University of LjubaljanaFaculty of Mathematics and Physics

Department of Physics

Seminar Ib, £etrti letnik, stari program

Stochastic motion ofmolecular motor dynein

Author: Miha JurasMentor: 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.

Contents

1 Introduction 2

2 Molecular motor dynein 2

2.1 Dynein structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Chemomechanical conformational states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Dynein dimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Stochastic process 5

3.1 Markov process and its properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.1.1 Three state example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Stationary distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 Dynein cycle as Markov process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 Dimer coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Calculation and simulation vs. measurements 9

4.1 Velocity calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Velocity from simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Conclusion 12

1 Introduction

All cells from unicellular organisms to cells of organ in human body need energy to survive. Thisprinciple is "living" prove of thermodynamic principles where cell is a system actively sustaining entropywithin itself on the account of its surrounding environment. Basic principle can be described as: cell isconsuming substance with high inner energy-�food�, extracts inner energy from it, uses energy to sustainorder within it self and �nally releases food byproduct and excess of energy as heat into environment.

Eukaryotic cell1 converts �food� into energy within mitochondria. Energy is stored in energy storingmolecule ATP (adenosine triphosphate) which acts as cell unit of energy and contains approximately100 × 10−21J ≈ 25 kT where T is room temperature at 25◦C = 298◦K [1]. ATP is afterwards used inmay cellular functions. This include [2] synthesis of DNA, RNA and proteins, macromolecules transportacross cell membrane, maintaining cell structure by facilitating assembly and disassembly of elementsof cell skeleton (cytoskeleton) and �nally in inner cell transport such as muscle movement and activetransport within cellular plasma.

Within this seminar we will explore one speci�c cellular function which is inner cell transport facil-itated by molecular motor dynein. As any human made motor also molecular motor have more innerstates in one thermodynamic cycle. By molecular motors we speak of chemical, mechanical and confor-mational 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 catalyz-ing hydrolysis reaction (chemical reaction where chemical bonds are cleavage by addition of water) bywhich ATP's energy is converted into mechanical work. Therefore these enzymes are named molecularmotors.

In human body there are three main groups of molecular motors. Myosin is responsible for musclecontraction, kinesin is transporting cellular cargo from center of the cell to the cellular membrane anddynein is transporting cellular cargo from the membrane towards center of the cell and is also responsiblefor cilia an �agella movement.

1Eukaryotic cells have inner membrane which encapsulates its organelles such as nucleus and mitochondria.

2

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, linkeris the worker part (doing mechanical work) and leg part (strut, buttress and MTBD) is responsible tohold 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 numberedcircles) where only four are able to accept ATP and catalyze hydrolysis. Due to the ring like shapethe structure has a name AAA ring where AAA stand for ATPases Associated with diverse cellularActivities. AAA ring has a function of a fuel cell where fuel is burned. Purple liker is a working partof 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 AAAring. Yellow stalk and orange buttress are structures facilitating communication between AAA ring andgray microtubule binding domain (MTBD) which binds dynein to the microtubule - a cellular highwaynetwork.

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 howspatially separated structures synchronize their own actions, inner states, in relation to each other.Some necessary empirical conditions must be ful�lled for dynein cycle to work. Firstly, linker shouldnot pull the cargo (generate powerstroke) when its leg is not bound to microtubule as dynein needssomething to hold on while pulling. Secondly linker should not move to pre-powerstroke position beforeleg is still bound because linker movement to pre-powerstroke moves dynein's leg forward therefore shouldbe 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 bein relaxed position - post-powerstroke position where it is not able to do any mechanical work. Secondlinker position is in strained position - pre-powerstroke position where it has stored elastomechanicalenergy and is waiting to "�re". Leg can be bound to microtubule or unbound from microtubule. Finlayfor the ring we distinguish four states: ATP binds to hydrolysis site on AAA ring, ATP is hydrolyzed intoADP and Pi, Pi is released and �nally ADP is released which is based on the ATP hydrolysis chemicalreaction

ATP � ADP + Pi,

where ATP stands for adenosine triphosphate, ADP for adenosine diphosphate and Pi for hydratedinorganic phosphate. Chemical reaction as such can run to both sides as ATP hydrolysis from ATP to

3

ADP or as ATP synthesis from ADP to ATP. Synthesis is normally taking place in mitochondria but wewill keep this possibility open. Even more we will extend reverse principle to whole dynein cycle for themeans explained in next sections.

Based on the empirical necessary conditions described in the beginning of this section and fromnumerous 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 suc-cessful 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 frommicrotubule. 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 occupiedby ATP, �ADP.Pi� hydrolysis site occupied by ADP and Pi and �ADP� hydrolysis site occupied by ADPonly.

Dynein state Leg state Linker state Ring statemechanical state label conformation state label chemical state label

MT.D bound to MT MT post-stroke D empty hydrolysis siteMT.D.ATP bound to MT MT post-stroke D ATP ATPD.ATP released from MT post-stroke D ATP ATPD*.ADP.Pi released from MT pre-stroke D* ADP + Pi ADP.Pi

MT.D*.ADP.Pi bound to MT MT pre-stroke D* ADP + Pi ADP.Pi

MT.D.ADP bound to MT MT post-stroke D ADP ADP

Figure 2: Illustration of 6 dynein states as de�ned 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 Pi

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 notenough to achieve processive movement. Due to thermal movement of dynein surrounding it is veryunlikely that once unbound from microtubule dynein will bind back on microtubule very soon whichwould 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 incase of one dynein [8]. While one dynein is bound to microtubule the other can be unbounded frommicrotubule and can therefore undergo transitions through its states, stretches in front, binds back tomicrotubule and afterwards pulls cargo through cell plasma.

As important as communication within dynein to achieve synchronization of dynein's di�erent partsduring hydrolysis is also communication between two dyneins in dimer. One dynein should wait until

4

the other is progressing and then they can exchange roles. Here the intriguing question arises: do theychange turns regularly walking either in �hand-over-hand� fashion where dyneins step one pass anotheror as �inchworm� mechanism where the leading dynein always moves a step forward before the trailingdynein can follow. In case of yeast dynein dimer where each of dynein heads have been marked withquantum 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 redquantum 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. Wewill 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 de�ned with state from and stateto which transition is made. Let us de�ne transition rate also known as kinetic rate kji which correspondsto the transition to state j from state i where an Einstein notation is used j ← i. Additionally we canimagine large number of identical systems with the same number of states and transition rates betweenthem. Let us suppose that systems are independent from each other what assures that the state anysystem is in is independent of the states the other systems are in. Through this we can de�ne occupationprobability pi of a state i

pi =Nnumber of systems in state i

Nnumber of all systems

.

Each state occupation probability pi evolves through time. All other states j contribute to increment ofoccupation probability for state i and vice versa state i contributes to the occupation of all other statesj. This relation is known as master equation

dpidt

=∑j

(kijpj − kjipi) .

5

If transition rate is independent of time and dependent only on the involved state pair and not fromprevious history of states system was then we can model these transitions with Markov process.

In probability theory this is identi�ed as continuous-time Markov process where kij is conditionalprobability for system in state i to migrate to state j and is de�ned as

kji = P (Xn+1 = xj |Xn = xi) .

The crucial property of Markov process known as Markov property is memorylessness. Its meaning isthat future behavior of the system depends only on the current state of the system and not on the statesthe system has been in before. We can formulate this as probability to reach state xn+1 at time tn+1

from state xn at time tn that has been in state xn−1 at time tn−1 . . . is same as probability to reachstate xn+1 at time tn+1 from state xn at time tn without any knowledge of its history

P(Xtn+1 = xn+1|Xtn = xn, . . . , Xt1 = x1, Xt0 = x0

)= P

(Xtn+1 = xn+1|Xtn = xn

).

Furthermore the probability for transition in a period ∆t is dependent only on the length of the periodand not on when the period started

P (Xt=t0+∆t = xn+1|Xt=t0 = xn) = P (Xt=∆t = xn+1|Xt=0 = xn) ,

which we can rewrite in explicit time dependence as

P (t0 ≤ t ≤ t0 + ∆t) = P (0 ≤ t ≤ ∆t) .

The only continuous distribution function that has this continuous memoryless property is exponentialdistribution [11] with rate k

w(t) = ke−kt,

therefore transition from state i to state j with rate kji is distributed exponentially as

wji(t) = kjie−kjit.

3.1.1 Three state example

To get better understanding we can examine three state system. Master equations for three state Markovprocess where pi are occupation probabilities for each of three states are

dp1

dt= −(k21 + k31)p1 + k12p2 + k13p3,

dp2

dt= k21p1 − (k12 + k32)p2 + k23p3,

dp3

dt= k31p1 + k32p2 − (k13 + k23)p3,

which can also be written in matrix form as

dp

dt= pQ.

Q is transition rate matrix de�ned as

Q =

−k21 − k31 k21 k31

k12 −k12 − k32 k32

k13 k23 −k13 − k23

.

Transition matrix can be represented as directed graph (Figure 4) where each row represents one state,non diagonal element kji rate of transition from state i to state j and �nally diagonal element Qii is thenegative sum of all rates from state i. Diagonal element Qii also relates to system dwell time in state ias τi = 1/Qii.

6

p1

p2p3

k21

k31

k32

k12

k13

k23

p1

p2p3

k32

k12k13

k23

p1 p2

p3p4

k41 k32

k12

k13k23

k43

k34

Figure 4: Left: three state Markov process represented as directed graph which forms a bidirectionalcycle where rates k21, k32 and k13 form a forward directed cycle and rates k12, k23 and k31 a backwarddirected 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 k21 and k31 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 timelimit leads system to ends up in state 1, which makes this state an absorbing state. Right: one exampleof four state Markov process where also diagonal transitions are possible and rates k21, k31, k42, k14 andk24 were eliminated.

3.2 Stationary distribution

Any physical system is evolving through time to reach its stationary state within its self and withits surroundings. In n state system with some initial occupation probabilities pt=0 this is resulted asconvergence of occupation probability to stationary occupation probabilities pt=∞. In Markov processthis can be applied as following

pn+1 = pn + dpn = pn + pnQdt = pn (I + Qdt) = p0 (I + Qdt)n+1

,

where p0 is initial distribution at time t = 0, pn is current distribution at time t = n dt and pn+1 isdistribution at time t = (n + 1)dt. (I + Qdt) is a generator of discrete-time Markov process which isprogressing observed system by discrete time interval dt.

After long enough time2 system reaches stationary distribution p∞ which satis�es relation

p∞ = p∞ (I + Qdt) ,

which means that time progression for dt does not change stationary distribution p∞. From this itfollows to

p∞Q = 0,

i.e., p∞ is left eigenvector of Q with eigenvalue 0.

3.3 Dynein cycle as Markov process

In Markov process transition to next state depends only on the state in which the system currently isand is independent of the states in which system was. In our daily experience we ordinary do not seethis kind of behavior. If a ball is in front of a goal and its previous location (one second ago) was afew 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 sphereor 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

2Quanti�cation of relaxing time is not scope of this seminar, but reader can �nd numerous sources in Markov processliterature [12].

7

which completely randomizes sphere momentum. This example leads us to conclusion that Markovproperty of memorylessness is obeyed by dynein.

We have noted in section (2.2) that we will extend reverse principle to whole dynein cycle. This isno issue for Markov process as it encloses transition from state i to state j and also its reverse transitionfrom state j to state i. Having this in mind we can construct dynein cycle (Table 2) with 6 states de�nedin Table 1.

Table 2: Dynein cycle with transition rates between dynein states [3]. For dynein to move forward (minusend 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 physiologicalconcentrations of ATP and its byproducts are [ATP] = 10−3M, [Pi] = 10−3M, [ADP] = 10−5M. Index ifor rates will be used in section 4.1 for calculating average velocity.

i State Transition rate Transition Parameter value... Previous cycle

MT.D

1 k−ATP

x yk+ATPk+ATP ATP binding 2× 106 M−1s−1

k−ATP ATP release 50 s−1

MT.D.ATP

2 k+MT

x yk−MTk−MT MT release, D 500 s−1

k+MT MT binding, D 100 s−1

D.ATP

3 k−RS

x yk+RSk+RS ATP hydrolysis, linker swing to pre-stroke 1000 s−1

k−RS ATP synthesis, linker swing to post-stroke 10 s−1

D*.ADP.Pi

4 k′−MT

x yk′+MT

k′+MT MT binding, D* 2.4× 104 s−1

k′−MT MT release, D* 10 s−1

MT.D*.ADP.Pi

5 k−PS

x yk+PSk+PS Power stroke, Pi release 2500 s−1

k−PS Reverse stroke, Pi binding 10−4 M−1s−1

MT.D.ADP

6 k+ADP

x yk−ADP k−ADP ADP release 200 s−1

k+ADP ADP binding 2.7× 106 M−1s−1

MT.D... Next cycle

We have shown that in systems with more than 3 states also diagonal transitions are possible. Al-though model of dynein we are examining has 6 states we are limiting it to have only bidirectional cycleas possible transitions which for example excludes diagonal transitions from state D.ATP directly toMT.D.ADP.

3.4 Dimer coupling

In previous section (3.3) we have listed transition rates of one dynein motor having 6 unique states with12 transition rates which constitutes 6 state bidirectional cycle. With coupling the two dynein motorswe 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 isbound to MT and waits for its turn. When �rst dynein �nishes ATP cycle then the second starts with itsown ATP cycle. This gating mechanism diminishes dynein dimer states to 12 states which actually aretwo single dynein cycles, �rst for the left and second for the right dynein. When simulating such modelit 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 waywhich could result in fully dissociation from MT from which average run length can be modeled andcompared to measurements [3, 13].

8

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 willassume there are no di�erences in transition rates in relation to single dynein cycle. The only change iswhen trailing dynein overtakes the leading one and ends its own cycle in leading position with transitionfrom MT.D.ADP to state MT.D then the other dynein starts its own cycle from state MT.D. This wayhydrolysis 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 afterforward step is bound 8 nm in front. Because one dynein if �xed and the other moves for 16 nm dyneindimer 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: inchwormwalking mode where right (blue) dynein is leading and left (yellow) is trailing dynein. Both illustrationsshow 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 decreasingthe lag. One step size in this mode is 8 nm therefore in one step dynein dimer center and cargo aremoved for 4 nm.

The main di�erence 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 additionalsimplicity of hand-over-hand walking mode where on the other hand in inchworm walking mode distanceis or increased or decreased. By having this in mind and that two dyneins are connected with dimerisatedtails the two dynein cycles di�er from each other which suggests that we should incorporate this intokinetic rates.

In higher walking modes we might combine hand-over-hand and inchworm walking mode with in-troducing choices for example in forward movement we would choose between leading dynein makes aninchworm movement to increase dyneins' distance or trailing dynein makes hand-over-hand movementto take the lead.

To keep the simplicity of the topic we will only explore the simplest model as discussed in thebeginning as hand-over-hand walking mode where knowledge of single dynein cycle is su�cient [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 concen-tration and the second low concentration will be hundred times lower as [ADP]L = 10µM. The transitionrate matrix Q and stationary distribution p∞ thus are

QH =

−2050 2000 0 0 0 50

100 −600 500 0 0 00 10 −1010 1000 0 00 0 10 −25010 25000 00 0 0 10 −2510 2500

200 0 0 0 27 −227

, pH,∞ =

0.06490.21800.10800.00430.04910.5557

:MT.D:MT.D.ATP:D.ATP:D*.ADP.Pi

:MT.D*.ADP.Pi

:MT.D.ADP

QL =

−70 20 0 0 0 50100 −600 500 0 0 00 10 −1010 1000 0 00 0 10 −25010 25000 00 0 0 10 −2510 2500

200 0 0 0 27 −227

, pL,∞ =

0.71710.02410.01190.00050.00740.2390

:MT.D:MT.D.ATP:D.ATP:D*.ADP.Pi

:MT.D*.ADP.Pi

:MT.D.ADP

9

where each value of Q has unit s−1. Blue printed diagonal elements of Q are the smallest in absolutevalue which means that this state has the lowest rate to transfer to other states which results in thelongest dwell time (τi = −1/Qii) and in the highest occupancy in stationary distribution p∞ for thatstate (also printed blue). Green printed upper diagonal rate k+ATP is the one that depends on ATPconcentration. In case of high ATP concentration MT.D.ADP is the most occupied state (56%) withdwell time of 4.4 ms but in case of low ATP concentration MT.D is the most occupied state (72%) withdwell time 14.3 ms. Strong dependence of on ATP concentrations also results in velocity of dynein dimeras it will be described in following sections.

4.1 Velocity calculation

In case of bidirectional circular transitions where transition matrix has tridiagonal form then averagevelocity can be calculated [14] as

v = d(keff+ − keff

− ),

where d is step length, keff− and keff

+ are e�ective backward and forward rates de�ned as

keff+ = 1/RN ,

keff− =

N∏j=1

k−jk+j

/RN ,

with

RN =

N∑j=1

rj , rj =1

k+j

1 +

N−1∑k=1

j+k∏i=j+1

k−ik+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

<v>H≈ 863.23 nm/s <v>L≈ 95.46 nm/s.

Ratios of forward against backward cycle are(keff

+

keff−

)H

≈ 1011

(keff

+

keff−

)L

≈ 109,

which con�rms that forward cycle (ATP hydrolysis as keff+ ) is far more probable than backward cycle

(ATP synthesis as keff− ).

4.2 Velocity from simulation

We can con�rm calculated velocity value with simulation in Mathematica with help of Continuous-MarkovProcess and RandomFunction functions. ContinuousMarkovProcess function accepts two argu-ments: �rst argument is starting probability distribution p(t = 0) which I set as stationary distributionp∞, second argument is transition matrix Q. RandomFunction also accepts two arguments: �rst isthe result of function ContinuousMarkovProcess and the second argument is time interval that systemshould repeat the process given as the �rst argument. Result of RandomFunction is a list of pairs (tj , ij)there tj is time and ij is dynein state after jth transition.

Because dynein cycle is bidirectional, with forward cycle (ATP hydrolysis) highly dominant, wecan count cycles dynein makes. We increase the count when dynein makes transition from stateMT.D*.ADP.Pi to MT.D.ADP or decrease the count in case of reverse transition. Selected transition iswhen dynein generates power stroke which moves its cargo forward for 8 nm which is the same distanceas dynein dimer center moves. Simple multiplication of cycle count ncycles and single step dynein centermovement d1 = 8 nm results in total distance l dynein dimer moved from start of counting

l = ncyclesd1.

10

After m transitions tm time has passed and dynein dimer had average velocity of

<v>= ncyclesd1/tm.

With this method we implicitly allow dynein dimer to forward direction of stepping only. This does notprohibit dynein to transit in backward direction of the transitions between the states, it only prohibitsbackward direction of stepping.

I have run 1000 simulations. In case of high ATP concentration (Figure 6 middle left) dynein waswalking for 10 seconds and achieved < v >H= 862 ± 18 nm/s which is comparable to measured value<vm>H= 800 nm/s and in case of low ATP concentration (Figure 6 middle right) dynein was walkingfor 50 seconds and achieved <vm>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 wasalso expected.

0.1 0.2 0.3 0.4 0.5t @sD

100

200

300

400

l @nmDvH = 912

vL = 128

800 820 840 860 880 900 920v @nm�sD

0.005

0.010

0.015

0.020

0.025

dΡ�dv @s�nmD v = 862, Σv = 18

85 90 95 100 105v @nm�sD

0.02

0.04

0.06

0.08

0.10

dΡ�dv @s�nmD v = 95, Σv = 3

0.00 0.01 0.02 0.03 0.04ts @sD

20

40

60

80

100

120

d�dts @s-1D

k = 183 ± 4 s-1

0.00 0.05 0.10 0.15 0.20 0.25 0.30ts @sD

5

10

15

20

d�dts @s-1D

k = 18.5 ± 0.3 s-1

Figure 6: Top: a stepping trace of one simulation run displaying dynein dimer center movement within0.5 s for both ATP concentrations. Middle left and right: velocity distribution of 1000 simulations withmean velocity v̄H = 862 nm/s and v̄L = 95 nm/s. Bottom left and right: One cycle time distributionwhich has form of convolution of may exponential distributions with long time limit decreasing like thelowest forward rate in case of high ATP concentration as k−ADP = 200 s−1 and k+ATP = 20 s−1 for lowATP concentration.

11

5 Conclusion

System for which we are able to de�ne inner states and rates of transitions between them with additionalconstraints that these rates are time independent and are not dependent on the history in which systemwas can be modeled with Markov process.

In our study of a single dynein chemomechanical and conformational cycle and a simple dynein dimermodel we were able to simulate and calculate its velocity which are comparable to measurements for twodi�erent ATP concentrations.

Formalism described opens new possibilities to add new dependencies to dynein stochastic movementone of them to be velocity to force dependence or possibility to develop more complex dynein dimermodel to describe also inchworm walking mode or even combination of hand-over-hand and inchwormmodes.

References

[1] J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, Sinauer Associates, Sunderland(2001).

[2] http://en.wikipedia.org/wiki/Adenosine_triphosphate (November 2013).

[3] A. �arlah and A. Vilfan, The winch model can explain both coordinated and uncoordinated steppingof cytoplasmic dynein, submitted.

[4] M. Juras, Molekularni motor dinein, Seminar Ia (October 2013),http://mafija.fmf.uni-lj.si/seminar/files/2013_2014/seminar-1a_Miha_Juras.pdf.

[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 ofcytoplasmic dynein, Proc. Natl. Acad. Sci. 104, 16134 (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 BehaviorCell, 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] http://www.mast.queensu.ca/~stat455/lecturenotes/set5.pdf (November 2013).

[11] http://en.wikipedia.org/wiki/Exponential_distributionq#Memorylessness (November2013).

[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, 025101 (2009).

[14] B. Derrida Velocity and Di�usion 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 cytoplasmicdynein Proc. Natl. Acad. Sci. 103, 5741 (2006).

12