Thesis. Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University

Transcription

1 Mechanical Unfolding of Solvated α-synuclein Studied by Molecular Dynamics Simulations Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University By Roman V. Sloutsky, B.A. Biophysics Graduate Program The Ohio State University 2009 Thesis Committee: Michael E. Paulaitis, Advisor Sherwin J. Singer

2 Copyright by Roman V Sloutsky 2009

3 Abstract α-synuclein is a natively disordered protein localized at presynaptic nerve terminals, where it is thought to modulate vesicular release and trafficking. It is the main component of amyloid fibrils and Lewy bodies characteristic of Parkinson s Disease and other neurodegenerative diseases. Although α-synuclein has no characteristic structure in aqueous solution, it adopts α-helical secondary structure upon association with lipid membranes and micelles and β-sheet secondary structure characteristic of amyloid formations upon aggregation in fibrils and Lewy bodies. Starting with an experimentally determined structure of α-synuclein in an SDS micelle, equilibrium dynamics, mechanical unfolding, and solvation along the unfolding pathway were studied by molecular dynamics simulations. Ten unfolding transitions starting from an ensemble of ten folded conformations were simulated and the Jarzynski Equality was applied to estimate the change in free energy associated with unfolding. Subsequently, equilibrium simulations of the protein constrained at several points along an unfolding trajectory were carried out and Widom s Potential Distribution Theorem was applied to estimate the excess chemical potential (free energy of solvation). An apparent local maximum in the free energy of solvation of α-synuclein was identified at an intermediate end-to-end distance, possibly indicating increased susceptibility to aggregation. ii

4 Dedicated to my wife Hilary iii

5 Acknowledgements I wish to thank my advisor, Michael Paulaitis, for the guidance and support which made this thesis possible. I am indebted to Arturas Ziemys, who provided critical consultation on all aspects of molecular dynamics simulations throughout this project. I also wish to thank Nathan Baker for helping me with implicit solvent calculations. I thank Arturas Ziemys, Ghalib Bello, Sowmi Utiramerur, Ali Hassanali, Sherwin Singer, and Hamsa Priya Mohan for providing stimulating discussion and useful suggestions. I am grateful to my family and friends for their endless love and support. iv

6 Vita 1997 Upper Arlington High School 2001 B.A., Northwestern University Scientific programmer, Biological Sciences Division, University of Chicago Graduate Teaching and Research Associate, The Ohio State University Major Field: Biophysics Fields of Study v

7 Table of Contents Abstract ii Dedication...iii Acknowledgements.iv Vita...v List of Tables..... vii List of Figures... viii Chapter 1: Atomic Force Microscopy and Steered Molecular Dynamics : Molecule Atomic Force Microscopy Experiments : Steered Molecular Dynamics Simulations and Jarzynski Analysis..6 Chapter 2: I27 and α-synuclein : I27 domain of titin : α-synuclein..13 Chapter 3: Methods 17 Chapter 4: Results...27 Chapter 5: Conclusion 53 References..56 Appendix A: Starting Conformation Ensemble Details 66 Appendix B: Sample Configuration Files..67 vi

8 List of Tables Table 4.1: Summary of subpopulation division criteria...42 Table 4.2: Summary of excess chemical potential estimates 50 Table A.1: Summary of helical orientations of ensemble conformations.66 vii

9 List of Figures Figure 1.1: Illustration of single molecule unfolding by AFM Figure 1.2: Sample force profile of an AFM experiment....3 Figure 1.3: Schematic representation of SMD. 8 Figure 2.1: Structure of I27 12 Figure 2.2: Sequence of α-synuclein..14 Figure 2.3: Structure of α-synuclein imbedded in SDS micelle 15 Figure 3.1: Sample α-synuclein conformations from 5ns simulation 19 Figure 3.2: θ φ distribution of α-synuclein conformations from 5ns simulation...21 Figure 4.1: Force curves from I27 unfolding simulations.28 Figure 4.2: Force and work curves from α-synuclein unfolding simulations...29 Figure 4.3: Work and free energy vs extension curves for α-synuclein unfolding...34 Figure 4.4: Gaussian distribution generated from work measurements 35 Figure 4.5: Representative conformations of α-synuclein along an unfolding trajectory.37 Figure 4.6: Summary of conformations selected for 1ns constrained simulations 38 Figure 4.7: Gaussian protein-solvent interaction energy distributions..39 Figure 4.8: Conformations representative of subpopulations from simulations C and D.43 Figure 4.9: Gaussian and Gaussian mixture energy distributions from simulation C...46 Figure 4.10: Gaussian and Gaussian mixture energy distributions from simulation D.48 viii

10 Chapter 1: Atomic Force Microscopy and Steered Molecular Dynamics 1.1: Single Molecule Atomic Force Microscopy Experiments Since the first such experiment [1] twelve years ago, single-molecule forceinduced unfolding experiments using atomic force microscopy (AFM) and other techniques, including molecular tweezers and the biomembrane force probe [2-4], have become increasingly common as a tool to study the unfolding of proteins. These experiments allow researchers to study how an externally applied force, as opposed to heat or chemical denaturants, causes proteins to unfold, as well as their subsequent refolding [2, 3]. In the single-molecule force spectroscopy experiment, the tip of an AFM cantilever contacts a surface to which polypeptide chains consisting of multiple repeats of the protein domain of interest have been covalently attached, usually by one of the peptide chain termini [5]. After non-specific attachment of the chain to the cantilever tip [2], either the cantilever holder or the surface is retracted, usually at a constant rate [3], applying a force to the protein domains causing them to unfold. While the rate of retraction is held constant, the rate at which the cantilever tip moves away from the surface is not [3]. Because the cantilever acts as a Hookean spring [5], the relative velocity of its tip and the surface depends its spring constant and the length and rigidity 1

11 of the multi-domain, polypeptide chain spanning the distance between them [3, 6]. The applied force at any moment can be related to the deflection of the cantilever via its spring constant [5]. Figure 1.1 (reproduced from [6]): Force-induced unfolding of a single polypeptide chain consisting of multiple repeats of a single protein domain. Repeats unfold sequentially, resulting in a saw-tooth force profile like the one shown in Figure 2. The tip-surface separation (z) is increased by retracting the stage surface by z surface. The force applied to the protein can be calculated from cantilever deflection (z cantilever ) using its spring constant. The magnitude of this force varies over the course of unfolding of the domains. A schematic representation of a force-induced unfolding experiment is shown in Figure 1.1. The force required to unfold multiple repeats of a protein domain in a polypeptide chain as a function of chain extension typically exhibits a saw-tooth pattern 2

12 [3], as shown in Figure 1.2, indicating sequential unfolding of the individual protein domains. Each saw-tooth corresponds to a ramping of force necessary to unfold a single domain, followed by a brief drop in force, until another domain begins to unfold. AFM-based mechanical unfolding of proteins has a number of advantages compared with other well-established experimental techniques for studying protein unfolding. Unlike heat and chemical unfolding, which probe an ensemble of molecules in bulk solution, the mechanical unfolding experiments follow the unfolding of a single molecule at a time, making it possible to distinguish differences in unfolding from molecule to molecule [4]. Also, unfolding induced by force naturally proceeds along a trajectory defined by a well-defined reaction coordinate, the extension distance [4], which can be readily duplicated in a molecular simulation. It is also reasonable to consider that unfolding of individual protein domains occurs as isolated events, thereby avoiding the possibility of protein aggregation during the unfolding process. Figure 1.2 (reproduced from [3]): The black trace recorded over the course of unfolding a polypeptide chain consisting of multiple repeats of a single protein domain shows a 3

13 typical saw-tooth pattern. On the other hand, the gray trace recorded during refolding does not show any features related to structure formation. Measurements of force-induced unfolding make it possible to probe the potential energy landscape along the folding/unfolding trajectory for features that reflect the force response of a protein [5] and to identify energy barriers and intermediate states of unfolding not accessible to traditional methods [3]. Proteins unfold through a different ensemble of pathways under force than under heat [2, 7, 8], although the ensembles are thought to approach one another as the loading rate is decreased and the thermal ensemble becomes dominant [2]. Thus, the unfolding pathway of a protein under mechanical stress is impossible to interrogate without mechanically manipulating the protein. Furthermore, protein folding pathways can be probed by repeated cycles of mechanical unfolding followed by refolding. Therefore, the relationship between sequence/structural elements and the ensemble of mechanical folding/unfolding pathways can be interrogated [2]. AFM experiments assess the mechanical stability of a protein, but the question arises whether this process is related to thermodynamic stability. The observation that the force response to increasing pulling speed is generally linear indicates that a kinetic phenomenon is being observed [4]. In fact, for a number of proteins, the free energy of unfolding measured by chemical or heat denaturation has been shown to be unrelated to the force at which the protein unfolds [9-12], suggesting that mechanical unfolding is a non-equilibrium process. Accordingly, with the exception of coiled-coil secondary structures and a single immunoglobulin-type domain [2, 13, 14], proteins also tend to 4

14 unfold and refold under force by different pathways, indicating that those pathways are irreversible [2, 3]. Figure 1.2 is an example of different unfolding and refolding forceextension profiles for a domain-repeat polypeptide construct. While individual domains unfold sequentially, resulting in a saw-tooth profile, they refold by a concerted pathway, so the corresponding profile has no domain-specific features. Secondary structure appears to be a critical factor that determines the mechanical stability of a protein. In general, α-helical proteins require the least force to unfold, β- sheet proteins require the most, and proteins with mixed α + β secondary structure fall in the intermediate range [5]. This correlation of mechanical stability with secondary structure was first observed by simulating the unfolding of different proteins [15], with subsequent experimental results [12, 16-19] confirming the observations from simulations. The time required to sample possible interatomic interactions before settling on an energetically optimal folded structure increases rapidly with the number of distant contacts along the primary sequence of amino acids in the polypeptide chain. Whenever a protein domain is pulled faster than its slowest relaxation time, and, therefore, is not allowed to sample all its possible configurations, the unfolding process proceeds under non-equilibrium conditions [3, 20]. Of all secondary structure types, only for coiled coils (no distant contacts) are the slowest relaxation times short enough to allow equilibrium unfolding at experimentally accessible pulling rates. Thus, it makes sense that all secondary structures except coiled coils would unfold via irreversible pathways. Interestingly, unlike topology, protein function is not a determinant of mechanical stability [5]. 5

15 1.2: Steered Molecular Dynamics Simulations and Jarzynski Analysis The Steered Molecular Dynamics (SMD) technique was conceived by analogy with AFM force-mode single molecule experiments and can be used to obtain free energy differences between thermodynamic states, such as folded vs unfolded protein or bound vs dissociated protein-protein and protein-ligand pairs. SMD simulations were first conducted by Grubmüller and coworkers to study the rupture of the streptavidin-biotin complex [21], followed by Schulten and coworkers, who analyzed the reconstruction of potential energy functions from SMD simulations [22]. Since then SMD simulations have been performed to study force-induced unbinding of non-covalently bonded proteinligand and protein-protein complexes [21-24], extraction of lipids [25, 26] and proteins [27] from lipid membranes, and other processes, in addition to force-induced unfolding of proteins. Earliest SMD simulations of force-induced unfolding were conducted with domains from the proteins titin [28] and fibronectin [29], which perform mechanical functions related to muscle elasticity and extracellular matrix stability, respectively. More recently this technique has been applied to a heme-containing oxygen transporter (apomyoglobin) [30], a lock-and-key enzyme (bovine carbonic anhydrase II) [31], lightand ph- driven cross-membrane ion transporters (bacteriorhodopsin, halorhodopsin, and the sodium-proton antiporter NhaA) [32], and a membrane-bound electron transporter hemoprotein (apocytochrome b 5 ) [33]. 6

16 In the SMD simulation, a time-dependent external force is applied along a designated path to an atom or set of atoms. Rather than applying the external force directly, a harmonic restraint is used to couple this force through a dummy atom, to which the force is applied. This can be visualized as pulling the target atom(s) by a spring, as in Figure 1.3. The dummy atom travels along the reaction coordinate, pulling the attached atoms behind it by the spring. It can travel at constant or variable speed, controlled by restraints placed on the applied force (e.g., constant or ramped force). For some applications, such as unfolding macromolecules, the dummy atom can travel along a straight line. For others, such as pulling a ligand from its binding pocket buried within a protein, its trajectory may be more complicated. It is important to note that, although the trajectory of the dummy atom is well defined, the trajectory of the atoms attached to it is generally more complex, as these atoms experience other forces in addition to the one applied via the harmonic restraint. Assuming a single reaction coordinate z (e.g., distance traveled by the atom or atoms to which the dummy atom is coupled), the applied force is: F(z, t) = K(z 0 + vt z), (1.1) where z 0 is the initial position of the dummy atom, v is the velocity at which the dummy atom is traveling, and K is the harmonic spring constant[34]. 7

17 Figure 1.3 (reprinted from [35]): In this schematic representation of SMD red circle represents the dummy atom and blue circle represents the atom(s) experiencing force F. Unlike other free energy computational methods that have been used to estimate the change in free energy along a trajectory e.g., umbrella sampling and free energy perturbation SMD intrinsically considers the work done along an irreversible or nonequilibrium trajectory [34]. Work w i as a function of z for the ith trajectory is given by: w i (z) = 0 z F i (z )dz, (1.2) where F i (z ) is the SMD force applied to the dummy atom at reaction coordinate value z. For SMD simulations of protein unfolding z is defined as the difference between the 8

18 current end-to-end distance at a particular time minus the initial end-to-end distance of the protein, or protein extension. The free energy profile can be estimated from an ensemble of these trajectories by applying the following relationship between the irreversible work and the free energy difference, G(z), <exp[ w i (z) / k B T ]> = exp[ G(z) / k B T ], (1.3) where k B T is the thermal energy and the brackets denote averaging independent trajectories sampled from an equilibrium ensemble of initial conditions [36, 37]. Modified versions of this technique have been devised [38, 39] and applied to SMD unfolding of proteins [32]. The accuracy of equation 1.3 was experimentally tested on an RNA system where both reversible and irreversible unfolding regimes are experimentally also accessible [40]. An RNA molecule was repeatedly mechanically unfolded and allowed to refold in alternating cycles of reversible and irreversible trajectories. Irreversible unfolding was performed at two different switching rates: a slower rate for near-equilibrium unfolding and a faster rate for far-from-equilibrium unfolding. G(z) was estimated from the ensembles of reversible and irreversible unfolding actualizations by the work average, fluctuation-dissipation estimate (work average minus βσ 2 /2, where β=(k B T) -1 and σ 2 is the variance of work values), and equation 1.3. Note that in the ensemble of reversible unfolding actualizations all work values are within experimental error and equal to G(z). Free energy estimates obtained by each method from both reversible and irreversible unfolding were compared with G(z) values obtained from the work of reversible 9

19 unfolding. While, as theoretically predicted by the authors, for reversible unfolding both the fluctuation-dissipation estimate and equation 1.3 underestimated G(z) equally (within 0.1k B T of each other), equation 1.3 produced more accurate estimates than the work average and the fluctuation-dissipation estimate for both near-equilibrium and farfrom-equilibrium irreversible unfolding. 10

20 Chapter 2: I27 and α-synuclein In this study we performed SMD simulations of unfolding of the I27 domain of the muscle protein titin and of α-synuclein, a natively disordered neuronal protein of unknown function associated with Parkinson s disease and other neurodegenerative diseases. SMD unfolding of the I27 domain of titin has been studied extensively, which allows us to benchmark our SMD simulations by comparing the results to those from these previous studies. α-synuclein undergoes folding-unfolding transitions upon association with and dissociation from lipid membranes, as well as in the course of aggregation into amyloid fibrils and, eventually, inclusion bodies. Understanding these folding-unfolding transitions can provide insights into the role of α-synuclein in fibril formation and the neurodegenerative diseases associated with this phenomenon. 2.1: I27 domain of titin The giant protein titin is responsible for the build-up of passive tension in stretched muscle. Some 90% of the protein consists of 132 fibronectin type III-like (Fn-3) and 112 immunoglobulin-like (Ig) domains [41]. When individual titin molecules were repeatedly stretched by AFM, sequential unfolding of Ig domains, followed by refolding 11

21 upon protein relaxation, was observed [42]. Force-induced unfolding of these domains, the I27 domain from human cardiac muscle resolved by NMR [43] in particular, has been studied extensively both by AFM experiments [7, 44-47] and simulation [28, 48-52]. This 89 amino acid, 11 kda domain has a folded structure of two β-sheets packed against each other. Each sheet consists of four strands. A ribbon structure of I27 and a schematic of the structure of the β-sheets are shown in Figure 2.1. Figure 2.1 (reproduced from [28]): (a) The ribbon structure of cardiac I27 based on its structure resolved by NMR [43] with the two β-sheets colored green and orange. The green sheet consists of strands A, B, E, and D and the orange sheet consists of strands A, G, F, and C. (b) Schematic of the β-sheets structure with backbone hydrogen bonds indicated by dotted lines. 12

22 2.2: α-synuclein α-synuclein is a 14.5 kda neuronal protein shown by immunohistochemistry to be localized at presynaptic nerve terminals [53]. The exact function of α-synuclein is unknown, but it appears to modulate vesicular release and trafficking [54]. Its 140 amino acid sequence contains a series of seven similar 11-residue subsequences (imperfect repeats) at the N-terminus, followed by a hydrophilic region at the C-terminus (Figure 2.2). The protein is natively unfolded in aqueous solution [55], but partially folds into an α-helical structure upon association with lipid micelle surfaces. When the protein is localized at the micelle-solvent interface, the N-terminal repeat region forms two α- helices separated by a short unstructured linker (Figure 2.3). This α-helical structure of the membrane-bound α-synuclein has been resolved by NMR [56]. The hydrophilic C- terminal tail remains unstructured and in aqueous solution at all times [56, 57]. 13

23 Figure 2.2 (reproduced from [56]): Amino acid sequence of human α-synuclein, with imperfect repeats labeled with roman numerals and residues 2-7 of each repeat (region of greatest similarity) highlighted in red. Underlined residues form α-helices when micellebound. The association between α-synuclein and Parkinson s disease has been demonstrated in two ways. First, a missense mutation in the α-synuclein gene was shown to cause a rare, heritable form of the disease [58]. Second, the α-synuclein protein was found to be the main component of Lewy body inclusions present in the more common, sporadic form of Parkinson s. Lewy bodies bind large numbers of α-synuclein-specific antibodies [59]. However, inactivation of the α-synuclein gene, resulting in failure to produce the protein, does not lead to the severe neurological symptoms associated with Parkinson s [60], indicating that loss of function is not responsible for these symptoms, suggesting therefore, that the tendency of α-synuclein to aggregate must be responsible for its association with Parkinson s disease. 14

24 Figure 2.3 (reproduced from [57]): Structure of α-synuclein embedded in a micelle, unstructured tail not shown. Views A and B are separated by a 180 degree rotation about the vertical axis. Colors in the ribbon reflect different NMR signal intensities. Fibrils formed by α-synuclein in vitro have been shown by electron diffraction [61] and electron paramagnetic resonance spectroscopy [62] to contain amyloid-like β-sheet structures. They also possess bulk features similar to amyloid fibrils [63]. Residues constitute a region highly prone to amyloid structure formation [54]. Molecular dynamics simulations suggest that interactions between partially folded α-synuclein molecules involving this region are involved in the nucleation of aggregates. Tsigelny and coworkers [64] observed the formation in solution, near a membrane interface, of two types of α-synuclein dimers: a head-to-tail non-propagating dimer in which the N- terminal region of one molecule interacts with the C-terminal region of another, and a head-to-head propagating dimer in which the two molecules interact via their residue regions. The propagating dimer allows consecutive docking of further copies of 15

25 the protein with a similarly low binding energy. A strong correlation between enhanced formation of α-synuclein fibrils and the presence of a partially folded intermediate (inducible by a decrease in ph or an increase in temperature) has also been experimentally demonstrated [65]. 16

26 Chapter 3: Methods NAMD[66] software, the CHARMM27 [67] force field, and the TIP3P water model implemented in the force field were used to perform equilibrium molecular dynamics (MD) simulations and non-equilibrium Steered Molecular Dynamics (SMD) simulations. All equilibrium simulations were carried out in the NPT ensemble at 298K and 1atm. Temperature was maintained using Langevin dynamics with a damping coefficient of 1ps -1 applied to non-hydrogen atoms. Pressure was maintained using a Nosé-Hoover Langevin piston with a period of 200fs and decay set to 100fs. Non-bonded van der Waals interactions were truncated at 12Å with switching distance beginning at 10Å. Electrostatic interactions were computed using particle-mesh-ewald (PME) method with a real space cutoff of 12Å and a grid point spacing of ~1Å. Periodic boundary conditions (PBC) were used. Equations of motion were integrated every 1fs with trajectories stored every 5ps during the 5ns equilibrium simulation of α-synuclein and every 100fs during the 1ns equilibrium simulations. These techniques were also used during SMD simulations, with the exception that temperature control was turned off in order to prevent the temperature control mechanism from modifying the velocities of the atoms subjected to translation. As a result the temperature increased slightly due to the acceleration of the translated atoms, but remained within 3K of 298K throughout all SMD simulations. In these simulations the α- 17

27 carbons of the C-terminal residue of I27 and the N-terminal residue of α- synuclein were attached by a harmonic spring to a dummy atom (moving point), which was translated in the long dimension of the water box at constant velocity. For the I27 domain, our SMD results were compared to previous results [28], and thus identical values for harmonic spring constant (6.0 kcal/mol/å 2 ) and pulling speed ( Å/timestep = 0.5 Å/ps) to those in [28] were used. The I27 simulation was performed for 700ps for a total extension of ~350Å more than sufficient to fully unfold the protein. SMD simulations of α-synuclein also used a harmonic spring constant of 6.0 kcal/mol/å 2, but a pulling speed of Å/timestep = 1 Å/ps. These simulations were performed for 400ps for a total extension of ~395Å, again sufficient to fully unfold the protein. I27 was pulled along the vector passing through the α-carbons of the two terminal residues. All α-synuclein pulls were along the same vector defined by α-carbons of the terminal residues. SMD force was recorded every 20fs for I27 and every 100fs for α-synuclein. Trajectories were stored every 1ps. Starting configurations were generated from the NMR-determined structures of the I27 domain (PDB ID: 1TIT [43]) and micelle-bound α-synuclein (PDB ID: 1XQ8 [56]). Initial positions of hydrogen atoms were assigned based on the CHARMM27 topology file. The structures were solvated with TIP3P[68] water molecules using the solvate utility in VMD[69]. To ensure charge neutrality, counterions (sodium ions) were added to both systems with the autoionize utility in VMD. The resulting I27 system has a water box with dimensions 56Å x 56Å x 430Å and contains 129,938 atoms. The resulting α-synuclein system has a water box with dimensions 125Å x 225Å x 110Å and contains 298,368 atoms. A 125Å x 318Å x 110Å box of 145,386 TIP3P water molecules (436,158 18

28 atoms) was also created using solvate in VMD. This extension box was equilibrated separately and combined with the α-synuclein system prior to SMD simulations in order to accommodate the fully extended protein. A 5ns equilibrium simulation of α-synuclein was performed to generate an ensemble of initial configurations for SMD. Ten configurations from this simulation were selected as starting points for SMD simulations. Relative orientation of the two major α- helices was used as the selection criterion. Helical axes were identified based on residues for one helix and residues for the other. Three examples are shown in Figure 3.1. Figure 3.1: Examples of α-synuclein conformations at (a) 0ps, (b) 2200ps, and (c) 4300ps during the 5ns equilibrium simulation. Blue arrows represent helical axis vectors. The distribution of relative orientations ( θ, φ) of these vectors over the equilibrium simulation was used to select initial conformations for SMD. Graphics were generated using VMD. 19

29 The vectors defined by the axes (as shown in Figure 3.1) were transformed into the spherical coordinate system and translated to a common origin. Relative θ ( θ) and φ ( φ) angles between these vectors were computed for each frame collected during the equilibration. The θ- φ distribution over 5ns is shown in Figure 3.2. Ten ( θ, φ) points distributed throughout the distribution were selected as initial conformations for unfolding by SMD. A procedure was devised to align the long coordinate of the water box with the line passing through the terminal α-carbons. A box with dimensions identical to the original system, containing the protein roughly at its center, and with its long coordinate aligned with the terminal α-carbons line is designated. All water molecules and ions in the original system and their copies in the replicated periodic boxes that fall into the designated box are included in the new system. Since the distribution of ions in the original system and its copies is generally not uniform, the number of ions that land in the designated box need not be equal to the number of ions in the original system. If this is the case, ions are translated into or out of the designated box as necessary. The new system is then rotated about the center of mass of the protein to align its coordinates with the overall coordinate system. Because the designated box has the protein at its approximate center, only the original copy of the protein itself and molecules within ~10Å land in the box and are included in the re-oriented system. Because the original system was equilibrated, the density of water molecules throughout the box is equal. Therefore, any box with dimensions identical to the original system and containing the protein also contains roughly the same number of water molecules as the original system. Such a system, provided it contains the appropriate number of ions to achieve charge 20

30 neutrality, is equivalent to the original, although it does require additional minimization and equilibration. θ- φ distribution over 5ns equilibration φ (degrees) x θ (degrees) Figure 3.2: Distribution of θ and φ angles between vectors defined by two helical axes over 5ns equilibration (blue) and conformations selected for SMD (red) This procedure was applied to the ten configurations of the original α-synuclein system selected for SMD. The resulting ten realigned systems were minimized and equilibrated. The psfgen package in VMD was then used to join each system with the extension box into an extended system with a 125Å x 535Å x 110Å water box and containing ~ atoms. The y dimension of each extended system (535Å) is slightly smaller than the sum of the y dimensions of the two systems which were combined to create it (318Å + 225Å = 543Å) because the two systems overlap slightly at their 21

31 interface. Extended systems were again minimized and equilibrated. As a result, pulling was performed along the central axis of the water box for all ten α-synuclein systems. All minimizations used the conjugate gradient method. Original systems of both proteins underwent 3ps of all-atom minimization. The ten realigned α-synuclein systems underwent a minimum of 5ps of minimization. The ten extended systems also underwent a minimum of 5ps of minimization. During these two rounds of minimizations the α- carbons of α-synuclein terminal residues were fixed in order to maintain the line passing through the α-carbons aligned with the long dimension of the water box. Following minimization and prior to equilibrium simulations, systems were initialized in NAMD by generating random distributions of initial velocities for all atoms such that the systems were at 298K. The I27 system was equilibrated for 465ps prior to the SMD production run. The ten realigned α-synuclein systems underwent a minimum of 60ps of equilibration and the ten extended systems underwent a minimum of 100ps of equilibration. The extension water box was equilibrated for 154ps. Throughout these equilibrations the α-carbons of both terminal residues of α-synuclein were fixed to maintain the alignment of the protein with the water box. The same atoms were fixed during each 1ns equilibrium simulation in order to keep the protein in constant position, as defined by end-to-end distance, along the unfolding trajectory. Equations of motion were not calculated for these atoms, though the forces between them continued to be calculated. Six 1ns equilibrium simulations of α-synuclein were performed to explore configurations at intermediate points along an unfolding trajectory predicted by SMD. Configurations with end-to-end distances of 102Å, 161Å, 181Å, 201Å, 221Å, and 241Å 22

32 were selected from an α-synuclein SMD trajectory to be used as initial configurations for the six 1ns constrained simulations. These configurations span the SMD-predicted unfolding trajectory from fully folded structure to an extended structure that is nearly linear and lacks all tertiary structure. Protein-solvent interaction energies consisting of contributions from electrostatic and van der Waals interactions between the solute (protein) and the solvent (water and ions) were computed. Contributions from non-bonded van der Waals interactions were computed atom-by-atom using the NAMDEnergy extension in NAMD, where interactions were truncated at 12Å with switching distance beginning at 10Å. Contributions from electrostatic interactions were computed using the Poisson- Boltzmann implicit solvent representation a continuum approximation of solvent molecules and co-solvated ions. This representation averages solute-solvent electrostatic interactions over all solvent configurations, given a particular solute configuration [70, 71], and has the advantage of circumventing long-range (finite box size effects) corrections needed in explicit-solvent electrostatic calculations as a function of solute configurations along the SMD trajectory [72-74]. The Poisson-Boltzmann (PB) equation is derived from the Poisson equation for a set of fixed charges (solute) in a dielectric continuum (solvent) by incorporating a Boltzmann distribution of mobile charges (co-solvated ions). The result is a second order non-linear partial differential equation for the electrostatic potential, φ(x), as a function of position x: ε(x) φ(x) = ρ fixed (x) + (m) q m c m exp[-β( q m φ(x) + V m (x) )], 23

33 where ε(x) is the position-specific dielectric constant, ρ fixed (x) is the fixed distribution of explicit charges in the solute, the sum is over m ion species present in the solvent, and q m, c m, and V m (x), are, respectively, the charge of ion species m, concentration of ion species m, and the steric interaction potential between the solute and ion species m [70, 75]. The summation term accounts for the distribution of mobile ion species present in the solvent. The electrostatic energy of the system can be calculated from φ(x) [71]. The Adaptive Poisson-Boltzmann Solver (APBS) [76] was used to carry out implicit solvent calculations. APBS discretizes the problem by defining a grid on which φ(x) will be evaluated and solves the PB equation numerically for each grid point. The grid spacing is adaptive, starting uniformly coarse and then becoming finer in areas of high error. The solution processes can be parallelized efficiently by assigning the adaptive refinement of the grid over each local domain to a separate processor, thereby minimizing the need for costly inter-processor communication [70, 76]. For every frame in the trajectories of the 1 ns constrained simulations intraprotein non-bonded electrostatic interaction energies were calculated analytically by applying Coulomb s law using the coulomb utility in APBS. Additionally, APBS was used to calculate protein-solvent electrostatic interaction energies in implicit solvent for every frame in these trajectories. This calculation is performed by subtracting the energy of charging the protein in vacuum from the energy of charging the protein in solvent [71]. A uniform dielectric equal to the internal dielectric of the protein without a mobile charge distribution is used for calculating the energy of charging the protein in vacuum. For calculating the energy of charging the protein in solvent the internal dielectric of the 24

34 protein is used for the protein s interior, the solvent s dielectric is used for the rest of the system volume, and a mobile charge distribution is used. The internal dielectric of globular proteins is estimated to be between 2 and 4, depending to the number of polarizable groups in the interior of the protein and the degree of their polarizability. However, as there are very few or no inter-helical contacts in any conformations of α-synuclein observed over the course of the 1 ns constrained simulations, the protein effectively has no interior. Because of this, in our calculations 1 was used for the internal dielectric of the protein. A dielectric constant of was used for the solvent (water). The solute-solvent interaction energies computed as described above were used to estimate the excess chemical potentials, or free energies of solvation, of α-synuclein configurations corresponding to the six end-to-end distances of the constrained simulations. According to the inverse form of the Potential Distribution Theorem (PDT) [77, 78], the excess chemical potential equals: µ ex = k B T ln < exp[ U / k B T ] >, (3.1) where is k B T is the thermal energy, U is the solute-solvent interaction energy computed above, and the brackets indicate averaging over all solute-solvent configurations for the coupled system of protein and solvent. The average can be written explicitly as: < exp[ U / k B T ] > = exp[ U / k B T ] P( U) d( U), (3.2) 25

35 where P( U) is the probability density function of U for a given protein end-to-end distance. The integral is difficult to evaluate due to the required sampling of highly improbable protein-solvent configurations with large U values that contribute significantly to the integral because of the Boltzmann weighting in equation (3.2). If P( U) is described by a Gaussian probability distribution with mean < U > and variance σ 2, then the excess chemical potential equals: µ ex = < U > + σ 2 / 2 k B T. (3.3) The Gaussian mixture generalization of P( U) is given by P( U) = n ω n (1/ (2πσ n 2 )) exp( ( U < U n>) 2 / 2σ n 2 ), (3.4) where < U n> is the mean value of the nth Gaussian distribution and ω n is the corresponding weight of that distribution, such that 0 ω n 1 for all n and n ω n = 1, and the excess chemical potential equals: µ ex = n ω n (< U n> + σ n 2 / 2 k B T). (3.5) 26

36 Chapter 4: Results Figure 4.1 compares the force-extension curve from our SMD simulation of I27 with a typical curve obtained by the Schulten group [28]. In both cases an initial peak with a height between 2000pN and 2500pN is observed during the first 20Å of extension, followed by a drop in force below 1000pN. This peak corresponds to the breakage of all hydrogen bonds between A G and AB strand pairs (Figure 2.1) from 9Å to 15Å of extension, described in detail in [28]. The rise in force observed in our profile from ~90Å to ~120Å of extension is not seen in the SMD simulation of the Schulten group, and appears to coincide with the separation of strands C and F (Figure 2.1). Afterward, force levels off just below 1000pN and continues to oscillate around this level until ~250Å, after which a sharp rise is observed in both simulations. This sharp rise is due to pulling on an essentially linear polypeptide that has almost no remaining secondary structure. 27

37 Figure 4.1: (a) Force vs extension for I27 from [28] and (b) force vs extension for I27 from our simulation. Both force profiles exhibit a sharp spike associated with the separation of A G and AB strand pairs (Figure 2.1), followed by a drop-off during the rest of unfolding and a second rise associated with pulling on a fully linearized protein. For α-synuclein SMD simulations, work over each pulling trajectory was calculated according to equation 1.2 by numerically computing the area under the force vs extension curve using the trapezoidal rule. Force-extension and work-extension profiles for the ten pulling trajectories are shown in Figure

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40

41 Unlike the I27 extension profiles in Figure 4.1, force increases almost linearly with extension in every α-synuclein unfolding trajectory. Also unlike I27, α-synuclein has no stable tertiary structure. Therefore, any change in force associated with change in protein structure results from modifications of whatever helical secondary structure is present, including bending of helices and loss of helical structure, and from reorientation of helices relative to each other. All of these happen concurrently during each pull, until all secondary structure is lost. The entire force trajectory of α-synuclein pulls corresponds to the segment of the I27 pull trajectory after all tertiary structure has been lost the post- 250Å rise in force. In the case of α-synuclein the increase in force is much more gradual because all helical structure must be lost before the point of pulling on a completely linearized protein is reached. Consistent with a linear increase in force, work, the integral of force, accumulates at an accelerating rate over the course of each pull. We apply equation 1.3 to estimate the change in free energy as a function of the extension, G(z), from our ensemble of ten actualizations, where <exp[ w i (z) / k B T ]> = (1/10)* i=1 10 exp[ w i (z) / k B T ] (4.1) is the average over the ensemble of the 10 trajectories. In order to perform this calculation, the work function along each trajectory was interpolated to obtain a uniform set of z values for all trajectories. G was then evaluated at each of these values. Work performed during each pull and the G estimate over the course of unfolding are shown in Figure 4.3. We found that G is a lower bound for the work profiles of the ten unfolding trajectories. Although it is possible for the work of an individual trajectory to 32

42 be lower, at some, or even all points along the trajectory, such unfolding trajectories are extremely rare and were not observed in our ensemble. In fact, the work profiles for all ten unfolding trajectories in the ensemble appear nearly identical, with a slight increase in slope between 150Å and 200Å of extension, when the final, rapid unzipping of helices begin. 33

43

44 At the nine extensions designated by vertical gray lines in Figure 4.3, covering the full range of extension, the average and standard deviation of the ten work values were used to approximate Gaussian distributions of work required to transition from a folded conformation to that extension. These distributions are shown in Figure 4.4. Figure 4.4: Gaussian distributions of work generated from the average and standard deviation of data at nine extension values: 40Å, 80Å, 120Å, 160Å, 200Å, 250Å, 300Å, 350Å, and 395Å. Representative conformations are shown in Figure 4.5. Due to small sample size, an accurate estimate of the variance was not possible, and therefore, these Gaussian distributions must be viewed as only a qualitative 35

45 description of the work distribution for the ten trajectories along the unfolding transition. Nonetheless, we see that the average value increases as expected, and the variance in the distribution of work required increases, which reflects an increasing contribution of the work due to dissipation. Representative conformations at each of the nine extension values are shown in Figure 4.5. Protein free energies of solvation were computed from six 1ns constrained simulations of protein conformations at protein end-to-end distances of (A) 102Å, (B) 161Å, (C) 181Å, (D) 201Å, (E) 221Å, and (F) 241Å along the unfolding trajectory. Six configurations from the trajectory of pull 10 were used as starting points for these simulations. These configurations span the transition from the pre-unfolding configuration A to a nearly linear conformation with intact α-helices F. The starting configurations are shown in Figure

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50 Solute (protein)-solvent (water + ions) van der Waals and electrostatic interaction energies from the six constrained simulations were combined, binned, and normalized by the total number of observations from each simulation (10,000), generating a histogram of total solute-solvent interaction energy frequencies. The average and standard deviation (square root of variance) from each data set was used to generate a Gaussian distribution. Probability distribution functions of the solute-solvent interaction energies corresponding to solute configurations A, B, E, and F are well approximated by these Gaussians, as shown in Figure 4.7. Probability distribution functions corresponding to solute configurations C and D are not described by single Gaussian distributions, so Gaussian mixtures describing these probability distributions were derived based on structural properties of the protein over the course of each simulation. Roughly one hundred configurations with extreme energy values were selected. Structural distinctions between extremely low and extremely high energy groups were identified and a single structural parameter was selected to describe the range of variation. The value of this parameter was used as the criterion for the bifurcation of the entire data set. Probability distributions for each subset were fit with Gaussians defined by the mean values and standard deviations of that subset. This procedure was repeated recursively until the probability distribution of each subset was well described by the Gaussian defined by its mean and standard deviation. Conformations from simulations C and D were divided into subpopulations C 1, C 2, C 3, C 4, D 1, and D 2 such that the solvation energy distribution of each subpopulation is well described by a Gaussian. The following structural parameters were defined for simulation C: 1) secondary structure conformation of residues helical or 41

51 turn/coil, as identified by STRIDE implemented in VMD, and 2) the distance between α- carbons of residues 21 and 47. The second parameter is a proxy for the overall distance between sometimes-helical residues and the coil consisting for residues (Figure 4.8 a, b). A single structural parameter was defined for simulation D: the relative φ angle ( φ) between axes of helices containing residues and (Figure 4.8 c). This criterion is very similar to the criteria used to select starting configurations for SMD, as described in chapter 3. Table 4.1 summarizes the split of C and D into subpopulations based on these parameters. Subpop. Division Criteria Res helicity 21 to 47 distance C 1 α- or 3-10 helical Under 14Å C 2 α- or 3-10 helical Over 14Å C 3 π-helical, turn, or coil Under 14Å C 4 π-helical, turn, or coil Over 14Å φ b/w helices and D 1 Under 0 D 2 Over 0 Table 4.1: Summary of criteria according to which conformations from simulations C and D were divided. Criteria were defined based on the quality of approximation of the resulting solvation energy distributions with Gaussians. 42

52 Figure 4.8: α-synuclein conformations representative of subpopulations (a) C 1 (cyan) and C 2 (orange), (b) C 3 (cyan) and C 4 (orange), (c) D 1 (cyan) and D 2 (orange). Red regions in (a) and (b) designate residues 20 23, which are helical in (a) and turn/coil in (b). Labels in (a) and (b) designate α-carbon atoms shown as VdW spheres. Dashed lines and corresponding labels describe distances between these atoms. Arrows in (c) represent helical axis vectors. Continued 43

53 Figure 4.8: Continued 44

54 Gaussian distributions were generated from means and standard deviations of solvation energy distributions of each subpopulation and combined into Gaussian mixture distributions according to equation 3.4. Distributions corresponding to each subpopulation were weighed by assigned parameter ω equal to the fraction of conformations belonging to that subpopulation. Resulting Gaussian mixture distributions approximate the overall energy distributions from simulations C (Figure 4.9) and D (Figure 4.10). Although the fit is poor in the kcal/mol region of distribution D, in the low energy tail of the distribution, the high energy tail of the distribution contributes much more significantly to the estimate of µ ex. Therefore, the impact of the poor fit in this region is expected to be small. 45

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59 Given Gaussian or Gaussian mixture approximations for the probability distributions of solute-solvent interaction energies for the constrained protein configurations, excess chemical potentials (free energies of solvation), µ ex, for each protein end-to-end distance were estimated according to equation 3.3 for Gaussian approximations and equation 3.5 for Gaussian mixture approximations. Utilizing this technique, µ ex values were calculated for the six solvation energy distributions (Table 4.2). E-to-E distnace <U> σ µ ex Distr. ω (Å) (kcal/mol) (kcal/mol) (kcal/mol) 102 A B C C C C D D E F Table 4.2: Average protein-solvent interaction energies (<U>), standard deviations (σ) of total protein-solvent interaction energy, and estimates of free energy of solvation (µ ex ) for α-synuclein by inverse PDT for protein-solvent interaction energy distributions at six end-to-end distance values. 50

60 Because the estimate of µ ex is sensitive to the magnitude of variance in the corresponding energy distribution, distributions with greater variance tend to result in larger µ ex estimates, sometimes despite lower mean values. Distribution F has a lower mean value than distribution E, but, because of its greater variance, has the larger µ ex. Similarly, B has a lower mean value, but a larger µ ex than A. For Gaussian mixture distributions, the weighted average of the variances of the constituent Gaussian distributions [eq. 3.5] makes the equivalent contribution to µ ex. Because the overall width of a mixture distribution depends on the distances between the mean values of the constituent distributions, as well as their variances, it correlates poorly with the µ ex estimate resulting from the mixture distribution. Thus distribution C, despite being wider than all four of the single Gaussian distributions, results in a smaller µ ex estimate than distributions A, B, and F the result of Gaussians accounting for the majority of the mixture having less variance than the single Gaussians of A, B, and F. Also, despite having nearly the same width, distributions C and D result in very different µ ex estimates. This highlights the fact that the division of populations of conformations with non- Gaussian energy distributions into subpopulations is critical, because a poor variance estimate for the energy distribution of any subpopulation will result in a poor estimate of µ ex. Extensive sampling of such populations is critical for the same reason. The characterization of subpopulations within populations from simulations C and D is very different. While each subpopulation in C consists of highly similar conformations, with distinct differences between subpopulations residues are either helical or not in concert, helical residues are either close to coil residues or far away the subpopulations in D are not nearly as well defined. This parallels the 51