Fürthauer et al. modeled cortical actin as thin films of active chiral fluids - with chiral stresses being generated by nematic simple motors. These motors are derived from elementary torque dipoles, which mimic those generated by myosin motors bound to two antiparallel actin fibers. Using such a hydrodynamic approach to defining actin flows, the authors demonstrate large-scale fluid flow, and chiral dynamics derived from molecular torques [28,29]. This suggests that such asymmetries in flow can lead to cellular level-chirality - as observed within developing the dividing Caenorhabditis elegans embryos [30]. Naganathan et al., first described the chiral dynamics of the actomyosin cytoskeletal machinery in the C. elegans embryo with the help of a thin chiral fluid flow model of the actin cytoskeleton [11]. The authors observe a biased flow of fluorescently labeled actin structures across the AP axis of the embryo and develop a physical model to describe observed chiral flow behavior - by modeling the actomyosin cortex as an active, thin chiral fluid (Fig. 1A) [11]. They define the AP (v_x) and the orthogonal (v_y) flow velocity components as being respectively driven by active tension and active chiral torque generated by ATP driven actomyosin activity as described below.

Here, the tension T along the AP axis and the torque density τ define a chiral index c=τ/T, η is the two-dimensional viscosity of the cortex, and γ refers to the friction in the cell membrane and cytosol [11]. The authors assume that torque and active tension are both linearly proportional to local concentrations of myosin. The tension force is derived from the pull generated by myosin heads connecting neighboring actin filaments, while the torque force is generated by the twisting of the actin filaments during the ATP-driven myosin head contraction. At scale, these molecular forces combine to cause a contraction and rotation of the cortex. To verify the accuracy of their model, they quantified the flow of the actomyosin cortex using particle image velocimetry and showed that while most flow occurs in the AP axis, there is a small orthogonal component of the flow field as well with a counter-rotating behavior between the anterior and posterior axes (Fig. 1A). Furthermore, the authors noted that this molecular symmetry breaking precedes the observed rotation of these cells - suggesting that the chiral molecular flows drive chiral cellular rotations. The authors went on to show the myosin motor and RhoA dependence of this process. Myosin perturbation causes a decrease in the overall flow rate, but no changes in the chiral index are observed, while Rho signaling regulates only the chiral torque component thereby changing the value of c [11]. The authors then investigate the role of these chiral flows in multicellular organization and embryonic symmetry breaking. They observe the chiral alignment of cells in the 4-cell stage embryo as well. This bias is similarly regulated by Rho activity. Finally, perturbation of the overall body-axis determining Wnt signaling pathway revealed a reduction of chiral index within individual cells, suggesting that global LR signaling may regulate chirality at the cytoskeletal level. Similar concepts have been used to describe the ATP-driven motion of in vitro actin clusters [31].

Using a very similar model, Pimpale et al., discovered a lineage-specificity of the chiral flows within C. elegans embryos [32]. They tracked actomyosin flow in all cells during the first nine cell divisions post-fertilization. The authors show that chiral flows appear only on the anterior founder cells during cytokinesis, but no chiral counter-rotation is observed in the posterior stem cells or the endomesodermal precursor founder cell lineages at the same timeframe. Furthermore, with RNAi and fluorescence imaging experiments, the authors show that not only are the chiral flows quantified based on the chiral thin-film theory lineage-specific but they are also associated with skews in spindle formation and cell division orientation [32]. In fact, the counter-rotation velocity deterministically regulates the cell alignment skew - in accordance with a directional friction-based model where the rate of angular skew is proportional to the counter-rotation velocity, while the directionality depends on the relative value of the friction on either side of the inner eggshell walls. Interestingly, our group characterized distinct chiral biases cells from the three different germ lineages derived from human embryonic stem cells (hESCs), suggesting a strong phenotype dependence of cell chirality [6].

Tee et al. described single-cell chiral cytoskeletal dynamics in vitro using human foreskin fibroblasts micropatterned on circular fibronectin islands [16]. Under isotropic confinement within the micropatterns, the cells developed robust focal adhesions (FAs) with the fibronectin substrate and formed a distinct, and initially isotropic network of radial and transverse F-actin fibers (RFs and TFs). Over time, these networks were observed to undergo asymmetric self-organization which lead to the chiral counterclockwise (CCW) “swirling” of the RFs, in tandem with a centripetal flow of the TFs [16]. They also show that the actin crosslinking protein α-actinin-1 regulates the directionality of the twirling, with a significant increase in the number of clockwise (CW) swirling cells upon overexpression of the crosslinker. The authors also ruled out any role played by the microtubule network, in accordance with previous evidence [9]. How actin crosslinking and polymerization can lead to chiral swirling of the RFs, however, is not obvious. The authors thus develop a physical model to describe the actin fiber dynamics [16].

The model consists of a symmetric arrangement of radially aligned RFs as flexible rods, and TFs bound to adjacent RFs as contractile elastic rods (Fig. 1B). The contractile stress within the TFs is held constant for all TFs throughout the simulations. The RFs elongate radially by Formin-dependent actin polymerization, while new TFs are probabilistically generated during the simulations at the Formin end of the RFs. The centripetal component of the TF contractility moves the TF-RF contacts along the RFs, while the tangential component (in the circumferential direction) can cause RF bending (Fig. 1B). In the initial isotropic arrangement, the tangential forces are balanced. However, slight deviation of RFs from the radial alignment causes rearrangements of its associated TFs, leading to a progressively compounding symmetry-breaking event.

The authors further model the chiral rotation of growing RFs. In the wild-type regime, Formin is capped and immobilized - preventing rotation of the RF at the FAs but allowing for its rotation at the free cytosolic end. A+14° rotation occurs with each actin dimer addition and promotes a CCW RF alignment. If actin crosslinkers are present, the rotation at the cytosolic end is prevented, resulting in the accumulation of torsional stress which is relieved every 12th polymerization event by a −166° rotation, causing a CW RF alignment. The rotation of RFs causes friction generation at the RF-TF binding locations similar to a rack and pinion system, causing the TFs to slide tangentially, leading to a chiral RF swirling [16]. These two physical models of actin swirling lay the foundation for actin-driven torque generation within eukaryotic cells and hold significant implications for the chiral morphogenesis of tissues and organs.

Li and Chen subsequently model the rotation of RFs because of actin polymerization, and the role of α-actinin cross-linkers in switching swirling direction. In this molecular clutch model, single actin filaments are represented by rods which grow and rotate, along with the formin interface which is modeled as a torsional spring. This rotation is transferred to the RFs through α-actinin cross linkers which are also modeled as torsional springs. When the filament forms cross-links with the RF, both rotate in a positive direction. Subsequently, the filament is capped, detached from Formin, and incorporated completely into the RF. This causes the stored elastic energy to be released and causes the RF and filament to rotate negatively. The difference between the two angles of rotation determines the overall rotation and swirling of the RF.

The authors show that high crosslinker attachment rate, high crosslinker concentration, and high actin monomer concentrations all lead to an overall CW swirling - consistent with in vitro findings and previous computational predictions [27]. Li and Chen further developed this model by introducing a sliding mode of crosslinker mobility along the growing filament, i.e., the crosslinker is allowed to move along the filament [33]. They compare this model to one where the crosslinkers remain fixed to show that the sliding motility mode leads to CW, CCW and no swirling cells whereas the fixed mobility mode does not lead to CW swirling - suggesting a sliding motility may be more realistic [33]. Finally, the authors extend the model to investigate three different modes of torque relaxation at the Formin-end of filaments. However, none of the modes yield satisfactory predictions based on experimental data [34]. Further studies may reveal more specific molecular mechanics of chiral RF swirling.

These models provide a more detailed physical understanding of the role of actin polymerization in generating intracellular torque and describe the specific role of actin crosslinkers and their mechanisms of action in determining the directionality of actin swirling. However, they still have limitations. Firstly, the rotation of RFs is not used to model the swirling of the overall actin structure itself in the Tee et al. model. Secondly, both the Tee et al. and Li and Chen models consider actin fibers as simple rods, which allows for a straightforward model for actin-generated cellular torque. However, more detailed models of actin dynamics are available in the field. One such example is the Cosserat Theory-based model of actin polymers reported by Floyd et al. [35]. The group further enhance their model by including an ellipsoid model of actin polymers which considers actin monomers (the ellipsoids) and their specific chemical bonds with neighboring monomers in the follow-up study by Gunaratne et al. [36]. Incorporating such detailed actin dynamics into the models of swirling would more accurately represent the rotation of actin RFs and their subsequent contribution to overall swirling dynamics - with results comparable to mechanical measurements performed in vitro [20].

Biological tissues often demonstrate behavior analogous to inanimate physical systems. A common example of this is that biological systems often involve signaling molecules that follow reaction-diffusion dynamics. Chen et al., describe a reaction-diffusion model that predicts the formation of chiral, parallel cell aggregation patterns following growth in alternating stripe-shaped micropatterns [15]. Upon removal of intermediate non-adherent stripes, the cells demonstrate chiral migration and wound-healing behavior. Upon long-term culture following chiral alignment, coherent and parallel cell aggregation is observed, compared to labyrinthian patterns observed in non-patterned control cultures [15]. This behavior is accurately recapitulated in a combined reaction-diffusion and chemotaxis model driven by the migration activator BMP-2 and its faster diffusing inhibitor MGP - with the assumption being that cells preferentially migrate towards areas of the higher activator morphogen. The authors use previously described reaction diffusion dynamics to determine morphogen patterns and combine it with a chemotaxis model which introduces two tunable vector variables that describe the differential migration in the two principal axes [15]. When anisotropies are introduced by setting the migration along one axis several orders of magnitude larger than the orthogonal axis, and biased by an angle θ, the resulting cellular aggregates form parallel aggregates as observed in the micropatterned in vitro samples (Fig. 2A). In contrast, an isotropic model with equal migration on both axes leads to a labyrinthian pattern of cell aggregation compared to the biased model (Fig. 2A) The authors further show that the observed chiral behavior is dependent on the accumulation of stress fibers at the interface boundaries - and that prevention of these interactions attenuates LR polarity [15]. They show that the microtubule organizing center polarizes at the boundary edge, like the leading-edge cells of wound healing assays, and that the cell-substrate interactions are required for this polarity and the subsequent chiral cell cluster formation [15]. Similar results were obtained by Li et al., who showed chiral symmetry breaking in a reaction-diffusion system under the influence of an external chiral electric field [40]. While different from the chemotaxis reaction-diffusion model, this study also demonstrates that an active external asymmetry can induce symmetry breaking within otherwise symmetric reaction-diffusion systems.

To study the collective behavior of cells, biophysicists often draw inspiration from non-living collective systems [41,42,43]. These non-living systems and their associated theories have proven quite useful for predicting collective tissue behaviors. Active matter systems of liquid crystals display similarities to developing cellular monolayers. In liquid crystals, rod-like particles are arranged in a fashion such that their orientational order is preserved. Further studies of chiral active matter revealed differing forms of asymmetry arising from antisymmetric stresses [28,44].

Duclos et al. study the spontaneous flow of cells in confined patterns as they relate to stresses generated by liquid crystals [45]. They highlight that cells follow a conserved alignment direction within the pattern boundaries and theorize that the degree of tilting is controlled by the strength of chiral stresses induced by biased cellular motion. A continuum model is utilized for the purposes of predicting this behavior. These models assume that cellular motion is very slow and viscous, generating an overall incompressible flow regime (i.e., $\nabla \cdot v=0$). For an incompressible fluid flow, the Navier–Stokes equation for cellular motion can be simplified so that the stresses within the monolayer can be better extrapolated. Studies attempting to model cellular motion as a flow of aligned particles require a robust definition of the active stresses being generated and the Frank free energy of the system. The Frank free energy describes the alignment and polarization interactions between elongated particles within the system [46]. Hoffman et al. build upon the work completed by Duclos et al. to generate a thorough description of the active stresses experienced by cells within a monolayer below [47]:

Here, σ_∥ and σ_? represent the stresses experienced by some volume element in the parallel and perpendicular directions in respect to nematic director n respectively. The new value τ describes the contribution of the chiral cellular rotation to the active stress arising from an intrinsic force dipole (Fig. 2B). With this chiral component to the active stress added, examination of the alignment angle within stripe patterns with increasing widths sees improved comparison with experimental data [45].

Another hallmark of liquid crystals are topological defects in which the perfect alignment of particles within a layer is disrupted. Cellular monolayers of Neural Progenitor cells (NPC) and Madin-Darby Canine Kidney (MDCK) cells, for example, have displayed such topological defects [48,49]. Hoffman et al. continue with their description of chiral active stress to describe the motion of +1/2 (“comet like”) disclinations. In monolayers of extensile or contractile particles, +1/2 defects will move in the direction of their head or tail respectively. In the presence of a chiral particle, as cells have been observed to behave as, it is predicted that the defect will rotate along its path of motion. Hoffman et al. provide an equation below to relate angle of biased migration to the active stress and the chiral torque:

The angle of defect rotation is related to the overall chiral torque if the overall active forces are known. This framework provides many similarities to one study completed by Maitra and Lenz [50]. Maitra and Lenz simulate positive half-integer defects of a nematic material to migrate and rotate in respect to negative half-integer defects within the system, leading to a recombination event (Fig. 2C). Yashunksy et al., utilizing a similar model to Hoffman et al., demonstrate that +1/2 topological defects in close proximity to boundaries will preferentially rotate while defects far away do not (Fig. 2D) [51]. Chiral flow on edges of patterned surfaces has been observed on stripe shaped domain for several different cell types (i.e. MDCK and NPC) [10,52]. Yamauchi et al. specifically apply stochastic modeling of active nematic particles on boundaries. They determined that steady state flows induced by geometry are hallmarks of chiral active nematics.

In contrast to half-integer defects, chiral events of integer defects have also been studied extensively in the presence of boundary conditions. Blanch-Mercader et al. examine the mechanics and flows of integer defects created by C2C12 cells grown on circular micropatterns [53]. Here, cells align in spirals on these micropatterns in directions consistent with previous alignment studies of C2C12s [9]. Blanch-Mercader et al. additionally apply continuum dynamics assuming incompressible flow of cells. While their model does not assume the presence of an active chiral stress, initial conditions of the model assume that cells are already aligned in a biased fashion. In another study by Hoffman et al., the model is expanded to examine the 3D morphogenesis of spiral aligned cells [54]. Here, researchers were able to recapitulate the morphogenesis observed in vitro by Guillamat et al. [55].

While nematic modeling of cellular motion has proved effective for predicting cellular behavior, there are situations when it fails. Since cells are only modeled as rods, the amorphous structure of the boundaries and cytoskeleton are lost. This leaves these models better suited for studying very elongated cells or very large populations of cells, while mechanical properties of individual cells are largely ignored.

Like nematic models, vertex models commonly used to simulate the dynamics of non-living systems like foam have gained popularity with its use in multicellular tissue dynamics [43,56,57,58]. In this model, instead of nematic rods, cells are modeled as polygons defined by their vertices and connected by edges which represent the cell-cell junctions. The dynamics of the system are then defined by defining energies based on polygon geometry and applying forces on the polygon vertices. These forces are then used to displace the vertices - usually using a gradient descent energy minimization algorithm or a simple Euler method, depending on the nature of the simulations. Generally speaking, these models incorporate polygon area and perimeter elasticities, to represent the forces generated by the actomyosin cortex and cell-cell adhesions. Some models also incorporate a line-tension which represents the energy associated with the cell membranes. It is possible, however, to include further intricacies by adding novel energies, forces, or dynamic regulations of their associated parameters. Details of various such methods are reviewed elsewhere [43,56]. In what follows, we discuss several implementations of chiral biases using some variations of the vertex model described above.