Cytoplasmic diffusion: molecular motors mix it up

To elucidate the nature of fluctuating motion in cells, many studies have analyzed the “passive” fluctuating motion of micrometer-sized spherical probe particles. If such particles were embedded in a viscous liquid driven by thermal Brownian fluctuations, they would exhibit random, diffusive motion, as shown schematically in Fig. 1 a (blue particle). Quantitatively, this means that the distance the particle has moved, Δx, after some time interval, τ, is described by

⁠, where the angled brackets indicate an average over many particles, and the diffusion coefficient, D, is given by the Stokes-Einstein equation:

which depends on the viscosity η and particle radius a (Einstein, 1905). This reflects the fundamentally thermal origin of diffusion in an equilibrium liquid, depending as well on the temperature (T) and Boltzmann's constant (k_B). This diffusive time dependence is shown schematically by the blue line in Fig. 1 a. Such motion is in stark contrast with steady particle motion in one direction with constant velocity v, illustrated by the black curve in Fig. 1 a; because

⁠, this motion is described by

\({\langle}{\Delta}x^{2}{\rangle}{=}v^{2}\mathrm{{\tau}}^{2}\)

⁠. In principle, one can track the motion of inert tracer particles in cells to determine whether their motion is diffusive with the appropriate diffusion coefficient. However, inside cells this picture is complicated by the fact that the cytoplasm is generally not a simple viscous liquid but rather a structured viscoelastic material (Luby-Phelps et al., 1987; Fabry et al., 2001). In a viscoelastic material, thermal fluctuations do not lead to ordinary diffusion, but rather “subdiffusive” motion, characterized by a different time-dependence:

\({\langle}{\Delta}x^{2}{\rangle}{\propto}\mathrm{{\tau}}^{\mathrm{{\alpha}}}\)

⁠, where α < 1, as shown schematically by the red curve in Fig. 1 a. Thus, studies showing diffusive, or even “superdiffusive” motion (α > 1), within the viscoelastic cytoplasm suggested this random motion is not thermally induced (Caspi et al., 2000). However, other studies assume that random intracellular motion is thermally induced, and use this assumption to extract the mechanical properties of the cell from tracer particle motion (Tseng et al., 2002; Panorchan et al., 2006). To quantitatively clarify this, several recent studies have analyzed both measurements of the “passive” random motion of tracer particles, as well as direct, active measurements of the viscoelastic properties of the cytoplasm, for example, using a magnetic tweezer to pull on cells (Lau et al., 2003; Bursac et al., 2005). These studies have concluded that the random motion not only has an unexpected time dependence but is also significantly larger in magnitude than would be expected for purely thermal fluctuations, suggesting that biological activity can indeed give rise to random fluctuating motion that dominates over thermal fluctuations.

Considerable new insight into the underlying physical origin of these motor-driven fluctuations was obtained from studies of a reconstituted actin network incorporating myosin II motors (Mizuno et al., 2007), oligomerized into processive motor assemblies similar to those found in the cellular cytoskeleton (Cai et al., 2006). The mechanical resistance of the actin network was precisely characterized using optical tweezers to actively pull on particles within the network. The fluctuating motion of the particles was also measured, both with and without motors present. By comparing these two measurements, the contribution of the thermal motion could be distinguished from that of the nonthermal motion, showing clearly that myosin motors can give rise to strong random fluctuating motion within the network. Interestingly, these random myosin-generated forces can also lead to a pronounced stiffening of the actin network, increasing the rigidity by as much as 100-fold (Mizuno et al., 2007; unpublished data).

The random fluctuating motion arising from the activity of cytoskeletal motor proteins can have important biophysical consequences. In an attempt to directly measure the underlying fluctuating forces, a different but complementary approach has recently been developed. It uses endogenous cytoskeletal microtubules as probes. Microtubules are stiff biopolymer filaments that are present in almost every animal cell and are physically linked to other components of the cytoskeleton (Rodriguez et al., 2003; Rosales-Nieves et al., 2006). Thus, as with spherical probe particles, their motion reflects forces and fluctuations of the network (Waterman-Storer and Salmon, 1997; Odde et al., 1999). However, in contrast to spherical probes, microtubules also exhibit a local bending motion, whose amplitude can, like a simple elastic spring, be used to determine the applied force (Fig. 1 b). Microtubules in cells indeed appear highly bent, as can be seen, for example, in the fluorescence image of the microtubule network within an adherent CHO cell, shown in Fig. 1 c. These bends fluctuate dynamically in time, which can be seen by subtracting sequential images, as shown in the top inset of Fig. 1 c.

This motion was studied in cells by tracking individual fluorescent microtubules, using both Cos7 and CHO cells (Brangwynne et al., 2007b). The curves defined by the microtubules are analyzed using a Fourier analysis technique developed to characterize the bending fluctuations of isolated biopolymers in thermal equilibrium (Gittes et al., 1993; Brangwynne et al., 2007a). This analysis is based on the fact that each curve can be represented as a sum of simpler sinusoidal curves, each of a different amplitude (a_q) and wavelength (λ). The wavelength is typically written in terms of its wave vector, q = 2π/λ, as sketched in the bottom inset of Fig. 1 c. Using this approach, intracellular motion was analyzed by calculating the mean-squared difference in amplitude using

⁠, as a function of lag time τ.

This quantity is analogous to that calculated for fluctuating particles,

⁠, but here effectively measures the fluctuating motion of the microtubule at different wavelengths. In thermal equilibrium, the amplitude difference will grow with τ up to a maximum value given by

where k_BT is the thermal energy scale and κ is the microtubule bending rigidity. The ratio of these defines the persistence length:

which represents the length scale at which thermal fluctuations completely change the direction of the filament; for microtubules, l_p is on the order of 1 mm (Gittes et al., 1993). This establishes the maximum amplitude of bending fluctuations that can be induced by thermal agitation and allows thermal and motor-induced fluctuations within the cell to be distinguished.

For microtubules in cells, the amplitude of the fluctuations was found to grow roughly linearly in time, the behavior expected for simple Brownian diffusion (Fig. 1 b). However, strikingly, the maximum bending amplitude in cells is much larger than that expected for thermally induced bends for l_p ≈ 1 mm. This is most apparent for small wavelength bends, q > 1 μm⁻¹, as shown by the blue triangles in Fig. 2 a, which are significantly above the maximum thermal amplitude shown by the solid line. Thus, although random intracellular motion can exhibit features similar to random Brownian motion, it appears inconsistent with a purely thermal origin.

In these experiments in living cells, there are several unknown variables that could play a role. For example, the persistence length of microtubules may vary within the cell, caused by a possible length dependence (Pampaloni et al., 2006) or arising from the effects of microtubule-associated proteins (Felgner et al., 1997). A simplified in vitro cytoskeleton was therefore developed, incorporating purified microtubules in a model actin network (Brangwynne et al., 2008). In the absence of motor proteins or other sources of nonequilibrium activity, microtubules embedded in the actin network are subject to only thermal forces that result in small bending fluctuations; as with the motion of spherical probe particles, the viscoelasticity of the surrounding network leads to subdiffusive behavior of the bending amplitudes,

⁠, as shown in Fig. 2 b. These fluctuations display a small maximum amplitude corresponding to l_p ≈ 1 mm, as shown by the red squares in Fig. 2 a. When myosin II motor assemblies are added to the actin network, the embedded microtubules show dramatically different behavior: they bend significantly more, and the bends are highly localized. However, although this behavior is driven by processive motor activity, the microtubule bends fluctuate randomly in time, with localized bends growing and shrinking rapidly, as illustrated by the typical time series shown in the inset of Fig. 2 b; this behavior is similar to that found using spherical probe particles (Mizuno et al., 2007). These microtubule bends appear to result from localized transverse forces (Landau and Lifshitz, 1986), as sketched in the bottom inset of Fig. 2 c. Analogous to a simple spring (force proportional to displacement, F = kΔx), the maximum force was directly determined from the amplitude of these bends, revealing force pulses on the order of 10 pN, which is consistent with that of a few myosin motors acting together (Finer et al., 1994). Interestingly, short wavelength bends can also arise from compressive forces acting within cells (Brangwynne et al., 2006).

The dynamic, localized microtubule bends observed in vitro lead to fluctuations in the bending amplitudes that are large and distinctly nonthermal at short wavelengths (large q), as shown by the comparison of thermal (Fig. 2 a, red squares) and motor-driven (blue squares) fluctuations for q ≥ 0.2 μm⁻¹. At longer wavelengths, however, the fluctuations are indistinguishable from those of thermally excited filaments, as expected for microtubules whose lateral motion is restricted by the surrounding elastic environment. Surprisingly, with added myosin motors, the actively driven bends fluctuate in a diffusive-like manner,

⁠, as shown in Fig. 2 b (blue squares). This nonthermal diffusive behavior can be understood in terms of steplike or on–off dynamics in the forces applied by individual motor assemblies as they bind and unbind to cytoskeletal filaments (Mizuno et al., 2007; MacKintosh and Levine, 2008). Thus, even processive motor activity has stochastic features that can give rise to diffusive-like motion of cytoplasmic components. Although it is likely that myosin II is not the only motor contributing to this behavior in cells, these experiments help elucidate the underlying biophysical processes.

These random nonthermal force fluctuations appear to be a ubiquitous feature of living cells. They can, therefore, play an important role in a variety of cellular processes. For example, as a result of large microtubule bending fluctuations, the tips of growing microtubules also undergo large fluctuations in directional orientation, leading to highly curved microtubule shapes that appear to be “frozen-in” by the surrounding elastic cytoskeleton, as shown in the example in Fig. 3 a. These random fluctuations in tip orientation are analogous to random thermal fluctuations and give rise to microtubule bends with a thermal-like dependence on q, as shown in Fig. 3 b. However, the corresponding nonequilibrium persistence length is reduced to ∼30 μm, ∼100-fold less than the thermal persistence length. Thus, the microtubule network is more bent by ∼100-fold in these cells as compared with isolated microtubules in solution. Although the effects of microtubule binding proteins or defects in the tubulin lattice could also contribute, tip fluctuations alone are sufficient to explain these large bends. Driven fluctuations can thus play a significant role in determining cytoskeletal architecture.

The enhanced diffusive dynamics that result from motor activity may also affect the rates of biochemical reactions that take place on the cytoskeletal scaffold (Forgacs et al., 2004), as well as those usually thought to be limited by thermal diffusion. Thus, “active” cytoplasmic diffusion could represent a kind of microscopic mixing that enables rapid diffusion of vesicles and small molecules.

Thermal fluctuations have been known to play a fundamental role in the behavior of all nonliving matter since Einstein's seminal work in 1905 (Einstein, 1905), in a paper explaining how thermal forces give rise to the diffusive motion first observed by Brown in 1828 (Brown, 1828). We propose that active intracellular force fluctuations represent the biological analogue, controlling cell behavior while subject to biochemical regulation. Interestingly, this may actually be closer to Brown's initial concept of a vital microscopic activity, in contrast to the nonliving, thermal motion that bears his name (Brown, 1828). This active motion is clearly a ubiquitous and important phenomenon in living cells; indeed, it may be that active processes contribute to virtually all randomly fluctuating, diffusive-like motion in cells.

C.P. Brangwynne acknowledges the hospitality of the Vrije Universiteit.

This work was supported by the National Science Foundation (DMR-0602684 and CTS-0505929), the Harvard Materials Research Science and Engineering Center (DMR-0820484), and the Harvard Interdisciplinary Graduate Education and Research Training program on Biomechanics (DGE-0221682). F.C. MacKintosh was partially supported by the Foundation for Fundamental Research on Matter (FOM). G.H. Koenderink is supported by a European Marie Curie Fellowship (FP6-2002-Mobility-6B, Contract No. 8526) and by FOM.