c 2006 by Prasanth Sankar. All rights reserved.

Transcription

1 c 2006 by Prasanth Sankar. All rights reserved.

2 PHENOMENOLOGICAL MODELS OF MOTOR PROTEINS BY PRASANTH SANKAR M. S., University of Illinois at Urbana-Champaign, 2000 DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate College of the University of Illinois at Urbana-Champaign, 2006 Urbana, Illinois

3 Abstract Motor proteins produce directional movement and mechanical output in cells by converting chemical energy into mechanical work. Innovative applications of physics experimental techniques to the study of motor proteins have generated a wealth of data. By careful analysis and modeling of this data, we try to gain a quantitative understanding of this biological system. We have proposed a way to incorporate the biochemical data on motor proteins into the modeling and thus minimize the number of fitting parameters and increase the relevance of the models. Mechanical understanding of motor proteins are obtained by using an experimental technique of observing the motion of a large cargo elastically attached to it. On modeling the effect of the elastic motor-cargo link on the motion of the cargo, we see that the conclusions on motor mechanism derived from such techniques are dependent on the size of the cargo as well as the stiffness of the link. By proposing a new modeling scheme, we devise a unified description of such experiments for various motor proteins. This description is based on the picture that the motor thermal fluctuations are rectified in a preferred direction by means of barriers whose heights are modulated by the chemical reactions taking place at the motor. iii

4 Acknowledgments I would like to thank my advisor Yoshitsugu Oono for his constant encouragement, guidance, and support. I would also like to thank Satwik Rajaram for carefully reading the manuscript and suggesting improvements, and Bojan Tunguz for being a nice office mate all these years. This work was supported in part by teaching assistantship from University of Illinois, and summer research assistance from University of Illinois Campus Research Board. iv

5 Table of Contents Chapter 1 Introduction Motivation Motor Proteins ATP: The Energy Source Types of Motor Proteins Experimental Studies of Motor Proteins Biochemical Studies Structural Studies Single Molecule Studies Theoretical approaches Our Point of View Road Map Chapter 2 Phenomenology of Motor Protein Kinesin Introduction Theoretical Approaches General Considerations on Phenomenological Modeling Empirical Facts Proposed Kinetic Scheme Kinetic Steps and States Modeling Details Explanation of the Mechanochemical Data Force-velocity relation Velocity-ATP Concentration Relation Force-Randomness Relation Randomness-ATP Concentration Relation Force-Run Length Relation Run Length-ATP Concentration Relation Velocity and Run Length Variation with ADP Discussion Uniqueness of Kinetic Scheme Motor Energetics Is There a Power Stroke in Kinesin Force Production? How Unique is the model? v

6 Chapter 3 Effects of the Elastic Motor-Cargo Link on Motor Transport Introduction Motor-Cargo System Variation of Motor Velocity with Link Stiffness Stokes Efficiency of Motor-Cargo System Discussion Chapter 4 Interpretation of Single Molecule Experiments of Motor Proteins Introduction Motor-Cargo System An Effective Cargo Only Representation of a Coupled Motor-Cargo System Mathematical Formulation of Effective Cargo Only Representation Explanation of Mechanochemical Data F 1 -ATPase Kinesin Myosin V Discussion Chapter 5 Affinity Switch Model for Rotary Motor F 1 -ATPase Introduction Model Algorithmic Description of the Model Comparison With Analytically Solvable Model Comparison of Model Predictions with Empirical Results Discussion Chapter 6 Summary and Open Questions Appendix A Solution of the Kinetic Model Appendix B Stochastic Energetics References Author s Biography vi

7 Chapter 1 Introduction 1.1 Motivation Ever since the application of X-ray crystallography to the determination of the structure of DNA, incorporation of physical techniques into biological research have produced a steady stream of advances in the field of cellular and molecular biology. This laid the foundation for one of the most exciting insights into the machinery of life, the structure-function relation which postulated that the function of nucleic acids and proteins are coded in their three dimensional structure. Attempts to determine the static structure of proteins (the protein folding problem) and understanding the structural contribution to function dominated the attention of physicists initially contributing to biological understanding. At the same time, physical techniques and theoretical tools which were developed to solve complex problems in a quantitative way were also incorporated into the available tools of biological research. This produced a new realm of quantitative understanding which was a break from the qualitative descriptions which characterized biological research. Development of these new experimental techniques and their application to the study of function of proteins have generated a wealth of quantitative data. With this, a new consensus has emerged that to have a complete understanding of a biological phenomenon, theoretical and experimental collaboration between biologists and physicists is needed. Accumulated experimental data allow proposals of qualitative hypotheses. These proposals are substantiated by using quantitative models based on the understanding of underlying physics and quantitative agreement of such models with available data. Such models are useful in 1

8 extracting further information from experimental data, making predictions regarding the unknowns and thus directing further experimental studies. While proposing such models, we must pay due attention to certain peculiarities of biological systems, nature of required understanding, and the way physical approaches should be modified while approaching a biological problem. Understanding of a biological system is complete only if its function is explained in terms of underlying physics and the interaction of the system with its environment. Even the simplest constituents of biological systems such as individual protein molecules are made up of many atoms. This makes application of direct reductionist methods of physics which seeks to explain the a phenomena in terms of known interactions of underlying atomic constituents less successful. It is difficult to bridge the timescale of atomic motions occurring in picoseconds or less to the biologically relevant dynamic processes of the system as a whole which occur in microseconds or slower. Another traditional physics approach of explaining relevant phenomena in terms of coarse-grained statistical descriptions also needs modification when applied to biological systems. Since the function is coded in relevant substructures of the system, coarse-graining should retain these structural domains which may be unique to individual systems. In addition, biological systems are in nonequilibrium. Though the statistical descriptions of equilibrium systems are well developed, the nonequilibrium extensions are not. Motor proteins such as myosins and kinesins which are used for directional transport in cells constitute one such biological system in which development and application of new experimental techniques have generated a wealth of information at various time and length scales [1]. At a fundamental physical level, these are energy transducers which convert available chemical energy in cells to mechanical energy which is used for directional motion. The mesoscopic size of the system forces them to do this energy conversion while being influenced by interactions with the thermal environment. The physically relevant quantities are the speed at which these motors move, the mechanical force generated, the role of structural elements in the energy conversion, the efficiency of chemical to mechanical energy 2

9 conversion, etc. An important question is how the system uses or minimizes the effects of thermal fluctuations in various processes of molecular machines. It is also interesting to know whether the function of different motors can be explained by a unified mechanism at a coarse-grained level. In this thesis, by incorporating relevant experimental information on the chemical processes, biochemical and structural information and the dynamics of relevant functional subdomains, we propose models which attempt to explain the working of motor proteins in a quantitative way. While proposing these models, attempts are made to minimize the number of fitting parameters by utilizing available biochemical and structural data on motors. 1.2 Motor Proteins Proteins are a class of macromolecules. Individually or in association with other macromolecules they perform a variety of biological functions needed to sustain life. Many proteins act as enzymes catalyzing chemical reactions. Others have structural and mechanical roles in the cellular machinery. Proteins are made of amino acids. Depending on the sequence of amino acids constituting them they can have unique three dimensional structures in cellular environment. Of the many required cellular functions, some involve directional movement and transport of macromolecules and supramolecular complexes. Without specialized motors, efficient transport of large complexes over eucaryotic cells is impossible. Motor proteins are such molecular motors which produce directional movement in cells and mechanical output by converting chemical energy into mechanical work [1] ATP: The Energy Source Motor proteins produce work by converting the free energy liberated by the hydrolysis of adenosine triphosphate (ATP) to mechanical energy. ATP is an organic compound com- 3

10 posed of adenine, the sugar ribose, and three phosphate groups. The breaking of the bond connecting the last phosphate ion to the rest of the molecule by the following hydrolysis reaction creates an adenosine diphosphate (ADP) and inorganic phosphate Pi. ATP + H 2 O ADP + Pi, (1.1) The breaking of the phosphate bond has a high negative free energy of reaction. The total free energy released depends on the standard free energy, G 0, and the concentrations of ATP, ADP and Pi. It is given as G = G 0 + k B T log [ADP][Pi], (1.2) [ATP] where G 0 = J/mol is the standard free energy change at ph 7, and [ATP], [ADP] and [Pi] are the concentrations of ATP, ADP and Pi, respectively, in molarity, k B is the Boltzmann constant and T is the absolute temperature. Since the typical length scales involved in motor proteins are nanometers (nm), and the forces generated are of the order of piconewtons (pn), it is convenient to use these units instead of SI units. In these units, the standard free energy change G 0 = 54 pnnm, k B T = 4.1 pnnm at room temperature. Under physiological conditions, [ATP] 10 3 M, [ADP] 10 4 M, and [Pi] 10 3 M, G 90 pnnm. G is negative under all practical conditions, implying that the reaction always proceed in the direction of hydrolysis, and that spontaneous net synthesis of ATP never happens in solution. However in the absence of any enzymes the reaction rate to the right for the above reaction is very low and it is this stability that makes the phosphate bond such an ideal high-energy source. 4

11 1.2.2 Types of Motor Proteins Diversity of cellular functions are realized through the evolution of many different classes of motor proteins. Latest estimate suggests there are 100 different motor proteins in eucaryotic cells performing various tasks. Different motor proteins can be classified according to their specific functions [2]. Kinesins are a family of motor proteins involved in cargo transport along microtubule tracks [3]. Microtubules are formed by the polymerization of αβ tubulin heterodimers. They have a hollow cylinder shape with diameter of 24 nm with tubulin dimers oriented parallel to the cylinder axis. In addition, they have a polarity which allows fixing the direction of motor motion with respect to the track as moving towards the plus end or the minus end of microtubule. Of the many members of kinesin family, some of them such as kinesin-1 form dimers and move processively (motor takes many steps before detaching from the track) toward the plus end of microtubule. Ncd is another kinesin which dimerizes and moves toward the minus end of microtubule nonprocessively. There are monomers such as Unc104, KIF1A which are also processive motors. In addition to transporting cargo such as vesicles, they are also involved in cell division and microtubule assembly. Defective functioning of kinesins are implicated in neurological disorders, cancer and various other diseases. Myosins are a family of motor proteins involved in muscle contraction as well as cargo transport along actin filaments [4]. An actin filament is a left-handed helix of actin monomers with a periodicity of 36 nm. Similar to microtubule they also have a polarity. The thick filaments in a muscle are composed of myosin II, and by sliding with the thin filaments of actin, they produce the muscle contraction force. Myosin V dimerizes and moves processively toward the plus end of actin whereas myosin VI dimers move towards minus end. Both are involved in cellular transport. Dyneins are another class of motor proteins [5]. One type of dynein powers cilia and 5

12 flagella (structures with a core made of microtubules which oscillate at high frequency by virtue of force generated by dynein molecules arrayed along microtubule). Another form of dynein, cytoplasmic dynein, is involved in the transport of organelles along microtubules. Polymerases such as DNA and RNA polymerases, move along the strands of DNA in order to replicate them and to transcribe them into RNA, respectively [6]. In addition to the linear motors above there is a class of rotary motors known as ATPases which are involved in transport of ions across membranes. A prominent member of rotary motors is F 1 -ATPase [7] which is involved in the synthesis of ATP from ADP and Pi using the proton flow across the membrane as the energy source. Similarly, flagellar motors involved in the directed motion of bacteria also use ion gradient across membrane as the energy source. Another class of motors are involved in packaging of viral DNA into protein capsids [8]. In this thesis we will be considering kinesin, myosin V, and F 1 -ATPase. 1.3 Experimental Studies of Motor Proteins Experimental characterization of motor proteins can be categorized into the general categories of biochemical, structural, mechanical and optical studies Biochemical Studies Biochemical studies assume that, while converting chemical energy of ATP to mechanical work, motors cycle through a series of long lived states with different positions along the track. After completion of a cycle, the motor advances by one step. These states depend on the state of the nucleotide (ATP or its hydrolysis intermediates) bound to it and whether the motor is attached to the track. The state transitions are represented by chemical transitions of the form, X 0 k 1 k i X 1 X i 1 X i X n, (1.3) k 1 k i 6

13 where X i is an intermediate state of the motor and k i are the rates at which one state transforms to another. In addition to identifying these long lived states, biochemical studies also seek to determine the rate constants of state transition. A summary of application of biochemical methods to the study of motor proteins is given in [9] Structural Studies Successful crystallization of protein molecules allows the determination of three dimensional structures using X-ray crystallography. Structural characterization of nucleotide and track binding regions and the regions connecting these two have contributed greatly to the understanding of motor function. The goal of such studies is to identify the possible structural transformations that the motor undergoes while hydrolyzing ATP, and generating force and movement. For this, it is sometimes possible to crystallize the intermediates with different nucleotide bound states and determine their structures [10]. Difficulties with crystallization mean there is not enough structural data to give a complete picture of motor conformational changes. Sometimes low resolution cryo-em images [11] are used to fill the gaps in structural knowledge. Kinesin: Figure 1.1 shows the crystal structure of kinesin dimer with both heads bound to ADP [12]. The structure comprises two motor heads connected through a coiled-coil domain. The head is attached to this coiled coil through a structurally important region called the neck linker (β9 and β10 in the figure). Structures of kinesin bound to microtubule are not yet determined and the docking of known structure to low resolution cryo-em images are used to infer the structure of microtubule bound states and possible structural changes associated with microtubule binding. Myosin: Structural studies of myosin are more advanced than kinesin since more hydrolysis intermediate structures available. Crystal structures of myosin II with no nucleotide or bound to ADP or ADP-vanadate have been reported. Structures of myosin bound to actin have not yet been reported. Here also docking of known structures to low resolution cryo- 7

14 Figure 1.1: Crystal structure of kinesin dimer [12]. EM images are used to infer the possible changes associated with actin binding. Fig. 1.2 shows the known crystal structures of myosin II and the structural changes associated with nucleotide changes [13]. One of the most important insights gained from crystal structures of myosin is that there is a lever like region (grey helix in Fig. 1.2) which rotates in response to the changes in nucleotide states. F 1 -ATPase: Another motor whose crystal structure determination has provided further insights into its force generation is the F 1 -ATPase motor. As shown in Fig. 1.3, the motor structure consists of α 3 β 3 γδǫ subunits [14]. The central γ subunit can rotate inside a cylinder made of three α and three β subunits arranged alternately. ATP binding sites are located primarily on a β subunit. Crystal structures show that, depending on the state of ATP present at the pocket, β subunit can be in either a closed state or open state. In addition to knowing the structures of individual motors, comparative studies of struc- 8

15 Figure 1.2: Known crystal structures of myosin II and a proposed schematic representation of possible structural changes during the ATP hydrolysis [13]. tures of different motors are also useful. It is seen that the core of myosin and kinesin structures have similar structure. This structural similarity suggests nucleotide binding, hydrolysis, and release of hydrolysis products may trigger similar motions in motor proteins. Thus it is possible that both motors have similar mechanisms. Such structural similarities are encouraging. It is possible that understanding the function of one motor will help in identifying the functional mechanism of other structurally similar motors Single Molecule Studies The advent of in vitro motility assays [15] allowed the study of motion produced under biochemically defined conditions using one or a few proteins. Such single molecule studies of motor proteins can be classified into two general categories. Mechanical studies monitors the motion of the motor against an external force applied to it. Optical studies with fluorescence 9

16 Figure 1.3: Schematic representation of F 0 F 1 -ATPase structure. Crystal structure is known only for F 1 part (the top part of the figure) [14]. tags use fluorescent probes attached to the motor to determine distances involved in the motion of the motor as a whole or structural changes within the motor. Mechanical studies: Motor proteins typically produce piconewton forces. To observe the motion of motors against a load, such piconewton forces can be applied to it using an optical trap (a focused laser beam) or by attaching the motor to a very fine glass needle. In a typical optical trap experiment shown in Fig. 1.4, the motor is attached to a large bead which is trapped in the laser beam. The motion of the bead is tracked at high resolution (to have accurate detection of bead position, beads which are many times motor size are used in experiments) [16]. To move the bead away from the trap center, motor has to exert a force on it to overcome the force exerted on the bead by the trap. Optical trap studies allow the determination of the variation of motor velocity with external load acting on it, the maximum work that the motor can produce, and the step size of motor displacements. 10

17 Another variation of this set up uses a large cargo attached to the motor and the motion of this cargo is observed. In this case, the motor work is measured as the work done against the viscous drag produced by the cargo. Application of optical trapping experiments on kinesins have shown that they move on microtubule by taking 8 nm steps, and generates a maximum force of 8 pn [16]. For myosin V, the step size is determined as 36 nm and the maximum force produced is 2 pn [17]. Figure 1.4: Optical trap setup used to study the load dependence of kinesin motion [16] Optical studies with fluorescence tags: Fluorescence resonance energy transfer (FRET) studies [18] are useful in characterizing the structural changes involved in motor operation. Here a donor and an absorber fluorophore are attached at two different locations on the proteins. If there is a structural change which modifies the distance between these fluorophores, the changes in emission spectra of the donor can be used to determine the distance change involved. Recent advances have allowed tracking of a single fluorophore attached to the 11

18 motor with very high resolution. For example, fluorescence imaging at nanometer accuracy (FIONA) [19] is able to track the location of a fluorophore to nanometer resolution. Application of FRET studies to kinesin shows that the neck linker undergoes a conformational change. Application of FIONA to kinesin and myosin V has shown that the dimers move in a hand-over-hand fashion in which the two dimers alternate their relative position. 1.4 Theoretical approaches The goal of any theoretical approach is to understand the mechanism by which motors convert chemical energy from ATP binding, hydrolysis and product release into mechanical work. The pioneering work was done by A. F. Huxley [20] to explain muscle contraction. A cross-bridge model involving relative sliding of actin and myosin filaments was proposed as causing muscle contraction. In the model, myosin molecules in the filament moves back and forth about an equilibrium position as a result of thermal fluctuations. On reaching near a binding site on actin, the strained myosin binds to it. The relaxation of the strain causes the sliding of the actin filament. On completion of actin sliding, myosin detaches from it and the cycle repeats. The determination of crystal structures and the identification of possible structural changes occurring during ATP hydrolysis has given rise to newer approaches to modeling motor operation. At the microscopic level, molecular dynamics simulations seek to explain motor operation in terms of the atomic structures and forces involved. Atomic resolution structures are used for detailed calculations of molecular energetics and dynamics. The ultimate aim is to understand how the conformational changes that engender the motion are produced by ATP and its hydrolysis. The presently available computing power is not yet sufficient to bridge the time scale of atomic motions to functionally relevant timescales. However, there is a simulation method which seeks to accelerate microscopic dynamics by the application of external forces [21]. This approach is ideally suited to the study of chem- 12

19 ical reaction paths. It has clarified the details of the energetics of ATP hydrolysis at the nucleotide pockets of known motor crystal structures [22]. Still, it is difficult to study the actual conformational changes during the motion cycle of motors. The time scale limitations of full molecular dynamics simulations have given rise to intermediate mesoscopic models which use some sort of abstraction of structural details as the starting point. In a class of models known as power stroke models, the protein structure is abstracted as a machine made up of a spring-like or elastic element to produce force, a lever to amplify the force, and a latch to regulate nucleotide binding or release. Fig. 1.5 shows such a representation of the operation of kinesin motors [23]. Figure 1.5: Kinesin motor as a molecular machine [23]. These power stroke models postulate the storage of energy of ATP hydrolysis in the spring. The relaxation of this stored energy through the motion of the lever is postulated as the mechanism driving the motor forward. β closing models in F 1 -ATPase [24], neck linker docking models of kinesin [25], and lever arm models of myosin [10] are examples of this modeling approach. Another class of models identify a few long-lived states in the mechanochemical cycle 13

20 of the motor and assume stochastic transitions with mean kinetic rates between them [26]. These models assume the whole motor as a particle that can go through a certain number of states with different positions along the track. After completion of the cycle the motor advances by one step. All the stochastic transitions between states are assumed to be reversible. In presence of an external load acting on the motor, the kinetic rates are modified as dependent exponentially on the load. The load dependence and the rate constants are determined by curve fitting to the experimentally determined mechanochemical data on the motor such as its load dependent velocity. 1.5 Our Point of View As summarized above, the presently available models capable of explaining the mechanochemical data, the power stroke models, imagine motors as miniature versions of macroscopic engines working by means of levers, springs, etc. In this picture, ATP binding, hydrolysis, or product release induces conformational changes in the protein that under load create strain. The strain drives movement of any load attached to the motor, and this movement which is referred as working stroke of the motor, relieves strain. For such a model, the free energy difference between the pre- and post-work stroke states must be comparable to motor work output. Though this is possible, there are no clear experimental confirmations of this assertion. For the putative work stroke of kinesin, the conformational change of neck linker, experiments show that the associated free energy change of nearly 8 pnnm [27] is much less than the motor output which is greater than 60 pnnm. For myosin, the work stroke is associated with lever arm rotation, but as of now there are no clear experimental observations which show that there is a significant free energy change associated with this. Similar is the case for the assigned work stroke of F 1 -ATPase, the closing of the β structure. On the other hand, it is well recognized that physics of small particles in solution are strongly affected by viscous drag and thermal noise which dominate the inertial forces which 14

21 determine system behavior of macroscopic systems. In ratchet systems (for a review, see [28]), the thermal fluctuations are rectified in a preferred direction by maintaining the system at nonequilibrium provided there is an asymmetry built in the system. For track binding motors, it is assumed that the interaction between motor and track produces an asymmetric potential. Application of such simple ratchets to the quantitative explanation of mechanochemical data of motor proteins have been less successful till now. Velocity and energy conversion efficiency obtainable from simple ratchet models are much lower than that of motor proteins. Nor are they able to explain the crucial role of the structural changes in motor force generation. In this thesis we will show that models without any explicit motor driving can explain the mechanochemical data on motor proteins quantitatively with a suitable chemically tightbound ratchet model. The ingredients needed in the models are, (i) incorporation of biochemical experimental data on motors, (ii) role of an elastic link that connects the motor to load, (iii) reinterpretation of the role of conformational changes as modulating the attachment and detachment of motor to the track in a preferred direction. 1.6 Road Map In Chapter 2, a phenomenological scheme to incorporate the experimental data on motors into modeling is proposed. The resulting model is used to explain the mechanochemical data on kinesin motor. It is found that mechanochemical data alone cannot be used to determine the unique motor force production mechanism. Indirect experimental observation of using a large cargo elastically attached to the motor is identified as the possible explanation for the inadequacy of mechanochemical data to identify the motor mechanism. In Chapter 3 the role of the elasticity of the link connecting motor to cargo is studied using simple motor models. It is shown that motor velocity and energetics are dependent on the link flexibility. Chapter 4 proposes a unified scheme incorporating motor biochemical data and motor-cargo link. 15

22 The resulting scheme is applied to explaining the mechanochemical data on motor proteins kinesin, myosin V, and F 1 -ATPase. In Chapter 5 incorporating the effects of motor-cargo link and its flexibility, a simple model (a null model) is proposed for the F 1 -ATPase motor which can explain the available mechanochemical data. Chapter 6 summarizes the results of this thesis. 16

23 Chapter 2 Phenomenology of Motor Protein Kinesin 2.1 Introduction As introduced in the previous chapter, active transport of molecules and molecular complexes are needed inside cells because of their sheer size. Motor proteins serve as the engines for this intracellular transport. Kinesins are one such family of motor proteins that move uni-directionally along the microtubule while hydrolyzing ATP. An interdisciplinary effort involving different experimental fields and theoretical approaches are needed to have an understanding of the most interesting aspect of molecular motors: its mechanism of converting chemical energy gained from the fuel molecules such as ATP into mechanical energy. Experimental characterization of these motors can be divided into two general categories. One set of experiments mainly seek qualitative characterization of the motors. Structural studies involving crystallization, identifying structural changes during motor motion, biochemical characterization of intermediate motor states and their correlation with ATP hydrolysis processes and qualitative descriptions of the way motor moves during its operation fall in this category. The other set of experiments (mechanochemical experiments) measures the motor output, its velocity, force production, and the efficiency of energy conversion Theoretical Approaches A detailed understanding will involve describing the motor operation at a microscopic level in terms of atomic motions. Molecular dynamics (MD) simulations involving motor structure, 17

24 interaction between motor and ATP and its hydrolysis intermediate, chemical details of ATP hydrolysis process, and motor track interaction will give such a detailed description. Such an approach has not yet been very successful since the time scale of molecular motor motions is still much longer than the time scale routinely available by the current MD. At the other end of description are the abstract models. Here, structural and biochemical details of motor operation are neglected and motors are modeled as point particles. In ratchet models [28], rectification of thermal fluctuations of motors by asymmetric potentials produced by motor-track interaction is identified as responsible for motor motion and force generation. In another class of models [26, 29, 30, 31, 32], a set of abstract chemical states localized along different track positions are introduced. The motor motion is identified with stochastic transitions between these states. An example is the modeling by Fisher and Kolomeisky [26], which tries to explain the mechanochemical data on kinesin. This presents an abstract kinetic scheme with four intermediate states with exponentially load dependent rate constants. Although the scheme is able to fit the data with the aid of more than fifteen fitting parameters, it is difficult to translate such a scheme into biochemically relevant statements concerning the internal mechanism of the motor operation. Besides, it is hard to incorporate structural or other biological information into the scheme General Considerations on Phenomenological Modeling As described above, theoretical contributions to the understanding of motor proteins, other than detailed molecular dynamics simulations, try to propose abstract models. Ability to fit the output from the quantitative experiments are used as a measure of success of these models. There are two chief obstacles in making a phenomenological description. One is that the mechanochemical data on motors alone do not adequately constrain the space of possible models as explicitly noted by Duke and Leibler [33]. The other difficulty is the number of parameters: if there were many fitting parameters (say, more than 10), even if we could fit the kinetic scheme to empirical results, it would be difficult to check whether the obtained 18

25 set of parameters is the best choice for a given kinetic scheme (let alone the verification of the scheme itself). Therefore, we wish to minimize the number of adjustable parameters as much as possible before trying to explain the empirical mechanochemical data quantitatively. To this end, based on the intermediate states and substeps of motor stepping inferred from structural and other experimental data, a possible kinetic scheme can be constructed first (without referring to the availability of rate constants). Then the biochemical experimental results can be used to determine the phenomenological rate constants appearing in the scheme. The descriptive power of such a scheme can be tested by trying to explain the mechanochemical data quantitatively. In essence, if successful, this will produce a kinetic scheme which may be regarded as a phenomenological summary (or a minimal model, so to speak) of biochemical and mechanochemical empirical results. Therefore, it is desirable to construct good mesoscopic models of molecular motors that can facilitate understanding of single molecule experiments (e.g., [34, 35, 36]) at the time scale of microseconds or longer. We feel the first step in this direction is the incorporation of qualitative information on motors into modeling. The general approach can be summarized as follows. Identify the relevant qualitative information on motors. Propose a model which incorporates the identified information, and the additional assumptions regarding the motor force production, and check the agreement of model results with quantitative measurements on motors. The methodology of phenomenological modeling of kinesin is developed in this chapter. While formulating the phenomenological model it is assumed that all the empirical data are really reliable. As we will see later, some structure-related empirical results are not very reliable. This makes phenomenological models less effective in giving a clear understanding of motor mechanism. 19

26 2.2 Empirical Facts As already discussed in the introduction, our strategy is to make a kinetic description maximally utilizing currently available biochemical and structural information, and then to try to reproduce the mechanochemical experimental results quantitatively with minimal fitting parameters. Therefore, we first outline the experimental results relevant to the kinesin dimer procession along the microtubule (Mt). (i) Observation of kinesin motion using optical traps while it is transporting a cargo shows that a kinesin dimer takes rapid 8 nm forward steps [35] spaced with (often long) waiting periods. The periodicity of microtubule track is also 8 nm. There can be occasional backward steps whose probability increases with increasing external load on the cargo [36, 37]. For low loads each 8 nm step takes place with consumption of a single ATP molecule [38, 39]. The kinesin dimer processivity (i.e., the capability to take many consecutive steps on the track before detaching from it) is due to the coordination of two monomer heads [40, 41]. Direct observation of stepping of each head by fluorescent tags shows that kinesin moves by a hand-over-hand mechanism in which the two heads switches their relative position on the track [42]. There are strong evidences indicating that the two consecutive steps are left-right asymmetric (or without left-right symmetry) [43]. (i) is the basic observation and our kinetic scheme is based on: the coordination of two heads. (ii) The 8 nm step can be resolved into fast and slow substeps, each corresponding to a (cargo) displacement of 4 nm. This was first demonstrated by Higuchi et al. [44] and has been fully confirmed by Nishiyama et al. [36]: a. The duration of the faster substeps is about 50 µs and is insensitive to the force (for 3-8 pn). b. The second substep duration is variable ( µs for higher loads). It is known that each head has four major states depending on the state of nucleotide 20

27 ATP or its hydrolysis intermediates that are attached to the kinesin head: E: without any nucleotide, D: with an ADP, T: with an ATP, and DP: with a hydrolyzed ATP (before releasing the phosphate ion Pi). We can approximately describe the state of a doubleheaded kinesin as (A, B), where state A is the state of the trailing head (closer to the minus end of Mt) and state B that of the advanced head. Also let us denote by the underline the binding to Mt. Thus, e.g., (DP, D) implies that the trailing head is with a hydrolyzed ATP and is attached to Mt, and the advanced head is with an ADP but not attached to Mt. We use X for a not specified state (this does not mean that any state is allowed). (iii) The reactivity of kinesin head with nucleotides may be summarized as follows: a. if a kinesin head is free from Mt, releasing ATP or accepting ATP is not easy. This exchange becomes much easier when the head is attached to Mt [40]. b. ATP ADP + Pi (without releasing the phosphate ion) becomes difficult when the head is attached to Mt; only when kinesin is a dimer is the release of phosphate ion possible [40]. c. Exchange of ADP is not strongly affected by the presence or absence of Mt. Under the presence of ADP, (E, E) is not observed [45]. d. The T and E states have nearly the same binding strength to Mt [46, 45]. D is significantly weaker than E or T [45]. DP is intermediate between T and D [46]. (iv) Transition from (X, E) to (X, T) triggers the force production. a. ATP binds to the motor with a second order rate constant 2±0.8µM 1 s 1 [47, 48] and this is followed by the 4 nm (cargo) displacement, which is completed within 50 µs [36]. b. It is strongly suggested [41] that X here is actually DP. (As yet there is no direct experimental demonstration, but many researchers assume the existence of such a state as (DP, T) [25]. (v) Detachment of one of the heads occurs in (DP, X). a. Hancock and Howard [41] have inferred that for a kinesin monomer the detachment occurs in the DP state. b. Hackney [40] suggests that phosphate ion dissociation precedes ADP dissociation. The 21

28 measured phosphate ion release rate from the monomeric kinesin is very small compared with the case of dimers [41, 48]. Therefore, the phosphate release process must be assisted by the other head. c. Study of the kinetics of a mutant defective in ATP hydrolysis [49] suggests that the hydrolysis of ATP in the rear head is required for the strong attachment of the front head to Mt. d. The step (forward or backward) is tightly coupled to the consumption of one ATP and the waiting state for both seems the same. The stalling state is interpreted to be where the probabilities of forward and backward steps are equal [36]. e. The measurement by Nishiyama et al. [36] and Carter and Cross [37] of the ratio of forward to rear steps as a function of load shows that the number of backward steps increase as a function of load. (vi) a. The ATP binding leads to two sequential isomerizations, the second of which reorients the neck linker relative to the Mt surface [50]. b. The neck linker docks, pointing in the forward direction, to the kinesin catalytic core in the T or DP state, but is not docked in the E or D state; this transition has an enthalpy change of H 200 pnnm [51]. It is also known that the ATP hydrolysis is not a prerequisite of neck linker docking [51]. This fact combined with a suggests that neck linker docking precedes ATP hydrolysis [50]. c. For a related protein KIF1A the neck linker docks in the T state but not in the D state [52]. d. Under the condition that both heads are attached to Mt, if the neck linker of the trailing head is docked, it is sterically impossible for the front head to have a docked neck linker [25, 53]. e. Tomishige and Vale [54, 55] have shown that when both heads are bound to Mt, the two neck linkers (each with length 4 nm) must be in opposite-directing conformations, pointing backward in the front head and forward in the rear. When the length of the neck linker is 22

29 decreased by two residues, processivity is lost. Inserting a 4 nm polyproline segment between the neck linker and the coiled coil allows the motor to move processively with 16 nm steps [54]. All these imply that (DP, T) assumed above must have transient substates (denoted with 1 and 2) with respect to the neck linker conformation. (vii) Transition from (X, D) to (X, E) is the transition from the singly to doubly bound conformation; there is a consensus about this process [41, 47, 56]. a. The Kawaguchi-Ishiwata experiments [57] have shown that with a head in the T state and the other in the D state the motor is singly attached to Mt. 2.3 Proposed Kinetic Scheme Kinetic Steps and States Based on the summary in the preceding section, we propose the following kinetic scheme for the kinesin dimer processivity (Fig. 2.1): (DP, E) k 1[ATP] k 1 (DP, T) k 2 k 2 [P] (D, T) k3 k 3 (T/DP, D)1 k 4 k 4 (T/DP, D)2 (DP, E). k 5 [ADP] k 5 Here, [ATP] denotes the ATP concentration, [ADP] the ADP concentration, and [P] the phosphate ion concentration. Detailed justification will follow the outline of our scenario. Our scenario may be outlined as follows. Supporting empirical facts summarized in the previous section are specifically mentioned with key steps. (1) After ATP goes into the front empty head attached to Mt, state (DP, T) is formed (cf. (iva)). However, the neck linker does not dock to the motor core of the front head with ATP immediately (cf. (vid)). 23

30 Figure 2.1: Kinetic scheme for kinesin dimer processivity. The nucleotide state of each head is denoted by, E: nucleotide free; T: with ATP; DP: hydrolyzed ATP, D: with ADP. Docked and undocked portions of neck linker are denoted by thick and thin lines respectively. (2) Next, release of a phosphate ion from the rear head (cf. (va,b)) and its detachment from microtubule occur in a concerted way, and (D, T) is reached. The cooperativity needed for accelerated phosphate release in the dimer (vb) comes from the attempts of the neck linker of the front head in T state to dock. (3) This allows the completion of docking of the neck linker of the front head to the core (cf. (vid)) and the rear head moves a distance of 8 nm forward and is poised to move forward further (perhaps slightly ahead of the previous front head that is still attached to Mt; the cryo-electron microscopy studies of Hoenger et al. [58] suggests such a state). This is our interpretation of the power stroke proposed by Vale et al. (cf. (vi)), and is correspondent to the fast substep (iia). The still Mt-bound head (the previous front head) is in the T or DP state with the docked neck linker (cf. (vib)) (likely to be in the DP state according to (iv)). 24

31 This is the state (T/DP, D)1. (4) From this state the unattached head in the D state (the previous rear head) diffuses forward a distance of another 8 nm to state (T/DP, D)2, which is just before reattaching of the new front head to the new position on Mt. Due to the steric constraint, the neck (or load) diffusion is required for the diffusing head to reach the new position on Mt. The still attached now-rear head is in the T or DP state (with the docked neck linker; very likely to be in DP [41, 49]). This corresponds to the second substep (iib). Strictly speaking, this is our interpretation, but is the simplest interpretation of the facts summarized in (ii). (5) The release of ADP and reattachment to Mt occurs to produce (DP, E), which corresponds to the same state as the starting state but having advanced 8 nm. This is the waiting state for ATP. (6) The backward steps at higher loads is taken into account by assuming that once ATP goes in (step (1)) the attempts of the front head linker to dock produce an internal tension and either of the heads can detach from microtubule (vd). There is a higher probability for the rear head to detach (ve). A possible kinetic scheme for backward steps is given in Discussion. Once the rate constants are known, with the aid of the standard procedure summarized in [59] the velocity, randomness, and their [ATP] and [ADP] as well as force dependences can be determined. The detailed explanation and justification of the states and processes follow. (DP, E) Supporting facts for this state to be the waiting state for a step have already been summarized in (iv) and (va) in the preceding section. The transition from this state to the next state (DP, T) involves the binding of ATP to the front head with a second order rate constant k 1. (DP, T) In this state the neck linker of the rear head is being docked but that of the front head in the 25

32 T state has not yet been docked (cf. (vid)). The binding of ATP is assumed to be contingent to a displacement of some part(s) of the front head. This displacement may correspond to the one observed in KIF1A due to Kikkawa et al. [52]. Another observation that may have relevance to this displacement is that the kinesin head has an intrinsic bias towards forward displacement independent of the neck linker [25]. If the monomer velocity v measured by Inoue et al. [60] is interpreted as due to a biasing displacement of δ associated with ATP binding followed by diffusion, then we have v = kδ, where k is the ATP turnover rate. Their data are approximately compatible with δ =1 nm. Observations by Higuchi et al. [44] have demonstrated that there is a waiting step after ATP binds the front head and before the motor starts to produce force. An interpretation is that the initial binding of ATP is a kind of collision complex, and it requires some displacements of the parts of the head to come into the (DP, T) state which is the state that allows the subsequent phosphate ion release from the rear head and its detachment from microtubule (the change to (D, T) ). This state is likely to be identified with the starting point of the neck linker docking (see Discussion). From this state (D, T) is formed with a rate constant k 2, and a reverse transition to the initial state (DP, E) that releases the bound ATP [41, 47, 56] with a rate constant k 1. Another process that can happen especially under high load is the detachment of the front head in T state and kinesin taking a backward step (vd, e). (D, T) This state is obtained by the release of a phosphate ion from the rear DP state (cf. (vb)), and detachment of the rear head from microtubule into the D state assisted by the front head (cf. (va,b)). The driving force for this change is interpreted to come from the initial stage of neck linker docking to the front head core. Further neck linker docking happening at the front head causes a forward transition to (T/DP, D)1 state with a rate constant k 3. Reattachment to microtubule and phosphate ion 26