Free Energy Perturbations of ATP in ATPase

Transcription

1 University College London Department of Physics and Astronomy M4201: Final Year Project Free Energy Perturbations of ATP in ATPase Author: Mirna Kovacevic Mathematics & Physics MSci Supervisors: Dr David Bowler Dr Veronika Brázdová March 2009

2 Abstract Previous theoretical and experimental studies have investigated the structure and function of various ATPase molecules. This theoretical project considers F1-ATPase from bovine heart mitochondria. The primary investigation is into the effect of the size of the model ATPase molecule on the accuracy of the results achieved. It concluded that the isolated unconstrained catalytic sites are the minimal model of the ATPase complex that can be used to model the function of the whole molecule. Further, free energy perturbations of ATP within the ATPase molecule in the catalytic β T P and β DP sites are considered to gain insight into the relative stabilities of the states and the difference between the catalytic sites. Due to discrepancies in results of the free energy calculations caused by errors in the parameters chosen, further free energy perturbation calculations were carried out on ATP solvated by water. These calculations determined the correct initial conditions necessary for the achievement of accurate results.

3 Contents 1 Introduction 3 2 Theoretical Background Structure and Properties of Molecules Concerned ATP ATP Hydrolysis ATPase Modelling the System Methods Molecular Dynamics Empirical Force Fields Minimization and Structure Optimization Equilibration Free Energy Perturbations Computer Code NAMD VMD System Setup Minimization & Equilibration Free Energy Perturbations Results Minimization & Equilibration β T P Site β DP Site Free Energy Perturbations Observations Data Analysis Further FEP Calculations Conclusions 47 7 Plans for Future Work 48 Appendix 50 Nomenclature 52 List of Figures 54 References 56

4 1 INTRODUCTION 1 Introduction The development of a scientific discipline that deals on the scales of nanometers, called nanotechnology, has made it possible to visualise and investigate structures at the atomic scale. Nanotechnology continues to push the boundaries of existing tools and knowledge, encompassing research fields from all areas of science. The fusion of molecular biology, chemical physics and scientific computing has made it possible to model and understand biological molecules essential to life. Among these molecules are proteins, including the enzyme ATPase, and understanding the processes by which these function is an important step to understanding the molecular machines which drive fundamental life processes. Previous theoretical [2, 3, 4, 6, 7] and experimental [5, 8] studies have investigated the structure and function of both ATP and ATPase, but the process by which the ATPase enzyme catalyzes the hydrolysis of ATP and harnesses the energy released during this reaction to drive other chemical processes is not entirely understood. This project considers ATPase and uses molecular dynamics to investigate further the chemical activity of this molecule. The primary aim is to determine and justify the minimal model of the ATPase complex that can be used to model the function of the whole molecule. The purpose of this is to minimize the length of time required in running molecular dynamics simulations, whilst ensuring that the model used is a precise enough approximation to yield accurate results. The ultimate aim is to understand fully the free energy changes occurring between the initial, transition and final states of reactions taking place in ATPase, thus providing a further insight into the process taking place in the ATPase molecule. 3

5 2 THEORETICAL BACKGROUND 2 Theoretical Background 2.1 Structure and Properties of Molecules Concerned ATP Adenosine triphosphate, usually abbreviated ATP, is an organic nucleotide molecule consisting of a nitrogenous adenine base, attached to a sugar ribose backbone, which is in turn attached to a chain of three phosphate groups linked by covalent bonds, as shown in Figure 1a. ATP is widely considered to be the molecular currency of energy transfer as the energy stored in the molecule is the source which drives many chemical processes necessary to maintain life [1]. This energy is stored in the covalent bonds joining the three phosphate groups and is released during the process of ATP hydrolysis. (a) ATP (b) ATPase: Top View (c) ATPase: Side View Figure 1: ATP and ATPase molecules ATP Hydrolysis The reaction between ATP and water, catalyzed by the enzyme ATPase, shown in Figures 1b and 1c, is known as ATP hydrolysis. This reaction consists of the catalytic H 2 O molecule attacking ATP, which causes the covalent bond between the terminal phosphate group and the oxygen joining this group to the rest of the molecule to be broken, and chemical energy stored in this bond released. When this happens, ATP breaks down into adenosine diphosphate, abbreviated to ADP, an inorganic phosphate group, 4

6 2 THEORETICAL BACKGROUND P i and a free proton [1]. The chemical reaction can be summarised as: AT P + H 2 O ADP + HP O4 ( P i) + H + (1) The breakdown of ATP into ADP and inorganic phosphate occurs in the presence of a magnesium ion, Mg 2+, located in close proximity of the phosphate chain, which has a catalytic role in the reaction [4]. ADP can further be hydrolysed to produce adenosine monophosphate, known as AMP, another inorganic phosphate group and proton, and a further release of energy ATPase ATP synthase, more commonly referred to as ATPase, belongs to a group of enzymes which catalyse ATP hydrolysis. In doing so, ATPase harnesses the energy released in the breaking of the covalent bond described in Section 2.1.2, and uses it to pump the protons released in the hydrolysis reaction against their thermodynamic gradient, thereby driving other chemical processes. When the enzyme works in reverse, it harnesses the energy of a transmembrane proton gradient as an energy source for adding an inorganic phosphate group to an ADP molecule to synthesise ATP. ATPase, shown in Figures 1b and 1c, is a large multisubunit complex composed of an integral membrane protein domain, F 0, attached to the peripheral catalytic domain, F 1 [5, 6, 7, 8]. This project is concerned with the catalytic F 1 domain, which can be separated from the F 0 membrane domain and it retains the ability to hydrolyze ATP. This particular form of ATPase is found in bovine heart mitochondria. The F 1 domain sits outside the mitochondrial membrane, and is composed of 3α and 3β subunits, represented in Figure 1b by the different coloured segments. The catalytic subinits are arranged in alternation around the central stalk, which consists of the subunits γδɛ [2], shown in Figure 1c. The interfaces between the α and catalytic β subunits form pockets, and the asymmetry of the γ subunit causes these to adopt different conformations, each containing a different structure of ATP. Two of these have similar conformations. One, called the β DP site, contains the transition state struc- 5

7 2 THEORETICAL BACKGROUND ture of ATP, and the other, called the β T P site, contains the bound start of reaction ATP structure. The third pocket has no bound ATP structure and is thus known as the empty β E site [8]. The β DP site is thought to be the highest energy site. The three catalytic sites are shown in Figure 2. The hydrolysis of ATP is believed to lead to a rotation of the central stalk [2, 6, 7], although the detailed mechanism is not entirely understood. It is thought that a mechanical rotation of the γ-subunit through 360, in steps of 120 [7], causes each of the β catalytic sites to go through the tight, loose and open states, thereby hydrolysing three ATP molecules [8]. In Figure 2, β E site is in the open, β T P site in the loose and β DP site in the tight state. (a) β E site (b) β T P site (c) β DP site Figure 2: Catalytic sites of ATPase molecule 6

8 2 THEORETICAL BACKGROUND 2.2 Modelling the System The structure of ATPase that is considered in this project is protein data bank (PDB) entry 2JDI. It is the ground state structure of F 1 -ATPase from bovine heart mitochondria at 1.9Å resolution [8]. The structure was obtained via X-ray diffraction and was chosen because it contains the data for both the ATP and ATPase molecules in the same file. This meant that it was possible to consider the ATP molecule in the same local environment as it would naturally occur, in order to generate results which are representative of the actual reactions and conformation changes taking place in the system. X-ray diffraction is used to determine the geometry a molecule by elastically scattering X-rays from crystalline structures. As the molecules considered in this project do not naturally occur in crystalline form, this means that the starting structure used is not necessarily the structure of the molecule found in the in vivo environment. In addition, X-ray diffraction experiments omit the coordinates of hydrogen atoms in structures due to their low scattering power and lack of core electrons. Hydrogen atoms therefore have to be added before using the structure as a model for the molecule. This in itself is a speculative technique, and equilibration of the system, described in section 3.3, is used to allow the system to reach an equilibrium point which may be more representative of its natural state. Due to this, the model of the molecule used in this project may yield results that are not entirely representative of what happens in living cells. In order to determine the minimal model of the ATPase complex that can be used to model the function of the whole molecule, several different variations of the original 2JDI structure were considered. The whole molecule was considered as one system. The molecule was then dissected so that each of the catalytic sites was considered as a separate isolated system, as shown in Figure 2. This allowed the comparison of the behaviour of the whole structure to the behaviour of each of the catalytic sites, to show the effect of reducing the size of the system such that only the catalytic site is considered. Subjecting each of the systems described above to a set of constraints allowed further examination of the effect of the size of the system on the accuracy of the function of the model molecule. The constraints were used 7

9 2 THEORETICAL BACKGROUND to specify which atoms in the system would be static and which would be free during molecular dynamics simulations. Each system was constrained such that all residues that have at least one atom within a distance, specified in A, of the ATP residue in the catalytic site were allowed to be free, and all other residues in the system were fixed. This effectively defined a volume within which the normal molecular dynamics simulation was performed. The remainder of the system, which was fixed during the simulation, acted as a boundary to preserve the global density and overall structure of the molecule. Distances of 10, 20 and 30 A, as shown in Figure 3, were specified in order to determine the optimum between the size of the system and the length of time required to run molecular dynamics simulations. (a) 10A from ATP in pocket free (b) 20A from ATP in pocket free (c) 30A from ATP in pocket free (d) Free system Figure 3: Constraints on the βt P site. Atoms represented in red are free and those represented in blue are fixed. 8

10 2 THEORETICAL BACKGROUND For the dissected systems, examining the βt P and βdp sites, setting the constraints meant that four systems, as shown in Figure 3, were considered for each of the sites. The whole molecule was also subjected to the above constraints for the βt P and βdp sites separately. This meant that four configurations of the whole molecule were considered for each of the sites, with the free system being the same in both cases. The constraints on the βt P site in the whole molecule system are shown in Figure 4. (a) 10A from ATP in βt P site free (b) 20A from ATP in βt P site free (d) Free system (c) 30A from ATP in βt P site free Figure 4: Constraints on the βt P site in whole ATPase molecule. Atoms represented in red are free and those represented in blue are fixed. 9

11 3 METHODS 3 Methods 3.1 Molecular Dynamics The method of molecular dynamics (MD) simulations is one of the dominant tools in the theoretical study of biological molecules. MD simulations are used to calculate the time dependent behaviour of a molecular system. The results, in conjunction with experimental studies, are used to understand the structure, dynamics and thermodynamic properties of systems [9, 10]. The method of MD simulations is based on classical mechanics. Each atom is considered to be a point mass whose motion is determined by the forces exerted on it by all other atoms, as described by the equations of motions [11]. MD simulations therefore involve solving Newton s Second Law for the many body problem, which must be done by numerical methods due to the large system of coupled equations.. In a system of N atoms, where atom i has mass m i, its position is given by the vector r i and the force exerted on atom i by all other atoms in the system is F i, the equation of motion is given by: F i = m i 2 r i t 2 (2) where t represents time. If the force on each atom F i is known, it is possible to determine the acceleration 2 r i t 2 of each atom in the system. Integrating Equation (2) then yields a trajectory that describes the position, velocity and acceleration of each atom as they vary with time. Several algorithms have been developed for this purpose, including the Verlet algorithm, velocity Verlet algorithm and Beeman s algorithm [11]. NAMD, the simulation program used in this project, uses the velocity Verlet algorithm [9, 10]. The calculated trajectory can then be used to determine the average values of properties of the whole system, such as energy. The method of MD simulations is deterministic, thus, if Equation (2) has been solved for each atom, the state of the system can be predicted at any time t Empirical Force Fields For the MD simulations described in Section 3.1 it is assumed that every atom experiences a force F i exerted on it by all other atoms [10, 11]. This 10

12 3 METHODS force is specified by a model force field which describes the interaction of atom i with the rest of the system. The force field, also referred to as the potential energy function for the system, has the form: V = V bond + V angle + V dihedral + V vdw + V Coulomb (3) The first term represents the stretching interactions between covalently bonded atoms, the second term represents the deformation energy of angles between covalent bonds to a particular atom bending and the third term represents the intrinsic energy for torsional deformation through covalently bonded atoms. These terms are given by Equations (4), (5) and (6), respectively, and together represent the variations in the covalent binding energy of the whole molecule [11]. V bond = K b (b b 0 ) 2 (4) bonds V dihedral = V angle = bond angles dihedral angles K θ (θ θ 0 ) 2 (5) K φ [1 + cos(nφ δ)] (6) In Equation (4), bonds counts every covalent bond in the system and in Equation (5), bond angles are the angles between every pair of covalent bonds that share a single atom. In Equation (6), dihedral angles describes atom pairs separated by three covalent bonds such that the central bond is subject to the torsion angle φ [10]. For each atom in a given context of bonds, the parameters K b, K θ and K φ are obtained from experimental studies of small model compounds, together with ab initio quantum calculations, which means that the potential energy function is an empirical quantity. The parameters b 0 and θ 0 correspond to the equilibrium bond length and equilibrium covalent bond angle respectively. In Equation (6), n=1,2,3 represents the symmetry in the dihedral angle and δ is a correction factor for the phase. These parameters are described in the force field parameter files used in the input for MD simulations. 11

13 3 METHODS The last two terms in Equation (3) describe the interactions between nonbonded atom pairs. The V vdw term is given by Equation (7). It corresponds to the van der Waals forces approximated by the Lennard-Jones potential. In this expression, the power 12 term represents the core repulsion, the power 6 term the attraction and ɛ ij and σ ij are the depth of the potential well and the distance at which the interparticle potential between atoms i and j is zero, respectively. The last term of Equation (3), given by Equation (8), describes the Coulombic interactions between a pair of atoms i and j separated by a distance r, where q i and q j are the respective charges of the two atoms and ɛ 0 is the permittivity of free space. V vdw = i [ (σij ) 12 4ɛ ij j>i V Coulomb = i r ij j>i ( σij r ij ) 6 ] (7) q i q j 4πɛ 0 r ij (8) The empirical potential energy function V, given in Equation (3), is differentiable with respect to the atomic coordinates. Equation (9) below expresses the force F i on atom i in terms of that potential energy function. F i = V r i (9) Thus, differentiation of the potential energy function V (described in Equations (3) - (8)) gives the size and direction of the force F i as described in Section 3.1. Equating equations (2) and (9) then gives: F i = m i 2 r i t 2 = V r i (10) Equation (10) then relates the derivative of the potential energy V to the changes in position as a function of time. Solving the potential energy equation therefore yields information of the position and velocity of each atom, as required by MD simulations to calculate each atom s trajectory [10, 11]. 12

14 3 METHODS 3.2 Minimization and Structure Optimization Given the structure of a molecule and a potential energy function as described in Section 3.1.1, there are several methods which can be used to study the dynamics of the molecule. Prior to implementing these, energy minimization is used to relax delocalized stresses in the system. Minimizing the energy of a model system corresponds to solving a non-linear optimization problem. This means that, given a set of independent variables x = (x 1, x 2, x 3,..., x n ) and an objective function V = V (x), the task is to find the set of values for the independent variables, denoted x, for which the function has its minimum value V (x ) = min(v (x)) [11]. If the molecule considered contains N atoms, the 3N components of x are the atomic coordinates and V is the potential energy, calculated from the potential energy function given by Equations (3) - (8). In an optimization problem, it is extremely difficult to find the global minimum of a general non-linear function with ten or more independent variables. [11]. Due to this, for molecules such as those considered in this project, only a small fraction of conformational space can be explored. Energy minimization therefore acts to refine molecular structures by eliminating steric conflicts and adjusting bond lengths and bond angles to values near their optimum values [11]. Steric effects arise from the spatial arrangements of atoms in a molecule, thus if atoms are too close together, overlapping electron clouds may affect the conformation of the molecule. Any continuous, differentiable function of the independent variable x can be expanded as a Taylor series about the point x 0 : f(x) = f(x 0 ) ( x x 0 )f (x 0 ) + (x x 0 ) 2 f ( x 0 2 ) +... (11) This is the equation for the one dimensional case. In the case of many dimensions, x is replaced by the vector x and matrices are introduced for the various derivatives. An optimization method is classified by its order, which is defined by the highest order derivative that is used in the method. The algorithm used for the task of non-linear optimization in NAMD is the conjugate gradient algorithm. The conjugate gradient method is a first order iterative descent technique. 13

15 3 METHODS In this method, starting from a point A on the potential surface, where the gradient is g 1, the first search direction, s 1, is taken to be along the negative gradient: s 1 = g 1 (12) Once the minimum value of the energy along this direction is found, the algorithm takes the second search direction to be a combination of the current gradient and the previous search direction: s k = g k + b k s k 1 (13) where the parameter b k is a weighting factor. For the n dimensional case, the method will not converge in n steps unless the potential surface is quadratic [11]. This is not generally the case and so numerical errors can accumulate in the determination of successive search directions since these depend on previous search directions as specified in Equation (13). In order to prevent the accumulation of errors, the algorithm is interrupted and reset every m steps, with m n, by setting b m = 0 in Equation (13). This bases the next search direction only on the current gradient [11]. Energy minimizations are simulated at a temperature of 0K for a specified length of time. The length of time required is judged by how long it takes to find the set of values for the independent variables x such that the function has its minimum value V (x ) = min(v (x)). Once these values have been found, the delocalized stresses in the system have been eliminated and the structure can be studied using molecular dynamics. 14

16 3 METHODS 3.3 Equilibration Molecular dynamics, as described in Section 3.1, is used to equilibrate the system prior to data collection during simulations. This ensures that errors in the initial configuration of the system are erased and the results collected during molecular dynamics simulations, such as the free energy perturbation calculations in this project, are representative of the system and are not skewed by initial conditions in the input files. Equilibration is performed on the system after the initial minimization described in Section 3.2. Since the minimization stage considers the system at 0K, the temperature of the system needs to be increased during equilibration to the required simulation temperature. The temperature is calculated from the atomic velocities [11], so that: 3k B T = N i=1 m i v i vi N (14) where k B is the Boltzmann constant, T is the absolute temperature of the system in Kelvin and v i is the velocity of atom i. The initial velocities assigned to the system are such that the temperature of the system is near 0K. The equilibration is then run for a number of iterations before the temperature is scaled upwards. In NAMD, the temperature scaling is done via velocity scaling [10], where the velocity of each atom is scaled in order to achieve the target temperature. This is done using the relation: v new = ( Ttarget T system ) 1 2 vcurrent (15) where v new and v old are the new and current velocities of the atoms respectively, and T target and T system are the required and current temperatures of the system. The increase to the required simulation temperature is done systematically during the equilibration stage. Once the required temperature has been reached, it is kept constant using the Langevin equation: m i 2 r i t 2 = F i γ i r i t m i + R i (16) 15

17 3 METHODS where γ i is the frictional damping coefficient applied to atom i. The Langevin equation is a version of Newton s Second Law, given by Equation (2), modified with the addition of two extra terms. The γ i r i t m i term represents a frictional damping that is applied to atom i, and the R i term represents the random forces acting on atom i as a result of solvent interaction. The extra terms are used to control the kinetic energy of the system, which allows the temperature of the system to be kept at a constant value. Simulations are run so that minimization is performed on the system first, followed by equilibration. Equilibration simulations are run on a system until there are no systematic drifts of potential energy and no significant changes occur in the large-scale structure of the model [11]. 16

18 3 METHODS 3.4 Free Energy Perturbations Molecular dynamics can also be used to generate an ensemble of configurations which can be used to compute thermodynamic quantities [10]. Among the most important thermodynamic quantities are free energies, which provide fundamental measures of the stability of a system [11]. The free energy of a system is measured relative to an arbitrary zero point, so the sign of the measurement does not have any significance. Thus only free energy differences between different configurations are relevant in yielding information on relative stabilities of the states [11]. The theory of free energy perturbations (FEP) is used in this project to gain insight into the relative stability of different states of ATP in ATPase. The fundamental equation in this theory is derived from the definition of Helmholtz free energy, F: F = k B T ln ( ) e βv (Γ) dγ N!Λ 3N (17) where the bracketed quantity is the partition function. In the partition function, β = 1 k B T, Γ is a state of the system in configuration space, V (Γ) is the potential energy for the configuration Γ (calculated from the potential energy function given by Equations (3) - (8)), and Λ is a function of the temperature. The Helmholtz free energy is the free energy in a canonical ensemble. This is an ensemble where the number of molecules, N, temperature, T, and the volume, v, of the system are kept constant, often denoted (N,T,v). In the theory of free energy perturbations, the free energy change in the transition from an initial state A to a final state B of a system is defined, using Equation (17), as: F A B = F B F A ( ) ( ) e βv B (Γ) dγ e βv A (Γ) dγ = k B T ln N!Λ 3N + k B T ln N!Λ 3N [ ] e βv B (Γ) dγ = k B T ln e βv A (Γ) dγ 17

19 3 METHODS = k B T ln [ ] e β(v B V A ) e βv AdΓ e βv AdΓ F A B = k B T ln exp{ β [V B V A } A (18) where A indicates an average over a Boltzmann sample of configurations governed by the potential energy function V (Γ) [11]. The difference between states A and B may be in the type of atoms involved, where F corresponds to mutating a molecule into a different one, or a difference in the geometry of the molecule, where F represents the difference in energy between different configurations of the molecule. Equation (18) is exact in the limit of infinite sampling, however, in the limit of finite length simulations, accurate estimates of free energy differences can only be obtained if the conformational changes between states A and B are small enough [10]. This means that, in the case of large differences in free energy, the pathway connecting state A to state B needs to be broken down into M intermediate states, λ k, in order to achieve the desired accuracy. Then A λ 1 = 0 corresponds to the initial state and B λ M = 1 corresponds to the final state. The total free energy difference between the initial and final states is calculated via the calculation of the free energy differences between the intermediate states. The potential energies in Equation (18) are calculated using the hybrid potential energy of the system, V λ. V λ is a function of the coupling parameter λ, that relates the initial state to the final state, defined as: V λ = V 0 + (1 λ)v A + λv B (19) where λ varies from 0 to 1 during the course of the simulation. V 0 is the potential energy of atoms in the system that do not undergo any transformation during the simulation, and V A and V B are the potential energies of the initial and final states respectively [11]. Using this equation, the change in energy is calculated for each configuration, and then the average indicated by Equation (18) is performed. In this project, FEP calculations were set up to calculate the relative 18

20 3 METHODS stabilities of two states of the ATP molecule in the catalytic β T P and β DP sites. The free energies were calculated to gain insight into the relative stabilities of the two states, but also the effect of the catalytic site chosen on the energy differences. In NAMD, the topologies for the initial state A and final state B coexist, without interacting. The atoms in the molecular topology are separated into three groups: (i) atoms which do not change during the simulation, (ii) atoms describing the initial state, A, of the system, and (iii) atoms describing the final state, B, of the system. This setup is referred to as the dual topology paradigm [10]. Here, the initial state was chosen to be the equilibrium state of ATP, shown in Figure 5a. In this state, the atoms in the phosphate chain correspond to λ = 0. The final state was set up such that the phosphate chain in the ATP molecule was stretched into a straight line, as shown in Figure 5b. The atoms in the phosphate chain in this state correspond to λ = 1. (a) Initial State (b) Final State Figure 5: States of ATP molecule in the FEP calculations The energy and forces in the FEP calculations are defined as a function of λ, such that the interaction of the phosphate chain in the initial state with the rest of the system is effective at the beginning of the simulation (λ = 0), and the interaction of the phosphate chain in the final state with the rest of the system is effective at the end of the simulation (λ = 1). For intermediate values of λ, both the initial and final state phosphate chains participate in the non-bonded interactions with the rest of the system, scaled on the current value of λ. The chains do not interact with each other. Due to this, it is necessary to explicitly define those atoms that are appearing and those that are dissappearing during the course of the FEP calculation, which is done in the setup of the system. 19

21 3 METHODS 3.5 Computer Code NAMD NAnoscale Molecular Dynamics (NAMD) is a computing program developed for calculations on large biomolecular systems [10], such as the one considered in this project. NAMD employs the methods described in Section 3 to solve particular structures. It is a complex package used for MD simulations which provide extensive information about the system. The type of simulation and the output of information is controlled by a range of parameters specified in the configuration file. The configuration file also contains names of PDB (protein data bank) input files, which specify the input coordinates for each atom in the system, PSF (protein structure file) input files, which specify the structure of the system, and parameter files, which specify the force fields described in Section to be used during the simulation. NAMD is based on the Charm++ package, a parallel programming system which allows simulations to be run on multiple processors, reducing the time required to run simulations. Both NAMD and Charm++ are written in C++. In this project, the simulations were run in the Unix environment VMD Visual Molecular Dynamics (VMD) is a molecular visualization program, used in conjunction with NAMD to visualise and analyse the system considered. VMD uses 3-D graphics and can be used to animate the trajectory of a molecular dynamics simulation. It provides a wide variety of methods for rendering and colouring a molecule, and makes use of Tcl (tool command language) scripts, both of which are useful when analysing different aspects of the system. Several VMD plugins were used in this project. For the setup of the system, plugins psfgen and solvate were used. For results analysis, RMSD calculator and RMSD trajectory tool were used. Several Tcl scripts were also used, specifically fep.tcl, distdiffs.tcl and centerofmass.tcl. 20

22 4 SYSTEM SETUP 4 System Setup The information for the starting structure, as described in Section 2.2, is contained in a protein data bank (PDB) file. This file specifies the type and position of each atom, and also the residue and chain that each atom belongs to. The first step in the setup of the system was to add the omitted hydrogen atoms to the initial structure, and generate a new PDB file which would be used to model the function of the molecule. This was done using the psfgen tool in VMD, which positions the missing hydrogen atoms with respect to the positions of the rest of the atoms in the system. After the hydrogen atoms were added to the whole, β T P and β DP systems, they contained 55649, and atoms respectively. Figure 6: ATPase molecule in the unit cell Each of the systems was set in a unit cell of dimension 200Å 200Å 200Å, centered so that the entire system was contained in the unit cell, as shown in Figure 6. Periodic boundary conditions, which surround the system considered with identical virtual unit cells, were used. Using periodic boundary conditions helps to eliminate surface interaction at the boundary of the system and creates a more accurate representation of the in vivo environment. 21

23 4 SYSTEM SETUP The CHARMM (Chemistry at HARvard Macromolecular Mechanics) force field [11] was used to specify the potential energy function, described in Section In particular, CHARMM22 and CHARMM27 force fields were used, which specify the potential energy functions for proteins and lipids, respectively. PME (Particle Mesh Ewald) method was used to deal with electrostatic interactions in the system. The cutoff distance, which sets the separation between long and short range forces for the PME method, was set at 12Å. Simulations used a 2fs integration step and rigid bonds were used for all linear bonds involving hydrogen and any other atoms. All calculations were run at 310K. This temperature was chosen because the considered ATPase molecule functions at body temperature in living organisms. The temperature of the system was kept constant by using Langevin dynamics for all non-hydrogen atoms, with the damping coefficient set at 5ps 1. In this project, the simulations on the systems were carried out in two phases. The first phase consisted of minimization and equilibration simulations, which aimed to determine the minimal model of the ATPase complex that can be used to model the function of the whole molecule. The second phase of the simulations consisted of free energy perturbations, which calculated the free energy changes taking place in the catalytic sites of the ATPase molecule. 22

24 4 SYSTEM SETUP 4.1 Minimization & Equilibration For the minimization and equilibration simulations, 4 atoms in each of the catalytic sites were fixed in order to anchor the systems and prevent them from drifting in space. The atoms were selected so that two atoms near the top and two near the bottom of each catalytic sites were fixed, with the top and bottom referring to the orientation of the sites as shown in Figure 2. Energy minimizations were run for 1.5ps for each system described in Section 2.2, with the exception of the unconstrained whole system. For this system, the energy minimization was extended to 5ps as the number of free atoms in the structure was approximately three times the maximum number of atoms in the other structures. Equilibrations of the systems were run for 0.1ns for systems with 10Å from the ATP molecule free, 0.125ns for systems with 20Å from the ATP molecule free and 0.15ns for systems with 30AA free and the free systems. These times were chosen to ensure simulations were long enough for the systems to reach an equilibrium state and the RMSD of the molecule to become converged. The results of the run on each system were written in trajectory files, which can be analysed using VMD. Table 1 shows the length of the minimization and equilibration runs for each system considered. Table 1: Length of time in ps of minimization and equilibration runs for each system considered 23

25 4 SYSTEM SETUP 4.2 Free Energy Perturbations FEP calculations consisted of four runs, one in the forward and one in the reverse direction for each of the catalytic β T P and β DP sites. The forward direction corresponds to the transition of the ATP molecule from the initial state in Figure 5a, which is the equilibrium state of ATP, to the final state, in which the phosphate chain is stretched into a straight line as shown in Figure 5b. The reverse direction is the transition from the final to the initial state. The β T P and β DP sites were considered isolated from the rest of the ATPase molecule. Each FEP calculation was split into 25 intermediate states and run, from λ = 0 to λ = 1, for a total of 0.2ns. At each intermediate state, equilibration was run for 0.5ps to equilibrate the structure, followed by 7.5ps of FEP data collection. This equates to a total of 8ps per intermediate state. These times were chosen based on the time scales for FEP calculations suggested by the tutorial for FEP in NAMD [12].The free energy change between each state was written in the fepout output file. Table 2 shows the λ value of each intermediate step. Table 2: λ values for each intermediate step λ increment

26 5 RESULTS 5 Results 5.1 Minimization & Equilibration To check that the molecules had reached equilibrium at the end of the equilibration simulations, the root mean square deviation (RMSD) of each trajectory was plotted using the RMSD trajectory tool in VMD. The system was considered to be in equilibrium if there were no significant drifts in the RMSD, which was determined by visual inspection of the plots. These plots are shown in the Appendix for the unconstrained equilibrated structures of the isolated catalytic β T P and β DP systems and the whole molecule system. To investigate the effect of the constraints on the function of the molecule, the RMSD of each system was considered with respect to the starting structure and the equilibrated free system structure. The latter comparison was performed to determine the minimum system which would sufficiently approximate the free system β T P Site Figure 7 shows the deviation in position of each atom from its initial position in the isolated β T P system. Figure 8 shows the deviation in position of each atom from its initial position in the whole molecule system for this site. The minimum and maximum position changes of the atoms are tabulated in both figures. As explained in Section 2.1.3, the β T P is the loose site in the ATPase molecule. For the structures with 30Å of the molecule from the ATP in the site free and the entirely free structures, shown in Figures 7c and 7d respectively, the equilibration of the system causes the loose site to become more closed, in both the isolated β T P and whole molecule systems. The closing motion can be explained by the molecule reaching an energy state closer to equilibrium if the pocket is more closed. The RMSD of these position changes in each constrainted structure considered is plotted in Figure 11, for both the isolated β T P and whole molecule systems. For the isolated β T P system with 20Å from the ATP molecule in the pocket free, Figure 7b shows that the pocket has closed compared to the structure with the same constraint in the whole molecule system, shown in Figure 8b. This implies that considering the catalytic site away from the rest of the molecule affects its behaviour. This is possible since the structure considered in the isolated β T P system is not a physical state of the molecule. 25

27 5 Amount of Molecule Free free Minimum Movement RESULTS Maximum Movement (a) 10A from ATP in pocket free (b) 20A from ATP in pocket free (c) 30A from ATP in pocket free (d) Free system Figure 7: Deviation in atom positions in the βt P site obtained from the equilibration simulation of the isolated system 26

28 5 Amount of Molecule Free free Minimum Movement RESULTS Maximum Movement (a) 10A from ATP in pocket free (b) 20A from ATP in pocket free (c) 30A from ATP in pocket free (d) Free system Figure 8: Deviation in atom positions in the βt P site obtained from the equilibration simulation of the whole molecule system 27

29 5 RESULTS For the β T P site, Figure 9 shows the change in position of the ATP molecule centre of mass from its initial position, both for the isolated β T P and whole molecule systems. The graph shows that, in the whole molecule system, the level of change increases with decreasing restraint on the molecule. This is expected since decreasing the restraint on the system allows the ATP molecule to move more freely. For the isolated β T P system, the level of change decreases with increasing constraints. This is not expected, since an increase in deviation of the rest of the molecule should lead to an increase in the deviation of the ATP molecule. This result may be due to a conformational deformation of the isolated β T P system. Although the system is anchored in space to prevent drifting, it is possible that the molecule contorts due to a lack of constraint at the boundaries of the system, which would usually be provided by the rest of the ATPase molecule. This explantion is supported by the minimum position changes of the atoms for the β T P system tabulated in Figure 7. These suggest that the changes in position of the atoms which have moved least during the calculation are greater than zero for the systems with 20Å and 30Å of the molecule free. This should not be the case since large sections of the system in both cases were fixed. Figure 10 shows the change in position of the ATP molecule centre of mass in the various equilibrated constrained systems, compared to the equilibrated free system. The graph indicates that, for the whole molecule system, the level of change in position decreases with decreasing restraint. This is expected since the comparison is made to the free structure. Thus, the less constrained the system, the smaller the deviation between the two ATP molecules. For the isolated β T P system, the level of change in position increases with decreasing constraint on the system. In this case, it is expected that this change would decrease. The unexpected results may be due to the fact that, in the isolated β T P system, the closing of the pocket, observed in Figure 7 for all constrainted and the free system, may cause the ATP molecule to deviate further than it would do in the whole molecule system. The closing of the pocket also occurs in the whole molecule system, but in this case the pocket first closes for the system with 30Å of molecule free, as shown in Figure 8. 28

30 5 RESULTS Figure 9: Change of the position of the centre of mass of ATP molecule in β T P site from its position in the starting structure Figure 10: Change of the position of the centre of mass of ATP molecule in β T P site from its position in the equilibrated free structure 29

31 5 RESULTS Figure 11 shows the RMSD of the atom positions in the catalytic site from the starting structure, in both the isolated β T P and whole molecule systems. In the isolated β T P system, the value of 37Å free corresponds to the average distance of the system edge from the ATP molecule in the pocket. At this value it is assumed that the isolated β T P system corresponds to the structure without constraints. For the whole molecule system, it was assumed that if all atoms that are within 50Å of the ATP molecule in the β T P site are free, then the structure considered is the free structure. 50Å is the average distance of the whole molecule system edge from its centre. For the comparison of the free structures to be valid, the whole molecule system free structure should have in fact been considered with only the catalytic β T P site unconstrained and the rest of the molecule fixed. Since simulations carried out on the free structure of the whole molecule system considered the ATPase molecule without any constraints, the line between the system with 30Å free and the free structure is interpolated. The 37Å constraint is considered in this case to be the whole ATPase molecule with the catalytic β T P site unconstrained and the rest of the molecule fixed. The value of the standard deviation of the RMSD at this constraint was also interpolated. The graph indicates that the whole molecule system has smaller RMSD than the isolated β T P system, for all constraints considered. This is plausible, since, in the whole molecule system, the presence of the rest of the ATPase molecule around the β T P site is expected to suppress the motion of the catalytic site, thus showing a smaller RMSD. Figure 12 shows the RMSD of the atom positions for the equilibrated constrained systems with respect to the equilibrated free system. In the case of the isolated β T P system, the RMSD values of the constrained structures were compared to the RMSD value of the free structure. In the case of the whole molecule system, the free structure was considered to be at 37Å as explained above. The RMSD values of the constrained structures were then compared with the value of the RMSD interpolated at 37Å to show the deviation of the constrained structures from the free structure. This graph indicates that, for both the isolated β T P and whole molecule systems, the deviation between the two systems decreases with the decrease in constraint on the molecule. This result is expected, since the less constrained the system, the smaller the deviation from the free structure should be. 30

32 5 RESULTS To determine the smallest structure that can be used to model the function of the molecule, an RMSD that does not change with a decrease in constraint on the system is desired. Since the RMSD from the free structure decreases in going from a smaller system to a larger one in both the isolated β T P and whole molecule systems, the minimal model that can be used in either case is the free structure. In light of the time required to run simulations, the unconstrained isolated β T P system was chosen to model the function of the molecule as it has fewer atoms. This model was then used to carry out free energy perturbation calculations on the ATP molecule in the β T P site. The error bars on the RMSD graphs show the standard deviation of the atom position changes in each case. The standard deviation increases with the decrease in constraint of the system when the equilibrated constrained systems are compared to their respective initial structures. This is expected, since the number of atoms that move is greater in systems which are less constrained. The error bars decrease in the comparison of the constrained systems to the free structure, which is expected since the decrease in constraints on the system corresponds to that structure tending towards the free structure. Both Figures 11 and 12 show that the RMSD of the whole molecule systems is smaller than that of the isolated β T P systems, for all constraints. From these results, it can be concluded that the entire ATPase molecule, even if it is constrained, dampens the motion of the catalytic β T P site. Thus, including the whole ATPase molecule in the simulation is important to achieve accurate results. 31

33 5 RESULTS Figure 11: Plot of RMSD of the atom positions from the starting structure in the β T P site obtained from the equilibration simulation of the isolated and whole molecule systems Figure 12: Plot of RMSD of the atom positions from the equilibrated free structure in the β T P site obtained from the equilibration simulation of the isolated and whole molecule systems 32

34 5 RESULTS β DP Site Figure 13 shows the change in position of the ATP molecule centre of mass from the initial structure in the isolated β DP system for each constrained system. Figure 14 shows change in position of the ATP molecule centre of mass from the initial structure in the β DP site in the whole molecule system, also for each constrained system. The minimum and maximum position changes of the atoms are again tabulated in both figures. The β DP site is the tight site in the ATPase molecule and is considered to be higher in energy than the β T P site, as explained in Section It is therefore expected that during equilibration, the site would open to some extent in order to reach an equilibrium energy. This was not entirely obvious in the trajectories of the equilibration simulations of the β T P site, although a very slight opening of parts of the site in comparison to the starting structure is observed. The tabulated minimum and maximum position changes suggest that the isolated β DP system does not deviate as much as the whole molecule system, which is not expected since the ATPase molecule is thought to dampen the motion of the site. However, when the RMSD of these position changes is calculated, the results, plotted in Figure 17, indicate that the isolated β DP system deviates more from its starting structure than the free molecule system, as expected. Figure 15 shows the change in position of the ATP molecule centre of mass from its initial position for the isolated β DP system. The calculated results show expected trends, since a decrease of the restraint on the system would allow the ATP molecule to deviate more. It is unusual that in the free whole molecule system, the ATP molecule deviates more than that in the isolated β DP system. It is expected that the whole molecule system would have a smaller deviation, as the presence of the whole ATPase molecule is suspected to dampen the motion of the catalytic site. The sudden increase in the change in position of the ATP molecule centre of mass between the system with 30Å constrained and the free system is not entirely clear and should be investigated further by constraining the system at smaller intervals between these two systems. The curves in the graph are not extended to these values as they seem to be anomalies in the results and would skew the trend in ATP deviation in the three constrained structures. 33

35 5 Amount of Molecule Free free Minimum Movement RESULTS Maximum Movement (a) 10A from ATP in pocket free (b) 20A from ATP in pocket free (c) 30A from ATP in pocket free (d) Free system Figure 13: Deviation in atom positions in the βdp site obtained from the equilibration simulation of the isolated system 34

36 5 Amount of Molecule Free free Minimum Movement RESULTS Maximum Movement (a) 10A from ATP in pocket free (b) 20A from ATP in pocket free (c) 30A from ATP in pocket free (d) Free system Figure 14: Deviation in atom positions in the βdp site obtained from the equilibration simulation of the whole molecule system 35

37 5 RESULTS Figure 16 shows the change in position of the ATP molecule centre of mass in the various equilibrated constrained systems, compared to the equilibrated free system. For the whole molecule system, there is a decrease in deviation with a decrease in constraint. This is expected, and the result is similar in trend to that in the β T P site, although the relative deviation of the molecule in the β DP site is higher. The results for the isolated β DP system show a slight unexpected increase in deviation with a decrease in constraint, which is again similar to the trend of the corresponding results in the β T P site, and can be explained by the fact that, if the system is isolated from the rest of the ATPase molecule, it can deviate more due to a lack of constraints at the boundary. Figure 17 shows the RMSD of the atom positions in the catalytic β DP site from the starting structure, in both the isolated β DP and whole molecule systems. Similarly to the β T P site, it is assumed that the isolated β T P system corresponds to the structure without constraints at 37Å. For the whole molecule system, it was also again assumed that if all atoms that are within 50Å of the ATP molecule in the β DP site are free, then the structure considered is the free structure. The value of the deviation at 37Å was again interpolated, as was the value of the standard deviation of the RMSD at this constraint. This graph indicates that the whole molecule system has smaller RMSD than the isolated β DP system in the case of all constraints considered. This is reasonable since the presence of the whole ATPase molecule around the β DP site is expected to suppress the motion of the catalytic site. Figure 18 shows the RMSD of the atom positions for the equilibrated constrained systems with respect to the equilibrated free system. For the isolated β DP system, the RMSD values of the constrained structures were compared to the RMSD value of the free structure. In the case of the whole molecule system, the free structure was again considered to be at 37Å. The interpolated value of the RMSD at 37Å was compared to the RMSD values of the constrained structures to show the deviation of the constrained structures from the free structure. This result is similar to that for the β T P site, shown in Figure 12, and implies that the deviation from the free structure decreases with the decrease in constraint on the molecule, for both the isolated β DP and whole molecule systems. Similarly to the β T P systems, 36

38 5 RESULTS Figure 15: Change of the position of the centre of mass of ATP molecule in β DP site from its position in the starting structure Figure 16: Change of the position of the centre of mass of ATP molecule in β DP site from its position in the starting structure 37

39 5 RESULTS the RMSD of the catalytic β DP site in the whole molecule system is smaller than that for the isolated β DP system. This is expected since the presence of the ATPase moelcule is thought to dampen the motion of the catalytic site. Similarly to the β T P site, the smallest structure that can be used to model the function of the molecule is found by searching for an RMSD that does not change with a decrease in constraint on the system. Since the RMSD from the free structure decreases in going from a smaller system to a larger one in both the isolated β DP and whole molecule systems, the minimal model that can be used in either case is again the free structure. In light of the time required to run simulations, the unconstrained isolated β DP system was chosen to model the function of the molecule as it has fewer atoms. This model was then used to carry out free energy perturbation calculations on the ATP molecule in the β DP site. Similarly again to the β T P site results, for the β DP site, the RMSD of the whole molecule systems is smaller than that of the isolated β DP systems for all constraints, as shown in Figures 17 and 18. From these results, it can be concluded that the entire ATPase molecule, even if it is constrained, dampens the motion of the catalytic β DP site. Thus, including the whole ATPase molecule in the simulation is important to achieve accurate results. 38

40 5 RESULTS Figure 17: Plot of RMSD of the atom positions from the starting structure in the β DP site obtained from the equilibration simulation of the isolated and whole molecule systems Figure 18: Plot of RMSD of the atom positions from the equilibrated free structure in the β DP site obtained from the equilibration simulation of the isolated and whole molecule systems 39

41 5 RESULTS 5.2 Free Energy Perturbations The FEP calculations were performed in the β T P and β DP sites, the results for which are plotted in Figure 19 and Figure 20, respectively. As explained in Section 4, the forward direction corresponds to the transition from the equilibrium state of ATP to the state in which the phosphate chain is stretched into a straight line. The reverse direction corresponds to the transition from the state with the phosphate chain stretched into a straight line back to the equilibrium state of ATP Observations For the β T P site, the results in Figure 19 indicate that the total free energy change in the forwards direction is kcal/mol. The maximum free energy change is kcal/mol, at the intermediate λ value of 0.9. This state corresponds to a transition state, which is defined to be the state of highest energy along the reaction pathway. Results for the reverse direction indicate that the molecule goes through another transition state, this time at the intermediate λ value of 0.7, which corresponds to the maximum energy of kcal/mol. The total free energy change in the reverse direction is expected to have the same absolute value as that in the forwards direction. However, the total free energy change in the reverse direction is only kcal/mol. Therefore the calculation in the reverse direction does not converge on the expected value of 0. For the β DP site, the results in Figure 20 show that, in the forward direction, the total free energy change is kcal/mol. The maximum free energy change is kcal/mol and occurs at the intermediate λ value of 0.6. In the reverse direction, the molecule goes through a transition state with energy kcal/mol at the intermediate λ value of 0.4. The calculation in the reverse direction does not converge to the expected value of 0. The total free energy change in the reverse direction is kcal/mol. 40

42 5 RESULTS Figure 19: Forwards and reverse FEP calculation results for β T P site Figure 20: Forwards and reverse FEP calculation results for β DP site 41

43 5 RESULTS Data Analysis The free energy change in the forwards direction was kcal/mol and kcal/mol for the β T P and β DP sites, respectively. This indicates that the equilibrium state is more stable in both sites than the state in which the phosphate chain is stretched into a straight line, as expected. The results also indicate that the stability of the states is not greatly affected by the catalytic site, as the total free energy changes are approximately equal in both cases. In both the β T P and β DP sites, it was expected that the start point of the forwards calculation should be the same as the end point of the reverse calculation, and vice versa. This was not the case in either site, and poses a problem in terms of the convergence of the FEP calculations. The problematic convergence at the end points may be due to the fact that the intermediate λ states were chosen so that there were more intermediate states between states λ=0 and λ=0.1 than there were between states λ=0.9 and λ=1. On reflection, these should not have differed, as the same convergence problems arise in the two areas. The incorrect convergence may partly be due to the dual topology setup of the calculation. In this setup, as explained in Section 3.4, atoms in the initial state corresponding to λ=0 disappear during the simulation and those in the final state corresponding to λ=1 appear. Due to the appearing and disappearing atoms, van der Waals clashes occur and result in end point catastrophes, which prevent correct convergence of the calculation. The convergence may be improved by increasing the number of intermediate λ states in the calculation and, hence, decreasing the conformational difference between consecutive λ states. This is a computationally costly process. The increase in the total free energy difference of the calculation in the reverse direction in the β DP site was not expected. Although an increase in energy through a transition state was a possibility, the calculation was expected to show an overall decrease in the free energy difference towards the value of 0. It is possible that the rest of the molecule would have an effect on the function of the catalytic site, so the fact that the catalytic site was considered isolated from the rest of the molecule may have caused this 42

44 5 RESULTS result. Another possibility for this result is that the 0.5ps equilibration stage at the beginning of each λ state was too short. Since the system considered is large, it is likely that the equilibration stage needs to be longer in order for the FEP calculation to converge Further FEP Calculations In order to investigate the effect of increasing the number of intermediate λ states and ensuring that the same number of intermediate states occurs at the beginning and the end of the calculation, the ATP molecule was considered out of the catalytic site. It was solvated by a rectangular box of TIP3 water molecules extending 15Å in all directions around the molecule. Considering this system, instead of the ATP molecule in one of the catalytic sites, reduced the number of molecules to 2690 and greatly increased the speed of FEP calculations. Table 3: New λ values for each intermediate step λ increment λ states chosen in increments of The ATP molecule was subjected to the same conformational change as in the FEP calculations in the catalytic sites and the calculation was again run in the forwards and reverse directions. The intermediate λ states were 43

45 5 RESULTS chosen as shown in Table 3. The extra intermediate steps increased the total length of the FEP calculation to 6.6ns. For a system as large as the catalytic β T P and β DP sites, a calculation of this length would be beyond the time limit of this project. Figure 21: Forwards and reverse FEP calculation results for ATP molecule in water box with extra intermediate λ states Figure 21 shows the results of these calculations. It is clear that the increase in number of intermediate λ states and their distribution at the beginning and the end of the calculation affects the results calculated. The curves for the forwards and reverse directions are of the same shape, however, in the forwards direction there is a sudden increase in the free energy plotted in the region λ=0 and λ=1. This implies the differences in size between λ states in this region, and therefore the conformational change of the molecule, is still too large. In order to calculate more accurately this change, a further increase in the number of λ states in this region would be required. The results obtained for the calculation in the reverse direction show that, if the λ states are chosen aaccurately, the total free energy change is significantly smaller. This implies that the choice of λ states greatly affects 44

46 5 RESULTS the results obtained. Although the setup of this calculation requires further modifictations to ensure accurately converged results are achieved, it does confirm the explanations for incorrect convergence proposed above to some extent. Therefore, in order to achieve convergence, it is important to run the FEP calculations on the catalytic β T P and β DP site systems with much smaller, and hence many more, intermediate λ steps. Although computationally very costly, the results of these calculations would provide a much more accurate understanding of the energy changes that take place in the catalytic sites. To investigate how the length of the equilibration stage affects data collected during the FEP calculation, the ATP molecule was again considered in the water box described above. The system was subjected to 5ps of equilibration prior to 7.5ps of FEP data collection in one case, and 0.5ps of equilibration prior to 7.5ps of FEP data collection in another case, for comparsion. Both of these were run in the forwards and reverse directions. The intermediate λ states were chosen as shown in Table 3, except those between λ=0.1 and λ=0.9, which were chosen in increments of 0.1. The extended equilibration at each intermediate λ state, and the increase in the number of λ states between λ=0.9 and λ=1, increased the total length of the FEP calculation to 0.4ns. The results for the FEP calculations with the equilibration stage extended to 5ps are shown in Figure 22, and those for the FEP calculation with the equilibration stage of 0.5ps are shown in Figure 23. Comparing the graphs in Figures 22 and 23, it is clear that the calculations in the forwards and reverse direction are in much closer agreement when the equilibration stage is of ample length for the system considered to reach an equilibrium point. Hence, in order to get results which demonstrate more accurately the energy changes occurring in the catalytic β T P and β DP sites, it would be necessary to increase the length of the equilibration stage in the FEP calculations so that the systems considered can adjust at each intermediate λ state. This process would be computationally expensive, but would provide an insight into the energy changes occurring in the catalytic sites. 45

47 5 RESULTS Figure 22: Forwards and reverse FEP calculation results for ATP molecule in water box with extended equilibration stage Figure 23: Forwards and reverse FEP calculation results for ATP molecule in water box 46

48 6 CONCLUSIONS 6 Conclusions The exact structure and function of the ATPase molecule is still unknown. The equilibration simulations carried out in this project revealed that the smallest structure that can be used to model the function of the molecule is the isolated free structure for both the catalytic β T P and β DP sites. These conclusions may not be correct, since they are drawn from comparisons of the RMSD of atom positions in the constrained structures to an interpolated RMSD value. Although the results obtained may not be entirely reliable, the simulations that have been carried out have advanced the understanding how the constraint imposed on the system affects the behaviour of the molecule. Based on the understanding gained from the achieved results, the next steps which need to be taken in order to verify the conclusions drawn have been clearly set out. Free energy perturbation calculations revealed that the equilibrium state of the ATP molecule in both catalytic sites is more stable than the state with the phosphate chain stretched into a straight line, which was expected. The results also suggested that the total free energy changes occurring in both catalytic sites are similar in value. These results are not representative of the systems considered as they are skewed by the choice of parameters in the setup of the calculation. The investigation into the effect of the parameters on the results showed that the number and distribution of intermediate λ states, and an increase in the length of the equilibration stage at each intermedate step, are vital for calculations to converge correctly. Combining the modifications in the number of λ states, their distribution at the beginning and end of the calculation, and the increased equilibration time, would ensure that the results collected during FEP calculations are representative of the systems considered. The information obtained from the investigation of the effect of the parameters on the results has provided a guide for the correct setup of future calculations of this kind. 47

49 7 PLANS FOR FUTURE WORK 7 Plans for Future Work There are many interesting simulations that can be run on both the ATP and ATPase molecules to increase the understanding of their beahviour. There are several modifications that can be made to the systems considered in this project to improve the results obtained. In the case of the equilibration simulations, the whole molecule system free structure should be considered with each of the β T P and β DP catalytic sites unconstrained, in turn, with the rest of the molecule fixed in both cases. This would provide simulation results for those interpolated above and allow a clearer comparison between the isolated catalytic site systems and the whole molecule systems. Further, the constraints on the system should be considered at smaller intervals, for example with 5Å between them. This would show more clearly the value of the constraint on each system which approximates the unconstrained system structure accurately enough. In the case of the free energy perturbation changes carried out in this project, modifications should be made to the number of λ states used, their distribution at the beginning and end of the calculation, and the length of the equilibration stages. FEP calculations should be run on the ATP molecule in the catalytic β T P and β DP sites with these modifications, in order to achieve results which are representative of the energy changes occurring in the sites, rather than of the setup of the calculation. There are several other FEP calculations that could be considered to gain furhter insight into the ATPase molecule. Understanding the ATP hydrolysis reaction, described in Section 2.1.2, is an important step in providing this insight. It would be informative to set up a FEP calculation that would calculate the energy difference in the transition from ATP to ADP and P i. This would calculate the energy released in the breaking of the covalent bond between the terminal phosphate group and the oxygen joining this group to the rest of the ATP molecule, and thus give an indication of how much chemical energy is stored in this bond. The setup of this calculation has been attempted in this project, however, there have been technical difficulties in simulating the breaking of the bond using methods based on 48

50 7 PLANS FOR FUTURE WORK classical mechanics. This problem may need to be approached using the hybrid technique of quantum mechanics and molecular dynamics (QM/MD). (a) 2JDI ATP (b) F1MgF ATP Figure 24: Configurations of ATP molecule Another interesting calculation would be to consider the energy differences in different structural conformations of the ATP molecule. The particular conformations that would be looked at are the equilibrium state of ATP, shown in Figure 24a, and the structure which experimentalists believe to be the transition state structure in ATP hydrolysis, shown in Figure 24b. This possible transition state structure has the last phosphate group in a planar configuration, while in the equilibrium structure the phosphate group is in a dihedral configuration. 49