Moothing techniques [21]. In this particular study, we chose to adopt the global sinusoidal representation above because the resulting curve is infinitely differentiable everywhere, thus suitable for the local linear approximation discussed below. ! Any conformation near the curve X ?above can be projected onto the curve. Specifically, we define projection function ! ! a Pa X , with the function value a Pa X corresponding to the ! ! X ?on the curve with the shortest distance to X . In this way we can project the entire accessible conformational space of the protein onto this one-dimensional curve in the 3N-dimensional coordinate space. A free energy, G ? can accordingly be defined as a function of the curve parameter a [25?7]. We carried out a set of umbrella-sampling simulations to compute the free energy profile G ? with a total of 30 umbrella windows. Each window i samples the vicinity of a corresponding reference ai . These references fai g (i = 1, …, 30) cover the range of [0,1], with a uniform spacing of 1/29. We adopted harmonic Ka umbrella potentials Ui {ai ? for each window. In the 2 coordinate space, this potential is of the form. i2 ! K h ! a Pa X {ai : Ui X 2 ??! In principle, the projection Pa X above is a nonlinear function. However, in the close vicinity of ai , the curve can be approximated by a straight line:Adenylate Kinase Conformationd ! ! ! X i zDa?X i Da: X i ? da??! As described earlier, by construction the arc length of X 18204824 ?is a linear function of a, such that the magnitude of the derivative, d ! D X , is a constant along the curve. Because the total curve da length is L between a = 0 and a = 1, we further have d ! D X L, and thus can rewrite Eq. 2 as da ! X i zDa?Ri zLDa:^i , r ??spring constant of 200 kcal/mol/degree2, on the orientation angle of the protein. These restraints eliminated the drift along the six degrees of order 117793 freedom for rigid-body translation/rotation, such that ! the 18055761 Cartesian coordinates X represent only the internal degrees of freedom, or the conformational state of the protein. To enhance the sampling, we implemented Hamiltonian replica exchange [38] in the umbrella-sampling simulations. At every 400 fs, two adjacent simulations i and j attempt to swap their restraining potentials, which would result in a change in the combined by h ! i Hamiltonian ! i h ! ! DU Ui X j zUj X i { Ui X i zUj X j . The attempt is accepted if DUv 0, or otherwise accepted with a probability of exp DU=kB T ? Each umbrella-sampling simulation was run for 40 ns, with the last 30 ns used for analysis. The free energy G ?was computed using the weighted histogram analysis method [39,40], with the statistical errors estimated from the uncertainties in the average sampled coordinate [40].! ! in which Ri :X i ? and ^i is the unit vector along the direction r d ! ! X i ? or the Nafarelin custom synthesis tangent of the curve at Ri . With this of da ! ! approximation, in the space near Ri , Pa X becomes a linear ! ! ! r projection: Pa X ai z X { Ri :^i =L. The umbrella potential (Eq. 1) thus becomes ! K h! ! i2 X { Ri :^i , r Ui X 2 ??ResultsIn this section, we first describe the results of our unrestrained simulations, in which some spontaneous transitions from the closed to the open conformation were observed. We then present the free energy profile along a transition pathway, as calculated from our biased sampling simulations.in which K:Ka =L2 . The resulting forces on the Ca atoms are then given by h ! !i ! ! F i X K.Moothing techniques [21]. In this particular study, we chose to adopt the global sinusoidal representation above because the resulting curve is infinitely differentiable everywhere, thus suitable for the local linear approximation discussed below. ! Any conformation near the curve X ?above can be projected onto the curve. Specifically, we define projection function ! ! a Pa X , with the function value a Pa X corresponding to the ! ! X ?on the curve with the shortest distance to X . In this way we can project the entire accessible conformational space of the protein onto this one-dimensional curve in the 3N-dimensional coordinate space. A free energy, G ? can accordingly be defined as a function of the curve parameter a [25?7]. We carried out a set of umbrella-sampling simulations to compute the free energy profile G ? with a total of 30 umbrella windows. Each window i samples the vicinity of a corresponding reference ai . These references fai g (i = 1, …, 30) cover the range of [0,1], with a uniform spacing of 1/29. We adopted harmonic Ka umbrella potentials Ui {ai ? for each window. In the 2 coordinate space, this potential is of the form. i2 ! K h ! a Pa X {ai : Ui X 2 ??! In principle, the projection Pa X above is a nonlinear function. However, in the close vicinity of ai , the curve can be approximated by a straight line:Adenylate Kinase Conformationd ! ! ! X i zDa?X i Da: X i ? da??! As described earlier, by construction the arc length of X 18204824 ?is a linear function of a, such that the magnitude of the derivative, d ! D X , is a constant along the curve. Because the total curve da length is L between a = 0 and a = 1, we further have d ! D X L, and thus can rewrite Eq. 2 as da ! X i zDa?Ri zLDa:^i , r ??spring constant of 200 kcal/mol/degree2, on the orientation angle of the protein. These restraints eliminated the drift along the six degrees of freedom for rigid-body translation/rotation, such that ! the 18055761 Cartesian coordinates X represent only the internal degrees of freedom, or the conformational state of the protein. To enhance the sampling, we implemented Hamiltonian replica exchange [38] in the umbrella-sampling simulations. At every 400 fs, two adjacent simulations i and j attempt to swap their restraining potentials, which would result in a change in the combined by h ! i Hamiltonian ! i h ! ! DU Ui X j zUj X i { Ui X i zUj X j . The attempt is accepted if DUv 0, or otherwise accepted with a probability of exp DU=kB T ? Each umbrella-sampling simulation was run for 40 ns, with the last 30 ns used for analysis. The free energy G ?was computed using the weighted histogram analysis method [39,40], with the statistical errors estimated from the uncertainties in the average sampled coordinate [40].! ! in which Ri :X i ? and ^i is the unit vector along the direction r d ! ! X i ? or the tangent of the curve at Ri . With this of da ! ! approximation, in the space near Ri , Pa X becomes a linear ! ! ! r projection: Pa X ai z X { Ri :^i =L. The umbrella potential (Eq. 1) thus becomes ! K h! ! i2 X { Ri :^i , r Ui X 2 ??ResultsIn this section, we first describe the results of our unrestrained simulations, in which some spontaneous transitions from the closed to the open conformation were observed. We then present the free energy profile along a transition pathway, as calculated from our biased sampling simulations.in which K:Ka =L2 . The resulting forces on the Ca atoms are then given by h ! !i ! ! F i X K.
