Journal of Theoretical Biology 380 (2015) 48–52
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Model for bidirectional movement of cytoplasmic dynein S. Sumathy, S.V.M. Satyanarayana n Department of Physics, Pondicherry University, R.Venkataraman Nagar, Kalapet, Puducherry 605 014, India
H I G H L I G H T S
Stochastic process models are developed for bidirectional motion of dynein motor. Probability for backward step uses Crook's like fluctuation theorem. Average motor velocity is negative beyond stall force. Backward motion beyond stall force is also powered by ATP hydrolysis.
art ic l e i nf o
a b s t r a c t
Article history: Received 20 October 2014 Received in revised form 2 March 2015 Accepted 22 April 2015 Available online 2 May 2015
Cytoplasmic dynein exhibits a directional processive movement on microtubule filaments and is known to move in steps of varying length based on the number of ATP molecules bound to it and the load that it carries. It is experimentally observed that dynein takes occasional backward steps and the frequency of such backward steps increases as the load approaches the stall force. Using a stochastic process model, we investigate the bidirectional movement of single head of a dynein motor. The probability for backward step is implemented based on fluctuation theorem of non-equilibrium statistical mechanics. We find that the movement of dynein motor is characterized with negative velocity implying backward motion beyond stall force. We observe that the motor moves backward for super stall forces by hydrolyzing the ATP exactly the same way as it does while moving forward for sub-stall forces. Movement of dynein is also simulated using a kinetic Monte Carlo method and the simulated velocities are in good agreement with velocities obtained using a stochastic rate equation model. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Dynein motor Force velocity relationship Fluctuation theorem
1. Introduction Molecular motors are nano-machines that do work by harnessing chemical energy (Chowdhury, 2013). Cytoplasmic dynein is a minus end directed motor protein that moves on microtubule. It is implicated in intracellular transport of vesicles, mRNA, protein complexes etc., from cell cortex to center of the cell (King, 2012). Dynein is a homo-dimer with each head consisting of a ring of six domains (Vale, 2000). Four of the six domains of the ring have sites with affinity for ATP binding. Dynein motor exhibits a gear like mechanism in controlling step size in response to the load force (Mallik et al., 2004). A model of single head of a dynein motor is simulated by Monte Carlo method (Singh et al., 2005). Singh et. al. computed force velocity relation, step size distribution from the simulation data and studied the ATP dependence of average velocity of the motor. Further, the load dependence of the
n
Corresponding author. E-mail address: [email protected] (S.V.M. Satyanarayana).
http://dx.doi.org/10.1016/j.jtbi.2015.04.029 0022-5193/& 2015 Elsevier Ltd. All rights reserved.
step size and ATP concentration dependence of the stall force are obtained by another model with a weak coupling between two reaction coordinates corresponding to chemical reactions and translocation of the motor (Gao, 2006). A complete mechanochemical model for a hand over hand stepping model of a homodimeric dynein is developed where the ATP hydrolysis cycle is coupled to a coarse grained structural model (Tyagankov et al., 2009, 2011). Dynein motor at a different spatio-temporal resolution is studied by multi-scale modeling (Serohijos et al., 2009). Stochastic process modeling of unidirectional movement of single head of a dynein motor is carried out systematically with one-, two- and three-step process (Sutapa Mukherji, 2008). While dynein moves processively toward the minus end of the microtubule, it is observed that dynein takes backward steps once in a while and the frequency of such backward steps increases as the load increases (Gennerich et al., 2007). Further, it was observed that single head of dynein is sufficient for processive motion (De Witt et al., 2012). In the present work, we develop a stochastic process model for bidirectional movement of cytoplasmic dynein's single head, along
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
the same lines of the model of Sutapa Mukherji (2008). We use Crook's like fluctuation theorem to define the ratio of probability of forward step to that of backward step. Dynein can take 8, 16, 24 or 32 nm step sizes depending on the load and ATP concentration. In this study we investigate bidirectional movement of dynein's single head for all the four step sizes. The paper is organized as follows. Description of stochastic process model, typical reaction scheme, different rate constants used in the study, a system of stochastic rate equations and the procedure to compute average velocity of the motor are presented in Section 2. Section 3 consists of results and discussion.
2. Model Microtubule is modeled as a passive one dimensional lattice on which single head of dynein moves. We assume that dynein's head is always attached to the microtubule site. Single head of dynein consists of one primary and three secondary ATP binding sites. It is assumed that out of four ATP binding sites, ATP hydrolysis takes place only at a primary site. However, the step length depends on the occupancy of the three secondary sites. If we denote that the minimum step size dynein can take as a, where a ¼ 8 nm, the step sizes are 4a, 3a, 2a and a when occupied secondary sites are 0, 1, 2 and 3 respectively. If we consider one primary and one secondary sites, the stochastic variable Skl j denotes the probability that a dynein's head is on the jth lattice site of microtubule with k; l ¼ 0; 1. Here k ¼ 0ð1Þ and l ¼ 0ð1Þ signify the corresponding primary or the secondary site being unoccupied (occupied) respectively. In this case, maximum step size of a single head of dynein is 2a and hence we call this a 2a model. If we have two and three secondary ATP binding sites considered, we refer to them as 3a model and 4a model respectively and denote the corresponding stochastic variable as Sklm and Sklmp . Since each index in the superscript of the stochastic j j variable can be 0 or 1 corresponding to the ATP binding site being unoccupied or occupied, we have number of stochastic variables as 4, 8 and 16 respectively for 2a, 3a and 4a models. The rate of ATP binding on the ith binding site of dynein's head is denoted as koni and the rate of ATP unbinding from the same site is denoted as koffi. Hydrolysis rate at the primary site must decrease with increasing load since we know that the motor stops at stall force. Further, it also depends on whether the secondary ATP sites are occupied. The load dependence of the hydrolysis rate is governed by the Boltzmann factor that is given by the following expression: kcat;i ¼ AðiÞkcat;0 exp½ αFdðiÞ=kB T
ð1Þ
Here, i¼ 1, 2, 3 and 4 and dðiÞ ¼ i a, kcat;0 is the hydrolysis rate for no load, α is the load distribution factor taken as positive since the hydrolysis rate should decrease with the increase in opposing external load and AðiÞ ¼ 1 if any of the secondary sites are occupied and is 0.01 if all the secondary sites are unoccupied. Along the lines of previous studies (Singh et al., 2005; Sutapa Mukherji, 2008), we assume that the ATP binding rates of secondary sites depend on the load as follows: kon;2 4 ðFÞ ¼ kon;2 4 exp½Fd0 =kB T
ð2Þ
d0 is an adjustable parameter in units of length. Cytoplasmic dynein from yeast and mammals shows difference in velocity and stall forces. Yeast dynein exhibits a higher stall force and smaller velocity compared to mammalian dynein (ReckPeterson et al., 2006; Gennerich et al., 2007; Ross et al., 2006; Mallik et al., 2004; Schroeder et al., 2010). Experiments by Mallik et al. (2004) reported that cytoplasmic dynein has a stall force of 1.1 pN at saturating ATP concentration and the stall force decreases with decrease in ATP concentration. On the other hand Toba et al.
49
(2006) observed a stall force of 7–8 pN which is independent of ATP concentration for mammalian cytoplasmic dynein. Further, Toba et al. reported a velocity of dynein motor around 800 nm/s. It is argued that the differences observed with respect to stall force and velocity in Yeast and mammalian dynein may be of structural origin (Hook and Vallee, 2012), that is, the absence of C terminal domain in Yeast may be responsible for higher force and lower velocity. In this model we consider the stall force of 7.5 pN as an input parameter and compute the velocity of the dynein motor. Molecular motors are known to take backward steps amidst their processive forward motion. It is observed that the directional movement of motors is governed by thermodynamics that restrict backward steps (Dean Astumian, 2010). This motivated us to examine the role of thermodynamic fluctuation theorem such as Crook's fluctuation theorem (Crooks, 1990) for motor taking backward steps. The probability ratio for forward and backward steps for kinesin motor is estimated from experiment (Carter and Cross, 2005; Nishiyama et al., 2002). Subsequently, the expression for the ratio of forward and backward step probabilities was derived from Crook's like fluctuation theorem of non-equilibrium thermodynamics for kinesin motor (Bier, 2008; Calzetta, 2009). In this work, we assume that the same expression is valid for dynein motor as well. The ratio of the probability for a forward step to backward step is given as PF d0 ½F s F ¼ exp ð3Þ PB 2kB T where FS is the stall force. Also note that P F þ P B ¼ 1 since dynein motor can take only forward or backward steps in our model. Since our interest is to understand the role of backward steps in the dynamics of dynein, in order to compare the results with previous model, we considered different rate constants and parameters in this study same as in the previous studies (Singh et al., 2005; Sutapa Mukherji, 2008). They are given in Table 1. Here kBT is the thermal energy and kB is the Boltzmann constant. In this study, we have computed the average velocity of single head of dynein molecule with 2a, 3a and 4a models with and without including backward steps. We investigate the velocity of the motor as a function of force both below (substall forces) and above (super stall forces) the stall force. We present the reaction scheme and stochastic rate equation model for bidirectional 3a model as a representative case. 2.1. The model In a 3a model, we have eight state variables for single head for dynein motor. The state variable vector on the jth lattice site of microtubule is given by
ρj ¼ ½S000 S001 S010 S100 S011 S101 S110 S111 T j j j j j j j j Table 1 Rate constants and parameters used in the study. Symbol
Value
kB T koff1 koff2 koff3 koff4 kon1
4.1 pN nm 10 s 1 250 s 1 250 s 1 250 s 1
kon2 ðF ¼ 0Þ
4 105 M 1 s 1 ½ATP kon2 ðF ¼ 0Þ=4 kon2 ðF ¼ 0Þ=6 6 nm 55 s 1 0.3
kon3 ðF ¼ 0Þ kon4 ðF ¼ 0Þ d0 kcat;0 α
4 105 M 1 s 1 ½ATP
ð4Þ
50
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
Sj001 kon1
Sj101
koff3 100
Sj-3
kcat3
Sj000
kon2
where the matrix Rðζ Þ is given by
Sj111
0
kon1 B 0 B B B 0 B B k B on1 ½Rðζ Þ ¼ B B 0 B B B 0 B B 0 @ 0
koff3 kon1 koff1
Sj100
kcat3
Sj+3
000
koff2 kon2
koff2
koff 3 ðk7 Þ 0 0 0 kon1 0 0
koff 2 0 ðk6 Þ 0 0 0 kon1 0
k1 0 0 ðk4 Þ 0 0 kon2 0
Sj-2
kcat2
Sj010
kon1
kon3
koff3 Sj-1111 kcat1
Sj011
Sj110
kon1
kcat2
Sj+2
kcat1
Sj+1011
Fig. 1. Reaction scheme for 3a model of single head of a dynein motor.
〈v〉 ¼ a The reaction scheme for the 3a model is presented in Fig. 1. In this scheme, when hydrolysis takes place on the primary site when none of the secondary sites are occupied, the stochastic 000 variable S100 changes to S000 j j þ 3 with probability PF and to Sj 3 with probability 1 P F . Further, hydrolysis at primary site with one (two) primary sites are occupied, the stochastic variable S110 ðS111 Þ j j 010 011 010 011 changes to Sj þ 2 ðSj þ 1 Þ with probability PF and Sj 2 ðSj 1 Þ with probability 1 P F . This model is similar to random walk in two dimensional space where one coordinate is the reaction coordinate and the other is the spatial coordinate (Bameta et al., 2013). The stochastic rate equations for the eight state variables of 3a model corresponding to the reaction scheme presented in Fig. 1 are given below 000 dSj
dt
¼ kon1 S000 þ koff 3 S001 þ koff 2 S010 þ koff 1 S100 þ ½P F S100 j j j j j3
001
dt
¼ ðkoff 3 þ kon1 ÞS001 j
010
dSj
dt dt dt
〈j〉 0 ¼ aλl ð1Þ t
111 ¼ ðkon1 þ koff 3 ÞS011 þ ½P F S111 j j 1 þ P B Sj þ 1 kcat1
101
dt
3. Results and discussion
3
[ATP] 5µM 50µM 500µM 1mM 2mM
2
10
ð5Þ
In matrix form these equations can be recast as dρ ¼ ½Aρj þ ½Bρj 1 þ ½Cρj þ 1 þ ½Dρj 2 þ½Eρj þ 2 þ ½Fρj 3 þ ½Hρj þ 3 dt ð6Þ Here A; B; C; D; E; F and H are matrices whose elements are various P j rate constants. Using a generating function, Gðζ ; tÞ ¼ 1 j ¼ 1 ζ ρj , the above equation can be recast as ! d 1 1 1 2 3 Gðζ ; tÞ ¼ ½A þ ζ ½B þ ½c þ ζ ½D þ 2 ½E þ ζ ½F þ 3 ½H Gðζ ; tÞ dt ζ ζ ζ ¼ ½Rðζ ÞGðζ ; tÞ
ð11Þ
10
¼ kon1 S010 þ kon2 S100 ðkcat2 þ kon3 þkoff 2 ÞS110 þ koff 3 S111 j j j j ¼ kon1 S011 þ kon2 S101 þ kon3 S110 ðkcat1 þkoff 3 ÞS111 j j j j
ð10Þ
For finding the derivative of the largest eigenvalue at ζ ¼ 1, it is 0 ″ 2 convenient to substitute ζ ¼ 1 þ δ and λ ¼ δλ ð1Þ þ ðδ =2Þλ ð1Þ in 0 Eq. (10) and λ ð1Þ is find out by equating the coefficients of δ to zero (Mogliner et al., 2013; Elston, 2000).
ð7Þ
U
111 dSj
ð9Þ
Det½Rðζ Þ λI ¼ 0
V (nm/s)
dt
2
where 〈j〉 is average index on the lattice where the motor is located at time t and λl ðζ Þ is the largest eigenvalue of ½Rðζ Þ and the primes denote the derivatives of λ with respect to ζ. For finding the largest eigenvalue, it is necessary to solve the characteristic equation
dSj ¼ kon1 S001 ðkon2 þ koff 3 ÞS101 j j dt 110 dSj
4
From the computed data of force velocity curves, we estimated the fit parameters V U ð0Þ ¼ 425:2 nm=s and γ ¼ 0:583 pN 1 . The observed force velocity relation in Eq. (11) is found to differ from the general form of force velocity relation for motors (Kunwar and Mogilner, 2010).
¼ kon1 S000 ðkcat3 þ kon2 þ koff 1 ÞS100 þ koff 3 S101 þ koff 2 S110 j j j j
011
dSj
3
V U ðFÞ ¼ V U ð0Þ expð γ FÞ
110 ¼ ðkon1 þ koff 2 ÞS010 þ koff 3 S011 þ ½P F S110 j j j 2 þ P B Sj þ 2 kcat2
100
dSj
1 0 C 0 C C C 0 C C 0 C C C k3 C C 0 C C C koff 3 A ðkcat1 þ koff 3 Þ
Force velocity relation VðFÞ is one of the important characteristics of the motor. We present in Fig. 2, the average velocity of unidirectional movement (corresponding to PF ¼1 in the model) of single head of dynein motor in 3a model. The force velocity relation for ATP concentration close to and higher than physiological concentrations (1 mM) is found to be
þ P B S100 j þ 3 kcat3 dSj
koff 3 0 ðkon2 þ koff 3 Þ 0 kon2
0 0 k2 koff 2 0 0 ðk5 Þ kon3
Here k1 ¼ koff 1 þ ½ð1 þ P F ðζ 1ÞÞ=ζ kcat3 , k2 ¼ ½ð1 þP F ðζ 1ÞÞ=ζ 2 kcat2 , k3 ¼ ½ð1 þ P F ðζ 1ÞÞ=ζ kcat1 , k4 ¼ koff 1 þ kon2 þ kcat3 , k5 ¼ kcat2 þ kon3 þ koff 2 , k6 ¼ kon1 þ koff 2 and k7 ¼ kon1 þ koff 3 . The transition matrix R½ζ ¼ 1 has the property that the sum of all elements in a column is zero. Thus, the largest eigenvalue of the matrix R½ζ ¼ 1 is zero. Our aim is to find the average velocity and it is found from the relation
010
koff3 Sj111
koff 3 0 ðk7 Þ 0 0 kon1
0 0 0
ð8Þ 6
110
0 0
1
10
0
10
-1
10
-2
10
0
5
10
15
F(pN) Fig. 2. Force velocity relation for unidirectional movement of single head of a dynein motor in 3a model.
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
600
51
1 1.5
B
V (nm/s)
400 300
1
5 µM 50 µM 500 µM 1 mM 2 mM
0 -0.5 -1
200
[ATP]
0.5
0.5
VB / VU
[ATP] 5µM 50µM 500µM 1mM 2mM
500
-1.5
8
10
12
14
0
16
100
-0.5
0 -100
-1
0
5
10
15
Average velocity for a bidirectional motion of single head of dynein in 3a model as a function of force for different ATP concentrations is shown in Fig. 3. It can be seen from Fig. 3 that at the stall force, the average velocity of the motor is zero. This is because the probability of forward step is equal to the probability of backward step, that is, on an average the number of forward steps is equal to the number of backward steps. This is in good agreement with observations made in single molecule experiments with force feedback optical tweezers at an ATP concentration of 1 mM (Gennerich et al., 2007). It can be seen from the inset of Fig. 3 that beyond stall force (7.5 pN) the velocity is negative, that is the motor walks backwards (in a direction opposite to its natural direction of motion in the absence of any load force). We observe from Fig. 3 and the inset that the average negative velocity as a function of force for super stall forces is independent of ATP concentration. This result is also in a good agreement with single molecule experiments (Gennerich et al., 2007). Further, as F increases beyond stall force, the negative average velocity decreases and approaches zero for high super stall force. Fig. 4 represents the ratio of velocities of bidirectional and unidirectional movement. It can be observed that this ratio is independent of ATP concentration in a range of ATP concentrations from 5 μM to 10 mM studied in this work for all forces. As F-0, V B ¼ V U , as can be expected. For F c F S , it is observed that V B ¼ V U . This implies for F c F S , the cargo acts like a rigid wall and the motor gets reflected with the same velocity in the opposite direction, in the average sense.The ratio given in Fig. 4 has the following functional form: VB ðF s FÞδ ð12Þ ¼ tanh VU 2kB T The value of δ is found to be 3 nm. For sub-stall forces, bidirectional velocity is a fraction of unidirectional velocity. This reduction in velocity is due to the number of backward steps that the motor head takes in a bidirectional motion. For super stall forces, the number of backward steps is more than the number of forward steps and as a result the average velocity is negative, with a magnitude equal to a fraction of the unidirectional velocity. For F c F S , almost all steps are backward steps. However, we observe that the average negative velocity of the bidirectional motor decreases in much the same way as the average positive velocity of the unidirectional motor approaching zero from negative and positive directions respectively. This can be understood as follows. Dynein motor hydrolyses ATP and the rates of ATP binding to different binding sites and ATP hydrolysis depend on the force. For every ATP hydrolysis event, a step forward or backward is taken
5
10
15
20
Force (pN)
F (pN) Fig. 3. Average velocity versus force for bidirectional movement of single head of a dynein motor for different ATP concentrations. The inset shows the average velocity beyond stall force (7.5 pN).
0
Fig. 4. Ratio of velocities corresponding to bidirectional and unidirectional movement of dynein motor as a function of force for different ATP concentrations.
based on probabilities obtained from fluctuation theorem. For super stall forces, the rates corresponding to ATP binding and hydrolysis are low such that number of ATP hydrolysis events become very less and tend to zero. The average bidirectional velocity is small negative for F c F S implies that ATP hydrolysis events, however small they are, lead to backward steps of the motor. During the backward movement the motor hydrolyzes ATP in exactly the same way as it does while moving forward for force F o F S . We observe that the 4a model where we have four ATP binding sites for single head of the dynein gives similar results. However, the magnitude of the average velocity is higher for 4a model compared to 3a model in both unidirectional as well as bidirectional cases. For example, the velocity of bidirectional movement of single head of dynein motor under no load is 461 nm/s in 3a model and 711 nm/s in 4a model. The velocity of dynein motor for 4a model is in reasonable agreement with experimental observations (Toba et al., 2006). In order to corroborate our stochastic rate equations model, we simulated the movement of cytoplasmic dynein motor with four ATP binding sites (4a model) using kinetic Monte Carlo method. We have 16 biochemical states of dynein motor participating in 34 reactions with respective rates of reactions, say Ri 34 i ¼ 1 . The state of the dynein motor is randomly chosen to be one of the 16 states. A reaction is randomly chosen and if the reactant corresponding to the chosen reaction coincides with the state of dynein motor, the reaction is implemented. The time is advanced according to the equation
Δt ¼
ln u RT
ð13Þ
where u is a uniform random number between 0 and 1 and RT is the total rate of all the reactions. Rate constant values used in the simulation are same as in Table 1. Backward steps are implemented with the probabilities obtained from Eq. (3). Fig. 5 presents the trajectory of the dynein motor corresponding to ATP concentration of 1 mM and zero load force. The trajectory exhibits saltatory movement, which is characteristic of biological molecules. The velocity of motor is computed and is given by 713 nm/s, which is in good agreement with the value obtained using stochastic rate equation 4a model. The inset of Fig. 5 shows the histogram of different steps dynein motor has take during the course of movement. Negative steps are insignificant since probability for backward step for zero load is very small. As can be seen from the figure, 24 nm steps are large in number followed by 16, 32 and 8 for 1 mM ATP concentration.
52
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
4000
Position (nm)
3000
2000
1000
0
0
1
2
3
4
5
time (s) Fig. 5. Trajectory of a dynein motor simulated using a kinetic Monte Carlo method for no load and an ATP concentration of 1 mM. The inset shows the histogram of different step sizes during the movement.
In conclusion, we have modeled the unidirectional and bidirectional movement of single head of dynein motor using stochastic rate equations. Backward steps are implemented using Crook's fluctuation theorem of nonequilibrium statistical mechanics. We model single head of dynein with one primary and one, two and three secondary ATP binding sites and presented the results for one primary and two secondary ATP binding sites, the 3a model. We find the magnitude of unidirectional or bidirectional velocity is larger when more number of secondary ATP binding sites are considered for low load forces. The computed velocity from 4a model of dynein motor is found to be in good agreement with measured velocities. We find that the motor moves backwards for super stall forces. The ratio of velocities corresponding to bidirectional movement and unidirectional movement exhibits an ATP concentration independent universal behavior. We find that for super stall forces, the motor moves backwards by hydrolyzing ATP exactly the same way as it does while moving forward for sub-stall forces. We also simulated the movement of dynein motor using the kinetic Monte Carlo method and the velocity of the simulated dynein motor is in good agreement with the velocity obtained using the stochastic rate equation model. References Bameta, Tripti, Padinhateeri, Ranjith, Inamdar, Mandar M., 2013. Force generation and step-size fluctuations in a dynein motor. J. Stat. Mech. P02030. Bier, M., 2008. Accounting for the energies and entropies of kinesin's catalytic cycle. Eur. Phys. J. B 65, 415–418. Calzetta, E.A., 2009. Kinesin and Crook's fluctuation theorem. Eur. Phys. J. B 68, 601–605.
Carter, N.J., Cross, R.A., 2005. Mechanics of the kinesin step. Nature 435, 308–312. Chowdhury, D., 2013. Stochastic mechano-chemical kinetics of molecular motors: a multidisciplinary enterprise from a physicists perspective. Phys. Rep. 529, 1–197. Crooks, G., 1990. Entropy production fluctuation theorem and nonequilibrium work relation for free energy differences. Phys. Rev. E 60, 2721. Dean Astumian, R., 2010. Thermodynamics and kinetics of molecular motors. Biophys. J. 98, 2401. DeWitt, Mark A., Chang, Amy Y., Combs, Peter A., Yildiz, Ahmet, 2012. Cytoplasmic dynein moves through uncoordinated stepping of the AAAþ ring domains. Science 335, 221–225. Elston, T.C., 2000. A macroscopic description of biomolecular transport. J. Math. Biol. 41, 189–206. Gao, Y.Q., 2006. A simple theoretical model explains dynein's response to load. Biophys. J. 90, 811–821. Gennerich, Arne, Carter, Andrew P., Reck-Peterson, Samara L., Vale, R.D., 2007. Force induced bidirectional stepping of cytoplasmic dynein. Cell 131, 952–965. Hook, Peter, Vallee, Richard, 2012. Dynein dynamics. Nat. Struct. Mol. Biol. 19, 467. Schroeder III, Harry W., Mitchell, Chris, Shuman, Henry, Holzbaur, Erika L.F., Goldman, Yale E., 2010. Motor number controls cargo switching at actinmicrotubule intersections in vitro. Curr. Biol. 20, 687. King, S.M., 2012. Dyneins: Structure, Biology and Disease, first ed. Elseiver. Kunwar, A., Mogilner, A., 2010. Robust transport by multiple motors with non linear force velocity relations and stochastic load sharing. Phys. Biol. 7, 16012. Mallik, R., Carter, B.C., Lex, S.A., King, S.J., Cross, S.P., 2004. Cytoplasmic dynein functions as gear in response to load. Nature 427, 649–652. Mogliner, A., Fisher, A.J., Baskin, R.J., 2001. Structural changes in the neck linker of kinesin explain the load dependence of the motor's mechanical cycle. J. Theor. Biol. 211, 143–157. Nishiyama, M., Huguchi, H., Yanagida, T., 2002. Chemomechanical coupling of the forward and backward steps of kinesin molecule. Nat. Cell. Biol. 4, 790–797. Reck-Peterson, Samara L., Yildiz, Ahmet, Carter, Andrew P., Gennerich, Arne, Zhang, Nan, Vale1, Ronald D., 2006. Single-molecule analysis of dynein processivity and stepping behavior. Cell 126, 335. Ross, Jennifer L., Wallace, Karen, Shuman, Henry, Goldman, Yale E., Holzbaur, Erika L.F., 2006. Processive bidirectional motion of dynein–dynactin complexes in vitro. Nat. Cell Biol. 8, 562. Mukherji, Sutapa, 2008. Model for the unidirectional motion of a dynein molecule. Phys. Rev. E 77, 051916. Serohijos, A.W.R., Tyagankov, D., Liu, S., Elston, T.C., Dokholyan, N.V., 2009. Multiscale approaches for studying energy transduction in dynein. Phys. Chem. Chem. Phys. 11, 4840–4850. Singh, M.P., Mallik, R., Gross, S.P., Wu, C.C., 2005. Monte Carlo modeling of single molecule cytoplasmic dynein. Proc. Natl. Acad. Sci. 102, 12059–12064. Toba, S., Watanabe, T.M., Yamaguchi-Okimoto, Lisa, Toyoshima, Yoko Yano, Higuchi, Hedeo, 2006. Overlapping hand-over-hand mechanism of single molecular motility of cytoplasmic dynein. Proc. Natl. Acad. Sci. 103, 5741. Tyagankov, D., Serohijos, A.W.R., Doholyan, N.V., Elston, T.C., 2009. Kinetic model for the coordinated stepping of cytoplasmic dynein. J. Chem. Phys. 130, 025101. Tyagankov, D., Serohijos, A.W.R., Doholyan, N.V., Elston, T.C., 2011. A physical model reveals the mechanochemistry responsible for dynein's processive motion. Biophys. J. 101, 144–150. Vale, R.D., 2000. AAA proteins: lord of the ring. J. Cell Biol. 150, F13–F19.
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Model for bidirectional movement of cytoplasmic dynein S. Sumathy, S.V.M. Satyanarayana n Department of Physics, Pondicherry University, R.Venkataraman Nagar, Kalapet, Puducherry 605 014, India
H I G H L I G H T S
Stochastic process models are developed for bidirectional motion of dynein motor. Probability for backward step uses Crook's like fluctuation theorem. Average motor velocity is negative beyond stall force. Backward motion beyond stall force is also powered by ATP hydrolysis.
art ic l e i nf o
a b s t r a c t
Article history: Received 20 October 2014 Received in revised form 2 March 2015 Accepted 22 April 2015 Available online 2 May 2015
Cytoplasmic dynein exhibits a directional processive movement on microtubule filaments and is known to move in steps of varying length based on the number of ATP molecules bound to it and the load that it carries. It is experimentally observed that dynein takes occasional backward steps and the frequency of such backward steps increases as the load approaches the stall force. Using a stochastic process model, we investigate the bidirectional movement of single head of a dynein motor. The probability for backward step is implemented based on fluctuation theorem of non-equilibrium statistical mechanics. We find that the movement of dynein motor is characterized with negative velocity implying backward motion beyond stall force. We observe that the motor moves backward for super stall forces by hydrolyzing the ATP exactly the same way as it does while moving forward for sub-stall forces. Movement of dynein is also simulated using a kinetic Monte Carlo method and the simulated velocities are in good agreement with velocities obtained using a stochastic rate equation model. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Dynein motor Force velocity relationship Fluctuation theorem
1. Introduction Molecular motors are nano-machines that do work by harnessing chemical energy (Chowdhury, 2013). Cytoplasmic dynein is a minus end directed motor protein that moves on microtubule. It is implicated in intracellular transport of vesicles, mRNA, protein complexes etc., from cell cortex to center of the cell (King, 2012). Dynein is a homo-dimer with each head consisting of a ring of six domains (Vale, 2000). Four of the six domains of the ring have sites with affinity for ATP binding. Dynein motor exhibits a gear like mechanism in controlling step size in response to the load force (Mallik et al., 2004). A model of single head of a dynein motor is simulated by Monte Carlo method (Singh et al., 2005). Singh et. al. computed force velocity relation, step size distribution from the simulation data and studied the ATP dependence of average velocity of the motor. Further, the load dependence of the
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Corresponding author. E-mail address: [email protected] (S.V.M. Satyanarayana).
http://dx.doi.org/10.1016/j.jtbi.2015.04.029 0022-5193/& 2015 Elsevier Ltd. All rights reserved.
step size and ATP concentration dependence of the stall force are obtained by another model with a weak coupling between two reaction coordinates corresponding to chemical reactions and translocation of the motor (Gao, 2006). A complete mechanochemical model for a hand over hand stepping model of a homodimeric dynein is developed where the ATP hydrolysis cycle is coupled to a coarse grained structural model (Tyagankov et al., 2009, 2011). Dynein motor at a different spatio-temporal resolution is studied by multi-scale modeling (Serohijos et al., 2009). Stochastic process modeling of unidirectional movement of single head of a dynein motor is carried out systematically with one-, two- and three-step process (Sutapa Mukherji, 2008). While dynein moves processively toward the minus end of the microtubule, it is observed that dynein takes backward steps once in a while and the frequency of such backward steps increases as the load increases (Gennerich et al., 2007). Further, it was observed that single head of dynein is sufficient for processive motion (De Witt et al., 2012). In the present work, we develop a stochastic process model for bidirectional movement of cytoplasmic dynein's single head, along
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
the same lines of the model of Sutapa Mukherji (2008). We use Crook's like fluctuation theorem to define the ratio of probability of forward step to that of backward step. Dynein can take 8, 16, 24 or 32 nm step sizes depending on the load and ATP concentration. In this study we investigate bidirectional movement of dynein's single head for all the four step sizes. The paper is organized as follows. Description of stochastic process model, typical reaction scheme, different rate constants used in the study, a system of stochastic rate equations and the procedure to compute average velocity of the motor are presented in Section 2. Section 3 consists of results and discussion.
2. Model Microtubule is modeled as a passive one dimensional lattice on which single head of dynein moves. We assume that dynein's head is always attached to the microtubule site. Single head of dynein consists of one primary and three secondary ATP binding sites. It is assumed that out of four ATP binding sites, ATP hydrolysis takes place only at a primary site. However, the step length depends on the occupancy of the three secondary sites. If we denote that the minimum step size dynein can take as a, where a ¼ 8 nm, the step sizes are 4a, 3a, 2a and a when occupied secondary sites are 0, 1, 2 and 3 respectively. If we consider one primary and one secondary sites, the stochastic variable Skl j denotes the probability that a dynein's head is on the jth lattice site of microtubule with k; l ¼ 0; 1. Here k ¼ 0ð1Þ and l ¼ 0ð1Þ signify the corresponding primary or the secondary site being unoccupied (occupied) respectively. In this case, maximum step size of a single head of dynein is 2a and hence we call this a 2a model. If we have two and three secondary ATP binding sites considered, we refer to them as 3a model and 4a model respectively and denote the corresponding stochastic variable as Sklm and Sklmp . Since each index in the superscript of the stochastic j j variable can be 0 or 1 corresponding to the ATP binding site being unoccupied or occupied, we have number of stochastic variables as 4, 8 and 16 respectively for 2a, 3a and 4a models. The rate of ATP binding on the ith binding site of dynein's head is denoted as koni and the rate of ATP unbinding from the same site is denoted as koffi. Hydrolysis rate at the primary site must decrease with increasing load since we know that the motor stops at stall force. Further, it also depends on whether the secondary ATP sites are occupied. The load dependence of the hydrolysis rate is governed by the Boltzmann factor that is given by the following expression: kcat;i ¼ AðiÞkcat;0 exp½ αFdðiÞ=kB T
ð1Þ
Here, i¼ 1, 2, 3 and 4 and dðiÞ ¼ i a, kcat;0 is the hydrolysis rate for no load, α is the load distribution factor taken as positive since the hydrolysis rate should decrease with the increase in opposing external load and AðiÞ ¼ 1 if any of the secondary sites are occupied and is 0.01 if all the secondary sites are unoccupied. Along the lines of previous studies (Singh et al., 2005; Sutapa Mukherji, 2008), we assume that the ATP binding rates of secondary sites depend on the load as follows: kon;2 4 ðFÞ ¼ kon;2 4 exp½Fd0 =kB T
ð2Þ
d0 is an adjustable parameter in units of length. Cytoplasmic dynein from yeast and mammals shows difference in velocity and stall forces. Yeast dynein exhibits a higher stall force and smaller velocity compared to mammalian dynein (ReckPeterson et al., 2006; Gennerich et al., 2007; Ross et al., 2006; Mallik et al., 2004; Schroeder et al., 2010). Experiments by Mallik et al. (2004) reported that cytoplasmic dynein has a stall force of 1.1 pN at saturating ATP concentration and the stall force decreases with decrease in ATP concentration. On the other hand Toba et al.
49
(2006) observed a stall force of 7–8 pN which is independent of ATP concentration for mammalian cytoplasmic dynein. Further, Toba et al. reported a velocity of dynein motor around 800 nm/s. It is argued that the differences observed with respect to stall force and velocity in Yeast and mammalian dynein may be of structural origin (Hook and Vallee, 2012), that is, the absence of C terminal domain in Yeast may be responsible for higher force and lower velocity. In this model we consider the stall force of 7.5 pN as an input parameter and compute the velocity of the dynein motor. Molecular motors are known to take backward steps amidst their processive forward motion. It is observed that the directional movement of motors is governed by thermodynamics that restrict backward steps (Dean Astumian, 2010). This motivated us to examine the role of thermodynamic fluctuation theorem such as Crook's fluctuation theorem (Crooks, 1990) for motor taking backward steps. The probability ratio for forward and backward steps for kinesin motor is estimated from experiment (Carter and Cross, 2005; Nishiyama et al., 2002). Subsequently, the expression for the ratio of forward and backward step probabilities was derived from Crook's like fluctuation theorem of non-equilibrium thermodynamics for kinesin motor (Bier, 2008; Calzetta, 2009). In this work, we assume that the same expression is valid for dynein motor as well. The ratio of the probability for a forward step to backward step is given as PF d0 ½F s F ¼ exp ð3Þ PB 2kB T where FS is the stall force. Also note that P F þ P B ¼ 1 since dynein motor can take only forward or backward steps in our model. Since our interest is to understand the role of backward steps in the dynamics of dynein, in order to compare the results with previous model, we considered different rate constants and parameters in this study same as in the previous studies (Singh et al., 2005; Sutapa Mukherji, 2008). They are given in Table 1. Here kBT is the thermal energy and kB is the Boltzmann constant. In this study, we have computed the average velocity of single head of dynein molecule with 2a, 3a and 4a models with and without including backward steps. We investigate the velocity of the motor as a function of force both below (substall forces) and above (super stall forces) the stall force. We present the reaction scheme and stochastic rate equation model for bidirectional 3a model as a representative case. 2.1. The model In a 3a model, we have eight state variables for single head for dynein motor. The state variable vector on the jth lattice site of microtubule is given by
ρj ¼ ½S000 S001 S010 S100 S011 S101 S110 S111 T j j j j j j j j Table 1 Rate constants and parameters used in the study. Symbol
Value
kB T koff1 koff2 koff3 koff4 kon1
4.1 pN nm 10 s 1 250 s 1 250 s 1 250 s 1
kon2 ðF ¼ 0Þ
4 105 M 1 s 1 ½ATP kon2 ðF ¼ 0Þ=4 kon2 ðF ¼ 0Þ=6 6 nm 55 s 1 0.3
kon3 ðF ¼ 0Þ kon4 ðF ¼ 0Þ d0 kcat;0 α
4 105 M 1 s 1 ½ATP
ð4Þ
50
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
Sj001 kon1
Sj101
koff3 100
Sj-3
kcat3
Sj000
kon2
where the matrix Rðζ Þ is given by
Sj111
0
kon1 B 0 B B B 0 B B k B on1 ½Rðζ Þ ¼ B B 0 B B B 0 B B 0 @ 0
koff3 kon1 koff1
Sj100
kcat3
Sj+3
000
koff2 kon2
koff2
koff 3 ðk7 Þ 0 0 0 kon1 0 0
koff 2 0 ðk6 Þ 0 0 0 kon1 0
k1 0 0 ðk4 Þ 0 0 kon2 0
Sj-2
kcat2
Sj010
kon1
kon3
koff3 Sj-1111 kcat1
Sj011
Sj110
kon1
kcat2
Sj+2
kcat1
Sj+1011
Fig. 1. Reaction scheme for 3a model of single head of a dynein motor.
〈v〉 ¼ a The reaction scheme for the 3a model is presented in Fig. 1. In this scheme, when hydrolysis takes place on the primary site when none of the secondary sites are occupied, the stochastic 000 variable S100 changes to S000 j j þ 3 with probability PF and to Sj 3 with probability 1 P F . Further, hydrolysis at primary site with one (two) primary sites are occupied, the stochastic variable S110 ðS111 Þ j j 010 011 010 011 changes to Sj þ 2 ðSj þ 1 Þ with probability PF and Sj 2 ðSj 1 Þ with probability 1 P F . This model is similar to random walk in two dimensional space where one coordinate is the reaction coordinate and the other is the spatial coordinate (Bameta et al., 2013). The stochastic rate equations for the eight state variables of 3a model corresponding to the reaction scheme presented in Fig. 1 are given below 000 dSj
dt
¼ kon1 S000 þ koff 3 S001 þ koff 2 S010 þ koff 1 S100 þ ½P F S100 j j j j j3
001
dt
¼ ðkoff 3 þ kon1 ÞS001 j
010
dSj
dt dt dt
〈j〉 0 ¼ aλl ð1Þ t
111 ¼ ðkon1 þ koff 3 ÞS011 þ ½P F S111 j j 1 þ P B Sj þ 1 kcat1
101
dt
3. Results and discussion
3
[ATP] 5µM 50µM 500µM 1mM 2mM
2
10
ð5Þ
In matrix form these equations can be recast as dρ ¼ ½Aρj þ ½Bρj 1 þ ½Cρj þ 1 þ ½Dρj 2 þ½Eρj þ 2 þ ½Fρj 3 þ ½Hρj þ 3 dt ð6Þ Here A; B; C; D; E; F and H are matrices whose elements are various P j rate constants. Using a generating function, Gðζ ; tÞ ¼ 1 j ¼ 1 ζ ρj , the above equation can be recast as ! d 1 1 1 2 3 Gðζ ; tÞ ¼ ½A þ ζ ½B þ ½c þ ζ ½D þ 2 ½E þ ζ ½F þ 3 ½H Gðζ ; tÞ dt ζ ζ ζ ¼ ½Rðζ ÞGðζ ; tÞ
ð11Þ
10
¼ kon1 S010 þ kon2 S100 ðkcat2 þ kon3 þkoff 2 ÞS110 þ koff 3 S111 j j j j ¼ kon1 S011 þ kon2 S101 þ kon3 S110 ðkcat1 þkoff 3 ÞS111 j j j j
ð10Þ
For finding the derivative of the largest eigenvalue at ζ ¼ 1, it is 0 ″ 2 convenient to substitute ζ ¼ 1 þ δ and λ ¼ δλ ð1Þ þ ðδ =2Þλ ð1Þ in 0 Eq. (10) and λ ð1Þ is find out by equating the coefficients of δ to zero (Mogliner et al., 2013; Elston, 2000).
ð7Þ
U
111 dSj
ð9Þ
Det½Rðζ Þ λI ¼ 0
V (nm/s)
dt
2
where 〈j〉 is average index on the lattice where the motor is located at time t and λl ðζ Þ is the largest eigenvalue of ½Rðζ Þ and the primes denote the derivatives of λ with respect to ζ. For finding the largest eigenvalue, it is necessary to solve the characteristic equation
dSj ¼ kon1 S001 ðkon2 þ koff 3 ÞS101 j j dt 110 dSj
4
From the computed data of force velocity curves, we estimated the fit parameters V U ð0Þ ¼ 425:2 nm=s and γ ¼ 0:583 pN 1 . The observed force velocity relation in Eq. (11) is found to differ from the general form of force velocity relation for motors (Kunwar and Mogilner, 2010).
¼ kon1 S000 ðkcat3 þ kon2 þ koff 1 ÞS100 þ koff 3 S101 þ koff 2 S110 j j j j
011
dSj
3
V U ðFÞ ¼ V U ð0Þ expð γ FÞ
110 ¼ ðkon1 þ koff 2 ÞS010 þ koff 3 S011 þ ½P F S110 j j j 2 þ P B Sj þ 2 kcat2
100
dSj
1 0 C 0 C C C 0 C C 0 C C C k3 C C 0 C C C koff 3 A ðkcat1 þ koff 3 Þ
Force velocity relation VðFÞ is one of the important characteristics of the motor. We present in Fig. 2, the average velocity of unidirectional movement (corresponding to PF ¼1 in the model) of single head of dynein motor in 3a model. The force velocity relation for ATP concentration close to and higher than physiological concentrations (1 mM) is found to be
þ P B S100 j þ 3 kcat3 dSj
koff 3 0 ðkon2 þ koff 3 Þ 0 kon2
0 0 k2 koff 2 0 0 ðk5 Þ kon3
Here k1 ¼ koff 1 þ ½ð1 þ P F ðζ 1ÞÞ=ζ kcat3 , k2 ¼ ½ð1 þP F ðζ 1ÞÞ=ζ 2 kcat2 , k3 ¼ ½ð1 þ P F ðζ 1ÞÞ=ζ kcat1 , k4 ¼ koff 1 þ kon2 þ kcat3 , k5 ¼ kcat2 þ kon3 þ koff 2 , k6 ¼ kon1 þ koff 2 and k7 ¼ kon1 þ koff 3 . The transition matrix R½ζ ¼ 1 has the property that the sum of all elements in a column is zero. Thus, the largest eigenvalue of the matrix R½ζ ¼ 1 is zero. Our aim is to find the average velocity and it is found from the relation
010
koff3 Sj111
koff 3 0 ðk7 Þ 0 0 kon1
0 0 0
ð8Þ 6
110
0 0
1
10
0
10
-1
10
-2
10
0
5
10
15
F(pN) Fig. 2. Force velocity relation for unidirectional movement of single head of a dynein motor in 3a model.
S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
600
51
1 1.5
B
V (nm/s)
400 300
1
5 µM 50 µM 500 µM 1 mM 2 mM
0 -0.5 -1
200
[ATP]
0.5
0.5
VB / VU
[ATP] 5µM 50µM 500µM 1mM 2mM
500
-1.5
8
10
12
14
0
16
100
-0.5
0 -100
-1
0
5
10
15
Average velocity for a bidirectional motion of single head of dynein in 3a model as a function of force for different ATP concentrations is shown in Fig. 3. It can be seen from Fig. 3 that at the stall force, the average velocity of the motor is zero. This is because the probability of forward step is equal to the probability of backward step, that is, on an average the number of forward steps is equal to the number of backward steps. This is in good agreement with observations made in single molecule experiments with force feedback optical tweezers at an ATP concentration of 1 mM (Gennerich et al., 2007). It can be seen from the inset of Fig. 3 that beyond stall force (7.5 pN) the velocity is negative, that is the motor walks backwards (in a direction opposite to its natural direction of motion in the absence of any load force). We observe from Fig. 3 and the inset that the average negative velocity as a function of force for super stall forces is independent of ATP concentration. This result is also in a good agreement with single molecule experiments (Gennerich et al., 2007). Further, as F increases beyond stall force, the negative average velocity decreases and approaches zero for high super stall force. Fig. 4 represents the ratio of velocities of bidirectional and unidirectional movement. It can be observed that this ratio is independent of ATP concentration in a range of ATP concentrations from 5 μM to 10 mM studied in this work for all forces. As F-0, V B ¼ V U , as can be expected. For F c F S , it is observed that V B ¼ V U . This implies for F c F S , the cargo acts like a rigid wall and the motor gets reflected with the same velocity in the opposite direction, in the average sense.The ratio given in Fig. 4 has the following functional form: VB ðF s FÞδ ð12Þ ¼ tanh VU 2kB T The value of δ is found to be 3 nm. For sub-stall forces, bidirectional velocity is a fraction of unidirectional velocity. This reduction in velocity is due to the number of backward steps that the motor head takes in a bidirectional motion. For super stall forces, the number of backward steps is more than the number of forward steps and as a result the average velocity is negative, with a magnitude equal to a fraction of the unidirectional velocity. For F c F S , almost all steps are backward steps. However, we observe that the average negative velocity of the bidirectional motor decreases in much the same way as the average positive velocity of the unidirectional motor approaching zero from negative and positive directions respectively. This can be understood as follows. Dynein motor hydrolyses ATP and the rates of ATP binding to different binding sites and ATP hydrolysis depend on the force. For every ATP hydrolysis event, a step forward or backward is taken
5
10
15
20
Force (pN)
F (pN) Fig. 3. Average velocity versus force for bidirectional movement of single head of a dynein motor for different ATP concentrations. The inset shows the average velocity beyond stall force (7.5 pN).
0
Fig. 4. Ratio of velocities corresponding to bidirectional and unidirectional movement of dynein motor as a function of force for different ATP concentrations.
based on probabilities obtained from fluctuation theorem. For super stall forces, the rates corresponding to ATP binding and hydrolysis are low such that number of ATP hydrolysis events become very less and tend to zero. The average bidirectional velocity is small negative for F c F S implies that ATP hydrolysis events, however small they are, lead to backward steps of the motor. During the backward movement the motor hydrolyzes ATP in exactly the same way as it does while moving forward for force F o F S . We observe that the 4a model where we have four ATP binding sites for single head of the dynein gives similar results. However, the magnitude of the average velocity is higher for 4a model compared to 3a model in both unidirectional as well as bidirectional cases. For example, the velocity of bidirectional movement of single head of dynein motor under no load is 461 nm/s in 3a model and 711 nm/s in 4a model. The velocity of dynein motor for 4a model is in reasonable agreement with experimental observations (Toba et al., 2006). In order to corroborate our stochastic rate equations model, we simulated the movement of cytoplasmic dynein motor with four ATP binding sites (4a model) using kinetic Monte Carlo method. We have 16 biochemical states of dynein motor participating in 34 reactions with respective rates of reactions, say Ri 34 i ¼ 1 . The state of the dynein motor is randomly chosen to be one of the 16 states. A reaction is randomly chosen and if the reactant corresponding to the chosen reaction coincides with the state of dynein motor, the reaction is implemented. The time is advanced according to the equation
Δt ¼
ln u RT
ð13Þ
where u is a uniform random number between 0 and 1 and RT is the total rate of all the reactions. Rate constant values used in the simulation are same as in Table 1. Backward steps are implemented with the probabilities obtained from Eq. (3). Fig. 5 presents the trajectory of the dynein motor corresponding to ATP concentration of 1 mM and zero load force. The trajectory exhibits saltatory movement, which is characteristic of biological molecules. The velocity of motor is computed and is given by 713 nm/s, which is in good agreement with the value obtained using stochastic rate equation 4a model. The inset of Fig. 5 shows the histogram of different steps dynein motor has take during the course of movement. Negative steps are insignificant since probability for backward step for zero load is very small. As can be seen from the figure, 24 nm steps are large in number followed by 16, 32 and 8 for 1 mM ATP concentration.
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S. Sumathy, S.V.M. Satyanarayana / Journal of Theoretical Biology 380 (2015) 48–52
4000
Position (nm)
3000
2000
1000
0
0
1
2
3
4
5
time (s) Fig. 5. Trajectory of a dynein motor simulated using a kinetic Monte Carlo method for no load and an ATP concentration of 1 mM. The inset shows the histogram of different step sizes during the movement.
In conclusion, we have modeled the unidirectional and bidirectional movement of single head of dynein motor using stochastic rate equations. Backward steps are implemented using Crook's fluctuation theorem of nonequilibrium statistical mechanics. We model single head of dynein with one primary and one, two and three secondary ATP binding sites and presented the results for one primary and two secondary ATP binding sites, the 3a model. We find the magnitude of unidirectional or bidirectional velocity is larger when more number of secondary ATP binding sites are considered for low load forces. The computed velocity from 4a model of dynein motor is found to be in good agreement with measured velocities. We find that the motor moves backwards for super stall forces. The ratio of velocities corresponding to bidirectional movement and unidirectional movement exhibits an ATP concentration independent universal behavior. We find that for super stall forces, the motor moves backwards by hydrolyzing ATP exactly the same way as it does while moving forward for sub-stall forces. We also simulated the movement of dynein motor using the kinetic Monte Carlo method and the velocity of the simulated dynein motor is in good agreement with the velocity obtained using the stochastic rate equation model. References Bameta, Tripti, Padinhateeri, Ranjith, Inamdar, Mandar M., 2013. Force generation and step-size fluctuations in a dynein motor. J. Stat. Mech. P02030. Bier, M., 2008. Accounting for the energies and entropies of kinesin's catalytic cycle. Eur. Phys. J. B 65, 415–418. Calzetta, E.A., 2009. Kinesin and Crook's fluctuation theorem. Eur. Phys. J. B 68, 601–605.
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