The biological world presents a stunning wealth of many-body phenomena for condensed matter physicists. Many of these phenomena are challenging to physicists because they involve complex building blocks and because they occur far from equilibrium. Far-from-equilibrium phenomena, even in simple systems, tend to resist the standard tools of condensed matter theory, and therefore remain largely uncharted territory for physicists.
Our research focuses on the physical mechanism underlying cell crawling. Specifically, we are interested in a type of biological motility, used in a form of cell crawling and by intracellular pathogens such as Listeria monocytogenes, that is driven not by motor proteins but by far-from-equilibrium self-assembly of the protein actin. During this process, ATP hydrolysis and activation of the protein complex Arp2/3 supply energy to the system to drive actin self-assembly from monomers (G-actin) to branched networks of filaments (F-actin) to push a bacterium or a cell forward. This driven self-assembly process is regulated by a cadre of proteins. The system has proven appealing to physicists because it is now possible to drive a latex bead through a buffer solution containing only the minimal set of these proteins. Such beads travel through solution propelled by a dense, branched actin network at their rear, demonstrating that non-equilibrium self-assembly of F-actin is sufficient to drive motility. These results make it easier to strip away nonessentials to develop the fundamental level of physical understanding that is typically achieved for inanimate systems. Our work during the last few years has led to a new answer to the question: how does driven self-assembly of actin lead to motion and force generation?
When a cell crawls, it must reorganize the cytoplasmic network of biopolymers that controls its shape. The shape and motion of the leading edge (the lamellipodium) of a crawling cell are determined primarily by a dynamic network of actin filaments (F-actin). The standard biological model for the regulation of this process is the dendritic nucleation model shown schematically in Fig. 1. Actin filaments are living polymers made up of actin "monomers," namely the globular protein G-actin. The filaments have a definite polarity such that monomers tend to be added to the plus (or "barbed") end and tend to fall off at the minus (or "pointed" end). This polarity is induced by ATP in an ingenious manner. In order for actin to polymerize, it must bind ATP or ADP. For ATP-bound actin, the polymerization rate is higher than the depolymerization rate, leading to net addition of ATP-bound actin. Over time, however, ATP bound to actin in a filament hydrolyzes to ADP. For ADP-bound actin, the depolymerization rate is higher than the polymerization rate. As a result, there is net growth at the fresh, ATP end and net loss at the stale, ADP end of a filament.
In order for a cell to crawl, it must concentrate the fresh, growing ends of F-actin near the membrane at the leading edge. Several proteins work in concert to accomplish this. A key component is Arp2/3 complex, which, when activated, binds to an existing filament and nucleates a branch with a new growing end. Arp2/3 complex is activated at the leading edge by WASP/Scar proteins attached to the membrane. One possibility is that it diffuses away but is caught before long by an actin filament, so that it creates a new branch, or new growing end, near the surface. Another possibility is that when activated and bound at the surface, it acts as a little machine that grabs a filament and attaches monomers to it to form a branch. In either case, it leads to branch growth near the leading edge.
Another important protein is capping protein, which binds to growing ends and prevents further growth. If one assumes that there is some rate at which growing ends are capped, depending on the concentration of capping protein, then older growing ends have a higher probability of being capped. Since older growing ends tend to be further behind the moving edge, capping protein has the effect of killing off growing ends that are not near the surface.
The final two critical proteins are actin-depolymerization factor (ADF) or severing protein, and profilin. ADF severs filaments in two, thus speeding up depolymerization by increasing the number of depolymerizing ends, while profilin converts ADP-bound actin monomers into ATP-bound monomers so that they can be recycled. Thus, filaments just behind the moving surface at the front of the branched network tend to grow due to Arp2/3, WASP and profilin, while filaments at the far end of the network depolymerize away due to ADF and capping protein.
The dendritic nucleation model also describes motility in a different geometry, in which the back end of the ellipsoidal Listeria bacterium is coated with a protein, ActA, which activates Arp2/3 complex. In this case, the growing ends of filaments accumulate just behind the back end of the bacterium to propel the bacterium. Similarly, polystyrene spheres, disks, and other objects coated with ActA, N-WASP or VCA, which all activate Arp2/3, also exhibit motility. We will refer generically to a "moving surface," which could be the back end of a particle or the leading edge of the cell membrane.
Details of the simulation model are published in our paper. Briefly, we use Brownian dynamics to describe the motion of all particles in our system (free monomers, treated as spheres; monomers in filaments, treated within a bead-spring model; Arp2/3, treated as a ghost particle that diffuses but does not interact with other particles and so is incapable of exerting forces; and the moving surface, modeled as a disk). We include the self-assembly as follows: if the center of a free monomer is close enough to that of a monomer at the growing end of a filament, and lies within an angular cone around the bond angle at the filament tip, then it has some probability of being captured. Branching is included in a similar way. Monomers at depolymerizing ends have some probability of falling off in each time step.
Forces in our simulation arise from interactions between particles. All monomers, whether free or in filaments, repel each other with a short-ranged, soft repulsion (excluded volume repulsion). Monomers in filaments are held together by a harmonic potential with a spring constant of 100kBT/σ, where σ = 5 nm is the monomer diameter. We introduce a bending potential along filaments to make them semiflexible. Finally, the monomers and disk repel each other with a short-ranged, excluded volume repulsion. Note that in the basic model we have not included an attraction between the disk and the filaments. In experiments, it is known that the branched actin network (the "actin comet tail") is attached to the moving surface. This issue will be discussed below.
We break symmetry by emitting activated Arp2/3 from the back side of the disk, only. Our disk is constrained to move only in the +z and −z directions, and we have periodic boundary conditions in all directions. We begin the simulation with only monomers and dimers (protofilaments). Over time, the filaments lengthen and, triggered by the activated Arp2/3, branched filaments self-assemble behind the disk. Eventually, the particle moves forward with a well-defined average velocity of order μm/s. This speed is more than an order of magnitude higher than experiments, but this is because we have increased the depolymerization rate and actin concentration.
We have calculated the speed as a function of the concentrations of the three key regulatory proteins, namely Arp2/3 complex, capping protein and severing protein (in our case, depolymerization rate). Experiments on solutions of purified proteins show that the dependence on these proteins is non-monotonic. We also observe non-monotonicity for all three proteins. This is a fairly stringent test for our model; indeed, our simulations are the first to observe these behaviors.
Before embarking on a discussion of the mechanism for motility in our simulation, one must understand how Newton's third law works in this context to give rise to motility. The force exerted by actin on the disk satisfies Fdisk = ζdiskvdisk while the force exerted by the disk on the actin satisfies Factin = ζactinvactin. If the drag coefficient of the branched actin structure is much larger than the drag on the disk, then Newton's third law yields vactin = −ζdiskvdisk/ζactin << −vdisk. Thus, the speed of the actin backwards is very slight compared to the motion of the disk forwards.
What is the origin of Fdisk in the first place? Activation of Arp2/3 leads to a build-up of actin concentration behind the disk. Fig. 3 shows the steady-state actin density profile measured in the frame of the moving disk in our simulation. Just behind the disk, the actin concentration is low because there is a repulsive interaction between the disk and the actin. As a function of increasing distance behind the disk, the actin concentration increases, reaches a maximum and then decays due to depolymerization.
The mechanism we observe from our simulations is very simple. If the interaction between the moving surface and actin is net repulsive, then the moving surface will scoot forwards, away from the actin accumulating behind it. Thus, the speed of the moving surface is set by the net polymerization speed, or the rate at which actin builds up behind the moving surface. When we add a non-specific attractive interaction with the surface, we find that the speed decreases with increasing strength of attraction, and can even reverse sign. Thus, if the actin concentration preferred at the surface is higher than the concentration built up by polymerization, then the disk will move backwards towards the actin.
We can only speculate as to the origin of the repulsion between actin and the moving surface in real systems. In the case of the cell membrane, Listeria bacterium and polystyrene beads, it seems possible that the origin is electrostatic, since actin, the cell membrane, the Listeria bacterium and polystyrene beads are all negatively charged, leading to a short-ranged, screened repulsion. However, we note that there are always excluded volume interactions, which are sufficient.
Should the mechanism apply to the real system?
There is no reason why our mechanism should fail in a system with realistic rate constants. It has been estimated that the actin concentration in the real comet tail is of order mM. This is much higher than the average concentration in solution, which is of order mM. In other words, the effect of building up actin concentration in the comet tail is even stronger in reality. (Note, however, that an increased concentration gradient relative to our simulation would not give rise to an increase of speed according to our mechanism, since the speed is determined by the rate at which actin is accumulating behind the moving surface.) It is hard to imagine how the experimentally-observed tail concentration could not give rise to motility via the mechanism we have proposed. The only question is whether binding of the tail to the surface is enough to disrupt the mechanism. This is the focus of the following proposed experiment on polystyrene beads in cell extracts or purified protein solutions.
The suggestion is to coat the beads with two proteins, one that both activates Arp2/3 and binds actin, such as N-WASP, VCA or ActA, and one that only binds actin, such as a bundling or crosslinking protein. (Such beads would need to be coated on one side only, like Janus particles; otherwise symmetry breaking might not occur and each bead might just be surrounded by a spherical halo of actin.) The addition of the second protein should give rise to an attraction of actin to the surface, which would work against the repulsion. However, we showed in our recent preprint that, contrary to our initial expectation, such an experiment could show a decrease in the forward velocity of actin-based-motility, but never lead to backwards motion.
Additionally, in numerical simulations with a short-ranged, specific binding interaction between actin and the disk, we find that self-diffusiophoretic mechanism of forward motility is not perturbed, provided the binding energy and binding site concentration are not too large. At physiological levels of ActA, and plausible binding energies, the forward velocity is nearly identical to the velocity without binding. By further increasing binding energies or concentrations, we can stall forward motion through a novel mechanism of control; when actin is strongly attracted to the disk, it remains attached for a long duration so that most of the time, actin filaments are sterically inhibited from polymerizing. Thus, attractive interactions do not disrupt the mechanism of motility, but strong binding can affect motility by a biophysical capping mechanism.
Our mechanism provides a simple explanation of an evolutionarily important feature of actin-driven motility, namely the magnitude of the stall force. For in vitro experiments on micron-sized beads, the force needed to drive the bead through a solution with a viscosity of 2.4 cP at a speed of 0.2 mm/s is of order tens of femtonewtons (fN). One might therefore expect that an opposing force of that order would suffice to stall the system. The actual measured stall force, however, is many orders of magnitude higher, in the nN range, even for crawling cells. This change has been attributed to a change of morphology or density of the actin comet tail, or to the dependence of the polymerization rate on stress.
At sufficiently low loads, such as the small drag force exerted by a 2.4cP solution on a moving bead, we find that the speed is independent of drag force, whether we vary the disk radius or the viscosity. It is set by the net polymerization rate, which depends on the free monomer concentration at the surface and the polymerization and depolymerization rate constants. Since the bead moves to avoid the accumulating actin, the force that is required, Flow ~ ηRvdisk, adjusts with disk size R or viscosity h so that the bead can move at the fixed speed, vdisk, at which actin builds up. This result is in good agreement with experiments on VCA-coated beads in purified protein solutions and with experimental measurements of actin comet tails self-assembled on an AFM cantilever. However, experiments on ActA-coated beads in cell extracts see very different behavior, possibly because of self-selection of motile beads. The beads are coated uniformly and initially build up a radially-symmetric cloud of branched actin networks. They must break this symmetry to develop steady-state motility by breaking through weak spots in the cloud. This appears to be more difficult in cell extracts (probably due to the existence of crosslinking and bundling proteins), and may bias the experiments. Experiments on beads near surfaces also show different results, possibly due to alterations in the comet tail due to the surface.
At high loads, we find that the velocity decreases with load and eventually reverses sign at the stall force. By varying the disk radius, we have established that the system exhibits a characteristic stall pressure. In other words, the stall force in our simulations increases as the square of the disk radius: Fstall ~ pstallR2. This is consistent with our mechanism: we find that at high loads, the actin concentration profile deforms but eventually saturates above some threshold load pressure. Above this threshold, the driving pressure remains fixed with increasing load, so the bead eventually reverses direction at sufficiently high loads. The stall pressure in our simulations is in good agreement with that measured experimentally. It is clear that Flow and Fstall are controlled by very different physics and can be different by many orders of magnitude, depending on the values of R and h. Our mechanism therefore provides a very natural explanation of the difference between these two forces.
Does the same physics apply to the real force-velocity relation?
Based on the insight outlined above, we suggest that the magnitude of the stall pressure in the real system is set by the energy cost of deformation of the concentration gradient, which is ultimately set by the compression modulus of the actin comet tail. If the comet tail is crosslinked, the compression modulus should be comparable to the Young's modulus. The Young's modulus has been measured to be of the order of kPa. This corresponds to nN/mm2, in excellent agreement with measurements of the stall pressure. This provides some level of confidence in our ideas.
The insight into the force-velocity relation suggests some experiments that could be done. It would be interesting to increase the compression modulus by adding additional crosslinking proteins to a cell extract. An alternate experiment would be to measure the stall force for the bundled system studied by Brieher, et al., in which the action of Arp2/3 is suppressed and fascin is added. We would expect this system to have a larger compression modulus and therefore a higher stall pressure.Further reading:
- A. Gopinathan, K. C. Lee, J. M. Schwartz, and A. J. Liu, Branching, capping, and severing in dynamic actin structures, Phys. Rev. Lett. 99 058103 (Aug. 2007).
- K.-C. Lee and A. J. Liu, New proposed mechanism of actin-polymerization-driven motility, Biophys. J. 95 4529 (Nov. 2008).
- K.-C. Lee and A. J. Liu, Force-velocity relation for actin-polymerization-driven motility from Brownian dynamics simulations, Biophys. J. 97 1295 (Sept. 2009).
- G. P. Alexander and A. J. Liu, Self-Diffusiophoresis in the advection dominated regime, (July, 2011).
When a cell divides, it must replicate its DNA and pass the replicated chromosomes and plasmids along to its daughter cell. Although this process is fundamental to all cells, the force-generating mechanisms underlying chromosome segregation are not well-understood. At a glance, the mechanism of chromosome segregation in Caulobacter crescentus is counterintuitive (see the cartoon below). Once DNA replication begins at one cell pole, a segment of the replicated chromosome (parS) is coated with the ParB protein. Meanwhile, a structure of actin-like ParA protein grows from the opposite cell pole until contacting the ParB-decorated chromosome. ParB binds to ParA and begins to disassemble it. As ParA retracts by depolymerization, it pulls the ParB-coated chromosome across the cell. No other components are known to be necessary for this process. How can ParA pull the chromosome at the very points that seem to be disintegrating?
In our paper with Ned Wingreen and Zemer Gitai at Princeton, we explain that self-diffusiophoresis is sufficient for robust chromosome pulling. Similar to the case of actin-polymerization-driven motility, described above, this mechanism for chromosome motility relies on self-generated and self-sustaining concentration gradients. However, unlike in the case of actin, attractive interactions and disassembly dominate (rather than repulsions and assembly). Since the ParB attached to the chromosome depolymerizes ParA, it creates a concentration gradient of ParA; since it ParB is attracted to ParA, it moves up the gradient towards higher ParA concentrations. This steady-state process pulls the DNA across the cell, as can be seen in our Brownian dynamics simulations.Further reading:
- E. J. Banigan, M. A. Gelbart, Z. Gitai, N. S. Wingreen, and A. J. Liu, Filament depolymerization can explain chromosome pulling during bacterial mitosis, PLoS Comput. Biol. 7 e1002145 (Sept. 2011).