD and advanced enhanced sampling (including aMD and ABF) simulations of the cytochrome P450cam. By examining the dihedral transition rates, we obtained a quantitative measure of the enhanced sampling power of aMD. Dihedral aMD accelerates the protein dihedral transitions by a factor of ~2 on average and there is a ~3.5 speedup for the dual-boost aMD relative to cMD. However, note that the acceleration of dihedral transitions from cMD to aMD exhibits significant variations among different secondary structures of the protein backbone and sidechains of different length and chemical type. For example, relative to cMD, the dual-boost aMD accelerates dihedral transitions in the flexible Bend, Coil and Turn regions of the protein backbone by ~12–45 times (Table 1) and ~7–63 times in the protein sidechains (Table 2).

Furthermore, several general trends of dihedral conformational transitions in the globular protein were revealed as follows. A general trend was first identified for the transition rate of protein backbone dihedrals (ϕ, ψ), which decreases when the protein backbone goes from the flexible Bend, Coil and Turn regions to the β bridge and β sheet, and then the helices. In a previous dihedral transition analysis of a non-redundant dataset of 459 high-resolution pairs of PDB protein structures with different conformations, it was also suggested that Turn and Coil regions tend to favor transitions much more than the β strand and α helix¹⁰. Furthermore, by dividing the protein residues into three classes of native secondary structures, Betancourt and Skolnick determined that the fraction of residues that adopt a native conformation is 0.89 for the helices, 0.47 for the β strand and 0.48 for the others (including Bridge, Turn and Coil)¹⁶. This implies that the β strand exhibits higher probability to undergo dihedral transitions than the helices. Previous MD simulations of an α/β protein (crambin) showed that at room temperature the mean-square displacement of dihedral angles, defined by 4 successive Cα atoms, increases as power law of time, t^α and residues with greater exponents α (i.e., higher mobility) are found in the loops/turns and chain ends, followed by the α helices⁹. Notably, in the Ramachandran plot the β-strand region covers a greater conformational space than the α-helix regions⁶², which involves steric, electrostatic and hydrogen-bonding interactions⁶³. This might potentially allow backbone dihedral angles in β strands more flexibility than in α helices. These findings are in good agreement with the first trend we identified on the differential flexibility of distinct protein backbone secondary structures.

For dihedrals in the protein sidechains, although the individual dihedral transitions also exhibit highly heterogeneous dynamics, two further trends were identified. First, sidechains of greater length exhibit higher average dihedral transition rates. Second, sidechains of the same length (particularly N_χ=2) exhibit decreasing transition rates on average with the protein residues going from the hydrophobic to the polar, charged and aromatic.

The above trends of sidechain dihedral transitions are supported by previous protein studies as well. Notably, sidechain conformational changes upon protein-protein association were examined using the DOCKGROUND non-redundant benchmark set of 233 protein complexes⁶⁴. Detailed analysis indicated that long sidechains with three or more dihedral angles are often subject to large conformational transition, but not for shorter residues with one or two dihedrals. Most of the protein sidechains undergo larger conformational changes in the dihedral most distant from the backbone. Moreover, the hydrophobic sidechains possess higher contribution to the increased surface areas during protein-protein binding than the polar residues. These are consistent with our findings that longer sidechains (N_χ=4 and 3) exhibit higher transition rates than the shorter ones (N_χ=2 and 1) and the hydrophobic sidechains appear more flexible than the others. The sidechain rearrangements upon ligand binding were also analyzed using two non-redundant sets (980 and 353 entries) of paired protein structures in the complexed and uncomplexed forms as constructed from the PDB database¹¹. No correlation was found either between the backbone movement of a residue upon ligand binding and the flexibility of its sidechain¹¹, which is in good agreement with our findings on the sidechain dihedral transition rates (Table S3). Further analysis showed that the flexibility of protein sidechains exhibits the following order: LYS > ARG, GLN, MET > GLU, ILE, LEU > ASN, THR, VAL, TYR, SER, HIS, ASP > CYS, TRP, PHE. In particular, residues with N_χ=4 (LYS and ARG) is more flexible than residues GLN, MET and GLU (N_χ=3), followed by those with N_χ=2 (ILE, LEU and ASN) and then those with N_χ=1 (THR, VAL, SER and CYS), with exceptions on the HIS, ASP, TRP and PHE residues. Moreover, for residues with N_χ=2, hydrophobic sidechains (ILE and LEU) appear more flexible than the polar sidechains (ASN), followed by the charged (ASP) and then finally the aromatic residues (TRP and PHE). This is consistent with our previous finding that in hydrated P450cam at T ≥ 220 K the nonexchangeable hydrogen atoms, which are probed by incoherent neutron scattering, exhibit more jumps between wells in hydrophobic sidechains than in hydrophilic, with the least jumps occurring in the aromatic sidechains ⁴⁰. Thus the electrostatic and hydrogen-bonding interactions in the hydrophilic (charged and polar) sidechains and steric hindrance of the aromatic sidechains play important roles in their dihedral transition rates.

To further understand the dependence of dihedral transition rates on the chemical type of residues, we have performed ABF simulations to calculate the free energy barriers for the χ₁ sidechain dihedral transitions in protein residues with N_χ=2. The ABF free energy calculations identified an intense peak at ~ 5 kcal/mol in the probability distribution of the dihedral energy barriers. However, the energy barriers are broadly distributed between 0.3–11.5 kcal/mol. Such dynamical heterogeneity was also found in a previous study of the methyl-containing dihedral transitions in the protein sidechains¹⁵. Moreover, our detailed analysis of the ABF free energy profiles showed that increased free energy barriers are involved for transitions of the χ₁ sidechain dihedrals going from the hydrophobic to the polar, charged and aromatic residues. The decreasing transition rates are thus correlated with the increasing energy barriers for dihedrals in the four distinct chemical types. However, it is important to note that, similar to the dihedral transition rates, the free energy barriers calculated for individual dihedral transitions exhibit high heterogeneity, as shown in Table S5. The above trends, obtained through statistical analysis, provide ensemble-averaged information on the protein dihedral conformational transitions. In summary, with the high-resolution view obtained from the present atomistic simulations, these findings provide further insights into the hierarchical dynamics and complex free energy landscapes of the protein dihedral conformational transitions as observed through extensive experimental studies^14,17.

IV. Conclusions

General trends of dihedral conformational transitions in cytochrome P450cam have been examined through combined cMD, aMD and ABF simulations. The aMD simulations were performed at two acceleration levels, using the dihedral and dual boost algorithms, respectively. Relative to cMD, dihedral aMD samples the protein dihedral transitions ~2 times faster on average, and dual-boost aMD ~3.5 times faster. In the protein backbone, a significantly higher dihedral transition rate is found in the Bend, Coil and Turn flexible regions, followed by the β bridge and β sheet, and then the helices. Protein sidechains of greater length generally undergo higher dihedral transition rates in the aMD enhanced sampling simulations. Additionally, sidechains of the same length (particularly N_χ = 2) exhibit decreasing transition rates from the hydrophobic to the polar, charged and aromatic. Further ABF free energy calculations identified correlations between the apparent increasing free energy barriers and the decreasing dihedral transition rates in the four chemical types of protein residues. However, the effects of friction on the sidechain dihedral transition rates^65,66 remain unclear and the dependence of sidechain friction on the residue chemical type would be an interesting subject of future investigation. It is important to note that the present aMD simulations of P450cam still suffer from large noise in the energy, preventing accurate reweighting even using cumulant expansion to the 2^nd order³⁵. The corresponding analysis of dihedral transitions is thus derived from the aMD-modified potential energy surfaces. With our recently developed Gaussian accelerated molecular dynamics (GaMD) method⁶⁷, in which the boost potential, constructed using a harmonic function, follows Gaussian distribution and accurate energetic reweighting can be achieved through cumulant expansion to the 2^nd order. It is a subject for future study to perform GaMD enhanced sampling simulations of P450cam and reweight the simulations to recover the canonical ensemble of the protein.

The above trends, obtained through statistical analysis, describe ensemble-averaged behavior of the protein dihedral transitions. There exists high dynamical heterogeneity in the dihedral transition rates and free energy barriers within the protein backbone of any given secondary structural type, and residue sidechains of any certain length or chemical type, and this heterogeneity has been discussed in a number of previous studies^10,11,64. Nevertheless, the general trends of dihedral conformational transitions revealed in this study provide important insights into the hierarchical dynamics and complex free energy landscapes of functional proteins. This will likely facilitate further development of enhanced sampling methods for simulations of large proteins (such as, for example, rotatable aMD that accelerates transitions of dihedrals that are relevant to system conformational changes^68,69). The present results also provide important information that will usefully be taken into account in the development of algorithms for induced-fit docking of small-molecule compounds to flexible protein targets in computer-aided drug design^18,70.