Nucleus-Targeted Drug Delivery: Theoretical Optimization of

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Nucleus-Targeted Drug Delivery: Theoretical Optimization of Nanoparticles Decoration for Enhanced Intracellular Active Transport Ohad Cohen† and Rony Granek†,‡,* †

The Stella and Avram Goren-Goldstein Department of Biotechnology Engineering and ‡The Ilse Katz Institute for Meso and Nanoscale Science and Technology, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel S Supporting Information *

ABSTRACT: A rational design for a nanoparticle is suggested, which will maximize its arrival efficiency from the plasma membrane to the nuclear surrounding. The design is based on grafting the particle surface with polymer spacers, each ending with a motor protein associating molecule, for example, nuclear localization signal peptide. It is theoretically shown that the spacer polymer molecular weight can be adjusted to significantly increase the effective particle processivity time. This should lead to appreciable enhancement of active transport of the nanocarrier, and consequently drug delivery, to the nucleus. KEYWORDS: Drug delivery, active transport, processivity time, intracellular transport, nanoparticles, nanocarriers

D

The motion of large molecules and organelles along the MT network is an active transport process mediated by motor proteins dynein and kinesin. Both of these motors move along microtubules with kinesin moving toward the plus (+) end and dynein toward the minus (−) end.12 Hence, dynein mediates transport toward the center of the cell (i.e., centrosome and nucleus regions) while kinesin mediates transport in the opposite direction toward the cell periphery. Thus, it is mainly dynein that should be harnessed when considering membraneto-nucleus transport. One of the major limitations of using dynein for membraneto-nucleus transport is its relatively short period of association with the MT, usually referred to as “processivity time”. Once associated with the MT, dynein molecules spend roughly one second before dissociating and diffusing away from the MT and into the cytoplasm. During this period of association, it moves with a velocity of ∼0.8 μm/sec, implying a mean displacement of ∼0.8 μm.13 However, mammalian cell radius is in the range 10−100 μm, thus implying that the motion of cargo carried by a single dynein molecule is highly intermittent, comprised of intervals of active and diffusive motion. As recently shown, low efficiency of membrane-to-nucleus transport is obtained in this way.14 Conversely, effectively increasing processivity time greatly enhances transport efficiency. In the present work, we propose a model for collective transport of a NP that is carried by several motor proteins on a MT. To achieve this, we suggest a rational design comprised of a spacer polymer (e.g., PEG) that is grafted to the particle

rug delivery and gene transfection systems often use solid nanoparticles (NPs) as carriers.1,2 While this is not as efficient as the use of viral vectors, there is an increasing effort to improve such systems in order to avoid both risk of disease infection and patient discomfort. Upon entry into the cell, usually via endocytosis, and release from the endosome, the NP carrier must arrive at its intracellular target where the drug is to be released. Thus, enhancing the efficiency of drug delivery systems should include improvement of each of these steps. This may be achieved through rational particle design. There have been reports of enhanced endocytosis through ligation of cell-penetrating peptides to the particle surface.3 Other studies have focused on optimization of the drug release kinetic processes to achieve maximal cellular uptake.4 Little work has been published on enhancement of the NP’s intracellular transport toward its targeted compartment. Of such compartments, the nucleus is considered a popular target for drug delivery systems.5,3 An obvious way to improve the NP’s transport efficiency (from the plasma membrane to the nucleus) would be to harness the cell’s natural active transport system, in particular the microtubule (MT) network and its associated motor proteins, for this purpose. This is reminiscent of the mechanism adopted by adenoviruses6−9 and retroviruses, for example, HIV8,9 and Herpes Simplex virus.5,7,10 Indeed, localization of these viruses in the centrosome region has been observed. As these viruses are known to be highly efficient vectors for gene therapy, mimicking their working mechanisms is a very promising route. While a lot of effort has been devoted in the past to achieve similar efficiencies with synthetic particles (e.g., polyplexes11), viral vectors have remained much more efficient. © 2014 American Chemical Society

Received: January 21, 2014 Revised: March 12, 2014 Published: March 20, 2014 2515

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periods). Therefore, in the present work we do not distinguish between the association and processivity times. Motor velocity may also vary under high loads.15,18 In the case of kinesin, the step size does not change but the stall duration between steps and its frequency both increase under high load.19 For dynein, the step size changes under different loads.18,20 In fact, these issues are of minor importance as NPs considered here produce low loads (Stokes friction forces), for example, ∼10−2 pN for particle radius R = 50 nm (using η = 3.2 × 10−3 Pa s), which is negligible relative to the stall force of 6 pN. A great deal of theoretical effort has also been assigned to describing the cooperative and antagonistic transport of multiple dynein and kinesin molecules associated with the same cargo.21−28 However, we emphasize that we do not study here any cooperative behavior of the motor proteins while they simultaneously carry the NP. Rather, we assume that they associate with, dissociate from, and move along the MT independently of each other. Hence, the effect we obtain for the processivity (association) time is purely collective in nature rather than cooperative. We define the processivity time τp as the mean first passage time (MFPT) for complete NP detachment from the MT (i.e., no motor association), where the initial condition is attachment of exactly one motor, implying that active motion has begun to take place. We begin by presenting a stochastic model for multiple motor association−dissociation kinetics where all motors have equal association−dissociation rates, and calculate τp for this model. Next, we describe the case where proteins have different rates, depending on relative position on the particle surface with respect to the MT. For this purpose, we first calculate these modified association−dissociation rates accounting for the free-energy changes of the spacer polymer upon association with the MT. The resulting τp is calculated for different spacer chain lengths, spacer graft densities, motor protein association energy parameters, and NP sizes. For illustrative purposes, we consider a particle with a total of n proteins available for attachment. In this simplified model, we assume that all proteins have identical association−dissociation kinetics. Thus all motors have the same association and dissociation rates, k+ and k−, respectively. Let Fi(t) be the probability for the particle to be attached by i proteins at time t. The kinetics is therefore described by the following set of equations

surface at one end and is linked to a motor protein at the other (see Figure 1A and Figure S1A in the Supporting Information). Although the main focus is on dynein, our treatment applies equally well to kinesin. For example, in vitro experiments with kinesin achieve such linking via specific chemistry,15 while in vivo cellular systems (where the target is the nucleus region) achieve this for dynein by coupling the free end of the polymer to a nuclear localization signal (NLS) peptide, which is known to be able to recruit cellular dynein from the cytosol via an α/βimportin complex,5,16,17 see Figure 1A.

dF1 = − [k1− + k1+]F1 + k 2−F2 dt Figure 1. (A) Schematic representation of the proposed particle covered with NLS-ended PEGs (green curvy lines). As shown for one of the PEG-NLS, the NLS recruits cytoplasmic α/β-importin proteins (orange and green ellipses) to form a complex with a dynein motor, that moves toward the (−) end of a MT filament. Figure not to scale. (B) The polymer chain dependence on the grafting angle θ.

i=1

dFi = − [ki− + ki+]Fi + ki−+ 1Fi + 1 + ki+− 1Fi − 1 2 ≤ i ≤ n − 1 dt dFn i=n = − kn−Fn + kn+− 1Fn − 1 dt (1)

k−i

−

where = ik is the transition rate between states i → i − 1 attached proteins and k+i = (n −i)k+ is the transition rate between states i → i + 1 attached proteins. If p and q = 1 − p are the fraction of times where a single protein is associated and dissociated, respectively, and τ is a characteristic time scale, these rates may be written as

Our focus is the mean “association time” of the NP with the microtubule; the particle is defined as associated with a given MT when at least one of its ligated motor proteins is associated with the same MT. In principle, the association time may be somewhat different from the processivity time. Motors can be associated with the MT but be stalled. Hence, to deduce a mean run length from the association time one must use a mean velocity (averaged over several walking and stalling

p ki+ = (n − i) , τ 2516

ki− = i

q τ

(2)

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We can now define the NP “survival probability” Ψ(t) as the probability of the NP to be associated with the MT at any given time by at least one motor. In terms of {Fi(t)}

one end to the spherical NP surface and attached to the motor by the other, such that the additional flexibility might enhance motor association rate. In reality, this is mediated by additional complexes, such as NLS-α−β in the case of dynein, but this does not alter the analysis.5 One such design of the spherical NP is shown in Figure 1A. Particle surface is covered by several grafted polymer molecules (e.g., PEG) that end with an NLS sequence. The latter recruits cytoplasmic dynein via the α/β-importin complex, leading to a particle decorated with multiple motors (see also Figure S1A in the Supporting Information). However, it is clear that freeenergy configurational penalty, and thus association−dissociation kinetics,21−26 should depend on the relative position of the grafted chain with respect to the MT. For side chains to reach the MT, one may effectively describe two free-energy penalties: The first is related to the reduction in the number of configurations when the free end attaches to the MT, and the second is associated with polymer stretching required to achieve attachment. In order to calculate these effects, consider the scheme presented in Figure 1B. Planes tangent to the NP surface at the various graft positions are also drawn. We denote the NP radius by R and the chain radius of gyration by Rg. For sufficiently large NPs, such that R ≫ Rg, we may ignore local curvature at the graft position and take the tangent plane to be an infinite, impenetrable flat wall. Consider for simplicity only Gaussian statistics of the chains, related to θ-solvent conditions.31 We take N to be the effective polymer index, that is, the number of polymer Kuhn segments, and a to be the Kuhn length (e.g., for PEG it is 0.76 nm32,33). In the case of a free chain suspended in a bulk solution and N ≫ 1, the end-to-end probability distribution function (PDF) is

n

Ψ(t ) =

∑ Fi(t )

(3)

i=1

We assume that once the NP is completely detached from the MT, the NP quickly diffuses away from it as the Stokes− Einstein diffusion coefficient is large. Thus the return of the NP to the origin and its reattachment may be considered as a separate process. Also, we neglect any transition between nonadjacent states, as simultaneous detachment of two or more proteins is highly unlikely. The MFPT for complete detachment can be calculated from: τp =

∫0

∞

Ψ(t )dt

(4)

The solution of the entire set of stochastic equations becomes more complex as n increases. However, a simple evaluation of the processivity time τp(i), that is related to exactly i motors being attached at t = 0, may be obtained (as commonly done in birth-death first passage time problems) via the backward equation (Gardiner Section 7.429) ki+[τp(i + 1) − τp(i)] − ki−[τp(i) − τp(i − 1)] = −1 (5)

with τp(0) = 0 (by definition) and τp(n) = τp(n + 1) implying that no more than a total of n proteins may participate. Solving eq 5 by means of recursion yields for the dimensionless time τp(i)k− τp(i)k− =

1 nw

i−1

n

1

∑ ⎛ n − 1⎞ l=0

⎜ ⎝

l

⎟w l ⎠

∑ j=l+1

⎛n⎞ j ⎜ ⎟w ⎝ j⎠

P( r ⃗ , R 0) = (6)

where w = p/q. In particular, given that one motor is initially associated, the solution for τp ≡ τp(1) further simplifies to τpk− =

q1 − n − q n(1 − q)

1

2

3/2

( 23π R 02)

e−3/2( r ⃗/ R 0)

(8)

where r ⃗ is the end-to-end vector and R0 = √Na is the rootmean-square (RMS) end-to-end distance.31 Consider now a polymer chain grafted at one end to an infinite planar wall situated at z = 0. The presence of the wall is accounted for by imposing absorbing boundary conditions for the chain random walk on the wall surface and taking the position of the first monomer to be a single Kuhn length away, z = a. Using the method of images (in a cylindrical coordinate system) it may be shown that34,35

(7) 30

Equation 7 is supported by the work of Lipowsky et al. It shows that there is a strong dependence of processivity time on the number n of proteins available for attachment, especially if q ≪ 1. Thus, under such conditions and if n ≥ 2 one may reach a processivity time more than an order of magnitude greater than the single protein processivity time. The discussion above describes a simple case where the collective dynamics are dictated by single motor kinetics, that is, when the particle surface covered by motors is flat. However, for a realistic NP geometry the first attached motor dictates the relative position of all other motors with respect to the MT, and the kinetics of each individual motor may differ. We consider specifically the case of a spherical NP. Clearly, its surface curves away from the MT and one may expect a decrease in the association rate for motors located farther away from the MT. To overcome this limitation, motors must exhibit a certain degree of extensibility and flexibility. Kinesin is considered to be a semiflexible polymer, which allows for some degree of freedom in attachment.25 Dynein, however, appears much more rigid12 and thus requires an additional flexible “handle” to allow side-motor association with the MT. We therefore suggest the addition of a spacer polymer, grafted at

P(ρ , z , R 0) =

2 1 −3/2(ρ / R 0)2 −3/2[(z − a)/ R 0]2 e e − e−3/2[(z + a)/ R 0] Q

(

) (9)

where ρ is the 2D cylindrical radius, and Q = (2π/3)3/2R03erf [(a/((2/3)1/2R0)] serves as the normalization factor in half space. This PDF signifies the ratio of the number of polymer configurations that end at (ρ⃗, z), to the total number of configurations, both in the presence of the wall. For z ≪ Na one obtains34 P(ρ , z , R 0) ≅

9z −3/2[(ρ2 + z 2)/(R 02)] e 2πR 0 4

(10)

Suppose an adhesion site exists at a point (ρ⃗, z) away from the wall, with adhesion energy ε > 0. Because we assume that only the free end may associate with the adhesion site, the free energy difference between the nonadhering and adhering states may be written as 2517

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ΔF = −ε − kBT ln[a3P(x , y , z)]

(11)

leading to ⎡ ⎛ 9za3 ⎞ 3 ⎛ ρ 2 + z 2 ⎞⎤ ⎜ ⎟⎥ ⎟ ΔF = −ε − kBT ⎢ln⎜ − ⎢⎣ ⎝ 2πR 0 4 ⎠ 2 ⎝ R 0 2 ⎠⎥⎦

(12)

The shortest distance between the NP and MT is denoted by d. We assume that attachments occur at minimal stretching, namely when the polymer end-to-end vector is perpendicular to the MT. In terms of R and the angle θ describing the polymer graft position (Figure 1B), this free-energy difference becomes ⎛ 9 cos θ[R(1 − cos θ ) + d]a3 ⎞ ⎟ ΔFθ = −kBT ln⎜ 2πR 0 4 ⎠ ⎝ ⎛ R(1 − cos θ ) + d ⎞2 3 + kBT ⎜ ⎟ −ε 2 R0 ⎝ ⎠

(13)

The minimal value of ΔFθ is at θ = 0, associated with the grafted chain closest to the MT. This implies that the most probable positioning of the first polymer is with θ = 0. Therefore, this chain is the one that determines the distance d, so that ΔFθ=0 may be regarded as a potential of mean force, and d may be found by minimizing ΔFθ=0 with respect to d, obtaining d = R0/√3. We distinguish between two major states {i}: a chain with its other end free (σ = 0), and a chain with its other end associated with the MT (σ = 1). The Boltzmann probability p(θ) of the associated state is given by e−ΔFθ / kBT 1 + e−ΔFθ / kBT

p (θ ) =

Figure 2. Probability p(θ) of chains to adhere to the MT, against grafting angle θ. (A) Fixed NP radius R = 50 nm at varying R0. (B) Fixed chain end-to-end distance R0 = 3 nm at varying NP radii. ε = 8kBT, a = 0.76 nm in both cases.

Because of this dependence on θ, we cannot use eq 1 to describe the stochastic process, and a more extended calculation is required. To this point, we defined a particle state merely by the number of its connecting motors. Now, however, we need to distinguish between different states having the same number of connecting motors but with different arrangements. Furthermore, because the particle is relatively small, its rotation and translation times are significantly shorter than the association and dissociation times discussed above. The rate at which a particle rotates over an angle θ is roughly

(14)

thereby 2

e{(ε / kBT ) − 3/2[(R(1 − cos θ) + d) / R 0] }

p(θ) =

2π R 0 4

2

9cos θ[R(1 − cos θ) + d]a3

+ e{(ε / kBT ) − 3/2[(R(1 − cos θ) + d) / R 0] }

k rot ≈

(15)

and that of the nonassociated state is q = 1 − p. Therefore, given an estimate for the apparent adhesion energy ε, distance from the MT d, polymer index N, and NP radius R, one can estimate the probability of any grafted chain end to adhere. Figure 2 shows p against θ for different values of R0 (Figure 2A) and R (Figure 2B), taking the equilibrium value of d = R0/ √3. We observe a plateau at low values of grafting angle θ, followed by a sharp decline. Increase of particle size at constant chain length leads to a narrower plateau with no effect on its value. Chain length dependence is more involved. As chain length increases the plateau value drops, resulting from an entropy loss upon end fixation, however, the decline from the plateau tends to be smoother, reflecting the side chains ability to stretch with a reduced entropy penalty. This dependence should affect processivity time, which we calculate next. We now turn to the problem of multimotor attachment kinetics for the spherical particle geometry. Equation 15 shows the effect of graft position on MT association probability. Using detailed balance we may deduce the position dependence of the association and dissociation rates as k+(θ ) =

p(θ ) , τ

k−(θ ) =

1 − p(θ ) τ

kBT 2πηθ 2R3

(17)

where η is solvent viscosity. For cytoplasmic conditions, given a particle with R ≤ 100 nm, the rotation rate is krot ≈ 103 s−1, that is, several orders of magnitude greater than the association and dissociation rates ∼1 s−1. We therefore assume that for each state of associated motors, the particle equilibrates around the minimal configurational total free-energy of adhered polymers. (The opposite case is studied in the Supporting Information, Section S1, for comparison.) An example of such a particle is presented in Figure 3 for the case of three proteins available for attachment (1,0 represent the state of a motor, where 1 represents association and 0 dissociation). The collective motor connectivity state is indexed by α. Let Pα(t) be the probability of the system to be in state α at time t. The complete set of differential master equations for this system may be written as 36

d ⃗ Pα(t ) = −M⃡ × Pα⃗ (t ) (18) dt ⃡ where M is the (v × v) transition rate matrix and v is the number of bound states. The elements of the matrix M⃡ are calculated in the Supporting Information, Section S1, on the basis of individual motor association−dissociation kinetics with no reattachment allowed after complete detachment.

(16) 2518

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Ψ(t ) =

∑ Pα(t ) = α

∑ Cje−λ t j

j=1

(19)

Here {λj} are the eigenvalues of the matrix M⃡ (λj > 0), and Cj =

∑ Cj̃ (α) α

(20)

where C̃ j(α) are coefficients determined by initial conditions. Accordingly, using eq 4 the processivity time is given by v

τp = Figure 3. The kinetic process of a NP with n = 3 chains available for attachment. The dashed line represents the particle symmetry axis. Because of fast rotations, each state equilibrates around the minimal free-energy, and each chain association and dissociation rates are determined by the appropriate p(R,R0,θ,d,ε), see Supporting Information.

∑ j=1

Cj λj

(21)

We define the interanchor distance on the NP surface ξ, which may be calculated from the surface anchor number density ϕ as ξ ≈ 1/√ϕ . Hence, we assume that anchors are positioned at angle intervals δθ = ξ/R. The typical values we shall choose for ξ (>15 nm) are such that steric interactions between bound proteins are expected to be nonsignificant. The sharp decline of p(θ) allows the introduction of a cutoff angle θco for which p(θ) < 10−3 × p(0). Chains with grafting angles θ > θco effectively do not participate in the association of the NP; lowering this lower limit on p(θ) essentially does not change the results for processivity time (see Supporting Information Section S4).

We assume that the initial condition corresponds to a single motor attachment with θ = 0 (see state α = 1 in Figure 3). Solving eq 18 the attachment survival probability is calculated as

Figure 4. Normalized processivity time τpk− for the spherical NP against: (A) chain end-to-end distance R0 for R = 50 nm and ξ = 15 nm at varying energy of association ε, and (B) normalized chain end-to-end distance R0/ξ for ε = 8kBT at varying particle radius R and interanchor distance ξ but with a fixed ratio ξ/R = 0.30. Dissociation rate is roughly k− ≅ 1 s−1. 2519

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accurately known. Conversely, the value of the maximum processivity time is strongly sensitive to this parameter. Estimates appearing in the literature suggest that this parameter is probably in the range of 7−10kBT, yielding a processivity time ranging from a few seconds to several hundred seconds. We emphasize that this is a collective effect rather than a cooperative one, in which all connecting motors are assumed to operate independently of each other. Moreover, motor velocity may fluctuate greatly with time and temporal stalling may even occur. Such effects have been recently addressed by Berger et al.,23,26 where it was claimed that when more than one motor are simultaneously attached to the MT, temporal stalling and unbinding processes are enhanced and are sensitive to the rigidity of linkers and their nonlinear elasticity. In fact, our results in Figure 4 offer a possibility to overcome such difficulties by increasing R0 beyond R0,max in order to reduce the effective spring constant ∼ kBT/R02 while maintaining a relatively high processivity time. Thus, while mean NP velocity may be somewhat reduced, the enhancement of processivity time, and thereby run length, should remain appreciable. Moreover, as shown in recent experiments, when larger NPs in the range R = 0.2−0.5 μm are considered, enhancement of mean velocity was observed and interpreted as a load-sharing effect,38,39 which might even suggest a minor increase in mean velocity for the particle sizes considered here. Such systems are ideal for the administration of nucleus targeting drugs, as the mean run length becomes comparable to cell dimensions. Upon entry into the cell via endocytosis and escape from the endosome, NPs carrying drugs must quickly arrive at their corresponding target and accumulate near it. Indeed, NPs designed according to our suggestion are expected to rapidly travel toward the centrosomal region and strongly localize there. Thus, even if too large to enter through the nuclear pore complex (i.e., larger than 40 nm in diameter), the drug will be released near the nuclear envelope and is more likely to penetrate the nucleus. In addition, our model may apply to the transport of adenoviruses, HIV and herpes simplex virus, for which strong dynein mediated centrosomal localization has been observed.8,40 It has also been suggested that similar transport of Golgi complexes41 is the mechanism behind their centrosomal localization. Apart from enhancement of transport, additional decoration technics might be applied in order to overcome other barriers, such as cell penetration and endosomal escape.3,42,43 Our analysis suggests a promising future for rational design of drug delivery nanocarriers.

Equations 19−21 were solved numerically for several NP and chain sizes. In Figure 4A, we show results for a particle of radius R = 50 nm. We plot τpk− against R0 for several values of ε in the range 7.5kBT < ε < 10 kBT, which is assumed to be the physical range.37 We observe a clear maximum at a certain chain size R0,m. In addition, there is a tremendous increase in τp as ε increases. The value of R0,m slowly decreases as ε increases. Also, as ε increases, a second maximum appears, that might even surpass the first one. This occurs due to the availability of more motors for attachment. To quantify this issue, we present in the Supporting Information the fraction of time that 1, 2, 3... motors are simultaneously bound in a single run, and the resulting mean number of bound motors. We find that the first maximum corresponds to a mean of two proteins, whereas the second maximum corresponds to an average of three (see Figure S7 in the Supporting Information). Figure 4B shows processivity time against the normalized chain end-to-end distance R0/ξ for different values of R and ξ, keeping the ratio ξ/R constant. Interestingly, the optimal value of R0/ξ (corresponding to the maximum in τp) remains roughly constant. (A small shift to the right is observed as R decreases.) As a supplementary plot, we show in the Supporting Information the dependence of τp on molecular weight, taking PEG as the spacer polymer (Figure S5). In this Letter, we presented the effect of collective active transport of a NP by several motor proteins. By considering a hypothetical case of a flat particle surface adjacent to an MT, we have shown that the presence of several motor proteins that are simultaneously available to carry the particle leads to strong enhancement of effective particle processivity. Our main focus was the case of a spherical particle. The key to our innovation is the use of flexible polymer ligands anchored to the particle surface, which mediate between the NP and the motors. The flexible polymers allow, by stretching, for motors originating from the particle sides to easily reach the microtubule, thus increasing the number of available motors. Yet, there exists an optimal chain length due to conflicting effects. Excessively short chains require increased stretching of side chains, thus decreasing their probability of attachment. This leads to reduction in the effective number of motors that are available for binding (and transport), and therefore a reduced processivity time. Conversely, excessively long chains reduce the probability of attachment of middle chains, due to a configurational entropy loss upon free-end fixation, which leads again to a reduction in the processivity time. We performed detailed analytical calculations for the motorMT attachment probability as a function of the polymeranchoring angle. We combined these calculations with a scheme for motor attachment-detachment kinetics that was numerically solved. For a given particle radius and polymer graft surface density, effective particle processivity time was calculated at varying chain lengths. We find that there is a very clear optimum for which a long processivity time is achieved. For instance, for a particle of radius 50 nm with approximately 140 chain grafts per particle (mean distance between grafts about 15 nm), a clear maximal processivity time is reached for a chain with a free end-to-end distance of about 5−6 Kuhn lengths, which in the case of PEG corresponds to molecular weight of about 450 Da (see Supporting Information Section S3). Different optimal lengths are obtained for different particle sizes and graft densities. Importantly, however, the optimal length is only weakly sensitive to the motor-MT association energy, which is not

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ASSOCIATED CONTENT

S Supporting Information *

Detailed explanation of the kinetic model, along with further analysis of processivity time and survival probability are provided. The material is available free of charge via the Internet http://pubs.acs.org

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Anne Bernheim, Michael Elbaum, Leah Gheber, Carlos Marques and Ro’ee Orland for useful discussions. This 2520

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(31) Gennes, P. G. d., Scaling concepts in polymer physics. Cornell University Press: Ithaca, N.Y., 1979; p 324 p. (32) Scheidler, P.; Kob, W.; Binder, K. J Phys Iv 2000, 10 (P7), 33− 36. (33) Lee, H.; Venable, R. M.; MacKerell, A. D.; Pastor, R. W. Biophys. J. 2008, 95 (4), 1590−1599. (34) Dimarzio, E. A. J. Chem. Phys. 1965, 42 (6), 2101−&. (35) Bickel, T.; Jeppesen, C.; Marques, C. M. Eur Phys J E 2001, 4 (1), 33−43. (36) Fushimi, K.; Verkman, A. S. J. Cell Biol. 1991, 112 (4), 719−725. (37) Bameta, T.; Padinhateeri, R.; Inamdar, M. M. J. Stat. Mech. 2013, P02030. (38) Macosko, J. C.; Cayhuti, T.; Townsend, B.; Shtridelman, Y.; Baysinger, K.; Holzwarth, G.; Harrington, D.; DeWitt, D.; Fallesen, T. Biophys. J. 2007, 497a−497a. (39) Shtridelman, Y.; Cahyuti, T.; Townsend, B.; DeWitt, D.; Macosko, J. C. Cell Biochem Biophys 2008, 52 (1), 19−29. (40) Kelkar, S.; Crystal, R. G.; Leopold, P. L. Mol Ther 2004, 9, S287−S287. (41) Thyberg, J.; Moskalewski, S. Experimental cell research 1999, 246 (2), 263−79. (42) Hatakeyama, H.; Akita, H.; Harashima, H. Adv Drug Deliver Rev 2011, 63 (3), 152−160. (43) Kogure, K.; Akita, H.; Harashima, H. J. Controlled Release 2007, 122 (3), 246−251.

research was supported by the Focal Technological Area Program of the Israeli National Nanotechnology Initiative (INNI).

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dx.doi.org/10.1021/nl500248q | Nano Lett. 2014, 14, 2515−2521