Stochastic motion of molecular motor dynein

Transcription

1 University of Ljubaljana Faculty of Mathematics and Physics Department of Physics Seminar I b, etrti letnik, stari program Stochastic motion of molecular motor dynein Author: Miha Juras Mentor: asist. dr. Andreja arlah Ljubljana, November 2013 Abstract In rst part I present molecular motor dynein and from its structure and ATP hydrolysis role its six dominant states. In second part I explore stochastic motion of general system within its discrete inner states as Markov process together with its memoryless property and stationary distribution. I apply Markov process to six dominant dynein states and furthermore how dynein dimer could be modeled. In nal part I compare calculation and simulation results on simplest dynein dimer model against dynein velocity measurements.

2 Contents 1 Introduction 2 2 Molecular motor dynein Dynein structure Chemomechanical conformational states Dynein dimer Stochastic process Markov process and its properties Three state example Stationary distribution Dynein cycle as Markov process Dimer coupling Calculation and simulation vs. measurements Velocity calculation Velocity from simulation Conclusion 12 1 Introduction All cells from unicellular organisms to cells of organ in human body need energy to survive. This principle is "living" prove of thermodynamic principles where cell is a system actively sustaining entropy within itself on the account of its surrounding environment. Basic principle can be described as: cell is consuming substance with high inner energy-food, extracts inner energy from it, uses energy to sustain order within it self and nally releases food byproduct and excess of energy as heat into environment. Eukaryotic cell 1 converts food into energy within mitochondria. Energy is stored in energy storing molecule ATP (adenosine triphosphate) which acts as cell unit of energy and contains approximately J 25 kt where T is room temperature at 25 C = 298 K [1]. ATP is afterwards used in may cellular functions. This include [2] synthesis of DNA, RNA and proteins, macromolecules transport across cell membrane, maintaining cell structure by facilitating assembly and disassembly of elements of cell skeleton (cytoskeleton) and nally in inner cell transport such as muscle movement and active transport within cellular plasma. Within this seminar we will explore one specic cellular function which is inner cell transport facilitated by molecular motor dynein. As any human made motor also molecular motor have more inner states in one thermodynamic cycle. By molecular motors we speak of chemical, mechanical and conformational states. Transitions between these states can be modeled by stochastic (random) mechanism [3]. Stochastic model and resulting motion will be the main topic. 2 Molecular motor dynein Cell proteins catalyzing (helping with) chemical reactions are called enzymes. Some of them are catalyzing hydrolysis reaction (chemical reaction where chemical bonds are cleavage by addition of water) by which ATP's energy is converted into mechanical work. Therefore these enzymes are named molecular motors. In human body there are three main groups of molecular motors. Myosin is responsible for muscle contraction, kinesin is transporting cellular cargo from center of the cell to the cellular membrane and dynein is transporting cellular cargo from the membrane towards center of the cell and is also responsible for cilia an agella movement. 1 Eukaryotic cells have inner membrane which encapsulates its organelles such as nucleus and mitochondria. 2

3 2.1 Dynein structure Dynein is the biggest molecular motor. Its mass is 10 times bigger than the mass of its relative kinesin. It is composed of three main parts (Figure 1): Head (AAA ring) is the fuel (ATP) burning part, linker is the worker part (doing mechanical work) and leg part (strut, buttress and MTBD) is responsible to hold dynein in place while pulling the cargo attached to its tail [4]. Figure 1: Illustration of dynein structure [5]. Head is composed of six similar structures (six numbered circles) where only four are able to accept ATP and catalyze hydrolysis. Due to the ring like shape the structure has a name AAA ring where AAA stand for ATPases Associated with diverse cellular Activities. AAA ring has a function of a fuel cell where fuel is burned. Purple liker is a working part of a dynein which changes positions by which pulls the white tail on which dynein cargo is attached. Linker position and its power stroke is dependent on the state in the cycle of hydrolysis cycle in AAA ring. Yellow stalk and orange buttress are structures facilitating communication between AAA ring and gray microtubule binding domain (MTBD) which binds dynein to the microtubule - a cellular highway network. 2.2 Chemomechanical conformational states All the three main dynein functional parts linker, ring and leg are correctly synchronized in one dynein cycle to enable dynein to do mechanical work. Due to the large dynein's size it is not obvious how spatially separated structures synchronize their own actions, inner states, in relation to each other. Some necessary empirical conditions must be fullled for dynein cycle to work. Firstly, linker should not pull the cargo (generate powerstroke) when its leg is not bound to microtubule as dynein needs something to hold on while pulling. Secondly linker should not move to pre-powerstroke position before leg is still bound because linker movement to pre-powerstroke moves dynein's leg forward therefore should be unbound from microtubule. Conditions above are not strictly obeyed by dynein in living cell but they are conditions in the model [6] to describe dynein states which are a combination of inner states for each dynein part. Linker can be in relaxed position - post-powerstroke position where it is not able to do any mechanical work. Second linker position is in strained position - pre-powerstroke position where it has stored elastomechanical energy and is waiting to "re". Leg can be bound to microtubule or unbound from microtubule. Finlay for the ring we distinguish four states: ATP binds to hydrolysis site on AAA ring, ATP is hydrolyzed into ADP and P i, P i is released and nally ADP is released which is based on the ATP hydrolysis chemical reaction ATP ADP + P i, where ATP stands for adenosine triphosphate, ADP for adenosine diphosphate and P i for hydrated inorganic phosphate. Chemical reaction as such can run to both sides as ATP hydrolysis from ATP to 3

4 ADP or as ATP synthesis from ADP to ATP. Synthesis is normally taking place in mitochondria but we will keep this possibility open. Even more we will extend reverse principle to whole dynein cycle for the means explained in next sections. Based on the empirical necessary conditions described in the beginning of this section and from numerous observations [6] all the inner states of each dynein part can be combined into dynein states. Bellow are listed the most dominant dynein states (Table 1) which can also be represented as illustration (Figure 2). Table 1: All the inner states of each dynein part generate 16 possible combinations ( 2 leg states 2 linker states 4 ring states = 16 dynein states) of dynein states, 6 of them being dominant in successful dynein processive movement [6]. Dynein state label is a combination of dynein part states labels. Leg has two states labeled as MT - bound to microtubule - and - (empty label) as released from microtubule. Linker has two states labeled as D - post-stroke position - D* - pre-stroke position. Ring has four states labeled as - (empty label) empty hydrolysis site, ATP hydrolysis site occupied by ATP, ADP.P i hydrolysis site occupied by ADP and P i and ADP hydrolysis site occupied by ADP only. Dynein state Leg state Linker state Ring state mechanical state label conformation state label chemical state label MT.D bound to MT MT post-stroke D empty hydrolysis site MT.D.ATP bound to MT MT post-stroke D ATP ATP D.ATP released from MT post-stroke D ATP ATP D*.ADP.P i released from MT pre-stroke D* ADP + P i ADP.P i MT.D*.ADP.P i bound to MT MT pre-stroke D* ADP + P i ADP.P i MT.D.ADP bound to MT MT post-stroke D ADP ADP Figure 2: Illustration of 6 dynein states as dened in Table 1 (adopted and adjusted from [7]). Arrows' orientation show cycle orientation where dynein makes forward cycle with ATP hydrolyzing to ADP. Cycle can also be reversed where dynein makes backward cycle with ATP synthesis from ADP and P i for which the probability is almost zero (Section 4.1). 2.3 Dynein dimer One dynein molecule (monomer) can undergo chemomechanical and conformational cycle but it is not enough to achieve processive movement. Due to thermal movement of dynein surrounding it is very unlikely that once unbound from microtubule dynein will bind back on microtubule very soon which would enable linker to release stored elastomechanical energy by pulling the cargo. With two dynein molecules connected with their tails as dimer, processivity is much grater as in case of one dynein [8]. While one dynein is bound to microtubule the other can be unbounded from microtubule and can therefore undergo transitions through its states, stretches in front, binds back to microtubule and afterwards pulls cargo through cell plasma. As important as communication within dynein to achieve synchronization of dynein's dierent parts during hydrolysis is also communication between two dyneins in dimer. One dynein should wait until 4

5 the other is progressing and then they can exchange roles. Here the intriguing question arises: do they change turns regularly walking either in hand-over-hand fashion where dyneins step one pass another or as inchworm mechanism where the leading dynein always moves a step forward before the trailing dynein can follow. In case of yeast dynein dimer where each of dynein heads have been marked with quantum dots measurements show that dynein can walk both modes (Figure 3). Figure 3: Stepping trace of the yeast dynein which is one of the slowest walking dyneins ( 100 nm/s). From the trace above we can determine velocity of 30 nm/s. Right head has been marked with red quantum dot (QD-655) and left with blue quantum dot (QD-585). Traces show no clear distinction of hand-over-hand or inchworm walking modes but a combination of both [9]. 3 Stochastic process As illustrated on example of dynein one can wonder which tool to use to describe such dynamics. We will examine Markov process and try to apply it to dynein. 3.1 Markov process and its properties This section is an extract of [10]. Let us imagine arbitrary system which has several states. In time system evolves through these states. Changing a state is transition between a pair of states. Pair of states is dened with state from and state to which transition is made. Let us dene transition rate also known as kinetic rate k ji which corresponds to the transition to state j from state i where an Einstein notation is used j i. Additionally we can imagine large number of identical systems with the same number of states and transition rates between them. Let us suppose that systems are independent from each other what assures that the state any system is in is independent of the states the other systems are in. Through this we can dene occupation probability p i of a state i p i = N number of systems in state i N number of all systems. Each state occupation probability p i evolves through time. All other states j contribute to increment of occupation probability for state i and vice versa state i contributes to the occupation of all other states j. This relation is known as master equation dp i dt = j (k ij p j k ji p i ). 5

6 If transition rate is independent of time and dependent only on the involved state pair and not from previous history of states system was then we can model these transitions with Markov process. In probability theory this is identied as continuous-time Markov process where k ij is conditional probability for system in state i to migrate to state j and is dened as k ji = P (X n+1 = x j X n = x i ). The crucial property of Markov process known as Markov property is memorylessness. Its meaning is that future behavior of the system depends only on the current state of the system and not on the states the system has been in before. We can formulate this as probability to reach state x n+1 at time t n+1 from state x n at time t n that has been in state x n 1 at time t n 1... is same as probability to reach state x n+1 at time t n+1 from state x n at time t n without any knowledge of its history P ( X tn+1 = x n+1 X tn = x n,..., X t1 = x 1, X t0 = x 0 ) = P ( Xtn+1 = x n+1 X tn = x n ). Furthermore the probability for transition in a period t is dependent only on the length of the period and not on when the period started P (X t=t0+ t = x n+1 X t=t0 = x n ) = P (X t= t = x n+1 X t=0 = x n ), which we can rewrite in explicit time dependence as P (t 0 t t 0 + t) = P (0 t t). The only continuous distribution function that has this continuous memoryless property is exponential distribution [11] with rate k w(t) = ke kt, therefore transition from state i to state j with rate k ji is distributed exponentially as Three state example w ji (t) = k ji e kjit. To get better understanding we can examine three state system. Master equations for three state Markov process where p i are occupation probabilities for each of three states are dp 1 dt dp 2 dt dp 3 dt which can also be written in matrix form as = (k 21 + k 31 )p 1 + k 12 p 2 + k 13 p 3, = k 21 p 1 (k 12 + k 32 )p 2 + k 23 p 3, = k 31 p 1 + k 32 p 2 (k 13 + k 23 )p 3, dp dt = pq. Q is transition rate matrix dened as k 21 k 31 k 21 k 31 Q = k 12 k 12 k 32 k 32. k 13 k 23 k 13 k 23 Transition matrix can be represented as directed graph (Figure 4) where each row represents one state, non diagonal element k ji rate of transition from state i to state j and nally diagonal element Q ii is the negative sum of all rates from state i. Diagonal element Q ii also relates to system dwell time in state i as τ i = 1/Q ii. 6

7 p 1 p 1 p 1 p 2 k 12 k 13 k 21 k 31 k 12 k 13 k 12 k 41 k 23 k 32 k 13 k 23 k 23 p 3 p 2 p 3 p 2 p 4 k 34 p 3 k 32 k32 k 43 Figure 4: Left: three state Markov process represented as directed graph which forms a bidirectional cycle where rates k 21, k 32 and k 13 form a forward directed cycle and rates k 12, k 23 and k 31 a backward directed cycle (naming of forward and backward cycle are used in consistency with naming in Figure 2). In general Markov processes are not two direction cycles. Middle: rates k 21 and k 31 have been eliminated (set to 0) meaning no transitions from state 1 to state 2 and from state 1 to stare 3 which in long time limit leads system to ends up in state 1, which makes this state an absorbing state. Right: one example of four state Markov process where also diagonal transitions are possible and rates k 21, k 31, k 42, k 14 and k 24 were eliminated. 3.2 Stationary distribution Any physical system is evolving through time to reach its stationary state within its self and with its surroundings. In n state system with some initial occupation probabilities p t=0 this is resulted as convergence of occupation probability to stationary occupation probabilities p t=. In Markov process this can be applied as following p n+1 = p n + dp n = p n + p n Qdt = p n (I + Qdt) = p 0 (I + Qdt) n+1, where p 0 is initial distribution at time t = 0, p n is current distribution at time t = n dt and p n+1 is distribution at time t = (n + 1)dt. (I + Qdt) is a generator of discrete-time Markov process which is progressing observed system by discrete time interval dt. After long enough time 2 system reaches stationary distribution p which satises relation p = p (I + Qdt), which means that time progression for dt does not change stationary distribution p. follows to From this it p Q = 0, i.e., p is left eigenvector of Q with eigenvalue Dynein cycle as Markov process In Markov process transition to next state depends only on the state in which the system currently is and is independent of the states in which system was. In our daily experience we ordinary do not see this kind of behavior. If a ball is in front of a goal and its previous location (one second ago) was a few centimeters before the goal we know with certainty that in next second the ball will hit the goal. Momentum of the ball is the key player our certainty is betting on. In microscopic environment of a cell where every molecule is constantly bombarded by water molecules (thermal collisions) and molecules/proteins quickly forget their history. In water at 20 C a sphere or radius of 3 nm and mass of 100 ku is hit by water molecule (water mass is 18 u) every 2.8 ps. Comparing this to dynein transition rates sphere is hit several million times during every transition 2 Quantication of relaxing time is not scope of this seminar, but reader can nd numerous sources in Markov process literature [12]. 7

8 which completely randomizes sphere momentum. This example leads us to conclusion that Markov property of memorylessness is obeyed by dynein. We have noted in section (2.2) that we will extend reverse principle to whole dynein cycle. This is no issue for Markov process as it encloses transition from state i to state j and also its reverse transition from state j to state i. Having this in mind we can construct dynein cycle (Table 2) with 6 states dened in Table 1. Table 2: Dynein cycle with transition rates between dynein states [3]. For dynein to move forward (minus end direction of MT), transitions between states should be downward. Transition rates k +ATP, k PS and k +ADP depend also on of molar concentration of substance used in transition. The physiological concentrations of ATP and its byproducts are [ATP] = 10 3 M, [P i ] = 10 3 M, [ADP] = 10 5 M. Index i for rates will be used in section 4.1 for calculating average velocity. i State Transition rate Transition Parameter value. Previous cycle MT.D k 1 k ATP k +ATP ATP binding M 1 s 1 +ATP k ATP ATP release 50 s 1 MT.D.ATP k 2 k +MT k MT MT release, D 500 s 1 MT k +MT MT binding, D 100 s 1 D.ATP k 3 k RS k +RS ATP hydrolysis, linker swing to pre-stroke 1000 s 1 +RS k RS ATP synthesis, linker swing to post-stroke 10 s 1 D*.ADP.P i 4 k MT k k +MT MT binding, D* s 1 +MT k MT MT release, D* 10 s 1 MT.D*.ADP.P i 5 k PS k +PS k +PS Power stroke, P i release 2500 s 1 k PS Reverse stroke, P i binding 10 4 M 1 s 1 MT.D.ADP 6 k +ADP k ADP k ADP ADP release 200 s 1 k +ADP ADP binding M 1 s 1 MT.D. Next cycle We have shown that in systems with more than 3 states also diagonal transitions are possible. Although model of dynein we are examining has 6 states we are limiting it to have only bidirectional cycle as possible transitions which for example excludes diagonal transitions from state D.ATP directly to MT.D.ADP. 3.4 Dimer coupling In previous section (3.3) we have listed transition rates of one dynein motor having 6 unique states with 12 transition rates which constitutes 6 state bidirectional cycle. With coupling the two dynein motors we get 6 6 = 36 possible dynein dimer states. The simplest model of dynein dimer is that when one dynein in undergoing ATP cycle the other is bound to MT and waits for its turn. When rst dynein nishes ATP cycle then the second starts with its own ATP cycle. This gating mechanism diminishes dynein dimer states to 12 states which actually are two single dynein cycles, rst for the left and second for the right dynein. When simulating such model it is enough to model only one dynein cycle with exchanging roles between the two dyneins in dimer. More complicated models are when two dyneins undergo their own cycles in semi coordinated way which could result in fully dissociation from MT from which average run length can be modeled and compared to measurements [3, 13]. 8

9 Within the simplest model dynein dimer can walk in hand-over-hand and inchworm modes. In the rst hand-over-hand walking mode (Figure 5 left) trailing dynein steps forward to take the lead. We will assume there are no dierences in transition rates in relation to single dynein cycle. The only change is when trailing dynein overtakes the leading one and ends its own cycle in leading position with transition from MT.D.ADP to state MT.D then the other dynein starts its own cycle from state MT.D. This way hydrolysis cycles regularly alternate from one dynein to another. One step size in this mode is 16 nm; trailing dynein is bound 8 nm behind leading dynein and after forward step is bound 8 nm in front. Because one dynein if xed and the other moves for 16 nm dynein dimer center moves for 8 nm which is the same distance cargo is pulled for. Figure 5: Left: hand-over-hand walking mode where dyneins are exchanging the lead. Right: inchworm walking mode where right (blue) dynein is leading and left (yellow) is trailing dynein. Both illustrations show only forward movement [7]. In the second inchworm walking mode (Figure 5 right) we could set right dynein as leading dynein. First the leading dynein steps forward increasing the lead. Then trailing dynein steps forward decreasing the lag. One step size in this mode is 8 nm therefore in one step dynein dimer center and cargo are moved for 4 nm. The main dierence of the two modes are in the resulting distance between the two dyneins. In rst hand-over-hand walking mode the distance between dyneins is not changed which is additional simplicity of hand-over-hand walking mode where on the other hand in inchworm walking mode distance is or increased or decreased. By having this in mind and that two dyneins are connected with dimerisated tails the two dynein cycles dier from each other which suggests that we should incorporate this into kinetic rates. In higher walking modes we might combine hand-over-hand and inchworm walking mode with introducing choices for example in forward movement we would choose between leading dynein makes an inchworm movement to increase dyneins' distance or trailing dynein makes hand-over-hand movement to take the lead. To keep the simplicity of the topic we will only explore the simplest model as discussed in the beginning as hand-over-hand walking mode where knowledge of single dynein cycle is sucient [13]. 4 Calculation and simulation vs. measurements We will examine dynein velocity for two ADP concentrations for which most measurements are done. High (standard or saturating) concentrations will be [ATP] H = 1 mm which is also the living cell concentration and the second low concentration will be hundred times lower as [ADP] L = 10 µm. The transition rate matrix Q and stationary distribution p thus are Q H = Q L = , p H, =, p L, = :MT.D :MT.D.ATP :D.ATP :D*.ADP.P i :MT.D*.ADP.P i :MT.D.ADP :MT.D :MT.D.ATP :D.ATP :D*.ADP.P i :MT.D*.ADP.P i :MT.D.ADP 9

10 where each value of Q has unit s 1. Blue printed diagonal elements of Q are the smallest in absolute value which means that this state has the lowest rate to transfer to other states which results in the longest dwell time (τ i = 1/Q ii ) and in the highest occupancy in stationary distribution p for that state (also printed blue). Green printed upper diagonal rate k +ATP is the one that depends on ATP concentration. In case of high ATP concentration MT.D.ADP is the most occupied state (56%) with dwell time of 4.4 ms but in case of low ATP concentration MT.D is the most occupied state (72%) with dwell time 14.3 ms. Strong dependence of on ATP concentrations also results in velocity of dynein dimer as it will be described in following sections. 4.1 Velocity calculation In case of bidirectional circular transitions where transition matrix has tridiagonal form then average velocity can be calculated [14] as v = d(k eff + k eff ), where d is step length, k eff and k eff + are eective backward and forward rates dened as k eff + = 1/R N, / N k eff k j = R N, k +j j=1 with R N = N j=1 r j, r j = 1 N k +j j+k k i k k=1 i=j+1 +i If we re-label rates as k +1 = k +MT.D, k 1 = k MT.D,... and note periodicity as k ±i = k ±(i+n) with N = 6 and d = 8 nm then calculated average velocity for the two ATP concentrations are Ratios of forward against backward cycle are ( k eff ) + k eff <v > H nm/s <v> L nm/s. H ( k eff ) + k eff 10 9, L which conrms that forward cycle (ATP hydrolysis as k+ eff ) is far more probable than backward cycle (ATP synthesis as k eff ). 4.2 Velocity from simulation We can conrm calculated velocity value with simulation in Mathematica with help of Continuous- MarkovProcess and RandomFunction functions. ContinuousMarkovProcess function accepts two arguments: rst argument is starting probability distribution p(t = 0) which I set as stationary distribution p, second argument is transition matrix Q. RandomFunction also accepts two arguments: rst is the result of function ContinuousMarkovProcess and the second argument is time interval that system should repeat the process given as the rst argument. Result of RandomFunction is a list of pairs (t j, i j ) there t j is time and i j is dynein state after j th transition. Because dynein cycle is bidirectional, with forward cycle (ATP hydrolysis) highly dominant, we can count cycles dynein makes. We increase the count when dynein makes transition from state MT.D*.ADP.P i to MT.D.ADP or decrease the count in case of reverse transition. Selected transition is when dynein generates power stroke which moves its cargo forward for 8 nm which is the same distance as dynein dimer center moves. Simple multiplication of cycle count n cycles and single step dynein center movement d 1 = 8 nm results in total distance l dynein dimer moved from start of counting l = n cycles d 1. 10

11 After m transitions t m time has passed and dynein dimer had average velocity of <v >= n cycles d 1 /t m. With this method we implicitly allow dynein dimer to forward direction of stepping only. This does not prohibit dynein to transit in backward direction of the transitions between the states, it only prohibits backward direction of stepping. I have run 1000 simulations. In case of high ATP concentration (Figure 6 middle left) dynein was walking for 10 seconds and achieved < v > H = 862 ± 18 nm/s which is comparable to measured value < v m > H = 800 nm/s and in case of low ATP concentration (Figure 6 middle right) dynein was walking for 50 seconds and achieved < v m > L = 95 ± 3 nm/s is also comparable to the measured value < v > L = 130 nm/s [15]. Both simulated values completely agree with the calculated in previous section which was also expected. l nm 400 vh 912 vl t s dρ dv s nm v 862,Σv dρ dv s nm v 95,Σv v nm s v nm s dρ dts s 1 dρ dts s k 183± 4 s 1 20 k 18.5± 0.3 s ts s ts s Figure 6: Top: a stepping trace of one simulation run displaying dynein dimer center movement within 0.5 s for both ATP concentrations. Middle left and right: velocity distribution of 1000 simulations with mean velocity v H = 862 nm/s and v L = 95 nm/s. Bottom left and right: One cycle time distribution which has form of convolution of may exponential distributions with long time limit decreasing like the lowest forward rate in case of high ATP concentration as k ADP = 200 s 1 and k +ATP = 20 s 1 for low ATP concentration. 11

12 5 Conclusion System for which we are able to dene inner states and rates of transitions between them with additional constraints that these rates are time independent and are not dependent on the history in which system was can be modeled with Markov process. In our study of a single dynein chemomechanical and conformational cycle and a simple dynein dimer model we were able to simulate and calculate its velocity which are comparable to measurements for two dierent ATP concentrations. Formalism described opens new possibilities to add new dependencies to dynein stochastic movement one of them to be velocity to force dependence or possibility to develop more complex dynein dimer model to describe also inchworm walking mode or even combination of hand-over-hand and inchworm modes. References [1] J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associates, Sunderland (2001). [2] (November 2013). [3] A. arlah and A. Vilfan, The winch model can explain both coordinated and uncoordinated stepping of cytoplasmic dynein, submitted. [4] M. Juras, Molekularni motor dinein, Seminar Ia (October 2013), [5] H. Schmidt et al., Insights into dynein motor domain function from a 3.3-Å crystal structure, Nat. Struct. Mol. Biol. 19, 492 (2012). [6] K. Imamula et al., The coordination of cyclic microtubule association/dissociation and tail swing of cytoplasmic dynein, Proc. Natl. Acad. Sci. 104, (2007). [7] A. P. Carter et al., Crystal clear insights into how the dynein motor moves, J. Cell Sci. 126, 705 (2013). [8] S. Reck-Peterson et al., Single-Molecule Analysis of Dynein Processivity and Stepping Behavior Cell, 126, 335 (2006). [9] M. A. DeWitt et al., Cytoplasmic Dynein Moves Through Uncoordinated Stepping of the AAA+ Ring Domains Science 335, 221 (2012). [10] (November 2013). [11] (November 2013). [12] D. Gillespie, Markov Processes: An Introduction for Physical Scientists, Academic Press, San Diego, California (1992). [13] D. Tsygankov et al., Kinetic models for the coordinated stepping of cytoplasmic dynein J. Chem. Phys., 130, (2009). [14] B. Derrida Velocity and Diusion Constant of a Periodic One-Dimensional Hopping Model J. Stat. Phys., 31, 443 (1983). [15] S. Toba et al., Overlapping hand-over-hand mechanism of single molecular motility of cytoplasmic dynein Proc. Natl. Acad. Sci. 103, 5741 (2006). 12