BROWNIAN DYNAMICS SIMULATION OF A FIVE-SITE MODEL FOR A MOTOR PROTEIN ON A BEAD-SPRING SUBSTRATE. A Thesis. Presented to

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1 BROWNIAN DYNAMICS SIMULATION OF A FIVE-SITE MODEL FOR A MOTOR PROTEIN ON A BEAD-SPRING SUBSTRATE A Thesis Presented to The Graduate Faculty of The University of Akron In Partial Fulfillment of the Requirements for the Degree Master of Science Nabina Paudyal December, 2014

2 BROWNIAN DYNAMICS SIMULATION OF A FIVE-SITE MODEL FOR A MOTOR PROTEIN ON A BEAD-SPRING SUBSTRATE Nabina Paudyal Thesis Approved: Accepted: Advisor Dr. Jutta Luettmer-Strathmann Dean of the College Dr. Chand Midha Faculty Reader Dr. Yu-Kuang Hu Dean of the Graduate School Dr. Rex D. Ramsier Faculty Reader Dr. Alper Buldum Date Department Chair Dr. David N. Steer ii

3 ABSTRACT Motor proteins play an important role in many biological processes. For example, kinesin molecules are responsible for the transport of vesicles in nerve cells and their malfunction has been linked to neurodegenerative diseases. Unfortunately, the complexity of motor proteins and their environment makes it difficult to model the detailed dynamics of molecular motors over long time scales. In this work, we develop a simple coarse-grained model for a motor protein on a bead-spring substrate under tension. In our model, different pair potentials describe interactions between substrate and motor, motor components and substrate components. The movement of motor proteins entails ATP hydrolysis, which is modeled in terms of mechano-chemical states that couple positional and chemical degrees of freedom. We have identified model parameters corresponding to protein and chain conformations of a walking protein as well as transition rates for the states and apply our model to simulate a motor protein walker with cargo. iii

4 ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my advisor Dr. Jutta Luettmer Strathmann for her valuable support and guidance throughout this journey. I would also like to thank my dear friend Maral Adeli Koudehi for her priceless help and company. I am thankful to Dr. Hu and Dr. Buldum for being on my committee and providing me valuable advice and encouragement. Special thanks to my research group and my friends Binod Dhakal, Kiran Khanal, Shavyata Acharya, Dipika Parajuli, Chinta Mani Aryal, and Sajeevi Withanage for their valuable guidance. I am grateful to the Department of Physics and all my colleagues for their direct and indirect support during my study and research. Finally I would like to dedicate all my works and my thesis to my parents and my brother Nabin who have always believed in me and have always been there for me to lift me up. iv

5 TABLE OF CONTENTS Page LIST OF TABLES LIST OF FIGURES vii viii CHAPTER I. INTRODUCTION Outline of thesis II. MODEL Conservative Forces Mechanical State Chemical Steps III. METHOD Brownian Dynamics Simulation Behavior of an isolated protein Conformation of the protein in contact with the substrate Kinetic energy IV. RESULTS Force-extension relation of the substrate Motor protein taking steps on the substrate without cargo v

6 4.3 Effect of changing the interaction of the attached head of a motor protein walking without cargo Motion of protein with cargo Five steps for a protein with cargo V. CONCLUSIONS BIBLIOGRAPHY vi

7 LIST OF TABLES Table Page 2.1 Model units of physical quantities Interaction parameters Interaction parameters for cargo Parameters for the equilibrium states Assigned parameters for the change of states to maintain the desired amplitudes vii

8 LIST OF FIGURES Figure Page 2.1 Coarse grained model of a motor protein on a bead spring substrate. Sites 1, 2, 3 and 4 represents protein heads and 5 represents the central site, which is bonded to the light chain domain for the attachment of cargo structures Graphs of spring potentials as a function of distance for three sets of parameters that describe bonds between different types of particles: BB for bead-bead interactions along the substrate chain, HH for bonds between protein sites in the same head domain, HC for bonds between protein head sites and the central protein sites; values for the parameters are presented in Table Graphs of four interparticle potentials as a function of distance between the particles. The solid lines represent the bead-bead harmonic spring (magenta), soft repulsive Lennard-Jones (black), Lennard-Jones (blue), and Morse (red) potentials, respectively. The parameters of the potentials are shown in Table Illustration of the generalized dihedral planes of our motor protein model. The dihedral angle α is the angle between the normal unit vectors of the planes Graph of the dihedral potential as a function of the angle α for three different dihedral amplitudes V Illustration of the bond angle θ formed by two unit vectors â and ˆb along the bonds between the central protein site and head sites 2 and Graph of the bond angle potential as a function of the bond angle θ. The equilibrium bond angle is θ 0 = in our model Illustration of the interactions between the components of the protein.. 16 viii

9 2.9 Illustration of the interactions between protein and substrate sites. Solid red arrows represent the Morse interaction between protein heads and the substrate, whereas dotted black arrows represent the soft repulsive Lennard-Jones interaction between the central protein and the substrate Illustration of the interactions between substrate sites. The heavy lines connecting the beads represent harmonic springs; the curved arrows indicate Lennard-Jonnes interactions between non bonded beads. An external tension force, represented by F pull, is applied to the end beads of the chain with equal magnitude and opposite directions Chemical state as a function of time for a two-step simulation. The initial state, S 1 = 1, S 2 = S 3 = S 4 = 0 is maintained during an equilibration period at the start of the simulation. The probability S 1 to be in state 1 (red line) falls off rapidly as S 2 (blue line) first increases. As S 2 decreases S 3 (purple line) first increases and then decreases as S 4 starts to grow. At the end of the first step, S 1 =S2 =S2 = 0 and S1 = 1. In the second step the system goes from S 4 = 1 to S 1 through the states S 2 and S Effect of K θ and V 0 on conformations of an isolated protein Illustration of the mechano-chemical states (1, 2, and 3) of a protein walker on a substrate. The potential parameters of the states are presented in Table Instantaneous (red) and segment-averaged (blue) kinetic energy as a function of time for simulations of the protein-vesicle-chain system, which has a total of N = 26 particles. Panel A and B show results for fixed and time-varying interaction parameters, respectively. The length of one time segment in the averaging process is steps, corresponding to a simulation time of 500. In both simulations, the average kinetic energy has a value of 39, in agreement with the equipartition theorem and a reduced temperature of one Force extension results for an isolated chain. The symbols represent simulation results for the extension, defined in equation (4.1), as a function of the pulling force for a chain of length N c = 20. The error bars indicate the standard deviation of block averages (see equation (3.16)). Two chain conformations for intermediate (left) and high (right) extensions are also shown ix

10 4.2 Two steps of a motor protein walker on a chain substrate. Panel A shows the smallest distance between the chain and head site 1 on head one (d 1, red) and the chain and head site 3 on head two (d 3, pink) together with the indices of the beads closest to the head sites (C 1, blue and C 3, cyan). Panel B shows the value of the Morse potential parameters D e1 and D e3 as a function of time Two steps of a motor protein on a substrate with increased attraction for the attached head site. Panel A shows the smallest distance between the chain and head site 1 on head one (d 1, red) and the chain and head site 3 on head two (d 3, pink) together with bead indices closest to head site 1 (C 1, blue) and head site 3 (C 3, cyan). Panel B shows the value of the Morse potential parameters D e1 and D e3 as a function of time Two steps of a motor protein walking with cargo on a substrate. The graph shows the smallest distance between the chain and head site 1 (d 1, red), head site 3 (d 3, pink) and the cargo (ves, orange). The change in Morse parameters is as shown in panel B of graph 4.2. The closest beads are C 1 (blue), C 3 (cyan) and C ves (yellow) for head sites 1, 3, and the cargo, respectively. In these simulations, the Morse parameters of the non-released head remained constant, as shown in panel B of figure Two steps of a motor protein walking with cargo. The lines represent the same quantities as in figure 4.4; the Morse amplitudes for this simulation are shown in figure 4.3 B Results for the same system as in figure 4.5 from a separate simulation step simulation with the new protocol. The lines represent the same quantities as in figure x

11 CHAPTER I INTRODUCTION A machine is called a motor if its output involves mechanical movement [1, 2]. Molecular motors play a role in complex processes such as cell motility, cell division, cell development and also the transport of biomolecules [3]. Axonal transport is the transport of independent cargo structures with the help of motor proteins in the axon of nerve cells [4]. To ensure proper functioning of the nerve cells, cargoes should be delivered to the correct destination in appropriate quantity. Failure of such transport has been linked to several neurodegenerative diseases [4]. The cargo structures most often detected are vesicles, mitochondria, proteins and other cellular structures. In general, cargoes may be membranous or non membranous. However, the ones found in axons are typically vesicles, which consist of a lipid membrane enclosing the molecules to be transported. Microtubules and microfilaments are the most common parts of the cytoskeleton present in axons of nerve cells. They act as tracks for axonal transport. Motor proteins such as kinesin and dynein are found on microtubules and microfilaments, while the motor protein myosin interacts with actin filaments in muscles [4, 5, 6]. All motor proteins consist of heavy and light chain domains. The heavy chain domains have a globular head that interacts with the substrate, whereas the light chain domains attach to cargo and other structures. Although motor pro- 1

12 teins are very small, they are very complex structures and interactions between motor proteins and their substrate involve multiple degrees of freedom [7]. Motors operate by adjusting their conformation in response to chemical changes that fuel the motor proteins. The movement of motor proteins is fueled by ATP (adenosine triphosphate) hydrolysis. To describe how a chemical reaction generates mechanical motion of a motor protein, a number of theoretical models have been developed [7, 8, 9, 10, 11, 12]. Except for Atzberger and Peskin [12], who developed a three-dimensional model for kinesin consisting of two sites in contact with a hard, patchy surface, these are onedimensional mechano-chemical models. In our modeling of a motor protein we consider four issues: (i) degrees of freedom, (ii) forms of interaction, (iii) dynamical equations and (iv) methods of solutions. In this work, we develop a coarse-grained three-dimensional model of a motor protein that transports cargo by walking along a bead spring substrate. In our model, different pair potentials describe interactions between substrate and motor, motor components and substrate components. To couple the mechanical degrees of freedom to chemical states, we learn from the one-dimensional model of Lan and Sun [11]. Lan and Sun investigated the dynamics of the motor protein myosin and the simultaneous muscle contraction [11]. They used a simple one-dimensional model of a single myosin structure. The conformation of myosin was described by the angle between the light chain domain and the motor domain. Mathematically, the myosin 2

13 dynamics is described by the following coupled Langevin equations η θ = E(θ, S) F + f b (t) (1.1) θ s = K(θ) t S (1.2) where η is the friction coefficient, θ is the angle between light chain domain and motor domain, E(θ, S) is the elastic energy of myosin, F is an external force, f b (t) is the Brownian random force, S is the chemical state of the motor and K(θ) is a matrix of chemical transition rates. The chemical states, S i, are identified as ATP hydrolysis, phosphate release, power-stroke, and ADP (adenosine diphosphate) release along with binding and detachment steps of myosin. ATP and ADP release are dependent on the conformation of myosin, therefore the transition rates for these states depend on θ. Lan and Sun determined values of the chemical transition rates from experimental data. In this work, we apply the concept of coupled equations as described in equation (1.1) and (1.2) to our three dimensional, coarse-grained model of a molecular motor on a substrate. We combine Brownian dynamics simulations for the positional degrees of freedom with a numeric solution of the equations of motion for the chemical variables to simulate a protein taking steps on a substrate. The cargo is attached to the motor protein with a harmonic spring. 3

14 1.1 Outline of thesis This thesis is organized as follows: In chapter II we present the model for the protein, the cargo, and the substrate and we define the mechano-chemical states of our model. The simulation method is described in chapter III and results are presented in chapter IV. We give a brief summary and conclusions in chapter V. 4

15 CHAPTER II MODEL We develop a coarse grained model of a motor protein walking on a bead spring substrate under tension. The walker has a total of five sites, with two binding sites in each of two head domains and a central site. The two head sites and the central sites are bonded to each other with harmonic springs. Equilibrium conformation and motion of the motor protein and a substrate are determined by interactions between the constituents. The surrounding medium is treated as an implicit solvent and provides a random force that we discuss in chapter 3. Figure 2.1: Coarse grained model of a motor protein on a bead spring substrate. Sites 1, 2, 3 and 4 represents protein heads and 5 represents the central site, which is bonded to the light chain domain for the attachment of cargo structures. 5

16 We use Lennard Jones potential parameters as units for the energy and length in our model. To complete the units, we set the mass m of a bead of the chain to one. With these choices, the units for temperature, time, etc are also fixed. Table 2.1 shows model units and SI units. Table 2.1: Model units of physical quantities Parameters Model SI Length [L] σ m Energy [E] ɛ J Temperature [T] ɛ k B K Mass [M] m kg Time [t] mσ 2 ɛ s Viscosity [η] m tσ Pa.s To make the connection to biological systems, we assume a temperature of 300 K, which implies ɛ = J. If we set σ = 5 nm, about times the size of a kinesin head domain, and set the density to that of water, ρ = 1000 kg m 3, we find m = ρπ σ3 6 = kg, about one third of the mass of a kinesin head [5, 6]. This gives s for the model unit of time. 6

17 2.1 Conservative Forces Interactions between the sites of the model are described by conservative forces. The total conservative force on particle i at position r i is given by F conservative ( r i ) = j { F hs ( r ij ) + F lj ( r ij ) + F m ( r ij )+ (2.1) F dih ( r ij ) + F bond ( r ij ) + F pull ( r ij )}, where F hs ( r ij ) represents harmonic spring forces, Flj ( r ij ) and F m ( r ij ) are due to Lennard-Jones and Morse potentials, respectively, and F dih ( r ij ) and F bond ( r ij ) are the forces related to the dihedral and bond angle, respectively. Fpull ( r ij ) represents an external tension applied in equal amounts but opposite directions to the end beads of the substrate chain to keep the substrate extended. The interaction forces are calculated from the negative gradients with respect to the particle s coordinates of the corresponding potential energy contributions. Let, U α be a potential energy contribution, then its corresponding force is calculated as, F α ( r i ) = ri U α. (2.2) The potentials describing interactions between the particles are now discussed in detail. Harmonic Spring: Harmonic springs represent bonds between two sites. With the compression or expansion of a bond, the bond length deviates from its equilibrium value. The potential 7

18 energy of the spring is the harmonic potential U hs (r ij ), [13] U hs (r ij ) = 1 2 K s(r ij B 0 ) 2, (2.3) where K s is the spring constant, r ij is the distance between the sites, and B 0 is the equilibrium bond length. Our model has three different sets of parameters to describe bonds between different types of particles. Section 2.2 explains which sites are connected by harmonic springs and Table 2.2 contains the parameter values. In figure 2.2 we represent graphs of the spring potentials used in this work. The force on particle i due to a harmonic spring is calculated as, F hs,i = K s (r ij B 0 )ˆr ij, (2.4) where ˆr ij = ( r j r i ) r ij is the unit vector pointing from particle i to particle j. Lennard-Jones Potential: The Lenard-Jones (LJ) potential is generally used to describe interactions between non-bonded pairs of neutral particles, [14] [ ( ) 12 ( ) ] 6 σ σ U lj (r ij ) = 4ɛ, (2.5) r ij where ɛ is the potential depth, r ij is the distance between the sites and σ sets the length scale. A graph of the LJ potential is included in Fig 2.3. The force on particle i due to the LJ potential is given by r ij [ ( F lj,i = 48ɛ 1 ) 12 σ 1 ( ) ] 6 σ ˆr ij. (2.6) r ij r ij 2 r ij 8

19 Figure 2.2: Graphs of spring potentials as a function of distance for three sets of parameters that describe bonds between different types of particles: BB for beadbead interactions along the substrate chain, HH for bonds between protein sites in the same head domain, HC for bonds between protein head sites and the central protein sites; values for the parameters are presented in Table

20 The LJ potential is a one length-scale potential with σ, the distance at which the potential value is zero, as its adjustable length. The attractive part of the potential is given by the r 12 term and the repulsive part is given by the r 6 term [14]. The cut-off value for the Lennard-Jones potential in our model is 3σ and the potential is shifted to make sure that its value is zero at the cut-off distance. Soft repulsive Lennard-Jones Potential: The soft repulsive Lennard-Jones (SLJ) potential is a purely repulsive Lennard-Jones potential, (U lj + ɛ) for r lj R c, U lj(r ij ) = 0 for r lj > R c, (2.7) where R c = 2 1/6 σ is the cut-off distance. A graph of the SLJ potential is included in Fig Morse Potential: A Morse potential describes interactions from the bond equilibrium to the detachment state, [13] U m (r ij ) = D e [ (1 e κ(r ij B m) ) 2 1 ], (2.8) where D e is the depth of the potential energy minimum, κ is a positive number which has the units of inverse of length and is used to control the decay length of the exponential term, r ij is the distance between the two sites and B m is the equilibrium bond length. The negative differentiation of the Morse potential with respect to distance yields the Morse force on particle i, 10

21 F m,i = 2κD e ( e κ(r ij B m) ) [ 1 e κ(r ij B m) ] ˆr ij. (2.9) The Morse potential has two length scales, B m and κ, which control the distance where the force becomes repulsive and the width of the attractive well, respectively. A graph of a Morse potential is included in Fig. 2.3; potential parameters are presented in Table 2.2 Figure 2.3: Graphs of four interparticle potentials as a function of distance between the particles. The solid lines represent the bead-bead harmonic spring (magenta), soft repulsive Lennard-Jones (black), Lennard-Jones (blue), and Morse (red) potentials, respectively. The parameters of the potentials are shown in Table 2.2. Dihedral Potential: A dihedral potential models the change in energy due to a change in the dihedral angle α, which is caused by a rotation of the dihedral planes [13, 14]. We use a generalized dihedral potential to control the conformation of the walker on the bead 11

22 spring substrate. Fig 2.4 shows the dihedral planes of our walker model. The dihedral potential is given by, U dih (r ij ) = V 0 (1 cos 2 α), (2.10) where V 0 is the dihedral amplitude and α is the angle between the dihedral planes. Fig 2.5 shows a graph of the dihedral potential; values for the parameters are included in Table 2.2. A dihedral force acts on the protein head and the central sites and is calculated from F dih,i = V 0 sin 2α j α r ij ˆr ij. (2.11) Figure 2.4: Illustration of the generalized dihedral planes of our motor protein model. The dihedral angle α is the angle between the normal unit vectors of the planes. Bond Angle Potential: Just as the harmonic spring potential, the bond angle potential is based on Hooke s law but the deviation from the equilibrium configuration is due to a change in the bond angle θ rather than a distance [13]. Fig 2.6 shows the angle θ in our walker 12

23 Figure 2.5: Graph of the dihedral potential as a function of the angle α for three different dihedral amplitudes V 0. model that we control with the bond angle potential, U bond (θ) = 1 2 K θ(θ θ 0 ) 2, (2.12) where K θ is the force constant, θ is the bond angle, and θ 0 is the equilibrium bond angle. Fig 2.7 shows a graph of the potential. The force is calculated from F bond,i = K θ (θ θ 0 ) j θ r ij ˆr ij. (2.13) 13

24 Figure 2.6: Illustration of the bond angle θ formed by two unit vectors â and ˆb along the bonds between the central protein site and head sites 2 and 3. Figure 2.7: Graph of the bond angle potential as a function of the bond angle θ. The equilibrium bond angle is θ 0 = in our model. 14

25 2.2 Mechanical State The mechanical state of the system is described by the positions of all the particles in the model and controlled by the potentials introduced in Section 2.1. Here we describe how we apply the potentials to define the interactions between the constituents of our model Intra protein interactions The sites of the protein interact with both bonded and non-bonded interactions as shown in Fig 2.8. Bonded interactions are represented by harmonic springs, which connect the two sites of each head and also the head sites to the central site. Non-bonded interactions between sites on different heads are described by the soft-repulsive Lennard-Jones potential. The bond angle interaction between two nonbonded head sites is used to control the distance between the two heads. The dihedral interaction controls the orientation of the planes. Figure 2.8 illustrates all the intra protein interactions Protein-substrate interactions Interactions between protein sites and substrate sites are described by two kinds of potentials, Morse and soft-repulsive Lennard-Jones. The Morse interaction between head sites of the protein and chain beads sets the attraction of the head to the substrate and allows us to control the release and attachment of the head to the substrate. The soft repulsive Lennard-Jones potential between the central site and the 15

26 Figure 2.8: Illustration of the interactions between the components of the protein. chain beads prevents overlap of the central site with the chain. Figure 2.9 illustrates the protein substrate interactions Substrate-substrate interactions Bonded interactions between the chain beads are represented by harmonic springs, while non-bonded interactions are described by a Lennard-Jones potential. An external tensile force, given by F pull, is introduced to keep the chain elongated for the protein walk. Figure 2.10 illustrates the substrate-substrate interactions. After defining the form of the interaction forces a parametrization process is required to complete the model [13]. We determined the interaction parameters of our model by evaluating trajectories and energy values for different trial parameters. The potential parameters for our model are presented in table

27 Figure 2.9: Illustration of the interactions between protein and substrate sites. Solid red arrows represent the Morse interaction between protein heads and the substrate, whereas dotted black arrows represent the soft repulsive Lennard-Jones interaction between the central protein and the substrate Figure 2.10: Illustration of the interactions between substrate sites. The heavy lines connecting the beads represent harmonic springs; the curved arrows indicate Lennard-Jonnes interactions between non bonded beads. An external tension force, represented by F pull, is applied to the end beads of the chain with equal magnitude and opposite directions. 17

28 Table 2.2: Interaction parameters. Protein Bonded Head-Head Harmonic Spring K s = 80ɛ/σ 2, B 0 = 0.9 σ Head-Central Harmonic Spring K s = 80ɛ/σ 2, B 0 = 1.8σ σ = 1, ɛ = 1, Non-Bonded Head-Central Soft Repulsive LJ R c = σ σ = 1, ɛ = 1, Head-Head Soft Repulsive LJ R c = σ Dihedral Planes Dihedral V 0 = 100 Head-Head Bond Angle θ 0 = Chain-Chain Bonded Harmonic Spring K s = 320ɛ/σ 2, B 0 = σ Non-Bonded Lennard Jones σ = 1, ɛ = 1, R c = 3σ External Tension F pull = 40ɛ/σ D e = 4ɛ, κ = 4/σ, Protein-Chain Chain-Head Morse B 0 = 0.9σ, B m = B 0 + ln(2)/κ σ = 1, ɛ = 1, Chain-Central Soft Repulsive LJ R c = σ 18

29 To simulate the transport of a cargo, like a vesicle, we add one more site to the model. This cargo site is bonded to the central site of the motor protein with a harmonic spring and interacts with non-bonded sites through the soft repulsive Lennard-Jones potential. Fig 2.1 includes an illustration of the motor protein with cargo. The cargo particle has four times the size and 64 times the mass of a protein or chain bead site. The parameters used to define the cargo interactions are given in table 2.3. Table 2.3: Interaction parameters for cargo. Cargo Head and Chain Soft Repulsive LJ Size = 4σ, σ c = 1 ( Size +σ), ɛ = 1, 2 R c = (σ c + σ) Central Harmonic Spring K s = 80ɛ/σ 2, B 0 = 4σ 19

30 2.3 Chemical Steps In the biological system, the chemical steps regulating the dynamics of a motor protein include ATP hydrolysis, phosphate release, powerstroke and ADP release [11]. In the Introduction, we discussed how Lan and Sun [11] mapped the biological system onto a one-dimensional model for a walker to obtain a mathematical description of the walking process for myosin. They used a model with eight states and experimental values for the transition rates. For our three dimensional model of a walker, we define n s = 4 chemical states. They represent the equilibrium protein conformations that serve as intermediate steps along the walk. The states are associated with different interaction parameter values during the simulations. Mathematically, the chemical state of the system is represented by a vector, S, with n s components. The change of the chemical state of the motor protein is defined by the first order differential equation S t = K S (2.14) where, K is an n s n s matrix of transition rates. States 1 and 4 in our model are identical and represent a conformation where both heads are in contact with the chain. To simulate a motor that can make a step with either head one or head two, 20

31 we define two transition matrices, K and K1. k fast k fast k slow 0 0 K =, (2.15) 0 k slow k slow k slow k slow 0 0 k slow 0 k fast K1 =, (2.16) 0 k slow k slow k fast where k fast and k slow represent fast and slow processes, respectively. They have the values k fast = and k slow = in our model. According to equation (2.14), the set of coupled differential equations for chemical state n is given by S n t = j K nj S j, (2.17) where S n is the probability to find the system in state n so that S n = 1. (2.18) n The chemical state is coupled to the mechanical state through the values of the potential parameters, Q( S) = n Q n S n, (2.19) where Q is any of the parameters (D e, K θ, V 0 ) that change during the simulation. This is further discussed in chapter 4. The system of equations (2.17), (2.18), (2.19) 21

32 is used to advance the chemical state during the simulations. Figure 2.11 shows the evolution of the states. Figure 2.11: Chemical state as a function of time for a two-step simulation. The initial state, S 1 = 1, S 2 = S 3 = S 4 = 0 is maintained during an equilibration period at the start of the simulation. The probability S 1 to be in state 1 (red line) falls off rapidly as S 2 (blue line) first increases. As S 2 decreases S 3 (purple line) first increases and then decreases as S 4 starts to grow. At the end of the first step, S 1 =S2 =S2 = 0 and S 1 = 1. In the second step the system goes from S4 = 1 to S 1 through the states S 2 and S 3. 22

33 CHAPTER III METHOD 3.1 Brownian Dynamics Simulation Since motor proteins are small particles in a liquid environment they undergo Brownian motion when they are not actively changing conformation. Brownian motion is the result of collisions with molecules in the surrounding solvent [15]. Since our motor protein model has an implicit solvent we use Brownian dynamics simulations to describe the motion. The net force on particle i of our model is given by F net,i = F conservative,i + F frictional,i + F random,i. (3.1) where F conservative,i is the conservative force defined in equation (2.1), F frictional,i is the frictional force due to the viscous medium, F frictional,i = ζ i v i. (3.2) where ζ i is the friction coefficient and v i is the velocity of the i th particle. Frandom,i is the random force that describes Brownian motion. From Newton s second law, F net,i = m i a i, (3.3) where m i is the mass of the i th particle and a i its acceleration, we find, m i a i = F conservative,i + F frictional,i + F random,i. (3.4) 23

34 As the inertial term is very small [1], we ignore the left-hand-side of equation (3.4) and, substituting F frictional,i from equation (3.2), we obtain ζ i v i = F conservative,i + F random,i (3.5) v i = v D i + v B i, (3.6) where v D i (t) is the deterministic velocity given by, v D i (t) = 1 ζ i Fconservative,i (t), (3.7) and v B i (t) is the random velocity v B i (t) = 1 ζ i Frandom,i. (3.8) The random force has an average value of zero, F random,i = 0. Its correlations are related to the friction coefficient F random,iα (t)f random,iβ (t ) = γ i δ(t t ), (3.9) where γ i = 2k B T ζ i and δ is the Dirac delta function. In thermal equilibrium, the theorem of equipartition of energy gives, K.E. = 3 2 Nk BT, (3.10) where K.E. is the average kinetic energy of N particles in three dimensions. In a Brownian dynamics simulation with time step δt, the random velocity is calculated from, v α B (t) = 2T δtζ n rα, (3.11) 24

35 where T is the reduced temperature and n r is a random number in Gaussian distribution, n rα = 0, n rα n rβ = δ αβ. The friction coefficient of a spherical particle ζ is given by, ζ = 6πηa, (3.12) where η is the shear viscosity and a is the radius of the particle. Since the random velocity determines the temperature, equations (3.11) and (3.12) help us to find a useful time step for Brownian dynamics simulations, δt = 2m 6πηa, (3.13) where m is the mass of the particle [15]. In our simulations, we use m = 1, a = σ/2 and δt = with η = to satisfy equation (3.13). Equations (3.7) and (3.11) with (3.12) are valid when all particles have the same mass. To simulate the cargo, which has larger mass, we set ζ c = ( mc ) ( ) σ m σ c 3πησ c, (3.14) where m c and σ c are the mass and diameter of the cargo, while m and σ are the mass and diameter of all the other particles. During the simulation, we collect data on quantities such as the kinetic energy, the potential energy and distances between particles. To obtain error estimates, we calculate the average of a quantity ξ i in a block of simulation steps and calculate the average according to equation ξ = 1 N b 25 N b i=1 ξ i, (3.15)

36 where N b is the number of blocks. The standard deviation of the block values is an estimate for the statistical uncertainty and calculated from σ(ξ) = 1 N b ξ i ξ N b 1 2. (3.16) i=1 3.2 Behavior of an isolated protein. To learn to control its conformations, we studied the behavior of the isolated protein for different values of the potential parameters K θ and V 0. Figure 3.1 shows protein conformations for relevant K θ and V 0 values. The figures illustrate that the distance Figure 3.1: Effect of K θ and V 0 on conformations of an isolated protein. between the heads decreases when the value of K θ is changed from 0 to 20. For different V 0 values the dihedral planes have different orientations relative to each 26

37 other. When V 0 = 100, the planes are parallel to each other, for V 0 = 100, on the other hand, the planes are perpendicular in orientation. 3.3 Conformation of the protein in contact with the substrate. We investigated the effect of the model parameters on the protein and chain system to identify the intermediate states for a protein taking a step. Figure 3.2 shows conformations of the chain and walker for the three identified states. The parameters Figure 3.2: Illustration of the mechano-chemical states (1, 2, and 3) of a protein walker on a substrate. The potential parameters of the states are presented in Table

38 of each state described in figure 3.2 are listed in table 3.1. State 1 is the initial state of the protein walk, where both heads are attached to the substrate. The dihedral planes are parallel in orientation and the bond-angle potential is zero. State 2 is the second state in a step; the Morse potential for one head is turned off, the dihedral planes are at right angles, and K θ is zero to help the protein head move round. In state 3 both heads are attached to the substrate and the bond angle interaction is turned on to achieve a small distance between head sites. We used these states to construct a 4-step model for the walk, where state 4 is the same as state 1. During the walk, when head one completes one step, the fourth state becomes the initial state for head 2 for the next step and so on. For head two to take a step the values of D e1, D e2, D e3 and D e4 in state 2 becomes 4, 4, 0 and 0, respectively. Table 3.1: Parameters for the equilibrium states Parameters D e1 D e2 D e3 D e4 K θ V 0 State State State State In our simulations the probabilities S 1, S 2, S 3 and S 4 to be in the states evolve in time as shown in figure Since S 2 and S 3 never come close to a value of one, we increase the range of the potential parameters when we calculate their 28

39 time-dependent values with equation (2.17). In table 3.2, we express the potential Table 3.2: Assigned parameters for the change of states to maintain the desired amplitudes. Parameters D e1 D e2 D e3 D e4 K θ V 0 State 1 State 2 State 3 State 4 D e,max D e,max D e,max D e,max K θ,min V pos D e,min D e,min D e,max D e,max K θ,min V neg D e,max D e,max D e,max D e,max K θ,max V pos D e,max D e,max D e,max D e,max K θ,min V pos parameters in terms of minimum and maximum values that, when used in equation (2.17), reproduce the values in Table 3.1. For the state evolution shown in figure 2.11, we use the values D e,max = 4ɛ, D e,min = -0.95ɛ, K θ,max = 53.5, K θ,min = 0, V pos = 100ɛ and V neg = 148ɛ. 3.4 Kinetic energy The instantaneous kinetic energy of the system is calculated from K.E. = 1 2 N m i v 2 i, (3.17) i=1 where N is the total number of particles and m i and v i are their masses and speeds, respectively. The instantaneous kinetic energy fluctuates rapidly due to the random forces. To examine the kinetic energy over larger time scales, we average its values 29

40 over segments of fixed length. In figure 3.3 we present graphs of the instantaneous and the segment-averaged kinetic energy for two simulations. In both cases, the system consists of the chain, the protein, and the cargo with a total number of N = 26 particles. For the reduced temperature k B T = 1.0 used in the simulations, we expect an average kinetic energy of K.E. = 39 according to the principle of equipartition of energy, Eq (3.10). This is exactly what we observe. Panel A of figure 3.3 shows results for fixed interaction parameters, whereas panel B shows results for changing parameters. During the time period in the figure, head one was detached from the chain substrate. This period was chosen to see if any variation in the segment-averaged kinetic energy occurs due to the change in the parameters. This is not the case, which shows that the system remains in thermal equilibrium. We expected this result because, in our simulations, the kinetic transition rates are very slow compared to the time scale of thermal fluctuations. 30

41 Figure 3.3: Instantaneous (red) and segment-averaged (blue) kinetic energy as a function of time for simulations of the protein-vesicle-chain system, which has a total of N = 26 particles. Panel A and B show results for fixed and time-varying interaction parameters, respectively. The length of one time segment in the averaging process is steps, corresponding to a simulation time of 500. In both simulations, the average kinetic energy has a value of 39, in agreement with the equipartition theorem and a reduced temperature of one. 31

42 CHAPTER IV RESULTS 4.1 Force-extension relation of the substrate The extension of the substrate chain is the distance between the first and last chain bead along the direction of the applied force. Since we apply the external force in the x-direction the extension X is given by X = (X Nc X 1 ), (4.1) where N c is the number of chain beads, N c = 20, and X Nc and X 1 are the x-coordinates of the last and first bead, respectively. In figure 4.1 we present results of the extension as a function of the pulling force. Due to the strong attractive interactions between the chain beads, the chain is found to be in a collapsed state in the absence of external tension. With the increase in the pulling force the extension of the substrate increases. At a particular pulling force, there is a rapid change in the extension as the chain is stretched from its globule state. Then the extension increases more slowly with the force. In our simulations with the protein walker we apply a tension force of 40 so that the chain is highly stretched. 32

43 Figure 4.1: Force extension results for an isolated chain. The symbols represent simulation results for the extension, defined in equation (4.1), as a function of the pulling force for a chain of length N c = 20. The error bars indicate the standard deviation of block averages (see equation (3.16)). Two chain conformations for intermediate (left) and high (right) extensions are also shown. 33

44 4.2 Motor protein taking steps on the substrate without cargo. We have performed simulations for a motor protein without cargo for steps corresponding to a simulation time of We also added an equilibration time of 10 7 steps so that the protein comes to thermal equilibrium before the chemical variables start to change. To investigate if the change in parameter values has the desired effect, we calculate the distance d 1 between site 1 on head one and the closest bead on the chain and the corresponding distance d 3 for site 3 on head two. In figure 4.2 we present simulation results for d 1 and d 3 in panel A together with the Morse potential amplitudes D e1 and D e3 in panel B. At time the value of the Morse potential amplitude D e1 starts decreasing to zero. Almost immediately, the distance d 1 between head one and the chain increases. It reaches a maximum value of about 1.8 before it decreases again to about one when D e1 goes back to its original, attractive value. The distance d 3 is nearly constant during this step. C 1 and C 3 represent the closest chain beads for head sites 1 and 3 respectively. During the time when head site 1 is released the bead index C 1 increases, showing that head one moves forward along the chain. At the same time, head two slides backward along the chain. When head one is attached again, both C 1 and C 3 are slowly sliding backward. At time the Morse amplitude D e3 starts decreasing and head two is released and reaches a maximum distance of about 1.6. The bead index C 3 at that time increases which suggests that the protein is taking a step forward. But at the same time the increase of the bead index C 1 suggest that the protein glides along the 34

45 chain. Since the protein moves along the chain even when both heads are attached, we decided to adjust the interaction parameters. 35

46 Figure 4.2: Two steps of a motor protein walker on a chain substrate. Panel A shows the smallest distance between the chain and head site 1 on head one (d 1, red) and the chain and head site 3 on head two (d 3, pink) together with the indices of the beads closest to the head sites (C 1, blue and C 3, cyan). Panel B shows the value of the Morse potential parameters D e1 and D e3 as a function of time. 36

47 4.3 Effect of changing the interaction of the attached head of a motor protein walking without cargo. In figure 4.2 we saw that both heads are moving when one head is released. keep the non-released head fixed, we change the Morse amplitudes D e3 and D e4 To to 2 D e,max when head one is released and we change D e1 and D e2 to 2 D e,max when head two is released. Figure 4.3 B shows the Morse amplitudes as a function of time while results of a simulation with these parameters are shown in figure 4.3 A. The results for C 1 and C 3 in panel A show that immediately after the attachment of head one and head two, the protein glides backward. 37

48 Figure 4.3: Two steps of a motor protein on a substrate with increased attraction for the attached head site. Panel A shows the smallest distance between the chain and head site 1 on head one (d 1, red) and the chain and head site 3 on head two (d 3, pink) together with bead indices closest to head site 1 (C 1, blue) and head site 3 (C 3, cyan). Panel B shows the value of the Morse potential parameters D e1 and D e3 as a function of time. 38

49 4.4 Motion of protein with cargo. We also studied a walking motor with attached cargo. In figure 4.4 we show results for the distances to the chain and nearest bead indices for head sites and cargo. The Morse amplitudes for this case are the same as in panel B of figure 4.2. The results in figure 4.4 show that when head one reattaches the protein stays in place for a while and then slides forward. After the attachment of head two, the protein starts moving backward along the chain even though both heads are attached to the chain. Figure 4.4: Two steps of a motor protein walking with cargo on a substrate. The graph shows the smallest distance between the chain and head site 1 (d 1, red), head site 3 (d 3, pink) and the cargo (ves, orange). The change in Morse parameters is as shown in panel B of graph 4.2. The closest beads are C 1 (blue), C 3 (cyan) and C ves (yellow) for head sites 1, 3, and the cargo, respectively. In these simulations, the Morse parameters of the non-released head remained constant, as shown in panel B of figure

50 In figure 4.5 we present simulation results for the case that the non-released head is more strongly attached as shown in figure 4.3 B. To find out how reproducible the results are, we performed simulations with identical parameters and initial configurations as in figure 4.5 but a different seed of the random number generator. The results shown in figure 4.6 have qualitatively the same behavior as those in figure 4.5. Figure 4.5: Two steps of a motor protein walking with cargo. The lines represent the same quantities as in figure 4.4; the Morse amplitudes for this simulation are shown in figure 4.3 B. 40

51 Figure 4.6: Results for the same system as in figure 4.5 from a separate simulation. 41

52 4.5 Five steps for a protein with cargo. Considering the results presented in figures , we concluded that two issues prevented the protein from walking effectively. For one, the protein sometimes moves the wrong foot, for example when the heads switched positions during the wait period. For two, the protein tends to slide while waiting for the next step to start. To address the first issue, we consider the position of the heads before a step is taken: the head farthest from the target chain end is determined and that head is released in the next step. To address the second issue, the wait time before the next step is reduced. To illustrate the protocol, we show the Morse amplitudes as a function of time for a five step walk in figure 4.7 B. The results for closest chain beads in panel A show that the motor protein is moving with its cargo in the forward direction. For the first three steps the movement of the motor protein is as intended. After, the third step, the protein is close to the end of the chain and the position of the head sites and the cargo fluctuate about a final position. 42

53 Figure 4.7: 5-step simulation with the new protocol. The lines represent the same quantities as in figure

54 CHAPTER V CONCLUSIONS In this work we have developed a five-site model for a motor protein in contact with a bead-spring substrate and performed Brownian dynamics simulations of the model. We have investigated the effect of model parameters on protein and chain conformations and used this information to determine suitable interaction parameters for our model. To describe a motor taking steps on the substrate, we have identified equilibrium states that serve as chemical states for the walking protein and evolve in time according to coupled rate equations. Since motor proteins are responsible for carrying cargo from one part of the cell to another, we have added a cargo structure in the form of a large sphere that is bonded by a harmonic spring to the central site of the motor protein. To investigate mechanical properties of the substrate and validate our code, we performed a series of simulations for an isolated chain under tension. A force extension curve for the isolated chain shows the expected behavior for stretching a chain from a globule to an elongated state. We have investigated the motor protein taking steps and found that we were able to control the attachment of the protein to the substrate. To investigate the effect of cargo on the protein s walk we have performed simulations with cargo and found that a large sphere is dragged very 44

55 slowly. Since thermal fluctuations play a large role in the system, we tested the reproducibility of our results by repeating simulations with identical conditions but different seeds for the random number generator. We found that the results produced agree qualitatively. Since the protein has a tendency to step back and forth we developed a protocol that allows the walker to keep going in the same direction on average. Our results show that the model and simulations developed in this work can be used as a starting point for a more detailed investigation of cargo-carrying motor protein walkers. 45

56 BIBLIOGRAPHY [1] Debashish Chowdhury. Stochastic mechano-chemical kinetics of molecular motors: A multidisciplinary enterprise from a physicist s perspective. Physics Reports, 529:1 197, [2] Debashish Chowdhury. Resource letter PBM-1: Physics of biomolecular machines. American Journal of Physics, 77: , [3] Philip Nelson. Biological Physics: Energy, Information, Life. W. H. Freeman, New York, [4] Anthony Brown. Axonal transport. In Donald W. Pfaff, editor, Neuroscience in the 21st Century From Basic to Clinical, chapter 10, pages Springer, New York, [5] H. Lodish, A. Berk, S. L. Zipursky, P. Matsudaira, D. Baltimore, and J. Darnell. Molecular Cell Biology. W. H. Freeman, New York, 4th edition, [6] R. Mallik and S. P. Gross. Molecular motors: Strategies to get along. Current Biology, 14:R971 R982, [7] D. Keller and C. Bustamante. The mechanochemistry of molecular motors. Biophys. J., 78: , [8] I. Derényi and T. Vicsek. The kinesin walk: A dynamic model with elastically coupled heads. Proc. Natl. Acad. Sci. USA, 93: , [9] C. Bustamante, D. Keller, and G. Oster. The physics of molecular motors. Acc. Chem. Res, 34: , [10] A. B. Kolomeisky and H. Phillips III. Dynamic properties of motor proteins with two subunits. J. Phys.: Condens. Matter, 17:S3887 S3899, [11] G. Lan and X. Sun. Dynamics of myosin-driven skeletal muscle contraction: I. Steady-state force generation. Biophys. J., 88: ,