Axis for the i-th dimension [24]. In this study e we adopted P = 1 such that the fitted curve has a relatively small curvature. We minimized the objective function !2 M{1 ! Avg P ! X k X k through a multidimensional nonlinear x2kFigure 1. The simulation system. AdK was initially in the closed conformation in this particular simulation. The AMPbd, LID, and CORE domains of the protein are colored red, yellow, and blue, respectively. K+ and Cl2 ions are drawn as blue and red spheres, respectively. The image was rendered using the VMD software [45]. doi:10.1371/journal.pone.0068023.g1 atm was achieved using the Nose-Hoover Langevin piston method [36], with the volume of the periodic box allowed to fluctuate but the cubic geometry strictly fixed. During the equilibration, the length of the periodic box was stabilized at ?,67 A.Free Energy SamplingTo further explore the thermodynamics of the conformational space, we first obtained a conformational pathway from the trajectories of the unrestrained simulations, as described below, and then carried out a set of restrained simulations to calculate the free energy profile along this pathway. Throughout this study, the protein conformation is represented by the positions of its N = 214 Ca atoms, or the 3N = 642 Cartesian coordinates. Any particular ! conformation i is thus denoted by a vector X i in this 3Ndimensional space. Applying the concept of principal curve [25], we obtained a pathway that represents the conformational space visited in the unrestrained simulations. Specifically, we first defined a straight line, in the 3N-dimensional space, that connects the open and ! ! closed AdK crystal structures, X OP and X CL , after proper alignment. For each frame i in the trajectories of the unrestrained simulations, we performed a rigid-body alignment with respect to ! the open-state crystal structure X OP , thus removing the overall translation and 301353-96-8 rotation of the protein. The protein coordinate ! after the alignment, X i , was then orthogonally projected onto the ! line XOP XCL above, with the projected point denoted by Y i . The ! entire range covered by all the fY i g on the line was then evenly ! divided into 100 segments. The entire set of coordinates fX i g was thus also 1407003 classified into 100 groups, according to the line segment ! each corresponding Y i lies in. Then for each group k, we !Avg calculated the average coordinate, X k , over all the coordinatesoptimization with respect to both fwij g and ftk g [24]. With the ! !Avg obtained fwij g, X ?thus defines a hPTH (1-34) continuous curve from X 0 Avg ! to X M , as the curve parameter t varies from 0 to 1. We use s ?to ! !Avg denote the arc length [24] between X 0 and X ?along this curve, thus with s??= 0, and s? L denoting the entire length !Avg !Avg of the curve between X 0 and X M in the 3N-dimensional space. In general, s ?is not a linear function of t. We may, however, define a new curve parameter a(t):s(t)=L. Because a ?is a monotonic function, the inverse function t ?is well defined, with each a[?,1 corresponding to a unique t ?,1, and thus a ! ! unique conformation X ?X ?on the curve. In this new parametrization, the arc length of the curve is a linear function of a. We have thus obtained a uniformly parametrized smooth ! pathway X ? a[?,1, that represents the sampled conformational space in the unrestrained simulations. We note that a curve can also be obtained by simple piece-wise linear or spline interpolation after applying some s.Axis for the i-th dimension [24]. In this study e we adopted P = 1 such that the fitted curve has a relatively small curvature. We minimized the objective function !2 M{1 ! Avg P ! X k X k through a multidimensional nonlinear x2kFigure 1. The simulation system. AdK was initially in the closed conformation in this particular simulation. The AMPbd, LID, and CORE domains of the protein are colored red, yellow, and blue, respectively. K+ and Cl2 ions are drawn as blue and red spheres, respectively. The image was rendered using the VMD software [45]. doi:10.1371/journal.pone.0068023.g1 atm was achieved using the Nose-Hoover Langevin piston method [36], with the volume of the periodic box allowed to fluctuate but the cubic geometry strictly fixed. During the equilibration, the length of the periodic box was stabilized at ?,67 A.Free Energy SamplingTo further explore the thermodynamics of the conformational space, we first obtained a conformational pathway from the trajectories of the unrestrained simulations, as described below, and then carried out a set of restrained simulations to calculate the free energy profile along this pathway. Throughout this study, the protein conformation is represented by the positions of its N = 214 Ca atoms, or the 3N = 642 Cartesian coordinates. Any particular ! conformation i is thus denoted by a vector X i in this 3Ndimensional space. Applying the concept of principal curve [25], we obtained a pathway that represents the conformational space visited in the unrestrained simulations. Specifically, we first defined a straight line, in the 3N-dimensional space, that connects the open and ! ! closed AdK crystal structures, X OP and X CL , after proper alignment. For each frame i in the trajectories of the unrestrained simulations, we performed a rigid-body alignment with respect to ! the open-state crystal structure X OP , thus removing the overall translation and rotation of the protein. The protein coordinate ! after the alignment, X i , was then orthogonally projected onto the ! line XOP XCL above, with the projected point denoted by Y i . The ! entire range covered by all the fY i g on the line was then evenly ! divided into 100 segments. The entire set of coordinates fX i g was thus also 1407003 classified into 100 groups, according to the line segment ! each corresponding Y i lies in. Then for each group k, we !Avg calculated the average coordinate, X k , over all the coordinatesoptimization with respect to both fwij g and ftk g [24]. With the ! !Avg obtained fwij g, X ?thus defines a continuous curve from X 0 Avg ! to X M , as the curve parameter t varies from 0 to 1. We use s ?to ! !Avg denote the arc length [24] between X 0 and X ?along this curve, thus with s??= 0, and s? L denoting the entire length !Avg !Avg of the curve between X 0 and X M in the 3N-dimensional space. In general, s ?is not a linear function of t. We may, however, define a new curve parameter a(t):s(t)=L. Because a ?is a monotonic function, the inverse function t ?is well defined, with each a[?,1 corresponding to a unique t ?,1, and thus a ! ! unique conformation X ?X ?on the curve. In this new parametrization, the arc length of the curve is a linear function of a. We have thus obtained a uniformly parametrized smooth ! pathway X ? a[?,1, that represents the sampled conformational space in the unrestrained simulations. We note that a curve can also be obtained by simple piece-wise linear or spline interpolation after applying some s.
