Exploring the free-energy landscapes of biological systems with steered molecular dynamics

Transcription

1 Exploring the free-energy landscapes of biological systems with steered molecular dynamics Guodong Hu and L. Y. Chen* Department of Physics, University of Texas at San Antonio, One UTSA Circle, San Antonio, Texas We perform steered molecular dynamics (SMD) simulations and use the Brownian dynamics fluctuation-dissipation-theorem (BD -FDT) to accurately compute the free-energy profiles for several biophysical processes of fundamental importance: hydration of methane and cations, binding of benzene to T4-lysozyme L99A mutant, and permeation of water through aquaglyceroporin. For each system, the center-of-mass coordinates of the small molecule (methane, ion, benzene, and water, respectively) is steered (pulled) at a given speed over a period time, during which the system transitions from one macroscopic state/conformation (State A) to another one (State B). The mechanical work of pulling the system is measured during the process, sampling a forward pulling path. Then the reverse pulling is conducted to sample a reverse path from B back to A. Sampling a small number of forward and reverse paths, we are able to accurately compute the free-energy profiles for all the afore-listed systems that represent various important aspects of biological physics. The numerical results are in excellent agreements with the experimental data and/or other computational studies available in the literature Wt, Rs, Cd, Jc, Ln *Liao.Chen@utsa.edu 1

2 I. Introduction Accurate computation of free-energy is of essential importance in quantitative studies of biological systems because most biophysical/chemical processes are driven by the free-energy gradients. 1-8 Methods to calculate free-energy differences between two states (conformations) A and B fall into two classes, equilibrium methods and, developed more recently, non-equilibrium methods. A recent review can be found in Ref. 6. The equilibrium approaches such as the thermodynamic integration method require more computing resources because they are based on full sampling of the equilibrium ensembles involved in a given biophysical process. The non-equilibrium approaches exploit the stochastic dynamics of the system driven with some applied forces and aim to map out the free-energy landscape in terms of the potential of mean force (PMF). 9 The free-energy difference between States A and B is extracted from the measurements of work along the transition paths connecting the two states. In Refs. 10, a non-equilibrium PMF type of approach has been developed on the basis of the Brownian dynamics fluctuation-dissipation theorems (BD-FDT) 11, extracting the equilibrium free-energy differences from irreversible (non-equilibrium) work measurements in steered molecular dynamics (SMD) simulations 12. The precision and efficiency of the BD-FDT approach have been demonstrated with the coil-helix transition of deca-alanine peptide. Preliminary results have been obtained for water transport through aquaglyceroporin GlpF. In this paper, we further BD-FDT study to three types of systems of fundamental biological importance: hydration of methane and sodium and potassium ions 13-17, binding of benzene in the L99A mutant of T4-lysozyme (T4L) 18-24, and water transport through 2

3 aquaglyceroporin embedded in a cell membrane In each case, we perform SMD simulations to sample five (or ten) forward pulling paths and the same number of reverse paths. With these small sets of paths sampled, we use BD-FDT to compute the freeenergy landscapes between States A and B. In the case of hydration free-energies, we are able to reproduce accurately the well-accepted results obtained using sophisticated methods incorporating various correction terms. In the case of binding benzene to T4L, the BD-FDT result for the absolute binding energy is in perfect agreement with the experimental data and with other computational studies in the literature. In the case of water permeation through GlpF, the BD-FDT results give the first complete map of the free-energy landscape along the entire channel length and are consistent with the existing studies. The rest of the paper is organized as follows: In Section II, we outline the theoretical formulation. In Section III, we compute the hydration free-energies for methane, sodium ion, and potassium ion. In Section IV, we compute the absolute binding free-energy of benzene to T4L, a unique representative of the ligand-protein complexes. In Section V, we extend our preliminary study in Ref. 10 to give an accurate free-energy profile of water along the entire channel of GlpF. In Section VI, we give a brief summary and discussion. 3

4 II. Theoretical formulation In an SMD study 12, 36, we pull (steer) a group of atoms with appropriate forces so that they go along a desired direction to explore a certain manifold of the system s phase space. In the current literature, the steering is mostly accomplished by pulling the centreof-mass of the steered group with a spring (elasticity constant k ) whose other end is advanced along a specified direction (taken as along the z-axis in this paper) at a constant speed v. Pulling the system from State A to State B, the following amount of work is done to the system (excluding the spring): W = k[ λ ( t) + zx ( ) zx ( )] dz. (1) A B F A A B Here the pulling control parameter λ F () t = vt. x is the set of the coordinates of all the atoms of a given system. zxis ( ) the z -coordinate of the centre-of-mass of the pulled molecule/group of atoms that is a linear function of x. Along a reverse path when the system is pulled from State B to State A, the work done to the system WB Ais a similarly defined force-displacement integral, W = k[ λ ( t) + zx ( ) zx ( )] dz (2) B A R B A B where λ R () t = vt. Note that the mechanical work defined in Eq. (1) or (2) is not identical to the thermodynamic work often used in the literature. 9 We assume each of the systems of our interest is governed by the Brownian dynamics. Therefore, we employ the Brownian dynamics fluctuation-dissipation theorem (BD-FDT) 11 to extract equilibrium free energy from irreversible work measurements. In 4

5 terms of mechanical work, the BD-FDT relates the equilibrium free energy to the nonequilibrium work as follows: exp( β G) = exp( βw ) exp( βw ) A B F 2 B A R. (3) Here G = GB GA is the difference between the free energies of States A and B. β = 1/ kt B with kb and T being the Boltzmann constant and the absolute temperature respectively. The statistical average over the forward paths is f W N 1 F = f W (4) ( p) ( ) ( ) F A B NF p= 1 where the index p indicates the p-th of the The statistical average over NF forward paths from State A to State B. NR reverse paths, from State B to State A, is N 1 R =. (5) ( p) ( ) f( WB A) R NR p= 1 f W In order to improve the statistics, we use infinitely stiff spring in the SMD simulations. In the limit of infinite stiffness, k, the pulling is accomplished by fixing the centre-of-mass coordinate of the pulled molecule, zx, ( ) to the control parameter λ () t ( ) at each time step. In this limit, dz = vdt and the pulling force then is F R equal to the net force on the pulled molecule exerted by the rest of the entire system, namely, k[ λ( t) z( x) + zx ( )] H( x) z. (6) A 0 Here H ( ) 0 x is the Hamiltonian of the entire system in absence of the pulling force. Then the mechanical work done to the system along a forward (reverse) path can be expressed as 5

6 W H ( x) H ( x) = dz, W = dz. (7) 0 0 A B B A z A B z=λ z B A z= λ Note that the work in Eq. (7) is stochastic in nature as the system is subject to random forces of the Brownian dynamics. In following three sections, Eq. (3) will be used in conjunction with Eq. (7) to compute the free-energy differences between various relevant states. 6

7 III. Hydration of methane and cations Needless to say, hydration free energy is fundamentally relevant to biological systems as all biochemical processes happen in aqueous environment. 37 In this section we employ the BD-FDT to compute the hydration free energy of methane, sodium cation, and potassium cation that have been very well studied with other approaches in the literature Our model system consists of a 40Å 40Å 40Åcube of TIP3 waters plus a molecule (methane, sodium ion, or potassium ion) to be hydrated. The water cube is placed on the xy-plane, 0 < z < 40Å, and periodically replicated on the x- and y- dimensions. Namely, we use periodic boundary conditions along the x- and y-axes but not along the z-axis. A reflective boundary is placed at z=0 for waters only (not for the molecule to be hydrated) so that all waters remain above the xy-plane. A small constant force field along the negative z-axis is applied to waters to keep the body of water within a well defined layer at z 0. This small constant force field ( kcal/mol amu Å ) mimics the gravity that exerts a constant force on every atom of the water molecules in proportion to its mass. The molecule to be hydrated is initially placed at (0, 0, z A ). From there it will be pulled into the body of water in SMD simulations. The intra- and inter- molecular interactions are represented by the CHARMM 27 force field. 38 The cut-off for long-range interactions are set to 14 Å with a switching distance of 12 Å. The SMD simulations are implemented with NAMD source code adapted for infinitely stiff spring pulling. The damping constant is chosen to be γ = 5/ ps. The temperature is set to T = 300K. The pulling speed v = 0.05 Å/ ps. We define State A when the molecule is outside the body of water ( z = 14.6Å for methane and z = 10Åfor cations) and State B as when the molecule is inside the A A 7

8 cube ( z = 15.4Å for methane and z = 10Å for cations). Then the hydration free energy B of a molecule can be split into two parts: B ( G G ) ( G G ) G = + (8) hydr B A A where G is the free energy of the system when the molecule is separated at an infinite distance from the body of water, z =. We use Eq. (3) in conjunction with Eq. (7) to determine GB GA and we use the following analytical approximation for GA G. From z = to z A, the molecule to be hydrated is well outside the range of van der Waals interaction with the waters. For the Coulomb interaction, when the molecule is charged, we can well approximate the body of water as a continuous medium (dielectric constant ε ) occupying the upper half space ( z 0 ). Moving the molecule to be hydrated at an infinitely (reversibly) slow speed from to z A, we analytically carry out the force-displacement integral and obtain the following result: G A 2 q 2ε 0 G = 1 16 πε 0 za ε 0 + ε (9) where ε 0 is the vacuum dielectric constant. q is the net charge carried by the molecule to be hydrated. For methane, q = 0 and, for sodium/potassium cation, q= + e. Taking the water dielectric constant asε = 80ε0, we have for sodium/potassium cation, GA = 8.1kcal/mol, and for methane, G = 0, of course. Now we will have the G total hydration free energy [Eq.(8)] once we determine G B G A through Eq.(3). A Plotted in Fig. 1 is a representative set of the work curves along five forward paths (pulling methane into water) and five reverse paths (pulling methane out of water). 8 G

9 From Fig. 1, we observe that the pulling at a speed of v = 0.05 Å/ ps is irreversible on the basis of the following details. The work W = W ( z) is over a partial path of pulling A z A B from za to z. The work WB z = WB A( z) is over a partial path of pulling from zb to z. Then work over a partial path from z to za is Wz A = WB A( za) WB A( z). It is clear from Fig.1 that W + W 0. It would be zero if the pulling were reversible. A z z A For the purpose of accurately determining the hydration free energy of methane, we have sampled five sets of pulling paths. Each set consists of five forward and five reverse paths. We apply Eq. (3) to each set to obtain a set of data for the free energy of the methane-water system as a function of the z-coordinate of methane s centre of mass. We perform statistical average over the five set of data for the free energy profiles and compute the root mean square deviations from the average as the error bars. The average free energy and the error bars are plotted in Fig. 2. In this manner, the hydration free energy of methane is determined to be = 2.4 ± 0.2 kcal/mol on the basis of G hydr CHARMM 27 force fields. This result is in perfect agreement with Shirts et al s value of 2.44 ± 0.02 from their extremely precise free energy calculation 14 and Henin and Chipot s result of calculation with the adaptive force biasing method 13. For cation hydration, we note that, in State A, the cation (sodium or potassium) is at z = 10Å and, in State B, z = 10Å. We further note that the water-vacuum interface A B is the xy-plane ( z = 0). Considering that cation can disturb the body of water over a long range, we divide the 20Å separation between A and B into 20 sections of 1Å each. We perform SMD over each section to sample five forward and five reverse paths. Using Eq. (3), we compute the free energy landscape over each section. The overall free energy 9

10 landscape is given Fig. 3. Sampling only five forward and five reverse pulling paths, we are able to accurately determine the free energy landscape of cation hydration. In fact, the hydration energy of sodium cation so determined, kcal/mol (with an estimated error ± 1kcal/mol), is perfect agreement with Warren and Patel s carefully corrected result of kcal/mol. 15 The potassium hydration energy of kcal/mol obtained here with the BD-FDT also perfectly agrees with Warren and Patel s result of kcal/mol

11 IV. Ligand-protein binding free-energy Ligand-protein binding is another biophysical problem of fundamental importance. 7, In this section, we explore the free-energy landscape between the dissociated state and the bound state of the benzene-t4l complex in solution. Binding of a ligand to a protein can be adhesion to the protein surface or a shallow hole on the protein surface, and it can be deep burial inside a cavity of the protein. The benzene-t4l system belongs to the latter case. In fact, the benzene molecule is completely buried inside the protein. This system represents a very severe test for the accuracy of the BD- FDT, Eq. (3). Even though the binding free-energy is only a function of states and independent of the paths to follow from State A to State B, it is believed that the PMF type of approach is not applicable to systems like benzene-t4l that has no obvious preferred binding pathways connecting the bound and dissociated states 7. In another word, the PMF type of approach, if successful, will not only reveal the statistical differences between the dissociated and bound states in terms of binding free energy but also map out the free-energy landscape along the entire transition path from the dissociated to the bound state. Indeed, for the benzene-t4l system, the BD-FDT approach does accurately predict the binding free energy and yield the free-energy landscape between the bound and the dissociated states. We start with the crystallographic structure of the benzene-t4l complex (PDB code: 181L) 18 and solvate it in a box of TIP3 waters and neutralize it with sodium and chlorine ions. The dimensions of the solvated system are 67Å 61Å 65Å. The temperature is kept at 300K and pressure at 1 bar. The damping constant is 5/ps. We use the CHARMM 27 force field 38 for intra and inter molecular interactions and NAMD

12 for stochastic dynamics simulations. We use periodic boundary conditions for all three dimensions. We employ the particle-mesh Ewald for electrostatic interactions. SMD 12, 36 simulations are performed by pulling the centre of mass of benzene out of (noted as forward) and back to (noted as reverse) the binding site that is completely inside the protein. In order to avoid causing too drastic distortions to the protein, we divide the pulling path between the bound and the dissociated states into ten section of 1Å each. Over each of the ten sections, we sample ten forward (away from the binding site) paths and ten reverse (back towards the binging site) paths. These pulling paths are shown in Fig. 4(a). Then we use Eq.(3) to estimate the free energy differences in each of the ten sections. The overall free energy landscape is shown in Fig. 4(b). The prediction of this BD-FDT approach is an absolute binding free energy of 5.7 kcal/mol that is consistent with Deng and Roux s result of 5.96 kcal/mol and the experimental measurement of 5.19 kcal/mol. 7 12

13 V. Water permeation through GlpF In this section, we explore the free-energy landscape of water permeation through aquaglyceroporin GlpF. GlpF possesses a homotetrameric structure: Each monomer forms an independently functional pore (channel). The core of this protein consists of two half-membrane spanning repeats related by an essentially two-fold symmetry. The N termini of the α -helical repeats convene approximately at the center of the channel, where the Asn Pro Ala (NPA) motifs are located. The selectivity filter (SF), the narrowest section of the pore, is about 8 Å from the center of the channel, namely, from the central plane of the lipid bilayer. In this work, the z axis of the Cartesian coordinates is chosen to be normal to the water-membrane interface, pointing from periplasm to cytoplasm. The origin of the coordinates is set so that z = 7.7Å at the midpoint between the N-termini of Asn 68 and Asn 203. This work uses the all-atom model of the GlpF-membrane system studied in Ref. 30, consisting of GlpF homotetramer embedded in a fully hydrated palmitoyloleylphosphatidyl-ethanolamine (POPE) bilayer. The GlpF-POPE-bilayer system is explicitly solvated with TIP3 waters. Sodium and Chlorine ions are added and the whole system is neutralized. The system consists of 106,189 atoms. The pressure and the temperature are maintained at 1 bar and 300 K respectively. The Langevin damping coefficient is chosen to be 5 ps. The particle-mesh Ewald method is utilized to compute electrostatic interactions. The time step of 1 fs is used for short-range interactions and 4 fs for long-range forces. Covalent bonds of all hydrogen atoms are fixed to their equilibrium length. The periodic boundary conditions are applied all three dimensions. 13

14 Starting from the fully equilibrated structure of Ref. 30, we run equilibrium Langevin dynamics for 4 ns after removing the four glycerols from the system. Then we perform SMD simulations to pull four waters through the four channels of GlpF simultaneously, one water molecule through each of them. We divide the 40Å length of the whole channel into eight sections of 5Å each. Over each of the eight sections, two sets of pulling paths are sampled: Each set has five forward paths (from periplasm to cytoplasm) and five reverse paths (from cytoplasm to periplasm). One of the two sets of pulling paths is plotted in Fig. 5(a). In this case, the order parameter is chosen as the center-of-mass z-coordinate of one pulled water molecule. The pulling speed is once again v = 0.05 Å/ ps. Following the same argument in Section III, we observe that the pulling is irreversible. We apply the BD-FDT to each set of the five forward and five reverse paths and obtained a set of free-energy differences over each of the eight sections. The overall freeenergy landscape is shown in Fig. 5(b). The average free energy is the statistical mean between the two sets of data and the error bars represent the deviations between them. It gives a complete map of the entire channel of water permeation. The free-energy profile and the PMF barrier are in agreement with de Groot et al s computational study 28, 35 and consistent with the experiments in the literature. 26 It is interesting to note that water permeation through the channels of GlpF is collective diffusion. Multiple water molecules form single files inside each pore and permeate through a channel in a concerted manner. In our in silico experiments, only one water per channel is pulled. Yet our non-equilibrium SMD study is actually in full 14

15 28, 35 agreement with the equilibrium simulations of the highly correlated diffusion of multiple water molecules. It indicates that the path averages in the numerator and the denominator of BD-FDT [Eq. (3)] cancel out the irreversible, non-equilibrium factors (the dissipative work terms in particular) in a very efficient manner even for a small number of pulling paths sampled. VI. Summary and discussion In summary, we have furthered the BD-FDT approach and explored the free-energy landscapes of three very different types of biological systems. Sampling five to ten forward and reverse pulling paths, we were able to accurately determine the free-energy profiles of the systems from the irreversible work measurements in SMD simulations. The accuracy of the BD-FDT method has been demonstrated by the perfect agreement with the existing experimental and/or computational results in the literature for hydration of methane and sodium and potassium cations. This BD-FDT study also has produced an accurate map of the free-energy landscape along the entire channel of GlpF for water permeation. It is reasonable to expect that BD-FDT will be successfully applied to more biological as well as chemical systems. Acknowledgments The authors acknowledge support from an NIH SC3 grant (GM084834), the UTSA Computational Biology Initiative, and the Texas Advanced Computing Center. 15

16 Figure captions Fig. 1 Work done along the forward (red) and reverse (green) pulling paths. For forward paths, the work value, W ( z), is what is done to the system when the center of mass of A B methane is pulled from z = 15.4Å to z. For reverse paths, the work value, W ( z) A is what is done to the system when the center of mass of methane is pulled from B A, zb = 15.4Å to z. The pulling speed is 0.05 Å / ps. The pulling is clearly irreversible because W ( z ) + W ( z ) 0. A B B B A A 16

17 Fig. 2 Free energy of the water-methane system as a function of methane s position. The free energy curve is the average over five sets of data. The error bars are the corresponding root mean square deviations from the average. Each set of the free-energy data is obtained through the BD-FDT from five forward and five reverse pulling paths. 17

18 Fig. 3 The free energy landscape of sodium cation (solid) and that of potassium cation (dashed). The zero point of free energy is set to when the cation is infinitely far from the body of water, z =. Note that the body of water occupies the space of 0 z 40Å. 18

19 (a) (b) Fig. 4 (a) Work done along the forward (spreading out towards right) and reverse (spreading out towards left) pulling paths. For forward paths, the work value, W ( z) A B, is what is done to the system when the centre of mass of benzene is pulled from the left end of a section to z. For reverse paths, the work value, W ( z), is what is done to the B A system when the centre of mass of benzene is pulled from the right end of the same 19

20 section to z. The pulling speed is 0.05 Å / W ps. The pulling is irreversible because + W 0. (b) Free energy profile estimated with the Brownian Dynamics FDT. A z z A 20

21 (a) (b) Fig. 5 (a) Work done along the forward (spreading out towards right) and reverse (spreading out towards left) pulling paths. The work is averaged over the four waters 21

22 pulled through the four channels simultaneously. For forward paths, the work value, W ( ) A B z, is what is done to the system when the centre of mass of a water molecule is pulled from the left end of a section to z. For reverse paths, the work value, W ( z), is what is done to the system when the centre of mass of a water molecule is pulled from the right end of the same section to z. The pulling speed is 0.05 Å / B A ps. The pulling is irreversible becausew + W 0. (b) Free energy profile estimated using the A z z A Brownian Dynamics FDT and the corresponding error bars. The midpoint between the N δ atoms of Asn-68 and Asn-203 of the NPA motifs is located at z = 7.7Å. The SF is located around z = 0and the NPA is around z = 7.7Å. 22

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