† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11874309, 11474237, and 11574310) and the 111 Project, China (Grant No. B16029).
Two-state folding and down-hill folding are two kinds of protein folding dynamics for small single domain proteins. Here we apply molecular dynamics (MD) simulation to the two-state protein GB1 and down-hill folding protein gpW to reveal the relationship of their free energy landscape and folding/unfolding dynamics. Results from the steered MD simulations show that gpW is much less mechanical resistant than GB1, and the unfolding process of gpW has more variability than that of GB1 according to their force–extension curves. The potential of mean force (PMF) of GB1 and gpW obtained by the umbrella sampling simulations shows apparent difference: PMF of GB1 along the coordinate of extension exhibits a kink transition point where the slope of PMF drops suddenly, while PMF of gpW increases with extension smoothly, which are consistent with two-state folding dynamics of GB1 and downhill folding dynamics of gpW, respectively. Our results provide insight to understand the fundamental mechanism of different folding dynamics of two-state proteins and downhill folding proteins.
The folding and unfolding of proteins are fundamental biological processes in the cell. Fully understanding of the protein folding and unfolding dynamics at atomic resolution is of great importance for facilitating the prediction of protein structure from amino acid sequence,[1] the protein design,[2,3] and the study of molecular mechanism of neuron degenerative diseases which are due to protein misfolding and aggregation.[4]
Dynamics of the protein folding and unfolding processes can be studied by biochemical denaturation experiment[5] or single molecule manipulation techniques.[6] On the other hand, atomistic molecular dynamics (MD) simulation is a powerful theoretical approach to study the folding and unfolding dynamics of proteins.[7] At the end of last century, detailed simulation studies were first applied to characterize protein folding/unfolding pathways. The basic idea behind MD is to apply Newtons laws of motion to every atom of the system using empirical potential energy functions.[8] However, due to the limitation of computational power, the study of protein folding/unfolding by the atomistic MD simulation is usually feasible to only small proteins.
From physical point of view, multiple dimensional conformation space of protein can be described by a funnel-shape free energy landscape.[9,10] From both single molecule manipulation experiment and MD simulation, the free energy landscape or potential of mean force (PMF) of proteins need to be constructed. Usually a protein is either at native state or in unfolded state. There is an energy barrier (transition state) which separates the native state and unfolded state. This kind of proteins is called two-state proteins.[11] If the barrier is very small, at physiological condition the protein can fold to native state very fast almost barrierlessly. Such proteins are called downhill proteins.[12,13]
To build PMF from MD simulation, conformational space of protein needs to be sampled sufficiently. But normal MD simulation will spend long time in local traps of the free energy landscape. Using replica exchange method, extensive all-atom simulation has been performed to get the PMF of a downhill folding protein BBL as a function of four different reaction coordinates.[14] A folding free energy barrier of 1–2 kBT was obtained from the PMF, and low folding cooperativity was confirmed for the BBL protein.
Typical two-state protein GB1 is the streptococcal B1 immunoglobulin-binding domain of protein G (Fig.
![]() | Fig. 1. The three-dimensional structures of proteins GB1 (a) and gpW (b) from two opposite perspectives. |
Representative down-hill protein gpW is a 62-residue α + β protein that participates in virus head morphogenesis (Fig.
In this paper, we compare the unfolding force-extension curves of GB1 and gpW by the steered molecular dynamics (SMD) simulation and find that GB1 is more mechanical resistant than gpW. PMF obtained from umbrella sampling reveals big difference between proteins GB1 and gpW, which gives physical foundation of their folding mechanism.
The steered molecular dynamics (SMD) simulations mimic force spectroscopy measurement in single molecule manipulation experiment like AFM, but proceed on much shorter time scale.[26] In a SMD simulation, one terminal end of the protein is fixed, while the other end is coupled to one end of a dummy spring with elastic constant k. When the end of the dummy spring moves away from the protein with constant speed v, its position ξ is a linear function of time
During the pulling process, the force on the end of the protein, f, is calculated from the extension of the dummy spring,
One of the main challenges in computational biology is the calculation of free energy difference. Here we use umbrella sampling to construct PMF of proteins.[28] In umbrella sampling, a system is driven to conformations with high free energy by a bias potential along a reaction coordinate.[29] The conformations are covered by a series of overlapping windows (Fig.
![](cpb_29_7_078701/cpb_29_7_078701_ieqn1.gif)
An accurate calculation of PMF requires an adequate number of windows (separate simulations) with enough overlap in conformational space.[28,31] Based on SMD simulation, a series of conformations were extracted from the SMD trajectory at regular interval of extension. Then each extracted conformation was used as the initial conformation to perform the umbrella sampling simulation in each window. When all simulations were done, the weighted histogram analysis method (WHAM), an extension of Ferrenberg and Swendsens multiple histogram technique, was applied to properly merge the data from all simulations to obtain the global PMF profile.[32, 33]
Here we list detail parameters of SMD and umbrella simulations. Proteins GB1 and gpW with explicit waters were simulated in the NPT ensemble using GROMACS package 5.1.4, with the GROMOS96 53A6 all-atom force field at the temperature of 310 K. A time step of 2 fs was selected. Periodic boundary conditions were applied in all directions. Particle-mesh-Ewald summation was selected for the long-range Coulomb interactions. The cut-off distances for electrostatics and vdW interactions were set at 1.4 nm. The simulations started from an energy-minimized average NMR structure, deposited as entry 1PGA for protein GB1 and 2L6Q for protein gpW in the RCSB protein data bank. The structures of both proteins were solvated with SPC model water before simulations. The pulling velocity of the restraint point in the SMD simulation was 0.001 nm/ps, and the spring constant was 100 kJ/mol/nm2. The umbrella simulations were performed in 95 extension windows for GB1 and 60 windows for gpW with spring constant of 500 kJ/mol/nm2. Each simulation lasted for 15 ns, and the trajectory from 5 ns to 15 ns was used for the analysis while the first 5 ns simulations were applied for reaching equilibrium.
SMD simulation of GB1 and gpW began with an equilibrated folded structure and stopped when reaching an extension of ∼ 7.1 nm after ∼ 5 ns of simulation. GB1 and gpW show different capabilities to sustain the stretching force in SMD simulation with the same pulling velocity and spring constant.
Figure
Figure
To obtain a more statistic and quantitative understanding of the unfolding process of GB1, the free energy profile of GB1 unfolding was calculated through a series of umbrella sampling simulations. Neighboring simulations have enough overlapping to build PMF over extension range of ∼ 2.4 nm to ∼ 4.1 nm for GB1 and ∼ 1.3 nm to ∼ 4.5 nm for gpW.
The PMF profiles of GB1 and gpW are shown in Fig.
The PMF of gpW is almost a linear function of extension when the extension is longer than 2 nm. The slope changes smoothly along extension from 1.4 nm to 4.5 nm. This smooth PMF of gpW is consistent with its downhill folding dynamics obtained in biochemistry study.
Different from gpW, the PMF profile of GB1 has a transition point at 3.2 nm where the slope changes from 292 pN to 74 pN, i.e., the slope drops to about one quarter of its value at extension smaller than 3.2 nm. This transition point is very close to the force peak position in the SMD simulations.
When a constant stretching force is applied to the end of protein in single molecule manipulation experiment or MD simulations, the free energy profile will tilt. Therefore, force-dependent free energy profile Gf(x) can be obtained from PMF (free energy profile at zero force, G0(x)) according to the equation
The Gf(x) at different forces is also shown in Fig.
As the PMF of gpW is almost a linear function of extension (Fig.
Small single domain proteins usually fold through two-state transition. But some fast folding proteins fold to their native states downhill the free energy landscape without big free energy barrier. We use GB1 as a typical two-state protein, and gpW as a typical downhill folding protein to study their different mechanical stability and free energy landscape by MD simulation.
In summary, firstly we found that GB1 can sustain much larger force than gpW by SMD simulation, which might come from the shearing pulling geometry of GB1. To understand the two-state property of GB1 and downhill folding mechanism of gpW, we focused our effort to build the free energy landscapes of GB1 and gpW from the PMF obtained by umbrella sampling MD simulation. PMF of GB1 increases with extension much faster than that of gpW, which is consistent with the different force resistance properties of GB1 and gpW obtained by SMD simulation. PMF of GB1 shows a kink point where the slope drops abruptly, corresponding to the conformation at the force peak during SMD simulation. In contrast to that of GB1, PMF of gpW increases smoothly with extension. Our simulation results are generally consistent with the two-state folding phenomena of protein GB1 and downhill folding behavior of gpW, and indicate that different protein folding dynamics is a natural consequence of their PMF.
In this paper, we only build PMFs of GB1 and gpW from their native states to partially extended conformations due to limited computational capability. Beyond the kink point of PMF of GB1, the free energy does not drop when there is no stretching force. Therefore, though the kink point is transition state of unfolding at high stretching forces, the transition state of unfolding at zero force might have longer extension. In the future if PMF from native state to fully unfolded peptide can be constructed, a whole picture of protein folding can be expected.
Free energy landscape can be also constructed from force-dependent folding and unfolding rates of proteins measured by single molecule stretching experiments.[34,35] Equilibrium folding and unfolding dynamics can be recorded by magnetic tweezers. At equilibrium critical force, the transition rates of downhill proteins are usually several orders of magnitude larger than those of two-state proteins, which is consistent with the free energy landscapes with low energy barrier for downhill folding proteins and high energy barrier for two-state proteins. Recently we found that the transition state of downhill protein gpW is located at longer extension than two-state protein GB1 from magnetic tweezers experiment, which agrees with the PMF of gpW from umbrella sampling MD simulation and indicates that the native state of gpW is very flexible. Single molecule experiments and MD simulations can work together to reveal the mechanism of protein folding.
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