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Effect of fuel concentration on cargo transport by a team of Kinesin motors
Physica A 467 (2017) 395–406
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Effect of fuel concentration on cargo transport by a team of Kinesin motors Anjneya Takshak 1 , Nirvantosh Mishra 1 , Aditi Kulkarni, Ambarish Kunwar ∗ Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Mumbai 400076, India
highlights • • • • • • • •
A team of Kinesin motor proteins transports cellular cargos using ATP as fuel. How fuel concentration affects transport by the team largely remains unexplored. Mechano-chemical models of cargo transport developed here fill up this knowledge gap. Mechano-chemical models include effect of both force and fuel on cargo transport. Models predict very large cargo travel distances at limiting fuel concentrations. Models predict Michaelis-Menten dependence of cargo velocity on fuel concentration. These predictions from modeling can be directly tested by in-vitro experiments. Our results may be used to regulate and fine-tune transport by motor-based shuttles.
article
info
Article history: Received 28 January 2016 Received in revised form 21 August 2016 Available online 12 October 2016 Keywords: Molecular motors Kinesin Mechano-chemical enzymes Monte-Carlo simulations
∗
Corresponding author. E-mail address: [email protected] (A. Kunwar).
1 First Co-authors. http://dx.doi.org/10.1016/j.physa.2016.10.022 0378-4371/© 2016 Elsevier B.V. All rights reserved.
abstract Eukaryotic cells employ specialized proteins called molecular motors for transporting organelles and vesicles from one location to another in a regulated and directed manner. These molecular motors often work collectively in a team while transporting cargos. Molecular motors use cytoplasmic ATP as fuel, which is hydrolyzed to generate mechanical force. While the effect of ATP concentration on cargo transport by single Kinesin motor function is well understood, it is still unexplored, both theoretically and experimentally, how ATP concentration would affect cargo transport by a team of Kinesin motors. For instance, how does fuel concentration affect the travel distances and travel velocities of cargo? How cooperativity of Kinesin motors engaged on a cargo is affected by ATP concentration? To answer these questions, here we develop mechano-chemical models of cargo transport by a team of Kinesin motors. To develop these models we use experimentally-constrained mechano-chemical model of a single Kinesin motor as well as earlier developed mean-field and stochastic models of load sharing for cargo transport. Thus, our new models for cargo transport by a team of Kinesin motors include fuel concentration explicitly, which was not considered in earlier models. We make several interesting predictions which can be tested experimentally. For instance, the travel distances of cargos are very large at limited ATP concentrations in spite of very small travel velocity. Velocities of cargos driven by multiple Kinesin have a Michaelis–Menten dependence on ATP concentration. Similarly, cooperativity among the engaged Kinesin
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motors on the cargo shows a Michaelis–Menten type dependence, which attains a maximum value near physiological ATP concentrations. Our new results can be potentially useful in controlling artificial nano-molecular shuttles precisely for targeted delivery in various nano-technological applications. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Molecular motors are specialized proteins which perform the long-range and short range transport of cargo particles (vesicles, mitochondria, mRNA particles, liposomes, pigment granules, etc.) along cytoskeletal filaments in eukaryotic cells [1]. The commonly employed proteins which perform this cargo trafficking inside cells belong to three families: Kinesin superfamily, Dynein and Myosin families [1,2]. Most Kinesin superfamily members are (+) end directed, which means that they carry cargo from the cell interior to the cell periphery. Several in-vitro single molecule studies have revealed the details about the mode of action of Conventional Kinesin (Kinesin-1) [2,3]. It is a highly processive motor which takes several steps on the microtubule [4] prior to detaching. It has two motor domains through which it attaches to the microtubule and travels in discrete steps of 8 nm in a coordinated hand over hand mechanism. During the process of cargo transport, a single Kinesin motor can move processively against an opposing force (load) in the range of up to 5 to 7 pN [5]. Additionally, Kinesin motors can also detach from microtubule with load-dependent unbinding rate and reattach [6–8]. Since the amount of force generated by a single Kinesin motor is usually insufficient for effective long-range intra-cellular transport, multiple Kinesin motors often work together as team to achieve this [9,8]. Several models have been proposed to study cargo transport by a team of molecular motors [10–16]. Klumpp et al. [6] proposed a model considering that the applied load on the cargo is distributed equally among all engaged motors. Models later developed by Kunwar et al. [17,18] incorporated stochastic load sharing among multiple motors engaged on a cargo. However, these models cannot be used to understand the effect of fuel concentration on cargo transport by a team of motors, as underlying single motor characteristics do not incorporate fuel dependence. In this paper, we use an experimentally-constrained model of single Kinesin motor to study cargo transport by a team of multiple Kinesin motors using earlier developed Mean-field [6] and Stochastic models [17,18] of load sharing. Hereafter, we shall refer to Mean-field model of load sharing [6] as Mean-field model, and Stochastic model of load sharing [17,18] as Stochastic model. 2. Experimentally-constrained model of single Kinesin motor Schnitzer et al. [5] proposed a load dependent Composite State Model to describe the stepping behavior of a single Kinesin motor over a microtubule which incorporates effect of both load and fuel (ATP) concentration. When an ATP molecule is bound to one of the two heads of a Kinesin motor, the motor is said to be in Composite State. According to this model, Kinesin head can detach from the microtubule, with a smaller ATP concentration-dependent probability, before ATP binding (State I). Kinesin head can also undergo detachment with a greater load-dependent probability, after ATP binding (State II). The experimental velocity curves of single Kinesin motors can be fitted using a Michaelis–Menten equation [5] as a function of ATP concentration (denoted by [ATP ]), at different loads as given by the following relation
v (F , [ATP]) =
dKcat (F ) [ATP] Vmax [ATP ] = K F [ATP ] + KM [ATP] + Kcat((F ))
(2.1)
b
where Vmax (= dKcat ) is the velocity at saturating ATP concentration in nm/s, d is the step size of a single Kinesin motor in nm, Kcat (F ) is the load-dependent catalytic turnover constant, and Kb (F ) is the load dependent second order rate constant for ATP binding to the Kinesin motor. Both the rate constants exhibit an inverse exponential type Boltzmann relationship with load as follows: Kcat (F ) =
o Kcat
pcat + qcat exp Kb ( F ) =
F .δcat kB T
Kbo pb + qb exp
F δb kB T
o where Kcat and Kbo are the respective rate constants, measured in s−1 and µM −1 s−1 respectively, in the absence of external load. kB T is the thermal energy at temperature T , δcat and δb represent the characteristic motor distances in nm over which the load acts during catalysis and binding respectively. pcat and qcat (= 1 − pcat ) are the fractions of the unloaded catalytic
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cycle required for biochemical transitions [5], whereas, pb and qb (= 1 − pb ) are the fractions of the unloaded catalytic cycle required for mechanical transitions of Kinesin motor [5]. The theoretical basis for the particular choice of the forms of Kcat (F ) and Kb (F ) can be understood as follows: Each of the two events of kinesin stepping cycle i.e. ATP binding and ATP hydrolysis can be modeled as a sequence of a load-independent biochemical transition followed by a load-dependent mechanical transition over a free energy barrier. The time required for mechanical transition over the free energy barrier increases with load following a Boltzmann-type relation given below: Ko
K (F ) =
p + q exp
Fδ kB T
where K o is the unloaded rate constant, kB T is the thermal energy at temperature T , δ represents the characteristic motor distance in nm over which the load acts. p and q (= 1 − p) are the fractions of the unloaded catalytic cycle required for biochemical transition and mechanical transition respectively [5]. The measured runlength of a single Kinesin motor also exhibits a Michaelis–Menten behavior as a function of ATP concentration, and an exponential Arrhenius–Eyring Kinetics with respect to load as follows [5] L (F , [ATP]) =
Lo [ATP ] [ATP ] + LM dA exp
=
−F δ L
kB T
[ATP]
[ATP] + B 1 + A exp
−F δ L
(2.2)
kB T
) is the runlength at saturating ATP concentration. δL is the characteristic distance over which BT k −F δ the load acts. LM = B 1 + A exp( k TL ) is that value of ATP concentration at which the motor runlength becomes half B
where Lo = dA exp(
−F δL
of its maximum value at a particular load. This equation was derived considering detachment of Kinesin head from either of two distinct states—State I and State II as described above. Here, (B/ [ATP]) is the probability of motor head detachment when it is in state I, which reduces with increasing ATP concentration as B is a constant. To include the exponential decline in runlengthwith load, motor detachment under State II was assumed to exhibit a load-dependent probability given by 1 A
F δL BT
exp( k
) , where A is a constant whose value is equal to average number of Kinesin catalytic cycles in the absence of
load prior to detachment. Velocity of single Kinesin motor starts to decrease with increasing load and motor finally stalls at high loads. The minimum opposing force (load) required to stall the motor is called stall force (Fs ). Motors experiencing forces greater than Fs are said to be in super-stall regime. In addition, Kinesin motor can also detach from the microtubule and its detachment rate for F ≤ Fs can be obtained by the following argument. One can think of average velocity of single motor given by Eq. (2.1) as average number of steps taken by the motor before detachment, multiplied by step size and divided by average time taken to dissociate from the microtubule. Similarly, the average runlength given by Eq. (2.2) is essentially the average number of steps taken by single motor before detachment from microtubule, multiplied by step size. Hence, one can obtain detachment rate of single motor by dividing Eq. (2.1) by Eq. (2.2). Therefore, detachment rate of a motile single Kinesin motor is given by
ϵ (F , [ATP]) =
Vmax ([ATP] + LM ) Lo ([ATP] + KM )
,
F ≤ Fs .
(2.3)
The detachment rate of single Kinesin motor for loads greater than stall force has been measured experimentally by Kunwar et al. [19] and is given by the following relation.
ϵ (F ) = 1.07 + (0.186F ) ,
F > Fs .
(2.4)
The dependence of detachment rate on ATP concentration has not been reported by anyone in the super-stall regime i.e. F > Fs. Therefore, we assume this dependence to be of Michaelis–Menten type as a function of ATP concentration for the following two reasons: (i) Kinesin motors spend longer time attached to microtubules at low ATP concentrations, and time spent decreases with increasing ATP concentration [5]. (ii) Kinesin is a mechano-chemical enzyme and most enzymes show Michaelis–Menten behavior as a function of substrate concentration. This assumption leads to the following relation for detachment of single Kinesin for loads in super-stall regime
ϵ (F , [ATP]) =
(1.07 + (0.186F )) [ATP] , [ATP] + KMϵ
F > Fs
(2.5)
where KMϵ is the Michaelis constant for motor detachment. We propose that discontinuity in detachment rate of motor at stall force (which is the interface of sub-stall and super-stall regimes) may arise from the fact that the motor moves with at least one head attached in the sub-stall regime at all times, but
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Fig. 1. Schematic of (A) Mean-field Model of load sharing and (B) Stochastic model of load sharing for a cargo being transported by n = 3 engaged motors out of total available N = 4 motors. In Mean-field model of load sharing, applied load F is shared equally among n engaged motors; whereas in Stochastic Model of load sharing, n engaged motors share load unequally.
does not move/step in the super-stall regime, where both of its heads remain bound to the microtubule. Thus, the response of the motor to an external force can be completely different in above two regimes which results in a discontinuity in the detachment rate at stall force. A number of models have been proposed to understand the biophysics of such systems which behave differently under low and high loads [20]. One recent hypothesis suggests that allosteric structural changes might occur in both motor and microtubule binding domains under high loads. These structural changes might lead to locking or tight coupling between motor head and microtubule binding site [21]. Experimental data suggest the tight coupling between motor head and microtubule binding domain persists roughly up to two times the stall force [19]. In the next sections, we describe how we used earlier developed Mean-field Model [6] and Stochastic Model [17,18] to understand cargo transport by a team of Kinesin motors using above experimentally-constrained model of single Kinesin motor which captures its mechano-chemistry i.e. effect of load as well as fuel concentration on its transport properties. 3. Models of cargo transport by a team of Kinesin motors 3.1. Mechano-chemical mean-field model of cargo transport To develop mechano-chemical mean-field model of cargo transport, we used the popular mean-field model proposed by Klumpp et al. [6] (summarized in Appendix A). In this model, a cargo is cooperatively transported by N molecular motors along a filament, where number of bound/engaged motors (n) changes due to motor detachment with rate ϵn and motor reattachment with rate πn , such that 0 < n ≤ N. The cargo in state n travels with velocity vn . To include the effect of both ATP and load in the mean-field model, we replaced linear-force–velocity relation in mean-field model by new force–velocity relation given in Eq. (2.1); i.e. Eq. (A.1) in Appendix A was replaced by Eq. (2.1). In addition, experimentallyconstrained force–dissociation relations given by Eqs. (2.3) and (2.5) (for F ≤ Fs and F > Fs respectively) were used to replace exponentially growing force–dissociation relation in mean-field model; i.e. Eq. (A.3) in Appendix A was replaced by Eqs. (2.3) and (2.5). In mean-field model, any load F experienced by cargo is shared equally among n bound motors i.e. each motor experiences a force of F /n (Fig. 1(A)). Therefore, corresponding expressions for vn and ϵn for mechano-chemical meanfield model were obtained by replacing F by F /n in above-mentioned equations. The average runlengths and velocities of the cargo for various load and ATP concentrations were calculated using probabilities Pn (n = 0, 1, . . . , N ) of cargo being in different states, as previously done in mean-field model [6]. Parameters used in calculations and their sources are given in Table 1. 3.2. Mechano-chemical stochastic model of cargo transport To develop mechano-chemical stochastic model of cargo transport, we use stochastic model developed by Kunwar et al. [17,18]. In this model, load among engaged motors is shared unequally due to stochastic stepping of motors, as well as their stochastic detachment and reattachment (Fig. 1(B)). The magnitude of load shared by a motor is determined by its stiffness and its relative position in the team with respect to cargo. Dividing Eq. (2.1) by step size d yields stepping rate σ for bound motors in our model (Eq. (C.1) in Appendix C). The detachment rate in our model was obtained by replacing F by |F | in Eqs. (2.3) and (2.5). |F | incorporates the fact that forward and backward loads have same effect on motor detachment rates as assumed in earlier works [17,18]. It is possible that some motor proteins may not have identical detachment rates under forward and backward loads e.g. yeast dynein [22]. A recent work by Takshak et al. has explored the effect such as anisotropy in detachment rate on cargo transport by a team of motor proteins [23]. The probabilities of motor binding, unbinding and stepping at any time step in the simulations were obtained by multiplying the respective rates of these events with time step ∆t. Time step ∆t was chosen in such a way that fastest event happens at most once during the interval ∆t. Detailed simulation procedure is given in Appendix B. Parameters used in simulations and their sources are given in Table 1.
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Table 1 Parameter values used for both Mechano-chemical Mean-field and Mechano-chemical Stochastic model. Parameter
Symbol
Value [Source(s)]
Step size of Kinesin motor Rest length of Kinesin motor Stiffness of Kinesin motor Stall force for Kinesin motor Detachment force for Kinesin motor Average number of catalytic cycles at zero load prior to detachment Constant for detachment probability prior to ATP bonding Characteristic distance associated with ATP binding Characteristic distance associated with ATP hydrolysis Characteristic distance associated with load dependence Rate constant for ATP hydrolysis at zero load Rate constant for ATP binding at zero load Probability of ATP non-hydrolysis at zero load Probability of ATP non-binding at zero load Probability of ATP hydrolysis at zero load Probability of ATP binding at zero load Michaelis constant for motor detachment Kinesin detachment rate under zero load Kinesin reattachment rate Time step used in simulations
d l k Fs Fd A B
8 nm [5] 110 nm [17–19] 0.32 pN/nm [17–19] 6 pN [6,17] 3 pN [6,17] 107 [5] 2.9 × 10−2 µM [5] 3.7 nm [5] 3.7 nm [5] 1.3 nm [5] 103 s−1 [5] 1.3 µM−1 s−1 [5] 6.2 × 10−3 [5] 4.0 × 10−2 [5] 1 − qcat [5] 1 − qb [5] 100 µM 1/s [6,17] 5/s [6,17] 10−4 s
δb δcat δL
o Kcat Kbo qcat qb pcat pb KMϵ
ϵo πo ∆t
4. Results We used our new mechano-chemical mean-field and stochastic models to obtain the collective transport properties of cargo driven by maximum N Kinesin motors (N = 1, 2, 3 and 4). Results obtained are discussed in the subsequent sections. 4.1. Mechano-chemical models predict very large cargo runlengths at limiting fuel concentrations The average runlengths from our new mechano-chemical models as a function of ATP concentration are shown in Fig. 2. The runlength increases with the number of motors (N ) in both models. The runlength increases in a Michaelis–Menten manner for a single motor as ATP concentration is increased. However, for multiple motor driven cargos (N > 1) the runlengths are quite large at low/limiting ATP concentrations (few µM) in comparison to saturating ATP concentrations (mM). These predictions are consistent with experimentally measured runlengths of multiple motor driven cargos, at low ATP concentrations and zero load, by Xu et al. [24]. At low ATP concentrations, runlength increases as detachment rates of individual motors decrease (Appendix C Fig. C.1(A)), which in turn increases overall residence time of cargo. Both mechano-chemical models predict a non-monotonic variation of runlength with ATP concentrations in low ATP regime at low forces (1.1 pN), where it first increases up to 3 µM and decreases thereafter. To understand this, we calculated ratio of cargo stepping rate to cargo detachment rate. Both of these rates increase as ATP concentration is increased from 1 µM to 3 µM. However, stepping rate increases more rapidly than detachment rate, which results in increased runlength. Beyond 3 µM, although both stepping rate and detachment rate increase, yet their ratio decreases. This results in monotonic decrease in runlength beyond 3 µM. At higher force (7.2 pN), the ratio of stepping to detachment rate always decreases monotonically with increasing ATP concentration. This results in monotonic decrease of runlength with ATP concentration at higher loads. In short, variations of ratio of cargo stepping to cargo detachment rates are reflected in variations of runlength with ATP concentration (Fig. D.1 in Appendix D). Interestingly, increase in runlength with number of motors at different forces was similar to earlier studies [6,17,18] and is shown in Fig. 3. For single Kinesin motor, the runlength decreases exponentially as load increases. However, in multiple Kinesin driven cargos, such monotonous decrease is not observed with respect to load (Fig. 3). In the sub-stall regime (F ≤ Fs ), the runlength first decreases, and then starts to increase as applied load approaches Fs . For (F > Fs ), the cargo runlength decreases monotonically with force. Such variation of runlength with applied load predicted, from both our mechano-chemical models, are similar to earlier published results [19]. These non-monotonic variations of runlengths have their origin in non-monotonic behavior of detachment rate of single Kinesin with force as reported earlier [17,19]. 4.2. Mechano-chemical stochastic model predicts lower cargo velocities at high loads at all fuel concentrations Both mechano-chemical mean-field and mechano-chemical stochastic models predict that cargo velocity has Michaelis–Menten dependence on ATP concentration. At low loads (1.1 pN), velocities of multiple motor driven cargos, predicted by mechano-chemical mean-field model, are roughly equal to single motor velocity. However, mechano-chemical stochastic model predicts a smaller velocity than single motors as reported earlier [17,18], for multiple motor driven cargos. At low loads (1.1 pN), velocities predicted by mechano-chemical stochastic model are consistent with limited experimental observation, at low ATP concentrations and zero load, by Xu et al. [24]; where a decrease in cargo velocity was observed
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Fig. 2. Variation of cargo runlength with ATP concentration at (A) low load and (B) high load. Figure legend MMM refers to data from Mechano-chemical Mean-field Model and legend MSM refers to data from Mechano-chemical Stochastic Model. In case of single motor driven cargo (N = 1) both models give similar results.
Fig. 3. Variation of cargo runlength with applied load at (A) low and (B) high ATP concentration. Figure legends are similar to Fig. 2.
Fig. 4. Variation of Cargo Velocity with ATP concentration at (A) low load and (B) high load. Figure legends are similar to Fig. 2.
when motor number was changed from N = 1 to N = 2 (Fig. 4(A) inset). At higher loads (7.2 pN), both mechano-chemical models predict that multiple motor driven cargo would be moving faster than single motor driven cargo. Cargo velocity decreases monotonically in a convex-up manner for a single motor driven cargo as applied load increases (Fig. 5). However, mechano-chemical stochastic model predicts almost linear decrease of cargo velocity with load, at high as well as low ATP concentrations. Such variation of cargo velocity with load have been reported earlier too [17,18]. In contrast, mechano-chemical mean-field model predicts convex-up nature of cargo velocity with load, at high as well as low ATP concentrations. We find that for the high-loads, the mechano-chemical mean-field model predicts a consistently higher average velocity than mechano-chemical stochastic model for N = 2 and N = 3. The deviations between the results of two models do not appear to be so prominent in the low-load regime (Fig. 4(A)). The origin of this difference in average velocity in low-load and high-load regimes can be understood as follows: In Mechano-chemical Mean-field Model (MMM), the load is equally
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Fig. 5. Variation of cargo velocity with applied load at (A) low and (B) high ATP concentration. Figure legends are similar to Fig. 2.
shared among the engaged motors. However, in Mechano-chemical stochastic model (MSM) the load shared by each engaged motor fluctuates due to their stochastic dynamics of stepping and detachment. In MSM, for low load regime (e.g. 1.1 pN) the differences among the fluctuating forces are small whereas they are very large for high load regime (7.2 pN). This results in large difference in average velocity in high load regime. 4.3. Cooperativity among motors also shows a Michaelis–Menten dependence on fuel concentration Cooperativity can be interpreted as tendency of multiple motors to work collectively as an efficient team during cargo transport. We expect team motors to show greater cooperativity if they spend greater time together on the microtubule so that the applied load can be shared, i.e. their residence times on the microtubule significantly overlap. To quantify cooperativity of the team of engaged motors, we propose a dimensionless parameter-Cooperativity Index λ. Mathematically,
N i=1 λ=
Tmi −Tc
N
⟨Tm ⟩
Tc v = 1 − N vo
Tmi
v vo
(4.1)
i =1
N
T
where ⟨Tm ⟩ = i=N1 mi . Tmi is the residence time of ith motor on microtubule, while Tc is the residence time of cargo onto microtubule. λ = 0 for a cargo driven by single Kinesin motor whereas λ ∈ (0, 1) for cargo driven by multiple Kinesin motors. Cooperativity Index essentially measures the ratio of average overlapping residence time to average residence time of motors in the team, multiplied by ratio of average velocity of team to maximum average velocity of single motor. We consider residence time of motors to overlap if they are attached simultaneously to the microtubule during cargo transport. v is essentially a weight factor to evaluate cargo velocity v , resulting from residence of motors on microtubule, against vo maximum possible average velocity vo for cargo. Maximum possible average velocity for cargos cannot be greater than single motor velocity vo . As N increases, λ increases (Fig. 6), implying that the cooperativity increases due to recruitment of more motors in the team. We find that λ increases in a Michaelis–Menten manner with ATP concentration attaining a maximum at around physiological ATP concentrations for all N. The bulk cytoplasmic ATP concentration in various mammalian cells is ∼1 mM [25]. It was found to be 1.0 mM in β -cells [26], 0.9 mM in HeLa cells [27] and 0.8–1.2 mM in COS-1 cells [28]. Therefore, we hypothesize that values of physiological ATP concentrations have been evolutionarily tuned to extract maximum cooperativity from a team of Kinesin motors engaged in cargo transport. The cooperativity index decreases with load due to more frequent motor detachments at higher loads (Fig. 7). This means that the tendency of individual motors on a cargo to work as an efficient team reduces as applied load on cargo is increased. This is a direct consequence of decay of motor residence times as well as cargo velocity, with increasing load values (Fig. 5). 5. Conclusions A quantitative understanding of complex dynamics of interacting motor proteins has been a fundamental physics challenge [10–16,29,30]. Here, we have studied cargo transport by a team of Kinesin motors by developing mechanochemical mean-field and stochastic models which included mechano-chemistry of single Kinesin motors. Our results highlight the effect of fuel concentration on transport properties of a team of motors engaged on a cargo. Our models make several predictions which may be confirmed using in-vitro experiments, such as drastic increase in cargo runlength
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Fig. 6. Variation of Cooperativity Index λ predicted by mechano-chemical stochastic model, at (A) low and (B) high loads, as a function of ATP concentration.
Fig. 7. Variation of Cooperativity Index λ with applied load at (A) low and (B) high ATP concentrations.
at limiting ATP concentrations in spite of reduced velocity, and Michaelis–Menten dependence of cargo velocity on ATP concentration. We find that qualitative variations of runlengths and velocities predicted by both models for many motor cases are identical. However, quantitative values of runlengths and velocities predicted by mechano-chemical stochastic model start to deviate significantly from the ones predicted by mechano-chemical mean-field model in high load regime. Deviation between the predicted values from both models increases with number of motors on the cargo. Our model can have potential applications in designing nanoscale transport systems called ‘‘molecular shuttles’’ which are artificial hybrid replicas of natural motor based transport systems in synthetic environments [31]. The three important characteristics of an efficient nano-molecular shuttle are the adequate directionality of transport, proper speed of material delivery, and faultless loading/unloading of material at source/sink [31]. The ATP concentration can be used to fine tune transport properties of ‘‘molecular shuttles’’ as per the requirement [31,32]. Our formalism can be easily extended to other important molecular motor proteins, in particular, cytoplasmic Dynein, which hauls cargos in a direction opposite to Kinesins [33]. Transport by a team of Dynein motors plays a crucial role in generating large forces in diverse cellular processes, such as movement of nuclei in migrating neurons, migration of fibroblast cells in wounded monolayers and transport of large nuclei in multinucleate muscle fibers [21]. Earlier developed experimentally-constrained Monte-Carlo model of single cytoplasmic Dynein [34] combined with its experimentally measured dissociation kinetics [19] can be a starting point for such a model of cargo transport by multiple Dynein motors. Such mechano-chemical model for cargo transport by multiple Dynein motors may uncover many novel properties of intracellular cargo transport, as Dynein motors are known to take steps of different sizes depending on load and fuel concentration [33,34]. It is known that many physiological cargos are transported by a team where opposite polarity motors (both Kinesin and Dynein) are simultaneously engaged [8]. This results in a frequent tug-of-war among the engaged opposite polarity motors, and the net transport direction is governed by the stronger team. Since the runlengths and velocity of both kinesin and dynein motors strongly depend on ATP concentration, it would be interesting to explore whether the net direction of cargo transport can be tuned by changing the ATP concentration.
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Acknowledgments Ambarish Kunwar was supported by Innovative Young Biotechnologist Award of DBT (Grant Number BT/06/IYBA/2012). Aditi Kulkarni was supported by IITB Research Internship Award 2012-2013 (Project Code 12IRINTRN003). Appendix A. Mean-field model of load sharing for cargo transport Klumpp et al. [6] proposed a transition rate model for cargo transport by a team of motors, which is also referred to as Mean-field model of load sharing [17–19]. In this model, a cargo is transported cooperatively by a maximum of N motors, where n is the number of motors attached to the filament at any time instant, such that 0 < n ≤ N. Motors can reattach to and detach from the microtubule stochastically. Thus, the cargo can probabilistically switch from state n (with n number of instantaneously bound/engaged motors) to either state n + 1 with reattachment rate πn , or state n − 1 with detachment rate ϵn . This model assumed a linear force–velocity relation for a single motor given by
F v(F ) = vo 1 − ;
0 ≤ F ≤ Fs
Fs
(A.1)
where vo is motor velocity at zero load. This model makes an important assumption that an opposing force (load) F applied to cargo is shared equally among the n engaged motors out of total N motors available on the cargo. This induces an effective interaction among engaged motors, due to which their transport parameters start to depend on the presence of other motors. Thus, in this model the velocity of engaged motors decreases linearly with applied load
F ; vn (F ) = vo 1 − nFs
0 ≤ F ≤ nFs .
(A.2)
The factor F /n in the above equation captures equal load sharing among n engaged motors. It was assumed that motors move with their unloaded velocity for forward loads, and hence vn (F ) = v for F ≤ 0. Motors stop moving for loads greater than stall force, and therefore vn (F ) = 0 for F > Fs . The detachment rate for a single motor was assumed to be exponentially growing with applied load
ϵ (F ) = ϵo exp
F
(A.3)
Fd
and, the force dependence of detachment rate in state n is given by
ϵn (F ) = nϵo exp
F
(A.4)
nFd
where ϵo is detachment rate under zero load and Fd is the detachment force for motor. Fd = Bδ , where kB T is the thermal energy term and δ is the characteristic length associated with transition from attached to detached states. Again, factor F /n here captures equal load sharing among n bound motors. The reattachment rate πn in state n was assumed to be independent of force, as motors in detached state do not feel any applied load. The reattachment rate is given by k T
πn = (N − n)πo
(A.5)
where πo is the reattachment rate of a single motor. Klumpp et al. [6] wrote a master equation for Pn , where Pn denotes the probability of cargo being in state n. Following analytical solutions for steady state were obtained by Klumpp et al. [6]
N −1 n πi Po = 1 + ϵ n =0 i =0 i +1
−1 ,
Pn = Po
n −1 πi ϵ i=0 i+1
(A.6)
where Po is the probability of the cargo not being attached to the microtubule. One can obtain average properties such as average velocity and average number of bound motors using above steady state probabilities. Thus, the average velocity of cargo is given by
⟨v⟩ =
N vn Pn 1 − Po n=1
(A.7)
and average number of bound motors are given by
⟨N ⟩ =
N nPn . 1 − Po n =1
(A.8)
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Fig. C.1. (A) Detachment rate and (B) stepping rate, for a single Kinesin motor as a function of ATP concentration at low load (1.1 pN) and high load (7.2 pN) values. (C) Detachment rate and (D) stepping rate, for a single Kinesin motor as a function of load at low (5 µM) ATP concentration and high (2 mM) ATP concentration.
An expression for average unbinding time in steady state can be obtained by assuming that the net reattachment and detachment rates balance each other. The average detachment/unbinding rate is given by
⟨ϵ⟩ = ϵ1
N −1 n πi 1+ ϵ n=1 i=1 i+1
−1 .
(A.9)
The average runlength (processivity) can be obtained from above equation by replacing rates which are in units of inverse time by units of inverse distance.
N −1 n v1 vi+1 πi ⟨x⟩ = 1+ . ϵ1 vi ϵi+1 n=1 i=1
(A.10)
Appendix B. Stochastic model of load sharing for cargo transport The Stochastic model of load sharing for cargo transport by multiple motors [17,18] is fundamentally different from the above mentioned mean-field model of load sharing. It makes the assumption that when an external force F is applied to a cargo which is being driven by n bound/engaged motors out of total N motors available on the cargo, F is shared unequally among n engaged motors [17,18]. Motors were modeled as special linkages of stiffness k, such that they exert a restoring force (according to Hooke’s law) only when they are stretched beyond their rest length l; and they buckle without any resistance when compressed. In stochastic model, the amount of load shared by a motor is determined by relative position of that motor with respect to cargo and motor stiffness. Simulations were started with all motors attached to the microtubule. At each time step, the engaged motors can either step on microtubule or can detach from microtubule. Additionally, the detached motors on cargo can also reattach to microtubule with certain reattachment rate. The equilibrium position of the cargo at each time step was obtained by using force balance. The cargo continues to move until all motors detach from microtubule. Appendix C. Stepping and detachment of a single Kinesin motor We use experimentally constrained model of single Kinesin motor by Schnitzer et al. [5] to obtain stepping and detachment rate of a single kinesin motor. The stepping rate σ can be obtained by dividing expression for velocity v by
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Fig. D.1. Ratio of cargo stepping to cargo detachment rate for multiple motor driven cargo as a function of ATP concentration at (A) low load (1.1 pN) and (B) at high load (7.2 pN). Figure legends are similar to Fig. 2.
the motor’s step size d. Thus, the stepping rate σ at a particular force F and ATP concentration (denoted by [ATP]) is given by the following relation
ϵσ (F , [ATP]) =
v d
kocat [ATP ]
=
[ATP ] pcat + qcat exp
F δcat kB T
+
o Kcat Kbo
pb + qb exp
F δb kB T
.
(C.1)
The detachment rate for single motor for sub-stall regime F ≤ Fs can be obtained by dividing the velocity v by runlength L. Thus, detachment rate ϵ of a single motor for F ≤ Fs and ATP concentration [ATP] is given by
ϵ ϵ (F , [ATP]) =
v L
o [ATP] + B 1 + A exp Kcat
= A exp
−F δ L kB T
[ATP] pcat + qcat exp
F δcat kB T
−F δ L
kB T
+
o Kcat Kbo
pb + qb exp
F δb kB T
.
(C.2)
The detachment rate in the super-stall regime i.e. F > Fs is given by Eq. (2.5) in the main text. Variation of stepping and detachment rate with load and ATP concentration is plotted in Fig. C.1. Appendix D. Stepping and detachment of cargo driven by a team of Kinesin motors Both mechano-chemical models of cargo transport by multiple Kinesin motors predict a non-monotonic variation of the ratio of cargo stepping to cargo detachment rate as a function of ATP concentration at low forces (Fig. D.1(A)). At low forces, ratio first increases and then starts decreasing. However, the ratio decreases monotonically as a function of ATP concentration at higher forces (Fig. D.1(B)). Same patterns are observed for cargo runlength as a function of ATP concentration described in Main Text (Fig. 2). It is worth pointing out that both cargo stepping and detachment rates increase individually as ATP concentration is increased, yet variation observed in runlength with ATP concentration (Fig. D.1) is decided by the ratio of these rates. Cargo stepping and detachment rates increase due to increase in individual motor stepping and detachment rates (Fig. C.1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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Physica A journal homepage: www.elsevier.com/locate/physa
Effect of fuel concentration on cargo transport by a team of Kinesin motors Anjneya Takshak 1 , Nirvantosh Mishra 1 , Aditi Kulkarni, Ambarish Kunwar ∗ Department of Biosciences and Bioengineering, Indian Institute of Technology Bombay, Mumbai 400076, India
highlights • • • • • • • •
A team of Kinesin motor proteins transports cellular cargos using ATP as fuel. How fuel concentration affects transport by the team largely remains unexplored. Mechano-chemical models of cargo transport developed here fill up this knowledge gap. Mechano-chemical models include effect of both force and fuel on cargo transport. Models predict very large cargo travel distances at limiting fuel concentrations. Models predict Michaelis-Menten dependence of cargo velocity on fuel concentration. These predictions from modeling can be directly tested by in-vitro experiments. Our results may be used to regulate and fine-tune transport by motor-based shuttles.
article
info
Article history: Received 28 January 2016 Received in revised form 21 August 2016 Available online 12 October 2016 Keywords: Molecular motors Kinesin Mechano-chemical enzymes Monte-Carlo simulations
∗
Corresponding author. E-mail address: [email protected] (A. Kunwar).
1 First Co-authors. http://dx.doi.org/10.1016/j.physa.2016.10.022 0378-4371/© 2016 Elsevier B.V. All rights reserved.
abstract Eukaryotic cells employ specialized proteins called molecular motors for transporting organelles and vesicles from one location to another in a regulated and directed manner. These molecular motors often work collectively in a team while transporting cargos. Molecular motors use cytoplasmic ATP as fuel, which is hydrolyzed to generate mechanical force. While the effect of ATP concentration on cargo transport by single Kinesin motor function is well understood, it is still unexplored, both theoretically and experimentally, how ATP concentration would affect cargo transport by a team of Kinesin motors. For instance, how does fuel concentration affect the travel distances and travel velocities of cargo? How cooperativity of Kinesin motors engaged on a cargo is affected by ATP concentration? To answer these questions, here we develop mechano-chemical models of cargo transport by a team of Kinesin motors. To develop these models we use experimentally-constrained mechano-chemical model of a single Kinesin motor as well as earlier developed mean-field and stochastic models of load sharing for cargo transport. Thus, our new models for cargo transport by a team of Kinesin motors include fuel concentration explicitly, which was not considered in earlier models. We make several interesting predictions which can be tested experimentally. For instance, the travel distances of cargos are very large at limited ATP concentrations in spite of very small travel velocity. Velocities of cargos driven by multiple Kinesin have a Michaelis–Menten dependence on ATP concentration. Similarly, cooperativity among the engaged Kinesin
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motors on the cargo shows a Michaelis–Menten type dependence, which attains a maximum value near physiological ATP concentrations. Our new results can be potentially useful in controlling artificial nano-molecular shuttles precisely for targeted delivery in various nano-technological applications. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Molecular motors are specialized proteins which perform the long-range and short range transport of cargo particles (vesicles, mitochondria, mRNA particles, liposomes, pigment granules, etc.) along cytoskeletal filaments in eukaryotic cells [1]. The commonly employed proteins which perform this cargo trafficking inside cells belong to three families: Kinesin superfamily, Dynein and Myosin families [1,2]. Most Kinesin superfamily members are (+) end directed, which means that they carry cargo from the cell interior to the cell periphery. Several in-vitro single molecule studies have revealed the details about the mode of action of Conventional Kinesin (Kinesin-1) [2,3]. It is a highly processive motor which takes several steps on the microtubule [4] prior to detaching. It has two motor domains through which it attaches to the microtubule and travels in discrete steps of 8 nm in a coordinated hand over hand mechanism. During the process of cargo transport, a single Kinesin motor can move processively against an opposing force (load) in the range of up to 5 to 7 pN [5]. Additionally, Kinesin motors can also detach from microtubule with load-dependent unbinding rate and reattach [6–8]. Since the amount of force generated by a single Kinesin motor is usually insufficient for effective long-range intra-cellular transport, multiple Kinesin motors often work together as team to achieve this [9,8]. Several models have been proposed to study cargo transport by a team of molecular motors [10–16]. Klumpp et al. [6] proposed a model considering that the applied load on the cargo is distributed equally among all engaged motors. Models later developed by Kunwar et al. [17,18] incorporated stochastic load sharing among multiple motors engaged on a cargo. However, these models cannot be used to understand the effect of fuel concentration on cargo transport by a team of motors, as underlying single motor characteristics do not incorporate fuel dependence. In this paper, we use an experimentally-constrained model of single Kinesin motor to study cargo transport by a team of multiple Kinesin motors using earlier developed Mean-field [6] and Stochastic models [17,18] of load sharing. Hereafter, we shall refer to Mean-field model of load sharing [6] as Mean-field model, and Stochastic model of load sharing [17,18] as Stochastic model. 2. Experimentally-constrained model of single Kinesin motor Schnitzer et al. [5] proposed a load dependent Composite State Model to describe the stepping behavior of a single Kinesin motor over a microtubule which incorporates effect of both load and fuel (ATP) concentration. When an ATP molecule is bound to one of the two heads of a Kinesin motor, the motor is said to be in Composite State. According to this model, Kinesin head can detach from the microtubule, with a smaller ATP concentration-dependent probability, before ATP binding (State I). Kinesin head can also undergo detachment with a greater load-dependent probability, after ATP binding (State II). The experimental velocity curves of single Kinesin motors can be fitted using a Michaelis–Menten equation [5] as a function of ATP concentration (denoted by [ATP ]), at different loads as given by the following relation
v (F , [ATP]) =
dKcat (F ) [ATP] Vmax [ATP ] = K F [ATP ] + KM [ATP] + Kcat((F ))
(2.1)
b
where Vmax (= dKcat ) is the velocity at saturating ATP concentration in nm/s, d is the step size of a single Kinesin motor in nm, Kcat (F ) is the load-dependent catalytic turnover constant, and Kb (F ) is the load dependent second order rate constant for ATP binding to the Kinesin motor. Both the rate constants exhibit an inverse exponential type Boltzmann relationship with load as follows: Kcat (F ) =
o Kcat
pcat + qcat exp Kb ( F ) =
F .δcat kB T
Kbo pb + qb exp
F δb kB T
o where Kcat and Kbo are the respective rate constants, measured in s−1 and µM −1 s−1 respectively, in the absence of external load. kB T is the thermal energy at temperature T , δcat and δb represent the characteristic motor distances in nm over which the load acts during catalysis and binding respectively. pcat and qcat (= 1 − pcat ) are the fractions of the unloaded catalytic
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cycle required for biochemical transitions [5], whereas, pb and qb (= 1 − pb ) are the fractions of the unloaded catalytic cycle required for mechanical transitions of Kinesin motor [5]. The theoretical basis for the particular choice of the forms of Kcat (F ) and Kb (F ) can be understood as follows: Each of the two events of kinesin stepping cycle i.e. ATP binding and ATP hydrolysis can be modeled as a sequence of a load-independent biochemical transition followed by a load-dependent mechanical transition over a free energy barrier. The time required for mechanical transition over the free energy barrier increases with load following a Boltzmann-type relation given below: Ko
K (F ) =
p + q exp
Fδ kB T
where K o is the unloaded rate constant, kB T is the thermal energy at temperature T , δ represents the characteristic motor distance in nm over which the load acts. p and q (= 1 − p) are the fractions of the unloaded catalytic cycle required for biochemical transition and mechanical transition respectively [5]. The measured runlength of a single Kinesin motor also exhibits a Michaelis–Menten behavior as a function of ATP concentration, and an exponential Arrhenius–Eyring Kinetics with respect to load as follows [5] L (F , [ATP]) =
Lo [ATP ] [ATP ] + LM dA exp
=
−F δ L
kB T
[ATP]
[ATP] + B 1 + A exp
−F δ L
(2.2)
kB T
) is the runlength at saturating ATP concentration. δL is the characteristic distance over which BT k −F δ the load acts. LM = B 1 + A exp( k TL ) is that value of ATP concentration at which the motor runlength becomes half B
where Lo = dA exp(
−F δL
of its maximum value at a particular load. This equation was derived considering detachment of Kinesin head from either of two distinct states—State I and State II as described above. Here, (B/ [ATP]) is the probability of motor head detachment when it is in state I, which reduces with increasing ATP concentration as B is a constant. To include the exponential decline in runlengthwith load, motor detachment under State II was assumed to exhibit a load-dependent probability given by 1 A
F δL BT
exp( k
) , where A is a constant whose value is equal to average number of Kinesin catalytic cycles in the absence of
load prior to detachment. Velocity of single Kinesin motor starts to decrease with increasing load and motor finally stalls at high loads. The minimum opposing force (load) required to stall the motor is called stall force (Fs ). Motors experiencing forces greater than Fs are said to be in super-stall regime. In addition, Kinesin motor can also detach from the microtubule and its detachment rate for F ≤ Fs can be obtained by the following argument. One can think of average velocity of single motor given by Eq. (2.1) as average number of steps taken by the motor before detachment, multiplied by step size and divided by average time taken to dissociate from the microtubule. Similarly, the average runlength given by Eq. (2.2) is essentially the average number of steps taken by single motor before detachment from microtubule, multiplied by step size. Hence, one can obtain detachment rate of single motor by dividing Eq. (2.1) by Eq. (2.2). Therefore, detachment rate of a motile single Kinesin motor is given by
ϵ (F , [ATP]) =
Vmax ([ATP] + LM ) Lo ([ATP] + KM )
,
F ≤ Fs .
(2.3)
The detachment rate of single Kinesin motor for loads greater than stall force has been measured experimentally by Kunwar et al. [19] and is given by the following relation.
ϵ (F ) = 1.07 + (0.186F ) ,
F > Fs .
(2.4)
The dependence of detachment rate on ATP concentration has not been reported by anyone in the super-stall regime i.e. F > Fs. Therefore, we assume this dependence to be of Michaelis–Menten type as a function of ATP concentration for the following two reasons: (i) Kinesin motors spend longer time attached to microtubules at low ATP concentrations, and time spent decreases with increasing ATP concentration [5]. (ii) Kinesin is a mechano-chemical enzyme and most enzymes show Michaelis–Menten behavior as a function of substrate concentration. This assumption leads to the following relation for detachment of single Kinesin for loads in super-stall regime
ϵ (F , [ATP]) =
(1.07 + (0.186F )) [ATP] , [ATP] + KMϵ
F > Fs
(2.5)
where KMϵ is the Michaelis constant for motor detachment. We propose that discontinuity in detachment rate of motor at stall force (which is the interface of sub-stall and super-stall regimes) may arise from the fact that the motor moves with at least one head attached in the sub-stall regime at all times, but
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Fig. 1. Schematic of (A) Mean-field Model of load sharing and (B) Stochastic model of load sharing for a cargo being transported by n = 3 engaged motors out of total available N = 4 motors. In Mean-field model of load sharing, applied load F is shared equally among n engaged motors; whereas in Stochastic Model of load sharing, n engaged motors share load unequally.
does not move/step in the super-stall regime, where both of its heads remain bound to the microtubule. Thus, the response of the motor to an external force can be completely different in above two regimes which results in a discontinuity in the detachment rate at stall force. A number of models have been proposed to understand the biophysics of such systems which behave differently under low and high loads [20]. One recent hypothesis suggests that allosteric structural changes might occur in both motor and microtubule binding domains under high loads. These structural changes might lead to locking or tight coupling between motor head and microtubule binding site [21]. Experimental data suggest the tight coupling between motor head and microtubule binding domain persists roughly up to two times the stall force [19]. In the next sections, we describe how we used earlier developed Mean-field Model [6] and Stochastic Model [17,18] to understand cargo transport by a team of Kinesin motors using above experimentally-constrained model of single Kinesin motor which captures its mechano-chemistry i.e. effect of load as well as fuel concentration on its transport properties. 3. Models of cargo transport by a team of Kinesin motors 3.1. Mechano-chemical mean-field model of cargo transport To develop mechano-chemical mean-field model of cargo transport, we used the popular mean-field model proposed by Klumpp et al. [6] (summarized in Appendix A). In this model, a cargo is cooperatively transported by N molecular motors along a filament, where number of bound/engaged motors (n) changes due to motor detachment with rate ϵn and motor reattachment with rate πn , such that 0 < n ≤ N. The cargo in state n travels with velocity vn . To include the effect of both ATP and load in the mean-field model, we replaced linear-force–velocity relation in mean-field model by new force–velocity relation given in Eq. (2.1); i.e. Eq. (A.1) in Appendix A was replaced by Eq. (2.1). In addition, experimentallyconstrained force–dissociation relations given by Eqs. (2.3) and (2.5) (for F ≤ Fs and F > Fs respectively) were used to replace exponentially growing force–dissociation relation in mean-field model; i.e. Eq. (A.3) in Appendix A was replaced by Eqs. (2.3) and (2.5). In mean-field model, any load F experienced by cargo is shared equally among n bound motors i.e. each motor experiences a force of F /n (Fig. 1(A)). Therefore, corresponding expressions for vn and ϵn for mechano-chemical meanfield model were obtained by replacing F by F /n in above-mentioned equations. The average runlengths and velocities of the cargo for various load and ATP concentrations were calculated using probabilities Pn (n = 0, 1, . . . , N ) of cargo being in different states, as previously done in mean-field model [6]. Parameters used in calculations and their sources are given in Table 1. 3.2. Mechano-chemical stochastic model of cargo transport To develop mechano-chemical stochastic model of cargo transport, we use stochastic model developed by Kunwar et al. [17,18]. In this model, load among engaged motors is shared unequally due to stochastic stepping of motors, as well as their stochastic detachment and reattachment (Fig. 1(B)). The magnitude of load shared by a motor is determined by its stiffness and its relative position in the team with respect to cargo. Dividing Eq. (2.1) by step size d yields stepping rate σ for bound motors in our model (Eq. (C.1) in Appendix C). The detachment rate in our model was obtained by replacing F by |F | in Eqs. (2.3) and (2.5). |F | incorporates the fact that forward and backward loads have same effect on motor detachment rates as assumed in earlier works [17,18]. It is possible that some motor proteins may not have identical detachment rates under forward and backward loads e.g. yeast dynein [22]. A recent work by Takshak et al. has explored the effect such as anisotropy in detachment rate on cargo transport by a team of motor proteins [23]. The probabilities of motor binding, unbinding and stepping at any time step in the simulations were obtained by multiplying the respective rates of these events with time step ∆t. Time step ∆t was chosen in such a way that fastest event happens at most once during the interval ∆t. Detailed simulation procedure is given in Appendix B. Parameters used in simulations and their sources are given in Table 1.
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Table 1 Parameter values used for both Mechano-chemical Mean-field and Mechano-chemical Stochastic model. Parameter
Symbol
Value [Source(s)]
Step size of Kinesin motor Rest length of Kinesin motor Stiffness of Kinesin motor Stall force for Kinesin motor Detachment force for Kinesin motor Average number of catalytic cycles at zero load prior to detachment Constant for detachment probability prior to ATP bonding Characteristic distance associated with ATP binding Characteristic distance associated with ATP hydrolysis Characteristic distance associated with load dependence Rate constant for ATP hydrolysis at zero load Rate constant for ATP binding at zero load Probability of ATP non-hydrolysis at zero load Probability of ATP non-binding at zero load Probability of ATP hydrolysis at zero load Probability of ATP binding at zero load Michaelis constant for motor detachment Kinesin detachment rate under zero load Kinesin reattachment rate Time step used in simulations
d l k Fs Fd A B
8 nm [5] 110 nm [17–19] 0.32 pN/nm [17–19] 6 pN [6,17] 3 pN [6,17] 107 [5] 2.9 × 10−2 µM [5] 3.7 nm [5] 3.7 nm [5] 1.3 nm [5] 103 s−1 [5] 1.3 µM−1 s−1 [5] 6.2 × 10−3 [5] 4.0 × 10−2 [5] 1 − qcat [5] 1 − qb [5] 100 µM 1/s [6,17] 5/s [6,17] 10−4 s
δb δcat δL
o Kcat Kbo qcat qb pcat pb KMϵ
ϵo πo ∆t
4. Results We used our new mechano-chemical mean-field and stochastic models to obtain the collective transport properties of cargo driven by maximum N Kinesin motors (N = 1, 2, 3 and 4). Results obtained are discussed in the subsequent sections. 4.1. Mechano-chemical models predict very large cargo runlengths at limiting fuel concentrations The average runlengths from our new mechano-chemical models as a function of ATP concentration are shown in Fig. 2. The runlength increases with the number of motors (N ) in both models. The runlength increases in a Michaelis–Menten manner for a single motor as ATP concentration is increased. However, for multiple motor driven cargos (N > 1) the runlengths are quite large at low/limiting ATP concentrations (few µM) in comparison to saturating ATP concentrations (mM). These predictions are consistent with experimentally measured runlengths of multiple motor driven cargos, at low ATP concentrations and zero load, by Xu et al. [24]. At low ATP concentrations, runlength increases as detachment rates of individual motors decrease (Appendix C Fig. C.1(A)), which in turn increases overall residence time of cargo. Both mechano-chemical models predict a non-monotonic variation of runlength with ATP concentrations in low ATP regime at low forces (1.1 pN), where it first increases up to 3 µM and decreases thereafter. To understand this, we calculated ratio of cargo stepping rate to cargo detachment rate. Both of these rates increase as ATP concentration is increased from 1 µM to 3 µM. However, stepping rate increases more rapidly than detachment rate, which results in increased runlength. Beyond 3 µM, although both stepping rate and detachment rate increase, yet their ratio decreases. This results in monotonic decrease in runlength beyond 3 µM. At higher force (7.2 pN), the ratio of stepping to detachment rate always decreases monotonically with increasing ATP concentration. This results in monotonic decrease of runlength with ATP concentration at higher loads. In short, variations of ratio of cargo stepping to cargo detachment rates are reflected in variations of runlength with ATP concentration (Fig. D.1 in Appendix D). Interestingly, increase in runlength with number of motors at different forces was similar to earlier studies [6,17,18] and is shown in Fig. 3. For single Kinesin motor, the runlength decreases exponentially as load increases. However, in multiple Kinesin driven cargos, such monotonous decrease is not observed with respect to load (Fig. 3). In the sub-stall regime (F ≤ Fs ), the runlength first decreases, and then starts to increase as applied load approaches Fs . For (F > Fs ), the cargo runlength decreases monotonically with force. Such variation of runlength with applied load predicted, from both our mechano-chemical models, are similar to earlier published results [19]. These non-monotonic variations of runlengths have their origin in non-monotonic behavior of detachment rate of single Kinesin with force as reported earlier [17,19]. 4.2. Mechano-chemical stochastic model predicts lower cargo velocities at high loads at all fuel concentrations Both mechano-chemical mean-field and mechano-chemical stochastic models predict that cargo velocity has Michaelis–Menten dependence on ATP concentration. At low loads (1.1 pN), velocities of multiple motor driven cargos, predicted by mechano-chemical mean-field model, are roughly equal to single motor velocity. However, mechano-chemical stochastic model predicts a smaller velocity than single motors as reported earlier [17,18], for multiple motor driven cargos. At low loads (1.1 pN), velocities predicted by mechano-chemical stochastic model are consistent with limited experimental observation, at low ATP concentrations and zero load, by Xu et al. [24]; where a decrease in cargo velocity was observed
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Fig. 2. Variation of cargo runlength with ATP concentration at (A) low load and (B) high load. Figure legend MMM refers to data from Mechano-chemical Mean-field Model and legend MSM refers to data from Mechano-chemical Stochastic Model. In case of single motor driven cargo (N = 1) both models give similar results.
Fig. 3. Variation of cargo runlength with applied load at (A) low and (B) high ATP concentration. Figure legends are similar to Fig. 2.
Fig. 4. Variation of Cargo Velocity with ATP concentration at (A) low load and (B) high load. Figure legends are similar to Fig. 2.
when motor number was changed from N = 1 to N = 2 (Fig. 4(A) inset). At higher loads (7.2 pN), both mechano-chemical models predict that multiple motor driven cargo would be moving faster than single motor driven cargo. Cargo velocity decreases monotonically in a convex-up manner for a single motor driven cargo as applied load increases (Fig. 5). However, mechano-chemical stochastic model predicts almost linear decrease of cargo velocity with load, at high as well as low ATP concentrations. Such variation of cargo velocity with load have been reported earlier too [17,18]. In contrast, mechano-chemical mean-field model predicts convex-up nature of cargo velocity with load, at high as well as low ATP concentrations. We find that for the high-loads, the mechano-chemical mean-field model predicts a consistently higher average velocity than mechano-chemical stochastic model for N = 2 and N = 3. The deviations between the results of two models do not appear to be so prominent in the low-load regime (Fig. 4(A)). The origin of this difference in average velocity in low-load and high-load regimes can be understood as follows: In Mechano-chemical Mean-field Model (MMM), the load is equally
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Fig. 5. Variation of cargo velocity with applied load at (A) low and (B) high ATP concentration. Figure legends are similar to Fig. 2.
shared among the engaged motors. However, in Mechano-chemical stochastic model (MSM) the load shared by each engaged motor fluctuates due to their stochastic dynamics of stepping and detachment. In MSM, for low load regime (e.g. 1.1 pN) the differences among the fluctuating forces are small whereas they are very large for high load regime (7.2 pN). This results in large difference in average velocity in high load regime. 4.3. Cooperativity among motors also shows a Michaelis–Menten dependence on fuel concentration Cooperativity can be interpreted as tendency of multiple motors to work collectively as an efficient team during cargo transport. We expect team motors to show greater cooperativity if they spend greater time together on the microtubule so that the applied load can be shared, i.e. their residence times on the microtubule significantly overlap. To quantify cooperativity of the team of engaged motors, we propose a dimensionless parameter-Cooperativity Index λ. Mathematically,
N i=1 λ=
Tmi −Tc
N
⟨Tm ⟩
Tc v = 1 − N vo
Tmi
v vo
(4.1)
i =1
N
T
where ⟨Tm ⟩ = i=N1 mi . Tmi is the residence time of ith motor on microtubule, while Tc is the residence time of cargo onto microtubule. λ = 0 for a cargo driven by single Kinesin motor whereas λ ∈ (0, 1) for cargo driven by multiple Kinesin motors. Cooperativity Index essentially measures the ratio of average overlapping residence time to average residence time of motors in the team, multiplied by ratio of average velocity of team to maximum average velocity of single motor. We consider residence time of motors to overlap if they are attached simultaneously to the microtubule during cargo transport. v is essentially a weight factor to evaluate cargo velocity v , resulting from residence of motors on microtubule, against vo maximum possible average velocity vo for cargo. Maximum possible average velocity for cargos cannot be greater than single motor velocity vo . As N increases, λ increases (Fig. 6), implying that the cooperativity increases due to recruitment of more motors in the team. We find that λ increases in a Michaelis–Menten manner with ATP concentration attaining a maximum at around physiological ATP concentrations for all N. The bulk cytoplasmic ATP concentration in various mammalian cells is ∼1 mM [25]. It was found to be 1.0 mM in β -cells [26], 0.9 mM in HeLa cells [27] and 0.8–1.2 mM in COS-1 cells [28]. Therefore, we hypothesize that values of physiological ATP concentrations have been evolutionarily tuned to extract maximum cooperativity from a team of Kinesin motors engaged in cargo transport. The cooperativity index decreases with load due to more frequent motor detachments at higher loads (Fig. 7). This means that the tendency of individual motors on a cargo to work as an efficient team reduces as applied load on cargo is increased. This is a direct consequence of decay of motor residence times as well as cargo velocity, with increasing load values (Fig. 5). 5. Conclusions A quantitative understanding of complex dynamics of interacting motor proteins has been a fundamental physics challenge [10–16,29,30]. Here, we have studied cargo transport by a team of Kinesin motors by developing mechanochemical mean-field and stochastic models which included mechano-chemistry of single Kinesin motors. Our results highlight the effect of fuel concentration on transport properties of a team of motors engaged on a cargo. Our models make several predictions which may be confirmed using in-vitro experiments, such as drastic increase in cargo runlength
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Fig. 6. Variation of Cooperativity Index λ predicted by mechano-chemical stochastic model, at (A) low and (B) high loads, as a function of ATP concentration.
Fig. 7. Variation of Cooperativity Index λ with applied load at (A) low and (B) high ATP concentrations.
at limiting ATP concentrations in spite of reduced velocity, and Michaelis–Menten dependence of cargo velocity on ATP concentration. We find that qualitative variations of runlengths and velocities predicted by both models for many motor cases are identical. However, quantitative values of runlengths and velocities predicted by mechano-chemical stochastic model start to deviate significantly from the ones predicted by mechano-chemical mean-field model in high load regime. Deviation between the predicted values from both models increases with number of motors on the cargo. Our model can have potential applications in designing nanoscale transport systems called ‘‘molecular shuttles’’ which are artificial hybrid replicas of natural motor based transport systems in synthetic environments [31]. The three important characteristics of an efficient nano-molecular shuttle are the adequate directionality of transport, proper speed of material delivery, and faultless loading/unloading of material at source/sink [31]. The ATP concentration can be used to fine tune transport properties of ‘‘molecular shuttles’’ as per the requirement [31,32]. Our formalism can be easily extended to other important molecular motor proteins, in particular, cytoplasmic Dynein, which hauls cargos in a direction opposite to Kinesins [33]. Transport by a team of Dynein motors plays a crucial role in generating large forces in diverse cellular processes, such as movement of nuclei in migrating neurons, migration of fibroblast cells in wounded monolayers and transport of large nuclei in multinucleate muscle fibers [21]. Earlier developed experimentally-constrained Monte-Carlo model of single cytoplasmic Dynein [34] combined with its experimentally measured dissociation kinetics [19] can be a starting point for such a model of cargo transport by multiple Dynein motors. Such mechano-chemical model for cargo transport by multiple Dynein motors may uncover many novel properties of intracellular cargo transport, as Dynein motors are known to take steps of different sizes depending on load and fuel concentration [33,34]. It is known that many physiological cargos are transported by a team where opposite polarity motors (both Kinesin and Dynein) are simultaneously engaged [8]. This results in a frequent tug-of-war among the engaged opposite polarity motors, and the net transport direction is governed by the stronger team. Since the runlengths and velocity of both kinesin and dynein motors strongly depend on ATP concentration, it would be interesting to explore whether the net direction of cargo transport can be tuned by changing the ATP concentration.
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Acknowledgments Ambarish Kunwar was supported by Innovative Young Biotechnologist Award of DBT (Grant Number BT/06/IYBA/2012). Aditi Kulkarni was supported by IITB Research Internship Award 2012-2013 (Project Code 12IRINTRN003). Appendix A. Mean-field model of load sharing for cargo transport Klumpp et al. [6] proposed a transition rate model for cargo transport by a team of motors, which is also referred to as Mean-field model of load sharing [17–19]. In this model, a cargo is transported cooperatively by a maximum of N motors, where n is the number of motors attached to the filament at any time instant, such that 0 < n ≤ N. Motors can reattach to and detach from the microtubule stochastically. Thus, the cargo can probabilistically switch from state n (with n number of instantaneously bound/engaged motors) to either state n + 1 with reattachment rate πn , or state n − 1 with detachment rate ϵn . This model assumed a linear force–velocity relation for a single motor given by
F v(F ) = vo 1 − ;
0 ≤ F ≤ Fs
Fs
(A.1)
where vo is motor velocity at zero load. This model makes an important assumption that an opposing force (load) F applied to cargo is shared equally among the n engaged motors out of total N motors available on the cargo. This induces an effective interaction among engaged motors, due to which their transport parameters start to depend on the presence of other motors. Thus, in this model the velocity of engaged motors decreases linearly with applied load
F ; vn (F ) = vo 1 − nFs
0 ≤ F ≤ nFs .
(A.2)
The factor F /n in the above equation captures equal load sharing among n engaged motors. It was assumed that motors move with their unloaded velocity for forward loads, and hence vn (F ) = v for F ≤ 0. Motors stop moving for loads greater than stall force, and therefore vn (F ) = 0 for F > Fs . The detachment rate for a single motor was assumed to be exponentially growing with applied load
ϵ (F ) = ϵo exp
F
(A.3)
Fd
and, the force dependence of detachment rate in state n is given by
ϵn (F ) = nϵo exp
F
(A.4)
nFd
where ϵo is detachment rate under zero load and Fd is the detachment force for motor. Fd = Bδ , where kB T is the thermal energy term and δ is the characteristic length associated with transition from attached to detached states. Again, factor F /n here captures equal load sharing among n bound motors. The reattachment rate πn in state n was assumed to be independent of force, as motors in detached state do not feel any applied load. The reattachment rate is given by k T
πn = (N − n)πo
(A.5)
where πo is the reattachment rate of a single motor. Klumpp et al. [6] wrote a master equation for Pn , where Pn denotes the probability of cargo being in state n. Following analytical solutions for steady state were obtained by Klumpp et al. [6]
N −1 n πi Po = 1 + ϵ n =0 i =0 i +1
−1 ,
Pn = Po
n −1 πi ϵ i=0 i+1
(A.6)
where Po is the probability of the cargo not being attached to the microtubule. One can obtain average properties such as average velocity and average number of bound motors using above steady state probabilities. Thus, the average velocity of cargo is given by
⟨v⟩ =
N vn Pn 1 − Po n=1
(A.7)
and average number of bound motors are given by
⟨N ⟩ =
N nPn . 1 − Po n =1
(A.8)
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Fig. C.1. (A) Detachment rate and (B) stepping rate, for a single Kinesin motor as a function of ATP concentration at low load (1.1 pN) and high load (7.2 pN) values. (C) Detachment rate and (D) stepping rate, for a single Kinesin motor as a function of load at low (5 µM) ATP concentration and high (2 mM) ATP concentration.
An expression for average unbinding time in steady state can be obtained by assuming that the net reattachment and detachment rates balance each other. The average detachment/unbinding rate is given by
⟨ϵ⟩ = ϵ1
N −1 n πi 1+ ϵ n=1 i=1 i+1
−1 .
(A.9)
The average runlength (processivity) can be obtained from above equation by replacing rates which are in units of inverse time by units of inverse distance.
N −1 n v1 vi+1 πi ⟨x⟩ = 1+ . ϵ1 vi ϵi+1 n=1 i=1
(A.10)
Appendix B. Stochastic model of load sharing for cargo transport The Stochastic model of load sharing for cargo transport by multiple motors [17,18] is fundamentally different from the above mentioned mean-field model of load sharing. It makes the assumption that when an external force F is applied to a cargo which is being driven by n bound/engaged motors out of total N motors available on the cargo, F is shared unequally among n engaged motors [17,18]. Motors were modeled as special linkages of stiffness k, such that they exert a restoring force (according to Hooke’s law) only when they are stretched beyond their rest length l; and they buckle without any resistance when compressed. In stochastic model, the amount of load shared by a motor is determined by relative position of that motor with respect to cargo and motor stiffness. Simulations were started with all motors attached to the microtubule. At each time step, the engaged motors can either step on microtubule or can detach from microtubule. Additionally, the detached motors on cargo can also reattach to microtubule with certain reattachment rate. The equilibrium position of the cargo at each time step was obtained by using force balance. The cargo continues to move until all motors detach from microtubule. Appendix C. Stepping and detachment of a single Kinesin motor We use experimentally constrained model of single Kinesin motor by Schnitzer et al. [5] to obtain stepping and detachment rate of a single kinesin motor. The stepping rate σ can be obtained by dividing expression for velocity v by
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Fig. D.1. Ratio of cargo stepping to cargo detachment rate for multiple motor driven cargo as a function of ATP concentration at (A) low load (1.1 pN) and (B) at high load (7.2 pN). Figure legends are similar to Fig. 2.
the motor’s step size d. Thus, the stepping rate σ at a particular force F and ATP concentration (denoted by [ATP]) is given by the following relation
ϵσ (F , [ATP]) =
v d
kocat [ATP ]
=
[ATP ] pcat + qcat exp
F δcat kB T
+
o Kcat Kbo
pb + qb exp
F δb kB T
.
(C.1)
The detachment rate for single motor for sub-stall regime F ≤ Fs can be obtained by dividing the velocity v by runlength L. Thus, detachment rate ϵ of a single motor for F ≤ Fs and ATP concentration [ATP] is given by
ϵ ϵ (F , [ATP]) =
v L
o [ATP] + B 1 + A exp Kcat
= A exp
−F δ L kB T
[ATP] pcat + qcat exp
F δcat kB T
−F δ L
kB T
+
o Kcat Kbo
pb + qb exp
F δb kB T
.
(C.2)
The detachment rate in the super-stall regime i.e. F > Fs is given by Eq. (2.5) in the main text. Variation of stepping and detachment rate with load and ATP concentration is plotted in Fig. C.1. Appendix D. Stepping and detachment of cargo driven by a team of Kinesin motors Both mechano-chemical models of cargo transport by multiple Kinesin motors predict a non-monotonic variation of the ratio of cargo stepping to cargo detachment rate as a function of ATP concentration at low forces (Fig. D.1(A)). At low forces, ratio first increases and then starts decreasing. However, the ratio decreases monotonically as a function of ATP concentration at higher forces (Fig. D.1(B)). Same patterns are observed for cargo runlength as a function of ATP concentration described in Main Text (Fig. 2). It is worth pointing out that both cargo stepping and detachment rates increase individually as ATP concentration is increased, yet variation observed in runlength with ATP concentration (Fig. D.1) is decided by the ratio of these rates. Cargo stepping and detachment rates increase due to increase in individual motor stepping and detachment rates (Fig. C.1). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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