SIAM News Blog

Living Matter and Active Cells

By Michael Shelley

Living things assemble themselves to a remarkable degree in a coordinated and hierarchical manner. A driving question in biology seeks to explore the factors that underlie and guide this astonishing capacity for self-assembly and self-organization, which is also the animating force for the multidisciplinary field of active matter. The back-and-forth conversation between cell biology and active matter physics has been fruitful and intense.

In their study of multicellular life, biologists naturally turn to its earliest stages and investigate the lively processes through which egg and embryonic cells develop. Figure 1a shows how a nematode embryo—composed of only a single cell—moves, combines, and segregates genetic material as it proceeds towards its first division. This dance is choreographed by elements of the spindle complex: a self-assembled organelle that comprises stiff biopolymers (microtubules), microtubule nucleating sites (centrosomes), molecular machines (motor proteins), and other specialized proteins (see Figure 1b). The spindle complex fulfills several tasks, and its form adapts and changes as it moves through each one. Its centrosomes negotiate the joining of the male and female nuclei and properly position the subsequent conjoined nuclei. The complex then elongates, separating the duplicated chromosomes on opposite sides of the soon-to-be divided cell.

All of this activity takes place over 10 or 15 minutes, but the complex’s structural elements—its microtubules—constantly disassemble and are replaced approximately every 20 seconds. While such transience might seem weird and counterintuitive, this ephemeral quality provides the complex with some of its necessary adaptivity. Because the spindle complex moves through the cell’s fluidic cytoplasm, the sequence is also a wonderfully complicated fluid-structure interaction problem. It involves mobile structures—many of which are transitory and flexible—that interact with each other hydrodynamically and through motor proteins in a confined space. Very specialized computational fluid dynamics methods, which assume that the cytoplasm is a simple Stokesian fluid, have helped unravel some of these mysteries (see Figure 1c).

Yet where does active matter come into play? First, the cell assembles the spindle complex—which is maintained and moved by a constant expenditure of energy—from many copies of the same molecules. The microtubules are coupled by molecular motors that walk along them, dragging along anything to which they are attached. For example, one motor might be attached to another molecular motor that is walking along another microtubule (see Figure 1d). At any rate, the motors need energy in order to move; the hydrolysis of adenosine triphosphate, a primary fuel source for the cell, provides this energy. Finally, Newton’s third law has to be obeyed in that the total force that a motor exerts must be zero. If a motor exerts a force \(\boldsymbol{F}\) on the microtubule, it must also exert a force \(-\boldsymbol{F}\) somewhere else — perhaps on the cytoplasm by the drag of a payload. This puts cytoskeletal assemblies on the turf of active matter, which in its purest form concerns itself with multiscale systems whose mobile microstructure converts a local energy source into mechanical work on the system. Passive matter systems are equilibrium systems where work is performed on the system from the outside, while active matter systems perform work on themselves.

Figure 1. Active dynamics and structures within cells, and the dynamics of their extracted components. 1a. The spindle complex’s dynamics in a single-cell nematode embryo. 1b. Tomographic reconstruction of the spindle complex and chromosomes (blue) in a nematode embryo. 1c. Simulation of a payload that is positioned by peripherally bound motors, which pull on microtubules. 1d. A schematic of aligned microtubules that are pulled past each other by double-headed kinesin motor complexes. 1e. An active in vitro assembly of microtubules and molecular motors. 1f. The steady rotational flow that can arise from confinement of an active suspension. The left side illustrates mean particle alignment and the right side illustrates particle streaklines. 1g. Detail from a large-scale simulation of active turbulence. Figure 1a adapted from [9], 1b adapted from [4], 1c adapted from [5], 1d adapted from [3], 1e adapted from [8], 1f adapted from [12], and 1g adapted from [11].

Figure 1e depicts the outcome of the extraction and purification of these cellular ingredients [8]. Freed from the confines and regulation of the cell, microtubules and motors organize to form biologically active materials that undergo self-driven, complex, and large-scale dynamics called active turbulence. Applied mathematicians and physicists model these systems, sometimes based on symmetry principles and sometimes via micro-to-macro (and back again) coarse-graining methods. One fundamental idea of the physics that governs the systems is that of an “active stress” — particularly an extensile active stress from collective microscale activity.

Continuum models for suspensions of active rod-like particles illustrate this concept [3, 6]. We describe the system’s state with a distribution function \(\Psi(\boldsymbol{x},\boldsymbol{p},t)\) of particle positions \(\boldsymbol{x}\) and orientations \(\boldsymbol{p}(|\boldsymbol{p}|=\boldsymbol{1})\), which evolve through a Fokker-Planck equation. This evolution requires conformational fluxes \(\dot{\pmb{x}}\) and \(\dot{\pmb{p}}\) that capture the microscopic particle dynamics; here, they are composed of active particles that swim, or stretch, or stretch the surrounding fluid — all while they are translated and rotated by a background flow \(\boldsymbol{u}\), which results from their own ensemble activity. Each active particle contributes a tensorial stress that is proportional to \(\boldsymbol{pp}\). Consequently, the background velocity \(\boldsymbol{u}\) solves a Stokes equation that is forced by the distributional average of \(\boldsymbol{pp}\) (among other things). That is,

\[-\nabla q+\Delta\boldsymbol{u}= -\nabla \cdot(\alpha\boldsymbol{D} + ...) \quad \textrm{and} \quad \nabla \cdot \boldsymbol{u}=0,\]

where \(\pmb{D}(\pmb{x},\pmb{t})=\int_{|p|=1}dS\pmb{pp}\Psi\) and \(\alpha\) measures the strength of particle activity. An extensile active stress has \(\alpha<0\), which corresponds to fluid being stretched along the particle axis by particle activity. This stretching flow causes nearby active particles to align — eventually yielding the large-scale, self-organized flows that are commonly associated with active turbulence.

The kinetic theory outlined here first described experiments on suspensions of swimming bacteria that demonstrated similarly complex dynamics [2]. It has provided a first-principles basis for the analysis and simulation of active suspensions, shedding light on their peculiar flow instabilities [7] and the way in which confinement sculpts their behaviors (see Figure 1f). Recently, scientists have used reductions that are based on thermodynamically consistent moment closures to simulate large-scale active turbulence (see Figure 1g). Looping back to the cell, similar active matter theories that stem from symmetry principles have successfully described the internal structure of the spindles themselves [1].

Another example with the same ingredients of microtubules and molecular motors occurs in egg cells, or oocytes: the largest cells that animals produce. Raising an egg is a community effort, and its production can require the movement of specialized proteins—supplied by other cells—around and across the oocyte. Molecular diffusion does so efficiently in small cells, but diffusive transport can be exceedingly slow in large ones. How might an oocyte overcome this supply chain issue? The shortcut is flow. Figure 2a shows the streamlines of a cell-spanning cytoplasmic vortex in the 300 micrometer-scale oocyte of the common fruit fly. With speeds of roughly 100 nanometers per second, these flows can transport proteins from one end of the cell to the other in 30 minutes; diffusion alone takes about a day.

What drives this little hurricane in a nearly microscopic egg? Fluorescent microscopy reveals that microtubules are attached to the cell wall and bent sideways, like seaweed in a running tide. Here, however, the seaweed pushes the flow. Figure 2b provides the basic physics. Kinesin-1 motor proteins carry payloads along microtubules towards their free "\(+\)" ends. According to Newton’s third law, the motors must push down on the microtubules—perhaps causing them to bend—and simultaneously push the cargo up through the fluid, causing it to flow. 

Figure 2. Internal flows within the developing egg (oocyte) of the fruit fly. 2a. Measured streamlines of cytoplasmic flows in the oocyte. 2b. A schematic of kinesin-1 motors carrying cargos up peripherally bound microtubules and generating flow. 2c. A bed of motor-laden microtubules undergoes a collective bending instability and produces a steady streaming flow. 2d. Phase diagram that shows the nature of flow instabilities as a function of dimensionless motor forcing \(\sigma\) and microtubule density \(\rho\). Microscopy data for 2a supplied by Wen Lu and Vladimir Gelfand and analyzed by Reza Farhadifar and Sayantan Dutta. Figure 2b courtesy of Reza Farhadifar, and Figures 2c and 2d adapted from [10] and courtesy of David Stein.

To better understand how this phenomenon might work, we developed a continuum model of the microtubule bed as an active porous medium [10]. A microtubule with position \(\boldsymbol{X}(s,t)\) (\(s\) is the arclength from the base) evolves relative to a background flow \(\boldsymbol{u}\) under its internal elastic forces and under a tangentially aligned motor load \(-\sigma\boldsymbol{X}_s\), which is a distributed follower force. Assuming that the microtubule bed is locally well aligned yields a coarse-grained bed velocity \(\boldsymbol{v}\), which itself creates the background fluid flow \(\boldsymbol{u}\) as the solution to a Brinkman-Stokes equation:

\[-\nabla q + \Delta\pmb{u}=\rho J[\pmb{\Lambda}(\pmb{u}-\pmb{v})-\sigma\pmb{X}_s] \quad \textrm{and} \quad \nabla \cdot \pmb{u}=0.\]

Here, \(\rho\) is the areal density of anchored microtubules, \(\bf{\Lambda}\) is a geometric tensor that captures the effect of microtubule orientation on fluid drag, and \(J\) is a Jacobian that handles the transition from the Lagrangian bed frame to the Eulerian fluid frame. Our model shows that the motors’ compressive load upon the microtubule bed can drive a novel, collectively organized buckling instability of the bed wherein the microtubules all bend as one, inclining themselves and letting the motor proteins drive the flow (see Figure 2c). The instability’s collective nature is revealed by the fact that it can only occur at sufficiently high microtubule density \(\rho\) (see Figure 2d).

While this fluid-structure problem yields new physics, models, and simulations, our contributions still leave many questions unanswered. What is the effect of the three-dimensional cell shape and bed inhomogeneity? Which cellular determinants initiate a transition to flow? What precise biological purpose is being served? We are working with experimentalists on these queries and hope to ultimately solve them. Many of the tools that we discuss here—i.e., multiscale modeling and coarse graining, continuum mechanics, partial differential equations that evolve in evolving domains, large-scale simulations, and stability analyses—have applications in other problems where cell biology and development meet active matter physics.

[1] Brugués, J., & Needleman, D. (2014). Physical basis of spindle self-organization. Proc. Natl. Acad. Sci., 111, 18496.
[2] Dombrowski, C., Cisneros, L., Chatkaew, S., Goldstein, R.E., & Kessler, J.O. (2004). Self-concentration and large-scale coherence in bacterial dynamics. Phys. Rev. Lett., 93, 098103.
[3] Gao, T., Betterton, M., Jhang, A.-S., & Shelley, M. (2017). Analytical structure, dynamics, and coarse graining of a kinetic model of an active fluid. Phys. Rev. Fluids, 2, 093302.
[4] Lindow, N., Redemann, S., Brünig, F., Fabig, G., Müller-Reichert, T., & Prohaska, S. (2018). Quantification of three-dimensional spindle architecture. Methods Cell Biol., 145, 45-64.
[5] Nazockdast, N., Rahimian, A., Needleman, D., & Shelley, M. (2017). Cytoplasmic flows as signatures for the mechanics of mitotic positioning. Mol. Biol. Cell, 28(23), 3261-3270.
[6] Saintillan, D., & Shelley, M. (2008). Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations. Phys. Rev. Lett., 100, 178103.
[7] Saintillan, D., & Shelley, M. (2013). Active suspensions and their nonlinear models. Comp. Rend. Phys., 14(6), 497-517.
[8] Sanchez, T., Chen, D.T.N., DeCamp, S.J., Heymann, M., & Dogic, Z. (2012). Spontaneous motion in hierarchically assembled active matter. Nature, 491, 431-434.
[9] Shinar, T., Mana, M., Piano, F., & Shelley, M.J. (2011). A model of cytoplasmically driven microtubule-based motion in the single-celled Caenorhabditis elegans embryo. Proc. Natl. Acad. Sci., 108(26), 10508-10513.
[10] Stein, D.B., De Canio, G., Lauga, E., Shelley, M.J., & Goldstein, R.E. (2021). Swirling instability of the microtubule cytoskeleton. Phys. Rev. Lett., 126, 028103.
[11] Weady, S., Shelley, M.J., & Stein, D.B. (2022). A fast Chebyshev method for the Bingham closure with application to active nematic suspensions. J. Comp. Phys., 457, 110937.
[12] Woodhouse, F.G., & Goldstein, R.E. (2012). Spontaneous circulation of confined active suspensions. Phys. Rev. Lett., 109, 168105.

Michael Shelley is an applied and computational mathematician at New York University—where he co-directs the Applied Mathematics Laboratory—and at the Simons Foundation’s Flatiron Institute, where he leads the Center for Computational Biology.

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