Molecular motions inside the cell
A paper in Science this week describes the use of carbon nanotubes to pinpoint the movements of the living cell in fine detail, making for a really nice study in quantitative/mathematical biology.
Noninvasive tracking was accomplished by imaging highly stable near-infrared luminescence of single-walled carbon nanotubes targeted to kinesin-1 motor proteins in COS-7 cells. We observed a regime of active random “stirring” that constitutes an intermediate mode of transport, different from both thermal diffusion and directed motor activity. High-frequency motion was found to be thermally driven. At times greater than 100 milliseconds, nonequilibrium dynamics dominated. In addition to directed transport along microtubules, we observed strong random dynamics driven by myosins that result in enhanced nonspecific transport. We present a quantitative model connecting molecular mechanisms to mesoscopic fluctuations.
The “mesoscopic" scale is more often seen in the context of pure and applied physics (microelectronics, nanofabrication and nanotechnology), though journals such as Soft Matter present research articles giving the same ‘condensed matter’ treatment to biological systems (“Where physics meets chemistry meets biology”).
From ancient Greek μέσος it refers simply to a ‘middle’/intermediate between the molecular and macroscopic scale, where neither atomistic/quantum nor classical physics/bulk models best describe observed behaviour, and novel effects may be described — from interference effects, to quantum confinement (giving rise to band gaps) and charging effects (such as the Coulomb blockade/staircase).
Although often presented as a water-based solvent, the cytosol is more accurately described as a “highly dynamic composite material ” with mechanical properties dominated by microtubules (MTs), F-actin and intermediate filaments; all driven by metabolism-energised polymerisation of actin and tubulin and from motor proteins (specifically nucleotide triphosphate hydrolysis).
The traditional technique to observe cells in motion is fluorescence microscopy, though long-term tracking of single molecules has been hindered by fluorophores’ instabilities and the fluorescence background in cells.
Though biological networks have been termed ‘scale-free’ or ‘-invariant’, and metabolic rate for example is well known to follow a power law, the internal structure of the cell itself is far from self-similar across scales.
At short times (microseconds to milliseconds), thermal motions should dominate. Between milliseconds and seconds, thermal diffusion might still be relevant, but there is mounting evidence, both in vitro and in vivo, that the motion of larger objects couples to myosin-driven stress fluctuations in the cytoskeleton.
» Mizuno (2007) Nonequilibrium mechanics of active cytoskeletal networks.
» Brangwynne (2008) Cytoplasmic diffusion: Molecular motors mix it up.
Here, temporal fluctuations, reminiscent of thermal diffusion in liquids, can arise from nonequilibrium dynamics in the viscoelastic cytoskeleton. On longer time scales, from minutes to hours, directed transport and larger-scale collective motions typically dominate. The motion of probe particles tracked inside cells has been classified as subdiffusive, diffusive, or superdiffusive. Such classifications, however, obscure the distinction between thermally driven and nonequilibrium fluctuations and are inadequate to identify intracellular material properties.
Motor proteins direct a whole host of molecular motions, kinesins and myosins being among the most heavily studied in vitro. Using fluorescence microscopy to track individual motor proteins is not only limited by instability of fluorophores, but the quality of the images taken (“signal to noise”) and efficiency of targetting probes to specific molecules.
Modern optical equipment and carefully designed fluorescent dyes have enabled experiments tracking single molecules at a time, though in living cells the authors note these experiments’ timeframes have been limited to around a second.
Their solution was to use single-walled carbon nanotubes (SWNTs), “stiff quasi–one-dimensional tubular all-carbon nanostructures with diameters of ~1 nm and persistence lengths above 10 μm” — which have the convenient property of luminescence in the near-infrared, a region ‘virtually free of autofluorescence in biological tissues’. Not only this, but the excitation time is ~100ps, such that high excitation can give ~1 ms resolution (1 ms = 109 ps).
The nanotubes were dispersed throughout the cell wrapped in short DNA oligonucleotides, with HaloTag protein fusion tags covalently attaching them specifically onto kinesin motor proteins (see Fig. 1, above).
Besides observing directed kinesin-driven transport on MTs, it is possible to directly observe fluctuations of the MT network because a moving kinesin must be bound to a MT. The MT tracks are embedded in the viscoelastic actin cytoskeleton, which in turn fluctuates as a result of stresses generated by cytoplasmic myosins.
With just 100 per cell, the group could track kinesin for up to an hour and a half, observing ~30% of them moving with some sense of direction; the rest locally constrained and moving in a random (stochastic) manner.
Some of the kinesins moved the whole length of the cell, suggesting they had cargo vesicles [along with other motor proteins] attached. Calculating mean squared displacement, MSD, of the molecules’ trajectories showed it grew over time following a power law which could be used to characterise the motion, 〈Δr2(τ)〉∝ τα(where r is distance travelled in the focal plane and τ the lag time). The exponent α shifted from ¼ to 1 between 5 ms and 2.5 s, indicating clear scale variance to the motion.
After this, the group acquired the nanotubes’ fluorescent signal at a rate of four frames per second ‒ using this 250 ms window to observe an intermediate between the thermal diffusion seen on the short timescales and directed motor activity.
With a well-designed control or two, they showed that the transverse motion of the nanotube-marked microtubules was not due to kinesin motors, but reflecting intrinsic dynamics of the cytoskeleton.
The way the relatively rigid MTs report these dynamics depends on two restoring forces: the elastic force of bent MTs and the force exerted by the strained cytoskeletal matrix in which the MTs are embedded. Because it is hard to bend an elastic rod on short length scales, the surrounding matrix yields to the MT when it is deformed on short length scales. By contrast, the MT yields to matrix forces for deflections of wavelength larger than ~1 μm. The shorter-wavelength MT deflections relax faster than our 5-ms frame rate. Therefore, we assume that the transverse MT motion we observe reflects the (active or passive) strain fluctuations of the surrounding matrix.
The MSD power-law exponent α generally reflects the randomness of motion. More precisely, in any medium, the MSD of an embedded probe particle is governed both by the material properties of the medium and the temporal characteristics of the forces driving the particle. For thermally driven Brownian motion in simple liquids, the MSD exponent α = 1. For thermal motion in viscoelastic media, which exhibit time- and frequency-dependent viscosity and elasticity, α < 1 strictly holds. For viscoelastic materials, the stiffness G(ω) typically increases with a power of frequency ω: G(ω) ∝ ωβ. This is observed in polymer solutions, where the viscoelastic exponent β ≈ 0.5 to 0.8, as well as in cells, where β ≈ 0.1 to 0.2 on time scales on the order of seconds. This value of the exponent is close to what is expected for purely elastic materials, where β = 0.
The nearly elastic behavior of cells can be understood as a consequence of strong cross-linking in the cytoskeleton.
Knowing the driving forces, it is possible to construct a relation between MSD exponent α and viscoelastic exponent β. For thermal driving forces, the MSD exponent α = β. Thermal fluctuations can therefore never appear as “superdiffusive” motion with α > 1. Nonthermal driving, by contrast, can result in superdiffusive motion. Theory provides a specific prediction for motion in nearly elastic solids driven by random stress fluctuations with long correlation times and sudden transitions: α = 1 + 2β. This prediction is expected to apply for cytoskeletal stress fluctuations caused by randomly distributed cytoplasmic myosin minifilaments. Myosin locally contracts the actin network with an attachment time of several seconds, followed by sudden release. Some hints of this predicted scaling have been reported for cells and reconstituted acto-myosin model systems. When β = 0 (i.e., in the elastic limit), the resulting MSDs can look deceptively like Brownian motion in a simple liquid, although the physical reason is entirely different. For observation times τ longer than the correlation time of the driving forces, the MSD is predicted to level off, as we observed. In our experiments, the stress correlation time should correspond to typical cytoplasmic myosin motor engagement times, which are indeed reported to be ~10 s in cells.
Still attached to microtubules, the kinesin molecules exhibit vigorous random (Brownian-like) motion as they are buffeted by myosins as described ‒ likely thrusting MTs into the path of other cellular particles. Tubulin forms strong tubular filaments embedded in a more flexible actin network. Nonmuscle myosin II exerts mechanical stress on it, which is released ‘suddenly’ as random stirring of the whole filament network, including the microtubules.
We observed a transition between thermal dynamics in the dominantly elastic cytoskeleton at short times to strongly nonequilibrium power-law dynamics, likely driven by myosin activity, at intermediate times. When the time exceeded the correlation time of the random stress generators, the intermediate regime was followed by a saturation to a maximum MSD, nearly constant over time. Note that in this regime, the MSD amplitude corresponds to a root mean square displacement of ~500 nm, which is larger than the estimated mesh size of the actin network, and thus larger than the expected spacing of obstacles in the crowded cytoplasm.
The authors lastly used myosin inhibitor blebbistatin to block myosin from the actin network, confirming their hypothesis with a dose-dependent reduction in what they call the amplitude of active stirring, and exponent α, “establishing nonmuscle myosin II as the dominant driving factor for random cytoskeletal stirring”.
We can explain the regimes we observe by a quantitative model of cytoskeletal fluctuations and directed motor motion that describes the transition from thermal motion to nonequilibrium stirring dynamics driven by myosin, as well as the transition from stirring dynamics to directed transport driven by kinesin. Our observations were made possible by the use of SWNT labels for broadband molecular tracking in cells. Many questions concerning motor transport in cells will now be addressable using this approach. We have focused here on the stirring dynamics, which constitute an important mode of active intracellular transport between the limits of random thermal diffusion and directed transport, accelerating nonspecific transport through the nanoporous cytoskeleton.
Lead author Nikta Fakhri will soon leave the Göttingen Institute for Biophysics to join the faculty at MIT as assistant professor of physics. Fakhri gave a talk in Massachussets last year on the topic, to the Chemical Engineering department in which some of the details of this paper made their debut:
The discovery of fullerenes provided exciting insights into how highly symmetrical structures of pure carbon can have remarkable physical properties. Single-walled carbon nanotubes (SWNTs) are the vanguard of such architectures. The organization of the hexagonal honeycomb carbon lattice into high-aspect-ratio cylinders with a variety of helical symmetries creates very unusual macromolecular structures representing an emerging research area in condensed matter physics and materials science: traditionally hard materials appearing in new soft matter applications and environments.
… the dynamics of SWNTs in liquids are essentially polymer-like. By exploiting the intrinsic near-infrared fluorescence of semiconducting SWNTs, we have imaged the Brownian motion of individual nanotubes in water and have measured directly the bending stiffness of SWNTs. The semiflexible chain model represents accurately the configurational dynamics of SWNTs suspended in water. Interestingly, the persistence length of SWNTs is comparable to that of biopolymers. This finding paves the way for using SWNTs as a model system for semiflexible polymers to answer long-standing fundamental questions in polymer physics.
… the confined dynamics of stiff macromolecules in crowded environments [are] a common feature of polymer composites and the cell cytoskeleton. In fixed porous networks, we find that even a small bending flexibility strongly enhances SWNTs’ motion. This ends a 30-year-old debate in polymer physics: the rotational diffusion constant is proportional to the filament bending compliance and counter-intuitively, independent of the network porosity. The dynamics of SWNTs in equilibrium and non-equilibrium biopolymer networks is more complex.
At long times, SWNTs reptate in the networks. At short times SWNTs can sample the spectrum of local stresses in equilibrium networks. In the non-equilibrium networks we observe strong local shape fluctuations driven by force generating molecular motors. I will discuss a newly developed microrheology technique in which we use nanotubes as “stealth probes” to measure viscoelastic properties of the host media. Finally, I will introduce a new single-molecule technique based on ultra-stable near-infrared fluorescence of short SWNTs, to study intracellular transport dynamics in living cells and in whole organisms. The combination of long-time stability and high signal-to-noise ratio enables the accurate long-term tracking of single motor proteins tagged with SWNTs traversing the entire cell. Remarkably, we can distinguish the motor protein’s motion along its microtubule track from the track’s underlying random non-thermal fluctuations.
She envisions the technology as applicable beyond probing biophysical questions, in the design of 'active' technical materials.
“Imagine a microscopic biomedical device that mixes tiny samples of blood with reagents to detect disease or smart filters that separate squishy from rigid materials.”
Fakhri will join the Physics of Living Systems group, seemingly on such a bio-materials science project. MIT lab colleague Jeremy England, known for work showing that E. coli reproduction is close to thermodynamic limits of efficiency, spoke of common interest in the cytosol and diffusive processes.
“We’re interested in the non-equilibrium thermodynamics of biological organization, so that could be construed to be about evolution and the origins of life or just about how you make or design self-replicators with desired properties.”
“Increasingly there are now instruments where you can make quantitative measurements on fluorescently labeled proteins in live cells,” England explains. “The cell biologists have their language and their frame of analysis that they’re most comfortable with for describing the phenomenon, but if there are interesting phenomena that are only going to be identifiable if you do the right quantitative analysis on all these numbers that you can now measure in the cell, then it’s useful to have people who are a bit more theoretically minded or physics minded who are there, when rubber meets road, when the data is being generated and helping to influence what kind of experiments get done.”
“We’re looking, for example, at diffusion of proteins in cells. Diffusion as a qualitative phenomenon is just things spreading out over space, but as a quantitative phenomenon, you can look at things like how rapidly a protein that’s labeled over here in the cell will wander over to another region of the cell that’s a certain distance away, and if you can make measurements of that, then you can start to say things that are more specific about characteristics of the diffusion that you are observing than simply seeing it spread out. And in those quantitative measurements, you can sometimes then see differences perhaps between different cells, or different conditions for the same type of cell, that may have biological relevance but that you wouldn’t have necessarily identified without the quantitative analysis,” England says.
⌇ Fakhri et al. (2014) High-resolution mapping of intracellular fluctuations using carbon nanotubes. Science, 344(1687), 1031-5
⌇ Levine and MacKintosh (2009) The mechanics and fluctuation spectrum of active gels. J Phys Chem B, 113, 3820–3830
⌇ MacKintosh and Levine (2008) Nonequilibrium mechanics and dynamics of motor-activated gels. Phys Rev Lett, 100, 018104
⌇ Lau et al. (2003) Microrheology, stress fluctuations, and active behavior of living cells. Phys Rev Lett, 91, 198101
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